Hazra Tania, Ahmed Ullah Sheik, Wang Siwen, Alexov Emil, Zhao Shan
Department of Mathematics, Misericordia University, Dallas, PA, 18612, USA.
Department of Mathematics, University of Alabama, Tuscaloosa, AL, 35487, USA.
J Math Biol. 2019 Jul;79(2):631-672. doi: 10.1007/s00285-019-01372-1. Epub 2019 Apr 27.
Calculations of electrostatic potential and solvation free energy of macromolecules are essential for understanding the mechanism of many biological processes. In the classical implicit solvent Poisson-Boltzmann (PB) model, the macromolecule and water are modeled as two-dielectric media with a sharp border. However, the dielectric property of interior cavities and ion-channels is difficult to model realistically in a two-dielectric setting. In fact, the detection of water molecules in a protein cavity remains to be an experimental challenge. This introduces an uncertainty, which affects the subsequent solvation free energy calculation. In order to compensate this uncertainty, a novel super-Gaussian dielectric PB model is introduced in this work, which devices an inhomogeneous dielectric distribution to represent the compactness of atoms and characterizes empty cavities via a gap dielectric value. Moreover, the minimal molecular surface level set function is adopted so that the dielectric profile remains to be smooth when the protein is transferred from water phase to vacuum. An important feature of this new model is that as the order of super-Gaussian function approaches the infinity, the dielectric distribution reduces to a piecewise constant of the two-dielectric model. Mathematically, an effective dielectric constant analysis is introduced in this work to benchmark the dielectric model and select optimal parameter values. Computationally, a pseudo-time alternative direction implicit (ADI) algorithm is utilized for solving the super-Gaussian PB equation, which is found to be unconditionally stable in a smooth dielectric setting. Solvation free energy calculation of a Kirkwood sphere and various proteins is carried out to validate the super-Gaussian model and ADI algorithm. One macromolecule with both water filled and empty cavities is employed to demonstrate how the cavity uncertainty in protein structure can be bypassed through dielectric modeling in biomolecular electrostatic analysis.
计算大分子的静电势和溶剂化自由能对于理解许多生物过程的机制至关重要。在经典的隐式溶剂泊松-玻尔兹曼（PB）模型中，大分子和水被建模为具有清晰边界的两种电介质。然而，在二维电介质环境中，内部腔和离子通道的介电特性很难逼真地建模。事实上，检测蛋白质腔中的水分子仍然是一个实验挑战。这引入了一个不确定性，影响了后续的溶剂化自由能计算。为了补偿这种不确定性，本文引入了一种新颖的超高斯介电PB模型，该模型设计了一种非均匀介电分布来表示原子的紧密程度，并通过间隙介电值来表征空穴。此外，采用最小分子表面水平集函数，使得当蛋白质从水相转移到真空时，介电分布保持平滑。这个新模型的一个重要特征是，随着超高斯函数的阶数趋近于无穷大，介电分布简化为二维电介质模型的分段常数。在数学上，本文引入了有效介电常数分析来对介电模型进行基准测试并选择最佳参数值。在计算上，使用伪时间交替方向隐式（ADI）算法来求解超高斯PB方程，发现在平滑介电环境中该算法是无条件稳定的。对柯克伍德球和各种蛋白质进行了溶剂化自由能计算，以验证超高斯模型和ADI算法。使用一个既有填充水的腔又有空腔的大分子来展示如何在生物分子静电分析中通过介电建模绕过蛋白质结构中的腔不确定性。
