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
Computational cost
Level of detail
– 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
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
ΔG1  ΔG2  ΔG3  ΔG4
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
qi
  x  
i  x  xi
Solvent
dielectric
Charge
locations
Coulomb law
• Simplest implicit solvent model
• Assumptions:
–
–
–
–
Solvent = homogeneous dielectric
Point charges
No mobile ions
Infinite domain (no boundaries)
Point charge
distribution
• Solution to Poisson equation
   x  
2
4

   0
q  x  x 
i
i
Boundary
condition
i
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
qi q j
1
G    
2 i j i  x j  xi
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)
 x  xi
qi e
  x  
i  x  xi
 4
 
  kT
Inverse
screening
length
1/ 2

nmQ 

m

2
i
Mobile ion
bulk density
Debye-Hückel law
3.5
3
2.5
Coulomb
Debye
2
Huckel
1.5
1
0.5
r
1
2
3
4
5
Debye-Hückel law
• Similar to Coulomb’s law
• Assumptions:
– Solvent = homogeneous
dielectric
– Point charges
– Non-interacting mobile ions
with linear response
2
– Infinite domain (no
boundaries)
• Solution to Helmholtz
equation
 x     x 
2
4

   0
q  x  x 
i
i
i
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
 xi  x j
qi q j e
1
G    
2 i j i  x j  xi
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

1  1   qi2
 solvG  1   

2    i  Ri j i f

2

r
ij
f  rij , Ri , R j   rij2  Ri R j exp  
 4 Ri R j







xi  x j , Ri , R j 

qi q j

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
    x    x   f  x 
  qi  x  xi 
• Assumes:
– Linear and local solvent
response
– No mobile ions
i
   0
Dielectric
function
Poisson equation: energies
• Total energies obtained
from
– Integral of polarization
energy
1
G    
4
1

8
1

8

2

  f   2    dx
    dx
1
  i qi  x  xi dx   8
q x 
i
i
i
Poisson equation: energies
• Total energies obtained
from
– Integral of polarization
energy
– Sum of charge-potential
interactions
1
G    
4
1

8
1

8

2

  f   2    dx
    dx
1
  i qi  x  xi dx   8
q x 
i
i
i
Poisson equation: energies
• Total energies obtained
from
– Integral of polarization
energy
– Sum of charge-potential
interactions
• Energies contain selfinteraction terms:
– Infinite (for analytic solution)
– Very unstable (for numerical
solution)
• Self-interactions must be
removed
For Coulomb law
1
G     qi  xi 
2 i
qi q j
1
 
2 i j  xi  x j
qi q j
1
 
2 i j i  xi  x j
qi2
1
  lim
2 i x  xi  x  xi
The reaction field
• The potential due to
inhomogeneous polarization of
1



x


x

  1 
the solvent
4
• 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…
  xi    p
1
 p  2  x  
4
2
q x  x 
i
i
q x  x 
i
i
  x   1  x   2  x 
Reaction field
i
i
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
 solvG  G 2   G 1 
1
  qi 2  xi   1  xi  
2 i
1
  qi  xi 
2 i
+ -+
- +
+ + +
+
-+
- + + -- +
++ +-+ + + + - + +- +
- +
- +
 ( x)   s
+ -+
+
+ + +
+
-+
- + + -- +
++ +-+ + + + - + +- +
- +
- +
-
 ( x)   p
A continuum description
of 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
A continuum description
of 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
Low
dielectric
High
dielectric
Point charge
A continuum description
of ion solvation
A continuum description
of ion desolvation
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
    x    x   4  qi  x  xi   4  x 
   0
Dielectric
function
i
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
  x   Qmnme
  Qm  x Vm ( x )
m
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
    x    x   4  Qm nm e   Qm  x Vm  x   4  qi  x  xi 
m
   0
i
Equation coefficients: charge distribution
   x    x   4  Qm nme  Qm  xVm  x  4  qi  x  xi 
• Charges are delta mfunctions:
hard to model
• Often discretized as splines
to “smooth” the problem
• What about higher-order
charge distributions?
i
+ -+
- +
+ + +
- + -++ + -- +
++ +-+ + + + - + +- +
- +
- +
Equation coefficients: mobile ion distribution
   x    x   4  Qm nme  Qm  xVm  x  4  qi  x  xi 
• Provides:
m
– Bulk ionic strength
– Ion accessibility
• Usually constructed
based on “inflated van
der Waals radii”
i
Equation coefficients: dielectric function
   x    x   4  Qm nme  Qm  xVm  x  4  qi  x  xi 
m
•
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!!!
i
Poisson-Boltzmann special cases
• 1:1 electrolyte (NaCl)
– Assume similar steric interactions for each species with
protein
– Simplify two-term series to hyperbolic sine
  x   4  Qm nm e   Q   x V
m
m
 x
m
 4 ec ne V  x  e   ec ( x )  e  ec  x  
 8 ec2 ne V  x  sinh   ec  x  
  2  x  sinh   ec  x  
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
•
   x    x     x  sinh  ec  x   4  qi  x  xi 
i
Small charge-potential interaction
– Linearized Poisson-Boltzmann
4  Qmnme  Qm  xVm  x  4 eV  x Qm2 nm   x    2  x   x 
m
m
   x    x    2  x   x   4  qi  x  xi 
i
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
Non-specific salt effects: screening
mAChE at 150 mM NaCl
mAChE at 0 mM NaCl
Poisson-Boltzmann energies
• Similar to Poisson equation
• Functional = integral over solution domain
• Solution extremizes free energy
1 


