Fixation Probability Research Articles

Environmental spatial and temporal variability and its role in non-favoured mutant dynamics.

Understanding how environmental variability (or randomness) affects evolution is of fundamental importance for biology. The presence of temporal or spatial variability significantly affects the competition dynamics in populations, and gives rise to some counterintuitive observations. In this paper, we consider both birth-death (BD) or death-birth (DB) Moran processes, which are set up on a circular or a complete graph. We investigate spatial and temporal variability affecting division and/or death parameters. Assuming that mutant and wild-type fitness parameters are drawn from an identical distribution, we study mutant fixation probability and timing. We demonstrate that temporal and spatial types of variability possess fundamentally different properties. Under temporal randomness, in a completely mixed system, minority mutants experience (i) higher than neutral fixation probability and a higher mean conditional fixation time, if the division rates are affected by randomness and (ii) lower fixation probability and lower mean conditional fixation time if the death rates are affected. Once spatial restrictions are imposed, however, these effects completely disappear, and mutants in a circular graph experience neutral dynamics, but only for the DB update rule in case (i) and for the BD rule in case (ii) above. In contrast to this, in the case of spatially variable environment, both for BD/DB processes, both for complete/circular graph and both for division/death rates affected, minority mutants experience a higher than neutral probability of fixation. Fixation time, however, is increased by randomness on a circle, while it decreases for complete graphs under random division rates. A basic difference between temporal and spatial kinds of variability is the types of correlations that occur in the system. Under temporal randomness, mutants are spatially correlated with each other (they simply have equal fitness values at a given moment of time; the same holds for wild-types). Under spatial randomness, there are subtler, temporal correlations among mutant and wild-type cells, which manifest themselves by cells of each type 'claiming' better spots for themselves. Applications of this theory include cancer generation and biofilm dynamics.

Evolutionary dynamics of complex multiple games.

Evolutionary game theory has been successful in describing phenomena from bacterial population dynamics to the evolution of social behaviour. However, it has typically focused on a single game describing the interactions between individuals. Organisms are simultaneously involved in many intraspecies and interspecies interactions. Therefore, there is a need to move from single games to multiple games. However, these interactions in nature involve many players. Shifting from 2-player games to multiple multiplayer games yield richer dynamics closer to natural settings. Such a complete picture of multiple game dynamics (MGD), where multiple players are involved, was lacking. For multiple multiplayer games—where each game could have an arbitrary finite number of players and strategies, we provide a replicator equation for MGD having many players and strategies. We show that if the individual games involved have more than two strategies, then the combined dynamics cannot be understood by looking only at individual games. Expected dynamics from single games is no longer valid, and trajectories can possess different limiting behaviour. In the case of finite populations, we formulate and calculate an essential and useful stochastic property, fixation probability. Our results highlight that studying a set of interactions defined by a single game can be misleading if we do not take the broader setting of the interactions into account. Through our results and analysis, we thus discuss and advocate the development of evolutionary game(s) theory, which will help us disentangle the complexity of multiple interactions.

Dominant Cubic Coefficients of the ‘1/3-Rule’ Reduce Contest Domains

Antagonistic exploitation in competition with a cooperative strategy defines a social dilemma, whereby eventually overall fitness of the population decreases. Frequency-dependent selection between two non-mutating strategies in a Moran model of random genetic drift yields an evolutionary rule of biological game theory. When a singleton fixation probability of co-operation exceeds the selectively neutral value being the reciprocal of population size, its relative frequency in the population equilibrates to less than 1/3. Maclaurin series of a singleton type fixation probability function calculated at third order enables the convergent domain of the payoff matrix to be identified. Asymptotically dominant third order coefficients of payoff matrix entries were derived. Quantitative analysis illustrates non-negligibility of the quadratic and cubic coefficients in Maclaurin series with selection being inversely proportional to population size. Novel corollaries identify the domain of payoff matrix entries that determines polarity of second order terms, with either non-harmful or harmful contests. Violation of this evolutionary rule observed with non-harmful contests depends on the normalized payoff matrix entries and selection differential. Significant violations of the evolutionary rule were not observed with harmful contests.

Quantifying Microsatellite Mutation Rates from Intestinal Stem Cell Dynamics in Msh2-Deficient Murine Epithelium.

