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Speciation Dynamics of an Agentbased 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 2nd nearest neighbor is blue (hashed white line). Therefore, white’s cluster seed includes: white, yellow, and blue. Generations 00.40 00.44 µ 00.50 01.20 1 50 1000 2000 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 Control case: Random mating Generations 02.00 µ 07.00 12.00 1 50 1000 2000 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: A2A – Coalescence: 2AA – Death: A0 N N+1 Production (A→2A) Coalescence (2A →A) Death (A →ᴓ) • Offspring only coalesce from neighboring parent particles Chapter 2: Self-coalescence • Not explicitly considered in basic 1+1 DP lattice model • Mimics diffusion process 2 • 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 1 Chapter 2: Neighbor Coalescence • Offspring from neighboring parents Coalescence coalesce 2 (2A →A) 1 2 1 Assuming Directed Percolation • Simple mean-field equation (essentially logistic) – Density as order parameter • 𝜕𝑡 𝜌 = 𝜏𝜌 − 𝑔𝜌2 – τ is the new control parameter • should depend on mutability and coalescence radius • 𝜏 = 𝜎𝑝 −𝜎𝑑 – 𝜎𝑝 is effective production rate (fitness & self-coalescence) – 𝜎𝑑 is effective death rate (random death) – g is a coupling term • g = 𝜎𝑐 , the effective coalescence rate (”neighbor rivalry”) Chapter 2: Neutral Bacterial Meanfield • • • • 𝑁𝑔𝑒𝑛 𝑓 𝑗=1 𝑗 Birth: = 2𝑁𝑔𝑒𝑛 Coalescence: 1 − 𝑃𝑐 = 1 − 𝑃𝑠𝑐 + 𝑃𝑛𝑐 Random death: (1 − 𝑃𝑟𝑑 ) ∆𝜌 = 𝜌 1 − 2𝑃𝑠𝑐 − 2𝑃𝑟𝑑 − 𝜌2𝑃𝑛𝑐 – Effective production rate = 𝜎𝑝 = 1 − 2𝑃𝑠𝑐 – Effective death rate = 𝜎𝑑 = 2𝑃𝑟𝑑 – Effective coalescence rate = 𝜎𝑐 ≈ 2𝑃𝑛𝑐 𝜌 ∝ 𝜌? 𝜏 = 𝜎𝑝 −𝜎𝑑 • Possibly a coupled dynamical equation for nearest neighbor spacing • 𝜏 = 1 − 2𝑃𝑠𝑐 − 2𝑃𝑟𝑑 & g𝜌 = 2𝑃𝑟𝑑 • ∆𝜌 = 𝜏𝜌 − 𝑔𝜌2 • Without nc, current prediction for critical mutability (~0.30) is <10% from simulation (~0.33) 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 reactiondiffusion • Clusters can produce n>1 offspring clusters • AnA (production) • Clusters go extinct • A0 (death) • m>1 or more clusters mix • mAA (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, 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. • 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. Dees & Bahar (2010) µ = 0.38 µ = 0.40 slope ~ -3.4 • Power law distribution of cluster sizes µ = 0.42 • 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 Clark & Evans Nearest Neighbor Test Asexual Fission • Clustered <= 0.38 (peak) • Dispersed >= 0.44 • Better than 1% significance Assortative Mating • Clustered <= 0.46 (peak) • Dispersed >= 0.54 • Better than 1% significance Temporal Percolation Spatial Percolation • 𝑁𝑖+1 = 𝑁𝑖 𝑗=1 𝑓𝑗 1 − 𝑃𝑠𝑐 − 𝑃𝑛𝑐 1 − 𝑃𝑟𝑑 • 𝑁𝑖+1 = 2𝑁𝑖 − 2𝑁𝑖 𝑃𝑠𝑐 − 2𝑁𝑖 𝑃𝑛𝑐 1 − 𝑃𝑟𝑑 • 𝑁𝑖+1 = 2𝑁𝑖 − 2𝑁𝑖 𝑃𝑠𝑐 − 2𝑁𝑖 𝑃𝑛𝑐 −2𝑁𝑖 𝑃𝑟𝑑 + (2𝑁𝑖 𝑃𝑠𝑐 + 2𝑁𝑖 𝑃𝑛𝑐 )𝑃𝑟𝑑 • 𝑁𝑖+1 − 𝑁𝑖 = 2𝑁𝑖 − 2𝑁𝑖 𝑃𝑠𝑐 − 2𝑁𝑖 𝑃𝑛𝑐 − 2𝑁𝑖 𝑃𝑟𝑑 − 𝑁𝑖 • ∆𝑁 = 𝑁 − 2𝑁𝑃𝑠𝑐 − 2𝑁𝑃𝑛𝑐 − 2𝑁𝑃𝑟𝑑
