Accurate geometries for “Mountain pass” regions of the Ramachandran plot using quantum chemical calculations

Unusual local arrangements of protein in Ramachandran space are not well represented by standard geometry tools used in either protein structure refinement using simple harmonic geometry restraints or in protein simulations using molecular mechanics force fields. In contrast, quantum chemical computations using small poly‐peptide molecular models can predict accurate geometries for any well‐defined backbone Ramachandran orientation. For conformations along transition regions—ϕ from −60 to 60°—a very good agreement with representative high‐resolution experimental X‐ray (≤1.5 Å) protein structures is obtained for both backbone C⁻¹‐N‐C_α angle and the nonbonded O⁻¹…C distance, while “standard geometry” leads to the “clashing” of O…C atoms and Amber FF99SB predicts distances too large by about 0.15 Å. These results confirm that quantum chemistry computations add valuable support for detailed analysis of local structural arrangements in proteins, providing improved or missing data for less understood high‐energy or unusual regions.     

Free‐energy function based on an all‐atom model for proteins

We have developed a free‐energy function based on an all‐atom model for proteins. It comprises two components, the hydration entropy (HE) and the total dehydration penalty (TDP). Upon a transition to a more compact structure, the number of accessible configurations arising from the translational displacement of water molecules in the system increases, leading to a water‐entropy gain. To fully account for this effect, the HE is calculated using a statistical‐mechanical theory applied to a molecular model for water. The TDP corresponds to the sum of the hydration energy and the protein intramolecular energy when a fully extended structure, which possesses the maximum number of hydrogen bonds with water molecules and no intramolecular hydrogen bonds, is chosen as the standard one. When a donor and an acceptor (e.g., N and O, respectively) are buried in the interior after the break of hydrogen bonds with water molecules, if they form an intramolecular hydrogen bond, no penalty is imposed. When a donor or an acceptor is buried with no intramolecular hydrogen bond formed, an energetic penalty is imposed. We examine all the donors and acceptors for backbone‐backbone, backbone‐side chain, and side chain‐side chain intramolecular hydrogen bonds and calculate the TDP. Our free‐energy function has been tested for three different decoy sets. It is better than any other physics‐based or knowledge‐based potential function in terms of the accuracy in discriminating the native fold from misfolded decoys and the achievement of high Z‐scores. Proteins 2009. © 2009 Wiley‐Liss, Inc.     

Classical cumulant dynamics for statistical chemical physics

Abstract

We developed classical cumulant dynamics for statistical mechanics in order to evaluate thermal equilibrium properties of a given system. The equations of motion (EOMs) for momentum and position were formulated together with those for second-order cumulant variables, which are functions of second-order moments. From the Kramers equation, and simplified EOMs were obtained by assuming a stationary state limit. The present method combined with the umbrella integration method was applied to evaluate free energy surface of a seven-particle Morse cluster. With low computational costs, the present approach gave almost equivalent free energy barrier those by conventional classical molecular dynamics.     

Calculating the free energy of association of transmembrane helices

A large number of experimental studies have been devoted to quantifying the interaction between transmembrane (TM) helices in detergent micelles and, more recently, in bilayers. Theoretical calculation of association free energy of TM helices would be useful for predicting the propensity of given sequences to oligomerize and for understanding the difference between association in micelles and in bilayers. In this article, the theoretical foundation for calculating the standard association free energy of TM helices is laid out and is applied to glycophorin A in both micelles and bilayers. The standard association free energy is decomposed into the effective energy, translational, rotational, and conformational entropy terms. The effective energy of association is obtained by molecular dynamics simulations in an implicit membrane model. The translational and rotational entropy of association is calculated from the probability distribution of the translational and rotational degrees of freedom obtained from the molecular dynamics simulations. The side-chain conformational entropy of association is estimated from the probability distribution obtained by rigid rotation of all side-chain dihedral angles. The calculated standard association free energy of glycophorin A in N-dodecylphosphocholine micelles is in good agreement with the experimental value. The translational entropy cost is larger, whereas the rotational entropy cost is smaller in bilayers than in micelles. The standard association free energy in 1,2-dimyristoyl-sn-glycero-3-phosphocholine bilayers is calculated to be approximately 1.3 kcal/mol more favorable than in N-dodecylphosphocholine micelles, consistent with available experimental data.     

