Nonlinear kinetics and new approaches to complex reaction mechanisms

Transient chirality inversion during racemization of a helical cobalt(III) complex

SignificanceWe first observed a transient chirality inversion on a simple unimolecular platform during the racemization of a chiral helical complex [LCo₃A₆]³⁺, i.e., the helicity changed from P-rich (right-handed) to M-rich (left-handed), which then racemized to a P/M equimolar mixture in spite of the absence of a reagent that could induce the M helix. This transient chirality inversion was observed only in the forward reaction, whereas the reverse reaction showed a simple monotonic change with an induction time. Consequently, the M helicity appeared only in the forward reaction. These forward and reverse reactions constitute a hysteretic cycle. Compounds showing such unique time responses would be useful for developing time-programmable switchable materials that can control the physical/chemical properties in a time-dependent manner.

Predicting a Kind of Unusual Multiple-States Dimerization-Modes Transformation in Protein PD-L1 System by Computational Investigation and a Generalized Rate Theory

The new cancer immunotherapy has been carried out with an almost messianic zeal, but its molecular basis remains unclear due to the complexity of programmed death ligand 1 (PD-L1) dimerization. In this study, a new and integral multiple dimerization-modes transformation process of PD-L1s (with a new PD-L1 dimerization mode and a new transformation path discovered) and the corresponding mechanism are predicted using theoretical and computational methods. The results of the state analysis show that 5 stable binding states exist in system. A generalized inter-state transformation rate (GITR) theory is also proposed in such multiple-states self-assembly system to explore the kinetic characteristics of inter-state transformation. A “drug insertion” path was identified as the dominant path of the PD-L1 dimerization-modes transformation. Above results can provide supports for both the relative drug design and other multiple-states self-assembly system from the theoretical chemistry perspective.

Dynamics of Bacterial Gene Regulatory Networks

The ability of bacterial cells to adjust their gene expression program in response to environmental perturbation is often critical for their survival. Recent experimental advances allowing us to quantitatively record gene expression dynamics in single cells and in populations coupled with mathematical modeling enable mechanistic understanding on how these responses are shaped by the underlying regulatory networks. Here, we review how the combination of local and global factors affect dynamical responses of gene regulatory networks. Our goal is to discuss the general principles that allow extrapolation from a few model bacteria to less understood microbes. We emphasize that, in addition to well-studied effects of network architecture, network dynamics are shaped by global pleiotropic effects and cell physiology.

Probabilistic density function method for nonlinear dynamical systems driven by colored noise

We present a probability density function (PDF) method for a system of nonlinear stochastic ordinary differential equations driven by colored noise. The method provides an integrodifferential equation for the temporal evolution of the joint PDF of the system's state, which we close by means of a modified large-eddy-diffusivity (LED) closure. In contrast to the classical LED closure, the proposed closure accounts for advective transport of the PDF in the approximate temporal deconvolution of the integrodifferential equation. In addition, we introduce the generalized local linearization approximation for deriving a computable PDF equation in the form of a second-order partial differential equation. We demonstrate that the proposed closure and localization accurately describe the dynamics of the PDF in phase space for systems driven by noise with arbitrary autocorrelation time. We apply the proposed PDF method to analyze a set of Kramers equations driven by exponentially autocorrelated Gaussian colored noise to study nonlinear oscillators and the dynamics and stability of a power grid. Numerical experiments show the PDF method is accurate when the noise autocorrelation time is either much shorter or longer than the system's relaxation time, while the accuracy decreases as the ratio of the two timescales approaches unity. Similarly, the PDF method accuracy decreases with increasing standard deviation of the noise.

