8,119 research outputs found
Host-Parasite Co-evolution and Optimal Mutation Rates for Semi-conservative Quasispecies
In this paper, we extend a model of host-parasite co-evolution to incorporate
the semi-conservative nature of DNA replication for both the host and the
parasite. We find that the optimal mutation rate for the semi-conservative and
conservative hosts converge for realistic genome lengths, thus maintaining the
admirable agreement between theory and experiment found previously for the
conservative model and justifying the conservative approximation in some cases.
We demonstrate that, while the optimal mutation rate for a conservative and
semi-conservative parasite interacting with a given immune system is similar to
that of a conservative parasite, the properties away from this optimum differ
significantly. We suspect that this difference, coupled with the requirement
that a parasite optimize survival in a range of viable hosts, may help explain
why semi-conservative viruses are known to have significantly lower mutation
rates than their conservative counterparts
Anderson Localization, Non-linearity and Stable Genetic Diversity
In many models of genotypic evolution, the vector of genotype populations
satisfies a system of linear ordinary differential equations. This system of
equations models a competition between differential replication rates (fitness)
and mutation. Mutation operates as a generalized diffusion process on genotype
space. In the large time asymptotics, the replication term tends to produce a
single dominant quasispecies, unless the mutation rate is too high, in which
case the populations of different genotypes becomes de-localized. We introduce
a more macroscopic picture of genotypic evolution wherein a random replication
term in the linear model displays features analogous to Anderson localization.
When coupled with non-linearities that limit the population of any given
genotype, we obtain a model whose large time asymptotics display stable
genotypic diversityComment: 25 pages, 8 Figure
Existence and Stability of a Spike in the Central Component for a Consumer Chain Model
We study a three-component consumer chain model which is based on Schnakenberg type kinetics. In this model there is one consumer feeding on the producer and a second consumer feeding on the first consumer. This means that the first consumer (central component) plays a hybrid role: it acts both as consumer and producer. The model is an extension of the Schnakenberg model suggested in \cite{gm,schn1} for which there is only one producer and one consumer. It is assumed that both the producer and second consumer diffuse much faster than the central component. We construct single spike solutions on an interval for which the profile of the first consumer is that of a spike. The profiles of the producer and the second consumer only vary on a much larger spatial scale due to faster diffusion of these components. It is shown that there exist two different single spike solutions if the feed rates are small enough: a large-amplitude and a small-amplitude spike. We study the stability properties of these solutions in terms of the system parameters. We use a rigorous analysis for the linearized operator around single spike solutions based on nonlocal eigenvalue problems. The following result is established: If the time-relaxation constants for both producer and second consumer vanish, the large-amplitude spike solution is stable and the small-amplitude spike solution is unstable. We also derive results on the stability of solutions when these two time-relaxation constants are small. We show a new effect: if the time-relaxation constant of the second consumer is very small, the large-amplitude spike solution becomes unstable. To the best of our knowledge this phenomenon has not been observed before for the stability of spike patterns. It seems that this behavior is not possible for two-component reaction-diffusion systems but that at least three components are required. Our main motivation to study this system is mathematical since the novel interaction of a spike in the central component with two other components results in new types of conditions for the existence and stability of a spike. This model is realistic if several assumptions are made: (i) cooperation of consumers is prevalent in the system, (ii) the producer and the second consumer diffuse much faster than the first consumer, and (iii) there is practically an unlimited pool of producer. The first assumption has been proven to be correct in many types of consumer groups or populations, the second assumption occurs if the central component has a much smaller mobility than the other two, the third assumption is realistic if the consumers do not feel the impact of the limited amount of producer due to its large quantity. This chain model plays a role in population biology, where consumer and producer are often called predator and prey. This system can also be used as a model for a sequence of irreversible autocatalytic reactions in a container which is in contact with a well-stirred reservoir
The phase transition in random catalytic sets
The notion of (auto) catalytic networks has become a cornerstone in
understanding the possibility of a sudden dramatic increase of diversity in
biological evolution as well as in the evolution of social and economical
systems. Here we study catalytic random networks with respect to the final
outcome diversity of products. We show that an analytical treatment of this
longstanding problem is possible by mapping the problem onto a set of
non-linear recurrence equations. The solution of these equations show a crucial
dependence of the final number of products on the initial number of products
and the density of catalytic production rules. For a fixed density of rules we
can demonstrate the existence of a phase transition from a practically
unpopulated regime to a fully populated and diverse one. The order parameter is
the number of final products. We are able to further understand the origin of
this phase transition as a crossover from one set of solutions from a quadratic
equation to the other.Comment: 7 pages, ugly eps files due to arxiv restriction
