Entrainment and unit velocity: surprises in an accelerated exclusion process

Collective polymerase dynamics emerge from DNA supercoiling during transcription

All biological processes ultimately come from physical interactions. The mechanical properties of DNA play a critical role in transcription. RNA polymerase can over or under twist DNA (referred to as DNA supercoiling) when it moves along a gene, resulting in mechanical stresses in DNA that impact its own motion and that of other polymerases. For example, when enough supercoiling accumulates, an isolated polymerase halts, and transcription stops. DNA supercoiling can also mediate nonlocal interactions between polymerases that shape gene expression fluctuations. Here, we construct a comprehensive model of transcription that captures how RNA polymerase motion changes the degree of DNA supercoiling, which in turn feeds back into the rate at which polymerases are recruited and move along the DNA. Surprisingly, our model predicts that a group of three or more polymerases move together at a constant velocity and sustain their motion (forming what we call a polymeton), whereas one or two polymerases would have halted. We further show that accounting for the impact of DNA supercoiling on both RNA polymerase recruitment and velocity recapitulates empirical observations of gene expression fluctuations. Finally, we propose a mechanical toggle switch whereby interactions between genes are mediated by DNA twisting as opposed to proteins. Understanding the mechanical regulation of gene expression provides new insights into how endogenous genes can interact and informs the design of new forms of engineered interactions.

Biologically motivated asymmetric exclusion process: Interplay of congestion in RNA polymerase traffic and slippage of nascent transcript

We develop a theoretical framework, based on an exclusion process, that is motivated by a biological phenomenon called transcript slippage (TS). In this model a discrete lattice represents a DNA strand while each of the particles that hop on it unidirectionally, from site to site, represents a RNA polymerase (RNAP). While walking like a molecular motor along a DNA track in a step-by-step manner, a RNAP simultaneously synthesizes an RNA chain; in each forward step it elongates the nascent RNA molecule by one unit, using the DNA track also as the template. At some special "slippery" position on the DNA, which we represent as a defect on the lattice, a RNAP can lose its grip on the nascent RNA and the latter's consequent slippage results in a final product that is either longer or shorter than the corresponding DNA template. We develop an exclusion model for RNAP traffic where the kinetics of the system at the defect site captures key features of TS events. We demonstrate the interplay of the crowding of RNAPs and TS. A RNAP has to wait at the defect site for a longer period in more congested RNAP traffic, thereby increasing the likelihood of its suffering a larger number of TS events. The qualitative trends of some of our results for a simple special case of our model are consistent with experimental observations. The general theoretical framework presented here will be useful for guiding future experimental queries and for analysis of the experimental data with more detailed versions of the same model.

Multispecies exclusion process with fusion and fission of rods: A model inspired by intraflagellar transport

We introduce a multispecies exclusion model where length-conserving probabilistic fusion and fission of the hard rods are allowed. Although all rods enter the system with the same initial length ℓ=1, their length can keep changing, because of fusion and fission, as they move in a step-by-step manner towards the exit. Two neighboring hard rods of lengths ℓ_{1} and ℓ_{2} can fuse into a single rod of longer length ℓ=ℓ_{1}+ℓ_{2} provided ℓ≤N. Similarly, length-conserving fission of a rod of length ℓ^{'}≤N results in two shorter daughter rods. Based on the extremum current hypothesis, we plot the phase diagram of the model under open boundary conditions utilizing the results derived for the same model under periodic boundary condition using mean-field approximation. The density profile and the flux profile of rods are in excellent agreement with computer simulations. Although the fusion and fission of the rods are motivated by similar phenomena observed in intraflagellar transport (IFT) in eukaryotic flagella, this exclusion model is too simple to account for the quantitative experimental data for any specific organism. Nevertheless, the concepts of "flux profile" and "transition zone" that emerge from the interplay of fusion and fission in this model are likely to have important implications for IFT and for other similar transport phenomena in long cell protrusions.

