Statistical Mechanicsand

Evolutionary Theory

Lloyd DemetriusHarvard University, Cambridge, Mass., USA

And

Max Planck Institute, Berlin, Germany

Evolutionary changes in morphological complexity

a. Ecological time scale (Single evolving lineage)Increases and decreases in adult body size

b. Geological time scale (Phyletic lineages)Increases in mean body size

c. Geological time scale (Clades)Increases in maximum body size

Evolution of the horse family

Changes in body size within the equid lineages

1. Increase in body size : North America2. Decrease in body size : Europe

Increase in mean body size within the equid taxon

Increase in maximum body size over the history of life

Problem

What is the evolutionary basis for the changes in body size over evolutionary time ?

Darwinian argument

Individuals differ in terms of their morphology, behavior and other phenotypic characteristics (variation)

Different phenotypes are characterized by differences in the acquisition and transformation of resources (natural selection)

There exists a correlation between the characteristics of parents and their offspring (heredity)

Darwinian fitness

The efficiency with which organisms transform resources into net offspring production

Levels of biological organization

1. Populational: Changes in the phenotypic composition of a population by a natural selection

2. Phyletic lineage: Changes in the species composition of a lineage by speciation and background extinction

3. Clade: Changes in the species composition of a clade by speciation and mass extinction

Darwinian modelOrganic diversity and changes in complexity can be explained

in terms of the following tenetSelection tenet

Resident type X1 ; Fitness W1

Variant type X2 ; Fitness W2

If W2 > W1 : then X2 replaces X1

FitnessThe efficiency to transform respurces into net-offspring production

X1

X2

Darwin‘s theory

Evolutionary Principle: Evolution by natural selection results in an increase in fitness

Explanatory Power

1. Variation in life history, body size, life span within and between species

2. The adaptation of species to their habitat

3. The changes in morphological complexity over time

Problem

Can Darwin‘s argument be translated into an analytical theory which will explain:

The diversity of species in space and time The adaptation of species to their environment The increase in complexity within lineages

Does there exist a demographic characterization of fitness which will predict the outcome of competition between variants and incumbents in a population of organisms ?

Characterizations ofDarwinian Fitness

Malthusian parameter (1930) Fisher‘s theory

Evolutionary entropy (1974) Directionality theory

The theory of evolution by natural selection

is the doctrine of Malthus applied to plants

and animals.

Darwin (1859)

Demographic model

Population described by d age-classes

bi = Probability of surviving from age-class (i) to age-class (i+1)

mi = Mean number of offspring produced by individual in age-class (i)

lj = b1,b2,...,bj-1 = Survivorship to age-class (j)

Vj = lj mj = Net-reproduction at age j

Malthusian parameter as Darwinian fitness

Matrix Representation of Graph

λr

vAv

log eigenvalue Dominant ,

00

00

00

1

2

1

21

d

d

b

b

b

mmm

A

)()1( tAutu

rtNt

tutN j

)(log1

lim)()(

rate growth Population r

Characterization of r : rj

jV

e1 jjj mlV

Fisher‘s Theory Growth rate r characterizes Darwinian Fitness:

Malthusian Principle: r predicts the outcome of competition between variant and incumbent types

X

r

X*

r*

X X*

0

*:*

Δr

Xrr wins

0

:*

Δr

Xrr wins

r*

r

rj

jV

e1

rrr *

Fisher‘s evolutionary theory

Population growth rate

Mean Fitness

Fisher‘s principle: Evolution by natural selection results in an increase in the mean malthusian parameter

0d

d rV

t

r

ji

ijij prr,

r

rj

jV

e1

The Malthusian Parameter as Darwinian Fitness

Critique

Computational studies: In Competition between mutants and the

resident population the growth rate is not always a good predictor of invasion success

Empirical studies:Invasion success is highly correlated with

body size and is contingent on the resource constraints

Darwin‘s theory of evolution by natural selection is the doctrine of Gibbs, Boltzmann and Clasius applied to plants and animals.

