Statistical Thermodynamics Lecture 12: Solvation Models: Molecular Mechanics Modeling of Hydration Effects Dr. Ronald M. Levy ronlevy@temple.edu
Bare Molecular Mechanics Atomistic Force Fields: torsion stretching E stretch = bonds K r r r 0 2 bending E bend = angles K q q q 0 2 non-bonded E torsion = torsions V n 2 [ 1±cos n n ] E non bonded = ij [ q i q j r ij 4 ij 12 ij r ij ] 6 ij 12 6 r ij
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. Explicit solvation models. Implicit solvation models. Two main approaches
Explicit Solvation Each solvent molecule is represented with a set of atomic interaction centers (just as for the solute). Most accurate/detailed. Computationally expensive. Requires averaging over solvent coordinates. Difficult to obtain relative free energies of solute conformations.
Implicit Solvation The solvent is represented by a continuum described by macroscopic parameters such as the dielectric constant, density, surface tension, etc. 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.
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: Δx min =Δx L nint Δx L 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 = 1 2 i q i φ r i φ r i = j,n q j r ij nl 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 methods The 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: c, i = q i / 3 /2 e r 2 ij 1 r ij erfc r r ρ k is the Fourier transform of the charge distribution: k ρ k 2 exp k 2 /4α k 2 k = i q i e i k r i measures the softness of the canceling distribution - the larger the more convergent is the direct sum (but less convergent is the reciprocal sum).
Particle mesh Ewald methods Straightforward implementations of the Ewald sum scale at best as N 3/2 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 P 3 M 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 longrange 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 Q xy 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). where 1 O x =Q xy dxdyo x exp { β [U x U x, y U y ] } O( x) = 1 Q x dxo( x)exp { β [U ( x)+w ( x )] } 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: x 2 x 1 ΔG 21 = kt ln P( x 2 ) P( x 1 ) = ΔU 21 + ΔW 21 Solvation (fixed solute conformation): y x Q sep =exp [ βu x ]Q y y x Q sol =exp [ βu ( x)] dy exp { β [U ( x, y)+u ( y )] } =exp { β [U ( x)+w ( x)] }Q y ΔG solv ( x )= kt ln Q sol Q sep = kt ln exp [ βw ( x)]=w ( x)
Polar/Non-polar decomposition of the solvent PMF ( v) U ch Δ G solv ( x)=w ( x) G np G cav ( w ) +U ch Δ G vdw G solv = G elec G np G np = G cav G vdw ( Δ G elec =U w) ( v ) ch U ch
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 PE for a non-homogeneous dielectric: [ ε x ϕ x ]= ρ x Electrostatic free energy of solute: G elec = 1 2 q i i ϕ( x i ) Electrostatic solvation free energy: ε=1 ε= ε slv + - W elec =G elec ( ε in =1, ε out =ε slv ) G elec (ε in =1, ε out =1) W elec = 1 2 i q i ϕ ( x i ) 1 2 i q i ϕ q ( x i )= 1 2 i q i ϕ rf ( x i ) direct Coulomb The reaction field potential φ rf is due to the polarization of the medium.
The PB equation takes into account the effect of free ions in solution [ x x ]= x = i q i x x i ions c j z j e z j x Debye-Hückel ion distribution - ε=1 + - When βz j ϕ x << 1 (otherwise ion adsorption): ε= ε slv + linearized form of the PB equation. [ ε x ϕ x ] k 2 x ε x ϕ x = q i δ x x i atoms where k 2 x = 2I kt ε x with I = 1 2 j c j z j 2 (ionic strength).
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 The basic idea is that a dielectric model of hydration should describe these two basic effects: 1.Dielectric polarization around polar groups Favorable interaction between exposed charged atoms and the polarized dielectric. Born model of ion hydration: ΔG elec = 1 2 1 1 ε w z2 R - - - + 1.Dielectric screening of electrostatic interactions The dielectric weakens the interactions between charges Distance-dependent dielectric models u ij = q i q j ε r ij r ij ε r + - + + - - - + r
Generalized Born Model W elec = 1 2 ( 1 1 ε w ) ij q i q j f ij (r ij ) f ij =[r 2 ij B i B j exp r 2 ij /4 B i B j ] 1/2 B i is the Born radius of atom i defined by: i W single = 1 ( 1 1 ) q 2 i 1 2 ε w B i 8π ( 1 1 ε w ) V q i 2 r r i 4 d3 r 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 W elec = 1 2 ( 1 1 ε w ) ij q i q j f ij (r ij ) The GB model works because it describes both dielectric polarization and dielectric screening effects. Polarization i=j ( self energy): i W single = 1 ( 1 1 ) q 2 i 2 ε w B i Dielectric screening i j (pair energies): u ij = q i q j ε r ij r ij ε ij r =S S x = [ 1 1 1 ε 1 Favors the solvent exposure (small B i ) of polar groups (large q). r B i B j 1 x 2 exp x 2 /4 ] 1 S x
Born Radii: Pairwise Descreening Scheme Born radius calculated as a sum of pairwise atomic contributions: 1 = 1 B i 4 dielectric 1 r i r d 3 r= 1 1 4 R i 4 solute-i 1 r i r 4 d 3 r Solute i i i Solvent Dielectric Summing over all j s: 1 B i 1 R i 1 4 j 1 = 1 1 = 1 1 B i R i B i R i 4 atom j Q ij j 1 r i r 4 d 3 r Q ij : pairwise descreening function. Forms the basis for most analytical GB models used in molecular simulations.
Pairwise Descreening Gotchas Q ij is not simply the integral of 1/ r i - r 4 over atom j because: 1. Overlaps with atom j and other atoms would be over-counted j i Reduce descreening contribution from atom j 2. Atoms i and j may overlap j i Integrate only over portion of j not 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) ΔG np =γa 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): G np = G cav G vdw
Example of an analytical NP model (the NP in AGBNP) DG np = G cav G vdw Δ G np = i [ γ i A i +α i ω( B i )] A i ω( B i ) i,α i : Surface area of atom i : Geometrical predictor based on Born radius : Surface tension and van der Waals adjustable parameters 6-4ϵ ω i ρ w i σ i slv. r r i =-16 π ρ 6 wϵ i σ i 6 3 3C i C i= 3 4 slv. 1 r r i 6 1 /3 B i
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. Solute-solvent van der Waals energy changes upon binding. NP explicit solvent Non-monotonic behavior can not be reproduced with a surface area model. Large scatter relative to surface area model due to residual interactions of buried atoms.