Impact of structured heterogeneities on reactive two-phase porous flow

Two-phase flow through heterogeneous media leads to scale-free distributions of irregularly shaped pockets of one fluid trapped within the other. Although reactions within these fluids are often modeled at the homogeneous continuum scale, there exists no current framework for upscaling from the pore scale that accounts for the complex and scale-free geometry of the bubbles. In this paper, we apply a linear-kinetics reaction-diffusion model to characterize the steady-state chemical environment inside the irregular pockets. Using a combination of theory and invasion-percolation simulations, we derive scaling laws describing the distribution of diffusion times within bubbles. We show that chemical concentrations within the bubbles are determined by the Laplace transform of the entire distribution of diffusion times from each location. This serves as a means to compute average concentrations of reactant within a bubble of unique geometry and size. Furthermore, the overall system size imposes upper bounds on the distribution of bubble sizes, thereby imposing a system-size dependence on the statistics and average concentrations. These conclusions have profound implications for continuum models of porous reactive flow, where kinetic and equilibrium parameters are often chosen from laboratory measurements made at centimeter scales.

Characterizing multistationarity regimes in biochemical reaction networks

Switch like responses appear as common strategies in the regulation of cellular systems. Here we present a method to characterize bistable regimes in biochemical reaction networks that can be of use to both direct and reverse engineering of biological switches. In the design of a synthetic biological switch, it is important to study the capability for bistability of the underlying biochemical network structure. Chemical Reaction Network Theory (CRNT) may help at this level to decide whether a given network has the capacity for multiple positive equilibria, based on their structural properties. However, in order to build a working switch, we also need to ensure that the bistability property is robust, by studying the conditions leading to the existence of two different steady states. In the reverse engineering of biological switches, knowledge collected about the bistable regimes of the underlying potential model structures can contribute at the model identification stage to a drastic reduction of the feasible region in the parameter space of search. In this work, we make use and extend previous results of the CRNT, aiming not only to discriminate whether a biochemical reaction network can exhibit multiple steady states, but also to determine the regions within the whole space of parameters capable of producing multistationarity. To that purpose we present and justify a condition on the parameters of biochemical networks for the appearance of multistationarity, and propose an efficient and reliable computational method to check its satisfaction through the parameter space.

Common structure in the heterogeneity of plant-matter decay

Carbon removed from the atmosphere by photosynthesis is released back by respiration. Although some organic carbon is degraded quickly, older carbon persists; consequently carbon stocks are much larger than predicted by initial decomposition rates. This disparity can be traced to a wide range of first-order decay-rate constants, but the rate distributions and the mechanisms that determine them are unknown. Here, we pose and solve an inverse problem to find the rate distributions corresponding to the decomposition of plant matter throughout North America. We find that rate distributions are lognormal, with a mean and variance that depend on climatic conditions and substrate. Changes in temperature and precipitation scale all rates similarly, whereas the initial substrate composition sets the time scale of faster rates. These findings probably result from the interplay of stochastic processes and biochemical kinetics, suggesting that the intrinsic variability of decomposers, substrate and environment results in a predictable distribution of rates. Within this framework, turnover times increase exponentially with the kinetic heterogeneity of rates, thereby providing a theoretical expression for the persistence of recalcitrant organic carbon in the natural environment.

Stochastic dynamics and non-equilibrium thermodynamics of a bistable chemical system: the Schlögl model revisited

Schlögl's model is the canonical example of a chemical reaction system that exhibits bistability. Because the biological examples of bistability and switching behaviour are increasingly numerous, this paper presents an integrated deterministic, stochastic and thermodynamic analysis of the model. After a brief review of the deterministic and stochastic modelling frameworks, the concepts of chemical and mathematical detailed balances are discussed and non-equilibrium conditions are shown to be necessary for bistability. Thermodynamic quantities such as the flux, chemical potential and entropy production rate are defined and compared across the two models. In the bistable region, the stochastic model exhibits an exchange of the global stability between the two stable states under changes in the pump parameters and volume size. The stochastic entropy production rate shows a sharp transition that mirrors this exchange. A new hybrid model that includes continuous diffusion and discrete jumps is suggested to deal with the multiscale dynamics of the bistable system. Accurate approximations of the exponentially small eigenvalue associated with the time scale of this switching and the full time-dependent solution are calculated using Matlab. A breakdown of previously known asymptotic approximations on small volume scales is observed through comparison with these and Monte Carlo results. Finally, in the appendix section is an illustration of how the diffusion approximation of the chemical master equation can fail to represent correctly the mesoscopically interesting steady-state behaviour of the system.

