Chemical Langevin Equation for Complex Reactions

The chemical Langevin equation (CLE) is a popular simulation method to probe the stochastic dynamics of chemical systems. The CLE's main disadvantage is its break down in finite time due to the problem of evaluating square roots of negative quantities whenever the molecule numbers become sufficiently small. We show that this issue is not a numerical integration problem, rather in many systems it is intrinsic to all representations of the CLE. Various methods of correcting the CLE have been proposed which avoid its break down. We show that these methods introduce undesirable artefacts in the CLE's predictions. In particular, for unimolecular systems, these correction methods lead to CLE predictions for the mean concentrations and variance of fluctuations which disagree with those of the chemical master equation. We show that, by extending the domain of the CLE to complex space, break down is eliminated, and the CLE's accuracy for unimolecular systems is restored. Although the molecule numbers are generally complex, we show that the "complex CLE" predicts real-valued quantities for the mean concentrations, the moments of intrinsic noise, power spectra, and first passage times, hence admitting a physical interpretation. It is also shown to provide a more accurate approximation of the chemical master equation of simple biochemical circuits involving bimolecular reactions than the various corrected forms of the real-valued CLE, the linear-noise approximation and a commonly used two moment-closure approximation.

The stochastic dynamical behavior of a well-stirred mixture of N molecular species that chemically interact through M reaction channels is accurately described by the chemical master equation. It is shown here that, whenever two explicit dynamical conditions are satisfied, the microphysical premise from which the chemical master equation is derived leads directly to an approximate time-evolution equation of the Langevin type. This chemical Langevin equation is the same as one studied earlier by Kurtz, in contradistinction to some other earlier proposed forms that assume a deterministic macroscopic evolution law. The novel aspect of the present analysis is that it shows that the accuracy of the equation depends on the satisfaction of certain specific conditions that can change from moment to moment, rather than on a static system size parameter. The derivation affords a new perspective on the origin and magnitude of noise in a chemically reacting system. It also clarifies the connection between the stochastically correct chemical master equation, and the deterministic but often satisfactory reaction rate equation.

As a fast and reliable technology with applications in diverse biological studies, cell-free protein synthesis has become popular in recent decades. The cell-free protein synthesis system can be considered a complex chemical reaction system that is also open to exogenous manipulation, including that which could otherwise potentially harm the cell's viability. On the other hand, since the technology depends on the cell lysates by which genetic information is transformed into active proteins, the whole system resembles the cell to some extent. These features make cell-free protein synthesis a valuable addition to synthetic biology technologies, expediting the design-build-test-learn cycle of synthetic biology routines. While the system has traditionally been used to synthesize one protein product from one gene addition, recent studies have employed multiple gene products in order to, for example, develop novel bacteriophages, viral particles, or synthetic metabolisms. Thus, we would like to review recent advancements in applying cell-free protein synthesis technology to synthetic biology, with an emphasis on multiple gene expressions.

Functional gels as microreactors

The chemical Fokker-Planck equation and the corresponding chemical Langevin equation are commonly used approximations of the chemical master equation. These equations are derived from an uncontrolled, second-order truncation of the Kramers-Moyal expansion of the chemical master equation and hence their accuracy remains to be clarified. We use the system-size expansion to show that chemical Fokker-Planck estimates of the mean concentrations and of the variance of the concentration fluctuations about the mean are accurate to order Ω(-3∕2) for reaction systems which do not obey detailed balance and at least accurate to order Ω(-2) for systems obeying detailed balance, where Ω is the characteristic size of the system. Hence, the chemical Fokker-Planck equation turns out to be more accurate than the linear-noise approximation of the chemical master equation (the linear Fokker-Planck equation) which leads to mean concentration estimates accurate to order Ω(-1∕2) and variance estimates accurate to order Ω(-3∕2). This higher accuracy is particularly conspicuous for chemical systems realized in small volumes such as biochemical reactions inside cells. A formula is also obtained for the approximate size of the relative errors in the concentration and variance predictions of the chemical Fokker-Planck equation, where the relative error is defined as the difference between the predictions of the chemical Fokker-Planck equation and the master equation divided by the prediction of the master equation. For dimerization and enzyme-catalyzed reactions, the errors are typically less than few percent even when the steady-state is characterized by merely few tens of molecules.

Plug-in context providers for reaction systems

Following Occam's principle, a proposed reaction mechanism should not contain assumptions about the existence of reactive intermediates and reaction paths that are unnecessary for a full description and interpretation of the available facts. A mechanism refers, in this paper, to a proposed reaction scheme or network that represents the reactions supposed to be going on in a complex reaction system with observable species as well as unobservable reactive intermediates. The scope is limited here to (pseudo) first-order reactions and the steady-state approximation is invoked in order to relate unknown mechanistic rate constants to experimentally determined ones, and, when available, theoretically calculated quantities. When the resulting, nonlinear system of equations admits a unique solution within a physically reasonable domain, it is concluded that the reaction mechanism fulfills Occam's principle. Otherwise, there are many or no solutions. No subjective or qualitative arguments enter the procedure and the outcome is not negotiable.

