To request any journal reprints, please contact us.
Dynamical Theory of Activated Processes in Globular Proteins
Scott H. Northrup, Michael R. Pear, Chyuan-Yih Lee, J. Andrew McCammon and Martin Karplus
A method is described for calculating the reaction rate in globular proteins of activated processes such as ligand binding or enzymatic catalysis. The method is based on the determination of the probability that the system is in the transition state and of the magnitude of the reactive flux for transition-state systems. An "umbrella sampling" simulation procedure is outlined for evaluating the transition-state probability. The reactive flux is obtained from an approach described previously for calculating the dynamics of transition-state trajectories. An application to the rotational isomerization of an aromatic ring in the bovine pancreatic trypsin inhibitor is presented. The results demonstrate the feasibility of calculating rate constants for reactions in proteins and point to the importance of solvent effects for reactions that occur near the protein surface.
Molecular Dynamics of Ferrocytochrome c: Anharmonicity of Atomic Displacements
Boryeu Mao, Michael R. Pear, J.A. McCammon and S.H. Northrup
The atomic position distributions obtained from a 32-ps molecular-dynamics simulation of tuna ferrocytochrome c at 297 K are analyzed in terms of their second, third, and fourth moments. Non-Gaussian relations among these moments are found for the majority of atoms in the molecule, indicating anharmonicity in the effective potential functions for the atomic motions. Many atoms exhibit only slightly anharmonic mobility during the 32-ps period, but about half of the atoms exhibit sizeable anharmonicity. For a typical atom, the anharmonic effects are largest for motions in the direction along which the largest displacements occur. Two classes of significantly anharmonic atoms are apparent: those whose effective potentials are distorted toward a square-well shape and those whose effective potentials have secondary minima corresponding to conformational substates.
Hinge-bending in L-Arabinose-binding Protein: The "Venus's-Flytrap" Model
Boryeu Mao, Michael R. Pear, J. Andrew McCammon and Florante A. Quiocho
Theoretical conformational energy calculations show that large changes in the width of the binding-site cleft in the L-arabinose-binding protein involve only modest changes in the protein internal energy. Solvation energy changes associated with such variations of the cleft width and with protein-ligand interactions are estimated to be significantly larger than the internal energy changes. These results indicate that the binding-site cleft is open in the unliganded protein and is induced to close upon ligation. This picture is consistent with experimental data on the structure and binding kinetics of the L- arabinose-binding protein and provides a physical framework for interpreting such data.
Surface Temperature Effects in Molecule-Surface Collisions
M. Berkowitz, D.J. Kouri and J.A. McCammon
An approximate, analytic technique is outlined for estimating the effect of surface temperature on the probabilities of molecular quantum state transitions induced by molecule-surface collisions. Applications to vibrational and electronic transitions are made. The results for the vibrational case may be useful in the problem of isotope separation.
Rate Theory for Gated Diffusion-Influenced Ligand Binding to Proteins
Scott H. Northrup, Fahimeh Zarrin and J. Andrew McCammon
The rate of binding of ligands to proteins may be determined not only by the relative diffusion rate of species through the solvent medium but also by the accessibility of the binding site. Because of the inherent flexibility and internal motion of proteins, this accessibility may fluctuate on the time scale of reaction, thereby causing the intrinsic reactivity of the protein to be a time-dependent quantity. Here, we present a general formulation of the kinetics of such gated reactions. Approximate analytical expressions for the rate constant are obtained for important limiting cases. These compare favorably with exact numerically obtained values for the gated reaction rate constant over a wide range of system parameters.
Molecular Dynamics with Stochastic Boundary Conditions
Max Berkowitz and J. Andrew McCammon
We present and illustrate a simple approach for carrying out molecular dynamics simulations subject to stochastic boundary conditions. Methods of this type are expected to be useful in the study of chemical reactions and other localized processes in dense media.
