Overview Solvation and Macromolecular Structure The structure and dynamics of biological macromolecules are strongly influenced by water: Electrostatic effects: charges are screened by water molecules and counterions. Hydrophobic effect: Entropic forces favor conformations that sequester non-polar hydrophobic domains within the interior of the molecule. Hydrodynamic effects: collision with solvent molecules imparts kinetic energy to macromolecules. Jay Taylor (ASU) APM 530 - Lecture 6 Fall 2010 1 / 32
Overview Solvation of Nucleic Acids Solvent effects on DNA and RNA are especially pronounced: Screening of the negatively-charged phosphates by water and counterions stabilizes helical and other tertiary structural motifs. 76% of the charge is reduced by water; 24% is reduced by divalent counterions. The transition between A-form and B-form DNA is partly controlled by hydration: B-DNA has 18-30 waters per nucleotide. A-DNA has 10-15 waters per nucleotide. DNA bending is promoted by hydration of the phosphates. Jay Taylor (ASU) APM 530 - Lecture 6 Fall 2010 2 / 32
Overview DNA is surrounded by multiple layers of water. Water density is increased up to six-fold in the first layer, especially around the phosphates and in the minor and major groove. A less stable second hydration shell extends out to 10 Å. A spine of hydration runs through the minor groove. Jay Taylor (ASU) APM 530 - Lecture 6 Fall 2010 3 / 32
Overview Explicit Solvation Models Solvent effects must also be included in molecular simulations. One approach is to explicitly include a large number of solvent molecules surrounding the macromolecule. Boundary conditions are usually periodic. Counterions can also be explicitly modeled. A large number of solvent molecules is generally required: Approximately 3000 waters are needed in a simulation of a 10 bp DNA duplex. Solvent viscosity leads to slower sampling of conformational space. Jay Taylor (ASU) APM 530 - Lecture 6 Fall 2010 4 / 32
Overview Implicit Solvation Models Implicit solvent models represent the solvent and counterions as a continuous medium. Implicit simulations can usually be run more quickly than explicit simulations. We are usually not interested in the distribution of individual water molecules in the solvent-solute interface. Residence times of water in DNA vary from hundreds of picoseconds to nanoseconds in the minor groove of A-tracts. Several methods are available: Solvent accessible surface area models Poisson-Boltzmann equation Generalized Born models Jay Taylor (ASU) APM 530 - Lecture 6 Fall 2010 5 / 32
SASA models Solvent Accessible Surface Area SASA models express the free energy of solvation as a sum G solv = i σ i ASA i where the sum is over all atoms in the macromolecule; ASA i is the surface area of atom i accessible to the solvent; σ i is an atom-specific surface tension parameter. The parameters σ i have been estimated from empirical hydration free energies for various organic compounds in water. Jay Taylor (ASU) APM 530 - Lecture 6 Fall 2010 6 / 32
SASA models Accessible Surface Area Calculations Atoms and solvent molecules are both modeled as spheres. The ASA of an atom is the area of those points on the surface of a sphere of radius R which can contact the center of a spherical solvent molecule that intersects no other atoms. R is the sum of the van der Walls radius of the atom and the radius of the solvent molecule (1.4 Å for water). Jay Taylor (ASU) APM 530 - Lecture 6 Fall 2010 7 / 32
SASA models Shrake-Rupley Algorithm The Shrake-Rupley algorithm (1973) uses a discretization of the molecular surface to estimate ASA: 92 points are distributed uniformly along a sphere centered at each atom with radius R (defined on the previous slide). The ASA is estimated by calculating the proportion of points that can be contacted by a solvent molecule that intersects no other atoms. Related methods approximate the surface using polyhedra. Gradients of discretized ASA-estimates must be evaluated numerically. Jay Taylor (ASU) APM 530 - Lecture 6 Fall 2010 8 / 32
