Speciation Dynamics of an Agent-based Evolution Model in Phenotype Space

Adam D. ScottCenter for Neurodynamics

Department of Physics & AstronomyUniversity of Missouri – St. Louis

Oral Comprehensive Exam5*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

BasisBiological• 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)

• ApplicabilityPhysics & 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

– SympatricCluster 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.

1 50 1000 2000

00.40

00.44

00.50

01.20

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

1 50 1000 2000

02.00

07.00

12.00

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: A2A– Coalescence: 2AA– Death: A0

• 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

• Simple mean-field equation (essentially logistic)– Density as order parameter

• – τ is the new control parameter

• should depend on mutability and coalescence radius

– is effective production rate (fitness & self-coalescence)– is effective death rate (random death)

– g is a coupling term• g = , the effective coalescence rate (”neighbor rivalry”)

Chapter 2: Neutral Bacterial Mean-field

• Birth: • Coalescence: • Random death:

– Effective production rate = – Effective death rate = – Effective coalescence rate = ?

• Possibly a coupled dynamical equation for nearest neighbor spacing

• &

• Without nc, current prediction for critical mutability (~0.30) is <10% from simulation (~0.33)

𝜏=𝜎𝑝−𝜎 𝑑

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• 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

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 TestAsexual Fission Assortative Mating

Temporal Percolation

Spatial Percolation