AN INTEGRATIVE THEORY OF BIO-ENERGETIC
GROWTH, MORTALITY AND EVOLUTIONARY
OPTIMIZATION
By
ZHONGLEI WANG

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Forestry
2012

ABSTRACT
AN INTEGRATIVE THEORY OF BIO-ENERGETIC GROWTH, MORTALITY AND
EVOLUTIONARY OPTIMIZATION

By
ZHONGLEI WANG
Growth is a universal energetic phenomenon for all living organisms, and is
undertaken by meristemetic and stem cells through energy metabolism. Most growth
processes can be modeled with a sigmoidal function, meaning that early growth is near
exponential and later growth declines with age or time until at a certain point (i.e.,
maturity), when evolutionary optimization is realized and a species should turn all or part
of its growth energy into reproduction and then death occurs, consequently. This is
usually called a life history process. Evolutionary optimization means that fitness should
be supposedly maximized during this process for all species, but how it is realized for all
species is still a challenge or gap in our knowledge. This dissertation is dedicated to fill
this gap.
First, a general bio-energetic growth theory is presented here that assumes a
dynamic universal energy pattern: a constant energy income rate deceleration with each
cell reproduction, and a species- or clade-constant maintenance cost of cells. This pattern
can produce the observed power relationships between energy metabolism and body size
as described in Kleiber’s law and the von Bertalanffy growth function. Secondly, the new
growth theory and a related new bio-energetic definition of mortality are combined with a
typical evolutionary optimization process to predict that maturity should begin when net
energy income rate equals to the maintenance cost for all living activities and that

maintenance cost per mass integrated till maturity should be a defined constant. Two fish
life history datasets were used to test the new theory, and the result supported the
predicted relationships well. Thirdly, because leaves are the basic functional modules of
plants, a comparable evolutionary life history theory for the leaf is developed. The theory
predicted a similar energy equilibrium status between energy income (photosynthesis)
and expenditure (maintenance) for mature leaves. A global leaf physiological trait dataset
(GLOPNET) was used to test this prediction, and a large proportion of variation in
2

photosynthesis of diverse mature leaves is explained (R = 87% on mass-based metrics).
Lastly, it is suggested that through an established relationship between leaf
photosynthesis and seed energy income, there should also be an allometric relationship
between leaf photosynthesis and seed mass. Using a large database of leaf traits and
matched seed mass for the same species, it was shown that though a relatively small
proportion (20%) of seed mass variation was explained by leaf photosynthesis, the
predicted relationship is still generally supported.

TABLE OF CONTENTS

LIST OF TABLES ....................................................................................... VII
LIST OF FIGURES ................................................................................... VIII
INTRODUCTION .......................................................................................... 1
1 Mass growth and energy metabolism are two complementary aspects of life in plants
and animals ......................................................................................................................... 1
2 Growth or energy production is under evolutionary optimization................................... 2
3 Current growth and life history evolution theories and their limitations ......................... 5
Outline................................................................................................................................ 7
REFERENCES................................................................................................................ 10

CHAPTER 1. FROM CELLULAR BIOENERGETICS TO INDIVIDUAL
GROWTH: A UNIFYING ENERGETIC GROWTH THEORY ................ 14
Abstract ............................................................................................................................ 14
Introduction ..................................................................................................................... 14
New Theory...................................................................................................................... 16
1 Energy budget of cells ................................................................................................... 16
2 A pure energetic cell growth theory............................................................................... 17
3 Linking the new energetic growth theory to traditional VBGF models ........................ 20
Discussion......................................................................................................................... 21
REFERENCES................................................................................................................ 27

CHAPTER 2. BIO-ENERGETIC EVOLUTION OF MATURITY:
THEORY, ANALYSIS AND VALIDATION IN FISH .............................. 30
Abstract ............................................................................................................................ 30
Introduction ..................................................................................................................... 30
New Theory...................................................................................................................... 34
Fitness concept and quantification .................................................................................... 34
Bio-energetic size/mass growth ........................................................................................ 35
Bio-energetic survivorship ................................................................................................ 37
Fitness optimization .......................................................................................................... 37
Methods ............................................................................................................................ 39
Application to fish............................................................................................................. 39
Fish dataset........................................................................................................................ 40
Data analysis ..................................................................................................................... 40
Results .............................................................................................................................. 41
1 The relationship between Lr and Linf ....................................................................... 41
2 The relationship between tr and K .............................................................................. 43
Discussion......................................................................................................................... 45
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1 Natural mortality is standardized bio-energetic maintenance cost ................................ 46
2 Energy equilibrium at maturity and its significance ...................................................... 47
3 Life history invariant at maturity rule may define a fast-slow life history spectrum .... 49
Conclusion ....................................................................................................................... 51
REFERENCES................................................................................................................ 54

CHAPTER 3. A NEW EVOLUTIONARY LIFE HISTORY THEORY FOR
LEAVES CAN EXPLAIN THE VARIANCE OF LEAF
PHOTOSYNTHETIC RATES AND CORRELATION WITH LEAF
TRAITS......................................................................................................... 58
Abstract ............................................................................................................................ 58
Introduction ..................................................................................................................... 58
Theory .............................................................................................................................. 62
Fitness quantification ........................................................................................................ 62
Net output of assimilated energy is proportional to leaf mass .......................................... 63
Survivorship of net energy output to maturity .................................................................. 65
Fitness optimization for leaves ......................................................................................... 65
Data and methods ........................................................................................................... 66
GLOPNET dataset ............................................................................................................ 66
Analytical Methods ........................................................................................................... 67
Model combination with climate factors .......................................................................... 69
Statistical analyses ............................................................................................................ 71
Results .............................................................................................................................. 72
1 Best models with and without climate factors and their performance ........................... 72
2 Performance of mass-based models in three sub-groups ............................................... 75
3 A pattern in the relative ratios of composite maintenance costs to photosynthesis ....... 83
Discussion......................................................................................................................... 85
1 An evolutionary life history theory for leaves has significant meaning for plant
functional traits studies ..................................................................................................... 85
2 A photosynthesis variation continuum across species with two contrasting driving
patterns .............................................................................................................................. 87
3 Lifetime structural mass cost varies little between different leaves .............................. 89
4 Temperature moderately influences photosynthesis and its association with other traits
........................................................................................................................................... 90
Conclusion ....................................................................................................................... 92
REFERENCES................................................................................................................ 94

CHAPTER 4. A LIFE HISTORY PERSPECTIVE ON THE LINKAGE
BETWEEN SEED MASS AND LEAF PHOTOSYNTHESIS.................... 99
Abstract ............................................................................................................................ 99
Introduction ..................................................................................................................... 99
Theory ............................................................................................................................ 101
Methods .......................................................................................................................... 102
Analysis of seed mass and leaf traits .............................................................................. 102
Results ............................................................................................................................ 103
v

Discussion....................................................................................................................... 106
REFERENCES.............................................................................................................. 110

SUMMARY AND FUTURE DIRECTIONS............................................. 113
REFERENCES.............................................................................................................. 116

vi

LIST OF TABLES

Table 2-1. Summary of parameters. .................................................................................. 39
2

Table 2-2. Parameter values for Figure 2-1. R is Pearson’s correlation coefficient. ...... 42
Table 2-3. Parameter values for Figure 2-2. The parameters are from the RMA procedure.
........................................................................................................................................... 44
Table 2-4. Comparisons between the new theory and the existing rules*. Each is discussed,
in turn, below. ................................................................................................................... 45
Table 3-1. Summary of regression analyses of photosynthetic rate in relation to 5 climate
factors. ............................................................................................................................... 70
Table 3-2. Comparison of models ..................................................................................... 72
Table 3-3. Parameter values for Figure 3-2. ..................................................................... 75
Table 3-4. Parameter values for figure 3-2. ...................................................................... 77
Table 3-5. Parameter values for Figure 3-4. ..................................................................... 79
Table 3-6. Parameter values for Figure 3-5. ..................................................................... 81

vii

LIST OF FIGURES

Figure 1-1. Instantaneous growth efficiency (IGE) vs cumulative growth efficiency for
four different values of  . IGE|α=2/3 represent instantaneous growth efficiency at
α=2/3, and the same for other legends. ............................................................................. 20
Figure 2-1. The relationship between Lr and Linf. Lr: length at first reproduction, Linf:
maximum length. n=85, females only. Red line is the theoretical line with a slope of 0.5
and zero intercept, blue line is the fitted line with parameters in Table 2-2. Data is from
He et al. (2001). ................................................................................................................ 42
Figure 2-2. Relationship between log(K) and log(Tr). K is the von Bertalanffy growth
coefficient. Tr (annotated as tr in the text) is age at first reproduction, n=78 (sample size),
females only. The red line is the predicted line with slope of -1 and intercept of
-0.16(which equals to log (ln2)), and the blue line is the fitted line with parameters in
Table 2-3. Data from He et al. (2001). .............................................................................. 44
Figure 3-1. An illustrative figure showing how eq. (3.7) is formulated in plant leaves as
shown in the text according to energy decomposition principles…………………………68
Figure 3-2. Predicted vs. measured Amass for all species pooled together. (a) by eq.
(3.8a) . (b) by eq. (3.8a-3). (c) by eq. (3.8b). (d) by eq. (3.8b-3). The red line is the 1:1
line through the origin, the blue line is the RMA regression line between measured and
predicted values. All related parameter values are in Table 3-3. ...................................... 74
Figure 3-3. Predicted vs. measured Amass for grass/herbs by (a) eq. (3.8b) and (b)
eq.(3.8b-3),and shrub/trees by (c) eq. (3.8b) and (d) eq.(3.8b-3). The red line is the 1:1
line through origin, the blue line is the RMA regression line. All related parameter values
are in Table 3-4. ................................................................................................................ 76
Figure 3-4. Predicted vs. measured Amass for evergreens by (a) eq. (3.8b) and (b)
eq.(3.8b-3),and non-evergreens by (c) eq. (3.8b) and (d) eq.(3.8b-3). The red line is the
1:1 line through origin, the blue line is the RMA regression line. All related parameter
values are Table 3-5. ......................................................................................................... 78
Figure 3-5. Predicted vs. measured Amass for temperate forest species by (a) eq. (3.8b)
and (b) eq.(3.8b-3),and woodland species by (c) eq. (3.8b) and (d) eq.(3.8b-3). The red
line is the 1:1 line through origin, the blue line is the RMA regression line. All related
parameter values are in Table 3-6. .................................................................................... 80

viii

Figure 3-6. The relationship between Km and bm. Data points are for eq. (3.8b) from
2

Table 3-4,5 and 6. The blue dashed line is the fitted curve with equation and R shown in
the graph............................................................................................................................ 82
Figure 3-7. The relative ratios with 95% confidence intervals of respiration,
nitrogen-related and leaf structural mass maintenance costs to photosynthesis,
corresponding to 2*Rdmass/Amass, bm  N mass / Amass , and K m /( Amass  LL) ,
respectively, for three sets of sub-groups: shrub/trees vs grass/herbs, evergreen vs
non-evergreens, woodland vs temperate forest species. ................................................... 84
Figure 4-1. Correlation between photosynthetic rate and seed mass. n=102. The 95%
confidence intervals (CIs) of the intercept and slope are (2.04, 2.20) and (-0.29, -0.41),
respectively. The red line is the regression line from the regression equation displayed in
the figure. ........................................................................................................................ 104
Figure 4-2. Correlations between seed mass and three leaf traits with 95% CIs of the
intercept and slope: (a) respiration (Rdmass), (1.07, 1.20) and (-0.23, -0.33), (b) LL, (0.60,
0.83) and (0.43, 0.61), (c) nitrogen content (Nmass), (0.24, 0.34) and (-0.17, -0.24),
respectively.. n=102. The red line is the regression line from the regression equation
displayed in the figure. .................................................................................................... 105

ix

INTRODUCTION
1 Mass growth and energy metabolism are two complementary aspects of life in
plants and animals
Growth is a fundamental activity of any living individual. Two types of growth
have been recognized: determinate and indeterminate. Determinate growth is used to
describe species that invest all growth energy into reproduction at maturity and are
approaching their maximum body size at maturation, while indeterminate growth is for
species that invest part of growth energy into reproduction and continue to grow after
maturity with a smaller maturation size relative to maximum size (West et al. 2001;
Kozlowski et al. 2004). Determinate growth is for all birds, some plants, insects and most
mammals, and indeterminate growth are for fish, most plants and others (Karkach 2006).
These two types may be mixed within one individual. For example, a plant’s trunk and
branches may grow indeterminately, while leaves, flowers and seeds grow determinately.
Changes in growth rate within the lifetime of an individual are called ontogenetic
growth, and are essentially a result of cell level processes, because growth is
fundamentally undertaken by only a few types of cells. Meristematic tissue in plants is
responsible for the girth and height augmentation of stem, branch growth and the mostly
planar expansion of leaves, while embryonic and adult stem cells in animals are
responsible for all kinds of tissue differentiation and maturation. Meristematic and stem
cell activity can be formulated not only as a mass growth process, but more importantly
in terms of universal energy dynamics, such as energy production, metabolism and
maintenance. The energy production units in plants are chloroplasts in leaves or stems,
and photosynthesis is the universal metabolic process for capturing light energy and
carbon in chloroplasts to produce all other types of structures and tissues in plants.
1

Animals depend on external resources to have energy production and metabolism. There
is always a maintenance cost related to the above processes. For example, respiration of
leaves is required to sustain a photosynthesis reaction, and other supporting costs are also
inevitable for plants to achieve a structure and height to effectively capture light, while
animals must spend certain amount of energy to maintain body temperature (i.e.,
endotherms) or to sustain basic functions of all tissues (measured by basal metabolic
rate).
This bio-energetic perspective of growth has been noticed in some pioneering
studies of ecology and a few more recent works. For example, growth in animals was
often considered as a process of energy balance between build-up and break-down of
tissues; von Bertalanffy (1938) specifically developed his well-known growth equation
on this concept, and Kooijman (2010) had a dynamic energy budget theory on it. For
plants, the energy budget of the whole plant or plant organs has been used to model plant
or leaf growth (Kikuzawa 1991; Kikuzawa et al. 2006). However, better bio-energetic
representations of growth are fundamental to advancing biological theory.
2 Growth or energy production is under evolutionary optimization
Since the discovery of Darwin’s theory of evolution, the focus of ecology and
evolutionary studies has been understanding how growth and reproductive processes are
modulated by natural selection to define different life histories of various species. The
central concept in evolutionary theory is fitness, which is defined as the lifetime number
of offspring that can reproduce themselves as to maintain the continuity of life cycle, and
it is generally assumed that this fitness has been maximized by evolutionary processes

2

during the long time of species formation and existence (Roff 1992; Stearns 1992;
Kozlowski 1996).
The result of evolutionary optimization is optimized life history traits for separate
species, i.e., life history traits should be closely coordinated with each other to achieve
the overall goal of maximum fitness for the individual as a whole. In practice, it means
that there should be general trade-off relations among traits of mass, energy and age for
plants and animals as well.
The simplest trait relations in biology are bivariate correlations. Most bivariate
relationships are defined by linear regressions on log transformed data. For example,
traits of animals, such as basal metabolic rate (BMR), age of female at first reproduction,
life expectancy at birth, instantaneous mortality rate, clutch size, reproduction output and
individual lifespan are related to body mass raised to a power of  , and the values of 
are usually around 2/3 to 3/4 or 1/3 to 1/4 (Sæther 1988; Harvey and Zammuto 1985;
Ricklefs 2006; Charnov 1991 and 2005). Plant traits, such as growth rate, total pigment
concentration, plant height and foliage/root/stem biomass are related to plant size in
similar power relations (Niklas 2004; Niklas and Enquist 2001). Further, there are also
allometric bivariate relations within and between plant organs, such as leaves and seeds.
For example, leaf photosynthetic rate is related to respiration, leaf lifespan (LL), leaf
mass per area (LMA), and leaf nitrogen (Reich et al. 1998a and 1998b; Poorter and
Garnier 1999; Reich et al. 1997), and seed mass is associated with plant height, leaf
photosynthetic rate, and relative growth rate (Moles et al. 2005). The many bivariate
correlations within one species could be better understood when put together. For
1/4

instance, because age at first reproduction scales as m

3

, and instantaneous mortality rate

-1/4

as m

, their product is a constant close to 2.1(Beverton and Holt 1959; Charnov et
1/4

al.1991); and as lifespan scales as m

3/4

, and reproductive output as m

, life time

reproductive energy per mass≈1to 2 for several animal species (Charnov 2005).
Another way of study trait relationships is to use a multivariate approach,
typically reducing the dimensions of “multiple trait space” to facilitate biological
interpretation. Such examples include a recent study that used principle component
analysis to check the inter-correlations of up to six critical leaf traits, and it was found
that there was a single major principle axis called “a leaf economics spectrum” that runs
from slow carbon gain on investments of nutrients and dry mass to the fast opposite
(Wright et al. 2004). Correspondingly, this latter work suggests a “fast-slow life history
continuum” defined for animal species, where key life history traits of animals are
distributed between two opposite trends: fast growth, early maturation, high fecundity
and low adult survival as one extreme, and slow growth, late maturation, low fecundity
and high adult survival as the other extreme (Bronikowski and Arnold 1999; Wiersma et
al. 2007a; Wiersma et al. 2007b; Wright et al. 2004; Reznick et al. 1990; Roff 2007).
Overall, given the seemingly great differences between plants and animals, it is striking
to see how similar fundamental allometric scaling relationships actually are among them.
This similarity leads the development of a recent theoretical framework called Metabolic
Scaling Theory (MST) that refers to mainly biophysical reasons as the mechanism and
resources of these universal patterns. However, the foundation and conclusions of this
theory are vague, problematic and often rejected by reality (Delong et al. 2010; Price et al.
2012; O’connor et al. 2007; Kozlowski et al. 2005).

