Group Report

An image-based reaction field method for electrostatic interactions in molecular dynamics simulationsPresented By: Yuchun Lin

Department of Mathematics & StatisticsDepartment of Physics & Optical ScienceUniversity of North Carolina at Charlotte

International Workshop on Continuum Modeling of Biomolecules September 14-16, 2009 in Beijing, China12Molecular Dynamics SimulationSimulation of biological macromolecules is a key area of interest:Understand the dynamic mechanisms of macromolecular function (protein folding, enzymatic catalysis)Predict the energetics of various biological processes (ligand association, protein stability)Design novel molecules with particular properties (drug design, protein engineering)Introduction & Background It still has some issues.Accurate simulations require the solvent to be treated carefully.Long range interaction: Truncation of electrostatic interaction leads to artifacts.23

ExplicitMore Accurate & Less Efficient

ImpliciteMore Efficient & Less AccurateHybrid implicit/explicit

Reaction FieldeIntroduction & Background

3Hybrid Solvation Models

Numerical Solution: W. Im, et al., J. Chem. Phys. 114(2001) 2924Generalized Born Model: M. S. Lee, et al, J. Comput. Chem. 25 (2004) 1967D. Bashford, et al., Annu. Rev. Phys. Chem. 51 (2000) 129152Numerical Solution: D. Beglov, et al, J. Chem. Phys. 100(1994) 9050H. Alper, et al, J. Chem. Phys.,99(1993) 9847G. Brancato, et al, J. Chem. Phys. 122(2005) 154109Image Approximation:P. K. Yang, et al, J. Phys. Chem. B, 106 (2002) 2973.G. Petraglio, et al, J. Chem. Phys. 123(2005) 044103A. Wallqvist, Mol. Sim. 10(1993) 1317.Arbitrary geometry

Exact solution of PB in particular geometries4Kirkwood expansion --- slow convergence at boundary Friedman image expression --- approximated & less accurateRepulsive potential applied --- strong surface effect

accurate up to O(1/) 4Basic Idea5Image-based method to compute reaction fieldFriedman expression for reaction field is approximateSurface effects are non-negligible or not removed easilyMultiple image charges methodPeriodic boundary conditions for non-electrostaticKnown DrawbacksOur SolutionsY. Lin, A. Baumketner, S. Deng, Z. Xu, D. Jacob, W. Cai, An image-based reaction field method for electrostatic interactions in molecular dynamics simulations of aqueous solutions, J. Chem. Phys., 2009, under review5

Theory: RF in multiple-image charges approach6Poisson-Boltzmann equations:

H. L. Friedman, Mol. Phys. 29 (1975) 15331543W. Cai, S. Deng, D. Jacobs, J. Comput. Phys., 223(2007), 846-864S. Deng, W. Cai, Comm. Comput. Phys., 2(2007), 1007-1026

67With Kirkwood expansion on pure solution caseFor using image method, let , , First series is the potential of Kelvin image:Using the integral identity and rewrite second series as:

Theory: RF in multiple-image charges approach

Where and

78Now the reaction field inside the cavity is:Next, we construct discrete image charge by Gauss-Radau quadrature:Here are the Gauss-Radau quadrature weights and points. Since s1=-1 and then x1=rK, the classical Kelvin image charge and the first discrete image charge can be combined, leading to:

Theory: RF in multiple-image charges approach

89Theory: Integration of the RF model with MD

Role of a buffer layer between explicit and implicit solvents :A. Wallqvist, Mol. Sim. 10(1993) 1317 L.Wang, J. Hermans, J. Phys. Chem. 99(1995) 12001910Choice of boundary conditions:Theory: Integration of the RF model with MD

d

Choice of box type: For Cubic Box:For L = 45, = 5 box, Cube allows only 2 for dL10Model11

Three parameters:

Number of image charge (Ni) Ni=0, 1, 2, 3

Thickness of buffer layer () =2, 4, 6, 8

Box size (L) L=30, 45, 60

dFor Truncated Octahedron:For L=45, =5 box, TO allows 17 for dFast Multipole Method is appliedL. Greengard, V. Rokhlin, J. Comput. Phys. 73 (1987) 3253481112

A buffer layer of at least 6 is required to yield uniform density. Large surface effect at low .Results: Relative Density#Ni=2 = 2 = 4 = 6 = 8L=300.0560.0110.0030.002L=45 0.0600.0070.0020.002L=60 0.0550.0090.0020.003Standard DeviationL=30L=45L=60

1213Number of image charges (Ni) is not criticalEffect of buffer layer thickness () is unnoticeable Effect of box size (L), converges on L=60 with PMEResults: Radial Distribution Function

13Results: Diffusion Coefficient14Reaction field is critical for the proper description of diffusionNi=1 = 4 = 6 = 8 L = 30 6.40(0.26)6.28(0.11)6.16(0.12)L = 45 6.21(0.08)6.20(0.10)6.16(0.14)L = 60 6.02(0.06)6.02(0.07)6.02(0.04)L = 80 5.98(0.02)5.98(0.02)5.99(0.03)Ni=2 = 4 = 6 = 8 L = 30 6.32(0.12)6.33(0.24)6.23(0.12)L = 45 6.16(0.09)6.16(0.10)6.15(0.08)L = 60 6.02(0.04)6.01(0.05)6.01(0.03)L = 80 5.96(0.02)5.98(0.02)5.98(0.02)Ni=3 = 4 = 6 = 8 L = 30 6.34(0.17)6.29(0.25)6.24(0.15)L = 45 6.18(0.11)6.19(0.10)6.16(0.07)L = 60 6.01(0.03)6.00(0.05)6.03(0.07)L = 80 5.98(0.02)6.00(0.04)5.98(0.03)PME 5.98(0.05)(Unit: 10-9m2s-1)14

15The convergence with the number of image charges occurs at Ni = 1Results: Dielectric Constant

L=60, =4

V. Ballenegger, J. P. Hansen, J. Chem. Phys. 122(2005) 11471115

16The dependence of on the thickness of buffer layer is weekResults: Dielectric Constant16

17Dielectric properties require large simulation boxes and RF correctionsResults: Dielectric ConstantPME: = 90 1017Summary & Conclusions18Summary:

Large enough buffer layer is important

Large box size produces good bulk properties of simulated water

Reaction field is essential for proper description of dielectric permittivity

Conclusion:

A new solvation model is proposed. Static, structural and dynamic properties of water are well reproduced compared to PME.

Applications to biological system are our future work.

Optimal parametersL = 60, = 6, Ni = 1.W. Cai, S. Deng, D. Jacobs, J. Comput. Phys., 223(2007), 846-864S. Deng, W. Cai, Comm. Comput. Phys., 2(2007), 1007-1026S. Deng, W. Cai, J. Comput. Phys. 227 (2007) 12461266.Y. Lin, A. Baumketner, S. Deng, Z. Xu, D. Jacobs, W. Cai, J. Chem. Phys., 2009, under review18Acknowledgement19

Funding byAdvisors: Dr. Andrij Baumketner Dr. Wei Cai Dr. Shaozhong Deng Dr. Don JacobsGroup Members: Dr. Xia JiDr. Haiyan JiangDr. Boris NiDr. Zhenli Xu Ms. Katherine Baker Mr. Wei SongMs. Ming Xiang19