Elimination of fast variables in chemical Langevin equations

Internal and external fluctuations are ubiquitous in cellular signaling processes. Because biochemical reactions often evolve on disparate time scales, mathematical perturbation techniques can be invoked to reduce the complexity of stochastic models. Previous work in this area has focused on direct treatment of the master equation. However, eliminating fast variables in the chemical Langevin equation is also an important problem. We show how to solve this problem by utilizing a partial equilibrium assumption. Our technique is applied to a simple birth-death-dimerization process and a more involved gene regulation network, demonstrating great computational efficiency. Excellent agreement is found with results computed from exact stochastic simulations. We compare our approach with existing reduction schemes and discuss avenues for future improvement.

Inhibition of Amyloid Fibrillization of Hen Egg-White Lysozymes by Rifampicin and p-Benzoquinone

It has been reported that more than 20 different human proteins can fold abnormally, resulting in the formation of pathological deposits and several lethal degenerative diseases. Despite extensive investigations on amyloid fibril formation, the detailed molecular mechanism remained rather elusive. The current research, utilizing hen egg-white lysozymes as a model system, is aimed at exploring inhibitory activities of two potential molecules against lysozyme fibril formation. We first demonstrated that the formation of lysozyme amyloid fibrils at pH 2.0 was markedly enhanced by the presence of agitation in comparison with its quiescent counterpart. Next, via numerous spectroscopic techniques and transmission electron microscopy, our results revealed that the inhibition of lysozyme amyloid formation by either rifampicin or its analogue p-benzoquinone followed a concentration-dependent fashion. Furthermore, while both inhibitors were shown to acquire an anti-aggregating and a disaggregating activity, rifampicin, in comparison with p-benzoquinone, served as a more effective inhibitor against in vitro amyloid fibrillogenesis of lysozyme. It is our belief that the data reported in this work will not only reinforce the findings validated by others that rifampicin and p-benzoquinone serve as two promising preventive molecules against amyloid fibrillogenesis, but also shed light on a rational design of effective therapeutics for amyloidogenic diseases.

Weibull Mixture Model for Modeling Nonisothermal Kinetics of Thermally Stimulated Solid-State Reactions: Application to Simulated and Real Kinetic Conversion Data

The possibility of applying Weibull mixture model for the fitting of the nonisothermal kinetic conversion data has been investigated. It has been found that the kinetic conversion data at different heating rates can be successfully described by one or the linear combination of few Weibull distribution functions. Several simulated and real kinetic conversion traces have been analyzed. An optimal fitting of the kinetic conversion data has been obtained by a mixture of Weibull distribution functions. The results obtained have shown that the obtained conversion curves calculated by the model proposed in this paper are in agreement with the raw kinetic conversion data.

Bayesian maximum entropy (two-dimensional) lifetime distribution reconstruction from time-resolved spectroscopic data

Time-resolved spectroscopy is often used to monitor the relaxation processes (or reactions) of physical, chemical, and biochemical systems after some fast physical or chemical perturbation. Time-resolved spectra contain information about the relaxation kinetics, in the form of macroscopic time constants of decay and their decay associated spectra. In the present paper we show how the Bayesian maximum entropy inversion of the Laplace transform (MaxEnt-iLT) can provide a lifetime distribution without sign-restrictions (or two-dimensional (2D)-lifetime distribution), representing the most probable inference given the data. From the reconstructed (2D) lifetime distribution it is possible to obtain the number of exponentials decays, macroscopic rate constants, and exponential amplitudes (or their decay associated spectra) present in the data. More importantly, the obtained (2D) lifetime distribution is obtained free from pre-conditioned ideas about the number of exponential decays present in the data. In contrast to the standard regularized maximum entropy method, the Bayesian MaxEnt approach automatically estimates the regularization parameter, providing an unsupervised and more objective analysis. We also show that the regularization parameter can be automatically determined by the L-curve and generalized cross-validation methods, providing (2D) lifetime reconstructions relatively close to the Bayesian best inference. Finally, we propose the use of MaxEnt-iLT for a more objective discrimination between data-supported and data-unsupported quantitative kinetic models, which takes both the data and the analysis limitations into account. All these aspects are illustrated with realistic time-resolved Fourier transform infrared (FT-IR) synthetic spectra of the bacteriorhodopsin photocycle.

