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dissertation entitled

ASSESSING THE VIABILITY
OF THE
SHIFTING BALANCE PROCESS

presented by

Francis E . —G Moore

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ASSESSING THE VIABILITY OF THE SHIFmIG BALANCE PROCESS
By

Francis B.-G. Moore

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Zoology

1996

ABSTRACT
ASSESSING THE VIABILITY OF THE SHIFTING BALANCE PROCESS
By

Francis B.-G. Moore

For over sixty-five years evolutionary biologists have debated the relative
importance of drift, selection and genetic constraints. One view contends that with
population sub- structure drifi and selection interact such that genetic constraints are more
easily overcome than in panmictic populations. In that view the shifting balance of
selective and genetic factors within a subdivided population can overcome many genetic
constraints. An alternate view holds that drift is an unimportant factor in evolution. The
two views continue to fuel major debates within evolutionary biology. This dissertation
attempts to assess the relevance of the shifting balance view.

Progress in shifting balance theory is reviewed in chapter one. That review
concludes that the shifting balance process is viable under many scenarios, but that the
efficacy of the process is highly model dependent. Especially important are the types of
gene interaction that produce genetic constraints. Chapter two reviews evidence for
epistatic effects on fitness. It concludes that there is overwhelming evidence that gene
interactions between loci create genetic constraints, but that few specific examples are
thoroughly understood.

Chapters three through five use spatially, genetically and individually specific
Monte Carlo simulations to assess the shifting balance process. The shifting balance

process requires small efl‘ective deme size in order for drift to occur and new gene

combinations to be found. Migration between demes increases effective population size
and allows export of new gene combinations. Migration's influences on these two aspects
of the shifting balance process are therefore in conflict. Chapter 3 investigates the effect of
migration rate on the shifting balance process. It concludes that there is a small window of
migration rates over which the process can operate effectively. Chapter four analyzes
several alternate gene interaction models and concludes that the efficacy of the shifting
balance process is highly dependent on specifics of gene interaction. Chapter five
investigates the eflicacy of the shifting balance process under variable migration rates. It
concludes that alternating periods of high and low migration can radically increase the

efficacy of the shifting balance process.

To the Winnebago Trickster and the Crooked Life Road.

ACKNOWLEDGEMENTS

I have never known a graduate student who has had as much access to their
advisor as I have. Steve Tonsor has provided countless insights through our many
animated discussions. I hope he can forgive me for taking up every unbooked minute in his
planner for four years nmning. I would also like to thank the other members of my
committee Don Hall, Sue Kalisz, Bob Newman and Alan Tessier who provided useful
encouragement and criticism throughout my research.

I thank J. Gorentz, S. Ozminski, and T. Holtsford for assistance with
programming. Discussions with faculty and graduate students at KBS and the BBB
program at MSU provided many ideas and honed many arguments. Out of that group
special thanks go to P. Smith, B. Foster, C. Huckins, D. Theide, D. Jenkins-Klus, K.
Geedey, M. McPeek, M. Travisano, T. Getty, D. Straney, A Davelos, L. Horth, P. Batra
and J. White for extensive discussion of relevant subjects and continued interest in my
work, and to George Laufl‘ for having built KBS into a fantastic learning center. My
thanks to D. Hall, R Hill and P. Smith, I doubt that I would ever have had the confidence
to enter graduate school without them Thanks to my family for appreciating the esoteric,
and to my wife for supporting me through the research and the writing.

This work was supported through a NSF Graduate Research Fellowship, and the

Research Training Group at KBS.

TABLE OF CONTENTS

LIST OF TABLES ......................................................................................................... viii

LIST OF FIGURES .......................................................................................................... ix

INTRODUCTION ............................................................................................................. 1

CHAPTER 1

THE SHIFTING BALANCE PROCESS: A REVIEW ..................................................... 2
Introduction ......................................................................................................... 2
Adaptive peaks- what they are .............................................................................. 3
Adaptive landscapes- why they matter .................................................................. 7
The balance in the shifting balance ........................................................................ 9
Progress in shifting balance theory ...................................................................... 10

Chapter 2

EMPIRICAL APPROACHES TO INVESTIGATING MULTIPEAKED SURFACES:

A REVIEW ................................................................................................................... 20
Introduction ....................................................................................................... 20
Methods of detecting gene interactions ............................................................... 20

evidence of gene interaction from classical genetics .................................. 21
evidarce of gene interaction fiom quantitative genetics ............................ 25
evidence of gene interaction from molecular genetic studies ..................... 27
Natural selection as a non-arbitrary ruler ............................................................. 29
new sources of gene interaction ............................................................... 29
gene interactions revisited ........................................................................ 34
Where to look for multiple peak epistasis ............................................................ 35
evidence from classical and population genetics ....................................... 37
optimizing, disruptive and correlational selection ..................................... 40
evidarce from structured populations and hybrid breakdown ................... 42
Summary ............................................................................................................ 44

Chapter 3

A SIMULATION OF WRIGHT S SHIFTING BALANCE PROCESS:

MIGRATION AND THE THREE PHASES .................................................................. 47
Introduction. ....................................................................................................... 47
Methods .............................................................................................................. 51

The components of the simulations ......................................................... 51
Analysis of the simulations ....................................................................... 57

vi

confidence limits for the data ................................................................... 59

Results ............................................................................................................... 59
Results of the panmixia model ................................................................. 59
Peak shifi propmsity ................................................................................ 59
Prerequisites for phase III ....................................................................... 60
Phase three propensity ............................................................................. 6O
SBP as a whole ........................................................................................ 65
Population size ........................................................................................ 65
Discussion. .......................................................................................................... 65
Chapter 4
SIMULATION OF THE SHIFTING-BALANCE PROCESS
IN OLIGOGENTC TRAITS ........................................................................................... 80
Introduction. ....................................................................................................... 80
Methods ............................................................................................................. 84
Results ............................................................................................................. 100
Two locus models ................................................................................. 100
Four locus models ................................................................................. 104
Eight locus models ................................................................................. 104
Sixteen locus models ............................................................................. 105
The overall efficacy of the SBP .............................................................. 108
Discussion ............................................................................................ 108
Chapter 5
FLUCTUATING MIGRATION AND THE SHIFTING BALANCE PROCESS ............ 124
Introduction ..................................................................................................... 124
Methods ........................................................................................................... 126
Results ............................................................................................................. 133
Extreme constant migration rates .......................................................... 136
Frequency of peak shifts ........................................................................ 136
Phase three propensity ........................................................................... 141
The SBP as a whole .............................................................................. 141
Discussion ........................................................................................................ 146
SUMMARY ................................................................................................................. 153
LIST OF REFERENCES .............................................................................................. 154

LIST OF TABLES

Table 3.1 - The mean number of individuals in demes of different mean fitness ............. 61
Table 4.1 - Summary of the 8 and 16 locus models ....................................................... 97
Table 4.2 - Summary of results fiom the 2, 4, 8, and 16 locus models ......................... 101
Table 4.3 - The eflicacy of the SBP in four different two locus models ....................... 114
Table 5.1 - Summary of the fluctuating migration models .......................................... 128

Table 5.2. Summary of results for the fluctuating and constant migration rate models. 137

LIST OF FIGURES

Figure 1.1 - Fitness sru'faces of four difl‘erent types .......................................................... 5

Figure 2.1 - Chicken combs produced by the combined action of two alleles
at each of two loci. ............................................................................................. 25

Figure 2.2 - Epistasis for fitness produced by optimizing selection (a) and

disruptive selection (b) acting on an additive trait ............................................... 33
Figure 3.1 - Population fitness surface ........................................................................... 56

Figure 3.2 - The percentage of trials in which at least one deme shifted versus
the migration rate .............................................................................................. 63

Figure 3.3 - The percentage of those populations with a deme shift, that
experienced a fixation of the highest fitness genotype throughout the

entire population versus migration rate ............................................................... 65
Figure 3.4 - Frequency of the highest fitness genotype versus migration rate .................. 68
Figure 3.5 - The percentage of trials in which the highest fitness genotype

was fixed throughout the population versus migration rate ................................. 70
Figure 3.6 - Percentage of demes remaining after 12000 generations .............................. 72
Figure 4.1 - Phenotypic fitness surface for the 2 locus models ........................................ 88
Figure 4.2 - Phenotypic fitness surface for the 4 locus models ........................................ 91
Figure 4.3 - Phenotypic fitness surface for the 8 locus models ........................................ 93
Figure 4.4 - Percentage of trails in which at least one deme shifted ............................... 103

Figure 4.5 - Percentage of those trails with a deme shift that
experienced a fixation of the highest fitness genotype ....................................... 107

Figure 4.6 - The percentage of trials in which the highest fitness genotype
was fixed throughout the entire population ....................................................... 110

Figure 4.7 - The fitness of each possible genotype in four different two
locus two allele models .................................................................................... 112

Figure 5.1 - Four examples of the type of migration rate cycles used in
these simulations .............................................................................................. l3 1

Figure 5.2 - Genotypic fitness surface .......................................................................... 135

Figure 5.3 - Percentage of trials in which at least one deme shifted versus
average Nm ..................................................................................................... 140

Figure 5.4 - Percentage of those trials with a deme shift that experienced a
fixation of the highest fitness genotype throughout the entire
population versus average Nm ......................................................................... 143

Figure 5.5 - Percentage of all trials in which the highest fitness genotype was
fixed throughout the population versus average Nm ......................................... 145

INTRODUCTION

Sixty five years after its introduction, Sewall Wright's shitting balance process
(Wright 1931) remains a controversial centerpiece of evolutionary theory. The theory
predicts that drift and selection may interact in a structured population so that adaptation
can proceed in ways not available to populations in panmixia. This dissertation reviews the
literature on the shifting balance process as well as evidence for multiple peak epistasis. It
also presents several novel theoretical results based on Monte Carlo simulations which test
the viability of the shifting balance theory under a variety of scenarios.

The purpose of the simulation tests of the shifting balance process is first to
determine whether there are any circumstances under which the shifting balance process
might work. Secondly, a diversity of models are compared to see if the process is equally
viable under all models. Finally, a model which relaxes one of the ecological
simplifications in previous models is used to determine if increased complexity in the

shifting balance process is likely to increase or decrease the eficacy of the process.

Chapter 1

THE SHIFTING BALANCE PROCESS: A REVIEW OF THE THEORY

Introduction

Sewall Wright's Shifting Balance Process (SBP) (Wright 1932; Wright 1982b)
incorporates the major microevolutionary forces of drifi, mass selection, and group
selection. For this reason it is a central paradigm of evolutionary biology (Wade and
Goodnight 1991). In the SBP drift, mass selection and group selection are tied together in
the context of genetic and developmental constraints. Such constraints are increasingly
viewed as major evohrtionary factors (Gould and Lewontin 1979; Steams 1980; Vrba and
Eldredge 1984). SBP is the most synthetic theory of microevolutionary dynamics. The
result of this synthesis is a combined body of theory which implies that historical
contingency is an important factor in the adaptation and diversification of organisms. This
microevolutionary view, in which constraints and historical contingency are integral,
anticipates recent trends in the interpretation of macroevolutionary data (Eldredge and
Gould 1972; Gould and Eldredge 1977; Derrickson and Ricklefs 1988; McLennan et al.
1988). Despite these implications of the SBP the complexity of interacting genetic,
selective and population dynamic factors involved has prevented rigorous investigation or
universal acceptance of the shifting balance view. Assessment of the roles of the SBP and

its components are still in their infancy. This chapter reviews the current state of

knowledge about shifting balance theory.

Adaptive Peaks- What They Are

The concept of the adaptive peak was first used by Wright (1932) as an area of
higher fitness on an adaptive landscape. There are, however, three fimdamentally different
types of adaptive landscapes. Provine (1986) has pointed out that some confirsion has
arisen fiom Wright's original ambiguity between two of these fimdamental types. Wright's
(1932) original diagrams (figure 1.1a) represented individual fitness surfaces in a
nonquantitative way (Provine 1986; Wright 1988). These diagrams represent the possible
genotypes plotted against the fitness of an individual of that genotype. The actual
diagrams that Wright used to discuss adaptive landscapes in 1932 and in many subsequent
papers, are not fully interpretable because multiple genetic dimensions are compressed into
two graphical dimensions (Provine 1986; Wright 1988).

It is possible to accurately describe an adaptive surface for a two locus, two allele
system (figure 1.1b). Ifmultiple peaks are present on such a surface, offspring of crosses
between those two optimal genotypes will be of lower fitness than either parent. This type
of adaptive surface can be called a genotypic fitness surface because the expected fitness
of an individual with a given genotype is what is plotted.

A second type of adaptive landscape is produced by plotting allele frequencies of a
population at multiple loci against the mean fitness of a population of individuals. Figure
1.1c is this type of surface. This surface can be called an allele frequency surface because
the allele fi'equency is being used to predict the expected mean fitness with in a population.

The population mean fitness surface is the sru'face of mean fitnesses associated with a

4

Figure 1.1 - Fitness surfaces of four different types.

Figure 1. la is the type of fitness surface originally used by Wright (1932). The axes are
unmarked and represent some undefined scaling of genotype so that each point on the
surface represents one genotype. Contour lines represent changes in fitness. Fitness peaks
and pits are marked by + and - respectively. 1. lb is a genotypic fitness surface with the
expected fitness of an individual (W) plotted against the number of doses of a mutant
allele at two loci. 1. 10 is a population fitness surface. The surface is created by plotting the
expected mean fitness of a population against the fiequency of mutant alleles at two loci
1.1d is a phenotypic fitness surface which plots the expected fitness of an individual (W)
against the phenotype of that individual

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multi-peaked genotypic fitness surface. This type of graph has several advantages over
Wright's original drawings. The SBP requires consideration of selective effects within a
subpopulation as well as metapopulation dynamics. The integrated fitness across a
population with the allele frequency fitness surface gives a comparison of the
expectedfitness of populations at different fiequencies. The shape of this allele frequency
by mean fitness graph gives not only the expected fitness of a sub-population with a given
set of allele frequencies, but also gives the direction in which selection is expected to push
the population at any set of allele frequencies. In this type of diagram the slope of the
fitness surface indicates the strength of selection on a population at those frequencies.
Genotypic fitness diagrams do not provide this insight. Population fitness surfaces give
expected mean fitnesses (actual values for a population's mean fitness will often deviate
from expectations when population size is small).
The third type of adaptive landscape is a phenotypic fitness surface (figure 1.1d).

In this surface, fitness is plotted against the phenotypic value of some trait (Simpson
1953). As long as variance around the mean is small this could represent either the mean
phenotype in a population or an individual's fitness. Unlike the previous types of adaptive
surfaces, the existence of gene interactions (epistasis) is not explicit. However this type of
surface does imply epistasis for fitness (Lande 1976). The phenotypic surface has the
advantage of dealing with phenotype rather than genotype, and it is the phenotype not the
genotype on which selection acts.

The problem of graphically representing more than two genetic dimensions is
inherent with the allele frequency and genotypic fitness surfaces. The phenotypic fitness

surface avoids some of the problems of graphing more than two loci at a time. The

7

phenotypic fitness surface has the disadvantage of not allowing a direct examination of the
gene interactions involved in fitness. This is problematic since a phenotype may have more
than one possible genetic architecture, each architecture associated with a different point
on an allele frequency surface (see figure 1.1a). In the phenotypic fitness surface,
multiple population fitness peaks may be indistinguishable by phenotype. All three types of
adaptive landscapes described here can represent the existence of multiple stable genetic
states. Each is usefirl and necessary for understanding difl‘erent facets of the SBP.
Adaptive landscapes - Why they matter

Adaptive peaks are evolutionary constraints. Despite the common genetic
machinery of all organisms, constraints on evolution vary between lineages. The reason
that constraints vary between lineages is that lineages can exist on different adaptive peaks
due to historic difl‘erences. If constraints were genetic alone then any two populations
exposed to the same selective environment would converge on the same result. The
interaction between genetics and selection is what varies between lineages. Selection
optimizes within a peak's domain, but prevents movements between peaks. This is because
selection limits the variation within a population and creates local constraints to adaptation
within a lineage that are contingent on history.

This emphasis on genetic constraints and their interaction with the natural selective
environment also implies that historical contingencies are important. Natural selection
acts on populations whose genetic architecture is contingent on past environmental
selection and past introductions of genetic variation through migration and mutation. This
means that while adaptation towards a local adaptive peak (e.g. suboptimum) may occur it

will rarely optimize the population fitness in a more global sense. Instead, one of many

8

stable genetic sub-optima that allows the population to persist in an environment may be
reached. The SBP is therefore a process by which populations can occasionally move to a
higher stable point. This process is contingent on variation in allele frequencies which is
stochastic in origin. The adaptive peak concept and the SBP are important because they
define the mechanisms whereby contingencies become an important factor in adaptation.

Ifadaptive peaks exist, their presence will affect the tempo and mode of evolution
(Wright 1982a, 1982b). At the microevolutionary scale population structure and drift are
more important when adaptive peaks exist (Wright 1982b). At the macroevolutionary
scale allopatric speciation might not be just the gradual accumulation of diflerences
through drift and selection. Instead allopatric speciation within the context of movement
between adaptive peaks will usually be a threshold event. On occasion suflicient variation
will exist so that a population will enter the domain of a new optimum Once this happens
selection can quickly re-optimize that population. This re-optimization results in post
mating barriers to migration between populations which are at different optima. The
predicted pattern of change in the SBP is one of periods of relative stasis, punctuated with
periods of rapid divergence (Wright 1982b).

The ability of lineages to move between the peaks is the most important aspect of
the existence of multiple peaks. Peak shifis can create phenotypic differences and breeding
barriers between populations, species and higher taxonomic groups. Adaptive valleys are
boundaries within which the immediate evolutionary potential of a lineage is constrained.
Without boundaries to the immediate evolutionary potential of a lineage the phylogenetic
distinctions between groups are unimportant for fixture dynamics. Yet if those immediate

bounds are considered static then it is diflicult to explain the radiation of lineages that has

9

occurred in the past. The question to be answered is, "how does a lineage constrained to
one selective domain reach a new selective domain?". The SBP provides a mechanism for

such movement.

The Balance in Shifting Balance

Wright viewed drift as an important evolutionary factor largely because of its
effect on the adaptive dynamics of populations (Provine 1986). When gene interactions
affect fitness, drift alters the balance between intrademic and interdemic selection. Ifdrift
is pervasive in a system then the opportunity for interdemic selection may arise. With drift
and gene interaction even demes under the same selective forces can differentiate. The
genetic differentiation between demes due to drift can create selective differences between
demes. This selection then forces firrther divergence. Ifdrift is insufficient to change which
alleles are favored then its effects will be quickly ameliorated by migration between demes
in combination with selection within demes.

Because of the role of drift, population structure can influence the course of
evolutionary change. This incorporation of spatial genetic structure is unique to the SBP.
Wright viewed the optimal adaptive scenario as including a balance of drift, intrademic
selection, and interdemic selection (Wright 1931; Wright 1932). Population structure
through its influence on drift can shift the balance of these factors (Wright 1931).

The SBP allows the exploration of multiple adaptive peaks through the combined
processes of drift, mass selection, and interdemic selection. Wright (1932; 1978a) saw
these processes acting in three phases in a structured population. In phase one, genetic

drift within subpopulations allows some demes to enter the domain of new adaptive peaks.

10

Next, mass selection within those demes would push them towards the new adaptive peak
(phase two). Finally if the new adaptive peak is of higher fitness than the other demes in
the population that deme may export more individuals than its neighbors. This differential
export of individuals under certain circumstances should allow those demes on the higher

peak to convert the demes on the lower peaks to the higher fitness peak (phase three).

Progress in Shifting Balance Theory

Since it was first proposed the SBP has provided a qualitative model for
researchers studying evolution in structured populations. Wright's quantitative analysis of
evolution previous to 1935 was restricted to single locus models (Wright 1931). The
SBP, however, was founded firmly in the belief that inter-locus interactions were
important in evohrtion. Wright therefore made a bridging of his qualitative theory and
quantitative analysis a lifelong goal (Provine 1986). He made several fimdamental
advances in this area.

First, Wright determined a method for studying the probability distributions of
alleles at multiple interacting loci (Wright 193 5). He accomplished this by analyzing traits
under the purely additive control of multiple loci. These additive traits were
mathematically tractable using his earlier integral approach (Wright 1931). By applying
non-linear fitness surfaces to the additive traits Wright modeled systems with multiple
adaptive peaks involving multiple loci. This permitted Wright to generate equilibrium
expressions for the distribution of allele frequencies across two loci

Wright (1935) firrther improved his analysis of the SBP by applying difl‘erential

equations derived from Kolmogorov's (1931) equations for continuous diffusion efl‘ects to

11

the problem Wright's forward Kohnogorov equation had independently been arrived at
several years earlier by physicists and is therefore referred to as the Fokker-Planck
equation. Application of this equation by Wright allowed investigations of non-equilibrium
processes. These techniques have formed the basis for most of the subsequent analysis of
the SBP. In models of this type, drift is modeled as a diffusion process which, combined
with deterministic factors, provides the probability of a given outcome. While diflirsion
models require assumptions that often restrict their generality (Gillespie 1989) they have
proved the most powerfirl approach yet in understanding the dynamics of peak shifts.

