# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2018155

## A simple and efficient technique to accelerate the computation of a nonlocal dielectric model for electrostatics of biomolecule

 1 School of Mathematics and Statistics, Changsha University of Science and Technology, Changsha, Hunan 410114 China 2 School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083 China

* Corresponding author

Received  March 2018 Revised  June 2018 Published  September 2018

The nonlocal modified Poisson-Boltzmann equation (NMPBE) is one important variant of a commonly-used dielectric continuum model, Poisson-Boltzmann equation (PBE). In this paper, we use a nonlinear block relaxation method to develop a new nonlinear solver for the nonlinear equation of $\Phi$ and thus a new NMPBE solver, which is then programmed as a software package in $\texttt{C}\backslash\texttt{C++}$, $\texttt{Fortran}$ and $\texttt{Python}$ for computing the electrostatics of a protein in a symmetric 1:1 ionic solvent. Numerical tests validate the new package and show that the new solver can improve the efficiency by at least $40\%$ than the finite element NMPBE solver without compromising solution accuracy.

Citation: Jiao Li, Jinyong Ying. A simple and efficient technique to accelerate the computation of a nonlocal dielectric model for electrostatics of biomolecule. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2018155
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##### References:
Time speedup $S_p$ defined in (23) achieved by our new NMPBE solver for the 12 protein tests on two mesh sets. Initial mesh sets denote the meshes used in Table 2 and Refined mesh sets mean the ones used in Table 3.
Newton Steps and the iteration numbers of the linear solver to solve $\Phi$ for case 1A63 in the new and the finite element NMPBE program packages. The left plot presents the number of block relaxation iteration (the x-axis), the Newton steps in each block iteration (the numbers above the x-axis), and the concrete/average iteration number (the points/the solid lines) of the linear solver in the new program package. The right plot presents the Newton steps (the x-axis) and the iteration numbers of the linear solver in the finite element one.
Basic information of the 12 proteins used for numerical tests. Here $n_{p}$ is the number of atoms.
 Index PDB ID $n_{p}$ Index PDB ID $n_{p}$ 1 2LZX 488 7 1A63 2065 2 1AJJ 513 8 1CID 2783 3 1FXD 811 9 1A7M 2803 4 1HPT 852 10 2AQ5 6024 5 4PTI 892 11 1F6W 8243 6 1SVR 1433 12 1C4K 11439
 Index PDB ID $n_{p}$ Index PDB ID $n_{p}$ 1 2LZX 488 7 1A63 2065 2 1AJJ 513 8 1CID 2783 3 1FXD 811 9 1A7M 2803 4 1HPT 852 10 2AQ5 6024 5 4PTI 892 11 1F6W 8243 6 1SVR 1433 12 1C4K 11439
Comparison of the performance of our new NMPBE solver (New) with that of the finite element NMPBE solver (FE) [19] in CPU time measured in seconds. Here Iter. Number denotes the iteration number needed in the nonlinear block relaxation method and $E_h$ is computed by (22), and residual norm means the norm of Equation (7)'s residual.
 PDB ID Number of Mesh Nodes Iter. Number Find $\Phi$ Total Time Relative error $E_{h}$ Residual norm New FE New FE 2LZX 26349 11 15.23 27.79 27.61 40.17 $2.1\times 10^{-8}$ $1.42\times10^{-4}$ 1AJJ 31910 11 26.60 48.25 45.29 66.94 $3.9\times 10^{-8}$ $3.69\times10^{-5}$ 1FXD 34469 12 23.19 42.48 49.74 69.03 $1.2\times 10^{-8}$ $8.03\times10^{-4}$ 1HPT 48229 10 32.33 58.78 58.08 84.53 $3.3\times 10^{-8}$ $1.11\times10^{-4}$ 4PTI 39468 10 25.85 46.52 47.14 67.81 $1.5\times 10^{-8}$ $8.95\times10^{-5}$ 1SVR 61074 11 55.17 90.59 96.88 132.30 $2.6\times 10^{-8}$ $4.25\times10^{-5}$ 1A63 22054 11 13.52 27.09 27.19 40.76 $1.6\times 10^{-8}$ $1.82\times10^{-4}$ 1CID 19872 10 11.07 21.68 23.10 33.71 $1.9\times 10^{-8}$ $1.09\times10^{-3}$ 1A7M 20883 10 11.63 22.42 24.53 35.33 $3.2\times 10^{-8}$ $3.16\times10^{-4}$ 2AQ5 38151 11 29.58 53.69 69.88 93.99 $2.8\times 10^{-8}$ $1.61\times10^{-4}$ 1F6W 49006 11 46.77 86.41 94.47 134.11 $2.3\times 10^{-8}$ $7.05\times10^{-4}$ 1C4K 72046 11 70.04 118.93 172.47 221.36 $3.7\times 10^{-8}$ $1.69\times10^{-3}$
