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An image-based reaction field method for electrostatic interactions in molecular dynamics simulations Presented By: Yuchun Lin Department of Mathematics & Statistics Department of Physics & Optical Science University of North Carolina at Charlotte International Workshop on Continuum Modeling of Biomolecules September 14-16, 2009 in Beijing, China

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2 Molecular Dynamics Simulation Simulation 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.

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3 Explicit More Accurate & Less Efficient Implicit More Efficient & Less Accurate Hybrid implicit/explicit Reaction Field Introduction & Background

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Hybrid Solvation Models Numerical Solution: W. Im, et al., J. Chem. Phys. 114(2001) 2924 Generalized Born Model: M. S. Lee, et al, J. Comput. Chem. 25 (2004) 1967 D. Bashford, et al., Annu. Rev. Phys. Chem. 51 (2000) 129152 Numerical Solution: D. Beglov, et al, J. Chem. Phys. 100(1994) 9050 H. Alper, et al, J. Chem. Phys.,99(1993) 9847 G. Brancato, et al, J. Chem. Phys. 122(2005) 154109 Image Approximation: P. K. Yang, et al, J. Phys. Chem. B, 106 (2002) 2973. G. Petraglio, et al, J. Chem. Phys. 123(2005) 044103 A. Wallqvist, Mol. Sim. 10(1993) 1317. Arbitrary geometry Exact solution of PB in particular geometries 4 Kirkwood expansion --- slow convergence at boundary Friedman image expression --- approximated & less accurate Repulsive potential applied --- strong surface effect accurate up to O(1/)

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Basic Idea 5 Image-based method to compute reaction field Friedman expression for reaction field is approximate Surface effects are non-negligible or not removed easily Multiple image charges method Periodic boundary conditions for non-electrostatic Known Drawbacks Our Solutions Y. 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 review

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Theory: RF in multiple-image charges approach 6 Poisson-Boltzmann equations: H. L. Friedman, Mol. Phys. 29 (1975) 15331543 W. Cai, S. Deng, D. Jacobs, J. Comput. Phys., 223(2007), 846-864 S. Deng, W. Cai, Comm. Comput. Phys., 2(2007), 1007-1026

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7 With Kirkwood expansion on pure solution case For 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

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8 Now 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 s 1 =-1 and then x 1 =r K, the classical Kelvin image charge and the first discrete image charge can be combined, leading to: Theory: RF in multiple-image charges approach

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9 Theory: 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) 12001

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10 Choice 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 d L

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Model 11 Three parameters: Number of image charge (N i ) N i =0, 1, 2, 3 Thickness of buffer layer ( ) = 2, 4, 6, 8 Box size (L) L=30, 45, 60 d For Truncated Octahedron: For L=45, =5 box, TO allows 17 for d Fast Multipole Method is applied L. Greengard, V. Rokhlin, J. Comput. Phys. 73 (1987) 325348

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12 A buffer layer of at least 6 is required to yield uniform density. Large surface effect at low . Results: Relative Density #N i =2 = 2 = 4 = 6 = 8 L=300.0560.0110.003 0.002 L=45 0.0600.0070.002 L=60 0.0550.0090.002 0.003 Standard Deviation L=30 L=45 L=60

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13 Number of image charges ( N i ) is not critical Effect of buffer layer thickness ( ) is unnoticeable Effect of box size (L), converges on L=60 with PME Results: Radial Distribution Function

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Results: Diffusion Coefficient 14 Reaction field is critical for the proper description of diffusion N i =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) 5.99(0.03) N i =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) N i =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 -9 m 2 s -1 )

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15 The convergence with the number of image charges occurs at N i = 1 Results: Dielectric Constant L=60, =4 V. Ballenegger, J. P. Hansen, J. Chem. Phys. 122(2005) 114711

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16 The dependence of on the thickness of buffer layer is week Results: Dielectric Constant

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17 Dielectric properties require large simulation boxes and RF corrections Results: Dielectric Constant PME: = 90 10

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Summary & Conclusions 18 Summary: 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 parameters L = 60, = 6, N i = 1. W. Cai, S. Deng, D. Jacobs, J. Comput. Phys., 223(2007), 846-864 S. Deng, W. Cai, Comm. Comput. Phys., 2(2007), 1007-1026 S. 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 review

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Acknowledgement 19 Funding by Advisors: Dr. Andrij Baumketner Dr. Wei Cai Dr. Shaozhong Deng Dr. Don Jacobs Group Members: Dr. Xia Ji Dr. Haiyan Jiang Dr. Boris Ni Dr. Zhenli Xu Ms. Katherine Baker Mr. Wei Song Ms. Ming Xiang