Electrostatic and hydrodynamic orientational steering effects in enzyme-substrate association

On the possibility of the existence of orienting hydrodynamic steering effects in the kinetics of receptor-ligand association

In the vast majority of biologically relevant cases of receptor-ligand complex formation, the binding site of the receptor is a small part of its surface, and moreover, formation of a biologically active complex often requires a specific orientation of the ligand relative to the binding site. Before the formation of the initial form of the complex, only long-range, electrostatic and hydrodynamic interactions can act between the ligand approaching the binding site and the receptor. In this context, the question arises whether as a result of these interactions, there is a pre-orientation of the ligand towards the binding site, which to some extent would accelerate the formation of the complex. The role of electrostatic interactions in the orientation of the ligand relative to the binding site of the receptor is well documented. The analogous role of hydrodynamic interactions, although assessed as very significant by Brune and Kim (PNAS 91, 2930-2934, (1994)), is still debatable. In this article, I present the current state of knowledge on this subject and consider the possibilities of demonstrating the orienting effect of hydrodynamic interactions in the processes of receptor-ligand association, in an experimental way supported by computer simulations.

Searching for Hydrodynamic Orienting Effects in the Association of Tri-N-acetylglucosamine with Hen Egg-White Lysozyme

Using stopped-flow fluorometry, we determined rate constants for the formation of diffusional encounter complexes of tri-N-acetylglucosamine (NAG₃) with hen egg-white lysozyme (k_a^WT) and its double mutant Asp48Asn/Lys116Gln (k_a^MT). We defined binding anisotropy, κ ≡ (k_a^WT – k_a^MT)/(k_a^WT + k_a^MT), and determined its ionic strength dependence. Our goal was to check if this ionic strength dependence provides information about the orienting hydrodynamic effects in the ligand-binding process. We also computed ionic strength dependence of the binding anisotropy from Brownian dynamics simulations using simple models of the lysozyme–NAG₃ system. The results of our experiments indicate that in the case of lysozyme and NAG₃ such hydrodynamic orienting effects are rather negligible. On the other hand, the results of our Brownian dynamics simulations prove that there exist molecular systems for which such orienting effects are substantial. However, the ionic strength dependence of the rate constants for the wild-type and modified systems do not exhibit any qualitative features that would allow us to conclude the presence of hydrodynamic orienting effects from stopped-flow experiments alone. Nevertheless, the results of our simulations suggest the presence of hydrodynamic orienting effects in the receptor–ligand association when the anisotropy of binding depends on the solvent viscosity.

Numerical Methods for Modeling Enzyme Kinetics

Differential equations are used to describe time-dependent changes in enzyme kinetics and pharmacokinetics. Analytical and numerical methods can be used to solve differential equations. This chapter describes the use of numerical methods in solving differential equations and its applications in characterizing the complexities observed in enzyme kinetics. A discussion is included on the use of numerical methods to overcome limitations of explicit equations in the analysis of metabolism kinetics, reversible inhibition kinetics, and inactivation kinetics. The chapter describes the advantages of using numerical methods when Michaelis-Menten assumptions do not hold.

NERDSS: A Nonequilibrium Simulator for Multibody Self-Assembly at the Cellular Scale

Currently, a significant barrier to building predictive models of cellular self-assembly processes is that molecular models cannot capture minutes-long dynamics that couple distinct components with active processes, whereas reaction-diffusion models cannot capture structures of molecular assembly. Here, we introduce the nonequilibrium reaction-diffusion self-assembly simulator (NERDSS), which addresses this spatiotemporal resolution gap. NERDSS integrates efficient reaction-diffusion algorithms into generalized software that operates on user-defined molecules through diffusion, binding and orientation, unbinding, chemical transformations, and spatial localization. By connecting the fast processes of binding with the slow timescales of large-scale assembly, NERDSS integrates molecular resolution with reversible formation of ordered, multisubunit complexes. NERDSS encodes models using rule-based formatting languages to facilitate model portability, usability, and reproducibility. Applying NERDSS to steps in clathrin-mediated endocytosis, we design multicomponent systems that can form lattices in solution or on the membrane, and we predict how stochastic but localized dephosphorylation of membrane lipids can drive lattice disassembly. The NERDSS simulations reveal the spatial constraints on lattice growth and the role of membrane localization and cooperativity in nucleating assembly. By modeling viral lattice assembly and recapitulating oscillations in protein expression levels for a circadian clock model, we illustrate the adaptability of NERDSS. NERDSS simulates user-defined assembly models that were previously inaccessible to existing software tools, with broad applications to predicting self-assembly in vivo and designing high-yield assemblies in vitro.

Diffusional Encounter Rate Constants for Xanthone and 2-Naphthoic Acid by Flash Photolysis Experiments and Brownian Dynamics Simulations: Substantial Effects of Polarizability of the Triplet State

Diffusional encounter rate constants, for xanthone and 2-naphthoic acid molecules in their triplet states with xanthone or 2-naphthoic acid molecules in their triplet or singlet states, were determined using nanosecond laser flash photolysis spectroscopy. Simultaneously, Brownian dynamics simulations were used to compute these rate constants for assumed models of encountering molecules. Altogether, a global fit to transient absorption progress curves, reporting populations of triplet state xanthone and triplet state 2-naphthoic acid molecules, allowed us to determine six diffusional encounter rate constants from our experiments. The most important result of this study is the detection of substantial effects of the electric polarizability of molecules in their triplet state, visible for xanthone triplet and 2-naphthoic acid ground states, a homo triplet-triplet annihilation of 2-naphthoic acid, and a hetero triplet-triplet annihilation for xanthone and 2-naphthoic acid.

General principles of binding between cell surface receptors and multi-specific ligands: A computational study

The interactions between membrane receptors and extracellular ligands control cell-cell and cell-substrate adhesion, and environmental responsiveness by representing the initial steps of cell signaling pathways. These interactions can be spatial-temporally regulated when different extracellular ligands are tethered. The detailed mechanisms of this spatial-temporal regulation, including the competition between distinct ligands with overlapping binding sites and the conformational flexibility in multi-specific ligand assemblies have not been quantitatively evaluated. We present a new coarse-grained model to realistically simulate the binding process between multi-specific ligands and membrane receptors on cell surfaces. The model simplifies each receptor and each binding site in a multi-specific ligand as a rigid body. Different numbers or types of ligands are spatially organized together in the simulation. These designs were used to test the relation between the overall binding of a multi-specific ligand and the affinity of its cognate binding site. When a variety of ligands are exposed to cells expressing different densities of surface receptors, we demonstrated that ligands with reduced affinities have higher specificity to distinguish cells based on the relative concentrations of their receptors. Finally, modification of intramolecular flexibility was shown to play a role in optimizing the binding between receptors and ligands. In summary, our studies bring new insights to the general principles of ligand-receptor interactions. Future applications of our method will pave the way for new strategies to generate next-generation biologics.

