Coevolution of quasispecies: B-cell mutation rates maximize viral error catastrophes

Coevolutionary transitions emerging from flexible molecular recognition and eco-evolutionary feedback

Highly mutable viruses evolve to evade host immunity that exerts selective pressure and adapts to viral dynamics. Here, we provide a framework for identifying key determinants of the mode and fate of viral-immune coevolution by linking molecular recognition and eco-evolutionary dynamics. We find that conservation level and initial diversity of antigen jointly determine the timing and efficacy of narrow and broad antibody responses, which in turn control the transition between viral persistence, clearance, and rebound. In particular, clearance of structurally complex antigens relies on antibody evolution in a larger antigenic space than where selection directly acts; viral rebound manifests binding-mediated feedback between ecology and rapid evolution. Finally, immune compartmentalization can slow viral escape but also delay clearance. This work suggests that flexible molecular binding allows a plastic phenotype that exploits potentiating neutral variations outside direct contact, opening new and shorter paths toward highly adaptable states.

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A scale-crossing framework identifies key determinants of viral-immune coevolution

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Fast specific response influences slow broad response by shaping antigen dynamics

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Antibody footprint shift enables breadth acquisition and viral clearance

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Model explains divergent kinetics and outcomes of HCV infection in humans

Periodic versus intermittent adaptive cycles in quasispecies coevolution

We study an abstract model for the coevolution between mutating viruses and the adaptive immune system. In sequence space, these two populations are localized around transiently dominant strains. Delocalization or error thresholds exhibit a novel interdependence because immune response is conditional on the viral attack. An evolutionary chase is induced by stochastic fluctuations and can occur via periodic or intermittent cycles. Using simulations and stochastic analysis, we show how the transition between these two dynamic regimes depends on mutation rate, immune response, and population size.

Running loose or getting lost: how HIV-1 counters and capitalizes on APOBEC3-induced mutagenesis through its Vif protein

Human immunodeficiency virus-1 (HIV-1) dynamics reflect an intricate balance within the viruses’ host. The virus relies on host replication factors, but must escape or counter its host’s antiviral restriction factors. The interaction between the HIV-1 protein Vif and many cellular restriction factors from the APOBEC3 protein family is a prominent example of this evolutionary arms race. The viral infectivity factor (Vif) protein largely neutralizes APOBEC3 proteins, which can induce in vivo hypermutations in HIV-1 to the extent of lethal mutagenesis, and ensures the production of viable virus particles. HIV-1 also uses the APOBEC3-Vif interaction to modulate its own mutation rate in harsh or variable environments, and it is a model of adaptation in a coevolutionary setting. Both experimental evidence and the substantiation of the underlying dynamics through coevolutionary models are presented as complementary views of a coevolutionary arms race.

Short-time evolution in the adaptive immune system

We exploit a simple model to numerically and analytically investigate the effect of enforcing a time constraint for achieving a system-wide goal during an evolutionary dynamics. This situation is relevant to finding antibody specificities in the adaptive immune response as well as to artificial situations in which an evolutionary dynamics is used to generate a desired capability in a limited number of generations. When the likelihood of finding the target phenotype is low, we find that the optimal mutation rate can exceed the error threshold, in contrast to conventional evolutionary dynamics. We also show how a logarithmic correction to the usual inverse scaling of population size with mutation rate arises. Implications for natural and artificial evolutionary situations are discussed.

Impacts of mutation effects and population size on mutation rate in asexual populations: a simulation study

Background

In any natural population, mutation is the primary source of genetic variation required for evolutionary novelty and adaptation. Nevertheless, most mutations, especially those with phenotypic effects, are harmful and are consequently removed by natural selection. For this reason, under natural selection, an organism will evolve to a lower mutation rate. Overall, the action of natural selection on mutation rate is related to population size and mutation effects. Although theoretical work has intensively investigated the relationship between natural selection and mutation rate, most of these studies have focused on individual competition within a population, rather than on competition among populations. The aim of the present study was to use computer simulations to investigate how natural selection adjusts mutation rate among asexually reproducing subpopulations with different mutation rates.

