Stochastic model reduction using a modified Hill-type kinetic rate law

Universally valid reduction of multiscale stochastic biochemical systems using simple non-elementary propensities

Biochemical systems consist of numerous elementary reactions governed by the law of mass action. However, experimentally characterizing all the elementary reactions is nearly impossible. Thus, over a century, their deterministic models that typically contain rapid reversible bindings have been simplified with non-elementary reaction functions (e.g., Michaelis-Menten and Morrison equations). Although the non-elementary reaction functions are derived by applying the quasi-steady-state approximation (QSSA) to deterministic systems, they have also been widely used to derive propensities for stochastic simulations due to computational efficiency and simplicity. However, the validity condition for this heuristic approach has not been identified even for the reversible binding between molecules, such as protein-DNA, enzyme-substrate, and receptor-ligand, which is the basis for living cells. Here, we find that the non-elementary propensities based on the deterministic total QSSA can accurately capture the stochastic dynamics of the reversible binding in general. However, serious errors occur when reactant molecules with similar levels tightly bind, unlike deterministic systems. In that case, the non-elementary propensities distort the stochastic dynamics of a bistable switch in the cell cycle and an oscillator in the circadian clock. Accordingly, we derive alternative non-elementary propensities with the stochastic low-state QSSA, developed in this study. This provides a universally valid framework for simplifying multiscale stochastic biochemical systems with rapid reversible bindings, critical for efficient stochastic simulations of cell signaling and gene regulation. To facilitate the framework, we provide a user-friendly open-source computational package, ASSISTER, that automatically performs the present framework.

Organic Carbonate Production Utilizing Crude Glycerol Derived as By-Product of Biodiesel Production: A Review

As a promising alternative renewable liquid fuel, biodiesel production has increased and eventually led to an increase in the production of its by-product, crude glycerol. The vast generation of glycerol has surpassed the market demand. Hence, the crude glycerol produced should be utilized effectively to increase the viability of biodiesel production. One of them is through crude glycerol upgrading, which is not economical. A good deal of attention has been dedicated to research for alternative material and chemicals derived from sustainable biomass resources. It will be more valuable if the crude glycerol is converted into glycerol derivatives, and so, increase the economic possibility of the biodiesel production. Studies showed that glycerol carbonate plays an important role, as a building block, in synthesizing the glycerol oligomers at milder conditions under microwave irradiation. This review presents a brief outline of the physio-chemical, thermodynamic, toxicological, production methods, reactivity, and application of organic carbonates derived from glycerol with a major focus on glycerol carbonate and dimethyl carbonate (DMC), as a green chemical, for application in the chemical and biotechnical field. Research gaps and further improvements have also been discussed.

Experimental conditions affecting the kinetics of aqueous HCN polymerization as revealed by UV-vis spectroscopy

HCN polymerization is one of the most important and fascinating reactions in prebiotic chemistry, and interest in HCN polymers in the field of materials science is growing. However, little is known about the kinetics of the HCN polymerization process. In the present study, a first approach to the kinetics of two sets of aqueous HCN polymerizations, from NH₄CN and NaCN, at middle temperatures between 4 and 38°C, has been carried out. For each series, the presence of air and salts in the reaction medium has been systematically explored. A previous kinetic analysis was conducted during the conversion of the insoluble black HCN polymers obtained as gel fractions in these precipitation polymerizations for a reaction of one month, where a limit conversion was achieved at the highest polymerization temperature. The kinetic description of the gravimetric data for this complex system shows a clear change in the linear dependence with the polymerization temperature for the reaction from NH₄CN, besides a relevant catalytic effect of ammonium, in comparison with those data obtained from the NaCN series. These results also demonstrated the notable influence of air, oxygen, and the saline medium in HCN polymer formation. Similar conclusions were reached when the sol fractions were monitored by UV-vis spectroscopy, and a Hill type correlation was used to describe the polymerization profiles obtained. This technique was chosen because it provides an easy, prompt and fast method to follow the evolution of the liquid or continuous phase of the process under study.

