Theory of the diffusion-influenced substrate binding rate to a buried and gated active site

Diffusion control in biochemical specificity

Biochemical specificity is critical in enzyme function, evolution, and engineering. Here we employ an established kinetic model to dissect the effects of reactant geometry and diffusion on product formation speed and accuracy in the presence of cognate (correct) and near-cognate (incorrect) substrates. Using this steady-state model for spherical geometries, we find that, for distinct kinetic regimes, the speed and accuracy of the reactions are optimized on different regions of the geometric landscape. From this model we deduce that accuracy can be strongly dependent on reactant geometric properties even for chemically limited reactions. Notably, substrates with a specific geometry and reactivity can be discriminated by the enzyme with higher efficacy than others through purely diffusive effects. For similar cognate and near-cognate substrate geometries (as is the case for polymerases or the ribosome), we observe that speed and accuracy are maximized in opposing regions of the geometric landscape. We also show that, in relevant environments, diffusive effects on accuracy can be substantial even far from extreme kinetic conditions. Finally, we find how reactant chemical discrimination and diffusion can be related to simultaneously optimize steady-state flux and accuracy. These results highlight how diffusion and geometry can be employed to enhance reaction speed and discrimination, and similarly how they impose fundamental restraints on these quantities.

Multiscale Approach for Computing Gated Ligand Binding from Molecular Dynamics and Brownian Dynamics Simulations

We develop an approach to characterize the effects of gating by a multiconformation protein consisting of macrostate conformations that are either accessible or inaccessible to ligand binding. We first construct a Markov state model of the apo-protein from atomistic molecular dynamics simulations from which we identify macrostates and their conformations, compute their relative macrostate populations and interchange kinetics, and structurally characterize them in terms of ligand accessibility. We insert the calculated first-order rate constants for conformational transitions into a multistate gating theory from which we derive a gating factor γ that quantifies the degree of conformational gating. Applied to HIV-1 protease, our approach yields a kinetic network of three accessible (semi-open, open, and wide-open) and two inaccessible (closed and a newly identified, "parted") macrostate conformations. The parted conformation sterically partitions the active site, suggesting a possible role in product release. We find that the binding kinetics of drugs and drug-like inhibitors to HIV-1 protease falls in the slow gating regime. However, because γ = 0.75, conformational gating only modestly slows ligand binding. Brownian dynamics simulations of the diffusional association of eight inhibitors to the protease─having a wide range of experimental association constants (∼10⁴-10¹⁰ M^-1 s^-1)─yields gated rate constants in the range of ∼0.5-5.7 × 10⁸ M^-1 s^-1. This indicates that, whereas the association rate of some inhibitors could be described by the model, for many inhibitors either subsequent conformational transitions or alternate binding mechanisms may be rate-limiting. For systems known to be modulated by conformational gating, the approach could be scaled computationally efficiently to screen association kinetics for a large number of ligands.

Trapping of particles diffusing in two dimensions by a hidden binding site

We study trapping of particles diffusing in a two-dimensional rectangular chamber by a binding site located at the end of a rectangular sleeve. To reach the site a particle first has to enter the sleeve. After that it has two options: to come back to the chamber or to diffuse to the site where it is trapped. The main result of the present work is a simple expression for the mean particle lifetime as a function of its starting position and geometric parameters of the system. This expression is obtained by an approximate reduction of the initial two-dimensional problem to the effective one-dimensional one which can be solved with relative ease. Our analytical predictions are tested against the results obtained from Brownian dynamics simulations. The test shows excellent agreement between the two for a wide range of the geometric parameters of the system.

DFMD: Fast and Effective DelPhiForce Steered Molecular Dynamics Approach to Model Ligand Approach Toward a Receptor: Application to Spermine Synthase Enzyme

Here we report a novel approach, the DelPhiForce Molecular Dynamics (DFMD) method, for steered molecular dynamics simulations to model receptor-ligand association involving charged species. The main purpose of developing DFMD is to simulate ligand's trajectory toward the receptor and thus to predict the "entrance" of the binding pocket and conformational changes associated with the binding. We demonstrate that the DFMD is superior compared with molecular dynamics simulations applying standard cut-offs, provides correct binding forces, allows for modeling the ligand approach at long distances and thus guides the ligand toward the correct binding spot, and it is very fast (frequently the binding is completed in <1 ns). The DFMD is applied to model the binding of two ligands to a receptor (spermine synthase) and it is demonstrated that it guides the ligands toward the corresponding pockets despite of the initial ligand's position with respect to the receptor. Predicted conformational changes and the order of ligand binding are experimentally verified.

