Solvation
• What is the role of solvation
• Models for solvation
• Models for protein environments
Need a
solvent to
model a
zwitterion!
+
e =72
Solvation
•
•
•
•
•
Solvation influences:
Structure
Energetics
Spectroscopy
Equilibria…
Free Energy of Solvation
• To Solvate a Species:
• Making space for the solute molecule in the solvent; the
enthalpy and entropy change needed to make a cavity in
the solvent
• Coupling the solute and the solvent energetically, ie
including solute/solvent interaction terms through van
der Waals and electrostatic interactions (and hydrogen
bonding)
• Relaxing the solvent and solute molecules
Free Energy of Solvation
• DGsol  the free energy change to transfer a molecule
from vacuum to solvent.
• DGsol = DGelec + DGvdw + DGcav (+ DGhb)
Electrostatic
component.
Van der Waals
interaction
between solute
and solvent.
An explicit
hydrogen
bonding term.
Free energy required to form the
solute cavity. Is due to the
entropic penalty due to the
reorganization of the solvent
molecules around the solute and
the work done in creating
the cavity.
Models for Solvation
•
•
•
•
•
Not at all
Implicit or continuum models
Explicit quantum models of hydration
Classical dynamical models of hydration
A combination of models
Continuum Methods
• Solvent interactions are dominated by electrostatics
• Treat electrostatics classically
• Turn a quantum electron density into a classical charge
distribution and solve the Poisson equation for
interaction with a dielectric continuum
Continuum Models
• In a dielectric medium the charge distribution of the
solute polarises the dielectric and this induces a Reaction
Field in the solute cavity
• the Reaction Field interacts with the dipolar molecule by:
– perturbing the rotational motion of
the molecule causing it to have a
preferred orientation
– enlarging the dipole moment of the
molecule by elastic displacement of
its constituent charges. This induced
dipole moment is denoted af, where
a is the polarizability of the molecule
and f is the electric field acting on the
molecule due to all sources except the
molecule itself
e
m
a
Dielectric
The dielectric constant of a solvent is defined as the ability
of the solvent to respond to an applied electric field:
Dielectric
dielectric constant can be considered as the solvent’s ability
to interact with a charged (or dipolar) solute :
Charge in a nonpolar solvent
e =2-4
Charge in a polar solvent
e(water)=78; e(methanol)=40
Dielectric
dielectric constant can be considered as the solvent’s ability
to interact with a charged (or dipolar) solute :
Dipole in a nonpolar solvent
e =2-4
Dipole in a polar solvent
e(water)=78; e(methanol)=40
The Born Model for Ion Solvation
• Born, 1920: the electrostatic component of the free energy of
solvation for placing a charge in a spherical solvent cavity.
• The solvation energy is equal to the work done to transfer the ion
from vacuum to the medium. This is the difference in work to
charge the ion in the two environments.
q 
1
DG elec  1 - 
2a 
e
e
2
q a
• Ionic radii from crystal structures is used.

• Only
relevant for species with a formal charge.
Gruesome Details
• Mathematical Formulas that can be applied
• A rather complex procedure is used to determine the
Born radii in Still’s implementation. In short,
The Born radius of an atom corresponds to the radius
that would return the electrostatic energy of the system
according to the Born equation if all of the other
molecules in the system were uncharged.
• In Cramer and Truhlar’s QM approach the radius of the
atom is a function of the charge on the atom.
The Onsager Model
• Onsager, 1936: considers a polarizable dipole with
polarizability a at the center of a sphere.
• The solute dipole induces a reaction field in the
surrounding medium which in turn induces an electric
field in the cavity (reaction field) which interacts with the
dipole.
The Onsager Model
Onsager’s model assumes:
• A molecule occupies a sphere of radius a, its
polarizability, a, is isotropic, and no saturation
(ie frequency dependent) effects can take place.
• The short range molecular interaction energy is
negligible compared to kBT.
• The Onsager reaction field R is proportional to m and in
the same direction as m
Classical Onsager Equation
R 
2 e - 1 m
2 e
+ 1 a