Macromolecular Conformational Energy Minimization: An Algorithm Varying Pseudodihedral Angles
Stephen C. Harvey and J. Andrew McCammon
The algorithms which have been traditionally used in the minimization of the conformational energies of macromolecules do not move groups of atoms collectively and thus ignore one of the obvious aspects of intramolecular motion. We present here an algorithm which is designed to do that and apply it to a test study on the hinge-bending motion in phenylalanine transfer RNA. Based on the observation that dihedral angles (torsions about bonds) are much more easily deformed than bond lengths or bond angles, it rotates groups of atoms in a manner that leaves bond lengths unchanged and makes minimal changes in bond angles. Each axis of rotation connects two atoms and is similar to a virtual bond, so we call the rotations pseudodihedrals. The algorithm is physically plausible, and it is efficient in the sense that when structures that have been refined by this method are subjected to more rigorous refinement by steepest descent minimization, lower final energies are obtained than when steepest descent is used alone and total computational time is reduced by about 40%.
Stochastically Gated Diffusion-Influenced Reactions
Attila Szabo, David Shoup, Scott H. Northrup and J. Andrew McCammon
The theory of diffusion-influenced reactions is extended to cases where the reactivity of the species fluctuates in time (e.g., the accessibility of a binding site of a protein is modulated by a gate). The opening and closing of the gate is assumed to be a stationary Markov process [i.e., it is described by the kinetic scheme (open) a ⇔ b (closed)]. When the reaction is described by suitable boundary conditions, by solving the appropriate reaction-diffusion equations, it is shown that the stochastically gated association rate constant (kSG) is given by kSG-1 = k∞-1 + [a-1b(a+b)κu(a+b)]-1, where κu is the Laplace transform of the time-dependent rate constant of the ungated problem and k∞ is the corresponding steady-state rate constant. The limits when the relaxation time for gate fluctuations is larger or smaller than the characteristic time for diffusion are considered. The relation to previous work is discussed. The theory is applied to three models: (i) a gated sphere, (ii) a gated disk on an infinite plane (e.g., a channel in a membrane), and (iii) a gated localized axially symmetric reactive site on the surface of a spherical macromolecule.
The Dynamic Picture of Protein Structure
J. Andrew McCammon and Martin Karplus
An important change is occurring in our picture of globular proteins. These molecules have traditionally been described in static terms. The high specificity of enzymes for their substrates has, for example, been likened to the complementarity of two pieces of a jigsaw puzzle. The static view of protein structure is now being replaced by a dynamic picture. It is recognized that the protein atoms are in a state of constant motion. The average positions correspond to what may be seen in an X-ray structure, but the atoms exhibit fluidlike motions of sizable amplitude around these average positions. The new dynamic picture subsumes the static picture in that the average positions allow for interpretation of many aspects of protein function in the classical language of structural chemistry. The recognition of the importance of fluctuations opens the way for more sophisticated and accurate discussions of protein function.
Generalized Langevin Dynamics Simulations with Arbitrary Time-Dependent Memory Kernels
M. Berkowitz, J.D. Morgan and J.A. McCammon
An algorithm is described that allows dynamical simulations to be performed based on generalized Langevin equations with arbitrary, time-dependent memory kernels. Test simulations show that good results are obtained for kernels with distinctly different forms (e.g., exponential and Gaussian).
Molecular Dynamics of Ferrocytochrome c: Time Dependence of the Atomic Displacements
John D. Morgan, J. Andrew McCammon and Scott H. Northrup
The thermal motions of the atoms in a dynamical simulation of ferrocytochrome c are geometrically decomposed into local and highly collective components, and the contributions of these components to the net motion are determined for different intervals of time. It is found that the atomic displacement magnitudes and anisotropies are governed by local motions for times < 10-12s, but that the highly collective motions tend to be dominant at longer times. Variations in this behavior are noted among different groups of atoms. Orientational correlations between the preferred directions of atomic displacement and elements of the protein structure are analyzed as a function of time scale. Finally, several sinificant implications of these results with respect to protein structure and function are considered.