In general the use of diflhsion equations in population genetics flourished in the
1940's and 1950's. The application to the SBP however was generally restricted to
defining the probability distributions of allele fiequencies for single locus (heterozygote
disadvantage) or polygenic adaptive landscapes (See Kimura 1964 for review). Lande
(Lande 1976; Lande 1985; Lande 1986), however, changed the focus of this approach by
applying diffusion equations to a more macroevolutionary question. He used diflirsion
equations to demonstrate that the time between peak shifts may be large even on a
geologic time scale. Lande concluded that when peak shifts do occur they should appear
very suddenly. Ptmctuated equih’brium is, therefore, the expected pattern of morphological
change when nnrltiple adaptive peaks exist (Lande 1985; Lande 1986).

Wright clearly recognized the implications of the shifting balance model for
macroevolutionary dynamics and was quick to respond (Wright 1982a; Wright 1982b)
when the ability of microevolutionary theory to explain punctuated evolution was
questioned (Gould and Eldredge 197 7). Wright's general prediction that evolution on a

rugged adaptive landscape will lead to a macroevolutionary pattern of stasis prmctuated by

12

rapid change was later quantified for a number of specific cases. In the case of polygenic
traits in a population large enough to guarantee a constant source of variance and
heritability Lande (Lande 1976; Lande 1985; Lande 1986) found that peak shifts would be
highly punctuated. Under moderate selection and with moderate variance and heritability
and an effective population size (N,) of 200, peak shifts are expected approximately every
1010 or 1011 generations. When N, is between 102 and 10‘, the residence time within the
adaptive trough would be between 102 and 103 generations. The expected time to a peak
shift increases exponentially with increasing N, while the residence time of the
intermediate form increases logrithmically in theses models (Lande 1985). Numerical
simulations (Newman et al. 1985) agree with Lande's general conclusions, as do analytical
results based on changes in phenotypic variance and environment (i. e. selection) as the
cause of peak shifts (Kirkpatrick 1982) .

Wright (1982b) points out that the main difference between neodarwinian models
and Eldridge and Gould's (1972) model of macroevolution is not one of tempo but of
mode. Rapid character change followed by long periods of stasis may be adequately
explained by traditional microevolutionary forces within a species, or as a process which is
tied to speciation. Wright noted that peak shifts could occur under both scenarios (Wright
1982a). One mechanism whereby speciation and adaptive peak shifts might be tied
together is through founder efl‘ect speciation (see Barton and Charlesworth 1984 for a
review).

Considerable analytical work has concentrated on the special case of peak shifts in
founder populations (Barton and Charlesworth 1984; Charlesworth and Rouhani 1988;

Rouhani and Barton 1987a). These models investigate a process which has been

l3

conceptually linked to speciation (Barton and Charlesworth 1984; Carson 1982; Mayr
1963; Templeton 1980), but which is not necessarily contingent on speciation. The models
investigate the probability of peak shifts in founder populations which are drawn from
large parent populations. Those diffusion approximations indicate that when heritabilities
are moderately high, founder event peak shifts will occur very rarely if the phenotypic
distance to be shifted is greater than several times the square root of the additive genetic
variance. When the phenotypic change required for a peak shift is relatively small,
however, the probability of peak shifts is greatly increased by founder events (Rouhani and
Barton 1987a). Growth rate and size of the founding population are in this case important
factors in determining the probability of those peak shifts.

Using simulations to test the assumptions of the Rouhani and Barton (1987a)
model Charlesworth and Rouhani (1988) found that the approximation methods were
accurate only when the size of the two adaptive peaks was approximately equal. Up to ten
percent of the formded populations shifted under some of these models. The size of the
selection coefficient was relatively unimportant in diflirsion approximations (Rouhani and
Barton 1987a), but was important in the simulations (Charlesworth and Rouhani 1988).
The degree to which the population is released fi'om selection during rapid growth after
colonization is critical in determining the degree to which the selection coeficient
influences the probability of peak shifts (Charlesworth and Rouhani 1988).

The probability of peak shifts in isolated demes or founder populations discussed
above constitute only a portion of the SBP. The process was originally envisioned in large
spatially structured populations. With near total isolation between demes the overall

probability of at least one peak shift occurring should increase (Lande 1985). What is not

14

clear is what should happen in populations where isolation between demes is not so
complete. The effect of migration in reducing or increasing the probability of peak shifts
therefore demands attention.

Migration is critical to the third phase of the SBP. Phase three (interdemic
selection) has received increased attention recently (Barton and Rouhani 1993; Crow et a1.
1990; Moore 1996 chapter 3; Moore and Tonsor 1994). The general result of these
studies is that phase three proceeds fairly readily when the number of loci involved is low
but is slowed significantly by increased numbers of loci (Moore 1996 chapter 3, Crow et
al 1990) . However it is also clear that without a reasonable amount of gene flow phase
three will not occur (Barton and Rouhani 1993; Moore and Tonsor 1994; Rouhani and
Barton 1987b).

Studies have just begun to look at the combined SBP including all three phases.
This melding of the process is critical since there are conflicting factors causing phases one
and three. Specifically migration rates need to be low enough to allow demes to drift into
the domain of attraction of new peaks. Migration rates must also be high enough to allow
the conversion of demes at the lower peak through interdemic selection. Recent studies
have demonstrated that there are certain migration rates which allow all three phases of
the process to proceed in several different genetic models (Moore 1996 chapter 3, Moore
and Tonsor 1994, Barton and Rouhani 1993, Rouhani and Barton 1987b ). Previous to
those studies it had not been clear that there would be any particular migration rate which
would readily allow both phase one and phase three for any genetic system

Rouhani and Barton ( 1987b) have compared the probability of peak shifts

occurring in a single panmictic population, a one dimensional continuously distributed

15
population and a two dimensional continuously distributed population. The comparisons
were based on a double gaussian fitness surface applied to a continuous character.
Expanding on analytical methods previously used in studying peak shift probabilities and
statistical mechanics they demonstrated that peak shifts can occur in a population even
when there are no discrete barriers to gene flow in the one dimensional case. As in the
panmictic case peak shift probability in the one dimensional model was exponentially
related to the strength of selection (Rouhani and Barton 1987b). Provided that the
neighborhood size is less than about 30, the probability of peak shifts becomes moderately
likely in the two dimensional case. In a two dimensional continuous population the size of
the neighborhood becomes the overriding factor influencing the probability of peak shifts,
and the strength of selection is no longer the dominant factor. It is not clear exactly why
these one and two dimensional models behave differently but spatial segregation in
genetic variation may be quite different in two dimensional systems. Increased parapatric
differentiation may allow more gradual transitions into the domains of new peaks when
multiple dimensions exist (Rouhani and Barton 1987b).

Barton and Rouhani (1993) have also studied shifting balance analytically in
structured demes. They modeled migration between demes using an island model in which
the effect of migration may be treated as a diflirsion process. They compared the typical
polygenic trait under a double gaussian fitness sruface with a chromosomal rearrangement
model in which heterozygotes have a disadvantage. They found that, " below a certain
critical number of migrants, the demes scatter towards difl‘erent adaptive peaks, and the
allele frequency amongst migrants evolves towards a single intermediate value" (Barton

and Rouhani 1993 ). When, however, the number of migrants (Nm) is large the whole

16

population evolves together and is likely to be trapped on the lower peak. Barton and
Rouhani detected a sharp transition between the two extremes at a Nm of approximately
1. Just below that critical point adaptation was most efficient because peak shifts were
most fi'equently spread throughout the population. The above conclusions held for the
single locus and polygenic models both with and without dominance. This prompted
Barton and Rouhani to conclude that the similarity of behavior across models indicated
that their conclusions may be robust across many genetic systems. Despite this conclusion
it was recognized that adaptation via the shifting balance is more effective with disruptive
selection on discrete alleles than in polygenic traits.

Moore and Tonsor (1994) explored the dynamics of a two locus two allele system
using Monte Carlo simulations. In those simulations migration was a two dimensional
stepping stone model They found that the probability of peak shifts was again strongly
dependent on population structure. Even with deme sizes less than 30 individuals, demes
did not shift to new peaks when the per capita migration rates exceeded 0.05. Migration
rates below 0.001 were not effective in spreading peak shifts throughout the population.
In that study, however, within 12,000 generations over 25% of simulations resulted in a
population wide peak shift at the optimum migration rate. Both Moore and Tonsor
(1994), and Barton and Rouhani (1993) found that adaptation was most effective when
Nm was between 0.1 and 1. This indicates that this result is fairly general not just across
single locus and polygenic systems with an island model migration scheme but also in an
oligogenic system with stepping stone migration.

Diflirsion approximations which have traditionally used to study peak shifts within

an isolated population have recently been applied to the investigation of the SBP as a

17

whole. Barton and Rouhani (1987) contrast two distinct applications of these techniques.
They recognize that diffirsion models either have dealt with the single locus
underdominant allele or the case where the number of alleles (or loci) is nearly infinite so
that an assumption of a nearly gaussian distribution of frequencies may be made. These
two type of models represent extremes with respect to the rate of introduction of mutant
alleles (Barton and Rouhani 1987 ). Barton and Rouhani therefore compared a single
underdominant allele model with a model which assumes the distribution of probabilities
about a peak will be approximately gaussian. They demonstrated analytically that for a one
locus two allele model when the number of mutations is low (N it >> 1/4) the frequency of
peak shifts is a product of the probability of fixation of a single mutation and the mutation
rate. When the number of mutations is high the results converged with those for the
gaussian models generally used for quantitative traits.

While the conclusions of Moore and Tonsor (1994) and Barton and Rouhani
(1993) on the optimum level of structure for the SBP agree, there is no expected rate of
population wide peak shifts which seems to apply across models. Maore (1996 chapter 3)
compared the efficacy of the shifting balance process across several oligogenic models in
an attempt to determine whether any generalizations are apparent. In studying 2 through
16 locus models Moore concluded that the rate of population wide peak shifts was
inconsistent across models. He did find that increasing the number of allelic substitutions
necessary for a specific amount of trait change (i e. the number of loci involved) decreased
the efficacy of the SBP in a given phenotypic fitness surface. Despite this, the efl‘ect of
changes in the fitness surface was substantial enough that certain 8 locus models were

more effective than some 2 locus models.

18

The shape of the fitness surface in Moore's (1996 chapter 3) models strongly
influenced the rate of population wide peak shifts. The amount of genotypic variation
within and between demes is one factor which should change with changes in fitness
surface. This difference in the amormt of genetic variation between demes seems to
account for the overriding effect of the fitness surface's shape (Moore 1996 chapter 3).
Moore varied the initial conditions of a subset of his simulations to produce differences in
initial variance. As variance was increased the rate of population wide peak shifts also
increased. The amount of variation about the lower peak relative to the phenotypic
distance to a new domain of attraction seemed to be important as predicted by Rouhani
and Barton (1987a).

Whitlock (1995) has demonstrated that in general, high levels of variance within a
population can induce peak shifts between alternate phenotypic states with little or no
drift. He has shown that variance can induce higher rates of peak shifts in founder or
bottlenecked populations than drift alone. Whitlock recognizes that other mechanisms
which are capable of increasing phenotypic variance are also likely to increase the rate of
peak shifts. The additive genetic variance in a metapopulation overall will not determine
the amount of additive variance in the individual subpopulations or vice versa (Whitlock et
al. 1993 ). Population structure is one factor that can alter the amount of additive genetic
variance in a population. Previous studies of the SBP have generally not included
increased variance due to population sub-structuring (Whitlock 1995). The few studies
that have included these efl‘ects (Moore and Tonsor 1994, Moore 1996 chapters 3 and 4)
have had relatively high overall peak shifi rates.

The models of the SBP to date have found that the SBP can occur. The probability

19

of peak shifts, under certain models, is low enough that the importance of the process
continues to be questioned (Whitlock 1995). Additionally the range of migration rates
which allow the entire process is quite narrow in the models of Barton and Rouhani
(1993) and Moore and Tonsor (1994).

The small range of migration rates which allow peak shifts in the models discussed
above may partly be a product of the simplicity of the models used. Variability of certain
ecological parameters such as migration rate, carrying capacity and environmentally
induced variation may increase the rate of peak shifts in many cases. Moore (1996 chapter
4) has investigated the effects of variation in migration rate over time in a two dimensional
stepping stone model. Fluctuations in migration rate dramatically increased the eflicacy of
the SBP. Moderate fluctuations in migration rate are likely to be common on a short scale.
The geological time scale provides many possibilities for large scale changes in the
migration rates of many species (Cronin and Schneider 1990; Van der Spoel 1994). The
effects of variation in ecological factors inchrding the strength of selection, population
Sizes, and migration rates are all potentially important to the SBP on a geological scale.

The SBP provides a mechanism whereby adaptive changes and subsequent
reproductive isolation through hybrid breakdown are likely to occur (Rouhani and Barton
1987b). Although in many circumstances the probability of peak shifts within a deme is
low, the process seems likely to occur frequently on a geologic time scale. For this reason,
patterns of speciation may be strongly influenced by the SBP.

The three phase SBP is quickly becoming a well explored model which
incorporates major processes which are likely to influence macroevolutionary patterns of

radiation, change and stasis. It includes internal genetic constraints (i e. phylogenetic

20

inertia), isolation and drift as well as the possibility of reinforcement of isolation through
hybrid breakdown. Ecological realism also demands the recognition of variation in many
of the SBP'S parameters because the time between peak shifts is expected to be relatively

large.

Chapter 2
EMPIRICAL APPROACHES TO INVESTIGATING MULTIPEAKED

SURFACES: A REVEIW

Introduction

Adaptive topographies with multiple peaks are the backbone of Sewall Wright's
shifting balance process (SBP). When mean population fitness is dependent on allele
frequencies at multiple loci the pressure of individual selection to maintain historical gene
combinations may prevent increases in population wide fitness. The existence of multiple
adaptive peaks implies that internal genetic constraints to adaptation exist (Moore 1996,
chapter 1). The SBP, within the framework of adaptive peaks, is likely to influence
macroevolutionary patterns of radiation, change and stasis (Wright 1985a, 1985b).
Considerable efl‘ort has been focused on the exploration of shifting balance theory (Moore
1996, chapter 1). This review responds to a fundamental question that the theoretical
work on shifting balance theory cannot answer. That question is "how ubiquitous is the
evidence for multiple adaptive peaks?" The relevance of the shifting balance theory is
contingent on evidence for adaptive peaks. This review also assesses possible supporting
lines of evidence for the SBP which emerge from the studies which may involve multiple

adaptive peaks.

21

22

Methods of Detecting Gene Interactions
Because gene interactions are so critical to the SBP, the evidence for gene
interactions must be discussed. Evidence for gene interactions is elusive. This does not
however, imply that gene interactions are rare. In the broadest sense anytime two genes
act to produce a common trait there is a gene interaction. It is however only the
nonadditive effects that are of interest in the SBP. Adaptive surfaces can be viewed as
multiple fitness maxima and minima resulting fi‘om nonadditive gene action in the
production of a trait relevant to fitness. This is probably a reasonable view of adaptive
landscapes with one exception. The exception is that the atomization of organisms into
discrete traits eliminates many important possible nonadditive gene interactions (Gould
and Lewontin 197 9; Wright 1969). The ubiquity of pleiotropy would cause even additive
traits to have certain gene combinations which were more favorable than others due to the
summed effects on other traits (Wright 1969). The division of whole organisms into
subsidiary traits is therefore problematic because gene interactions rely on the definition of
a trait. Many gene interactions may be missed if traits are not properly defined.
Evidence of gene interaction from classical genetics
Despite the expectation that many types of gene interaction will be missed if
individual traits are the focus of study, strong evidence for epistatic interactions on
phenotype is ahnost as long standing as the study of genetics itself Bateson and Punnett
(Punnett 1923) observed strong gene interaction in a two-locus two allele system
controlling comb shape in chickens. In this case true breeding pea and rose comb chickens
when crossed will produce only walnut comb F 1S. F23 consist of a 9:3:3:1 phenotypic

ratio of walnut:pea:rose:single comb individuals. This is an example of discontinuous

23

variation where four specific phenotypic states can be identified (see figure 2.1). It is easy
to envision how selection on such discrete states could create a multi-peaked adaptive
landscape.

Introductory genetics texts provide many examples of gene interactions which
provide 9:3:3:1 ratios or modified two locus two allele systems. Some examples yield
discrete classes as in the case of chicken comb while others seem more continuous because
they are additive. It is important, however, to avoid viewing discrete classes which
produce an additive gradient of variation as being continuous. This error is easy to slip
into since the analogy between continuity and gradation is so strong. How a trait is
viewed determines whether multiple peak epistasis is found.

If discrete multilocus combinations produce discrete phenotypic classes then
arbitrary scalings can be deceptive. All such traits could produce a multiple peaked sru'face
given the right scaling. It can also be argued that all epistatic interactions may be removed
by adjusting the scaling. In the case of natural systems selection determines the scaling.
There is no reason to expect that scaling to be additive. The important point is that if there
are discrete interactions then the potential for epistatic interactions for fitness exist.

Even when the number of gene interactions increases beyond two loci with two
alleles, discrete variation is still found. Wheat grain color is determined by a three-locus
two allele system (East 1910; Nilsson-Ehle 1909). Trihybrid crosses for these loci produce
seven discrete phenotypic classes. Total pigment content acts additively and superficially
appears quantitative. There is, however, epistatic gene action if the trait of interest is the
absolute value of the deviation from medium red. In that case those allele combinations

which produce very high or very low pigment levels have the highest trait value, and

24

Figure 2.1 - Chicken combs produced by the combined action of two alleles at each of two
loci.

Four different chicken comb are produced by the interaction of alleles at two loci. Ifat
least one copy of the B allele and one copy of the A allele is present the pea Phenotype
will result (upper left comer). Double homozygote aabb individuals develop with a single
comb (lower right comer). Ifat least one copy of the A allele is present and the individual
is homozygous for the bb allele the individual will have the rose phenotype (lower left). If
at least one copy of the B allele is present and the individual is homozygous for the aa
allele the individual will have the walnut phenotype (upper right). Mating of two double
heterozygote (pea) individuals will produce all four phenotypes.

25

Figure 2.1

 

 

a

A

 

 

 

 

 

 

 

26

intermediate levels of pigment have lower trait values.

Classical genetics provides us with many examples of traits which are controlled by
multiple loci, and which do not easily fit a quantitative model A few good examples
include coat and eye variation in guinea pigs (Wright 1963; Wright 1968), mice (Russell
1949), dogs (Little 1957; Winge 1950), and other mammals (Castle 1940; Wright 1968) as
well as color patterns in Heliconius (Clark and Sheppard 1971; Mallet 1989; Sheppard et
al. 1985; Turner 1971). Examples in plants include flower color polymorphisms (Bateson
1909; Lawrence and Scott-Moncriefl‘ 1935; Onslow 1916), tristyly (De Nettancourt 1977)
and leaf morphology (Yu 1989). However, even when genes interact to produce a
conspicuously additive trait (as in wheat grain pigments) this may not be the 'natural' scale
for selection. In fact, all traits with a relatively few interacting loci have the potential to
produce a multi peak fitness surface. Genes that seem conspicuously non additive may not
produce a multi-peaked adaptive surface although they have that potential

Evidence of gene interaction fiom Quantitative Genetics

Quantitative traits may have both additive and nonadditive components
contributing to their variance. Heritabilities are determined by the additive variation extant
in a population. The residual genetic variance within a population may in part be due to
interactions between loci. Non-additive variance components indicate that non-additive
gene action is present. Small non-additive components, however, do not indicate the lack
of non-additive gene action. If many non-additive interactions exist but there is little
polymorphism at the contributing loci the non-additive variance component will be small
even though the number of potential interactions is large. Ifnon-additive variance is

significant this is a strong indication that non-additive gene action exists.

27

Optimal breeding regimes in domesticated organisms are often determined by the
relative importance of these components of population variance. A number of techniques
have therefore been developed to determine the proportion of variance explained by the
various sources of genetic variance. These techniques include, AN OVA techniques
(Hayman and Mather 1955; Kempthome 1966), generation means, and testcross designs
(Anderson and Kempthom 1954; Hayman 1958; Jinks 1956; Mather 1949).

Of the techniques mentioned above AN OVA techniques are the most problematic.
In the AN OVA method, the epistatic variance component estimates are based on a small
fraction of the actual epistatic variance that can actually be separated from other
nonadditive effects. This causes a reduction in the power of ANOVA techniques to detect
epistatic variance. Epistatic variance components are generally small in comparison with
their error estimates using ANOVA (Whitlock et al. 1995) although this is not always true
(e. g. seed weight in rice, Ram et al. 1989). Generation means tests and test cross studies
have a greater ability to detect epistatic interactions. Many such tests have detected some
level of epistasis. In reanalyzing cases of overdominance in 5 different species J'mks found
that in all cases interlocus effects caused an appreciable amount of the effects previously
attributed to overdominance (J'mks 1955).

The proportion of the variance in a population attributed to epistasis is aflected by
the number of potential epistatically interacting loci relative to the number of additive loci
and the allele frequencies at each epistatic locus. The expectation in a large population is
that selection on a given trait will reduce polymorphism at epistatically interacting loci
which produce that trait. The direction of selection on the epistatic alleles will be

determined by the frequency of the alleles at other loci. As the level of polymorphism is

28

reduced at one locus, all loci that could epistatically interact with it will act increasingly
additively. The possible epistatic interactions are what is important in the creation of
multiple adaptive peaks not the frequency of certain alleles in certain populations at any
given time.

