Lecture 12: Solvation Models: Molecular Mechanics

Modeling of Hydration Effects

Dr. Ronald M. Levyronlevy@temple.edu

Statistical Thermodynamics

“Bare” Molecular Mechanics Atomistic Force Fields:

torsion stretching

bending

non-bonded

E stretch= ∑bonds

K r r− r 0 2

E bend = ∑angles

K q q−q 0 2

E torsion= ∑torsions

V n

2[1±cos n−n ]

E non−bonded=∑ij [ qi q j

r ij

4ij ij12

r ij

12−

ij6

r ij

6 ]

Hydration has a large effect on the conformations of macromolecules

MD simulation, RMSD from native native:

… and on ligand binding

Distribution of complex decoy binding energies:

Challenge

• Model hydration conveniently, as accurately as needed by the application and with the least computational cost.

Two main approaches

• Explicit solvation models.• Implicit solvation models.

Explicit Solvation

• Most accurate/detailed.• Computationally expensive.• Requires averaging over solvent coordinates.• Difficult to obtain relative free energies of solute

conformations.

• Each solvent molecule is represented with a set of atomic interaction centers (just as for the solute).

Implicit Solvation

• Theoretical framework based on solvent PMF.• Not as accurate, especially for short-range solute-solvent

interactions.• Reduced dimensionality.• Relative solvation free energies from single point

effective potential energy calculations.

• The solvent is represented by a continuum described by macroscopic parameters such as the dielectric constant, density, surface tension, etc.

• Many recent developments in molecular modeling have focused on solvation models.– Explicit: long-range electrostatic models– Implicit: improve coverage, accuracy and efficiency.

• Sometimes it is hard to keep track of all the choices.

From a 2007 publication:

The peptide is simulated in TIP3P and several variations of the GB implicit solvent model: GBHCT, GBOBC, and GBNeck (igb = 1, 5, and 7, respectively, in Amber 9). […] For consistency, MBondi radii were used in both the GB REMD simulations and subsequent GB and PE energy calculations described below.

For TIP3P simulations, Ala10 was solvated in a truncated octahedral box with 983 solvent molecules […] long-range electrostatic interactions were calculated using periodic boundary conditions via the particle mesh Ewald (PME).

Roe, Wickstrom, et al. JPC B 111:1846 (2007)

Explicit Hydration: Water Models• A water model specifies the interaction sites of a water molecule and their

partial charges and LJ parameters.

• There is a large variety of models: rigid/flexible, 3-points/4-points or more, polarizable, dissociable, …

• The 3 rigid models most frequently used are SPC, TIP3P and TIP4P:

SPC TIP3P TIP4P

r(OH),Å 1.0 0.96 0.96

HOH, deg 109.5 104.5 104.5

σ(O), Å 3.16 3.16 3.15

ε(O),kcal/mol 0.1554 0.155 0.155

q(O) -0.82 -0.834 0.0

q(H) 0.41 0.417 0.52

q(M) -1.04

Periodic Boundary Conditions (PBC’s)• PBC’s are often used to simulate a

bulk solution with a relatively small number of molecules.

• The “central” simulation box is replicated in 3D space so that the system has no edges.

• The “minimum image convention” says that the actual distance between two particles is the smallest distance between all of the distances between their images:

Δxmin=Δx−Lnint ΔxL

where L is the box size and x is the distance (along x) between any two images of a pair of particles. nint()=nearest integer.

• A variety of space filling box shapes can be used. The “truncated octahedron” has particularly nice properties.

Treatment of long-range electrostatic interactions

Electrostatic energy (per cell) of a periodic system:

U c=12∑i

q i φ r i φ r i =∑j , n

q j

∣r ijnL∣

• The sum for the electrostatic potential φ includes all of the particles in the unit cell plus all of their images.

• Straight sum is “conditionally convergent”; for example it would diverge if the positive and negative charges are summed separately.

• Unlike LJ interactions, in this case truncation of the sum (with a distance cutoff) can be inaccurate.

• In explicit solvent molecular simulation programs the infinite sum problem is circumvented using the “Ewald summation” class of models.

Ewald summation methodsThe Ewald sum is based on a decomposition of the 1/r interaction into a “fast decaying” component (handled with distance cutoffs) and a “smooth” component (summed using a Fourier transformation in reciprocal space).

Think of surrounding each charge by a canceling Gaussian charge distribution:

∑ij

1r ∑ij

erfc r r

∑k

ρ k2 exp−k 2 /4α

k 2

ρk is the Fourier transform of the charge distribution:

measures the “softness” of the canceling distribution - the larger the more convergent is the direct sum (but less convergent is the reciprocal sum).

k=∑iqi e−i k⋅r i

c , i=−q i / 3 /2 e− r 2

Particle mesh Ewald methods• Straightforward implementations of the Ewald sum scale at best as N3/2

• More efficient implementations of complexity NlogN exploit FFT techniques by expressing the charge density on a regular grid.