In order to adapt to surrounding environments, multiple signaling pathways have been evolved in cells. The first step of these pathways is to detect external stimuli, which is conducted by the dynamic interactions between cell surface receptors and extracellular ligands. As a result, recognition of extracellular ligands by cell surface receptors is an indispensable component of many physiological or pathological activities. In both natural selection and drug design, the presence of multiple binding sites in extracellular ligand complexes (so-called multi-specific ligands) is a common strategy to target different receptors on surface of the same cell. Such spatial organization of ligand binding sites can elaborately modulate the downstream signaling pathways. However, our understanding to the interactions between multi-specific ligands and membrane receptors is largely limited by the fact that these interactions are difficult to quantify and they have only been successfully measured in a very small number of cases in vivo. Using a simple computational model, we can realistically simulate the binding process between specially designed multi-specific ligands and membrane receptors on cell surfaces. This study therefore provides a useful pathway to unravel basic mechanisms of ligand-receptor interactions and design principles for new drug candidates.

Hydrodynamic Steering in Protein Association Revisited: Surprisingly Minuscule Effects of Considerable Torques

We investigate the previously postulated hydrodynamic steering phenomenon, resulting from complication of molecular shapes, its magnitude and possible relevance for protein-ligand and protein-protein diffusional encounters, and the kinetics of diffusion-controlled association. We consider effects of hydrodynamic interactions in a prototypical model system consisting of a cleft enzyme and an elongated substrate, and real protein-protein complexes, that of barnase and barstar, and human growth hormone and its binding protein. The kinetics of diffusional encounters is evaluated on the basis of rigid-body Brownian dynamics simulations in which hydrodynamic interactions between molecules are modeled using the bead-shell method for detailed description of molecular surfaces, and the first-passage-time approach. We show that magnitudes of steering torques resulting from the hydrodynamic coupling of associating molecules, evaluated for the studied systems on the basis of the Stokes-Einstein type relations for arbitrarily shaped rigid bodies, are comparable with magnitudes of torques resulting from electrostatic interactions of binding partners. Surprisingly, however, unlike in the case of electrostatic torques that strongly affect the diffusional encounter, overall effects of hydrodynamic steering torques on the association kinetics, while clearly discernible in Brownian dynamics simulations, are rather minute. We explain this result as a consequence of the thermal agitation of the binding partners. Our finding is relevant for the general understanding of a wide spectrum of molecular processes in solution but there is also a more practical aspect to it if one considers the low level of shape detail of models that are usually employed to evaluate hydrodynamic interactions in particle-based Stokesian and Brownian dynamics simulations of multicomponent biomolecular systems. Results described in the current work justify, in part at least, such a low-resolution description.

Linking computation and experiments to study the role of charge-charge interactions in protein folding and stability

Over the past two decades there has been an increase in appreciation for the role of surface charge-charge interactions in protein folding and stability. The perception shifted from the belief that charge-charge interactions are not important for protein folding and stability to the near quantitative understanding of how these interactions shape the folding energy landscape. This led to the ability of computational approaches to rationally redesign surface charge-charge interactions to modulate thermodynamic properties of proteins. Here we summarize our progress in understanding the role of charge-charge interactions for protein stability using examples drawn from my own laboratory and touch upon unanswered questions.

Dynamics theory for molecular liquids based on an interaction site model

Dynamics theories for molecular liquids based on an interaction site model have been developed over the past few decades and proved to be powerful tools to investigate various dynamical phenomena.

Strong influence of periodic boundary conditions on lateral diffusion in lipid bilayer membranes

The Saffman-Delbrück hydrodynamic model for lipid-bilayer membranes is modified to account for the periodic boundary conditions commonly imposed in molecular simulations. Predicted lateral diffusion coefficients for membrane-embedded solid bodies are sensitive to box shape and converge slowly to the limit of infinite box size, raising serious doubts for the prospects of using detailed simulations to accurately predict membrane-protein diffusivities and related transport properties. Estimates for the relative error associated with periodic boundary artifacts are 50% and higher for fully atomistic models in currently feasible simulation boxes. MARTINI simulations of LacY membrane protein diffusion and LacY dimer diffusion in DPPC membranes and lipid diffusion in pure DPPC bilayers support the underlying hydrodynamic model.

Effects of Spatially Dependent Mobilities on the Kinetics of the Diffusion-Controlled Association Derived from the First-Passage-Time Approach

Brownian dynamics (BD) simulations and the first-passage-time approach are applied to investigate diffusion-controlled association in a biologically relevant model system consisting of a fixed receptor with an elongated cavity and a capsule-like ligand that fits this cavity precisely. Before the binding at the receptor cavity, the ligand undergoes translational and rotational diffusion, either free or under the influence of electrostatic interactions with the receptor. The spatial dependence of the translational and rotational mobilities of the ligand resulting from its hydrodynamic interactions (HIs) with the receptor is accounted for in BD simulations, and an accurate numerical approach is applied for the evaluation of the spatially dependent mobility tensor of the ligand. Different magnitudes of electrostatic interactions (either attraction or repulsion) between the ligand and receptor are considered. The effective range of receptor-ligand electrostatic interactions is varied to account for their screening under different conditions of ionic strength. The effects of HIs on the kinetics of the diffusion-controlled association, evaluated for different electrostatic properties of binding partners, are thoroughly analyzed and discussed.

Linking 3D and 2D binding kinetics of membrane proteins by multiscale simulations

Membrane proteins are among the most functionally important proteins in cells. Unlike soluble proteins, they only possess two translational degrees of freedom on cell surfaces, and experience significant constraints on their rotations. As a result, it is currently challenging to characterize the in situ binding of membrane proteins. Using the membrane receptors CD2 and CD58 as a testing system, we developed a multiscale simulation framework to study the differences of protein binding kinetics between 3D and 2D environments. The association and dissociation processes were implemented by a coarse-grained Monte-Carlo algorithm, while the dynamic properties of proteins diffusing on lipid bilayer were captured from all-atom molecular dynamic simulations. Our simulations show that molecular diffusion, linker flexibility and membrane fluctuations are important factors in adjusting binding kinetics. Moreover, by calibrating simulation parameters to the measurements of 3D binding, we derived the 2D binding constant which is quantitatively consistent with the experimental data, indicating that the method is able to capture the difference between 3D and 2D binding environments. Finally, we found that the 2D dissociation between CD2 and CD58 is about 100-fold slower than the 3D dissociation. In summary, our simulation framework offered a generic approach to study binding mechanisms of membrane proteins.