Environmental dependence of epistatic variation can allow polymorphisms to
persist in a population despite selection. Selection on a trait within a single environment
will reduce the polymorphism at loci that have strong interactions in that environment. If
the environment is then changed there may be a change in which interactions are
important. This will create interactions which have not been acted on by selection
previously. In this case exposure to extreme environments can release epistatic variance
(Dykheusen and Hart] 1980). Environmental dependence of epistatic variance measures
has been demonstrated in Nicotiana (Imks et al. 1973 ). The potential number of gene
interactions which are readily detectable in an undisturbed population may therefore be
small when compared with the total number of possible interactions which exist.

Overall, quantitative genetics provides a large amormt of evidence for epistatic
interactions. While many of these interactions are weak when compared to the overall
genetic variance this may often be due to the loss of high levels of polymorphism at
relevant loci through the action of selection. This is to be expected in a large population
when looking at alleles involved in an adaptive peak. Many important fitness peaks may
be obscured when quantitative traits are studied independently from each other. Despite
these problems there is still evidence that epistatic interactions exist with at least

moderate strength and frequency in quantitative traits.

29

Evidence of gene interaction from molecular genetic studies

One potential source of information about epistatic gene action is through
Quantitative Trait Locus (QTL) studies. In this type of study polymorphic markers allow
the mapping of a quantitative trait onto specific chromosomal regions (see Cheverud et al.
1993; Tanksley 1993) for reviews of these techniques). Multiple factor interactions
between map regions may also be identified. While some QTL studies have found little or
no evidence for epistatic interactions others have found rrmltiple factor interactions
(Whitlock et al. 1995). One of the best examples of interiocus interaction identified by
QTL work is in stemopleural bristle number in Drosophila melanogaster (Long et al.
1995). It is important to note that this study was based on a population produced from the
hybridization of lines that had previously been selected for either high or low bristle
number. Hybridization between lines can expose epistatic interactions which are not found
in either parental lineage.

In addition to the 'survey' approach to QTL work discussed above specific
candidate loci are fiequently identified for study a priori based on possible metabolic
importance. This type of approach has identified epistatic interactions in a number of traits
(reviewed in Whitlock et al. 1995). Combined with the survey techniques in QTL analysis,
this method shows great promise in identifying epistatic interactions. While these
techniques thus far have only produced weak evidence of epistatic interactions there are
several ways in which the number of these interactions identified may increase. The ability
to detect epistasis between tightly linked loci requires very high resolution maps which
have previously been very difficult to obtain (Cheverud et al. 1993; Tanksley 1993 ). Tight

linkage between coadapted gene complexes is, however, exactly what is expected in the

30

case of coadaptation of genes (Prakash and Lewontin 1968). Therefore, as QT'L maps
gain resohrtion the identification of epistatic loci may be accelerated. Additional examples
of epistasis may also arise as more studies use populations produced by crossing divergent
populations.

Classical genetics, quantitative genetics, and QTL work all indicate there is a
potential for gene interactions to be important. While the interactions actually elucidated
by these techniques may be less than overwhelming, they seem sufficient to indicate that
many adaptive peaks potentially exist. In fact, it is expected that the number of detectable
interactions will be low when selection is involved and crosses are within local
populations. As expected those QTL studies which demonstrate epistasis are those with
the most divergent crosses. It is surprising how many cases of gene interaction exist. What
remains unclear is whether any of these examples of gene interaction actually produces

adaptive peaks.

Natural Selection As a Non-Arbitrary Ruler

The presence or absence of epistasis as stressed above is dependent on the scaling
of traits. None of the examples given so far has referred to epistasis for fitness, the only
trait which is truly relevant to adaptive landscapes. The only appropriate and non-
arbitrary scaling is provided by natural selection. Not only are rugged adaptive landscapes
produced by epistatically controlled traits that have strong directional selection acting on
them, but also they are produced by nonlinear fitness functions acting on traits that have
predominantly additive genetic compositions. The use of natural selection as the ruler

upon which a non-arbitrary scale for adaptive landscapes is scribed is necessary but also

31

provides some problems.
New sources of gene interactions

Non-linear fitness firnctions applied to additive traits create epistasis for fitness.
Even optimizing selection in which there is a single phenotypic optima is a potential source
of epistasis for fitness. A trait that is determined by purely additive gene action will exhibit
only nonadditive variation for fitness if it has reached a selective optima (Falconer 1981),
p. 394). The absence of response to selection by a population at an optimal equilibrium
(i e. a heritability of 0) means that all genetic variation is nonadditive.

Even when optimizing selection has not yet reached eqruhbrium conditions there
will be multiple sets of alleles across all loci which will result in an optimum phenotype.
This means that adaptive peaks exist. One way of envisioning this is to recognize how
selection acts on an allele which has a fixed additive effect on trait value. Selection is
against an allele which increases trait value if the population mean is higher that the
optimum Selection favors the same substitution if the mean is lower than the optimum
(See figure 2.23).

In the case of wheat grain color two possible scalings were mentioned. The first
was based on total pigment content of the grain. This scale would be applied if the trait
were under directional selection. This scale does not result in any epistasis. Ifhowever
stabilizing selection favored the intermediate phenotype, the scaling is based on the
deviation from that medium red color. In this second case an adaptive landscape with five
peaks would be created. Wheat color is not a true quantitative trait but it demonstrates the
potential for conversion of additive trait variance into epistatic variance for fitness.

Optimizing selection on a purely additive quantitative trait will also result in nonadditive

32

Figure 2.2 - Epistasis for fitness produced by optimizing selection (a) and disruptive
selection (b) acting on an additive trait.

Alleles which increase the numeric value of the trait are indicated by a + while alleles that
reduce the numeric value of the trait are indicated by a -. The + alleles are favored only
when there is no more than one other + allele in that genotype. The curves represent
fitness plotted against phenotype. Note that there are many different combinations which
can provide the same phenotype.

33

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34

variance components of fitness (Tachida and Cockerham 1988).

Optimizing selection acting on an additive trait produces a qualitatively special
type of epistasis. Because the optimal trait may be built up from multiple possible
genotypic combinations the epistatic interactions are diffuse throughout the loci involved
in the trait. This 'diflirse epistasis' may have special characteristics. For instance, in the
case of optimizing selection on a quantitative trait, if the number of loci is large and the
phenotypic effect is relatively uniform across loci, n!/[(n/2)!]2 adaptive peaks may exist
with 11 pairs of alleles (Wright 1939). However, the valleys between any two peaks are
vanishingly small, and the peaks are eiqiected to have roughly the same mean fitness.
Despite these generalizations diflirse epistasis may be very important in producing
interesting adaptive topographies tmder certain circumstances. These circumstances will
be discussed when the topic of integration across traits is discussed.

Disruptive selection exists in two forms. Either both phenotypic extremes are
locally optimal, or some of the optima are not at the phenotypic extremes . In both cases
the selection on an allele is dependent on the frequency of the other alleles effecting that
trait. This is due solely to the relative positions of the two phenotypic optima compared to
the mean phenotype in a population. Depending on the position of the mean phenotype
relative to the phenotypic optima selection may force the population toward one or the
other optimum (see figure 2.2b). Selection therefore alters allele frequencies at one locus
in a way which is dependent on the fiequency of alleles at other loci In addition if one or
both phenotypic optima are not at the phenotypic extremes then multiple genotypic
solutions will also produce the same phenotypic optimum allowing genotypic variation

between populations on the same phenotypic peak to exist.

35

Integration across traits by selection can cause epistasis for fitness. Selection
acting on one trait may affect the trait mean of other traits due to the pleiotropic effects of
genes that effect both traits. The result of this correlated response between traits is that
many traits may be constrained by the opposing forces of selection acting on multiple traits
sinmltaneously. Wright recognized that when stabilizing selection acted on a trait, fitness
differences could exist between the multiple genotypic optima. These differences would be
due to pleiotropic effects of the particular allelic combinations at each particular optima
(Wright 1939). Even though different genotypic peaks may not difl‘er in their fitness
efl‘ects through a particular trait, the pleiotropic effects may cause fitness difi‘erences.
Selection may often work on pairs of traits. The selection on one trait can be dependent on
the state of a second trait. This correlational selection integrates across traits. Unlike
integration via metabolic or developmental pleiotropy this integration is imposed by the
environment. Even if two traits are completely genetically independent multiple adaptive
peaks will exist if the fitness of a given allele at one locus is dependent on the alleles
present at another locus. In the case of correlational selection certain alleles at loci
controlling a trait are only favored if appropriate alleles are present in loci affecting a
second trait.

Gene Interactions Revisited

The previous sections provided evidence of gene interaction in arbitrary traits.
These arbitrary traits indicate that there are many possible interactions that could provide
multiple peaks. Certain types of interaction have not, however, been addressed. Natural
selection works on populations of whole organisms and gives us a nonarbitrary scale for

measuring gene interaction. Gene interactions between traits are integrated by natural

36

selection. Therefore we must at least consider that evolutionary constraints on any given
trait of an organism exist due to selection on other traits. In the SBP these constraints are
not necessarily insurmountable. They should however be difiicult to detect since they are
expected to produce near fixation of particular genetic combinations across geological
time scales, thus disguising the very interactions which drive the SBP.

Where to look for multiple peak epistasis

Multiple adaptive peaks can exist even when genetic polymorphisms for multiple
peaks do not exist. Variation within a population will rarely span the domain of nnrltiple
adaptive peaks because selection will tend to maintain most population within the domain
of a single peak. Because variation within a population will usually not include multiple
adaptive peaks, evidence nnrst come from those special situations where variation
spanning multiple peaks is expected. Multiple peaks are most likely to exist when the third
phase of the SBP is incomplete due to isolation of demes, or when selection has had
insuficient time to remove variation (i. e. in novel environments).

Most previously mentioned examples of epistasis in the formation of traits exist in
domesticated species. Many traits in domesticated organisms are released from natural
selective forces and exposed to novel selective forces. For this reason variation which
would not be found in natural populations may be present. In addition, examples of
epistatic polymorphism may exist because the traits are not tmder strong selection even in
natural systems. Examples of adaptive peaks in a natural system will be much more
dificult to detect. Rarity of epistatic polymorphisms for fitness in natural populations is
an expected result of the SBP. The situation where multiple adaptive peaks will most

easily be detected is when either the external environment or the internal genetic

37
environment of an organism is variable.

Within a population, the genetic variation should remain within the domain of one
adaptive peak for long periods. Therefore, epistasis for fitness will be nearly nonexistent
within any given population. If, however, the external environment is changed, the
pleiotropic interactions and the correlated selection between traits may change. The
change in interactions is expected whenever environmental efl‘ects are not uniform across
all traits that a locus influences. In that case, when the environment is changed the
correlation between traits which is due to the pleiotropic effects of a locus will change.
The mapping of genotype onto phenotype has now changed Until a selective eqrrihhrium
has been reached the probability of detecting variation which spans multiple peaks is
increased. Within a population, periods during and shortly after large scale environmental
changes are the most conducive to the detection of epistasis for fitness.

Differences in coadapted gene complexes between populations are likely to
provide evidence of adaptive peaks. These differences can accumulate only when the SBP
is ineffective. Specifically, the third phase cannot have occurred if such variation is to be
preserved between populations. Multiple peak polymorphism across demes is most likely
in situations where multiple demes of at least periodically small size are highly isolated.
The isolation needed to prevent the third phase of the SBP requires immigration of less
than approximately one individual every 100 generations (Barton and Rouhani 1993,
Moore and Tonsor 1994). Crosses between such demes are most likely to expose variation
spanning multiple adaptive peaks.

Difl‘erences in internal genetic architecture should be most pronounced between

subspecies and species. Here the possibility of the third phase occurring is largely

38

eliminated In cases where species can produce hybrid offspring, multiple adaptive peaks
might be found in the variation between those species. In the case of between species (or
subspecies) hybrids the expectation is that each parent is the product of a somewhat
different selective environment. Lowering of fitness therefore must be relative to either
parent in either environment. This is true for both interpopulational and interspecific
crosses.
Evidence fiom classical and population genetics

Classical genetics has provided examples of epistatic interactions that are clearly
linked to fitness. Adaptive peaks have been clearly defined in few if any cases (e. g. Clark
and Sheppard 1971; Clark et al. 1968; Nagel et al. 1989). Several genes are known to
ameliorate the effects of sickle cell anemia (Nagel et al. 1989). There are also apparently
fitness costs associated with some of these modifying genes (Nagel et al. 1989). The exact
shape of the fitness surface for these genes is unknown and would be expected to change
in a spatially concordant way with malaria risk. If nnrltiple peaks exist in this system
population structure alone is unlikely to be capable of protecting the polymorphism
Presumably spatial and temporal variability in selection provide adequate explanation for
the polymorphism

Targeting particular proteins for study when they are suspected of being of
particular importance has also had some success in identifying epistatic genes which seem
to effect fitness components (e. g. Cavener and Clegg 1981; Dykhuizen and Hart] 1980;
Goolish and Burton 1989; McKechnie and Geer 1988). The presence of the
polymorphisms necessary for the identification of these interactions may be the result of

different factors in each case. Epistasis for fitness may be created by the introduction to a

39

new environment (Dykhuizen and Hartl 1980). It may be preserved by population
structure (Goolish and Burton 1989) , or be the product of many factors including spatial
and temporal variation in the environment (Cavener and Clegg 1981; McKechnie and Geer
1988).

One of the weakest pieces of evidence for adaptive peaks also is one of the more
ubiquitous. Selection for coadapted gene complexes will create aggregations of alleles
that are favorable only as a group. It is to be expected that these aggregations of alleles
will show up as linkage disequilibria between coadapted loci. While recombination may
break up such aggregations, population structure and physical linkage of genes may allow
favorable combinations to segregate. Therefore, many loci that interact epistatically are
tightly linked physically (see Hedrick et al. 197 8 for review).

Alternative explanations for the presence of linkage disequihhrium weaken this
type of evidence for multiple peak landscapes. Physical linkage within a chromosome
surely must be eliminated as a cause of the disequih‘brium Because the opportunity for
coadaptation of genes is increased by physical linkage the two factors may often be
difficult to separate. Despite these caveats some examples of disequih'brium between loci
(Barker 1979; Clegg et al. 1972; Hedrick et al. 1978; Klitz and Thomson 1987; Lewontin
and White 1960) seem to support the existence of multiple adaptive peaks.

One exceptional case of disequilibrium in a natural system is in shell banding and
color patterns in snails of the genus Cepaea. In C. nemoralis and C. hortensis nnrltiple
shell pattern and color loci are in diseqruhhrhrm (Jones et al. 1977). The loci are tightly
physically linked but also have been shown to strongly interact in the determination of

fitness. The most fit peak appears to vary between location. Local selection therefore acts

40
to preserve the polymorphisms between populations in this system

Chromosomal inversion can allow the independent evolution of groups of genes
within multiple lines of Drosophila pseudobscura. Population structure in this case results
in populations with different inversion types (Prakash and Lewontin 1968). The existence
of heterosis between karyotypes from within the same population without such heterosis
in inter-populational crosses has caused some to conclude that the inversion types carry
coadapted gene complexes (Dobzhansky 1951). Prakash and Lewontin (1968) detected
a pattern of allelic variation within the inversions that indicated coadaptation of at least
two loci This implies nurltiple adaptive peaks although again little can be said about the
shape of the surface. In this example variation spanning multiple adaptive peaks probably
is maintained because the coadapted gene combinations are protected from
recombinational decay by linkage and the inversion types.

Frequency dependent selection on chromosomal inversion types also generates a
multiple peaked adaptive surface even without coadaptations developing within the
inversions. This type of multiple peaked surface may be common (see Whitlock et al. 1995
for discussion). This is due to underdominance for fitness among the karyotypes.
Frequency dependent selection on a trait that is under control of multiple genes can also
create hybrid breakdown for fitness within a species. In some Heliconius butterflies local
selection favors phenotypes that resemble parental types in wing pattern (Mallet 1989;
Mallet et al. 1990). Hybrids that do not resemble either parental population are selected
against. Similar circumstances exist in other butterflies where mimicry is important
(Turner 1977). Whether frequency dependent selection causes reduced fitness in the F l or

the F2 depends on the dominance relationships in the two crossed populations.

4 l

Underdominance for fitness at a single locus produces the most simple type of
multi-peaked adaptive surface. As mentioned above chromosomal inversion types seem to
be common sources of underdominance although not all examples are clear cut. In some
cases of pericentric inversions, underdominance may not be present (Coyne et a1. 1993 ).
Underdominance for fitness exists in the snail Partula. Variation in the direction of shell
coiling within populations of Partula exists (Johnson 1982; Lipton and Murray 1979) and
is due to the action of a single gene (Neirnan et al. 1990). Dificulties in mating between
different coiling types have been shown (Gittenberger 1988; Lipton and Murray 1979).

Optimizing, disruptive, and correlational selection

Despite the difficulties in detecting optimizing selection (Endler 1986; Travis
1989) many examples of optimizing selection exist (see Travis 1989, Endler 1986 for
reviews). Many documented instances of optimizing selection are less than concrete. The
vast number of demonstrations, however, adds credence to the alleged ubiquity of
optimizing selection.

Two types of optimizing selection can exist (Endler 1986; Gavrilets and De Jong
1993; Travis 1989). In one type of optimizing selection, selection acts directly on the focal
trait to create an optimal phenotype. Selection on gall size in the Solidago-Eurosta system
(Weiss and Abrahamson 1986) is one example where this type of optimizing selection may
be going on. This type of optimizing selection directly creates diflirse epistasis between all
the alleles acting on the trait.

Indirect selection via pleiotropic genes that act on multiple traits that are directly
related to fitness creates the second type of optimizing selection (Gavrilets and De Jong

1993). This type of optimizing selection seems to be the predominant type in studies to

42

date. Selection on stemopleural bristle number in D. melanogaster is an example of this
mode of selection (Kearsey and Barnes 1970; McGill and Mather 1971).

Indirect optimizing selection is expected to only exhibit diffuse epistasis
temporarily (Whitlock et al. 1995). Selection will remove alleles that have negative
effects through nrultiple traits simultaneously. Similarly, it will fix alleles which increase
fitness through their effect on multiple traits. This leaves only alleles that have tradeoffs in
their effects on fitness through multiple traits (Wade 1992). Indirect optimizing selection
implies that adaptive peaks exist because the directional selection on individual
components of fitness is limited by the pleiotropic effect on other components of Mess.
Changing the trait value for such a trait requires changing the balance of tradeofl‘s across
traits. Genetic variance is maintained but the population is unable to respond to selection
because all the residual variation is non-additive. Here multiple peaks would be evidenced
by the existence of different optima between populations.

Most examples of optimizing selection are not clearly identifiable as either of the
above two types. What is clear is that there is a great deal of evidence that optimizing
selection is common (Travis 1990, Endler 1986). With indirect selection absolute proof of
multiple peak epistasis requires examples where the optirmrm is altered between
populations due to the genetic architecture within populations. Given this state of affairs
optimizing selection can be viewed as providing a common if not ubiquitous opportrmity
for multiple adaptive peaks to exist.

The number of examples of disruptive selection on polygenic traits in natural
systems is limited. While some good examples exist, few populations have variation that

extends over multiple phenotypic peaks. This does not mean that such adaptive peaks are

43

not available, just that they must be found by crossing individuals of divergent populations
or closely allied species. In the Afiican Pyrenestes finches there is disnrptive selection on
beak shape. Beak shapes determine the types of seeds most easily processed with certain
intermediate beak types being inferior on both seed types. These intermediate types have
lower Mess over certain periods (Smith 1993). Perhaps the most well known example
of disruptive selection is in Galapagos finches. Here selection on beak morphology
determines the food type upon which individuals feed (Schluter and Grant 1984). In this
case reproductive isolation allows the coexistence of multiple peaks. Instances where rapid
adaptive radiations have occrured in the relatively recent past should be some of the best
places to look for the coexistence of multiple adaptive peaks on separate islands.
Correlational selection, where the state of one trait determines the Mess of the
phenotype at another trait is conceptually alluring. Brodie ( 1992) has provided an
interesting example of this in the garter make Thamnophis ordinoides. These makes have
a polymorphism for color/pattem and also for anti predator behavior. Striped makes are
more susceptible to detection than spotted makes when they are stationary. When striped
makes are moving however it is more diflicult to judge their ground speed than it is for a
spotted make. Stripedness is negatively correlated with the propensity to reverse direction
and survivorship through the early juvenile stages is enhanced by the appropriate match of
behavior and pattern (Brodie 1992). Similar patterns have been shown between resting site
choice behavior and coloration in Biston beiularia (Kettlewell 1955). Correlational
selection between host preference and host specialization may be responsible for reduced

fitness in hybrids between two sibling species of Rhagoletis (Bierbaum and Bush 1992).

44

Evidence fiom structured populations and Hybrid breakdown

One striking result of a review of multi-peak epistasis is that without some peculiar
circumstances variation within a population will not include multiple peaks (Whitlock et
al. 1995). Population structure is a critical part of the SBP for this very reason. There are
two types of studies that specifically look at the efl‘ects of structure on genetic
architecture. The first type is through studies of outbreeding depression and hybrid
breakdown. This breakdown can occur either between populations or species. The second
type is fiom studies of the efficacy of evolution in populations with different levels of
structure.

A reduction in Mess between parents and ofl"spring when the parents come from
different populations may be the product of the breakup of coadapted genes. If hybrid
offspring (F l) have lower average Mess then either parent the results are consistent with
the expectations for the decay of epistatic interactions but may also be produced by
underdominance. This is termed outbreeding depression. If the effect increases with
subsequent generations (F2, F3, . . . ) there is unequivocal evidence for epistasis and this
will be called hybrid breakdown.

