Molecular Dynamics, Monte Carlo and Docking Lecture 21 Introduction to Bioinformatics MNW2

Allowed phi-psi angles Red areas are preferred, yellow areas are allowed, and white is avoided

2.3a Hamiltonian equations of motion Hamiltonian equations (one degree of freedom): H – Hamiltonian function, Hamiltonian, q, p –Canonical variables: generalized coordinate (q) and momentum [impulses] (p).

q = coordinates p = momentum

v(t) = (r(t +  t) - r(t -  t))/2  t

Molecular Dynamics Knowledge of the atomic forces and masses can be used to solve the position of each atom along a series of extremely small time steps (on the order of femtoseconds = seconds). The resulting series of snapshots of structural changes over time is called a trajectory. The use of this method to compute trajectories can be more easily seen when Newton's equation is expressed in the following form: The "leapfrog" method is a common numerical approach to calculating trajectories based on Newton's equation. The steps can be summarized as follows:

Force field The potential energy of a system can be expressed as a sum of valence (or bond), crossterm, and nonbond interactions: The energy of valence interactions comprises bond stretching (E bond ), valence angle bending (E angle ), dihedral angle torsion (E torsion ), and inversion (also called out-of- plane interactions) (E inversion or E oop ) terms, which are part of nearly all forcefields for covalent systems. A Urey-Bradley term (E UB ) may be used to account for interactions between atom pairs involved in 1-3 configurations (i.e., atoms bound to a common atom): E valence = E bond + E angle + E torsion + E oop + E UB Modern (second-generation) forcefields include cross terms to account for such factors as bond or angle distortions caused by nearby atoms. Crossterms can include the following terms: stretch-stretch, stretch-bend-stretch, bend-bend, torsion-stretch, torsion-bend-bend, bend-torsion-bend, stretch-torsion-stretch. The energy of interactions between nonbonded atoms is accounted for by van der Waals (E vdW ), electrostatic (E Coulomb ), and (in some older forcefields) hydrogen bond (E hbond ) terms: E nonbond = E vdW + E Coulomb + E hbond

Force field

f = a/r 6 - b/r 12 Van der Waals forces distance energy

F = kq 1 q 1 /r 2

Figure: Snapshots of ubiquitin pulling with constant velocity at three different time steps.

antibody HyHEL-63 (cyan) complexed with Hen Egg White Lysozyme The X-ray structure of the antibody HyHEL-63 (cyan) uncomplexed and complexed with Hen Egg White Lysozyme (yellow) has shown that there are small but significant, local conformational changes in the antibody paratope on binding. The structure also reveals that most of the charged epitope residues face the antibody. Details are in Li YL, Li HM, Smith-Gill SJ and Mariuzza RA (2000) The conformations of the X-ray structure Three-dimensional structures of the free and antigen-bound Fab from monoclonal antilysozyme antibody HyHEL-63. Biochemistry 39: Salt links and electrostatic interactions provide much of the free energy of binding. Most of the charged residues face in interface in the X-ray structure. The importance of the salt link between Lys97 of HEL and Asp27 of the antibody heavy chain is revealed by molecular dynamics simulations. After 1NSec of MD simulation at 100°C the overall conformation of the complex has changed, but the salt link persists. Details are described in Sinha N and Smith-Gill SJ (2002) Electrostatics in protein binding and function. Current Protein & Peptide Science 3: