Ji-Hyun Kim

Contribution of RNA polymerase concentration variation to protein expression noise.

Cell-to-cell variation in gene expression, or noise, is a general phenomenon observed within cell populations. Transcription is known to be the key stage of gene expression where noise is generated, however, how variation in RNA polymerase (RNAP) concentration contributes to gene expression noise is unclear. Here, we quantitatively investigate how variations in absolute amounts of RNAP molecules affect noise in the expression of two fluorescent protein reporters driven by identical promoters. We find that intrinsic noise is independent of variation in RNAP concentrations, whereas extrinsic noise, which is variation in gene expression due to varying cellular environments, scales linearly with variation in RNAP abundance. Specifically, the propagation of RNAP abundance variation to expressed protein noise is inversely proportional to the concentration of RNAP, which suggests that the change in noise that results from RNAP fluctuations is determined by the fraction of promoters that is not occupied by RNAP.

Diffusion-influenced reactions involving a reactant with two active sites

We consider the kinetics of diffusion-influenced reactions which involve a reactant species that can be modeled as a sphere with two reactive patches located on its surface at an arbitrary angular distance. An approximate analytic expression for the rate coefficient is derived based on the Wilemski-Fixman-Weiss decoupling approximation and a multivariable Padé approximation. The accuracy of the rate expression is evaluated against computer simulations as well as an exact analytic expression available for a special case. The present theory provides accurate estimates for the magnitude of diffusive interference effects between the two reactive patches. We also present an efficient Brownian dynamics method for calculating the time-dependent rate coefficient, which is applicable when the reactants involve multiple active sites.

A rigorous foundation of the diffusion-influenced bimolecular reaction kinetics.

We formulate a general theory of the diffusion-influenced kinetics of irreversible bimolecular reactions occurring in the low concentration limit. Starting from the classical Liouville equation for the reactants and explicit solvent molecules, a formally exact expression for the bimolecular reaction rate coefficient is derived; the structures of reactant molecules and the sink functions may be arbitrarily complicated. The present theoretical formulation shows clearly how the well-known Noyes and Wilemski-Fixman rate theories are related and can be improved in a systematic manner. The general properties of the rate coefficient such as the long-time behavior and the upper and the lower bounds are analyzed. When the reaction can occur at a range of distance, the non-Markovianity of repeated encounter events between a reactant pair becomes significant and either the Noyes theory or the Wilemski-Fixman theory fails. The present theory provides a practical method for calculating the rate expression for such reactions, which improves significantly on the Wilemski-Fixman theory.

Excluded Volume Effects on the Intrachain Reaction Kinetics

On the basis of the recently developed optimized Rouse-Zimm theory of chain polymers with excluded volume interactions, we calculate the long-time first-order rate constant k(1) for end-to-end cyclization of linear chain polymers. We first find that the optimized Rouse-Zimm theory provides the longest chain relaxation times tau(1) of excluded volume chains that are in excellent agreement with the available Brownian dynamics simulation results. In the free-draining limit, the cyclization rate is diffusion-controlled and k(1) is inversely proportional to tau(1), and the k(1) values calculated using the Wilemski-Fixman rate theory are in good agreement with Brownian dynamics simulation results. However, when hydrodynamic interactions are included, noticeable deviations are found. The main sources of errors are fluctuating hydrodynamic interaction and correlation hole effects as well as the non-Markovian reaction dynamic effect. The physical natures of these factors are discussed, and estimates for the magnitudes of required corrections are given. When the corrections are included, the present theory allows the prediction of accurate k(1) values for the cyclization of finite-length chains in good solvents as well as the correct scaling exponent in the long-chain limit.

