arXiv subjects

Martin A. Nowak

Publications and source records attributed to Martin A. Nowak.

At least 55 records · Page 3Linked to original sources

arXiv2016

Evolution of worker policing

Workers in insect societies are sometimes observed to kill male eggs of other workers, a phenomenon known as worker policing. We perform a mathematical analysis of the evolutionary dynamics of policing. We investigate the selective forces behind policing for both dominant and recessive mutations for different numbers of matings of the queen. The traditional, relatedness-based argument suggests that policing evolves if the queen mates with more than two males, but does not evolve if the queen mates with a single male. We derive precise conditions for the invasion and stability of policing alleles. We find that the relatedness-based argument is not robust with respect to small changes in colony efficiency caused by policing. We also calculate evolutionarily singular strategies and determine when they are evolutionarily stable. We use a population genetics approach that applies to dominant or recessive mutations of any effect size.

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arXiv2015

Spatial model predicts dispersal and cell turnover cause reduced intra-tumor heterogeneity

Most cancers in humans are large, measuring centimeters in diameter, composed of many billions of cells. An equivalent mass of normal cells would be highly heterogeneous as a result of the mutations that occur during each cell division. What is remarkable about cancers is their homogeneity - virtually every neoplastic cell within a large cancer contains the same core set of genetic alterations, with heterogeneity confined to mutations that have emerged after the last clonal expansions. How such clones expand within the spatially-constrained three dimensional architecture of a tumor, and come to dominate a large, pre-existing lesion, has never been explained. We here describe a model for tumor evolution that shows how short-range migration and cell turnover can account for rapid cell mixing inside the tumor. With it, we show that even a small selective advantage of a single cell within a large tumor allows the descendants of that cell to replace the precursor mass in a clinically relevant time frame. We also demonstrate that the same mechanisms can be responsible for the rapid onset of resistance to chemotherapy. Our model not only provides novel insights into spatial and temporal aspects of tumor growth but also suggests that targeting short range cellular migratory activity could have dramatic effects on tumor growth rates.

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arXiv2014

Spatial heterogeneity in drug concentrations can facilitate the emergence of resistance to cancer therapy

Acquired resistance is one of the major barriers to successful cancer therapy. The development of resistance is commonly attributed to genetic heterogeneity. However, heterogeneity of drug penetration of the tumor microenvironment both on the microscopic level within solid tumors as well as on the macroscopic level across metastases may also contribute to acquired drug resistance. Here we use mathematical models to investigate the effect of drug heterogeneity on the probability of escape from treatment and time to resistance. Specifically we address scenarios with sufficiently efficient therapies that suppress growth of all preexisting genetic variants in the compartment with highest drug concentration. To study the joint effect of drug heterogeneity, growth rate, and evolution of resistance we analyze a multitype stochastic branching process describing growth of cancer cells in two compartments with different drug concentration and limited migration between compartments. We show that resistance is more likely to arise first in the low drug compartment and from there populate the high drug compartment. Moreover, we show that only below a threshold rate of cell migration does spatial heterogeneity accelerate resistance evolution, otherwise deterring drug resistance with excessively high migration rates. Our results provide new insights into understanding why cancers tend to quickly become resistant, and that cell migration and the presence of sanctuary sites with little drug exposure are essential to this end.

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arXiv2014

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 $K u$. 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 $K \leq u$. 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.

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arXiv2014

Predicting the outcomes of treatment to eradicate the latent reservoir for HIV-1

Massive research efforts are now underway to develop a cure for HIV infection, allowing patients to discontinue lifelong combination antiretroviral therapy (ART). New latency-reversing agents (LRAs) may be able to purge the persistent reservoir of latent virus in resting memory CD4+ T cells, but the degree of reservoir reduction needed for cure remains unknown. Here we use a stochastic model of infection dynamics to estimate the efficacy of LRA needed to prevent viral rebound after ART interruption. We incorporate clinical data to estimate population-level parameter distributions and outcomes. Our findings suggest that approximately 2,000-fold reductions are required to permit a majority of patients to interrupt ART for one year without rebound and that rebound may occur suddenly after multiple years. Greater than 10,000-fold reductions may be required to prevent rebound altogether. Our results predict large variation in rebound times following LRA therapy, which will complicate clinical management. This model provides benchmarks for moving LRAs from the lab to the clinic and can aid in the design and interpretation of clinical trials. These results also apply to other interventions to reduce the latent reservoir and can explain the observed return of viremia after months of apparent cure in recent bone marrow transplant recipients and an immediately-treated neonate.