Bimolecular multistage diffusion-influenced chemical reactions proceeding from different sites in solutions. I. Rate constants

General matrix algebraic equations for calculating rate constants of multistage diffusion-influenced reactions (involving bimolecular exchange reactions as elementary stages) in liquid solutions that proceed from different active sites in the immediate vicinity of the contact of reactants have been obtained on the basis of the kinematic approximation developed by the authors earlier. The equations make it possible to express rate constants of any multistage multisite bimolecular reaction between non-identical reactants in terms of the defined reaction constants and stationary Green functions averaged over reaction sites and completely determined by molecular motion of reactants or their molecular groups. The asymptotic behavior of these rate constants as they attain their steady-state values on completion of the transient stage is established. It is shown that it coincides with the corresponding exact time asymptote. Calculations are made with some specific two-stage (three-channel) bimolecular reactions as an example.

Effect of surface curvature on diffusion-limited reactions on a curved surface

To investigate how the curvature of a reactive surface can affect reaction kinetics, we use a simple model in which a diffusion-limited bimolecular reaction occurs on a curved surface that is hollowed inward, flat, or extended outward while keeping the reactive area on the surface constant. By numerically solving the diffusion equation for this model using the finite element method, we find that the rate constant is a non-linear function of the surface curvature and that there is an optimal curvature providing the maximum value of the rate constant, which indicates that a spherical reactant whose entire surface is reactive (a uniformly reactive sphere) is not the most reactive species for a given reactive surface area. We discuss how this result arises from the interplay between two opposing effects: the exposedness of the reactive area to its partner reactants, which causes the rate constant to increase as the curvature increases, and the competition occurring on the reactive surface, which decreases the rate constant. This study helps us to understand the role of curvature in surface reactions and allows us to rationally design reactants that provide a high reaction rate.

Substrate tunnels in enzymes: structure-function relationships and computational methodology

In enzymes, the active site is the location where incoming substrates are chemically converted to products. In some enzymes, this site is deeply buried within the core of the protein, and, in order to access the active site, substrates must pass through the body of the protein via a tunnel. In many systems, these tunnels act as filters and have been found to influence both substrate specificity and catalytic mechanism. Identifying and understanding how these tunnels exert such control has been of growing interest over the past several years because of implications in fields such as protein engineering and drug design. This growing interest has spurred the development of several computational methods to identify and analyze tunnels and how ligands migrate through these tunnels. The goal of this review is to outline how tunnels influence substrate specificity and catalytic efficiency in enzymes with buried active sites and to provide a brief summary of the computational tools used to identify and evaluate these tunnels.

BDflex: a method for efficient treatment of molecular flexibility in calculating protein-ligand binding rate constants from brownian dynamics simulations

A method developed by Northrup et al. [J. Chem. Phys. 80, 1517 (1984)] for calculating protein-ligand binding rate constants (k(a)) from brownian dynamics (BD) simulations has been widely used for rigid molecules. Application to flexible molecules is limited by the formidable computational cost to treat conformational fluctuations during the long BD simulations necessary for k(a) calculation. Here, we propose a new method called BDflex for k(a) calculation that circumvents this problem. The basic idea is to separate the whole space into an outer region and an inner region, and formulate k(a) as the product of k(E) and η(d), which are obtained by separately solving exterior and interior problems. k(E) is the diffusion-controlled rate constant for the ligand in the outer region to reach the dividing surface between the outer and inner regions; in this exterior problem conformational fluctuations can be neglected. η(d) is the probability that the ligand, starting from the dividing surface, will react at the binding site rather than escape to infinity. The crucial step in reducing the determination of η(d) to a problem confined to the inner region is a radiation boundary condition imposed on the dividing surface; the reactivity on this boundary is proportional to k(E). By confining the ligand to the inner region and imposing the radiation boundary condition, we avoid multiple-crossing of the dividing surface before reaction at the binding site and hence dramatically cut down the total simulation time, making the treatment of conformational fluctuations affordable. BDflex is expected to have wide applications in problems where conformational fluctuations of the molecules are crucial for productive ligand binding, such as in cases where transient widening of a bottleneck allows the ligand to access the binding pocket, or the binding site is properly formed only after ligand entrance induces the closure of a lid.

