1. Lecture 11: Potential Energy Functions Dr. Ronald M. Levy ronlevy@temple.edu Originally contributed by Lauren Wickstrom (2011) Statistical Thermodynamics

2. Spring 2013 Microscopic/Macroscopic Connection The connection between microscopic interactions and macroscopic thermodynamic properties is encoded in the potential energy function U(x) A model of the system starts with the specification of its potential energy function

3. Statistical Thermodynamics Spring 2013 Terminology Potential Energy – non-kinetic part of the internal energy of a system. Molecular Mechanics - Classical mechanics description of the potential energy. Function of nuclear positions (electronic motion neglected) – Born-Oppenheimer approximation Force field – Parameterized, analytical function of the potential energy All-Atom Force Field – Function of the positions of all of the atoms (as opposed to a coarse-grained force field).

4. Statistical Thermodynamics Spring 2013 Some Common All-atom force fields for biomolecules OPLS (Optimized Potential for Liquid Simulations) – Jorgensen/Friesner AMBER (Assisted Model Building with Energy Refinement)* – Kollman/Case CHARMM (Chemistry at Harvard Molecular Mechanics)* – Karplus GROMOS (GROningen MOlecular Simulation package)* – Berendsen and van Gunsteren AMOEBA (Atomic Multipole Optimized Energetics for Biomolecular Applications) Ren & Ponder ECEPP (Empirical Conformational Energy Program for Peptides) - Scheraga *These are force fields as well as simulation packages.

5. Statistical Thermodynamics Spring 2013 Born-Oppenheimer formulation Molecular degrees of freedom: nucleii positions ( x ) For each configuration of the nucleii we have a series of electronic states at energies E 0, E 1, E 2,... Assume that electrons readjust instantaneously to new positions of the nucleii. System always remains in ground state at energy E 0 (x) c an in princible be evaluated at each x by QM methods (Gaussian, etc.) In practice need a simple function that mimics as best as possible the true E 0 (x)

6. Statistical Thermodynamics Spring 2013 Typical formulation of a Non-polarizable Non-dissociative Force Field: Energy termsInteractions V bond 1-2 V angle 1-3 V torsions 1-4 V LJ, V coul. Non-bonded Torsion Bond Bond Angle Non-Bonded

7. Statistical Thermodynamics Spring 2013 Bond stretching Morse Potential Hooke's law – harmonic approximation – non dissociative – reasonable for small displacements

8. Statistical Thermodynamics Spring 2013 Bond stretching parameters k obtained from vibrational spectra d 0 obtained from X-ray crystallography Hard degree of freedom Often constrained in MD If δd =.2 Å for carbonyl V bond = 11.4 kcal/mol Type k (kcal/mol/ Å 2 ) d 0 (Å) CA-N3371.44 C=O5701.22 CA N C O

9. Statistical Thermodynamics Spring 2013 Angle bending Hooke’s Law in angle coordinates TypeK θ (kcal/mol/ radian 2 )θ 0 (deg) N-CA-HA35109.5 CA-C-O80120.4 If δ θ = 4° for CA-C-0 V angle = 11.6 kcal/mol CA C N O HA N

10. Statistical Thermodynamics Spring 2013 Torsional terms Black: V n =4; n=2; γ=180 Red: V n =2; n=3;γ=0 Barriers of rotation Ethane – 9 dihedral angles (H-C-C-H) Butane – 27 dihedral angles (1 C-C-C-C, 10 H-C-C-C, 16 H-C-C-H) Fourier expansion

11. Statistical Thermodynamics Spring 2013 120 degrees periodicity Parameters are obtained by fitting to high level QM data Torsional parameters Torsion angle VnVn N-CA-CB-HB1.23 N-CA-CB-HB2.23 N-CA-CB-HB3.23

12. Statistical Thermodynamics Spring 2013 Improper torsions Describe out of plane motion This is often important to maintain planar structure. Examples: Peptide bond Benzene OPLS – adjusts force constants instead of using improper function

13. Statistical Thermodynamics Spring 2013 Non-bonded interactions: Electrostatic Important “directional” interaction energy term Charge distribution calculated from QM calculations Electrostatic interaction between two molecules:

14. Statistical Thermodynamics Spring 2013 Electrostatics  One representation: Multipole Expansion  Charge distribution represented in terms of its moments (charges, dipoles, quadrupoles, octupoles)

15. Statistical Thermodynamics Spring 2013 Different types of multipole interactions Interactions become weaker with higher multipole moments Attraction or repulsion - charge and orientation of the dipole Type of interaction Distance r Dependence Charge- charge 1/r Charge-dipole 1/r 2 Dipole/dipole 1/r 3 Dipole/ induced dipole 1/r 6 + δ+ δ- Charge Dipole UNFAVORABLE - δ+ δ- Charge Dipole FAVORABLE

16. Statistical Thermodynamics Spring 2013 Example: Benzene – Benzene interaction 144 charge-charge interactions First term: quadrupole-quadruple calculation (no monopole, no dipole) Only valid when the distance between two molecules is much larger than the internal dimensions. Multipolar expansions are computationally expensive

17. Statistical Thermodynamics Spring 2013 More common representation: partial charges Charge distribution described by delta functions at “charged sites” (usually atomic sites) Partial charges from : Fit to experimental liquid properties (OPLS) ESP charge fitting to reproduce electrostatic potentials of high level QM X-ray crystallographic electron densit y

18. Statistical Thermodynamics Spring 2013 Polarization Charges can induce other charge asymmetries by polarization Most common strategy is to employ pre-polarized charge distributions Example: Dipole of water in vacuum = 1.8 D Dipole fit to yield liquid properties = 2.2-2.5 D Sometimes overlooked work for creating polarized charge distribution: E = electric field μ = dipole α =point polarizablity

19. Statistical Thermodynamics Spring 2013 Explicit treatment of polarization Induced dipole Interactions between permanent charges, between charges and induced dipoles and between induced dipoles Induced dipoles and electrostatic field solved to self- consistency Polarization energy = Electric field from permanent charges Most commonly used “fixed-charge” force fields to not treat explicit polarization

20. Statistical Thermodynamics Spring 2013 σ Parameters fit to reproduce experimental liquid properties σ ε ε =Depth of potential well σ =collision diameter (potential energy = zero) Repulsive and van der Waals interactions Lennard Jones potential

21. Statistical Thermodynamics Spring 2013 1,4 interactions Electrostatics + LJ interactions 1-2, 1-3 interactions excluded 1-4 interactions scaled Inclusion in bonded terms of force field Force fieldElectrostatics(factor)van der Waals OPLS.5 AMBER.83.5 CHARMM11 1-2 1-3 1-4

22. Statistical Thermodynamics Spring 2013 Computational timing of different force field terms Different parts of the energy scale differently Bonded – linear Non-bonded interactions – N 2 Requires most computational time Cutoffs must be implemented to reduce this cost

23. Statistical Thermodynamics Spring 2013 Final thoughts Force fields rely on experimental data for parameterization Validation is also a key aspect of building these models (experimental observables) Time scale and sampling are important aspects are essential for force field evaluation One must know the limitations of particular models (force field biases)

24. Statistical Thermodynamics Spring 2013 Solvation: Solvent Potential of Mean Force W : Implicit solvent Approximate model Averages solvation effect (potential mean force) High dielectric = solvent; Low dielectric = solute Less friction than water (faster transitions) Fewer degrees of freedom ε=80 ε=1