Template-Directed Copolymerization, Random Walks along Disordered Tracks, and Fractals

A universal method for analyzing copolymer growth

Polymers consisting of more than one type of monomer, known as copolymers, are vital to both living and synthetic systems. Copolymerization has been studied theoretically in a number of contexts, often by considering a Markov process in which monomers are added or removed from the growing tip of a long copolymer. To date, the analysis of the most general models of this class has necessitated simulation. We present a general method for analyzing such processes without resorting to simulation. Our method can be applied to models with an arbitrary network of sub-steps prior to addition or removal of a monomer, including non-equilibrium kinetic proofreading cycles. Moreover, the approach allows for a dependency of addition and removal reactions on the neighboring site in the copolymer and thermodynamically self-consistent models in which all steps are assumed to be microscopically reversible. Using our approach, thermodynamic quantities such as chemical work; kinetic quantities such as time taken to grow; and statistical quantities such as the distribution of monomer types in the growing copolymer can be directly derived either analytically or numerically from the model definition.

Template-directed growth of copolymers

The theory of multistate template-directed reversible copolymerization is developed by extending the method based on iterated function systems to matrices, taking into account the possibility of multiple activation states instead of a single one for the growth process. In this extended theory, the mean growth velocity is obtained with an iterated matrix function system and the probabilities of copolymer sequences are given by matrix products defined along the template. The theory allows us to understand the effects of template heterogeneity, which include a fractal distribution of local growth velocities far enough from equilibrium, and a regime of sublinear growth in time close to equilibrium.

Error-Speed Correlations in Biopolymer Synthesis

Synthesis of biopolymers such as DNA, RNA, and proteins are biophysical processes aided by enzymes. The performance of these enzymes is usually characterized in terms of their average error rate and speed. However, because of thermal fluctuations in these single-molecule processes, both error and speed are inherently stochastic quantities. In this Letter, we study fluctuations of error and speed in biopolymer synthesis and show that they are in general correlated. This means that, under equal conditions, polymers that are synthesized faster due to a fluctuation tend to have either better or worse errors than the average. The error-correction mechanism implemented by the enzyme determines which of the two cases holds. For example, discrimination in the forward reaction rates tends to grant smaller errors to polymers with faster synthesis. The opposite occurs for discrimination in monomer rejection rates. Our results provide an experimentally feasible way to identify error-correction mechanisms by measuring the error-speed correlations.

Template-specific fidelity of DNA replication with high-order neighbor effects: A first-passage approach

DNA replication fidelity is a critical issue in molecular biology. Biochemical experiments have provided key insights on the mechanism of fidelity control by DNA polymerases in the past decades, whereas systematic theoretical studies on this issue began only recently. Because of the underlying difficulties of mathematical treatment, comprehensive surveys on the template-specific replication kinetics are still rare. Here we propose a first-passage approach to address this problem, in particular the positional fidelity, for complicated processes with high-order neighbor effects. Under biologically relevant conditions, we derived approximate analytical expressions of the positional fidelity which show intuitively how some key kinetic pathways are coordinated to guarantee the high fidelity, as well as the high velocity, of the replication processes. It is also shown that the fidelity at any template position is dominantly determined by the nearest-neighbor template sequences, which is consistent with the idea that replication mutations are randomly distributed in the genome.

Length and sequence relaxation of copolymers under recombination reactions

We describe the kinetics and thermodynamics of copolymers undergoing recombination reactions, which are important for prebiotic chemistry. We use two approaches: the first one, based on chemical rate equations and the mass-action law describes the infinite size limit, while the second one, based on the chemical master equation, describes systems of finite size. We compare the predictions of both approaches for the relaxation of thermodynamic quantities towards equilibrium. We find that for some choice of initial conditions, the entropy of the sequence distribution can be lowered at the expense of increasing the entropy of the length distribution. We consider mainly energetically neutral reactions, except for one simple case of non-neutral reactions.

Iterated function systems for DNA replication

The kinetic equations of DNA replication are shown to be exactly solved in terms of iterated function systems, running along the template sequence and giving the statistical properties of the copy sequences, as well as the kinetic and thermodynamic properties of the replication process. With this method, different effects due to sequence heterogeneity can be studied, in particular, a transition between linear and sublinear growths in time of the copies, and a transition between continuous and fractal distributions of the local velocities of the DNA polymerase along the template. The method is applied to the human mitochondrial DNA polymerase γ without and with exonuclease proofreading.

Thermodynamic Bounds on the Ultra- and Infra-affinity of Hsp70 for Its Substrates

The 70 kDa heat shock protein Hsp70 has several essential functions in living systems, such as protecting cells against protein aggregation, assisting protein folding, remodeling protein complexes, and driving translocation into organelles. These functions require high affinity for nonspecific amino acid sequences that are ubiquitous in proteins. It has been recently shown that this high affinity, called ultra-affinity, depends on a process driven out of equilibrium by ATP hydrolysis. Here, we establish the thermodynamic bounds for ultra-affinity, and further show that the same reaction scheme can in principle be used both to strengthen and to weaken affinities (leading in this case to infra-affinity). We show that cofactors are essential to achieve affinity beyond the equilibrium range. Finally, biological implications are discussed.

Optical tweezers studies of transcription by eukaryotic RNA polymerases

Transcription is the first step in the expression of genetic information and it is carried out by large macromolecular enzymes called RNA polymerases. Transcription has been studied for many years and with a myriad of experimental techniques, ranging from bulk studies to high-resolution transcript sequencing. In this review, we emphasise the advantages of using single-molecule techniques, particularly optical tweezers, to study transcription dynamics. We give an overview of the latest results in the single-molecule transcription field, focusing on transcription by eukaryotic RNA polymerases. Finally, we evaluate recent quantitative models that describe the biophysics of RNA polymerase translocation and backtracking dynamics.

Kinetics and thermodynamics of living copolymerization processes

Theoretical advances are reported on the kinetics and thermodynamics of free and template-directed living copolymerizations. Until recently, the kinetic theory of these processes had only been established in the fully irreversible regime, in which the attachment rates are only considered. However, the entropy production is infinite in this regime and the approach to thermodynamic equilibrium cannot be investigated. For this purpose, the detachment rates should also be included. Inspite of this complication, the kinetics can be exactly solved in the regimes of steady growth and depolymerization. In this way, analytical expressions are obtained for the mean growth velocity, the statistical properties of the copolymer sequences, as well as the thermodynamic entropy production. The results apply to DNA replication, transcription and translation, allowing us to understand important aspects of molecular evolution.This article is part of the themed issue 'Multiscale modelling at the physics-chemistry-biology interface'.