1 / 56

610 likes | 916 Views

Implicit solvent simulations. Nathan Baker (baker@biochem.wustl.edu) BME 540. Introduction to biomolecular electrostatics. Highly relevant to biological function Important tools in interpretation of structure and function

Download Presentation
## Implicit solvent simulations

**An Image/Link below is provided (as is) to download presentation**
Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.
Content is provided to you AS IS for your information and personal use only.
Download presentation by click this link.
While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.
During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

**Implicit solvent simulations**Nathan Baker (baker@biochem.wustl.edu) BME 540**Introduction to biomolecular electrostatics**• Highly relevant to biological function • Important tools in interpretation of structure and function • Electrostatics pose one of the most challenging aspects of biomolecular simulation • Long range • Divergent • Existing methods limit size of systems to be studied Acetylcholinesterase Fasciculin-2**Implicit solvent simulations: background**• Solute typically only accounts for 5-10% of atoms in explicit solvent simulation • Implicit methods: • Solvent treated as continuum of infinitesimal dipoles • Ions treated as continuum of charge • Some deficiencies: • Polarization response is linear and local • Mean field ion distribution ignores fluctuations and correlations • Apolar effects treated by various, heuristic methods**Modeling biomolecule-solvent interactions**• Solvent models • Explicit • Molecular dynamics • Monte Carlo • Integral equation • RISM • 3D methods • DFT • Primitive • Poisson equation • Phenomenological • Generalized Born • Modified Coulomb’s law • Ion models • Explicit • Molecular dynamics • Monte Carlo • Integral equation • RISM • 3D methods • DFT • Field theoretic • Poisson-Boltzmann • Extended PB, etc. • Phenomenological • Generalized Born • Debye-Hückel Computational cost Level of detail**Explicit solvent simulations**• Sample the configuration space of the system: ions, atomically-detailed water, solute • Sampling performed with respect to an ensemble: NpT, NVT, etc. • Algorithms: molecular dynamics and Monte Carlo • Advantages: • High levels of detail • Easy inclusion of additional degrees of freedom • All interactions considered explicitly • Disadvantages: • Slow (and uncertain) convergence • Time-consuming • Boundary effects • Poor scaling to larger systems • Some effects still not considered in many force fields…**Implicit solvent simulations**• Free energy evaluations: • Usually based on static solute structures or small number of conformational “snapshots” • Solvent effects included in: • Implicit solvent electrostatics • Surface area-dependent apolar terms • Useful for: • Solvation energies • Binding energies • Mutagenesis studies • pKa calculations**Implicit solvent simulations**• Stochastic dynamics • Usually based on Langevin or Brownian equations of motion • Solvent effects included in: • Implicit solvent electrostatics forces • Hydrodynamics • Random solvent forces • Useful for: • Bimolecular rate constants • Conformational sampling • Dynamical properties Animation courtesy of Dave Sept**Analytical models**• Include: • Coulomb • Debye-Hückel • Generalized Born • Other • Simple and fast • Do not accurately capture solvation behavior • Require parameterization…**Coulomb law**• Simplest implicit solvent model • Assumptions: • Solvent = homogeneous dielectric • Point charges • No mobile ions • Infinite domain (no boundaries) Charge magnitudes Solvent dielectric Charge locations**Coulomb law**• Simplest implicit solvent model • Assumptions: • Solvent = homogeneous dielectric • Point charges • No mobile ions • Infinite domain (no boundaries) • Solution to Poisson equation Point charge distribution Boundary condition**Coulomb law**• Simplest implicit solvent model • Assumptions: • Solvent = homogeneous dielectric • Point charges • No mobile ions • Infinite domain (no boundaries) • Solution to Poisson equation • Very simple energy evaluation**Debye-Hückel law**• Similar to Coulomb’s law • Assumptions: • Solvent = homogeneous dielectric • Point charges • Non-interacting mobile ions with linear response • Infinite domain (no boundaries) Inverse screening length Mobile ion bulk density**Debye-Hückel law**• Similar to Coulomb’s law • Assumptions: • Solvent = homogeneous dielectric • Point charges • Non-interacting mobile ions with linear response • Infinite domain (no