Transient amplifiers of selection and reducers of fixation for death-Birth updating on graphs

The spatial structure of an evolving population affects which mutations become fixed. Some structures amplify selection, increasing the likelihood that beneficial mutations become fixed while deleterious mutations do not. Other structures suppress selection, reducing the effect of fitness differences and increasing the role of random chance. This phenomenon can be modeled by representing spatial structure as a graph, with individuals occupying vertices. Births and deaths occur stochastically, according to a specified update rule. We study death-Birth updating: An individual is chosen to die and then its neighbors compete to reproduce into the vacant spot. Previous numerical experiments suggested that amplifiers of selection for this process are either rare or nonexistent. We introduce a perturbative method for this problem for weak selection regime, meaning that mutations have small fitness effects. We show that fixation probability under weak selection can be calculated in terms of the coalescence times of random walks. This result leads naturally to a new definition of effective population size. Using this and other methods, we uncover the first known examples of transient amplifiers of selection (graphs that amplify selection for a particular range of fitness values) for the death-Birth process. We also exhibit new families of "reducers of fixation", which decrease the fixation probability of all mutations, whether beneficial or deleterious.Comment: 51 pages, 5 figure

A General Modeling Framework for Describing Spatially Structured Population Dynamics

Variation in movement across time and space fundamentally shapes the abundance and distribution of populations. Although a variety of approaches model structured population dynamics, they are limited to specific types of spatially structured populations and lack a unifying framework. Here, we propose a unified network‐based framework sufficiently novel in its flexibility to capture a wide variety of spatiotemporal processes including metapopulations and a range of migratory patterns. It can accommodate different kinds of age structures, forms of population growth, dispersal, nomadism and migration, and alternative life‐history strategies. Our objective was to link three general elements common to all spatially structured populations (space, time and movement) under a single mathematical framework. To do this, we adopt a network modeling approach. The spatial structure of a population is represented by a weighted and directed network. Each node and each edge has a set of attributes which vary through time. The dynamics of our network‐based population is modeled with discrete time steps. Using both theoretical and real‐world examples, we show how common elements recur across species with disparate movement strategies and how they can be combined under a unified mathematical framework. We illustrate how metapopulations, various migratory patterns, and nomadism can be represented with this modeling approach. We also apply our network‐based framework to four organisms spanning a wide range of life histories, movement patterns, and carrying capacities. General computer code to implement our framework is provided, which can be applied to almost any spatially structured population. This framework contributes to our theoretical understanding of population dynamics and has practical management applications, including understanding the impact of perturbations on population size, distribution, and movement patterns. By working within a common framework, there is less chance that comparative analyses are colored by model details rather than general principles

Genetic Diversity of EBV-Encoded LMP1 in the Swiss HIV Cohort Study and Implication for NF-Κb Activation

Epstein-Barr virus (EBV) is associated with several types of cancers including Hodgkin's lymphoma (HL) and nasopharyngeal carcinoma (NPC). EBV-encoded latent membrane protein 1 (LMP1), a multifunctional oncoprotein, is a powerful activator of the transcription factor NF-κB, a property that is essential for EBV-transformed lymphoblastoid cell survival. Previous studies reported LMP1 sequence variations and induction of higher NF-κB activation levels compared to the prototype B95-8 LMP1 by some variants. Here we used biopsies of EBV-associated cancers and blood of individuals included in the Swiss HIV Cohort Study (SHCS) to analyze LMP1 genetic diversity and impact of sequence variations on LMP1-mediated NF-κB activation potential. We found that a number of variants mediate higher NF-κB activation levels when compared to B95-8 LMP1 and mapped three single polymorphisms responsible for this phenotype: F106Y, I124V and F144I. F106Y was present in all LMP1 isolated in this study and its effect was variant dependent, suggesting that it was modulated by other polymorphisms. The two polymorphisms I124V and F144I were present in distinct phylogenetic groups and were linked with other specific polymorphisms nearby, I152L and D150A/L151I, respectively. The two sets of polymorphisms, I124V/I152L and F144I/D150A/L151I, which were markers of increased NF-κB activation in vitro, were not associated with EBV-associated HL in the SHCS. Taken together these results highlighted the importance of single polymorphisms for the modulation of LMP1 signaling activity and demonstrated that several groups of LMP1 variants, through distinct mutational paths, mediated enhanced NF-κB activation levels compared to B95-8 LMP1

Formation of morphogen gradients: Local accumulation time

Spatial regulation of cell differentiation in embryos can be provided by morphogen gradients, which are defined as the concentration fields of molecules that control gene expression. For example, a cell can use its surface receptors to measure the local concentration of an extracellular ligand and convert this information into a corresponding change in its transcriptional state. We characterize the time needed to establish a steady-state gradient in problems with diffusion and degradation of locally produced chemical signals. A relaxation function is introduced to describe how the morphogen concentration profile approaches its steady state. This function is used to obtain a local accumulation time that provides a time scale that characterizes relaxation to steady state at an arbitrary position within the patterned field. To illustrate the approach we derive local accumulation times for a number of commonly used models of morphogen gradient formation

