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1 Subject of the Lecture: Conceptual basis for the development of force fields. Implementation/validation Water - a worked example Extensions - combining molecular mechanics and quantum mechanics (QM/MM)
2 Currently, complete quantum mechanical solution to most problems of chemical or biological interest is not possible
3 Difficulties are compounded by the fact that statistical and dynamical aspects of the problem must be included. In fact, they are the essence of biology.
4 Schrödinger s equation: h i Ψ(x,t) t = - h 2 Ψ(x,t) 2m x 2 + V(x,t)Ψ(x,t) F(x,t) = m d2 x dt 2 Newton s equations of motion ; F(x,t) = - V(x,t) x Can we consider molecular systems as classical?
5 Conceptual Basis Quantum mechanics provides clues to the form of the potential function. We will analyze the problem stepwise. Consider two interacting molecules A and B:
6 H A H B H = H A + H B + H int
7 Conceptual Basis - Electrostatics (Rayleigh-Schrodinger perturbation theory) R θ r ϕ(r) = q 4πεr r ( ) n R P n (cosθ) Multipole expansion - charge density is represented as a sum of point multipoles (charge, dipole, quadrupole)
8 Conceptual Basis (2) - Electrostatics ~R -1 _ charge-charge ~R -2 _ charge-dipole ~R -3 _ dipole-dipole, - charge-quadrupole q A q B q A =q B =0
9 Conceptual Basis - polarization A B Unperturbed charges A B Perturbed (polarized) charges In the absence of charge transfer the first non-vanishing term is dipole
10 Conceptual Basis - dispersion energy (electron correlation) - electron E 1 - nucleus E 2 < E 1 Dispersion energy is always negative E disp ~ R -6 (+ R -8 + )
11 Conceptual Basis - short-ranged potential Beyond the R-S perturbation formalism ( symmetry-adapted theory) Arises from Pauli s exclusion principle In the first-order approximation proportional to the electron overlap decays exponentially with the distance Always positive
12 van der Waals potential (electron correlation) Buckingham potential: E vdw = -Ar -6 + Bexp(-αr) Lennard-Jones potential: E vdw = -Ar -6 + Br -12 σ 6 E vdw = 4ε[- + ] σ 12 r 6 r 12 r min = 2 1/6 σ; E vdw (min) = ε
13 Van der Waals potential (cont.)
14 Van der Waals potential (cont.)
15 Bond-stretching
16 Angle-bending
17 Stretching or bending
18 Torsion
19 Torsions (cont.)
20 SUMMARY E = (E el + E vdw) + (E stretch + E bend + E tors + E it ) Electrostatic energy is represented using multipole expansion. In general, polarization energy is markedly smaller than unperturbed energy. van der Waals energy represents electron correlation. The dispersion term is always negative whereas short-range energy is always repulsive. Taken together, E vdw has a minimum. The remaining terms describe covalent bonding.
21 Implementation - a Lego force field Different atom types have different parameters. Same atom types have the same parameters.
22 'C a' : carbon in O-C=O acid 'C k' : carbon in C=O not bonded to N, and not an acid group 'C n' : carbon in N-C=O 'C*' : aromatic C in 5-membered ring next to two carbons 'CAxna' : special parameters for nucleotide bases. aromatic C in DAP C2,C3,C4 pyridine; Cytosine C4,C5,C6 'CBxna' : special parameters for nucleotide bases. Adenine C4,C5,C6; Guanine C2,C4,C5 'CB' : aromatic C at juntion of 5-membered and 6-membered rings 'CKxna' : special parameters for nucleotide bases. Adenine C8 Guanine 'CM' : alkene sp2 carbon (non-aromatic) 'CMxna' : special parameters for nucleotide bases. Carbon in Uracil C5,C6 'CO' : sp3 anomeric carbon (bonded to ether O and alcohol O); this type shows up in carbohydrates 'CQ' : sp2c in 6-membered ring between deprotonated N's 'CR' : aromatic C in 5-membered ring next to two nitrogens 'CT' : aliphatic sp3 hybrid carbon 'CV' : aromatic C in 5-membered ring next to C and deprotonated N 'CW' : sp2 aromatic C in 5-membered ring next to C and NH
23 * 'O' : oxygen in C=O, not an acidic site * 'O2' : oxygen double bonded to carbon in COO- or COOH * 'OHa' : oxygen bonded to H in RCOOH * 'OHm' : oxygen bonded to H in a mono-alcohol * 'OHp' : oxygen bonded to H in polyols or phenol * 'OS' : sp3 O in an ether or acetal
24 We need to assign and validate parameters Charges on atoms (for each atom type) Other electrostatic parameters (if needed) Van der Waals parameters (for each pair of atom types) Force constants and equilibrium distances (angles) for stretching (bending) Parameters for torsions
25 We need to assign and validate parameters Theory or experiment? Isolated molecules or dense phases? Large or small molecules? Structure? Spectroscopy? Thermodynamics? Pure substances or mixtures? Every force field should be well balanced
26 Typical sources/validation of other potentials Charges - electrostatic potential Van der Waals parameters - crystallography Stretching & bending parameters - spectroscopy Torsional parameters - quantum mechanics, NMR spectroscopy
27 Different Force Fields: AMBER ( Assisted Model Building with Energy Refinement). CHARMM (Chemistry at HARvard using Molecular Mechanics). GROMOS (GROenigen MOlecular Simulation) OPLS (Optimized Parameters for Large-scale Simulations) MMFF (the Merck Molecular Force Field) Other (MM3, Tripos, )
