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‘ , This is to certify that the
.32.

% thesis entitled

sioCHASTIc STUDIES OF AGING AND MORTALITY
.i‘ IN MULTICELLULAR ORGANISMS

  

presented by

IRA DAVID SKURNICK

has been accepted towards fulfillment
of the requirements for

_EH_._D_._degree in _—BIOPHYSICS

' /

fi/CQJHK rig-”N 9‘44
// Major professor ' /

 

Date OCTOBER 1z 1%;

0-7639

ABSTRACT
STOCHASTIC STUDIES OF AGING AND MORTALITY
IN MULTICELLULAR ORGANISMS

By

Ira David Skurnick

This study is concerned with the development of a quantitative
phenomenological theory of aging. Such a theory has two immediate
objectives: (1) To mathematically delineate the kinetics of mortality
starting from fundamental time-independent considerations, and (2) To
explain the temperature dependence of the mortality rate in poikilo-
therms.

Probabilistic clocks are studied and it is shown that only a
very small number of steps likely function as rate—limiting factors
in senescence. This is summarized in the derivation and discussion
of the Irk‘law. If the number of rate-limiting steps, k, is small,
the relative spread in age at death will be broad. This is consis-
tent with empirical observation.

The asymptotic development of order theory (a branch of the
theory of probability) is sketched, and its application to the de—
scription of a molecular theory of aging is discussed. It is shown,
in particular, that the two simplest empirically determined expres-
sions for the mortality rate (Gompertz law and the power law) derive

from a common root in order theory. The significance of this

Ira David Skurnick
observation is four-fold: (1) It becomes clear that observed kinetic
results have their fundamental explanation in general probabilistic
considerations and are not intrinsically biological in character;

(2) The kinetic results do not reflect, in any simple way, the under-
lying molecular mechanisms responsible for senescence; (3) To the ex-
tent that these simple empirical results provide an adequate kinetic
description, no rate-limiting molecular process can be uniquely cho—
sen on the basis of kinetic studies alone; and (A) The mathematical
character of the rate-limiting events can differ sharply from both
Gompertz law and the power law. Theorems are presented which enable
one to determine if a proposed molecular process could be consistent
with the simple empirical results.

Lastly, two specific models are pr0posed which treat kinetic and
thermodynamic considerations in a more demanding fashion. Both models
contain elementary steps consistent in character with the thermal de—
naturation of proteins. Both are in agreement with kinetic studies
conducted at constant temperature. The dramatic life shortening ef-
fect of elevated temperature on poikilotherms appears in a natural
way. Theoretical predictions compare favorably with experimental re-
sults obtained at a constant temperature, and provoke re-examination
of the question of temperature-memory effects in senescence. Physio-
logical vitality is shown to decline linearly with time and the mag-

nitude of this decline is consistent with empirical findings.

STOCHASTIC STUDIES OF AGING AND MORTALITY
IN MULTICELLULAR ORGANISMS

By

Ira David Skurnick

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of BiOphysics

IQTA

DEDICATION

This dissertation is dedicated to my parents
Jack and Nettie Skurnick
and to my wife

Miriam Carol Skurnick

Their encouragement has made this work possible and

their enthusiasm has made it a joyous undertaking.

ii

ACKNOWLEDGMENTS

Several individuals and institutions provided assistance through-
out the duration of the work presented in this dissertation and deserve
recognition.

Professors Gabor Kemeny, Barnett Rosenberg, Alfred Haug, and
Subhendra Mahanti each read the manuscript and provided thoughtful
questions and constructive comments.

Professor Edward Eisenstein is appreciated for his early support
and many kindnesses.

Professor Barnett Rosenberg, with his uncommon appreciation of the
role of theory in biology, has been a steady source of encouragement
and a fruitful source of guidance.

Any expression of appreciation to Professor Gabor Kemeny, my major
professor, would be inadequate despite its sincerity. His imprint is
felt throughout this dissertation; yet the true extent of his influence
can best be measured by the yardstick of time. I consider him both my
Teacher and my friend.

I am indebted to the Adult DevelOpment and Aging Branch of the
National Institute of Child Health and Human Development, and particu-
larly to Dr. Lester Smith, for the Opportunity to be an active partici-
pant in the Fifth Annual Biology of Aging "Summer Course" held at the
University of Minnesota/Duluth in August 1973. The meeting served to

improve my perspective and helped to sharpen some of the views expressed

iii

in this dissertation; views which I originally presented in somewhat
less polished form at that time.
I am grateful to Mrs. Mary Bandurski and Mrs. Bea Rabin for intro-

ducing me, in a practical sense, to Drosophila melanogaster, and for

 

the many hours both spent in the careful accumulation of empirical
data.

Particular thanks are due Miss Peggy McKelvey and Mrs. Marie
Betterly for undertaking the very difficult task of transcribing my
handwritten notes, and for doing so with care and good humor.

Lastly, I am.appreciative of support received from the National
Institutes of Health, through Grant GM—Olh22, and for funds provided
by the College of Human Medicine and the College of Osteopathic Med-

icine at Mighigan State University.

iv

TABLE OF CONTENTS

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . vi
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . vii
Chapter
1 :mmmmmeN ... ... ... ... ... ... .. l
2 EMPIRICAL BACKGROUND . . . . . . . . . . . . . . . . 5
3 PROBABILISTIC CLOCKS . . . . . . . . . . . . . . . . 30
h ASYMPTOTIC DISTRIBUTIONS OF ORDER THEORY . . . . . . 39
5 DOMAINS OF ATTRACTION . . . . . . . . . . . . . . . . 5h
6 THE CHAIN MODEL . . . . . . . . . . . . . . . . . . . 56
7 THE SEQUENTIAL MODEL . . . . . . . . . . . . . . . . 62
8 THE COMPETITIVE MODEL . . . . . . . . . . . . . . . . 79
9 DECLINE IN VITALITY . . . . . . . . . . . . . . . . . 87
10 CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . 92
APPENDIX . . . . . . . . . . . . . . . . . . . . . . . . . . 95
LIST OF REFERENCES . . . . . . . . . . . . . . . . . . . . . l3l

Table

LIST OF TABLES

Page
Comparison of compensation law constants Tc and b . . 9
Computer simulation for competitive model k6 = c6,
T = 25°C 0 o o o o o o o o o o o o o o o o o o 99
Computer simulation for competitive model k6 = c6,
T = 39°C 0 o o o o o o o a o o o o o o o o o o o o 107
Computer simulation for competitive model k7 = c ,
T = 25°C 0 O O O O O I O O O O O O O O O O O O '0? I O 115
Computer simulation for competitive model k7 = c7,
T = 39°C . . . . . . . . . . . . . . . . . . . . . . 123

vi

Figure

10
ll
l2
13
1h

15

LIST OF FIGURES

A plot of the activation entropy, [38% , versus the
activation enthalpy, [RBI , for thermal killing of
various organisms . . . . . . . . . . . . . . . . . .

A plot of the activation entrOpy, A 8* , versus the
activation enthalpy, A;H , for protein denaturation.

Characteristic survival curves for closed pOpulations
of multicellular organisms . . . . . . . . . . . . .

Plot of logarithm of mortality rate versus time
(Gompertz plot) for sample populations of DrOSOphila
melanogaster at 25°C . . . . . . . . . . . . . . .

 

 

Plot of double logarithm of reciprocal of the sur-
viving fraction versus lOgarithm of time (power law
plot) for Drosophila melanogaster at temperatures

of (l) 33°C, (2) 31°C, (3) 27°C, and (u) 25°C . . . .

 

Gompertz plots of human mortality data . . . . . . .
Power law plots of human mortality data . . . .

Arrhenius plot of the values of A for Drosophila
melanogaster . . . . . . . . . . . . . . . . . . . .

 

 

Decline in human functional capacity as a function
Of age 0 O O O O O O O O O O O O O O O O O O O O O 0

Power law plotzai(k)n._9 1%éf2) . . . . . . . . . .

Power law plot:§(k)n .9 V063) . . . . . . . . .

Gompertz plot: S§(k)n _4, 7(2) . . . . . . . . . . .

General power law plot . . . . . . . . . . . . . .
Schematic diagram of the sequential process . . . . .

Schematic diagram of the competitive process . . . .

vii

11

1h

17

19

21

23

26

28
A8
50
52
6h
67
81

CHAPTER I

INTRODUCTION

Brian Goodwin once remarked, facetiously, that theoretical bio-
logists had "nothing to start from and nothing to prove." All there is
in biology, he wrote,

Is a set of concepts such as organization, adaptation, regulation,
competence, homeostasis, etc., which must carry an enormous burden
of more or less intuitive understanding and experience about the
essential principles of biological structure and function.1
To justify the biologists' belief in the fundamental importance of
macromolecules, reductionists would require that there be deveIOped,
". . . from [a molecular] basis, a superstructure of derived theory
which will give . . . some insight into the basic formal organization
of the . . . forms of life which we commonly encounter."2 Statistical
mechanics is one example of a successful reductionist theory in
physics, in which phenomena at one level are explained in terms of dy-
namical events at a lower level of organization. Theoretical biologists
frequently face difficulties in satisfying the demands of reductionists
because biology often lacks adequate laws to describe molecular activ-
ities in cells, global prOperties of cells and organisms and, neces-
sarily then, the rules which relate such microsc0pic and macrOSCOpic
descriptions.

This rather general dilemma faces one again in attempts to an-

swer two fundamental questions which stand out in sharp relief. The

first asks how life emerges from aggregations of non-living atoms and

l

molecules. The second treats the inverse problem, begs for a physico-
chemical explanation of senescence, and is the focus of the present
study.

It has been stated, unequivocally, that researchers know of no
inherent factor(s) which automatically produce senescence in all lines
of animals and plants.3 Contrary to Weissman's view that aging is a
general property of all multicellular organisms,’4 it now appears that
there may be multicellular organisms which do not undergo senescence,
and unicellular organisms which do.

We can provide an operational definition for senescence; we can
define criteria which age—related changes must meet to be considered a
part of any basic aging process; and we can describe some of the bio-
chemical and physiological correlates of aging.3 In spite of such con-
siderable efforts, we remain at a primitive stage in our understanding
of senescence,and we are obliged to admit that further general princi-
ples have been notoriously difficult to articulate.

We do not appreciate the relationship between stochastic and
genetically programmed processes in senescence; we do not understand
the effects of localized molecular lesions on the performance of in-
teracting biological control systems; and.we are not at all clear
about the relationship between processes which determine the life
Span of organisms and\those which are responsible for the physio-
logical decline common in older members of a specie.

To this day no preponderance of experimental evidence yet
favors any one of the varied hypotheses which postulate primary molec—

ular mechanisms for senescence. Even granting that such molecular

events occur, no evidence supports, for any of them, a causal

association with aging. Moreover, the effects of events occurring on
the molecular level must be amplified to higher levels of biological
organization before an organism ultimately perishes. Which step(s) in
this overall process are slowest, and thus serve to define the charac-
ter of a Specie's survival curve and determine its lifespan, are
largely unknown. There is no evidence to suggest that those events
which trigger senescence are synonymous with rate—limiting step(s)

in aging. One should not be too quick to accept the pessimism im-
plicit in the view that primary mechanisms must be understood and
brought under control before the rate of aging can be slowed down

and lifespan significantly lengthened.

It might be better to start an inquiry from the Opposite per—
Spective and attempt to deduce a quantitative phenomenological theory
of aging. The importance of mortality in the develOpment of a general
theory of senescence has been recognized by biologists,5 and this pro-
vides a natural starting point for the present work. Survival or life
tables are available for a broad variety of both poikilothermic and
homeothermic organisms under a number Of external influences, and many
seemingly dissimilar species exhibit survival patterns which appear to
Share a common character. Thus, in Spite of the possibility that spe-
cific rate-limiting mechanisms might differ from Specie to specie, it
is plausible to believe that one might deduce valid general principles
from such a study.

A theory that focuses its efforts on understanding something of
the character of rate—limiting steps in senescence may or may not shed
much light on the nature of primary molecular processes. But it never-

theless becomes a yardstick against which such latter hypotheses must

eventually be judged. In this respect such a theory is like a dress—
maker's mannequin. If garments are the right Size the fit may be sat-
isfactory, but this in no way proves that other combinations would not
be as flattering. When they are the wrong Size, however, the fit is
poor and Obvious to all.

It is recognized that certain environmental influences produce

6

more dramatic effects on lifespan than others; it is simple to appre-
ciate that the former command attention here. It would be prudent as
well in the early develOpment of such a phenomenological theory to draw
inferences from facts, and avoid the pitfalls provided by nature to
those who draw inferences from inferences. Lastly, one should also be
keenly aware that the mathematical analysis of biological phenomena in-

volves a compromise between a desire to represent these phenomena real-

istically, and limitations imposed by one's mathematical artistry.

CHAPTER 2

EMPIRICAL BACKGROUND

Strehler has pointed out that individual organisms may cease to
exist in four ways:

They can give rise to two or more organisms by fission (bacteria);
the protoplasm of individuals may coalesce (slime molds); they
may engage in wholesale replacement of their constituent parts
(sea anemones); or they age and die.3

Only the last alternative is of interest here.

The mortality rate, )A(t), which describes the decline in a

closed population with age, is defined by the relation

DB3 = -\_\/Nu—\] 3; ML) (1)

N(t) represents the number of surviving members Of a population at time
t. In general fl(t) depends on factors both intrinsic and extrinsic to
the organism.

At prohibitive temperatures, viruses, yeasts, bacteria, and mam-
malian cells in culture commonly decline in number exponentially with
time.7 The mortality rate is thus independent of time although it de—
pends strongly on environmental temperature. For such organisms the
logarithm of the mortality rate constant is found to decline linearly
with increasing reciprocal absolute temperature. Such an Arrhenius re-
lationship suggests that the rate—limiting events underlying thermal

death occur on the molecular level, and may be treated by the absolute

rate-theory equation

ax? (Ag/R) 2x? (- AHVRT) (2)

(KT
A”) = “7:“

\C, the transmission coefficient, is approximately equal to unity. kB,
h, and R are the Boltzmann, Planck, and gas constants respectively. _T is
the absolute temperature. The activation enthalpy for thermal death,
.AHI’, can be determined from the Slope of the Arrhenius plot and, when
combined with empirically determined values for )A(t), yield the corres-
ponding activation entropy, ASA- .

