1. Lecture 7: Molecular Mechanics: Empirical Force Field Model Nanjie Deng Structural Bioinformatics II

2. Terminology Potential Energy – non-kinetic part of the internal energy of a system. Potential Energy Surface – a mathematical function that gives the potential energy of a molecule as a function of its structure. Molecular Mechanics - Classical mechanics description of the potential energy. 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).

3. Potential Energy Surface Determines Structure and Properties of a Molecular System A model of any molecular system starts with the specification of its potential energy as a function of its atomic coordinates. The potential energy of the structure of a molecule determines its stability: High potential energy is associated with a less stable structure, and conversely, a structure with low potential energy is more stable. Low energyHigh energy Alanine dipeptide dihedral angles    

4. Molecular Mechanics - Classical mechanics description of the potential energy. The potential energy is a function of nuclear positions (electronic motion neglected) – Born-Oppenheimer approximation Force field – Parameterized, analytical function of the potential energy: Potential energy of a molecular system can be computed using a molecular mechanics force field Because of their smaller mass, electrons readjust instantaneously to new positions of the nucleii.

5. 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

6. 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.

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

8. Bond stretching parameters k obtained by fitting to 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. 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. Torsional terms Black: V n =4; n=2; γ=180 Red: V n =2; n=3;γ=0 Barriers of rotation vary 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. 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. 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. Non-bonded interactions: Electrostatic Important “directional” interaction energy term Charge distribution calculated from QM calculations Electrostatic interaction between two molecules: i.e. Coulomb’s law on two charge distributions.

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

15. 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. 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. 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. Hydrogen bonds are a special type of dipole-dipole interactions H-bonds are extremely important for biomolecular structures Some force fields have special functional forms for treating H- bonding The majority of the force fields rely on combinations of eletrostatic and van der Waals terms to reproduce H-bonds. Hydrogen bond strength depends on the relative orientation of the interacting partners.

19. 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

20. 1,4 interactions are scaled in some force fields 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

21. Computational costs 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

22. Using force field calculation to understand molecular binding Guanine-CytosineUracil-2,6-Diaminepyridine The Gua-Cyt binding is ~1,000 times stronger even though the same number of hydrogen bonds are formed. Calculation using OPLS force field shows that E int (Gua-Cyt) = -22.1 kcal/mol, while E int (Ura-DAP) = -11.4 kcal/mol. Secondary interactions are key! 2 unfavorable, 2 favorable 4 unfavorable

23. Energy Minimization: Seeking the Local Minima on the Potential Energy Surface Global minimumHigh energy barrier   Energy minima correspond to the structures most likely to be observed experimentally. Energy minimized structures can be used for computing vibrational modes and frequencies. They are also used to remove strains in the system prior to running Molecular Dynamics simulations. Local minimum

24. Energy Minimization: Commonly Used Minimizers Steepest Descent. (Robust, but converges slowly) Conjugated Gradient. (Converges faster than SD) Methods that use first order derivatives of the energy function, memory requirement Newton Raphson Adopted-basis Newton Raphson Methods that use second order derivatives (Hessian Matrix, memory Methods that use second order derivatives are fast but may have problems with bad initial structures. Commonly used practice is to run a quick Steepest Descent to remove bad atomic contacts, and then run a more thorough minimization using Newton-Raphson.

25. Energy Minimization : the Steepest Descent Method 1.At each cycle of the Steepest Descent, the system moves in the direction of the force, i.e. the opposite of gradient of the energy function, downhill. 2.It uses line search to locate the minimum along the direction of the force. 3.Once at the new location (the minimum found in the line search), the calculation restart from step 1, until the change between current structure and the new structure is smaller than the convergence criteria.

26. Final thoughts Force fields rely on high level quantum mechanical and/or experimental data for parameterization Validation is also a key aspect of building these models (experimental observables) One must know the limitations of particular models (force field biases) The energy minimization methods only allow you to find the nearest local minima on the PES. To search for the global minimum, other methods such as Molecular Dynamics simulation (MD), Monte Carlo (MC) methods are needed.