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

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

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

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

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

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

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

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

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

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”

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.

µ Generations 

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

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)

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

µ Generations  Control case: Random mating

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

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

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

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

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

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 )

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

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

Assuming Directed Percolation

Chapter 2: Neutral Bacterial Mean- field

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?

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

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

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)

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

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 ?

Related Sources Dees, N.D., Bahar, S. Noise-optimized speciation in an evolutionary model. PLoS ONE 5(8): e11952, de Aguiar, M.A.M., Baranger, M., Baptestini, E.M., Kaufman, L., Bar-Yam, Y. Global patterns of speciation and diversity. Nature 460: , Young, W.R., Roberts, A.J., Stuhne, G. Reproductive pair correlations and the clustering of organisms. Nature 412: , 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, Perkins, E. Super-Brownian Motion and Critical Spatial Stochastic Systems. Solé, Ricard V. Phase Transitions. Princeton University Press, Yeomans, J. M. Statistical Mechanics of Phase Transitions. Oxford Science Publications, Henkel, M., Hinrichsen, H., Lübeck, S. Non-Equilibrium Phase Transitions: Absorbing Phase Transitions. Springer, 2009.

Dees & Bahar (2010)

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

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

Temporal Percolation

Spatial Percolation