How long does it take to establish a morphogen gradient

ABSTRACT A morphogen gradient is defined as a concentration field of a molecule that acts as a dose-dependent regulator of cell differentiation. One of the key questions in studies of morphogen gradients is whether they reach steady states on timescales relevant for developmental patterning. We propose a systematic approach for addressing this question and illustrate it by analyzing several models that account for diffusion and degradation of locally produced chemical signals. Received for publication 13 June 2010 and in final form 27 July 2010. *Correspondence: [email protected] One of the key questions in studies of morphogen gradients is whether they reach steady states on timescales relevant for developmental patterning (1-3). Although Crick posed this question 40 years ago (4), a general formalism for addressing this question is still lacking. Here we suggest a method for tackling this problem in an important class of biophysical models that account for diffusion and degradation of locally produced chemical signals. We begin with a simple model that is commonly used as a first step in the analysis of more complex mechanisms of morphogen gradient formation and interpretation (5). Let C(x,t) be the concentration at a distance x > 0 from the boundary, where a morphogen is produced at a constant rate Q. Signal production begins at t ¼ 0, when C(x,t) ¼ 0 throughout the system. The concentration satisfies Here D is the diffusivity and k is degradation rate constant. First, we consider the total amount of morphogen accumulated in the system by time t: One can show that N(t) starts from zero and exponentially approaches its steady-state value N s ¼ Q/k, NðtÞ ¼ N s ð1 À expðÀktÞÞ: To quantify the approach to the steady state, we introduce the relaxation function, denoted by R N (t). This function is defined as the ratio of the difference between the current and steady-state values of N(t) to this difference at t ¼ 0, where the decay time, t N ¼ 1/k, is independent of the diffusivity and the signal production rate. Thus, for the total amount of morphogen in the system, the relaxation to the steady state is exponential. Similarly, we introduce the local relaxation function, R(x,t), to analyze the approach of C(x,t) to C s (x), its steady-state value at a given x: Rðx; tÞ ¼ Cðx; tÞ À C s ðxÞ Cðx; 0Þ À C s ðxÞ ¼ 1 À Cðx; tÞ C s ðxÞ : This function monotonically decays with time from unity at t ¼ 0 to zero as time tends to infinity. The initial and final values of R(x,t) are independent of x, but the relaxation kinetics clearly depends on position. To characterize this kinetics by a single timescale we use the fact that the fraction of the steady-state concentration at point x accumulated between t and tþdt is given by ½ À vRðx; tÞ=vtdt: Therefore, the negative derivative of the relaxation function is the probability density for the time of the local accumulation process, We use this probability density to find the mean time t(x), which is the local relaxation time to steady state at point x. To show how this formalism works, we first use the known solution for C(x,t) (3): Editor: Andre Levchenko

Fixation probabilities in graph-structured populations under weak selection.

A population's spatial structure affects the rate of genetic change and the outcome of natural selection. These effects can be modeled mathematically using the Birth-death process on graphs. Individuals occupy the vertices of a weighted graph, and reproduce into neighboring vertices based on fitness. A key quantity is the probability that a mutant type will sweep to fixation, as a function of the mutant's fitness. Graphs that increase the fixation probability of beneficial mutations, and decrease that of deleterious mutations, are said to amplify selection. However, fixation probabilities are difficult to compute for an arbitrary graph. Here we derive an expression for the fixation probability, of a weakly-selected mutation, in terms of the time for two lineages to coalesce. This expression enables weak-selection fixation probabilities to be computed, for an arbitrary weighted graph, in polynomial time. Applying this method, we explore the range of possible effects of graph structure on natural selection, genetic drift, and the balance between the two. Using exhaustive analysis of small graphs and a genetic search algorithm, we identify families of graphs with striking effects on fixation probability, and we analyze these families mathematically. Our work reveals the nuanced effects of graph structure on natural selection and neutral drift. In particular, we show how these notions depend critically on the process by which mutations arise

The Molecular Clock of Neutral Evolution Can Be Accelerated or Slowed by Asymmetric Spatial Structure

Over time, a population acquires neutral genetic substitutions as a consequence of random drift. A famous result in population genetics asserts that the rate, K, at which these substitutions accumulate in the population coincides with the mutation rate, u, at which they arise in individuals: K = u. This identity enables genetic sequence data to be used as a “molecular clock” to estimate the timing of evolutionary events. While the molecular clock is known to be perturbed by selection, it is thought that K = u holds very generally for neutral evolution. Here we show that asymmetric spatial population structure can alter the molecular clock rate for neutral mutations, leading to either Ku. Our results apply to a general class of haploid, asexually reproducing, spatially structured populations. Deviations from K = u occur because mutations arise unequally at different sites and have different probabilities of fixation depending on where they arise. If birth rates are uniform across sites, then K ≤ u. In general, K can take any value between 0 and Nu. Our model can be applied to a variety of population structures. In one example, we investigate the accumulation of genetic mutations in the small intestine. In another application, we analyze over 900 Twitter networks to study the effect of network topology on the fixation of neutral innovations in social evolution