Kinetic measurements of hydrocarbon conversion reactions on model metal surfaces

Examples from recent studies in our laboratory are presented to illustrate the main tools available to surface scientists for the determination of the kinetics of surface reactions. Emphasis is given here to hydrocarbon conversions and studies that rely on the use of model systems, typically single crystals and controlled (ultrahigh vacuum) environments. A detailed discussion is provided on the use of temperature-programmed desorption for the determination of activation energies as well as for product identification and yield estimations. Isothermal kinetic measurements are addressed next by focusing on studies under vacuum using molecular beams and surface-sensitive spectroscopies. That is followed by a review of the usefulness of high-pressure cells and other reactor designs for the emulation of realistic catalytic conditions. Finally, an analysis of the power of isotope labeling and chemical substitutions in mechanistic research on surface reactions is presented.

Transition Event Statistics in Genetics and Disordered Kinetics. Theoretical Approaches for Extracting Rate Distributions from Experimental Data

We study the analogies between the theory of rate processes in disordered systems and the overdispersed molecular clocks in evolutionary biology. A biological "molecular clock" expresses the statistics of the number of amino acid or nucleotide substitutions during evolution. Random variations of the evolution rates lead to statistical (overdispersed) molecular clocks which are described by random point processes with random substitution rates. We find that the models for overdispersed molecular clocks are equivalent to those of the random-rate or random channel models used in disordered kinetics. The number of transport (reaction) events in disordered kinetics plays the same role as the number of substitution events in molecular biology. We study the connections between the (observed) statistics of the transition events and the statistics of random rate coefficients and random channels; a unified approach is developed which is valid both in molecular biology and in disordered kinetics. We develop methods for extracting statistical information about the variations of rate coefficients from experimental or observed data regarding the fluctuations of the numbers of substitution, reaction, or transport events. For systems with static disorder, the observed statistics of the number of reaction events, expressed in terms of probabilities at a given time or by the cumulants of the number of transition events at a given time, contains the information necessary for evaluating the cumulants or the probability density of the rate coefficients or the density of states for random channel kinetics. For dynamic disorder this is not possible; further information about multitime probability distributions of the reaction events is needed.

Study of Oscillations and Pattern Formation in the NO + CO Reaction on Pt(100) Surfaces through Dynamic Monte Carlo Simulation: Toward a Realistic Model

Oscillations and pattern formation driven by a surface reconstruction are studied for the catalytic reduction of NO by CO on Pt(100) single-crystal surfaces through dynamic Monte Carlo simulations at low pressure and relatively high temperatures conditions. This study incorporates recent experimental evidence obtained for the same reaction on a Rh(111) surface, which modifies the reaction scheme used in previous approaches. The main consequence of such experimental evidence is that the production of N(2) occurs through two parallel mechanisms: (a) the classical N + N recombination step; (b) the formation and subsequent decay of an (N-NO) intermediate species as the fastest pathway. Moreover, different factors influencing the NO dissociation rate, the key step in the whole reaction, such as the availability of neighboring vacant sites, the formation of N-islands, and the presence of other NO and CO adsorbed species in the neighborhood, are also taken into account and their effects discussed. Sustained, modulated, irregular, and damped oscillations are observed in our analysis as well as the formation of cellular structures and turbulent patterns. The effect and the importance of each elementary reaction step on the behavior of the system are discussed.

