Molecular Mechanics

• Studies involving covalent interactions (enzyme reaction): quantum mechanics; extremely slow

• Studies involving noncovalent interactions (conformational references, molecular recognition): classical mechanics; acceptable for a few structures

• Studies involving sequences only: statistical formalisms; extremely fast

Molecular Mechanics

• Study how protein/protein, protein/ligand, protein/NA interactions. Why they are specific? how to mimic them?

• Use them in structure-based drug design, docking.

• Study how proteins/NAs change conformations. How a specific function/mechanism is realized?

Theoretical Ground:Classical Mechanics

Building on the work of Galileo and others, Newton unveiled his laws of motion in 1686. According to Newton:

• I. A body remains at rest or in uniform motion (constant velocity - both speed and direction) unless acted on by a net external force.

• II. In response to a net external force, F, a body of mass m accelerates with acceleration a = F/m.

• III. If body i pushes on body j with a force Fij, then body j pushes on body i with a force Fji.

Theoretical Ground:Classical Mechanics

• How to obtain forces? Easy if an energy model is given.

Where to use Molecular Mechanics Energy Model?

• Molecules containing thousands of atoms

• Organics, oligonucleotides, and peptides

• Vacuum, implicit, or explicit solvent environments

• Ground state only

• Thermodynamic and kinetic via simulations.

Building Principles of Molecular Mechanics (Energy Model)

• Nuclei and electrons are lumped into atom-like particles

• Atom-like particles are spherical (radii obtained from measurements or theory) and have a net charge (obtained from theory)

• Interactions are based on springs and classical potentials

• Interactions must be preassigned to specific sets of atoms

• Interactions determine the spatial distribution of atom-like particles and their energies

Simplistic Molecular Mechanics Force Field

Van der Waals Charge - Charge

Bond

Angle

ImproperDihedral

Dihedral

Bond Stretching Energy

Bond Stretching Energy

Angle Bending Energy

Angle Bending Energy

Significance of Energy Parameters

Torsion Energy

The torsion energy is modeled by a simple periodic function:

Significance of Energy Parameters

The Roles of Torsion Energy

• The torsion energy in molecular mechanics is primarily used to correct the remaining energy terms rather than to represent a physical process.

• The torsional energy represents the amount of energy that must be added to or subtracted from the Stretching + Bending + Non-Bonded interaction terms to make the total energy agree with experiment or rigorous quantum mechanical calculation for a model dihedral angle (ethane, for example might be used a model for any H-C-C-H bond).

Cross Terms

Possible cross terms:

• stretch-stretch, stretch-bend, strech-torsion;

• bend-bend, bend-torsion;

• torsion-torsion. (Fig. 4.13, Leach)

Needed in studies of high-frequency motions, i.e. vibrational spectra.

Non-Bonded Energy

Van der Waals Energy

Significance of Energy Parameters

Electrostatic Energy

• The electrostatic contribution is modeled using a Coulombic potential.

• The electrostatic energy is a function of:o (a) charges on the non-bonded atoms;o (b) inter-atomic distance;o (c) molecular dielectric expression that

accounts for the attenuation of electrostatic interaction by the molecule itself.

Electrostatic Energy: Dielectrics

• The molecular dielectric is set to a constant value between 1.0 and 4.0. However, it has to be consistent with how a force field is designed. (not a free parameter)

• A linearly varying distance-dependent dielectric (i.e. 1/r) is sometimes used to account for the increase in the solvent (aka, water) dielectrics as the separation distance between interacting atoms increases. (This is being abandoned)

• When it is needed, the Poisson’s equation, or its approximation, has to be used. (This is gaining popularity)

Other Nonbonded Interactions: Hydrogen Bonding

• Hydrogen bonding term is usually wrapped into the electrostatic term in force fields widely used today. However it does not imply that hydrogen bonding is purely electrostatic in nature.

• Hydrogen bonding, if explicitly represented, uses a 10-12 Lennard-Jones potentials. This replaces the 6-12 Lennard-Jones term for atoms involved in hydrogen-bonding.

Other Nonbonded Interactions: Polarization

• Polarization is important when large environmental changes occur, i.e. from protein interior to water, or from membrane to water.

• Usually modeled as inducible dipole: μ = E• Note it is not free to induce a dipole: the work done is 1/2

E2.• Finally, electrostatic energy includes charge-charge,

charge-dipole, and dipole-dipole; or electrostatic field is from charge and dipole.

• No stable force fields with polarization available right now!

Scaling of Nonbonded Terms

• Scaling of electrostatic energy: charge-charge 1/r; charge-dipole 1/r2, dipole-dipole 1/r3.

• Scaling of van der Waals energy: 1/r6.• The example of two point charges on z-

axis.