Maximum Entropy Prediction of Non-Equilibrium Stationary Distributions for Stochastic Reaction Networks with Oscillatory Dynamics

Many chemical reaction networks in biological systems present complex oscillatory dynamics. In systems such as regulatory gene networks, cell cycle, and enzymatic processes, the number of molecules involved is often far from the thermodynamic limit. Although stochastic models based on the probabilistic approach of the Chemical Master Equation (CME) have been proposed, studies in the literature have been limited by the challenges of solving the CME and the lack of computational power to perform large-scale stochastic simulations. In this paper, we show that the infinite set of stationary moment equations describing the stochastic Brusselator and Schnakenberg oscillatory reactions networks can be truncated and solved using maximization of the entropy of the distributions. The results from our numerical experiments compare with the distributions obtained from well-established kinetic Monte Carlo methods and suggest that the accuracy of the prediction increases exponentially with the closure order chosen for the system. We conclude that maximum entropy models can be used as an efficient closure scheme alternative for moment equations to predict the non-equilibrium stationary distributions of stochastic chemical reactions with oscillatory dynamics. This prediction is accomplished without any prior knowledge of the system dynamics and without imposing any biased assumptions on the mathematical relations among species involved.

Data-Driven Method to Estimate Nonlinear Chemical Equivalence

There is great need to express the impacts of chemicals found in the environment in terms of effects from alternative chemicals of interest. Methods currently employed in fields such as life-cycle assessment, risk assessment, mixtures toxicology, and pharmacology rely mostly on heuristic arguments to justify the use of linear relationships in the construction of "equivalency factors," which aim to model these concentration-concentration correlations. However, the use of linear models, even at low concentrations, oversimplifies the nonlinear nature of the concentration-response curve, therefore introducing error into calculations involving these factors. We address this problem by reporting a method to determine a concentration-concentration relationship between two chemicals based on the full extent of experimentally derived concentration-response curves. Although this method can be easily generalized, we develop and illustrate it from the perspective of toxicology, in which we provide equations relating the sigmoid and non-monotone, or "biphasic," responses typical of the field. The resulting concentration-concentration relationships are manifestly nonlinear for nearly any chemical level, even at the very low concentrations common to environmental measurements. We demonstrate the method using real-world examples of toxicological data which may exhibit sigmoid and biphasic mortality curves. Finally, we use our models to calculate equivalency factors, and show that traditional results are recovered only when the concentration-response curves are "parallel," which has been noted before, but we make formal here by providing mathematical conditions on the validity of this approach.

Solution of Chemical Master Equations for Nonlinear Stochastic Reaction Networks

Stochasticity in the dynamics of small reacting systems requires discrete-probabilistic models of reaction kinetics instead of traditional continuous-deterministic ones. The master probability equation is a complete model of randomly evolving molecular populations. Because of its ambitious character, the master equation remained unsolved for all but the simplest of molecular interaction networks. With the first solution of chemical master equations, a wide range of experimental observations of small-system interactions may be mathematically conceptualized.

Modeling stochastic noise in gene regulatory systems

The Master equation is considered the gold standard for modeling the stochastic mechanisms of gene regulation in molecular detail, but it is too complex to solve exactly in most cases, so approximation and simulation methods are essential. However, there is still a lack of consensus about the best way to carry these out. To help clarify the situation, we review Master equation models of gene regulation, theoretical approximations based on an expansion method due to N.G. van Kampen and R. Kubo, and simulation algorithms due to D.T. Gillespie and P. Langevin. Expansion of the Master equation shows that for systems with a single stable steady-state, the stochastic model reduces to a deterministic model in a first-order approximation. Additional theory, also due to van Kampen, describes the asymptotic behavior of multistable systems. To support and illustrate the theory and provide further insight into the complex behavior of multistable systems, we perform a detailed simulation study comparing the various approximation and simulation methods applied to synthetic gene regulatory systems with various qualitative characteristics. The simulation studies show that for large stochastic systems with a single steady-state, deterministic models are quite accurate, since the probability distribution of the solution has a single peak tracking the deterministic trajectory whose variance is inversely proportional to the system size. In multistable stochastic systems, large fluctuations can cause individual trajectories to escape from the domain of attraction of one steady-state and be attracted to another, so the system eventually reaches a multimodal probability distribution in which all stable steady-states are represented proportional to their relative stability. However, since the escape time scales exponentially with system size, this process can take a very long time in large systems.