Automatic network coupling analysis for dynamical systems based on detailed kinetic models

We introduce a numerical complexity reduction method for the automatic identification and analysis of dynamic network decompositions in (bio)chemical kinetics based on error-controlled computation of a minimal model dimension represented by the number of (locally) active dynamical modes. Our algorithm exploits a generalized sensitivity analysis along state trajectories and subsequent singular value decomposition of sensitivity matrices for the identification of these dominant dynamical modes. It allows for a dynamic coupling analysis of (bio)chemical species in kinetic models that can be exploited for the piecewise computation of a minimal model on small time intervals and offers valuable functional insight into highly nonlinear reaction mechanisms and network dynamics. We present results for the identification of network decompositions in a simple oscillatory chemical reaction, time scale separation based model reduction in a Michaelis-Menten enzyme system and network decomposition of a detailed model for the oscillatory peroxidase-oxidase enzyme system.

Effects of patch temperature on spontaneous action potential train due to channel fluctuations: Coherence resonance

Based on the Hodgkin-Huxley (HH) model, the effects of patch temperature as a control parameter on the spontaneous action potentials for finite size of membrane patch are studied. With increasing patch temperature, it is found that the mean open rates of sodium and potassium channels of the HH neuron are decreased, and the mean duration of spikes of membrane potential is also decreased, which are qualitatively consistent with previous experimental results of single ion channel. Under moderate patch size, the mean interspike interval of membrane potential first decreases, reaches a minimum, and then increases with increasing patch temperature. It is shown that for both low and high temperatures, the channels fluctuation-induced spontaneous action potentials appear to be rather irregular, while for moderate patch temperature, relatively coherent oscillations observed. By defining a measure parameter beta, we show that there is a maximal region for the measure beta in the patch temperature and patch size parameter plane where the coherence resonance phenomena are very remarkable, and the characteristic correlation time of the output also confirm our result.

A chemical waveform synthesizer

Algorithms and methods were developed to synthesize complex chemical waveforms in open volumes by using a scanning-probe microfluidic platform. Time-dependent variations and oscillations of one or several chemical species around the scanning probe, such as formation of sine waves, damped oscillations, and generation of more complex patterns, are demonstrated. Furthermore, we show that intricate bursting and chaotic calcium oscillations found in biological microdomains can be reproduced and that a biological cell can be used as a probe to study receptor functionalities as a function of exposure to time-dependent variations of receptor activators and inhibitors. Thus, the method allows for studies of biologically important oscillatory reactions. More generally, the system allows for detailed studies of complex time-varying chemical and physical phenomena in solution or at solution/surface interfaces.

Kinetic oscillations in the NO+CO reaction on the Pt(100) surface: An alternative reaction mechanism

Kinetic oscillations in the catalytic reduction of NO by CO on a reconstructing Pt(100) surface are simulated by using a dynamic Monte Carlo method. The simulation is based on the HS model and takes into account an alternative reaction mechanism arising from recent experimental findings for the catalytic reduction of No on Rh(111), which replaces the classical N+N recombination step by the formation of a (N-NO)(*) intermediary species for the production of molecular nitrogen. A synchronized mechanism and spatiotemporal patterns are observed during the oscillations. Oscillations are analyzed in terms of the controlling parameters involved in the reaction mechanism. Different values of these parameters lead to sustained, attenuated, and modulated oscillations.

Thermodynamics of stoichiometric biochemical networks in living systems far from equilibrium

The principles of thermodynamics apply to both equilibrium and nonequilibrium biochemical systems. The mathematical machinery of the classic thermodynamics, however, mainly applies to systems in equilibrium. We introduce a thermodynamic formalism for the study of metabolic biochemical reaction (open, nonlinear) networks in both time-dependent and time-independent nonequilibrium states. Classical concepts in equilibrium thermodynamics-enthalpy, entropy, and Gibbs free energy of biochemical reaction systems-are generalized to nonequilibrium settings. Chemical motive force, heat dissipation rate, and entropy production (creation) rate, key concepts in nonequilibrium systems, are introduced. Dynamic equations for the thermodynamic quantities are presented in terms of the key observables of a biochemical network: stoichiometric matrix Q, reaction fluxes J, and chemical potentials of species mu without evoking empirical rate laws. Energy conservation and the Second Law are established for steady-state and dynamic biochemical networks. The theory provides the physiochemical basis for analyzing large-scale metabolic networks in living organisms.

