Theory on the rate equation of Michaelis–Menten type single-substrate enzyme catalyzed reactions

Analytical solution to the Michaelis–Menten (MM) rate equations for single-substrate enzyme catalysed reaction is not known. Here we introduce an effective scaling scheme and identify the critical parameters which can completely characterize the entire dynamics of single substrate MM enzymes. Using this scaling framework, we reformulate the differential rate equations of MM enzymes over velocity-substrate, velocity-product, substrate-product and velocity-substrate-product spaces and obtain various approximations for both pre- and post-steady state dynamical regimes. Using this framework, under certain limiting conditions we successfully compute the timescales corresponding to steady state, pre- and post-steady states and also compute the approximate steady state values of velocity, substrate and product. We further define the dynamical efficiency of MM enzymes as the ratio between the reaction path length in the velocity-substrate-product space and the average reaction time required to convert the entire substrate into product. Here dynamical efficiency characterizes the phase-space dynamics and it would tell us how fast an enzyme can clear a harmful substrate from the environment. We finally perform a detailed error level analysis over various pre- and post-steady state approximations along with the already existing quasi steady state approximations and progress curve models and discuss the positive and negative points corresponding to various steady state and progress curve models.     

Discrete state model and accurate estimation of loop entropy of RNA secondary structures

Conformational entropy makes important contribution to the stability and folding of RNA molecule, but it is challenging to either measure or compute conformational entropy associated with long loops. We develop optimized discrete k-state models of RNA backbone based on known RNA structures for computing entropy of loops, which are modeled as self-avoiding walks. To estimate entropy of hairpin, bulge, internal loop, and multibranch loop of long length (up to 50), we develop an efficient sampling method based on the sequential Monte Carlo principle. Our method considers excluded volume effect. It is general and can be applied to calculating entropy of loops with longer length and arbitrary complexity. For loops of short length, our results are in good agreement with a recent theoretical model and experimental measurement. For long loops, our estimated entropy of hairpin loops is in excellent agreement with the Jacobson-Stockmayer extrapolation model. However, for bulge loops and more complex secondary structures such as internal and multibranch loops, we find that the Jacobson-Stockmayer extrapolation model has large errors. Based on estimated entropy, we have developed empirical formulae for accurate calculation of entropy of long loops in different secondary structures. Our study on the effect of asymmetric size of loops suggest that loop entropy of internal loops is largely determined by the total loop length, and is only marginally affected by the asymmetric size of the two loops. Our finding suggests that the significant asymmetric effects of loop length in internal loops measured by experiments are likely to be partially enthalpic. Our method can be applied to develop improved energy parameters important for studying RNA stability and folding, and for predicting RNA secondary and tertiary structures. The discrete model and the program used to calculate loop entropy can be downloaded at http://gila.bioengr.uic.edu/resources/RNA.html.     

Rate‐Reactivity Model: A New Theoretical Basis for Systematic Kinetic Characterization of Heterogeneous Catalysts

Non‐steady‐state kinetic measurements contain a wealth of information about catalytic reactions and other gas–solid chemical interactions, which is extracted from experimental data via kinetic models. The standard mathematical framework of microkinetic models, which are typically used in computational catalysis and for advanced modeling of steady‐state data, encounters multiple challenges when applied to non‐steady‐state data. Robust phenomenological models, such as the steady‐state Langmuir–Hinshelwood–Hougen–Watson equations, are presently unavailable for non‐steady‐state data. Herein, a novel modeling framework is proposed to fulfill this need. The rate‐reactivity model (RRM) is formulated in terms of experimentally observable quantities including the gaseous transformation rates, concentrations, and surface uptakes. The model is linear with respect to these quantities and their pairwise products, and it is also linear in terms of its parameters (reactivities). The RRM parameters have a clear physicochemical meaning and fully characterize the kinetic behavior of a specific catalyst state, but unlike microkinetic models that rely on hypothetical surface intermediates and specific reaction networks, the RRM does not require any assumptions regarding the underlying mechanism. The systematic RRM‐based procedure outlined in this paper enables an effective comparison of various catalysts and the construction of more detailed microkinetic models in a rational manner. The model was applied to temporal analysis of products pulse‐response data as an example, but it is more generally applicable to other non‐steady‐state techniques that provide time‐resolved rates and concentrations. Several numerical examples are given to illustrate the application of the model to simple model reactions.     