Significantly, ASAP and AHA: appear to be linearly related over

the range 25 kcal/mole 4 Mi é 200 kcal/mole. This linear relation,

2% \ =\=
AS '-“ "1T: A“ 4- b (3)

C.

is a manifestation of an extra-thermodynamic relationship known as the
"compensation law." Tc, is the compensation temperature, is approximate—
ly 325°K and b is reasonably approximated as -66 cal/mole-°K (Figure l).
The thermal denaturation of virtually all proteins,8 obey similar ex-
ponential kinetics, satisfy the compensation law, and yield values for
the constants TC and b closely identical to those appr0priate for the
thermal death of the unicellular "organisms" described above (Table l
and Figure 2). This has led to the provocative hypothesis that the
thermal denaturation of proteins may serve as the rate-limiting step in
the thermal death of unicellular organisms.T Indeed, Rahn has sug-
gested that the exponential decline in pOpulations of unicellular or-
ganisms may trace to the destruction of only a single protein molecule

per cell.9

Figure l. A plot of the activation entrOpy, 138:,B , versus the
activation enthalpy, A H , for thermal killing of various organisms.
Yeasts are denoted by ('); bacteria by O ; and viruses byA. For a
fuller discussion of the data see Reference 7.

 

500 '

400

(N
O
O

 

200

AS“ col/mole deg

IOO

 

 

1

l

600 r x I w u . . / ,

 

 

"GOO ' 5O 1 IOO I50

AkaCOI/mqle

Figure l

200

_ jiv...‘

TABLE I

Comparison Of compensation law constants, Tc and b.

 

 

from: A8* = aAH*-+ b ; Tc = 1/3
TC(°K) b(calJmole XIX)
Proteins 329 -64.9
Virus 330 -64
Yeasts 325 -64.5

Bacteria 331 -65

10

Figure 2. A plot of the activation entrOpy, AS+ , versus the
activation enthalpy, AHT , for protein denaturation. Data taken from
Reference 8 and discussion at length in Reference 7. The regression
line is a least squares fit to the data. The constants for the line
are: T0 = 329°K and b = -6h.9 cal/mole-°K.

' 400

300

IOO

AS1F col/mole Cdeg

ll

 

N
O
O .

 

 

 

 

5O IOO I50 I
AH“ kcol/mole

 

Figure 2

12

As early as 1912 Eikjmann reported studies in which the mor-
tality rate for thermal killing of bacteria was found to be time de-
pendent.10 This is a common finding for multicellular organisms in-
cluding man.ll If /A(t) is an increasing function of time, the specie
is said to age. For multicellular organisms the characteristic popu-
lation survival curve is shown in Figure 3(a). An initial lag phase
is followed by a decline which tails Off for longer lived members of
the cohort. The number of members of the cohort dying at any given
age is shown in Figure 3(b). It is important to notice that the
spread in age at death is broad when compared to the mean value, and
that the decline about the mean is a continuous but not symmetric func-
tion of time.

Historically, kinetic studies received their principal impetus
from actuaries concerned with obtaining precise numerical expressions
for survival data appearing in human life tables. NO compelling effort
was made to understand the nature of the physico-chemical processes in—
herent in senescence.

Gompertz originally proposed that the mortality rate increased

exponentially with time11

_ M:
(1605 ‘ Be (A)

This was modified first by Makeham, who added a constant term to

Gompertz' relation,l2

mm = A we“ <5)

13

Figure 3. Characteristic survival curves for closed popula-
tions of multicellular organisms. Surviving fraction of population is
plotted versus time in curve (a). Proportion of population dying at a
specific age is exhibited by curve (b). Note that this latter distri-
bution is continuous, relatively broad ( A~0.3), and asymmetric. The
power law constants for the curves are: C*.= S and 1: = 70 time units.

114

“o;

8.2 km
:32 o

m Ohsmflm

v
AH.»

 

 

 

 

 

 

. no

no;

 

15

and subsequently by Perks, who added a fourth constant13

Be)t

 

I+De‘

Presumably, one could obtain an arbitrarily good fit to the empirical
data by imaginatively incorporating additional constants. Doing so
makes it progressively more difficult to understand where these con—
stants originate, and what they truly represent. We are also sobered
by Einstein's rebuke that with three constants, one could fit an
elephant!

It is understandable that most early theoretical efforts set
the derivation of Gompertz' expression as a major objective. This
seemed reasonable since the relation appeared at least moderately suc-
cessful when applied to human mortality.

Recently, it was Shown that the mortality rate of a number of
poikilothermic and homeothermic organisms could be fairly represented

6

by a simple two—constant "power" law of the form

CK
Mt): At (7)

In Figures h and 5, the Gompertz law and the power law are contrasted

using empirical data obtained from survival studies of Drosophila

 

melanogaster at different temperatures. A similar comparison is made

 

in Figures 6 and 7 for data taken from life tables apprOpriate for

in In the

white males, provided by the U.S. Bureau of Vital Statistics.
latter case, it would seem that the power law representation is super-

ior, particularly when non-aging related causes of death are subtracted

from the total number of deaths in each age category and redistributed

16

Figure A. Plot of logarithm Of mortality rate versus time
(Gompertz plot) for sample pOpulations of Drosophila melanogaster at
25°C. Note that data points cannot be fit by a Single straight line.
See Reference 27 for discussion of this experiment which compared
survivorship and mortality rates of flies reared and housed under
sterile conditions with those living in cultures containing the yeasts
which normally accompany Drosophila and which serve as one of their

immediate food sources.

LOG MORTALITY RATE

0 I

OI

OOI

EFFECT OF STERILE CONDITIONS ON MORTALITY RATE

17

TOTA LtMALE + FE MALE)

 

 

 

AGE IN DAYS

Figure A

I I I I I I I I I
0
000630 .
o 0
«x9 Ifflb °
Q
g.» o o
....e ‘09 °' '
c)
a. o
‘r q;n
o. ‘0 ° 0
o dx>
o
0 @CD
' o
' (9
cc)
(b
($9
0
o
o
" 0 oo 0 Central IIthS'I “
o N-3.740
. 0° 0 Storm "IQHZS‘T
N-Lozo
o
O
.0
o
o
1 l 1 1 1 l I l
s :2 IS 24 30 36 42 48 54

 

18

Figure 5. Plot of double logarithm of reciprocal of the sur-
viving fraction versus logarithm of time (power law plot) for
Drosophila melanogaster at temperatures of (l) 33°C, (2) 31°C,

(3) 27°C, andf(h7 25°C. According to the power law }A(t) = At
from which it follows that

anm: A A+o<9me

By comparison

9w (LEW/Nah: 9n (f;\ + (om 9M7:

l9

2
[I'M]! A m
III [[0 o
3 {oil-loo"!!-
IIOI'I’ ’0 0 0.7
I If
Om, I. o O o
ll/lolo/ o
In Ollie /

I03

IO'

 

 

I03 ’

no”

2
w

«:2: >¢<mtm¢< .23ch m.zm¢0>.>¢3m

A95 9°“?

Figure 5

20

Figure 6. Gompertz plots of human mortality data. Data are
for white males dying in the United States in 1967. Curve (1) con-
siders all causes of death. Curve (2), by comparison, treats only
"age-related" causes of death. See Reference 6 for discussion.

21

10° -
IO

80

60

4O

20

 

h

I 2 ..o 4

 

 

m

I -,_Soo.oo_\m12m9 LEE C3450:

AGE (YEARS) , _

Figure 6

22

Figure 7. Power law plot of human mortality data. Data is
the same as that used in construction of Figure 6. Curve (1) con-
siders all causes of death. Curve (2) treats only "age—related"
causes of death. See Reference 6 for further discussion.

MORTALITY RATE (DEATHS/IO0,000)

IO" -

I0”?- I-

 

IOTA

23

2.

1 l l J

20 40 so so IOO
_AOE(YEAB§lI_

Figure 7

 

2h

proportionately. The power-law representation is particularly useful

with respect to analysis of Drosophila data for another reason: it

 

readily permits thermodynamic analysis. Since }L(t) has dimensions of

reciprocal time, A can be written as

(8)

Betweeen 25°C and 33°C 0K: h.5 i 0.3. While oLiS effectively un-
changed, the time constant ’Z’changes by a factor of h.53 (Figure 8).
That is, only one constant is markedly temperature dependent over the
interval cited. Indeed, A satisfies an Arrhenius relationship too, and
the calculated value for AHT is of the order of 200 kcal/mole.6

This large value for the activation enthalpy is consistent with
that expected for the thermal denaturation of proteins, and suggests
that the rate—limiting step(s) in the senescence Of multicellular
poikilotherms may be just such a process.

Earlier theorists demanded more than just the derivation of the
Gompertz equation. They required also that some variable, to be iden—
tified ultimately with physiological vitality, decline linearly with
time. This condition was imposed since neural conduction velocity,
basic metabolic rate, standard renal plasma flow, and maximum breathing
capacity--to name just a few parameters of gross vitality-—were all
known to behave roughly in this fashion (see Figure 9).

There is no need to discuss earlier theoretical efforts here

since they have been critically reviewed elsewhere.3

Generally speak—
ing, however, the earlier approaches are fairly described as not funda-

mental in the sense of deducing time-dependent behavior from time-in-

dependent considerations. Neither were they successful in deriving both

25

Figure 8. Arrhenius plot of the values of A for Drosophila
melanogaster. A values are computed from intercept points Of Fig—
ure 5. Activation enthalpy, AHT, as computed from the SIOpe of
this straight line is 190 kcal/mole.

 

 

26

ARRHENIUS PLOT OF A VALUES FROM

POWER LAW

IO'5

N)

VALUES
I TITIWT]

<IO

H)

ESI

I I IT11I1

l T TTTITII

 

I I I’I IIIII

 

I lrlIllIl

I55

1

Figure 8

27

Figure 9. Decline in human functional capacity (vitality) as
a function of age. Data collected by N. Shock and collaborators and
discussed by B. Strehler in Reference 3.

PROPERTY REMAINING

°/o

(AVERAGE)

I00—

90*-

80-

7OI-

50—-

40—

20-

 

1
O 40 SO 60 7O 80 90

28

\" “ \ Vito-... - CONDUCTION VELOCITY

S‘s-...,” .
.J.\“. w.-;13......./—BASAL METABOLIC RATE

\\\§\\'\"‘--.\\. LSTANDARD CELL WATER
\‘ ..... \ CARDIAC INDEX
' \
gr STAIIMTIRMSMSUN,
\\ VITAL CAPACITY

\

L—STAI~IOARO RENAL PLASMA
r FLOW (OIODRAST)

STANDARD RENAL PLASMA FLOW
(PAH)

MAXIMAL BREATHING
CAPACITY
I I I I I l

AGE (YE/:RS)

Figure 9

29
the Gompertz relation and the linear decline in vitality; generally,
one was assumed or postualted if the other was to be obtained. A
Eri ori, both should be treated on equal footing and appear as emergent
properties of some set of simpler considerations. Recognizing the

utility and comparative accuracy of the power law, it might be reason-

able to require that a theory yield the power law. Alternatively,

since both Gompertz law and the power law are at best only good approx-
imations, one could be more stringent and demand that a theory come

closer to fitting the empirical results themselves. By such an

approach, one might expect to learn what the power law and Gompertz

relation share in common. Lastly, of all the factors know to influ-

ence longevity, none make so profound an impact on poikilotherms as
environmental temperature; yet, not one of the earlier semi—quantita-

tive theories really introduces temperature in a natural way.

CHAPTER 3

PROBABILISTIC CLOCKS

The relatively broad spread in age at death commonly exhibited
by multicellular organisms (Figure 3(b)) suggests that the rate-
limiting factor(s) in senescence may be probabilistic in origin. This
should not be totally surprising. One hopes to understand senescence
in physico-chemical terms and the basic laws of physics and chemistry
are essentially statistical in character. The macrOSCOpic determin-
istic expressions frequently encountered in practice are really only
approximations. That they are often good approximations reflects the
fact that they describe the averaged behavior of a very large number of
elements. The greater the number of elements in a system, the more ac-
curate the deterministic approximations become.

The rate-limiting factor(s) in senescence define an aging
"clock" in the sense that some number of events must occur and be ef-
fectively registered by the system before the organism perishes. De-
terministic clocks, such as ordinary watches, are valuable precisely
because they keep time well. When they fail to do so, they are re-
paired or replaced. There is no reason to believe however, that prob-
abilistic clocks necessarily function with the same precision. Although
probabilistic clocks can be designed to Operate in some arbitrany
fashion, there are two very Simple counting methods worth exploring if
only because of their heuristic value.

Let there be n events, some combination k of which must occur

30

31

before the organism perishes. The events may occur either in a pre-
scribed sequence or in random order. In the latter instance it is
unimportant which specific events occur, just so long as there are k
of them. Certainly the distribution function describing the time of
occurrence of the required number of events would be expected to de-
pend markedly on the method of counting employed. Nevertheless, an
important conclusion can be deduced even without assuming distribu-
tion funutions for the occurrence of individual events.

Consider first, the sequential counting process. Let tj be
the elapsed time between the occurrence Of events j and j—l. The

total time required for completion of k events is

\z
Th 7’ LITE“ Itszz‘ki (9)
3:\

The tJ's are not fixed values. The best that can be said is that

there is some density function fj(t), which gives the probability den-

sity that tj = t. The mean value of Tk’ E(Tk) or T£, can then be

written as

-

IJEi = TX.

