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 {\it quasispecies} which consists of the fittest genotype and its closely related mutants. The quasispecies dynamics on rugged fitness landscape follow a punctuated (or step-like) 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.Comment: Proceedings of Statphys conference at IIT Guwahati, to be published in Praman

Locating the minimum : Approach to equilibrium in a disordered, symmetric zero range process

We consider the dynamics of the disordered, one-dimensional, symmetric zero range process in which a particle from an occupied site $k$ hops to its nearest neighbour with a quenched rate $w(k)$. These rates are chosen randomly from the probability distribution $f(w) \sim (w-c)^{n}$, where $c$ is the lower cutoff. For $n > 0$, this model is known to exhibit a phase transition in the steady state from a low density phase with a finite number of particles at each site to a high density aggregate phase in which the site with the lowest hopping rate supports an infinite number of particles. In the latter case, it is interesting to ask how the system locates the site with globally minimum rate. We use an argument based on local equilibrium, supported by Monte Carlo simulations, to describe the approach to the steady state. We find that at large enough time, the mass transport in the regions with a smooth density profile is described by a diffusion equation with site-dependent rates, while the isolated points where the mass distribution is singular act as the boundaries of these regions. Our argument implies that the relaxation time scales with the system size $L$ as $L^{z}$ with $z=2+1/(n+1)$ for $n > 1$ and suggests a different behaviour for $n < 1$.Comment: Revtex, 7 pages including 3 figures. Submitted to Pramana -- special issue on mesoscopic and disordered system

Persistence in Cluster--Cluster Aggregation

Persistence is considered in diffusion--limited cluster--cluster aggregation, in one dimension and when the diffusion coefficient of a cluster depends on its size $s$ as $D(s) \sim s^\gamma$. The empty and filled site persistences are defined as the probabilities, that a site has been either empty or covered by a cluster all the time whereas the cluster persistence gives the probability of a cluster to remain intact. The filled site one is nonuniversal. The empty site and cluster persistences are found to be universal, as supported by analytical arguments and simulations. The empty site case decays algebraically with the exponent $\theta_E = 2/(2 - \gamma)$. The cluster persistence is related to the small $s$ behavior of the cluster size distribution and behaves also algebraically for $0 \le \gamma < 2$ while for $\gamma < 0$ the behavior is stretched exponential. In the scaling limit $t \to \infty$ and $K(t) \to \infty$ with $t/K(t)$ fixed the distribution of intervals of size $k$ between persistent regions scales as $n(k;t) = K^{-2} f(k/K)$, where $K(t) \sim t^\theta$ is the average interval size and $f(y) = e^{-y}$. For finite $t$ the scaling is poor for $k \ll t^z$, due to the insufficient separation of the two length scales: the distances between clusters, $t^z$, and that between persistent regions, $t^\theta$. For the size distribution of persistent regions the time and size dependences separate, the latter being independent of the diffusion exponent $\gamma$ but depending on the initial cluster size distribution.Comment: 14 pages, 12 figures, RevTeX, submitted to Phys. Rev.

Nonequilibrium Statistical Mechanics of the Zero-Range Process and Related Models

We review recent progress on the zero-range process, a model of interacting particles which hop between the sites of a lattice with rates that depend on the occupancy of the departure site. We discuss several applications which have stimulated interest in the model such as shaken granular gases and network dynamics, also we discuss how the model may be used as a coarse-grained description of driven phase-separating systems. A useful property of the zero-range process is that the steady state has a factorised form. We show how this form enables one to analyse in detail condensation transitions, wherein a finite fraction of particles accumulate at a single site. We review condensation transitions in homogeneous and heterogeneous systems and also summarise recent progress in understanding the dynamics of condensation. We then turn to several generalisations which also, under certain specified conditions, share the property of a factorised steady state. These include several species of particles; hop rates which depend on both the departure and the destination sites; continuous masses; parallel discrete-time updating; non-conservation of particles and sites.Comment: 54 pages, 9 figures, review articl

Phases of a conserved mass model of aggregation with fragmentation at fixed sites

