Statistical Mechanics and Evolutionary Theory

1. Statistical MechanicsandEvolutionary 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. 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 ofDarwinian 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. 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. 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 FitnessCritique • 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 entropyDemographic model • Microstates: • Population growth rate:

28. P P Biological networksMacrostates from microstates Ann. App. Prob. (1974) 3.

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

33. Robustness 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. S S S* X X* X X* S* S S* S X X* X X* S* Invasion dynamicsEvolutionary entropy predicts the outcome of competition Limited Resources Variable Resources

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. 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 studyRelation between entropy and body size

47. Directionality theory predicts evolutionary changes in body size Changes in body size within a singleevolving 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 Meanbody size increases within a taxon ( Cope‘s Rule )

50. Increase in mean body size within the equid taxon