Finite Element Estimation of Protein-Ligand Association Rates with Post-Encounter Effects: Applications to Calcium binding in Troponin C and SERCA

We introduce a computational pipeline and suite of software tools for the approximation of diffusion-limited binding based on a recently developed theoretical framework. Our approach handles molecular geometries generated from high-resolution structural data and can account for active sites buried within the protein or behind gating mechanisms. Using tools from the FEniCS library and the APBS solver, we implement a numerical code for our method and study two Ca(2+)-binding proteins: Troponin C and the Sarcoplasmic Reticulum Ca(2+) ATPase (SERCA). We find that a combination of diffusional encounter and internal 'buried channel' descriptions provide superior descriptions of association rates, improving estimates by orders of magnitude.

Mechanistic insight into the blocking of CO diffusion in [NiFe]-hydrogenase mutants through multiscale simulation

[NiFe]-hydrogenases are fascinating biological catalysts with potential application in biofuel cells. However, a severe problem in practical application is the strong sensitivity of hydrogenase to gaseous inhibitor molecules such as CO and O(2). Recently, a number of successful protein engineering studies have been reported that aimed at lowering the access of diatomic inhibitors to the active site pocket, but the molecular mechanism conferring increased resistance remained unclear. Here we use a multiscale simulation approach combining molecular dynamics with a master equation formalism to explain the steady drop in CO diffusion rate observed for the mutants V74M L122A, V74M L122M, and V74M of Desulfovibrio fructosovorans [NiFe]-hydrogenase. We find that diffusion in these variants is controlled by two gates, one between residues 74 and 476 and the other between residues 74 and 122. The existence of two control points in different locations explains why the reduction in the experimental diffusion rate does not simply correlate with the width of the main gas channel. We also find that in the more effective mutation (V74M) CO molecules are still able to reach the active site through transitions that are gated by the microsecond dihedral motions of the side chain of R476 and the thermal fluctuations of the width of the gas channel defined by M74 and L122. Reflecting on the molecular information gained from simulation, we discuss future mutation experiments that could further lower the diffusion rates of small ligands inhibiting [NiFe]-hydrogenase.

Diffusion-influenced ligand binding to buried sites in macromolecules and transmembrane channels

We consider diffusion-influenced binding to a buried binding site that is connected to the surface by a narrow tunnel. Under the single assumption of an equilibrium distribution of ligands over the tunnel cross section, we reduce the calculation of the time-dependent rate coefficient to the solution of a one-dimensional diffusion equation with appropriate boundary conditions. We obtain a simple analytical expression for the steady-state rate that depends on the potential of mean force in the tunnel and the diffusion-controlled rate of binding to the tunnel entrance. Potential applications of our theory include substrate binding to a buried active site of an enzyme and permeant ion binding to an internal site in a transmembrane channel.

Theory and simulation of diffusion-influenced, stochastically gated ligand binding to buried sites

We consider the diffusion-influenced rate coefficient of ligand binding to a site located in a deep pocket on a protein; the binding pocket is flexible and can reorganize in response to ligand entrance. We extend to this flexible protein-ligand system a formalism developed previously [A. M. Berezhkovskii, A, Szabo, and H.-X. Zhou, J. Chem. Phys. 135, 075103 (2011)] for breaking the ligand-binding problem into an exterior problem and an interior problem. Conformational fluctuations of a bottleneck or a lid and the binding site are modeled as stochastic gating. We present analytical and Brownian dynamics simulation results for the case of a cylindrical pocket containing a binding site at the bottom. Induced switch, whereby the conformation of the protein adapts to the incoming ligand, leads to considerable rate enhancement.

A theory for the proton transport of the influenza virus M2 protein: extensive test against conductance data

A theory for calculating the proton flux through the influenza virus M2 channel is tested here against an extensive set of conductance data. The flux is determined by the rate constants for binding to the His(37) tetrad from the two sides of the membrane and the corresponding unbinding rate constants. The rate constants are calculated by explicitly treating the structure and dynamics of the protein. Important features revealed by previous studies, such as a gating role for Val(27) at the entrance to the channel pore, and channel activation by viral exterior pH, are incorporated in this theory. This study demonstrates that the conductance function of the M2 proton channel can now be quantitatively rationalized by the structure and dynamics of the protein.