4

3 Current growth and life history evolution theories and their limitations
Clearly, the universality of relations and life histories in plants and animals should
be understood by similar bio-energetic optimization processes, because energy is the
universal currency and metric for them especially plant leaves, and evolutionary
optimization is a common constraining force for most if not all living organisms. Hence,
modeling how the bio-energetic optimization process is undertaken to produce all kinds
of trait associations or life histories for diverse species is a major challenge in ecology
(Roff 1984; Stears and Koella 1986; Kozlowski 1996).
Currently, a typical optimization process in animals defines fitness as the product
of two critical components: fecundity and survival rate. Fecundity is the lifetime
offspring number, survival rate is the rate of the offspring surviving to its own
reproduction stage. As fecundity is a function of body size, variation of body size with
time, i.e., growth, is required to model fecundity. But the present growth functions are
mainly empirical and focus on mass variation. Survival rate is mainly modeled from
empirically fitted instantaneous mortality rate. As photosynthesis capacity in plants and
the growth rate in animals usually decline with age (Kitajima et al. 2002), and the risk of
mortality from mainly physiological and related factors increases with age, a growth
process is thus optimized at certain age when reproduction should take over pure
semantic growth to have the maximum number of offspring. This age is then determined
by species-specific parameters from mass growth process and individual’s independently
estimated survival rate (Roff 1984). However, mass growth and survival rate are usually
strongly correlated such that fast growth would correspond to low survival and vice versa
(Charnov 1991), confounding possibly simpler but more fundamental constraint for

5

determining optimum maturity. Out present understanding to optimum age of
reproduction therefore are mainly empirical, and life history constraint is also limited to
various strong inter-correlations (Charnov 1993; Roff 2007).
Growth and life history theories for plants are especially rare and not well
developed, because there are significant differences between plants and animals (Stearns
2000). The life history of whole individual of most tree species is difficult to quantify.
Many plant species live many to hundreds of years, and contain large amount of living or
dead mass that show great functional plasticity to outside conditions and thus are hard to
precisely measure over a long period (He et al. 2009; Tjoelker et al. 2005; Wright et al.
2005). For example, most species traits tend to shift along climatic gradients (Wright et al.
2004; Mauseth 2009; Jin et al. 2008). More importantly, unlike unitary animals, plants
are modular so that the concept of “individual” is often not strictly applicable. The above
mentioned bivariate correlation and multivariate associations are usually also held within
leaves and between leaves and seeds. There are models assuming that leaves are energy
production unit and the net assimilation rate of a leaf should be optimized to have highest
energy production rate (Kikuzawa 1991). However, such models are not compatible with
the fact that leaves actually have largely varying net photosynthetic rates. For these
reasons, plant ecologists are now suggesting that the functional trait variation of leaves
and seed are themselves representing and defining whole plant life history strategies
(Bonser et al. 2010; Poorter et al. 2006).
The specialty of plants compared to animals further necessitates a bio-energetic
framework of growth and life history, because plant growth, especially that of leaves, is
more related to energy budget that can be calculated from photosynthesis, respiration,

6

dead mass investment and related cost, and mortality may be more relevant to relative
energy expenditure rate for functional maintenance. In other words, new insights in
ecology can be raised if life history theory of animals and the energetic nature of plants
are combined to provide a common mechanism for explaining their great similarity in
growth-energy allometry and life histories.

Outline
In summary, there is a need for new theories to integrate three importnts aspects
of plant and animal biology: (i) new bio-energetic growth process and survival rate with
explicit relationship between them; (ii) simplified life history theory based on
evolutionary optimization on them that can be applied to animals, plant or plant organs to
provide explanations for their inter-specific life history variations; (iii) analyses of
datasets that support the new framework. This dissertation research was conducted to fill
the above-mentioned gaps. The following is an outline of the research:
In chapter 1, a bio-energetic growth model is developed. It is based on basic
characteristics of energy dynamics, and is a universal bio-energetic form of various
empirical von Berterlanffy growth functions (VBGFs). In chapter 2, the new growth
model and a related energetic definition of mortality are integrated into a typical
evolutionary optimization process. This produces a new life history theory in terms of
energetic for animals. A large fish trait dataset was used to test it. Chapter 3 further
extends the above study method by developing a similar growth and evolutionary
optimization process for leaves. A global dataset (Wright et al. 2004) containing field
measurements of critical mature leaf traits was used to test the leaf life history theory.
Finally in chapter 4, seed mass is then linked to mature leaf photosynthesis because seeds

7

have a similar growth process from a general relationship between energy dynamics and
mass growth. This prediction was tested by a large leaf trait and coupled seed mass
dataset.

8

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9

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syndromes among 39 grassland and savannah species. New Phytologist 167, 493–508
(2005).
von Bertalanffy, L. A quantitative theory of organic growth. Human biology 10, 181–213
(1938).
West, G. B., Brown, J. H. & Enquist, B. J. A general model for ontogenetic growth. Nature
413, 628–631 (2001).
12

Wiersma, P., Chappell, M. A. & Williams, J. B. Cold-and exercise-induced peak metabolic
rates in tropical birds. Proceedings of the National Academy of Sciences 104, 20866
(2007a).
Wiersma, P., Muñoz-Garcia, A., Walker, A. & Williams, J. B. Tropical birds have a slow
pace of life. Proceedings of the National Academy of Sciences 104, 9340–9345 (2007b).
Wright, I. J. et al. The worldwide leaf economics spectrum. Nature 428, 821–827 (2004).
Wright, I. J. et al. Modulation of leaf economic traits and trait relationships by climate.
Global Ecology and Biogeography 14, 411–422 (2005).

13

CHAPTER 1. FROM CELLULAR BIOENERGETICS TO
INDIVIDUAL GROWTH: A UNIFYING ENERGETIC GROWTH
THEORY
Abstract
Inspired by an ontogenetic growth model (OGM) proposed in 2001 by West et al., a
general bio-energetic growth theory was developed that assumes a universal energy
dynamic pattern: a constant energy influx deceleration with each cell growth event in
meristematic and stem tissues and a species or clade-constant maintenance cost of cells.
This new growth theory links mass explicitly with energy, deduced a general form of the
von Berterlanffy mass growth function and also explained the difficulty of determining a
fixed power coefficient for the relationship between energy influx rate and body mass.
This study helps improve understanding of power-scaling relations between metabolism
and body mass.

Introduction
Growth is a fundamental biological process. Growth rates are rarely constant and
changes in growth rate within an organism’s lifetime are often referred as ontogenetic
growth; one of the most fundamental processes in biology. This process has been studied
mainly by modeling an individual’s overall body size variation over its lifetime with
empirical equations (Karkach, 2006). One such model that has been widely used is the
von Bertalanffy growth function (VBGF) (von Bertalanffy, 1938):

dm
 a  m  b  m 
dt

(1.1)

14

where m is body mass at time t , a and b are positive proportional coefficients, and

 and  are two exponents relating energy influx (anabolism) and energy dissipation
(catabolism) with body mass, respectively (Banavar et al. 2002).
While it has been generally accepted that energy dissipation, also referred to as
‘maintenance cost’, is proportional to body mass, i.e.,   1 (He and Stewart 2001;
West et al. 2001), the selection of a specific value for  has been a subject of debate.
For example, many empirical studies suggested from surface area versus volume scaling
relationship that  

2
, while a new ontogenetic growth model (OGM) suggested that
3

basal metabolic rate (BMR) is the source of anabolism and consequently suggested that

3
  , because there is a general 3/4-power relationship between BMR and body mass
4
(known as “Kleiber’s law”) (Mcmahon 1973; Nunn and Barton 2000). However, this new
OGM-based explanation of BMR to be energy influx rate is not strictly true (Makarieva et
al. 2004), and the validity of Kleiber’s law itself is controversial because the power
parameter of 3/4 is not supported in many cases (Dodds et al. 2001; O’Connor et al.
2007). It seems unlikely, based on these latter studies, that the value of α should be
invariant. Thus, it is still unclear why there is a power relationship between energy influx
and body mass and what specific value the power parameter  should assume under
any given condition.
Here, we suggest that although VBGFs are useful models, new models should
consider cell-level growth and energy dynamics, because all growth is undertaken
principally by certain types of cells. Meristematic cells in plants are responsible for the
girth and height augmentation of stems and branches, and stem cells in animals are the
15

ultimate sources of all types of tissues and organs. So we suppose a mechanistic link
between energy influx and growth for these cell types. This link is scaled up to define a
general VBGF form that itself represents a universal pattern of cell energy dynamics over
an organism’s lifespan and that its key coefficients are species-specific cell-level
properties. This new theory also explains why it is difficult to distinguish a specific
power parameter value of 2/3 from 3/4 in common VBGF models.

New Theory
1 Energy budget of cells
Living cells have two counterbalancing activities and functions: growth (including
reproduction and other productive activities) and maintenance. Cell growth means
reproducing a new differentiated or reproductive cell, and maintenance means using the
cell’s energy income to maintain its own viability and functionality. As a result, the
energy budget of cells at certain time t can be generally separated into two parts
(adapted from West et al. 2001):

dN
B
 N t  Bc  Ec  t
dt
dt
where

(1.2)

B
is the energy income rate of all cells, N t is the number of cells, Bc is the
dt

maintenance cost or energy dissipation rate(for metabolic loss, mechanical support, and
other energy costs dissipated irreversibly) of a cell, E c is the energy required to
reproduce a new cell, and dNt is the number of newly produced cells.
Eq. (1.2) can be rearranged to make it more general as:

dNt
B Bc


Nt
E c Ec

(1.3)
16

where B is the energy income rate of a cell, B  N t  B .
Integrating eq. (1.3) over a certain time period 0 to t , the following function
emerges:

Nt  N0  e

(  Bc )

(1.4)

Where N 0 is the initial cell number at the beginning time 0, usually 1 if cell division
t

begins from one cell,   
0

B
represents the total energy income over time t for a cell,
Ec

t

B
and  Bc   c represents the total maintenance cost over time t for a cell.
Ec
0

If the maintenance cost is zero, eq. (1.4) can be used to calculate cumulative
energy income N ttot in terms of number of cells as:

N ttot  N 0  e

(1.5)

and cumulative growth efficiency can then be calculated as growth divided by the
cumulative energy income:

Nt
 e Bc .
N ttot

(1.6)

Obviously, if there were no maintenance cost such that  Bc  0 , the cumulative growth
efficiency would be 1.
2 A pure energetic cell growth theory
As mentioned above, cell maintenance cost

Bc
is usually viewed as constant for
Ec

a species (West et al. 2001; Makarieva et al. 2008). The remaining critical relationship in

17

eq. (1.3) lies between

dNt
B
and
, which is a proxy of relative growth rate (RGR).
Ec
Nt

One common observation for most if not all living organism is that RGR decreases with
increasing body size (Falster et al.2008), suggesting that cell replication as the main part
of body size growth has a negative feedback to cell energy influx rate. In the simplest
case, this observation can be transformed into a linear relationship as this: energy income
rate has a relative change of (  1) each time a new cell is produced, and   1. This
assumption is similar to Newton’s second law of motion regarding deceleration.
Mathematically, it can be expressed as:

d B dN t

   1
Nt
B

(1. 7)

Eq. (1.7) indicates a simple, but critical, link between energy and mass.
Combining it with eq. (1.3) produces a new, purely energetic growth function, removing
mass from the relationship:

B
d (1  c )
B  Bc  (  1)
B
Ec
(1  c )
B

(1.8)

This new bio-energetic growth eq. (1.8) describes the variation of instantaneous
growth efficiency, defined as net growth energy ( B  Bc ) divided by gross energy

B
income rate B , i.e. (1  c ) , with time, or with each cell reproduction (in the latter case
B
time is kept by tallying discrete events). It says that instantaneous growth efficiency has a
relative declining rate of

Bc
 (  1) with time or with each new cell reproductive event.
Ec

18

If we assume that the initial instantaneous growth efficiency is nearly identical to 1 at
time zero, which is reasonable, then the integrative form of eq. (1.8) can be written as:

1 t
Bc
Ec
1e
B
Bc

(1.9)

Eq. (1.9) suggests a negative exponential curve in growth efficiency variation
Bc

 t
Nt
with time. As cumulative growth efficiency
is e Ec (see eq.(1.6)), eq.(1.9) can
N ttot

be transformed as:

 N 
B
1- c   t 
B  N ttot 



1 
(1.10)

Eq. (1.10) means that an energetic growth process may also represent a power
relationship between instantaneous and cumulative growth efficiency with the power
parameter 1 -   being the only critical variable.
To test for the sensitivity of the relationship in eq. (1.10) to variation of 1 -   ,
four commonly considered values of  : 1/2, 2/3, 3/4 and 4/5, and a common range of (0,
1) values for instantaneous and cumulative growth efficiency were used to produce four
hypothetical growth relationships (Figure 1-1).