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arXiv2014

Universality of fixation probabilities in randomly structured populations

The stage of evolution is the population of reproducing individuals. The structure of the population is know to affect the dynamics and outcome of evolutionary processes, but analytical results for generic random structures have been lacking. The most general result so far, the isothermal theorem, assumes the propensity for change in each position is exactly the same, but realistic biological structures are always subject to variation and noise. We consider a population of finite size $n$ under constant selection whose structure is given by a wide variety of weighted, directed, random graphs; vertices represent individuals and edges interactions between individuals. By establishing a robustness result for the isothermal theorem and using large deviation estimates to understand the typical structure of random graphs, we prove that for a generalization of the Erdős-Rényi model the fixation probability of an invading mutant is approximately the same as that of a mutant of equal fitness in a well-mixed population with high probability. Simulations of perturbed lattices, small-world networks, and scale-free networks behave similarly. We conjecture that the fixation probability in a well-mixed population, $(1-r^{-1})/(1-r^{-n})$, is universal: for many random graph models, the fixation probability approaches the above function uniformly as the graphs become large.

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arXiv2014

Risk management of solitary and eusocial reproduction

Social insect colonies can be seen as a distinct form of biological organization because they function as superorganisms. Understanding how natural selection acts on the emergence and maintenance of these colonies remains a major question in evolutionary biology and ecology. Here, we explore this by using multi-type branching processes to calculate the basic reproductive ratios and the extinction probabilities for solitary versus eusocial reproductive strategies. In order to derive precise mathematical results, we use a simple haploid, asexual model. In general, we show that eusocial reproductive strategies are unlikely to materialize unless large fitness advantages are gained by the production of only a few workers. These fitness advantages are maximized through obligate rather than facultative eusocial strategies. Furthermore, we find that solitary reproduction is `unbeatable' as long as the solitary reproductive ratio exceeds a critical value. In these cases, no eusocial parameters exist that would reduce their probability of extinction. Our results help to explain why the number of solitary species exceeds that of eusocial ones: eusociality is a high risk, high reward strategy, while solitary reproduction is better at risk management.

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arXiv2014

Optional games on cycles and complete graphs

We study stochastic evolution of optional games on simple graphs. There are two strategies, A and B, whose interaction is described by a general payoff matrix. In addition there are one or several possibilities to opt out from the game by adopting loner strategies. Optional games lead to relaxed social dilemmas. Here we explore the interaction between spatial structure and optional games. We find that increasing the number of loner strategies (or equivalently increasing mutational bias toward loner strategies) facilitates evolution of cooperation both in well-mixed and in structured populations. We derive various limits for weak selection and large population size. For some cases we derive analytic results for strong selection. We also analyze strategy selection numerically for finite selection intensity and discuss combined effects of optionality and spatial structure.

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arXiv2014

Zero-determinant alliances in multiplayer social dilemmas

Direct reciprocity and conditional cooperation are important mechanisms to prevent free riding in social dilemmas. But in large groups these mechanisms may become ineffective, because they require single individuals to have a substantial influence on their peers. However, the recent discovery of the powerful class of zero-determinant strategies in the iterated prisoner's dilemma suggests that we may have underestimated the degree of control that a single player can exert. Here, we develop a theory for zero-determinant strategies for multiplayer social dilemmas, with any number of involved players. We distinguish several particularly interesting subclasses of strategies: fair strategies ensure that the own payoff matches the average payoff of the group; extortionate strategies allow a player to perform above average; and generous strategies let a player perform below average. We use this theory to explore how individuals can enhance their strategic options by forming alliances. The effects of an alliance depend on the size of the alliance, the type of the social dilemma, and on the strategy of the allies: fair alliances reduce the inequality within their group; extortionate alliances outperform the remaining group members; but generous alliances increase welfare. Our results highlight the critical interplay of individual control and alliance formation to succeed in large groups.