Energy of a dipole in an E field  R = -  R m
Work done assembling
3
charge distributi
on =
R m
2
D G elec 
- R m
2

e - 1 m 2
2 e + 1 a 3
• If the species is charged an appropriate Born term must be added.
• Other Models: A point dipole at the center of a sphere (Bell), A
quadrupole at the center of a sphere (Abraham), multipole
expansion to represent the solute, ellipsoidal and molecular cavities.
Quantum Onsager – Self Consistent
Reaction Field
• The reaction field is a first-order perturbation of the
Hamiltonian.
H tot  H 0 + H RF
H RF  - mˆ

2 e - 1
2 e + 1 a
3
 mˆ 
D G elec   H tot  - 0 H 0 0 +

2 e - 1
2 2 e + 1 a
3
m
2
Correction factor corresponding
to the work done in creating the
charge distribution of the solute
within the cavity in the medium
The Cavity in the Onsager Model
• Spherical and ellipsoidal cavities may be used
• Advantage: analytical expressions for the first and second
derivatives may be obtained
• Disadvantage: this is rarely true!
• How does one define the radius value?
– For a spherical molecule the molecular volume, Vm can be found:
a
3

3V m
4 N A
; Vm 
MW

– Estimate by the largest interatomic distance
– Use an electron density contour
– Often the radius obtained from these procedures is increased to
account for the fact that a solvent particle can not approach right up to
the molecule
Surfaces
Isodensity surface +
van d er W aals
surface
S o lvent accessib le
suface
P ro b e S p here
R e-entrant surface
M o lecular surface
C o ntact surface
The Polarizable Continuum Method (PCM)
• Scaled van der Waals radii of the atoms are used to determine the
cavity surface.
• The solute charge distribution is projected onto this surface as
either a discretised distribution or a continuous distribution
• For example: the surface is divided into a number of small surface
elements with area DS.
• If Ei is the electric field gradient at pt i due to the solute then an
initial charge, qi is assigned to each element via:
H  H o +  s (r )
• The potential due to the point charges, s(r) is found, giving a new
electric field gradient. The charges are modified until they converge.
• The solute Hamiltonian is modified:
• After each SCF new values of qi and s(r) are computed.
The PCM Approach
D G el    H  d  -   0 H 0  0 d  -
1
2
  ( r )  ( r ) dr
Work done in creating the charge
distribution within the cavity in
the dielectric medium.
Problems:
– Using a discretised charge is problematic
– The wavefunction extends outside the cavity so the sum of the charges
on the surface is not equal and opposite to the charge of the solute 
outlying charge error
– The charge distribution may be scaled so that this is true
– The result depends upon the nature of the cavity and the nature of the
surface chosen
The Conductor-Like Screening Model (COSMO)
•
The dielectic is replaced with a conductor:
D E el ( dielectric ) 
•
e - 1
D E el ( conductor
e + x 
); 0  x  2
For the classical case the energy of the system is:
E (q ) 
1
QCQ + QBq +
2
1
qAq
2
 E ( q ) = Bq + Aq  0
q  -A
DE  -
1
-1
BQ
QBA
-1
BQ
2
•
•
C is the Coulomb matrix,
Bim represents the interaction between two unit charges placed at the
position of the solute charge Qi and the apparent charge qm, and Amn
represents the interaction between two unit charges at qm and qn.
SMD (Marenich, Cramer and Truhlar)
• PCM model based on the generalized Born approximation
• solvent cavity defined by superpositions of atom-centred spheres.
• The “D” stands for “density” to denote that the full solute electron
density is used without defining partial atomic charges.
• This model includes “surface tension” terms at the
solute−solvent boundary that are used model short-range
interactions between the solute and solvent molecules in the first
solvation shell.
• The radii and the atomic surface tensions have been parametrized
with a training set of 2821 solvation data including nonaqueous
solvents
• It gives significantly better results (especially for non-aqueous
solvents) than the original PCM formalism.
Explicit inclusion of Solvent Molecules
• the simplest and most accurate solvation model is to use collections
of the actual solvent molecules for the solvent interactions
performing a “supermolecule” calculation
• BUT: typically many solvent molecules (100–1000) are required to