Free-Propagator Reweighting Integrator for Single-Particle Dynamics in Reaction-Diffusion Models of Heterogeneous Protein-Protein Interaction Systems

We present a new algorithm for simulating reaction-diffusion equations at single-particle resolution. Our algorithm is designed to be both accurate and simple to implement, and to be applicable to large and heterogeneous systems, including those arising in systems biology applications. We combine the use of the exact Green's function for a pair of reacting particles with the approximate free-diffusion propagator for position updates to particles. Trajectory reweighting in our free-propagator reweighting (FPR) method recovers the exact association rates for a pair of interacting particles at all times. FPR simulations of many-body systems accurately reproduce the theoretically known dynamic behavior for a variety of different reaction types. FPR does not suffer from the loss of efficiency common to other path-reweighting schemes, first, because corrections apply only in the immediate vicinity of reacting particles and, second, because by construction the average weight factor equals one upon leaving this reaction zone. FPR applications include the modeling of pathways and networks of protein-driven processes where reaction rates can vary widely and thousands of proteins may participate in the formation of large assemblies. With a limited amount of bookkeeping necessary to ensure proper association rates for each reactant pair, FPR can account for changes to reaction rates or diffusion constants as a result of reaction events. Importantly, FPR can also be extended to physical descriptions of protein interactions with long-range forces, as we demonstrate here for Coulombic interactions.

A molecule-centered method for accelerating the calculation of hydrodynamic interactions in Brownian dynamics simulations containing many flexible biomolecules

Inclusion of hydrodynamic interactions (HIs) is essential in simulations of biological macromolecules that treat the solvent implicitly if the macromolecules are to exhibit correct translational and rotational diffusion. The present work describes the development and testing of a simple approach aimed at allowing more rapid computation of HIs in coarse-grained Brownian dynamics simulations of systems that contain large numbers of flexible macromolecules. The method combines a complete treatment of intramolecular HIs with an approximate treatment of the intermolecular HIs which assumes that the molecules are effectively spherical; all of the HIs are calculated at the Rotne-Prager-Yamakawa level of theory. When combined with Fixman's Chebyshev polynomial method for calculating correlated random displacements, the proposed method provides an approach that is simple to program but sufficiently fast that it makes it computationally viable to include HIs in large-scale simulations. Test calculations performed on very coarse-grained models of the pyruvate dehydrogenase (PDH) E2 complex and on oligomers of ParM (ranging in size from 1 to 20 monomers) indicate that the method reproduces the translational diffusion behavior seen in more complete HI simulations surprisingly well; the method performs less well at capturing rotational diffusion but its discrepancies diminish with increasing size of the simulated assembly. Simulations of residue-level models of two tetrameric protein models demonstrate that the method also works well when more structurally detailed models are used in the simulations. Finally, test simulations of systems containing up to 1024 coarse-grained PDH molecules indicate that the proposed method rapidly becomes more efficient than the conventional BD approach in which correlated random displacements are obtained via a Cholesky decomposition of the complete diffusion tensor.

Hydrodynamic effects on the relative rotational velocity of associating proteins

Hydrodynamic steering effects on the barnase-barstar association were studied through the analysis of the relative rotational velocity of the proteins. We considered the two proteins approaching each other in response to their electrostatic attraction and employed a method that accounts for the long-range and many-body character of the hydrodynamic interactions, as well as the complicated shapes of the proteins. Hydrodynamic steering effects were clearly seen when attractive forces were applied to the geometric centers of the proteins (resulting in zero torques) and the attraction acted along the line that connects centers of geometry of proteins in their crystallographic complex. When we rotated barstar relative to barnase around this line by an angle in the range from -90° to 60°, the rotational velocity arising solely from hydrodynamic interactions restored the orientation of the proteins in the crystal structure. However, because, in reality, both electrostatic forces and torques act on the proteins and these forces and torques depend on the protein-protein distance and the relative orientation of the binding partners, we also investigated more realistic situations employing continuum electrostatics calculations based on atomistic protein models. Overall, we conclude that hydrodynamic interactions aid barnase and barstar in assuming a proper relative orientation upon complex formation.

IMPORTANCE OF EXCLUDED VOLUME AND HYDRODYNAMIC INTERACTIONS ON MACROMOLECULAR DIFFUSION IN VIVO

The interiors of all living cells are highly crowded with macromolecules, which results in a considerable difference between the thermodynamics and kinetics of biological reactions in vivo from that in vitro. To begin to elucidate the principles of intermolecular dynamics in the crowded environment of cells, employing Brownian dynamics (BD) simulations, we examined possible mechanism(s) responsible for the great reduction in diffusion constants of macromolecules in vivo from that at infinite dilution. In an E. coli cytoplasm modelcomprised of 15 different macromolecule types at physiological concentrations, where macromolecules were represented by spheres with their Stokes radii, BD simulations were performed with and without hydrodynamic interactions (HI). Without HI, the calculated diffusion constant of green fluorescent protein (GFP) is much larger than experiment. On the other hand, when HI were considered, the in vivo experimental GFP diffusion constant is almost reproduced without adjustable parameters. In addition, HI give rise to significant, size independent intermolecular dynamic correlations. These results suggest that HI play an important role on macromolecular dynamics in vivo.

Attractive hydration forces in DNA-dendrimer interactions on the nanometer scale

The energetic contribution of attractive hydration forces arising from water ordering is an interesting but often neglected aspect of macromolecular interactions. Ordering effects of water can bring about cooperativity in many intermolecular transactions, in both the short and long range. Given its high charge density, this is of particular importance for DNA. For instance, in nanotechnology, highly charged dendrimers are used for DNA compaction and transfection. Hypothesizing that water ordering and hydration forces should be maximal for DNA complexes that show charge complementarity (positive-negative), we present here an analysis of water ordering from molecular dynamics simulations and free energy calculations of the interaction between DNA and a nanoparticle with a high positive charge density. Our results indicate not only that complexation of the dendrimer with DNA affects the local water structure but also that ordered water molecules facilitate long-range interactions between the molecules. This contributes significantly to the free energy of binding of dendrimers to DNA and extends the interaction well beyond the electrostatic range of the DNA. Such water effects are of potentially substantial importance in cases when molecules appear to recognize each other across sizable distances, or for which kinetic rates are too fast to be due to pure diffusion. Our results are in good agreement with experiments on the role of solvent in DNA condensation by multivalent cations and exemplify a microscopic realization of mean-field phenomenological theories for hydration forces between mesoscopic surfaces.

On the importance of hydrodynamic interactions in lipid membrane formation

Hydrodynamic interactions (HI) give rise to collective motions between molecules, which are known to be important in the dynamics of random coil polymers and colloids. However, their role in the biological self-assembly of many molecule systems has not been investigated. Here, using Brownian dynamics simulations, we evaluate the importance of HI on the kinetics of self-assembly of lipid membranes. One-thousand coarse-grained lipid molecules in periodic simulation boxes were allowed to assemble into stable bilayers in the presence and absence of intermolecular HI. Hydrodynamic interactions reduce the monomer-monomer association rate by 50%. In contrast, the rate of association of lipid clusters is much faster in the presence of intermolecular HI. In fact, with intermolecular HI, the membrane self-assembly rate is 3-10 times faster than that without intermolecular HI. We introduce an analytical model to describe the size dependence of the diffusive encounter rate of particle clusters, which can qualitatively explain our simulation results for the early stage of the membrane self-assembly process. These results clearly suggest that HI greatly affects the kinetics of self-assembly and that simulations without HI will significantly underestimate the kinetic parameters of such processes.