Evidence for hybrid breakdown is common when lines or populations are provided
ample time to diverge (Burton 1990a; Burton 1990b; Hard et al. 1992; King 1955;
Vetukiv 1956, and see reviews in Gieger 1988; Wright 1977). Interspecific crosses are
inherently prone to hybrid breakdown. Interspecific hybrid breakdown is especially
interesting because it may often represent the effects of breaking up suites of traits that
have coadapted to different selective peaks. (Grant 197 5; Stebbins 1955).

The final evidence for the existence of adaptive peaks is that when several

45

researchers have attempted to test for the effects of population structure on evolutionary
dynamics they have formd that population structure can affect the ability of a population to
adapt to an environment. While some such studies have failed to detect increased or
divergent adaptation with population subdivision (Katz and Enfield 197 7; Madalena and
Robertson 1975; Rathie and Nicholas 1980) others have seen just such results (Cohan et
al. 1989; Enfield 1977 ; King 1955; Wade and Goodnight 1991). The critical factor is the
mixing rate between subpopulations. Those experiments that demonstrated increased
adaptation or divergent modes of adaptation had low or nonexistent mixing rates.
Adaptive peaks combine selection and genetic constraints into a single
evolutionary viewpoint. In this view selection acts within the limits of the genetic
architecture caused by interactions of genes in determining fitness. The ultimate scale upon
which gene interactions are measured is therefore selection. Adaptive peaks may be
produced by most models incorporating selection on natural populations, and there is a
great deal of empirical evidence for their existence. Because adaptive peaks constrain
evolution by natural selection, the ability of populations to shift between adaptive peaks

via the SBP may decide the tempo of adaptation in many natural populations.

Summary
This review has emphasized adaptive peaks as a critical feature of the SBP. The
reason for this emphasis is two fold. First adaptive peaks are the essential basis of the
SBP. Without them drift is just noise in a purely deterministic system Secondly there is a
multiplicity of types of adaptive peaks which may exist and not all of these types will

necessarily act identically in the SBP. The evidence that multiple adaptive peaks exist is

46

persuasive. The evidence for their existence, however, is persuasive mostly by sheer
volume however. Examples that suggest the existence of adaptive peaks abound but the
clearly delineated fitness effects are rarely available . More concrete examples of multi-
peaked fitness effects are needed. This is especially true because the influence of the SBP
may be somewhat reliant on the ubiquity of different types of epistasis for Mess.

The search for adaptive peaks should emphasize those areas where peaks are most
likely to appear. Population structure is one factor that may preserve multiple adaptive
peaks. In order to preserve variation that spans multiple peaks, migration rates between
demes must be consistently low enough to prevent interdemic selection. This will prevent
the most favorable gene combinations from spreading through a population quickly once a
shift has occurred. Other factors which may allow multiple peak polymorphisms to persist
include variable selection and introduction to novel environments. If anything less than the
most favorable situations for the existence of multiple adaptive peaks are studied then
failure to find such epistasis can be claimed as a failure to look in the right place.

Examination of multiple types of genetic models in the SBP leads to the
conclusion that population structure is generally critical to the ability of populations to
reach new optima. Migration rate between demes, is critical to both the probability of
shifts to occur and the ability of shifts to spread throughout a population. Population size
is critical to the probability of peak shifts. Variation in migration is likely to increase the
eflicacy‘of the process. Even though there are these general conclusions about what
conditions are most adaptive, the probability of peak shifts is highly dependent on the
genetic models. Some adaptive landscapes will be much more prone to peak shifts than

others.

47

There are additional factors whose effect on the SBP needs to be studied. Variance
in phenotype which can temporarily allow the traversing of adaptive valleys could greatly
increase the efficacy of the SBP. Variance in phenotype and its effect on fitness surfaces
needs empirical study. Finally, the possibility of bridging many of the gaps between micro
and macroevolutionary theory should be a logical goal of researchers throughout
evolutionary biology. The SBP provides a framework within which those bridges might be
built. The empirical evidence for the existence of adaptive peaks as well as the theoretical

viability the SBP is strong enough that process is probably of major evolutionary

significance.

Chapter 3
A SIMULATION OF WRIGHT'S SHIFTING BALANCE PROCESS:

MIGRATION AND THE THREE PHASES

Introduction

Sewall Wright's shifting balance process (SBP) is a mechanism by which complex
genetic traits can evolve despite the pressure of individual selection to maintain a historical
genetic arrangement (Wright 1982a). A critical concept in the SBP is that of the adaptive
topography. In such a topography mean population Mess is dependent on allele
frequencies at multiple loci Wright was interested in topographies with multiple mean
Mess optima (peaks) with intervening mean Mess minima (troughs). For this reason
gene-gene interactions are of central importance to the SBP (Wright 1977, 1978b, 19823).
Although multiple peaks may arise due to underdominance or multiallelic overdominance
at a single locus, Wright saw adaptive topographies as arising primarily from interlocus
interactions.

Wright viewed epistasis as pervasive. Kacser and Burns (1981) explicate a
mechanism by which epistasis would be as pervasive as dominance for genes whose
products fimction in metabolic chains or networks. Jinks (1983) suggests that epistasis
may explain all or most cases of overdominance reported from combining ability

experiments. Even for genes with purely additive effects on the phenotype , when the

48

49
Mess firnction is non-linear, phenotypically additive genes will exhibit epistasis for Mess.
For example, with optimizing selection, an allele with a positive phenotypic value can have
positive or negative Mess effects, depending on the sum of genotypic values for all loci
affecting the trait. Because of Wright's interest in epistasis, and because there is mounting
evidence for its inrportance as a mode of gene action, we focus on epistasis as a
component of the SBP.

When epistatic interactions exist, the can constrain the ability of a large population
to evolve in the direction of the most-fit genotypes. These constraints are the fabled Mess
valleys of the multilocus, multipeaked adaptive topography of the type Wright envisioned.

Underlying a nurlti-peaked adaptive topography is a multi-locus genotypic Mess
surface in which the relative Mess of any allele depends on the genotype in which it is
manifested (see Provine, 1986, pp. 307-317 for a discussion of the confirsion surrounding
Wright's original Mess surfaces). The allele fiequencies within a (random-mating) deme
determine the predominate genotypes within which an allele is manifested. The direction
and magnitude of selection acting on epistatically interacting loci is thus determined by the
deme's allele fiequencies. Likewise, the mean Mess of a deme depends on the fi'equency
of interacting alleles, and this can result in multiple Mess optima in a Wrightian adaptive
topography.

In an infinitely large population, evolution by natural selection will bring the
population to the local fitness optimum, leaving the remaining Mess surface tmexplored.
In spatially structured populations with small deme sizes, random genetic drift can remit in
an exploration of the entire surface. Migration among demes limits the extent of stochastic

divergence in allele frequency among demes, and limits the ability of the population as a

50

whole to move among peaks. To understand the evolutionary process in structured
populations, Wright therefore believed that one needed to rmderstand the interactions of
random genetic drift, epistatic Mess effects at drifting loci, and the homogenizing effects
of migration among demes.

Wright (1977, pg. 455) partitioned the shifting balance process (SBP) into three
phases (Wade and Goodnight, 1991). Phase I is the stochastic drift of allele fi'equencies
within demes, which can shift a deme into the attractive domain of an alternative peak.
Phase II is the shift of allele fiequencies towards the optimum of the ‘new' peak through
individual selection within the shifted deme. Phase III is the conversion of surrounding
demes to a higher peak through immigration from a previously peak-shifted deme (ie.
interdemic selection).

Phase I is most effective with low migration among demes and small effective
population size. Phase III occurs most readily with high rates of migration among demes.
Only with some intermediate level of migration can both phases I and 1]] occur (Wade and
Goodnight, 1991). However, the range of migration rates between 0.0 and 1.0 that allow
the SBP remains almost entirely unknown. Wade and Goodnight (1991) found that mean
Mess increased at levels of migration below 0.05. However, the genetic causes of
changes in mean Mess have not been established. It has been by no means clear that for
any given epistatic system there is any constant migration rate which will allow all three
phases of the SBP to occur (Hartl and Clark, 1989, pp. 323-324). The purpose of this
study was to explore the interaction of migration and the three phases in determining the
domain of migration rates for which the SBP is likely. We focus here on the role of

migration as a factor governing the time to and frequency of population-wide peak shifts.

51

When Wright first proposed the SBP (1931), very little was understood about
drift, interdemic selection, effective population sizes, or epistatic variation in natural
populations. Since then, considerable progress has been made in understanding the
potential role of drift in natural populations (Kerr and Wright, 1954, Buri, 1956, Epling,
Lewis and Ball, 1960, Bowden, 1982), epistasis in determining Mess differences (Burton
1990, Wade 1985, and for a review see Barker, 1979), and interdemic selection
(McCauley and Wade, 1980, Wade, 1977, Wade and McCauley, 1980, 1984, Goodnight,
1985). However, the only empirical test of the SBP as a whole was made by Wade and
Goodnight (1991), which indicates that Mess changes in complex traits can be influenced
by the population structure in a way that broadly corresponds to the expectations of
shifting balance theory.

The necessarily large scale of investigations like that of Wade and Goodnight
(1991) slows progress towards an understanding of the role of the SBP in nature. Until
more experimental results are available, and as a theoretical underpinning providing
increasingly explicit expectations for firture empirical studies, a more mechanistic
rmderstanding of the interaction of relevant factors is needed.

In order to gauge the relative importance of the shifting balance process we need
to know what conditions (if any) are conducive to the SBP, and the ubiquity of those
conditions. Because of the complexity of the SBP and the role of stochastic processes, a
comprehensive mathematical description has been elusive. Barton and Rouhani (1993)
provide the most comprehensive view to date, but because of their need for mathematical
tractability their work is limited to purely additive genes or to intralocus allelic

interactions. In lieu of a mathematical model we have used a computer simulation to

52
explore the process. We use a sinmlation which specifies an individual's genotype, Mess
and dispersal behavior. By keeping track of individuals in this way, we have avoided
simplifications which prevent insight into the role of migration in the SBP. Our overall
goal was to describe the relationship between migration rate and the probability of a
population-wide peak shift. We looked for the domain of migration rates in which peak
Shifts occur with any frequency. We asked if one can expect an optimum migration rate for
the SBP under the conditions of any particular population. We also examined the effect of
migration rate on the propensity for peak shifts through a combination of phases I and H,
and migration rate's effect on phase 111. Finally, we examined the relationship between
migration rate and the extent of deme extinction/recolonization, and its effect on the

propensity for peak shifts.

Methods
The Components of the Simulation

We modeled a diploid, obligately sexual, semelparous species. Density independent
mortality (hard selection) took place in the juvenile phase prior to migration and mating.
Selection was based on genotype-dependent smvival probabilities. Migration was followed
by random mating of adults.

A critical aspect of the SBP is the variation of deme (sub-population) size. The
variation in deme size (N) allows variation in the absolute number of individuals
emigrating from demes (Nm), without requiring differential per-capita migration rates (m)
between demes. This can drive the third phase of the SBP (Crow et al, 1990). In addition,

variation in deme size will also create variation in the rate of random genetic drift. This

53

afl‘ects phase I as well as altering the relative contribution of immigrant genotypes to the
genotype and allele frequencies in the mating pool

In real populations it is not unreasonable to assume that some type of resource
limitation often sets an upper bound (K) on the number of individuals which can survive
within a population, and excess reproductive capabilities will tend to force populations
towards that K (Verhulst, 1838). Because, in this case, all populations are expected to
remain near their carrying capacity, variation in size amongst demes near K must be
regenerated each generation by hard selection within demes. In these simulations hard
selection associated with the genotypic composition of the demes results in variation in
mean Mess among demes.

A finite maximum deme size was imposed by incorporating a carrying capacity (K
= 30), and intrinsic growth capabilities into a logistic fimction (N(,,,) = N0) +
rN(,)(K-N(,) /K, r = 1.1). This function determined the maximum number of individuals that
could be born into that deme during the next generation (Nam)- For each offspring,
random male and female parents were chosen from within the deme and a randomly drawn
copy of the gene at each locus was pulled from each of the chosen parents to produce an
offspring (i e. assortment was fully independent). This procedure was repeated until the
necessary (Nam) number of offspring were produced. Because all demes have the same r
and K, during each generation all demes tend towards the same carrying capacity. This
demographic model tends to reduce divergence in deme size based on genotype
frequencies and therefore reduce the opportunity for interdemic selection to take place.
This is a relatively unfavorable scenario for the SBP.

In the event that at least one individual of each sex was not present that deme was

54

allowed to go extinct. Recolonization of extinct populations occurred only when
individuals of both sexes chanced to migrate into the extinct deme in the same generation.

We simulated a population which contained three difl‘erent Mess-phenotypes.
This allowed variably-sized demes whose average deme size was related to phenotype.
Each phenotype had a predefined absolute juvenile survivorship. There was no mortality
outside of the juvenile phase. Using two loci, each with two alleles, provided us with nine
genotypes. The nine genotypes provided three phenotypes based on the model of epistatic
interaction employed by Crow et al (1990). The average 3115511th Messes were as
follows; double homozygote wild type = 0.636, individuals which were heterozygous at
one locus but homozygous wild type at the other locus = 0.620, and individuals which had
at least one copy of the mutant genotype at each locus = 0.700.

In Crow et al's formula the three phenotypes have Messes that would be
described by the equations:

W1: 1, W2=(1- s), andW,=(1+ks)
In these sinnrlations s determines the strength of selection and k determines the relative
heights of the multiple peaks. For these sinurlations s was set to .025 and k was set at 4
yielding remix; Messes of 0.909, 0.886, and 1.000 for the three phenotypes respectively.
When random mating and infinite population sizes are assumed this system produces the
Wrightian adaptive topography shown in figure 3.1.

These simulations followed single meta-populations (hereafter referred to as
populations) which were composed of multiple subpopulations (hereafter termed demes).
The migration rates (m) between demes were the per capita probabilities of leaving the

parent deme during the dispersal phase. The probability of a given individual migrating to

55

Figure 3.1 - Population Mess surface.

This surface is created by plotting the equected mean population Mess against the
frequency of the mutant alleles at 2 loci

56

Figure 3.1

N¢_h0 HUZWDONMM

m~.o om.o

mh.o oo.fi

 

 

 

 

 

 

 

 

:Z::::

 

 

 

 

 

 

 

 

L.
m

m
L
L

 

\ : A

It IE it E3

a 1mm.”
mmm.o
mmm.o
mwm.o
Mm.“

 

 

 

l3

57
a new deme a given distance away was described by the gamma function:
Pr0b(X) = ("0x (1-m)
Where X is the number of demes away the migrant would be traveling. Individuals were
allowed to migrate up to 5 demes away from their parent deme. Any individual migrating
X demes away was placed in a randomly chosen deme at that distance.

Genetic variation was introduced via mutation. A fixed mutation rate of 5 x 10*5
mutations per copy, per generation, per capita introduced mutants at each locus. This
nurtation rate was held constant throughout all simulations. For the original sinmlations
only forward mutations were allowed. Subsequent simulations with both forward and back
mutations were then run over the range of migration rates in which population wide peak
shifts were likely to occur (m = .001, .0025, ..., 0.1).

The initial conditions of the simulated population are as follows: A 10 by 10 matrix of
demes was arranged in a torus in order to reduce edge effects. Each deme originally
consisted of 15 individuals each of whose gender was randomly chosen. Every individual
in the initial population was genotypically identical Hence all demes were fixed at the
local optimum corresponding to the lower peak in figure 3.1. The intent of these
sinmlations was to determine the efl‘ect of migration on the movement of demes and
eventually populations to the higher peak.

Separate runs were conducted for thirteen migration rates from m = 0.0 to m =
0.5. Thirty trials were run at each migration rate. In addition to the 13 different migration
rates, 40 trials were run in a large (K = 3000) panmictic population. This allowed the
comparison of different levels of population structure to a totally unstructured population

of equivalent size. Each simulation was run for 12000 generations. Data from the 6000th

58

and 12000th generation were used in the analysis of these simulations.
Analysis of Simulations

The SBP is primarily concerned with events which can alter a population's
genotypic composition and therefore its mean fitness. Because the population as a whole is
the focal point of investigation our results consist largely of frequencies taken across a
population, or of the percentage of trials for which the population falls into a given class.
The propensity of a given deme to shifi to a higher mean fitness peak is a measure of the
combined efiicacy of phases I and H of the SBP. We were interested in the propensity for
phases I and II to lead to peak shifts. We therefore determined the percentage of trials at
given migration rate in which the population had at least one deme shift.

The recruitment scheme used in these simulations tended to produce recruits near
carrying capacity. Yet when the per capita migration rate is fixed for all genotypes
differences in deme size are a necessary driving force behind the third phase. In order to
determine the size difference between demes with different genotypic compositions we
placed demes into three difi‘erent categories based on mean fitness. 'High-peak' demes
were defined as demes which were fixed for the highest fitness genotype. 'Low-peak'
demes were defined as demes which have an expected mean juvenile fitness above 0.900
but below 1.000. 'Trough' demes were defined as all demes with expected juvenile fitness
below 0.900. An ANOVA was performed to test for differences in mean deme size
between these deme types.

We were interested in exactly what range of migration rates allowed interdemic
selection to convert the entire population to the domain of the higher peak. We therefore

calculated, for those trials which had at least one deme sbifi, the percentage that ended in

59

the fixation of the higher fitness genotype throughout the population (i. e. the tendency for
phase 3 to occur once phases 1 and 2 have occurred) for each migration rate.

One expected characteristic of the shitting balance process is that the global
fiequency of the highest fitness genotype will depend on the rate of migration between
demes. Therefore, the frequency of the highest fitness genotype was determined for each
nm, from this the mean frequency of the highest fitness genotype was calculated for each
migration rate.

The percentage of all trials which led to the fixation of the highest fitness genotype
throughout the population is a direct measure of the efficacy of the SBP as a whole. Here
the emphasis is on a complete transition fiom fixation of one genotype to the fixation of
another genotype. We therefore calculated the percentage of all 30 populations (i e. trials)
at each migration rate which shifted entirely to the higher adaptive peak

These sinnrlations allowed the extinction and recolonization of demes. Differential
extinction of demes based on genotype, and the differential export of individuals based on
deme size are two potentially important sources of interdemic selection which can drive
the third phase of the SBP. The mean percent of the original demes remaining at the end
of a sinnrlation was compared for simulations which had at least one deme shift versus
sinnrlations in which no deme shifis took place. The comparison was made for each
migration rate. This provided a comparison of the predominance of extinction over
recolonization between populations in which the SBP had been initiated and those in

which it had not.

60

Confidence limits for the data.

For each migration rate (treatment) we performed 30 trials, each of which
consisted of 1 population of 100 demes. Because computational constraints restricted the
number of trials to 30 per treatment, bootstrapped 95% confidence intervals were
estimated by multiple resampling of the available trials. These confidence intervals were
computed by bootstrapping 2000 random samples fiom the 30 trials and excluding the
highest and lowest 2.5 percent of the bootstraps. This provides an unbiased estimate of the
mean at the cost of biasing the error estimates (Weir 1990). The confidence intervals were

1miversally larger than 95% confidence intervals arrived at using parametric assumptions.

Results

The results of the 6000th generation are qualitatively similar to those fi'om the
12,000th generation. The results of simulations which include back mutation are
indistinguishable from those with only forward mutations. Therefore, only results from the
12,000 generation of trials which did not include back mutation are presented.

Results of the Panmixia model
In 40 trials no individuals with the highest fitness genotype were recorded from runs based
on a large population with random mating.
Peak shrft propensity

Figure 3.2 is a plot of the propensity of demes to shift as a function of migration
rate. Demic peak shift propensity is herein defined by the percentage of trails in which at
least one deme shifts to the higher fitness peak. In the simulations we ran, migration rates

above m = 0.05 showed no peak shifts. For all treatments below m = 0.0075, zero peak

61

shift propensity lies outside the 95% confidence intervals.
Prerequisites for phase III
In these simulations the size of a deme depended significantly (p < 0.0001) on the
expected mean juvenile survivorship of the population (table 3.1). Fitness dependent mean

deme size differences occurred despite a recruitment scheme which tended to produce

recruits at carrying capacity.

Table 3.1 - The mean number of individuals in demes of different mean fitness.

High peak demes are fixed for the highest fitness genotype, low peak demes Have mean
fitness that is less than the high peak demes but equal or greater than the original
population mean fitness. trough demes have lower fitness than the original population
mean.

 

Mean Deme Size

 

 

Deme Type Mean N SD

High Peak 26.9528 2973 1.4730

Low Peak 24.3537 17,737 2.6201

Trough 23.4408 152 2.6085
Phase III propensity

Figure 3.3 depicts the percent of trials with a deme shift present that led to the
fixation of the higher fitness genotype throughout the population (i. e. the tendency for
phase 3 to occur once phases 1 and 2 have occurred). When m is less than or equal to
0.001 phase three never occurred. The tendency for fixation to occur climbs rapidly fiom
0.0% to 100% between m = 0.001 and 0.0075. Above this point, population-wide fixation

of genotypes once the first deme has shifted is 100% until the migration rates increase to

62

Figure 3.2 - The percentage of trials in which at least one deme shifted versus the
migration rate.

Shaded areas are 95% confidence intervals.