19

Instantaneous growth efficiency (IGE)

1
0.9
0.8
0.7
0.6
0.5
0.4

IGE|α=0.5

0.3

IGE|α=2/3

0.2

IGE|α=3/4

0.1

IGE|α=4/5

0
0

0.2

0.4
0.6
Cumulative growth efficiency

0.8

1

Figure 1-1. Instantaneous growth efficiency (IGE) vs cumulative growth efficiency for
four different values of  . IGE|α=2/3 represent instantaneous growth efficiency at
α=2/3, and the same for other legends. For interpretation of the reference to color in this
and all other figures, the reader is referred to the electronic version of this dissertation.
From Figure 1-1, it is clear that eq. (1.10) with different  values would
produce quite different growth curves with similar forms. This is a mathematical property
of the power relationship that has a power coefficient in the range of (0,1).
3 Linking the new energetic growth theory to traditional VBGF models
Because eq. (1.7) links energy with mass, the above pure energetic growth theory
can be back-transformed into a traditional mass growth function. First, integration of eq.
(1.7) gives:

B  B0  Nt  1

(1.11)

where B0 is the initial cell energy influx rate at time 0.
Eq. (1.11) has another form for an individual of Nt cells as:

20

B  B0  N t

(1.12)

B
Because maximum or asymptotic size N T is reached when 1 - c is zero or
B

B  Bc , which means zero growth energy availability and thus a stabilized size, a new
relation at maximum size from eq. (1.11) is:

Bc  B0  NT  1

(1.13)

Taking eqs. (1.11) and (1.13) together, we have:

Bc  N t

B  NT







1 
(1.14)

Substituting eq. (1.14) into eq. (1.9), a general mass growth function is deduced
as:

N
1-  t
N
 T






1 

B
 c 1 t
Ec
e

(1.15)

Eq. (1.15) is a general form of VBGF, with the traditional VBGF growth
coefficient K being replaced by

Bc
 1    .
Ec

Discussion
The new growth theory presented here suggested that eqs. (1.9), (1.10), (1.12) and
(1.15) are four inter-convertible ways for modeling the same growth process.
First, eq. (1.9) suggested a universal trend of exponential decreasing for
instantaneous growth efficiency. This trend is observed in animals (West et al.2001;
Banavar et al. 2002). Because this trend is also commonly observed in tree growth,
dendrochronological studies have chosen it as the natural growth trend in trees to detrend
21

a ring-width growth pattern (Naurzbaev et al. 2004; Fritts 1963; Cook et al. 1990 p109),
supporting the validity of eq. (1.9) in plants.
Second, it was deduced that this pure energetic trend can also be translated into a
relationship between instantaneous and cumulative growth efficiency that was
constrained only by a power parameter (1   ) (see eq. (1.10)). A higher  such as 3/4
means that cumulative value would be 1/16 for a specific instantaneous growth efficiency
such as 1/2, while a lower  such as 2/3 would have a value of 1/8 for the same 1/2.
Because instantaneous growth efficiency represents the relative proportion of total energy
income available for growth, reproduction, and other related activities, the parameter 
has a clear meaning: compared to species with lower  , species with higher  achieve
higher excess energy for life history activities even at a later life stage with lower
cumulative (overall) growth efficiency, or they achieve higher excess energy for life
history activities at the same overall efficiency (Figure 1-1). A higher excess energy
could then bring more flexibility, faster growth and more reproduction output to a species,
which in the extreme case of   1 produces exponential growth (see eq. (1.1)). This is
also obvious from eq. (1.7), as   1 the extent of negative feedback of growth on
energy influx is lessened, and thus helps maintain a higher growth rate for a longer time.
Conversely, lower  values such  

2
should be rare as they would significantly
3

reduce growth efficiency. However, even if energy use efficiency relations systematically
vary between taxa, eq. (1.10) and  values may need to be quite different to bring this
variation to certain detectable difference level (Fig. 1.1), which is difficult when higher

 values should be favored over lower ones in an evolutionary context.

22

Third, a general power relationship between energy influx rate and body mass was
generated in eq. (1.12). When the power parameter  

3
, this relation is connected to
4

Kleiber's law because energy influx rate and BMR are in fixed proportion in many
cases(Rolfe and Brown 1997).
Fourth, it was shown in eq. (1.15) that the above pure energetic growth model
about allometric relations between energy dynamics and mass is interchangeable with a
general form of VBGF. Because VBGF is universally realized in various species, it is
suggested that corresponding allometric rules about body mass and energy should also be
quite universal, which has been shown by many studies (Hoppeler and Weibel 2005;
Marquet et al. 2005; Agutter and Wheatley 2004). This new theory then provides a new
universal basis for variability in allometric relationships between growth, mass, and
energy scaling of biology.
This unification of several different mechanistic growth models helps to clarify
apparent differences and universal energetic theme in general growth processes, such as
various VBGFs in the form of eq. (1.1) that are widely accepted in biology (Chen et al.
1992). But since traditional FBGF coefficient K can be decomposed into two variables in
the form of

Bc
 (1   ) , it is inappropriate to validate a pre-determined  value by a
Ec

statistical fitting of K like West et al. did (2001). It is then a challenge to determine which

 is more appropriate for a specific species from growth data only.
Further, Figure 1-1 indicate that if the commonly argued values of 2/3 and 3/4 for

 are considered, the (1 

Bc
) values would have a difference of less than 11% across
B

23

the growth processes. And for possible cumulative growth efficiency of larger than 20%
(Straile 1997; Reeve 1963), this difference would shrink to about 8%, which makes it
difficult to be detected. For example, according to eq. (1.14), this variance between
growth functions is the same as for the ratio between mass and maximum mass to have a
variance of 25% if  

2
3
to 35% if  
within a growth function. In practice, that
3
4

ratio could easily vary by 25-35%. For instance, Forsman (1991) found that maximum
size is usually affected by prey body size and prey availability, and extrapolated
maximum size is usually a function of growth curves which are affected by individual
growth history, geographic location, disease, reproduction and other stresses. Therefore,
it is really not easy to determine an exact  value from available growth or macro-level
measurements on individuals.
As cumulative growth efficiency was quantified in equation (1.6), smaller cell
maintenance cost

Bc
would lead to higher cumulative growth efficiency over a certain
Ec

period. Environmental factors such as cold temperature that may reduce maintenance cost
should able animals to attain higher growth efficiency and larger body size under similar
food resources. This conclusion might be underlying a phenomenon that most
ecototherms grow slower but reach a larger maximum size when growing in colder
temperature environments than in higher temperature ones (the “temperature-size rule” )
(Kozlowski et al. 2004; Atkinson 1994)).
Finally, the new theory of growth could provide a more mechanistic
understanding to other important aspects of biology that are related with growth, such as
reproduction, mortality, life span and other important life history traits (Reich 2001;
24

Shmidt-Nielsen 1984; Peters 1986; Niklas and Enquist 2001; Qu et al., 2004). This will
be shown in the next several chapters. Future studies also need to address an outstanding
question: why should energy influx rate or growth efficiency decline a certain amount
with each cell generation? Answers to this question may tap the more fundamental
secrets about longevity and growth than are hidden within cells.

25

REFERENCES

26

REFERENCES

Agutter, P. S. & Wheatley, D. N. Metabolic scaling: consensus or controversy? Theoretical
Biology and Medical Modelling 1, 13 (2004).
Atkinson, D. Temperature and Organism Size--A Biological Law for Ectotherms?
Advances in ecological research 25, 1–58 (1994).
Banavar, J. R., Damuth, J., Maritan, A. & Rinaldo, A. Ontogenetic growth
(Communication arising): Modelling universality and scaling. Nature 420, 626–626
(2002).
Chen, Y., Jackson, D. A. & Harvey, H. H. A comparison of von Bertalanffy and
polynomial functions in modelling fish growth data. Canadian Journal of Fisheries and
Aquatic Sciences 49, 1228–1235 (1992).
Cook, E. R. & Kairiukstis, L. A. Methods of dendrochronology: applications in the
environmental sciences. (Springer: 1990).
Dodds, P. S., Rothman, D. H. & Weitz, J. S. Re-examination of the ‘3/4-law’ of
metabolism. Journal of Theoretical Biology 209, 9–27 (2001).
Falster, D. S., Moles, A. T. & Westoby, M. A general model for the scaling of offspring
size and adult size. The American Naturalist 172, 299–317 (2008).
Forsman, A. Variation in sexual size dimorphism and maximum body size among adder
populations: effects of prey size. The Journal of Animal Ecology 253–267 (1991).
Fritts, H. C. Computer programs for tree-ring research. Tree-Ring Bulletin 25, 2–7 (1963).
He, J. X. & Stewart, D. J. Age and size at first reproduction of fishes: predictive models
based only on growth trajectories. Ecology 82, 784–791 (2001).
Hoppeler, H. & Weibel, E. R. Scaling functions to body size: theories and facts. Journal of
Experimental Biology 208, 1573–1574 (2005).
Karkach, A. Trajectories and models of individual growth. Demographic Research 15,
347–400 (2006).
Kozlowski, J., Czarnoleski, M. & Danko, M. Can optimal resource allocation models
explain why ectotherms grow larger in cold? Integrative and Comparative Biology 44, 480
(2004).

27

Makarieva, A. M., Gorshkov, V. G. & Li, B. L. Ontogenetic growth: models and theory.
Ecological modelling 176, 15–26 (2004).
Makarieva, A. M. et al. Mean mass-specific metabolic rates are strikingly similar across
life’s major domains: evidence for life’s metabolic optimum. Proceedings of the National
Academy of Sciences 105, 16994 (2008).
Marquet, P. A. et al. Scaling and power-laws in ecological systems. Journal of
Experimental Biology 208, 1749–1769 (2005).
McMahon, T. Size and shape in biology. Science 179, 1201–1204 (1973).
Naurzbaev, M. M., Hughes, M. K. & Vaganov, E. A. Tree-ring growth curves as sources of
climatic information. Quaternary Research 62, 126–133 (2004).
Niklas, K. J. & Enquist, B. J. Invariant scaling relationships for interspecific plant biomass
production rates and body size. Proceedings of the National Academy of Sciences 98,
2922–2927 (2001).
Nunn, C. L. & Barton, R. A. Allometric slopes and independent contrasts: a comparative
test of Kleiber’s law in primate ranging patterns. The American Naturalist 156, 519–533
(2000).
O’Connor, M. P. et al. Reconsidering the mechanistic basis of the metabolic theory of
ecology. Oikos 116, 1058–1072 (2007).
Peters, R. H. The ecological implications of body size. (Cambridge Univ. Press: Cambridge,
1986).
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mathematical modeling study. Journal of cell science 117, (2004).
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Reich, P. B. Body size, geometry, longevity and metabolism: do plant leaves behave like
animal bodies? Trends in Ecology & Evolution 16, 674–680 (2001).
Rolfe, D. & Brown, G. C. Cellular energy utilization and molecular origin of standard
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28

Straile, D. Gross growth efficiencies of protozoan and metazoan zooplankton and their
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(1938).
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29

CHAPTER 2. BIO-ENERGETIC EVOLUTION OF MATURITY:
THEORY, ANALYSIS AND VALIDATION IN FISH
Abstract
Theories of life history evolution describe how physiological and demographic traits of
an organism are optimized to produce the largest number of offspring that can reproduce
themselves (maximizing fitness). One of the traits that is the focus of many optimization
models is maturity in terms of age or size, because knowing how maturity is set by
evolution may help determine most life history traits such as growth, lifespan, mortality
rate and reproduction output (Charnov 1990; 1993; Wiersma et al. 2007; Roff 2007).
However, present life history models on maturity are mainly empirical and fragmented
(He et al. 2001; Charnov 2008), and a universal mechanism for detailing how maturity
evolved is still lacking. Here, a new life history theory that integrates principles of
bio-energetic growth and mortality processes with evolutionary optimization is advanced
to provide a mechanism for determining maturity. This new theory proposes that maturity
should occur when net growth energy and maintenance cost are at an equilibrium status,
and thus the integrated maintenance cost per mass till this point should be a defined
constant. One large growth and life history dataset for fish species provided support for
the new theory. Finally, it is also discussed how this new understanding of maturity may
help explain the variation of other life history traits such as lifespan, mortality, basal
metabolic rate, reproductive effort and body size.

Introduction
Maturity is the time or size when reproduction begins. The time at maturity is
measured as age, the size mass or length, and these two measurements are termed as age

30

and size at first reproduction, respectively. At maturity, the life history of an individual
goes from immature growth into a reproduction-and-then-death stage, which makes
maturity a critical turning point from an evolutionary perspective. Thus, understanding
how this turning point is determined remains an important subject in life history and
evolutionary studies (Charnov 1993; Kozlowski 1992).
Empirical studies indicate that maturity is closely related to other important life
history traits such as growth, life-span, mortality and body size. For example, He et al.
(2001) found that growth coefficients, such as the von Bertalanffy growth function
(VBGF) coefficient K and asymptotic body length (Linf) can empirically predict both age
and size at first reproduction. Beverton and Holt (1959) supposed that length at first
reproduction (Lr) is a fixed empirical ratio of Linf for a group of fish species. The age of
mammalian females at first reproduction (tr) is strongly positively correlated (R= 0.98)
with life expectancy at birth, a measure of mortality (Harvey and Zammuto 1985).
Charnov (2005) and Calder (1984) both theorize that tr and Lr are closely associated with
adult life-span, basal metabolic rate (BMR, a measure of the pace of growth activities),
and body size (M) in various forms. These empirical relationships have not been shown
to be robust or universal, and not much theory was developed to explain the underlying
mechanism that may help define and predict maturity in a mechanistic and robust way
and explain its complex correlations with other life history traits. Ecological and
evolutionary scientists then developed life history theory to fill these gaps gap (Stearns
1992; Roff 1992; Charnov 1993).

31

Life history theory deals with how evolutionary forces forged coordinated life
history traits such as growth, maturity, mortality, life-span and size for a species to
achieve maximum evolutionary success (Stearns 2000). This success is quantified by
fitness. Traditional fitness definition is the overall lifetime mature offspring number. The
maximization of this number is then defined as an optimization process that all living
organisms should have fulfilled over their long evolutionary time. Hence, maturation puts
a universal constraint on the variation and association of life history traits.
A typical optimization process in animals defines fitness as the product of
fecundity and survival rate. Fecundity is the lifetime offspring number reproduced and is
quantified as a function of body growth, survival rate is the rate of the offspring surviving
to its own reproduction stage and is empirically estimated. As growth potential declines
while mortality risk accumulates with age or size, maturity is a certain age when
immature growth should be transformed into reproductive growth to have the maximum
fitness. A resolution for maturity from such an evolutionary optimization process has
been presented in Roff's classical life history theory framework as following (1992 and
2007):

tr 

1
3K
 ln(
 1)
K
Z

(2.1)

where tr is maturity measured as age at first reproduction, Z is natural mortality (instant),
and K is VBGF growth coefficient. Obviously, tr is determined by growth and natural
mortality together.
When both K and Z are estimated from empirical growth functions and field data,
eq. (2.1) can provide quite a precise prediction about tr (Roff 2007). But several questions
32

complicate use of eq. (2.1). First, K and Z are unstable and hard to get empirical
parameters. In particular, though K has been thought as an empirical growth coefficient
with no explicit biological meaning, it has the tendency to increase with all factors
causing "stress" and an increase of oxygen consumption, such as increasing temperatures
(Pauly 1980). Similarly, while Z is defined as all mortality from physiological/genetic
reasons such as aging, disease, and the competition and predation without fishing, direct
reports of Z are scarce and various convenient estimations are used instead (Pauly 1980;
He et al. 2001). Second, there have been suggestions that K and Z should be biologically
connected in some way (Charnov 1993; Jensen 1996). For example, both K and Z have
the same unit of 1/t where t is time, both increase with stress and temperature, and
empirical studies often found stable linear correlations between them (Jensen 1996;
Charnov 1991). This induces a possibility of further simplifying eq. (2.1) to facilitate new
understanding. But questions on how the simplification should be and what’s the
mechanism for doing that are still unanswered. Lastly, observed relations between
maturity and many other life history traits as mentioned above have not been fully
incorporated into and then explained by eq. (2.1).
So, here I ask several questions: (i) can we have a biological meaning for K and Z?
(ii) how should they then be connected by possible mechanism? (iii) what's the new
simplified form of eq. (2.1) and its biological significance when K and Z are related? (iv)
how would the new form provide a unified explanation to maturity and its multiple
correlations with other traits?
In this chapter, a new life history theory is presented that begins with the concept
and quantification of fitness, then develops general bio-energetic and mechanistic growth

33

and mortality models, next establishes their relationship, and finally applies the fitness
optimization process to produce a unified bio-energetic function for determining maturity.
This function and a deduced constraint are both tested using a large fish dataset. It lastly
discusses possible mechanistic interpretations to many maturity-related correlations.

New Theory
Fitness concept and quantification
A common definition of fitness defines it as lifetime mature offspring number for
a species at a demographically equilibrium or stable state (Kozlowski 1996; Lande 1982;
Charnov 1990; Charnov and Skúladóttir 2000). From a bio-energetic perspective, it
means that, over its life-span, an individual should transfer as much reproductive output
(fecundity) to next mature generation as possible so that the energy flow within the
species will be maximized. In general, fitness can quantified as the product of fecundity
and the survival to a mature state (Roff 1992 and 2007; Roff et al. 2006; Stearns 1992):

R0 (t )  f (t )  l (t )

(2.2)

where R0 (t ) is fitness at time t, f (t ) is fecundity measured as reproductive output and

l (t ) is the possibility of the output surviving to maturity. Note that these three terms in eq.
(2.2) are all cumulative values because they are related with the fitness of an individual
during its entire life-span.
Fecundity is usually proportional to body mass M as c  M (Sopow and
Quiring 2004; Barnes and Hughes 1999), where c is a proportionality constant, so
fecundity is a function of mass over time (i.e., growth).

