A new FFT-based algorithm to compute Born radii in the generalized Born theory of biomolecule solvation

Abstract

In this paper, a new method for calculating effective atomic radii within the generalized Born (GB) model of implicit solvation is proposed, for use in computer simulations of bio-molecules. First, a new formulation for the GB radii is developed, in which smooth kernels are used to eliminate the divergence in volume integrals intrinsic in the model. Next, the Fast Fourier Transform (FFT) algorithm is applied to integrate smoothed functions, taking advantage of the rapid spectral decay provided by the smoothing. The total cost of the proposed algorithm scales as O(N(3)logN + M) where M is the number of atoms comprised in a molecule, and N is the number of FFT grid points in one dimension, which depends only on the geometry of the molecule and the spectral decay of the smooth kernel but not on M. To validate our algorithm, numerical tests are performed for three solute models: one spherical object for which exact solutions exist and two protein molecules of differing size. The tests show that our algorithm is able to reach the accuracy of other existing GB implementations, while offering much lower computational cost.

Fig. 1

An illustration of how integration in eqs.( 2.7) and ( 3.7) is carried out for atoms at grid points in the interior of a solute and near its surface. A small sphere S_i is drawn around each grid point that is: (a) fully inside the molecule and (b) partially outside of the molecule where A_i is the outlying part of S_i

Fig. 2

Smoothed f(x) (top left), smoothed kernel G(r) (bottom left) and their Fourier transforms (right). Gn refers to a smoothed kernel using the smoother $W_a^n\left(r\right)$, a = 0.2

Fig. 3

CPU time vs. number of atoms for three settings of our new FFT-based algorithm with a smoother $W_a^2\left(r\right)$, a = 0.2 and the average relative error less than 1.0%. Grid-based: h = 0.0625; FFT 1: h = 0.0625; FFT 2: h = 0.03125. All tests were done for a model spherical solute of radius 1.

Fig. 4

Born radii derived using our new FFT-based algorithm vs. grid-based radii computed for a model spherical solute of radius 1 using the smoother $W_a^2\left(r\right)$, a = 0.2. The correlation coefficients are 0.9936, 0.9994 and 0.9999 for h = 0.125, 0.0625 and 0.03125, respectively.

Fig. 5

Born radii computed using our new FFT-based algorithm vs. grid-based radii obtained in CHARMM [39] for immunoglobulin binding protein (PDB access code 3GB1 ). In FFT calculations, 128³ grid points in domain [−32 Å, 32 Å]³ were used. Results for different smoothers $W_a^2\left(r\right)$ are shown in (a) a = 3.0 Å; (b) a = 4.0 Å; (c) a = 4.5 Å; (d) a = 5.0 Å

Fig. 6

Same as Fig. 5 but for the human cyclophilin A (PDB access code 1OCA). In the FFT calculations, 128³ grid points in domain [−40 Å, 40 Å]³ were used. Results for different smoothers $W_a^2\left(r\right)$ are shown in (a) a = 4.0 Å; (b) a = 5.0 Å; (c) a = 5.5 Å; (d) a = 6.0 Å