 

l ' ' ' ' l ' ' fl ‘ l ' ' ' ' l ' ' ' ' l ' ' ‘ ' ' i T Y T
0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10
NHGRATION RATE

 

 

g Om HM~<Am

64

Figure 3.3 - The percentage of those populations with a deme shift, that experienced a
fixation of the highest fitness genotype throughout the entire population versus migration

rate.

Shaded areas are 95% confidence intervals.

65
Figure 3.3

 
 

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MIGRATION RATE

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66

the point where no deme shifts occur (m = 0.05).
SBP as a whole
The fiequency of the highest fitness genotype in these sinurlations was dependent
on the migration rate (figure 3.4). The maximum frequency of the highest fitness genotype
was 0.35 when m was equal to 0.0025. When m exceeded 0.10 there was a 0.00 frequency
of the highest fitness genotype. The frequency of the highest fitness genotype also dropped
below 0.10 when the migration rate dropped below 0.0005.
The percentage of trials which lead to the fixation of the highest fitness genotype
(figure 3.5) throughout the population is maximized at 30.0 percent fixation of the higher
fitness genotype when m = 0.005. No whole-population peak shifts were seen below m =
0.001 or above m = 0.1.
Population size
Figure 3.6 demonstrates that deme extinctions are related to migration rate. Deme
extinction rates for trials which have had at least one deme peak shift and trials which have
had no peak shifts are similar. However, populations in which demes have shifted have

fewer extinctions throughout the range of migration rates.

Discussion
A fimdamental outcome of the shifting balance process is that when gene
interactions affect fitness, population substructuring can increase the mean absolute fitness
of a population. This occurs through the interaction of drift, selection, and migration. This
study indicates that the migration rates between demes are important in determining

genotype frequencies at fitness-related loci when epistasis is involved. These simulations

67

Figure 3.4 - Frequency of the highest fitness genotype versus migration rate.

Shaded areas are 95% confidence intervals.

    

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69

Figure 3.5 - The percentage of trials in which the highest fitness genotype was fixed
throughout the population versus migration rate.

Shaded areas are 95% confidence intervals.

 

 

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10
MIGRATION RATE

 

71

Figure 3.6 - Percentage of demes remaining after 12000 generations.

Percentage of demes remaining after 12000 generations versus migration rate for
populations which have (diamonds), and have not (squares) had at least one deme which
shifted peaks.

72

Figure 3.6

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73

demonstrate that there is a non-zero optinnrm migration rate for the SBP. While the
optimum migration rate for the fixation of the highest fitness genotype is low (m = 0.005),
extremely low rates of migration actually decrease the fiequency of the highest fitness
genotype. Below m = 0.0001 population wide fixation of the highest fitness genotype
declines to zero. There is actually a range of migration rates in which the shifting balance
process is relatively likely to change the genotype frequencies in the direction of increased
absolute fitness. In the case of the system examined in these sinnrlations, this range of
migration rates does not extend to zero.

The shifting balance process as envisioned by Wright is the result of three phases
occurring simultaneously within a deme-structured population. Migration rate influences
all three of these phases. Because migration rate has opposing efi‘ects on drift and selective
diflirsion, there is a lower limit on the 'benefit' of lowered migration rate on population
fitness.

A number of analytical investigations of the first two phases of shifiing balance
theory (Lande, 1985, Barton and Rouhani, 1987, Rouhani and Barton, 1987a, 1987b,
Charlesworth and Rouhani, 1988) have been made. Recently Barton and Rouhani (1993)
have also investigated the three phases combined. All of these studies have centered
around a quantitative (i. e. continuously distributed) polygenic trait with optimizing
selection. Our simulation did not use a large number of loci each having equal (and
additive) effects on the trait of interest.

In the case of a polygenic trait under stabilizing selection we have a fitness
difl‘erence between two phenotypic peaks of size k with all variation in the phenotype

produced by the additive affects of many loci. As mentioned in the introduction, this sets

74
up epistatic interactions among 'additive loci when the population is near the optimum
phenotype. The magnitude of the interaction between two loci is extremely small and all
such interactions are equal This provides quite weak epistasis for fitness between loci, and
an astronomical number of approximately optimal combinations of allele frequencies for
each of the phenotypic peaks. In a second system the same difference in fitness (H)
between two optima is created by a two locus two allele interaction. In this case epistasis
for fitness is quite strong and there may only be one set of allelic frequencies which can
produce the optimum mean fitness.

Despite these differences, the general conclusions are that peak shifts can occur
and that the probability of these peak shifts is dependent on the level of structuring in the
population. This agrees with work on quantitative polygenic traits in continuous (Barton
and Rouhani, 1993, Rouhani and Barton, 1987b), and discreet populations (Lande, 1985,
Barton and Rouhani, 1987, Charlesworth and Rouhani, 1988, Rouhani and Barton 1987a,
1987b). Barton and Rouhani (1993) analyze of all three phases of the SBP for both
optimizing selection on an additive polygenic trait and selection against heterozygotes
within a locus. They conclude from this that the SBP is likely to produce similar results
without regard the type of genetic system which produces an adaptive landscape. Our
results support this conclusion by demonstrating that interlocus interactions between a
discrete number of loci act in a manner similar to their two models.

Phase III of the shifting balance process has been far less analyzed than the
processes underlying phases I and II. Crow et al. (1990) have demonstrated that under a
relatively wide set of circumstances phase III would be expected to proceed readily

despite the barriers presented by hybrid breakdown. Their model starts with 2 demes at

75

two different peaks, and examines the effect of migration between the demes on the
propensity of the low-peak deme to shift. They conclude that very little migration is
necessary to affect the phase III shift. They state that "whatever weaknesses it [SBP] may
have are not in the third phase." It should be noted that the strong influence of migration
should prevent a deme from shifting to a higher peak when surrounding demes are all on
the lower peak. The same sensitivity to migration which makes phase HI pervasive over a
wide range of migration rates restricts phase I to very low migration rates.

Although our model was restricted to the two locus case, the genetic system used
in our sinnrlation was modeled after Crow et al. (1990). As in Crow et al (1990) our
study showed a wide range of migration values over which phase III was effective. All
migration rates above 0.001 allowed phase III to occur. This means that in our sirmrlations
phase III was fi'equently successful when m was two orders of magnitude less than s. This
occurred despite the following ways in which our sinurlations differed from Crow et al.'s;
l) the number of demes involved, 2) the initial genotype frequencies, 3) the nature of
differences in migrant numbers between demes, 4) the inclusion of phases 1, and 1].

Barton (1992) has recently presented an alternative interpretation of Crow et al's
(1990) sinmlations. Barton points out that differential migration rates between two demes
alone can allow the higher migration rate deme to overwhelm the lower migration rate
deme's genotype. Given suflicient differences in per capita migration rate, migration and
not selective advantage will allow one deme to dominate. However, for two reasons our
sinnrlations favor the interpretation of Crow et al. that selection is an important factor.
First, we generated all of our variation in genotype fiequency and deme size while in the

presence of migration. This implies that migration is not successfully swamping out all the

76

effects of selection and drift. Secondly, because our model included no difference in per
capita migration rate between demes, any deme with a novel genotype must have been
exporting many fewer individuals than the combined total of all surrormding demes. In this
case only the relatively greater resistance of the higher fitness demes to invasion can
explain the success of the third phase.

The Crow et al ( 1990) model of selective diffusion between demes relies only on
difference in migration rates between demes. Our model allows differential extinction and
recolonization of demes as well The ability of remaining demes to re-colonize extinct
demes should be more sensitive to migration rate than is the extinction rate of extant
demes. This results from the probability of colonization's dependence on two colonists (i e.
a fimction of m2) of opposite sex arriving simultaneously. This will lead to the increasing
loss of demes to extinction, and a disproportionate decrease in the replacement of these
demes by re-colonization as m decreases. This pattern was born out in these simulations
(see figure 3.6). A decrease in the number of extant demes at low It: will decrease the
probability of a peak shift occurring at low migration rates. In addition, as demes go
extinct they create holes which are barriers to the exchange of individuals between the
remaining demes under isolation by distance or stepping stone migration. For this reason
extremely low migration rates probably cause an escalating decrease in the efficacy of the
shifting balance as a population persists for long periods without a peak shift.

The mean deme size for low peak demes was less than the mean size for high-peak
demes. The extinction rate within populations which had high-peak demes was therefore
less than the extinction rate for populations which had no high-peak demes. This allows

differential extinction and colonization to become a potential force in the shifting balance

77

process. The biggest differences in the extinction rate for shifted versus non-shifted demes
seem to occur when m is less than .025 (see figure 3.6). Ifdifl‘erent assumptions were
made about recolonization, for example, if sets of migrants from a single deme colonized
vacant sites, the role of differential recolonization as a cause of internd selection could
be greatly enhanced. It must be emphasized that the effect of colonization and extinction
on the difi‘erentiation or homogenization of demes may be quite specific to a given model
of propagule movement and composition (Wade and McCauley, 1988).

Periodic fluctuations in migration rate may increase the propensity for fixation of
peak shifts throughout a population by decoupling phase 111 from phase I and II (c. g.
Wright 1977, pg. 473 ). Figure 3.6 demonstrates that there is a threshold below which little
or no effective export of high fitness genotypes might be expected. However, it is below
this level of migration that one expects to have the highest frequency of demes which will
shift. Occasional increases in migration rate could therefore rapidly spread peak shifts
which are most likely to have occurred during periods of low migration rate.

Biased migration, fluctuating migration rate, and an increase in the number of
demes can all increase the likelihood of the SBP occurring. The model of migration which
we have used has been based on equal per capita rates of migration for all demes. The
range of migration rates conducive to the shifting balance process should be erqranded if
migration rate fluctuates, is genotype-sensitive (e.g. m increases as deme productivity
increases), triggered by extinction, or targeted towards demes with low population
density.

The period of stasis between population wide peak shifts may be affected by the

number of demes available. Wright envisioned the SBP as being most likely to occur over

78

tens of thousands of demes (M. J. Wade, pers. com 1991). We have included only 100
demes in our simulations. Increasing the number of demes increases the probability of
having at least one peak shift occur. This will increase the frequency of population-wide
peak shifts at relatively high migration rates.

The generality of any model is limited by assumptions made in the name of
tractability. Most of the assumptions made in this analysis were unfavorable for the SBP.
Our results can therefore be expanded in that the eficacy of the SBP should be greater
under less restrictive assumptions. We discussed above how some more favorable
assumptions than those we made would improve the eficacy of the SBP relative to these
results. It is more diflicult to generalize across different adaptive landscapes (i. e. different
genetic models). At present it is impossible to investigate the entire range of possible types
of gene interactions. This problem will not be overcome until we have a detailed
understanding of what types of epistasis are common in the empirical world. Until such
information is available the generality of all models of the SBP will be restricted.

Despite these restrictions we have demonstrated that phases one, two, and three
can occur together under one genetic model with very unfavorable population dynamic
models. Because we used a genetic model similar to Crow et al we can also predict that
increasing the number of loci involved or decreasing the relative difference in peak heights
(k) will increase the critical migration rate necessary for phase three (Crow et al 1990).
Although we can't predict the effect of these factors on phases one and two.

Dominance and the strength of selection should also affect the critical migration
rate (Phillips 1993, Crow et al 1990). In general our model (dominant genotype favored)

allows phase three to occur at lower critical migration rates than the reverse mode]

79

(recessive genotype favored) (Phillips 1993 ). The effect of changes in dominance on phase
one, however, may be to allow a greater effect of drift due to the ability of the recessive
alleles to persist in low peak demes while being protected from selection in heterozygotes.
This should increase the range of migration rates which allow peak shifts to occur, so that
there may be an offsetting effect of dominance on phases one and three. The effect of
changing selection strength on critical migration rates is dependent on the model of
dominance used and the recombination rate (Crow et al 1990).

In conclusion, a simulation relying on a strictly mechanistic model of two epistatic
loci in a structured population of finite size demonstrates the eficacy of the shifting
balance process. These simulations demonstrate that there is a fundamental conflict
between the demands on population structure for success of the first verses the third phase
of Wright's theory. The conflicting requirements of these two phases leave a window of
migration rates which allow peak shifts to occur and subsequently spread in these
sinurlations. Eflicacy of the shifting balance process can therefore be bounded by

maximum and minimum migration rates.

Chapter 4

SllVIULATION OF THE SHIFTING-BALANCE PROCESS IN OLIGOGENIC TRAITS

Introduction

Sewall Wright's shifting balance process (SBP) is a centerpiece of evolutionary
theory in the twentieth centtuy. Its components have been the focus of many recent
theoretical studies (Barton 1992; Barton and Rouhani 1993; Barton and Rouhani 1987;
Charlesworth and Rouhani 1988; Crow et al. 1990; Kirkpatrick 1982; Lande 1985; Moore
and Tonsor 1994; Phillips 1993; Rouhani and Barton 1987a; Rouhani and Barton 1987b;
Rouhani and Barton 1993; Whitlock 1995). Understanding the SBP may be critical to
understanding evolution in a metapopulation context and therefore may have implications
for conservation biology (McCauley 1993). The SBP may also allow microevolutionary
forces to explain macroevolutionary patterns (Kirkpatrick 1982; Lande 1985; Lande 1986;
Wright 1982a; Wright 1982b). Many factors related to the SBP have been investigated
theoretically. These inchrde migration rates, population size and founder efl‘ect speciation.

Genie interactions and the multi-peaked adaptive landscapes that result from them
are fundamental to the SBP. The evidence for the existence of multi-peaked adaptive
landscapes is substantial, and there are many ways in which multi-peaked surfaces can
manifest themselves (Whitlock et al. 1995). Much of the previous theoretical work on

SBP has concentrated on additive polygenic traits with multiple phenotypic optima

80

81

(Charlesworth and Rouhani 1988; Lande 1985; Phillips 1993; Rouhani and Barton 1987a;
Rouhani and Barton 1987b; Rouhani and Barton 1993; Whitlock 1995). Other studies of
the SBP have looked at single locus systems or systems of several (oligogenic) loci (Crow,
Engles and Denniston 1990; Moore and Tonsor 1994; Phillips 1993; Wright 1941) which
produce a multi-peaked surface. A primary goal of this study is to compare the eflicacy of
the SBP under several oligogenic models in an attempt to determine whether
generalizations about the rate of peak shifts across systems are reasonable.

Barton and Rouhani (1993) studied the complete SBP including interdemic
selection, and found that both a chromosomal inversion model with homozygote
advantage, and polygenic models showed optimal adaptation when the number of migrants
(Nm) is slightly less than one. They viewed their results as suggesting "that adaptation via
the 'shifting balance' is more effective with disruptive selection on discrete alleles than with
disruptive selection on a quantitative trait" (Barton and Rouhani 1993 ). They also
concluded that the pattern of effectiveness of the SBP which they found may be extended
to any form of multi-peaked adaptive surface. This conchrsion agrees with the simulation
results of (Moore and Tonsor 1994) which studied the efficacy of the entire SBP across a
wide range of migration rates for a two locus case. They concluded that the entire process
occurred readily when Nm ranged between 0.05 and 1.

Barton and Rouhani (1987) had previously compared a single locus models of
underdominance to an additive polygenic models of peak shift probabilities in a single
deme. They concluded that the probability of adaptive shifts depended mainly on the depth
of the adaptive valley. They found that the two models behaved similarly when the deme

wide mutation rate (N u) in the single locus model was high (N u>> 1/4) but not when it

82
was low (N u<<1/4).

Two different types of genetic models have been used in studying the SBP.
Nonpolygenic models use relatively few loci with discrete effects on fitness. In this case
each locus may have tmique interactions with other loci in the production of fitness. In the
other major model epistasis is generated by the stabilizing selection acting on a trait with a
purely additive polygenic basis. This results in a large number of possible genotypic
optima for any given phenotypic optimum All loci experience weak epistatic interactions
when the number of loci is large and the effect of each locus on overall trait value is
uniform This weak and uniform epistasis will be referred to as 'diflirse epistasis'
throughout this paper. In the intermediate cases where the number of loci is larger than
two and less then a very large number the complete SBP is largely unexplored. Two
studies of the third phase of the process have looked at the effects of the number of loci
using analytical models (Phillips 1993, and Crow et al 1990). They have demonstrated
that the third phase is more rapid when the number of loci is small (Crow et al 1990) and
when dominant genes are favored (Phillips 1993). In the case of the two locus model of
Moore and Tonsor (1994), where the entire SBP was modeled, only one genetic model
was explored. Previous studies have left the effectiveness of the complete SBP across
oligogenic systems largely unexplored. In this study I attempt to determine how consistent
the rates of peak shifts and adaptation are across oligogenic systems. This study
investigates the eflicacy of the SBP in systems when the expectation for deviations from
an additive polygenic model are most likely to occur (i e. when Nu<<1/4).

Phenotypic and genetic variance are of critical importance to the SBP. Increases in

83

phenotypic variance alone are capable of inducing peak shifts (Whitlock 1995, Kirkpatrick
1982). In addition, epistatic genes may alter the additive genetic variance when
populations are subdivided (Goodnight 1995; Whitlock et al. 1993). Ifthe number of loci
interacting is large it may be reasonable to approximate drift, and its resulting effect on
variance, as a diflirsion process. When the number of interacting loci is relatively low or if
a relatively few loci afl‘ect a trait disproportionately, stochastic changes in variance which
are not easily modeled by a diffusion process may be critical

In the two locus model of Moore and Tonsor (1994) no variation in genotype
existed at the onset of the sinnrlations. Variation was generated by nurtation during the
course of the simulations. Because the genotypic distance between peaks required allelic
substitutions at only two loci this did not provide too severe an impediment to the SBP.
When substitutions must occur at a greater number of loci, however, mutation may not
create new allelic combinations quickly enough to create a detectable rate of peak shifts. If
mutation within demes does not create a suflicient number of polymorphic loci then there
is still a possibility that migration will create favorable gene combinations by mixing. This
however may require migration rates which would overwhelm the local differentiation to
the extent that only one peak may be explored. Therefore, the particular shape of the
fitness function for a trait may be critical in determining the effectiveness of the SBP.

It is likely that in many cases trait distributions that seem continuous may actually
result fiom the action of relatively few loci (East 1910; Tanksley 1993 ). Even in traits that
have continuous distributions resulting fi'om a large number of interacting loci the majority
of genes may be of small effect while a small number of genes have large effects on trait

value. Because drift induced variance is critical in the SBP it is not clear how the process

84

will work in these intermediate cases. This study compares efficacy of the SBP in several
genetic systems of two to sixteen loci. Epistasis for fitness is produced by combining
nonlinear phenotypic fitness functions for a trait with additive gene action on the same
trait. In this regard these models nrirror the predominant additive polygenic models of
disruptive selection used in most previous studies. They are not, however, systems of large
number of loci. They model intermediate numbers of interacting genes. A comparison with

the non-additive two locus model of Moore and Tonsor (1994) is also included.

Methods

Computer simulations were used to explore the SBP and extend the models of
Moore and Tonsor (1994). Both studies model a diploid, obligate sexual, semelparous
species with density independent mortality prior to migration, and mating based on
genotype-dependent survival probabilities. Variation in deme size (N) allowed variation in
the absohrte number of individuals emigrating from demes (Nm), without requiring
differential per-capita migration rates (m) between demes. The simulations specified an
individual's genotype, fitness and dispersal behavior. Keeping track of individuals in this
way avoids simplifications which prevent insight into the SBP.

As in Moore and Tonsor (1994) the sinnrlations followed single meta-populations
(hereafter referred to as populations) for 1200 generations. Each population was
composed of a 10 by 10 matrix of subpopulations (hereafter referred to as demes) which
was arranged in a torus in order to reduce edge effects. The migration rate (m) between
demes was the per capita probability of leaving the parent deme during a generation. The

probability of a given individual migrating to a new deme a given distance away was

85
described by the gamma fimction:
PIOMX) = (”0x ‘1‘")
where X is the number of demes away the migrant would be traveling. Individuals were
allowed to migrate up to 5 demes away fiom their parent deme. Any individual migrating
X demes away was placed in a randomly chosen deme at that distance. All simulations
were run with m = .005. This is near the optimum migration rate in the sinurlations of
Moore and Tonsor (1994).
A finite maximum deme size was imposed by incorporating a carrying capacity (K

= 30), and intrinsic growth capabilities into a logistic function (Non) = Na) +
rN(,)(K-N(,))/K r = 1.1). This function determined the maximum number of individuals that
could be born into that deme during the next generation (N (rm)- Because all demes had the
same r and K, during each generation all demes tended towards the same carrying
capacity. Stochastic extinction of demes was allowed and recolonization occurred only
when individuals of both sexes migrated into an extinct deme in the same generation.

Sexes were separate and mating was random within demes. Genetic variation was
introduced via mutation. A fixed mutation rate of 5 x 10*5 nurtations per copy, per
generation, per capita introduced mutants at each locus. This mutation rate was held
constant throughout all simulations. Forward and back mutation rates were identical
throughout these sinnrlations.

Each phenotype in these simulations had a predefined absolute juvenile

survivorship. There was no mortality outside of the juvenile phase. Multiple peaks in the
genetic adaptive landscape in all of the sinnrlations were generated by the additive effect of

nnrltiple loci on a two-peaked phenotypic fitness surface. Each locus had two alleles, one

86

of which increased trait value, the other of which decreased trait value. All loci had
equivalent and additive effects on trait value within a given sinnrlation and there was no
dominance of alleles. Epistasis for fitness was generated by applying a nonlinear fitness
firnction to a genotypes phenotype. In all simulations there were two different phenotypic
optima separated by an intervening fitness trough.

