Collaborative Research: A Regularized Poisson Boltzmann Model for Fast Computation of the Ensemble Average Polar Solvation Energy

该合作研究项目将开发数学模型和模拟工具，用于研究大生物分子（例如蛋白质）与周围水分子之间的相互作用，模拟水环境。用于表征这种相互作用的能量计算是复杂的，因为生物分子的结构不是完全固定或刚性的，并且周围的水分子也在不断运动。因此，为了提供与实验可测量的能量相当的量，必须在相应的数学描述中考虑这些构象变化。 一个新的理论模型将在这个项目中制定适当的生物物理学的考虑与数学的进步相结合，使模拟模拟大分子和水原子的构象变化的影响。拟议的数学发展将使分子生物科学和生物物理学的研究人员受益。此外，所提出的模型和算法将在DelPhi中实现，DelPhi免费分发给学术用户，以确保数学，化学，物理和生物学从业者的广泛使用。此外，本项目将为本科生和研究生提供生物建模、计算和数学分析方面的跨学科研究和培训机会。然而，这种能量的直接泊松-玻尔兹曼（PB）计算需要以数千个快照的形式产生代表性的结构系综，这在计算上是非常昂贵的。如果可以通过采用单一结构来计算系综平均溶剂化能，则可以实现计算时间的巨大节省，所述单一结构通过模拟经由异质介电分布的大分子构象变化的效果来实现。 在这个项目中，一个新的超高斯PB模型将体现三个关键的创新：（1）在连续体静电偏微分方程（PDE）中纳入大分子的环境依赖性原子特征;（2）开发一个新的正则化公式来处理奇异电荷，通过严格的数学分析开发新的椭圆偏微分方程：部分电荷和水分子（在空腔内，与蛋白质结合，以及在本体溶剂中）将不再像当前的高斯模型那样被建模为均匀的空间区域，而是将反映整个溶质-溶剂系统的灵活性;以及（3）对于真空和水环境中的大分子，消除了确定尖锐分子表面的需要。 将进行模型基准测试和生物应用测试进行验证，得到的工具将被纳入广泛使用的德尔菲程序包计算整体平均溶剂化能。该项目由数学科学部数学生物学计划、化学部化学理论、模型和计算方法计划以及刺激竞争性研究的既定计划（EPSCoR）共同支持。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

This collaborative research project will develop mathematical models and simulation tools for studying the interactions between large biological molecules, for example proteins, and surrounding water molecules modeling an aqueous environment. The energy computation for characterizing such interactions is complex because the structures of biomolecules are not completely fixed or rigid, and the surrounding water molecules are also in constant motion. Thus, to deliver quantities that are comparable with experimentally-measurable energies, one must account for these conformational changes in the corresponding mathematical description. A new theoretical model will be formulated in this project by combining appropriate biophysical considerations with mathematical advances, allowing simulations to mimic the effect of conformational changes in both macromolecule and water atoms. The proposed mathematical development will benefit researchers in molecular biosciences and biophysics. Moreover, the proposed models and algorithms will be implemented in DelPhi, which is distributed free of charge to academic users, to ensure extensive usage by practitioners from mathematics, chemistry, physics, and biology. In addition, this project will provide interdisciplinary research and training opportunities for undergraduate and graduate students in biological modeling, computation and mathematical analysis.Experimentally-observable solvation energies are ensemble averaged. However, direct Poisson-Boltzmann (PB) calculations of such energies require the generation of a representative ensemble of structures in terms of thousands of snapshots, which is computationally very expensive. Tremendous savings in computational time can be achieved if one can calculate the ensemble average solvation energy by employing a single structure by mimicking the effect of conformation changes of macromolecules via heterogeneous dielectric distributions. In this project, a novel super-Gaussian PB model will be formulated embodying three key innovations: (1) incorporation of environment-dependent atomic characteristics of macromolecules within the continuum electrostatic partial differential equation (PDE); (2) development of a novel, regularized formulation to treat singular charges, with new elliptic PDEs developed through rigorous mathematical analysis: partial charges and water molecules (inside cavities, bonded to the protein, and in bulk solvent) will no longer be modeled as homogeneous spatial regions as in the current Gaussian model, but will reflect the flexibility of the entire solute-solvent system; and (3) elimination of the need to determine a sharp molecular surface, for macromolecules in both vacuum and water environments. Model benchmarking and biological applications tests will be carried out for validation, and the resulting tool will be incorporated into the widely-used DelPhi program package for computing ensemble-average solvation energies. This project is supported jointly by the Division of Mathematical Sciences Mathematical Biology Program, the Division of Chemistry Chemical Theory, Models and Computational Methods Program, and the Established Program to Stimulate Competitive Research (EPSCoR).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.