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arXiv2013

Forgiver triumphs in alternating Prisoner's Dilemma

Cooperative behavior, where one individual incurs a cost to help another, is a wide spread phenomenon. Here we study direct reciprocity in the context of the alternating Prisoner's Dilemma. We consider all strategies that can be implemented by one and two-state automata. We calculate the payoff matrix of all pairwise encounters in the presence of noise. We explore deterministic selection dynamics with and without mutation. Using different error rates and payoff values, we observe convergence to a small number of distinct equilibria. Two of them are uncooperative strict Nash equilibria representing always-defect (ALLD) and Grim. The third equilibrium is mixed and represents a cooperative alliance of several strategies, dominated by a strategy which we call Forgiver. Forgiver cooperates whenever the opponent has cooperated; it defects once when the opponent has defected, but subsequently Forgiver attempts to re-establish cooperation even if the opponent has defected again. Forgiver is not an evolutionarily stable strategy, but the alliance, which it rules, is asymptotically stable. For a wide range of parameter values the most commonly observed outcome is convergence to the mixed equilibrium, dominated by Forgiver. Our results show that although forgiving might incur a short-term loss it can lead to a long-term gain. Forgiveness facilitates stable cooperation in the presence of exploitation and noise.

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arXiv2013

TTP: Tool for Tumor Progression

In this work we present a flexible tool for tumor progression, which simulates the evolutionary dynamics of cancer. Tumor progression implements a multi-type branching process where the key parameters are the fitness landscape, the mutation rate, and the average time of cell division. The fitness of a cancer cell depends on the mutations it has accumulated. The input to our tool could be any fitness landscape, mutation rate, and cell division time, and the tool produces the growth dynamics and all relevant statistics.

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arXiv2012

The Evolution of Extortion in Iterated Prisoner's Dilemma Games

Iterated games are a fundamental component of economic and evolutionary game theory. They describe situations where two players interact repeatedly and have the possibility to use conditional strategies that depend on the outcome of previous interactions. In the context of evolution of cooperation, repeated games represent the mechanism of reciprocation. Recently a new class of strategies has been proposed, so called 'zero determinant strategies'. These strategies enforce a fixed linear relationship between one's own payoff and that of the other player. A subset of those strategies are 'extortioners' which ensure that any increase in the own payoff exceeds that of the other player by a fixed percentage. Here we analyze the evolutionary performance of this new class of strategies. We show that in reasonably large populations they can act as catalysts for the evolution of cooperation, similar to tit-for-tat, but they are not the stable outcome of natural selection. In very small populations, however, relative payoff differences between two players in a contest matter, and extortioners hold their ground. Extortion strategies do particularly well in co-evolutionary arms races between two distinct populations: significantly, they benefit the population which evolves at the slower rate - an instance of the so-called Red King effect. This may affect the evolution of interactions between host species and their endosymbionts.

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arXiv2011

Prosperity is associated with instability in dynamical networks

Social, biological and economic networks grow and decline with occasional fragmentation and re-formation, often explained in terms of external perturbations. We show that these phenomena can be a direct consequence of simple imitation and internal conflicts between 'cooperators' and 'defectors'. We employ a game-theoretic model of dynamic network formation where successful individuals are more likely to be imitated by newcomers who adopt their strategies and copy their social network. We find that, despite using the same mechanism, cooperators promote well-connected highly prosperous networks and defectors cause the network to fragment and lose its prosperity; defectors are unable to maintain the highly connected networks they invade. Once the network is fragmented it can be reconstructed by a new invasion of cooperators, leading to the cycle of formation and fragmentation seen, for example, in bacterial communities and socio-economic networks. In this endless struggle between cooperators and defectors we observe that cooperation leads to prosperity, but prosperity is associated with instability. Cooperation is prosperous when the network has frequent formation and fragmentation.