Standardizing the free energy change of transmembrane helix-helix interactions

Side-to-side associations of transmembrane alpha-helices are integral components of the structure and function of helical membrane proteins. A fundamental unknown in the understanding of the chemical principles driving the lateral interactions between transmembrane alpha-helices is the balance of forces arising from the polypeptide sequence versus the hydrophobic solvent. To begin to address this question, a consideration of basic thermodynamic principles has been applied to assess the experimental free energy change associated with transmembrane helix dimerization in micelles. This analysis demonstrates the ability to partition the apparent free energy of transmembrane helix-helix association into two components. The first component is a statistical energy term, which arises from the fact that there are an unequal number of reactants and products. The second component is a standard state free energy change, which informs on the molecular details of the transmembrane helix self-association reaction. The advantage of separating these two energy terms arises from the fact that extrapolation to the standard state free energy change normalizes the statistical energy term so that it applies equivalently in all experimental systems. Accompanying experimental results for the glycophorin A transmembrane alpha-helix dimer measured in micelles are well described by these theoretical components assuming an ideal-dilute solution.     

The statistical-thermodynamic basis for computation of binding affinities: a critical review.

Although the statistical thermodynamics of noncovalent binding has been considered in a number of theoretical papers, few methods of computing binding affinities are derived explicitly from this underlying theory. This has contributed to uncertainty and controversy in certain areas. This article therefore reviews and extends the connections of some important computational methods with the underlying statistical thermodynamics. A derivation of the standard free energy of binding forms the basis of this review. This derivation should be useful in formulating novel computational methods for predicting binding affinities. It also permits several important points to be established. For example, it is found that the double-annihilation method of computing binding energy does not yield the standard free energy of binding, but can be modified to yield this quantity. The derivation also makes it possible to define clearly the changes in translational, rotational, configurational, and solvent entropy upon binding. It is argued that molecular mass has a negligible effect upon the standard free energy of binding for biomolecular systems, and that the cratic entropy defined by Gurney is not a useful concept. In addition, the use of continuum models of the solvent in binding calculations is reviewed, and a formalism is presented for incorporating a limited number of solvent molecules explicitly.     

Computational delineation of the catalytic step of a high‐fidelity DNA polymerase

The Bacillus fragment, belonging to a class of high‐fidelity polymerases, demonstrates high processivity (adding ～115 bases per DNA binding event) and exceptional accuracy (1 error in 10⁶ nucleotide incorporations) during DNA replication. We present analysis of structural rearrangements and energetics just before and during the chemical step (phosphodiester bond formation) using a combination of classical molecular dynamics, mixed quantum mechanics molecular mechanics simulations, and free energy computations. We find that the reaction is associative, proceeding via the two‐metal‐ion mechanism, and requiring the proton on the terminal primer O3′ to transfer to the pyrophosphate tail of the incoming nucleotide before the formation of the pentacovalent transition state. Different protonation states for key active site residues direct the system to alternative pathways of catalysis and we estimate a free energy barrier of ～12 kcal/mol for the chemical step. We propose that the protonation of a highly conserved catalytic aspartic acid residue is essential for the high processivity demonstrated by the enzyme and suggest that global motions could be part of the reaction free energy landscape.     

Escapement mechanisms: Efficient free energy transduction by reciprocally-coupled gating

Conversion of the free energy of NTP hydrolysis efficiently into mechanical work and/or information by transducing enzymes sustains living systems far from equilibrium, and so has been of interest for many decades. Detailed molecular mechanisms, however, remain puzzling and incomplete. We previously reported that catalysis of tryptophan activation by tryptophanyl-tRNA synthetase, TrpRS, requires relative domain motion to re-position the catalytic Mg²⁺ ion, noting the analogy between that conditional hydrolysis of ATP and the escapement mechanism of a mechanical clock. The escapement allows the time-keeping mechanism to advance discretely, one gear at a time, if and only if the pendulum swings, thereby converting energy from the weight driving the pendulum into rotation of the hands. Coupling of catalysis to domain motion, however, mimics only half of the escapement mechanism, suggesting that domain motion may also be reciprocally coupled to catalysis, completing the escapement metaphor. Computational studies of the free energy surface restraining the domain motion later confirmed that reciprocal coupling: the catalytic domain motion is thermodynamically unfavorable unless the PPi product is released from the active site. These two conditional phenomena—demonstrated together only for the TrpRS mechanism—function as reciprocally-coupled gates. As we and others have noted, such an escapement mechanism is essential to the efficient transduction of NTP hydrolysis free energy into other useful forms of mechanical or chemical work and/or information. Some implementation of both gating mechanisms—catalysis by domain motion and domain motion by catalysis—will thus likely be found in many other systems.     