J.A. McCammon and B. Mao
McGraw-Hill Yearbook of Science and Technology, 1984 (Supplement to Encyclopedia of Science and Technology, 5th Ed.), pp. 363-365 (1983, Invited review)
Dynamics of Proteins: Elements and Function
M. Karplus and J.A. McCammon
The classic view of proteins has been static in character, primarily because of the dominant role of the information provided by high-resolution X-ray crystallography for these very complex systems. The intrinsic beauty and remarkable detail of the drawings of protein structures led to an image in which each protein atom is fixed in place; an article on lysozyme by Phillips , the books by Dickerson & Geis, and by Perutz & Fermi, the review by Richardson give striking examples. Stating clearly the static viewpoint, Tanford suggested that as a result of packing considerations "the structure of native proteins must be quite rigid." Phillips wrote recently "... the period 1965-75 may be described as the decade of the rigid macromolecule. Brass models of double helical DNA and a variety of protein molecules dominated the scene and much of the thinking."
Side-Chain Rotational Isomerization in Proteins: A Mechanism Involving Gating and Transient Packing Defects
J.A. McCammon, C.Y. Lee and S.H. Northrup
Tyrosine ring rotational isomerization trajectories from a dynamical simulation are analyzed to clarify the involvement of structural fluctuations in the surrounding protein matrix. The correlation of matrix atom displacements with ring rotation is determined by examination of both ensemble averages and time sequences of protein configurations. Transient packing defects are quantitatively assessed by the Voronoi polyhedron method. The results show that the isomerization is a gated process, in which the ring rotation is systematically preceded by the spontaneous displacement of a section of adjacent backbone. This displacement creates a transient packing defect (~ 10 Å3) that helps to initiate the transition and reduces the energy barrier for the transition proper by relieving unfavorable van der Waals contacts. The results are discussed in the context of current models for motion in proteins, liquids, and solids.
Diffusion-Controlled Reactions: A Variational Formula for the Optimum Reaction Coordinate
Max Berkowitz, J.D. Morgan, J.A. McCammon and S.H. Northrup
The preferred path for a diffusion-controlled reaction depends, in general, upon global properties of the potential surface and the frictional resistance to motion upon this surface. A variational formula for this path is derived. The corresponding Euler-Lagrange equations are examined for two important special cases.
Theoretical Study of Hinge-bending in L-Arabinose-Binding Protein: Internal Energy and Free Energy Changes
Boryeu Mao and J. Andrew McCammon
The L-arabinose-binding protein of Escherichia coli is a periplasmic component of the L-arabinose transport system. Its three-dimensional structure has been determined by x-ray diffraction and shown to have two globular domains and a connecting hinge. These structural features enclose a cleft in which the L-arabinose-binding site is located. The flexibility of the protein hinge that allows hinge-bending motion is investigated here by theoretical analysis of the changes in conformational energy and molecular structure that accompany the opening and closing of the cleft. The hinge of the molecule is found to be quite permissive in that only moderate increases in the internal energy occur upon opening the cleft. Solvation changes of charged groups on the cleft-facing surfaces of the lobes are estimated to make important contributions to the overall energetics of the system. The results indicate that an open conformation for the unliganded protein is stabilized by the exposure and solvation of charged groups in the cleft, and that the cleft is induced to close upon ligand binding. This picture is consistent with experimental data on the structure and the binding kinetics of L-arabinose-binding protein, and provides a physical framework for interpreting such data.
Saddle-Point Avoidance in Diffusional Reactions
Scott H. Northrup and J. Andrew McCammon
An important concept in chemical reactions is the reaction coordinate. The identification of a preferred reaction pathway on a multidimensional energy surface is essential both for definition of the mechanism and efficient calculation of the rate of a reaction. In certain cases, such as the rotational isomerization of polymers in solution, one encounters heavily damped diffusive motion on soft potential surfaces. The identification of reaction paths in such cases requires consideration not only of the potential energy surface, but also of the inherent diffusion-preferred motion. In an important recent paper, van der Zwan and Hynes correctly identified the preferred direction of motion through a saddle point as the direction minimizing the work done against both potential and frictional forces. In this communication, we point out that cases can occur in which frictional effects cause the preferred reaction pathway to bypass saddle points completely. We also show that such cases can be analyzed approximately by contracted models of the reaction kinetics.
Molecular Dynamics of Phenylalanine Transfer RNA
M. Prabhakaran, S.C. Harvey, B. Mao and J.A. McCammon
Journal of Biomolecular Structure and Dynamics, Vol. 1, pp. 357-369 (1983)