I0.
: E: ' :
E (TR) 3'; (*8 In. (10)

I'M/5’

where T: is just the mean value of tj' The variance in Tk’ a measure
J

of the Spread in age at death, is

IL
2. -n
VOW-(Tn) == PI. =Z 65+ «1 Z CNCPQIE)
33‘ 135

(11)
k! _
The last sum in Equation ll extends over each of the ETTETETT'paIrS
2
(tr, ts) with r (,s. CV3 , the variance in t3, is given by

32

0';- : E[ (*3"1j)1] (12)

The covariance is defined to be

CW ( L», is) = E[(tr"Pr)(k$“-£STK (13)

For simplicity it can be assumed that the k events are independently

and identically distributed in time. Then

__ __ a l
Ari—A: IUS :0 (III)

for j = l, 2, . . . k. Since the k events are independently distributed

in time

Coxr (flaky: O

(15)

for all r and s. It thus follows that

E (R) 7' 8% (I6)

and

VM (Th): R01 (17)

The standard deviation in Tk is given by the positive square root of

the variance. The ratio of the standard deviation to the mean, [5(k),

then gives the relative spread in age at death and is just

33

AW ICC/A) w

[X(k) thus varies inversely as the square root of the number Of events
which must be registered by the aging clock. For k large, the relative
spread is small and the accuracy of such a probabilistic clock can be
quite sharp. For k small, however, the probabilistic clock is leppy.
For elementary distribution functions, such as exponential functions,
(PVT-"’l. In Figure 3(b) [er 0.3. Thus, for a sequential counting
process k :$ 10.

Alternatively, the events could occur in random order. If F(t)
is the cumulative probability that a given event occurs by time t, the

probability that at least k out of n events occur by t is given by

n o
n 3 “'3
t = . ._
PM E i F (DB PHI] (19>
d=k
which can be rewritten in integral form as
“ FOB I2
yea v\—
a3 t: -—
”(WE3 k ( It) 0 1 <1 7L) C” (20)

Here too, it has been assumed that the individual events are independ—
ently and identically distributed in time. In order to calculate means
and variances the corresponding probability density function C?(k)n(t)

is needed.

(21)

kg fir“
1“ A
(96098: iémgf‘zkh) F “QUIT—i 21%)

The mean value for the time at which the kEE event occurs is then given

by

3h

00
O
The variance is likewise
cm
3 l (tack ERA
mGft): 071%“) I It” (23)

To facilitate integration in Equations 22 and 23 it is convenient to

rewrite k?(k)n(t) as

-Iz
(Joana: %E( t.) Th“) (14:03)“ iiidl (21+)

where the term in brackets in the last equation is recognized as the
binomial distribution for the probability of k successes in n independ—
ent trials. The probability of success in a given trial is just F(t).
The binomial distribution, abbreviated as b[k;n,F(t)], can be expanded
about that value of k for which the function is a maximum, that is,

about k = nF(t). Letting k = nF(t) + 2, it follows that

9%? (—- fiianFL-b) (\~F 0*)»
32m Papa—F(afi (25)

This approximation is quite good as n grows large provided that k <LKn,

 

\o YtWflfi; nféQ] ~

where Kn3/n2 -€>O as n—%><>0. F(t) can be expanded in a Taylor series

about t = t0 where F(to) = k/n. Thus

WI = View itFRIPeritifi-ACIIRAIP' (26>
t’fia (“7%

For n large only the first order correction term need be kept, thus

35

FL‘t\ "J k/fl 3‘ SOC] (27)
where
59‘) z 0‘ (Etc) (28)
and
<1 = i POW
0“: (29)
into
th

The mean value for the time of occurrence of the k——-event thus

becomes

I—k/n

 

(,MPL-‘kiwv—T SIAM
9°) ° FAIL KI: _ -
Eh +8} lQMG': +SXI 7‘ 5)

.49!“

 

 

 

(30)

The larger the value of n, the smaller 5 mmst be in order that contri-
butions to the integrand are non-negligible. For example, if n = 100
and k/n = 1/2, the integral is fi 10'5 for 5 Z/O.2. The expression

for the mean can thus be accurately approximated by

00

 

 

It \(ts _9‘87’
Ek("=\= 20")? _oe 0+; 9 d8 (31)
where
w k : Rf ._
ZL ) n) AQW“< (A <1 Va) (32)

and

36

P1: “/9~(%—)(I-%\ (33)

Similarly, the variance can be expressed as

00
z

_ 1
(t:+a%+%:_) Q,F 8d8~ E:(‘Q (3h)

IR
Van. dfl = _Z-Zn—ég)

The ratio of the standard deviation to the mean thus reduces to

Mk)" "a: 7:1“ (RAIUJR) (35)

where the product (atO) is some function of k/n. For k some fixed

'~oD

 

fraction of n it follows that A (k) varies as l/‘ik. A Simple example

might serve to illustrate the point. Let

F (ii) = (CL-L)“ (36)

then
It
(are) 2' “77 (37)
and
\ T-"'—_I
Am: ”AT, ' (‘/°()A‘"T\ (38)

Once again the relative accuracy of the probabilistic clock is
seen to vary as the inverse square-root of the number Of independent
events counted. In practice, the clock could count a large number of

events. The a[R‘law requires that the subset which occurs most slowly,

37

and thus determines the time scale on which the clock functions, must
be small in number in order to explain the Observed broad spread in
age at death.

It is important, as well, to understand what the KRTlaw does
not mean. It does not mean that procesSes involving large numbers of
molecules or large numbers Of cells can be summarily disregarded.
Consider the following example. Suppose that a large number Of mole—
cules, n, have to be synthesized before an organism can perish. Let
synthesis be by means of an autocatalytic reaction initiated by k
molecules in which the mean number Of molecules synthesized grows

exponentially in time. That is

«— I:
VI = h eL (39)

There will be some distribution in the number of molecules synthesized
by any given time, and Delbrfick has shown that the ratio of the stand—

ard deviation to the mean varies as

cm“) /E (“I N VAT; (no)

for n ‘)f> k, Thus a relatively broad spread in the number of mole-
cules synthesized is seen to be due to the small number of molecules
needed to initiate the autocatalytic process. Equivalently there will
be some distribution in the time at which a critical level of molecules,
n*, is formed. The peak in the distribution function occurs at t*

given by

Y\
(I ZWQ’WM (III)

38

The standard deviation about t*, in units of l/c, is easily Shown to be

6° N VAT: (12)

which approximation becomes more accurate as k increases.

CHAPTER A

ASYMPTOTIC DISTRIBUTIONS OF ORDER THEORY

The results of the previous chapter Show that probabilistic
aging clocks can function in a manner markedly different from that of
their deterministic counterparts. Aging clocks have been discussed
elsewhere16 in the context of programmed aging where it was implicitly
assumed that the aging of the individual had some positive adaptive
value for the Specie. Such a clock could function even if the mortal-
ity of the individual derived, in a proximate sense, from a breakdown
in the integrity of biological processes. A discussion of such ques-
tions is beyond the scope of the present paper. The point to be em-
phasized is that if a prObabilistic clock depends on some small number
of events, then it may provide for the relatively broad spread in age
at death.

This, of course, does not mean that any process involving small
numbers Of events will be realistic. For example, the amplification
process discussed by Delbrfick in his study of fluctuations provides for
the broad Spread desired. However, the distribution function giving the
time at which the critical number of molecules is synthesized is
skewed in the direction opposite to that found in practice for mortal-
ity as shown in Figure 3(b). In fact as the number of molecules initi-
ating the amplification process is increased the distribution function
becomes a (symmetric) Gaussian. A second example is provided by

l
Szilard in his study of the nature of the aging process. 7 Szilard

39

IIO
assumes a typical cohort to be heterogeneous in the number of genetic
faults each member inherits. Subsequent "hits" render genes ineffec-
tive and death occurs when the number of functional cells falls below
some critical fraction. The inherited faults are distributed in the
pOpulation at random, and the mean number of faults inherited per per—
son must be between 2 and h to account for the relative spread in age

at death. Szilard's approach has been faulted by Maynard-Smith18

and
the latter's criticisms will not be repeated here. One flaw, however,
was appreciated by Szilard himself. In the crude form of the theory,
members of a cohort would die only in certain years with no deaths in
the intervening period. This prediction contrasts sharply with
reality.

Apparently then, survival or mortality curves must be derived.
Of the counting mechanisms discussed in Chapter 3 for probabilistic
aging clocks, the one which assumes that rate-limiting events may occur
in random order is of particular interest for three reasons: First, it
focuses attention directly on the character of the rate-limiting events.
The assumption implicit in this approach is that amplification occurs
on a time scale much Shorter than that required for the occurrence Of
the initiating events. Second, it permits considerable further analy-
sis without the need to specify the cumulative distribution function,
F(t). In this respect the conclusions to be drawn are relatively model-
independent. And lastly, this approach yields kinetic results which

are striking in their agreement with empirical findings.

Let the events be labeled as E E

l’ 2, . . . En for purposes of

identification. The time at which event E Occurs is designated by t,

J jn'

Since the events occur at random, one could well have

hl

IA

is“ 5 Laws inn '4' Tina (A3)

It makes more sense in the present context to label events by the order
in which they occur in time. The time at which the jig-temporally—

ordered events occurs is t(j)n' Thus

Am“ 5 (“(2)-As E thM (MI)

Clearly, t(j)n is some function of both n and the set of unordered var-

iables Itjni' By definition then

it”. “ W‘s, it.) in,” AMI (A5)

The set of ordered variables §t(j)nflgis referred to frequently as the
variational series of the unordered random variables. The behavior of
these "ordered" random variables is described by "order theory."

The probability that at least j events occur by time t is just

ADA—k6)“ {AK 7 @Lj)‘\£\ (1&6)

Consistent with the original assumption that at least k events must
occur for an organism to perish, it follows from the law Of large num-
berslg that the decline in a pOpulation with time Should be given by

N(t)/Nb) ‘2 A- 233‘,

EA)
k\‘*

(W)

N(t) is the Size of the pOpulation at time t, and it is implicit in
this last expression that no new members are allowed to enter the

cohort.

A2

Thus a population survival curve can reflect the probability of
failure of individual organisms, and the problem Of describing the ki-
netics of mortality reduces, in principle, to the determination of the
distribution functions §(k)n(t)° The latter should depend upon k, n,
and the precise functions which describe the occurrence Of the under-
lying events themselves.

For the sake of simplicity it can be assumed that the events
occur independently of one another, and that they can be described by
a common probability density function. This is likely an oversimpli-
fication of the problem. These restrictions can be relaxed, but they
are used here because they lead to quite powerful results in their own
right.

Let f(t) be the probability density that an individual event
occurs at time t. The probability that the event occurs by time t is

then just
“t

Ht} = It“) An
(AB)
The probability that the event has not yet occurred is just l—F(t).
By any instant only two outcomes are possible with respect to
a given event; it either has or has not yet occurred. The n independ—
ent events thus constitute a set of Bernoulli trials. The prObability
that exactly k of n events occur by time t is then given by the bino-

l9

mial distribution

 

' N(t) [WAIT—h (1:9)

\
L(IEV“IFQ£I)':' \C:(hrk3\

A3

The probability that at least k events occur by time t is

TL

icky: Z (A) FEED—F(tflfl
- H

 

(19)
which can be written in integral form as
T \‘ Fm I: T ‘2
'£)=' 1-1) I A
IRAQ I _, (g 3% M (20)
(Rafi IR). 6

This last equivalence can be verified by carrying through the indicated
integration "by parts."
As n-? ob , i. (t) may approach a stable limiting form
(k)n
gfi(k)(t). If so, the two distributions can differ, at most, by a linear

transformation in their arguments. That is,

Team») -«~-> abs» (so)

as n—57cx) for some choice of the constants an and bn. This is referred
to as the stability criterion,20’21 In general, for arbitrary F(t),
such stable limit distributions do not exist. The set of functions
§F(t)~g which possess a common asymptotic distribution, 36%)“), de—
fine the latter's "domain of attraction." If k/n-? 0 or 1 as n-=7‘><>
the resultant stable distributions are called extreme-value functions.
If, on the other hand, k/n—Tv' A (O 4 7\ 4 l) as n-—?OO , the limiting
functions are referred to as asymptotic distributions of the central
terms. The latters' respective domains of attraction are termed "norm?
al" if the asymptotic distributions are independent of the rate at

which k/n’?) .

There are only three extreme-value distributions.21 They are:

IIII

Qt)
WAG?) :2 o 3 iéo
‘ if: (51)
...—...— ——?L Ira—I
(Ia—OAS Q [X d“ ') ‘l:>o,<=(7o
a» f
.. I _. - Va
1 (It) '” m) g Q%1 (AK) Wli4°0 (52)
and _xx
0%) I -I
O
z ‘ ) 7t >0 (53)

There are four asymptotic distribution' for the central terms

with normal domains of attraction.21 They are:
o(
Q) \ C“: “1.11
”‘6: (fl: PIE-dg 0' 61 d9: 3 A320) [)0
“‘ (5h)
1' O ) t 40
~6Ifld .
$62) ‘ QIfifxd i4 £>
: b— ' 0 6
" 0(6) (ARE 9L ) ’
--00
(55)
:2 ”I ') 35>O
o(
a It
$5) ,3... .I \_ fl,
- (XOR: I355; 0“ d8 ) A40)L\>o
....ao d,
gjt l 7. (56)
I I- ”axa
'—" Z 4“ Eli-II Q2 7“, "1170) £170

AS

and

u
C?

Tm) dc} 3 “I: é: ——I

II

l~
6L 3 «1 47h §,I

"-3 A 3 9C>|

On should already suspect that the distributions for the
central terms will prove unsatisfactory (see discussion Of the lSRTlaw
in Chapter 3). @051), 935:2), and §3(h)are discontinuous and in
view of Equations A7 and 50 can be rejected immediately. In fact, the
discontinuity arises in the vicinity of where ETN(t) is itself a maxi-
mum. There is no corresponding discontinuity in the empirical data.