Results

The competition results for the different subpopulations showed that a population could evolve to an "optimum" mutation rate during long-term evolution, and that this rate was modulated by both population size and mutation effects. A larger population could evolve to a higher optimum mutation rate than could a smaller population. The optimum mutation rate depended on both the fraction and the effects of beneficial mutations, rather than on the effects of deleterious ones. The optimum mutation rate increased with either the fraction or the effects of beneficial mutations. When strongly favored mutations appeared, the optimum mutation rate was elevated to a much higher level. The competition time among the subpopulations also substantially shortened.

Conclusions

Competition at the population level revealed that the evolution of the mutation rate in asexual populations was determined by both population size and mutation effects. The most striking finding was that beneficial mutations, rather than deleterious mutations, were the leading force that modulated the optimum mutation rate. The initial configuration of the population appeared to have no effect on these conclusions, confirming the robustness of the simulation method developed in the present study. These findings might further explain the lower mutation rates observed in most asexual organisms, as well as the higher mutation rates in some viruses.

Detailed analysis of an Eigen quasispecies model in a periodically moving sharp-peak landscape

The Eigen quasispecies model in a periodically moving sharp-peak landscape considered in previous seminal works [M. Nilsson and N. Snoad, Phys. Rev. Lett. 84, 191 (2000)] and [C. Ronnewinkel, in, edited by L. Kallel, B. Naudts, and A. Rogers (Springer-Verlag, Heidelberg, 2001)] is analyzed in greater detail. We show here, through a more rigorous analysis, that results in those papers are qualitatively correct. In particular, we obtain a phase diagram for the existence of a quasispecies with the same shape as in the above cited paper by C. Ronnewinkel, with upper and lower thresholds for the mutation rate between which a quasispecies may survive. A difference is that the upper value is larger and the lower value is smaller than the previously reported ones, so that the range for quasispecies existence is always larger than thought before. The quantitative information provided might also be important in understanding genetic variability in virus populations and has possible applications in antiviral therapies. The results in the quoted papers were obtained by studying the populations only at some few genomes. As we will show, this amounts to diagonalizing a 3×3 matrix. Our work is based instead in a different division of the population allowing a finer control of the populations at various relevant genetic sequences. The existence of a quasispecies will be related to Perron-Frobenius eigenvalues. Although huge matrices of sizes 2(ℓ), where ℓ is the genome length, may seem necessary at a first look, we show that such large sizes are not necessary and easily obtain numerical and analytical results for their eigenvalues.

Optimal mutation rates in dynamic environments: The Eigen model

We consider the Eigen quasispecies model with a dynamic environment. For an environment with sharp-peak fitness in which the most-fit sequence moves by k spin-flips each period T we find an asymptotic stationary state in which the quasispecies population changes regularly according to the regular environmental change. From this stationary state we estimate the maximum and the minimum mutation rates for a quasispecies to survive under the changing environment and calculate the optimum mutation rate that maximizes the population growth. Interestingly we find that the optimum mutation rate in the Eigen model is lower than that in the Crow-Kimura model, and at their optimum mutation rates the corresponding mean fitness in the eigenmodel is lower than that in the Crow-Kimura model, suggesting that the mutation process which occurs in parallel to the replication process as in the Crow-Kimura model gives an adaptive advantage under changing environment.

Semiconservative quasispecies equations for polysomic genomes: the general case

This paper develops a formulation of the quasispecies equations appropriate for polysomic, semiconservatively replicating genomes. This paper is an extension of previous work on the subject, which considered the case of haploid genomes. Here, we develop a more general formulation of the quasispecies equations that is applicable to diploid and even polyploid genomes. Interestingly, with an appropriate classification of population fractions, we obtain a system of equations that is formally identical to the haploid case. As with the work for haploid genomes, we consider both random and immortal DNA strand chromosome segregation mechanisms. However, in contrast to the haploid case, we have found that an analytical solution for the mean fitness is considerably more difficult to obtain for the polyploid case. Accordingly, whereas for the haploid case we obtained expressions for the mean fitness for the case of an analog of the single-fitness-peak landscape for arbitrary lesion repair probabilities (thereby allowing for noncomplementary genomes), here we solve for the mean fitness for the restricted case of perfect lesion repair.