A Comparison of Seven Fast but Approximate Methods to Compute the Free Energy of Deprotonation for Amino Acids in Aqueous Solution

Abstract

In recent years, a variety of methods based on statistical mechanics have been successfully applied to calculate free energy differences of chemical reactions from molecular simulation. The accuracy and computational efficiency vary strongly between these methods. Seven approximate but fast methods to calculate free energy differences are compared in terms of accuracy and efficiency with the accurate but expensive thermodynamic integration method as reference, using 28 protonation and deprotonation reactions of aspartic acid in aqueous solution as test cases. At least two simulations are required to obtain an accurate free energy difference between two states of the system. Both, the averaged one-step perturbation method and the linear response method yield the most accurate results, while the latter method shows the fastest convergence.     

Thermodynamics and Kinetics of Molecular Motors

Molecular motors are first and foremost molecules, governed by the laws of chemistry rather than of mechanics. The dynamical behavior of motors based on chemical principles can be described as a random walk on a network of states. A key insight is that any molecular motor in solution explores all possible motions and configurations at thermodynamic equilibrium. By using input energy and chemical design to prevent motion that is not wanted, what is left behind is the motion that is desired. This review is focused on two-headed motors such as kinesin and Myosin V that move on a polymeric track. By use of microscopic reversibility, it is shown that the ratio between the number of forward steps and the number of backward steps in any sufficiently long time period does not directly depend on the mechanical properties of the linker between the two heads. Instead, this ratio is governed by the relative chemical specificity of the heads in the front-versus-rear position for the fuel, adenosine triphosphate and its products, adenosine diphosphate and inorganic phosphate. These insights have been key factors in the design of biologically inspired synthetic molecular walkers constructed out of DNA or out of small organic molecules.     

Discrimination of near-native protein structures from misfolded models by empirical free energy functions

Free energy potentials, combining molecular mechanics with empirical solvation and entropic terms, are used to discriminate native and near-native protein conformations from slightly misfolded decoys. Since the functional forms of these potentials vary within the field, it is of interest to determine the contributions of individual free energy terms and their combinations to the discriminative power of the potential. This is achieved in terms of quantitative measures of discrimination that include the correlation coefficient between RMSD and free energy, and a new measure labeled the minimum discriminatory slope (MDS). In terms of these criteria, the internal energy is shown to be a good discriminator on its own, which implies that even well-constructed decoys are substantially more strained than the native protein structure. The discrimination improves if, in addition to the internal energy, the free energy expression includes the electrostatic energy, calculated by assuming non-ionized side chains, and an empirical solvation term, with the classical atomic solvation parameter model providing slightly better discrimination than a structure-based atomic contact potential. Finally, the inclusion of a term representing the side chain entropy change, and calculated by an established empirical scale, is so inaccurate that it makes the discrimination worse. It is shown that both the correlation coefficient and the MDS value (or its dimensionless form) are needed for an objective assessment of a potential, and that together they provide much more information on the origins of discrimination than simple inspection of the RMSD-free energy plots.     

Does Flory–Huggins theory help in interpreting solute partitioning experiments?

The development of Flory–Huggins (FH) theory is reviewed, particularly with regard to the molecular significance of the interaction parameter that scales the contact interaction of solute and solvent. The chemical potential given by FH theory for an “idealute” solute is then compared with that provided by a more general, statistical thermodynamic approach. It is found that the FH contact term does not directly correspond to the solvation free energy. The significance of this result for the interpretation of free energies of transfer of a solute from one solvent to another is examined. It is shown that neither the earlier recommended standard free energy change for the process (using the infinitely dilute reference state, mole fraction units) nor the recently recommended FH-corrected standard free energy change provides the solvation energy desired. Instead, the standard free energy using the infinitely dilute reference state and molarity units, as long advocated by Ben-Naim, provides the desired solvation free energy. Correction of extant values, based on mole fraction units, is easily made. However, application of such results to problems of protein folding is not straightforward. © 1994 John Wiley & Sons, Inc.     