34

Bio-energetic size/mass growth
Living organisms have two contrasting functions for consuming the energy
acquired: growth (including reproduction) and maintenance. Growth is a process of using
assimilated energy to produce new tissues, including the reproductive tissue.
Maintenance is a process of maintaining the viability and effective functioning of whole
body, and represents the share of acquired energy that is dissipated irreversibly. Thus, the
energy budget of a living organism at time t can be generally quantified as (modified
from West et al. 2001):

BC

dM
 Em
dt

(2.3)

where B is the energy income, M is body mass, C is the corresponding maintenance
cost rate,

dM
is the new mass production rate, and Em is the energy required to
dt

produce one unit mass and is usually constant for a species(West et al. 2001).
If B and C vary very little with time, eq. (2.3) becomes a traditional
exponential growth equation, denoting that early growth follows an exponential trajectory
(Karkach 2006). However, actual rates of these parameters can vary with time, making
general sigmoidal curves more appropriate for describing growth. The VBGF is such a
curve and has been widely used as one of the best in describing the whole body growth
process (Chen et al. 1992). VBGF assumes that energy income rate (also “anabolism”) is
proportional to mass raised to certain power and maintenance cost (also "catabolism") is
linearly proportional to mass (von Bertalanffy 1938; Banavar et al. 2002; also see
Chapter 1), as follows:

35

B  B0  M 


C  Cm  M


(2.4)

where B0 is the initial energy income rate per mass

(1-  )

,  is a power

parameter that is less than 1, meaning a declining energy income per mass, and Cm is a
proportional constant that represents the maintenance cost per mass.
Taking eq. (2.4) into eq. (2.3), a general von Bertalanffy growth equation can be
expressed as:
B
C
dM
 0  M  m  M
dt
Em
Em

where

(2.5)

Cm
is the maintenance cost(energy dissipating rate) Cm per mass standardized
Em

by the energy contained within that mass, E m .
The integrative form of eq. (2.5) is:
Cm

 (1- )
t
M (1- )
Em
1- (
)
e
M max

(2.6)

where M max represents the maximum body size when

dM
0.
dt

C
Eq. (2.6) is a typical VBGF, with the term 1     m being VBGF growth
Em
coefficient K.
It can be deduced from eqs. (2.4) and (2.5) that:

(

M (1- ) C
)

M max
B

(2.7)

36

where

C
represents the relative proportion of energy income that is used for
B

maintenance.
Eq. (2.6) is used for fecundity quantification in eq. (2.2).
Bio-energetic survivorship
Because a standardized maintenance cost

Cm
Em

is an instantaneous rate

equivalent to the relative energy dissipating rate for any bio-energetic process within
living cells or tissues, it is assumed here that the standardized energy dissipation rate

Cm
is also the bio-energetic instantaneous mortality rate for an individual. This pure
Em
energetic definition of survival is derived from the perspective of energy production and
its sustainability, and can reflect the more general underlying origin of natural mortality
Z.
So, similar to a method in population ecology for determining cumulative survival
rate from instant mortality rate (Roff 2007), the cumulative survival rate of reproduction
energy output can be quantified as following:
C
 m t
l (t )  e E m

(2.8)

This cumulative survival rate means that the reproductive output itself also has a
maintenance cost or mortality rate until the next generation matures.
Fitness optimization
Substituting eqs. (2.6) and (2.8) into eq. (2.2), the optimization of equation (2.2)
for fitness maximization is realized when:

37

B  2C

(2.9)

The implication of eq. (2.9) is that when net energy income rate

dM
 Em
dt

(  B  C ) equals to energy maintenance cost (or “mortality rate” in analog sense) C , the
lifetime reproductive energy output to next generation of mature individuals should be
maximized and reproduction should ensue at this time. Before this moment, growth is
more energy-efficient because the net energy income per mass is larger than the possible
maintenance cost of that mass; after this moment, reproduction, and then death, is more
energy efficient because otherwise the possible maintenance cost would be larger than the
net energy income itself. Therefore, a new physiologically-based bio-energetic definition
of maturity is advanced here.
Also, a new rule is realized by substituting eqs. (2.9) and (2.7) into eq. (2.6):

1     Cm  tr  ln 2

(2.10)

Em

where tr is the time point when maximum fitness is achieved, which also defines the
optimal age at first reproduction.

1
C
Because 1     m is the VBGF coefficient K, tr   ln 2 occurs (from eq.
K
Em
(2.10)), which can be tested using real data if K and t r are both available for the same
species or population.

38

Table 2-1. Summary of parameters.
Parameters
Linf

Description
Asymptotic body length

Units
m

Lr

Length at first reproduction

m

tr

Age at first reproduction

year

K

VBGF growth coefficient

year 1

Z
t
Ro(t)
f(t)
l(t)
M

Natural mortality (instant)
Age
Fitness at age t
Fecundity at age t
Survival rate at age t
Body mass or size at age t

c

Proportional constant

B

Energy income rate

C

Maintenance cost

Em

Energy for production of per new mass

B0

Unitless

Cm

Energy income per mass at time zero
Power parameter
Maintenance cost per mass

Mmax

Maximum body size

kg



year 1
year
number
number
Unitless
kg
1

1  kg

J  s 1
J  s 1
J  kg 1
J  kg (1 )  s-1
J  kg 1  s-1

Methods
Application to fish
For fish, L  b  M  , where  is a power parameter, L is body length, and b
is a proportional constant. At the time of maturity, since

C
 0.5 , eq. (2.7) can be
B

written for fish as:


Lr
 0.5（1-）
Linf

(2.11)

39

Empirical studies showed that 1 -   and  both vary within the similar narrow
range of 1/3 to 1/4 (Enquist et al. 1999; Essington et al. 2001, Roff et al. 2006 and 2007;
Lester et al. 2004). Assuming a value of 1/3 for all fish (He et al. 2008; Kalaycı et al.
2007):

Lr
 0.5
Linf

(2.12)

Fish dataset
Abundant life-history data are available for many fish species, especially for
commercially important ones. Fish data are usually fitted with the VBGF, and the critical
parameter-VBGF coefficient K is reported together with other life history data (Essington
et al. 2001). He et al. (2001) collected such a comprehensive data set, including VBGF
coefficient K, tr , Lr , and Linf , for 215 fish species from published studies; this data
set was used to test the new theory.
Data analysis
The R package "smatr" for reduced major axis (RMA) was used for estimating the
relationship between tr and K, because it is appropriate when variance in dependent and
independent variables are similar and the goal is the regression slope (Warne and
Charnov 2008; O’Connor et al. 2007), as was the case here. Because maximum or
asymptotic size is more precisely measured when compared to size at first reproduction,
OLS was used for regression between length at first reproduction and maximum length.
Data points with K  1.0 were not used (7 data points excluded in total) because
the theory confines relative maintenance cost to be less than 1 and very high mortality
rate as represented by K might not be due to physiological mortality defined above, and
40

only female data points were used as males and females are following different
evolutionary routes (Roff 1992). After analyzing the available data with these constraints,
it was determined that 78 points were available for testing the correlation between K and

tr , and 85 points were available for length correlation analysis.

Results
1 The relationship between Lr and Linf
A statistically significant strong correlation between Lr and Linf was found
2

with a R of 0.75 (Figure 2-1 and Table 2-2). The fitted slope of 0.49 was very close to
the predicted one of 0.5, and its 95% confidence interval (CI) of 0.43 to 0.56 includes 0.5,
indicating that eq. (2.12) can produce a good estimate of Lr if Linf

is reliably

measured. However, the positive intercept for the fitted line is not zero, indicating a
possible “nugget effect” that might cause the proposed model to underestimate actual

Lr .

41

200
150
100
0

50

Lr

0

100

200

300

400

L i nf
Figure 2-1. The relationship between Lr and Linf. Lr: length at first reproduction, Linf:
maximum length. n=85, females only. Red line is the theoretical line with a slope of 0.5
and zero intercept, blue line is the fitted line with parameters in Table 2-2. Data is from
He et al. (2001).
2

Table 2-2. Parameter values for Figure 2-1. R is Pearson’s correlation coefficient.

intercept
slope

Predicted coefficients
0
0.50

Fitted coefficients*
7.42 (2.45,12.38)
0.49 (0.43,0.56)

R

2

0.75

*number in parenthesis is the 95% lower and upper confidence limit, respectively.

42

2 The relationship between tr and K
2

The regression between tr and K showed support for eq. (2.10), with a R of
0.70 (Figure 2-2 and Table 2-3). The actual fitted slope of -1.06 (95% CI is -0.93 to -1.20)
was not different from the expected -1. More importantly, the predicted intercept of 0.69
(= ln2) was included in the back-transformed 95% CI (0.66=10

-0.18

, 1.10=10

0.04

) of the

fitted intercept. However, the most likely value for the intercept (the mean) was 0.85,
which means that the theoretical value is at the lower bound of actual values (see
discussion).

43

0.0
-0.5
-1.0
-1.5

Log K

0.0

0.5

1.0

1.5

Log T r
Figure 2-2. Relationship between log(K) and log(Tr). K is the von Bertalanffy growth
coefficient. Tr (annotated as tr in the text) is age at first reproduction, n=78 (sample size),
females only. The red line is the predicted line with slope of -1 and intercept of
-0.16(which equals to log (ln2)), and the blue line is the fitted line with parameters in
Table 2-3. Data from He et al. (2001).
Table 2-3. Parameter values for Figure 2-2. The parameters are from the RMA procedure.

Intercept
slope

Predicted coefficients
-0.16
-1

Fitted coefficients*
-0.07 (-0.18, 0.04)
-1.06 (-0.93,-1.20)

2

R
0.70

*number in parenthesis is the 95% lower and upper confidence limit, respectively.
44

Discussion
The theory described here provides a new framework that unifies previous
empirical and theoretical understanding of maturity and the maturity-related life history
rules. It thus may greatly advance our understanding of the evolution of maturity.
Table 2-4. Comparisons between the new theory and the existing rules*. Each is discussed,
in turn, below.
names of new rules
1 Natural mortality is
standardized
bio-energetic
maintenance cost
2 Energy equilibrium at
maturity

new life history
theory
C
Z m
Em

current rules and the reference
second Beverton and
Holt
invariant
(Beverton and Holt
1959).

Z / K  c1

B  2C at maturity

None

3 Length at maturity
and asymptotic length

1
Lr   Linf
2

third Beverton and
Lr / Linf  c2 Holt
invariant
(Beverton and Holt
1959).

4 Metabolic rate and
body size

1
 B0  M  ,
2
for mature
individuals only.

5 RE

RE  BMR

3
4
RE  M

6 Life history invariant
at maturity

1     Cm  tr  ln 2

Z  tr  c3

BMR 

Em

*

BMR  M

3
4

Kleiber’s law (West et
al. 2001)
Charnov et al. 2001b;
Charnov 2001a
first Beverton and
Holt
invariant(Beverton
and
Holt
1959;Charnov
1991b,1993
and
2001b)).

RE is reproductive effort, which is defined as the reproductive energy (or mass)
devoted to reproduction, c1 , c2 and c3 are undefined constants that are specific for
certain populations, and other symbols and parameters are as mentioned in the text.

45

1 Natural mortality is standardized bio-energetic maintenance cost
Fecundity and natural mortality are two interconnected themes of living
organisms, which are inextricably linked to fitness measures. Fecundity is usually
proportional to mass which is easy to measure and describe (Roff 2007). As Einstein
noted (Einstein 1935), mass and energy are essentially the same, so the VBGF , a widely
used growth equation, can be easily combined with the first principle of energy
conservation (eq. (2.3)) to have a bio-energetic mechanism for it, as presented here.
Natural mortality, on the other hand, is previously viewed as an important empirical
parameter, though it has some kind of intrinsic target range or value (Hewitt and Hoenig
2005). The present study provides a new definition for natural mortality: the standardized
maintenance cost of energy. This new definition can be considered as an inherent
energetic characteristic. This same energy depreciation rate is also explicitly present in
the growth process, defining a factor limiting growth rate. It is then suggested that the
finest level maintenance cost probably occurs intrinsically at cellular and tissue level, but
it has an individual-level consequence: it may also determine an individual or
population's natural mortality Z.
These new definitions simplify life history modeling and produced important life
history constraints that can be validated by real data, as shown above. Therefore,

Cm
Em

might be a better proxy of Z than traditional population mortality rate. From

C
K  (1   )  m it is obvious that K  (1   )  Z . This may provide a mechanistic
Em
reason for the previously-viewed coincidence that Z has the same dimension of 1/time as

46

K and also for the “second Beverton and Holt invariant” which states a statistical
relationship between Z and K (Beverton and Holt, 1959; Charnov, 2008).
2 Energy equilibrium at maturity and its significance
The new theory suggested that maturity means a special energetic relationship
quantified by eq. (2.9), which leads to three important consequences:
2.1 maturity length is half of asymptotic body length
Measured maturity length is half of asymptotic body length just as predicted
(Figure 2-1 and Table 2-2). However, there was a “nugget effect”, which represents a
non-zero intercept value. This might be that one very large maximum length (Linf) value
of over 300 (cm) weighed too much. Excluding this value reduced the intercept to zero,
and the slope was still close to 0.5 for fish.
Beverton and Holt (1959) proposed the concept of "third Beverton and Holt
invariant" which states that Lr is certain fixed ratio of Linf. This was accepted by Charnov
(1993) as one of the life history invariants. Jensen (1996) further provided an explanation
to it by assuming that reproduction should occur at the time of maximum mass
production rate. However, his explanation is itself based on an assumption that was not
validated. In contrast, the life history theory in this chapter deduced a definite
evolutionary constraint for length at maturity, and the exact value for the ratio is
established as 0.5.
2.2 BMR and M are related by life history evolution
The new relationship in eq. (2.9) means that there is a fixed energy relationship
between maintenance cost (C) and energy income (B). Because B is empirically taken by

47

VBGF to be proportional to body size (M) raised to a power of  (see eq. (2.4)), and

C

is quantified mainly as BMR for animals, a parameter that measures the energy used to
just maintain basic physiological activities (Rønning et al. 2007; Mckechnie and Wolf
2004), there should also be a relationship between BMR and M among species in the

1
form of BMR   B0  M  .
2
The exact value of this power parameter  may vary, but two widely used
2

3

dm
dm
VBGF has the following forms:
 a1  m 3  b1  m and
 a2  m 4  b2  m ,where
dt
dt
a1, b1, a2 and b2 are proportional coefficients. Therefore, 2/3 and 3/4 are two possible
values of  . Previous studies found that BMR and M really has a power relationship
with power parameter ranging in 2/3, 3/4 or values between and around them (O'Connor
et al. 2007; White et al. 2003).
Specifically, because  

3
is a convenient average of possible values (Glazier
4

3
4 , is called Kleiber's
2005; Dodds et al. 2001), this special form of relation, BMR  M

law (West et al. 2001). This law is deemed by some scientists as universal and attracted
many recent research interests (Kozlowski and Weiner 1997; West et al. 1997; Beuchat
1997; Enquist et al. 1999). The new relationship shown above suggests that Kleiber's law
is only the average case for a diverse situation, and its validity lies in life history
constraints that are reflected in the form and validity of VBGF. So these two quite
general biological patterns are actually from the same origin (see Chapter 1 for further
study).

48

2.3 Reproductive effort (RE) is proportional to BMR
If the net growth energy is to be distributed all (determinant growth) or
proportionally (indeterminate growth) to reproduction at maturity, reproductive effort
(RE) would be proportional to net growth energy that is equal to maintenance cost for
mature individuals. As maintenance cost is BMR in animals, which itself proportional to
M raised to certain powers such as 2/3 or 3/4, RE would also be connected with M raised
to a power. The new theory hence defines mechanistic relationships among RE , BMR

and M. Charnov et al. found that RE is proportional to M

3
4 for mammals(Charnov et al.