End-to-end vs interior loop formation kinetics in unfolded polypeptide chains

The conformational search for favorable intramolecular interactions during protein folding is limited by intrachain diffusion processes. Recent studies on the dynamics of loop formation in unfolded polypeptide chains have focused on loops involving residues near the chain ends. During protein folding, however, most contacts are formed between residues in the interior of the chain. We compared the kinetics of end-to-end loop formation (type I loops) to the formation of end-to-interior (type II loops) and interior-to-interior loops (type III loops) using triplet-triplet energy transfer from xanthone to naphthylalanine. The results show that formation of type II and type III loops is slower compared to type I loops of the same size and amino acid sequence. The rate constant for type II loop formation decreases with increasing overall chain dimensions up to a limiting value, at which loop formation is about 2.5-fold slower for type II loops compared to type I loops. Comparing type II loops of different loop size and amino acid sequence shows that the ratio of loop dimension over total chain dimension determines the rate constant for loop formation. Formation of type III loops is 1.7-fold slower than formation of type II loops, indicating that local chain motions are strongly coupled to motions of other chain segments which leads to faster dynamics toward the chain ends. Our results show that differences in the kinetics of formation of type I, type II, and type III loops are mainly caused by differences in internal flexibility at the different positions in the polypeptide chain. Interactions of the polypeptide chain with the solvent contribute to the kinetics of loop formation, which are strongly viscosity-dependent. However, the observed differences in the kinetics of formation of type I, type II, and type III loops are not due to the increased number of peptide-solvent interactions in type II and type III loops compared to type I loops as indicated by identical viscosity dependencies for the kinetics of formation of the different types of loops.     

Closing the side-chain gap in protein loop modeling

The success of structure-based drug design relies on accurate protein modeling where one of the key issues is the modeling and refinement of loops. This study takes a critical look at modeled loops, determining the effect of re-sampling side-chains after the loop conformation has been generated. The results are evaluated in terms of backbone and side-chain conformations with respect to the native loop. While models can contain loops with high quality backbone conformations, the side-chain orientations could be poor, and therefore unsuitable for ligand docking and structure-based design. In this study, we report on the ability to model loop side-chains accurately using a variety of commercially available algorithms that include rotamer libraries, systematic torsion scans and knowledge-based methods. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

Steady-state fluctuations of a genetic feedback loop: an exact solution

Genetic feedback loops in cells break detailed balance and involve bimolecular reactions; hence, exact solutions revealing the nature of the stochastic fluctuations in these loops are lacking. We here consider the master equation for a gene regulatory feedback loop: a gene produces protein which then binds to the promoter of the same gene and regulates its expression. The protein degrades in its free and bound forms. This network breaks detailed balance and involves a single bimolecular reaction step. We provide an exact solution of the steady-state master equation for arbitrary values of the parameters, and present simplified solutions for a number of special cases. The full parametric dependence of the analytical non-equilibrium steady-state probability distribution is verified by direct numerical solution of the master equations. For the case where the degradation rate of bound and free protein is the same, our solution is at variance with a previous claim of an exact solution [J. E. M. Hornos, D. Schultz, G. C. P. Innocentini, J. Wang, A. M. Walczak, J. N. Onuchic, and P. G. Wolynes, Phys. Rev. E 72, 051907 (2005), and subsequent studies]. We show explicitly that this is due to an unphysical formulation of the underlying master equation in those studies.     

Phase-death mode in two-coupled chemical oscillators studied with reactors of different volume and by simulation

The behavior of the phase-death mode (cessation of oscillations and transition to a steady state) in the two-coupled Belousov–Zhabotinsky reactions is studied in the asymmetric coupling. In an experiment, the types of synchronized modes and their regions were investigated as an extension of our previous study. This experiment is compared to a simulation using the two-coupled three-variable Oregonator models. The results confirm that the phase-death mode changes from “bistable” to “monostable” and reveal the stable region of the latter.     

Experimental study of embedding of attractors in concentration space

The embedding of attractors and their stable and unstable manifolds can be studied experimentally by controlled addition of chemical species to bring about a particular response. For stable small amplitude oscillations near a Hopf bifurcation from a steady state the embedding can be completely determined even in systems where two of the species are not observable. A quenching of the oscillations by dilution candetermine the steady state concentrations. A species that cannot quench the oscillations almost certainly cannot be an essential component of the oscillation. The method can be extended to a study of attractor associated with subharmonic and quasiperiodic bifurcations and of attractors corresponding to nonperiodic motion. We present preliminary results for a subharmonic bifurcation.     