Selecting near-native conformations in homology modeling: the role of molecular mechanics and solvation terms

A free energy function, combining molecular mechanics energy with empirical solvation and entropic terms, is used for ranking near-native conformations that occur in the conformational search steps of homology modeling, i.e., side-chain search and loop closure calculations. Correlations between the free energy and RMS deviation from the X-ray structure are established. It is shown that generally both molecular mechanics and solvation/entropic terms should be included in the potential. The identification of near-native backbone conformations is accomplished primarily by the molecular mechanics term that becomes the dominant contribution to the free energy if the backbone is even slightly strained, as frequently occurs in loop closure calculations. Both terms become equally important if a sufficiently accurate backbone conformation is found. Finally, the selection of the best side-chain positions for a fixed backbone is almost completely governed by the solvation term. The discriminatory power of the combined potential is demonstrated by evaluating the free energies of protein models submitted to the first meeting on Critical Assessment of techniques for protein Structure Prediction (CASP1), and comparing them to the free energies of the native conformations.     

Coupling of fast and slow modes in the reaction pathway of the minimal hammerhead ribozyme cleavage

By employing classical molecular dynamics, correlation analysis of coupling between slow and fast dynamical modes, and free energy (umbrella) sampling using classical as well as mixed quantum mechanics molecular mechanics force fields, we uncover a possible pathway for phosphoryl transfer in the self-cleaving reaction of the minimal hammerhead ribozyme. The significance of this pathway is that it initiates from the minimal hammerhead crystal structure and describes the reaction landscape as a conformational rearrangement followed by a covalent transformation. The delineated mechanism is catalyzed by two metal (Mg(2+)) ions, proceeds via an in-line-attack by CYT 17 O2' on the scissile phosphorous (ADE 1.1 P), and is therefore consistent with the experimentally observed inversion configuration. According to the delineated mechanism, the coupling between slow modes involving the hammerhead backbone with fast modes in the cleavage site appears to be crucial for setting up the in-line nucleophilic attack.     

Lee Pedersen's work in theoretical and computational chemistry and biochemistry

Nature at the lab level in biology and chemistry can be described by the application of quantum mechanics.In many cases,a reasonable approximation to quantum mechanics is classical mechanics realized through Newton’s equations of motion.Dr.Pedersen began his career using quantum mechanics to describe the properties of small molecular complexes that could serve as models for biochemical systems.To describe large molecular systems required a drop-back to classical means and this led surprisingly to a major improvement in the classical treatment of electrostatics for all molecules,not just biological molecules.Recent work has involved the application of quantum mechanics for the putative active sites of enzymes to gain greater insight into the key steps in enzyme catalysis.     

Molecular Dynamics as a Mathematical Mapping. II. Partial Derivatives in the Microcanonical Ensemble

Abstract

Performing molecular dynamics in a fully continuous and differentiable framework can be viewed as a deterministic mathematical mapping between, on one side, the force field parameters that describe the potential energy interactions and input macroscopic conditions, and, on the other, the calculated corresponding macroscopic properties of the bulk molecular system.

Within this framework, it is possible to apply standard methods of variational calculus for the computation of the partial derivatives of the molecular dynamics mapping based on the integration of either the adjoint equations or the sensitivity equations of the classical Newtonian equations of motion. We present procedures for these computations in the standard microcanonical (N, V, E) ensemble, and compare the computational efficiency of the two approaches. The general formulations developed are applied to the specific example of bulk ethane fluid.

With these procedures in place, it is now possible to compute the partial derivatives of any property determined by molecular dynamics with respect to any input property and any potential parameter. Moreover, these derivatives are computed to essentially the same level of numerical accuracy as the output properties themselves.     