Microsatellite sequences have an enhanced susceptibility to mutation, and can act as sentinels indicating elevated mutation rates and increased risk of cancer. The probability of mutant fixation within the intestinal epithelium is dictated by a combination of stem cell dynamics and mutation rate. Here, we exploit this relationship to infer microsatellite mutation rates. First a sensitive, multiplexed, and quantitative method for detecting somatic changes in microsatellite length was developed that allowed the parallel detection of mutant [CA]n sequences from hundreds of low-input tissue samples at up to 14 loci. The method was applied to colonic crypts in Mus musculus, and enabled detection of mutant subclones down to 20% of the cellularity of the crypt (∼50 of 250 cells). By quantifying age-related increases in clone frequencies for multiple loci, microsatellite mutation rates in wild-type and Msh2-deficient epithelium were established. An average 388-fold increase in mutation per mitosis rate was observed in Msh2-deficient epithelium (2.4 × 10−2) compared to wild-type epithelium (6.2 × 10−5).

First-order effect of frequency-dependent selection on fixation probability in an age-structured population with application to a public goods game

In this paper, we deduce the first-order effect of frequency-dependent viability and fertility selection on the probability of fixation of a mutant in a large finite haploid population with a fixed age structure by applying a direct small perturbation method to the neutral two-timescale genealogical process. This effect is expressed in terms of fixation-fitness coefficients times ancestry coefficients that are related to the effective population size. In the case of constant selection, the fixation-fitness coefficients are functions of the coefficients of viability and fertility selection weighted by reproductive values and population-structure coefficients for the different age classes. This explains the difference between the effects of viability selection and fertility selection on fixation probability in age-structured populations. With frequency-dependent selection in the form of a public goods game, the fixation-fitness coefficients depend also on the public good allocation strategy of the population and the resource allocation strategies of the individuals. In this case, the results show that weak selection may favor the fixation of cooperation if the cooperators allocate more resources to reproduction versus survival than the defectors do.

Computation and Simulation of Evolutionary Game Dynamics in Finite Populations

The study of evolutionary dynamics increasingly relies on computational methods, as more and more cases outside the range of analytical tractability are explored. The computational methods for simulation and numerical approximation of the relevant quantities are diverging without being compared for accuracy and performance. We thoroughly investigate these algorithms in order to propose a reliable standard. For expositional clarity we focus on symmetric 2 × 2 games leading to one-dimensional processes, noting that extensions can be straightforward and lessons will often carry over to more complex cases. We provide time-complexity analysis and systematically compare three families of methods to compute fixation probabilities, fixation times and long-term stationary distributions for the popular Moran process. We provide efficient implementations that substantially improve wall times over naive or immediate implementations. Implications are also discussed for the Wright-Fisher process, as well as structured populations and multiple types.

The surface-active properties of waste processing of tin ore raw materials

The article presents the results of a study of the surface-active properties of tin ore wastes due to their impact on the environment and public health. The object of the study is the ore dressing waste from the closed mining enterprise “Solnechny GOK” of the Khabarovsk region of the Far Eastern Federal District. Sodium oxalate, Trilon B, methylene blue, sodium humate were used as sorbates. Methods of establishing the fact of interaction of waste with sorbates focused on the discoloration of colored complexing agents in the presence of waste. The use of titrimetric methods allowed us to establish the number of reaction centers of waste. To determine the nature of the active centers of the waste, foam quenching formed by anionic and non-ionic surfactants was applied. The effect of the molar mass and the structure of sorbates on the nature of their interaction with the surface of the waste is shown. Low molecular weight sorbates promote the formation of soluble complexes of heavy metal compounds and contamination of groundwater. High molecular polydentate complexing agents form surface films. The revealed physicochemical properties will make it possible to predict an increase in the probability of contamination of groundwater with low molecular weight organic complexing agents, as well as the probability of a better fixation of soil on waste in the presence of high molecular weight organic substances of the soil absorption complex.