Interference of two codirectional exclusion processes in the presence of a static bottleneck: A biologically motivated model

We develop a two-species exclusion process with a distinct pair of entry and exit sites for each species of rigid rods. The relatively slower forward stepping of the rods in an extended bottleneck region, located in between the two entry sites, controls the extent of interference of the codirectional flow of the two species of rods. The relative positions of the sites of entry of the two species of rods with respect to the location of the bottleneck are motivated by a biological phenomenon. However, the primary focus of the study here is to explore the effects of the interference of the flow of the two species of rods on their spatiotemporal organization and the regulations of this interference by the extended bottleneck. By a combination of mean-field theory and computer simulation, we calculate the flux of both species of rods and their density profiles as well as the composite phase diagrams of the system. If the bottleneck is sufficiently stringent, then some of the phases become practically unrealizable, although not ruled out on the basis of any fundamental physical principle. Moreover, the extent of suppression of flow of the downstream entrants by the flow of the upstream entrants can also be regulated by the strength of the bottleneck. We speculate on the possible implications of the results in the context of the biological phenomenon that motivated the formulation of the theoretical model.

Increasing Protein Production Rates Can Decrease the Rate at Which Functional Protein Is Produced and Their Steady-State Levels

The rate at which soluble, functional protein is produced by the ribosome has recently been found to vary in complex and unexplained ways as various translation-associated rates are altered through synonymous codon substitutions. To understand this phenomenon, here, we combine a well-established ribosome-traffic model with a master-equation model of cotranslational domain folding to explore the scenarios that are possible for the protein production rate, J, and the functional-nascent protein production rate, F, as the rates of various translation processes are altered for five different E. coli proteins. We find that while J monotonically increases as the rates of translation-initiation, -elongation, and -termination increase, F can either increase or decrease. We show that F's nonmonotonic behavior arises within the model from two opposing trends: the tendency for increased translation rates to produce more total protein but less cotranslationally folded protein. We further demonstrate that under certain conditions these nonmonotonic changes in F can result in nonmonotonic variations in post-translational, steady-state levels of functional protein. These results provide a potential explanation for recent experimental observations in which the specific activity of enzymatic proteins decreased with increased synthesis rates. Additionally our model has the potential to be used to rationally design transcripts to maximize the production of functional nascent protein by simultaneously optimizing translation initiation, elongation, and termination rates.

Exponential decay of spatial correlation in driven diffusive system: A universal feature of macroscopic homogeneous state

Driven diffusive systems have been a paradigm for modelling many physical, chemical, and biological transport processes. In the systems, spatial correlation plays an important role in the emergence of a variety of nonequilibrium phenomena and exhibits rich features such as pronounced oscillations. However, the lack of analytical results of spatial correlation precludes us from fully understanding the effect of spatial correlation on the dynamics of the system. Here we offer precise analytical predictions of the spatial correlation in a typical driven diffusive system, namely facilitated asymmetric exclusion process. We find theoretically that the correlation between two sites decays exponentially as their distance increases, which is in good agreement with numerical simulations. Furthermore, we find the exponential decay is a universal property of macroscopic homogeneous state in a broad class of 1D driven diffusive systems. Our findings deepen the understanding of many nonequilibrium phenomena resulting from spatial correlation in driven diffusive systems.

Totally asymmetric exclusion process fed by using a non-Poissonian clock

In this article we consider the one-dimensional totally asymmetric open-boundary exclusion process fed by a process with power-law-distributed waiting times. More specifically, we use a modified Pareto distribution to define the jump rate for jumps into the system. We then characterize the propagation of fluctuations through the system by kinetic Monte Carlo simulations and by numerical evaluation of the steady-state partition function.

Theoretical Analysis of Dynamic Processes for Interacting Molecular Motors

Biological transport is supported by collective dynamics of enzymatic molecules that are called motor proteins or molecular motors. Experiments suggest that motor proteins interact locally via short-range potentials. We investigate the fundamental role of these interactions by analyzing a new class of totally asymmetric exclusion processes where interactions are accounted for in a thermodynamically consistent fashion. It allows us to connect explicitly microscopic features of motor proteins with their collective dynamic properties. Theoretical analysis that combines various mean-field calculations and computer simulations suggests that dynamic properties of molecular motors strongly depend on interactions, and correlations are stronger for interacting motor proteins. Surprisingly, it is found that there is an optimal strength of interactions (weak repulsion) that leads to a maximal particle flux. It is also argued that molecular motors transport is more sensitive to attractive interactions. Applications of these results for kinesin motor proteins are discussed.