Reaction systems, introduced by Ehrenfeucht and Rozenberg, are computational models inspired by biochemical reactions transpiring within the living cells. Numerous studies focus on rs functions specified by minimal reaction systems, due to their simplicity and rich generative power. Manzoni et al [1] showed that every reaction system can be simulated by a minimal reaction system over an expanded background set. Motivated by their work, we introduce hybrid reaction systems as generalizations of reaction systems. Every rs function is then canonically expressible as a composition of two functions specified by minimal hybrid reaction systems. Using our new approach, the result of Manzoni et al [1] is revisited and strengthened.

Reaction systems are a formal model for computational processing in which reactions operate on sets of entities (molecules) providing a framework for dealing with qualitative aspects of biochemical systems. This paper is concerned with reaction systems in which entities can have discrete concentrations, and so reactions operate on multisets rather than sets of entities. The resulting framework allows one to deal with quantitative aspects of reaction systems, and a bespoke linear-time temporal logic allows one to express and verify a wide range of key behavioural system properties. In practical applications, a reaction system with discrete concentrations may only be partially specified, and the possibility of an effective automated calculation of the missing details provides an attractive design approach. With this idea in mind, the current paper discusses parametric reaction systems with parameters representing unknown parts of hypothetical reactions. The main result is a method aimed at replacing the parameters in such a way that the resulting reaction system operating in a specified external environment satisfies a given temporal logic formula.This paper provides an encoding of parametric reaction systems in smt, and outlines a synthesis procedure based on bounded model checking for solving the synthesis problem. It also reports on the initial experimental results demonstrating the feasibility of the novel synthesis method.

A distributed reaction system consists of a finite set of reaction systems that either interact with a common environment or interact with each other by communicating products or reactions. A reaction system is a well-known qualitative formal model of interactions between biochemical reactions. A reaction is a triplet of nonempty sets representing chemicals, called the set of reactants, the set of inhibitors, and the set of products. A reaction corresponds to a chemical reaction performed on a set of chemicals, and a reaction system is a finite nonempty set of reactions. In this paper, we examine two variants of distributed reaction systems. We introduce the notion of a distributed reaction system with communication by request (a qDRS for short), where sets of products are communicated between the component reaction systems by queries. First, we show that every qDRS can be represented by a reaction system. After that we compare distributed reaction systems with communication by request to extended distributed reaction systems (EDRSs), models that were introduced in a previous paper. We prove that extended distributed reaction systems, where a context automaton provides input for the component reaction systems, simulate distributed reaction systems with communication by request and distributed reaction systems with communication by request simulate special variants of extended distributed reaction systems. Furthermore, we assign languages to these two variants of distributed reaction systems. We prove that the class of agreement languages of extended distributed reaction systems is equal to the class of languages of nondeterministic multihead finite automata and the agreement language of every distributed reaction system with communication by request is an element of a certain subregular language class.

Modelling of complex liquid–solid reaction systems in semibatch reactors: Claisen condensation in industrial scale

Determination of kinetic parameters of polymerizations by differential scanning calorimetry

Currently, the development of a new method for the analysis of glyoxylic acid in selective oxidation of glyoxal system has become a key subject in its new industrial process development. Since solid phase extraction (SPE) could isolate and concentrate desired analytes from complex matrices, in this paper we tested the possibility of coupling SPE technique with HPLC for simultaneous determination of glyoxylic acid, formic acid, and oxalic acid in this complex chemical reaction system. Results demonstrated that the developed method could be successfully applied to this system, after samples were passed through SAX cartridges and yielded recoveries from 96.7% to 103%. The limits of detection were 2.6 × 10−6, 3.6 × 10−6, and 3.5 × 10−7 mol L−1, respectively; and their linear ranges between 1 × 10−6 and 1 × 10−2 mol L−1.

The scale-up of chemical processes involves substantial changes in reactor size, operational modes, and data characteristics, leading to significant challenges in predicting product distribution across scales. This study presents a unified modeling framework that integrates the mechanistic model with deep transfer learning to accelerate chemical process scale-up. The framework is demonstrated through a case study on naphtha fluid catalytic cracking. A molecular-level kinetic model was developed from laboratory-scale experimental data, and a deep neural network was designed and trained to represent complex molecular reaction systems. To address the challenge of discrepancies in data types at various scales, a property-informed transfer learning strategy was developed by incorporating bulk property equations into the neural network. This approach enabled automated prediction of pilot-scale product distribution with minimal data. Moreover, process conditions of the pilot plant were optimized using a multi-objective optimization algorithm.

Reaction systems are a formal framework for modeling processes driven by biochemical reactions. They are based on the mechanisms of facilitation and inhibition. A main assumption is that if a resource is available, then it is present in sufficient amounts and as such, several reactions using the same resource will not compete concurrently against each other; this makes reaction systems very different as a modeling framework than traditional frameworks such as ODEs or continuous time Markov chains. We demonstrate in this paper that reaction systems are rich enough to capture the essential characteristics of ODE-based models. We construct a reaction system model for the heat shock response in such a way that its qualitative behavior correlates well with the quantitative behavior of the corresponding ODE model. We construct our reaction system model based on a novel concept of dominance graph that captures the competition on resources in the ODE model. We conclude with a discussion on the expressivity of reaction systems as compared to that of ODE-based models.