Krylov subspace methods for computing hydrodynamic interactions in brownian dynamics simulations

Hydrodynamic interactions play an important role in the dynamics of macromolecules. The most common way to take into account hydrodynamic effects in molecular simulations is in the context of a brownian dynamics simulation. However, the calculation of correlated brownian noise vectors in these simulations is computationally very demanding and alternative methods are desirable. This paper studies methods based on Krylov subspaces for computing brownian noise vectors. These methods are related to Chebyshev polynomial approximations, but do not require eigenvalue estimates. We show that only low accuracy is required in the brownian noise vectors to accurately compute values of dynamic and static properties of polymer and monodisperse suspension models. With this level of accuracy, the computational time of Krylov subspace methods scales very nearly as O(N(2)) for the number of particles N up to 10 000, which was the limit tested. The performance of the Krylov subspace methods, especially the "block" version, is slightly better than that of the Chebyshev method, even without taking into account the additional cost of eigenvalue estimates required by the latter. Furthermore, at N = 10,000, the Krylov subspace method is 13 times faster than the exact Cholesky method. Thus, Krylov subspace methods are recommended for performing large-scale brownian dynamics simulations with hydrodynamic interactions.

A method for computing association rate constants of atomistically represented proteins under macromolecular crowding

In cellular environments, two protein molecules on their way to form a specific complex encounter many bystander macromolecules. The latter molecules, or crowders, affect both the energetics of the interaction between the test molecules and the dynamics of their relative motion. In earlier work (Zhou and Szabo 1991 J. Chem. Phys. 95 5948-52), it has been shown that, in modeling the association kinetics of the test molecules, the presence of crowders can be accounted for by their energetic and dynamic effects. The recent development of the transient-complex theory for protein association in dilute solutions makes it possible to easily incorporate the energetic and dynamic effects of crowders. The transient complex refers to a late on-pathway intermediate, in which the two protein molecules have near-native relative separation and orientation, but have yet to form the many short-range specific interactions of the native complex. The transient-complex theory predicts the association rate constant as k(a) = k(a0)exp(-ΔG*(el)/k(B)T), where k(a0) is the 'basal' rate constant for reaching the transient complex by unbiased diffusion, and the Boltzmann factors captures the influence of long-range electrostatic interactions between the protein molecules. Crowders slow down the diffusion, therefore reducing the basal rate constant (to k(ac0)), and induce an effective interaction energy ΔG(c). We show that the latter interaction energy for atomistic proteins in the presence of spherical crowders is 'long'-ranged, allowing the association rate constant under crowding to be computed as k(ac) = k(ac0)exp[-(ΔG*(el) + ΔG*(c))/k(B)T]. Applications demonstrate that this computational method allows for realistic modeling of protein association kinetics under crowding.

Contributions of far-field hydrodynamic interactions to the kinetics of electrostatically driven molecular association

We simulated the diffusional encounters in periodic systems of model isotropic and anisotropic molecules using Brownian dynamics. We considered the electrostatic, excluded volume, and far-field hydrodynamic forces between diffusing molecules. Our goal was to estimate to what extent the hydrodynamic interactions influence the association kinetics when the associating partners are oppositely charged and their direct electrostatic interactions are screened by small mobile ions of dissolved salt. Overall, including hydrodynamic interactions decreases the association rate constants. The relative magnitude of this decrease does not depend on the ionic strength for the association of isotropic charged objects. This also holds true for nonspecific association (i.e., without restrictions regarding the relative orientation of binding partners in an encounter complex) of anisotropic objects. However, such dependence is visible for orientation-specific association of anisotropic objects. Moreover, we observe that some orientations of anisotropic molecules are hydrodynamically favorable during their mutual approach, and that such molecules can be hydrodynamically steered toward a particular relative orientation. This hydrodynamic orientational steering is impeded in case of strong electrostatic interactions or steric hindrance.

The binding process of a nonspecific enzyme with DNA

Protein-DNA recognition of a nonspecific complex is modeled to understand the nature of the transient encounter states. We consider the structural and energetic features and the role of water in the DNA grooves in the process of protein-DNA recognition. Here we have used the nuclease domain of colicin E7 (N-ColE7) from Escherichia coli in complex with a 12-bp DNA duplex as the model system to consider how a protein approaches, encounters, and associates with DNA. Multiscale simulation studies using Brownian dynamics and molecular-dynamics simulations were performed to provide the binding process on multiple length- and timescales. We define the encounter states and identified the spatial and orientational aspects. For the molecular length-scales, we used molecular-dynamics simulations. Several intermediate binding states were found, which have different positions and orientations of protein around DNA including major and minor groove orientations. The results show that the contact number and the hydrated interfacial area are measures that facilitate better understanding of sequence-independent protein-DNA binding landscapes and pathways.

Effect of fluidic transport on the reaction kinetics in lectin microarrays

Lectins are the proteins which can distinguish glycosylation patterns. They are frequently used as biomarkers for progressions of several diseases including cancer. As the lectin microarray based prognosis devices miniaturize the process of glycoprofiling, it is anticipated that their performance can be augmented by integration with microfluidic framework. This is analogous to microfluidics based DNA arrays. However, unlike small oligonucleotide microarrays, it remains uncertain whether the binding reaction-kinetic parameters can be considered invariant of imposed hydrodynamics, for relatively larger and structure sensitive molecules such as lectins. Here we show, using two standard lectins namely Concanavalin A and Abrus Agglutinin, that the steady state binding efficiency unexpectedly declines beyond a critical shear rate magnitude. This observation can be explained only if the associated reaction constants are presumed to be functions of hydrodynamic parameters. We methodically deduce the shear rate dependence of association and dissociation constants from the comparison of experimental and model-simulation trends. The aforementioned phenomena are perceived to be the consequences of strong hydrodynamic perturbations, culminating into molecular structural distortion. The exploration, therefore, reveals a unique coupling between reaction kinetics and hydrodynamics for biomacromolecules and provides a generic scheme towards futuristic microfluidics-coupled biomedical assays.

Diffusion of hydrophobin proteins in solution and interactions with a graphite surface

Background

Hydrophobins are small proteins produced by filamentous fungi that have a variety of biological functions including coating of spores and surface adhesion. To accomplish these functions, they rely on unique interface-binding properties. Using atomic-detail implicit solvent rigid-body Brownian dynamics simulations, we studied the diffusion of HFBI, a class II hydrophobin from Trichoderma reesei, in aqueous solution in the presence and absence of a graphite surface.

