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Implicit Solvation Models

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Title: Implicit Solvation Models


1
Implicit Solvation Models
  • Xiaolin Cheng
  • UT/ORNL Center for Molecular Biophysics
  • 02/18/2008

2
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
3
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
4
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
5
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.
6
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.
7
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?

8
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.
9
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.

10
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.
11
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.
12
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.
13
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.
14
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.

15
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.
16
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)

17
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,
18
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
19
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
20
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
21
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.
22
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
23
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
24
Delphi
25
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/
26
APBS
27
Diverse Applications
1. surface potentials, solvent transfer energies,
binding energies, pKa shifts, and effective
dielectric constants, and solvent forces
28
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
29
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
30
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)
31
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)
32
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)
33
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)
34
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.
35
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
36
More on pKa Calculations
assuming electrostatics dominate
37
Generalized Born Model
The solvation free energy for an arbitrary charge
distribution of N charges
38
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.
39
Microscopic Definition of Born Radii
Polarized charges occur mainly at the boundary
(around Rion).
Nina, Beglov, Roux, J. Phys. Chem. B 101, 5239
(1997)
40
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
41
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.
42
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

43
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.
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