Characterizing Phase Transitions in a Model of Neutral Evolutionary Dynamics

Adam David ScottDepartment of Physics & Astronomy

University of Missouri at St. LouisAPS March Meeting – Baltimore, MD

18 March 2013

Acknowledgements

• Dr. Sonya Bahar• Dawn King• Dr. Nevena Marić

• McDonnell Foundation Complex Systems Grant

• UMSL Department of Physics & Astronomy

• Model• Phase transitions– Evidence for two separate behaviors

• Obtaining critical exponents for each

Model

• ONLY phenotypic evolution– No genetics & no physical space– 2D Continuous with absorbing boundaries

• Asexual• Branching & coalescing random walks– Reaction-diffusion process

• Mutability, μ– Maximum possible offspring mutation from parent– Control parameter

• Clustering

Dawn’s: Random death percentage for each organismControl parameter is random death percentage

THIS: Random death percentage of populationControl parameter is mutability

Model: Clustering

• Phenetic species– Speciation by phenotypes

• Cluster seeds– Reference• Nearest neighbor• Second nearest neighbor

• Closed sets of cluster seeds

Scott et al. (submitted)

Generations

Mutability

Phase Transitions: Order Parameters

Scott et al. (submitted)

0.34 0.38 0.34 0.38

• <Population> • <Number of Clusters>

Evidence: Times to extinction

Evidence: Nearest Neighbor Index

0.34 0.38

• Clark & Evans – Average nearest neighbor distance / random expectation

Scott et al. (submitted)

Phase Transitions

• Temporal• 2nd order

non-equilibrium– Absorbing state:

extinction• Directed percolation

• Spatial• 2nd order

equilibrium• Isotropic percolation

Directed Percolation: Critical Exponents

• Density scaling, – Density decay rate,

• Correlation length, – Decay rate,

• Correlation time, • Need 2 for classification–

(Henkel, Hinrichsen, Lübeck)

0.310 0.315 0.320

0.325

0.330

0.335

0.340

μ 0.310 0.315 0.320 0.325 0.330α 3.489 3.326 2.589 1.773 0.462

Directed Percolation Density Decay Rate

• This model

• Compared with others– (Hinrichsen)• 2.6%

– (Lauritsen)• 2.4%

Isotropic Percolation: Critical Exponents

• Static measures– Fisher exponent• Distribution of cluster mass, ns, for size, s

– Cluster fractal dimension • Cluster mass (N)

– Number of organisms in cluster

• Cluster diameters (L)– Longest of the shortest paths

• Still running!

(Lesne)

Conclusions

• Two 2nd order phase transitions– Non-equilibrium (directed percolation)– Equilibrium (isotropic percolation)

• Possibility for this behavior in real biology?– First experimental directed percolation observed

in 2007 for liquid crystals (Takeuchi et al.)

Absorbing StateExtinction

Surviving Surviving

𝜇 0.335 𝜇 0.38

Aggregated Spanning

Thank you!

Absorbing StateExtinction

Surviving Surviving

𝜇 0.335 𝜇 0.38

Aggregated Spanning

References• Henkel, Hinrichsen, Lübeck, Non-Equilibrium Phase Transitions:

Vol 1: Absorbing Phase Transitions, 2009• Hinrichsen, “Nonequilibrium Critical Phenomena and Phase

Transitions into Absorbing States”, arXiv, 2000• Lauritsen, Sneppen, Marošová, Jensen, ”Directed percolation with

an absorbing boundary”, 1997.• Lesne, Renormalization Methods: Critical Phenomena, Chaos,

Fractal Structures, 1998• Scott, King, Marić, Bahar, “Clustering and Phase Transitions on a

Neutral Landscape”, submitted• Takeuchi, Kuroda, Chaté, Sano, “Directed percolation criticality in

turbulent liquid crystals”, 2007, 2009