Results

In the simulations, HFBI exists in solution as a mixture of monomers in equilibrium with different types of oligomers. The oligomerization state depends on the conformation of HFBI. When a Highly Ordered Pyrolytic Graphite (HOPG) layer is present in the simulated system, HFBI tends to interact with the HOPG layer through a hydrophobic patch on the protein.

Conclusions

From the simulations of HFBI solutions, we identify a tetrameric encounter complex stabilized by non-polar interactions between the aliphatic residues in the hydrophobic patch on HFBI. After the formation of the encounter complex, a local structural rearrangement at the protein interfaces is required to obtain the tetrameric arrangement seen in HFBI crystals. Simulations performed with the graphite surface show that, due to a combination of a geometric hindrance and the interaction of the aliphatic sidechains with the graphite layer, HFBI proteins tend to accumulate close to the hydrophobic surface.

Brownian dynamics simulation of protein solutions: structural and dynamical properties

The study of solutions of biomacromolecules provides an important basis for understanding the behavior of many fundamental cellular processes, such as protein folding, self-assembly, biochemical reactions, and signal transduction. Here, we describe a Brownian dynamics simulation procedure and its validation for the study of the dynamic and structural properties of protein solutions. In the model used, the proteins are treated as atomically detailed rigid bodies moving in a continuum solvent. The protein-protein interaction forces are described by the sum of electrostatic interaction, electrostatic desolvation, nonpolar desolvation, and soft-core repulsion terms. The linearized Poisson-Boltzmann equation is solved to compute electrostatic terms. Simulations of homogeneous solutions of three different proteins with varying concentrations, pH, and ionic strength were performed. The results were compared to experimental data and theoretical values in terms of long-time self-diffusion coefficients, second virial coefficients, and structure factors. The results agree with the experimental trends and, in many cases, experimental values are reproduced quantitatively. There are no parameters specific to certain protein types in the interaction model, and hence the model should be applicable to the simulation of the behavior of mixtures of macromolecules in cell-like crowded environments.

Absolute protein-protein association rate constants from flexible, coarse-grained Brownian dynamics simulations: the role of intermolecular hydrodynamic interactions in barnase-barstar association

Theory and computation have long been used to rationalize the experimental association rate constants of protein-protein complexes, and Brownian dynamics (BD) simulations, in particular, have been successful in reproducing the relative rate constants of wild-type and mutant protein pairs. Missing from previous BD studies of association kinetics, however, has been the description of hydrodynamic interactions (HIs) between, and within, the diffusing proteins. Here we address this issue by rigorously including HIs in BD simulations of the barnase-barstar association reaction. We first show that even very simplified representations of the proteins--involving approximately one pseudoatom for every three residues in the protein--can provide excellent reproduction of the absolute association rate constants of wild-type and mutant protein pairs. We then show that simulations that include intermolecular HIs also produce excellent estimates of association rate constants, but, for a given reaction criterion, yield values that are decreased by ∼35-80% relative to those obtained in the absence of intermolecular HIs. The neglect of intermolecular HIs in previous BD simulation studies, therefore, is likely to have contributed to the somewhat overestimated absolute rate constants previously obtained. Consequently, intermolecular HIs could be an important component to include in accurate modeling of the kinetics of macromolecular association events.

On the contributions of diffusion and thermal activation to electron transfer between Phormidium laminosum plastocyanin and cytochrome f: Brownian dynamics simulations with explicit modeling of nonpolar desolvation interactions and electron transfer events

The factors that determine the extent to which diffusion and thermal activation processes govern electron transfer (ET) between proteins are debated. The process of ET between plastocyanin (PC) and cytochrome f (CytF) from the cyanobacterium Phormidium laminosum was initially thought to be diffusion-controlled but later was found to be under activation control (Schlarb-Ridley, B. G.; et al. Biochemistry 2005, 44, 6232). Here we describe Brownian dynamics simulations of the diffusional association of PC and CytF, from which ET rates were computed using a detailed model of ET events that was applied to all of the generated protein configurations. The proteins were modeled as rigid bodies represented in atomic detail. In addition to electrostatic forces, which were modeled as in our previous simulations of protein-protein association, the proteins interacted by a nonpolar desolvation (hydrophobic) force whose derivation is described here. The simulations yielded close to realistic residence times of transient protein-protein encounter complexes of up to tens of microseconds. The activation barrier for individual ET events derived from the simulations was positive. Whereas the electrostatic interactions between P. laminosum PC and CytF are weak, simulations for a second cyanobacterial PC-CytF pair, that from Nostoc sp. PCC 7119, revealed ET rates influenced by stronger electrostatic interactions. In both cases, the simulations imply significant contributions to ET from both diffusion and thermal activation processes.

Fruitful and futile encounters along the association reaction between proteins

The association reaction between pairs of proteins proceeds through an encounter complex that develops into the final complex. Here, we combined Brownian dynamics simulations with experimental studies to analyze the structures of the encounter complexes along the association reaction between TEM1-beta-lactamase and its inhibitor, beta-lactamase-inhibitor protein. The encounter complex can be considered as an ensemble of short-lived low free-energy states that are stabilized primarily by electrostatic forces and desolvation. For the wild-type, the simulation showed two main encounter regions located outside the physical binding site. One of these regions was located near the experimentally determined transition state. To validate whether these encounters are fruitful or futile, we examined three groups of mutations that altered the encounter. The first group consisted of mutations that increased the experimental rate of association through electrostatic optimization. This resulted in an increase in the size of the encounter region located near the experimentally determined transition state, as well as a decrease in the energy of this region and an increase in the number of successful trajectories (i.e., encounters that develop into complex). A second group of mutations was specifically designed to either increase or decrease the size and energy of the second encounter complex, but either way it did not affect k(on). A third group of mutations consisted of residues that increased k(on) without significantly affecting the encounter complexes. These results indicate that the size and energy of the encounter regions are only two of several parameters that lead to fruitful association, and that electrostatic optimization is a major driving force in fast association.

Rational stabilization of enzymes by computational redesign of surface charge-charge interactions

Here, we report the application of a computational approach that allows the rational design of enzymes with enhanced thermostability while retaining full enzymatic activity. The approach is based on the optimization of the energy of charge-charge interactions on the protein surface. We experimentally tested the validity of the approach on 2 human enzymes, acylphosphatase (AcPh) and Cdc42 GTPase, that differ in size (98 vs. 198-aa residues, respectively) and tertiary structure. We show that the designed proteins are significantly more stable than the corresponding WT proteins. The increase in stability is not accompanied by significant changes in structure, oligomerization state, or, most importantly, activity of the designed AcPh or Cdc42. This success of the design methodology suggests that it can be universally applied to other enzymes, on its own or in combination with the other strategies based on redesign of the interactions in the protein core.

