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

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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

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length

Mobile ionbulk density

Debye-Hückel law

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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

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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

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1

2

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1lim

Coulomb

2

1

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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

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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