Nonlinear (Fluctuational) dynamics and mathematical modeling of homogenous oxidation of biological substrates

Nonlinear dynamics and mathematical modeling approaches were presented for studying chemical oscillations during homogeneous oxidation of biological substrates in the presence of oxygenated complexes of iron(II). It was shown that examination of fluctuational dynamics is only possible via consistent use of several algorithms for time series processing and that flicker-noise spectroscopic technique is best suited to deriving the most exhaustive information from the patterns of temporal variation of signals. The qualitative analysis was performed, and numerical solution was found for the system of differential equations modeling the reactions kinetics. The nature of the steady state was elucidated, and the possibility of Andronov-Hopf type bifurcation with realization of an oscillatory mode was demonstrated.     

Experimental strategy for characterization of essential dynamical variables in oscillatory systems: effect of double-layer capacitance on the stability of electrochemical oscillators

An experimentally accessible algorithm for changing the time scale associated with a dynamical variable is proposed. In general, a differential controller can be applied to (a) identify the essential species in oscillatory systems and (b) explore their role in the feedback loops. Here, we report on classifying electrochemical oscillators by changing the time scale over which the electrode potential varies; the type of different electrochemical oscillators is identified based on whether the controlled modification of pseudo-capacitance induces or suppresses current oscillations.     

Specific Features of a Two-Phase Bimolecular Chemical Reaction in the Case of the Association of Both Reactants

The effect of the association of both reactants on the kinetics of their bimolecular reaction in the liquid phase is studied. The mathematical modeling of chemical reactions that are described by nonlinear differential equations is performed. The steady states, the conditions for the emergences of intermediates, and the nature of their concentration oscillations in the reaction system are described. It is found that the concentration of the intermediates has two types of oscillations (harmonic and relaxation oscillations) characterized by significantly different times. The relationship between the observed rate constant of the process, the rate constants for the elementary stages, and the reactant concentrations is found.     

The Model of “Minimal Oscillator” Derived from the Kinetic Mechanism of the H2O2–NaSCN–NaOH–CuSO4 Dynamical System

Dissipative chemical reactions, which involve oscillatory variations of the concentrations of the intermediates in time, are usually characterized with complicated kinetic mechanisms. However, the essential source of the oscillations can often be reduced to only a few reaction steps providing the alternative domination of the positive and negative feedback loops. In an extreme case such a reduction leads to the so–called “minimal oscillator,” the concept used in the past for the well‐known Belousov‐Zhabotinsky (BZ) reaction. In the present work, we construct such a minimal system for the (discovered by M. Orbán) H₂O₂–NaSCN–NaOH–CuSO₄ homogeneous oscillator, in which instabilities originate from kinetic mechanism substantially different from that proposed for the BZ system. The methodology involves intuitive analysis of the reaction mechanism, supported by numerical calculations and spectrophotometric measurements. We show how the actual, only three‐variable model evolves from our previously elaborated: nine‐ and five–variable mechanisms and prove that its further reduction to two–variable one is not possible. Thus the present work is a final step in our searches for the “minimal Orbán oscillator”.     

微圆盘电极上耦合一级均相反应的稳态电流

在电极表面反应物浓度均匀的近似假设下, 得到了微圆盘电极上暂态可逆反应的一般解. 然后利用该一般解得到了微圆盘电极上耦合一级均相反应的可逆电极反应CE、EC、EC′和ECE 的稳态电流计算公式.     

Steady flow generation in microcirculatory systems

In this paper we explore the mechanical generation of steady-non pulsatile-flow in microfluidic systems. The rationale of the paper is inspired in the example of cardiovascular systems where at the microscale (i.e. capillaries) the flow is steady rather than pulsatile to optimize performance. We present a solution to the generation of steady flow in engineered microfluidic systems either in open or closed loop configurations via the use of disc pumps. The disc pump consists of a flat rotating disc and utilizes both viscous drag and centrifugal force to achieve pumping. Experiments using single loop and double loop microfluidic systems are presented to characterize the disc pump. Continuous flow generated by the disc pumps can be used to separate particles based on size using recirculating loops and for extraction of small particles without disturbing the concentration of bigger particles. The potential impact of this technology includes sample separation and extraction techniques into portable microfluidic labs-on-a-chip, and long term culture systems for cells in suspension.     

Three-residue loop closure in proteins: a new kinematic method reveals a locus of connected loop conformations

The closure of a three-residue loop was studied using a developed kinematic method. It was shown that there are infinite number of three-residue loops (a locus of conformations), which can connect two segments of a polypeptide. This adds to the current understanding of a finite number of conformations for three-residue loop-closure. In the developed method, some of the equations can be solved analytically to reduce the computation cost. Benefiting from the reduced computation time, we determined all the relative positions of two polypeptide segments that can be connected by a three-residue loop.     