Equivalence of two approaches for modeling ion permeation through a transmembrane channel with an internal binding site

Ion permeation through transmembrane channels has traditionally been modeled using two different approaches. In one approach, the translocation of the permeant ion through the channel pore is modeled as continuous diffusion and the rate of ion transport is obtained from solving the steady-state diffusion equation. In the other approach, the translocation of the permeant ion through the pore is modeled as hopping along a discrete set of internal binding sites and the rate of ion transport is obtained from solving a set of steady-state rate equations. In a recent work [Zhou, J. Phys. Chem. Lett. 1, 1973 (2010)], the rate constants for binding to an internal site were further calculated by modeling binding as diffusion-influenced reactions. That work provided the foundation for bridging the two approaches. Here we show that, by representing a binding site as an energy well, the two approaches indeed give the same result for the rate of ion transport.

Theory and simulation on the kinetics of protein-ligand binding coupled to conformational change

Conformational change during protein-ligand binding may significantly affect both the binding mechanism and the rate constant. Most earlier theories and simulations treated conformational change as stochastic gating with transition rates between reactive and nonreactive conformations uncoupled to ligand binding. Recently, we introduced a dual-transition-rates model in which the transition rates between reactive and nonreactive conformations depend on the protein-ligand distance [H.-X. Zhou, Biophys. J. 98, L15 (2010)]. Analytical results of that model showed that the apparent binding mechanism switches from conformational selection to induced fit, when the rates of conformational transitions increase from being much slower than the diffusional approach of the protein-ligand pair to being much faster. The conformational-selection limit (k(CS)) and the induced-fit limit (k(IF)) provide lower and upper bounds, respectively, for the binding rate constant. Here we introduce a general model in which the energy surface of the protein in conformational space is coupled to ligand binding, and present a method for calculating the binding rate constant from Brownian dynamics simulations. Analytical and simulation results show that, for an energy surface that switches from favoring the nonreactive conformation while the ligand is away to favoring the reactive conformation while the ligand is near, k(CS) and k(IF) become close and, thus, provide tight bounds to the binding rate constant. This finding has significant mechanistic implications and presents routes for obtaining good estimates of the rate constant at low cost.

A solvable model for the diffusion and reaction of neurotransmitters in a synaptic junction

BACKGROUND

The diffusion and reaction of the transmitter acetylcholine in neuromuscular junctions and the diffusion and binding of Ca2+ in the dyadic clefts of ventricular myocytes have been extensively modeled by Monte Carlo simulations and by finite-difference and finite-element solutions. However, an analytical solution that can serve as a benchmark for testing these numerical methods has been lacking.

RESULT

Here we present an analytical solution to a model for the diffusion and reaction of acetylcholine in a neuromuscular junction and for the diffusion and binding of Ca2+ in a dyadic cleft. Our model is similar to those previously solved numerically and our results are also qualitatively similar.

CONCLUSION

The analytical solution provides a unique benchmark for testing numerical methods and potentially provides a new avenue for modeling biochemical transport.

Gated Diffusion-controlled Reactions

The binding and active sites of proteins are often dynamically occluded by motion of the nearby polypeptide. A variety of theoretical and computational methods have been developed to predict rates of ligand binding and reactivity in such cases. Two general approaches exist, "protein centric" approaches that explicitly treat only the protein target, and more detailed dynamical simulation approaches in which target and ligand are both treated explicitly. This mini-review describes recent work in this area and some of the biological implications.

Diffusion-Influenced Transport of Ions across a Transmembrane Channel with an Internal Binding Site

We develop a microscopic theory for ion transport across a transmembrane channel with an internal binding site. The ion flux is determined by the rate constants for binding to the internal site from the two sides of the membrane and the corresponding unbinding rate constants. The binding rate constants are formulated in terms of diffusion-influenced reactions. The theory allows a reconciliation of experimental data on the conductance of the influenza A M2 proton channel with the structure and dynamics of the protein.

From induced fit to conformational selection: a continuum of binding mechanism controlled by the timescale of conformational transitions

In receptor-ligand binding, a question that generated considerable interest is whether the mechanism is induced fit or conformational selection. This question is addressed here by a solvable model, in which a receptor undergoes transitions between active and inactive forms. The inactive form is favored while unbound but the active form is favored while a ligand is loosely bound. As the active-inactive transition rates increase, the binding mechanism gradually shifts from conformational selection to induced fit. The timescale of conformational transitions thus plays a crucial role in controlling binding mechanisms.