2001; Charnov and Ernest 2006; Charnov 1993), which supports this new relation for RE.
3 Life history invariant at maturity rule may define a fast-slow life history spectrum
The energy equilibrium status at maturity combined with universal growth
equation (2.6) further defines a new life history invariant rule, which is eq. (2.10). This
expected relationship between tr and K in eq. (2.10) was supported by fish data (Figure
2-2 and Table 2-3), supposing a universal constraint that relates maturation with the pace
of life for animals.
Further, if   2 / 3 in many cases (Glazier 2007), then

Cm
 tr  2.08 .Because
Em

Cm
is also defined as mortality, there is a chance to experimentally test this relation.
Em
Two studies showed the product of mortality and age at first production t r to be 2.1 for
parasitic nematodes and Pandalid shrimp (Charnov 2001) and 2.2 for some other groups
(Charnov 1991). This relationship between mortality and age at first reproduction was
also empirically recorded as “the first Beverton and Holt invariant” (Beverton and Holt

49

1959; Charnov 1991, 1993 and 2001). However, since they did not establish the value of
the product and a mechanism as in eq. (2.10)., the “invariant”, up to this point, is only an
empirical observation.
Eq. (2.10) also defines a possible "fast-slow life history continuum" through
relationships between maturity and many other aspects of life. For example, larger t r or
smaller

Cm
means later age at first reproduction and/or longer life-span, lower
Em

mortality rate, lower BMR (a slower life pace because any fast-pace physiological activity
would require higher metabolic rate), and relatively lower RE per mass (  M ( 1) ),
while smaller t r or larger

Cm
represents a faster life pace and other corresponding
Em

life history syndrome. This "fast-slow life history spectrum" was empirically observed
(Bronikowski and Arnold 1999; Wiersma et al. 2007a). For example, it has been shown
that tropical birds have later maturation, slower growth and metabolic rates, lower
fecundity and higher adult survival rates than their temperate counterparts (Wiersma et al.
2007), while shorter / smaller species of fish have earlier maturation, faster growth and
metabolism, higher fecundity and lower adult survival rate than longer ones (Roff 2007).
It is also worth discussing that the predicted value of 0.69 was not very close to
the mean of 0.85 from real data (Figure 2-2 and Table 2-3). This means that the actual
product of tr and K is higher than predicted, and then either t r or K or both tend to be
measured higher. This is possible due to several reasons. First, the detection probability
for capturing a first reproduction is pretty less than 1, and later detection is more likely to
result in an increased age at first reproduction. Second, the estimation of K is negatively
related with asymptotic size (Munro 1982), while asymptotic size is estimated by
50

maximum size when growth rate equals to zero under quite ideal natural
conditions(Stamps et al. 1998; Hopkins 1992; Jennings et al. 1999). However, this
maximum size when using observed maximum value will be a minimum possible
asymptotic value due to detection problem and growth retardation from mostly inevitable
environmental stresses and reproduction, leading to a underestimation of asymptotic size
and overestimation of K. Therefore, the expected product value 0.69 of t r and K should
be a lower bound of the one in real world. This might explain the difference between 0.85
of real data and the expected 0.69.

Conclusion
In summary, the new theory suggests that a standardized bio-energetic
maintenance cost is equivalent to the natural mortality rate Z, and the VBGF growth
coefficient K equals to (1   )  Z , where  is a power parameter connecting energy
income rate with body mass. Therefore, eq. (2.1) can be simplified into interchangeable
eqs. (2.9) or (2.10). These new equations are universal bio-energetic and evolutionary
constraints. They contain a physiological and bio-energetic determination of maturity in
eq. (2.9), and a eq. (2.10) that can further help explain most observed life history
associations between maturity and other traits. In this way, previously empirical life
history rules were explained with coherent and consistent mechanisms.
It is striking that Z is a key parameter identifying a species' characteristics and life
history. This parameter encapsulates concepts from growth and energy metabolism to
reproduction and evolution, and can be used to explain why growth, energy metabolism,
reproduction and life history are all closely associated with each other to produce diverse
life history syndromes or niches among a vast continuum of biological taxa. The work
51

presented here could be a demonstrated research approach that starts from macro-level
phenomena, such as life history diversity, to arrive at micro level mechanisms, such as
cell metabolism dynamics.

52

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53

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CHAPTER 3. A NEW EVOLUTIONARY LIFE HISTORY THEORY
FOR LEAVES CAN EXPLAIN THE VARIANCE OF LEAF
PHOTOSYNTHETIC RATES AND CORRELATION WITH LEAF
TRAITS
Abstract
The fundamental physiological processes in leaves involve two opposing functions:
energy production by photosynthesis and energy dissipation from leaf maintenance costs
such as dark respiration, mass investment, and protein turnover. Here, a mechanistic
understanding of how these two functions of a leaf are optimized to have maximum
energy output over its lifespan can advance out knowledge of plant ecology and evolution
is provided. Specifically, the new theory predicts that the maximum energy output over a
leaf’s lifespan is achieved when there is an energetic equilibrium between net
photosynthetic rate and leaf maintenance cost for mature leaves. Leaf trait data for C3
species from the Global Plant Trait Network (GLOPNET) are used to test this prediction.
The result showed that 87% (mass-based) and 58% (area-based) of inter-specific
variation in leaf photosynthetic rates can be explained by the predicted relationship.
Further analyses on sub-groups of species suggest a leaf life history continuum that runs
from “nitrogen-driven fast photosynthesis” type to “carbon-driven slow photosynthesis”
type. The results also indicated that climate moderately affects the predicted
relationships.

Introduction
Leaves are the most evolutionarily conserved functional units of life for diverse
plant species that span eleven orders of magnitude in size (Niklas and Enquist 2001). In

58

particular, leaves undertake the fundamental processes of leaf energy metabolism
(photosynthesis and respiration), gas and water vapor exchange between plants and
environment, and material storage of nutrients, water and food for plants(though there are
a few plants which have lost their capacity for photosynthesis and persist in mutualistic
relationships with other plants, e.g., Epifagus viginiana). So, while plants are very
different in appearance, their leaf is a proxy of the plant in many critical functional
aspects, and there are theories that view the plant as tube or pipe bundles that begin from
the root and end in the leaves where various energetic, hydraulic and nutritional processes
are undertaken and/or modulated (the "pipe model"; see Shinozaki et al. 1964a and 1964b
and Waring et al. 1982).
Leaves are also the basic compositional units of a plant species. For example,
flowers and fruits are transformed leaves from a phylogenic perspective (Eames 1961),
and other parts of plants such as stems, roots and branches are extended tissues of leaves
for the function of mechanical support, water and nutrient exchange, and energy storage
(Evert and Eichhorn 2004).
A leaf has its own life history that may be different than that of the whole plant: it
grows as described by various sigmoidal growth equations, then reaches a stage where its
size and form are stabilized and fixed (i.e., it exhibits determinate growth), and finally
dies after a functioning period of photosynthesis (A), which captures light energy through
carbon dioxide, oxygen and water vapor exchanges between cells and atmosphere, to
finish what is called a leaf life-span (LL) (Groot and Meicenheimer 2000). The carbon
energy a leaf harvested from photosynthesis has a maintenance cost that includes
respiration (R), leaf structural mass investment per area (LMA), and leaf nitrogen (N)

59

related maintenance cost as N is critical to important photosynthetic proteins such as
Rubiscos that are central in most metabolic processes within plant tissues (Wright et al.
2004; Atkin and Tjoelker 2003; Westoby et al. 2002). But these life history traits of
leaves still represent important whole plant characteristics, indicating that the leaf is in
certain extent a proxy of plants. For example, thin leaves with high photosynthetic rate
are more easily decomposable, compared to think leaves (Santiago 2007). Photosynthetic
rate, LMA, R, and N were found to be positively correlated with RGR (R= 0.90), which is
negatively correlated (R=-0.9) with LL (Reich et al. 1998a and 1998b; Poorter and
Garnier 1999; Niinemets 2001; Poorter and Bongers 2006).
Life history traits of leaves are tightly correlated with each other. For example, R
is critical to net assimilation rates of carbon dioxide (Hagihara and Hozumi 1991;
Sprugel et al. 1995), a high LMA is usually related with longer LL, lower N and lower A
(Wright et al. 2002), and A is strongly correlated with SLA, LL, and N (Reich et al. 1997;
Shipley et al. 2006). When these leaf trait variation and associations are considered
together across the whole plant biota, there is a universal leaf economics spectrum
running from leaves with low A, long LL, high LMA, low N, and low R to leaves with
opposite traits syndrome (Wright et al. 2004). Therefore, there seems to have universal
constraints on bi- and multiple-leaf trait associations. But a mechanistic understanding to
what the constraints are, why and how they are evolved for all plant leaves is still lacking
(but see Shipley et al. 2006).
This challenge of understanding leaf-level trait variation and associations, as well
as the functional and compositional importance of leaves as proxy of plants (Reich 2001),
lead to the new idea here that as the plant is often clonal or modular with a population of

60

leaves with axillary bud and the attendant stem internode (Begon et al. 1996;
Vallejo-Marín et al 2010; Herben 1994), selective evolutionary processes may have
worked on leaves quite directly so that there should be a life history theory for leaves.
Life history theory is centered at the axiom that natural variation and patterns of
traits, such as those related to development, growth, reproduction and mortality, are
constrained by evolutionary optimization, which is the maximization of fitness measured
as lifetime number of mature offspring. Though this theory has been successfully applied
to the animal kingdom to provide a coherent and tractable mechanistic understanding of
how individual trait variation and the tightly correlated trait syndrome among species are
forged by evolution (Roff 1992; Stearns 1992; Chapter 2), it is suggested that the life
history theory for unitary animals can't be directly applied to plants as they are modular
(Vuorisalo and Mutikainen 1999; Stearns 1992). Thus, the above new idea also represents
a new way for solving this difficulty. However, there has been very little research, to our
knowledge, on how a life history theory for leaves should be developed. So the two
specific questions are: (1) Can we have an evolutionary life history theory for leaves? (2)
How would such a theory explain leaf life history, especially the variance of
photosynthesis and correlation with other traits?
In this chapter, the above two questions are answered. First, a new and detailed
life history theory for leaves is advanced. Second, this new theory is tested using the
Global Plant Trait Network, or GLOPNET, database that contains field measurements of
critical leaf traits and the associated climate factors for many species across the globe
(Wright et al. 2004). The test results are analyzed and climate effects are also studied.

61

Finally, there is a discussion on the ecology and evolutionary significance of the new
theory.

Theory
Fitness quantification
If a leaf is considered as an evolutionary selection unit like an individual
organism, it must survive and reproduce and thus its evolutionary fitness can be described
by the lifetime net output of assimilated energy standardized as the potential number of
mature offspring (leaves/seeds) it produces. This definition is similar to the one for
animals, where the fitness is quantified as the lifetime number of mature individuals
(Roff 1992 and 2007; Stearns 1992). Under this definition, the fitness of a leaf can be
quantified as the product of leaf fecundity and its survivorship to the leaf maturation
stage, when the maximum fitness of a leaf is achieved and it's then optimal to transfer the
net assimilated energy from its own growth to next generation in the form of assimilated
energy:

R0 (t )  f (t )  l (t )

(3.1)

where R0 (t ) is the leaf fitness, f (t ) is leaf fecundity measured as net output of
assimilated energy and l (t ) is its possibility of surviving to maturity. Note that these
three terms in eq. (3.1) are all cumulative values because they are related with the fitness
of a leaf over its entire life-span (i.e., t is the time since the leaf bud breaks).

62

Net output of assimilated energy is proportional to leaf mass
Studies have demonstrated that f (t ) is usually proportional to mass (M) (Sopow
and Quiring 1998; Barnes and Hughes 1999). Hence, there should be a descriptive
function to model leaf mass variation with time, that is, a leaf growth function. Since leaf
function is mainly quantified in energy terms, the leaf growth equation is to be modeled
as a bio-energetic equation incorporating the dynamics of energetic traits such as
photosynthesis, respiration, and other maintenance energy costs. Photosynthesis is the
energy income-generating process of a leaf, and there are two contrasting channels for
consuming this generated carbon energy: growth and maintenance.

Growth uses energy

and materials to produce new tissues and organs and maintenance maintains the viability
and effective functioning of existing leaf tissues. In particular, the maintenance cost
represents the assimilated energy that is dissipated irreversibly in the form of degradation
or death of certain living cell parts or whole cells. Thus, the energy budget of a leaf at
time t can be modeled as following (modified from West et al. 2001):

BC

dM
 Em
dt

(3.2)

where B is the assimilated energy income of a leaf from photosynthesis, C is the
maintenance cost, Em is the energy required to reproduce one unit leaf mass, and

dM
dt

is the rate of new leaf mass production.
If B and C vary very little with time, we would have a traditional exponential
growth equation.

Studies have shown that early leaf growth usually follows an

exponential trajectory (Groot and Meicenheimer 2000; Lamoreaux et al. 1978). However,
overall leaf photosynthetic rate B shows a declination with time (Kitajima et al. 2002;

63

Kikuzawa and Ackerly 1999). Since leaves are determinate growers, general logistic
functions, such as the von Bertalanffy growth function (VBGF), are appropriate in
describing the mass growth process of leaves (Jackson 1984; Yin et al. 2003; Karadavut
et al. 2010; Kirby 1988; Dennett et al. 1978).
The VBGF assumes that energy income rate (also “anabolism”) is proportional to
mass raised to certain power, and maintenance cost (also "catabolism") is linearly
proportional to mass (von Bertalanffy 1938; also see Chapter 1). In terms of a leaf, these
assumptions can be written as:
B  B0  M 


C  Cm  M


(3.3)

where B0 is the initial photosynthetic rate at time 0, M is leaf mass,  is a power
parameter that typically varies between 2/3 and 3/4 (Banavar 2002; Dodds et al. 2001;
O'Connor et al. 2007), and Cm is the maintenance cost per leaf mass, usually a constant
for a specific-taxa (West et al. 2001).
Taking eq. (3.3) into eq. (3.2), a general VBGF is:

C
dM B0

 M  m  M
dt
Em
Em
where

(3.4)

Cm
is the maintenance cost (energy dissipating rate) per mass standardized by
Em

the produced energy per mass.
The integrated form of eq. (3.4) is:
Cm

 (1- )
t
M (1- )
Em
1- (
)
e
M max

(3.5)

64

where M max represents the maximum leaf size when

that (

M (1- ) C
, where
)

M max
B

C
B

dM
 0 , and it can be deduced
dt

represents the relative proportion of leaf

photosynthesis that is used for maintenance.
Eq. (3.5) is a general bio-energetic mass growth equation (but still a VBGF) for
leaves, and is used for fecundity quantification.
Survivorship of net energy output to maturity
The survivorship of the net output of assimilated energy to next mature generation
can also be defined in energetic terms just as (f(t)) was (eq. (3.5), above). From eq. (3.4),
any mass production process should have a standardized maintenance cost

Cm
, which is
Em

the instantaneous energy dissipating rate relative to the produced energy, just like
mortality rate for individuals. So the cumulative survival rate of net output can be
quantified analogous to a method in population ecology for determining cumulative
survival rate from instantaneous mortality rate:
C
 m t
l (t )  e E m

(3.6)

where l (t ) is the cumulative survival rate of net output of assimilated energy to a
certain time t in a leaf's life-span.
Fitness optimization for leaves
Taking eqs. (3.5) and (3.6) into eq. (3.1), the optimization of equation (3.1),
which means the maximization of leaf fitness, is achieved when:
B  2C

(3.7)
65

The implication of eq. (3.7) is that when net energy income rate (B-C) =

dM
 Em (i.e. the growth rate of mass) equals to the energy dissipating rate, (i.e., the
dt
maintenance cost rate of mass), the lifetime net leaf energy output to next mature
generation should be maximized, and a leaf should at this time stop its own growth (thus
determinant growth) and transport all available energy to produce new leaves or flowers
and seeds. Thus, an evolutionary definition of leaf maturity in terms of bio-energetics is
clearly specified by eq. (3.7). This equation can be tested with life history trait data from
mature leaves.

