Implicit Solvation Models

- Xiaolin Cheng
- UT/ORNL Center for Molecular Biophysics
- 02/18/2008

Water is important

Water has importance as a solvent, a solute, a

reactant and a biomolecule, structuring proteins,

nucleic acids and even cells. Water unique

properties High melting point Water

shrinks on melting High viscosity High

dielectric constant Its unique hydration

properties towards biological macromolecules

(particularly proteins and nucleic acids)

determine their three-dimensional structures, and

hence their functions, in solution. In

molecular modeling studies, the water environment

can be represented either explicitly or

implicitly as we will discuss below.

http//en.wikipedia.org/wiki/Water_model

Water Molecule Structure

V-shaped, approximately tetrahedrally arranged,

sp3-hybridized electron pairs, two of which are

bonded with hydrogen atoms leaving the two

remaining lone pairs. gaseous water molecule

O-H length 0.95718 Å, H-O-H angle 104.474

liquid water O-H length 0.991 Å, H-O-H angle

105.5 (ab initio) O-D length 0.970 Å, D-O-D

angle 106 (neutron diffraction) These bond

lengths and angles are likely to change, due to

polarization shifts, in different hydrogen-bonded

environments and when the water molecules are

bound to solutes and ions. Commonly used

molecular models use O-H lengths of between 0.957

Å and 1.00 Å and H-O-H angles of 104.52 to

109.5.

http//en.wikipedia.org/wiki/Water_model

Explicit Water Models

A water model is defined by its geometry,

together with other parameters such as the atomic

charges and Lennard-Jones parameters.

Many different models have been proposed they

can be classified by the number of points used to

define the model (atoms plus dummy sites),

whether the structure is rigid or flexible, and

whether the model includes polarization effects.

http//en.wikipedia.org/wiki/Water_model

Explicit Water Models

The simplest water models treat the water

molecule as rigid and rely only on non-bonded

interactions. The electrostatic interaction is

modeled using Coulomb's law and the dispersion

and repulsion forces using the Lennard-Jones

potential.

Electrostatic

Lennard-Jones

Note In most water models, the Lennard-Jones

term applies only to the interaction between the

oxygen atoms.

Explicit Water Model Examples

?(Å) ?(kjmol-1) l1(Å) l2(Å) q1(e) q2(e) ?? ??

SPC/E 3.166 0.650 1.00 - 0.4238 -0.8476 109.47 -

TIP3P 3.1506 0.6364 0.9572 - 0.4170 -0.8340 104.52

TIP4P 3.15365 0.6480 0.9572 0.15 0.5200 -1.0400 104.52 52.26

TIP5P 3.1200 0.6694 0.9572 0.70 0.2410 -0.2410 104.52 109.47

The SPC/E model adds an average polarization

correction to the potential energy function

better density, diffusion constant CHARMM

version of the TIP3Pmodel places Lennard-Jones

parameters on the hydrogen atoms.

Explicit Water Models

parameterized by running Monte Carlo or molecular

dynamics simulations and adjusting the parameters

until the bulk properties (density and heat of

vaporization) are reproduced well enough.

Dipole moment Dielectric constant Self diffusion, 10-5cm2/s Average configurational energy, kjmol-1 Density Maximum, ?C Expansion coefficient, 10-4?C-1

SPC/E 2.35 71 2.49 -41.5 -38 5.14

TIP3P 2.35 82 5.19 -41.1 -91 9.2

TIP4P 2.18 53 3.29 -41.8 -25 4.4

TIP5P 2.29 81.5 2.62 -41.3 4 6.3

Expt. 2.95 78.4 2.30 -41.5 3.984 2.53

- Radial distribution function (neutron

scattering), phase transition, vibrational

spectrum, biomolecular solvation - Bridge microscopic model ? macroscopic

properties - Homework
- How are dipole moment, radial distribution

function, heat of vaporization, expansion

coefficient dielectric constant, self diffusion

coefficient calculated in molecular dynamics

simulation? - How can these properties be measured

experimentally?

Explicit Water Models

Explicit water models can provide a realistic

picture of how biomolecules behave in a

biological environment structured water

molecules, solvation, hydrophobic effect but

the large number of water molecules in addition

to the biomolecule adds significant computational

costs -- the computational cost of a water

simulation increases with the number of

interaction sites in the water model when using

rigid water models in molecular dynamics, there

is an additional cost associated with keeping the

structure constrained. Water molecule is a

flexible molecule with electronic

polarization nonpolarizable models have been

shown to be inherently unable to simultaneously

predict certain physical properties, such as

melting temperature and the temperature of

maximum density.