51. (3) - . . _
dt EEOL (t) 18 continuous, but only If cl - c2. It would fol-

low that.%EN(t) should be symmetric about its maximum, but this is con—
tradicted by empirical findings (Figure 3(b)).
The extreme value distributions are more interesting. In fact,

for k = l
at

(I)
:= 1- .
960, (7(2) 6 ) '11 70) 000 (58)

From Equation A7 and the definition of the mortality rate, Equation 1,

it follows that

c&~I
MAM (2 0LT: (59)
I
«A:
Similarly

t:
e

A) “'
°/\ (H '—’ 1,. 6 (6o)

II6

from which the analogous mortality rate can be shown to be

)A (*3) 2 2+ (61)
A“)
Thus, apart from time—scale factors, both the power law and Gompertz
law have been recovered. The distributions dpiF)(t) are not of value
in the present context.
For k small, but no longer equal to unity, departures from lin-
earity appear in the characteristic power law and Gompertz plots (see

Figures 10-12). When k grows large the very character of the asymptotic

distributions are modified. For example, when k = n

..L-‘h °‘
3;“ “q: Q ) 3 £50,4>0
:: A -’ £70 ) 0‘70 (62)
and
Q—‘t
0‘) .. ‘ . -004i< co
7: (t -— Q
\ > (63)

In the first instance the corresponding mortality rate becomes

d-\ -.(-i)d -L- t)“
)th)\ Swift) 2 /‘*Q 3 tea (61+)

W
at
In the latter case
_ Q-t
-1; ..e. - co
= a 4-6 ‘ ”“4“
RUN m P / ’ (65)

A

Neither of these results are satisfactory.

1&7

. <2) ,
Figure 10. Power law plot: §?(k)(£):7 Eb UT The mortality
n A

rate is given by

Qd-I
F (34‘) \ “((1, : 0‘ (3*) \+ (34:50:

Only the asymptotes are shown.

 

1&8

VNG—I —"'>

‘—" V «I Za-I

 

ln_(3t)

Figure 10

h9

Figure ll. Power law plot: @(kfiH Vuég), The mortality
d.

rate is given by

3d“ :4
Mm. -= W. /LwMJ

o(

Only the asymptotes are shown.

In);

 

50

V'f‘uvc,'-'l I

4—— V~30-I

 

In (31)

Figure ll

51

2
Figure 12. Gompertz plot: éé(k) (£)s,zicg. The mortality
rate is given by n

2521:
“MW = ge Luefl

Only the asymptotes are shown.

In)!

 

52

......
V~2

 

91

Figure 12

53

Thus both of the simple empirically determined expressions for
the mortality rate derive from a common origin in the asymptotic dis-
tributions of order theory. In fact, both are deduced for the special
case where k = l. Apparently then the survival curves apprOpriate for
both homeothermic and poikilothermic multicellular organisms have their
fundamental explanation in general probabilistic considerations, and

are not intrinsically biological in character after all.

CHAPTER 5

DOMAINS OF ATTRACTION

Since for each asymptotic distribution there exists a domain of
attraction, that is a set of functions RF,(t)?‘all of which yield to the
same limit distribution, it is. immediately clear that kinetic studies
alone cannot uniquely specify the underlying molecular rate-limiting
Processes. It would be of considerable value, nevertheless, to be able
to state quickly and confidently whether a proposed mechanism could be
Consistent with observed kinetic data. This would be particularly ad-
VaJttrtsaigeous since for most arbitrarily chosen functions, F(t), no stable

asymptotic distributions exist at all.

Several authors have established criteria by which the limit
distribution, appropriate to F(t), can be determined.2l’22’23’2h In-
deed, Smirnov has proven that if F(t) belongs to the domain of attrac-
tiOn of any one of the proper limit laws described in Chapter ’4 for
some particular value of k, then it belongs to the same domain of at-
t'raction for all values of k.21

For F(t), suitably standardized (i.e., for some choice of the

constants a and bn), to belong to the domain of the limit distribu-
tion >001(k)(t) it is both necessary and sufficient21 that:

(a) There exist some tO such that F(tO) = O and

F(tO + 2,) > O for each E> O, and

(b) For each ’t’)O

5h

9m flieflfl. a ,1,“ (66)
—);—~>o+ F(to+t)

Likewise, for F(t) to belong to the domain of attraction of the limit

distribution X(k)(t), it is necessary and sufficient21 that

Slim n F(adr +\o.n\ -_—. Qt <67)

«V900

for each t. The constant bn is defined to be the smallest value of t

for which the inequalities
\
F _ é / i F +0
U: o) r\ (i: ) (68)

hold. The constant an is defined as the smallest value of t '> 0 such

‘5 (Milli—”01) é ‘/ne g F( at jib—0]) (69)

Gnedenko has shown that for F(t) to belong to the domain of >\(k)(t) it

is fully equivalent to require that22

mm ——\wF ( Mt) : MOK
fig 00 1- ? (Keg mi)

for allm>0and oLrako.

(70)

Additional and equivalent sets of criteria have been described

by Von Mises23 and Uzgorenflu

CHAPTER 6

THE CHAIN MODEL

As an abstraction one could represent a biological organism by
a chain which consists of a very large number of links. Let the num—
ber of links be denoted by n. The results of Chapter h suggest that
the chain be assumed to break (and the organism to perish) when the
first link breaks, irrespective of which link breaks first. In pre-
vious discussions no specific functional forms were chosen for F(t).
This is equivalent to not assuming a specific process for the breakage
of individual links. This was done to preserve the model-independent
character of earlier conclusions. It is worth assuming a specific
failure mechanism here, if only to illustrate application of some of
the theorems governing domains of attraction for asymptotic limit
distributions.

Let failure of a link be described by a sequential process.
(See Chapter 3 for a discussion of sequential processes in connection
with the sk‘law.) Transformation of a link from one state to the next
is equivalent to deterioration. After <i-units of deterioration the
link breaks. This is most likely an oversimplification since it is
reasonable that some non—zero probability of breakage should be associ-
ated with each level of wear. Assume further that the time spent in
state j is independent of the time spent in other states and can be de-
scribed by some probability density function fj(t)° The corresponding

cumulative distribution function, Fj(t), gives the probability that
56

57

transformation from state 3 to state 3 + l is accomplished in a time
interval of length t. If it is assumed that the probability of im-
mediate completion of a transformation known to be incomplete after an
interval t is itself a constant with respect to time, then the pro-
posed failure mechanism is built up of states which, taken individually,
show none of the characteristics of an aging system. This can be ex-

pressed mathematically by writing

\

———————- . —- F m: ,0
\—F$'L-l:\ it (71)
from which it follows that
5’53:

U
’5-
I
(B l
1?
'3

Fa (+5

and

(73)

§, (9

The time required for a link to fail, Tck’ is then simply the sum of

H
L/Q'
L».

the time spent in the OL stages

OK
1:2 2‘35 m)
i=1

where the tj's form a set of mutually independent random variables.
The distribution in time for'Tok can be found by the use of

Laplace transforms. Note that

- 27* -s _ —s a
{1251‘}: E< 25“" 3)-’-‘E( 12 tie“: Q t) (75)

58

Since the tj's are all mutually independent random variables it follows

-st
that the same is true for the set ie 31. Thus

EEG“ : ECéSfi\'E(Q—Shim Egfid) (76)

But 00

E (fix: [mm = M W

(D

(77)

where fj(t) is the probability density for completion of the jEELstage.
Letting f(t) denote the probability density function describing comple-

tion of all Ci stages, and combining Equations 76 and 77 yields

Ififlak = i“ Mguflfllgfimr (78)

where the last equality holds only if the time Spent in each of the
states is identically distributed. Assuming this to be true, Equation

73 becomes

$36.» = g e“ 9*

(79)

for j = l, 2, . . . 0L. Thus

iifiiflk :2 §/§+S (80)

and

MW»? (8/9651 (81)

59

Taking the inverse transform gives the distribution in time for Td .

{0.3 = mad" e“ ”yet—m. (82)

The probability that a given link fails by some time t is given by the

cumulat ive di st r ibut ion

1:
FOE): f “Md“ (83)

O

which reduces to

W)

(oi— _3\\6‘(d) 5*» (81+)

by use of Equation 82. 76‘ (oi , ?t) is the incomplete gamma function de-

fined by the integral
37+.
F(d We) - Six 75H A
I * V‘ (85)
O

and differs from the (complete) gamma function, F(“), in that the
upper limit goes to infinity in the latter instance.
For (gt) small the incomplete gamma function can be expanded

in ascending powers of (9t). Thus

 

24> :3," W3 ows
. l
F(‘Q: (ed— \3\ ZS: F(d-x-SH] (‘83 (86)
4:: fit)“
01‘.

(87)

Clearly F(O) = O and F(E'.) > O for each 5 > 0. Likewise

'

6O

9km F(‘rgfl ‘ ’Z’DL

_‘

89% Y (m ‘88)

for all ”TV 0.

Thus there exist constants an and bn such that F(ant + bn) be-
longs to the domain of attraction of the limit distribution 1/0610“).
Under such conditions the kinetics of chain breakage (and thus the de—
cline in a cohort) would be described by the power law.

On the other hand, for (9t) large, the incomplete gamma function

may be expanded in decreasing powers of ( 9t):

OH 3}, 31—}
F(flJ‘Qfl— (gt) a: 2 £531“:— + 19(\5’7L\.M)]

W0 09*)“ (89)
Where
M: \,a,3,. (90)
~17? <CUVEC7L) 4 547C
‘9‘ f a” (91)
and

RE" V’ (”HO/Tm (92)

Thus as (9t) grows large

F(i:\, x 1- %)d~é—f (93’

(on- s) ‘.

61

But

-\o() -
30am 2 M Noam“

Qtpob \’F ( \03 Ma“) (Sm-9w ibiwwflnsj ugly-) (91+)

 

i

‘ (95)

by repeated application of L'Hospital's rule. Thus by Gnedenko's
criterion, it appears that there exist constants an and bn such that
F(ant + bn) belongs to the domain of ‘>ék((t). The kinetics of chain
breakage, under these conditions, would obey Gompertz law.

Not only does it appear then that the simplest empirical find-
ings have their root in a common mathematical principle, but both can
apparently be derived from a common schematic model. By now it should
be clear that while failure mechanisms may lead to unique asymptotic
limit distributions, the converse is not true. Thus there is no way,
resting on kinetic results alone, to show that the above schematic
model is uniquely valid. Furthermore, it should be apparent as well
that if biological considerations are to enter the kinetic equations
for mortality, they must do so through the constants that appear in

those equations.

CHAPTER 7

THE SEQUENTIAL MODEL

The use of order theory in conjunction with a simple sequential
failure mechanism has come very close to describing observed kinetic re-
sults. There are, however, at least three limitations to this approach:
First, it may be unrealistic to assume that n—€><>0 in practice. Second,
the rate at whichwfigkk)n(t) approaches its asymptotic form depends
markedly on F(t) itself. In fact, for certain distributions, F(t), the
error involved in using the corresponding limit distribution is neglig-
ible for n as small as 10. For others the error would still be signifi—

6

cant when n /~’lO . Lastly, the qualitative character ofH§§(k)n(t) can
change dramatically as n increases. Certain distributions which tend
to )\(k)(t) as n-€?<x9 are better approximated by '}é¥(k)(t) for small
n. The classic example involves extremes chosen from a Gaussian dis-
tribution and is discussed by Fisher and Tippett.2O
It is also true that deviations from simple power—law and

Gompertz plots appear in practice. The most general power law plot is
that shown in Figure 13. A nonlinearity (saturation) is occasionally
evident for large ($7t). As n decreases one would expect to see more
of this since the cohort survives to large values of (9 t). On the
other hand, there is no explanation for the non-linearity observed at
small (g’t) that is entirely satisfactory. It may be that small fluc-

tuations in percent survivors in this range are magnified out of pro-

portion on a double logarithmic plot. It may also be true that this
62

63

Figure 13. General power law plot. Note non—linear regions
for both small and large value of reduced time where lepe of curve
grows small. The flattening of the curve at large values of reduced
time is often referred to in the text as saturation.

In):

 

61+

 

|n(31)

Figure 13

65
non-linearity is an intrinsic characteristic of aging systems.

It is reasonable, then, to develOp explicit kinetic models and
obtain formal solution to §§(k)n(t) for finite n. Equivalently, the
corresponding mortality rate may be deduced. Given such explicit
models thermodynamic parameters may be obtained and compared with ex-
periment. It makes good sense to adhere closely to models that yield
the power law or Gompertz law in the asymptotic limit. It is worth ob-
serving that the rate constants introduced in Chapter 6 (Equation 71)
are explicitly time-independent permitting the introduction of the
theory of rate processes (and hence temperature) in a natural and ex—
plicit manner. Heretofore no quantitative theory of senescence has
permitted this, and yet temperature is the single most significant var-
iable affecting the lifespan of poikilothermic organisms.

The simplest place to start would be with the sequential pro-
cess described in Chapter 6. That is, it is assumed that the chain
consists of n links which are identical but independent. The breakage
of a link is described by a sequential process in which (XLstates are
lifted to a special importance because they are slowest. This is shown
schematically in Figure lh. The deterioration rate constant cj describes
transformation of a link from state J to state J + 1. Nothing need be
said at this point concerning the explicit temperature dependence of
these rate constants.

Assume a Markov process for the breakage of a link. Thus the
develOpment in time for the breakage of a link depends only on the in-
stantaneous state of the link and not on the past history of the failure
process. That is, it does not depend on the manner by which the instan-

taneous state was reached.

66

Figure 1%. Schematic diagram of the sequential process. Chain
has n links. Deterioration rate constant c governs transformation of
a link from state 3 to state 3 + 1. If link is in any state between
3 = l and j = c1, inclusive, the link is considered unbroken. These
states all lie within the dashed lines shown in the figure. The link
breaks upon transformation to state cX.+ l.

 

 

 

 

68

If pJ(t) is the probability that a link is in state j at time
t, the temporal evolution of the failure of the link can be described

by the following set of differential—difference equations.

3
S; )3. Om : C54 ’Vs-l (A “ H F5 (H (96)

where J = l, 2, . . . o( + l, and subject to the restrictions that

(MOB :\ 3 (=1

(97)

((30 ('kfi 2 o (98)

for all t.

