1. Speciation Dynamics of an Agent- based Evolution Model in Phenotype Space Adam D. Scott Center for Neurodynamics Department of Physics & Astronomy University of Missouri – St. Louis Oral Comprehensive Exam 5*31*12

2. Proposed Chapters Chapter 1: Clustering and phase transitions on a neutral landscape (completed) Chapter 2: Simple mean-field approximation to predict universality class & criticality for different competition radii Chapter 3: Scaling behavior with lineage and clustering dynamics

3. Basis Biological Modeling – Phenotype space with sympatric speciation Phenotype = traits arising from genetics Sympatric = “same land” / geography not a factor Possibility vs. prevalence – Role of mutation parameters as drivers of speciation Evolution = f(evolvability) Applicability Physics & Mathematics Branching & Coalescing Random Walk – Super-Brownian – Reaction-diffusion process Mean-field & Universality – Directed &/or Isotropic Percolation

4. Broader Context/ Applications Bacteria Example: microbes in hot springs in Kamchatka, Russia Yeast and other fungi – Reproduce sexually and/or asexually – Nearest neighbors in phenotype space can lead naturally to assortative mating Partner selection and/or compatibility most likely – MANY experiments involve yeast

5. Model: Overview Agent-based, branching & coalescing random walkers – “Brownian bugs” (Young et al 2009) Continuous, two-dimensional, non-periodic phenotype space – traits, such as eye color vs. height Reproduction: Asexual fission (bacterial), assortative mating, or random mating – Discrete fitness landscape Fitness = # of offspring Natural selection or neutral drift Death: coalescence, random, & boundary

6. Model: “Space” Phenotype space (morphospace) – Planar: two independent, arbitrary, and continuous phenotypes – Non-periodic boundary conditions – Associated fitness landscape

7. Model: Fitness Natural Selection Darwin Varying fitness landscape over phenotype space – Selection of most fit organsims – Applicable to all life Fitness = 1-4 – (Dees & Bahar 2010) Neutral Theory Hubbell – Ecological drift Kimura – Genetic drift Equal (neutral) fitness for all phenotypes – No deterministic selection – Random drift – Random selection Fitness = 2

8. Model: Mutation Parameter Mutation parameter -> mutability – Ability to mutate about parent(s) Maximum mutation All organisms have the same mutability Offspring uniformly generated Example of assortative mating assuming monogamous parents

9. Model: Reproduction Schemes Assortative Mating – Nearest neighbor is mate Asexual Fission – Offspring generation area is 2µ*2µ with parent at center Random Mating – Randomly assigned mates

10. Model: Death Coalescence – Competition – Offspring generated too close to each other (coalescence radius) Random – Random proportion of population (up to 70%) – “Lottery” Boundary – Offspring “cliff-jumping”

11. Model: Clusters Clusters seeded by nearest neighbor & second nearest neighbor of a reference organism – A closed set of cluster seed relationships make a cluster = species Speciation – Sympatric Cluster seed example: The white organism has nearest neighbor, yellow (solid white line). White’s 2 nd nearest neighbor is blue (hashed white line). Therefore, white’s cluster seed includes: white, yellow, and blue.

12. 15010002000 00.40 00.44 00.50 01.20 µ Generations 

13. Chapter 1: Neutral Clustering & Phase Transitions Non-equilibrium phase transition behavior observed for assortative mating and asexual fission, not for random mating Surviving state clustering observed to change behavior above criticality

14. Assortative Mating Potential phase transition – Extinction to Survival – Non-equilibrium Extinction = absorbing – Critical range of mutability Large fluctuations Power-law species abundances Peak in clusters  Quality (Values averaged over surviving generations, then averaged over 5 runs)

15. Asexual Fission Slightly smaller critical mutability Same phase transition indicators Same peak in clusters Similar results for rugged landscape with Assortative Mating

16. 15010002000 02.00 07.00 12.00 µ Generations  Control case: Random mating

17. Random Mating Population peak driven by mutability & landscape size comparison No speciation Almost always one giant component Local birth not guaranteed!

18. Conclusions Mutability -> control parameter – Population as order parameter – Continuous phase transition extinction = absorbing state – Directed percolation universality class? Speciation requirements – Local birth/ global death (Young, et al.) – Only phenotype space (compare de Aguiar, et al.) – For both assortative mating and asexual fission

19. Chapter 1: Progress Manuscript submitted to the Journal of Theoretical Biology on April 16 Under review as of May 2 No update since

20. Chapter 2 Goal: to have a tool which predicts critical mutability and critical exponents for a given coalescence radius = Mean-field equation – Directed percolation (DP) & Isotropic percolation (IP) Neutral landscape with fitness = 2 for all phenotypes – May extend to arbitrary fitness if possible Asexual reproduction – Will attempt extension to assortative mating

21. Temporal & Spatial Percolation Temporal  Survival – Time to extinction becomes computationally infinite – DP Spatial  “Space filling” – Largest clusters span phenospace – IP

22. 1+1 Directed Percolation Reaction-diffusion process of particles – Production: A  2A – Coalescence: 2A  A – Death: A  0 Offspring only coalesce from neighboring parent particles N N+1 Production (A→2A) Coalescence (2A →A) Death (A → ᴓ )

23. Chapter 2: Self-coalescence Not explicitly considered in basic 1+1 DP lattice model Mimics diffusion process May act as a correction to fitness, giving effective birth rate “Sibling rivalry” – Probability for where the first offspring lands in the spawn region – Probability that the second offspring lands within a circle of a given radius whose center is offspring one and its area is also in the spawn region 2 1

24. Chapter 2: Neighbor Coalescence Offspring from neighboring parents coalesce 1 Coalescence (2A →A) 2 1 2

25. Assuming Directed Percolation

26. Chapter 2: Neutral Bacterial Mean- field

27. Chapter 2: Neighbor Coalescence Increased rate with larger mutability & coalescence radius – Varies amount of overlapping space for coalescence Should depend explicitly on nearest neighbor distances May be determined using a nearest neighbor index or density correlation function Possibility of a second dynamical equation of nearest neighbor measure coupled with density?

28. Chapter 2: Progress Have analytical solution for sibling rivalry Have method in place to estimate neighbor rivalry Waiting for new data for estimation Need to finish simple mean-field equation Need data to compare mean-field prediction of criticality for different coalescent radii Determine critical exponents – Density, correlation length, correlation time

29. Chapter 3: Scaling Can organism behavior predict lineage behavior? – Center of “mass”  center of lineage (CL) – Random walk Path length of descendent organisms & CL – Branching & (coalescing) behavior Can organism behavior predict cluster behavior? – Center of species (centroids) – Clustering clusters – Branching & coalescing behavior May determine scaling functions & exponents – Population  # of Clusters? Fractal-like organization at criticality? – Lineage branching becomes fractal? – Renormalization: organisms  clusters

30. Chapter 3: Cluster level reaction- diffusion Clusters can produce n>1 offspring clusters A  nA (production) Clusters go extinct A  0(death) m>1 or more clusters mix mA  A(coalescence)

31. Chapter 3: Predictions Difference of clustering mechanism by reproduction – Assortative mating: organisms attracted (sink driven) Greater lineage convergence (coalescence) – Bacterial: clusters from blooming (source driven) Greater lineage branching (production) Greater mutability produces greater mixing of clusters & lineages Potential problem: far fewer clusters for renormalization

32. Chapter 3: Progress Measures developed for cluster & lineage behavior Extracted lineage and cluster measures from previous data Need to develop concrete method for comparing the BCRW behavior between reproduction types ?

33. Related Sources Dees, N.D., Bahar, S. Noise-optimized speciation in an evolutionary model. PLoS ONE 5(8): e11952, 2010. de Aguiar, M.A.M., Baranger, M., Baptestini, E.M., Kaufman, L., Bar-Yam, Y. Global patterns of speciation and diversity. Nature 460: 384-387, 2009. Young, W.R., Roberts, A.J., Stuhne, G. Reproductive pair correlations and the clustering of organisms. Nature 412: 328-331, 2001. Hinsby Cadillo-Quiroz, Xavier Didelot, Nicole Held, Aaron Darling, Alfa Herrera, Michael Reno, David Krause and Rachel J. Whitaker. Sympatric Speciation with Gene Flow in Sulfolobus islandicus. PLoS Biology, 2012. Perkins, E. Super-Brownian Motion and Critical Spatial Stochastic Systems. http://www.math.ubc.ca/~perkins/superbrownianmotionandcriticalspatialsystems.pdf. http://www.math.ubc.ca/~perkins/superbrownianmotionandcriticalspatialsystems.pdf Solé, Ricard V. Phase Transitions. Princeton University Press, 2011. Yeomans, J. M. Statistical Mechanics of Phase Transitions. Oxford Science Publications, 1992. Henkel, M., Hinrichsen, H., Lübeck, S. Non-Equilibrium Phase Transitions: Absorbing Phase Transitions. Springer, 2009.

34. Dees & Bahar (2010)

35. µ = 0.38µ = 0.40 µ = 0.42 slope ~ -3.4 Power law distribution of cluster sizes Scale-free Large fluctuations near critical point (Solé 2011) Characteristic of continuous phase transition Near criticality parabolic distributions change gradually Mu < critical  concave down Mu > critical  concave up

36. Clustered <= 0.38 (peak) Dispersed >= 0.44 Better than 1% significance Clustered <= 0.46 (peak) Dispersed >= 0.54 Better than 1% significance Clark & Evans Nearest Neighbor Test Asexual FissionAssortative Mating

37. Temporal Percolation

38. Spatial Percolation