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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 - PowerPoint PPT Presentation
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Kinetic Monte Carlo Simulations of Statistical-mechanical Models of
Biological Evolution
Per Arne Rikvold and Volkan SevimSchool of Computational Science,
Center for Materials Research and Technology, and Department of Physics,
Florida State University
R.K.P. ZiaCenter for Stochastic Processes in Science and Engineering,
Department of Physics, Virginia TechSupported 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).
DynamicsProbability that an individual of genotype I has F
offspring in generation t before dying is PI({nJ(t)}).
Probability of dying without offspring is (1PI).
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 [1,+1], except MII = 0.
]/)()(/)(exp[1
1)})(({
0tottot NtNtNtnMtnP
JJIJ
JI
Deterministic approximation]1)})[(({)()1( tnFPtntn JIII
)(
2)()( )()})(({)()/(
IKJIKIK OtnPtnL
]/)()(/)(exp[1
1)})(({
0tottot NtNtNtnMtnP
JJIJ
JI
: mutation rate per individual)exp(1
1)(
xxP
Mutations
Each individual offspring undergoes mutation to a different genotype with probability /L per gene and individual.
Fixed points for = 0
)})(({)()1( tnFPtntn JIII
Without mutations the equation of motion reduces to
such that the fixed-point populations satisfy
*Jn
*Jn
1)})(({ * tnFP JI
This yields the total population for an N-species fixed point:
IJIJJ
J FNnN10
**tot ~
1)1ln(
M
where is the inverse of the submatrix of MIJ in N-species space.There are also expressions for the individual .
IJ1~ M*Jn
Stability of fixed pointsThe internal stability of the fixed point is determined
by the eigenvalues of the community matrix
The stability against an invading mutant i is given by the invader’s invasion fitness:
IJIJ
IJI
IJ
nJ
IIJ FM
N
n
Ftn
tn
I
1*tot
*
~2
1ln1
1)(
)1(~
* M
JKJK
JKJKiJ
i
i
MF
F
tn
tn
11 ~~1exp11
ln)(
)1(ln
MM
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
• = 103
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(F1)]
• Diversity, D(t), gives the number of heavily populated species. Obtained as D(t) = exp[S(t)]
where
S(t) = I [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 > 1000nI [101,1000]nI [11,100]nI [2,10]nI = 1
Quasi-steady states (QSS) punctuated by active periods. Self-similarity.
Stability of Quasi-steady States (QSS)Multiplication rate of small-population mutant i in
presence of fixed point of N resident species, J, K:
JKJK
JKJKiJ
i
i
MF
F
tn
tn
11 ~~1exp11
)(
)1(
MM
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(F1)]
Species’ lifetime distributions
Stationarity of diversity measures
• 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
Running time and ensemble averages.
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 1/2 distribution of QSS lifetimes leads to 1/f-like flicker noise
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, including competition and adaptive foraging