Collaborative Research: A new diffuse-interface approach to ensemble average solvation energy: modeling, analysis and computation

在生物过程的定量分析中，溶质和溶剂之间复杂的相互作用通常用溶剂化能来描述：将溶质或生物分子（如蛋白质、DNA、RNA）从真空转移到感兴趣的溶剂环境（如具有一定离子强度的水）的自由能。与将溶质和溶剂视为单独分子的显式溶剂模型相比，隐式溶剂模型将溶剂相的作用平均化为连续介质，效率更高，因此可以处理更大的系统。隐式溶剂模型的核心是一个分离溶质和溶剂域的界面。然而，通常使用的界面定义是特别划分，因此不可忽视地高估或低估了溶剂化自由能。变分隐式溶剂化模型（VISM）通过优化溶剂化能泛函耦合、生物分子的离散描述和溶剂的连续描述，已经成为计算溶质-溶剂界面配置的一种成功方法。本合作研究项目的目的是提高用VISM方法预测溶剂化能的精度和计算效率。该项目将涉及数学模型和数值算法的新发展，可以更好地反映生物大分子与周围离子环境之间的相互作用。此外，该项目将为学生提供参与合作研究的机会。研究表明，忽略由原子振动或热力学波动引起的溶质-溶剂界面的固有随机性，会导致在预测溶剂化能时出现重大误差。由于实验观察到的溶剂化能是系综平均，本研究的主要目标是开发一种能够捕获系综平均溶剂化能（EASE）的扩散界面VISM。EASE表示从溶剂-溶剂体系的所有可能的微观状态计算出的溶剂化能的加权平均值。在EASE的常规计算中，需要进行显式溶剂模拟，例如分子动力学（MD），以获得数千种溶质-溶剂构型（快照），并对每个快照进行能量计算。利用统计力学和几何测量理论的工具，该项目旨在开发一种创新的扩散界面主义，通过利用单个扩散界面剖面而不是数千个单独的快照来严格再现EASE。此外，虽然与使用预定界面的隐式溶剂模型相比，扩散界面vism提供了更高的精度，但与确定界面相关的计算成本带来了重大挑战。这一限制阻碍了扩散界面vism在大分子体系中的应用。因此，本研究的第二个目标是在扩散界面vism框架内开发用于溶剂化能计算的加速数值算法。通过利用所提出的扩散界面VISM泛函的结构，将设计一种结合增广拉格朗日方法和不精确牛顿格式的新型优化算法。这种创新的方法将显著提高计算效率。最后，对该模型的分析将为研究总变差最小化问题解的规律性提供新的技术。在本研究项目范围内开发的分析技术也可能在数学界具有更广泛的相关性和适用性。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

In the quantitative analysis of biological processes, the complex interactions between the solute and solvent are typically described by solvation energies: the free energy of transferring the solute or biomolecule (e.g., proteins, DNA, RNA) from the vacuum to a solvent environment of interest (e.g., water at a certain ionic strength). Compared with explicit solvent models, which treat both the solute and the solvent as individual molecules, implicit solvent models, which average the effect of solvent phase as continuum media, are much more efficient and thus can handle much larger systems. Central to the construction of implicit solvent models is an interface separating the solute and solvent domains. However, commonly used interface definitions are ad hoc partitions and thus either non-negligibly overestimate or underestimate the solvation free energies. Variational implicit solvation models (VISM) have emerged as a successful approach to calculate the disposition of the solute-solvent interface by optimizing a solvation energy functional coupling the discrete description of the biomolecule and the continuum description of the solvent. The objective of this collaborative research project is to enhance the precision and computational efficiency of solvation energy prediction by means of VISM. The project will involve novel developments in mathematical models and numerical algorithms that can better reflect the interactions between biological macromolecules and the surrounding ionic environment. In addition, the project will provide opportunities for students to be involved in this collaborative research.Research has shown that neglecting the inherent randomness associated with solute-solvent interfaces, resulting from atom vibrations or thermodynamic fluctuations, can lead to substantial errors in predicting solvation energies. Since experimentally observed solvation energies are ensemble averages, the primary objective of this research is to develop a diffuse-interface VISM capable of capturing the ensemble average solvation energy (EASE). EASE represents the weighted average of solvation energies computed from all possible microstates of the solute-solvent system. In the routine calculation of EASE, one needs to carry out explicit solvent simulations, such as molecular dynamics (MD), to obtain thousands of solute-solvent configurations (snapshots) and perform energy calculations for each snapshot. Leveraging tools from statistical mechanics and geometric measure theory, this project aims to develop an innovative diffuse-interface VISM that rigorously reproduces EASE by utilizing a single diffuse interface profile instead of thousands of individual snapshots. Furthermore, while diffuse-interface VISMs offer improved accuracy compared to implicit solvent models utilizing predetermined interfaces, the computational cost associated with determining the interface poses a significant challenge. This limitation hampers the application of diffuse-interface VISMs to large molecular systems. Therefore, the second goal of this research is to develop accelerated numerical algorithms for solvation energy computation within the framework of diffuse-interface VISMs. By capitalizing on the structure of the proposed diffuse-interface VISM functional, a novel optimization algorithm will be devised that combines the augmented Lagrangian method with an inexact Newton scheme. This innovative approach will markedly enhance computational efficiency. Lastly, the analysis of the proposed model will provide novel techniques for studying the regularity of solutions to total variation minimization problems. The analytic techniques developed within the scope of this research project may also have broader relevance and applicability within the mathematical community.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.