Computing minimal entropy production trajectories: an approach to model reduction in chemical kinetics

Advanced experimental techniques in chemistry and physics provide increasing access to detailed deterministic mass action models for chemical reaction kinetics. Especially in complex technical or biochemical systems the huge amount of species and reaction pathways involved in a detailed modeling approach call for efficient methods of model reduction. These should be automatic and based on a firm mathematical analysis of the ordinary differential equations underlying the chemical kinetics in deterministic models. A main purpose of model reduction is to enable accurate numerical simulations of even high dimensional and spatially extended reaction systems. The latter include physical transport mechanisms and are modeled by partial differential equations. Their numerical solution for hundreds or thousands of species within a reasonable time will exceed computer capacities available now and in a foreseeable future. The central idea of model reduction is to replace the high dimensional dynamics by a low dimensional approximation with an appropriate degree of accuracy. Here I present a global approach to model reduction based on the concept of minimal entropy production and its numerical implementation. For given values of a single species concentration in a chemical system all other species concentrations are computed under the assumption that the system is as close as possible to its attractor, the thermodynamic equilibrium, in the sense that all modes of thermodynamic forces are maximally relaxed except the one, which drives the remaining system dynamics. This relaxation is expressed in terms of minimal entropy production for single reaction steps along phase space trajectories.     

Repro-Modelling Based Generation of Intrinsic Low-Dimensional Manifolds

Effective procedures for the reduction of reaction mechanisms, including the intrinsic low-dimensional manifold (ILDM) and the repro-modelling methods, are all based on the existence of very different time scales in chemical kinetic systems. These two methods are reviewed and the advantages and drawbacks of them are discussed. An algorithm is presented for the repro-modelling based generation of ILDMs. This algorithm produces an unstructured table of ILDM points, which are then fitted using spline functions. These splines contain kinetic information on the behaviour of the chemical system. Combustion of hydrogen in air is used as illustrative example. Simulation results using the fitted model are compared with the outcome of calculations based on the detailed reaction mechanism for homogeneous explosions and 1D laminar flames.     

An inverse problem in reaction kinetics

In this paper we investigate the problem of extracting information about chemical reactions involving multiple species from the time history of the concentration of each species. The mathematical model of the kinetic system leads to a system of ordinary differential equations. Our focus is to examine whether the species’ concentrations as functions of time are sufficient to determine what chemical reactions, and at what reaction rates, have occurred. We show that within the limitation of our model, there may be many candidate reaction systems that could explain the data. Using the notion of sparsity, we provide a quantitative assessment of the question of distinguishability. We further demonstrate that sparsity enforcing approaches, such as minimizing the ℓ ₁ or the ℓ ₀ norms are not reliable. Our conclusion is that additional knowledge about the kinetic system will be necessary to reliably solve this inverse problem.     

Reduction of chemical reaction networks using quasi-integrals

We present a numerical method to identify possible candidates for quasi-stationary manifolds in complex reaction networks governed by systems of ordinary differential equations. Inspired by singular perturbation theory, we examine the ratios of certain components of the reaction rate vector. Those ratios that rapidly approach a nearly constant value define a slow manifold for the original flow in terms of quasi-integrals, that is, functions that are nearly constant along the trajectories. The dimensionality of the original system is thus effectively reduced without reliance on a priori knowledge of the different time scales in the system. We also demonstrate the relation of our approach to singular perturbation theory which, in its simplest form, is just the well-known quasi-steady-state approximation. In two case studies, we apply our method to oscillatory chemical systems: the 6-dimensional hemin-hydrogen peroxide-sulfite pH oscillator and a 10-dimensional mechanistic model for the peroxidase-oxidase (PO) reaction system. We conjecture that the presented method is especially suited for a straightforward reduction of higher dimensional dynamical systems where analytical methods fail to identify the different time scales associated with the slow invariant manifolds present in the system.     

