Analytical theory of the nonequilibrium spatial distribution of RNA polymerase translocations

Analytic delay distributions for a family of gene transcription models

Models intended to describe the time evolution of a gene network must somehow include transcription, the DNA-templated synthesis of RNA, and translation, the RNA-templated synthesis of proteins. In eukaryotes, the DNA template for transcription can be very long, often consisting of tens of thousands of nucleotides, and lengthy pauses may punctuate this process. Accordingly, transcription can last for many minutes, in some cases hours. There is a long history of introducing delays in gene expression models to take the transcription and translation times into account. Here we study a family of detailed transcription models that includes initiation, elongation, and termination reactions. We establish a framework for computing the distribution of transcription times, and work out these distributions for some typical cases. For elongation, a fixed delay is a good model provided elongation is fast compared to initiation and termination, and there are no sites where long pauses occur. The initiation and termination phases of the model then generate a nontrivial delay distribution, and elongation shifts this distribution by an amount corresponding to the elongation delay. When initiation and termination are relatively fast, the distribution of elongation times can be approximated by a Gaussian. A convolution of this Gaussian with the initiation and termination time distributions gives another analytic approximation to the transcription time distribution. If there are long pauses during elongation, because of the modularity of the family of models considered, the elongation phase can be partitioned into reactions generating a simple delay (elongation through regions where there are no long pauses), and reactions whose distribution of waiting times must be considered explicitly (initiation, termination, and motion through regions where long pauses are likely). In these cases, the distribution of transcription times again involves a nontrivial part and a shift due to fast elongation processes.

The balance between intrinsic and ecological fitness defines new regimes in eco-evolutionary population dynamics

Selection upon intrinsic fitness differences is one of the most basic mechanisms of evolution, fundamental to all biology. Equally, within macroscopic populations and microscopic environments, ecological interactions influence evolution. Direct experimental evidence of ecological selection between microscopic agents continues to grow. Whilst eco-evolutionary dynamics describes how interactions influence population fitness and composition, we build a model that allows ecological aspects of these interactions to fall on a spectrum independent of the intrinsic fitness of the population. With our mathematical framework, we show how ecological interactions between mutating populations modify the estimated evolutionary trajectories expected from monoculture fitnesses alone. We derive and validate analytical stationary solutions to our partial differential equations that depend on intrinsic and ecological terms, and mutation rates. We determine cases in which these interactions modify evolution in such ways as to, for example, maintain or invert existing monoculture fitness differences. This work discusses the importance of understanding ecological and intrinsic selection effects to avoid misleading conclusions from experiments and defines new ways to assess this balance from experimental results. Using published experimental data, we also show evidence that real microbiological systems can span intrinsic fitness dominant and ecological-effect dominant regimes and that ecological contributions can change with an environment to exaggerate or counteract the composite populations' intrinsic fitness differences.

Kinetics and thermodynamics of exonuclease-deficient DNA polymerases

A kinetic theory is developed for exonuclease-deficient DNA polymerases, based on the experimental observation that the rates depend not only on the newly incorporated nucleotide, but also on the previous one, leading to the growth of Markovian DNA sequences from a Bernoullian template. The dependencies on nucleotide concentrations and template sequence are explicitly taken into account. In this framework, the kinetic and thermodynamic properties of DNA replication, in particular, the mean growth velocity, the error probability, and the entropy production are calculated analytically in terms of the rate constants and the concentrations. Theory is compared with numerical simulations for the DNA polymerases of T7 viruses and human mitochondria.

Stationary distribution of self-organized states and biological information generation

Self-organization, where spontaneous orderings occur under driven conditions, is one of the hallmarks of biological systems. We consider a statistical mechanical treatment of the biased distribution of such organized states, which become favored as a result of their catalytic activity under chemical driving forces. A generalization of the equilibrium canonical distribution describes the stationary state, which can be used to model shifts in conformational ensembles sampled by an enzyme in working conditions. The basic idea is applied to the process of biological information generation from random sequences of heteropolymers, where unfavorable Shannon entropy is overcome by the catalytic activities of selected genes. The ordering process is demonstrated with the genetic distance to a genotype with high catalytic activity as an order parameter. The resulting free energy can have multiple minima, corresponding to disordered and organized phases with first-order transitions between them.

Energetics of the single-headed kinesin KIF1A

KIF1A is a single-headed molecular motor that moves processively and unidirectionally along a microtubule by using the chemical energy released by hydrolyzing adenosine triphosphate (ATP) into adenosine diphosphate (ADP) and inorganic phosphate (P(i)). Although the movement of KIF1A seems to have successfully been explained by a simple Brownian motor model of the flashing ratchet type, this model is not suited to discuss the energetics of KIF1A. We introduce an elaborated model of the ratchet type to investigate how the chemical free energy is converted into mechanical work by taking account of the binding and release of reactant (ATP) and product (ADP and P(i)) molecules to and from the motor. The efficiency of energy transduction, the power output, and other quantities are calculated from the analytically obtained steady-state solution of the Fokker-Planck equations. It turns out that the concentrations of the reactant and product molecules that optimize both the efficiency and the power are close to those in the cell.

Fitting experimental transcription data with a comprehensive template-dependent modular kinetic model

In the companion article, we developed a modular scheme for representing the kinetics of transcription elongation by RNA polymerase. As an example of how to use these approaches, in this article we use a comprehensive modular model of this sort to fit experimental transcript elongation results obtained on the canonical tR2 template of phage λ by means of complementary bulk gel electrophoresis and surface plasmon resonance assays. The gel electrophoresis results, obtained in experiments quenched at various times after initiation of transcription, provide distributions of RNA lengths as a function of time. The surface plasmon resonance methods were used to monitor increases and decreases in the total mass of transcription elongation complexes in the same experiments. The different measures of transcription dynamics that these methods provide allow us to use them in combination to obtain a set of largely robust and well-defined kinetic parameters. The results show that our modular approach can be used to develop and test predictive kinetic schemes that can be fit to real transcription elongation data. They also suggest that these approaches can be extended to simulate the kinetics of other processes that involve the processive extension or shortening of nucleic acid chains and related systems of sequential branching reaction events.