Rate theories for biologists

Some of the rate theories that are most useful for modeling biological processes are reviewed. By delving into some of the details and subtleties in the development of the theories, the review will hopefully help the reader gain a more than superficial perspective. Examples are presented to illustrate how rate theories can be used to generate insight at the microscopic level into biomolecular behaviors. An attempt is made to clear up a number of misconceptions in the literature regarding popular rate theories, including the appearance of Planck's constant in the transition-state theory and the Smoluchowski result as an upper limit for protein-protein and protein-DNA association rate constants. Future work in combining the implementation of rate theories through computer simulations with experimental probes of rate processes, and in modeling effects of intracellular environments so that theories can be used for generating rate constants for systems biology studies is particularly exciting.

The gates of ion channels and enzymes

Protein dynamics are essential for virtually all protein functions, certainly for gating mechanisms of ion channels and regulation of enzyme catalysis. Ion channels usually feature a gate in the channel pore that prevents ion permeation in the closed state. Some bifunctional enzymes with two distant active sites use a tunnel to transport intermediate products; a gate can help prevent premature leakage. Enzymes with a buried active site also require a tunnel for substrate entrance; a gate along the tunnel can contribute to selectivity. The gates in these different contexts show distinct characteristics in sequence, structure and dynamics, but they also have common features. In particular, aromatic residues often appear to serve as gates, probably because of their ability, through side chain rotation, to effect large changes in cross section.

Dissection of the high rate constant for the binding of a ribotoxin to the ribosome

Restrictocin belongs to a family of site-specific ribonucleases that kill cells by inactivating the ribosome. The restrictocin-ribosome binding rate constant was observed to exceed 10(10) M(-1) s(-1). We have developed a transient-complex theory to model the binding rates of protein-protein and protein-RNA complexes. The theory predicts the rate constant as k(a) = k(a0) exp(-DeltaG(el)*/k(B)T), where k(a0) is the basal rate constant for reaching the transient complex, located at the outer boundary of the bound state, by random diffusion, and DeltaG(el)* is the average electrostatic interaction free energy of the transient complex. Here, we applied the transient-complex theory to dissect the high restrictocin-ribosome binding rate constant. We found that the binding rate of restrictocin to the isolated sarcin/ricin loop is electrostatically enhanced by approximately 300-fold, similar to results found in other protein-protein and protein-RNA complexes. The ribosome provides an additional 10,000-fold rate enhancement because of two synergistic mechanisms afforded by the distal regions of the ribosome. First, they provide additional electrostatic attraction with restrictocin. Second, they reposition the transient complex into a region where local electrostatic interactions of restrictocin with the sarcin/ricin loop are particularly favorable. Our calculations rationalize a host of experimental observations and identify a strategy for designing proteins that bind their targets with high speed.

Uncovering channels in photosystem II by computer modelling: current progress, future prospects, and lessons from analogous systems

Even prior to the publication of the crystal structures for photosystem II (PSII), it had already been suggested that water, O(2) and H(+) channels exist in PSII to achieve directed transport of these molecules, and to avoid undesirable side reactions. Computational efforts to uncover these channels and investigate their properties are still at early stages, and have so far only been based on the static PSII structure. The rationale behind the proposals for such channels and the computer modelling studies thus far are reviewed here. The need to take the dynamic protein into account is then highlighted with reference to the specific issues and techniques applicable to the simulation of each of the three channels. In particular, lessons are drawn from simulation studies on other protein systems containing similar channels.

Gated binding of ligands to HIV-1 protease: Brownian dynamics simulations in a coarse-grained model

The internal motions of proteins may serve as a "gate" in some systems, which controls ligand-protein association. This study applies Brownian dynamics simulations in a coarse-grained model to study the gated association rate constants of HIV-1 proteases and drugs. The computed gated association rate constants of three protease mutants, G48V/V82A/I84V/L90M, G48V, and L90M with three drugs, amprenavir, indinavir, and saquinavir, yield good agreements with experiments. The work shows that the flap dynamics leads to "slow gating". The simulations suggest that the flap flexibility and the opening frequency of the wild-type, the G48V and L90M mutants are similar, but the flaps of the variant G48V/V82A/I84V/L90M open less frequently, resulting in a lower gated rate constant. The developed methodology is fast and provides an efficient way to predict the gated association rate constants for various protease mutants and ligands.