Equation 96 does not explicitly contain term(s) to account for

repair processes. This is likely an oversimplification of the situa-

tion and is equivalent to treating damage repaired as damage that has

not occurred.
Letting fj(s) = fipjfifls, taking the Laplace transform of both

sic1es of Equation 96, and solving for the general term yields

€35): C\C1”'Csi (75430 /S*C§“‘ (' /S+C-Q (99)

The inverse transform gives

(DJ-GA]: C, C2"'C$-\i iC/SKHC/SJ‘CAWC/SKDR (100)

69
The link breaks when it goes from state d to state d + 1. Thus the

probability that the link is still intact is

TVA =1-F(fl = >:\ “(H (101)
3:

Similarly, the probability that the chain is still intact is

Shh) = q: “(fl : <2: 395 (15)“ (102)

Consequently the age-Specific failure rate, the mortality rate, is

AW (“/i’aX'i‘imt) ”03’

The instantaneous lepe of the power-law plot, a sensitive indicator of ‘

the character of survival curve, is given by

v(t) = 3:39;: Swim: if; mum (at

Experimental evidence suggests that v(t) is generally not an

integral number. This is in contrast with results obtained when
n -->oo . It was shown (see Chapter 6) that the sequential model de-

seribed above satisfied Smirnov's criteria and could yield to the

k .
asmptotic distribution 0(L )(t). For k = 1 it has already been

Shown that

#(fi) 1' mick) (105)

Thus

”Um .... a” (106)

70

More generally, for finite n, v(t) can be written as

site) 5941+)
“(it)”? A’s—...... d‘t

d rv N
2‘; +6) Ht)

which clearly does not depend upon n. For any given:F(t), increasing n

 

(107)

only serves to bring S(t) to zero more rapidly. Thus a plot of the
mortality rate versus time stops at progressively smaller values of
time as n is increased. Increasing n then effectively serves to mask
the intrinsic character of the mortality curve. This can be demon—
strated for the sequential model rather easily.

Letting cl = c = . . . = c = g , it follows from Equation

2 Ci
87 that

”Twae1— Eff

()0 (108)

for (gt) small. In this regime

—(
3111 W9) : "“9 Qfl. (109)
(0‘4)!

and

9: z __ ‘k air-3.
(gt) 91 £11

(110 )
(oz-2.) \.

Therefore
om ( glad/(01-x)!
vat=t)——-i . =«4
i: )- ggfl (111)
01‘.

Thus the result originally obtained when n—+><30 is recovered as

(gt) -—> 0. However, when (3t) >> 1,

71

-\

—— i:
”Hm as as) (SPOOL
(ct-0‘.

(112)

Thus

_—

N 5’
iii) L— (~— (gt) +(o4 003%) (

(ck-Q (113)

and

33:6»ng g1 (Sgt) 26* 083-19 We”) 1*) “(32 11h)

From which it follows that

5-1- - r‘v 1' ELL
dkxwy+m\ (it + 9* (115)

[ gem) LO“ Mam]
1‘ 9* (it)

(«‘11) (116)
81:

and

 

 

dz ‘ _ ..
311.16%) 350*) _ 3)

Expanding the denominator in a power series, multiplying through, and

keeping terms only as far as (1/ 9t)2 gives

A?»

od-l

8: L4“); (117)

 

       

Thus

N 01"
(f(gg ~ 34: (118)

72
Thus v(gt) ...», O as (gt) 4700 . But this is the saturation effect.
To what extent it is seen in practice clearly depends on n.

It makes good sense that each of the 0( rate constants 1C3?)
should be of the same order of magnitude. If any of the stages in the
process went to completion appreciably faster than the others it should
not appear as a component in the rate-limiting sequence. Conversely,
if one rate constant were significantly smaller than the others it
should be the only term seen in practice.

Let C1: C2 = . . . = cd-l= (1% c“. It follows from

Equat ion 99 that

1% (S) = C34 (‘/s+c3 (119)

for,j=l,2,...,o(-l,and

{(fi = :4 (y5+c)°1"( x/sa-Cu) (120)

for j = o( . Taking inverse transforms gives

—dc '. .
(9503: e (at)A WM) (121)

 

forj —l, 2, . . ., ol—l, and
oL-\ €_C t
C a K‘ffid _H) < V;
: C“ C 122
for j = 0( . F is the incomplete gamma function described earlier.
Thus

0(-\

(123)

73

 

d.-\ 5+d- I
A, (“i at) c:;___2- 6‘ * (ab-[(C Codi—l
A’VQA 4% Q (J \.)\ + (c: QA‘ army“) (12h)

N(ZE «48410:: 2:”? (CQW (2:21) [(6 CA1] (125)

for small (ct). Letting co(‘54 c gives

46%"; e d (126)

as should be expected.

Clarke and Maynard—Smith have suggested that there are two
components to the senescence process.25 One part is temperature-
independent and is associated with aging. The dying "phase" is re-
garded as temperature-dependent. This view is supported by Hollings-
worth.26 Put into the context of the sequential model, only 9x would
be markedly temperature sensitive. Since, from the theory of rate-

processes, one can write

_Aui/m

C': C
5 '3 e (127)
this is equivalent to asserting that only AtHj is of considerable

value. If this view is correct, then i:(t) and consequently S(t)
should approach exponential form as the temperature is lowered. It
should be possible to estimate the temperature decrease required to
produce this effect and then compare the results with experiment. The

Av r»
ratio of 4-(t) to its exponential counterpart ( i‘exp) is

7h

 

diiffz «A r
Mid/”Ia? 2: (is); 9 (C CM Ef<c*)r’(fs)f(°w] 928)

For cat = 1, ca 44 c, and n >/ 10, (f-Fexp)n é 1+ X 10-5. It is
reasonable then to restrict attention to the range 9% t a; 1. For

d25andco4/c «'10-2

1.00 S 4(19/xwé ‘15” (129)

The lower limit for the inequality corresponds to fix t = O and the

upper limit to qg‘t = 1. Thus within the range of practical interest

/\/

+03} 4 ‘04 2):”? (130)

From Equation 127 it is clear that

Q“ ( 91.33:” A%:(_\._—._)_)
C0403) E '1“ T2, (131)

2

If coL(Tl) NC and co((T2) ~10- c
..L _ _L.. :3 __ A”) 1.3....
T. '11 Mr: (132)

Let Tl be 29°C, the middle of a representative temperature

a
range commonly encountered in practice. If [&H°< rxa 2OO kcal/mole,
then by T2 = 2h.8°C the characteristic survival curve should be expo-

. “—t o .
nential. For [XHOL /\J 100 kcal/mole, T should be 20.h C. Surv1val

2

curves for wild-type Drosophila are not observed to become exponential

 

for temperatures as low as 17.5°C. In fact the general analytical form

75

of the survival curve remains that described by the power law, and the
exponent,c¥ , is held roughly constant over a wide range in temperature.
Thus A H: 4 100 kcal/mole. Yet Rosenberg, et al report measured
activation enthalpies of approximately 190 heal/mole;6 this then must
somehow be shared out among the several steps in the sequential process.
Clarke and Maynard—Smith's division of senescence into the two
phases described earlier implies an absence of temperature-memory ef-
fects in aging. Thus if flies are exposed to a high temperature for
some period of time and, surviving this, are shifted to a second lower
temperature, they should be expected to die off as if they had been ex—
posed to only the lower temperature all along. In the sequential model
all ci rate constants are of comparable value and this relationship must
be maintained as the temperature is changed if the SIOpe of the power

6 It follows that allc>L rate constants

law plot is to remain constant.
should be temperature dependent with comparable activation enthalpies.
Thus one should expect to see some manifestation of temperature memory.
One way to detect the presence of temperature memory would be as
follows: Divide a cohort of wildetype Drosophila melanogaster into 5
groups upon emergence of the imago. The entire population should be
collected in some short period, say within 2h hours. Let 3 of these
groups serve as controls, one at each of three temperatures. Temperae
tures of 31°C, 3h°C, and 37°C would be reasonable. Set one of the re-
maining test groups at 3h°C and the other test group at 37°C. In prin~
ciple all 5 groups should be started at the same time. After some
period of time, sufficient to allow for a small decline (£820%) in the

test pOpulation at the highest temperature, shift the test populations

at 3h°C and 37°C to the incubator housing the 31°C control group. If

76

Clarke and Maynard-Smith are correct, those flies surviving exposure to
the elevated temperatures should die off as if they had been in the in-
cubator at 31°C all along. Such an experiment has been performed with
individually housed flies and preliminary results suggest that rather
than mirroring the decline exhibited by the 31°C control group, the fur—
ther expectation of life of the test group originally at 37°C is ap-
proximately 1/2 that of the test group originally at 3h°C. This
clearly hints at the possible presence of temperature-memory. 'One has
to be careful though. Differences in adult lifespan on the order of a
factor of two can be explained by differences in the temperature of the
pre-imaginal environment or by differences in larval density. Flies
used in the preliminary experiments were distributed among the 5 pop-
ulation subgroups at random, thus minimizing the effects of differ-
ences in larval crowding. Pre-imaginal environmental temperature was
the same for all flies.

Strehler has considered the effects of high temperature shocks
of comparatively short duration (1/2 hour - 3 hours) on the subsequent

27

mortality of adult Drosophila melanogaster. His results suggest the

 

'presence of temperature memory as reflected by elevated mortality rates
as late as 25 days after exposure to thermal shock. Similarly, Lints
and Lints have explored the effects of pre-imaginal temperature on
adult lifespan of the same specie, and report that adult lifespan is
increased with decreasing pre-imaginal temperature.28 The three obser-
vations, taken together, suggest that the question of the presence of
temperature memory is still open.

If the C1 rate constants are of comparable magnitude a further

interesting observation can be made. In the limit where ( gt) grows

77

small Equations 103 and 108 combine to give the following expression

for the mortality rate.

%A(9£)== (\§<9£f:>/EK”D(<1_.Q§§?\ (133)

2: ’Y\ Y (SD—Lyd—VQX—W] \.

(131+)

for (9t) AA at ! which is reasonable for large n. Thus

W/A (97g 2 o< Qw 8 + ,va [Org—(@0031 (135)

‘9 is the only temperature—dependent quantity on the right-hand side of

Equation 135. This can be re-written as

(my 3% e \<— (a Alf/g) :11: (136)

where use has been made of Equation 127 replacing c with S).

K 1' QWEQQN afield/(“-011 (137)

The quantity (c{ZXH# ) is now an "effective" activation enthalpy and
should be compared with that determined from experimental measurement.
Although it gives the appearance of representing a single activated
step it really is the accumulation of’cx activated processes each with
an enthalpy of 11H°F. The fact that the activation enthalpies of the
individual steps sum in this manner is not really the consequence of
any physical or chemical characteristic of the underlying molecular

processes. It is, instead, a consequence of the statistics. If the

78

apprOpriate time scale were (Ot) >7 1, use of Equations 93 and

103 would give

#(it) 2 f (138)

Thus the effective activation enthalpy would be smaller by a factor of
04 than that observed when (gt) is very small. The point to be ap-
preciated is that very large values of effective activation enthalpy
may be observed in practice and still be consistent with our general
knowledge of molecular reactions. Rosenberg, et al have found that

(d AH") ) ’V 190 kcal/mole over the temperature range from 25°C to

33°C, and that at - l ’”’h.5 i 6.6% over the same interval.6 Thus

at
AH N Bets kaA/Maa (139)

from which it follows that a drOp in environmental temperature from
33°C to 25°C should produce an increase in lifespan, calculated from

the equation

MT23/416“) : W):— QWEZ?) 1):“1563] (1110)

by a factor of approximately el'Sl' or h.53. This is fully consistent

with the experimental data Rosenberg and his colleagues present.
t(T2)/t(Tl) is the ratio of the time required for the population to

fall to any prescribed fraction at the temperatures T2 and T1 respec-
tively. So long as there are no critical temperatures for the individual
steps, the plot of QM» vs l/T should be a single unbroken line with

a slope of (- dAH-‘h /R).

CHAPTER 8

THE COMPETITIVE MODEL

There are two limitations with the simple sequential model.

One is its inability to reproduce the non—linearity seen in the mor-
tality rate curves at small ((gt). The other is that it does not ex-
plicitly permit a link to fail at various levels of wear. If movement
from one state to the next were analogous to deterioration, as would
be reasonable, failure of the link should be more likely at each suc-
ceeding state. In a sense this is implicit in the sequential model
since movement from state j to j + 1 brings the link one step closer
to failure. Both weaknesses are removed in the competitive model
which is outlined schematically in Figure 15.

Once again the deterioration rate constant Cj represents trans—
formation of a link from state 3 to state j + l. The set of depletion
rate constants% kjkare appropriate to breakage of the link at the cor-
responding levels of wear. It is assumed explicitly that the rate con-
stants refer to reactions on the molecular level. The ratio (kJ/cj) is
assumed to increase as J increases. This is consistent with the as-
sumption that breakage is more likely at more advanced stages of deter-
ioration.

Assuming a Markov process, as was done in Chapter 7, the set of

differential-difference equations can now be written as

a . \
r57; 710:) = C11 (95.169 - ((1821) (W (12.1)

79

80

Figure 15. Schematic diagram of the competitive process.
Chain has n links. Allowable states are shown for a single link.
The set of rate constants c. refer to deterioration. The set of

rate constants {k 1 refer tonreakage of the link at the correspond-
ing levels of weal.

82

subject to the same restrictions as were expressed earlier (Equations
97 and 98). Taking Laplace transforms of both sides, and solving for

the general term gives

£6) 2 C‘CZ'HCA-l 0/5+MM'/5+MJ“'(‘/5+MD (1112)

where m = c + k . For the general case, where all the mj's are

3 J J

unequal,

3 ”NA:

77 e
t3<l§z (:(C1”(%-KZL

5'
r:\ 3:} (Ma—MA
abr-

This result cannot often be expressed in closed form. One situation in

 

(11(3)

which closed form solutions can be obtained is as follows. Let cj = c
for all j. Let k - k = A for all j. Then
3+1
-N\ _k
/ \
(916) r 2 (1M)

(P1 (1} : éM1t<iv><('é-*A)

ls

(1’45)

0

emrt ( i“
- ——-—— LC; ~tts
(30C) “ e @a)‘. KAXLQ ) (1116)

from which it follows that the probability of the link beingsatill
unbroken is
(43)

(11+?)