28 Who is the fairest of them all?
29 Force Fields - a worked example: water models Water is the biological solvent Water has many unusual properties Bulk, bound, crystal and droplet water is important
30 Different representations of water
31 Models of water: Model Type σ Å 6 ε kj mol -1 6 l 1 Å l 2 Å q 1 q 2 θ φ SSD [511] SPC [94] a SPC/E [3] a SPC/HW (D 2 O) [220] a TIP3P [180] a PPC 1, 2 [3] b TIP4P [180] 10 c TIP4P-FQ [197] c SWFLEX-AI 2 [201] c four terms used , COS/G3 [704] 11 c TIP5P [180] d POL5/TZ 2 [256] d varies Six-site [491] c/d OO HH OO HH L M L M
32 Parameterizing a water model Simple averages: density (ρ) enthalpy of vaporization ( Hvap) structure (O-O RDF) dipole moment Higher-order averages: heat capacity (C p ) dielectric constant (ε) isothermal compressibility (κ) coefficient of thermal expansion (α) Kinetic observables: diffusion constant (D) rotational autocorrelation (τ rot ) hydrogen bond lifetimes (τ HB ) A particular challenge is getting the temperature dependence of these right, e.g., the density maximum of water around 4 C.
33 Some of the calculated physical properties of the water models. Model Dipole moment Dielectric constant Self diffusion, 10-5 cm 2 /s Average configurational energy, kj mol - 1 Density maximum, C Expansion coefficient, 10-4 C -1 SSD 2.35 [511] 72 [511] 2.13 [511] [511] -13 [511] - SPC 2.27 [181] 65 [185] 3.85 [182] [185] [704] ** SPC/E 2.35 [3] 71 [3] 2.49 [182] [3] -38 [183] - PPC 2.52 [3] 77 [3] 2.6 [3] [3] +4 [184] - TIP3P 2.35 [180] 82 [3] 5.19 [182] [180] -13 [180] 9.2 [180] TIP4P 2.18 [3,180] 53 a [3] 3.29 [182] [180] -25 [180] 4.4 [180] TIP4P-FQ 2.64 [197] 79 [197] 1.93 [197] [201] +7 [197] - SWFLEX-AI 2.69 [201] 116 [201] 3.66 [201] [201] - - COS/G3 ** 2.57 [704] 88 [704] 2.6 [704] [704] [704] TIP5P 2.29 [180] 81.5 [180] 2.62 [182] [180] +4 [180] 6.3 [180] POL5/TZ [256] 98 [256] 1.81 [256] [256] +25 [256] - Six-site* 1.89 [491] 33 [491] [491] 2.4 [491] Expt. 2.65, [180]
34 Ewald, 298K, 1 atm O-O RDF comparison g(r) SPC SPC/E TIP3P TIP4P TIP5P Angstrom The radial distribution function represents the average density as a function of distance from a particular particle. It can be measured experimentally by a variety of means. Many thermodynamic properties can be derived from the RDF.
35 What s missing from these water models? Polarizability Bond flexibility Dissociation Multibody effects Purely quantum effects Dipole Moments Gas phase TIP4P SPC Liquid water 1.85 D 2.18 D 2.27 D ~2.5 D An ongoing challenge: Water in first solvation shell of protein (or trapped in an active site or the interior) is rather different than bulk water.
36
37 An Example - a Diels-Alder Reaction
38 The Diels-Alder cyclo-addition reaction catalyzed by 1E9, a catalytic antibody that was raised against a transition-state analogue compound
39
40 Application of QM/MM
41 Treatment of QM/MM boundaries
42
43
44
45 A simplification
46 Other approaches: General Valence Bonds Rappé-Goddard Car-Parrinello Path integral methods
47
48 For more info Andrew Pohorille at
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