1. Statistical Mechanics and Evolutionary Theory
Lloyd Demetrius
Harvard University, Cambridge, Mass., USA
And
Max Planck Institute, Berlin, Germany

2. Evolutionary changes in morphological complexity
Ecological time scale (Single evolving lineage)
Increases and decreases in adult body size
Geological time scale (Phyletic lineages)
Increases in mean body size
Geological time scale (Clades)
Increases in maximum body size

3. Evolution of the horse family

4. Changes in body size within the equid lineages
Increase in body size : North America
Decrease in body size : Europe

5. Increase in mean body size within the equid taxon

7. Increase in maximum body size over the history of life

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

9. 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)

10. Darwinian fitness
The efficiency with which organisms transform resources into net offspring production

11. Levels of biological organization
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

12. If W2 > W1 : then X2 replaces X1
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
If W2 > W1 : then X2 replaces X1
Fitness
The efficiency to transform respurces into net-offspring production X1 X2

13. Darwin‘s theory
Evolutionary Principle: Evolution by natural selection results in an increase in fitness
Explanatory Power
Variation in life history, body size, life span within and between species
The adaptation of species to their habitat
The changes in morphological complexity over time

14. 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

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

16. Characterizations of Darwinian Fitness
Malthusian parameter (1930) Fisher‘s theory
Evolutionary entropy (1974) Directionality theory

17. The theory of evolution by natural selection
is the doctrine of Malthus applied to plants
and animals.
Darwin (1859)

18. Population described by d age-classes
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

19. Malthusian parameter as Darwinian fitness
Matrix Representation of Graph
Characterization of r :

20. Growth rate r characterizes Darwinian Fitness:
Fisher‘s Theory
Growth rate r characterizes Darwinian Fitness:
Malthusian Principle: r predicts the outcome of competition between variant and incumbent types r r* r X* X X* X r*

21. 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

22. 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

23. Darwin‘s theory of evolution by natural selection is the doctrine of Gibbs, Boltzmann and Clasius applied to plants and animals.
Directionality theory
(1974)

24. Directionality theory
Evolutionary entropy, S , characterizes Darwinian Fitness

25. Evolutionary principles
Evolutionary dynamics within a single evolving lineage
(Mutation and Selection)
Directionality Principle for Entropy
Limited Resources: Evolution increases entropy
Variable Resources: Evolution decreases entropy
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
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

26. 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

27. Origin of evolutionary entropy Demographic model
Microstates:
Population growth rate:

28. Biological networks Macrostates from microstatesP
Ann. App. Prob. (1974) P 3.

29. Demographic networks Macrostates from microstates
Entropy:
Reproductive potential:
Generation time:

30. Properties of entropy Measure of uncertainty Measure of diversity
Measure of robustness

31. 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)

32. Robustness Genealogies: Set of paths of the graph Path:
Matrix associated with the graph 1 3 2 d
...

33. Robustness Theorem: Annals. App. Prob.(1994)
Prob. that the sample mean
Theorem:
Annals. App. Prob.(1994)

34. Reproduction potential and resource constraints
Proposition: In Populations in dynamical equilibrium with resource conditions
E<0: Constant resource
E>0: Variable resource

35. 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

36. Invasion dynamics Evolutionary entropy predicts the outcome of competition
Limited Resources X X* S S* X X* S* S
Variable Resources X X* S S*

37. Predictions of directionality theory
Based on the entropic principes of selection we predict the evolutionary changes at three different levels of biological organization.
Single evolving lineage – Mutation and selection
Aggregate of phyletic lineages – Speciation and background extinction
An ensemble of clades – Speciation and mass extinction

38. 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:
Entropy increases in equilibrium species
Entropy decreases in opportunistic species

39. 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

40. 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.

41. Main tenets of the evolutionary process
Evolutionary dynamics within a single evolving lineage
Equilibrium species: Entropy increases
Opportunistic species: Entropy decreases
Evolutionary dynamics within a taxon
The rate of change of mean entropy is equal to the variance in entropy
Evolutionary dynamics within a clade
The upper entropic limit increases as one clade replaces another

42. Implications of the evolutionary tenets
Evolution of life history
Evolution of body size
Evolution of senescence

43. Body size and physiological time
Allometric relations
Body size and physiological time
Physica A.
(2003)
Physiological time, Body size
Physiological time
Cycle time of metabolic processes
Generation time
Life span

44. Entropy and generation time
Theorem

45. The evolution and distribution of species body size
Relation between entropy S
and body size W

46. Empirical study Relation between entropy and body size

47. 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

48. Changes in body size within the equid lineages
Increase in body size : North America
Decrease in body size : Europe

49. 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 )

50. Increase in mean body size within the equid taxon

51. Evolutionary changes in the upper limit of bodysize
The upper limit of body size increases as one clade replaces another over geological time.

52. Changes in the upper limit of body size

53. The evolution of life span
Evolutionary entropy is analytically related to life span L
Directionality theory predicts species variation in life span

54. Empirical observation Entropy and life span

55. 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

56. Intensity of natural selection

57. Conclusion Darwinian Fitness is characterized by evolutionary entropy
Diversity of species and evolutionary change in complexity can be described in terms of the following tenets:
Population level:
Equilibrium species: Entropy increases
Opportunistic species: Entropy decreases
Phyletic level:
Mean entropy increases
Clade:
The upper entropic limit increases

58. Relation between thermodynamic variables and evolutionary parameters
Free energy,
Thermodynamic entropy,
Temperature,
Mean energy,
Evolutionary parameters
Growth rate,
Demographic rate, Reciprocal generation time,
Reproductive potential

59. 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.

60. 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.