Transcript Clustering and Phase Transitions on a Neutral

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: A2A
– Coalescence: 2AA
– Death: A0
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
• 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, 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