195219391: e”
A

83

 

where
4A
4. 4—90 — - t
MU (A ‘(1 6 J m‘ (1148)
Thus
. ~11
"Kim C _(‘L‘IQQ ‘)
:H‘ «‘t
M “£92 A J (1119)

M 1

from which it is clear that the slope is independent of n and goes to

zero as (t A)‘9’°D . That is, the saturation effect is recovered. The
behavior of the slope for small (til) depends on the value of c/ml.
For c/ml < l
v(tM -—-—> o
(150)

as (t A) -> O. This is the non-linearity for small (tA) referred to

earlier. If c/ml = l

 

- - (41m
9*") “RM : 9”“ t1. (151)
btbh‘? C) 6i§sn> Q’ —\
:: Qua)»- ‘ (14:15 (152)
amvo

= ) (153)

c/ml cannot be greater than unity by definition. Thus for kl )7 O the
character of the general mortality curve is recreated. The lepe starts
out at some small value, rises to a maximum, and then decreases toward

zero once again. The instantaneous value of the slope is generally

811

not equal to an integral number; and this, too, is consistent with ex-
perimental observation.

A related example of some interest would be where successive
kj's (depletion rate constants) differ by a constant multiplicative
factor. This would correspond to successive activation enthalpies
differing by equal increments. Unfortunately, this does not yield a
closed-form solution. A computer program, capable of treating arbi—
trary values for the respective rate constants has been prepared, and
is presented in the Appendix with sample results. Once again, non-
linearities appear for both small and large values of reduced time.
The curvature and length of the approximately linear segment clearly
change as the rate constants are varied. It is explicitly assumed in
the computer analysis that kl < k2 < . . .(kj < . . ., and that
cl = c2 = . . . = cj = . . . = c. Of coumse, wholly arbitrary values
could be chosen too.

If the set of deterioration rate contstants % chare assumed
to be temperature independent the model would be analogous to the scheme
proposed by Clarke and Maynard-Smith. The main contribution to deple-
tion would come at that state for which kJ :: c3. As the temperature
is increased, the ration kj/Cj increases as well; thus the principal
contribution to depletion would occur at progressively earlier states.
Likewise, the slope on the power law plot should decrease. As the
temperature is increased further a point should quickly be reached
where kl '>1j7 cl, and the survival curve should reduce to exponential
form. However, no substantial change in the character of the survival

curve is seen between l7.5°C and 39°C as determined by experimental

measurement. Since deterioration rate constants have been assumed to

85
be temperature independent over this range, their corresponding activa-
tion enthalpies must be negligibly small. Thus the measured activation
enthalpy would reflect contributions coming primarily from the deple-
+ )k

tion constants. But (AH4: )kj—l > (AH . Thus the measured activa-

J
tion enthalpy should increase as the temperature is increased.

In the same context, one should not expect to see many conse-
quences of temperature memory. Clearly the degree of wear experienced
would be independent of temperature. Starting at some high temperature
the prOportion of deaths by any given level of deterioration would be
greater than that at a lower temperature. If the temperature is shifted
to a lower value after some interval of time, the surviving population
would be at the same average state of wear as a control group run for
an equal period of time at the lower temperature. Thus the further ex-
pectation of life of the two groups should be the same.

If the deterioration rate constants are temperature dependent,

the situation is different. In the computer analysis it was arbitrar-

ily assumed that kj/kj l 2: 10 at 25°C. Since

hi / ‘21-) = any BAT)“;- MAKER—.12) /fT (1511)

the incremental difference in activation enthalpy for the depletion
constants is approximately 1h.2 kcal/mole. Raising the temperature

from 25°C to 39°C reduces the ratio kj/kj to approximately 3.3 with

l
the consequence that the number of states contributing to depletion is
broadened. For the simulation presented in the Appendix, two results
become apparent. First, the maximum possible lepe changes by an incre—

ment of only about —1.7. This is well within the range observed exper-

imentally. Second, although the number of states contributing to

86

depletion increases, the states making the predominant contribution do
not change over a broad range in reduced—time. Thus an Arrhenius plot
should remain essentially linear over this temperature range. Since
the deterioration rate constants are temperature dependent, it neces-
sarily follows that the system will reflect the existence of tempera—
ture memory. The life-shortening effect of elevated temperatures is
achieved primarily by the increased values for the deterioration con-
stants, and secondarily by corresponding changes in the depletion
constants.

It is possible to find a sharp break in the Arrhenius plots with
higher values of AHI' at higher temperatures and lower values of AHI‘
at lower temperatures. Consider the following situation. At some tem-
perature Tl assume that only states j and j-+ l contribute significantly

to depletion. If, as the temperature is increased to T the system

2,

passes through the compensation temperature apprOpriate to the pro-

tein molecules postulated to be involved in these reactions, then k

j—l
suddenly becomes larger than kj' How much larger it becomes depends
on the activation enthalpies appropriate to k and k . The greater

J-1 3
this difference, the sharper the break one would expect to see. The

important point to recognize is that the change in measured activation
enthalpy would, at least qualitatively, be in the direction suggested
by Hollingsworth's experiments.29 Similar results have been obtained

experimentally by Hettinger, and Rosenberg, et a1.30

CHAPTER 9

DECLINE IN VITALITY

In the previous sections the concept of wear was introduced
with respect to individual links. But a chain consists of several
links and each can undergo deterioration before the chain breaks. The
net deterioration of the chain must reflect the wear of its constituent
links. There is some maximum deterioration the chain could likely sus—
tain before failure would follow with virtual certainty. This would be
the case, for example, if each link was in its last state prior to
breakage. At the other extreme it is possible for the chain to break
because one link failed although the remaining links had suffered no
wear. It is natural to relate the ratio of the deterioration of the
chain at some instant to the maximum possible wear it could suffer,
with the fractional loss in vitality experienced by a biological organ-
ism as it undergoes senescence.

In practice, there really is no single measure of vitality.
Instead, there are several gross parameters of physiological and bio-
chemical function, and as illustrated in Figure 9, all seem to decline
linearly with advancing age. By the time a human cohort has been re-
duced to between 5% and 10% of its original number, the decline in these
various indices is between 20% and 60%.

As described earlier, a unit of damage is assumed to be sus-
tained when a link undergoes a transformation from one state to the

succeeding state. The molecular events dealt with then are the

87

88

rate-limiting events in loss of vitality as well as mortality. Thus
both aspects of aging systems are treated here on equal footing. The
extent of deterioration can be calculated by the use of generating
functions.19 Let p(j;t) be the probability that a given link has sus-

tained j units of damage but is still unbroken. Thus

WW) ‘1 (3511650 (155)

If, on the average, a link can sustain Ci.units of damage before it
breaks, then the probability that the link is still unbroken by time t

is simply

cX-\
I“) I 1+ H8340 (156)

j=c>
The generating function for the damage sustained by an unbroken link is
then just
iii 4—(
GAS): Z (3(5310§ Z (”(110
jFo =0 (157)

where the subscript 1 identifies the ith link in the chain. The gener-

ating function for the entire chain, G(s), is given by

V\

7\
G(S\= )\ 61(3)? [61(5)] (158)

where the n links are treated as identical. The average deterioration,

E[D(t)] sustained by the chain is then

(5)3653]: I“ 33% (159)

S):

89

and the variance in the deterioration is simply

Z _. 2.
V1161]: .4- G(s\+_cL . .. 1.4 6.
( As." ) 1360 85 C) (l )
sex sfl 5:(
The mean fractional damage sustained by the chain is

55st (31%)] /1. (om) (1611

which reduces simply to

0(\ a1

I

8m(10=(‘,/°U) Z?) 53W) 21 199*) (162)
3.0 =0

and is explicitly independent of n. §h(t) is the negative of the frac-
tional change in vitality. The length of the chain only comes into
consideration in determining the time by which the cohort has been re-
duced to some fraction of its original number.
For the simple sequential model discussed in Chapter 7, it fol-
lows that
. s '1.
10(38): a {Ca/5" (163)

where it has been explicitly assumed that deterioration rate constant

c = c for all j. Thus

J J-1
2 66/J
.2 deg
MA) (”3; J " ;()°°‘) (1611)
is)/(
jzo
Clearly the second factor on the right hand side, B(t), is a fraction

 

equal to or less than unity and must decline as (ct) increases.

If n = 750 the cohort declines to between 5% and 10% of its

90
orignal number by the time ct = 2. If n = 50 the same is true by
(ct) = h. Letting CK = 8, a large but not unreasonable value, it is
possible to calculate the change in B(t) over a time range of practical
interest. Between ct = O and ct = h, B(t) changes from 1.0 to 0.9h, a
decline of only 6%. Thus for all practical purposes the fractional

change in vitality, given by - gn(t), is just

lieu : _— ds/a—\ (165)

Thus the vitality declines linearly. The mean decline calculated from
Equation 165, is -28.5% for n = 750 and -57.1% for n = 50. This com—
pares with corresponding values of -28.h% and -53.6% obtained by use
of the more exact expression, Equation 16h. For an intermediate value,
n = 250, the mean change is -h2%. The standard deviation can be calcu-
lated by use of Equation 160 and that is generally quite small (e.g.,
4Eé50 : -0.h2 i .012). Thus the decline in vitality is linear over
the range of interest and for reasonable values of n, yields percent
declines within the range of experimental observation.

In the competitive model the exact results depend, of course,
on the values chosen for the respective rate constants. To show that
a linear result can be obtained, and that the calculated changes in
vitality fall within reasonable limits one need only refer to the com-
puter simulation. For the case where 07 = k7, kj/kj—l = 10, and
Ci: 7, the ratio of the damage (deterioration) sustained increases by
a factor of approximately 2.81 between ct = 1 and ct = 3. The depar-
ture from strict linearity is thus approximately 6.3%. Since for

n j; 52, Fn(3)f§ .05, it is safe to say that over a reasonable range

both in n and ct, the decline in vitality is linear. By ct = 3 the

91
decline is approximately h6.6%, a reasonable value.
Using this approach, the decline in vitality and the kinetic
results for mortality are treated on equal footing. One does not have
to be assumed for the other to be derived as has been the case in

3

earlier quantitative treatments.

CHAPTER 10

CONCLUSIONS

Examination of probabilistic molecular clocks suggest that they
may be used to account for the kinetics of mortality of multicellular
organisms, and permit the effects of temperature to be introduced in
a natural manner. Three mechanisms are proposed by which such an aging
clock could conceivably function. While this does not rule out more
complex processes, the simple mechanisms serve to suggest that only a
very small number of events likely serve as rate—limiting factor(s) in
senescence. This is summarized by the {galaw which suggests that if
the number of such steps is small the spread in age at death will be
broad. This is consistent with empirical findings. The rate-limiting
step(s) thus determine the characteristic lifespan of biological organ-
isms and may serve to initiate amplification processes which lead to
the proximate causes of death. The time scale for amplification is
likely much more rapid than that appropriate for the occurrence of the
initiating events themselves. Comments on the character of amplifica-
tion processes are beyond the sc0pe of this paper, but interesting re—
marks have been made by Orgel both for the senescence of whole organ-
isms as well as for the demise of cells grown in culture.31 It is im-
jportant to recognize, however, that even when large fractions of cells
or organ system constituents are required to malfunction certain tem-
jporal aspects of the organism's response are governed by the small num~

'ber of events which initiate amplification. Care should also be taken

92

93

to distinguish steps which initiate amplification from primary events
whose occurrence may serve to make senescence inevitable.

One specific mechanism upon which a probabilistic aging clock
could function is as follows: From some large set of posSible events
occurring in random order, some small number are required to occur
without preference shown to any specific subset. If only one such
event is needed two interesting limit distributions are obtained. The
distributions are interesting specifically because they yield Gompertz
law and the power law, the two simple kinetic expressions previously de—
rived from empirical studies. The virtue of such an analysis is that
it is essentially model-independent, and it reveals that Observed
kinetic results have their fundamental explanation in rather general
probabilistic considerations. The results are not intrinsically bio-
logical in character. An earlier mathematical treatment by Sacher and

32

Trucco which lead to Gompertz law is thus seen to be just one of many
plausible mechanisms when judged solely against kinetic data.
ReCOgnizing these limitations two specific models are prOposed
which treat kinetic and thermodynamic considerations in a more demand-
ing fashion. The sequential and competitive models contain component
steps consistent in character with the thermal denaturation of proteins.
Both models are consistent with kinetic studies conducted at constant
temperature and provide for the dramatic life shortening effect ele—
vated temperature has on poikilotherms. The presence or absence of
temperature memory is determined, respectively, by whether or not the
deterioration rate constants are themselves temperature dependent. Ex—

perimental resolution of this point would be of considerable interest.

In the absence of temperature memory the effect of decreasing core body

9h

temperature would be predominantly to extend that portion of life most
apprOpriate to senility. The competitive model reproduces non-linear-
ities seen in the mortality rate curve both for small and large values
of time. The sequential model only provides for the latter result. The
competitive model appears more realistic in its assumptions and is thus
more appealing. It also has the virtue of being able to account for the
existence of critical temperatures. High values of activation enthalpy
would be apparent at high temperatures, and low values of activation
would be effective at low temperatures. Such a phenomena is beyond the
SCOpe of the sequential model as described herein. Both the sequential
and competitive models provide for linear decline in physiological vi-

tality and treat this variable on an equal footing with mortality.

APPENDIX

APPENDIX

In the discussion of the competitive model (see Chapter 8) it

was remarked that the equation

“Q”! C}: mvt
it“): C‘C2"'CA-) Z 3 (1)13)
3311’ (“u-"‘8
=1

=%v
could not generally be expressed in closed—form. pj(t) is understood

 

r:\

to be the probability (density) that a given link is in state j at
time t. m = c + k.. cj is the deterioration rate constant corres—

J J J

ponding to transformation of the link from state j to state j + 1. k3,
on the other hand, is the depletion rate constant appropriate for
breakage of the link at the corresponding level of wear. The probabil-
ity that a given link is unbroken by time t is given by summing pJ(t)
over the allowed states and is denoted by 17(t) in the text.