Population structure determines the tradeoff between fixation probability and fixation time

The rate of biological evolution depends on the fixation probability and on the fixation time of new mutants. Intensive research has focused on identifying population structures that augment the fixation probability of advantageous mutants. But these amplifiers of natural selection typically increase fixation time. Here we study population structures that achieve a tradeoff between fixation probability and time. First, we show that no amplifiers can have an asymptotically lower absorption time than the well-mixed population. Then we design population structures that substantially augment the fixation probability with just a minor increase in fixation time. Finally, we show that those structures enable higher effective rate of evolution than the well-mixed population provided that the rate of generating advantageous mutants is relatively low. Our work sheds light on how population structure affects the rate of evolution. Moreover, our structures could be useful for lab-based, medical, or industrial applications of evolutionary optimization.

Exploring and mapping the universe of evolutionary graphs identifies structural properties affecting fixation probability and time

Population structure can be modeled by evolutionary graphs, which can have a substantial influence on the fate of mutants. Individuals are located on the nodes of these graphs, competing to take over the graph via the links. Applications for this framework range from the ecology of river systems and cancer initiation in colonic crypts to biotechnological search for optimal mutations. In all these applications, both the probability of fixation and the associated time are of interest. We study this problem for all undirected and unweighted graphs up to a certain size. We devise a genetic algorithm to find graphs with high or low fixation probability and short or long fixation time and study their structure searching for common themes. Our work unravels structural properties that maximize or minimize fixation probability and time, which allows us to contribute to a first map of the universe of evolutionary graphs.

Behavior of different numerical schemes for random genetic drift

In the problem of random genetic drift, the probability density of one gene is governed by a degenerated convection-dominated diffusion equation. Dirac singularities will always be developed at boundary points as time evolves, which is known as the fixation phenomenon in genetic evolution. Three finite volume methods: FVM1-3, one central difference method: FDM1 and three finite element methods: FEM1-3 are considered. These methods lead to different equilibrium states after a long time. It is shown that only schemes FVM3 and FEM3, which are the same, preserve probability, expectation and positiveness and predict the correct probability of fixation. FVM1-2 wrongly predict the probability of fixation due to their intrinsic viscosity, even though they are unconditionally stable. Contrarily, FDM1 and FEM1-2 introduce different anti-diffusion terms, which make them unstable and fail to preserve positiveness.

Description of the Donor—Acceptor Bond in Terms of the Theory of Generalized Charges

Previously, a universal relation for the energy and length of homopolar covalent bond has been derived with the use of the theory of generalized charges, which represents an asymptotic approximation for interatomic forces of the quantum-statistical model of a multicomponent electron gas. The article discusses the new implications of the theory, obtained to describe the donor-acceptor bond. The bond energy includes the contribution of the donor atom characteristic of a homopolar bond and the Coulomb interaction of the atoms, with allowance for the probability of fixation of the bond electrons on the acceptor. Regularities for the donor-acceptor bond length and energy have been discovered. In particular, for the particular case realized in hydrides, expressions are obtained for the length and bond energy versus the atomic number of the acceptor.

Dynamics of multiplayer games on complex networks using territorial interactions.

The modeling of evolution in structured populations has been significantly advanced by evolutionary graph theory, which incorporates pairwise relationships between individuals on a network. More recently, a new framework has been developed to allow for multiplayer interactions of variable size in more flexible and potentially changing population structures. While the theory within this framework has been developed and simple structures considered, there has been no systematic consideration of a large range of different population structures, which is the subject of this paper. We consider a large range of underlying graphical structures for the territorial raider model, the most commonly used model in the new structure, and consider a variety of important properties of our structures with the aim of finding factors that determine the fixation probability of mutants. We find that the graphical temperature and the average group size, as previously defined, are strong predictors of fixation probability, while all other properties considered are poor predictors, although the clustering coefficient is a useful secondary predictor when combined with either temperature or group size. The relationship between temperature or average group size and fixation probability is sometimes, however, nonmonotonic, with a directional reverse occurring around the temperature associated with what we term "completely mixed" populations in the case of the hawk-dove game, but not the public goods game.

Reading Shakespeare Sonnets: Combining Quantitative Narrative Analysis and Predictive Modeling -an Eye Tracking Study.

As a part of a larger interdisciplinary project on Shakespeare sonnets’ reception (1, 2), the present study analyzed the eye movement behavior of participants reading three of the 154 sonnets as a function of seven lexical features extracted via Quantitative Narrative Analysis (QNA). Using a machine learning-based predictive modeling approach five ‘surface’ features (word length, orthographic neighborhood density, word frequency, orthographic dissimilarity and sonority score) were detected as important predictors of total reading time and fixation probability in poetry reading. The fact that one phonological feature, i.e., sonority score, also played a role is in line with current theorizing on poetry reading. Our approach opens new ways for future eye movement research on reading poetic texts and other complex literary materials(3).