Further Readings

- 0. http//en.wikipedia.org/wiki/Water_model
- 1. Jorgensen, W. L. Quantum and statistical

mechanical studies of liquids. 10. Transferable

intermolecular potential functions for water,

alcohols, and ethers. Application to liquid

water. J. Am. Chem. Soc. 1981, 103, 335-340. - 2. P. Ren and J. W. Ponder, Polarizable Atomic

Multipole Water Model for Molecular Mechanics

Simulation, J. Phys. Chem. B 107, 5933-5947

(2003) - 3. Darden T, Perera L, Li L and Pedersen L.

(1999) "New tricks for modelers from the

crystallography toolkit the particle mesh Ewald

algorithm and its use in nucleic acid

simulations", Structure 7, R55-R60. - 4. L. Greengard, V. Rokhlin, J. Comput. Phys. 73

(1987) 325.

Implicit Solvation Models

Implicit Solvent

Average Density

Explicit Solvent

http//feig.bch.msu.edu/main-research-methodology.

html

It is possible to construct an implicit solvent

model by approximating the medium outside the

water-excluded volume as a continuum with

electrostatic, entropic, and viscous properties

that match water.

Implicit Solvation Models

Consider the ensemble average of property A

dependent on only the solute degrees of freedom X

Integration over the solvent degrees of freedom,

and define the potential of mean force W(X)

Now the expression for ltAgt above becomes,

where the integration is only on the solute

degrees of freedom and solvent effects are

accounted implicitly by W(X). The definition of

W(X) does not immediately simply the calculation

of ltAgt. But W(X) is available from other sources,

such as empirical description of the solvent.

Implicit Solvation Models

The gradient of W(X) with respect to X is the

force F(X) exerted on the solute by the solvent

averaged over the ensemble of solvent

configurations when the solute degrees of freedom

fixed at X

The function W(X) describes a free energy

potential not a mere potential energy the

solvation free energy on the solute due to the

solvent averaged over many solvent

configurations. It is noteworthy that the

solvation free energy is the free energy required

to transfer a solute molecule from the vacuum

(gas phase) to the solvent.

Implicit Solvation Models

An alternative approach, we can write the

solvation free energy of the solute in

conformation X

where ?Gnp is the change of solvation free energy

in going from nothing to the non-polar solute and

?Gelec is the change of free energy in going from

the non-polar (uncharged) form of the solute to

the polar (charged) form of the solute.

Nonpolar Solvation

The solvation of a non-polar solute in water is

disfavored by the disruption of the hydrogen

bonding network of water and also by the loss of

entropy associated with the fact that water

molecules cannot occupy the volume occupied by

the solute. The solvation of non-polar solute is

however favored by attractive van der Waals

interactions between the solute and solvent, even

though the van der Waals interactions are usually

not as strong as the polar interactions.

The non-polar solvation free energy of a solute

is proportional to the solvent accessible surface

area (SASA) of the solute

where A(X) is the SASA of conformation X, ? is an

adjustable parameter that is interpreted as

surface tension. For hydrocarbons ? 5 cal/(mol

Å2) The formulation becomes strictly valid in

macroscopic thermodynamics - the creation of a

phase boundary between two macscopic phases is

associated with a free energy increase

proportional to the area of the boundary surface.

Nonpolar Solvation

The van der Waals surface area is the surface of

the union of the spherical atomic surfaces

defined by the van der Waals radius of each

component atom in the molecule. The solvent

accessible surface area (SASA) is the surface

area of a biomolecule that is accessible to a

solvent. ASA is typically calculated using a

sphere (of solvent) of a particular radius to

'probe' the surface of the molecule. The

solvent-excluded surface (also known as the

molecular surface or Connolly surface), which is

imagined as a cavity in bulk solvent.

Nonpolar Solvation

The solvent accessible surface area (SASA)

partially accounts for the hydrophobic effect.

The hydrophobic effect is the property that

non-polar molecules tend to form intermolecular

aggregates in water, an entropic effects arising

from solute-imposed constraints on the

organization of the water or solvent molecules.

At the molecular level, the hydrophobic effect is

an important driving force for biological

structures and responsible for protein folding,

protein-protein interactions, formation of lipid

bilayer membranes, nucleic acid structures, and

protein-small molecule interactions. SASA

enjoys many successful applications solvation

energy, binding free energy (MMPB/SA), and

protein dynamics (GB/SA) but not accurate can

not capture the "specific" distance-dependent

interactions this surface area pertains to the

solute, while the hydrophobic effect is entropic

in nature and occurs on the side of the

solvent. Further readings Chandler, D.,

"Insight Review Interfaces and the driving force

of hydrophobic assembly, Nature 437, 640-647

(2005). For a recent discussion, see PNAS, 103,

8331 (2006)

Polar Solvation

Consider the free energy of placing a charge q at

the center of a spherical cavity in a solvent.

