Spin Coherent State Representation of the Crow-Kimura and Eigen Models of Quasispecies Theory

We present a spin coherent state representation of the Crow-Kimura and Eigen models of biological evolution. We deal with quasispecies models where the fitness is a function of Hamming distances from one or more reference sequences. In the limit of large sequence length N, we find exact expressions for the mean fitness and magnetization of the asymptotic quasispecies distribution in symmetric fitness landscapes. The results are obtained by constructing a path integral for the propagator on the coset SU(2)/U(1) and taking the classical limit. The classical limit gives a Hamiltonian function on a circle for one reference sequence, and on the product of 2 ^m −1 circles for m reference sequences. We apply our representation to study the Schuster-Swetina phenomena, where a wide lower peak is selected over a narrow higher peak. The quadratic landscape with two reference sequences is also analyzed specifically and we present the phase diagram on the mutation-fitness parameter phase space. Furthermore, we use our method to investigate more biologically relevant system, a model of escape from adaptive conflict through gene duplication, and find three different phases for the asymptotic population distribution.     

随机化的Eigen模型研究

为了使Eigen模型能够更真实地描述物种的演化过程,将确定性Eigen模型改造成随机模型.以Eigen模型为理论框架,把基因序列中每一个位点的突变率看作一个高斯随机变量,从而导出随机性Eigen模型.对于此随机性Eigen模型,当突变率的涨落强度较小时,准物种的误差阈位置几乎没有改变,仍是个相变点;而当突变率的涨落强度变大时,误差阈由一个相变点变为一个转变区域.在真实的物种演化过程中,误差阈应是一个转变区域,而且在解决实际问题时应考虑该转变区域的上限.     

Adaptation dynamics of the quasispecies model

We study the adaptation dynamics of an initially maladapted population evolving via the elementary processes of mutation and selection. The evolution occurs on rugged fitness landscapes which are defined on the multi-dimensional genotypic space and have many local peaks separated by low fitness valleys. We mainly focus on the Eigen’s model that describes the deterministic dynamics of an infinite number of self-replicating molecules. In the stationary state, for small mutation rates such a population forms a quasispecies which consists of the fittest genotype and its closely related mutants. The quasispecies dynamics on rugged fitness landscape follow a punctuated (or steplike) pattern in which a population jumps from a low fitness peak to a higher one, stays there for a considerable time before shifting the peak again and eventually reaches the global maximum of the fitness landscape. We calculate exactly several properties of this dynamical process within a simplified version of the quasispecies model.      

Dynamical phase transition in a model for evolution with migration

We study a simple quasispecies model for evolution in two different habitats, with different fitness landscapes, coupled through one-way migration. Our key finding is a dynamical phase transition at a critical value of the migration rate, at which the time to reach the steady state diverges. The genetic composition of the population is qualitatively different above and below the transition. Using results from localization theory, we show that the critical migration rate may be very small-demonstrating that evolutionary outcomes can be very sensitive to even a small amount of migration.     

Speeding up Evolutionary Search by Small Fitness Fluctuations

We consider a fixed size population that undergoes an evolutionary adaptation in the weak mutation rate limit, which we model as a biased Langevin process in the genotype space. We show analytically and numerically that, if the fitness landscape has a small highly epistatic (rough) and time-varying component, then the population genotype exhibits a high effective diffusion in the genotype space and is able to escape local fitness minima with a large probability. We argue that our principal finding that even very small time-dependent fluctuations of fitness can substantially speed up evolution is valid for a wide class of models.     

Dynamic fitness landscapes: expansions for small mutation rates

We study the evolution of asexual microorganisms with small mutation rate in fluctuating environments, and develop techniques that allow us to expand the formal solution of the evolution equations to first order in the mutation rate. Our method can be applied to both discrete time and continuous time systems. While the behavior of continuous time systems is dominated by the average fitness landscape for small mutation rates, in discrete time systems it is instead the geometric mean fitness that determines the system's properties. In both cases, we find that in situations in which the arithmetic (resp. geometric) mean of the fitness landscape is degenerate, regions in which the fitness fluctuates around the mean value present a selective advantage over regions in which the fitness stays at the mean. This effect is caused by the vanishing genetic diffusion at low mutation rates. In the absence of strong diffusion, a population can stay close to a fluctuating peak when the peak's height is below average, and take advantage of the peak when its height is above average.     

