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

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  • 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.

    Mathematics Subject Classification: Primary: 65N30, 92-08; Secondary: 92C40.


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  • Figure 1.  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.

    Figure 2.  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.

    Table 1.  Basic information of the 12 proteins used for numerical tests. Here $n_{p}$ is the number of atoms.

    IndexPDB ID$n_{p}$IndexPDB ID$n_{p}$
     | Show Table
    DownLoad: CSV

    Table 2.  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 IDNumber of Mesh NodesIter. NumberFind $\Phi $Total TimeRelative error $E_{h}$Residual norm
    2LZX263491115.2327.7927.6140.17$2.1\times 10^{-8}$$1.42\times10^{-4}$
    1AJJ319101126.6048.2545.2966.94$3.9\times 10^{-8}$$3.69\times10^{-5}$
    1FXD344691223.1942.4849.7469.03$1.2\times 10^{-8}$$8.03\times10^{-4}$
    1HPT482291032.3358.7858.0884.53$3.3\times 10^{-8}$$1.11\times10^{-4}$
    4PTI394681025.8546.5247.1467.81$1.5\times 10^{-8}$$8.95\times10^{-5}$
    1SVR610741155.1790.5996.88132.30$2.6\times 10^{-8}$$4.25\times10^{-5}$
    1A63220541113.5227.0927.1940.76$1.6\times 10^{-8}$$1.82\times10^{-4}$
    1CID198721011.0721.6823.1033.71$1.9\times 10^{-8}$$1.09\times10^{-3}$
    1A7M208831011.6322.4224.5335.33$3.2\times 10^{-8}$$3.16\times10^{-4}$
    2AQ5381511129.5853.6969.8893.99$2.8\times 10^{-8}$$1.61\times10^{-4}$
    1F6W490061146.7786.4194.47134.11$2.3\times 10^{-8}$$7.05\times10^{-4}$
    1C4K720461170.04118.93172.47221.36$3.7\times 10^{-8}$$1.69\times10^{-3}$
     | Show Table
    DownLoad: CSV

    Table 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 IDNumber of Mesh NodesIter. NumberFind $\Phi $Total TimeRelative error $E_{h}$Residual norm
    2LZX53540010167.0291.2311.2435.3$3.5\times 10^{-8}$$4.07\times10^{-4}$
    1AJJ53832110223.9437.3387.4600.8$3.5\times 10^{-8}$$1.33\times10^{-4}$
    1FXD54084911201.5346.2363.1507.8$1.7\times 10^{-8}$$3.22\times10^{-4}$
    1HPT5432209186.8399.6332.3545.0$3.3\times 10^{-8}$$2.68\times10^{-4}$
    4PTI5413299173.4328.7319.5474.7$2.3\times 10^{-8}$$3.7\times10^{-4}$
    1SVR55017010229.7411.0415.3596.7$2.0\times 10^{-8}$$1.29\times10^{-4}$
    1A6355801011253.3442.8573.1762.7$2.4\times 10^{-8}$$2.62\times10^{-3}$
    1CID55837410203.0389.0409.4595.4$2.7\times 10^{-8}$$4.13\times10^{-4}$
    1A7M56391911242.9471.7442.6671.4$4.8\times 10^{-8}$$3.09\times10^{-4}$
    2AQ557782110296.7566.8637.7907.8$3.5\times 10^{-8}$$1.41\times10^{-4}$
    1F6W57468611332.1597.3707.8973.0$3.7\times 10^{-8}$$1.06\times10^{-3}$
    1C4K57311114396.6698.3940.81242.5$3.5\times 10^{-8}$$1.18\times10^{-3}$
     | Show Table
    DownLoad: CSV
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