On Phase Transitions to Cooperation in the Prisoner's Dilemma
Game theory formalizes certain interactions between physical particles or
between living beings in biology, sociology, and economics, and quantifies the
outcomes by payoffs. The prisoner's dilemma (PD) describes situations in which
it is profitable if everybody cooperates rather than defects (free-rides or
cheats), but as cooperation is risky and defection is tempting, the expected
outcome is defection. Nevertheless, some biological and social mechanisms can
support cooperation by effectively transforming the payoffs. Here, we study the
related phase transitions, which can be of first order (discontinous) or of
second order (continuous), implying a variety of different routes to
cooperation. After classifying the transitions into cases of equilibrium
displacement, equilibrium selection, and equilibrium creation, we show that a
transition to cooperation may take place even if the stationary states and the
eigenvalues of the replicator equation for the PD stay unchanged. Our example
is based on adaptive group pressure, which makes the payoffs dependent on the
endogeneous dynamics in the population. The resulting bistability can invert
the expected outcome in favor of cooperation.Comment: For related work see http://www.soms.ethz.ch
Finite-size scaling of the error threshold transition in finite population
The error threshold transition in a stochastic (i.e. finite population)
version of the quasispecies model of molecular evolution is studied using
finite-size scaling. For the single-sharp-peak replication landscape, the
deterministic model exhibits a first-order transition at , where is the probability of exact replication of a molecule of length , and is the selective advantage of the master string. For
sufficiently large population size, , we show that in the critical region
the characteristic time for the vanishing of the master strings from the
population is described very well by the scaling assumption \tau = N^{1/2} f_a
\left [ \left (Q - Q_c) N^{1/2} \right ] , where is an -dependent
scaling function.Comment: 8 pages, 3 ps figures. submitted to J. Phys.
Schwinger Boson Formulation and Solution of the Crow-Kimura and Eigen Models of Quasispecies Theory
We express the Crow-Kimura and Eigen models of quasispecies theory in a
functional integral representation. We formulate the spin coherent state
functional integrals using the Schwinger Boson method. In this formulation, we
are able to deduce the long-time behavior of these models for arbitrary
replication and degradation functions.
We discuss the phase transitions that occur in these models as a function of
mutation rate. We derive for these models the leading order corrections to the
infinite genome length limit.Comment: 37 pages; 4 figures; to appear in J. Stat. Phy
Error threshold in finite populations
A simple analytical framework to study the molecular quasispecies evolution
of finite populations is proposed, in which the population is assumed to be a
random combination of the constiyuent molecules in each generation,i.e.,
linkage disequilibrium at the population level is neglected. In particular, for
the single-sharp-peak replication landscape we investigate the dependence of
the error threshold on the population size and find that the replication
accuracy at threshold increases linearly with the reciprocal of the population
size for sufficiently large populations. Furthermore, in the deterministic
limit our formulation yields the exact steady-state of the quasispecies model,
indicating then the population composition is a random combination of the
molecules.Comment: 14 pages and 4 figure
Physics, Stability and Dynamics of Supply Networks
We show how to treat supply networks as physical transport problems governed
by balance equations and equations for the adaptation of production speeds.
Although the non-linear behaviour is different, the linearized set of coupled
differential equations is formally related to those of mechanical or electrical
oscillator networks. Supply networks possess interesting new features due to
their complex topology and directed links. We derive analytical conditions for
absolute and convective instabilities. The empirically observed "bull-whip
effect" in supply chains is explained as a form of convective instability based
on resonance effects. Moreover, it is generalized to arbitrary supply networks.
Their related eigenvalues are usually complex, depending on the network
structure (even without loops). Therefore, their generic behavior is
characterized by oscillations. We also show that regular distribution networks
possess two negative eigenvalues only, but perturbations generate a spectrum of
complex eigenvalues.Comment: For related work see http://www.helbing.or
Mutation, selection, and ancestry in branching models: a variational approach
We consider the evolution of populations under the joint action of mutation
and differential reproduction, or selection. The population is modelled as a
finite-type Markov branching process in continuous time, and the associated
genealogical tree is viewed both in the forward and the backward direction of
time. The stationary type distribution of the reversed process, the so-called
ancestral distribution, turns out as a key for the study of mutation-selection
balance. This balance can be expressed in the form of a variational principle
that quantifies the respective roles of reproduction and mutation for any
possible type distribution. It shows that the mean growth rate of the
population results from a competition for a maximal long-term growth rate, as
given by the difference between the current mean reproduction rate, and an
asymptotic decay rate related to the mutation process; this tradeoff is won by
the ancestral distribution.
Our main application is the quasispecies model of sequence evolution with
mutation coupled to reproduction but independent across sites, and a fitness
function that is invariant under permutation of sites. Here, the variational
principle is worked out in detail and yields a simple, explicit result.Comment: 45 pages,8 figure
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