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Register a new userMathematical Analysis of Chemical Reaction Systems
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The use of mathematical methods for the analysis of chemical reaction systems has a very long history, and involves many types of models: deterministic versus stochastic, continuous versus discrete, and homogeneous versus spatially distributed. Here we focus on mathematical models based on deterministic mass-action kinetics. These models are systems of coupled nonlinear differential equations on the positive orthant. We explain how mathematical properties of the solutions of mass-action systems are strongly related to key properties of the networks of chemical reactions that generate them, such as specif
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The emergence of self-sustaining autocatalytic networks in chemical reaction systems has been studied as a possible mechanism for modelling how living systems first arose. It has been known for several decades that such networks will form within systems of polymers (under cleavage and ligation reactions) under a simple process of random catalysis, and this process has since been mathematically analysed. In this paper, we provide an exact expression for the expected number of self-sustaining autocatalytic networks that will form in a general chemical reaction system, and the expected number of these networks that will also be uninhibited (by some molecule produced by the system). Using these equations, we are able to describe the patterns of catalysis and inhibition that maximise or minimise the expected number of such networks. We apply our results to derive a general theorem concerning the trade-off between catalysis and inhibition, and to provide some insight into the extent to which the expected number of self-sustaining autocatalytic networks coincides with the probability that at least one such system is present.
The Bond Graph approach and the Chemical Reaction Network approach to modelling biomolecular systems developed independently. This paper brings together the two approaches by providing a bond graph interpretation of the chemical reaction network concept of complexes. Both closed and open systems are discussed. The method is illustrated using a simple enzyme-catalysed reaction and a trans-membrane transporter.
Chemical reactions occur in energy, environmental, biological, and many other natural systems, and the inference of the reaction networks is essential to understand and design the chemical processes in engineering and life sciences. Yet, revealing the reaction pathways for complex systems and processes is still challenging due to the lack of knowledge of the involved species and reactions. Here, we present a neural network approach that autonomously discovers reaction pathways from the time-resolved species concentration data. The proposed Chemical Reaction Neural Network (CRNN), by design, satisfies the fundamental physics laws, including the Law of Mass Action and the Arrhenius Law. Consequently, the CRNN is physically interpretable such that the reaction pathways can be interpreted, and the kinetic parameters can be quantified simultaneously from the weights of the neural network. The inference of the chemical pathways is accomplished by training the CRNN with species concentration data via stochastic gradient descent. We demonstrate the successful implementations and the robustness of the approach in elucidating the chemical reaction pathways of several chemical engineering and biochemical systems. The autonomous inference by the CRNN approach precludes the need for expert knowledge in proposing candidate networks and addresses the curse of dimensionality in complex systems. The physical interpretability also makes the CRNN capable of not only fitting the data for a given system but also developing knowledge of unknown pathways that could be generalized to similar chemical systems.
In most natural sciences there is currently the insight that it is necessary to bridge gaps between different processes which can be observed on different scales. This is especially true in the field of chemical reactions where the abilities to form bonds between different types of atoms and molecules create much of the properties we experience in our everyday life, especially in all biological activity. There are essentially two types of processes related to biochemical reaction networks, the interactions among molecules and interactions involving their conformational changes, so in a sense, their internal state. The first type of processes can be conveniently approximated by the so-called mass-action kinetics, but this is not necessarily so for the second kind where molecular states do not define any kind of density or concentration. In this paper we demonstrate the necessity to study reaction networks in a stochastic formulation for which we can construct a coherent approximation in terms of specific space-time scales and the number of particles. The continuum limit procedure naturally creates equations of Fokker-Planck type where the evolution of the concentration occurs on a slower time scale when compared to the evolution of the conformational changes, for example triggered by binding or unbinding events with other (typically smaller) molecules. We apply the asymptotic theory to derive the effective, i.e. macroscopic dynamics of the biochemical reaction system. The theory can also be applied to other processes where entities can be described by finitely many internal states, with changes of states occuring by arrival of other entities described by a birth-death process.
In this work, we design a type of controller that consists of adding a specific set of reactions to an existing mass-action chemical reaction network in order to control a target species. This set of reactions is effective for both deterministic and stochastic networks, in the latter case controlling the mean as well as the variance of the target species. We employ a type of network property called absolute concentration robustness (ACR). We provide applications to the control of a multisite phosphorylation model as well as a receptor-ligand signaling system. For this framework, we use the so-called deficiency zero theorem from chemical reaction network theory as well as multiscaling model reduction methods. We show that the target species has approximately Poisson distribution with the desired mean. We further show that ACR controllers can bring robust perfect adaptation to a target species and are complementary to a recently introduced antithetic feedback controller used for stochastic chemical reactions.
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