Possible trait vahres for all simulations were centered on the arbitrary trait value
(Z) of 0. The lower of the two phenotypic optima was always at Z = 1, and the higher
peak was at Z = -1. The fitness of phenotypes at Z = -l was constant throughout all the
simulations at 0. 700. Those individuals with phenotypes at Z = 1 had the same fitness
(0.636) throughout all the simulations. The fitness of individuals corresponding to the
phenotypic fitness trough was 0.620 throughout the sinmlations. These values are identical
to those in Moore and Tonsor (1994). The decrease in fitness from the optima to the
trough was linear in all these simulations.

The focus of this study was a comparison of the effectiveness of the SBP under
different genotypic adaptive landscapes. A total often different landscapes were
compared. Three two locus models were run for comparison with Moore and Tonsor's
(1994) model which included dominance of the higher fitness alleles. Dominance of the
alleles associated with the higher fitness peak has been shown to greatly increase the
propensity for phase three (Phillips 1993). The phenotypic fitness surface for the two
locus models is found in figure 4.1. In the two locus model each allele added either -1/2 or
1/2 to the phenotypic value of an individual The Trough locations for the three models
were Z = 1/2, Z = 0 and Z = - 1/2. The high and low peaks were located at the extremes of

Z = -l and Z = 1 respectively.

87

Figure 4.1 - Phenotypic fitness surface for the 2 locus models.

Individual fitness is plotted against trait value for 2 locus simulations with trough locations
at Z = 1/2, 0 and -1/2.

88

Figure 4.1

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89

Three different four locus models were run. In all four locus models each allele
contributed exactly 1/4 or -1/4 to the phenotype. The optima were again located at the
extremes of the phenotypic distribution. The three different four locus models correspond
to three difl‘erent trough locations. The troughs were again located at Z = 1/2, Z = 0 and Z
= - 1/2 (see figure 4.2). Because the effect of an allelic substitution in the four locus model
is half that of a substitution in the two locus model it requires twice as many allelic
substitutions to reach an equivalent trough location as it did in the two locus models. As in
the two locus models the starting condition was with all individuals on the lower
phenotypic optimum, so that all variation between demes in allele frequency had to be
generated by mutation and drift within the 12000 generation simulation. The four locus
models differs from the two locus model only in the number of allelic substitutions
required to change the trait value one unit. This number changed fiom two to four.

In the eight locus models each allele contributed either 1/4 or - 1/4 to the trait
value exactly as in the four locus models. Since the optima still remained at Z = l and Z =
-1 the same number of allelic changes were needed to move between peaks as in four
locus model This allowed an investigation of the effects of changing the number of loci
from four to eight loci without increasing the number of allelic substitutions needed to
move between peaks. In this model the locations of the optima were not at the extremes of
the trait distribution. In the eight locus model the trait distribution extends fiom -2 to 2. In
these models the fitness of individuals dropped off slowly as the optima was surpassed in
either the positive or negative direction. This decrease in fitness is linear and symmetrical
with the deviation in Z from either 20mm . The maximum reduction in fitness was set at

0.1 (see figure 4.3). As in the four locus models, simulations were followed for 12000

90

Figure 4.2 - Phenotypic fitness surface for the 4 locus models.

Individual fitness is plotted against trait value for 4 locus sinnrlations with trough locations
at Z = 1/2, 0 and -1/2.

91

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Figure 4.3 - Phenotypic fitness surface for the 8 locus models.

Individual fitness is plotted against trait value for 8 locus simulations with trough locations
at Z = 1/2, 0 and -1/2.

93

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94

generations.

The eight locus models increase the number of loci at which mutation can occur
relative to the four locus model This may influence the rate at which variation between
demes builds up. It should also increase the equih'brium variation between demes. These
are potentially important factors in the SBP because when variation exists between demes
immigrants may carry the new alleles necessary to push a deme into the domain of a new
peak The starting variation between demes in allele fi'equency may also influence the
outcome of the process. This study therefore looked at a variety of different starting
conditions.

In the eight locus models rmrltiple genotypes were capable of producing each
phenotypic optimum. Because there are nnrltiple genotypic fitness peaks which are hidden
in this type of phenotypic optima this will be referred to as 'a diflirse adaptive peak '. There
are several difl‘erent ways the simulations could start on or near the diffuse lower peak,
each with possible ramifications for the efficacy of SBP. One simple model would start
with the entire population fixed at Z=2, ie. to the right of the right -most peak. This
would mean that the population started with no variation, but variation was allowed to
arise between demes as different loci mutate in different populations and selection drives
each of the demes to its individual genotypic optima. Because these models began with the
population fixed at an extreme phenotype they will be referred to as the ' phenotypic
extreme' models and are labeled as the eight locus 'A' models. Fifty ms were made for
each of three trough locations (Z = 1/2, Z = 0, Z = -l/2) as in all other eight locus
models.

Another simple model would have the population start fixed at Z = l with all

95

demes fixed for the same alleles. This means that there will be no allelic or phenotypic
variation at the start and that selection will tend to keep it that way. Because this model
starts at the lower optima and there is no genotypic variance at the start of the trials this
model is referred to as the 'no variance' model and labeled the eight locus 'B' model Rims
were performed under the no variance model using the same set of trough locations as in
the phenotypic extreme model An important contrast exists between the phenotypic
extreme (A) and the no variance (B) models. In the phenotypic extreme models mutation
and selection should act to increase the genetic variance between demes, in the no
variance model selection should act to decrease genetic variation between demes.

In the eight locus phenotypic extreme model the time required for the phenotype to
evolve to the vicinity of the lower peak will reduce the time available for movement
between peaks relative to the no variance model For this reason simulations were run
where all individuals began with a phenotype of two and the mean phenotypic value of the
population was monitored. When the mean phenotypic value of the population approached
the lower peak the 12000 generation trial was started. No demes shified to the higher peak
until the 12000 generation trial was begrm. This allowed variation between demes to arise
by the combined action of selection and mutation without including the time necessary to
find the lower peaks in the 12000 generations of the nm. The first generation in these
simulations began when the mean trait vahre was reduced to Z = 5/4. Some demes in these
cases could start the trial at or near the phenotypic trough. Demes never, however, started
the trial in the domain of the higher peak. Because these trials allowed variance to build up
before the start of a trail they are referred to as the 'high variance' models, and are labeled

the eight locus 'C' models.

96

The eight locus 'Skewed variance' (D) model was run with the same starting
assumptions as the high variance model but in order to avoid the biasing effect of starting
the simulation with some demes on or near the trough no individual deme in these
simulations was allowed to have a phenotypic mean any less than that of the lower peak
(i. e. 1) at the start of a trial This means that none of the populations started near the
trough. This skews the phenotypic variation away from the troughs but also prevents the
bias of the third model This skewed distribution was created by allowing selection to
drive each demes mean trait value towards Z = 1. Once any deme reached Z = 1
truncation selection eliminated all individuals in that deme whose trait value was lower
than 1. The truncation selection was removed and the 12000 generation run was started
once the population mean dropped from Z = 2 to Z = 5/4.

The final eight locus model started with each deme fixed at Z = 1 as in the no
variance model In this model, however, each deme was fixed for a random set of four loci
which were chosen independently of the four loci which are chosen in any other deme.
This is the same as allowing each deme to arrive at the lower peak independently (i. c.
with out migrational input from other demes) but making sure there is no phenotypic
variation at generation 0. Because these models maximize the amount of genetic variance
between demes while each deme is fixed at the lower phenotypic optimum they are
referred to as the 'maximum variance' (E) models. The different eight locus models are
summarized in Table 4.1.

A history of evohrtionary and ecological events determines the genetic structure of a
population at any point in time. The starting point for each of the eight locus models

represents a different set of assumptions about what that history has been. The extreme

97

fixation (A) model represents a population which has not reached an equilibrium near any
peak This is realistic if the population is experiencing a relatively novel arvironment. The
extreme fixation model also begins with no genetic or phenotypic variation between

demes. These two conditions may be fairly common in a population which has recently

Table 4.1 — Summary of the 8 and 16 locus models.

Each 8 and 16 locus model is described by a mnemonic and initial variation. Some natural
factors which might induce the pattern of variation are listed. The expected change in
genetic variation under the sinnrlated conditions is also listed.

 

 

 

 

 

 

 

 

 

Model Mnemonic Initial Possible causes ofinitial Change in
variation variation pattern (variation
8 Locus Extreme None New niche and range H-t
A Phenotype (founder effect)
8 Locus No Variation None Old niche with new range Slow
B (founder efl‘ect) Buildup
8 Locus High Moderate Long term stabilizing - - -
C Variation selection with migration
8 Locus Skewed Low Same as above but with - - -
D Variation truncating selection agent
8 Locus Maximum ~ Highest Long term stabilizing - - -
E Variation possible selection with no migration
16 Locus Extreme None New niche and range + + +
A Phenotype (founder effect)
16 Locus High Moderate Long term stabilizing - - -
C Variation selection with migration
16 Locus Maximum Highest Long term stabilizing - - -
E Variation possible selection with no migration

 

 

 

 

 

 

 

expanded it's range or niche. This in not an equih'brium state for any natural population,

but might be quite common.

98

The no variance (B) model also starts with no variation between demes, but is at a
phenotypic optimum This is a model which could easily represent a population created by
the rapid expansion of a small founding population . In this case the population is near to
an equilibrium state for the alleles of interest. This may be a common circumstance when
new patches of suitable habitat become available to a population.

The high variance (C) models also starts roughly centered around an equihbfial
mean but with variance in both directions around that mean. This situation would result
from relatively uniform stabilizing selection across a structured population and is probably
a common situation in natural populations. The skewed variance model (D) begins with
little variance either genetic or phenotypic and a mean phenotype near equihhrium When a
population has evolved under directional selection up to a truncating point it would
temporarily resemble the skewed variance model if the source of trtmcation selection was
suddenly removed exposing a new adaptive surface. This model might easily result when
ecological stresses such as competitors, predators, drought or resource limitation are
suddenly removed.

The maximum variance (E) model will exist whenever multiple demes are allowed
to independently reach equilibrium under a stabilizing selection which is uniform across
demes. A maximum amount of genetic variance between demes arises in this model given
that all demes are fixed at the lower phenotypic optirmrm The one feature in this model
which is unlikely in natural populations is the lack of variation within demes as well as the
lack of any phenotypic variation between demes. The omission of these types of initial
variation will reduce the eflicacy of the SBP but allow a separate investigation of the

effects of inter-demic variation. The inter-demic variance in the E model represents a type

99

of variance that will arise under a relatively constant selective environment when demes
have been isolated for long periods of time. This is therefore a model which reflects a
sudden increase in migration rate. All of the eight locus models are reasonably close to
conditions which might frequently arise in natural populations. Ifthere is a fault in the
realism of these models it is that they all restrict the variation in predetermined ways in
order to increase understanding of the efi‘ects of variance.

Three sixteen locus models were run with each allele contributing either 1/8 or
- 1/ 8 and trough at 1/2. This preserved the general form of the eight locus models but
doubled the number of allelic substitutions necessary to shift between peaks. The sixteen
locus 'phenotypic extreme' model was started under the same condition as the eight locus
phenotypic extreme model A sixteen locus 'high variance' model was nm which was
equivalent to the eight locus 'high variance' model. In that model the 12000 generation run
was begun once the population mean dropped to 9/8. The sixteen locus 'maximum
variance' model was started under the same conditions as the eight locus maximum
variance model All demes were fixed for a specific allelic combination which resulted in a
trait value of -1 but each deme was allowed to find this combination independently.

The propensity of a given deme to shift to a higher mean fitness peak is a measure
of the combined eficacy of phases I and H of the SBP. The percentage of trials in which
the population had at least one deme shift to the higher peak was therefore recorded in
order to compare the eflicacy of phases one and two across genetic models. For those
trials which had at least one deme shift, the percentage which ended with the entire
population having achieved the higher fitness peak (i. e. the tendency for phase 3 to occur

once phases 1 and 2 have occurred) was determined for each genetic model The

100

percentage of all trials which led to all demes shifting to the highest fitness peak is a direct
measure of the eflicacy of the SBP as a whole. The percentage of trials for each genetic
model in which all demes shifted to the higher adaptive peak was therefore calculated.

Since fixation on the higher adaptive peaks is not expected in these models some
standard method of determining when a deme had shifted peaks was needed. For the
purposes of this study any deme whose mean fitness was higher than the fitness of a deme
fixed on the lower fitness adaptive peak was considered to have shifted peaks.

For each genetic model at least 50 trials were run, each of which consisted of 1
population of 100 demes. Bootstrapped 95% confidence intervals were estimated by
multiple resampling of the available trials. These confidence intervals were computed by
bootstrapping 2000 random samples fiom the trials and excluding the highest and lowest
2.5 percent of the bootstraps. Any population with at least 95% of its demes fixed on the

high peak was considered to have fixed on the high peak population wide.

Results

The percentage of trials which had at least one peak shift and the percentages of

those shifts which spread through the population are listed for each model in table 4.2.
Two locus Models

All the two locus models were reasonably likely to have at least one deme shift to
the higher peak within 12000 generations (figure 4.4). As the size of the domain of the
higher peak increased the propensity for peak shifts also increased. When the trough was
closest to the higher peak at Z = - 1/2, the percentage of trails with at least one peak shift

was 14%. When the trough was at Z = 0 this percentage was increased to 22%, and

101

Table 4.2 - Summary of results from the 2, 4, 8, and 16 locus models.

The percentage of trials with at least one shift (% with shift) and the percentage of trials
which experienced a peak shift which then had that shift spread throughout the population
(% shifts fixed) are listed for each model, and trough locations of Z = 1/2, 0, and -l/2. l6
locus models were only run with a trough location of Z = 1/2.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

l _Tlo_ugh Location _
Model Z = 1/2 Z = 0 Z = -1I2

% With % Shifts % With % Shifts % With % Shifts

a Shift Fixed a Shift Fixed a Shift Fixed

2 locus 98.0 94.0 1 22.0 64.0 14.0 72.0
4 locus 5.0 0.0 0.0 NA 0.0 NA
8 locusA 45.0 25.9 13.3 0.0 1.7 0.0
8 locus B 24.0 8.3 0.0 NA 2.0 0.0
8 locus C 34.0 64.7 16.0 50.0 6.0 100.0
8 locus D 22.0 36.4 4.0 0.0 0.0 NA
8 locus B 100.0 100.0 98 100.0 70.0 82.9
16 locus A 4.0 0.0 - - - -
l6 locus C 0.0 NA - - - -
16 locus E 20.0 40.0 - - - -

 

 

102

Figure 4.4 - Percentage of trails in which at least one deme shifted.

Listing is by number of loci, variation model and trough location. Error bars represent
95% confidence intervals. Asterisks indicate high variance models.

103

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further increased to 98% when the trough was located at Z = 1/2.

The propensity for phase three to occur, once phase one and two had occurred,
was highest (94%) when the trough was nearest to the lower peak at Z = 1/2. At trough
locations of Z = 0 or Z = -1/2 the probability of a shifi spreading dropped to 64% and
72% respectively. The difference between results when the trough was located at Z = O
and Z = 1/2 is not significant (see figure 4.4).

Four Locus Models

Very few peak shifts occurred in the four locus models. When the domain of
attraction of the higher peak was large (ie. when the trough was at Z = 1/2) only 5% of
trials resulted in a peak shifi. On those occasions when there was a shift to the higher peak
there was not a spread of that peak shifi throughout the population. When the trough was
located at Z< 1/2 there were no peak shifts at all

Eight Locus Models

The eight locus models were generally more prone to peak shifts than the four
locus models. This was true for all three trough locations. However, when the trough was
at Z = -1/2 or 0 the ability to detect difl’erences between models was poor due to the low
number of trials in which shifis occurred. The exception to this was the eight locus
maximum variance (E) model which consistently produced the highest number of peak
shifts and a very high rate of fixation of those shifts.

The phenotypic extreme (A) and high variance (C) models were more prone to
peak shifts at all trough locations than were the no variance (B) and skewed variance (D)
models. The high variance (C) model was indistinguishable from the phenotypic extreme

(A) model in this regard when the trough was located at O or -1/2, but was intermediate

105

between the phenotypic extreme (A) and the no variance (B) and skewed variance (D)
models when the trough was at Z = 1/2 (see figure 4.4). As the trough location moves
from 1/2 to -1/2 the probability of a peak shift decreases in all models. This occurred
primarily as the trough is moved from Z = 1/2 to Z = 0 (figure 4.3).

Those models which produced many peak shifts were not necessarily the best at
fixing those peak shifts (see figure 4.5). In the maxinmm variance (E), which had the
highest percentage of peak shifts at every trough value, 100% of the peak shifts were fixed
when the trough was at 1/2 and 0. When the trough was located at Z = -l/2 the
percentage of peak shifts that spread throughout the population dropped to approximately
83%. The high variance (C) model, which was efl‘ective at producing shifts in all three
trough locations, was the next most effective at fixing peak shifts. It fixed approximately
64% of all the peak shifts which occurred. The percentage of peak shifts which spread
through the population for the high variance (C) model increased as the domain of the
higher peak was reduced (i. e. when the trough was moved towards - 1/2). The phenotypic
extreme (A), no variance (B) and skewed variance (D) models were less likely to spread
peak shifts than the high variance (C) and maximum variance (E) models, and did not
result in any fixation across demes when the trough was less than 1/2.

16 Locus Models

The sixteen locus model was much less likely to produce peak shifts than the
equivalent eight locus models. The number of trials that produced a peak shift in the
sixteen locus phenotypic extreme (A) model was so low (2) that it is impossible to
determine whether it is any more or less prone to such shifts than the high variance (C)

model in which no peak shifts occurred. The sixteen locus maximum variance (E) model

106

Figure 4.5 - Percentage of those trails with a deme shift that experienced a fixation of the
highest fitness genotype.

Listing is by number of loci, variation model and trough location. Error bars represent
95% confidence intervals. Asterisks indicate high variance models.

107

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did show an appreciable number (20%) of peak shifts of which a large percentage were
fixed (40%). Despite the respectable percentage of shifts fixed in the sixteen locus
maxirmrm variance (E) model it still performed poorly in this regard compared with eight
locus E model (100% fixation.)
The Overall Eflicacy of the SBP

The overall eflicacy of the shifting balance process is best measured by the
frequency with which the entire population is shifted to the highest peak In general only
the eight locus and two locus models were capable of producing detectable population
wide peak shifts in these simulations (see figure 4.6). Only the high variance (E) model
produced population wide peak shifts in the sixteen locus systems. In those models where
population wide peak shifts did occur the propensity for such shifts dropped off rapidly as
the domain of the higher peak was narrowed. This trend, however was less noticeable in

the eight locus high variance (C) and maximum variance (E) models.

Discussion

Moore and Tonsor (1994) previously studied the behavior of a similar two locus
two allele model across a wide range of migration rates. The epistatic model used by
Moore and Tonsor included complete dominance of the alleles on the highest fitness peak.
The two locus model with a trough located at Z = 0 in this study differs from the model
of Moore and Tonsor only with respect to dominance (see figure 4.7 for a comparison of
the two locus epistatic models). The no dominance model in this study successfully
exported peak shifts only 64 percent of the time, while shifts were successfirlly exported

80 percent of the time in the Model of Moore and Tonsor. Phillips (1993) and Crow et

109

Figure 4.6 - The percentage of trials in which the highest fitness genotype was fixed
throughout the entire population.

Listing is by number of loci, variation model and trough location. Error bars represent
95% confidence intervals. Asterisks indicate high variance models.

110

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Figure 4.7 - The fitness of each possible genotype in four different two locus two allele
models.

The model of Moore and Tonsor (1994) is represented by 4.7a, 4.7b - 4.7d represent the
models used in this study with troughs at Z = 1/2, 0 and -1/2 respectively.

112

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al. (1990) have determined that dominance of the alleles which produce the higher peak
should lead to a greater propensity for the completion of phase three. This study bears out
those findings. In addition an added probability of peak shifts within a deme is seen when
dominance of the higher peak alleles is included (table 4.3).

The four locus models in this study were a great deal less likely to experience peak shifts
than in the equivalent two locus models (figure 4.4). This is partially due to the increased
genotypic drift required in order to enter the new domain of attraction. The two locus
model with a trough at Z = - 1/2 requires the same number of allelic substitutions as the
four locus model with a trough at Z=1/2. The four locus model in that case still had
significantly fewer shifts than the two locus model. This indicates that in addition to the
slope of the fitness surface on either side of the trough, the number of allelic substitutions
needed to push a deme into the domain of a new peak affects the probability of peak
shifts. Because the phenotypic fitness surfaces compared between the two locus and the
four locus models were exactly alike, the poor performance of the four locus models
relative to the two locus models indicates that increasing the number of loci while
decreasing the allelic effect on phenotype decreases the probability of peak shifts.

While it would be nice to draw general conclusions about the comparative eflicacy
of phase three across the two and four locus models the failure to produce peak shifts
when the domain of the higher peak was not large (i. e. when the trough was at Z = 0 or
less) prevented any exploration of this issue. The four locus model produced no
population wide peak shifts when the trough was located at Z = 1/2 while the two locus
model with the same trough location spread its peak shifts throughout the population 94

percent of the time. For these simulations the probability of adaptation of the entire

114

population via peak shifts dropped to zero in the change from the two to the four locus
models. The number of allelic substitutions necessary in order to produce a peak shift was
increased as the number of loci contributing to the trait increased in these models. The
efficacy of the SBP clearly can be reduced by increasing the number of loci contributing to

a trait in this situation.