Synthesizing within-host and population-level selective pressures on viral populations: the impact of adaptive immunity on viral immune escape

The evolution of viruses to escape prevailing host immunity involves selection at multiple integrative scales, from within-host viral and immune kinetics to the host population level. In order to understand how viral immune escape occurs, we develop an analytical framework that links the dynamical nature of immunity and viral variation across these scales. Our epidemiological model incorporates within-host viral evolutionary dynamics for a virus that causes acute infections (e.g. influenza and norovirus) with changes in host immunity in response to genetic changes in the virus population. We use a deterministic description of the within-host replication dynamics of the virus, the pool of susceptible host cells and the host adaptive immune response. We find that viral immune escape is most effective at intermediate values of immune strength. At very low levels of immunity, selection is too weak to drive immune escape in recovered hosts, while very high levels of immunity impose such strong selection that viral subpopulations go extinct before acquiring enough genetic diversity to escape host immunity. This result echoes the predictions of simpler models, but our formulation allows us to dissect the combination of within-host and transmission-level processes that drive immune escape.

Theoretical aspects of immunity

The immune system recognizes a myriad of invading pathogens and their toxic products. It does so with a finite repertoire of antibodies and T cell receptors. We here describe theories that quantify the dynamics of the immune system. We describe how the immune system recognizes antigens by searching the large space of receptor molecules. We consider in some detail the theories that quantify the immune response to influenza and dengue fever. We review theoretical descriptions of the complementary evolution of pathogens that occurs in response to immune system pressure. Methods including bioinformatics, molecular simulation, random energy models, and quantum field theory contribute to a theoretical understanding of aspects of immunity.

Maximum, minimum, and optimal mutation rates in dynamic environments

We analyze the dynamics of the parallel mutation-selection quasispecies model with a changing environment. For an environment with the sharp-peak fitness function in which the most fit sequence changes by k spin flips every period T , we find analytical expressions for the minimum and maximum mutation rates for which a quasispecies can survive, valid in the limit of large sequence size. We find an asymptotic solution in which the quasispecies population changes periodically according to the periodic environmental change. In this state we compute the mutation rate that gives the optimal mean fitness over a period. We find that the optimal mutation rate per genome, k/T , is independent of genome size, a relationship which is observed across broad groups of real organisms.

The quasispecies nature and biological implications of the hepatitis C virus

Many RNA viruses exist as a cloud of closely related sequence variants called a quasispecies, rather than as a population of identical clones. In this article, we explain the quasispecies nature of RNA viral genomes, and briefly review the principles of quasispecies dynamics and the differences with classical population genetics. We then discuss the current methods for quasispecies analysis and conclude with the biological implications of this phenomenon, focusing on the hepatitis C virus.

Mutagenic evidence for the optimal control of evolutionary dynamics

Elucidating the fitness measures optimized during the evolution of complex biological systems is a major challenge in evolutionary theory. We present experimental evidence and an analytical framework demonstrating how biochemical networks exploit optimal control strategies in their evolutionary dynamics. Optimal control theory explains a striking pattern of extremization in the redox potentials of electron transport proteins, assuming only that their fitness measure is a control objective functional with bounded controls.

Comparison of three replication strategies in complex multicellular organisms: asexual replication, sexual replication with identical gametes, and sexual replication with distinct sperm and egg gametes

This paper studies the mutation-selection balance in three simplified replication models. The first model considers a population of organisms replicating via the production of asexual spores. The second model considers a sexually replicating population that produces identical gametes. The third model considers a sexually replicating population that produces distinct sperm and egg gametes. All models assume diploid organisms whose genomes consist of two chromosomes, each of which is taken to be functional if equal to some master sequence, and defective otherwise. In the asexual population, the asexual diploid spores develop directly into adult organisms. In the sexual populations, the haploid gametes enter a haploid pool, where they may fuse with other haploids. The resulting immature diploid organisms then proceed to develop into mature organisms. Based on an analysis of all three models, we find that, as organism size increases, a sexually replicating population can only outcompete an asexually replicating population if the adult organisms produce distinct sperm and egg gametes. A sexual replication strategy that is based on the production of large numbers of sperm cells to fertilize a small number of eggs is found to be necessary in order to maintain a sufficiently low cost for sex for the strategy to be selected for over a purely asexual strategy. We discuss the usefulness of this model in understanding the evolution and maintenance of sexual replication as the preferred replication strategy in complex, multicellular organisms.

