Implicit solvent simulations

1
Implicit solvent simulations

• Nathan Baker
• (baker_at_biochem.wustl.edu)
• BME 540

2
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
3
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

4
Modeling biomolecule-solvent interactions

• 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

• Solvent models
• Explicit
• Molecular dynamics
• Monte Carlo
• Integral equation
• RISM
• 3D methods
• DFT
• Primitive
• Poisson equation
• Phenomenological
• Generalized Born
• Modified Coulombs law

Level of detail
Computational cost
5
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

6
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

7
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
8
Analytical models

• Include
• Coulomb
• Debye-Hückel
• Generalized Born
• Other
• Simple and fast
• Do not accurately capture solvation behavior
• Require parameterization

9
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
10
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
11
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

12
Debye-Hückel law

• Similar to Coulombs 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
13
Debye-Hückel law
14
Debye-Hückel law

• Similar to Coulombs law
• Assumptions
• Solvent homogeneous dielectric
• Point charges
• Non-interacting mobile ions with linear response
• Infinite domain (no boundaries)
• Solution to Helmholtz equation

15
Debye-Hückel law

• Similar to Coulombs 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

16
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

17
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

18
Poisson equation

• Describes electrostatic potential due to
• Inhomogeneous dielectric
• Charge distribution
• Assumes
• Linear and local solvent response
• No mobile ions

Dielectric function
19
Poisson equation energies

• Total energies obtained from
• Integral of polarization energy

20
Poisson equation energies

• Total energies obtained from
• Integral of polarization energy
• Sum of charge-potential interactions

21
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

22
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
23
Reaction field example

• Potentials near low dielectric bodies do not
  superimpose
• Contain
• Coulombic term
• Reaction field term

Total electrostatic potential
Reaction field
24
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

25
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

26
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

27
A continuum descriptionof ion solvation
28
A continuum descriptionof ion desolvation
29
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

30
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
31
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
32
Poisson-Boltzmann derivation step 3

• Substitute mobile charge distribution back into
  Poisson equation
• Result Nonlinear partial differential equation

33
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?

34
Equation coefficients mobile ion distribution

• Provides
• Bulk ionic strength
• Ion accessibility
• Usually constructed based on inflated van der
  Waals radii

35
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!!!

36
Poisson-Boltzmann special cases

• 11 electrolyte (NaCl)
• Assume similar steric interactions for each
  species with protein
• Simplify two-term series to hyperbolic sine

Modified screening coefficient zero inside
biomolecule
11 electrolyte charge distribution
37
Poisson-Boltzmann special cases

• 11 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

38
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

39
Non-specific salt effects screening
40
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
41
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

42
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

43
Electrostatic influenceson ligand binding
44
Electrostatic influenceson ligand binding
45
Poisson-Boltzmann equationforce 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
46
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

47
Solving the PE or PBE

• Determine the coefficients based on the
  biomolecular structure
• Discretize the problem
• Solve the resulting linear or nonlinear algebraic
  equations

48
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

49
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)
50
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

51
Basic numerical solution

• Iteratively solve matrix equations obtained by
  discretization
• Linear multigrid
• Nonlinear Newtons method and multigrid
• Multigrid solvers offer optimal solution
• Accelerate convergence
• Fine ? coarse projection
• Coarse problems converge more quickly
• Big systems are still difficult
• High memory usage
• Long run-times
• Need parallel solvers

52
Errors in numerical solutions

• Electrostatic potentials are very sensitive to
  discretization
• Grid spacings lt 0.5 Å
• Smooth surface discretizations
• Errors most pronounced next to biomolecule
• Large potential and gradients
• High multipole order
• Errors decay with distance
• Approximately follow multipole expansion behavior
• Coarse grid spacings will correctly resolve
  electrostatics far away from molecule

53
Poisson-Boltzmann equationagreement with
Coulombs law

• Energy consists of two components
• Coulombs law contribution often poorly
  approximated at short lengths scales and/or
  coarse grid spacings
• Solvation energy/reaction field contribution
  generally well-approximated at reasonable grid
  spacings
• Solution
• Use analytical methods to obtain Coulombic energy
• Slow scales as O(N ln N) to O(N2)
• Not always necessary
• Use approximate methods to obtain solvation energy

54
Poisson-Boltzmann Pros and Cons

• Advantages
• Compromise between explicit and GB methods
• Reasonably fast and accurate
• Linear scaling
• Applicable to very large systems
• Disadvantages
• Limited range of applicability
• Fails badly with highly-charged systems and/or
  high salt concentrations
• Neglects molecular details of solvent and
  solvation

55
PBE current solution methods

• Complicated geometries require numerical
  solutions
• Numerical methods
• Local vs. global basis functions
• Discretization
• Finite domain (usually) with appropriate boundary
  conditions
• PB methods usually use local basis functions
  spatial discretization
• Beware numerical artifacts!
• Convergence of the method
• Inappropriate spacings

56
Electrostatics Software