Extended Adaptive Biasing Force Algorithm. An On-the-Fly Implementation for Accurate Free-Energy Calculations

Proper use of the adaptive biasing force (ABF) algorithm in free-energy calculations needs certain prerequisites to be met, namely, that the Jacobian for the metric transformation and its first derivative be available and the coarse variables be independent and fully decoupled from any holonomic constraint or geometric restraint, thereby limiting singularly the field of application of the approach. The extended ABF (eABF) algorithm circumvents these intrinsic limitations by applying the time-dependent bias onto a fictitious particle coupled to the coarse variable of interest by means of a stiff spring. However, with the current implementation of eABF in the popular molecular dynamics engine NAMD, a trajectory-based post-treatment is necessary to derive the underlying free-energy change. Usually, such a posthoc analysis leads to a decrease in the reliability of the free-energy estimates due to the inevitable loss of information, as well as to a drop in efficiency, which stems from substantial read-write accesses to file systems. We have developed a user-friendly, on-the-fly code for performing eABF simulations within NAMD. In the present contribution, this code is probed in eight illustrative examples. The performance of the algorithm is compared with traditional ABF, on the one hand, and the original eABF implementation combined with a posthoc analysis, on the other hand. Our results indicate that the on-the-fly eABF algorithm (i) supplies the correct free-energy landscape in those critical cases where the coarse variables at play are coupled to either each other or to geometric restraints or holonomic constraints, (ii) greatly improves the reliability of the free-energy change, compared to the outcome of a posthoc analysis, and (iii) represents a negligible additional computational effort compared to regular ABF. Moreover, in the proposed implementation, guidelines for choosing two parameters of the eABF algorithm, namely the stiffness of the spring and the mass of the fictitious particles, are proposed. The present on-the-fly eABF implementation can be viewed as the second generation of the ABF algorithm, expected to be widely utilized in the theoretical investigation of recognition and association phenomena relevant to physics, chemistry, and biology.自适应偏置力（adaptive biasing force, ABF）算法在自由能计算中的合理应用需满足若干先决条件：即度量变换的雅可比（Jacobian）矩阵及其一阶导数需可获取，且粗变量需相互独立并完全脱离任何完整约束或几何约束，这极大地局限了该方法的应用范围。扩展型自适应偏置力（extended ABF, eABF）算法则通过将时变偏置施加于通过刚性弹簧与目标粗变量耦合的虚拟粒子上，规避了这些固有局限。然而，当前在主流分子动力学模拟引擎NAMD中实现的eABF版本，需通过基于轨迹的后处理步骤才能提取得到对应的自由能变化量。通常这类事后分析会因不可避免的信息丢失而降低自由能估计的可靠性，同时由于大量文件系统读写操作导致计算效率下降。我们开发了一款用户友好的运行时（on-the-fly）代码，可在NAMD中执行eABF模拟。在本研究中，我们通过八个示例测试了该代码的性能：一方面将该算法的性能与传统ABF进行对比，另一方面则与结合事后分析的原始eABF实现方案进行对比。我们的研究结果表明：（i）当所涉及的粗变量彼此耦合或与几何约束、完整约束绑定的关键场景下，运行时eABF算法可输出准确的自由能面；（ii）相较于事后分析结果，该算法大幅提升了自由能变化估计的可靠性；（iii）相较于常规ABF，该算法仅带来可忽略的额外计算开销。此外，在本次实现方案中，我们还提出了eABF算法两个关键参数的选取指南，即弹簧刚度与虚拟粒子质量。本运行时eABF实现可被视为第二代ABF算法，有望在物理、化学与生物学相关的识别与缔合现象理论研究中得到广泛应用。