To study the effect of quenched disorder in a class of reaction-diffusion systems, we introduce a conserved mass model of diffusion and aggregation in which the mass moves as a whole to a nearest neighbour on most sites while it fragments off as a single monomer (i.e. chips off) from certain fixed sites. Once the mass leaves any site, it coalesces with the mass present on its neighbour. We study in detail the effect of a \emph{single} chipping site on the steady state in arbitrary dimensions, with and without bias. In the thermodynamic limit, the system can exist in one of the following phases -- (a) Pinned Aggregate (PA) phase in which an infinite aggregate (with mass proportional to the volume of the system) appears with probability one at the chipping site but not in the bulk. (b) Unpinned Aggregate (UA) phase in which \emph{both} the chipping site and the bulk can support an infinite aggregate simultaneously. (c) Non Aggregate (NA) phase in which there is no infinite cluster. Our analytical and numerical studies show that the system exists in the UA phase in all cases except in 1d with bias. In the latter case, there is a phase transition from the NA phase to the PA phase as density is increased. A variant of the above aggregation model is also considered in which total particle number is conserved and chipping occurs at a fixed site, but the particles do not interact with each other at other sites. This model is solved exactly by mapping it to a Zero Range Process. With increasing density, it exhibits a phase transition from the NA phase to the PA phase in all dimensions, irrespective of bias. Finally, we discuss the likely behaviour of the system in the presence of extensive disorder.Comment: RevTex, 19 pages including 11 figures, submitted to Phys. Rev.

Activated Random Walkers: Facts, Conjectures and Challenges

We study a particle system with hopping (random walk) dynamics on the integer lattice $\mathbb Z^d$. The particles can exist in two states, active or inactive (sleeping); only the former can hop. The dynamics conserves the number of particles; there is no limit on the number of particles at a given site. Isolated active particles fall asleep at rate $\lambda > 0$, and then remain asleep until joined by another particle at the same site. The state in which all particles are inactive is absorbing. Whether activity continues at long times depends on the relation between the particle density $\zeta$ and the sleeping rate $\lambda$. We discuss the general case, and then, for the one-dimensional totally asymmetric case, study the phase transition between an active phase (for sufficiently large particle densities and/or small $\lambda$) and an absorbing one. We also present arguments regarding the asymptotic mean hopping velocity in the active phase, the rate of fixation in the absorbing phase, and survival of the infinite system at criticality. Using mean-field theory and Monte Carlo simulation, we locate the phase boundary. The phase transition appears to be continuous in both the symmetric and asymmetric versions of the process, but the critical behavior is very different. The former case is characterized by simple integer or rational values for critical exponents ($\beta = 1$, for example), and the phase diagram is in accord with the prediction of mean-field theory. We present evidence that the symmetric version belongs to the universality class of conserved stochastic sandpiles, also known as conserved directed percolation. Simulations also reveal an interesting transient phenomenon of damped oscillations in the activity density

Quantifying the Adaptive Potential of an Antibiotic Resistance Enzyme

For a quantitative understanding of the process of adaptation, we need to understand its “raw material,” that is, the frequency and fitness effects of beneficial mutations. At present, most empirical evidence suggests an exponential distribution of fitness effects of beneficial mutations, as predicted for Gumbel-domain distributions by extreme value theory. Here, we study the distribution of mutation effects on cefotaxime (Ctx) resistance and fitness of 48 unique beneficial mutations in the bacterial enzyme TEM-1 β-lactamase, which were obtained by screening the products of random mutagenesis for increased Ctx resistance. Our contributions are threefold. First, based on the frequency of unique mutations among more than 300 sequenced isolates and correcting for mutation bias, we conservatively estimate that the total number of first-step mutations that increase Ctx resistance in this enzyme is 87 [95% CI 75–189], or 3.4% of all 2,583 possible base-pair substitutions. Of the 48 mutations, 10 are synonymous and the majority of the 38 non-synonymous mutations occur in the pocket surrounding the catalytic site. Second, we estimate the effects of the mutations on Ctx resistance by determining survival at various Ctx concentrations, and we derive their fitness effects by modeling reproduction and survival as a branching process. Third, we find that the distribution of both measures follows a Fréchet-type distribution characterized by a broad tail of a few exceptionally fit mutants. Such distributions have fundamental evolutionary implications, including an increased predictability of evolution, and may provide a partial explanation for recent observations of striking parallel evolution of antibiotic resistance