American Institute of Mathematical Sciences

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January  2020, 16(1): 357-369. doi: 10.3934/jimo.2018155

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

Jiao Li 1, and  Jinyong Ying 2,,

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  January 2020 Early access  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.

Keywords: Nonlocal model, implicit continuum model, finite element method, block iteration method, biomolecular electrostatics.
Mathematics Subject Classification: Primary: 65N30, 92-08; Secondary: 92C40.
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 and Management Optimization, 2020, 16 (1) : 357-369. doi: 10.3934/jimo.2018155
References:
[1]

M. V. Basilevsky and D. F. Parsons, An advanced continuum medium model for treating solvation effects: Nonlocal electrostatics with a cavity, The Journal of Chemical Physics, 105 (1996), 3734-3746.  doi: 10.1063/1.472193.

[2]

I. Borukhov, D. Andelman and H. Orland, Steric effects in electrolytes: A modified Poisson-Boltzmann equation, Physical Review Letters, 79 (1997), 435. doi: 10.1103/PhysRevLett.79.435.

[3]

V. B. Chu, Y. Bai, J. Lipfert, D. Herschlag and S. Doniach, Evaluation of ion binding to DNA duplexes using a size-modified Poisson-Boltzmann theory, Biophysical Journal, 93 (2007), 3202-3209.  doi: 10.1529/biophysj.106.099168.

[4]

W. Deng, J. Xu and S. Zhao, On developing stable finite element methods for pseudo-time simulation of biomolecular electrostatics, Journal of Computational and Applied Mathematics, 330 (2018), 456-474.  doi: 10.1016/j.cam.2017.09.004.

[5]

R. Dogonadze and A. Kornyshev, Polar solvent structure in the theory of ionic solvation, Journal of the Chemical Society, Faraday Transactions 2: Molecular and Chemical Physics, 70 (1974), 1121-1132.  doi: 10.1039/f29747001121.

[6]

A. Hildebrandt, R. Blossey, S. Rjasanow, O. Kohlbacher and H.-P. Lenhof, Novel formulation of nonlocal electrostatics, Physical Review Letters, 93 (2004), 108104.

[7]

B. Honig and A. Nicholls, Classical electrostatics in biology and chemistry, Science, 268 (1995), 1144-1149.  doi: 10.1126/science.7761829.

[8]

J. Hu, S. Zhao and W. Geng, Accurate pKa computation using matched interface and boundary (MIB) method based Poisson-Boltzmann solver, Communication in Computational Physics, 23 (2018), 520-539. 

[9]

A. Kornyshev, A. Rubinshtein and M. Vorotyntsev, Model nonlocal electrostatics. Ⅰ, Journal of Physics C: Solid State Physics, 11 (1978), 3307. doi: 10.1088/0022-3719/11/15/029.

[10]

A. A. Kornyshev and G. Sutmann, Nonlocal dielectric saturation in liquid water, Physical Review Letters, 79 (1997), 3435. doi: 10.1103/PhysRevLett.79.3435.

[11]

J. Li and D. Xie, A new linear Poisson-Boltzmann equation and finite element solver by solution decomposition approach, Communications in Mathematical Sciences, 13 (2015), 315-325.  doi: 10.4310/CMS.2015.v13.n2.a2.

[12]

J. Li and D. Xie, An effective minimization protocol for solving a size-modified Poisson-Boltzmann equation for biomolecule in ionic solvent, International Journal of Numerical Analysis and Modeling, 12 (2015), 286-301. 

[13]

J. Li, J. Ying and D. Xie, On the analysis and application of an ion size-modified Poisson-Boltzmann equation, Nonlinear Analysis: Real World Applications, submitted.

[14]

J. L. Liu, Numerical methods for the Poisson-Fermi equation in electrolytes, Journal of Computational Physics, 247 (2013), 88-99.  doi: 10.1016/j.jcp.2013.03.058.

[15]

A. Logg and G. N. Wells, DOLFIN: Automated finite element computing, ACM Transactions on Mathematical Software (TOMS), 37 (2010), Art. 20, 28 pp. doi: 10.1145/1731022.1731030.

[16]

L. R. Scott and D. Xie, Analysis of a Nonlocal Poisson-Boltzmann Equation, Technical report, Research Report UC/CS TR-2016-1, Dept. Comp. Sci., Univ. Chicago, 2016.

[17]

M. Vorotyntsev, Model nonlocal electrostatics. Ⅱ. spherical interface, Journal of Physics C: Solid State Physics, 11 (1978), 3323. doi: 10.1088/0022-3719/11/15/030.

