88 research outputs found
Haldane linearisation done right: Solving the nonlinear recombination equation the easy way
The nonlinear recombination equation from population genetics has a long
history and is notoriously difficult to solve, both in continuous and in
discrete time. This is particularly so if one aims at full generality, thus
also including degenerate parameter cases. Due to recent progress for the
continuous time case via the identification of an underlying stochastic
fragmentation process, it became clear that a direct general solution at the
level of the corresponding ODE itself should also be possible. This paper shows
how to do it, and how to extend the approach to the discrete-time case as well.Comment: 12 pages, 1 figure; some minor update
Single--crossover recombination in discrete time
Modelling the process of recombination leads to a large coupled nonlinear
dynamical system. Here, we consider a particular case of recombination in {\em
discrete} time, allowing only for {\em single crossovers}. While the analogous
dynamics in {\em continuous} time admits a closed solution, this no longer
works for discrete time. A more general model (i.e. without the restriction to
single crossovers) has been studied before and was solved algorithmically by
means of Haldane linearisation. Using the special formalism introduced by Baake
and Baake (2003), we obtain further insight into the single-crossover dynamics
and the particular difficulties that arise in discrete time. We then transform
the equations to a solvable system in a two-step procedure: linearisation
followed by diagonalisation. Still, the coefficients of the second step must be
determined in a recursive manner, but once this is done for a given system,
they allow for an explicit solution valid for all times.Comment: J. Math. Biol., in pres
Biological evolution through mutation, selection, and drift: An introductory review
Motivated by present activities in (statistical) physics directed towards
biological evolution, we review the interplay of three evolutionary forces:
mutation, selection, and genetic drift. The review addresses itself to
physicists and intends to bridge the gap between the biological and the
physical literature. We first clarify the terminology and recapitulate the
basic models of population genetics, which describe the evolution of the
composition of a population under the joint action of the various evolutionary
forces. Building on these foundations, we specify the ingredients explicitly,
namely, the various mutation models and fitness landscapes. We then review
recent developments concerning models of mutational degradation. These predict
upper limits for the mutation rate above which mutation can no longer be
controlled by selection, the most important phenomena being error thresholds,
Muller's ratchet, and mutational meltdowns. Error thresholds are deterministic
phenomena, whereas Muller's ratchet requires the stochastic component brought
about by finite population size. Mutational meltdowns additionally rely on an
explicit model of population dynamics, and describe the extinction of
populations. Special emphasis is put on the mutual relationship between these
phenomena. Finally, a few connections with the process of molecular evolution
are established.Comment: 62 pages, 6 figures, many reference
Supercritical multitype branching processes: the ancestral types of typical individuals
For supercritical multitype branching processes in continuous time, we
investigate the evolution of types along those lineages that survive up to some
time t. We establish almost-sure convergence theorems for both time and
population averages of ancestral types (conditioned on non-extinction), and
identify the mutation process describing the type evolution along typical
lineages. An important tool is a representation of the family tree in terms of
a suitable size-biased tree with trunk. As a by-product, this representation
allows a `conceptual proof' (in the sense of Kurtz, Lyons, Pemantle, Peres
1997) of the continuous-time version of the Kesten-Stigum theorem.Comment: 23 pages, 1 figure; minor additions, added reference
A probabilistic view on the deterministic mutation-selection equation: dynamics, equilibria, and ancestry via individual lines of descent
We reconsider the deterministic haploid mutation-selection equation with two
types. This is an ordinary differential equation that describes the type
distribution (forward in time) in a population of infinite size. This paper
establishes ancestral (random) structures inherent in this deterministic model.
In a first step, we obtain a representation of the deterministic equation's
solution (and, in particular, of its equilibrium) in terms of an ancestral
process called the killed ancestral selection graph. This representation allows
one to understand the bifurcations related to the error threshold phenomenon
from a genealogical point of view. Next, we characterise the ancestral type
distribution by means of the pruned lookdown ancestral selection graph and
study its properties at equilibrium. We also provide an alternative
characterisation in terms of a piecewise-deterministic Markov process.
Throughout, emphasis is on the underlying dualities as well as on explicit
results.Comment: J. Math. Biol., in pres
On the role of invariants for the parameter estimation problem in Hamiltonian systems
Baake M, Baake E, Eich E. On the role of invariants for the parameter estimation problem in Hamiltonian systems. Physics letters. 1993;180(1-2):74-82.The parameter estimation problem is discussed for differential equations that describe a Hamiltonian system. Since the conserved total energy is an invariant which contains all parameters of the system, we can achieve parameter estimation without any numerical integration. This is demonstrated for data in the chaotic region of the Hénon-Heiles system and of the planar double pendulum. We show that the method works well for ideal as well as noisy data. In this context, an appropriate method for the generation of reliable time series in the presence of an invariant is discussed. Finally, it is shown that our method also provides a simple approach to global fitting in discrete dynamical systems with invariants
The general recombination equation in continuous time and its solution
Baake E, Baake M, Salamat M. The general recombination equation in continuous time and its solution. Discrete and Continuous Dynamical Systems. 2016;36(1):63-95.The process of recombination in population genetics, in its deterministic limit, leads to a nonlinear ODE in the Banach space of finite measures on a locally compact product space. It has an embedding into a larger family of nonlinear ODEs that permits a systematic analysis with lattice-theoretic methods for general partitions of finite sets. We discuss this type of system, reduce it to an equivalent finite-dimensional nonlinear problem, and establish a connection with an ancestral partitioning process, backward in time. We solve the finite-dimensional problem recursively for generic sets of parameters and briefly discuss the singular cases, and how to extend the solution to this situation
How T-cells use large deviations to recognize foreign antigens
A stochastic model for the activation of T-cells is analysed. T-cells are
part of the immune system and recognize foreign antigens against a background
of the body's own molecules. The model under consideration is a slight
generalization of a model introduced by Van den Berg, Rand and Burroughs in
2001, and is capable of explaining how this recognition works on the basis of
rare stochastic events. With the help of a refined large deviation theorem and
numerical evaluation it is shown that, for a wide range of parameters, T-cells
can distinguish reliably between foreign antigens and self-antigens.Comment: 16 pages, 6 figures; minor revision, new simulations; J Math Biol.,
in pres
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