Evolutionary dynamics of the most populated genotype on rugged fitness landscapes

We consider an asexual population evolving on rugged fitness landscapes which are defined on the multidimensional genotypic space and have many local optima. We track the most populated genotype as it changes when the population jumps from a fitness peak to a better one during the process of adaptation. This is done using the dynamics of the shell model which is a simplified version of the quasispecies model for infinite populations and standard Wright-Fisher dynamics for large finite populations. We show that the population fraction of a genotype obtained within the quasispecies model and the shell model match for fit genotypes and at short times, but the dynamics of the two models are identical for questions related to the most populated genotype. We calculate exactly several properties of the jumps in infinite populations, some of which were obtained numerically in previous works. We also present our preliminary simulation results for finite populations. In particular, we measure the jump distribution in time and find that it decays as t(-2) as in the quasispecies problem.

Asexual and sexual replication in sporulating organisms

Replication via sporulation is the replication strategy for all multicellular life, and may even be observed in unicellular life (such as with budding yeast). We consider diploid populations replicating via one of two possible sporulation mechanisms. (1) Asexual sporulation, whereby adult organisms produce single-celled diploid spores that grow into adults themselves. (2) Sexual sporulation, whereby adult organisms produce single-celled diploid spores that divide into haploid gametes. The haploid gametes enter a haploid "pool," where they may recombine with other haploids to form a diploid spore that then grows into an adult. We consider a haploid fusion rate given by second-order reaction kinetics. We work with a simplified model where the diploid genome consists of only two chromosomes, each of which may be rendered defective with a single point mutation of the wild-type. We find that the asexual strategy is favored when the rate of spore production is high compared to the characteristic growth rate from a spore to a reproducing adult. Conversely, the sexual strategy is favored when the rate of spore production is low compared to the characteristic growth rate from a spore to a reproducing adult. As the characteristic growth time increases, or as the population density increases, the critical ratio of spore production rate to organism growth rate at which the asexual strategy overtakes the sexual one is pushed to higher values. Therefore, the results of this model suggest that, for complex multicellular organisms, sexual replication is favored at high population densities and low growth and sporulation rates.

Self-organizing behavior in a lattice model for co-evolution of virus and immune systems

We propose a lattice model for the co-evolution of a virus population and an adaptive immune system. We show that, under some natural assumptions, both probability distribution of the virus population and the distribution of activity of the immune system tend during the evolution to a self-organized critical state.

Error thresholds for quasispecies on single peak Gaussian-distributed fitness landscapes

Based on the Eigen and Crow-Kimura models with a single peak fitness landscape, we propose that the fitness values of all molecules be Gaussian distributed random variables to incorporate the fluctuation effects of the fitness landscapes (noise of environments). And we investigate the quasispecies distribution and error threshold using ensemble average method within this theoretical framework. Numerical results show that a small fluctuation of the fitness landscape causes only a slight change in the concentration distribution and error threshold, which implies that the error threshold is stable against small perturbations. However, for a sizable fluctuation, quite different from the previous deterministic models, our statistical results reveal that the transition from quasi-species to error catastrophe is no longer so sharp, indicating the error threshold is located within a certain range and shifts toward a larger value.

Selective advantage for sexual reproduction

This paper develops a simplified model for sexual reproduction within the quasispecies formalism. The model assumes a diploid genome consisting of two chromosomes, where the fitness is determined by the number of chromosomes that are identical to a given master sequence. We also assume that there is a cost to sexual reproduction, given by a characteristic time tau(seek) during which haploid cells seek out a mate with which to recombine. If the mating strategy is such that only viable haploids can mate, then when tau(seek) = 0, it is possible to show that sexual reproduction will always out compete asexual reproduction. However, as tau(seek) increases, sexual reproduction only becomes advantageous at progressively higher mutation rates. Once the time cost for sex reaches a critical threshold, the selective advantage for sexual reproduction disappears entirely. The results of this paper suggest that sexual reproduction is not advantageous in small populations per se, but rather in populations with low replication rates. In this regime, the cost for sex is sufficiently low that the selective advantage obtained through recombination leads to the dominance of the strategy. In fact, at a given replication rate and for a fixed environment volume, sexual reproduction is selected for in high populations because of the reduced time spent finding a reproductive partner.