Surface plasmon resonance analysis of antifungal azoles binding to CYP3A4 with kinetic resolution of multiple binding orientations

The heme-containing cytochrome P450s (CYPs) are a major enzymatic determinant of drug clearance and drug-drug interactions. The CYP3A4 isoform is inhibited by antifungal imidazoles or triazoles, which form low-spin heme iron complexes via formation of a nitrogen-ferric iron coordinate bond. However, CYP3A4 also slowly oxidizes the antifungal itraconazole (ITZ) at a site that is approximately 25 A from the triazole nitrogens, suggesting that large antifungal azoles can adopt multiple orientations within the CYP3A4 active site. Here, we report a surface plasmon resonance (SPR) analysis with kinetic resolution of two binding modes of ITZ, and the related drug ketoconazole (KTZ). SPR reveals a very slow off-rate for one binding orientation. Multiphasic binding kinetics are observed, and one of the two binding components resolved by curve fitting exhibits "equilibrium overshoot". Preloading of CYP3A4 with the heme ligand imidazole abolishes this component of the antifungal azole binding trajectories, and it eliminates the conspicuously slow off-rate. The fractional populations of CYP3A4 complexes corresponding to different drug orientations can be manipulated by altering the duration of the pulse of drug exposure. UV-vis difference absorbance titrations yield low-spin spectra and K(D) values that are consistent with the high-affinity complex resolved by SPR. These results demonstrate that ITZ and KTZ bind in multiple orientations, including a catalytically productive mode and a slowly dissociating inhibitory mode. Most importantly, they provide the first example of a SPR-based method for the kinetic characterization of binding of a drug to any human CYP, including mechanistic insight not available from other methods.

Continuum diffusion reaction rate calculations of wild-type and mutant mouse acetylcholinesterase: adaptive finite element analysis

As described previously, continuum models, such as the Smoluchowski equation, offer a scalable framework for studying diffusion in biomolecular systems. This work presents new developments in the efficient solution of the continuum diffusion equation. Specifically, we present methods for adaptively refining finite element solutions of the Smoluchowski equation based on a posteriori error estimates. We also describe new, molecular-surface-based models, for diffusional reaction boundary criteria and compare results obtained from these models with the traditional spherical criteria. The new methods are validated by comparison of the calculated reaction rates with experimental values for wild-type and mutant forms of mouse acetylcholinesterase. The results show good agreement with experiment and help to define optimal reactive boundary conditions.

Finite element solution of the steady-state Smoluchowski equation for rate constant calculations

This article describes the development and implementation of algorithms to study diffusion in biomolecular systems using continuum mechanics equations. Specifically, finite element methods have been developed to solve the steady-state Smoluchowski equation to calculate ligand binding rate constants for large biomolecules. The resulting software has been validated and applied to mouse acetylcholinesterase. Rates for inhibitor binding to mAChE were calculated at various ionic strengths with several different reaction criteria. The calculated rates were compared with experimental data and show very good agreement when the correct reaction criterion is used. Additionally, these finite element methods require significantly less computational resources than existing particle-based Brownian dynamics methods.

A Brownian dynamics study: the effect of a membrane environment on an electron transfer system

During the past few years, three-dimensional crystal structures of many of the important integral membrane proteins responsible for the bioenergetic processes of photosynthesis and respiration have been determined. Moreover, a few crystal structures of protein-protein complexes have become available that characterize the interaction between those membrane proteins and the electron carrier protein cytochrome c. Here, we address the association kinetics for binding of cytochrome c to cytochrome c oxidase (COX) from Paracoccus denitrificans by Brownian dynamics simulations. The effects of ionic strength and protein mutations were studied for two different cytochrome c species: the positively charged, dipolar horse heart cytochrome c and the negatively charged physiological electron transfer partner cytochrome c(552). We studied association toward "naked" COX and toward membrane-embedded COX where the membrane is represented as an uncharged DPPC bilayer modeled in atomistic detail. For the nonnatural association toward "naked" COX, the association rates are >100 times larger for horse heart cytochrome c than for cytochrome c(552). Interestingly, the presence of the lipid bilayer leads to a dramatic decrease of the association rate of horse heart cytochrome c, but slightly enhances association of cytochrome c(552), leading to very similar association rates of both proteins to membrane-embedded COX. This finding from computational modeling studies may reflect the optimization of surface patches and of the total net charge on electron transfer pairs in nature.

Interpretation of NMR relaxation properties of Pin1, a two-domain protein, based on Brownian dynamic simulations

Many important proteins contain multiple domains connected by flexible linkers. Inter-domain motion is suggested to play a key role in many processes involving molecular recognition. Heteronuclear NMR relaxation is sensitive to motions in the relevant time scales and could provide valuable information on the dynamics of multi-domain proteins. However, the standard analysis based on the separation of global tumbling and fast local motions is no longer valid for multi-domain proteins undergoing internal motions involving complete domains and that take place on the same time scale than the overall motion. The complexity of the motions experienced even for the simplest two-domain proteins are difficult to capture with simple extensions of the classical Lipari-Szabo approach. Hydrodynamic effects are expected to dominate the motion of the individual globular domains, as well as that of the complete protein. Using Pin1 as a test case, we have simulated its motion at the microsecond time scale, at a reasonable computational expense, using Brownian Dynamic simulations on simplified models. The resulting trajectories provide insight on the interplay between global and inter-domain motion and can be analyzed using the recently published method of isotropic Reorientational Mode Dynamics which offer a way of calculating their contribution to heteronuclear relaxation rates. The analysis of trajectories computed with Pin1 models of different flexibility provides a general framework to understand the dynamics of multi-domain proteins and explains some of the observed features in the relaxation rate profile of free Pin1.

Differences in electrostatic properties at antibody-antigen binding sites: implications for specificity and cross-reactivity