2
G   
 f         cosh    1dx


4 
2

Fixed chargepotential 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 influences
on 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
Electrostatic influences
on ligand binding
Balanol
Protein Kinase A
Electrostatic influences
on 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
kT 

F [u ] 
i


  2 
1   
2
- f u  
  u   
 cosh  u   1 dx
2  i
4 ec 
2  xi 

 xi 
2
Reaction field
Dielectric boundary
Osmotic pressure
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
1. Determine the coefficients based on the
biomolecular structure
2. Discretize the problem
3. 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., DebyeHückel, 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
Basic numerical solution
• Iteratively solve matrix equations
obtained by discretization:
– Linear: multigrid
– Nonlinear: Newton’s 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…
Errors in numerical solutions
•
Electrostatic potentials are very
sensitive to discretization:
– Grid spacings < 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

  x  
l 0
M l  , 
x
l 1
Poisson-Boltzmann equation:
agreement with Coulomb’s law
• Energy consists of two components:
– Coulomb’s law contribution: often poorly approximated at short
lengths scales and/or coarse grid spacings
– Solvation energy/reaction field contribution: generally wellapproximated 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
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
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
Electrostatics Software
Software
package
Description
URL
Availability
APBS
Solves PBE in parallel with FD MG and FE AMG solvers.
Provides limited GB support
http://agave.wustl.edu/apbs/
Windows, All Unix. Free,
open source.
DelPhi
Solves PBE sequentially with highly optimized FD GS
solver.
http://trantor.bioc.columbia.edu/delphi/
SGI, Linux, AIX. $250
academic.
GRASP
Visualization program with emphasis on graphics; offers
sequential calculation of qualitative PB potentials.
http://trantor.bioc.columbia.edu/grasp/
SGI. $500 academic.
MEAD
Solves PBE sequentially with FD SOR solver.
http://www.scripps.edu/bashford
Windows, All Unix. Free,
open source.
UHBD
Multi-purpose program with emphasis on SD; offers
sequential FD SOR PBE solver.
http://mccammon.ucsd.edu/uhbd.html
All Unix. $300 academic.
MacroDox
Multi-purpose program with emphasis on SD; offers
sequential FD SOR PBE solver.
http://pirn.chem.tntech.edu/macrodox.htm
l
SGI. Free, open source.
Jaguar
Multi-purpose program with emphasis on QM; offers
sequential FE MG, SOR, and CG PBE solvers.
Offers GB support.
http://www.schrodinger.com/Products/jag
uar.html
Most Unix. Commercial.
CHARMM
Multi-purpose program with emphasis on MD; offers
sequential FD MG PBE solver and can be linked with
APBS. Offers GB support.
http://yuri.harvard.edu
All Unix. $600 academic.
AMBER
Multi-purpose program with emphasis on MD; offers GB
support.
http://www.amber.ucsf.edu/amber/
All Unix. $400 academic