Molecular dynamics (MD) simulations  A deterministic method based on the solution of Newton’s equation of motion F i = m i a i for the ith particle; the acceleration at each step is calculated from the negative gradient of the overall potential, using F i = - grad V i - = -  V i

 V i = Gradient of potential?  Derivative of V with respect to the position vector r i = (x i, y i, z i ) T at each step  a xi ~ -  V/  x i a yi  ~ -  V/  y i  a zi  ~  V/  z i V i =  k (energies of interactions between i and all other residues k located within a cutoff distance of R c from i)

Non-Bonded Interaction Potentials  Electrostatic interactions of the form E ik (es) = q i q k /r ik  Van de Waals interactions E ij (vdW) = - a ik /r ik 6 + b ik /r ik 12 Bonded Interaction Potentials  Bond stretching E i (bs) = (k bs /2) (l i – l i 0 ) 2  Bond angle distortion E i (bad) = (k  /2) (  i –  i 0 ) 2  Bond torsional rotation E i (tor) = (k  /2) f(cos  i ) Interaction potentials include;

Example 1: gradient of vdW interaction with k, with respect to r i  E ik (vdW) = - a ik /r ik 6 + b ik /r ik 12  r ik = r k – r i  x ik = x k – x i  y ik = y k – y i  z ik = z k – z i  r ik = [ (x k – x i ) 2 + (y k – y i ) 2 + (z k – z i ) 2 ] 1/2   V/  x i =  [- a ik /r ik 6 + b ik /r ik 12 ] /  x i where r ik 6 = [ (x k – x i ) 2 + (y k – y i ) 2 + (z k – z i ) 2 ] 3

Example 2: gradient of bond stretching potential with respect to r i  E i (bs) = (k bs /2) (l i – l i 0 ) 2  l i = r i+1 – r i  l ix = x i+1 – x i  l iy = y i+1 – y i  l iz = z i+1 – z i  l i = [ (x i+1 – x i ) 2 + (y i+1 – y i ) 2 + (z i+1 – z i ) 2 ] 1/2  E i (bs) /  x i = - m i a ix (bs) (induced by deforming bond l i ) = (k bs /2)  [ (x i+1 – x i ) 2 + (y i+1 – y i ) 2 + (z i+1 – z i ) 2 ] 1/2 – l i 0  2 /  x i = k bs (l i – l i 0 )  [ (x i+1 – x i ) 2 + (y i+1 – y i ) 2 + (z i+1 – z i ) 2 ] 1/2 – l i 0  /  x i = k bs (l i – l i 0 ) (1/2) (l i -1 )  (x i+1 – x i ) 2 /  x i = - k bs (1 – l i 0 / l i ) (x i+1 – x i )

The Verlet algorithm Perhaps the most widely used method of integrating the equations of motion is that initially adopted by Verlet [1967].The method is based on positions r(t), accelerations a (t), and the positions r(t -  t) from the previous step. The equation for advancing the positions reads as r(t+  t) = 2r(t)-r(t-  t)+  t 2 a(t) There are several points to note about this equation. It will be seen that the velocities do not appear at all. They have been eliminated by addition of the equations obtained by Taylor expansion about r(t): r(t+  t) = r(t) +  t v(t) + (1/2)  t 2 a(t)+... r(t-  t) = r(t) -  t v(t) + (1/2)  t 2 a(t)- The velocities are not needed to compute the trajectories, but they are useful for estimating the kinetic energy (and hence the total energy). They may be obtained from the formula v(t)= [r(t+  t)-r(t-  t)]/2  t

Periodic boundary conditions

Initial velocities ( v i ) using the Boltzmann distribution at the given temperature v i = (m i /2  kT) 1/2 exp (- m i v i 2 /2kT)

How to generate MD trajectories?  Known initial conformation, i.e. r i (0) for all atom i  Assign v i (0), based on Boltzmann distribution at given T  Calculate r i (  t) = r i (0) +  t v i (0)  Using new r i (  t) evaluate the total potential V i on atom I  Calculate negative gradient of V i to find a i (  t) = -  V i /m i  Start Verlet algorithm using r i (0), r i (  t) and a i (  t)  Repeat for all atoms (including solvent, if any)  Repeat the last three steps for ~ 10 6 successive times (MD steps)

Limitations of MD simulations  Full atomic representation  noise  Empirical force fields  limited by the accuracy of the potentials  Time steps constrained by the fastest motion (bond stretching of the order of femptoseconds  Inefficient sampling of the complete space of conformations  Limited to small proteins (100s of residues) and short times (subnanoseconds)