Change in Recessive Lethal Alleles Frequency in Inbred Populations
Arindam RoyChoudhury
(Submitted on 10 Apr 2013)

In a population practicing consanguineous marriage, rare recessive lethal alleles (RRLA) have higher chances of affecting phenotypes. As inbreeding causes more homozygosity and subsequently more deaths, the loss of individuals with RRLA decreases the frequency of these alleles. Although this phenomenon is well studied in general, here some hitherto unstudied cases are presented. An analytical formula for the RRLA frequency is presented for infinite monoecious population practicing several different types of inbreeding. In finite diecious populations, it is found that more severe inbreeding leads to quicker RRLA losses, making the upcoming generations healthier. A population of size 10,000 practicing 30% half-sib marriages loses more than 95% of its RRLA in 100 generations; a population practicing 30% cousin marriages loses about 75% of its RRLA. Our findings also suggest that given enough resources to grow, a small inbred population will be able to rebound while losing the RRLA.

The causal meaning of Fisher’s average effect
James J. Lee, Carson C. Chow
(Submitted on 6 Apr 2013)

In order to formulate the Fundamental Theorem of Natural Selection, Fisher defined the average excess and average effect of a gene substitution. Finding these notions to be somewhat opaque, some authors have recommended reformulating Fisher’s ideas in terms of covariance and regression, which are classical concepts of statistics. We argue that Fisher intended his two averages to express a distinction between correlation and causation. On this view the average effect is a specific weighted average of the actual phenotypic changes that result from physically changing the allelic states of homologous genes. We show that the statistical and causal conceptions of the average effect, perceived as inconsistent by Falconer, can be reconciled if certain relationships between the genotype frequencies and non-additive residuals are conserved. There are certain theory-internal considerations favoring Fisher’s original formulation in terms of causality; for example, the frequency-weighted mean of the average effects equaling zero at each locus becomes a derivable consequence rather than an arbitrary constraint. More broadly, Fisher’s distinction between correlation and causation is of critical importance to gene-trait mapping studies and the foundations of evolutionary biology.

Minimal clade size in the Bolthausen-Sznitman coalescent
Fabian Freund, Arno Siri-Jégousse
(Submitted on 14 Jan 2013 (v1), last revised 6 Mar 2013 (this version, v2))

This article shows the asymptotics of distribution and moments of the size $X_n$ of the minimal clade of a randomly chosen individual in a Bolthausen-Sznitman $n$-coalescent for $n\to\infty$. The Bolthausen-Sznitman $n$-coalescent is a Markov process taking states in the set of partitions of $\left\{1,\ldots,n\right\}$, where $1,\ldots,n$ are referred to as individuals. The minimal clade of an individual is the equivalence class the individual is in at the time of the first coalescence event this individual participates in.\\ The main tool used is the connection of the Bolthausen-Sznitman $n$-coalescent with random recursive trees introduced by Goldschmidt and Martin (see \cite{goldschmidtmartin}). This connection shows that $X_n-1$ is distributed as the number $M_n$ of all individuals not in the equivalence class of individual 1 shortly before the time of the last coalescence event. Both functionals are distributed like the size $RT_{n-1}$ of an uniformly chosen table in a standard Chinese restaurant process with $n-1$ customers.We give exact formulae for these distributions.\\ Using the asymptotics of $M_n$ shown by Goldschmidt and Martin in \cite{goldschmidtmartin}, we see $(\log n)^{-1}\log X_n$ converges in distribution to the uniform distribution on [0,1] for $n\to\infty$.\\ We provide the complimentary information that $\frac{\log n}{n^k}E(X_n^k)\to \frac{1}{k}$ for $n\to\infty$, which is also true for $M_n$ and $RT_n$.

The consequences of gene flow for local adaptation and differentiation: A two-locus two-deme model
Ada Akerman, Reinhard Bürger
(Submitted on 6 Mar 2013)

We consider a population subdivided into two demes connected by migration in which selection acts in opposite direction. We explore the effects of recombination and migration on the maintenance of multilocus polymorphism, on local adaptation, and on differentiation by employing a deterministic model with genic selection on two linked diallelic loci (i.e., no dominance or epistasis). For the following cases, we characterize explicitly the possible equilibrium configurations: weak, strong, highly asymmetric, and super-symmetric migration, no or weak recombination, and independent or strongly recombining loci. For independent loci (linkage equilibrium) and for completely linked loci, we derive the possible bifurcation patterns as functions of the total migration rate, assuming all other parameters are fixed but arbitrary. For these and other cases, we determine analytically the maximum migration rate below which a stable fully polymorphic equilibrium exists. In this case, differentiation and local adaptation are maintained. Their degree is quantified by a new multilocus version of $\Fst$ and by the migration load, respectively. In addition, we investigate the invasion conditions of locally beneficial mutants and show that linkage to a locus that is already in migration-selection balance facilitates invasion. Hence, loci of much smaller effect can invade than predicted by one-locus theory if linkage is sufficiently tight. We study how this minimum amount of linkage admitting invasion depends on the migration pattern. This suggests the emergence of clusters of locally beneficial mutations, which may form `genomic islands of divergence’. Finally, the influence of linkage and two-way migration on the effective migration rate at a linked neutral locus is explored. Numerical work complements our analytical results.