Four-State Quantum Chain as a Model of Sequence Evolution

A variety of selection-mutation models for DNA (or RNA) sequences, well known in molecular evolution, can be translated into a model of coupled Ising quantum chains. This correspondence is used to investigate the genetic variability and error threshold behaviour in dependence of possible fitness landscapes. In contrast to the two-state models treated hitherto, the model explicitly takes the four-state nature of the nucleotide alphabet into account and allows for the distinction of mutation rates for the different base substitutions, as given by standard mutation schemes of molecular phylogeny. As a consequence of this refined treatment, new phase diagrams for the error threshold behaviour are obtained, with appearance of a novel phase in which the nucleotide ordering of the wildtype sequence is only partially conserved. Explicit analytical and numerical results are presented for evolution dynamics and equilibrium behaviour in a number of accessible situations, such as quadratic fitness landscapes and the Kimura 2 parameter mutation scheme.     

Maternal effects in molecular evolution

We introduce a model of molecular evolution in which the fitness of an individual depends both on its own and on the parent's genotype. The model can be solved by means of a nonlinear mapping onto the standard quasispecies model. The dependency on the parental genotypes cancels from the mean fitness, but not from the individual sequence concentrations. For finite populations, the position of the error threshold is very sensitive to the influence from parent genotypes. In addition to biological applications, our model is important for understanding the dynamics of self-replicating computer programs.     

Eigen model version of evolution with directed migration

We consider the two-habitat quasispecies model, which describes evolutionary process with migration on the basis of the Eigen model. In the first habitat there is only one genotype, and here is an influx of the replicators from the first habitat to the second one with the rate h. We solve exactly the case of a single-peak fitness landscape in both habitats, when in the first habitat there are no mutations. The Eigen model version of the model is more adequately describes the real biological experiments than the Crow-Kimura model, as can be related to the serial transfer experiments in chemical reactor.     

Deterministic group selection model for the evolution of altruism

We study the evolution of an infinite population of asexually reproducing individuals, each of which can be either altruist or non-altruist, subdivided into reproductively isolated groups (demes) of finite size under the action of two opposed selective pressures, namely, differential individual reproduction and differential deme extinction. We derive a recursion equation for the deterministic, discrete time evolution of the frequencies of the different types of demes, classified according to the number of altruistic individuals they have. We give emphasis to the detrimental effects of mutation and migration on the stability of the altruistic demes, which are the only stable demes in the absence of these processes. Furthermore, we draw an analogy between the proposed deterministic group selection model and the quasispecies model for molecular evolution. Received: 22 May 1998 / Accepted: 18 August 1998     

Schwinger Boson Formulation and Solution of the Crow-Kimura and Eigen Models of Quasispecies Theory

We express the Crow-Kimura and Eigen models of quasispecies theory in a functional integral representation. We formulate the spin coherent state functional integrals using the Schwinger Boson method. In this formulation, we are able to deduce the long-time behavior of these models for arbitrary replication and degradation functions. We discuss the phase transitions that occur in these models as a function of mutation rate. We derive for these models the leading order corrections to the infinite genome length limit.     

Evolving complex dynamics in electronic models of genetic networks

Ordinary differential equations are often used to model the dynamics and interactions in genetic networks. In one particularly simple class of models, the model genes control the production rates of products of other genes by a logical function, resulting in piecewise linear differential equations. In this article, we construct and analyze an electronic circuit that models this class of piecewise linear equations. This circuit combines CMOS logic and RC circuits to model the logical control of the increase and decay of protein concentrations in genetic networks. We use these electronic networks to study the evolution of limit cycle dynamics. By mutating the truth tables giving the logical functions for these networks, we evolve the networks to obtain limit cycle oscillations of desired period. We also investigate the fitness landscapes of our networks to determine the optimal mutation rate for evolution.     