[18]

S. Weggler, V. Rutka and A. Hildebrandt, A new numerical method for nonlocal electrostatics in biomolecular simulations, Journal of Computational Physics, 229 (2010), 4059-4074.  doi: 10.1016/j.jcp.2010.01.040.

[19]

D. Xie and Y. Jiang, A nonlocal modified Poisson-Boltzmann equation and finite element solver for computing electrostatics of biomolecules, Journal of Computational Physics, 322 (2016), 1-20.  doi: 10.1016/j.jcp.2016.06.028.

[20]

D. Xie, Y. Jiang and L. R. Scott, Efficient algorithms for a nonlocal dielectric model for protein in ionic solvent, SIAM Journal on Scientific Computing, 35 (2013), B1267-B1284.  doi: 10.1137/120899078.

[21]

D. Xie and J. Li, A new analysis of electrostatic free energy minimization and Poisson-Boltzmann equation for protein in ionic solvent, Nonlinear Analysis: Real World Applications, 21 (2015), 185-196.  doi: 10.1016/j.nonrwa.2014.07.008.

[22]

D. Xie, H. W. Volkmer and J. Ying, Analytical solutions of nonlocal poisson dielectric models with multiple point charges inside a dielectric sphere, Physical Review E, 93 (2016), 043304. doi: 10.1103/PhysRevE.93.043304.

[23]

D. Xie and J. Ying, A new box iterative method for a class of nonlinear interface problems with application in solving Poisson-Boltzmann equation, Journal of Computational and Applied Mathematics, 307 (2016), 319-334.  doi: 10.1016/j.cam.2016.01.005.

[24]

J. Ying and D. Xie, A new finite element and finite difference hybrid method for computing electrostatics of ionic solvated biomolecule, Journal of Computational Physics, 298 (2015), 636-651.  doi: 10.1016/j.jcp.2015.06.016.

[25]

J. Ying and D. Xie, A hybrid solver of size modified Poisson-Boltzmann equation by domain decomposition, finite element, and finite difference, Applied Mathematical Modelling, 58 (2018), 166-180.  doi: 10.1016/j.apm.2017.09.026.

show all references

References:
[1]

M. V. Basilevsky and D. F. Parsons, An advanced continuum medium model for treating solvation effects: Nonlocal electrostatics with a cavity, The Journal of Chemical Physics, 105 (1996), 3734-3746.  doi: 10.1063/1.472193.

[2]

I. Borukhov, D. Andelman and H. Orland, Steric effects in electrolytes: A modified Poisson-Boltzmann equation, Physical Review Letters, 79 (1997), 435. doi: 10.1103/PhysRevLett.79.435.

[3]

V. B. Chu, Y. Bai, J. Lipfert, D. Herschlag and S. Doniach, Evaluation of ion binding to DNA duplexes using a size-modified Poisson-Boltzmann theory, Biophysical Journal, 93 (2007), 3202-3209.  doi: 10.1529/biophysj.106.099168.

[4]

W. Deng, J. Xu and S. Zhao, On developing stable finite element methods for pseudo-time simulation of biomolecular electrostatics, Journal of Computational and Applied Mathematics, 330 (2018), 456-474.  doi: 10.1016/j.cam.2017.09.004.

[5]

R. Dogonadze and A. Kornyshev, Polar solvent structure in the theory of ionic solvation, Journal of the Chemical Society, Faraday Transactions 2: Molecular and Chemical Physics, 70 (1974), 1121-1132.  doi: 10.1039/f29747001121.

[6]

A. Hildebrandt, R. Blossey, S. Rjasanow, O. Kohlbacher and H.-P. Lenhof, Novel formulation of nonlocal electrostatics, Physical Review Letters, 93 (2004), 108104.

[7]

B. Honig and A. Nicholls, Classical electrostatics in biology and chemistry, Science, 268 (1995), 1144-1149.  doi: 10.1126/science.7761829.

[8]

J. Hu, S. Zhao and W. Geng, Accurate pKa computation using matched interface and boundary (MIB) method based Poisson-Boltzmann solver, Communication in Computational Physics, 23 (2018), 520-539. 

[9]

A. Kornyshev, A. Rubinshtein and M. Vorotyntsev, Model nonlocal electrostatics. Ⅰ, Journal of Physics C: Solid State Physics, 11 (1978), 3307. doi: 10.1088/0022-3719/11/15/029.

[10]

A. A. Kornyshev and G. Sutmann, Nonlocal dielectric saturation in liquid water, Physical Review Letters, 79 (1997), 3435. doi: 10.1103/PhysRevLett.79.3435.