Table 4.3 - The efficacy of the SBP in four different two locus models.

Four models are compared. The Moore and Tonsor (1994) (dominant favored) model is
contrasted with the three two locus models in this study (Z = 1/2, 0 or - 1/2). The
percentage of trials with at least one peak shift (% with a shift), the percentage of peak
shifts that then spread throughout the population (%of shifts fixed) and the percentage of
trials which end in population wide peak shifts (% trials fixed) are presented for each
model.

 

 

 

 

 

2 Locus Model I % With a Shift % of Shifts Fixed % of Trials Fixed

High Peak Dominant 80 32 26

Trough at Z = 1/2 98 94 92

Trough at Z = 0 22 64 14
_'I_‘r£1_1_ghatZ=-l/2 14 71 10

 

 

 

The eight locus results indicate that increasing the number of loci that contribute to
a trait does not necessarily reduce the eficiency with which the SBP proceeds. The eight
locus models clearly were more effective than the four locus models in both producing
peak shifts (figure 4.4) and in exporting those new combinations throughout the entire
population (figure 4.5). In the eight locus models the relationship between peaks in terms
of the number of mutant alleles which need to be substituted in order to affect a change in

the domain is the same as in the four locus model The phenotypic effect of any single

115

allelic substitution has remained constant between the eight locus and the four locus
models. The number of loci contributing to the trait, however has doubled in the eight
locus model The number of loci at which nnrtations can occur has therefore been doubled.
This increased rate at which mutations are introduced will increase the rate at which
variation between demes is produced as well as the equih‘brium level of variation within
and between demes.

Unlike the four locus models, in the eight locus models there are multiple equally
fit genetic peaks at the lower phenotypic peak. Selection may therefore maintain new
variation as it arises in the 8 locus model In the four locus models there is only one
genetic peak at the lower phenotypic peak and selection will therefore swiftly remove
variation between demes. This explains why the four locus models are less likely to
ermerience peak shifts than the two locus, while the eight locus models are more likely to
experience peak shifts than the four locus models. Whitlock (1995) has demonstrated that
variance and changes in variance can be extremely important in the generation of peak
shifts. The general increase in the probability for peak shifts in the eight locus models is
probably attributable to this increased variation present.

In the four locus model there is only one most favored genotype within the domain
of either peak. In this case selection will tend to reduce the variation between demes as
well as within demes. This is not always the case in the eight locus models where there are
multiple equally favored genotypes within the domain of either phenotypic peak With
these diffuse phenotypic peaks selection can act either to increase or to decrease the
variance between demes. Ifa population consists of demes which are all close to the same

genetic optimum then selection should push all those demes towards the same genetic

116

optimum. This tends to reduce the variance between demes. This model is the starting
condition for the eight locus no variance (B) models in these sinnrlations. If however the
demes are uniformly fixed for the same genotype which is far fi'om optinmm, then
selection will favor those beneficial allelic substitutions which arise locally and selection
will increase the variance between demes. In the eight locus A, C, D and E models
selection should tend to increase the between-deme genetic variance relative to the no
variance (B) model Because migration between demes homogenizes alleles across demes
the lower the migration rate the more selection will tend to increase the variance between
demes. The eight locus maximum variance (E) simulations model the highest genetic
variance between demes since all demes rest on independent genetic optima at the
beginning of the simulation. This is the expectation for populations which have adapted
without migration between demes.

The A, C and D models should have lower variance between demes because
migration is always homogenizing the variance across demes. The high variance (C) and
skewed variance (D) models restrict access to the higher peak until the entire population's
mean is near the lower peak. This allows migration to reduce the between demes genetic
variance so that these models have lower expected genetic variance at generation 1 than
the maximum variance (E) model Because of the restrictions on phenotypic variance the
skewed variance (D) models require more generations to reach their starting conditions.
The genetic variance between demes should therefore be lower in those models than in the
high variance (C) models.

Ultimately the difference between the phenotypic extreme (A) models and both the

high variance (C) and skewed variance (D) models is what happens during migration. In

117

all three models the migration events homogenize the alleles across demes. In the high
variance (C) and skewed variance (D) models migration when the lower peaks are being
found only homogenizes demes. Any phenotypic variation beyond the lower peak and
towards the higher peak which is generated is removed by truncation selection. The higher
peak only exists after the entire population mean reaches the lower peak in the high
variance (C) and skewed variance (D) models. In the phenotypic extreme (A) models
migration between demes will always create favored genotypes if hybrid individuals fall firr
enough into the domain of the higher peak. This seems to be a realistic model and it allows
for greater variance than the high variance (C) and skewed variance (D) models. However
the phenotypic extreme (A) model allows very little time for the exploration of new peaks.
The maximum variance (E) model should have the highest between-deme variance

initially, followed by the high variance (C) and skewed variance (D) models respectively.
The phenotypic extreme (A) and no variance (B) models both begin with no genetic
variance between demes. The phenotypic extreme (A) model, however, should generate
higher levels of variance than even the high variance (C) model eventually. The no
variance (B) model should remain low in genetic variance relative to all the other eight
locus models throughout each 12000 generation run.

Migration between demes can introduce new alleles to demes. When the genetic
variance between demes is high this effect is much more important in creating new allelic
combinations than is the introduction of new alleles via mutation. Only at very small
migration rates is mutation likely to dominate over migration in creating new allelic
combinations. The efi‘ect of these new allelic combinations is to increase phenotypic and

genetic variance temporarily within a deme. These periods of higher phenotypic variance

118

are when peak shifts are most likely to occur (Whitlock 1995). The eight locus models,
which had the highest expected variance were the ones most likely to produce peak shifts.
The probability was highest for the maximum variance (E) models followed by the
phenotypic extreme (A) and high variance (C) models with the no genetic variance (B)
and skewed variance (D) models both producing relatively few shifis (figure 4.4). These
simulations therefore confirm the importance of variance induced peak shifts (Whitlock
1995) in the SBP.

Of the three models that were most effective at producing peak shifts (E, A, C) the
phenotypic extreme (A) model proved the least effective at phase three of the SBP. The
generations needed to create variation in this model most likely do not leave enough time
for phase three to be completed. The rate of phase three has been shown to depend on the
number of loci involved (Crow. et al 1990). With eight loci and many generations passing
before a peak shift occurs the phenotypic extreme (A) model may often be unable to
successfirlly reach the higher peak throughout the population. When peak shifts do
actually occur the phenotypic extreme (A) models are however, still significantly more
effective at spreading peak shifts throughout the population than the no variance (B)
models (see figure 4.4). The maximum variance models (B) were almost always successful
at spreading peak shifts.

The sixteen locus models were nm with all parameters exactly the same as their
corresponding eight locus models except that each allelic substitution had half as much
phenotypic effect as it did in the eight locus models. The sixteen locus models were
dramatically less likely to produce peak shifts and to have those shifts propagate

throughout the population (see table 4.2). These results indicate that the number of loci is

119

important in determining the probability of a peak shift occurring in a given amount of
time. While this result mirrors the comparison of the two and four locus models it is much
more surprising. Unlike the two and four locus models which have only two genetic
optima the eight and sixteen locus models both contain many genetic optima. The sixteen
locus models used allow genetic variance to build up between demes using exactly the
same mechanisms to generate that variance as in the eight locus models. Despite this the
sixteen locus model was much less effective in finding the optimal phenotype in the 12000
generations of these simulations. It therefore may be that in general, with a given fitness
fimction, increasing the number of loci contributing to a trait may reduce the probability of
a peak shift.

Barton and Rouhani (1987) analytically compared the behavior of single locus
models and additive polygenic models of the SBP. They found that two types of models
had similar peak shift probabilities when the per population mutation rates were high
(N u>>1/4) but not when they were low. In the present study two through sixteen locus
models of the SBP with a high mutation rate indicate that the efiicacy of the process is
highly dependent on the particular model The probability of peak shifts seems to be
dependent on both the number of loci involved and the amount of genetic variance which
is likely to be generated within and between demes. Of the factors investigated by Barton
and Rouhani (1987) peak shifts depended mainly on the depth of the adaptive valley.
These sinnrlations, in which the depth of the valley was not varied, indicate that the
relative size of the domains of the higher and lower peaks also can strongly influence the
probability of peak shifts. Models with troughs that were closer to the lower peak were in

general much more likely to experience peak shifts (see table 4.2). Rouhani and Barton

120

(1987) found that in a polygenic character peak shifts that require crossing a trough that is
more than a few times the square root of the additive genetic variance of the parent
population may be unlikely in founder populations. In these sinmlations the affect of allelic
substitution is clearly a factor in the balance between the distance to a new domain and the
average amount of variation.

Previous analysis of phase three by Crow et al (1990) predicts that the rate at
which phase three occurs is increased as the number of loci is increased The trend in these
sinmlations generally fits those predictions. Dominance of the higher fitness alleles
improves the rate of the third phase in the sinnrlations of Moore and Tonsor (1994)
relative to these sinrulations. This agrees with the predictions of Phillips (1993) and Crow
et al (1990). In addition the dominance efi‘ects seem to be part of a generally increased
efficacy of the third phase when the domain of attraction for the higher fitness peak is
increased. In this study models with troughs closer to the low peak in general were more
effective at spreading peak shifts (see figure 4.4). However, for those eight locus
simulations where the probability of peak shifts was highest (models E and C) the
effectiveness of phase three remained constant or even increased as the domain of the
higher peak was reduced.

In comparing single locus and polygenic models of the SBP Barton and Rouhani
(1993) concluded that the most favorable conditions for the SBP probably remain constant
across genetic systems. This conclusion is reaffirmed by the two locus model of Moore
and Tonsor (1994). It is important to note, however, that while the optimal population
structure for most genetic systems may be relatively consistent, the actual effectiveness of

the process may vary widely between genetic models. In these simulations the eflicacy of

121

the SBP as a whole was highly variable between models (figure 4.5).

The genetic variation which was present at the beginning of each 12000 generation
trial seems to have had a large effect on the probability of a global peak shift. Despite this,
probabilities of global peak shifts ranged from 0.0% to 92.1% when there was no initial
genetic variance. The shape of the fitness surface is therefore important in determining the
probability of a global peak shift. The phenotypic effect of each allelic substitution in the
four and eight locus models was equivalent. The only difference between these models
was the capability for the production of phenotypes which are not intermediate between
the optima. A comparison of the four locus and eight locus models demonstrates that
models of disruptive selection may be much more conducive to the SBP when the optima
are not on the phenotypic extremes. The eight locus models, where the optima are not on
the phenotypic extremes, allow the generation of greater variation between demes which
increases the probability of peak shifts.

As predicted by Barton and Rouhani (1993), as models moved from several
discrete loci towards a polygenic trait the probability of population wide peak shifts
decreases if all other factors are held constant. Clearly the shape of the fitness fimction
alone does not determine the probability of a peak shift. Not smprisingly, however,
models with larger domains of attraction for the higher peak had higher probabilities of a
peak shift in these simulations when all other factors were held constant. When all other
factors were not held constant some systems of sixteen loci produced global peak shifts
readily as did some models in which the domain of attraction for the higher peak was
small Many models produced peak shifts in the 12000 generations that°were sinnrlated.

The eight locus models were intended to investigate the effect of different amounts

122

of initial variance on the SBP. A comparison of the maximum variance (E) model with the
other models also gives some insight into the effect of variable migration rates on the SBP.
In effect the maximum variance (E) model represents an extreme model where total lack
of migration allows each deme to independently find the lower adaptive peak, and
subsequent increases in migration rate allow mixing of alleles It has previously been
suggested that variable migration rates can increase the efl‘ectiveness of the SBP by
segregating the first and second phase which occur best at very low migration rates fiom
the third phase which occurs only at higher migration rates (Moore and Tonsor 1994).
The results of the maximum variance (E) model when compared to the other eight locus
models suggest that even when phase 1 and 2 have not occtured during periods of very
low migration, the variance between demes which results from low gene flow may be
critical When migration between the differentiated demes is reinitiated it creates new gene
combinations which seem to trigger peak shifts with a very high frequency as predicted by
Whitlock (1993 ). Ultimately, high variance is only needed for brief periods in order to
trigger peak shifts. Variable migration rates allow variance to build between demes during
low migration periods and to appear within demes during periods of high migration.
While model B demonstrates how variation of migration rates might influence the
probability of peak shifts, the extreme nature of the model seems limiting. The extreme
variance model is, however, one of many realistic ecological scenarios. The amount of
phenotypic variation within a population is a product of the isolation of local breeding
units, the history of selection, and past colonization patterns. Such ecological factors in
natural populations are dynamic. The ecological history of a lineage may result in a

population which is in any of the circumstances listed in table 4.1. The evolutionary

123

consequences of that history extend into the future by influencing the probability of peak
shifts between adaptive peaks (table 4.2).

In conclusion this study indicates that the overall eficacy of the SBP is not
consistent across genetic models. Increasing the number of loci contributing to a trait with
a specific fitness surface can decrease the rate at which the SBP progresses. This is only
true, however, within a given fitness surface. The shape of the fitness surface itself;
through its effect on variance, can override the changes in the number of loci acting on the
trait. The variance within and between demes is determined by the selective surface and
the phenotypic and genotypic starting point of the demes within a population. Higher
genetic variance triggers peak shifts. Ecological conditions which produce periodic
increases in variance increase the likelihood of adaptation via the SBP. Specifically the
history of migration between demes alters the SBP. Overall the probability of peak shifts
in these simulation was high enough to indicate that the SBP may often be an important

evolutionary factor.

Chapter 5

FLUCTUATING MIGRATION AND THE SHIFTING BALANCE PROCESS

Introduction

Recent work on the shifting balance process (SBP) suggests that the process is
effective at producing meta-populational shifts between adaptive peaks under certain
circumstances (Barton and Rouhani 1993; Moore 1996 chapter 3; Moore and Tonsor
1994). The SBP requires a balance of factors that may severely restrict the conditions
under which the process increases the rate of adaptive evolution. Specifically migration
rates that are optimal for the spread of peak shifts are too high to allow many peak shifts
to occur. Consequently the SBP only seems to be an efl‘ective means of adaptive evolution
over a small range of Nm values (Barton and Rouhani 1993; Moore and Tonsor 1994).
However, it has been suggested that periodic changes in migration rates may greatly
enhance the efficacy of the process by decoupling the first and third phases (Moore 1996
chapter 3; Moore and Tonsor 1994, Slatkin 1985).

Moore and Tonsor ( 1994) and Barton and Rouhani (1993) found that a Nm of a
little less than one provided the optimal level of population structure for the SBP.
However, the probability of peak shifts within a deme increases dramatically when Nm is
decreased by several orders of magnitude below one (Moore and Tonsor 1994). At very

low levels of gene flow demes can effectively drift into the domain of new peaks.

124

125

Subsequent periods of high migration should then allow the spread of favorable allele
combinations throughout a population. In this way fluctuations in migration rate may
decouple the first and third phases of the SBP.

Fluctuations in migration rate may also trigger peak shifts through its effect on the
distribution of variance within and among demes. Increases in phenotypic variance are a
potentially important factor in creating peak shifts (Whitlock 1995). Migration rates and
efl‘ective population sizes determine the distribution of genetic variation within and among
demes. Low migration rates allow the buildup of genetic variation among demes via drift.
Subsequent increases in migration may then allow variation to be redistributed so that the
variation within many demes is increased. Even when the mean phenotype of a deme is not
altered by drift, genetic variation among demes can build up over time in nnrltilocus
models of optimizing selection. If sufficient genetic variation among demes develops,
subsequent migration may then trigger peak shifts between phenotypic optima (Moore,
1996 chapter 3). Ifthe migration rates remain high peak shifts spread readily between
demes.

Stochastic variation around a mean migration rate has been incorporated in
previous models of the SBP (Moore and Tonsor 1994, Barton and Rouhani 1993). Those
models have not included large fluctuations in mean migration rates. Wright thought that
the SBP was likely to occur over long periods (Wright 1982a; Wright 1982b). Biotic and
abiotic factors can radically alter the distribution and movements of organisms both over a
small number of generations and over geologic time. Because migration can change over
geologic time (Cronin and Schneider 1990; Van der Spoel 1994) and because geologic

time scales are involved in the SBP large changes in mean migration rate have the

126

potential to alter the eflicacy of the SBP. This study includes fluctuations in migration rate
on several time scales into a model of the SBP. The purpose of this study is two-fold. First
it is intended to determine the extent to which fluctuations in migration might influence
adaptive evohrtion by decoupling phase one from phase three of the SBP. The study also
investigates the patterns of fluctuations in migration rate which allow such decoupling to

take place.

Methods

In order to compare the SBP with fluctuating migration to the SBP with a constant
expected migration rate Monte Carlo sinnrlations were used. This study extends the
models of Moore and Tonsor (1994). Both studies model a diploid, obligate sexual,
semelp arous species. Density independent mortality before migration and mating was
based on genotype-dependent survival probabilities. The model tracked individual
genotype, fitness and dispersal behavior. Variation in deme size (N) allowed variation in
the absolute number of individuals emigrating from demes (Nm), without requiring
difl‘erential per-capita migration rates (m) between demes.

As in Moore and Tonsor (1994) the simulations followed single meta-populations
(hereafter called populations) for 12000 generations. Each population was composed of a
10 by 10 matrix of subpopulations (demes) which was arranged in a torus to reduce edge
effects. The migration rate (m) between demes was the per capita probability of leaving
the parent deme during a generation. Because of the probabilistic nature of the model
there is expected to be some stochastic variation around m even in those models that had

no intentional fluctuations. The probability of a given individual migrating to a location X

127
demes away was described by the gamma fimction: Prob(X) = (m)x ‘1'"). Any individual
migrating X demes away was placed in a randomly chosen deme at that distance.
Simulations were run with 9 different constant migration rates between m = 0.0005 and m
= 0.05. These constant migration rate sinnrlations provided a point of comparison for
models with fluctuating migration rates.

It may take substantial time for demes to drift into the domain of a new peak. The
drift phase is most likely to occur at low migration rates. Under subsequent high migration
rates demes that have drifted will quickly return to the vicinity of a traditional peak. It was
therefore important to look at the period of the high migration as well as the period of the
low migration rate as possible factors influencing the adaptive efliciency of the SBP. In the
fluctuating migration rate simulations the initial migration rate was always m = 0.0005.
This migration rate was used because it produced no population wide peak shifts in the
sinnrlations of Moore and Tonsor (1994) despite allowing demes to experience peak
shifts. After a fixed period the migration rate was increased to m = .05. A migration rate
of m = 0.05 prevented peak shifts from occurring within demes in a previous study
(Moore and Tonsor, 1994). Three different high migration rate models were used. High
migration rates lasted for periods of 10, 100 or 1000 generations depending on the model.
Each of those three models was run with' several diflerent periods of low migration rate.
Migration rates cycled between the high and the low migration rates after the predefined
periods described in table 5.1. Cycling continued until each 12,000 generation simulation
was finished. Figure 5.1 diagrams examples of several of the different migration models
used in these simulations.

The average number of migrants and the average migration rate calculations are

128

Table 5.1 - Summary of the fluctuating migration models.

Each row represents a different migration model The first column lists the number of
consecutive low migration rate generations the population experiences before a high
migration period. The second cohunn lists the number of generations of high migration
rate that the population experiences before beginning a new cycle of low and then high
migration. Column three lists the number of cycles in a given sinnrlation. The mean
migration rate across all 12,000 generations is listed in cohrmn four. Cohmm five is the

average number of migrants per deme per generation.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

# of # of Number Mean m Mean
Low Migration High Migration of Cycles Nm
Generations/Cycle Generations/ Cycle
1 1000 1 0.0046 0.1126
5000 1000 2 0.0088 0.2131
3000 3 0.0129 0.3135
1000 6 0.0253 0.61484
1 1900 1 0.0009 0.0222
5900 2 0.0013 0.0323
3900 3 0.0017 0.0423
1900 100 6 0.003 0.0724
900 12 0.0055 0. 1327
400 24 0.0104 0.2532
100 60 0.0253 0.6148
11990 1 0.0005 0.0132
5990 2 0.0006 0.0142
3990 10 3 0.0006 0.0152
1990 6 0.0008 0.0182
990 12 0.001 0.0242

 

 

Table 5.1 (cont'd).

129

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

390 30 0.0017 0.0423
190 60 0.003 0.072

90 10 120 0.0055 0.1327
40 240 0.0104 0.2532
10 600 0.0253 0.6148
0 (Constant @ m = 0.00025) 0 0 0.0003 0.0061
0 (Constant @ m = 0.0005) 12000 0 0.0005 0.0122
0 (Constant @ m = 0.00075) 0 0 0.0008 0.0183
0 (Constant @ m = 0.001) 0 o 0.001 0.0244
0 (Constant @ m = 0.0025) 0 0 0.0025 0.0609
0 (Constant @ m = 0.005) 0 0 0.005 0.1218
0 (Constant @ m = 0.0075) 0 0 0.0075 0.1826
0 (Constant @ m = 0.01) 0 0 0.01 0.2435
0 (Constant @ m = 0.025) 0 0 0.025 0.6088
0 (Constant @ m = 0.05) 0 12000 0.05 1.2175

 

 

 

 

 

 

130

Figure 5.1 - Four examples of the type of migration rate cycles used in these sinmlations.