A computer program has been written, based on Equation 1&3, to
evaluate pJ(t) at arbitrary values of time for arbitrary sets of rate
constantsqacjls and )mjk. The values of these rate constants are read
by the computer and are denoted in the program by the symbols C(J) and
M(J), respectively. The number of allowed states for a given link is
denoted by K and is read by the computer along with the values of time,
T, at which the dependent variable is to be evaluated. The latter ap—

pears in the program as P(J). DP(J) and DDP(J) correspond, respectively,

to the first and second derivatives of pj(t) with respect to time. The

95

96
sum of terms P(J) over the allowed states j = l, 2, . . . K is denoted
in the computer program by F and is evaluated at each selected value of
time. DF and DDF correspond respectively to the first and second deriv-
atives of :F(t) with respect to time. Lastly, the lepe v(t) is calcur
lated according to Equation 107 and is designated by the symbol SLP.
The computer program appears on the following page.

Sample results, obtained on a CDC 6500 computer, are presented
in Tables 2 through 5. For comparative purposes the deterioration rate
constants are adjusted to unity. It is arbitrarily assumed that at
25°C the ratio kj/k _ : 10 for all allowed values of j. Assuming

j 1
Tc == 325°K, it follows from Equation 15h that consecutively numbered
depletion rate constants have activation enthalpies which differ by
1h.2 kcal/mole. At 39°C the ratio kj/kj-l becomes 3.3. In Tables 2
and 3, depletion and deterioration rate constants become equal at
state 6. In Tables 9 and 5 these same constants do not become equal
until state 7. Tables 2 and h refer to computer simulations at 25°C

while Tables 3 and 5 refer to computer simulation at 39°C. Results

are discussed in the main text.

97

--'-:—'r .w:»—

      

s FonMAT(I3)

. PRINT 6

, '6 FORMAT(* *) e
PRE(1)=1

r - 00 100 J=1.K ‘

* READ 25. C(J)9M(J) ‘

as F09MAT(010.4.010.4)

 

  

... pRE(J+1)=PRE(J)*C(J) f‘i'fh'f‘_.
100 CONTINUE - ~ }fi.1
PRINT 35 .I’f?¥ififa
K 35 FOQMAT(* J *.8X.*P(J)*o16X9*DP(J)*.14x,§oop(J)§,, J“ a
010x.«C(J)*.8x.*M(J)*) is
10] READ 75. T 5’ i

75 FODMAT(D9.3) a, 9'2»,

: IF(T)5000.30.30 - v

30 00 2000 J=1.K

' SUM=0

5 05uw=0
DDSUM=O

P0 00 1000 L=19J

.‘ ML=M(L)

. PPOD=1

20 00 202 1:1.J
DIFF= M(I)~ML
IF(01FF)201.202.201

 

201 PPOD=PpoD*DIFF .-

202 CONTINUE -. 1:
Q = -(ML*T) c1
0N=DEXP(O) ~ 1. ”
TEQM(L)=DN/PROD ' ~ -.
DTEPM(L)=TERM(L)*(-ML) <1:f.:§ra;:1

DDTERM(L)=TERM(L)*(ML**2) ,
SUM=SUM+TEPM(L) ’
DSUM=DSUM+DTERM(L)
DDGUM=DDSUM+DDTERM(L)

1000 CONTINUE
PRINT 6
P(J)=PRE(J)*SUM
DP(J)=PRE(J)*DSUM
DDD(J)=PRE(J)*DDSUM
PRINT 859 J9 P(J). DP(J). DDP(J)- C(J),

89 F0RMAT(I39 5X9 010.30 IOX9 010.39 10X9

€011.59 2X9 011.5)

2000 CONTINUE
PRINT 6.‘ _ , A
iPRINT-4§“_" , h ,1 , 0

 

 

98

3 4R FOQMAT(4X0*T*o14Xo*F*914X9*OF*0l?X9*DDF*912X9*SLP*)
1 F=0
" 0F=0
;‘ 00F=0
DO 3000 J:19K
F=F+D(J)
OF=0F+OP(J)
DOF=ODF+OOP(J)
3000 CONTINUE
SLP=T*(DDF/DF-DF/F)
DPTNT 959 T9F9DFQDDFQSLp
9% FODMAT(010.395X9010.305X9 010.305X-DIO.395X9010.3)
GO TO 101
5000 CONTINUE
" Fm

99

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100

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101

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102

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103

         

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105

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A.v.pcoov m mange

106

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I1

A.u.pqoov m manwe

107

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. mcuocmm.u moccoqm.u ~6+C¢o~. chcocfi.
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comm n B .00 u ox Hmdoa m>fipwpmmaoo h0% COflpdeeflm Ampdeoo .m mange

108

    

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4

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A.u.paoov m mange

112

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116

  

 

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A.u.pqoov : wagua

117

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A.w_paouv J mapwe

118

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u

Mfiucmfim.
NOICNBN.
:ouompm.
moucmpm.

.doucmrm.
yoncmmo.
oo.c~o~.
oo.c¢«m.
oo.c¢mm.

oo+coq~.

     

m0...

x3..\.W
.
m
fl
~c.oocm.
h
“A
o“
o
m
ed
m.“
¢
m.
.m,

A.v.paoov : wands

120

mc+aoa¢dd.

Horocccam.

“mo.occo-.
r u .. .

”fia}ooofla~.

.flwnoofioofi.

n¢.ogooo~.

”Ho.occco~.
fi
”flotocooofi.
m ;. I..

.0.,-_ ..

km «.282.

¢pwoo~oo~.
. n.

...

       

mornoc~b~.
mowndoo__.

~o+oo¢ccmc

aall.:znnwfl]

mo.cccamw. fl,_o+cpoocwk

.Hoocoooow.
—o+ccoco~.
~o+cooco~.
no+coooo~o
”coaccocy.
#O¢cocoo—.
~oococoo~.

~OICcooo—.

- ~¢IQ9*~I.
aJm

”cocococ—.
~o+cccoc~.
~o+ooooo~o
Hooccooo~o

~0¢cocoo_o

wnpbphbv.-th.ooooc~.

1&7; rLf.

#o-cmmfl.-
Noncbmucl
~0Iomrm.l
“OIocNMLI
~OIQNON.I
moIo¢®mo
~OICOMN.
HOICNoN-

Nelc¢ho-

“clomflmol

use
finlomo~ol

helcmo~ol

.cclomo_.I

mclco-.I

Helcnm~.l

acnccbh.I--.

moccmom. coocfimo.
.molcmON. NoIcomo..
NOICNHB. ~0ICOCh.
mcuommm.4 oo.ooofi.
ficncfimm.I oo+c¢-.
"onoucm.I oo+Coa~.
Houomom.u ~o:cm¢m.
Houoosm.u fioIcsmm.
mouc¢po.u _monc¢~o.
oc+ocm~.l . ooooomm.
no u
gunamnm. gaucmmm.
NOIcmmm. woocmmm.
¢ouomam. concomm.
mcuo¢mm. moucmrm.
floucmrm. floucocc.
HcIomrm. oo+cm¢~.

vllll

~cocco¢o
h

.
~=

ofi

A.u.pnoov : manna

121

,mo+c~coo~.
¢C+Dcmco—o
mcocco—C—o
Nc+ccoo-.
HOIDccccmo
~o+occc-o
~O+Ccc~c—o
,~o+:o~cc—o
~o+e~ooo~o
,HO+Ccccc~o

~o+cccco~o

mo+cnccoao
r .

geowocflwcw.

oo+cm~>o

aqm
~o.Cococ—w
~o+cccoo—.
~0¢Cccoopo
~o+coooo~o
~0+Cccoo~.
~0+Ccooo~o
~C¢Coooc~.
go+cccocgo
«Coaccoo_o
~0§Cccoo—o

~o+cccoo~.

~o+Co>~o

agm
~c+Coooo—o

Fowcoooofi.

ficlo_mm.

uc:
m—Icomm.
moIeomm.
moICCMM.
MOIcrmm.
moucwcm.
moIcomo.
welshom.
NOIc~¢m.
moIc~¢~.
MOIcrcm.

¢OICJmca

dolommfi.l

uCC
-I00m~.l

honacm~.-

1

~cIC~ocol
uc

NHIc¢mo.I
melc¢mo.I
mctcmmo.l
moIofiqo.I
ficIooo~.I
~oIc¢¢~.I
~oI:-#.I
Nclcmmm.I
moIcma~.I
MOIQOQ¢.I

¢OIC¢m¢oI

OOIOQN~.I

no
mfilcmCm.

melcmox.

~OIC¢rxo

u

u

A~I0>o~.
hoocso~.
:oIc>o~.
mochofi.
flcncacm.
~0IC-m.

Holcqu.

moICnmN.
molc¢m¢o

¢OIC¢m¢.

oo+Ccho

HfiIComco

helcpqc.

A.u.pcoov : magma

mo+cccfi.
h

-

c~

Hoocccm.
h

-

V

cw

122

me

#o

+Q~coo~.

+oo~oo~.

_vFCIQCo.c~.

d

Na

+£ooo—m.

.~o+oocccmo

m~o+ccoo_~.

~c+ooc~c~.

~c+Cc~oc~.

~o+o~ooc~.

~o

~o

+ooooo_.

+coooo~.

III!- ‘Ihabl ,I

oo+egm¢.
a4m

_o+Coooc~.
~o+Ccooc~.
~o.coooo~.
—o+ccooo~.
~o+Ccocc~.
~o+Cocoo_.
~o+¢cooc~.
~o+ccooo~.
_o+cQooc~.
HoIcooQOM.

_o+oooood.

melodc_o
nae

m_uco¢¢.
ooucc¢¢.
couccqa.
«ou:m¢¢.
moucncq.
mouccco.
MOImsmm.
moucsog.
¢cucmmm.

melcwom.

celccom.

NeIChdmol

uc
M~IC¢~N.I
0cIo¢~>.I
ocIcm—~.I
doI5~H>.I
McIO¢dh.I
moIcmc~.I
melcomcol
nono~m~.I
¢OIcmcm.I
mclcmm¢.l

Celooom.I

melocmm.

mfiIchM.
moucxcfi.
molcmcd.
moucmofi.
NoI:m_~.
malacma.
McIC¢mo.
moIcfisfi.
¢ouc¢¢m.
moIoom¢.

CoICOOM.

Nc+comfi.

HH

0“

A.u.pcoov J magma

Q11" 61... .1. . I .

123

 

.¢;woflcoooag. . flmwom>~.u oo+o¢~a. ,mmucmcm. . IIINJ

. “o.ccooogvm.a.a.o@+ooooq oo+omco.u cc+cmco.. ...

_ouommm. .:mm¢ro¢sfi.u nonco¢~.- #o.coo~. "cuoooqu

.qu moo me L p

, go.coooo~. ONICMON. mmncmom. emucmmm. H"

r .mmocmymr ~o+Cco°o_. >~-o¢m~. omncomm. NNICNoA. om
#mo+occHw_. "c.coooo_. ¢~-c¢m~. NHIQNoH. omucmmm. o
”_o+oocmm¢. Ho.coooc~. Nfluacdm. «#IQOFH. pfiuooofi. x
185235. ~o.¢ccoo.. ocuco_¢. mfinemmm. 4~-c~m~. p
.~o+oocom.. Ho.cGooc~. ocucqc_. couc-¢. m~-o¢mm. c
,wc.ooowQ~. “o‘ooooofi. ¢o-cmo¢. oOIchfi. oo-cm~¢. m
._cwoopmc~. Ho.ccooo~. moucomo. ¢ouomo¢. co-cmo~L ¢.

_Ho+o~mocp. .c.¢oocc~. co.co>o. NCIQmao. ¢oncmo¢. «w

~o+ammoc_. ~c.¢ococfi. ~o+c~o_.u oo+oomo. NOICOoo. m.
"~o+o~ooo~. ~o.¢cooo~. oo+c~oo. oo+o~oo.- oc.¢ooo. ~
.a.z .w.u Ravage Aw.ao .s.a a

._ . .M

- . . IE I- III?! «..l., I- . . . . 3w».

comm n 9 .Po n PM Hmuoa v>fipwpmm500 Mom qofipwasaflm hmpsmfioo .m wands

12h

1*"

~o—"~ -———u-—.

—.—.—.——

flowocccmm.
~o+QooOo—o
~o+oo~mo~.
~c+o~woo~.
~c+ocmcc—.

gooohooc—o

rc+0ommm~.
mc+occo>m.
mc+ooo~NHI

~o+cccmm¢.

.~c+ooocomo

~o+oo¢om~.
_o+oocoo~.
~c+oo~mo..

~c+o~moc~.

‘11,. l

\‘i’l 1‘ u j

_;~o»amq@m~.

~o+Coooo—.
~o+coooo~.
~o+coooc~.
~o+ccooo..

~o+cocoo~.

co+cmmm.
adv

_c+cooco~.
~o+Coooc~.
~o+ccmoa~6
doocoooe~.
~o+coooc~.
~o+cocoo~.
~o¢Coooo~.

_o+cOooo~.

~o+ocooc~.

.H.NOI

\

gcccoom.
cc+cm©~.

~oloo¢h.

oo+co~o.l

oc+C~ooo

moIQm~moI
Lac

mfiucmom.
-Icmco.
ooIcfimo.
nelo¢no.
moucmmm.
MCICN¢~.
NOIcfimq.
fiouc¢~w.

oo+ewm~.

é...

NOImMMMI
Mo-c0o~.
ficuommc.
co+c~mm.
cc.omom.

oo+ohooo

Monacmo.I

be
mfiucm¢c.
mfiugfimfi.

onlcc¢—o

mOIO©~—.

NOIQ¢mh.
welco¢M6
moICNq~.
NOIScho

HCIcomm.

moucmmdu

moaccm~.
dolccma.
~OICNWN.
oo+cmcm.

oo+ccoo.