Two classes of quasi-steady-state model reductions for stochastic kinetics

The quasi-steady-state approximation (QSSA) is a model reduction technique used to remove highly reactive species from deterministic models of reaction mechanisms. In many reaction networks the highly reactive intermediates (QSSA species) have populations small enough to require a stochastic representation. In this work we apply singular perturbation analysis to remove the QSSA species from the chemical master equation for two classes of problems. The first class occurs in reaction networks where all the species have small populations and the QSSA species sample zero the majority of the time. The perturbation analysis provides a reduced master equation in which the highly reactive species can sample only zero, and are effectively removed from the model. The reduced master equation can be sampled with the Gillespie algorithm. This first stochastic QSSA reduction is applied to several example reaction mechanisms (including Michaelis-Menten kinetics) [Biochem. Z. 49, 333 (1913)]. A general framework for applying the first QSSA reduction technique to new reaction mechanisms is derived. The second class of QSSA model reductions is derived for reaction networks where non-QSSA species have large populations and QSSA species numbers are small and stochastic. We derive this second QSSA reduction from a combination of singular perturbation analysis and the Omega expansion. In some cases the reduced mechanisms and reaction rates from these two stochastic QSSA models and the classical deterministic QSSA reduction are equivalent; however, this is not usually the case.     

Solvent effects on the kinetics of gelation and the crosslink density of polysiloxane gels

A wide range of hydrocarbons were rapidly gelled by adding a polysiloxane copolymer in the presence of divinylbenzene and a platinum catalyst. The gel point was measured over a range of concentrations for hydrocarbons/solvents and organogels, using three separate methods: rheology, visual (tilt-tube) and FTIR. As the fraction of solvent was increased, the rate of reaction decreased, leading to an increase in the gelation time. The absolute value of the gel point depends upon the techniques used to measure it. For any particular system the gel point values always followed the order: rheology > visual > FTIR. The crosslink densities of the gel systems were determined using both swelling and thermomechanical analysis. The swelling measurements confirmed that the addition of large quantities of solvent markedly reduced the crosslink density of the obtained chemical gel networks, which helped in designing gels with suitable critical strength for effective field work. Also the effectiveness of gelation with the proposed gelling system for different hydrocarbons/solvents was evaluated, and discussed in relation to their dielectric properties.This paper is dedicated to Mike Owen on occasion of his winning the DeBruyn medal, the first silicon chemist to do so.     

Tutorial: The modelling of chemical processes

The modelling of chemical processes entails the computation of the concentration profiles of all reaction species as a function of the reaction time. The basis for the calculations is the system of differential equations (ODE's) that is defined by the reaction mechanism. Most textbooks on chemical kinetics concentrate on those few reaction mechanisms that lead to ODE's with explicit solutions. In this tutorial, we demonstrate that numerical integration is a viable alternative, that it can be applied to any mechanism, and that it is easy to do so. Matlab example programs illustrate the concepts and they allow the reader to explore the effects of changing conditions such as initial concentrations or rate constants etc. Example reaction mechanisms include a zero-th order enzymatic reaction and reactions at non-constant temperature and pH.     

A partial-propensity formulation of the stochastic simulation algorithm for chemical reaction networks with delays

Several real-world systems, such as gene expression networks in biological cells, contain coupled chemical reactions with a time delay between reaction initiation and completion. The non-Markovian kinetics of such reaction networks can be exactly simulated using the delay stochastic simulation algorithm (dSSA). The computational cost of dSSA scales with the total number of reactions in the network. We reduce this cost to scale at most with the smaller number of species by using the concept of partial reaction propensities. The resulting delay partial-propensity direct method (dPDM) is an exact dSSA formulation for well-stirred systems of coupled chemical reactions with delays. We detail dPDM and present a theoretical analysis of its computational cost. Furthermore, we demonstrate the implications of the theoretical cost analysis in two prototypical benchmark applications. The dPDM formulation is shown to be particularly efficient for strongly coupled reaction networks, where the number of reactions is much larger than the number of species.     