Deciding to look: revisiting the linking hypothesis for spoken word recognition in the visual world

ABSTRACTThe visual world paradigm (VWP) studies of spoken word recognition rely on a linking hypothesis that connects lexical activation to the probability of looking at the referent of a word. The standard hypothesis is that fixation probabilities track activation levels transformed via the Luce Choice Rule. Under this assumption, given enough power, any difference between positive activations should be detectable using VWP. We argue that looking at a referent of a word is a decision, made when the word’s activation exceeds a context-specific threshold. Subthreshold activations do not drive saccades, and differences among such activations are undetectable in VWP. Evidence is provided by VWP experiments on Japanese. Bayesian analyses indicate a relatively high threshold: saccades to cohort competitors do not exceed those to unrelated distractors unless the cohort competitor shares the initial CVC with the target. We argue that threshold setting constitutes an understudied source of variability in VWP data.

Evolution of cooperation in an epithelium.

Cooperation is prevalent in nature, not only in the context of social interactions within the animal kingdom but also on the cellular level. In cancer, for example, tumour cells can cooperate by producing growth factors. The evolution of cooperation has traditionally been studied for well-mixed populations under the framework of evolutionary game theory, and more recently for structured populations using evolutionary graph theory (EGT). The population structures arising due to cellular arrangement in tissues, however, are dynamic and thus cannot be accurately represented by either of these frameworks. In this work, we compare the conditions for cooperative success in an epithelium modelled using EGT, to those in a mechanical model of an epithelium—the Voronoi tessellation (VT) model. Crucially, in this latter model, cells are able to move, and birth and death are not spatially coupled. We calculate fixation probabilities in the VT model through simulation and an approximate analytic technique and show that this leads to stronger promotion of cooperation in comparison with the EGT model.

Methods for approximating stochastic evolutionary dynamics on graphs

Population structure can have a significant effect on evolution. For some systems with sufficient symmetry, analytic results can be derived within the mathematical framework of evolutionary graph theory which relate to the outcome of the evolutionary process. However, for more complicated heterogeneous structures, computationally intensive methods are required such as individual-based stochastic simulations. By adapting methods from statistical physics, including moment closure techniques, we first show how to derive existing homogenised pair approximation models and the exact neutral drift model. We then develop node-level approximations to stochastic evolutionary processes on arbitrarily complex structured populations represented by finite graphs, which can capture the different dynamics for individual nodes in the population. Using these approximations, we evaluate the fixation probability of invading mutants for given initial conditions, where the dynamics follow standard evolutionary processes such as the invasion process. Comparisons with the output of stochastic simulations reveal the effectiveness of our approximations in describing the stochastic processes and in predicting the probability of fixation of mutants on a wide range of graphs. Construction of these models facilitates a systematic analysis and is valuable for a greater understanding of the influence of population structure on evolutionary processes.

Allele fixation probability in a Moran model with fluctuating fitness landscapes.

Evolution on changing fitness landscapes (seascapes) is an important problem in evolutionary biology. We consider the Moran model of finite population evolution with selection in a randomly changing, dynamic environment. In the model, each individual has one of the two alleles, wild type or mutant. We calculate the fixation probability by making a proper ansatz for the logarithm of fixation probabilities. This method has been used previously to solve the analogous problem for the Wright-Fisher model. The fixation probability is related to the solution of a third-order algebraic equation (for the logarithm of fixation probability). We consider the strong interference of landscape fluctuations, sampling, and selection when the fixation process cannot be described by the mean fitness. Such an effect appears if the mutant allele has a higher fitness in one landscape and a lower fitness in another, compared with the wild type, and the product of effective population size and fitness is large. We provide a generalization of the Kimura formula for the fixation probability that applies to these cases. When the mutant allele has a fitness (dis-)advantage in both landscapes, the fixation probability is described by the mean fitness.

Binocular Eye Movements Are Adapted to the Natural Environment.