~o+coodo

m
-Ic~m>.
qfiuc¢mfi.
mflucoqfl.
ofincocfi.
oncmnfi.
noIco¢~.

. moIcowm.
moucfima.

Nolcmm¢o

oc+ccc~.

h

“a

CH

A.U.pqoov m mHQdB

125

'3‘!

wawocoosm.

Nc+ooQ~N~.

~o+occmmao

_~o+ccoccm.

~c+0ccom~.

~c+occoc~.
~0+Cosmopo

~o+o~mcc~.

~c+o¢mocmo

~o+Q~ooo—.

mc+eommm—.
NooCGOONM.
mcooocgmwo

.orocommc.

wfio.occocm.

.n. \ ...)“ I

 

.¢o+cccoo_.

~o+coooc~.
~o+Ccooc~.
~o+Cccoo_.
~o+Coooo~o
_o+ocooc~.
~o+coooc~.
~o+Ccooc~.
~o+cocoo~.

~o+coooo~o

~o+cm~—o
aJm

~c+Coooo_.
~o+Coooo~.
HoIocooo_.
_o4cOoooH.

~o+cocoomo

moIcawA.
delcmmc.
moIcOo~o
NCICO_~.
~CIcmomo
flolc¢mn.

~OICdmm.

oo+cmm~.I

oc+c~om.l

oo+cm©m.

mcIQNMmol
...Ec

OCICONO.
NGICMOO.
mCIO¢N¢o
Jelc¢ooo

molecgfio

onoxmm.
¢oIeom~.
moIcmrm.
moIomom.
~cIcm-.
~cIc¢qco
oc+c_mfi.
cc+cma~.
molcomm.

oc+smcro

Nclcmmm.l

u:
Galahwm.
mclc¢mh.
ooICCNm.
molc¢mmo

mouoomfi.

ncIcnmm.
moIccrm.
«olccm¢.
«oICcnc.
moIcch.
~oIcoqfi.
HoIcnoo.
oo+erfi.
oo+c~¢m.

oo+cmcm.

co+cooc.

m
HfiIcmmm.
OoIoumm.
NOIcmwm.
ecIcocwo

¢OICO~Ho

oc+Cocm.

flu

ca

A.w.pcoov m magma

126

 

...

OIacmnmfi.
Newcoeopm.
mo+o°¢~m~.
.~c+ocomm¢.
_«oIooooom.
H¢+oeoom~.
mROIQcoocH.
_w+oo>mo_.
.wo.owmeo~.
HpIQINOCH.
w_ero~coc..
T.
amo+ocmmm~.

I ...MF. II'.

I II-II-I'I I’IIIIIIII

»_.a.1w-1mw~w.Qmo¢. M.-.. HmIo~o¢uw
do.c¢cm. "oIoHo¢.Lm..u. floIcIQm.I
mam . use . ,M. ue
._o+cccoo~. FOIoemm. . NOIemo¢.
~o+coooo~. mOIamms. moIomom.
~o+ccocc_. mOIoopm. MoIomofi.
_o.ocooo~. NOIoo¢m. moIohom.
uo.cooco~. floIoomfi. ~oIocmH.
~o+ccoec~a “CIcmmm. HoIoo¢¢.
"c.9ocoofi. NOIo~m¢.I HoIocmm.
~o+coooc_. .OIomoo.I “cIomcm.
Ho.cocoo~. oo.o¢m~.I moIo~o~.I
~o.¢oooo~. ma-e~m¢. oo+omm_.I
~o.¢ocoo_. oo+omm~. oo+omm~.u
~o.oo-. doIom¢H.I moIomos.I
QJm mac kc
_o.ccocc.. scIcmmm. moIoomm.
- II I -IIII - .. .I

fioIm~o$qwquw. .~

.» a... ..

8.53. 25.3%,,

k h 3A
G

poIcfipfl. fim
moIommm. p

«oncmom. oi

.

NOIommH. m
NoIc~mm. s

_

#OIQ¢~m. o.

_

,k

_oICme. m.
oc+c»-. e
oo+cocm. . m
oo+oc~m. m
00.9mm~. _
oo+c~oo. “o.ooo~.

u »
ooIcm~¢. Ha
I IIL

A

u.

pqoov m «Hams

127

c _ .. . ..arul

~m+oococ_.:
_~c+oc~mcfi.

Fc.c~¢co_.

~c+0¢mcc~o

~c+c~ooo~o

rc+ocmmm~.
No+oocohmo
mo+oco~mfi.
~o+oocmm¢.
~o+Ccooom.
~o+occom~.
~o+oecoo—.
~o+coch—I

~c+c~moo~.

~c+o¢moc_.

~c+¢ccco_.
~o+cococ~.

fioocoooo—o

~o+coooc~I

~o+coocc~.

~o+ccmm.
a4m

_o+ccooc~.
~oICocoo_.
_o+cOcoo~.
~o+ccooc_.
~o+coooo~.
~o+coooo~.
~o+Ccooc~.
~o+ooooc..
_o+coooo~.

~o+Coooo~.

~oICo¢¢.I
~0IC¢NNII
~OICQ®~I
Melcccm.

~OIcwm~I

ficIQom©.I

ucc
NOIcowm.
mOIONMM.
MOICCN~I
NCICQHH.
NOIcmoH.
~0Icom~.I
HoIo¢¢m.I
~OIocm~.I
HoIc~¢m.I

~OIQOO¢I

NOIco¢¢II
~oIcqx¢.I
~cIC>m>.I
«oncsqm.I
AoIcmm~.I

ficIcqqo.I

uc
onocoo.
qu:m_~.
monommc.
moICorm.
~oIc_cm.
HoIemoq.

Housedc.

NcIQNoN.I

~OION¢NII

~OIcmoool

oo+cnm~I
oo+cmqa.
oo+c¢q~.
Hochmw.

~3ICma~.

.oo+co~o.

holcomm.
quCNCA.
mOIco~¢.
moIccqm.
HoIcomw.
_oIchr.
oo+ccw~.
oo+cm~m.
oo+cmmNI

oo+co¢~.

~9+Cocm.

"a

Cu

A.@.pcoov m mHDwB

128

MmmrQCCWMH.
”fic.ocomm¢.
afioo0ococm.
~o+ccoomgo
_:

aonccooc2.
mdc.co>mc_.
«~c+c~moc_.
qmo+c¢moo—I

.wc.o~ooc_.

.rc+eemmm_.
“NCIQQOoFm.
mNCICCcfimfi.
W~o+occmmq.
~o+CoccoNI

Ho+cccom~.

.Ao+ccooc~.

—o+Ccooo~.
~c+cococ—o
~o+Cocoo~I
~o+Ccooo~I
—o+ccooo~.
~o+Coooo~.
_o+ccoco~.
~o+Cccco~I

~c+ocm—I
aam

~o+Cccoc~I
—coCcooc—.
~0+Ccooc~o
~0+Cocoo~.
~o+Cccoo~.

~c+Cococ~I

moIeohm.I
NOIcrmm.I
~0IC¢4~.I
doIc¢:m.I
HOIcoc~.I
NOIcmmo.
~OIcmmm.
AoIoficm.

NOIcmhc.

~0IQ¢~¢II
moo

NOIemcm.I

molcmmdol

MOICI>~II

NCIC>mm.I

~OIobNHII

~oIcomm.I

moIo¢q~.
NcIacqw.
moIcfiom.
NOIcgom.I
flcIcmqm.I
McIomaw.I
HoIooo¢.I
~0IcmcmrI

molcmhc.l

oo+cmmHIl

u:
scIcoqa.
¢0IQ¢C~I
moIefior.
mouommc.
~oI9m>~.

Holeoam.

moIcmyfi.
~OIcama.
Holcfifio.
oo+cmn~.
oo+cmm~.
OOIC¢K~I
HoICNAm.
~oIc¢mm.

moIcfiho.

oo+cymbw

L
concepfi.
¢olcmmm.
moncfimr.
“cloned.
doncmcm.

OOICOFA.

Lf‘

fly

c~

A.c.pnoov m wands

129

 

 

2.2.1.... .I‘ III-’- .0 - ..Ifi.’

..x.. m.cmo .I...:.1 Hmub2.o2. 2c42mom.I 2oLcm2o. OIOOO222
a2m 2am uc.. 2 2.22
2o.ccooo2. thc¢m2. FOIoomm.I poIeocc. 2m
2o.oooco2. mcIcmm2. mOIo¢o¢. mcIa22m. o2
2o.c°coc2. coIomsm. moIomm2.I. monomon. _ o
2o.2ooco2. moICmoc. NoIom22.I moIcmmm. m
2o.cocoe2. moucoom. mcIooos. 2oIcmm2. s
2o.oooon2. 2o.cccco2. moIcmmm. 20Io222.I 2oIooo2. o.
_2fimmococ2. 2o.coooc2. moIoe2m. No-02no. 2OIcmq2. m.
2omoc-c2. 2o+ooeoo2. AOImcmm. NOIcmq¢.I moqumo. «U
2o.owmo22. 2o.aoooo2. No-9»m2. moIoo>2.I moIco2m. m”
2.8 o.o¢~oo2. 2o.coooo2. mOIcomm. moIcmo¢.I moIosac. N
‘m chooc2. 2o.coooc2. .20I02mc. ¢eIo2m¢.I quc2mc. .ww
m 2o.com2. moIom2m.I oo.ocp2.I @o.cm2c. 2c.ooom. , w
a2m uac no u 2 2 m
fimo.conm~2. 2o.eocoo2. NOIcmom.I soIawmm. ooIchm. 22 2
chmewwImnI 2onroooo2. moIcomn.w .I. -.Imeooom. I IoIcoom. .IIImH

A.v.p:OUV m manna

 

irawemmm”.
. ..W...

cacaoobm.

                            

. O

ccvcm@mo
me

~o.cc¢oa~.
_o.c°aoc..
_o.coocc~.
~¢¢cccoqmm
Hc+ccocc~.
~o.ccooc~.
Ho.c°oco~«
~o+cooca_.
~o.¢cacc_.

~0§cooco~o

NotoN-o
Lac

mo-on¢~.
co-om-.
mo-cmmo.
«o-co¢~.
mcuoohm.
moucmmm.
mouem¢m.
¢oncomo.
gono¢cm.

molcoom.

Noncom~.l

no
couch—m.n
oouosom.a
monomoo.

mo-o¢~_.u
mc-o»~¢.n
noncomm.s
mouqomm.u
mouanm~.-
«onomqm.u
mango—«.u

.:IQMcm.I

.Ifl.

 

Nonemam.

mc.ccmwa.
Fa .
moue¢~m. _t
coucmmm. o
¢o-omq‘. oL
.mo-c¢¢~. .u m_
mo-o¢om. s
mcucmph. c
moLomq¢. m,
mo:co¢a. ¢_
eoucmmm. mg.
monom¢¢. w
coucmom. n
..l...-r.rrkn.

A.u_paoov m mapwa

LIST OF REFERENCES

10.

ll.

l2.
13.
1h.

15.
16.

17.

LIST OF REFERENCES

B C. Goodwin, Temporal Organization in Cells, Academic Press,
London and New York, 1963, p. 2.

C. H. Waddington, Foreword, in B. C. Goodwin, Temgoral Organiza-

 

tion in Cells, Academic Press, London and New York, 1963.

 

B. L. Strehler, Time, Cells, and Aging, Academic Press, New
York and London, 1962.

A. Weissman, Essays Upon Heredity and Kindred Biological Prob-
lems, Oxford University Press, London and New York, 1891.

 

A. Comfort, Ageing, The Biology of Senescence, Rev. Ed.,
Routledge and K. Paul, London, 196k.

B. Rosenberg, G. Kemeny, L. Smith, I. D. Skurnick, and
M. Bandurski, Mech. Age Dev., 2 (1973) 275.

 

B. Rosenberg, G. Kemeny, R. C. Switzer, and T. C. Hamilton,
Nature, 232 (1971) h71.

M. Joly, A Physico-Chemical Approach to Denaturation of Pro-
teins, Academic Press, New York and London, 1965.

O. Rahn, Bact. Rev., 8 (l9hh) 1.

 

C. Eikjmann, Folia Microbiol., 1 (1912) 359.

 

B. Gompertz, Phil. Trans. Roy. Soc. (London), Series A, 115
(1825) 513.

 

W. M. Makeham, J. Inst. Actu., 13 (1867) 325.

 

w. Perks, J. Inst. Actu., 63 (1932) 12.

 

Vital Statistics of the United States, 1967, Vol. 2, Mortality
(Part A), U. s. Govt. Ptg. Office, 1969.

 

M. Delbrnck, J. Chem. Phys., 8 (19uo) 12o.

 

Z. A. Medvedev, Exp. Geront., T (1972) 227.

 

L. Szilard, Proc. Nat. Acad. Sci. (U. s.), h5 (1959) 30.

 

131

18.
19.

20.

21.

22.

23.

2h.

25.

26.
27.
28.
29.
30.
31.

32.

132

J. Maynard-Smith, Nature, 18h (1959) 956.

W. Feller, An Introduction to Probability Theory and Its
Applications, Vol. 1, 6rd Ed.) John Wiley and Sons, New York,
1968.

R. A. Fisher and L. H. C. Tippett, Proc. Camb. Phil. Soc., 2h
(1928) 180.

 

N. V. Smirnov. American Mathematical Society Translation
Number 67, 1952.

P. B. Gnedenko, Ann. of Math., 11 (19h3) M23 (in French).

 

R. de Mises, in Selected Papers of Richard von Mises, Vol. 2,
American Mathematical Society, Providence, l96h, 271 (in
French).

N. T. Uzgaren, in Studies in Mathematics and Mechanics, Pre-
sented to Richard von Mises, Academic Press, New York and
London, 195M, 3h6.

 

J. M. Clarke and J. MaynardrSmith, J. Exp. Biol., 38 (1961)
679.

 

M. J. Hollingsworth, Exp. Geront., l (1966) 259.

 

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