Antibodies HyHEL8, HyHEL10, and HyHEL26 (HH8, HH10, and HH26, respectively) recognize highly overlapping epitopes on hen egg-white lysozyme (HEL) with similar affinities, but with different specificities. HH8 binding to HEL is least sensitive toward mutations in the epitope and thus is most cross-reactive, HH26 is most sensitive, whereas the sensitivity of HH10 lies in between HH8 and HH26. Here we have investigated intra- and intermolecular interactions in three antibody-protein complexes: theoretical models of HH8-HEL and HH26-HEL complexes, and the x-ray crystal structure of HH10-HEL complex. Our results show that HH8-HEL has the lowest number and HH26-HEL has the highest number of intra- and intermolecular hydrogen bonds. The number of salt bridges is lowest in HH8-HEL and highest in HH26-HEL. The binding site salt bridges in HH8-HEL are not networked, and are weak, whereas, in HH26-HEL, an intramolecular salt-bridge triad at the binding site is networked to an intermolecular triad to form a pentad. The pentad and each salt bridge of this pentad are exceptionally stabilizing. The number of binding-site salt bridges and their strengths are intermediate in HH10-HEL, with an intramolecular triad. Our further calculations show that the electrostatic component contributes the most to binding energy of HH26-HEL, whereas the hydrophobic component contributes the most in the case of HH8-HEL. A "hot-spot" epitope residue Lys-97 forms an intermolecular salt bridge in HH8-HEL, and participates in the intermolecular pentad in the HH26-HEL complex. Mutant modeling and surface plasmon resonance (SPR) studies show that this hot-spot epitope residue contributes significantly more to the binding than an adjacent epitope residue, Lys-96, which does not form a salt bridge in any of the three HH-HEL complexes. Furthermore, the effect of mutating Lys-97 is most severe in HH26-HEL. Lys-96, being a charged residue, also contributes the most in HH26-HEL among the three complexes. The SPR results on these mutants also highlight that the apparent "electrostatic steering" on net on rates actually act at post-collision level stabilization of the complex. The significance of this work is the observed variations in electrostatic interactions among the three complexes. Our work demonstrates that higher electrostatics, both as a number of short-range electrostatic interactions and their contributions, leads to higher binding specificity. Strong salt bridges, their networking, and electrostatically driven binding, limit flexibilities through geometric constrains. In contrast, hydrophobic driven binding and low levels of electrostatic interactions are associated with conformational flexibility and cross-reactivity.

The Poisson-Boltzmann equation for biomolecular electrostatics: a tool for structural biology

Electrostatics plays a fundamental role in virtually all processes involving biomolecules in solution. The Poisson-Boltzmann equation constitutes one of the most fundamental approaches to treat electrostatic effects in solution. The theoretical basis of the Poisson-Boltzmann equation is reviewed and a wide range of applications is presented, including the computation of the electrostatic potential at the solvent-accessible molecular surface, the computation of encounter rates between molecules in solution, the computation of the free energy of association and its salt dependence, the study of pKa shifts and the combination with classical molecular mechanics and dynamics. Theoretical results may be used for rationalizing or predicting experimental results, or for suggesting working hypotheses. An ever-increasing body of successful applications proves that the Poisson-Boltzmann equation is a useful tool for structural biology and complementary to other established experimental and theoretical methodologies.

Protein-protein association: investigation of factors influencing association rates by brownian dynamics simulations

The rate of protein-protein association limits the response time due to protein-protein interactions. The bimolecular association rate may be diffusion-controlled or influenced, and in such cases, Brownian dynamics simulations of protein-protein diffusional association may be used to compute association rates. Here, we report Brownian dynamics simulations of the diffusional association of five different protein-protein pairs: barnase and barstar, acetylcholinesterase and fasciculin-2, cytochrome c peroxidase and cytochrome c, the HyHEL-5 antibody and hen egg lysozyme (HEL), and the HyHEL-10 antibody and HEL. The same protocol was used to compute the diffusional association rates for all the protein pairs in order to assess, by comparison to experimentally measured rates, whether the association of these proteins can be explained solely on the basis of diffusional encounter. The simulation protocol is similar to those previously derived for simulation of the association of barnase and barstar, and of acetylcholinesterase and fasciculin-2; these produced results in excellent agreement with experimental data for these protein pairs, with changes in association rate due to mutations reproduced within the limits of expected computational and modeling errors. Here, we find that for all protein pairs, the effects of mutations can be well reproduced by the simulations, even though the degree of the electrostatic translational and orientational steering varies widely between the cases. However, the absolute values of association rates for the acetylcholinesterase: fasciculin-2 and HyHEL-10 antibody: HEL pairs are overestimated. Comparison of bound and unbound protein structures shows that this may be due to gating resulting from protein flexibility in some of the proteins. This may lower the association rates compared to their bimolecular diffusional encounter rates.

Comparison of salt effects on the reactions of acetylcholinesterase with cationic and anionic inhibitors

The influence of inorganic salts on the inhibition of acetylcholinesterase by charged organophosphorous inhibitors has been studied. It has been shown that the salt effect on the reaction of acetylcholinesterase with anionic bis(p-nitrophenyl) phosphate is determined by the influence of added salts on the activity coefficient of the inhibitor. In contrast to the salt effects on the reaction of acetylcholinesterase with cationic compounds, it does not include contribution from the enzyme charges. The smaller salt effect in the case of anionic inhibitor can be explained assuming that the anionic inhibitor does not form a non-covalent complex with the enzyme before the phosphorylation step of the reaction. Comparison of salt effects on the substrate turnover showed that in the case of cholinesterases from natural sources they are larger than in the case of enzymes expressed in recombinant cell clones. The enhanced salt effects may result from post-translational modification of the enzyme.

Bimolecular reaction simulation using Weighted Ensemble Brownian dynamics and the University of Houston Brownian Dynamics program

We discuss here the implementation of the Weighted Ensemble Brownian (WEB) dynamics algorithm of Huber and Kim in the University of Houston Brownian Dynamics (UHBD) suite of programs and its application to bimolecular association problems. WEB dynamics is a biased Brownian dynamics (BD) algorithm that is more efficient than the standard Northrup-Allison-McCammon (NAM) method in cases where reaction events are infrequent because of intervening free energy barriers. Test cases reported here include the Smoluchowski rate for association of spheres, the association of the enzyme copper-zinc superoxide dismutase with superoxide anion, and the binding of the superpotent sweetener N-(p-cyanophenyl)-N'-(diphenylmethyl)-guanidinium acetic acid to a monoclonal antibody fragment, NC6.8. Our results show that the WEB dynamics algorithm is a superior simulation method for enzyme-substrate reaction encounters with large free energy barriers.

Pathways of ligand clearance in acetylcholinesterase by multiple copy sampling

The clearance of seven different ligands from the deeply buried active-site of Torpedo californica acetylcholinesterase is investigated by combining multiple copy sampling molecular dynamics simulations, with the analysis of protein-ligand interactions, protein motion and the electrostatic potential sampled by the ligand copies along their journey outwards. The considered ligands are the cations ammonium, methylammonium, and tetramethylammonium, the hydrophobic methane and neopentane, and the anionic product acetate and its neutral form, acetic acid. We find that the pathways explored by the different ligands vary with ligand size and chemical properties. Very small ligands, such as ammonium and methane, exit through several routes. One involves the main exit through the mouth of the enzyme gorge, another is through the so-called back door near Trp84, and a third uses a side door at a direction of approximately 45 degrees to the main exit. The larger polar ligands, methylammonium and acetic acid, leave through the main exit, but the bulkiest, tetramethylammonium and neopentane, as well as the smaller acetate ion, remain trapped in the enzyme gorge during the time of the simulations. The pattern of protein-ligand contacts during the diffusion process is highly non-random and differs for different ligands. A majority is made with aromatic side-chains, but classical H-bonds are also formed. In the case of acetate, but not acetic acid, the anionic and neutral form, respectively, of one of the reaction products, specific electrostatic interactions with protein groups, seem to slow ligand motion and interfere with protein flexibility; protonation of the acetate ion is therefore suggested to facilitate clearance. The Poisson-Boltzmann formalism is used to compute the electrostatic potential of the thermally fluctuating acetylcholinesterase protein at positions actually visited by the diffusing ligand copies. Ligands of different charge and size are shown to sample somewhat different electrostatic potentials during their migration, because they explore different microscopic routes. The potential along the clearance route of a cation such as methylammonium displays two clear minima at the active and peripheral anionic site. We find moreover that the electrostatic energy barrier that the cation needs to overcome when moving between these two sites is small in both directions, being of the order of the ligand kinetic energy. The peripheral site thus appears to play a role in trapping inbound cationic ligands as well as in cation clearance, and hence in product release.