 PDB ID Number of Mesh Nodes Iter. Number Find $\Phi$ Total Time Relative error $E_{h}$ Residual norm New FE New FE 2LZX 26349 11 15.23 27.79 27.61 40.17 $2.1\times 10^{-8}$ $1.42\times10^{-4}$ 1AJJ 31910 11 26.60 48.25 45.29 66.94 $3.9\times 10^{-8}$ $3.69\times10^{-5}$ 1FXD 34469 12 23.19 42.48 49.74 69.03 $1.2\times 10^{-8}$ $8.03\times10^{-4}$ 1HPT 48229 10 32.33 58.78 58.08 84.53 $3.3\times 10^{-8}$ $1.11\times10^{-4}$ 4PTI 39468 10 25.85 46.52 47.14 67.81 $1.5\times 10^{-8}$ $8.95\times10^{-5}$ 1SVR 61074 11 55.17 90.59 96.88 132.30 $2.6\times 10^{-8}$ $4.25\times10^{-5}$ 1A63 22054 11 13.52 27.09 27.19 40.76 $1.6\times 10^{-8}$ $1.82\times10^{-4}$ 1CID 19872 10 11.07 21.68 23.10 33.71 $1.9\times 10^{-8}$ $1.09\times10^{-3}$ 1A7M 20883 10 11.63 22.42 24.53 35.33 $3.2\times 10^{-8}$ $3.16\times10^{-4}$ 2AQ5 38151 11 29.58 53.69 69.88 93.99 $2.8\times 10^{-8}$ $1.61\times10^{-4}$ 1F6W 49006 11 46.77 86.41 94.47 134.11 $2.3\times 10^{-8}$ $7.05\times10^{-4}$ 1C4K 72046 11 70.04 118.93 172.47 221.36 $3.7\times 10^{-8}$ $1.69\times10^{-3}$
Comparison of the performance of our new NMPBE solver (New) on the refined meshes with that of the finite element NMPBE solver (FE) [19] in CPU time measured in seconds.
 PDB ID Number of Mesh Nodes Iter. Number Find $\Phi$ Total Time Relative error $E_{h}$ Residual norm New FE New FE 2LZX 535400 10 167.0 291.2 311.2 435.3 $3.5\times 10^{-8}$ $4.07\times10^{-4}$ 1AJJ 538321 10 223.9 437.3 387.4 600.8 $3.5\times 10^{-8}$ $1.33\times10^{-4}$ 1FXD 540849 11 201.5 346.2 363.1 507.8 $1.7\times 10^{-8}$ $3.22\times10^{-4}$ 1HPT 543220 9 186.8 399.6 332.3 545.0 $3.3\times 10^{-8}$ $2.68\times10^{-4}$ 4PTI 541329 9 173.4 328.7 319.5 474.7 $2.3\times 10^{-8}$ $3.7\times10^{-4}$ 1SVR 550170 10 229.7 411.0 415.3 596.7 $2.0\times 10^{-8}$ $1.29\times10^{-4}$ 1A63 558010 11 253.3 442.8 573.1 762.7 $2.4\times 10^{-8}$ $2.62\times10^{-3}$ 1CID 558374 10 203.0 389.0 409.4 595.4 $2.7\times 10^{-8}$ $4.13\times10^{-4}$ 1A7M 563919 11 242.9 471.7 442.6 671.4 $4.8\times 10^{-8}$ $3.09\times10^{-4}$ 2AQ5 577821 10 296.7 566.8 637.7 907.8 $3.5\times 10^{-8}$ $1.41\times10^{-4}$ 1F6W 574686 11 332.1 597.3 707.8 973.0 $3.7\times 10^{-8}$ $1.06\times10^{-3}$ 1C4K 573111 14 396.6 698.3 940.8 1242.5 $3.5\times 10^{-8}$ $1.18\times10^{-3}$
 PDB ID Number of Mesh Nodes Iter. Number Find $\Phi$ Total Time Relative error $E_{h}$ Residual norm New FE New FE 2LZX 535400 10 167.0 291.2 311.2 435.3 $3.5\times 10^{-8}$ $4.07\times10^{-4}$ 1AJJ 538321 10 223.9 437.3 387.4 600.8 $3.5\times 10^{-8}$ $1.33\times10^{-4}$ 1FXD 540849 11 201.5 346.2 363.1 507.8 $1.7\times 10^{-8}$ $3.22\times10^{-4}$ 1HPT 543220 9 186.8 399.6 332.3 545.0 $3.3\times 10^{-8}$ $2.68\times10^{-4}$ 4PTI 541329 9 173.4 328.7 319.5 474.7 $2.3\times 10^{-8}$ $3.7\times10^{-4}$ 1SVR 550170 10 229.7 411.0 415.3 596.7 $2.0\times 10^{-8}$ $1.29\times10^{-4}$ 1A63 558010 11 253.3 442.8 573.1 762.7 $2.4\times 10^{-8}$ $2.62\times10^{-3}$ 1CID 558374 10 203.0 389.0 409.4 595.4 $2.7\times 10^{-8}$ $4.13\times10^{-4}$ 1A7M 563919 11 242.9 471.7 442.6 671.4 $4.8\times 10^{-8}$ $3.09\times10^{-4}$ 2AQ5 577821 10 296.7 566.8 637.7 907.8 $3.5\times 10^{-8}$ $1.41\times10^{-4}$ 1F6W 574686 11 332.1 597.3 707.8 973.0 $3.7\times 10^{-8}$ $1.06\times10^{-3}$ 1C4K 573111 14 396.6 698.3 940.8 1242.5 $3.5\times 10^{-8}$ $1.18\times10^{-3}$
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