Adaptive biasing force method for scalar and vector free energy calculations

Abstract

In free energy calculations based on thermodynamic integration, it is necessary to compute the derivatives of the free energy as a function of one (scalar case) or several (vector case) order parameters. We derive in a compact way a general formulation for evaluating these derivatives as the average of a mean force acting on the order parameters, which involves first derivatives with respect to both Cartesian coordinates and time. This is in contrast with the previously derived formulas, which require first and second derivatives of the order parameter with respect to Cartesian coordinates. As illustrated in a concrete example, the main advantage of this new formulation is the simplicity of its use, especially for complicated order parameters. It is also straightforward to implement in a molecular dynamics code, as can be seen from the pseudocode given at the end. We further discuss how the approach based on time derivatives can be combined with the adaptive biasing force method, an enhanced sampling technique that rapidly yields uniform sampling of the order parameters, and by doing so greatly improves the efficiency of free energy calculations. Using the backbone dihedral angles Phi and Psi in N-acetylalanyl-N'-methylamide as a numerical example, we present a technique to reconstruct the free energy from its derivatives, a calculation that presents some difficulties in the vector case because of the statistical errors affecting the derivatives.

Analysis of the selectivity filter of the voltage-gated sodium channel NavRh

“…The probability is further transformed by negative logarithm (G = −RT log(P Na )) for convenient comparison with the free energy calculation described below ( Figure 1D, left panel and Supplementary information, Figure S3). For more detailed thermodynamics analysis, the free energy profile for ion diffusion along the pathway was generated by the adaptive biasing force (ABF) method [42][43][44][45]. The potential of mean force (PMF) curve also indicates two energetically favorable binding sites ( Figure 1D, right panel), the positions of which agree perfectly with sites 1 and 2 derived from equilibrium simulation.…”

“…The adaptive biasing force simulation PMF was calculated using the ABF method [42][43][44][45]. Distance between the target ion and the geometric center of all α-carbons in the SF (residue 178 to 183) projected on the axis perpendicular to the membrane was chosen as the reaction coordinate, which was further divided into six non-overlapping windows with the most extracellular one spanning 6 Å and the others 3 Å.…”

Exploring Multidimensional Free Energy Landscapes Using Time-Dependent Biases on Collective Variables

“…82 Furthermore, in several instances, NANMA has served as a discriminating test case for assessing the performance of novel numerical schemes. 8,83,96 In the present work, determination of the two-dimensional free energy landscape of NANMA only represents the necessary preamble to an independent series of computations based on variables of higher collectivity than the mere φ and ψ torsional angles of the backbone. However predictable, the results of these preliminary free energy calculations depicted in Figure 1 agree quantitatively with previous investigations; see, for instance, ref 96.…”

“…More recently, the same authors have described a new estimator for the free energy gradient, based on time derivatives of the reaction coordinates, and its use for multidimensional ABF calculations. 8 The NAMD 2.6 implementation of the ABF method using eqs 2 and 8 (in the original one-dimensional version of den Otter) has been described previously. 6 In comparison, the present implementation offers a greatly extended range of applications by allowing multidimensional free energy surfaces to be computed, and by handling linear combinations of predefined variables.…”

“…An implementation of the metadynamics approach of Laio and Parrinello 4 for mainstream simulation packages has been made available only very recently. 5 The previous publicly available implementation 6 of the adaptive biasing force method 7,8 (ABF), in version 2.6 of NAMD, 9 has been applied successfully to a number of challenging cases. The domains of application of ABF include the recognition and association of peptides or proteins, [10][11][12] peptide-or protein-lipid interactions, [13][14][15] small molecules interacting in a confined environment, 16 cyclodextrin association with cholesterol, 17 steroid drugs, 18 and molecular ions, 19 as well as cyclodextrin self-assembly.…”