The optimized Rouse–Zimm theory of excluded volume effects on chain dynamics

Based on the optimized Rouse-Zimm (ORZ) approximation to the Kirkwood diffusion equation, we investigate the effects of excluded volume interactions on the single chain dynamics. By incorporating the nonuniformly expanded moments of interbead distances into the expressions for the diffusion and structure matrices appearing in the ORZ diffusion equation, we obtain the general relaxation spectrum for flexible chains that is valid over the range from theta; solvents to good solvents. The present theory involves four parameters: the Kuhn statistical length b(0), the bead number N, the excluded volume parameter z, and the hydrodynamic interaction parameter h(*). These model parameters are determined from structural data of polymers with the aid of the quasi-two-parameter theory. The set of relaxation times of ORZ normal modes calculated with these bead-and-spring model parameters enables the theoretical prediction of various frictional and dynamical properties of polymers within a unified framework. The present ORZ theory generalizes the Ptitsyn-Eizner-type approaches by incorporating the nonuniform chain expansion effect into the structure matrix as well as the diffusion matrix.

An accurate expression for the rates of diffusion-influenced bimolecular reactions with long-range reactivity

By using the recently developed method for solving the Fredholm integral equations of the second kind, we derive a very accurate expression for the steady-state rate constant of diffusion-influenced bimolecular reactions involving long-range reactivity. We consider the general case in which the reactants interact via an arbitrary central potential and hydrodynamic interaction. The rate expression becomes exact in the two opposite limits of small and large reactivity, and also performs very well in the intermediate regime.

Internal Diffusion-Controlled Enzyme Reaction: The Acetylcholinesterase Kinetics.

Acetylcholinesterase is an enzyme with a very high turnover rate; it quenches the neurotransmitter, acetylcholine, at the synapse. We have investigated the kinetics of the enzyme reaction by calculating the diffusion rate of the substrate molecule along an active site channel inside the enzyme from atomic-level molecular dynamics simulations. In contrast to the previous works, we have found that the internal substrate diffusion is the determinant of the acetylcholinesterase kinetics in the low substrate concentration limit. Our estimate of the overall bimolecular reaction rate constant for the enzyme is in good agreement with the experimental data. In addition, the present calculation provides a reasonable explanation for the effects of the ionic strength of solution and the mutation of surface residues of the enzyme. The study suggests that internal diffusion of the substrate could be a key factor in understanding the kinetics of enzymes of similar characteristics.

Comment on “Nonrenewal Statistics in the Catalytic Activity of Enzyme Molecules at Mesoscopic Concentrations”

It is well known in enzyme kinetics that the Michaelis-Menten (MM) equation is applicable only to enzymes in the steady state. We show that the result obtained in the previous work [Phys. Rev. Lett. 107, 218301 (2011)] is inconsistent with the MM equation, not because the authors considered the enzyme system at mesoscopic concentrations but because they considered the enzyme system in the non-stationary state. The substrate concentration dependence of the mean turnover time is, in fact, consistent with the MM equation in the steady state, regardless of the number of enzymes in the system.

Nonclassical Kinetics of Clonal yet Heterogeneous Enzymes

Enzyme-to-enzyme variation in the catalytic rate is ubiquitous among single enzymes created from the same genetic information, which persists over the lifetimes of living cells. Despite advances in single-enzyme technologies, the lack of an enzyme reaction model accounting for the heterogeneous activity of single enzymes has hindered a quantitative understanding of the nonclassical stochastic outcome of single enzyme systems. Here we present a new statistical kinetics and exactly solvable models for clonal yet heterogeneous enzymes with possibly nonergodic state dynamics and state-dependent reactivity, which enable a quantitative understanding of modern single-enzyme experimental results for the mean and fluctuation in the number of product molecules created by single enzymes. We also propose a new experimental measure of the heterogeneity and nonergodicity for a system of enzymes.

Effects of chain stiffness on the quenching of an excited polymer by small quenchers

We present a theory for analyzing the effects of chain stiffness on the diffusion-influenced quenching kinetics of an excited polymer. We model the polymer as an optimized Rouse–Zimm chain and the quencher molecule as a spherical particle. The excitation is considered to be localized at any one monomer, or to move randomly along the chain backbone. In regard to the dependence on the chain stiffness, we found two distinctive kinetic regimes. When the excitation migration rate is small, the quenching rate decreases as the chain becomes stiffer. On the other hand, when the mobility of excitation is large, the opposite trend is observed. We also investigate the dependence of Stern–Volmer coefficient on the length and stiffness of the chain in the fast excitation migration limit.