Quasispecies in time-dependent environments

In recent years, quasispecies theory in time-dependent (that is, dynamically changing) environments has made dramatic progress. Several groups have addressed questions such as how the time scale of the changes affect viral adaptation and quasispecies formation, how environmental changes affect the optimal mutation rate, or how virus and host co-evolve. Here, we review these recent developments, and give a nonmathematical introduction to the most important concepts and results of quasispecies theory in time-dependent environments. We also compare the theoretical results with results from evolution experiments that expose viruses to successive regimes of replication in two or more different hosts.

Exact solution of the Eigen model with general fitness functions and degradation rates

We present an exact solution of Eigen's quasispecies model with a general degradation rate and fitness functions, including a square root decrease of fitness with increasing Hamming distance from the wild type. The found behavior of the model with a degradation rate is analogous to a viral quasispecies under attack by the immune system of the host. Our exact solutions also revise the known results of neutral networks in quasispecies theory. To explain the existence of mutants with large Hamming distances from the wild type, we propose three different modifications of the Eigen model: mutation landscape, multiple adjacent mutations, and frequency-dependent fitness in which the steady-state solution shows a multicenter behavior.

Semiconservative quasispecies equations for polysomic genomes: the haploid case

This paper develops the semiconservative quasispecies equations for genomes consisting of an arbitrary number of chromosomes. We assume that the chromosomes are distinguishable, so that we are effectively considering haploid genomes. We derive the quasispecies equations under the assumption of arbitrary lesion repair efficiency, and consider the cases of both random and immortal strand chromosome segregation. We solve the model in the limit of infinite sequence length for the case of the static single fitness peak landscape, where the master genome has a first-order growth rate constant of k>1, and all other genomes have a first-order growth rate constant of 1. If we assume that each chromosome can tolerate an arbitrary number of lesions, so that only one master copy of the strands is necessary for a functional chromosome, then for random chromosome segregation we obtain an equilibrium mean fitness of [equation in text] below the error catastrophe, while for immortal strand co-segregation we obtain kappa (t=infinity)=k[e(-mu(1-lambda/2))+e(-mulambda/2)-1] (N denotes the number of chromosomes, lambda denotes the lesion repair efficiency, and mu is identical with epsilonL, where epsilon is the per base-pair mismatch probability, and L is the total genome length). It follows that immortal strand co-segregation leads to significantly better preservation of the master genome than random segregation when lesion repair is imperfect. Based on this result, we conjecture that certain classes of tumor cells exhibit immortal strand co-segregation.

Genetic instability and the quasispecies model

Genetic instability is a defining characteristic of cancers. Microsatellite instability (MIN) leads to by elevated point mutation rates, whereas chromosomal instability (CIN) refers to increased rates of losing or gaining whole chromosomes or parts of chromosomes during cell division. CIN and MIN are, in general, mutually exclusive. The quasispecies model is a very successful theoretical framework for the study of evolution at high mutation rates. It predicts the existence of an experimentally verified error catastrophe. This catastrophe occurs when the mutation rates exceed a threshold value, the error threshold, above which replicative infidelity is incompatible with cell survival. We analyse the semiconservative quasispecies model of both MIN and CIN tumors. We consider the role of post-methylation DNA repair in tumor cells and demonstrate that DNA repair is fundamental to the nature of the error catastrophe in both types of tumors. We find that CIN introduces a plateau in the maximum viable mutation rate for a repair-free model, which does not exist in the case of MIN. This provides a plausible explanation for the mutual exclusivity of CIN and MIN.

Information catastrophe in RNA viruses through replication thresholds

RNA viruses are known to replicate at very high mutation rates. These rates are actually known to be close to their so-called error threshold. This threshold is in fact a critical point beyond which genetic information is lost through a so-called error catastrophe. However, the transition from a stable quasispecies to genetic drift and loss of information can also occur by crossing replication thresholds, below some replication rates, the viral population is suddenly unable to survive. Available data from hepatitis C virus population analysis [Mas, A., Ulloa, E., Bruguera, M., Furcić, I., Garriga, D., Fábregas, S., Andreu, D., Saiz, J.C., Díez, J., 2004. Hepatitis C virus population analysis of a single-source nosocomial outbreak reveals an inverse correlation between viral load and quasispecies complexity. J. Gen. Virol. 85, 3619-3626] can be interpreted through this theoretical view, providing evidence for such a replication threshold. Here a simple model is used in order to provide evidence for such a phenomenon, consistent with available data.