On the use of catalysis to bias reaction pathways in out-of-equilibrium systems

Catalysis is an essential function in living systems and provides a way to control complex reaction networks. In natural out-of-equilibrium chemical reaction networks (CRNs) driven by the consumption of chemical fuels, enzymes provide catalytic control over pathway kinetics, giving rise to complex functions. Catalytic regulation of man-made fuel-driven systems is far less common and mostly deals with enzyme catalysis instead of synthetic catalysts. Here, we show via simulations, illustrated by literature examples, how any catalyst can be incorporated in a non-equilibrium CRN and what their effect is on the behavior of the system. Alteration of the catalysts'' concentrations in batch and flow gives rise to responses in maximum conversion, lifetime (i.e. product half-lives and t90 – time to recover 90% of the reactant) and steady states. In situ up or downregulation of catalysts'' levels temporarily changes the product steady state, whereas feedback elements can give unusual concentration profiles as a function of time and self-regulation in a CRN. We show that simulations can be highly effective in predicting CRN behavior. In the future, shifting the focus from enzyme catalysis towards small molecule and metal catalysis in out-of-equilibrium systems can provide us with new reaction networks and enhance their application potential in synthetic materials, overall advancing the design of man-made responsive and interactive systems.

We show, via simulations, how catalytic control over individual paths in a fuel-driven non-equilibrium chemical reaction network in batch or flow gives rise to responses in maximum conversion, lifetime and steady states.     

A synthetic reaction network: chemical amplification using nonequilibrium autocatalytic reactions coupled in time

This article reports a functional chemical reaction network synthesized in a microfluidic device. This chemical network performs chemical 5000-fold amplification and shows a threshold response. It operates in a feedforward manner in two stages: the output of the first stage becomes the input of the second stage. Each stage of amplification is performed by a reaction autocatalytic in Co(2+). The microfluidic network is used to maintain the two chemical reactions away from equilibrium and control the interactions between them in time. Time control is achieved as described previously (Angew. Chem., Int. Ed. 2003, 42, 768) by compartmentalizing the reaction mixture inside plugs which are aqueous droplets carried through a microchannel by an immiscible fluorinated fluid. Autocatalytic reaction displayed sensitivity to mixing; more rapid mixing corresponded to slower reaction rates. Synthetic chemical reaction networks may help understand the function of biochemical reaction networks, the goal of systems biology. They may also find practical applications. For example, the system described here may be used to detect visually, in a simple format, picoliter volumes of nanomolar concentrations of Co(2+), an environmental pollutant.     

Solar‐Energy‐Driven Photoelectrochemical Biosensing Using TiO2 Nanowires

Photoelectrochemical sensing represents a unique means for chemical and biological detection, with foci of optimizing semiconductor composition and electronic structures, surface functionalization layers, and chemical detection methods. Here, we have briefly discussed our recent developments of TiO₂ nanowire‐based photoelectrochemical sensing, with particular emphasis on three main detection mechanisms and corresponding examples. We have also demonstrated the use of the photoelectrochemical sensing of real‐time molecular reaction kinetic measurements, as well as direct interfacing of living cells and probing of cellular functions.     

燃烧反应动力学研究进展

化学反应动力学是燃烧过程分析的重要工具。燃烧微观反应过程、复杂反应机理、燃烧实验测量和湍流燃烧数值模拟等方面的研究工作已经取得了长足进步。本文主要介绍燃烧反应动力学研究方法,包括电子结构方法、燃烧反应热力学和速率常数的计算方法、燃烧详细机理构建和简化、反应力场分子模拟以及燃烧中间体测量、燃料点火延迟和光谱诊断等方面的研究现状。燃烧反应动力学具有很强的应用背景,燃烧过程化学物种的反应速率计算是湍流燃烧数值模拟的一个中心任务。由于燃烧反应网络的高度复杂性,我们对燃烧机理的认识还远不清楚。化学反应和湍流相互作用研究的深入、燃烧反应动力学和计算流体力学的协同发展,将对新燃料设计、燃烧数值模拟、发动机内流道流场结构的准确描述产生深远影响。     

Eliminating fast reactions in stochastic simulations of biochemical networks: a bistable genetic switch

In many stochastic simulations of biochemical reaction networks, it is desirable to "coarse grain" the reaction set, removing fast reactions while retaining the correct system dynamics. Various coarse-graining methods have been proposed, but it remains unclear which methods are reliable and which reactions can safely be eliminated. We address these issues for a model gene regulatory network that is particularly sensitive to dynamical fluctuations: a bistable genetic switch. We remove protein-DNA and/or protein-protein association-dissociation reactions from the reaction set using various coarse-graining strategies. We determine the effects on the steady-state probability distribution function and on the rate of fluctuation-driven switch flipping transitions. We find that protein-protein interactions may be safely eliminated from the reaction set, but protein-DNA interactions may not. We also find that it is important to use the chemical master equation rather than macroscopic rate equations to compute effective propensity functions for the coarse-grained reactions.     

