Download Darwinian Fitness & Directionality Theory

Statistical Mechanics
and
Evolutionary Theory
Lloyd Demetrius
Harvard 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.
2.
Increase in body size : North America
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.
2.
3.
Populational: Changes in the phenotypic
composition of a population by a natural
selection
Phyletic lineage: Changes in the species
composition of a lineage by speciation and
background extinction
Clade: Changes in the species composition of
a clade by speciation and mass extinction
Darwinian model
Organic diversity and changes in complexity can be
explained in terms of the following tenet
Selection tenet
Resident type X1 ; Fitness W1
Variant type X2 ; Fitness W2
X1
X2
If W2 > W1 : then X2 replaces X1
Fitness
The efficiency to transform respurces into net-offspring production
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 of
Darwinian Fitness

Malthusian parameter (1930)

Evolutionary entropy (1974)
Fisher‘s theory
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
m1
b
 1
A 0


 0
md 
0

0 
u (t  1)  Au(t )
b2

0

 
 bd 1 0 
Av  v ,   Dominant eigenvalue
m2

r  log λ
N (t )   u j (t )
1
lim   log N (t )  r
t 
r  Population growth rate
Characterization of r :
1 
Vj
e
rj
Vj  l jmj
Fisher‘s Theory
Growth rate r characterizes Darwinian Fitness:
Malthusian Principle: r predicts the outcome of competition
between variant and incumbent types
1 
r
X
Vj
e rj
r*
r
X*
X
X*
r*
r  r * r
r*  r : X * wins
Δr  0
r*  r : X wins
Δr  0
Fisher‘s evolutionary theory
Population growth rate r
1 
Vj
e rj
Mean Fitness
r   rij pij
i, j
dr
 Vr  0
dt
Fisher‘s principle: Evolution by natural selection
results in an increase in the mean malthusian
parameter
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
S   p j log p j
j 1
pj 
Vj
e rj
Evolutionary principles
1.
Evolutionary dynamics within a single evolving lineage
(Mutation and Selection)
Directionality Principle for Entropy
•
•
2.
Limited Resources: Evolution increases entropy
Variable Resources: Evolution decreases entropy
Evolutionary dynamics within a taxon
(Speciation and Extinction)
Fundamental Theorem of Evolution
•
•
3.
The rate of change of mean entropy is equal to the variance in entropy
Mean entropy increases over geological time
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 entropy
Demographic model

Microstates:
m1
b
 1
A 0


 0
Net reproducti ve function : V j
m2

0
b2



bd 1
Survivorsh ip : l j
Fecundity : m j

Population growth rate:
r  log 
Vj
1   rj
e
r  Growth Rate
md 
0 
0

 
0 
Biological networks
Macrostates from microstates
A  (aij )  0
P
 { pij }
Au  λu
pij  0  aij  0
Ann. App. Prob. (1974)
Variational principle for log λ
Theorem 1. log   sup    ij pij log pij    ij pij log aij 
P
2. There exist a unique matrix Pˆ  ( pˆ ij ) for which
the sup is attained
pij 
3.
aiju j
u j
log    i pij log pij   i pij logaij
r  H 
Demographic networks
Macrostates from microstates
m1
b
 1
A 0


 0
m2

0
b2



bd 1
md 
0 
0

 
0 
pj 
 p1
1

P0


 0
Vj
e
rj
p2  pd 
0  0 
1  0

 
0 1 0 
r  log λ
E S
r 
T T
S   p j log p j
Entropy:
Reproductive potential: E   p j log V j
Generation time:
T   jp j
rT  E  S
Properties of entropy
S   p j log p j
1.
Measure of uncertainty
2.
Measure of diversity
3.
Measure of robustness
pj 
Vj
e rj
Uncertainty measure
d
S   p j log p j
j 1
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)
p1  0, p2  0,  , pd 1  0, pd  1
p1  0, p2  0,, pd  0
Robustness
2
1
3
...
d
Genealogies: Set of paths of the graph
Path:
( x x x ...) x  (1,2,...d )
0 1 2
i
A  (aij )  0
Matrix associated with the graph
  log a x
0 x1
S n  log a x0 x1  log a x1 x2  ...  a xn1 xn
1

  lim  S n ( x) 
n  n


1
Pn ( )  S n ( )  
n
Robustness
1
Pn ( )  S n ( n)  
n
1

Qn ( )  Prob. that the sample mean  S n  differs from the mean 
n

by more than  .
 1

R  lim  log Qn ( ) 
n 
 n

Theorem:
SR  0
Annals. App. Prob.(1994)
Reproduction potential and
resource constraints
E   p j log V j
E  rT  S
E  0  rT  S ; E  0  rT  S
Proposition: In Populations in dynamical equilibrium with
resource conditions
E<0: Constant resource
E>0: Variable resource
The Entropic Selection Principle
Entropy 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
Invasion dynamics
Evolutionary entropy predicts the outcome of
competition
Limited Resources
S
X
S
S*
X
X*
X*
S*
s*  s : X wins
s*  s : X * wins
Variable Resources
S
X
S*
X*
s*  s : X * wins
S
X
X*
S*
s*  s : X wins
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
S   p j log p j
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 species
2.
Entropy decreases in opportunistic species
Evolutionary dynamics within a taxon
Long run changes in mean entropy as one phyletic
lineage replaces another under speciation and
background extinction.
S   pi Si
dS
 VS
dt


The rate of change in mean entropy is equal to
variance in entropy
Mean entropy increases
Evolutionary dynamics within a clade
Long run changes in maximum entropy as one clade
replaces another under mass extinction
Sˆ  max (Si )
Sˆ  0
The upper entropic limit increases as
one clade replaces another over
geological time.
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 taxon
The rate of change of mean entropy is equal to the
variance in entropy
3. Evolutionary dynamics within a clade
The 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 time
T  W
T  Physiological time,

W  Body size
2   1
3
Physiological time
1.
2.
3.
Cycle time of metabolic processes
Generation time
Life span
Physica A.
(2003)
Entropy and generation time
S   p j log p j
T   jp j
Theorem
S  a  log T
a  Taxon specific constant
The evolution and distribution of
species body size
S   p j log p j
pj 
Vj
e rj
Relation between entropy S
and body size W
S  a1  b1 log W
Empirical study
Relation 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.
2.
Increase in body size : North America
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
S  a2  log L
Directionality theory predicts species variation in
life span
Empirical observation
Entropy 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.
2.
Darwinian Fitness is characterized by
evolutionary entropy
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
Evolutionary parameters
~
Free energy, F
Growth rate, r
~
Thermodynamic entropy, S
Demographic rate, S
~
Reciprocal generation time,T
Temperature, T
~
Reproductive potential 
Mean energy, E
~
~ ~~
F  E  ST
r S
T
Relation between thermodynamic
principles and evolutionary principles
Thermodynamic entropy:
~
S   ~
p j log ~
pj
Diversity of energy distribution
Demographic entropy:
S   p j log p j
Diversity of energy flow
The entropic principle for evolution is a non—equilibrium
analogue of the entropic principle for physical systems.
Relation between thermodynamic
principles and evolutionary principles
~ dQ
Thermodynamic entropy: dS  ~
T
Demographic entropy:
dS  T dP
Analytic relation between generation time, T and
~
Temeprature T :
 h 
  G * 
T   ~  exp 
 kT 

~
T


Theorem: The entropc principle for thermodynamic
systems is the limit (G*  0) of the entropic principle for
evolutionary processes.