Directionality theory

(1974)

Directionality theory

Evolutionary entropy, S , characterizes Darwinian Fitness

d

jjj ppS

1

log

rj

jj

Vp

e

Evolutionary principles

1. Evolutionary dynamics within a single evolving lineage(Mutation and Selection)

Directionality Principle for Entropy• Limited Resources: Evolution increases entropy• Variable Resources: Evolution decreases entropy

2. Evolutionary dynamics within a taxon (Speciation and Extinction)

Fundamental Theorem of Evolution• The rate of change of mean entropy is equal to the variance in entropy• Mean entropy increases over geological time

3. Evolutionary dynamics within a a clade ( Speciation, background and mass extinction )

Secondary Theorem of Evolution• The upper entropic limit of species in a clade increases as the claded replaces

another over geological time

Organization

The origin of evolutionary entropy: Its demographic basis

The directionality principles for evolution:Their mathematical basis

Implications of directionality theory for the study of• Life history evolution• Evolution of body size• Evolution of senescence

Origin of evolutionary entropyDemographic model

Microstates:

Population growth rate:

j

j

j

m

l

V

:Fecundity

:ipSurvivorsh

:function vereproducti Net

Rate Growth

r

V

r

rj

j

e1

log

00

00

00

1

2

1

21

d

d

b

b

b

mmm

A

Biological networksMacrostates from microstates

λuAuaA ij 0)(

Theorem

for principle lVariationa λlog

00}{ ijijij app P

j

jijij u

uap

ijijiijiji appp logloglog

Hr

attained is sup the

whichfor matrix unique a exist There

)ˆ(ˆ.2

loglogsuplog.1

ij

ijijijijijij

pP

appp

P

Ann. App. Prob. (1974)

3.

Demographic networksMacrostates from microstates

0100

010

00121

dppp

P

λrV

prj

jj log

e

Entropy:

Reproductive potential:

Generation time:

j

jj

jj

jpT

VpE

ppS

log

log

SErT

T

S

T

Er

00

00

00

1

2

1

21

d

d

b

b

b

mmm

A

Properties of entropy

1. Measure of uncertainty

2. Measure of diversity

3. Measure of robustness

jj ppS logrj

jj

Vp

e

Uncertainty measure

Uncertainty in the age of the mother a randomly chosen newborn

pj Probability that the mother of a randomly chosen newborn belongs to age class (j)

d

jjj ppS

1

log

1,0,,0,0 121 dd pppp 0,,0,0 21 dppp

Robustness

Genealogies: Set of paths of the graphPath:

Matrix associated with the graph

)(1

)(

)(1

lim

...loglog

log

12110

10

nn

nn

xxxxxxn

xx

Sn

P

xSn

aaaS

a

nn

1 32 d......

0)(

),...2,1(...)( 210

ij

i

aA

dxxxx

Robustness

)(1

)( nSn

P nn

.

1)(

than moreby

mean the from differs

nn Sn

Q

)(log

1lim nn

Qn

R

Theorem: 0 RSAnnals. App. Prob.(1994)

Prob. that the sample mean

Reproduction potential and resource constraints

SrTESrTE

SrTE

0;0

Proposition: In Populations in dynamical equilibrium with resource conditions

E<0: Constant resource

E>0: Variable resource

jj VpE log

The Entropic Selection PrincipleEntropy as darwinian fitness

Competition betweem variant and incumbent is a stochastic process determined by entropy (S) and contingent on the resource constraints (E)

Limited resources: (E<0)Mutants with increased entropy have increased robustness and will prevail (a.s)

Variable resources: (E>0)• Large population size:

Mutants with decreased entropy will have decreased robustness and will prevail (a.s)

• Small population size:The outcome of competition will be a stochastic process described by probabilities contigent on population size

X X*

S

S*

wins Xss :*

X X*

S*S

wins*:* Xss

X XX* X*

SSS*

S*

wins Xss :*wins *:* Xss

Invasion dynamicsEvolutionary entropy predicts the outcome

of competitionLimited Resources

Variable Resources

Predictions of directionality theory

Based on the entropic principes of selection we predict the evolutionary changes at three different levels of biological organization.