adequately solvate a solute molecule
– the expense of describing these molecules with quantum chemical
techniques generally far outweighs the effort required to describe the
substrate itself !
• Use MM for the solvent, giving a QM/MM method (BUT how many
polarizable MM methods do we have…)
• Embed the solute–solvent complex can be embedded in a periodic
cell
• Lots of solvent molecules introduce a large number of
degrees of freedom that need to be either minimized or
averaged over during the course of the calculation
Definitions of the First Solvation Shell…
• predetermined number of solvent molecules,
• a distance cutoff in the solute-solvent distribution
functions
• an energy cutoff in the size of the solute-solvent
interaction energy.
• those solvent molecules that make contact with the
exposed van der Waals surface area of the solute
• those components of the free energy that correlate
statistically with the solvent accessible surface area or
the van der Waals surface area
First Solvation Shell
• Cavitation and dispersion (and hydrogen bonding) are
the most important first solvation shell effects
• stabilizing
– a hydrogen bond can orient the solvent around a polar solute
leading to a favourable interaction
• destabilizing
– the loss of orientational freedom (due to hydrogen bonding in
the first solvation shell) around a nonpolar solute results in an
unfavourable, hydrophobic interaction
2 water molecules
• With 2 explicit water molecules, one to stabilize the
positive and one to stabilize the negative charge, the
minimum energy structures of the amino acids is
zwitterionic.
-
+
Molecular Mechanics
• Model the entire system classically using force fields
“Mathematical expression describing the dependence of the
energy of a molecule on all atomic coordinates”
• there is no universally applicable water force field
• the TIP4P water model is probably the most widely used
Force Fields for Water
Dipole moment, D
Dielectric constant
Self diffusion, 10-5
cm2/s
Average
configurational
energy, kJ/mol
Density maximum,
°C
Expansion
coefficient,
10-4 °C-1
SSD
2.35
72
2.13
-40.2
-13
-
SPC
2.27
65
3.85
-41.0
-45
7.3 **
SPC/E
2.35
71
2.49
-41.5
-38
5.14
SPC/Fw
2.39
79.63
2.32
-
-
4.98
PPC
2.52
77
2.6
-43.2
+4
-
TIP3P
2.35
82
5.19
-41.1
-91
9.2
TIP3P/Fw
2.57
193
3.53
-
-
7.81
TIP4P
2.18
53
3.29
-41.8
-25
4.4
TIP4P-FQ
2.64
79
1.93
-41.4
+7
-
Expt.
2.95
78.4
2.30
-41.5
+3.984
2.53
Model
Force Fields for Water
Dipole moment, D
Dielectric constant
Self diffusion, 10-5
cm2/s
Average
configurational
energy, kJ/mol
Density maximum,
°C
Expansion
coefficient,
10-4 °C-1
TIP4P/2005
2.305
60
2.08
-
+5
2.8
SWFLEX-AI
2.69
116
3.66
-41.7
-
-
COS/G3 **
2.57
88
2.6
-41.1
-
7.0
GCPM
2.723
84.3
2.26
-44.8
-13
-
SWM4-NDP
2.461
79
2.33
-41.5
-
-
TIP5P
2.29
81.5
2.62
-41.3
+4
6.3
TIP5P-Ew
2.29
92
2.8
-
+8
4.9
TTM2-F
2.67
67.2
1.4
-45.1
-
-
POL5/TZ
2.712
98
1.81
-41.5
+25
-
Six-site*
1.89
33
-
-
+14
2.4
Expt.
2.95
78.4
2.30
-41.5
+3.984
2.53
Model
Composite Methods
• Partition the system into QM/MM/Dielectric
Composite Methods
• How are the partitions defined?
• How is the coupling term between the partitions
defined?
• How is energy transferred between the partitions?
• How sensitive are the results to the partitioning?
• Gordon’s EFP method seems to be the most “selfconsistent”
Effects of Solvent
Solvent will preferentially stabilise:
• solutes with bigger dipole moments
• More polarisable solvents
– Transition states tend to be more stabilised by solvent than
reactants or produts
– Solvent lowers energy barrier and speeds up the reaction
• “Quantitative” accuracy requires accounting for first
solvation shell effects
• Typical accuracies are 10-20 kJ/mol
Solvent Effects on Spectra
• Solvent has 2 main effects on absorption spectra:
• absorption peaks become broader:
gas phase
•
solution
phase
position of lmax may differ in different solvents:
shift to longer l: ‘red shift’
shift to shorter l: ‘blue shift’
Protein Environments
The hydrophobic side chains of a protein, illustrated as balls
Protein Environments
the dipoles on the protein backbone
Protein Environments
rotatable hydropholic side chains of a protein on the surface of a protein
Protein Environments
a common partitioning of the dielectric