The process of reversibly placing the charge is

accomplished by introducing a charging parameter

? such that ?q is the charge along the charging

process. The solute-solvent potential energy of

the system at ? is

The mean force along the charging parameter ?,

The charging free energy is therefore given by,

Polar Solvation

An alternative would be to consider the solvent

to be a uniform dielectric medium with dielectric

constant ? (for water ? 80). The dielectric is

polarized by the charge at the center of the

spherical cavity and will produce a reaction

field that opposes the electric field produced by

the solute charge. The average reaction field

lt?gtv,?q at ? generated by the dielectric at the

center of the sphere is given by

Gausss theorem

Polar Solvation

For the general case of a solute of arbitrary

shape with several partial charge sites, the

electrostatic free energy is given by,

? satisfies the Poisson equation

analytical solution available for spherical,

cylindrical, or planar symmetry

Poisson-Boltzmann Theory

The electrostatic potential related to charge

density is given by Poissons law

Mobile ions and the Poisson-Boltzmann equation

Expand at low-salt concentration

where,

The Tanford and Kirkwood model for protein

Numerical Solution of PB

Numerical solution (FD, BE, FEM) The finite

difference formulation spatial derivatives are

approximated using neighboring points. A

successive overrelaxation method used to get

rapid convergence in solving the linear systems

obtained from the finite difference

discretization The boundary element method

utilizes analytical solutions obtained in terms

of Greens functions and discretization on the

domain surface (molecular surface) The finite

element method an adaptive multilevel approach

based on tetrahedral elements to create a dense

mesh to capture the dielectric discontinuity

across the molecular surface.

Numerical Solution of PB

Advantages

Disadvantages

Finite difference and uniform mesh methods Fast solvers Low memory overhead Cartesian mesh Non-adaptive Poor solution resolution Previous parallel methods complicated and inefficient

Boundary element methods Smaller numerical systems Easier interaction evaluation Less efficient solvers Only applicable to linear problem

Finite element methods Highly adaptive Relatively fast solvers Previous solver and adaptive methods inadequate Previous parallel methods complicated and inefficient

adapted from Nathan A. Bakers slides, North

Dakota State University, 2003

Delphi

A Finite Difference Poisson-Boltzmann

Solver DelPhi provides numerical solutions to the

Poisson-Boltzmann equation (both linear and

nonlinear form) for molecules of arbitrary shape

and charge distribution. It has nice features for

assigning different dielectric constants to

different regions of space and treating systems

containing mixed salt solutions.

http//wiki.c2b2.columbia.edu/honiglab_public/ind

ex.php/SoftwareDelPhi GRASP (Graphical

Representation and Analysis of Structural

Properties) GRASP is a molecular visualization

and analysis program. It is particularly useful

for the display and manipulation of the surfaces

of molecules and their electrostatic properties.

http//wiki.c2b2.columbia.edu/honiglab_public/ind

ex.php/SoftwareGRASP

Delphi

APBS

APBS Adaptive Poisson-Boltzmann Solver APBS is

a software package for the numerical solution of

the Poisson-Boltzmann equation (PBE), for

evaluating the electrostatic properties of

nanoscale biomolecular systems. APBS was

designed to efficiently evaluate electrostatic

properties for such simulations for a wide range

of length scales to enable the investigation of

molecules with tens to millions of atoms. APBS

uses FETK (the Finite Element ToolKit) to solve

the Poisson-Boltzmann equation numerically. http/

/apbs.sourceforge.net/ PMV (Python Molecular

Viewer) PMV is a powerful molecular viewer that

has a number of customizable features and comes

with many pluggable commands ranging from

displaying molecular surfaces to advanced volume

rendering. http//mgltools.scripps.edu/

APBS

Diverse Applications

1. surface potentials, solvent transfer energies,

binding energies, pKa shifts, and effective

dielectric constants, and solvent forces

Diverse Applications

1. surface potentials, solvent transfer energies,

binding energies, pKa shifts, and effective

dielectric constants, and solvent forces

adapted from Nathan A. Bakers slides, North

Dakota State University, 2003

Diverse Applications

1. surface potentials, solvent transfer energies,

binding energies, pKa shifts, and effective

dielectric constants, and solvent forces

Determine binding energies between 30S ribosomal

subunit and aminoglycoside antibiotics - suggests

importance of basic functional groups on Ring IV

adapted from Nathan A. Bakers slides, North

Dakota State University, 2003

Diverse Applications

1. solvent transfer energies, binding energies,

pKa shifts, solvent forces, surface potentials

and effective dielectric constants 2.