Functionality Versus “Typical Product” Measures of Technological Progress

Technological progress and adoption are fundamentally interconnected with environmental challenges faced by society. At the product level, researchers often explore the interplay between technological change and the environment by tracking trends in impacts per unit functionality—for example, gasoline consumed per distance traveled by a vehicle. In this article, we explore an alternative measure: “typical product.” A typical product measure accounts for changes in consumers’ demand and use of products as product quality improves—for example, gasoline consumed for a typical driving pattern for a vehicle. We compare and contrast functionality and typical product measures through a case study of electricity use to fabricate Intel desktop microprocessors from 1995 to 2006. The functionality normalization is measured in terms of electricity use per transistor produced. Results show rapid and sustained exponential decrease. The “typical product” measures electricity use per typical desktop microprocessor of a given year (e.g., a Pentium II in 1998, a Pentium IV in 2002). Results show that, despite fluctuations, energy use per typical microprocessor is roughly constant over the 12‐year period. The explanation of this result is that although technological progress dramatically reduces the energy needed per transistor, it also induces demand for more powerful chips, which contain many more transistors. The typical product measure has applications in defining functional units in life cycle assessment, characterizing rebound effects, and measuring energy efficiency trends.     

A molecular level study of selective cation capture by a host–guest mechanism for 25,26,27,28-tetramethoxycalix[4]arene in MClO4 solution (M = Na,K)

25,26,27,28-Tetramethoxycalix[4] arene selectively captures cations by changing the conformation. In this study, a hybrid approach of ab initio molecular orbital calculation and statistical mechanics for molecular liquid was utilised to understand the capture mechanism in electrolyte solution phase at molecular level. The association free energy and solvation structure were evaluated on the basis of statistical mechanics for molecular liquids. The selectivity is correctly reproduced by the computation, namely cone conformer captures Na⁺ while K⁺ is recognised by the partial-cone conformer.     

Solvent effect on binding thermodynamics of biopolymers

The indirect solvent-induced effect on the free energy of binding of biopolymers is examined within the framework of classical statistical mechanics. We focus specifically on the role of the solute-solvent hydrogen bonding. In particular, we have estimated the first order solvent effect on the indirect interaction between two biopolymers. We find that the solvent-induced interactions between two hydrophilic groups through water-bridged hydrogen bonds could significantly enhance the binding free energy. Some preliminary estimates indicate that this effect is significant and perhaps could be crucial in molecular recognition processes. Furthermore, we have calculated, from crystal structure data, the distance distribution between all the oxygens and nitrogens on the surface of some proteins that do not belong to the binding domain. In most cases we found an enhanced peak in the range of 4-5 A, which is where we expect to find strong solvent-induced interactions.     

Statistical thermodynamic analysis of peptide and protein insertion into lipid membranes.

A statistical thermodynamic approach is used to analyze the various contributions to the free energy change associated with the insertion of proteins and protein fragments into lipid bilayers. The partition coefficient that determines the equilibrium distribution of proteins between the membrane and the solution is expressed as the ratio between the partition functions of the protein in the two phases. It is shown that when all of the relevant degrees of freedom (i.e., those that change their character upon insertion into the membrane) can be treated classically, the partition coefficient is fully determined by the ratio of the configurational integrals and thus does not involve any mass-dependent factors, a conclusion that is also valid for related processes such as protein adsorption on a membrane surface or substrate binding to proteins. The partition coefficient, and hence the transfer free energy, depend only on the potential energy of the protein in the membrane. Expressing this potential as a sum of a "static" term, corresponding to the equilibrium (minimal free energy) configuration of the protein in the membrane, and a "dynamical" term representing fluctuations around the equilibrium configuration, we show that the static term contains the "solvation" and "lipid perturbation" contributions to the transfer free energy. The dynamical term is responsible for the "immobilization" free energy, reflecting the loss of translational and rotational entropy of the protein upon incorporation into the membrane. Based on a recent molecular theory of lipid-protein interactions, the lipid perturbation and immobilization contributions are then expressed in terms of the elastic deformation free energy resulting from the perturbation of the lipid environment by the foreign (protein) inclusion. The model is formulated for cylindrically shaped proteins, and numerical estimates are given for the insertion of an alpha-helical peptide into a lipid bilayer. The immobilization free energy is shown to be considerably smaller than in previous estimates of this quantity, and the origin of the difference is discussed in detail.