R. K. P. Zia

Totally asymmetric exclusion process with extended objects: A model for protein synthesis

The process of protein synthesis in biological systems resembles a one dimensional driven lattice gas in which the particles have spatial extent, covering more than one lattice site. We expand the well studied totally asymmetric exclusion process, in which particles typically cover a single lattice site, to include cases with extended objects. Exact solutions can be determined for a uniform closed system. We analyze the uniform open system through two approaches. First, a continuum limit produces a modified diffusion equation for particle density profiles. Second, an extremal principle based on domain wall theory accurately predicts the phase diagram and currents in each phase. Finally, we briefly consider approximate approaches to a nonuniform open system with quenched disorder in the particle hopping rates and compare these approaches with Monte Carlo simulations.

Probability currents as principal characteristics in the statistical mechanics of non-equilibrium steady states

One of the key features of non-equilibrium steady states (NESS) is the presence of non-trivial probability currents. We propose a general classification of NESS in which these currents play a central distinguishing role. As a corollary, we specify the transformations of the dynamic transition rates which leave a given NESS invariant. The formalism is most transparent within a continuous-time master equation framework since it allows for a general graph-theoretical representation of the NESS. We discuss the consequences of these transformations for entropy production, present several simple examples, and explore some generalizations, to discrete time and continuous variables.

Modeling Translation in Protein Synthesis with TASEP: A Tutorial and Recent Developments

The phenomenon of protein synthesis has been modeled in terms of totally asymmetric simple exclusion processes (TASEP) since 1968. In this article, we provide a tutorial of the biological and mathematical aspects of this approach. We also summarize several new results, concerned with limited resources in the cell and simple estimates for the current (protein production rate) of a TASEP with inhomogeneous hopping rates, reflecting the characteristics of real genes.

Towards a model for protein production rates

In the process of translation, ribosomes read the genetic code on an mRNA and assemble the corresponding polypeptide chain. The ribosomes perform discrete directed motion which is well modeled by a totally asymmetric simple exclusion process (TASEP) with open boundaries. Using Monte Carlo simulations and a simple mean-field theory, we discuss the effect of one or two “bottlenecks” (i.e., slow codons) on the production rate of the final protein. Confirming and extending previous work by Chou and Lakatos, we find that the location and spacing of the slow codons can affect the production rate quite dramatically. In particular, we observe a novel “edge” effect, i.e., an interaction of a single slow codon with the system boundary. We focus in detail on ribosome density profiles and provide a simple explanation for the length scale which controls the range of these interactions.

Inhomogeneous exclusion processes with extended objects: the effect of defect locations.

We study the effects of local inhomogeneities, i.e., slow sites of hopping rate q<1, in a totally asymmetric simple exclusion process for particles of size l>or=1 (in units of the lattice spacing). We compare the simulation results of l=1 and l>1 and notice that the existence of local defects has qualitatively similar effects on the steady state. We focus on the stationary current as well as the density profiles. If there is only a single slow site in the system, we observe a significant dependence of the current on the location of the slow site for both l=1 and l>1 cases. When two slow sites are introduced, more intriguing phenomena emerge, e.g., dramatic decreases in the current when the two are close together. In addition, we study the asymptotic behavior when q-->0. We also explore the associated density profiles and compare our findings to an earlier study using a simple mean-field theory. We then outline the biological significance of these effects.

Factorised Steady States in Mass Transport Models

We study a class of mass transport models where mass is transported in a preferred direction around a one-dimensional periodic lattice and is globally conserved. The model encompasses both discrete and continuous masses and parallel and random sequential dynamics and includes models such as the zero-range process and asymmetric random average process as special cases. We derive a necessary and sufficient condition for the steady state to factorize, which takes a rather simple form.

Nature of the condensate in mass transport models.

We study the phenomenon of real space condensation in the steady state of a class of one-dimensional mass transport models. We derive the criterion for the occurrence of a condensation transition and analyze the precise nature of the shape and the size of the condensate in the condensed phase. We find two distinct condensate regimes: one where the condensate is Gaussian distributed and the particle number fluctuations scale normally as L(1/2) where L is the system size, and the second regime where the particle number fluctuations become anomalously large and the condensate peak is non-Gaussian. We interpret these results within the framework of sums of random variables.

Onset of Spatial Structures in Biased Diffusion of Two Species

We consider a stochastic lattice gas with equal numbers of oppositely charged particles, diffusing under the influence of a uniform external electric field and the excluded-volume condition. Employing both Monte Carlo and analytic techniques, we discover a novel phase transition, controlled by particle density and field strength, separating a homogeneous phase from another with spatial inhomogeneities. We discuss the nature of this transition.

Canonical analysis of condensation in factorised steady states

We study the phenomenon of real space condensation in the steady state of a class of mass transport models where the steady state factorises. The grand canonical ensemble may be used to derive the criterion for the occurrence of a condensation transition but does not shed light on the nature of the condensate. Here, within the canonical ensemble, we analyse the condensation transition and the structure of the condensate, determining the precise shape and the size of the condensate in the condensed phase. We find two distinct condensate regimes: one where the condensate is gaussian distributed and the particle number fluctuations scale normally as L1/2 where L is the system size, and a second regime where the particle number fluctuations become anomalously large and the condensate peak is non-gaussian. Our results are asymptotically exact and can also be interpreted within the framework of sums of random variables. We further analyse two additional cases: one where the condensation transition is somewhat different from the usual second order phase transition and one where there is no true condensation transition but instead a pseudocondensate appears at superextensive densities.

A possible classification of nonequilibrium steady states

We propose a general classification of nonequilibrium steady states in terms of their stationary probability distribution and the associated probability currents. The stationary probabilities can be represented graph theoretically as directed labelled trees; closing a single loop in such a graph leads to a representation of probability currents. This classification allows us to identify all choices of transition rates, based on a master equation, which generate the same nonequilibrium steady state. We explore the implications of this freedom, e.g., for entropy production.