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Broken symmetry and critical phenomena in population genetics : the steppingstone model Duty, Timothy Lee
Abstract
In this thesis, I study the behaviour of the SteppingStone model: a stochastic model from theoretical population genetics. It was introduced by Kimura and Weiss [1][2] as a simple model to investigate the interplay of the evolutionary processes of random genetic drift, mutation, migration and selection. In particular, they were interested in the behaviour of spatiallystructured populations when these processes were mediated by local interactions. From the point of view of statistical physics, the SteppingStone model can be viewed as a 2species, nonequilibrium reactiondiffusion model with spatial degrees of freedom and unique fluctuations that arise from irreversible processes at the microscale. My thesis begins with a brief overview of the theory of stochastic processes that includes both the classical treatment using master equations, the FokkerPlanck equation and the Langevin equation, and modern formalisms that map these equations to operators and functional integrals. This is followed by a discussion of the steadystate and critical phenomena in the WrightFisher and Moran models, the nonspatial predecessors of the SteppingStone model. Critical phenomena in these models is shown to be associated with the breaking of a discrete symmetry that results in a discontinuous order parameter. Next, the SteppingStone model is introduced and reformulated using operators and pathintegrals. Kimura and Weiss were able to solve for the steadystate of the SteppingStone model, under certain conditions, but the dynamics of the model remained elusive. This model is related however, to the "Voter" model, which has been wellstudied and is known to have a critical spatial dimension of 2 in that only for d ≤ 2 does the system asymptotically reach one of the 2 degenerate absorbing states. I extend the results of Kimura and Weiss and obtain exact results for the dynamics of the SteppingStone model. Two regimes of the model are analyzed: 1) the neutral regime, where selection has vanished and mutation is viewed as the control parameter. Here a steady state exists, characterized by a correlation length that diverges as the mutation rate, μ, becomes zero; and 2) the broken symmetry or "fixed" state, where selection is small but finite, and mutations so rare that the relevant description concerns the dynamics of "avalanches" or "cascades" of new alleles induced by the initial mutation and perturbing the system from the absorbing state. A unique kind of dynamical critical phenomenon occurs when selective advantage and mutation become negligible. Like the Voter model, it is qualitatively different both above and below a critical dimension d[sub c] = 2. For spatial dimension d ≤ 2 , the critical behaviour is associated with the breaking of a discrete symmetry and corresponds to fixation of one of the two alleles (genotypes). Symmetry breaking in the SteppingStone model is shown to be a consequence of the asymptotic return probability of a random walk—identically one for d ≤ 2 and strictly less than one for d > 2. In addition to the correlation length, the steadystate of the neutral regime is further characterized by a measure of variance defined as the amplitude of the twopoint correlation function evaluated at vanishing separation. In genetics this measure of variance is known as F[sub ST], the fixation index. Exact results are derived for both the steadystate value of FST, and its asymptotic timedependence at the critical point, which approaches 1 in both d = 1 and d = 2. At the critical point, 1 — F[sub ST]~ t[sub 1/2] for d = 1. For d = 2, a much slower decay is found, 1 — F[sub ST] ~ 1/ln(t). The d = 3 critical behaviour of F[sub ST] is that it approaches a constant C < 1 as (C — F[sub ST]) ~ t[sub 1/2]. The constant C is nonuniversal and related to the return probability of a random walk. It approaches 1 for very large values of the dimensionless constant K = [Λ/(2 π² Dn[sub Tg)] where Λ⁻¹ is the spatial scale of the interaction, rg is the generation time, D is the diffusion constant and n is the population density. A related infinitealleles model was studied by both Sawyer [3] and Nagalaki[4]. These authors found similar asymptotics for the probability that two randomly choses individuals are genetically identical under certain assumptions. Sawyer[3] also proved that fixation in the infinite alleles model occurs iff the random walk followed by the individuals is recurrent. The broken symmetry or "fixed" regime of the SteppingStone model has been explored from the point of view of survival of rare mutant alleles, here parameterized by a coefficient of selection s. The exponent u, governing the divergence of the relaxation time as s —> 0 is calculated and found to be v = 2 for d — 1, while for d ≥ 2 it is given by the mean field value of v = 1. Two other exponents for critical spreading processes are determined and scaling arguments are presented and used to find the decay exponents that characterize the timedependent survival probability and its asymptotic value for very long times. These results also establish the upper critical dimension of the model, d[sub c] = 2. Finally, these results are rederived and supplemented by a dynamical renormalization group (RG) analysis. The critical behaviour in d = 1 is found in both regimes to be controlled by nontrivial fixed points. The critical behaviour of the broken symmetry regime for d ≥ 2 and its RG fixed point is that of a critical branching process. For the neutral regime, however, the renormalization group flow is qualitatively different in rf = 2 and 3, reflecting the existence of broken symmetry for rf = 2. The RG flow for the rf = 3 neutral regime contains a line of fixed points with the effective description at large scales given by a Gaussian version of the timedependent LandauGinsburg model. Finally, the renormalization results are found to be valid to all orders of perturbation theory.
Item Metadata
Title 
Broken symmetry and critical phenomena in population genetics : the steppingstone model