Computer simulations of actin polymerization can explain the barbed-pointed end asymmetry

Computer simulations of actin polymerization were performed to investigate the role of electrostatic interactions in determining polymerization rates. Atomically detailed models of actin monomers and filaments were used in conjunction with a Brownian dynamics method. The simulations were able to reproduce the measured barbed end association rates over a range of ionic strengths and predicted a slower growing pointed end, in agreement with experiment. Similar simulations neglecting electrostatic interactions indicate that configurational and entropic factors may actually favor polymerization at the pointed end, but electrostatic interactions remove this trend. This result would indicate that polymerization at the pointed end is not only limited by diffusion, but faces electrostatic forces that oppose binding. The binding of the actin depolymerizing factor (ADF) and G-actin complex to the end of a filament was also simulated. In this case, electrostatic steering effects lead to an increase in the simulated association rate. Together, the results indicate that simulations provide a realistic description of both polymerization and the binding of more complex structures to actin filaments.

A modular treatment of molecular traffic through the active site of cholinesterase

We present a model for the molecular traffic of ligands, substrates, and products through the active site of cholinesterases (ChEs). First, we describe a common treatment of the diffusion to a buried active site of cationic and neutral species. We then explain the specificity of ChEs for cationic ligands and substrates by introducing two additional components to this common treatment. The first module is a surface trap for cationic species at the entrance to the active-site gorge that operates through local, short-range electrostatic interactions and is independent of ionic strength. The second module is an ionic-strength-dependent steering mechanism generated by long-range electrostatic interactions arising from the overall distribution of charges in ChEs. Our calculations show that diffusion of charged ligands relative to neutral isosteric analogs is enhanced approximately 10-fold by the surface trap, while electrostatic steering contributes only a 1.5- to 2-fold rate enhancement at physiological salt concentration. We model clearance of cationic products from the active-site gorge as analogous to the escape of a particle from a one-dimensional well in the presence of a linear electrostatic potential. We evaluate the potential inside the gorge and provide evidence that while contributing to the steering of cationic species toward the active site, it does not appreciably retard their clearance. This optimal fine-tuning of global and local electrostatic interactions endows ChEs with maximum catalytic efficiency and specificity for a positively charged substrate, while at the same time not hindering clearance of the positively charged products.

Brownian dynamics simulations of protein folding: access to milliseconds time scale and beyond

Protein folding occurs on a time scale ranging from milliseconds to minutes for a majority of proteins. Computer simulation of protein folding, from a random configuration to the native structure, is nontrivial owing to the large disparity between the simulation and folding time scales. As an effort to overcome this limitation, simple models with idealized protein subdomains, e.g., the diffusion-collision model of Karplus and Weaver, have gained some popularity. We present here new results for the folding of a four-helix bundle within the framework of the diffusion-collision model. Even with such simplifying assumptions, a direct application of standard Brownian dynamics methods would consume 10,000 processor-years on current supercomputers. We circumvent this difficulty by invoking a special Brownian dynamics simulation. The method features the calculation of the mean passage time of an event from the flux overpopulation method and the sampling of events that lead to productive collisions even if their probability is extremely small (because of large free-energy barriers that separate them from the higher probability events). Using these developments, we demonstrate that a coarse-grained model of the four-helix bundle can be simulated in several days on current supercomputers. Furthermore, such simulations yield folding times that are in the range of time scales observed in experiments.

Influence of static and dynamic bends on the birefringence decay profile of RNA helices: Brownian dynamics simulations

Bends in nucleic acid helices can be quantified in a transient electric birefringence (TEB) experiment from the ratio of the terminal decay times of the bent molecule and its fully duplex counterpart (tau-ratio method). The apparent bend angles can be extracted from the experimental tau-ratios through the application of static (equilibrium-ensemble) hydrodynamic models; however, such models do not properly address the faster component(s) of the birefringence decay profile, which can represent up to 80% of the total birefringence signal for large band angles. To address this latter issue, the relative amplitudes of the components in the birefringence decay profile have been analyzed through a series of Brownian dynamics (BD) simulations. Decay profiles have been simulated for three-, five-, and nine-bead models representing RNA molecules with central bends of 30 degrees, 60 degrees, and 90 degrees, and with various degrees of associated angle dispersion. The BD simulations are in close agreement with experimental results for the fractional amplitudes, suggesting that both amplitudes and terminal tau-ratios can be used as a measure of the magnitudes of bends in the helix axis. Although the current results indicate that it is generally not possible to distinguish between relatively fixed and highly flexible bends from single tau-ratio measurements, because they can lead to similar reductions in terminal decay time and amplitude, measurements of the dependence of the fractional amplitudes on helix length may afford such a distinction.

Simulation of the diffusional association of barnase and barstar

The rate of protein association places an upper limit on the response time due to protein interactions, which, under certain circumstances, can be diffusion-controlled. Simulations of model proteins show that diffusion-limited association rates are approximately 10(6)-10(7) M-1 s-1 in the absence of long-range forces (Northrup, S. H., and H. P. Erickson. 1992. Kinetics of protein-protein association explained by Brownian dynamics computer simulations. Proc. Natl. Acad. Sci. U.S.A. 89:3338-3342). The measured association rates of barnase and barstar are 10(8)-10(9) M-1 s-1 at 50 mM ionic strength, and depend on ionic strength (Schreiber, G., and A. R. Fersht. 1996. Rapid, electrostatically assisted association of proteins. Nat. Struct. Biol. 3:427-431), implying that their association is electrostatically facilitated. We report Brownian dynamics simulations of the diffusional association of barnase and barstar to compute association rates and their dependence on ionic strength and protein mutation. Crucial to the ability to reproduce experimental rates is the definition of encounter complex formation at the endpoint of diffusional motion. Simple definitions, such as a required root mean square (RMS) distance to the fully bound position, fail to explain the large influence of some mutations on association rates. Good agreement with experiments could be obtained if satisfaction of two intermolecular residue contacts was required for encounter complex formation. In the encounter complexes, barstar tends to be shifted from its position in the bound complex toward the guanine-binding loop on barnase.