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

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.

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

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) 129–152 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) Image Approximation: P. K. Yang, et al, J. Phys. Chem. B, 106 (2002) G. Petraglio, et al, J. Chem. Phys. 123(2005) A. Wallqvist, Mol. Sim. 10(1993) 13–17. 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/ε)

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

Theory: RF in multiple-image charges approach 6 Poisson-Boltzmann equations: H. L. Friedman, Mol. Phys. 29 (1975) 1533–1543 W. Cai, S. Deng, D. Jacobs, J. Comput. Phys., 223(2007), S. Deng, W. Cai, Comm. Comput. Phys., 2(2007),

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

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

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) 13–17 L.Wang, J. Hermans, J. Phys. Chem. 99(1995) 12001

10 Choice of boundary conditions: Theory: Integration of the RF model with MD d Choice of box type: For Cubic Box: For L = 45Å, τ = 5Å box, Cube allows only 2 Å for d L

Model 11 Three parameters: Number of image charge (N i ) N i =0, 1, 2, 3 Thickness of buffer layer ( τ ) τ = 2Å, 4Å, 6Å, 8Å Box size (L) L=30Å, 45Å, 60Å d For Truncated Octahedron: For L=45Å, τ =5Å box, TO allows 17 Å for d Fast Multipole Method is applied L. Greengard, V. Rokhlin, J. Comput. Phys. 73 (1987) 325–348

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

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

Results: Diffusion Coefficient 14 Reaction field is critical for the proper description of diffusion N i =1 τ = 4 Å τ = 6 Å τ = 8 Å L = 30 Å6.40(±0.26)6.28(±0.11)6.16(±0.12) L = 45 Å6.21(±0.08)6.20(±0.10)6.16(±0.14) L = 60 Å6.02(±0.06)6.02(±0.07)6.02(±0.04) L = 80 Å5.98(±0.02) 5.99(±0.03) N i =2 τ = 4 Å τ = 6 Å τ = 8 Å L = 30 Å6.32(±0.12)6.33(±0.24)6.23(±0.12) L = 45 Å6.16(±0.09)6.16(±0.10)6.15(±0.08) L = 60 Å6.02(±0.04)6.01(±0.05)6.01(±0.03) L = 80 Å5.96(±0.02)5.98(±0.02) N i =3 τ = 4 Å τ = 6 Å τ = 8 Å L = 30 Å6.34(±0.17)6.29(±0.25)6.24(±0.15) L = 45 Å6.18(±0.11)6.19(±0.10)6.16(±0.07) L = 60 Å6.01(±0.03)6.00(±0.05)6.03(±0.07) L = 80 Å5.98(±0.02)6.00(±0.04)5.98(±0.03) PME 5.98(±0.05) (Unit: m 2 s -1 )

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

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

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

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 = 60Å, τ = 6Å, N i = 1. W. Cai, S. Deng, D. Jacobs, J. Comput. Phys., 223(2007), S. Deng, W. Cai, Comm. Comput. Phys., 2(2007), S. Deng, W. Cai, J. Comput. Phys. 227 (2007) 1246–1266. Y. Lin, A. Baumketner, S. Deng, Z. Xu, D. Jacobs, W. Cai, J. Chem. Phys., 2009, under review

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