Creator  
Publisher 
University of British Columbia

Date Issued 
2000

Description 
In this thesis, I study the behaviour of the SteppingStone model: a stochastic model
from theoretical population genetics. It was introduced by Kimura and Weiss [1][2]
as a simple model to investigate the interplay of the evolutionary processes of random
genetic drift, mutation, migration and selection. In particular, they were interested in
the behaviour of spatiallystructured populations when these processes were mediated
by local interactions. From the point of view of statistical physics, the SteppingStone
model can be viewed as a 2species, nonequilibrium reactiondiffusion model with spatial
degrees of freedom and unique fluctuations that arise from irreversible processes at the
microscale.
My thesis begins with a brief overview of the theory of stochastic processes that
includes both the classical treatment using master equations, the FokkerPlanck equation
and the Langevin equation, and modern formalisms that map these equations to operators
and functional integrals. This is followed by a discussion of the steadystate and critical
phenomena in the WrightFisher and Moran models, the nonspatial predecessors of the
SteppingStone model. Critical phenomena in these models is shown to be associated
with the breaking of a discrete symmetry that results in a discontinuous order parameter.
Next, the SteppingStone model is introduced and reformulated using operators and pathintegrals.
Kimura and Weiss were able to solve for the steadystate of the SteppingStone model,
under certain conditions, but the dynamics of the model remained elusive. This model is
related however, to the "Voter" model, which has been wellstudied and is known to have
a critical spatial dimension of 2 in that only for d ≤ 2 does the system asymptotically
reach one of the 2 degenerate absorbing states.
I extend the results of Kimura and Weiss and obtain exact results for the dynamics
of the SteppingStone model. Two regimes of the model are analyzed: 1) the neutral
regime, where selection has vanished and mutation is viewed as the control parameter.
Here a steady state exists, characterized by a correlation length that diverges as the
mutation rate, μ, becomes zero; and 2) the broken symmetry or "fixed" state, where
selection is small but finite, and mutations so rare that the relevant description concerns
the dynamics of "avalanches" or "cascades" of new alleles induced by the initial mutation
and perturbing the system from the absorbing state.
A unique kind of dynamical critical phenomenon occurs when selective advantage
and mutation become negligible. Like the Voter model, it is qualitatively different both
above and below a critical dimension d[sub c] = 2. For spatial dimension d ≤ 2 , the critical
behaviour is associated with the breaking of a discrete symmetry and corresponds to
fixation of one of the two alleles (genotypes). Symmetry breaking in the SteppingStone
model is shown to be a consequence of the asymptotic return probability of a random
walk—identically one for d ≤ 2 and strictly less than one for d > 2. In addition to
the correlation length, the steadystate of the neutral regime is further characterized
by a measure of variance defined as the amplitude of the twopoint correlation function
evaluated at vanishing separation. In genetics this measure of variance is known as F[sub ST],
the fixation index. Exact results are derived for both the steadystate value of FST, and
its asymptotic timedependence at the critical point, which approaches 1 in both d = 1
and d = 2. At the critical point, 1 — F[sub ST]~ t[sub 1/2] for d = 1. For d = 2, a much
slower decay is found, 1 — F[sub ST] ~ 1/ln(t). The d = 3 critical behaviour of F[sub ST] is that
it approaches a constant C < 1 as (C — F[sub ST]) ~ t[sub 1/2]. The constant C is nonuniversal
and related to the return probability of a random walk. It approaches 1 for very large
values of the dimensionless constant K = [Λ/(2 π² Dn[sub Tg)] where Λ⁻¹ is the spatial scale of the
interaction, rg is the generation time, D is the diffusion constant and n is the population
density.
A related infinitealleles model was studied by both Sawyer [3] and Nagalaki[4]. These
authors found similar asymptotics for the probability that two randomly choses individuals
are genetically identical under certain assumptions. Sawyer[3] also proved that
fixation in the infinite alleles model occurs iff the random walk followed by the individuals
is recurrent.
The broken symmetry or "fixed" regime of the SteppingStone model has been explored
from the point of view of survival of rare mutant alleles, here parameterized by
a coefficient of selection s. The exponent u, governing the divergence of the relaxation
time as s —> 0 is calculated and found to be v = 2 for d — 1, while for d ≥ 2 it is given by
the mean field value of v = 1. Two other exponents for critical spreading processes are
determined and scaling arguments are presented and used to find the decay exponents
that characterize the timedependent survival probability and its asymptotic value for
very long times. These results also establish the upper critical dimension of the model,
d[sub c] = 2.
Finally, these results are rederived and supplemented by a dynamical renormalization
group (RG) analysis. The critical behaviour in d = 1 is found in both regimes to be
controlled by nontrivial fixed points. The critical behaviour of the broken symmetry
regime for d ≥ 2 and its RG fixed point is that of a critical branching process. For the
neutral regime, however, the renormalization group flow is qualitatively different in rf = 2
and 3, reflecting the existence of broken symmetry for rf = 2. The RG flow for the rf = 3
neutral regime contains a line of fixed points with the effective description at large scales
given by a Gaussian version of the timedependent LandauGinsburg model. Finally,
the renormalization results are found to be valid to all orders of perturbation theory.

Extent 
7917538 bytes

Genre  
Type  
File Format 
application/pdf

Language 
eng

Date Available 
20090727

Provider 
Vancouver : University of British Columbia Library

Rights 
For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.

DOI 
10.14288/1.0085696

URI  
Degree  
Program  
Affiliation  
Degree Grantor 
University of British Columbia

Graduation Date 
200011

Campus  
Scholarly Level 
Graduate

Aggregated Source Repository 
DSpace

Item Media
Item Citations and Data
Rights
For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.