Iterated function systems for DNA replication

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.

Kinetic assays of DNA polymerase fidelity: A theoretical perspective beyond Michaelis-Menten kinetics

The high fidelity of DNA polymerase (DNAP) is critical for the faithful replication of DNA. There are several quantitative approaches to measure DNAP fidelity. Directly counting the error frequency in the replication products gives the true fidelity but it turns out very hard to implement in practice. Two biochemical kinetic approaches, the steady-state assay and the transient-state assay, were then suggested and widely adopted. In these assays, the error frequency is indirectly estimated by using kinetic theories combined with the measured apparent kinetic rates. However, whether it is equivalent to the true fidelity has never been clarified theoretically, and in particular there are different strategies using these assays to quantify the proofreading efficiency of DNAP but often lead to inconsistent results. In this paper, we make a comprehensive examination on the theoretical foundation of the two kinetic assays, based on the theory of DNAP fidelity recently proposed by us. Our studies show that while the conventional kinetic assays are generally valid to quantify the discrimination efficiency of DNAP, they are valid to quantify the proofreading efficiency of DNAP only when the kinetic parameters satisfy some constraints which will be given explicitly in this paper. These results may inspire more carefully-designed experiments to quantify DNAP fidelity.

Role of thermal fluctuations in biological copying mechanisms

During transcription, translation, or self-replication of DNA or RNA, information is transferred to the newly formed species from its predecessor. These processes can be interpreted as (generalized) biological copying mechanism as the new biological entities like DNA, RNA, or proteins are representing the information of their parent bodies uniquely. The accuracy of these copying processes is essential, since errors in the copied code can reduce the functionality of the next generation. Such errors might result from perturbations on these processes. Most important in this context is the temperature of the medium, i.e., thermal noise. Although a reasonable amount of experimental studies have been carried out on this important issue, theoretical understanding is truly sparse. In the present work, we illustrate a model study which is able to focus on the effect of the temperature on the process of biological copying mechanisms, as well as on mutation. We find for our paradigmatic models that, in a quite general scenario, the copying processes are most accurate at an intermediate temperature range; i.e., there exists an optimum temperature where mutation is most unlikely. This allows us to interpret the observations for some biological species with the aid of our model study.

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.

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.