Data and methods
GLOPNET dataset
Measurements of five critical leaf traits- LMA, LL, A mass or Aarea , and Rdmass
or Rdarea , N mass or N area were reported in the Global Plant Trait Network
(GLOPNET) dataset, which includes 2,548 species from 219 families, representing 175
sites distributed from the arctic to the tropics (Wright et al. 2004). 196 species had all five
traits measured, but 10 of them were C4 plants, i.e., do not use C3 carbon fixation as their
metabolic pathway for carbon fixation in photosynthesis. Given the small sample size
available for C4 plants and their very different photosynthetic physiology, only 186
species of C3 plants were used as the total sample frame for this study. Information on
growth form, habitat type and climate for each species was given in GLOPNET as well.
Three sets of sub-grouping were used to examine the predicted relationship of eq. (3.7) in
them: (1): grass/herbs vs shrub/trees, (2): evergreens vs non-evergreens, and (3):

66

temperate forest vs woodland species. Only seven species from the tropical rain forest
ecosystems were available, so data from this category were not analyzed as a sub-group
separately.
Analytical Methods
A mechanistic model was developed to analyze the data in terms of the presented
theory (Figure 3-1). Leaf energy influx was partitioned into two parts: the “working” part

B for growth and the “lost” part C for maintenance. While B can be well quantified
by photosynthetic rate (red arrow, Figure 3-1), the maintenance energy C was not
available for direct measurement, so it had to be represented using a conceptual model
(blue arrows, Figure 3-1): C is composed of a respiration cost for mobile tissue content
(represented by leaf dark respiration rate) and a structural maintenance cost for mobile
and immobile tissues. The latter is further decomposed into a lifetime mass investment
cost for immobile cell parts (structural carbohydrates) and a replacement cost that
compensates for the decay of mobile cell parts or critical live macromolecules, such as
DNAs, RNAs and proteins. The lifetime mass investment can be represented by leaf dry
mass since the mobile components of a leaf should be either eliminated or negligible in
dry status, and this investment should be divided by leaf life-span to become a cost of
rate because the immobile leaf dry mass functions over the whole life-span of a leaf. The
replacement cost for the decay or turn-over of mobile cell parts was assumed to be
proportional to nitrogen content, because nitrogen is the most important element in many
mobile cell proteins and other structures (Wright et al. 2004; Loomis 1997).

67

Photosynthetic rate ( energy influx rate)

Respiration rate

Growth rate

Lifetime mass investment cost

Cell or parts
replacement cost

equal
Maintenance cost (energy dissipating rate)
at maturity

Figure 3-1. An illustrative figure showing how eq. (3.7) is formulated in plant leaves as
shown in the text according to energy decomposition principles.

The GLOPNET variables were related to the conceptualization of eq. (3.7) in
Figure 3-1, so that the data could be fit to the new model. An area-based bio-energetic
relationship from eq. (3.7) is first derived according to Figure 3-1:

Aarea  2  Rdarea  Ka  LMA / LL  ba  N area
where Aarea is

(3.8a)

photosynthetic rate per area of leaf, Rdarea is dark respiration rate

per area, LMA/ LL represents the lifetime structural mass investment per area (LMA)
distributed over entire leaf life-span (LL), Narea represents leaf nitrogen content per area,
Ka is a parameter for the conversion of each unit structural mass investment into energy

68

cost, and ba is a proportional constant representing the replacement cost related to per
unit nitrogen content.
By dividing both sides of eq. (3.8a) with LMA, a mass-based equation is:

Amass  2  Rdmass  K m / LL  bm  N mass

(3.8b)

where Amass is the photosynthetic rate per unit leaf mass, Rdmass is the dark
respiration per unit leaf mass, Km converts one unit mass of structural mass investment
into energy units, Nmass represents leaf nitrogen content per mass, and bm is a
proportional constant relating replacement cost with per unit nitrogen content of leaves.
Because leaf nitrogen content is usually correlated with leaf lifespan (Wright et al.
2004; Reich et al. 1998a), two more models that did not include nitrogen content were
added to compare with eqs. (3.8a) and (3.8b) to see if simpler models were sufficient:

Aarea  2  Rdarea  Ka  LMA/ LL

(3.8a-1)

Amass  2 * Rdmass  Km / LL

(3.8b-1)

Model combination with climate factors
Leaf physiology is known to vary with climate as a well as from intrinsic
differences between species. Therefore, the models needed to account for possible effects
from climate. Five key climate factors were available from the GLOPNET data: solar
radiance (RAD), potential evapo-transpiration (PET), mean annual temperature (MAT),
vapor pressure deficit (VPD), and mean annual precipitation (MAP). To determine which
climate variables should be included in the model, generalized linear regression analysis
was conducted relating each of the five climate factors to photosynthetic rate (Table 3-1).

69

Table 3-1. Summary of regression analyses of photosynthetic rate in relation to 5 climate
factors.
2

R
Amass

Intercept

RAD

PET

0.26

489.0 ***

-2.87 *** -0.002

VPD

MAT

MAP

18.48

4.51*

0.004

Notes:*** P<0.00, ** P<0.001,*P<0.01.
The result suggested that RAD and MAT were significantly correlated with Amass ,
so these two climatic variables were added as linear terms to produce new equations
(3.8a-2) and (3.8b-2), assuming that climate effects are linear and additive effects and
that the distributions of these two climate factors across sites are approximately normal
(Wright et al. 2007). Notice that simpler eqs. (3.8a-1) and (3.8b-1) are not used in
combination with climate factors, the reasons are presented in the result section.

Aarea  2 * Rdarea  Ka  LMA / LL  ba  N area  K1 * MAT  K 2 * RAD (3.8a-2)
Amass  2 * Rdmass  Km / LL  bm  N mass  K 3 * MAT  K 4 * RAD

(3.8b-2)

where K1, K2, K3 and K4 are four new proportional constants.
In order to determine if only one of these two climate factors should be
incorporated as they are often inter-correlated, two new equations were developed: MAT
only and RAD only, for both area- and mass-based metrics:

Aarea  2 * Rdarea  Ka  LMA / LL  ba  N area  K 5 * MAT

(3.8a-3)

Amass  2 * Rdmass  Km / LL  bm  N mass  K 6 * MAT

(3.8b-3)

Aarea  2 * Rdarea  Ka  LMA/ LL  ba  N area  K 7 * RAD

(3.8a-4)

Amass  2 * Rdmass  Km / LL  bm  N mass  K8 * RAD

(3.8b-4)

where K5, K6, K7, and K8 are four proportional constants.

70

Statistical analyses
The nls2 package in R (version 2.12.0) was used to estimate the coefficients in the
above equations using non-linear, least-squares regression. To further analyze possible
patterns of different sub-groups in driving photosynthesis (according to eq. (3.7)), three
additional parameters were calculated as ratios relative to photosynthetic rate: (1)

2 * Rdmass / Amass (dark respiration rate/photosynthetic rate), representing relative
respiration cost, (2) bm  N mass / Amass , representing relative replacement cost, and (3)

K m /( Amass  LL) , representing relative lifetime structural mass cost (all expressed on
mass-base for simplicity). T-tests were used to test for significant differences in these
three ratios between the paired plant sub-groups.
Reduced major axis (RMA) regression is used to test the relationship between the
predicted and measured values, because variance in dependent and independent variables
are similar and the goal is the regression slope (Warne and Charnov 2008; Swenson and
Enquist 2008; Wright et al. 2005).
An AIC difference of 4-7 units can indicate distinguishable models, and
differences of 10 units or more are considered to indicate a substantial difference
(Burnham and Anderson 2002; Rose et al. 2009). So, two models were considered
different when their AIC values differed by more than 10. For similar models, the ones
2

with higher R or fewer parameters were selected to observe the principle of parsimony.

71

Results
1 Best models with and without climate factors and their performance
For models without climate variables (the left side of Table 3-2), eqs. (3.8a) and
(3.8b) were better than eqs. (3.8a-1) and (3.8b-1) in terms of lower AIC value and higher
2

R . For models with climate factors incorporated (Table 3-2, right side), eqs. (3.8a-2) and
2

(3.8b-2) had the same R and slightly lower but not significantly different AIC values
than eqs. (3.8a-3) and (3.8b-3) respectively, so the latter two were considered better
2

because they have only one climate variable included. Because eq. (3.8b-4) had lower R
2

and significant higher AIC value than eq. (3.8b-3), and eq. (3.8a-4) had lower R and
similar AIC values than eq. (3.8a-3), they were viewed as generally worse than eqs.
(3.8a-3) and (3.8b-3) in performance and were both excluded from further analysis to
keep consistency between area- and mass-based metrics. The four best models (in bold,
Table 3-2) were further analyzed (Figure 3-2 and Table 3-3).
Table 3-2. Comparison of models
3.8a
2

R
AIC

3.8b

3.8a-1

3.8b-1

3.8a- 3.8b2
2

3.8a3

3.8b3

3.8a4

3.8b4

0.58
1092

0.87
1883

0.57
1172

0.84
1966

0.63
1024

0.63
1031

0.88
1872

0.61
1029

0.87
1883

0.88
1864

For all plant species pooled together, eqs. (3.8a) and (3.8b) both fit the data well,
2

but the area-based eq. (3.8a) with a R value of 0.58 explained 29% less variation than
2

the mass-based eq. (3.8b) with a R value of 0.87 (Figure 3-2 (a) and (c)). The RMA

72

regression slope from eq. (3.8a) is the predicted 1.00, and the intercept of -0.95 is close to
0 with confidence intervals (CI) of (-2.42, 0.51) including 0. The RMA regression slope
of 0.89 from eq. (3.8b) is close but slightly different from 1.00 (CI is from 0.83 to 0.96),
and the intercept is close but different form zero (CI is from 0.44 to 16.1), indicating a
possible "nugget effect" from the model. The discrepancy of slope from model prediction
would be removed if the intercept is set to zero, meaning the slight bias is from non-zero
intercept. This was further explored in the study of sub-groups that follows.
The addition of two climate variables in eqs. (3.8a-3) and (3.8b-3) had a small
2

2

effect on the R values, improving the fit of the models from a R of 0.58 to 0.63 for
area-based metrics and 0.87 to 0.88 for mass-based metrics, but it also produced a bias.
This bias is reflected in both the higher than zero intercept values of 2.27 and 17.5 and
the lower than 1.00 slopes of 0.78 and 0.83, respectively (Table 3-3). Specifically, the
predicted photosynthetic rates were overestimated for leaves with smaller photosynthetic
rates (less than 100 (mass-based) or 10 (area-based)), indicating a possible overestimation
2

trend at this range. Similarly, area-based eq. (3.8a-3) had a 0.25 less R than mass-based
eq. (3.8b-3) (also see Figure 3-2 (b) and (d)). This suggests that though area- and
mass-based equations were derived from the same mechanism, area-based metrics seem
to be less closely related with leaf energetics and ecology than mass-based ones, which
has also been supposed by other studies (Wright et al.2004; Atkinson et al. 2010). Hence,
only mass-based eqs (3.8b) and (3.8b-3) (in bold in Table 3-3) were selected to be
applied to sub-groups to further explore their performance.

73

10 15 20 25 30 35

b

0

5

Predicted Aarea

10 15 20 25 30 35
0

5

Predicted Aarea

a

0

5

10 15 20 25 30 35

0

10 15 20 25 30 35

d

300
200
100
0

100

200

300

Predicted Amass

400

c

400

500

Observed Aarea

0

Predicted Amass

500

Observed Aarea

5

0

100

200

300

400

500

0

Observed Amass

100

200

300

400

500

Observed Amass

Figure 3-2. Predicted vs. measured Amass for all species pooled together. (a) by eq.
(3.8a) . (b) by eq. (3.8a-3). (c) by eq. (3.8b). (d) by eq. (3.8b-3). The red line is the 1:1
line through the origin, the blue line is the RMA regression line between measured and
predicted values. All related parameter values are in Table 3-3.

74

Table 3-3. Parameter values for Figure 3-2.
Km
a 0.22(0.02)

***
b 0.21(0.02)

***
c 212(19)**

*

K5 for b and
K6 for d

RMA
Intercept
-0.95(-2.42,0.
51)

RMA Slope
1.00
(0.89,1.12)

2

R

0.47(0.22)

0.28(0.03)*

*

**

24.4(2.4)*
14.8(3.5)*

1.07(0.29)*

**

**

2.27(1.18,3.35 0.78
)
(0.69,0.87)

0.58/0.3
3
0.63/0.3
9

8.26(0.44,16.1 0.89(0.83,0.9
)
6)

***

**

d 238(20)**

*

bm
1.83(0.18)

0.87/0.7
6

17.5(10.5,24.5 0.83(0.78,0.8
)
9)

0.88/0.7
7

2

Notes: n=186, R is the correlation coefficient of the regression model / the RMA
regression, number in parenthesis is standard error or the 95% lower and upper
confidence limit. ***: <0.000, **<0.001,*<0.05.
2 Performance of mass-based models in three sub-groups
The comparison of model performance of eq. (3.8b) and (3.8b-3) in three sets of
sub-groups is analyzed. For grass/herbs, the slope and intercept from eq. (3.8b) was not
2

different from 1 and 0, respectively, with a R of 0.73, and those from eq. (3.8b-3) had a
2

slope significantly smaller than 1 and an intercept larger than 0 with slightly higher R of
0.75, similarly indicating a biased result from including climate in the model for species
2

with slower photosynthetic rates. For shrub/trees, there are higher R values of 0.88 and
0.89 for eqs. (3.8b) and (3.8b-3), respectively, but the slopes of 0.92 and 0.87 are slightly
or significantly different from 1 (CIs is from 0.85 to 0.99 and 0.81 to 0.93, respectively).

75

500
400
300
200

Predicted Amass

b

0

100

500
400
300
200
100
0

Predicted Amass

a

0

100

200

300

400

500

0

200

300

400

500

d

300
200
100
0

100

200

300

Predicted Amass

400

c

400

500

Observed Amass

0

Predicted Amass

500

Observed Amass

100

0

100

200

300

400

500

0

Observed Amass

100

200

300

400

500

Observed Amass

Figure 3-3. Predicted vs. measured Amass for grass/herbs by (a) eq. (3.8b) and (b)
eq.(3.8b-3),and shrub/trees by (c) eq. (3.8b) and (d) eq.(3.8b-3). The red line is the 1:1
line through origin, the blue line is the RMA regression line. All related parameter values
are in Table 3-4.

76

Table 3-4. Parameter values for Figure 3-3.
n
30

c
d

bm
a

a
b

Km

156

195***
(42)
196***
(41.0)
b
379***
(33)
413***
(32)

2

R
RMA intercept RMA slope
24.5
0.84
(-21.3,70.2)
(0.64,1.09) 0.73/0.53

K6
a

22.1**
(7.8)
15.1*
(8.8)
a
15.9***
(2.6)
4.54
(3.74)

2.68
(1.72)

50.0
(16.6,91.4)

0.71
0.75/0.56
(0.55,0.91)

4.62
(-2.07,11.3)

0.92
0.88/0.78
(0.85,0.99)

1.04*** 10.9
(4.86,16.9)
(0.26)

0.87
0.89/0.80
(0.81,0.93)

2

Notes: n is sample size, R is the correlation coefficient of the regression model / the
RMA regression, the same letter indicates no statistical difference between sub-groups (α
= 0.05) for eq. (3.8b), while different letters indicate significant difference, number in
parenthesis is standard error or the 95% lower and upper confidence limit. ***: <0.00,

**<0.001,*<0.05
The pattern from Figure 3-3 and Table 3-4 are largely seen in the next two sets of
sub-groups (Figure 3-4 and Table 3-5; Figure 3-5 and Table 3-6). Specifically, the RMA
regression slopes and intercepts from eq. (3.8b) were 1 and 0 except for non-evergreens,
which had a slope of 0.80 (CI is 0.68 to 0.95) and an intercept of 32.1 (CI is 5.05 to 59.2).
Those from eq. (3.8b-3) were biased for small photosynthetic rates with a slope lower
2

than 1 and an intercept larger than zero. The R ranged from 0.69 to 0.88, depending on
2

sub-groups, but 6 out of 8 cases had a R value ranging from 0.77 to 0.88, suggesting
good fit of both equations to the data.