The migration rate is plotted against the number of generations since the start of the
simulation. Two different 1000 Generation Periods of Higher Migration (GPHM) models
are shown, one with a 4000 generation cycle (a), and one with a 2000 generation cycle
(b). One 10 GPHM model with a 12000 generation cycle (c) is shown, and one 100
GPHM model with a cycle of 4000 generations (d) is also shown.

131

Figure 5.1

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132

averages across all 12,000 generations. Because these simulations follow the SBP for only
12,000 generations the average Nm which could be investigated with the 1000 Generation
Period of High Migration (GPHM) model was 0.11262. Simulations of more than 12,000
generations would have been required to investigate lower Nm vahres and those values
would not have been comparable to the 12,000 generation models in this study.

A maximum deme size was imposed by incorporating a carrying capacity (K = 30),
and intrinsic grth capabilities into a logistic firnction (N (m) = Na) + rN(,)(K-N(,))/IQ r =
1.1). This function determined the maximum number of individuals that could be born into
that deme during the next generation (Nm)). All demes had the same r and K, during each
generation all demes therefore tended towards the same carrying capacity. Stochastic
extinction of demes was allowed and recolonization occurred only when individuals of
both sexes migrated into an extinct deme in the same generation.

Sexes were separate and mating was random within demes. All genetic variation
was introduced via mutation. A mutation rate of 5 x 10“ mutations per copy, per
generation, per capita introduced mutants at each locus. This mutation rate was constant
throughout all simulations. Forward and back mutation rates were identical

Each phenotype had a predefined absolute juvenile sru'vivorship. There was no
mortality outside the juvenile phase. A two locus two allele model, identical to Moore and
Tonsor's (1994), was used. The average absolute fitnesses of the genotypes were as
follows: double homozygote wild type = 0.636, individuals that were heterozygous at one
locus but homozygous wild type at the other = 0.620 and individuals that had at least one
copy of the mutant allele at each locus = 0.700. These values yield average relative

fitnesses of 0.909, 0.886 and 1.00 respectively. The fitness surface for individual

133

genotypes is shown in figure 5.2. All simulations were started with absolute fixation on the
lower peak (all homozygous wild type).

At least 200 independent sinnrlations (trials) were run for each fluctuating
migration model There were three variables of concern in these simulations. The first is a
measure of the efficacy of phases one and two. It is the number of trials in which at least
one deme shifted to the higher peak The second is a measure of the effectiveness of the
third phase. It is the percentage of trials that, after experiencing an initial peak shift within
a deme (deme shift), then fixed that shift throughout the population A population was
considered to have fixed on the new peak when at least 95% of its extant demes had
shifted to the new peak The final measure is the number of all trials that end in fixation of
the highest fitness genotype throughout the population. This represents the eficacy of the
SBP as a whole. Confidence intervals (95%) on all measures were bootstrapped as in
Moore and Tonsor (1994).

The Nm values used in the results and discussion assume a mean deme size of
24.35 individuals at the time of migration. This value is based on deme sizes for 17,737
demes which had not shifted to the higher peak in Moore and Tonsor (1994). The
parameters effecting deme size were exactly the same in this study and Moore and Tonsor
(1994). The primary purpose of this study was to determine the conditions which lead to

peak shifts, so the mean size of demes after peak shifts was not included.

Rama
The results for both the constant and fluctuating migration rate models are summarized in

table 5.2.

134

Figure 5.2 - Genotypic fitness surface.

The surface is a plot of the fitness of an individual against the number of copies of nurtant
alleles as each of two loci.

135

Figure 5.2

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136

Extreme Constant Migration Rates

No peak shifts occurred in the constant m = 0.05 sinnrlations (Nm - 1.2175). In
44.2% of the constant m = 0.0005 trials (Nm - 0.012175) at least one deme shifted to the
higher peak. In none of those cases did the peak shift then spread throughout the
population. At neither extreme of constant migration rate was there any population wide
peak shift.

Frequency of peak shifts

Figure 5.3 plots the percentage of trails in which at least one deme has shifted to
the domain of the higher peak as a function of average Nm. Lower Nm vahres represent
longer periods of low migration rate for a given period of high migration rate. Fluctuating
models with Nm > 0.03 decline in the number of peak shifts with increasing Nm. A plateau
seems to exist once Nm drops below approximately 0.03. The maxinnrm number of peak
shifts is reached near a Nm of 0.02, when the high migration period is 10 or 100
generations. Constant migration rates may not have reached a plateau at the lowest
migration rates run. The models with a high migration period of 1000 were not examined
for Nm values below .11 so it is impossible to tell whether there would be a plateau in the
Nm = .03 region.

The model with a 1000 GPHM and average Nm = 0.11262 produced nearly twice
as many shifts as any other models with similar Nm. The models with 100 generation
periods of high migration produced fewer peak shifts than models with 10 GPHM. The
10 GPHM migration models were not consistently diflerent from the static models with a

similar Nm. The 10 GPHM models tended towards higher peak shift rates than the

137

Table 5.2. Summary of results for the fluctuating and constant migration rate models.

Results are listed by the number of generations of high migration rate in each cycle (High
Migration Periods) and the number of migrants expected each generation fiom each deme
(Mean Nm). The percentage of trials in which at least one peak shift occurred, the
percentage of those trials which had a peak shift in which the shift spread throughout the
population, and the percentage of all trials in which a population wide peak shift occurred
are listed in columns 3, 4, and 5 respectively.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

High Migration Periods Mean Nm % of Trials % of Shifts % of Trials
(Generations) with at Least Fixed with Fixation
One Shift
0.11262 51.5 100 51.5
1000 GPHM 0.21306 37.5 98.7 37.0
0.31351 23.0 100.0 23.0
061484 2.0 100.0 2.0
0.02222 39.0 0.0 0.0
0.03226 42.5 7.1 3.0
0.04231 38.0 39.5 15.0
100 GPHM 0.07244 27.0 70.4 19.0
0.13271 11.5 91.3 10.5
0.25324 2.0 100 2.0
0.61484 1.5 100 1.5
0.01318 44.5 0.0 0.0
0.01418 49.0 0.0 0.0
0.01519 34.5 0.0 0.0
10 GPHM 0.01820 51.0 0.0 0.0
0.02423 49.5 7.1 3.5
0.04231 43.0 52.3 22.5
0.07244 34.5 81.2 28.0

 

 

138

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 5.2 (cont'd).

0.13271 25.0 80.4 20.5
10 GPHM 0.25324 7.5 86.7 6.5

0.61484 2.0 100 2.0
Constant @ m = 0.00025 0.00609 51 0.0 0.0
Constant @ m = 0.0005 0.01218 44.2 0.0 0.0
Constant @ m = 0.00075 0.01826 45 0.0 0.0
Constant @ m = 0.001 0.02435 38.5 0.0 0.0
Constant @ m = 0.0025 0.06088 27.0 22.2 6.0
Constant @ m = 0.005 0.12175 22.0 76.1 16.8
Constant @ m = 0.0075 0.18263 14.5 89.7 13.0
Constant @ m = 0.01 0.2435 11.0 77.3 85.0
Constant @ m = 0.025 0.60875 2.5 100.0 2.5
Constant @ m = 0.05 1.2175 0.0 NA 0.0

 

 

 

139

Figure 5.3 - Percentage of trials in which at least one deme shifted versus average Nm.

Lines are plotted separately for constant rate, 10, 100 and 1000 Generation Periods of
High Migration (GPHM). Error bars represent 95% confidence intervals. Error bars are
not presented for constant migration rate simulations in order to reduce clutter.

140

Figure 5.3

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141

constant models over an Nm range of between 0.02 and 0.2. When Nm was below 0.02
constant migration rate models were at least as prone to shifis as the 10 GPHM models.
The 100 GPHM models produced fewer shifts than the 10 GPHM throughout the entire
Nm range except near 0.6. When Nm was near 0.6 there Was very little difference between
any of the migration models.
Phase Three Propensity
Figure 5.4 plots the percentage of trials with a deme shifi that then led to the
fixation of the higher peak throughout the population (the tendency for phase three to
occur once phases one and two had occurred) as a function of the arithmetic mean of Nm
over time. For the models with a 1000 GPHM virtually all peak shifts were fixed
throughout the population. There were no trials nm in the 1000 GPHM model which had
a Nm of less than 0.11262. In the other models phase three was unsuccessfirl when Nm
was less than approximately 0.02 and the propensity for phase three increased with
increasing Nm. At Nm values higher than approximately 0.25 all models were nearly 100%
efficient at exporting peak shifts throughout the population. In the range between 0.05 and
0.3 the 10 and 100 GPHM were usually more effective at spreading peak shifls than the
constant migration models.
The SBP as a Whole
Figure 5.5 plots the percentage of trials in which there was population wide fixation of the
highest fitness genotype (the eflicacy of the process as a whole) as a function of mean
Nm. The 1000 GPHM models are significantly more effective than other models with
equivalent average Nms with the exception of Nm - 0.6. The 100 GPHM model is less

effective than the 10 GPHM model at Nm values less than 0.6. At an Nm of 0.6 the

142

Figure 5.4 - Percentage of those trials with a deme shift that experienced a fixation of the
highest fitness genotype throughout the entire population versus average Nm.

Lines are plotted separately for constant rate, 10, 100 and 1000 Generation periods of
High Migration (GPHM). Error bars represent 95% confidence intervals. Error bars are
not presented for constant migration rate simulations in order to reduce clutter.

143

Figure 5.4

SZ

    

 

 

 

 

 

 

 

No 0.0 md vd md Nd Pd 0
T__F_r____A__p_%___pLT_F_E_%r_W_____~_b_“___Fbt_e¢_______h_+_-~__nu_.Wo Z
....M E o
.‘N u 1W ON ~
...~. q [H] on r—L
8mm «gmGOUuiiii ..a m
same 2 LmF ....w ow <
same 2: so.-. .....< om
2:8 82 -.....eé ‘ -m. cm X
E _
............ ...,. om m
............... a... . x, x... . . m
..................... .u.\.........¢ 1m om
..... o.....................«.. ......r... .................... .-..o In 00—. ..\c

144

Figure 5.5 - Percentage of all trials in which the highest fitness genotype was fixed
throughout the population versus average Nm.

Lines are plotted separately for constant rate, 10, 100 and 1000 Generation Periods of
Higher Migration Rate (GPHM). Error bars represent 95% confidence intervals. Error
bars are not presented for constant migration rate simulations in order to reduce clutter.

145

Figure 5.5

No

 

  

52

od md v.0

H _

 

 

aéwgmfiooniiii
2:02 Lml
Emcee 32¢
amass .. ........ -

 

 

 

 

md

fl

 

Nd

u

 

to

d

 

Lr_PL_b_L———nnrr-—————r_—F—b—_—_—pC_-L——[_—Fbp___—Pb—_r.

o

      

o—

om
om

ov
om

om

OFF-I Edm~<ulm

c\°

146

constant migration and the 10, 100 and 1000 GPHM models all converge towards the
same low level of population wide peak shifis (- 2% ).

The optimum average Nm varies between models. The optimum Nm value for the
completion of the SBP is in the vicinity of 0.07 for both the 10 and 100 GPHM models.
The optimum Nm for the constant migration rate model is in the vicinity of 1.2. The
optimum Nm for the 1000 GPHM model was not found in this study. When the average
number of migrants is below 0.1 fluctuating models experience more population wide peak
shifts than do constant migration rate models because the optinmm migration rate for the

fluctuating models is shifted towards lower Nm values.

Discussion

Previous studies (Moore and Tonsor 1994; Barton and Rouhani 1993) have
indicated that an optimal level of population structure exists for the SBP. Around this
optimum the rate of evolution between adaptive peaks within a population is maximized.
Moore and Tonsor (1994) found that Nm in the vicinity of 0.1 produced reasonably high
probabilities of population-wide peak shifts. Barton and Rouhani (1993) found that Nm of
just below 1 should provide the maximum number of population-wide peak shifts. The
present study extends previous models by fluctuating the migration rates. The optimal
average Nm in this study agrees in general with the optimal Nm of Barton and Rouhani,
and with the optimal range of Nm from Moore and Tonsor. The maximum rate of adaptive
evolution occurs with an approximate average Nm of between 0.043 and 0.122. The
optimal migration rate appears to be lower when the period of high migration is either 10

or 100 generations than it is for a constant migration rate (figure 5.5).

147

The optimum average Nm for adaptation via the SBP appears to be in the range of
0.07 when 10 to 100 generation periods of high migration are interspersed amongst low
migration rate periods. When migration is constant the optimal Nm for adaptation via the
SBP increases slightly to approximately 0.12. Across fluctuating and non-fluctuating
migration models the rate of population wide peak shifts at that optimum is not constant.
This decrease in the optimal Nm for the SBP with 10 or 100 GPHM indicates that
fluctuation in migration can disrupt the relationship between phase I and phase III. Longer
periods of high migration rate are likely to completely decouple the two antagonistic
phases.

The present study demonstrates that even though the optimal Nm for adaptation in
the SBP does not seem to be changed greatly by fluctuating migration, there is a
substantial efl‘ect of fluctuating migration on the eflicacy of the process. Iffluctuations of
high migration are of the correct period a significantly higher efficacy of the SBP is
achieved than with any static migration rate. Fluctuation between very high and very low
migration rates allow population wide peak shifi propensities which are not possible with
either the high or low rate. In addition, if the periods of the high and low fluctuations are
of the right size it is possible to achieve higher populational peak shift rates than are
attainable even with the most emcient static migration rate.

Most ecological process are studied on short time scales. Short periods of high
migration (10 or 100 generations) are capable of producing population wide peak shift
rates similar to static migration when the average migration rates were roughly
equivalent. Ecological conditions are altered periodically on a geological time scale. Long

period fluctuations of this geological scale allow much higher rates of population wide

148

peak shifts than static migration models or short period fluctuations. The 1000 GPHM
models produced population wide peak shifts 51.5% of the time with an average Nm of
0.11262 The static migration rate model with an approximate Nm of 0. 12175 produced
population wide peak shifts in only 16.8% of trials. The three fold advantage in population
wide peak shifts of the 1000 GPHM model indicates that the SBP is more important if
long term fluctuations in migration rate occur than if migration rates remain stable around
the optimum Nm value.

The sinmlations with 100 GPHM produced significantly fewer population wide
peak shifts than sinmlations with 10 GPHM. There are two phases that are affected by
migration in the shifting balance process. The first is drifl which requires small Nm values.
The second is interdemic selection which requires relatively high migration rates. As the
period of high migration increases, the probability of successfully exporting peak shifts to
new demes is increased and there is a concurrent reduction in drift. The 10 GPHM trials
produce more peak shifis (figure 5.3) but are no more effective at exporting them (figure
5.4) than the 100 GPHM trials. It is possible that 100 generations of high migration allows
mass selection to eliminate the majority of variation which had previously been protected
from selection by fixation within demes. When only 10 generations of high migration
occur mass selection should be much less efl‘ective at removing that variation. Apparently
the 100 GPHM is long enough to severely retard the drift of demes into the domain of a
new peak, but not long enough to export peak shifls throughout the population. A 10
GPHM does not apparently disrupt the exploration of adaptive landscapes.

A 1000 GPHM is clearly adequate for interdemic selection to take place when m =

0.05. Nearly all peak shifls were fixed throughout the population in the 1000 GPHM

149

(figure 5.4). Since even 3000 generations of low migration can produce a substantial
number of peak shifts a noticeable fraction of trials result in population wide peak shifts
during the 1000 GPHM trials even at very low average migration rates. Because the first
and third phases have been decoupled in the 1000 GPHM the limiting parameter in
determining the overall efficacy of the SBP with this model is probably the number of peak
shifts which are likely to occur during the intervening periods of low migration. If 6000
GPHM models had been run it is predicted that high rates of populational peak shifts
would have been detected even at an average Nm of 0.6. This is expected because the
period of low migration in that model would be adequate to provide a large number of
deme shifts (Moore and Tonsor 1994) which would then be spread.

In addition to an exceptional ability to export peak shifts, the 1000 GPHM trials
also produced more peak shifts than would be expected for an equivalent constant Nm
(figure 5.3). The increase in peak shifis is most likely a product of remixing of alleles
between demes after many generations of independent drift. Demes which had drifted to
fixation in the adaptive valley may suddenly be supplied with the allele necessary to push
them into the domain of the higher peak once migration is increased. Increasing
phenotypic variance even without any change in trait means can induce peak shifts in a
deme experiencing disruptive selection (Whitlock 1995). Moore (1996, chapter 3) found
that remixing of alleles in demes that have been isolated for long periods increases the
propensity for peak shifts to occur. In the 1000 GPHM trials there was ample opportunity
for variation to build up between demes as well as periods of mixing which allow that
variation to produce novel phenotypes. The propensity for phase 1 to occur increased for

a given Nm in those models relative to the 10 or 100 GPHM models. The ability to export

150

those shifts was retained in those models as well Clearly the conflicting requirements of
the first and third phases of the SBP were decoupled in this model

In this study periods of high migration equal to 1000 generations with Nm values
below 0.1 were not explored. As Nm dropped to 0.11262 with a 1000 GPHM the number
of population wide peak shifts continued to increase. This is the expected trend if the
optimal average Nm is in the vicinity of 0.07 or less. Indications from the 10 and 100
GPHM trials are that the average Nm in a fluctuating migration model is interchangeable
with Nm for predicting what is the optimally adaptive migration rate between
demes.

The decoupling of the first and third phases by long periods of high migration
interspersed with long periods of low migration may lead to even greater populational
peak shift probabilities than seen in this study. The 1000 GPHM model may have even
higher rates of peak shifts if the average Nm is dropped lower. Although the 12,000
generation run time for these simulations limits this study to an Nm of 0. l 1262 or greater,
real populations may go for even longer periods with a low average rate of migration.
Longer cycles with lower Nm would have provided more deme shifts which will inevitably
be converted into populational shifts given long periods of high migration.

It is possible that given a different genetic model 1000 GPHM might be insufficient
to increase the efficacy of the SBP. Epistatic fitness models with large numbers of
interacting loci have been shown to proceed through phase three more slowly than models
with only a few loci (Crow et al. 1990; Phillips 1993). This could increase the period of
high migration necessary for the SBP to occur. It is also possible that under some models

more favorable to the creation and maintenance of variation that 100 GPHM might be

151

more than suflicient to increase the eficacy of the SBP. Certain oligogenic models of
disruptive selection are more efl‘ective at producing population wide peak shifts than some
two locus models (Moore, 1996 chapter 3). This study primarily indicates that there are
ways in which a slightly more complex model of the SBP can lead to a substantially higher
global peak shift probability. The SBP is driven by events involving rare interactions of
factors. Interacting ecological factors can occasionally create the correct environment for
peak‘shifts in a population that is not normally likely to experience such shifts. Ifthe SBP
is occurring in natural systems it is likely to occur in an environment with considerably
higher complexity than traditional models provide, and in most cases at higher rates than
traditional models predict.

These results have ramifications for the interpretation of inter-demic migration
rates measured in natural populations. Populations which seem to have stable and very low
rates of gene flow between demes are likely to experience some peak shifis within demes.
They are unlikely to have those shifts spread between demes. Over very long periods of
time, however, intervening periods of high migration can substantially increase the eflicacy
of interdemic selection. In this scenario neither high nor low migration rates provides a
necessary barrier to adaptation via the SBP. Instead the period of the high and low
migrations rates and the average Nm are important. Biotic and abiotic factors such as
predator and competitor presence, resource availability, temperature, rainfall and sea level
change over many temporal scales. Changes in these factors over annual and lunar periods
may well affect the adaptive ability of some organisms with short generation times.
Geologic processes such as glacial periodicity, continental drift and uplift may influence

evolution in other organisms through their effects on migration.

152

The measurement of gene flow in natural populations is dificult but some progress
has been made in that area. Reliable Nm values from natural populations range far above
and below 1 (Slatkin 1985). Given a broad range of migration across species some species
are likely to be near the optimum migration rate for the SBP. Measuring the level of gene
flow between populations over the short term will, however, give little indication of
whether the SBP is likely to be operating. Long periods of very low migration followed by
long periods of high migration will produce the maximum rate of peak shifts throughout a
population.

In conclusion, the most eficient average migration rate for adaptation via the SBP
appears to be slight lower for some fluctuating migration rate models than it is for
constant migration rate models. The decoupling of phases 1 and 3 via fluctuating
migration rates can increase the range of average Nm over which the SBP occurs. This
decoupling is also capable of greatly increasing the rate of population wide peak shifts
within that range of average Nm. What is most important to note is that short term studies
of population structure in a given species probably give little evidence of whether the SBP

is a common mode of adaptive evolution.

Summary

A review of literature on the presence of adaptive peaks indicates that adaptive
peaks are ubiquitous but that detailed descriptions of fitness surfaces are nearly
nonexistent. Several novel theoretical results are presented in this dissertation. A
sinnrlation study in chapter three is among the first to show that under certain
circumstances the shifting balance process is viable. Chapter four indicates that efiicacy of
the process is not uniform across models. Chapter five shows that ecological complexity
may also alter the efficacy of the shifting balance process. Models of the shifting balance
process that include fluctuations in migration rate may be more likely to produce adaptive
peak shifts throughout a population than models with fixed migration rates. The models
presented in this dissertation otherwise agree with the findings of previous theoretical
works on the shifting balance process which indicate that the optimal number of migrants
for adaptation via the shifting balance is slightly below 1.0 and that the generation of

variance between and within demes is a critical factor in the process.

153

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