Development of a "modular" scheme to describe the kinetics of transcript elongation by RNA polymerase

Transcript elongation by RNA polymerase involves the sequential appearance of several alternative and off-pathway states of the transcript elongation complex (TEC), and this complicates modeling of the kinetics of the transcription elongation process. Based on solutions of the chemical master equation for such transcription systems as a function of time, we here develop a modular scheme for simulating such kinetic transcription data. This scheme deals explicitly with the problem of TEC desynchronization as transcript synthesis proceeds, and develops kinetic modules to permit the various alternative states of the TECs (paused states, backtracked states, arrested states, and terminated states) to be introduced one-by-one as needed. In this way, we can set up a comprehensive kinetic model of appropriate complexity to fit the known transcriptional properties of any given DNA template and set of experimental conditions, including regulatory cofactors. In the companion article, this modular scheme is successfully used to model kinetic transcription elongation data obtained by bulk-gel electrophoresis quenching procedures and real-time surface plasmon resonance methods from a template of known sequence that contains defined pause, stall, and termination sites.

Model of gene transcription including the return of a RNA polymerase to the beginning of a transcriptional cycle

The gene transcription occurs via the RNA polymerase (RNAP) recruitment on the DNA promoter sequence, formation of a locally open DNA chain, promoter escape, steps of the RNA synthesis, and RNA and RNAP release after reading the final DNA base. Just after the end of the RNA synthesis, RNAP surrounds the closed DNA chain and may diffuse along DNA, desorb, or reach the promoter and start the RNA-synthesis cycle again. We present a generic kinetic model taking the latter steps into account and show analytically and by Monte Carlo simulations that it predicts transcriptional bursts even in the absence of explicit regulation of the transcription by master proteins.

Relaxation dynamics near nonequilibrium stationary states in Brownian ratchets

A comprehensive study of the static and dynamical properties of a representative stochastic model of Brownian ratchet effects for molecular motors is reported. The model describes Brownian motions on two periodic potentials under static and time-dependent forces, where there are two distinct locations of chemical reactions coupling the levels with reversible rates within a period. Complete stationary properties have been obtained analytically for arbitrary potentials under external force. Dynamical relaxation properties near nonequilibrium stationary states were examined by considering the response function of velocity upon time-dependent external force, expressed in terms of the conditional probability density of the model. The latter is fully calculated using a systematic numerical method using matrix diagonalization, which is easily generalized to more complicated models for studying both static and dynamical properties. The behavior of the time-dependent response examined for model potentials suggests that the characteristic relaxation time near stationary states generally decreases linearly with respect to increasing velocity as one goes away from equilibrium via an increase in chemical potential of fuel species, a prediction testable in single molecule experiments.

Molecular dynamics studies of the energetics of translocation in model T7 RNA polymerase elongation complexes

Translocation in the single subunit T7 RNA polymerase elongation complex was studied by molecular dynamics simulations using the posttranslocated crystal structure with the fingers domain open, an intermediate stable in the absence of pyrophosphate, magnesium ions, and nucleotide substrate. Unconstrained and umbrella sampling simulations were performed to examine the energetics of translocations. The extent of translocation was quantified using reaction coordinates representing the average and individual displacements of the RNA-DNA hybrid base pairs with respect to a reference structure. In addition, an unconstrained simulation was also performed for the product complex with the fingers domain closed, but with the pyrophosphate and magnesium removed, in order to examine the local stability of the pretranslocated closed state after the pyrophosphate release. The average spatial movement of the entire hybrid was found to be energetically costly in the post- to pretranslocated direction in the open state, while the pretranslocated state was stable in the closed complex, supporting the notion that the conformational state dictates the global stability of translocation states. However, spatial fluctuations of the RNA 3'-end in the open conformation were extensive, with the typical range reaching 3-4 A. Our results suggest that thermal fluctuations play more important roles in the translocation of individual nucleotides than in the movement of large sections of nucleotide strands: RNA 3'-end can move into and out of the active site within a single conformational state, while a global movement of the hybrid may be thermodynamically unfavorable without the conformational change.

Interacting RNA polymerase motors on a DNA track: effects of traffic congestion and intrinsic noise on RNA synthesis

RNA polymerase (RNAP) is an enzyme that synthesizes a messenger RNA (mRNA) strand which is complementary to a single-stranded DNA template. From the perspective of physicists, an RNAP is a molecular motor that utilizes chemical energy input to move along the track formed by DNA. In many circumstances, which are described in this paper, a large number of RNAPs move simultaneously along the same track; we refer to such collective movements of the RNAPs as RNAP traffic. Here we develop a theoretical model for RNAP traffic by incorporating the steric interactions between RNAPs as well as the mechanochemical cycle of individual RNAPs during the elongation of the mRNA. By a combination of analytical and numerical techniques, we calculate the rates of mRNA synthesis and the average density profile of the RNAPs on the DNA track. We also introduce, and compute, two different measures of fluctuations in the synthesis of RNA. Analyzing these fluctuations, we show how the level of intrinsic noise in mRNA synthesis depends on the concentrations of the RNAPs as well as on those of some of the reactants and the products of the enzymatic reactions catalyzed by RNAP. We suggest appropriate experimental systems and techniques for testing our theoretical predictions.