SASA models Linear Combination of Pairwise Overlaps Weiser et al. (1999) proposed an approximate analytical expression for the ASA based on an inclusion-exclusion-like formula: A i P 1 S i + P 2 j N(i) A ij + S i is the surface area of atom i. j,k N(i) k j (P 3 + P 4 A ij )A jk A ij is the surface area of sphere i buried inside sphere j. N(i) is the neighbor list of atom i. P 1 - P 4 were calculated using least squares regression. The relative error is in the range 0.1 7.8%. The resulting formula can be differentiated. Jay Taylor (ASU) APM 530 - Lecture 6 Fall 2010 9 / 32
SASA models Limitations of SASA Models SASA models have several important limitations: Solvation free energies are not linearly related to surface area, e.g., SASA overestimates hydration free energies of cyclic alkanes. The solvation free energy calculated using SASA ignores the electrostatic effects of the solvent. SASA does not account for interactions between the solvent and polar atoms that are buried in the interior of the macromolecule. Jay Taylor (ASU) APM 530 - Lecture 6 Fall 2010 10 / 32
Poisson-Boltzmann equation Continuum Electrostatics Continuum electrostatic models account for the effects of the solvent on the electrostatics of the macromolecule. The simplest approach is to introduce a distance-dependent dielectric function: ɛ(r) = D s + D 0 1 + ke κ(ds+d 0) r D 0. ɛ(r) mimics electrostatic screening by the solvent. This approach ignores the fact that the solute itself influences the distribution of solvent molecules: Solvent molecules are often excluded from the interior of the solute. Counterions and polar solvent molecules condense around charged atoms. Jay Taylor (ASU) APM 530 - Lecture 6 Fall 2010 11 / 32
Poisson-Boltzmann equation Potential of Mean Force More accurate results can be obtained by modeling the distribution of the solvent around the macromolecule. At equilibrium, the total charge density of the solvent and any dissolved salts is described by the Boltzmann distribution ρ solvent (r) = i q i c i e Ṽi (r)/k B T where q i and c i are the charge and concentration of the i th ion; The potential of mean force for ion i is approximately related to the bulk electrostatic field by Ṽ i (r) q i Φ(r). Jay Taylor (ASU) APM 530 - Lecture 6 Fall 2010 12 / 32
Poisson-Boltzmann equation The Poisson-Boltzmann Equation We also know that the bulk potential Φ is related to the charge distribution of the solvent and the solute via Poisson s equation [ɛ(r) Φ(r)] = 4πρ solute (r) 4πρ solvent (r). Then, by combining this result with the expression for ρ solvent from the previous slide, we obtain the Poisson-Boltzmann equation: [ɛ(r) Φ(r)] = 4πρ solute (r) 4π i q i c i e q i Φ(r)/k B T. Jay Taylor (ASU) APM 530 - Lecture 6 Fall 2010 13 / 32
Poisson-Boltzmann equation The Poisson-Boltzmann Equation When the solvent consists of two oppositely charged ions with the same concentration c, the Poisson-Boltzmann equation can be written in the form: [ɛ(r) Φ(r)] = 4πρ solute (r) 4πqc (e ) qφ(r)/kbt e qφ(r)/k BT = 4πρ solute (r) 8πqc sinh(qφ(r)/k B T ). Jay Taylor (ASU) APM 530 - Lecture 6 Fall 2010 14 / 32
Poisson-Boltzmann equation The Linearized Poisson-Boltzmann Equation In practice, the Poisson-Boltzmann equation is usually linearized about 0: [ɛ(r) Φ(r)] = 4πρ solute (r) 4π i q i c i (1 q i Φ(r)/k B T ) where we have assumed/defined = 4πρ solute (r) + γ 2 Φ(r), 0 = i c i q i and γ 2 = 8π k B T c i qi 2. i This linearized version is appropriate when the electrostatic potential is much smaller than the thermal energy: q i Φ(r) k B T. Jay Taylor (ASU) APM 530 - Lecture 6 Fall 2010 15 / 32
Poisson-Boltzmann equation Debye-Hückel Theory Analytical solutions to the linearized P-B equation are available for some special cases. The Debye-Hückel theory considers a single spherical ion in an electrolyte with radius a and uniform surface charge q; constant interior and solvent permittivities ɛ i and ɛ s. In this case, the LPB can be written as (ɛ s /ɛ i )κ 2 Φ(r) if r < a 2 Φ(r) = κ 2 Φ(r) if r > a, where κ 2 = γ 2 /ɛ s. Jay Taylor (ASU) APM 530 - Lecture 6 Fall 2010 16 / 32
Poisson-Boltzmann equation Debye-Hückel Theory For r a, this equation coincides with the Helmholtz equation and has a unique solution in C 1 : Φ(r) = [ ] q 1 4π ɛ i r 1 ɛ i a + 1 ɛ sa(1+κa) if r < a ( ) exp(κa) q exp( κ r ) 1+κa ɛ r if r a. The second part of the solution illustrates the exponential screening of the ion by counterions in the solvent. κ 1 is called the Debye screening length and is inversely proportional to the square root of the ionic strength. Jay Taylor (ASU) APM 530 - Lecture 6 Fall 2010 17 / 32