Exclusion processes with avalanches

In an exclusion process with avalanches, when a particle hops to a neighboring empty site which is adjacent to an island the particle on the other end of the island immediately hops, and if it joins another island this triggers another hop. There are no restrictions on the length of the islands and the duration of the avalanche. This process is well defined in the low-density region ρ < 1/2. We describe the nature of steady states (on a ring) and determine all correlation functions. For the asymmetric version of the process, we compute the steady state current, and we describe shock and rarefaction waves which arise in the evolution of the step-function initial profile. For the symmetric version, we determine the diffusion coefficient and examine the evolution of a tagged particle.

Accelerated transport and growth with symmetrized dynamics

In this paper we consider a model of accelerated dynamics with the rules modified from those of the recently proposed [Dong et al., Phys. Rev. Lett. 109, 130602 (2012)] accelerated exclusion process (AEP) such that particle-vacancy symmetry is restored to facilitate a mapping to a solid-on-solid growth model in 1+1 dimensions. In addition to kicking a particle ahead of the moving particle, as in the AEP, in our model another particle from behind is drawn, provided it is within the "distance of interaction" denoted by ℓ(max). We call our model the doubly accelerated exclusion process (DAEP). We observe accelerated transport and interface growth and widening of the cluster size distribution for cluster sizes above ℓ(max), when compared with the ordinary totally asymmetric exclusion process (TASEP). We also characterize the difference between the TASEP, AEP, and DAEP by computing a "staggered" order parameter, which reveals the local order in the steady state. This order in part explains the behavior of the particle current as a function of density. The differences of the steady states are also reflected by the behavior of the temporal and spatial correlation functions in the interface picture.

Interplay between finite resources and a local defect in an asymmetric simple exclusion process

When particle flux is regulated by multiple factors such as particle supply and varying transport rate, it is important to identify the respective dominant regimes. We extend the well-studied totally asymmetric simple exclusion model to investigate the interplay between a controlled entrance and a local defect site. The model mimics cellular transport phenomena where there is typically a finite particle pool and nonuniform moving rates due to biochemical kinetics. Our simulations reveal regions where, despite an increasing particle supply, the current remains constant while particles redistribute in the system. Exploiting a domain wall approach with mean-field approximation, we provide a theoretical ground for our findings. The results in steady-state current and density profiles provide quantitative insights into the regulation of the transcription and translation process in bacterial protein synthesis.

Emergent motion of condensates in mass-transport models

We examine the effect of spatial correlations on the phenomenon of real-space condensation in driven mass-transport systems. We suggest that in a broad class of models with a spatially correlated steady state, the condensate drifts with a nonvanishing velocity. We present a robust mechanism leading to this condensate drift. This is done within the framework of a generalized zero-range process (ZRP) in which, unlike the usual ZRP, the steady state is not a product measure. The validity of the mechanism in other mass-transport models is discussed.

Mass transport perspective on an accelerated exclusion process: analysis of augmented current and unit-velocity phases

In an accelerated exclusion process (AEP), each particle can "hop" to its adjacent site if empty as well as "kick" the frontmost particle when joining a cluster of size ℓ≤ℓ(max). With various choices of the interaction range, ℓ(max), we find that the steady state of AEP can be found in a homogeneous phase with augmented currents (AC) or a segregated phase with holes moving at unit velocity (UV). Here we present a detailed study on the emergence of the novel phases, from two perspectives: the AEP and a mass transport process (MTP). In the latter picture, the system in the UV phase is composed of a condensate in coexistence with a fluid, while the transition from AC to UV can be regarded as condensation. Using Monte Carlo simulations, exact results for special cases, and analytic methods in a mean field approach (within the MTP), we focus on steady state currents and cluster sizes. Excellent agreement between data and theory is found, providing an insightful picture for understanding this model system.