Protein-protein, protein-DNA, protein-ligand

interactions (association, dissociation)

Diverse Applications

1. solvent transfer energies, binding energies,

pKa shifts, solvent forces, surface potentials

and effective dielectric constants 2.

Protein-protein, protein-DNA, protein-ligand

interactions (association, dissociation) 3.

Electrostatic mediated diffusion processes (ion

permeation, etc)

Diverse Applications

1. solvent transfer energies, binding energies,

pKa shifts, solvent forces, surface potentials

and effective dielectric constants 2.

Protein-protein, protein-DNA, protein-ligand

interactions (association, dissociation) 3.

Electrostatic mediated diffusion processes (ion

permeation, etc)

Diverse Applications

1. solvent transfer energies, binding energies,

pKa shifts, solvent forces, surface potentials

and effective dielectric constants 2.

Protein-protein, protein-DNA, protein-ligand

interactions (association, dissociation) 3.

Electrostatic mediated diffusion processes (ion

permeation, etc) 4. Efficient sampling of

structures (not very popular yet) 5. Can apply

to not only solution but also mixed system such

as membrane 6. Hybrid implicit/explicit solvation

models - it is possible to include a layer or

sphere of water molecules around the solute, and

model the bulk with an implicit solvent.

Further readings M. Feig, C. Brooks, Curr. Opin.

Struct. Biol. 14, 217 (2004) T. Simonson, Rep.

Prog. Phys. 66, 737 (2004)

Practical Considerations

A PB model ignores the volume of ions in the

medium. Therefore, PB equation is valid for

dilute ionic solutions (i.e., concentration 0.15

M). Dielectric constant Radii (dielectric

interface, importance of the boundary) Pay

attention to parameterization details Ionization

states of protein side chains

Determining side-chain pKas

PDB2PQR is a Python software package that

automates many of the common tasks of preparing

structures for continuum electrostatics

calculations, providing a platform-independent

utility for converting protein files in PDB

format to PQR format. http//pdb2pqr.sourceforge.n

et/ H is an automated system that computes pK

values of ionizable groups in macromolecules and

adds missing hydrogen atoms according to the

specified pH of the environment.

http//biophysics.cs.vt.edu/H/ Whatif

http//swift.cmbi.kun.nl/whatif/ Further

readings http//enzyme.ucd.ie/Science/pKa/ D.

Bashford and M. Karplus Biochemistry, 29

10219--10225, 1990.

More on pKa Calculations

Exposed residues are trivial Buried residues

are more difficult to deal with because of

heterogeneous electrostatic interactions - many

residues with shifted pKa play an important role

More on pKa Calculations

assuming electrostatics dominate

Generalized Born Model

The solvation free energy for an arbitrary charge

distribution of N charges

Generalized Born Model

Important and Difficult effective Born radii

calculation

This functional form of the so-called Generalized

Born (GB) approximation has been used with

considerable success to efficiently evaluate

hydration energies for small molecules.

Parameterization of the method involves

accounting for the effects of neighboring solute

atoms in the determination of each atom's

effective Born radius a.

Microscopic Definition of Born Radii

Polarized charges occur mainly at the boundary

(around Rion).

Nina, Beglov, Roux, J. Phys. Chem. B 101, 5239

(1997)

Calculation of Born Radii

Self-energy

Pairwise descreening approximation

j

Rij

Hi is the fraction of the area of a sphere of

radius r centered at atom i that is shielded by a

sphere ?j at a distance of Rij

i

r

Generalized Born Model

A model based on further approximation to

PB analytical, simple, fast and reasonable

approximation Efficient sampling of structures

(dynamics is somewhat tricky to

interpret) Many successful applications

Trp-cage folding, Simmerling et al., 2002 JACS

DNA stability, Tsui et al., 2000, JACS

binding energies, MMGB/SA, P. Kollman et al.

2000, Accts. Chem. Res. constant pH

simulations J. Mongon et al. 2005, Curr. Opin.

Struct. Biol.

Limitations of Implicit Solvation Models

- The hydrophobic effect
- this surface area pertains to the solute, while

the hydrophobic effect is entropic in nature and

occurs on the side of the solvent. - Viscosity
- water molecules by randomly colliding and

impeding the motion of solutes - makes sampling

of configurations and phase space faster

misleading kinetics results - using Langevin dynamics with a damping constant
- Coupling between nonpolar and polar terms
- Recent discussion see, J. Dzubiella et al. J.

Chem. Phys. 2006 - Pay attention to parameterization details
- Over-stabilized salt-bridge

Limitations of Implicit Solvation Models

- Hydrogen-bonds with water
- The average energetic contribution of

protein-water hydrogen bonds may be reproduced

with an implicit solvent. However, the

directionality of these hydrogen bonds will be

missing.