Dynamic disorder in single-molecule Michaelis-Menten kinetics: The reaction-diffusion formalism in the Wilemski-Fixman approximation

Particle dynamics in viscoelastic media: Effects of non-thermal white noise on barrier crossing rates

The growing interest in the dynamics of self-driven particle motion has brought increased attention to the effects of non-thermal noise on condensed phase diffusion. Thanks to data recently collected by Ferrer et al. on activated dynamics in the presence of memory [Phys. Rev. Lett. 126, 108001 (2021)], some of these effects can now be characterized quantitatively. In the present paper, the data collected by Ferrer et al. are used to calculate the extent to which non-thermal white noise alters the time taken by single micron-sized silica particles in a viscoelastic medium to cross the barrier separating the two wells of an optically created bistable potential. The calculation-based on a generalized version of Kramers's flux-over-population approach-indicates that the added noise causes the barrier crossing rate (compared to the noise-free case) to first increase as a function of the noise strength and then to plateau to a constant value. The precise degree of rate enhancement may depend on how the data from the experiments conducted by Ferrer et al. are used in the flux-over-population approach. As claimed by Ferrer et al., this approach predicts barrier crossing times for the original silica-fluid system that agree almost perfectly with their experimental counterparts. However, this near-perfect agreement between theory and experiment is only achieved if the theoretical crossing times are obtained from the most probable values of a crossing time distribution constructed from the distributions of various parameters in Kramers's rate expression. If the mean values of these parameters are used in the expression instead, as would be commonly done, the theoretical crossing times are found to be as much as 1.5 times higher than the experimental values. However, these times turn out to be consistent with an alternative model of viscoelastic barrier crossing based on a mean first passage time formalism, which also uses mean parameter values in its rate expression. The rate enhancements predicted for barrier crossing under non-thermal noise are based on these mean parameter values and are open to experimental verification.

A Revisit to Turnover Kinetics of Individual Escherichia coli β-Galactosidase Molecules

Single-molecule experiments on β-galactosidase from Escherichia coli that catalyzes the hydrolysis of resorufin-β-d-galactopyranoside revealed important observations like fluctuating catalytic rate, memory effects arising from temporal correlations between the enzymatic turnovers and nonexponential waiting time distributions. The root cause of the observed results is intrinsic fluctuations among the different conformers of the active species, during the course of the reaction, thereby imparting dynamic disorder in the system under investigation. Originally, a multistate stochastic kinetic theory was employed that, despite satisfying the measured waiting time distributions and the mean waiting times at different substrate concentrations, yields a constant estimate of the randomness parameter. Inevitably, this manifests a strong disagreement with the substrate-concentration-dependent time variations of the said distribution, which at the same time misinterprets the measured magnitudes of the randomness parameter at lower concentrations. Here, we suggest a dual approach to the single-enzyme reaction, independently, making important improvements over the parent study and the recently suggested two-state stochastic analyses followed by quantitative rationalization of the experimental data. In the first case, an off-pathway mechanism satisfied the Michaelis-Menten equation under the circumstance of prevailing disorder while tested against the single-molecule data. However, recovery of randomness data in the lower-concentration regime, albeit primarily marks a significant refinement, a qualitative agreement at the growing concentrations seems to be reasoned by an account of switching among the limited numbers of discrete conformers. Consequently, in the second case, we circumvented the conventional way of approaching the enzyme catalysis and mapped the dynamics of structural transitions of the biocatalyst with the temporal fluctuations of the spatial distance between the different locations along a coarse-grained polymer chain. Exploiting a general mechanism for dynamic disorder, a reaction-diffusion formalism yielded an analytical expression for the waiting time distribution of the enzymatic turnovers, from which the mean waiting time and the randomness parameter were readily determined. Application of our results to the findings of the experiment on single β-galactosidase shows a quantitative agreement in each case. This soundly validates the usefulness of accounting for a more rigorous microscopic description pertinent to the conformational multiplicity in rationalizing the real-time data over the routine state-based sketch of the reaction system.