77

500
400
300
200

Predicted Amass

b

0

100

500
400
300
200
100
0

Predicted Amass

a

0

100

200

300

400

500

0

200

300

400

500

d

300
200
100
0

100

200

300

Predicted Amass

400

c

400

500

Observed Amass

0

Predicted Amass

500

Observed Amass

100

0

100

200

300

400

500

0

Observed Amass

100

200

300

400

500

Observed Amass

Figure 3-4. Predicted vs. measured Amass for evergreens by (a) eq. (3.8b) and (b)
eq.(3.8b-3),and non-evergreens by (c) eq. (3.8b) and (d) eq.(3.8b-3). The red line is the
1:1 line through origin, the blue line is the RMA regression line. All related parameter
values are in Table 3-5.

78

Table 3-5. Parameter values for Figure 3-4.

n
132

c
d

bm
b

a
b

Km

54

452***
(82)
408***
(80)
a
193***
(36)
201***
(33)

RMA
intercept
4.66
(-3.00,12.3)

K6

13.4***
(3.6)
5.01
(4.36)

a

27.7***
(5.6)
16.6**
(6.16)

a

2

R
RMA slope
0.88
(0.77,1.00) 0.69/0.47

0.94** 12.6
(5.92,19.3)
(0.29)
32.1
(5.05,59.2)

0.77
(0.68,0.88)

0.69/0.48

0.80
(0.68,0.95)

0.78/0.60

2.60** 57.8
(37.6,78.0)
(0.79)

0.68
(0.59,0.80)

0.83/0.69

2

Notes:n is sample size, R is the correlation coefficient of the regression model / the
RMA regression, the same letter indicates no statistical difference between sub-groups
for eq.(3.8b)(α = 0.05), while different letters indicate significant difference, number in
parenthesis is standard error or the 95% lower and upper confidence limit. ***: <0.00,

**<0.001,*<0.05.

79

500
400
300
200

Predicted Amass

b

0

100

500
400
300
200
100
0

Predicted Amass

a

0

100

200

300

400

500

0

200

300

400

500

d

300
200
100
0

100

200

300

Predicted Amass

400

c

400

500

Observed Amass

0

Predicted Amass

500

Observed Amass

100

0

100

200

300

400

500

0

Observed Amass

100

200

300

400

500

Observed Amass

Figure 3-5. Predicted vs. measured Amass for temperate forest species by (a) eq. (3.8b)
and (b) eq.(3.8b-3),and woodland species by (c) eq. (3.8b) and (d) eq.(3.8b-3). The red
line is the 1:1 line through origin, the blue line is the RMA regression line. All related
parameter values are in Table 3-6.

80

Table 3-6. Parameter values for Figure 3-5.

n
a
96

Km
172***
(24)

b

236***
(48)

a

191***
(24)

c

bm

d

83

28.1***
(3.5)

a

19.7***
(4.44)
b

264***
(44)

17.0***
(2.9)
2.67
(4.22)

RMA
intercept
5.22
(-8.00,18.4)

K6

18.0
(6.34,29.7)

1.46**
(0.50)

-2.49
(-12.6,7.67)

a

14.5
(6.79,22.2)

1.31***
(0.30)

RMA slope
0.92
(0.83,1.02
)
0.84
(0.76,0.93)
0.98
(0.85,1.12
)
0.78
(0.68,0.89
)

2

R

0.87/0.
75
0.88/0.
77
0.77/0.
60
0.79/0.
63

2

Notes: n is sample size, R is the correlation coefficient of the regression model / the
RMA regression, the same letter indicates no statistical difference between sub-groups
for eq.(3.8b)(α = 0.05), while different letters indicate significant difference, number in
parenthesis is standard error or the 95% lower and upper confidence limit. ***: <0.00,

**<0.001,*<0.05
It was shown here that although eq. (3.8b), which included Nmass is significantly
better than eq. (3.8b-1), without it, there was a possible correlation between MAT and
Nmass in some cases, because MAT caused the effect of Nmass to be insignificant in 3 out
of 6 cases (Table 3-4 (d), -5 (b) and -6 (d)). This might contribute to the bias in eq.
(3.8b-3)(see discussion).
For eq. (3.8b), Km values were significantly different between contrasting
sub-groups while bm values were not. Further, Km was negatively correlated with bm
across sub-groups (see Table 3-4,5 and 6 and Figure 3-6), i.e., a higher or lower Km
corresponded to lower or higher bm, respectively, indicating that they are complementary
(Figure 3-6). Therefore, plants can evolve to have either a lower-cost (lower Km) or
81

"cheap" lifetime structure investment with higher-cost (higher bm) or "expensive"
replacement cost, or the reverse. High-photosynthesis sub-groups such as grass/herbs,
non-evergreens, and temperate forest plants had "cheap" lifetime leaf structure and
"expensive" maintenance replacement cost, while the contrasting low-photosynthesis
ones had the opposite strategy. Analyses for eq. (3.8b-3) had similar results and are not
shown here.

20
15

bm

25

0.7

y=977x
2
R =0.84

200

250

300

350

400

450

Km
Figure 3-6. The relationship between Km and bm. Data points are for eq. (3.8b). The blue
2

dashed line is the fitted curve with the equation and R shown in the graph.

82

3 A pattern in the relative ratios of composite maintenance costs to photosynthesis
The bm*Nmass , Km /LL and dark respiration rate are three composites of leaf
maintenance cost that indirectly drive photosynthetic rates for mature leaves through the
energy balance requirement for leaf maturation (eq. 3.7). Their relative contribution to
the photosynthetic rate (Figure 3-7) demonstrated that high-photosynthesis leaves had
lower respiration/photosynthesis ratio (2*Rdmass/Amass), and a higher structural
maintenance /photosynthesis ratio (the sum of these two ratios is 1), although this pattern
was not significant for the grass/herb and shrub/tree sub-groups. The higher structural
investment is mainly due to higher cell or parts replacement cost (higher bm*Nmass/Amass)
but similar lifetime structural mass investment (Km / (Amass*LL). The low photosynthesis
leaves had the opposite characteristics. Therefore, the high and low-photosynthesis leaves
represented two contrasting photosynthesis models.

83

Shrub/Tree
Grass/Herb

0.5
0.4
0.3

ratio to photosynthesis

0.2

0.6

ratio to photosynthesis

ratio to photosynthesis

0.6

0.7

0.5

Evergreen
Non-Evergreen

0.4
0.3
0.2

0.6

Woodland
Temperate

0.5
0.4
0.3
0.2
respiration

nitrogen

leaf mass

Figure 3-7. The relative ratios with 95% confidence intervals of respiration,
nitrogen-related and leaf structural mass maintenance costs to photosynthesis,
corresponding to 2*Rdmass/Amass, bm  N mass / Amass , and K m /( Amass  LL) , respectively,

84

Figure 3-7 (cont’d). for three sets of sub-groups: shrub/trees vs grass/herbs, evergreen vs
non-evergreens, woodland vs. temperate forest species.

Discussion
Here, an evolutionary life history theory for leaves was developed. It was based
on bio-energetic leaf growth and maintenance/mortality processes, with a focus on how
evolution has modulated leaf life history traits to maximize the lifetime net energy/carbon
output to the next generation of leaves. The predictions of this theory were supported by a
global leaf trait database and the theory can be discussed in terms of four major insights
described below.
1 An evolutionary life history theory for leaves has significant meaning for plant
functional traits studies
A major thread of studies in plant functional ecology is about evolution and the
resulted trade-offs (Kikuzawa 1991; Reich et al. 2003; Donovan et al. 2011). For example,
Reich et al. (2003) stated that it is necessary to analyze the adaptive significance of plant
functional traits by modeling possible evolutionary constraints on them. However, not
much has been done besides assumptive suggestions until this research. Kikuzawa (1991)
and Kikuzawa et al. (1999) put forward a theory assuming that the net carbon
assimilation rate of a leaf should be maximized by evolution. However, Kikuzawa’s
theory cannot explain why leaves obviously have so many different net assimilation rates,
instead of just one optimized type with the highest net production rate for all plant
species (Westoby et al. 2002). Clearly, individuals can be successful with higher survival
rates as opposed to faster growth rates (Reich 2001), as evolution is generally more about
lifetime fitness than speed or rate (Shipley et al. 2005). Besides, leaves are not purely

85

dependant functional organs, rather they should be reviewed from a holistic perspective
of plant life history evolution (Reich et al. 2003).
The new theory presented here is based on a theory about leaf life history
evolution that is similar to the theory for individual life history (Roff 2007): net energy
output to the next mature generation of leaves (and by extension, flowers or seeds) over a
leaf's lifetime should be optimized. This optimization theory was tested by the
GLOPNET data, and its mechanism for each critical component is detailed: fecundity is
in growth eq. (3.5), and survival in eq. (3.6). For all species pooled, the new theory
2

without including climate effects provided a good fit to the GLOPNET data (R =0.58 and
0.87 on area- and mass-basis, respectively; see Figure 3-2). The predicted slope of 1.00
was realized on an area-basis but not on mass-basis although the latter has a slope of 0.89
with its upper 95% CI limit (0.96) close to 1.00 (Figure 3-2 (c)). Next, it was shown that
this deviation can be removed when the new theory was fitted separately for four out of
six sub-groups, and even for the two exceptions (shrub/trees and non-evergreens), the
upper limits of (0.99 and 0.95, respectively) of 95% CIs of slopes are quite close to 1.
Reasons for possible small deviations in two sub-groups could be from three
sources. First, there is a non-zero intercept value for deviated regression lines. If that
value is set to zero, this deviation from predicted slope of 1 would disappear. Secondly,
the Amass used here is the photosynthetic potential of mature leaves, which may not be
the same as the photosynthetic rate when leaves just mature because the concept of leaf
maturity is not well studied and quantified previously and photosynthetic potential
declines after the time of maturity (Kikuzawa and Ackerly 1999). Last, the coarse
sub-grouping and sometimes small sample sizes for some sub-groups may hide
86

underlying patterns. For example, some pioneering evergreen pine species have been
found to actually have high photosynthesis leaves (Reich et al. 2007), though evergreens
on average have leaves with lower photosynthesis rates. And the sub-group of shrub/trees
includes some fast-growing shrubs such as Manihot esculenta and Atriplex stipitata and
trees such as Solanum straminifolia, and Cecropia ficifolia that are not different in
functioning with grass/herbs. We hope that future studies that have more precisely
measurements of A, R, SLA and N for more species, and not just for mature leaves , could
allow for improved exploration of the new theory.
2 A photosynthesis variation continuum across species with two contrasting driving
patterns
Recognizing that photosynthetic rates vary along a continuum from slow to fast
with corresponding relationships to other leaf traits, the new life history theory for leaves
provide a framework for exploring leaf trait variation. When paired sub-groupsthat have
relatively contrasting leaf / plant life history characteristics (e.g., evergreens vs.
non-evergreens) were examined (Figure 3-6 and 3-7), two contrasting photosynthesis
types are obvious: the “nitrogen-limited, slow photosynthesis” type and the
“carbon-limited, fast photosynthesis” type.
Although there is obvious variability within the coarse sub-groups, the
slow-photosynthesis sub-groups: shrubs/trees, evergreens and woodland species,
represented a trend toward the nitrogen-limited slow photosynthesis type, where mature
leaves has a smaller nutrient-related maintenance cost relative to photosynthetic rates
(lower bm*Nmass/Amass), but higher respiration costs (mainly in the form of carbon)
driving photosynthetic rates (reflected in higher respiration/photosynthesis ratio), and leaf

87

structures made of more "expensive" materials (higher Km) (see Figure 3-7). This
photosynthesis pattern should be adaptive in areas with low nutrient concentration and
other adverse conditions, because high rates of photosynthesis may be nutritionally
unaffordable and energetically inefficient.
The other three sub-groups: grass/herbs, non-evergreens and temperate forest
plants, generally demonstrated the characteristics of the opposite carbon-limited fast
photosynthesis type, which occurs when, e.g., resource conditions are good and higher
photosynthetic activities are possible. In this case, a species should perform optimally
when sufficient nutrients, especially nitrogen, can facilitate fast photosynthesis, but when
dark respiration of leaves is more limited. For these three groups, the ratio of
nitrogen-related replacement maintenance cost to photosynthetic rate was higher, the
respiration/photosynthesis ratio was correspondingly lower, respiration appeared more
efficient (Figure 3-7), and leaf structures were made of “cheaper” material (lower Km)
(see Figure 3-6). A study by Reich et al. (1998b) compared evergreen tree taxa with
relatively lower photosynthetic rates, such as Pinus, Picea and Thuja, to deciduous tree
taxa with relatively higher photosynthetic rates, such as Populus and Betula, and showed
that the former are usually distributed at sites with a low potential rate of resource capture,
and the latter are usually colonizing high light early successional sites with high nitrogen
uptake availability. Low-photosynthesis taxa such as Pinus contain relatively more cell
wall materials such as lignin and cellulose and higher carbon level(more energetically
expensive), while high photosynthesis taxa such as Populus have relatively more
cytoplasmic elements such as proteins and sterols and a higher nitrogen:carbon ratio
(Niemann et al. 1992). These studies above also suggest that the slower growers are
88

functioning as nitrogen-limited slow photosynthesis types, while the fast ones are
operating under more of a carbon-limited fast photosynthesis type. Other studies have
also suggested that the advantages of evergreen leaves are both energetic and
nutrient-economic (Moore 1984; Moore 1980), and that slow-growing plant species are
characteristic of higher respiration to photosynthesis ratio (Loveys et al. 2002; Way and
Sage 2007).
Though the average trait values of the three paired sub-groups might be different,
the overlap of data range is obvious (Fig. 3, 4 and 5). This is not only consistent with
other empirical studies (Wright et al. 2004), but further supports that eq. (3.7) from the
evolutionary perspective quantified here should be universally observed both within and
among species groups. Thus, it represents a general constraint, meaning that an individual
leaf may adopt a same or quite different photosynthetic rate according to its extant
environments, as long as the leaf traits are associated in a way that satisfies the constraint.
This can explain why different leaf types usually coexist in many different ecosystems. It
also indicates that the widely used functional type classifications, which assume distinctly
separated life history traits among groups (Bonan et al. 2003; White et al. 2000), might
not be appropriate. However, the challenge is to determine the maintenance cost, which
should be a fundamental characteristic of a leaf. The conceptual model in Figure 3-1 did
not include possible costs such as internode, stem and roots, though there were evidence
that these costs may affect leaf energy budget as well (Kikuzawa et al.1999)
3 Lifetime structural mass cost varies little between different leaves
A striking finding is that the lifetime mass investment cost relative to
photosynthesis is actually similar or constant across all sub-groups, as indicated by

89

indifferent Km/(Amass*LL) values (Figure 3-7). Given a constant Km value within a
species group, there should be a significant trade-off relationship between Amass and LL.
2

In fact, the correlation between Amass and LL across all species is strong (R =-0.83),
which has also been shown by more general correlation analyses to this trait-pair (Wright
et al. 2004 and 2005), but the slope is not -1, though close to it (result now shown). This
might be that the value of Km, which is the energy cost per leaf mass, are different
between groups (Table 3-2 and -3), and hence the negative correlation can’t be
appropriately fitted with one equation for all. Nevertheless, this trade-off suggests that the
continuum of photosynthesis should be accompanied by a similar continuum running
from short (at high Amass ) to long (at low Amass) leaf life-span.
Also because this structural mass cost is relatively similar among sub-groups,
relative carbon cost (mainly for respiration) and nitrogen cost (for maintenance
replacement) are complementary so that the above "carbon-limited slow photosynthesis"
is nitrogen-driven and vice versa. This trade-off theme between nitrogen and carbon is
common in plant biology, but the theory presented here shows its possible evolutionary
reason.
4 Temperature moderately influences photosynthesis and its association with other
traits
MAT is selected in the model as an independent climate variable for explaining
the variance of Amass. It has been known that photosynthesis changes with temperature in
a characteristic bell-shaped curve (Ehleringer and Sandquist 2010), and Amass was

90

positively related to MAT (Reich et al. 2007). Adding temperature in eq.(3.8b-3)
improved the explained variation in photosynthesis by 1% for all species pooled, and by
1-2% in five sub-groups and 5% in one sub-group, indicating very little temperature
effects, as supposed by other studies (Wright et al. 2004). This could be that as long as a
species adapts certain position in the photosynthesis continuum, it is set to be equally
competitive to others in terms of fitness and largely independent of environmental
conditions, which makes it hard to directly relate macro-climate with individual
performance. However, including temperature in the model produced a overestimation
bias for smaller photosynthetic rates and also caused Nmass effect to be negligible in
several cases (Table 3-4 (d), -5 (b) and -6 (d)). The bias means a change of slope or
intercept or both for the regression between Amass and leaf trait association, which is in
agreement with a finding that changes in growth temperature altered the scaling
relationship between photosynthesis and leaf traits in certain leaves (Atkinson et al. 2010).
As MAT overwhelmed Nmass in determining Amass, its effect on non-linearly influencing
different Amass is possible. Specifically, species with higher photosynthetic rates may
respond less to temperature, i.e., correlations between higher photosynthesis species and
MAT should have lower slope than the ones between lower photosynthesis species and
MAT. This was supported by data because climate-included eq. (3.8b-3) produced
shallower slopes for high-photosynthesis sub-groups such as grass/herbs and
non-evergreens than low-photosynthesis subgroups such as shrub/trees and evergreens
(slopes were 0.71 and 0.68 vs. 0.87 and 0.77, respectively; see Table 3-4 and 3-5). Atkin
et al. (2006a and 2006b) supposed that fast-growing species may have a greater ability to

91

acclimate than slow-growing species, meaning that high photosynthesis leaves are more
likely to be less affected by temperature and have a higher degree of metabolic
homeostasis than low photosynthesis leaves. Under different global change scenarios, it
may further indicate that high-photosynthesis plant species should be able to cope with
possible temperature increase

better than the low-photosynthesis ones.