Selective advantage of topological disorder in biological evolution

We examine a model of biological evolution of Eigen's quasispecies in a so-called holey fitness landscape, where the fitness of a site is either 0 (lethal site) or a uniform positive constant (viable site). The evolution dynamics is therefore determined by the topology of the genome space which is modelled by the random Bethe lattice. We use the effective medium and single-defect approximations to find the criteria under which the localized quasispecies cloud is created. We find that shorter genomes, which are more robust to random mutations than average, represent a selective advantage which we call “topological”. A way of assessing empirically the relative importance of reproductive success and topological advantage is suggested. Received 9 August 2002 / Received in final form 7 November 2002 Published online 14 February 2003 RID="a" ID="a"e-mail: slanina@fzu.cz     

Phase transitions in unstable cancer cell populations

The dynamics of cancer evolution is studied by means of a simple quasispecies model involving cells displaying high levels of genetic instability. Both continuous, mean-field and discrete, bit-string models are analysed. The string model is simulated on a single-peak landscape. It is shown that a phase transition exists at high levels of genetic instability, thus separating two phases of slow and rapid growth. The results suggest that, under a conserved level of genetic instability the cancer cell population will be close to the threshold level. Implications for therapy are outlined.Received: 22 April 2003, Published online: 22 September 2003PACS: 87.10.+e Biological physics: General theory and mathematical aspects - 87.23.Kg Dynamics of evolution - 87.23.-n Ecology and evolution - 89.75.Fb Structures and organization in complex systems     

Self-organized critical model of biological evolution

A punctuated equilibrium model of biological evolution with relative fitness between different species being the fundamental driving force of evolution is introduced. Mutation is modeled as a fitness updating cellular automation process where the change in fitness after mutation follows a Gaussian distribution with mean x > 0 and standard deviation σ. Scaling behaviors are observed in our numerical simulation, indicating that the model is self-organized critical. Besides, the numerical experiment suggests that models with different x and σ belongs to the same universality class.     

A New Method for the Solution of Models of Biological Evolution: Derivation of Exact Steady-State Distributions

We investigate well-known models of biological evolution and address the open problem of how construct a correct continuous analog of mutations in discrete sequence space. We deal with models where the fitness is a function of a Hamming distance from the reference sequence. The mutation-selection master equation in the discrete sequence space is replaced by a Hamilton-Jacobi equation for the logarithm of relative frequencies of different sequences. The steady-state distribution, mean fitness and the variance of fitness are derived. All our results are asymptotic in the large genome limit. A variety of important biological and biochemical models can be solved by this new approach. PACS numbers: 87.10.+e, 87.15.Aa, 87.23.Kg, 02.50.-r     

Effects of spatial competition on the diversity of a quasispecies

The diversity harbored by populations of RNA viruses results from high mutation rates, as well as from the characteristics of the environment where they evolve. By means of a simple model for structured quasispecies, we quantify how competition for space among phenotypic types shapes their distribution at the mutation-selection equilibrium. We introduce a general framework to treat this problem and relate mutation rate and competition strength to the quasispecies composition. For diffusion limited competition, diversity typically increases and the asymptotic growth rate of the population diminishes as diffusion decreases. Limited mobility confers a relative advantage to worse competitors. The stationary state is characterized by an over-production of viral particles. Empirical data allow an estimation of mutation rates compatible with the diversity observed in viral populations infecting cellular monolayers.     

Biological evolution and statistical physics

This review is an introduction to theoretical models and mathematical calculations for biological evolution, aimed at physicists. The methods in the field are naturally very similar to those used in statistical physics, although the majority of publications have appeared in biology journals. The review has three parts, which can be read independently. The first part deals with evolution in fitness landscapes and includes Fisher's theorem, adaptive walks, quasispecies models, effects of finite population sizes, and neutral evolution. The second part studies models of coevolution, including evolutionary game theory, kin selection, group selection, sexual selection, speciation, and coevolution of hosts and parasites. The third part discusses models for networks of interacting species and their extinction avalanches. Throughout the review, attention is paid to giving the necessary biological information, and to pointing out the assumptions underlying the models, and their limits of validity.     

Quantum phase transition in spin glasses with multi-spin interactions

We examine the phase diagram of the p-interaction spin glass model in a transverse field. We consider a spherical version of the model and compare with results obtained in the Ising case. The analysis of the spherical model, with and without quantization, reveals a phase diagram very similar to that obtained in the Ising case. In particular, using the static approximation, reentrance is observed at low temperatures in both the quantum spherical and Ising models. This is an artifact of the approximation and disappears when the imaginary time dependence of the order parameter is taken into account. The resulting phase diagram is checked by accurate numerical investigation of the phase boundaries.