Poisson-Boltzmann equation Numerical Solutions to the Poisson-Boltzmann Equation In practice, both the PB and the LPB equations must be solved numerically. Several difficulties need to be addressed: Repetitive solution of the PBE is needed for energy minimization and MD. Solutions need to be smooth in space and time to be used in MD. Boundary conditions need to be accurately specified. The dielectric function changes rapidly at the solvent-solute interface. Finite-difference, finite-element and boundary element approaches have all been developed for this problem. Jay Taylor (ASU) APM 530 - Lecture 6 Fall 2010 18 / 32
Poisson-Boltzmann equation Finite Difference Methods for the LPBE Luo et al. (2002) describe an efficient implementation of the finite difference approach. A 1 Å cubic mesh is used for high-accuracy solutions. The atomic charges and dielectric function are interpolated onto the grid points. The linearized PBE is replaced by a finite-difference equation Ax = b where A is a 7-banded symmetric positive-definite matrix; b is the interpolated charge distribution. The FDPB equation is solved iteratively using the Modified Incomplete Cholsky Conjugate Gradient method. Jay Taylor (ASU) APM 530 - Lecture 6 Fall 2010 19 / 32
Poisson-Boltzmann equation Preconditioning The preconditioning matrix used in MICCG is M = (L + D)D 1 (D + U) where U and L are the strictly upper and lower parts of A and D is the diagonal matrix with diagonal entries d 1 i = A i,1 A 2 i 1,2(A 2 i 1,2 + αa 2 i 1,3 + αa 2 i 1,4)d i 1 A 2 i nx,3(αa 2 i nx,2 + A 2 i nx,3 + αa 2 i nx,4)d i nx A 2 i nxny,4(αa 2 i nxny,2 + αa 2 i nxny,3 + A 2 i nxny,4)d i nxny. Here A has dimension n x n y n z with diagonal element A i,1 and non-zero off-diagonal elements A i,2, A i,3, A i,4 at row i. Jay Taylor (ASU) APM 530 - Lecture 6 Fall 2010 20 / 32
Poisson-Boltzmann equation Acceleration of the FDBE Iterative solution of the FDBE can be accelerated by the following practices: The convergence criterion for the conjugate gradient algorithm can be taken to be 10 2 rather than 10 6. This leads to a 6-fold speedup with correlation coefficient of 0.99 compared to the 10 6 calculation. The current value the potential can be used as an initial guess for the subsequent conjugate gradient iteration. The PB potential can be updated intermittently (every 2-10 steps). There is a tradeoff between the frequency of updating and the number of iterations required per update. Jay Taylor (ASU) APM 530 - Lecture 6 Fall 2010 21 / 32
Poisson-Boltzmann equation Electrostatic Focusing Electrostatic focusing can be used to obtain accurate solutions to elliptic problems with non-periodic boundary conditions: A low-accuracy solution is first obtained on a coarse mesh spanning the entire space. This solution is then used to define boundary conditions for a higher accuracy calculation on a subdomain using a finer mesh. Example: Electrostatic focusing accelerates the FDBE core routine by a factor of 3-4 when applied to HIV-1 protease with an initial 2 Å grid of 100 Å 3 followed by a final 1 Å grid of 80 60 50 Å. Jay Taylor (ASU) APM 530 - Lecture 6 Fall 2010 22 / 32
Poisson-Boltzmann equation Electrostatic Focusing and Parallelization Baker et al. (2001) showed that electrostatic focusing can also be used to parallelize the FDBE algorithm. Each processor obtains a coarse solution over the whole domain. This coarse solution is then used to assign boundary conditions to each of a collection of overlapping subdomains. Each processor then solves the PBE on a finer mesh on each subdomain. This method keeps communication between processors to a minimum. Jay Taylor (ASU) APM 530 - Lecture 6 Fall 2010 23 / 32
Poisson-Boltzmann equation Application to Microtubule Electrostatics By combining a multigrid algorithm with parallel focusing, the authors are able to solve the linearized PBE for a microtubule containing 90 dimers and over 1.25 million atoms. The microtubule structure was inferred from X-ray data. Both the coarse global mesh and the finer subdomain meshes contained 97 3 grid points, with a final resolution of 0.54 Å. The LPBE was solved in less than 1 hour using 686 processors. Jay Taylor (ASU) APM 530 - Lecture 6 Fall 2010 24 / 32