Conclusion
Leaves are basic structural, functional and physiological modules of plants.
Though leaf characteristics, particularly functional and life history traits, are diverse,
there are strong universal association patterns among them. Explaining why and how
these patterns are formed remained an enduring challenge. Plant modularity studies have
suggested looking for evolutionary constraints on leaves for possible answers, but not
many testable theories have been advanced prior to this research. This paper confirmed
that evolutionary constraint can explain the variation of photosynthesis and its association
with other leaf traits across species. Analyses of the driving pattern for photosynthesis
indicated that nitrogen content and carbon respiration are the two key driving components
of photosynthesis that trades off with each other. Overall, this study could advance our
understanding of plant evolution and ecology further from plant to leaf scale.

92

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93

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CHAPTER 4. A LIFE HISTORY PERSPECTIVE ON THE LINKAGE
BETWEEN SEED MASS AND LEAF PHOTOSYNTHESIS
Abstract
Seed mass is an important functional trait that defines the life history strategy of a plant.
Variation in seed mass and leaf traits, especially photosynthetic rate, have been shown to
be correlated, but of the mechanisms that underlie this empirical correlation are not well
understood. To fill this gap, a new theory of seed growth based on a widely-used
sigmoidal growth function and an energetic relationship between photosynthesis of
mature leaves and seed energy income is proposed here. A mechanistic relationship
between seed mass and leaf photosynthesis is produced by the new theory. A dataset
including leaf trait and seed mass measurements of 102 species collected globally is used
to test this relationship. The results indicate that, as predicted, there was a negative
correlation between seed mass and leaf photosynthetic rate with a slope of -0.37, but only
20% of seed mass variation was explained. Additional proportions of variation in seed
mass were explained by other leaf traits.

Introduction
Seeds represent a critical life stage of most plant species. The majority of plant
types begin from, and reproduces by, seeds, including conifers, cycads, gnetophytes, and
angiosperms. Some notable exceptions are seedless vascular plants, including bryophytes
(mosses), pterophytes (ferns) and minor groups such as lycophytes (club mosses), but
these also have a seed-like (spore) stage that begins and ends their life-cycles. A critical
functional trait of seeds is their mass, because seed mass is directly connected with the
reproduction strategy of a species, i.e., a plant may evolve to have many smaller seeds or

99

few larger ones. Seed mass varies over a range of over 11 orders of magnitude among
species and 5-6 orders of magnitude within a given community (Moles et al. 2005b;
Westoby et al. 2002). This variation is connected with different plant regeneration types
that help define species niche (Perez-Ramos et al. 2010). In particular, large seeds have
more energy reserve for greater initial seedling mass that can help plants to endure many
different hazards, such as dryness, shade, poor soil nutrients and shallow soil, and thus to
dominate closed or shaded environments (Grubb and Metcalfe 1996; Hewitt 1998; Paz
and MartÍnez-Ramos 2003; Poorter and Rose 2005). On the other hand, small seeds are
lighter, often wind-dispersible for large distances, and more abundant per area than larger
ones, which may help plants to colonize new sites with abundant seedlings and confer
them a direct fitness advantage under certain conditions (Muller-Landau 2010; Moles and
Westoby 2006). Small seeds also tend to experience lower predation risks than the large
ones whose survival is often depressed by predation pressure from animals such as
rodents (Hulme 1998).
Leaf photosynthesis is another critical plant functional trait that helps define plant
niche and life histories (Chapter 3). Especially, higher photosynthetic rate is related to a
higher RGR and less shade-tolerance (Reich et al. 1998a). Photosynthesis is also related
to other important leaf traits such as leaf dark respiration (Rdmass), leaf nitrogen content
(Nmass) and lifespan (LL) (Chapter 3; Wright et al. 2004).
Because seed mass and leaf photosynthesis are two functional traits of the same
species, it is hence reasonable to expect a relationship between them. This idea is
supported by emerging evidence that variation in seed mass is indeed linked to leaf
photosynthesis for some species (Reich et al. 2003). For example, seed mass is negatively
100

correlated with photosynthetic rate in the same individuals of several temperate plant
2

species (R up to 81%) (Reich et al. 1998a and 1998b), and also with RGR and diameter
growth rates of adult trees as photosynthesis is in some taxa and habitats (Shipley and
Peters 1990; Castro et al. 2008). But how general and in what form this negative
relationship between seed mass and photosynthesis is across many species still remains a
largely unknown question.
Further, correlation and causality are not the same thing, so it is important to seek
a mechanistic explanation for the linkage between seed mass and leaf photosynthesis.
Currently, such explanations are still rare, although phylogenetic and cimparative
perspectives are often supposed (Moles 2006; Moles and Westoby2004b; Ackerly and
Donoghue1995; Reich et al. 2003; Reich et al. 1998a).
Here, new research was conducted to shed new light on seed mass-photosynthesis
relationships.

First, a sigmoidal growth process for seeds is proposed. Second, the

energy metabolism of seed mass growth is linked to leaf photosynthesis. Finally, a large
global dataset containing measurements on seed mass, photosynthesis and leaf
physiological traits of diverse plant species was analyzed to test the theory.

Theory
General logistic functions, such as the von Bertalanffy growth function (VBGF),
are appropriate in describing most growth processes. The VBGF assumes that energy
income rate (“anabolism”) is proportional to mass raised to certain power as the
following (von Bertalanffy 1938; Banavar et al. 2002; also see Chapter 1):

B  B0  M 

(4.1)

101

where B is the energy income rate of a seed, M is seed mass, B0 is constant for a
given species,  is a power parameter that may vary between 2/3 and 3/4 (Dodds et al.
2001; O'Connor et al. 2007a and 2007b).
At maturity, leaves should stop growing and export available net assimilated
energy to seed growth. This net assimilated energy can be calculated from the following
relationship (see Chapter 3):
Bl  2C

(4.2)

where Bl is gross photosynthetic rate or assimilated energy income, C is the leaf
maintenance cost. The net assimilated energy output Bl  C  therefore is

1
Bl , and is
2

used as the energy income rate, B , of a seed.
Next, seed mass should be related to leaf photosynthesis as the following in a
mass-basis:
Bl  k  2  B0  M  1

(4.3)

where Bl is mass-based photosynthetic rate, if seed mass is proportional to foliage mass
as reflected in a parameter k (Niklas and Enquist 2001). This relationship can be tested
with measurements of leaf photosynthesis and matched seed mass.

Methods
Analysis of seed mass and leaf traits
A global effort on collecting plant traits, namely Global Plant Trait Network
(GLOPNET), has records of leaf trait measurements for many plant species distributed
across the globe (Wright et al. 2004; see Chapter 3 for more details). These were matched

102

to a seed mass dataset provided by Moles et al. (2005b); 102 records of species were
identified from both databases with both leaf trait and seed mass information and these
data were used to test eq. (4.3). Because photosynthesis also correlates with Rdmass , LL
and N mass (Wright et al. 2004), possible correlations between seed mass and these
indirectly linked variables were also tested.
All correlation analysis was conducted using reduced major axis regression (RMA).
A dedicated software package, smatr, in R statistical software (ver. 2.15.0) was used to
2

get slopes, intercepts and R values.

Results
In general, photosynthetic rate explained 20% of variance in seed mass and the
slope parameter was -0.35 (Figure 4-1), indicating the hypothesized negative correlation
between them.

103

3.0
2.5
2.0
1.5
1.0

Log(A mass)

2

0.5

y=2.12-0.35x, R =0.20

-1

0

1

2

3

Log(seed mass)
Figure 4-1. Correlation between photosynthetic rate and seed mass. n=102. The 95%
confidence intervals (CIs) of the intercept and slope are (2.04, 2.20) and (-0.29, -0.41),
respectively. The red line is the regression line from the regression equation displayed in
the figure.
Rdmass, LL and Nmass each had certain predicting power for seed mass (Figure
2

4-2). Nmass was a relatively weaker predictor of seed mass (R =0.13), Rdmass and LL
2

explained higher amounts of variance in seed mass (R =0.22 and 0.25, respectively).
Also consistent with established inter-relationships between these three leaf traits and

104

photosynthesis (Wright et al. 2004), Rdmass, and Nmass were negatively related to seed
mass with slopes of -0.27 and -0.20, and LL was positively related to seed mass with a

0.0 0.5 1.0 1.5 2.0 2.5

1.0

Log(LL )

1.5

a

0.5

Log(R dm ass )

2.0

slope of 0.51.

2

y=1.13-0.27x, R =0.22
-1

0

1

2

-1.0

0.0

2

3

0

1

2

3

Log(seed mass)

c

-0.5

0.0

0.5

1.0

y=0.72+0.51x, R =0.25
-1

Log(seed mass)

Log(N m as s )

b

-1.0

2

y=0.29-0.20x, R =0.13
-1

0

1

2

3

Log(seed mass)
Figure 4-2. Correlations between seed mass and three leaf traits with 95% CIs of the
intercept and slope: (a) respiration (Rdmass), (1.07, 1.20) and (-0.23, -0.33), (b) LL, (0.60,
0.83) and (0.43, 0.61), (c) nitrogen content (Nmass), (0.24, 0.34) and (-0.17, -0.24),

105

respectively. n=102. The red line is the regression line from the regression equation
displayed in the figure.

Discussion
Seed mass varies locally or globally across species. Here, an allometric growth
model linking leaf photosynthesis with seed mass was presented. This model suggests an
explanation for the correlation between seed mass variation and leaf physiological traits
across a wide variety of plant species occurring globally.
Growth is a universal phenomenon for living organisms including seeds. A typical
growth process with time usually appears as a sigmoidal function. VBGF is such a
well-know growth function widely used in many growth studies (Burd et al. 2006;
Karkach 2006). The assumption of VBGF that energy income rate is related to mass
raised to a power has been supposed to be underlying observed allometric scaling
relations (Chapter 1 and 2), and are largely supported by real data from both animals and
plants (Niklas and Enquist 2001; Enquist et al. 1999). Like leaves (in Chapter 3), seeds
also have their own growth trajectories that can be described by sigmoidal growth curves
such as VBGF. Therefore, there should be a power relationship between the energy
income rate of a seed and it’s mass. The energy income rate for seed growth is from
leaves, and is especially tightly related to the mature leaf photosynthesis (eq.(4.2)). It was
then predicted that seed mass should be allometrically related to mature leaf
photosynthesis (eq.(4.3)). Data from 102 species collected throughout the globe
supported the relationship (Figure 4-1). The indirect correlation between seed mass and
other leaf traits were also supported (Figure 4-2). Though seed mass variation has been
explored from various aspects, the new insight presented in this paper is to our
106

knowledge its first kind to directly relate leaf traits with seed mass in a quantitative and
mechanistic model and to verify the generality of the relations in a large number of plant
species.
Studies on allometric scaling between mass and metabolism supposed various
values for the scaling parameter  that usually ranges around a span of 2/3 to 3/4
(Delong et al. 2010; Sears et al. 2012), though recent arguments that 3/4 is a better value
are highlighted (Enquist et al. 1998). The present study has the (  1) value of -0.35,
which seems to be close to -1/3 from a possible  

2
case, and its 95% confidence
3

interval of -0.41 to -0.29 did not include the value of -0.25 if  

3
. This is consistent
4

with a near 2/3 relationship between energy income rate and mass for plant species that is
derived from photosynthesis having a near 2/3 relationship with respiration which scales
isometrically with mass (Reich et al. 2006). However, this 2/3 might be more of a
convenient average of many possible values, and the actual  value should vary from
species to species, as VBGF can use from 2/3 to 3/4 scaling depending on different cases
(Banavar et al. 2002; Chapter 1).
The relatively low explained variance of 20% may be due to the fact that the seed
mass data were compiled from literature and averaged for a species. It has been
demonstrated that the correlation between seed mass and photosynthesis can be as high as
81% in more precise individual studies for the same plants of several species (Reich et al.
1998a and 1998b).
Photosynthetic rate in leaves has been used to define a possible "fast-slow
photosynthesis continuum" of leaf life history (Chapter 3). As seed mass is linked to
107

photosynthesis in a mechanistic way here, it may help further extend that continuum to
include seed mass. For example, higher-photosynthesis species should have smaller seeds,
higher RGR and maintenance cost or mortality rate, and short leaf lifespan, while
lower-photosynthesis species have the opposite trait syndrome. Through its link with
photosynthesis, linkages between seed mass and other plant traits, such as plant height,
stem diameter, time to first reproduction, plant mass, potential RGR, canopy area, and
canopy volume (Moles et al. 2005b; Leishman et al. 2000; Castro et al. 2008; Moles and
Westoby 2004b; Silvertown 1981) may also be explained. Together, they reflect the
concept that seed mass is really an important plant functional trait representing certain
ecologically important life history dimensions of plants (Westoby et al. 2002; Reich et
al.1998a, 1998b). But future studies are needed to test the suggested relationship here
using larger datasets of precise measurements from the same individuals. Local
environmental factors that might contribute to seed mass variation, such as climate and
geographic effects, should also be taken into consideration (Moles et al. 2005b; Leishman
et al. 1995). So far, these kinds of studies are still rare, and much is still needed to further
our understanding of seed mass variation.

108

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SUMMARY AND FUTURE DIRECTIONS

In this thesis, a systematic bio-energetic theory for growth and life history of
plants and animals was developed. It started with a new understanding to growth and
mortality in terms of energy metabolism dynamics, and finished with a new evolutionary
life history theory. From fishes to plant leaves and seeds, it has been shown that the new
theoretical framework was generally well supported. In particular, this framework is
centered on energy instead of mass, though both are tightly correlated. This idea of
energy as both building blocks and regulators of development and life history has
important biological significance. Meristem cells, for example, are supposed to be
activated by plant energy supply and correlated hormones (Uggla et al. 2001; Borisjuk et
al. 1998; Oribe et al. 2003; Druart et al. 2007), as well as prevailing physical conditions
such as temperature, light and water availability. Studies have also shown that biological
energy, in forms of glucose, ATP, protein or sugars, are critical developmental
signals(Begum et al. 2007).The stress of the weight generated by a plant’s own growth,
as well as environmental stressors, can serve as developmental signals (Ko et al. 2004；
Telewski 2006; Chehab et al. 2009). These all then point to the possibility of further
testing the framework presented here at cellular level. Especially, is there a molecular
mechanism within cells for the growth efficiency declination with age or each cell
production (in Chapter 1)? How is the most important life history trait, i.e., relative
maintenance cost or energetic mortality rate Cm , for a species (in Chapter 2) defined
Em

genetically or physiologically on cell level and affected by environmental factors?

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Answers to them might reveal even more fundamental rules and biological mechanisms
and pave the road for a grand unifying theory of ecology.

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REFERENCES

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