A hypothesis concerning diffusion-limited protein-ligand interactions

A simple assumption allows the prediction of the numerical value for a 'universal' limiting kinetic rate for wholly diffusion-limited reactions between small neutral molecules and macromolecules. This prediction is compared with appropriate experimental data for binding of ligands to myoglobin and to enzymes. It is shown that in the absence of electrostatic effects, this limit is approached but not exceeded. The model also makes very specific predictions concerning the viscosity and temperature dependence of such reactions.

Molecular recognition of oxygen by protein mimics: dynamics on the femtosecond to microsecond time scale

Molecular recognition by biological macromolecules involves many elementary steps, usually convoluted by diffusion processes. Here we report studies of the dynamics, from the femtosecond to the microsecond time scale, of the different elementary processes involved in the bimolecular recognition of a protein mimic, cobalt picket-fence porphyrin, with varying oxygen concentration at controlled temperatures. Electron transfer, bond breakage, and thermal "on" (recombination) and "off" (dissociation) reactions are the different processes involved. The reaction on-rate is 30 to 60 times smaller than that calculated from standard Smoluchowski theory. Introducing a two-step recognition model, with reversibility being part of both steps, removes the discrepancy and provides consistency for the reported thermodynamics, kinetics, and dynamics. The transient intermediates are configurations defined by the contact between oxygen (diatomic) and the picket-fence porphyrin (macromolecule). This intermediate is critical in the description of the potential energy landscape but, as shown here, both enthalpic and entropic contributions to the free energy are important. In the recognition process, the net entropy decrease is -33 cal mol(-1) K(-1); Delta H is -13.4 kcal mol(-1).

Kinetics of diffusion-assisted reactions in microheterogeneous systems

This review is focused on the basic theory of diffusion-assisted reactions in microheterogeneous systems, from porous solids to self-organized colloids and biomolecules. Rich kinetic behaviors observed experimentally are explained in a unified fashion using simple concepts of competing distance and time scales of the reaction and the embedding structure. We mainly consider pseudo-first-order reactions, such as luminescence quenching, described by the Smoluchowski type of equation for the reactant pair distribution function with a sink term defined by the reaction mechanism. Microheterogeneity can affect the microscopic rate constant. It also enters the evolution equation through various spatial constraints leading to complicated boundary conditions and, possibly, to the reduction of dimensionality of the diffusion space. The reaction coordinate and diffusive motion along this coordinate are understood in a general way, depending on the problem at hand. Thus, the evolution operator can describe translational and rotational diffusion of molecules in a usual sense, it can be a discrete random walk operator when dealing with hopping of adsorbates in solids, or it can correspond to conformational fluctuations in proteins. Mathematical formulation is universal but physical consequences can be different. Understanding the principal features of reaction kinetics in microheterogeneous systems enables one to extract important structural and dynamical information about the host environments by analyzing suitably designed experiments, it helps building effective strategies for computer simulations, and ultimately opens possibilities for designing systems with controllable reactivity properties.

Conformation gating as a mechanism for enzyme specificity

Acetylcholinesterase, with an active site located at the bottom of a narrow and deep gorge, provides a striking example of enzymes with buried active sites. Recent molecular dynamics simulations showed that reorientation of five aromatic rings leads to rapid opening and closing of the gate to the active site. In the present study the molecular dynamics trajectory is used to quantitatively analyze the effect of the gate on the substrate binding rate constant. For a 2. 4-A probe modeling acetylcholine, the gate is open only 2.4% of the time, but the quantitative analysis reveals that the substrate binding rate is slowed by merely a factor of 2. We rationalize this result by noting that the substrate, by virtue of Brownian motion, will make repeated attempts to enter the gate each time it is near the gate. If the gate is rapidly switching between the open and closed states, one of these attempts will coincide with an open state, and then the substrate succeeds in entering the gate. However, there is a limit on the extent to which rapid gating dynamics can compensate for the small equilibrium probability of the open state. Thus the gate is effective in reducing the binding rate for a ligand 0.4 A bulkier by three orders of magnitude. This relationship suggests a mechanism for achieving enzyme specificity without sacrificing efficiency.