This paper provides a full classification of the dynamics for continuous-time Markov chains (CTMCs) on the nonnegative integers with polynomial transition rate functions and without arbitrary large backward jumps. Such stochastic processes are abundant in applications, in particular in biology. More precisely, for CTMCs of bounded jumps, we provide necessary and sufficient conditions in terms of calculable parameters for explosivity, recurrence versus transience, positive recurrence versus null recurrence, certain absorption, and implosivity. Simple sufficient conditions for exponential ergodicity of stationary distributions and quasi-stationary distributions as well as existence and nonexistence of moments of hitting times are also obtained. Similar simple sufficient conditions for the aforementioned dynamics together with their opposite dynamics are established for CTMCs with unbounded forward jumps. Finally, we apply our results to stochastic reaction networks, an extended class of branching processes, a general bursty single-cell stochastic gene expression model, and population processes, none of which are birth–death processes. The approach is based on a mixture of Lyapunov–Foster-type results, the classical semimartingale approach, and estimates of stationary measures.

Detailed balance of a chemical reaction network can be defined in several different ways. Here we investigate the relationship among four types of detailed balance conditions: deterministic, stochastic, local, and zero-order local detailed balance. We show that the four types of detailed balance are equivalent when different reactions lead to different species changes and are not equivalent when some different reactions lead to the same species change. Under the condition of local detailed balance, we further show that the system has a global potential defined over the whole space, which plays a central role in the large deviation theory and the Freidlin–Wentzell-type metastability theory of chemical reaction networks. Finally, we provide a new sufficient condition for stochastic detailed balance, which is applied to construct a class of high-dimensional chemical reaction networks that both satisfies stochastic detailed balance and displays multistability.

Delayed production can substantially alter the qualitative behaviour of feedback systems. Motivated by stochastic mechanisms in gene expression, we consider a protein molecule which is produced in randomly timed bursts, requires an exponentially distributed time to activate and then partakes in positive regulation of its burst frequency. Asymptotically analysing the underlying master equation in the large-delay regime, we provide tractable approximations to time-dependent probability distributions of molecular copy numbers. Importantly, the presented analysis demonstrates that positive feedback systems with large production delays can constitute a stable toggle switch even if they operate with low copy numbers of active molecules.

The Hopf bifurcation from spike solutions for the classical Gierer–Meinhardt system in a onedimensional interval is considered. The existence of time-periodic solution near the Hopf bifurcation parameter for a boundary spike is rigorously proved by the classical Crandall–Rabinowitz theory. The criteria for the stability of the limit cycle are determined, and it is shown that the limit cycle is unstable.

Lipid vesicles appear ubiquitously in biological systems. Understanding how the mechanical and intermolecular interactions deform vesicle membranes is a fundamental question in biophysics. In this article we develop a fast algorithm to compute the surface configurations of lipid vesicles by introducing surface harmonic functions to approximate themembrane surface. This parameterization allows an analytical computation of the membrane curvature energy and its gradient for the efficient minimization of the curvature energy using a nonlinear conjugate gradient method. Our approach drastically reduces the degrees of freedom for approximating the membrane surfaces compared to the previously developed finite element and finite difference methods. Vesicle deformations with a reduced volume larger than 0.65 can be well approximated by using as small as 49 surface harmonic functions. The method thus has a great potential to reduce the computational expense of tracking multiple vesicles which deform for their interaction with external fields.

Mesh generation is a bottleneck for finite element simulations of biomolecules. A robust and efficient approach, based on the immersed boundary method proposed in [8], has been developed and implemented to generate large-scale mesh body-fitted to molecular shape for general parallel finite element simulations. The molecular Gaussian surface is adopted to represent the molecular surface, and is finally approximated by piecewise planes via the tool phgSurfaceCut in PHG [43], which is improved and can reliably handle complicated molecular surfaces, through mesh refinement steps. A coarse background mesh is imported first and then is distributed into each process using a mesh partitioning algorithm such as space filling curve [5] or METIS [22]. A bisection method is used for the mesh refinements according to the molecular PDB or PQR file which describes the biomolecular region. After mesh refinements, the mesh is optionally repartitioned and redistributed for load balancing. For finite element simulations, the modification of region mark and boundary types is done in parallel. Our parallel mesh generation method has been successfully applied to a sphere cavity model, a DNA fragment, a gramicidin A channel and a huge Dengue virus system. The results of numerical experiments show good parallel efficiency. Computations of electrostatic potential and solvation energy also validate the method. Moreover, the meshing process and adaptive finite element computation can be integrated as one PHG project to avoid the mesh importing and exporting costs, and improve the convenience of application as well.

Poisson-Nernst-Planck equations are a coupled system of nonlinear partial differential equations consisting of the Nernst-Planck equation and the electrostatic Poisson equation with delta distribution sources, which describe the electrodiffusion of ions in a solvated biomolecular system. In this paper, some error bounds for a piecewise finite element approximation to this problem are derived. Several numerical examples including biomolecular problems are shown to support our analysis.

This paper summarizes the mathematical and numerical theories and computational elements of the adaptive fast multipole Poisson-Boltzmann (AFMPB) solver. We introduce and discuss the following components in order: the Poisson-Boltzmann model, boundary integral equation reformulation, surface mesh generation, the nodepatch discretization approach, Krylov iterative methods, the new version of fast multipole methods (FMMs), and a dynamic prioritization technique for scheduling parallel operations. For each component, we also remark on feasible approaches for further improvements in efficiency, accuracy and applicability of the AFMPB solver to large-scale long-time molecular dynamics simulations. The potential of the solver is demonstrated with preliminary numerical results.

A classical result about Markov jump processes states that a certain class of dynamical systems given by ordinary differential equations are obtained as the limit of a sequence of scaled Markov jump processes. This approach fails if the scaling cannot be carried out equally across all entities. In the present paper we present a convergence theorem for such an unequal scaling. In contrast to an equal scaling the limit process is not purely deterministic but still possesses randomness. We show that these processes constitute a rich subclass of piecewise-deterministic processes. Such processes apply in molecular biology where entities often occur in different scales of numbers.

During DNA replication, small fragments of DNA are formed. These have been observed experimentally and the mechanism of their formation modelled mathematically. Using the stochastic model of Cowan and Chiu (1992), (1994), we find the probability distribution of the number of fragments. A new discrete distribution arises. The work has interest as an application of the recent theory on quasirenewal equations in Piau (2000).

The DNA of higher animals replicates by an interesting mechanism. Enzymes recognise specific sites randomly scattered on the molecule and establish a bidirectional process of unwinding and replication from these sites. We investigate the limiting distribution of the completion time for this process by considering related coverage problems investigated by Janson (1983) and Hall (1988).

The double-stranded molecule, DNA, has the unique property of replication and, because of this, it is the central molecule of life. The mechanism of replication for each single strand is intricate, involving enzymes which move along each of the single strands building a complementary copy. At the frontier of this action, the events have a strong stochastic character due to the random location on the DNA of key ‘sites' where copying commences. A model of this process is analysed. The central problem of interest is the mean length of certain ‘islands' of newly replicated DNA developed at the randomly located ‘sites'. These islands, which have been observed experimentally, are called Okazaki fragments.