Tradeoff between short-term and long-term adaptation in a changing environment

We investigate the competition dynamics of two microbial or viral strains that live in an environment that switches periodically between two states. One of the strains is adapted to the long-term environment, but pays a short-term cost, while the other is adapted to the short-term environment and pays a cost in the long term. We explore the tradeoff between these alternative strategies in extensive numerical simulations and present a simple analytic model that can predict the outcome of these competitions as a function of the mutation rate and the time scale of the environmental changes. Our model is relevant for arboviruses, which alternate between different host species on a regular basis.

A quasispecies on a moving oasis

A population evolving in an inhomogeneous environment will adapt differently to different areas. We study the conditions under which such a population can maintain adaptations to a particular region when that region is not stationary, but can move. In particular, we consider a haploid population living near a moving favorable patch ("oasis") in the middle of a large "desert." At one genetic locus, individuals may have one of a few gene sequences that convey an advantage while in the oasis at the cost of a disadvantage in the desert. The distribution of genetic states in the population, possibly localized in genome space around the oasis-adapted genotypes, is known as a quasispecies. We find that the ratio of oasis-adapted individuals to desert-adapted ones exhibits sharp transitions at particular oasis velocities. We calculate an extinction velocity, and a switching velocity above which the dominance switches from the oasis-adapted genotype to the desert-adapted one. This switching velocity is analogous to the quasispecies mutational error threshold. Above this velocity, the population cannot maintain adaptations to the properties of the oasis.

Quasispecies dynamics and RNA virus extinction

The extinction of foot-and-mouth disease virus (FMDV) is strongly influenced by mutation rates, types of mutations, relative viral fitness and virus population regimens during infection. Here we review experimental results and theoretical models that describe a contrast between the effective extinction of FMDV subjected to increased mutagenesis, and the remarkable resistance to extinction of the same and related FMDV clones subjected to serial bottleneck events. The results suggest procedures to master key parameters to develop effective antiviral strategies based on virus entry into error catastrophe.

Selective advantage for conservative viruses

In this article we study the full semiconservative treatment of a model for the coevolution of a virus and an adaptive immune system. Regions of viability are calculated for both conservatively and semiconservatively replicating viruses interacting with a realistic semiconservatively replicating immune system. The conservative virus is found to have a selective advantage in the form of an ability to survive in regions with a wider range of mutation rates than its semiconservative counterpart, as well as an increased replication rate where both species can survive. This may help explain the existence of a rich range of viruses with conservatively replicating genomes, a trait that is found nowhere else in nature.

Imperfect DNA lesion repair in the semiconservative quasispecies model: derivation of the Hamming class equations and solution of the single-fitness peak landscape

This paper develops a Hamming class formalism for the semiconservative quasispecies equations with imperfect lesion repair, first presented and analytically solved in Y. Brumer and E.I. Shakhnovich (q-bio.GN/0403018, 2004). Starting from the quasispecies dynamics over the space of genomes, we derive an equivalent dynamics over the space of ordered sequence pairs. From this set of equations, we are able to derive the infinite sequence length form of the dynamics for a class of fitness landscapes defined by a master genome. We use these equations to solve for a generalized single-fitness-peak landscape, where the master genome can sustain a maximum number of lesions and remain viable. We determine the mean equilibrium fitness and error threshold for this class of landscapes, and show that when lesion repair is imperfect, semiconservative replication displays characteristics from both conservative replication and semiconservative replication with perfect lesion repair. The work presented here provides a formulation of the model which greatly facilitates the analysis of a relatively broad class of fitness landscapes, and thus serves as a convenient springboard into biological applications of imperfect lesion repair.

