# Kinetic Monte Carlo Simulations of Statistical-mechanical Models

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```					Kinetic Monte Carlo Simulations of
Statistical-mechanical Models of
Biological Evolution
Per Arne Rikvold and Volkan Sevim
School of Computational Science,
Center for Materials Research and Technology,
and Department of Physics,
Florida State University
R.K.P. Zia
Center for Stochastic Processes in Science and Engineering,
Department of Physics, Virginia Tech
Supported by FSU (SCS and MARTECH), VT, and NSF
Biological Evolution and
Statistical Physics
• Complicated field with many
unsolved problems.
• Complex, interacting nonequilibrium
problems.
• Need for simplified models with universal
properties. (Physicist’s approach.)
Modes of Evolution
• Does evolution proceed uniformly or
in fits and starts?
• Scarcity of intermediate forms (“missing links”)
in the fossil record may suggest fits and starts.
• Fit-and-start evolution termed punctuated equilibria
by Eldredge and Gould.
• Punctuated equilibria dynamics resemble
nucleation and growth in phase transformations
and
stick-slip motion in friction and earthquakes.
Models of Coevolution
• Among physicists, the best-known coevolution
model is probably the Bak-Sneppen model.
• The BS model acts directly on interacting
species, which mutate into other species.
• But: in nature selection and mutation act
directly on individuals.
Individual-based Coevolution Model
• Binary, haploid genome of length L gives
2L different potential genotypes. 01100…101
• Considering this genome as coarse-grained, we
consider each different bit string a “species.”
• Asexual reproduction in
discrete, nonoverlapping generations.
• Simplified version of model introduced by
Hall, Christensen, et al.,
Phys. Rev. E 66, 011904 (2002);
J. Theor. Biol. 216, 73 (2002).
Dynamics
Probability that an individual of genotype I has F
offspring in generation t before dying is PI({nJ(t)}).
Probability of dying without offspring is (1-PI).
1
PI ({nJ (t )}) 
1  exp[ - M IJ nJ (t ) / N tot (t ) N tot (t ) / N 0 ]
J
N0: Verhulst factor limits total population Ntot(t).
MIJ : Effect of genotype J on birth probability of I.
MIJ and MJI both positive: symbiosis or mutualism.
MIJ and MJI both negative: competition.
MIJ and MJI opposite sign: predator/prey relationship.
Here: MIJ quenched, random e [-1,+1], except MII = 0.
Deterministic approximation
nI (t  1)  nI (t ) FPI ({nJ (t )})[1 -  ]
 ( / L)  nK ( I ) (t ) PK ( I ) ({nJ (t )})  O( 2 )
K (I )

P( x) 
1        : mutation rate
1  exp( x)
per individual

1
PI ({nJ (t )}) 
1  exp[ - M IJ nJ (t ) / N tot (t ) N tot (t ) / N 0 ]
J
Mutations
Each individual offspring undergoes
mutation to a different genotype with
probability /L per gene and individual.
Fixed points for  = 0
Without mutations the equation of motion reduces to
nI (t  1)  nI (t ) FPI ({nJ (t )})
such that the fixed-point populations           n*
J    satisfy
FPI ({n* (t )})  1
J

This yields the total population for an N-species fixed point:
                            
  n*  N 0 ln( F - 1)                 
1        n   *

 
J

*
N                                      ~ -1        
M
tot        J
J

             IJ
IJ


 
~ -1
where M       is the inverse of the submatrix of MIJ in N-species space.
IJ
*
There are also expressions for the individual n J .
Stability of fixed points
The internal stability of the fixed point is determined
by the eigenvalues of the community matrix
                                       
nI (t  1)                 1 n 
*

 M IJ - ln F - 1 -
~                                                                 2
 IJ                  -  IJ  1 - 
 
I
~ -1           
nJ (t ) n* 
I
 FN
*

tot

M
IJ
IJ


The stability against an invading mutant i is given by
                                                           
                                                           
 ni (t  1)                           F                                  
 n (t )   ln 
ln                                                                       
 i                                         
1  F - 1 exp1 -  M iJ M
~ -1      
JK    
~ -1
M      
JK  

                JK                     JK             
Monte Carlo algorithm:
3 layers of nested loops
1. Loop over generations t
2. Loop over genotypes I with nI > 0 in t
3a. Loop over individuals in I, producing F
offspring with probability PI({nJ(t)}), or
killing individual with probability 1-PI
3b. Loop over offspring to mutate with
probability 
Simulation parameters
• N0 = 2000
• F=4
• L = 13      213 = 8192 potential genotypes
•  = 10-3
This choice ensures that both Ntot and the
number of populated species are << the
total number of potential genotypes, 2L
Main quantities measured
• Normalized total population,
Ntot(t)/[N0 ln(F-1)]
• Diversity, D(t), gives the number of
heavily populated species. Obtained as
D(t) = exp[S(t)]
where
S(t) = - SI [nI(t)/Ntot(t)] ln [nI(t)/Ntot(t)]
is the information-theoretical entropy
(Shannon-Wiener index).
Simulation Results
Diversity,
D(t)
Ntot(t),
normalized

nI > 1000
nI e [101,1000]
nI e [11,100]
nI e [2,10]
nI = 1

Quasi-steady states (QSS) punctuated by active periods. Self-similarity.
Multiplication rate of small-population mutant i in
presence of fixed point of N resident species, J, K:
ni (t  1)                              F

                                        
ni (t )
1  F - 1 exp1 -  M iJ M 
~ -1      
JK    
~ -1
M      
JK 
 JK                     JK             
Active and Quiet Periods

Histogram of entropy changes   Histograms of period durations
Power Spectral Densities
(squared norm of Fourier transform)

PSD of D(t)       PSD of Ntot(t)/[N0 ln(F-1)]
Stationarity of diversity measures
Running time and ensemble averages.
• Total species richness, N(t)
• No. of species with nI > 1
• Shannon-Wiener D(t)
• Mean Hamming distance
between genotypes
• Total population Ntot(t)/N0ln3
• Standard deviation of
Hamming distance
Summary of completed work
• Simple model for evolution of haploid, asexual
organisms
• Based on birth/death process of individual
organisms
• Shows punctuated equilibria of quasi-steady states
(QSS) of a few populated species, separated by
active periods
• Self-similarity and 1/t2 distribution of QSS
P.A.R. and R.K.P.Z., Phys. Rev. E 68, 031913
(2003); J. Phys. A 37, 5135 (2004)
V.S. and P.A.R., arXiv:q-bio.PE/0403042
Current work and future plans
•   Predator/prey models
•   Community structure and food webs
•   Stability vs connectivity
•   Effects of different functional responses,