Ellen Baake

Biological evolution through mutation, selection, and drift: An introductory review

Motivated by present activities in (statistical) physics directed towardsbiological evolution, we review the interplay of three evolutionary forces:mutation, selection, and genetic drift. The review addresses itself tophysicists and intends to bridge the gap between the biological and thephysical literature. We first clarify the terminology and recapitulate thebasic models of population genetics, which describe the evolution of thecomposition of a population under the joint action of the various evolutionaryforces. Building on these foundations, we specify the ingredients explicitly,namely, the various mutation models and fitness landscapes. We then reviewrecent developments concerning models of mutational degradation. These predictupper limits for the mutation rate above which mutation can no longer becontrolled by selection, the most important phenomena being error thresholds,Mullers ratchet, and mutational meltdowns. Error thresholds are deterministicphenomena, whereas Mullers ratchet requires the stochastic component broughtabout by finite population size. Mutational meltdowns additionally rely on anexplicit model of population dynamics, and describe the extinction ofpopulations. Special emphasis is put on the mutual relationship between thesephenomena. Finally, a few connections with the process of molecular evolutionare established.

Modelling the fast fluorescence rise of photosynthesis

We construct an ODE model for the fast fluorescence rise of photosynthesis by combining the current reaction scheme of the PS II two-electron-gate with a quasi steady-state description of the fast processes of excitation energy transfer and primary charge separation. The model is fitted to measured induction curves with a multiple shooting algorithm, and remarkably good approximations of the data are obtained. Model refinements are discussed focusing on PS II heterogeneity, and on PS I.

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. We then focus on the case when the type is determined by a sequence of letters (like nucleotides or matches/mismatches relative to a reference sequence), and we ask how much of the above competition can still be seen by observing only the letter composition (as given by the frequencies of the various letters within the sequence). If mutation and reproduction rates can be approximated in a smooth way, the fitness of letter compositions resulting from the interplay of reproduction and mutation is determined in the limit as the number of sequence sites tends to infinity. 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. In this model, the fitness of letter compositions is worked out explicitly. In certain cases, their competition leads to a phase transition.

SUPERCRITICAL MULTITYPE BRANCHING PROCESSES: THE ANCESTRAL TYPES OF TYPICAL INDIVIDUALS

For supercritical multitype Markov 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 nonextinction), 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 et al.) of the continuous-time version of the Kesten-Stigum theorem.

Ising Quantum Chain and Sequence Evolution

A sequence space model which describes the interplay of mutation and selection in molecular evolution is shown to be equivalent to an Ising quantum chain. Observable quantities tailored to match the biological situation are then employed to treat three fitness landscapes exactly.

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.

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 et al. (J Theor Biol 209:465–486, 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.

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 discrete time, allowing only for single crossovers. While the analogous dynamics in continuous time admits a closed solution (Baake and Baake in Can J Math 55:3–41, 2003), this no longer works for discrete time. A more general model (i.e. without the restriction to single crossovers) has been studied before (Bennett in Ann Hum Genet 18:311–317, 1954; Dawson in Theor Popul Biol 58:1–20, 2000; Linear Algebra Appl 348:115–137, 2002) and was solved algorithmically by means of Haldane linearisation. Using the special formalism introduced by Baake and Baake (Can J Math 55:3–41, 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.

Diploid models on sequence space

The “decoupled” version of the selection mutation as adapted to (binary) sequence space is investigated. For diploid analogues of the single-peaked landscape, dominance effects on error thresholds are found both in the deterministic and the stochastic case. For more complicated landscapes, nonlinear oscillations may emerge during the transition to equidistribution.

A Differential Equation Model for the Description of the Fast Fluorescence Rise (O-I-D-P-Transient) in Leaves

In this abridged version of Ref. [1], we reinvestigate the fast fluorescence rise by means of a differential equation model which combines the current scheme of PS II electron transport with a full description of excitation energy transfer.