[11]

J. Li and D. Xie, A new linear Poisson-Boltzmann equation and finite element solver by solution decomposition approach, Communications in Mathematical Sciences, 13 (2015), 315-325.  doi: 10.4310/CMS.2015.v13.n2.a2.

[12]

J. Li and D. Xie, An effective minimization protocol for solving a size-modified Poisson-Boltzmann equation for biomolecule in ionic solvent, International Journal of Numerical Analysis and Modeling, 12 (2015), 286-301. 

[13]

J. Li, J. Ying and D. Xie, On the analysis and application of an ion size-modified Poisson-Boltzmann equation, Nonlinear Analysis: Real World Applications, submitted.

[14]

J. L. Liu, Numerical methods for the Poisson-Fermi equation in electrolytes, Journal of Computational Physics, 247 (2013), 88-99.  doi: 10.1016/j.jcp.2013.03.058.

[15]

A. Logg and G. N. Wells, DOLFIN: Automated finite element computing, ACM Transactions on Mathematical Software (TOMS), 37 (2010), Art. 20, 28 pp. doi: 10.1145/1731022.1731030.

[16]

L. R. Scott and D. Xie, Analysis of a Nonlocal Poisson-Boltzmann Equation, Technical report, Research Report UC/CS TR-2016-1, Dept. Comp. Sci., Univ. Chicago, 2016.

[17]

M. Vorotyntsev, Model nonlocal electrostatics. Ⅱ. spherical interface, Journal of Physics C: Solid State Physics, 11 (1978), 3323. doi: 10.1088/0022-3719/11/15/030.

[18]

S. Weggler, V. Rutka and A. Hildebrandt, A new numerical method for nonlocal electrostatics in biomolecular simulations, Journal of Computational Physics, 229 (2010), 4059-4074.  doi: 10.1016/j.jcp.2010.01.040.

[19]

D. Xie and Y. Jiang, A nonlocal modified Poisson-Boltzmann equation and finite element solver for computing electrostatics of biomolecules, Journal of Computational Physics, 322 (2016), 1-20.  doi: 10.1016/j.jcp.2016.06.028.

[20]

D. Xie, Y. Jiang and L. R. Scott, Efficient algorithms for a nonlocal dielectric model for protein in ionic solvent, SIAM Journal on Scientific Computing, 35 (2013), B1267-B1284.  doi: 10.1137/120899078.

[21]

D. Xie and J. Li, A new analysis of electrostatic free energy minimization and Poisson-Boltzmann equation for protein in ionic solvent, Nonlinear Analysis: Real World Applications, 21 (2015), 185-196.  doi: 10.1016/j.nonrwa.2014.07.008.

[22]

D. Xie, H. W. Volkmer and J. Ying, Analytical solutions of nonlocal poisson dielectric models with multiple point charges inside a dielectric sphere, Physical Review E, 93 (2016), 043304. doi: 10.1103/PhysRevE.93.043304.

[23]

D. Xie and J. Ying, A new box iterative method for a class of nonlinear interface problems with application in solving Poisson-Boltzmann equation, Journal of Computational and Applied Mathematics, 307 (2016), 319-334.  doi: 10.1016/j.cam.2016.01.005.

[24]

J. Ying and D. Xie, A new finite element and finite difference hybrid method for computing electrostatics of ionic solvated biomolecule, Journal of Computational Physics, 298 (2015), 636-651.  doi: 10.1016/j.jcp.2015.06.016.

[25]

J. Ying and D. Xie, A hybrid solver of size modified Poisson-Boltzmann equation by domain decomposition, finite element, and finite difference, Applied Mathematical Modelling, 58 (2018), 166-180.  doi: 10.1016/j.apm.2017.09.026.

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.
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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.
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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$n_{p}$
12LZX48871A632065
21AJJ51381CID2783
31FXD81191A7M2803
41HPT852102AQ56024
54PTI892111F6W8243
61SVR1433121C4K11439
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IndexPDB ID$n_{p}$IndexPDB ID$n_{p}$
12LZX48871A632065
21AJJ51381CID2783
31FXD81191A7M2803
41HPT852102AQ56024
54PTI892111F6W8243
61SVR1433121C4K11439
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
NewFENewFE
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}$
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PDB IDNumber of Mesh NodesIter. NumberFind $\Phi $Total TimeRelative error $E_{h}$Residual norm
NewFENewFE
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}$
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
NewFENewFE
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}$
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PDB IDNumber of Mesh NodesIter. NumberFind $\Phi $Total TimeRelative error $E_{h}$Residual norm
NewFENewFE
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}$
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