Insight into G-DNA structural polymorphism and folding from sequence and loop connectivity through free energy analysis

The lengths of G-tracts and their connecting loop sequences determine G-quadruplex folding and stability. Complete understanding of the sequence-structure relationships remains elusive. Here, single-loop G-quadruplexes were investigated using explicit solvent molecular dynamics (MD) simulations to characterize the effect of loop length, loop sequence, and G-tract length on the folding topologies and stability of G-quadruplexes. Eight loop types, including different variants of lateral, diagonal, and propeller loops, and six different loop sequences [d0 (i.e., no intervening residues in the loop), dT, dT(2), dT(3), dTTA, and dT(4)] were considered through MD simulation and free energy analysis. In most cases the free energetic estimates agree well with the experimental observations. The work also provides new insight into G-quadruplex folding and stability. This includes reporting the observed instability of the left propeller loop, which extends the rules for G-quadruplex folding. We also suggest a plausible explanation why human telomere sequences predominantly form hybrid-I and hybrid-II type structures in K(+) solution. Overall, our calculation results indicate that short loops generally are less stable than longer loops, and we hypothesize that the extreme stability of sequences with very short loops could possibly derive from the formation of parallel multimers. The results suggest that free energy differences, estimated from MD and free energy analysis with current force fields and simulation protocols, are able to complement experiment and to help dissect and explain loop sequence, loop length, and G-tract length and orientation influences on G-quadruplex structure.     

Developing Itô stochastic differential equation models for neuronal signal transduction pathways

Mathematical modeling and simulation of dynamic biochemical systems are receiving considerable attention due to the increasing availability of experimental knowledge of complex intracellular functions. In addition to deterministic approaches, several stochastic approaches have been developed for simulating the time-series behavior of biochemical systems. The problem with stochastic approaches, however, is the larger computational time compared to deterministic approaches. It is therefore necessary to study alternative ways to incorporate stochasticity and to seek approaches that reduce the computational time needed for simulations, yet preserve the characteristic behavior of the system in question. In this work, we develop a computational framework based on the It? stochastic differential equations for neuronal signal transduction networks. There are several different ways to incorporate stochasticity into deterministic differential equation models and to obtain It? stochastic differential equations. Two of the developed models are found most suitable for stochastic modeling of neuronal signal transduction. The best models give stable responses which means that the variances of the responses with time are not increasing and negative concentrations are avoided. We also make a comparative analysis of different kinds of stochastic approaches, that is the It? stochastic differential equations, the chemical Langevin equation, and the Gillespie stochastic simulation algorithm. Different kinds of stochastic approaches can be used to produce similar responses for the neuronal protein kinase C signal transduction pathway. The fine details of the responses vary slightly, depending on the approach and the parameter values. However, when simulating great numbers of chemical species, the Gillespie algorithm is computationally several orders of magnitude slower than the It? stochastic differential equations and the chemical Langevin equation. Furthermore, the chemical Langevin equation produces negative concentrations. The It? stochastic differential equations developed in this work are shown to overcome the problem of obtaining negative values.     

Introduction to Chemical Oscillations Using a Modified Lotka Model

In teaching chemical kinetics most textbooks use the Lotka-Volterra Model to introduce the concept of chemical oscillations. Unfortunately, the Lotka-Volterra Model yields neutrally stable limit cycles for any initial conditions, which is a nonphysical property not observed in chemical kinetics. A more physical, two-variable model with simple linear stability analysis is, therefore, desirable. Here, we consider a Modified Lotka-Volterra Model that shows multiple physical steady states and both damped and stable oscillations. We can also study a stable node bifurcation to a saddle point and a stable node bifurcation to a stable limit cycle. This dynamically richer model can be analyzed through a simple linear stability analysis and numerical integration of the system of ordinary differential equations. Both methods, in particular the analytical analysis, are accessible to undergraduate students.     

Loop-length-dependent folding of G-quadruplexes

Guanine-rich DNA sequences can form a large number of structurally diverse quadruplexes. These vary in terms of strand polarity, loop composition, and conformation. We have derived guidelines for understanding the influence of loop length on the structure adopted by intramolecular quadruplex-forming sequences, using a combination of experimental (using CD and UV melting data) and molecular modeling and simulation techniques. We find that a parallel-stranded intramolecular quadruplex structure is the only possible fold when three single residue loops are present. When single thymine loops are present in combination with longer length loops, or when all loops are longer than two residues, both parallel- and antiparallel-folded structures are able to form. Multiple conformations of each structure are likely to coexist in solution, as they were calculated to have very similar free energies.