• The original idea of Hockeny and Eastwood (1981) consisted of assigning fractional charges to grid points: PPPM or P3M

• Darden, York and Pedersen (1993) presented an (equivalent) approach based on Lagrange interpolation of the structure factors exp(ikr) on a grid: PME

• PME was later refined using B-spline interpolation by Essmann et al. (1995) to obtain gradients for MD: SPME − In most recent papers, “PME” is SPME

Fast Multipole Methods

• Based on representing distant charge distributions by multipole expansions.

• Very favorable asymptotic scaling of order N – but with high initial overhead.

• For values of N typical for biomolecular simulations FMM tends not to be as efficient as PME-based methods.

• Still the only efficient method to treat long-range interactions without periodic boundary conditions.

• Used routinely in simulations of million-body simulations of galaxies, etc.

• A future for FMM in the context of implicit solvent modeling?

Figuerido, Levy, Zhou, Berne. JCP 106:9835 (1997)

Implicit Solvation:the solvent potential of mean force (SPMF)

• Write the canonical partition function Qxy of the solution in terms of solute coordinates x, and solvent coordinates y.

• The average of any observable that depends only on x can be written in terms of an “effective solvation potential” W(x).

⟨O x ⟩=Q xy−1∫ dxdyO x exp {− β [ U x U x , y U y ] }

⟨O ( x )⟩=1

Q x∫ dxO ( x ) exp {− β [U ( x )+W ( x ) ] }

where

exp [−βW ( x ) ]=∫ dy exp {−β [U ( x , y )+U ( y ) ] }∫ dy exp {−βU ( y ) }

defines the solvent potential of mean force

• So called because the gradients of W(x) with respect to the solute coordinates are equal to the average forces of the solvent on the solute atoms at fixed solute conformation.

• W(x) implicitly contains all of the effects of solvation.

The solvent potential of mean force is a free energy

Conformational equilibria:

ΔG solv ( x )=− kT lnQ sol

Q sep

=−kT ln exp [− βW ( x ) ]=W ( x )

Solvation (fixed solute conformation):

y x Q sep= exp [− βU x ] Q y

y xQ sol=exp [− βU ( x ) ]∫ dy exp {−β [U ( x , y )+ U ( y ) ] } =exp {− β [U ( x )+W ( x ) ] }Q y

ΔG21=−kT lnP( x2 )P( x1 )

=ΔU 21+ ΔW 21x1x2

Polar/Non-polar decomposition of the solvent PMF

−U ch( v )

+U ch( w )

G solv ( x )=W ( x ) G np G cav

G vdW

G solv= G elec G np

G np= G cav G vdW G elec=U ch

( w )−U ch( v )

Typical Modern Implicit Solvent Model

Electrostatic Component: Continuum Dielectric• Poisson-Boltzmann solvers (accurate but numerical and

slow).• Generalized Born models (faster, can be expressed as

analytic function).

Non-Polar Component:• Solute surface area models• Cavity + van der Waals NP models.

The Poisson Equation (PE) and the Poisson-Boltzmann (PB) equations

−∇⋅[ ε x ∇ ϕ x ]= ρ x

ε=1

ε= εslv

PE for a non-homogeneous dielectric:

Electrostatic free energy of solute:

Gelec=12∑i

q i ϕ ( x i )

W elec=G elec( ε in= 1, ε out= ε slv )−G elec ( ε in= 1, ε out= 1 )

Electrostatic solvation free energy:

W elec=12∑i

q i ϕ ( x i )−12∑i

qi ϕ q ( xi)=12∑i

qi ϕ rf ( x i)

The reaction field potential φrf is due to the polarization of the medium.

+ -

direct Coulomb

The PB equation takes into account the effect of free ions in solution

linearized form of the PB equation.

ε=1

ε= εslv

+ -

+

-

When (otherwise ion adsorption):∣ βz j ϕ x ∣<< 1

−∇⋅[ ε x ∇ ϕ x ] k 2 x ε x ϕ x = ∑atoms

q i δ x− x i

Debye-Hückel ion distribution

k 2 x =2I

kT ε x with (ionic strength).where

−∇ [ x ∇ x ]= x =

∑iq i x−x i∑ions

c j z j e− z j x

I =12 ∑ j

c j z j2

Numerical solutions of the PB equation

• The PB equation is solved on a grid in both surface and volume formulations.

• Finite difference: solves the PB equation on a volume grid (APBS, Delphi, UHBD)

• Finite element: solves integral form of the equation on a volume grid (PBF)

• Boundary element: surface grid.• PB solvers often available in molecular simulation packages: Amber,

CHARMM, IMPACT, etc.• Main drawback: continuum dielectric models are not suitable for

specific short-range solute-solvent interactions, finite size effects, non-linear effects, high ionization states.

• Other limitations are dependence on atomic radii parameters, speed, lack of analytical derivatives, dependence on frame of reference.

Approximate continuum dielectric models

1.Dielectric polarization around polar groups

• Favorable interaction between exposed charged atoms and the polarized dielectric.

