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
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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
4321 ΔΔΔΔ GGGG
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)
i
i i
qx
x x
Chargemagnitudes
Chargelocations
Solventdielectric
Coulomb law• Simplest implicit solvent model• Assumptions:
– Solvent = homogeneous dielectric– Point charges– No mobile ions– Infinite domain (no boundaries)
• Solution to Poisson equation
2 4
0
i ii
x q x x
Point chargedistribution
Boundarycondition
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
1
2i j
i j i j i
q qG
x x
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)
1/ 2
24
ix xi
i i
m im
q ex
x x
n QkT
Inversescreening
length
Mobile ionbulk density
Debye-Hückel law
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r
0.5
1
1.5
2
2.5
3
3.5
Debye Huckel
Coulomb
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
2 2 4
0
i ii
x x q x x
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
1
2
i jx x
i j
i j i j i
q q eG
x x
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
2
22
1 11
2 , ,
, , exp4
i jisolv
i j ii i j i j
ijij i j ij i j
i j
q qqG
R f x x R R
rf r R R r R R
R R
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
0
i ii
x x f x
q x x
Dielectricfunction
21
4
1
8
2
1 1
8 8i i i ii i
G f dx
dx
q x x dx q x
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
21
4 2
1
81 1
8 8i i i ii i
f
q x x q x
G dx
dx
dx
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
2
1
2
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Coulomb
2
1
2
aw
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i
i ii
i j
i j i j
i
x xi i
i j
i j i i j
G q x
q q
x x
q q
x
q
x
x x
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…
2
1 2
1
2
1
4
1
4
i ii
i p
p i ii
x x q x
x
x
x
x q x
x
x
x
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
2 1
2 1
1
2
1
2
solv
i i ii
i ii
G G G
q x x
q x
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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
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
Point charge
Highdielectric
Lowdielectric
A continuum descriptionof ion solvation
A continuum descriptionof 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
4 4
0
i ii
x x q x x x
Dielectricfunction
Biomolecularcharge
distribution
Mobilecharge
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
( )m mQ x V xm m
m
x Q n e
Ioncharges
Ionbulk densities
Ion-protein stericinteractions
Poisson-Boltzmann derivation: step 3
• Substitute mobile charge distribution back into Poisson equation
• Result: Nonlinear partial differential equation
4 4
0
m mQ x V xm m i i
m i
x x Q n e q x x
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?
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-
4 4m mQ x V x
mi
im m ix x Q xn e q x
Equation coefficients: mobile ion distribution
• Provides:– Bulk ionic strength– Ion accessibility
• Usually constructed based on “inflated van der Waals radii”
4 4m mQi i
x xm
i
Vm
m
Q n ex x q x x
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!!!
4 4m mQ x V xm m i i
m i
Q n e q xx xx
Poisson-Boltzmann special cases
• 1:1 electrolyte (NaCl)– Assume similar steric interactions for each species with
protein– Simplify two-term series to hyperbolic sine
( )
2
2
4
4
8 sinh
sinh
m m
cc
Q x V xm m
m
V x e xe xc
V xc c
c
x Q n e
e ne e e
e ne e x
x e x
Modified screening coefficient:zero inside biomolecule
1:1 electrolytecharge 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
sinh 4c i ii
x x x e x q x x
2 24 4m mQ x V x V xm m m m
m m
Q n e e Q n x x x
2 4 i ii
x x x x q x x
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
21cosh 1
4 2fG dx
Fixed charge-potential interactions Dielectric
polarizationMobile 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
Electrostatic influenceson ligand binding
Protein Kinase A
Balanol
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
2 2
2
2
1[ ] - cosh 1 d
4 2i ic i i
kTF u f u u u x
e x x
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., 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
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
1
0
,ll
l
Mx
x
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 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
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
• 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
PBE: current solution methods
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.html
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/jaguar.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