Embed Size (px)
Citation preview
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