• Born model of ion hydration:

2.Dielectric screening of electrostatic interactions

• The dielectric weakens the interactions between charges

• Distance-dependent dielectric models

+-

--

-+-

-- +

+

+

The basic idea is that a dielectric model of hydration should describe these two basic effects:

ΔG elec=−1

2 1− 1

εw z2

R

u ij=q i q j

ε r ij r ij

ε r

r

W elec=− 12 (1− 1

εw )∑ij

qi q j

f ij (r ij )

f ij = [ r ij2 B i B j exp −r ij

2 /4 B i B j ]1 /2

Bi is the Born radius of atom i defined by:

W singlei =− 1

2 (1− 1εw )

q i2

Bi

≈− 18π (1− 1

ε w)∫

V

q i2

∣r−r i∣4 d3 r

Generalized Born Model

• Satisfies the Born model in the two limits of infinite separation and complete overlap of solute atoms.

• Basically it’s an interpolation formula.

Overall Features of Generalized Born Models

The GB model “works” because it describes both dielectric polarization and dielectric screening effects.

W elec=− 12 (1− 1

εw )∑ij

qi q j

f ij (r ij )

Polarization i=j (“self” energy):

W singlei =− 1

2 (1− 1εw )

q i2

B i

Favors the solvent exposure (small Bi) of polar

groups (large q).

Dielectric screening i≠j (pair energies):

u ij=q i q j

ε r ij r ij

εij r =S r

B i B j

S x =[1−1− 1ε 1

1− x−2 exp −x2 /4 ]−1

S x

Born Radii: Pairwise Descreening Scheme

Solute

Solvent Dielectrici i i

1B i

=1

4 ∫dielectric

1

∣r i− r∣4d 3 r=

1Ri

−1

4 ∫solute- i

1

∣r i− r∣4d 3 r

1B i

=1R i

1B i

=1R i

−1

4 ∫atom j

1

∣r i−r∣4d 3 r

j

Summing over all j’s:1B i

≈1R i

−1

4 ∑j

Q ij

Born radius calculated as a sum of pairwise atomic contributions:

Qij: pairwise descreening

function.Forms the basis for most analytical GB models used in molecular simulations.

Pairwise Descreening GotchasQij is not simply the integral of 1/|ri - r|4 over atom j because:

1. Overlaps with atom j and other atoms would be over-counted

2. Atoms i and j may overlap

i

j

ij

Reduce descreening contribution from atom j

Integrate only over portion of jnot overlapping with i

3. Both situations can and do occur simultaneously.

GB implementations• Most major biomolecular simulation packages (CHARMM, Amber,

IMPACT, Gromacs, etc.) include pairwise descreening GB implementations suitable for MD calculations.

• Key ingredients are the atomic radii and the description of the solute volume.

• The atoms overlap problem is generally addressed by empirical scaling coefficients parameterized with respect to higher level calculations – that is the geometric model is parameterized in addition to the energetic model (ACE, GB/SA, GBHCT, GBSW)

• Work on the AGBNP series of models shows that “geometric” parameterization is unnecessary.

• Some implementations (GBMV, SGB) perform numerical integration on a grid (volume or surface) – non-analytic, higher computational cost, difficulties with derivatives, dependence on coordinate frame.

• Some implementations differ in the choice of the GB distance function f(r)

• Many of the models include continuum dielectric “correction” terms.• Recent developments have focused on the “interstitial” volumes problem

(GBneck, GBMV, AGBNP2).

Non-Polar Hydration Free Energy• Defined as the work of introducing the “uncharged” solute into

solution. • In some ways a harder problem for modeling than electrostatics.• Often (and, nearly as often, incorrectly) ignored because appears

“small” in magnitude. • Traditionally modeled in terms of the solvent-exposed surface of the

solute by means of a surface tension parameter (SA models)

• Motivated by macroscopic interfacial models and theories of hydration of cavities (probability of spontaneous occurrence of voids) in water.

• Recent developments have moved beyond surface-only models adopting distinct geometric models for the cavity and van der Waals components of non-polar hydration (NP models):

ΔG np= γA

G np= G cav G vdW

Example of an analytical NP model (the “NP” in AGBNP)

G np=∑i

[ γ i A i+ α i ω ( B i ) ]

ωi≈ ρw∫slv .

-4ϵiσ i6

∣r−ri∣6

=-16 π ρwϵiσi

6

3Ci3

C i= 34∫slv .

1∣r−r i∣

6 1 /3

≈B i

: Surface area of atom i

: Geometrical predictor based on Born radius

: Surface tension and van der Waals adjustable parameters

A i

ω ( B i )

i , αi

DG np= G cav G vdW

Advantages of NP formulation• The free energy of cavity formation is defined by the solute geometry (by the

surface area, say).• Van der Walls interactions are longer-ranged and also depend on the

properties of individual atoms (C vs. O, say). • It makes sense to model these two components independently.

PMF of dimerization of uncharged alanine dipeptide.

Non-monotonic behavior can not be reproduced with a surface area model.

Solute-solvent van der Waals energy changes upon binding.

explicit solvent

NP

Large scatter relative to surface area model due to residual interactions of buried atoms.