A two-fold advantage of sex
Su-Chan Park, Joachim Krug
(Submitted on 27 Feb 2013)

The adaptation of large asexual populations is hampered by the competition between independently arising beneficial mutations in different individuals, which is known as clonal interference. In classic work, Fisher and Muller proposed that recombination provides an evolutionary advantage in large populations by alleviating this competition. Based on recent progress in quantifying the speed of adaptation in asexual populations undergoing clonal interference, we present a detailed analysis of the Fisher-Muller mechanism for a model genome consisting of two loci with an infinite number of beneficial alleles each and multiplicative (non-epistatic) fitness effects. We solve the deterministic, infinite population dynamics exactly and show that, for a particular, natural mutation scheme, the speed of adaptation in sexuals is twice as large as in asexuals. This result is argued to hold for any nonzero value of the rate of recombination. Guided by the infinite population result and by previous work on asexual adaptation, we postulate an expression for the speed of adaptation in finite sexual populations that agrees with numerical simulations over a wide range of population sizes and recombination rates. The ratio of the sexual to asexual adaptation speed is a function of population size that increases in the clonal interference regime and approaches 2 for extremely large populations. The simulations also show that recombination leads to a strong equalization of the number of fixed mutations in the two loci. The generalization of the model to an arbitrary number $L$ of loci is briefly discussed. For a particular communal recombination scheme, the ratio of the sexual to asexual adaptation speed is approximately equal to $L$ in large populations.

Mutation Rules and the Evolution of Sparseness and Modularity in Biological Systems
Tamar Friedlander, Avraham E. Mayo, Tsvi Tlusty, Uri Alon
(Submitted on 18 Feb 2013)

Biological systems show two structural features on many levels of organization: sparseness, in which only a small fraction of possible interactions between components actually occur; and modularity: the near decomposability of the system into modules with distinct functionality. Recent work suggests that modularity can evolve in a variety of circumstances, including goals that vary in time such that they share the same subgoals (modularly varying goals). Here, we studied the origin of modularity and sparseness focusing on the nature of the mutation process, rather than variations in the goal. We use simulations of evolution with different mutation rules. We find that commonly used sum-rule mutations, in which interactions are mutated by adding random numbers, do not lead to modularity or sparseness except for special situations. In contrast, product-rule mutations in which interactions are mutated by multiplying by random numbers, a better model for the effects of biological mutations, lead to sparseness naturally. When the goals of evolution are modular, in the sense that specific groups of inputs affect specific groups of outputs, product-rule mutations lead to modular structure; sum-rule mutations do not. Product-rule mutations generate sparseness and modularity because they keep small interaction terms small.

Fitness distributions in spatial populations undergoing clonal interference
Jakub Otwinowski, Joachim Krug
(Submitted on 18 Feb 2013)

Competition between independently arising beneficial mutations is enhanced in spatial populations due to the linear rather than exponential growth of the clones. Recent theoretical studies have pointed out that the resulting fitness dynamics is analogous to a surface growth process, where new layers nucleate and spread stochastically, leading to the build up of scale-invariant roughness. This scenario differs qualitatively from the standard view of adaptation in that the speed of adaptation becomes independent of population size while the fitness variance does not, in apparent violation of Fisher’s fundamental theorem. Here we exploit recent progress in the understanding of surface growth processes to obtain precise predictions for the universal, non-Gaussian shape of the fitness distribution for one-dimensional habitats, which are verified by simulations.

Muller’s ratchet with overlapping generations
Jakob J. Metzger, Stephan Eule
(Submitted on 14 Feb 2013)

Muller’s ratchet is a paradigmatic model for the accumulation of deleterious mutations in a population of finite size. A click of the ratchet occurs when all individuals with the least number of deleterious mutations are lost irreversibly due to a stochastic fluctuation. In spite of the simplicity of the model, a quantitative understanding of the process remains an open challenge. In contrast to previous works, we here study a model of the ratchet with overlapping generations. Employing an approximation which describes the fittest individuals as one class and the rest as a second class, we obtain closed analytical expressions of the ratchet rate in the rare clicking regime. As a click in this regime is caused by a rare large fluctuation from a metastable state, we do not resort to a diffusion approximation but apply an approximation scheme which is especially well suited to describe extinction events from metastable states. This method also allows for a derivation of expressions for the quasi-stationary distribution of the fittest class. Additionally, we show numerically that the formulation with overlapping generations leads to similar results as the standard model with non-overlapping generations and the diffusion approximation in the regime where the ratchet clicks frequently. For parameter values closer to the rare clicking regime, however, we find that the rate of Muller’s ratchet strongly depends on the microscopic reproduction model.

Dualities in population genetics: a fresh look with new dualities
Gioia Carinci, Cristian Giardina’, Claudio Giberti, Frank Redig
(Submitted on 13 Feb 2013)

We apply our general method of duality, introduced in [Giardina’, Kurchan, Redig, J. Math. Phys. 48, 033301 (2007)], to models of population dynamics. The classical dualities between forward and ancestral processes can be viewed as a change of representation in the classical creation and annihilation operators, both for diffusions dual to coalescents of Kingman’s type, as well as for models with finite population size. Next, using SU(1,1) raising and lowering operators, we find new dualities between the Wright-Fisher diffusion with $d$ types and the Moran model, both in presence and absence of mutations. These new dualities relates two forward evolutions. From our general scheme we also identify self-duality of the Moran model.