Continuous-time Markov chains are frequently used to model the stochastic dynamics of (bio)chemical reaction networks. However, except in very special cases, they cannot be analyzed exactly. Additionally, simulation can be computationally intensive. An approach to address these challenges is to consider a more tractable diffusion approximation. Leite and Williams (Ann. Appl. Prob. 29, 2019) proposed a reflected diffusion as an approximation for (bio)chemical reaction networks, which they called the constrained Langevin approximation (CLA) as it extends the usual Langevin approximation beyond the first time some chemical species becomes zero in number. Further explanation and examples of the CLA can be found in Anderson et al. (SIAM Multiscale Modeling Simul. 17, 2019).

In this paper, we extend the approximation of Leite and Williams to (nearly) density-dependent Markov chains, as a first step to obtaining error estimates for the CLA when the diffusion state space is one-dimensional, and we provide a bound for the error in a strong approximation. We discuss some applications for chemical reaction networks and epidemic models, and illustrate these with examples. Our method of proof is designed to generalize to higher dimensions, provided there is a Lipschitz Skorokhod map defining the reflected diffusion process. The existence of such a Lipschitz map is an open problem in dimensions more than one.

All vital functions of living cells rely on the production of various functional molecules through gene expression. The production periods are burst-like and stochastic due to the discrete nature of biochemical reactions. In certain contexts, the concentrations of RNA or protein require regulation to maintain a fine internal balance within the cell. Here we consider a motif of two types of RNA molecules – mRNA and an antagonistic microRNA – which are encoded by a shared coding sequence and form a feed forward loop (FFL). This control mechanism is shown to be perfectly adapting in the deterministic context. We demonstrate that the adaptation (of the mean value) becomes imperfect if production occurs in random bursts. The FFL nevertheless outperforms the benchmark feedback loop in terms of counterbalancing variations in the signal. Methodologically, we adapt a hybrid stochastic model, which has widely been used to model a single regulatory molecule, to the current case of a motif involving two species; the use of the Laplace transform thereby circumvents the problem of moment closure that arises owing to the mRNA–microRNA interaction. We expect that the approach can be applicable to other systems with nonlinear kinetics.

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.

We analyse the structure of equilibria of a coagulation–fragmentation–death model of silicosis. We present exact multiplicity results in the particular case of piecewise constant coefficients, results on existence and non-existence of equilibria in the general case, as well as precise asymptotics for the infinite series that arise in the case of power law coefficients.

In this paper, we study an adsorption model arising in the dynamics of several surfactants at the air-water interface, where the Langmuir isotherm is employed for modelling the time-dependent surface concentration, providing a nonlinear dynamical boundary condition. Existence of a weak solution is proved by using the Rothe method for a semi-discrete problem in time. After obtaining some a priori estimates and passing to the limit in the time discretization parameter, we conclude that the original Langmuir problem has a bounded solution. An uniqueness result is also given.

We analyse the steady-state operation of a continuous flow bioreactor in which the biochemical reaction is governed by noncompetitive substrate inhibition (Andrews kinetics). A generalized reactor model is used in which the well-stirred bioreactor and the idealized membrane bioreactor are special cases. As generic properties of systems subject to substrate inhibition have been obtained by other authors, we discuss reaction engineering features specific to Andrews kinetics.

Markovian systems with multiple interacting subsystems under the influence of a control unit are considered. The state spaces of the subsystems are countably infinite, whereas that of the control unit is finite. A recent infinite level-dependent quasi-birth-and-death model for such systems is extended by facilitating the automatic representation and generation of the nonzero blocks in its underlying infinitesimal generator matrix with sums of Kronecker products. Experiments are performed on systems of stochastic chemical kinetics having two or more countably infinite state space subsystems. Results indicate that, even though more memory is consumed, there are many cases where a matrix-analytic solution coupled with Lyapunov theory yields a faster and more accurate steady-state measure compared to that obtained with simulation.

In this paper we consider the generalized Hodgkin-Huxley model introduced in Austin (2008). This model describes the propagation of an action potential along the axon of a neuron at the scale of ion channels. Mathematically, this model is a fully coupled piecewise-deterministic Markov process (PDMP) in infinite dimensions. We introduce two time scales in this model in considering that some ion channels open and close at faster jump rates than others. We perform a slow-fast analysis of this model and prove that, asymptotically, this ‘two-time-scale’ model reduces to the so-called averaged model, which is still a PDMP in infinite dimensions, for which we provide effective evolution equations and jump rates.

Systems of stochastic chemical kinetics are modeled as infinite level-dependent quasi-birth-and-death (LDQBD) processes. For these systems, in contrast to many other applications, levels have an increasing number of states as the level number increases and the probability mass may reside arbitrarily far away from lower levels. Ideas from Lyapunov theory are combined with existing matrix-analytic formulations to obtain accurate approximations to the stationary probability distribution when the infinite LDQBD process is ergodic. Results of numerical experiments on a set of problems are provided.

Multi-component flow with chemical reactions is a common problem in different industrial applications: the mixing chamber of a reaction injection molding (RIM) machine; the dynamics of diesel soot particles interacting with a porous-ceramic particulate filter; reactive transport in porous media; bio-chemical processes involving enzyme-catalyzed kinetics. In all these cases, mass diffusion/convection and wall or volume chemical interactions among components play an important role. In the present paper we underline the importance of diffusion/convection/reaction mechanisms in bio-chemical processes using the Lattice Boltzmann (LB) technique. The bio-application where we studied diffusion/convection/reaction mechanisms is the quorum-sensing pathway for the bio-synthesis of the AI-2, a molecule that allows the bacteria to launch a coordinated attack on a host immune system (see [9,10] for more details of the bio-application). The overall goal is to create a micro-device to screen potential drugs that inhibit AI-2 bio-synthesis. The Michaelis-Menten saturation kinetic model is implemented at the reactive surface and the results are shown in terms of two dimensionless numbers: Damkohler (Da) and Peclet (Pe) number. For high Pe number a small conversion of reactants into products is obtained at the reactive surface, but the overall flux of products is high; moreover, a fast saturation of the conversion of reactants to products is obtained for high Da numbers. The trade-off for setting the Pe and Da numbers depends on the specific application and the technologies used in the micro-device (e.g., sensitivity of the detector, cost of reactants).

In this paper we establish limit theorems for a class of stochastic hybrid systems (continuous deterministic dynamics coupled with jump Markov processes) in the fluid limit (small jumps at high frequency), thus extending known results for jump Markov processes. We prove a functional law of large numbers with exponential convergence speed, derive a diffusion approximation, and establish a functional central limit theorem. We apply these results to neuron models with stochastic ion channels, as the number of channels goes to infinity, estimating the convergence to the deterministic model. In terms of neural coding, we apply our central limit theorems to numerically estimate the impact of channel noise both on frequency and spike timing coding.

We find explicit analytical formulae for the time dependence of the probability of the number of Okazaki fragments produced during the process of DNA replication. This extends a result of Cowan on the asymptotic probability distribution of these fragments.

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 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.