Humans and many animals make frequent saccades requiring coordinated movements of the eyes. When landing on the new fixation point, the eyes must converge accurately or double images will be perceived. We asked whether the visual system uses statistical regularities in the natural environment to aid eye alignment at the end of saccades. We measured the distribution of naturally occurring disparities in different parts of the visual field. The central tendency of the distributions was crossed (nearer than fixation) in the lower field and uncrossed (farther) in the upper field in male and female participants. It was uncrossed in the left and right fields. We also measured horizontal vergence after completion of vertical, horizontal, and oblique saccades. When the eyes first landed near the eccentric target, vergence was quite consistent with the natural-disparity distribution. For example, when making an upward saccade, the eyes diverged to be aligned with the most probable uncrossed disparity in that part of the visual field. Likewise, when making a downward saccade, the eyes converged to enable alignment with crossed disparity in that part of the field. Our results show that rapid binocular eye movements are adapted to the statistics of the 3D environment, minimizing the need for large corrective vergence movements at the end of saccades. The results are relevant to the debate about whether eye movements are derived from separate saccadic and vergence neural commands that control both eyes or from separate monocular commands that control the eyes independently.SIGNIFICANCE STATEMENT We show that the human visual system incorporates statistical regularities in the visual environment to enable efficient binocular eye movements. We define the oculomotor horopter: the surface of 3D positions to which the eyes initially move when stimulated by eccentric targets. The observed movements maximize the probability of accurate fixation as the eyes move from one position to another. This is the first study to show quantitatively that binocular eye movements conform to 3D scene statistics, thereby enabling efficient processing. The results provide greater insight into the neural mechanisms underlying the planning and execution of saccadic eye movements.

The evolutionary advantage of cultural memory on heterogeneous contact networks

Cultural processes, as well as the selection pressures experienced by individuals in a population over time and space, are fundamentally stochastic. Phenotypic variability, together with imperfect phenotypic transmission between parents and offspring, has been previously shown to play an important role in evolutionary rescue and (epi)genetic adaptation of populations to fluctuating temporal environmental pressures. This type of evolutionary bet-hedging does not confer a direct benefit to a single individual, but can instead increase the adaptability of the whole lineage.Here we develop a population-genetic model to explore cultural response strategies to temporally changing selection, as well as the role of local population structure, as exemplified by heterogeneity in the contact network between individuals, in shaping evolutionary dynamics. We use this model to study the evolutionary advantage of cultural bet-hedging, modeling the evolution of a variable cultural trait starting from one copy in a population of individuals with a fixed cultural strategy. We find that the probability of fixation of a cultural bet-hedger is a non-monotonic function of the probability of cultural memory between generations. Moreover, this probability increases for networks of higher mean degree but decreases with increasing heterogeneity of the contact network, tilting the balance of forces toward drift and against selection.These results shed light on the interplay of temporal and spatial stochasticity in shaping cultural evolutionary dynamics and suggest that partly-heritable cultural phenotypic variability may constitute an important evolutionary bet-hedging strategy in response to changing selection pressures.

Multiple infection of cells changes the dynamics of basic viral evolutionary processes.

The infection of cells by multiple copies of a given virus can impact viral evolution in a variety of ways, yet some of the most basic evolutionary dynamics remain underexplored. Using computational models, we investigate how infection multiplicity affects the fixation probability of mutants, the rate of mutant generation, and the timing of mutant invasion. An important insight from these models is that for neutral and disadvantageous phenotypes, rare mutants initially enjoy a fitness advantage in the presence of multiple infection of cells. This arises because multiple infection allows the rare mutant to enter more target cells and to spread faster, while it does not accelerate the spread of the resident wild‐type virus. The rare mutant population can increase by entry into both uninfected and wild‐type‐infected cells, while the established wild‐type population can initially only grow through entry into uninfected cells. Following this initial advantageous phase, the dynamics are governed by drift or negative selection, respectively, and a higher multiplicity reduces the chances that mutants fix in the population. Hence, while increased infection multiplicity promotes the presence of neutral and disadvantageous mutants in the short‐term, it makes it less likely in the longer term. We show how these theoretical insights can be useful for the interpretation of experimental data on virus evolution at low and high multiplicities. The dynamics explored here provide a basis for the investigation of more complex viral evolutionary processes, including recombination, reassortment, as well as complementary/inhibitory interactions.