Understanding the Reaction Dynamics on Heterogeneous Catalysts Using a Simple Stochastic Approach.

Recent experimental advances on investigating nanoparticle catalysts with multiple active sites provided a large amount of quantitative information on catalytic processes. These observations stimulated significant theoretical efforts, but the underlying molecular mechanisms are still not well-understood. We introduce a simple theoretical method to analyze the reaction dynamics on catalysts with multiple active sites based on a discrete-state stochastic description and obtain a comprehensive description of the dynamics of chemical reactions on such catalysts. We explicitly determine how the dynamics of catalyzed chemical reactions depend on the number of active sites, on the number of intermediate chemical transitions, and on the topology of underlying chemical reactions. It is argued that the theory provides quantitative bounds for realistic dynamic properties of catalytic processes that can be directly applied to analyze the experimental observations. In addition, this theoretical approach clarifies several important aspects of the molecular mechanisms of chemical reactions on catalysts.

Different time scales in dynamic systems with multiple outcomes.

Stochastic biochemical and transport processes have various final outcomes, and they can be viewed as dynamic systems with multiple exits. Many current theoretical studies, however, typically consider only a single time scale for each specific outcome, effectively corresponding to a single-exit process and assuming the independence of each exit process. However, the presence of other exits influences the statistical properties and dynamics measured at any specific exit. Here, we present theoretical arguments to explicitly show the existence of different time scales, such as mean exit times and inverse exit fluxes, for dynamic processes with multiple exits. This implies that the statistics of any specific exit dynamics cannot be considered without taking into account the presence of other exits. Several illustrative examples are described in detail using analytical calculations, mean-field estimates, and kinetic Monte Carlo computer simulations. The underlying microscopic mechanisms for the existence of different time scales are discussed. The results are relevant for understanding the mechanisms of various biological, chemical, and industrial processes, including transport through channels and pores.

Protein--DNA interactions: reaching and recognizing the targets.

Protein searching and recognizing the targets on DNA was the subject of many experimental and theoretical studies. It is often argued that some proteins are capable of finding their targets 10-100 times faster than predicted by the three-dimensional diffusion rate. However, recent single-molecule experiments showed that the diffusion constants of the protein motion along DNA are usually small. This controversy pushed us to revisit this problem. We present a theoretical approach that describes some physical-chemical aspects of the target search and recognition. We consider the search process as a sequence of cycles, with each cycle consisting of three-dimensional and one-dimensional tracks. It is argued that the search time contains three terms: for the motion on three-dimensional and one-dimensional segments, and the correlation term. Our analysis shows that the acceleration in the search time is achieved at some intermediate strength of the protein-DNA binding energy and it is partially "apparent" because it is in fact reached by parallel scanning for the target by many proteins. We also show how the complementarity of the charge patterns on a target DNA sequence and on the protein may result in electrostatic recognition of a specific track on DNA and subsequent protein pinning. Within the scope of a model, we obtain an analytical expression for the capturing well. We estimate the depth and width of such a potential well and the typical time that a protein spends in it.

Localization of shocks in driven diffusive systems without particle number conservation.

We study the formation of localized shocks in one-dimensional driven diffusive systems with spatially homogeneous creation and annihilation of particles (Langmuir kinetics). We show how to obtain hydrodynamic equations that describe the density profile in systems with uncorrelated steady state as well as in those exhibiting correlations. As a special example of the latter case, the Katz-Lebowitz-Spohn model is considered. The existence of a localized double density shock is demonstrated in one-dimensional driven diffusive systems. This corresponds to phase separation into regimes of three distinct densities, separated by localized domain walls. Our analytical approach is supported by Monte Carlo simulations.

Simple mechanochemistry describes the dynamics of kinesin molecules.

Recently, Block and coworkers [Visscher, K., Schnitzer, M. J., & Block, S. M. (1999) Nature (London) 400, 184--189 and Schnitzer, M. J., Visscher, K. & Block, S. M. (2000) Nat. Cell Biol. 2, 718--723] have reported extensive observations of individual kinesin molecules moving along microtubules in vitro under controlled loads, F = 1 to 8 pN, with [ATP] = 1 microM to 2 mM. Their measurements of velocity, V, randomness, r, stalling force, and mean run length, L, reveal a need for improved theoretical understanding. We show, presenting explicit formulae that provide a quantitative basis for comparing distinct molecular motors, that their data are satisfactorily described by simple, discrete-state, sequential stochastic models. The simplest (N = 2)-state model with fixed load-distribution factors and kinetic rate constants concordant with stopped-flow experiments, accounts for the global (V, F, L, [ATP]) interdependence and, further, matches relative acceleration observed under assisting loads. The randomness, r(F,[ATP]), is accounted for by a waiting-time distribution, psi(1)(+)(t), [for the transition(s) following ATP binding] with a width parameter nu identical with (2)/<(Delta t)(2)> approximately 2.5, indicative of a dispersive stroke of mechanicity approximately 0.6 or of a few ( greater than or similar to nu - 1) further, kinetically coupled states: indeed, N = 4 (but not N = 3) models do well. The analysis reveals: (i) a substep of d(0) = 1.8--2.1 nm on ATP binding (consistent with structurally based suggestions); (ii) comparable load dependence for ATP binding and unbinding; (iii) a strong load dependence for reverse hydrolysis and subsequent reverse rates; and (iv) a large ( greater than or similar to 50-fold) increase in detachment rate, with a marked load dependence, following ATP binding.

Force-velocity relation for growing microtubules.

Forces generated by microtubule polymerization and depolymerization are important for the biological functioning of cells. The mean growth velocity, V, under an opposing force, F, has been measured by; Science 278:856-860) for single microtubules growing in vitro, but their analysis of the data suggested that V decreased more rapidly with F than equilibrium (or "thermodynamic") theory predicted and entailed negative values for the dissociation rate and undefined (or unreasonable) values for the stall force, at which V vanishes. By contrast, considering the mean work done against the external load and allowing for load-distribution factors for the "on" and "off" rates, we find good agreement with a simple theory that yields a plausible stalling force. Although specific numerical results are sensitive to choice of fitting criteria, about 80% of the variation with load is carried by the "off" (or dissociation) rate, but, since that is small (in accordance with independent observations), the dominant force dependence comes from the "on" rate, which is associated with a displacement length, d(1), significantly longer than d(0) approximately 1/13(8.2 nm), the mean length increase per added tubulin dimer. Measuring the dispersion in length of the growing microtubules could provide a check. The theory implies that the stationary stall state (at V = 0) is not one of simple associative thermal equilibrium, as previously supposed; rather, it appears to be dissipative and kinetically controlled.

The force exerted by a molecular motor.

The stochastic driving force exerted by a single molecular motor (e. g., a kinesin, or myosin) moving on a periodic molecular track (microtubule, actin filament, etc.) is discussed from a general viewpoint open to experimental test. An elementary "barometric" relation for the driving force is introduced that (i) applies to a range of kinetic and stochastic models, (ii) is consistent with more elaborate expressions entailing explicit representations of externally applied loads, and (iii) sufficiently close to thermal equlibrium, satisfies an Einstein-type relation in terms of the velocity and diffusion coefficient of the (load-free) motor. Even in the simplest two-state models, the velocity-vs.-load plots exhibit a variety of contrasting shapes (including nonmonotonic behavior). Previously suggested bounds on the driving force are shown to be inapplicable in general by analyzing discrete jump models with waiting time distributions.