We consider molecular communication networks consisting of transmitters and receivers distributed in a fluidic medium. In such networks, a transmitter sends one or more signaling molecules, which are diffused over the medium, to the receiver to realize the communication. In order to be able to engineer synthetic molecular communication networks, mathematical models for these networks are required. This paper proposes a new stochastic model for molecular communication networks called reaction-diffusion master equation with exogenous input (RDMEX). The key idea behind RDMEX is to model the transmitters as time series of signaling molecule counts, while diffusion in the medium and chemical reactions at the receivers are modeled as Markov processes using master equation. An advantage of RDMEX is that it can readily be used to model molecular communication networks with multiple transmitters and receivers. For the case where the reaction kinetics at the receivers is linear, we show how RDMEX can be used to determine the mean and covariance of the receiver output signals, and derive closed-form expressions for the mean receiver output signal of the RDMEX model. These closed-form expressions reveal that the output signal of a receiver can be affected by the presence of other receivers. Numerical examples are provided to demonstrate the properties of the model.
For some time, RDME has been considered to be a phenomenological model because it diverges in certain cases @cite_11 . Fortunately, the problem has been resolved in @cite_7 and there is now a firm theoretic basis for RDME. There are many examples of work that use RDME to model natural molecular communication networks, see @cite_5 @cite_14 . However, these papers do not consider the transmitters. The use of RDME in studying synthetic molecular communication networks appear to be novel. To the best of our knowledge, our RDMEX model, which is formed by coupling time sequences of signalling molecule emission pattern with RDME, has not been proposed before. The proposed RDMEX model is one of the novel contributions of this paper.
@cite_5 @cite_14 @cite_7 @cite_11
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A method is developed for incorporating diffusion of chemicals in complex geometries into stochastic chemical kinetics simulations. Systems are modeled using the reaction-diffusion master equation, with jump rates for diffusive motion between mesh cells calculated from the discretization weights of an embedded boundary method. Since diffusive jumps between cells are treated as first order reactions, individual realizations of the stochastic process can be created by the Gillespie method. Numerical convergence results for the underlying embedded boundary method, and for the stochastic reaction-diffusion method, are presented in two dimensions. A two-dimensional model of transcription, translation, and nuclear membrane transport in eukaryotic cells is presented to demonstrate the feasibility of the method in studying cell-wide biological processes. Spontaneous separation of bi-stable biochemical systems into spatial domains of opposite phases Quantitative analysis of biochemical networks often requires consideration of both spatial and stochastic aspects of chemical processes. Despite significant progress in the field, it is still computationally prohibitive to simulate systems involving many reactants or complex geometries using a microscopic framework that includes the finest length and time scales of diffusion-limited molecular interactions. For this reason, spatially or temporally discretized simulations schemes are commonly used when modeling intracellular reaction networks. The challenge in defining such coarse-grained models is to calculate the correct probabilities of reaction given the microscopic parameters and the uncertainty in the molecular positions introduced by the spatial or temporal discretization. In this paper we have solved this problem for the spatially discretized Reaction-Diffusion Master Equation; this enables a seamless and physically consistent transition from the microscopic to the macroscopic frameworks of reaction-diffusion kinetics. We exemplify the use of the methods by showing that a phosphorylation-dephosphorylation motif, commonly observed in eukaryotic signaling pathways, is predicted to display fluctuations that depend on the geometry of the system. The reaction–diffusion master equation (RDME) is a model for chemical systems in which both noise in the chemical reaction process and the diffusion of molecules are important. It extends the chemical master equation for well-mixed chemical reactions by discretizing space into a collection of voxels. In this work, we show how the RDME may be rewritten as an equivalent 'particle tracking' model following the motion and interaction of individual molecules on a lattice. This new representation can be interpreted as a discrete version of the spatially-continuous 'probability distribution function' stochastic reaction–diffusion model studied by Doi. We show how this new representation can be mapped to a quantum field theory, complementing the existing work by Peliti mapping the RDME, and Doi mapping his spatially continuous model, to quantum field theories. The formal continuum limit, as the voxel size approaches zero, of the 'particle tracking' representation is studied to consider the question of whether the RDME approximates any spatially continuous model.