Importance of DNA repair in tumor suppression

The transition from a normal to cancerous cell requires a number of highly specific mutations that affect cell cycle regulation, apoptosis, differentiation, and many other cell functions. One hallmark of cancerous genomes is genomic instability, with mutation rates far greater than those of normal cells. In microsatellite instability (MIN tumors), these are often caused by damage to mismatch repair genes, allowing further mutation of the genome and tumor progression. These mutation rates may lie near the error catastrophe found in the quasispecies model of adaptive RNA genomes, suggesting that further increasing mutation rates will destroy cancerous genomes. However, recent results have demonstrated that DNA genomes exhibit an error threshold at mutation rates far lower than their conservative counterparts. Furthermore, while the maximum viable mutation rate in conservative systems increases indefinitely with increasing master sequence fitness, the semiconservative threshold plateaus at a relatively low value. This implies a paradox, wherein inaccessible mutation rates are found in viable tumor cells. In this paper, we address this paradox, demonstrating an isomorphism between the conservatively replicating (RNA) quasispecies model and the semiconservative (DNA) model with post-methylation DNA repair mechanisms impaired. Thus, as DNA repair becomes inactivated, the maximum viable mutation rate increases smoothly to that of a conservatively replicating system on a transformed landscape, with an upper bound that is dependent on replication rates. On a specific single fitness peak landscape, the repair-free semiconservative system is shown to mimic a conservative system exactly. We postulate that inactivation of post-methylation repair mechanisms is fundamental to the progression of a tumor cell and hence these mechanisms act as a method for the prevention and destruction of cancerous genomes.

An error catastrophe in cancer?

A comparison between the evolution of cancer cell populations and RNA viruses reveals a number of remarkable similarities. Both display high levels of plasticity and adaptability as a consequence of high degrees of genetic variation. It has been suggested that, as it occurs with RNA viruses, there is a threshold in the levels of genetic instability affordable by cancer cells in order to be able to overcome selection barriers (Trends Genet. 15 (1999) M57). Here we explore this concept by means of a simple mathematical model. It is shown that an error threshold exists in this model, which investigates both competition between cancer cell populations and its impact on overall tumor growth dynamics. Once the threshold is reached, the highly unstable tumor cell populations, which were sustaining malignant growth, become unable to maintain their genetic information, which in turn triggers a slowed down overall tumor growth regime.

Solution of the quasispecies model for an arbitrary gene network

In this paper, we study the equilibrium behavior of Eigen's quasispecies equations for an arbitrary gene network. We consider a genome consisting of N genes, so that the full genome sequence sigma may be written as sigma= sigma1sigma2...sigmaN, where sigma(i) are sequences of individual genes. We assume a single fitness peak model for each gene, so that gene i has some "master" sequence sigma(i,0) for which it is functioning. The fitness landscape is then determined by which genes in the genome are functioning and which are not. The equilibrium behavior of this model may be solved in the limit of infinite sequence length. The central result is that, instead of a single error catastrophe, the model exhibits a series of localization to delocalization transitions, which we term an "error cascade." As the mutation rate is increased, the selective advantage for maintaining functional copies of certain genes in the network disappears, and the population distribution delocalizes over the corresponding sequence spaces. The network goes through a series of such transitions, as more and more genes become inactivated, until eventually delocalization occurs over the entire genome space, resulting in a final error catastrophe. This model provides a criterion for determining the conditions under which certain genes in a genome will lose functionality due to genetic drift. It also provides insight into the response of gene networks to mutagens. In particular, it suggests an approach for determining the relative importance of various genes to the fitness of an organism, in a more accurate manner than the standard "deletion set" method. The results in this paper also have implications for mutational robustness and what C.O. Wilke termed "survival of the flattest."

Host-parasite coevolution and optimal mutation rates for semiconservative quasispecies

In this paper, we extend a model of host-parasite coevolution to incorporate the semiconservative nature of DNA replication for both the host and the parasite. We find that the optimal mutation rate for the semiconservative and conservative hosts converge for realistic genome lengths, thus maintaining the admirable agreement between theory and experiment found previously for the conservative model and justifying the conservative approximation in some cases. We demonstrate that, while the optimal mutation rate for a conservative and semiconservative parasite interacting with a given immune system is similar to that of a conservative parasite, the properties away from this optimum differ significantly. We suspect that this difference, coupled with the requirement that a parasite optimize survival in a range of viable hosts, may help explain why semiconservative viruses are known to have significantly lower mutation rates than their conservative counterparts.