Ostreolysin, a pore-forming protein from the oyster mushroom, interacts specifically with membrane cholesterol-rich lipid domains

Ostreolysin, a 15 kDa pore-forming protein from the edible oyster mushroom (Pleurotus ostreatus), is lytic to membranes containing both cholesterol and sphingomyelin. Its cytotoxicity to Chinese hamster ovary cells correlates with their cholesterol contents and with the occurrence of ostreolysin in the cells detergent resistant membranes. Moreover, ostreolysin binds to supported monolayers and efficiently permeabilizes sonicated lipid vesicles, only if cholesterol is combined with either sphingomyelin or dipalmitoylphosphatidylcholine. Addition of mono- or di-unsaturated phosphatidylcholine to the cholesterol/sphingomyelin vesicles dramatically reduces the ostreolysin's activity. It appears that the protein recognizes specifically a cholesterol-rich lipid phase, probably the liquid-ordered phase.

Experimental test of a method for determining causal connectivities of species in reactions

Theoretical analysis has shown the possibility of determining causal connectivities of reacting species and the reaction mechanism in complex chemical and biochemical reaction systems by applying pulse changes of concentrations of one or more species, of arbitrary magnitude, and measuring the temporal response of as many species as possible. This method, limited to measured and pulsed species, is given here an experimental test on a part of glycolysis including the sequence of reactions from glucose to fructose 1,6-biphosphate, followed by the bifurcation of that sequence into two branches, one ending in glycerol 3-phosphate, the other in glyceraldehyde 3-phosphate. Pulses of concentrations of one species at a time are applied to the open system in a non-equilibrium stationary state, and the temporal responses in concentrations of six metabolites are measured by capillary zone electrophoresis. From the results of these measurements and the use of the theory for their interpretation, we establish the causal connectivities of the metabolites and thus the reaction mechanism, including the bifurcation of one chain of reactions into two. In this test case of the pulse method, no prior knowledge was assumed of the biochemistry of this system. We conclude that the pulse method is relatively simple and effective in determining reaction mechanisms in complex systems, including reactants, products, intermediates, and catalysts and their effectors. The method is likely to be useful for substantially more complex systems.

Interaction of nominally soluble proteins with phospholipid monolayers at the air-water interface

The interactions of carbonmonoxyhemoglobin (HbCO), glyceraldehyde-3-phosphate dehydrogenase (GAPDH), and polyhistidine with phospholipid monolayers at the air-water interface were studied at physiological pH and ionic strength. HbCO and GAPDH both interact more strongly with monolayers containing negatively charged lipids. The interaction of HbCO and GAPDH with lipid monolayers decreases with increasing pH. Both the HbCO-monolayer and the GAPDH-monolayer interactions can be modeled as diffusion-limited processes, with kinetic data fit to a stretched exponential equation. The significance of these kinetics are discussed. Polyhistidine interacts only with monolayers containing lipids with negatively charged headgroups. In total, the results presented are consistent with an HbCO-lipid interaction with a large electrostatic component, a GAPDH-lipid interaction with comparable electrostatic and hydrophobic components, and a polyhistidine-lipid interaction that is solely electrostatic.

Determination of causal connectivities of species in reaction networks

We present an approach to the determination of causal connectivities and part of the kinetics of complex reaction systems. Our approach is based on analytical and computational methods for studying the effects of a pulse change of concentration of a chemical species in a reaction network, either at equilibrium or in a nonequilibrium stationary state. Such disturbances generally propagate through a few species, depending on the values of the kinetic coefficients, before being broadened and dissipated. This short range gives a local probe of the kinetics and connectivity of the reaction network. The range of propagation also indicates species to perturb in further experiments. From piecing together these local connectivities, the global structure of the network can be constructed. The experimental design allows deduction of both reaction orders and rate constants in many cases. An example of the usefulness of the approach is illustrated on a model of a part of glycolysis.