Chemical reaction-diffusion networks: convergence of the method of lines

We show that solutions of the chemical reaction-diffusion system associated to \(A+B\rightleftharpoons C\) in one spatial dimension can be approximated in \(L^2\) on any finite time interval by solutions of a space discretized ODE system which models the corresponding chemical reaction system replicated in the discretization subdomains where the concentrations are assumed spatially constant. Same-species reactions through the virtual boundaries of adjacent subdomains lead to diffusion in the vanishing limit. We show convergence of our numerical scheme by way of a consistency estimate, with features generalizable to reaction networks other than the one considered here, and to multiple space dimensions. In particular, the connection with the class of complex-balanced systems is briefly discussed here, and will be considered in future work.     

A reduction method for multiple time scale stochastic reaction networks

In this paper we develop a reduction method for multiple time scale stochastic reaction networks. When the transition-rate matrix between different states of the species is available, we obtain systems of reduced equations, whose solutions can successively approximate, to any degree of accuracy, the exact probability that the reaction system be in any particular state. For the case when the transition-rate matrix is not available, one needs to rely on the chemical master equation. For this case, we obtain a corresponding reduced master equation with first-order accuracy. We illustrate the accuracy and efficiency of both approaches by simulating several motivating examples and comparing the results of our simulations with the results obtained by the exact method. Our examples include both linear and nonlinear reaction networks as well as a three time scale stochastic reaction-diffusion model arising from gene expression.     

An equation-free probabilistic steady-state approximation: dynamic application to the stochastic simulation of biochemical reaction networks

Stochastic chemical kinetics more accurately describes the dynamics of "small" chemical systems, such as biological cells. Many real systems contain dynamical stiffness, which causes the exact stochastic simulation algorithm or other kinetic Monte Carlo methods to spend the majority of their time executing frequently occurring reaction events. Previous methods have successfully applied a type of probabilistic steady-state approximation by deriving an evolution equation, such as the chemical master equation, for the relaxed fast dynamics and using the solution of that equation to determine the slow dynamics. However, because the solution of the chemical master equation is limited to small, carefully selected, or linear reaction networks, an alternate equation-free method would be highly useful. We present a probabilistic steady-state approximation that separates the time scales of an arbitrary reaction network, detects the convergence of a marginal distribution to a quasi-steady-state, directly samples the underlying distribution, and uses those samples to accurately predict the state of the system, including the effects of the slow dynamics, at future times. The numerical method produces an accurate solution of both the fast and slow reaction dynamics while, for stiff systems, reducing the computational time by orders of magnitude. The developed theory makes no approximations on the shape or form of the underlying steady-state distribution and only assumes that it is ergodic. We demonstrate the accuracy and efficiency of the method using multiple interesting examples, including a highly nonlinear protein-protein interaction network. The developed theory may be applied to any type of kinetic Monte Carlo simulation to more efficiently simulate dynamically stiff systems, including existing exact, approximate, or hybrid stochastic simulation techniques.     

OVERVIEW ON SURFACE MICROSTRUCTURING BY PHOTODESORPTION ETCHING OF CHLORINATED SILICON

Developing a mechanistic interpretation of complex dynamical chemical systems such as halogen photoetching requires correlated microscopic data on the kinetics, dynamics, surface composition and microstructure of prototypical and real surfaces. This overview is concerned especially with two important variables which significantly influence the microetching mechanisms and structures; (I) the role of electronic point defects induced by substitutional doping in producing site-specific reactions and, (II) the quantum mechanical enhancement of chemical reaction induced by uv-radiation at low fluences and temperatures.

From uv-photoetching and photodesorption studies of heavily doped Si(100) and Si(111) with chlorine beams at low laser fluences, the mechanisms of photostimulated desorption is analyzed based primarily on the kinetics of chemisorption and surface layer microanalysis obtained from core-level photoemission. These results are coupled with time-of-flight dynamical measurements on the energetics of the photodesorption process to provide a more unified understanding of anisotropic photon-stimulated microetching.