boundaries) • Solution to Helmholtz equation**Debye-Hückel law**• Similar to Coulomb’s law • Assumptions: • Solvent = homogeneous dielectric • Point charges • Non-interacting mobile ions with linear response • Infinite domain (no boundaries) • Solution to Helmholtz equation • Simple energy evaluation**Generalized Born**• Used to calculate solvation energies (forces) • Modification of Born ion solvation energy: • Adjust effective radii of atoms based on environment • Differences between buried and exposed atoms • Fast to evaluate • Lots of variations • Hard to parameterize**Non-analytical continuum models**• Include: • Poisson • Poisson-Boltzmann • More realistic description of biomolecules: • Allow for variable dielectrics: • Interior (2-20) • Solvent (80) • Define regions of inaccessibility for ions • Complicated geometries require numerical solution • More computationally demanding**Poisson equation**• Describes electrostatic potential due to: • Inhomogeneous dielectric • Charge distribution • Assumes: • Linear and local solvent response • No mobile ions Dielectric function**Poisson equation: energies**• Total energies obtained from • Integral of polarization energy**Poisson equation: energies**• Total energies obtained from • Integral of polarization energy • Sum of charge-potential interactions**Poisson equation: energies**• Total energies obtained from • Integral of polarization energy • Sum of charge-potential interactions • Energies contain self-interaction terms: • Infinite (for analytic solution) • Very unstable (for numerical solution) • Self-interactions must be removed**The reaction field**• The potential due to inhomogeneous polarization of the solvent • The difference of potentials with: • Inhomogeneous dielectric • Homogeneous dielectric • Implicitly removes terms due to self-interactions: • Non-singular • Numerically-stable • Not available via simpler models… Reaction field**Reaction field example**• Potentials near low dielectric bodies do not superimpose • Contain: • Coulombic term • Reaction field term Total electrostatic potential Reaction field**Solvation energy**• Solvation energies obtained directly from reaction field • Difference of • Homogeneous • Inhomogeneous dielectric calculations • Self-energies removed in this process: • Numerical stability • Non-infinite results**A continuum descriptionof ion desolvation**• Two Born ions at varying separations • Solve Poisson equation at each separation • Increase in energy as “water” is squeezed out of interface • Desolvation effect • Less volume of polarized water • Important points • Non-superposition of Born ion potentials • Reaction field causes repulsion at short distances • Dielectric medium “focuses” field**Low**dielectric High dielectric Point charge A continuum descriptionof ion solvation • Born ion model • Non-polarizable ion • Point charge • Higher polarizability medium • “Reaction field” effects • Non-Coulombic potential inside ion due to polarization of solvent • Solvation energy • Simple model with analytical solutions**Poisson-Boltzmann equation**• Abbreviation = PBE • Describes electrostatic potential due to: • Inhomogeneous dielectric • Mobile counterions • “Fixed” (biomolecular) charge distribution • Assumes: • Linear and local solvent response • No explicit interaction between mobile ions**Poisson-Boltzmann derivation: step 1**• Start with Poisson equation to describe solvation • Supplement biomolecular charge distribution with mobile ion term Dielectric function Biomolecular charge distribution Mobile charge distribution**Poisson-Boltzmann derivation: step 2**• Choose mobile ion charge distribution form: • Boltzmann distribution no explicit ion-ion interaction • No detailed structure for atom (de)solvation Ion charges Ion bulk densities Ion-protein steric interactions**Poisson-Boltzmann derivation: step 3**• Substitute mobile charge distribution back into Poisson equation • Result: Nonlinear partial differential equation**Equation coefficients: charge distribution**• Charges are delta functions: hard to model • Often discretized as splines to “smooth” the problem • What about higher-order charge distributions?**Equation coefficients: mobile ion distribution**• Provides: • Bulk ionic strength • Ion accessibility • Usually constructed based on “inflated van der Waals radii”**Equation coefficients: dielectric function**• Describes change in dielectric response: • Low dielectric interior (2-20) • High dielectric solvent (80) • Many definitions: • Molecular (solid line) • Solvent-accessible (dotted line) • van der Waals (gray circles) • Inflated van der Waals (previous slide) • Smoothed definitions (spline-based and Gaussian) • Results can be very sensitive to the choice of surface!!!