Journal of Nonlinear Mathematical Physics

Fundamental questions posed in classical genetics since early 20th century are still fundamental in today post genomic age. What has changed is the availability of huge amount of molecular genetics information on a broad spectrum of species and a more powerful and rich methodological approach, particularly that one based on statistical mechanics and dynamical system theory which is providing unprecedented prediction power. Here we focus on the behavior of basic life forms such as bacteria and viruses which have small genomes and short generation times. We show that central issues of the evolutionary theory, i.e. how genotype, phenotype and fitness are related, the effect of positive and negative natural selection, the specie formation could be described by simple models which allow predictions and validation using experimental data.

The existence of solitonic excitations is a generic feature of a broad class of homogeneous models for nonlinear DNA internal torsional dynamics, but many properties of solitonic propagation depend on the actual model one is considering. In this paper we perform a detailed and comparative numerical investigation of the profiles and time evolution of solitons for two different models, the Yakushevich one and the more recent "composite" model of [1], and for two different choices of the potential describing the pairing interaction between bases (harmonic and Morse potential). We consider not only homogeneous DNA chains but also inhomogeneous ones (with sequence of bases corresponding to a real organism, the Human Adenovirus 2). We show that twist solitons can propagate in inhomogeneous chains over biologically significant distances. It is also shown that stable soliton propagation is possible for inhomogeneous chains when dissipation and an external force are present. On a more general level, our results indicate that solitonic propagation can take place in highly inhomogeneous nonlinear media.

We review models of biological evolution in which the population frequency changes deterministically with time. If the population is self-replicating, although the equations for simple prototypes can be linearised, nonlinear equations arise in many complex situations. For sexual populations, even in the simplest setting, the equations are necessarily nonlinear due to the mixing of the parental genetic material. The solutions of such nonlinear equations display interesting features such as multiple equilibria and phase transitions. We mainly discuss those models for which an analytical understanding of such nonlinear equations is available.

The Transfer-Integral (TI) operator is a powerful method to investigate the statistical physics of 1-dimensional models, like those used to describe DNA denaturation. At the cost of a certain number of approximations, the TI equation can be reduced to a Pseudo–Schrödinger Equation (PSE), according to which the DNA sequence is equivalent to a point particle moving in a potential well. In this paper, I check the validity of the standard PSE approximation for two different 1-dimensional DNA models, and show that it fails to provide correct results for both of them. I then propose a generalized PSE, which works well for one of the two models. Finally, I discuss the particle description of DNA denaturation that is derived from this generalized PSE.

In this paper we prove some results and detail some calculations published in a previous paper by us on the Eigen model with a periodically moving sharp-peak landscape. The model is concerned with evolution of a virus population in a time-dependent environment mimicking interaction of the viruses with the immune system of a host.

In this work we reexamine the unzipping and bubble formation of DNA in the context of the Peyrard–Bishop–Dauxois DNA model using the transfer integral operator method. After a brief overview of the method, we use it to calculate the probabilities that consecutive base-pairs are stretched beyond a threshold amplitude. We compare the continuous Peyrard–Bishop–Dauxois model with an Ising-type model and demonstrate their similarities. For the Peyrard–Bishop–Dauxois model, we derive an expression for the force that is required to open a double-stranded sequence at one end while the other end is kept closed. In the thermodynamic limit, we use this expression to study the dependence of the unzipping force on the temperature. We also calculate the opening probabilities in the case of zero force and in the case where a force is applied in the middle of the sequence. Analytically, the case where the force is applied in the middle is similar to the case of a homogeneous DNA sequence of strong GC pairs with a defect, in the form of a weaker AT base pair, in the middle. Numerical results verify this similarity.

In this paper we propose a Bayesian scheme for the determination of the unfolding and refolding kinetic rates of DNA hairpins under tension. This method is based on the hypothesis that the unfolding-refolding dynamics is well described by a Markov Chain. The results from the Bayesian method are contrasted to widely used techniques and good agreement is found. This work can be seen as a validation of the standard techniques from a statistical point of view.

A phenomenological model based on the three-dimensional theory of nonlinear elasticity is developed to describe the phenomenon of overstretching in the force-extension curve for double-stranded DNA (dsDNA). By using the concept of a material with multiple reference configurations a single formula is obtained to fit the force-extension curve.

The paper uses mesoscopic, nonlinear lattice dynamics based (Peyrard–Bishop–Dauxois, PBD) modeling to describe thermal properties of DNA below and near the denaturation temperature. Computationally efficient notation is introduced for the relevant statistical mechanics. Computed melting profiles of long and short heterogeneous sequences are presented, using a recently introduced reparametrization of the PBD model, and critically discussed. The statistics of extended open bubbles and bound clusters is formulated and results are presented for selected examples.

In the present paper we investigate the rotational oscillations of the nitrous bases that form a central base pair in a short DNA fragment consisting of three base pairs. For this purpose we use a simple mechanical model of the DNA fragment where the bases are imitated by pendulums, and the interactions between the bases are imitated by springs. It is suggested that the pendulums of the central pair oscillate, and the pendulums of neighboring pairs located at the edges of the fragment are immovable. The model takes into account the hydrogen bonds between the bases in pairs, the stacking interactions between neighboring base pairs and the helicity of the DNA structure. We derived the Lagrangian of the model system and the nonlinear equations of motions. Parameters of the equations were considered in details. We found solutions of the equations in the homogeneous case when the fragment contained only identical base pairs: Adenine-Thymine (AT) or Guanine-Cytosine (GC). The trajectories of the model system in the configuration space were also constructed in this approximation.

A nonlinear model describing DNA dynamics, called helicoidal Peyrard–Bishop model, is described. It is shown that the model can explain a local opening of a DNA helix during transcription. An impact of friction forces is also studied. It is pointed out that a role of viscosity is crucial for DNA-RNA transcription.