Substantial alterations of the etching mechanisms occur when selective surface molecular processes are driven quantum mechanically by low level photon radiation rather than thermally. This is clearly reflected in the dynamical mechanisms for photodesorption which become strongly site- and atomic process-selective illustrated by the energetics of the processes. Creation and transport of charged carriers, especially at high doping levels by photoionization coupled with field-induced charge transport, introduces new reaction channels into the surface chemistry leading to resultant changes in the microstructure on an atomic scale. The results from the kinetics, velocity dynamics and film composition measurements are combined in terms of the dependency of chlorine adsorption on doping at high carrier levels and low laser fluences, to provide an improved interpretation of the anisotropic microetching in terms of field-promoted electron-hole activation.     

The use of digital simulation to improve the cyclic voltammetric determination of rate constants for homogeneous chemical reactions following charge transfers

Cyclic voltammetry (CV) is a very useful electrochemical tool used to study reaction systems that include chemical steps that are coupled to electron transfers. This type of system generally involves the chemical reaction of an electrochemically generated free radical. Published methods exist that are used to determine the kinetics of electrochemically initiated chemical reactions from the measurements of the peak current ratio (i_pa/i_pc) of a cyclic voltammogram. The published method requires working curves to relate a kinetic parameter to the peak current ratio.In the presented work, a digital simulation package was used to obtain improved working curves for specific working conditions. The curves were compared with the published results for the first- and second-order chemical reactions following the charge transfer step mechanisms.According to the presented results, the previously published working curve is reliable for a mechanism with a first-order chemical reaction; however, a change in the switching potential requires a recalculation of the curve. In the case of mechanisms with a second-order step (dimerisation and disproportionation), several different views exist on how the second-order chemical term should be expressed so that different values of the constant are obtained. Parameters such as electrode type, electrode area, electroactive species concentration, switching potential, scan rate and method for peak current ratio calculation modify the working curves and must always be specified.We propose a standardised method to obtain the most reliable kinetic constant values.The results of this work will permit researchers who handle simulation software to construct their own working curves. Additionally, those who do not have the simulation software could use the working curves described here.The revelations of the presented experiments may be useful to a broad chemistry audience because this study presents a simple and low-cost procedure for the study of free radicals that otherwise should be studied with more sophisticated and expensive techniques, such as ESR or pulse radiolysis.     

Interdiffusion/reaction at the polymer/polymer interface in multilayer systems probed by linear viscoelasticity coupled to FTIR and NMR measurements

This paper describes the experimental investigation of the interdiffusion/reaction mechanisms of asymmetric polymer-polymer interfaces. The study deals with the assessment of the chemical reactions occurring at the interface between two reactive polymers. A focal point of the investigation was to study these interfacial reactions by an array of techniques at very different space scales: from macroscopic viscoelastic investigations to IR and NMR spectroscopies at the molecular scale. The studied material pairs include PE-GMA/PA6 as the reactive system (RS) and PE/PA6 as the non-reactive one (NRS) - of coextruded multilayer polymers, i.e., after processing. The linear viscoelastic properties of the reactive multilayer systems were determined and the mechanisms were analyzed by NMR and FTIR measurements. Substantial reactions occurred during the rheological measurements and the results indicated the preferential formation of a copolymer at the interface, triggered by the neighboring layers. Moreover, the contribution of an interface/interphase effect was investigated along with the increase in the number of layers. The results showed that the variation in dynamic modulus of the multilayer system was a result of both diffusion and chemical reaction. Specific experiments were carried out to follow-up on the physicochemical phenomena, and the results were rationalized by comparing the obtained data with theoretical models. The effect of this interphase was quantified at a specific welding time and oscillation frequency thanks to rheological modeling. Because of the coupling between rheology and spectroscopical tools, potential reactions between the GMA functions and the amine/carboxylic polyamide chain ends were explored. The results highlighted that the main reaction mechanism was constituted by the crosslinking reaction between the GMA and carboxylic acid units, and not by that between GMA and amine end functions.