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arXiv2010

The pace of evolution across fitness valleys

How fast does a population evolve from one fitness peak to another? We study the dynamics of evolving, asexually reproducing populations in which a certain number of mutations jointly confer a fitness advantage. We consider the time until a population has evolved from one fitness peak to another one with a higher fitness. The order of mutations can either be fixed or random. If the order of mutations is fixed, then the population follows a metaphorical ridge, a single path. If the order of mutations is arbitrary, then there are many ways to evolve to the higher fitness state. We address the time required for fixation in such scenarios and study how it is affected by the order of mutations, the population size, the fitness values and the mutation rate.

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arXiv2009

Accumulation of driver and passenger mutations during tumor progression

Major efforts to sequence cancer genomes are now occurring throughout the world. Though the emerging data from these studies are illuminating, their reconciliation with epidemiologic and clinical observations poses a major challenge. In the current study, we provide a novel mathematical model that begins to address this challenge. We model tumors as a discrete time branching process that starts with a single driver mutation and proceeds as each new driver mutation leads to a slightly increased rate of clonal expansion. Using the model, we observe tremendous variation in the rate of tumor development - providing an understanding of the heterogeneity in tumor sizes and development times that have been observed by epidemiologists and clinicians. Furthermore, the model provides a simple formula for the number of driver mutations as a function of the total number of mutations in the tumor. Finally, when applied to recent experimental data, the model allows us to calculate, for the first time, the actual selective advantage provided by typical somatic mutations in human tumors in situ. This selective advantage is surprisingly small, 0.005 +- 0.0005, and has major implications for experimental cancer research.

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arXiv2009

Evolutionary game dynamics in phenotype space

Evolutionary dynamics can be studied in well-mixed or structured populations. Population structure typically arises from the heterogeneous distribution of individuals in physical space or on social networks. Here we introduce a new type of space to evolutionary game dynamics: phenotype space. The population is well-mixed in the sense that everyone is equally likely to interact with everyone else, but the behavioral strategies depend on distance in phenotype space. Individuals might behave differently towards those who look similar or dissimilar. Individuals mutate to nearby phenotypes. We study the `phenotypic space walk' of populations. We present analytic calculations that bring together ideas from coalescence theory and evolutionary game dynamics. As a particular example, we investigate the evolution of cooperation in phenotype space. We obtain a precise condition for natural selection to favor cooperators over defectors: for a one-dimensional phenotype space and large population size the critical benefit-to-cost ratio is given by b/c=1+2/sqrt{3}. We derive the fundamental condition for any evolutionary game and explore higher dimensional phenotype spaces.

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arXiv2009

Mutation-selection equilibrium in games with multiple strategies

In evolutionary games the fitness of individuals is not constant but depends on the relative abundance of the various strategies in the population. Here we study general games among n strategies in populations of large but finite size. We explore stochastic evolutionary dynamics under weak selection, but for any mutation rate. We analyze the frequency dependent Moran process in well-mixed populations, but almost identical results are found for the Wright-Fisher and Pairwise Comparison processes. Surprisingly simple conditions specify whether a strategy is more abundant on average than 1/n, or than another strategy, in the mutation-selection equilibrium. We find one condition that holds for low mutation rate and another condition that holds for high mutation rate. A linear combination of these two conditions holds for any mutation rate. Our results allow a complete characterization of n*n games in the limit of weak selection.

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arXiv2008

Strategy abundance in 2x2 games for arbitrary mutation rates

We study evolutionary game dynamics in a well-mixed populations of finite size, N. A well-mixed population means that any two individuals are equally likely to interact. In particular we consider the average abundances of two strategies, A and B, under mutation and selection. The game dynamical interaction between the two strategies is given by the 2x2 payoff matrix [(a,b), (c,d)]. It has previously been shown that A is more abundant than B, if (N-2)a+Nb>Nc+(N-2)d. This result has been derived for particular stochastic processes that operate either in the limit of asymptotically small mutation rates or in the limit of weak selection. Here we show that this result holds in fact for a wide class of stochastic birth-death processes for arbitrary mutation rate and for any intensity of selection.

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