Poisson-Boltzmann equation Applications to Nucleic Acids Because of their high charge density, continuum electrostatic models of nucleic acids usually resort to the full non-linear PBE. Misra & Draper (2000) showed that the PBE quantitatively predicts the free energy of Mg 2+ binding to a yeast trna (right). Similar results have been obtained for salt effects on DNA-ligand binding and peptide-rna docking (Misra et al., 1994; Garcia-Garcia & Draper, 2003). Jay Taylor (ASU) APM 530 - Lecture 6 Fall 2010 25 / 32
Generalized Born models Generalized Born Models Generalized Born models approximate the solvation free energy by the formula: where ( G GB = 1 1 ) ɛ w i qi 2 1 ( 1 1 ) 2α i 2 ɛ w i j α i is the effective Born radius of atom i; ɛ w is the dielectric constant of water; f GB (r ij, α i, α j ) = rij 2 + α iα j exp( rij 2/4α iα j ). q i q j f GB (r ij, α i, α j ) Jay Taylor (ASU) APM 530 - Lecture 6 Fall 2010 26 / 32
Generalized Born models Solvation Free Energy of Isolated Charges The effective Born radius α i is chosen so that the quantity ( 1 1 ) q 2 i ɛ w 2α i is the electrostatic solvation energy of an isolated charge surrounded by a spherical shell that excludes the solvent. α i can be interpreted as the average distance of atom i from the solvent accessible surface of the molecule; This is an approximation since most atoms will not be surrounded by spherical cavities. Jay Taylor (ASU) APM 530 - Lecture 6 Fall 2010 27 / 32
Generalized Born models Calculation of the Effective Born Radii The effective Born radii can be estimated numerically but more often are evaluated using an approximate analytical expression (Hawkins et al. 1996): α 1 i ρ 1 i 1 j i ρ i r 2 H ij(r, ρ j )dr ρ 1 i j g(r ij, ρ i, ρ j ) where ρ i is the intrinsic radius of atom i (i.e., in isolation). H ij (r, ρ j ) is the fraction of the area of a sphere of radius r centered at atom i that is shielded by atom j. Jay Taylor (ASU) APM 530 - Lecture 6 Fall 2010 28 / 32
Generalized Born models Interaction Solvation Energy The second term that appears in the generalized Born model 1 2 ( 1 1 ) q i q j ɛ w f GB (r ij, α i, α j ) i j is the pairwise contribution of the charges in the molecule to the solvation energy. The interpolation function f GB has been chosen so that: lim r ij 0 f GB (r ij, α i, α j ) = α i α j lim r 1 r ij ij f GB (r ij, α i, α j ) = 1. Jay Taylor (ASU) APM 530 - Lecture 6 Fall 2010 29 / 32
Generalized Born models Generalized Born Models in Practice The Generalized Born model is often used in combination with the SASA estimate of the nonpolar solvation energy (GBSA models). Because the GBSA solvation energy can be calculated relatively quickly and can also be differentiated analytically, these models have seen more widespread use in MD than the Poisson-Boltzmann equation. On the other hand, GBSA is considered to be a cruder approximation for the solvation energy than PBE. Jay Taylor (ASU) APM 530 - Lecture 6 Fall 2010 30 / 32
Generalized Born models Generalized Born Simulations of DNA Tsui & Case (2001) compared the results of explicit solvent and generalized Born simulations of a 10 base-pair DNA helix for a 12 ns period. Both methods produce stable structures over the 12 ns trajectory. Hydrogen bond distances are shorter by 0.1 0.2 Å in the explicit solvent simulation. The atomic fluctuations of the inner base pairs are highly correlated between the two methods (right). Jay Taylor (ASU) APM 530 - Lecture 6 Fall 2010 31 / 32
References References Luo, R., David, L. and Gilson, M. K. (2002) Accelerated Poisson-Boltzmann Calculations for Static and Dynamic Systems. J. Comput. Chem. 23:1244-1253. Schlick, T. (2006) Molecular Modeling and Simulation. Springer. Tsui, V. and Case, D.A. (2000) Theory and applications of the generalized Born solvation model in macromolecular simulations. Biopolymers 56:275-291. Zacharias, M. (2006) Continuum Solvent Models to Study the Structure and Dynamics of Nucleic Acids and Complexes with Ligands. In Computational Studies of RNA and DNA, ed. by Jiri Sponer and F. Lankas. Springer. Jay Taylor (ASU) APM 530 - Lecture 6 Fall 2010 32 / 32