The mutation process on the ancestral line under selection

We consider the Moran model of population genetics with two types, mutation, and selection, and investigate the line of descent of a randomly-sampled individual from a contemporary population. We trace this ancestral line back into the distant past, far beyond the most recent common ancestor of the population (thus connecting population genetics to phylogeny), and analyse the mutation process along this line. To this end, we use the pruned lookdown ancestral selection graph (Lenz et al., 2015), which consists of a set of potential ancestors of the sampled individual at any given time. Relative to the neutral case (that is, without selection), we obtain a general bias towards the beneficial type, an increase in the beneficial mutation rate, and a decrease in the deleterious mutation rate. This sheds new light on previous analytical results. We discuss our findings in the light of a well-known observation at the interface of phylogeny and population genetics, namely, the difference in the mutation rates (or, more precisely, mutation fluxes) estimated via phylogenetic methods relative to those observed in pedigree studies.

Solving the migration-recombination equation from a genealogical point of view

We consider the discrete-time migration–recombination equation, a deterministic, nonlinear dynamical system that describes the evolution of the genetic type distribution of a population evolving under migration and recombination in a law of large numbers setting. We relate this dynamics (forward in time) to a Markov chain, namely a labelled partitioning process, backward in time. This way, we obtain a stochastic representation of the solution of the migration–recombination equation. As a consequence, one obtains an explicit solution of the nonlinear dynamics, simply in terms of powers of the transition matrix of the Markov chain. The limiting and quasi-limiting behaviour of the Markov chain are investigated, which gives immediate access to the asymptotic behaviour of the dynamical system. We finally sketch the analogous situation in continuous time.

Mutation-selection models solved exactly with methods of statistical mechanics

We reconsider deterministic models of mutation and selection acting on populations of sequences, or, equivalently, multilocus systems with complete linkage. Exact analytical results concerning such systems are few, and we present recent and new ones obtained with the help of methods from quantum statistical mechanics. We consider a continuous-time model for an infinite population of haploids (or diploids without dominance), with N sites each, two states per site, symmetric mutation and arbitrary fitness function. We show that this model is exactly equivalent to a so-called Ising quantum chain. In this picture, fitness corresponds to the interaction energy of spins, and mutation to a temperature-like parameter. The highly elaborate methods of statistical mechanics allow one to find exact solutions for non-trivial examples. These include quadratic fitness functions, as well as 'Onsager's landscape'. The latter is a fitness function which captures some essential features of molecular evolution, such as neutrality, compensatory mutations and flat ridges. We investigate the mean number of mutations, the mutation load, and the variance in fitness under mutation-selection balance. This also yields some insight into the 'error threshold' phenomenon, which occurs in some, but not all, examples.

Mutation and recombination with tight linkage

An exact solution of the mutation-recombination equation in continuous time is presented, with linear ordering of the sites and at most one mutation or crossover event taking place at every instant of time. The differential equation may be obtained from a mutation-recombination model with discrete generations, in the limit of short generations, or weak mutation and recombination. The solution relies on the multilinear structure of the dynamical system, and on the commuting properties of the mutation and recombination operators. It is obtained through diagonalization of the mutation term, followed by a transformation to certain measures of linkage disequilibrium that simultaneously linearize and diagonalize the recombination dynamics. The collection of linkage disequilibria, as well as their decay rates, are given in closed form.

Distance measures in terms of substitution processes

Phylogenetic reconstruction from DNA or amino acid sequences relies heavily on suitable distance measures. A number of new distance measures (asynchronous, LogDet, and paralinear distances) which possess the desired property of tree additivity under fairly general models of sequence evolution have been proposed recently, but they are not well understood from a mechanistic point of view. We review them here in a unifying framework, which is the substitution process in continuous time. The emerging interpretation will also clarify the relationship among these distance measures. We also tackle situations with site-to-site variation of substitution rates which is well known to cause non-additive distances and inconsistent branch lengths. For homogeneous, stationary, time-reversible models, this may be repaired provided that the distribution of rates is known. In contrast, we will show that, for non-stationary models, different tree topologies may produce identical joint distributions of letters in pairs of sequences, given the same distribution of rates. This precludes the existence of any tree-additive pairwise distance measure.

What can and what cannot be inferred from pairwise sequence comparisons?

We address questions of identifiability in molecular phylogeny, the art of reconstructing the history of a sample of sequences given just the sequences at the leaves of the phylogenetic tree. Here, the 'history' consists of the tree topology, plus the transition probabilities which define the Markov process of sequence evolution along the branches of the tree. It is assumed that sequences have infinite length, and the pairwise joint distributions of letters at the leaves is taken to be known. We focus on two cases: (1) If the sites of a sequence evolve identically and independently, the topology can be reconstructed, but the one-way edge transition matrices cannot. However, the return-trip transition matrices are reconstructible for every edge, up to conjugation in the case of internal edges. (2) If a rate factor varies from site to site, different topologies may produce identical pairwise joint distributions, even under the same distribution of rate factors. Consequently, identifiability of the topology is lost on the basis of pairwise sequence comparisons, even if the distribution of rate factors is known. The results are discussed in the context of additive measures of phylogenetic distance.

Error propagation in reproduction of diploid organisms. A case study on single peaked landscapes

Two versions of the diploid selection mutation equation as adapted to sequence space are studied. Focussing on diploid generalizations of the well-established single peaked landscape, quantitative effects of dominance on error thresholds in infinite populations are found, as well as unexpected qualitative features like multiple equilibria. Analogues of these phenomena are also recovered in stochastic versions for finite populations.

A quantitative description of fluorescence excitation spectra in intact bean leaves greened under intermittent light

We present a simple approach for the calculation of in vivo fluorescence excitation spectra from measured absorbance spectra of the isolated pigments involved. Taking into account shading of the pigments by each other, energy transfer from carotene to chlorophyll a, and light scattering by the leaf tissue, we arrive at a model function with 6 free parameters. Fitting them to the measured fluorescence excitation spectrum yields good correspondence between theory and experiment, and parameter estimates which agree with independent measurements. The results are discussed with respect to the origin and the interpretation of in vivo excitation spectra in general.