1. Single evolving lineage – Mutation and selection

2. Aggregate of phyletic lineages – Speciation and background

extinction

3. An ensemble of clades – Speciation and mass extinction

Evolutionary dynamics within an evolving lineage

Long run changes in entropy as one population type replaces another under mutation and natural selection

Equilibrium species: Species subject to limited resource conditions

Opportunistic species: Species subject to variable resource conditions

Evolutionary principles: 1. Entropy increases in equilibrium species2. Entropy decreases in opportunistic species

jj ppS log

Evolutionary dynamics within a taxon

Long run changes in mean entropy as one phyletic lineage replaces another under speciation and background extinction.

The rate of change in mean entropy is equal to variance in entropy

Mean entropy increases

SVdt

Sd

iiSpS

Evolutionary dynamics within a clade

Long run changes in maximum entropy as one clade replaces another under mass extinction

The upper entropic limit increases as one clade replaces another over

geological time.

)(maxˆiSS

0ˆ S

Main tenets of the evolutionary process

1. Evolutionary dynamics within a single evolving lineage

• Equilibrium species: Entropy increases

• Opportunistic species: Entropy decreases

2. Evolutionary dynamics within a taxonThe rate of change of mean entropy is equal to the

variance in entropy

3. Evolutionary dynamics within a cladeThe upper entropic limit increases as one clade

replaces another

Implications of the evolutionary tenets

Evolution of life history

Evolution of body size

Evolution of senescence

Allometric relations

Body size and physiological timePhysica A.

(2003)Physiological time, Body size

Physiological time1. Cycle time of metabolic processes2. Generation time3. Life span

WT T W

132

Entropy and generation time

Theorem

j

jj

jpT

ppS log

TaS log

constantspecific Taxon a

The evolution and distribution of species body size

rj

jj

jj

Vp

ppS

e

log

WbaS log11

Relation between entropy S

and body size W

Empirical studyRelation between entropy and

body size

Directionality theory predicts evolutionary changes in body size

Changes in body size within a single evolving lineage

Limited resource conditions

Increase in body size Variable resource conditions

Decrease in body size

Changes in body size within the equid lineages

1. Increase in body size : North America2. Decrease in body size : Europe

Directionality theory predicts evolutionary change in body size

within a taxon

The rate of change of the mean body size of species within a phyletic lineage is equal to the species variance in body size

Mean body size increases within a taxon

( Cope‘s Rule )

Increase in mean body size within the equid taxon

Evolutionary changes in the upper limit of bodysize

The upper limit of body size increases as one clade replaces another over geological time.

Changes in the upper limit of body size

The evolution of life span

Evolutionary entropy is analytically related to life span L

Directionality theory predicts species variation in life span

LaS log2

Empirical observationEntropy and life span

The evolution of senescence

Directionality theory explains variation in the rate of aging between equilibrium and opportunistic species.

Proposition: The intensity of natural selection is a convex function of age

Intensity of natural selection

Conclusion

1. Darwinian Fitness is characterized by evolutionary entropy

2. Diversity of species and evolutionary change in complexity can be described in terms of the following tenets:

a) Population level:Equilibrium species: Entropy increases

Opportunistic species: Entropy decreases

b) Phyletic level:Mean entropy increases

c) Clade:The upper entropic limit increases

Relation between thermodynamic variables and evolutionary parameters

Thermodynamic variables

Free energy,

Thermodynamic entropy,

Temperature,

Mean energy,

Evolutionary parameters

Growth rate,

Demographic rate, Reciprocal generation time,

Reproductive potentialE~

F~

S~

T~ T

r

S

TSr

TSEF

~~~~

Relation between thermodynamic principles and evolutionary principles

Thermodynamic entropy:

Diversity of energy distribution

Demographic entropy:

Diversity of energy flow

The entropic principle for evolution is a non—equilibrium analogue of the entropic principle for physical systems.

jj ppS ~log~~

jj ppS log

Relation between thermodynamic principles and evolutionary principles

Thermodynamic entropy:

Demographic entropy:

Analytic relation between generation time, and Temeprature :

Theorem: The entropc principle for thermodynamic systems is the limit of the entropic principle for evolutionary processes.

T

dQSd ~~

dPTdS

T~ T

T

G

Tk

hT ~

*exp~

)0*( G