We have applied the Transition Path Sampling algorithm to the reaction catalyzed by the enzyme Lactate Dehydrogenase. This study demonstrates the ease of scaling Transition Path Sampling for applications on many degree of freedom systems, whose energy surface is a complex terrain of valleys and saddle points. As a Monte Carlo importance sampling method, transition path sampling is capable of surmounting barriers in path phase space and focuses simulation on the rare event of enzyme catalyzed atom transfers. Generation of the transition path ensemble, for this reaction, resolves a paradox in the literature in which some studies exposed the catalytic mechanism of hydride and proton transfer by lactate dehydrogenase to be concerted and others stepwise. Transition path sampling has confirmed both mechanisms as possible paths from reactants to products. With the objective to identify a generalized, reduced reaction coordinate, time series of both donor-acceptor distances and residue distances from the active site have been examined. During the transition from pyruvate to lactate, residues located behind the transferring hydride collectively compress toward the active site causing residues located behind the hydride acceptor to relax away. It is demonstrated that an incomplete compression/relaxation transition across the donor-acceptor axis compromises the reaction.

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Theoretically based computational methods have been developed in our group to identify protein motions, symmetrically coupled to the reaction coordinate, which modulate the width and height of the barrier to reaction. Previous studies have applied the methods to horse liver alcohol dehydrogenase (HLADH), to help explain experimental kinetic isotope effects. In this paper the methods have been applied to the two isoforms of human lactate dehydrogense (LDH) enzymes which facilitate hydride transfer during the interconversion of pyruvate and lactate. LDH isoforms have evolved to accommodate substrate demand in different parts of the body. The active sites of the isoforms are identical in amino acid content yet the kinetics are distinct. We have performed molecular dynamics simulations for each isoform with either substrate bound. The signature of the protein promoting vibration (PPV) is distinct for each isoform due to differences in the donor-acceptor distance. We hypothesize that kinetic control of hydride transfer may be exerted via a decreased donoracceptor distance when lactate is bound to the heart isoform and when pyruvate is bound to the skeletal muscle isoform. The identity, frequency, and position of active site amino acid motions correlated to the donor-acceptor motion also vary for each isoform. These results demonstrate that even in almost identical enzymes, subtle differences in protein structure, remote from the active site, can have significant effects on reaction dynamics.

Background: Many functional proteins have a symmetric structure. Most of these are multimeric complexes, which are made of non-symmetric monomers arranged in a symmetric manner. However, there are also a large number of proteins that have a symmetric structure in the monomeric state. These internally symmetric proteins are interesting objects from the point of view of their folding, function, and evolution. Most algorithms that detect the internally symmetric proteins depend on finding repeating units of similar structure and do not use the symmetry information.Results: We describe a new method, called SymD, for detecting symmetric protein structures. The SymD procedure works by comparing the structure to its own copy after the copy is circularly permuted by all possible number of residues. The procedure is relatively insensitive to symmetry-breaking insertions and deletions and amplifies positive signals from symmetry. It finds 70% to 80% of the TIM barrel fold domains in the ASTRAL 40 domain database and 100% of the beta-propellers as symmetric. More globally, 10% to 15% of the proteins in the ASTRAL 40 domain database may be considered symmetric according to this procedure depending on the precise cutoff value used to measure the degree of perfection of the symmetry. Symmetrical proteins occur in all structural classes and can have a closed, circular structure, a cylindrical barrel-like structure, or an open, helical structure. Conclusions: SymD is a sensitive procedure for detecting internally symmetric protein structures. Using this procedure, we estimate that 10% to 15% of the known protein domains may be considered symmetric. We also report an initial, overall view of the types of symmetries and symmetric folds that occur in the protein domain structure universe.

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For a system in thermal equilibrium, described by classical statistical mechanics, we derive an unbiased estimator for the marginal probability distribution of a coordinate of interest, rho( x). This result provides a "binless" method for estimating the potential of mean force, Phi = -beta (-1) ln rho, eliminating the need to construct histograms or perform numerical thermodynamic integration. In our method, the distribution that we seek to compute is expressed as the sum of a reference distribution, rho 0(x)essentially an initial guess or estimate of rho( x)and a correction term. While the method is valid for arbitrary rho 0, we speculate that an accurate choice of the reference distribution improves the convergence of the method. Using a model molecule, simulated both in vacuum and in solvent, we validate our proposed approach and compare its performance with the histogram and thermodynamic integration methods. We also discuss and validate an extension in which our approach is used in combination with a biasing force, meant to improve uniform sampling of the coordinate of interest.

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