The equilibrium properties of a number of deterministic mutation-selection models of sequence evolution are investigated. Both two- and four-state sequences are considered, the mutation model is a ...single-step mutation model. Two types of fitness functions are studied, namely permutation-invariant fitness functions, where the fitness of a sequence depends only on the number of mutations, not on their location within the sequence, and Hopfield-type fitness functions, where the fitness of a sequence is determined by its similarity to a number of predefined patterns. Maximum principles to determine the population mean fitness in equilibrium are derived, where the maximiser gives also the ancestral mean genotype. These maximum principles are used to investigate the error threshold phenomenon, i.e., the phenomenon that for certain fitness functions the population changes at a critical mutation rate from a well localised to a delocalised distribution in sequence space. The error threshold phenomenon is investigated for a four-state model with permutation-invariant fitness functions and for a two-state model with Hopfield-type fitness functions. Both models yield ordered and disordered as well as partially ordered phases.
A deterministic mutation-selection model in the sequence space approach is
investigated. Genotypes are identified with two-letter sequences. Mutation is
modelled as a Markov process, fitness ...functions are of Hopfield type, where the
fitness of a sequence is determined by the Hamming distances to a number of
predefined patterns. Using a maximum principle for the population mean fitness
in equilibrium, the error threshold phenomenon is studied for quadratic
Hopfield-type fitness functions with small numbers of patterns. Different from
previous investigations of the Hopfield model, the system shows error threshold
behaviour not for all fitness functions, but only for certain parameter values.
We review some known properties of the "emptiness formation probability" correlation which we have calculated numerically for spin-1/2 XX chains with constant (homogeneous) or alternating (dimerized) ...nearest-neighbor coupling and an external field (in \(z\) direction) for arbitrary temperature. The long-distance asymptotic behavior of this correlation is known to be Gaussian at zero temperature and exponential at finite temperature for the homogeneous chain. By simple analytical arguments the exponential behavior at finite temperature extends to the dimerized system. Numerical results for the dimerized chain confirm the exponential decay at finite temperature and show Gaussian decay at zero temperature.
A deterministic mutation-selection model in the sequence space approach is investigated. Genotypes are identified with two-letter sequences. Mutation is modelled as a Markov process, fitness ...functions are of Hopfield type, where the fitness of a sequence is determined by the Hamming distances to a number of predefined patterns. Using a maximum principle for the population mean fitness in equilibrium, the error threshold phenomenon is studied for quadratic Hopfield-type fitness functions with small numbers of patterns. Different from previous investigations of the Hopfield model, the system shows error threshold behaviour not for all fitness functions, but only for certain parameter values.
JSTAT (2004) P07007 A four-state mutation-selection model for the evolution of populations of
DNA-sequences is investigated with particular interest in the phenomenon of
error thresholds. The ...mutation model considered is the Kimura 3ST mutation
scheme, fitness functions, which determine the selection process, come from the
permutation-invariant class. Error thresholds can be found for various fitness
functions, the phase diagrams are more interesting than for equivalent
two-state models. Results for (small) finite sequence lengths are compared with
those for infinite sequence length, obtained via a maximum principle that is
equivalent to the principle of minimal free energy in physics.
Bulletin of Mathematical Biology 66 (2004) 397-421 We study the equilibrium behaviour of a deterministic four-state
mutation-selection model as a model for the evolution of a population of
nucleotide ...sequences in sequence space. The mutation model is the Kimura 3ST
mutation scheme, and the selection scheme is assumed to be invariant under
permutation of sites. Considering the evolution process both forward and
backward in time, we use the ancestral distribution as the stationary state of
the backward process to derive an expression for the mutational loss (as the
difference between ancestral and population mean fitness), and we prove a
maximum principle that determines the population mean fitness in
mutation-selection balance.
A four-state mutation-selection model for the evolution of populations of DNA-sequences is investigated with particular interest in the phenomenon of error thresholds. The mutation model considered ...is the Kimura 3ST mutation scheme, fitness functions, which determine the selection process, come from the permutation-invariant class. Error thresholds can be found for various fitness functions, the phase diagrams are more interesting than for equivalent two-state models. Results for (small) finite sequence lengths are compared with those for infinite sequence length, obtained via a maximum principle that is equivalent to the principle of minimal free energy in physics.
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