Solvable biological evolution model with a parallel mutation-selection scheme

Based on the connection between a quantum spin model and an asexual biological evolution model with a single-peak fitness function in parallel mutation-selection scheme, we solve exactly both static and dynamics of the evolution model. We find that relaxation in such a parallel scheme is faster than that in a connected scheme of Eigen model. Our method can also be extended to other fitness functions.

A quasispecies approach to viral evolution in the context of an adaptive immune system

A deeper understanding of the mechanisms that determine viral evolution in the context of an adaptive immune system is vital for the development of efficient strategies to defeat viral infections. The problem of describing these mechanisms is discussed using the concept of quasispecies. Conditions for both an optimal immune response and for highest viral viability are derived from theoretical models and are supported by empirical data.

Eigen model as a quantum spin chain: exact dynamics

We map the Eigen model of biological evolution [Naturwissenschaften 58, 465 (1971)] into a quantum spin model with non-Hermitian Hamiltonian. Based on such a connection, we derive exact relaxation periods for the Eigen model to approach static energy landscape from various initial conditions. We also study a simple case of dynamic fitness function.

Digital evolution in time-dependent fitness landscapes

We study the response of populations of digital organisms that adapt to a time-varying (periodic) fitness landscape of two oscillating peaks. We corroborate in general predictions from quasi-species theory in dynamic landscapes, such as adaptation to the average fitness landscape at small periods (high frequency) and quasistatic adaptation at large periods (low frequency). We also observe adaptive phase shifts (time lags between a change in the fitness landscape and an adaptive change in the population) that indicate a low-pass filter effect, in agreement with existing theory. Finally, we witness long-term adaptation to fluctuating environments not anticipated in previous theoretical work.

Equilibrium distribution of mutators in the single fitness peak model

This Letter develops an analytically tractable model for determining the equilibrium distribution of mismatch repair deficient strains in unicellular populations. The approach is based on the single fitness peak model, which has been used in Eigen's quasispecies equations in order to understand various aspects of evolutionary dynamics. As with the quasispecies model, our model for mutator-nonmutator equilibrium undergoes a phase transition in the limit of infinite sequence length. This "repair catas-trophe" occurs at a critical repair error probability of epsilon(r)=L(via)/L, where L(via) denotes the length of the genome controlling viability, while L denotes the overall length of the genome. The repair catastrophe therefore occurs when the repair error probability exceeds the fraction of deleterious mutations. Our model also gives a quantitative estimate for the equilibrium fraction of mutators in Escherichia coli.

Increase in error threshold for quasispecies by heterogeneous replication accuracy

In this paper we investigate the error threshold for quasispecies with heterogeneous replication accuracy. We show that the coexistence of error-free and error-prone polymerases can greatly increase the error threshold without a catastrophic loss of genetic information. We also show that the error threshold is influenced by the number of replicores. Our research suggests that quasispecies with heterogeneous replication accuracy can reduce the genetic cost of selective evolution while still producing a variety of mutants.

Error threshold transition in the random-energy model

We perform a statistical analysis of the error threshold transition in quasispecies evolution on a random-energy fitness landscape. We obtain a precise description of the genealogical properties of the population through extensive numerical simulations. We find a clear phase transition and can distinguish two regimes of evolution: The first, for low mutation rates, is characterized by strong selection, and the second, for high mutation rates, is characterized by quasineutral evolution.

From HIV infection to AIDS: a dynamically induced percolation transition?

The origin of the unusual incubation period distribution in the development of AIDS is largely unresolved. A key factor in understanding the observed distribution of latency periods, as well as the occurrence of infected individuals not developing AIDS at all, is the dynamics of the long-lasting struggle between HIV and the immune system. Using a computer simulation, we study the diversification of viral genomes under mutation and the selective pressure of the immune system. In non-HIV infections, vast spreading of viral genomes in genome space usually does not take place. In the case of an HIV infection, this may occur, as the virus successively weakens the immune system by the depletion of CD4+ cells. In a sequence space framework, this leads to a dynamically induced percolation transition, corresponding to the onset of AIDS. As a result, we obtain a prolonged shape of the incubation period distribution, as well as a finite fraction of non-progressors that do not develop AIDS, comparing well with results from recent clinical research.