Building Water Models, A Different Approach

Simplified, classical models of water are an integral part of atomistic molecular simulations, especially in biology and chemistry where hydration effects are critical. Yet, despite several decades of effort, these models are still far from perfect. Presented here is an alternative approach to constructing point charge water models - currently, the most commonly used type. In contrast to the conventional approach, we do not impose any geometry constraints on the model other than symmetry. Instead, we optimize the distribution of point charges to best describe the "electrostatics" of the water molecule, which is key to many unusual properties of liquid water. The search for the optimal charge distribution is performed in 2D parameter space of key lowest multipole moments of the model, to find best fit to a small set of bulk water properties at room temperature. A virtually exhaustive search is enabled via analytical equations that relate the charge distribution to the multipole moments. The resulting "optimal" 3-charge, 4-point rigid water model (OPC) reproduces a comprehensive set of bulk water properties significantly more accurately than commonly used rigid models: average error relative to experiment is 0.76%. Close agreement with experiment holds over a wide range of temperatures, well outside the ambient conditions at which the fit to experiment was performed. The improvements in the proposed water model extend beyond bulk properties: compared to the common rigid models, predicted hydration free energies of small molecules in OPC water are uniformly closer to experiment, root-mean-square error < 1kcal/mol

Point Charges Optimally Placed to Represent the Multipole Expansion of Charge Distributions

We propose an approach for approximating electrostatic charge distributions with a small number of point charges to optimally represent the original charge distribution. By construction, the proposed optimal point charge approximation (OPCA) retains many of the useful properties of point multipole expansion, including the same far-field asymptotic behavior of the approximate potential. A general framework for numerically computing OPCA, for any given number of approximating charges, is described. We then derive a 2-charge practical point charge approximation, PPCA, which approximates the 2-charge OPCA via closed form analytical expressions, and test the PPCA on a set of charge distributions relevant to biomolecular modeling. We measure the accuracy of the new approximations as the RMS error in the electrostatic potential relative to that produced by the original charge distribution, at a distance the extent of the charge distribution–the mid-field. The error for the 2-charge PPCA is found to be on average 23% smaller than that of optimally placed point dipole approximation, and comparable to that of the point quadrupole approximation. The standard deviation in RMS error for the 2-charge PPCA is 53% lower than that of the optimal point dipole approximation, and comparable to that of the point quadrupole approximation. We also calculate the 3-charge OPCA for representing the gas phase quantum mechanical charge distribution of a water molecule. The electrostatic potential calculated by the 3-charge OPCA for water, in the mid-field (2.8 Å from the oxygen atom), is on average 33.3% more accurate than the potential due to the point multipole expansion up to the octupole order. Compared to a 3 point charge approximation in which the charges are placed on the atom centers, the 3-charge OPCA is seven times more accurate, by RMS error. The maximum error at the oxygen-Na distance (2.23 Å ) is half that of the point multipole expansion up to the octupole order

Accelerating Electrostatic Surface Potential Calculation with Multiscale Approximation on Graphics Processing Units

Tools that compute and visualize biomolecular electrostatic surface potential have been used extensively for studying biomolecular function. However, determining the surface potential for large biomolecules on a typical desktop computer can take days or longer using currently available tools and methods. This paper demonstrates how one can take advantage of graphic processing units (GPUs) available in today’s typical desktop computer, together with a multiscale approximation method, to significantly speedup such computations. Specifically, the electrostatic potential computation, using an analytical linearized Poisson Boltzmann (ALPB) method, is implemented on an ATI Radeon 4870 GPU in combination with the hierarchical charge partitioning (HCP) multiscale approximation. This implementation delivers a combined 1800-fold speedup for a 476,040 atom viral capsid

A partition function approximation using elementary symmetric functions.

In statistical mechanics, the canonical partition function [Formula: see text] can be used to compute equilibrium properties of a physical system. Calculating [Formula: see text] however, is in general computationally intractable, since the computation scales exponentially with the number of particles [Formula: see text] in the system. A commonly used method for approximating equilibrium properties, is the Monte Carlo (MC) method. For some problems the MC method converges slowly, requiring a very large number of MC steps. For such problems the computational cost of the Monte Carlo method can be prohibitive. Presented here is a deterministic algorithm - the direct interaction algorithm (DIA) - for approximating the canonical partition function [Formula: see text] in [Formula: see text] operations. The DIA approximates the partition function as a combinatorial sum of products known as elementary symmetric functions (ESFs), which can be computed in [Formula: see text] operations. The DIA was used to compute equilibrium properties for the isotropic 2D Ising model, and the accuracy of the DIA was compared to that of the basic Metropolis Monte Carlo method. Our results show that the DIA may be a practical alternative for some problems where the Monte Carlo method converge slowly, and computational speed is a critical constraint, such as for very large systems or web-based applications

Accuracy comparison.

<p>Accuracy for the direct interaction algorithm (DIA) and the Metropolis Monte Carlo (MC) method. Accuracy is calculated as the RMS error relative to the exact value. The number of steps is chosen such that the computation time for the MC method is at least 10 times the computation time for the DIA (<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0051352#pone-0051352-t001" target="_blank">Table 1</a>).</p

Accuracy as a function of system size.

<p>Accuracy for the direct interaction algorithm (DIA) and the Metropolis Monte Carlo (MC) method as a function of system size (log-log scale). Accuracy is calculated as the RMS error relative to the exact value. The number of steps is chosen such that the computation time for the MC method is at least 10 times the computation time for the DIA (<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0051352#pone-0051352-t001" target="_blank">Table 1</a>).</p

Partitioning of microstates.

<p>For a 4 particle system, the set of all possible microstates are partitioned into subsets of states as follows. With particle 1 as the selected particles, the microstates are first partitioned into two subsets, one with particle 1 in state and another with , with contributions to the partition function corresponding to and respectively (Eq. (3)). Each of these subsets are further partitioned into subsets with the same number of particles, , in state , with contributions to the partition function corresponding to and in Eq. (10) and (11) respectively.</p

Number of Monte Carlo (MC) steps.

<p>The number of MC steps is chosen such that the computation time (CPU time) for the MC method is at least 10 times the computation time for the direct interaction algorithm (DIA).</p

Direct and indirect interactions.

<p>For the five particle system shown here, with particle 1 as the selected particle, the direct interactions are shown as solid lines, and indirect interactions as dotted lines.</p

Split-merge algorithm.

<p>Consider an 8-particle system, with particle 1 being the selected particle. First, the split-merge algorithm recursively separates the particles, other than the selected particle, into a hierarchical binary tree. Next, the elementary symmetric function (ESF) for the leaf nodes, which consist of a single particle, are calculated. The ESF for all the other nodes, are then computed by recursively, starting from the bottom, merging the ESF from the two branches for each node using Eq. (21).</p