Uniform approximation of solutions by elimination of intermediate species in deterministic reaction networks

Chemical reactions often proceed through the formation and the consumption of intermediate species. An example is the creation and subsequent degradation of the substrate-enzyme complexes in an enzymatic reaction. In this paper we provide a setting, based on ordinary differential equations, in which the presence of intermediate species has little effect on the overall dynamics of a biological system. The result provides a method to perform model reduction by elimination of intermediate species. We study the problem in a multiscale setting, where the species abundances as well as the reaction rates scale to different orders of magnitudes. The different time and concentration scales are parameterized by a single parameter N. We show that a solution to the original reaction system is uniformly approximated on compact time intervals to a solution of a reduced reaction system without intermediates and to a solution of a certain limiting reaction systems, which does not depend on N. Known approximation techniques such as the theorems by Tikhonov and Fenichel cannot readily be used in this framework

Elimination of intermediate species in multiscale stochastic reaction networks

We study networks of biochemical reactions modelled by continuous time Markov processes. Such networks typically contain many molecular species and reactions and are hard to study analytically as well as by simulation. Particularly, we are interested in reaction networks with intermediate species such as the substrate-enzyme complex in the Michaelis-Menten mechanism. Such species are virtually in all real-world networks, they are typically short-lived, degraded at a fast rate and hard to observe experimentally. We provide conditions under which the Markov process of a multiscale reaction network with intermediate species is approximated by the Markov process of a simpler reduced reaction network without intermediate species. We do so by embedding the Markov processes into a one-parameter family of processes, where reaction rates and species abundances are scaled in the parameter. Further, we show that there are close links between these stochastic models and deterministic ODE models of the same networks

Product-form poisson-like distributions and complex balanced reaction systems

Stochastic reaction networks are dynamical models of biochemical reaction systems and form a particular class of continuous-time Markov chains on Nn. Here we provide a fundamental characterization that connects structural properties of a network to its dynamical features. Specifically, we define the notion of "stochastically complex balanced systems" in terms of the network's stationary distribution and provide a characterization of stochastically complex balanced systems, parallel to that established in the 1970s and 1980s for deterministic reaction networks. Additionally, we establish that a network is stochastically complex balanced if and only if an associated deterministic network is complex balanced (in the deterministic sense), thereby proving a strong link between the theory of stochastic and deterministic networks. Further, we prove a stochastic version of the "deficiency zero theorem" and show that any (not only complex balanced) deficiency zero reaction network has a product-form Poisson-like stationary distribution on all irreducible components. Finally, we provide sufficient conditions for when a product-form Poisson-like distribution on a single (or all) component(s) implies the network is complex balanced, and we explore the possibility to characterize complex balanced systems in terms of product-form Poisson-like stationary distributions

Addition of flow reactions preserving multistationarity and bistability

We consider the question whether a chemical reaction network preserves the number and stability of its positive steady states upon inclusion of inflow and outflow reactions. Often a model of a reaction network is presented without inflows and outflows, while in fact some of the species might be degraded or leaked to the environment, or be synthesized or transported into the system. We provide a sufficient and easy-to-check criterion based on the stoichiometry of the reaction network alone and discuss examples from systems biology

Node balanced steady states: Unifying and generalizing complex and detailed balanced steady states

We introduce a unifying and generalizing framework for complex and detailed balanced steady states in chemical reaction network theory. To this end, we generalize the graph commonly used to represent a reaction network. Specifically, we introduce a graph, called a reaction graph, that has one edge for each reaction but potentially multiple nodes for each complex. A special class of steady states, called node balanced steady states, is naturally associated with such a reaction graph. We show that complex and detailed balanced steady states are special cases of node balanced steady states by choosing appropriate reaction graphs. Further, we show that node balanced steady states have properties analogous to complex balanced steady states, such as uniqueness and asymptotic stability in each stoichiometric compatibility class. Moreover, we associate an integer, called the deficiency, to a reaction graph that gives the number of independent relations in the reaction rate constants that need to be satisfied for a positive node balanced steady state to exist. The set of reaction graphs (modulo isomorphism) is equipped with a partial order that has the complex balanced reaction graph as minimal element. We relate this order to the deficiency and to the set of reaction rate constants for which a positive node balanced steady state exists

Stationary distributions of systems with discreteness-induced transitions

We provide a theoretical analysis of some autocatalytic reaction networks exhibiting the phenomenon of discreteness-induced transitions. The family of networks that we address includes the celebrated Togashi and Kaneko model. We prove positive recurrence, finiteness of all moments and geometric ergodicity of the models in the family. For some parameter values, we find the analytic expression for the stationary distribution and discuss the effect of volume scaling on the stationary behaviour of the chain. We find the exact critical value of the volume for which discreteness-induced transitions disappear

Identification of hidden population structure in time-scaled phylogenies

Abstract Population structure influences genealogical patterns, however data pertaining to how populations are structured are often unavailable or not directly observable. Inference of population structure is highly important in molecular epidemiology where pathogen phylogenetics is increasingly used to infer transmission patterns and detect outbreaks. Discrepancies between observed and idealised genealogies, such as those generated by the coalescent process, can be quantified, and where significant differences occur, may reveal the action of natural selection, host population structure, or other demographic and epidemiological heterogeneities. We have developed a fast non-parametric statistical test for detection of cryptic population structure in time-scaled phylogenetic trees. The test is based on contrasting estimated phylogenies with the theoretically expected phylodynamic ordering of common ancestors in two clades within a coalescent framework. These statistical tests have also motivated the development of algorithms which can be used to quickly screen a phylogenetic tree for clades which are likely to share a distinct demographic or epidemiological history. Epidemiological applications include identification of outbreaks in vulnerable host populations or rapid expansion of genotypes with a fitness advantage. To demonstrate the utility of these methods for outbreak detection, we applied the new methods to large phylogenies reconstructed from thousands of HIV-1 partial pol sequences. This revealed the presence of clades which had grown rapidly in the recent past, and was significantly concentrated in young men, suggesting recent and rapid transmission in that group. Furthermore, to demonstrate the utility of these methods for the study of antimicrobial resistance, we applied the new methods to a large phylogeny reconstructed from whole genome Neisseria gonorrhoeae sequences. We find that population structure detected using these methods closely overlaps with the appearance and expansion of mutations conferring antimicrobial resistance