Microscopic insights into dynamic disorder in the isomerization dynamics of the protein BPTI

Understanding the dynamic disorder behind a process, i.e., the dynamic effect of fluctuations that occur on a timescale slower or comparable with the timescale of the process, is essential for elucidating the dynamics and kinetics of complicated molecular processes in biomolecules and liquids. Despite numerous theoretical studies of single-molecule kinetics, our microscopic understanding of dynamic disorder remains limited. In the present study, we investigate the microscopic aspects of dynamic disorder in the isomerization dynamics of the Cys14-Cys38 disulfide bond in the protein bovine pancreatic trypsin inhibitor, which has been observed by nuclear magnetic resonance. We use a theoretical model with a stochastic transition rate coefficient, which is calculated from the 1-ms-long time molecular dynamics trajectory obtained by Shaw et al. [Science 330, 341-346 (2010)]. The isomerization dynamics are expressed by the transitions between coarse-grained states consisting of internal states, i.e., conformational sub-states. In this description, the rate for the transition from the coarse-grained states is stochastically modulated due to fluctuations between internal states. We examine the survival probability for the conformational transitions from a coarse-grained state using a theoretical model, which is a good approximation to the directly calculated survival probability. The dynamic disorder changes from a slow modulation limit to a fast modulation limit depending on the aspects of the coarse-grained states. Our analysis of the rate modulations behind the survival probability, in relation to the fluctuations between internal states, reveals the microscopic origin of dynamic disorder.

Extensions to Michaelis-Menten Kinetics for Single Parameters

Biochemical transformation kinetics is based on the formation of enzyme-substrate complexes. We developed a robust scheme based on unit productions of enzymes and reactants in cyclic events to comply with mass action law to form enzyme-substrate complexes. The developed formalism supports a successful application of Michaelis-Menten kinetics in all biochemical transformations of single parameters. It is an essential tool to overcome some challenging healthcare and environmental issues. In developing the formalism, we defined the substrate [S]= [Product]3/4 and rate of reaction based on rate and time perspectives. It allowed us to develop two quadratic equations. The first, represents a body entity that gave a useful relationship of enzyme E = 2S0.33, and the second nutrients/feed, each giving [Enzymes] and [Enzyme-substrate complexes], simulating rate of reaction, [substrate], and their differentials. By combining [Enzymes] and [Enzyme-substrate complexes] values, this quadratic equation derives a Michaelis-Menten hyperbolic function. Interestingly, we can derive the proportionate rate of reaction and [Enzymes] values of the quadratics resulting in another Michaelis-Menten hyperbolic. What is clear from these results is that between these two hyperbolic functions, in-competitive inhibitions exist, indicating metabolic activities and growth in terms of energy levels. We validated these biochemical transformations with examples applicable to day to day life.

Dynamic disorder in quasi-equilibrium enzymatic systems

Conformations and catalytic rates of enzymes fluctuate over a wide range of timescales. Despite these fluctuations, there exist some limiting cases in which the enzymatic catalytic rate follows the macroscopic rate equation such as the Michaelis-Menten law. In this paper we investigate the applicability of macroscopic rate laws for fluctuating enzyme systems in which catalytic transitions are slower than ligand binding-dissociation reactions. In this quasi-equilibrium limit, for an arbitrary reaction scheme we show that the catalytic rate has the same dependence on ligand concentrations as obtained from mass-action kinetics even in the presence of slow conformational fluctuations. These results indicate that the timescale of conformational dynamics--no matter how slow--will not affect the enzymatic rate in quasi-equilibrium limit. Our numerical results for two enzyme-catalyzed reaction schemes involving multiple substrates and inhibitors further support our general theory.

Dynamic disorder-driven substrate inhibition and bistability in a simple enzymatic reaction

Conformations and catalytic rates of enzymes (biological catalysts) fluctuate over a wide range of time scales. Recent experimental and theoretical investigations demonstrated case studies where the enzymatic catalysis rate follows the Michaelis-Menten (MM) rate law despite molecular fluctuations. In this paper, we investigate deviations from MM law and their effects on the dynamical behavior of the enzymatic network. We consider a simple kinetic scheme for a single substrate enzymatic reaction in which the product release step is treated explicitly. We examine how conformational fluctuations affect the underlying rate law in the quasi-static limit when conformational dynamics is very slow in one of the states. Our numerical results and analytically solvable model indicate that slow conformational fluctuations of the enzyme-substrate complex lead to non-MM behavior, substrate inhibition, and possible bistability of the reaction network.