**Poisson-Boltzmann special cases**• 1:1 electrolyte (NaCl) • Assume similar steric interactions for each species with protein • Simplify two-term series to hyperbolic sine Modified screening coefficient: zero inside biomolecule 1:1 electrolyte charge distribution**Poisson-Boltzmann special cases**• 1:1 electrolyte (NaCl) • Assume similar steric interactions for each species with protein • Simplify two-term series to hyperbolic sine • Small charge-potential interaction • Linearized Poisson-Boltzmann**Non-specific salt effects: screening**• Lots of types of non-specific ion screening: • Variable solvation effects (Hofmeister) • Ion “clouds” damping electrostatc potential • Changes in co-ion and ligand activity coefficients • Condensation • Not all ion effects are non-specific! • Generally reduces effective range of electrostatic potential • Shown here for acetylcholinesterase • Illustrated by potential isocontours • Observed experimentally in reduced binding rate constants**mAChE at 0 mM NaCl**mAChE at 150 mM NaCl Non-specific salt effects: screening**Poisson-Boltzmann energies**• Similar to Poisson equation • Functional = integral over solution domain • Solution extremizes free energy Fixed charge- potential interactions Dielectric polarization Mobile charge energy**PBE: removing “self energies” and calculating**interesting stuff • Energy calculations must be performed with respect to reference system with same discretization: • Same differential operator: • Same charge representation • Reference systems implicit in • Solvation energies • Binding energies**Electrostatic influenceson ligand binding**• Examine inhibitor binding to protein kinase A: • Part of drug design project by McCammon and co-workers • Illustrates how electrostatics governs specificity and affinity • Look at complementarity between ligand and protein electrostatics • Verify with experimental data (relative binding affinities) • Use to guide design of improved inhibitors**Balanol**Protein Kinase A Electrostatic influenceson ligand binding**Poisson-Boltzmann equation:force evaluation**• Integral of electrostatic potential over solution domain • Assume solution fixed over atomic displacements • Differentiate with respect to atomic positions • Contains contributions from Osmotic pressure Dielectric boundary Reaction field**PBE: considerations with force evaluation**• Remove self-energies: two PB calculations to give “reaction field forces” • Inhomogeneous dielectric: non-zero fixed charge, dielectric boundary, and osmotic pressure forces • Homogeneous dielectric: only non-zero fixed charge forces • Coulombic interactions added in analytically • Uses: • Minimization • Single-point force evaluation • Dynamics • Need fast setup and calculation • Currently ~8 sec/calc for Ala2 1 day/ns with 10 fs steps**Solving the PE or PBE**• Determine the coefficients based on the biomolecular structure • Discretize the problem • Solve the resulting linear or nonlinear algebraic equations**Discretization**• Choose your problem domain: finite or infinite? • Usually finite domain • Requires relatively large domain • Uses asymptotically-correct boundary condition (e.g., Debye-Hückel, Coulomb, etc.) • Infinite domain requires appropriate basis functions • Choose your basis functions: global or local? • Usually local: map problem onto some sort of grid or mesh • Global basis functions (e.g. spherical harmonics) can cause numerical difficulties**Discretization: local methods**• Polynomial basis functions (defined on interval) • “Locally supported” on a few grid points • Only overlap with nearest-neighbors sparse matrices Boundary element (Surface discretization) Finite element (Volume discretization) Finite difference (Volume discretization)**Discretization: pros & cons**• Finite difference: • Sparse numerical systems and efficient solvers • Handles linear and nonlinear PBE • Easy to setup and analyze • Non-adaptive representation of problem • Finite element: • Sparse numerical systems • Handles linear and nonlinear PBE • Adaptive representation of problem • Not easy to setup and analyze • Less efficient solvers • Boundary element: • Very adaptive representation of problem • Surface discretization instead of volume • Not easy to setup and analyze • Less efficient solvers • Dense numerical system • Only handles linear PBE

More Related