1 Molecular Simulations Macroscopic EOS (vdW, PR) Little molecular detail Empirical parameters (  ) Seeking understanding of complex systems Surfactants Cell membranes RNA, DNA

Maxwell-Boltzmann Velocity Distribution 2

Speed Distribution 3 Read more about manipulation of integral: Fraction of molecules in velocity increment

Speed Distribution 4

Cumulative Speed Distribution (CH3 at 300K) 5

6 Calculating Energy Total Energy U = E KE + E PE Kinetic Energy N m - # sites/molec ( e.g. 3) m i – mass site i; N - # molec Independent of state of aggregation!

7 E KE for center of mass Some E KE is directionless (rotational, vibrational) E PE – configurational energy 4 pairs u 0 r/r/ Pair Potential

8 Ergodic Theorem Sampling small system over time Molecular Dynamics (MD) Sampling many small systems at one time Monte Carlo (MC), Statistical Mechanics Both methods provide the same expectation values, same results.

9 System size 1 cm 3 water E22 molecules Usually use ~1000 atoms of diameter  Methane at liquid density (~5 neighbors < 2.5  ) Need 5000 pair steps per time step Time step usually 1-2 fs (10E-15s) Need O(1E9) eval per ns (1E-9)!  s (1E-6) simulations challenging!

10 Using finite size systems Periodic Boundary Conditions e.g. if x i,new < 0, x i,new = x i,new + L Minimum Image Convention Use image if closer than actual site Cutoff Distance Use max cutoff = L/2 to avoid actual and image PE.

’ L L L 2L LL LL trial accepted 5’ L See

12 Waldo’s Walk Home Introduction to periodic boundaries and biased sampling (L/2, -L/2) (L/2, L/2) (-L/2, -L/2) (-L/2, L/2) If 10% trained, don’t use this 0.1(360)=36 o (Biased sampliing forces most steps toward the correct direction, but does not prevent errors.) home waldoperiodic.m

13 Molecular Monte Carlo Steps If  E step < 0, accept If  E step > 0 If (0 ≤ rand ≤ 1) < exp(-  E step /kT), accept exp(-  Estep/kT)  Estep accept reject Lots acceptedFew accepted

14 Molecular Dynamics Continuous MD Challenge – computer takes finite steps Position, velocity, acceleration out of synch Discontinuous MD (DMD) Hard spheres or step potential Velocities constant until PE changes u r/r/ Step potential

15 Mathematics of relative positions, velocities 2 1 r2r2 r1r1 r 12 = r 1 – r 2 v 12  r 12 v b = magnitude, r 12 v 12 cos  Particles are approaching (dr 12 /dt < 0) when b ≡ r ijv ij < 0

16 Dot product b = r 12v 12 = [r 12x, r 12y, r 12z ] [v 12x, v 12y, v 12z ] = r 12x v 12 x+ r 12y v 12y + r 12z v 12z = |r 12 ||v 12 |cos  Note: in Matlab b = dot(r,v) where r,v are vectors. Also, r 12r 12 = r 12 2, v 1v 1 = v 1 2 Kinetic energy is m i (v iv i )/2 Relative position at any time until collision: r ij = r ij o + v ij o (t - t o ) Collision occurs when r 12r 12 =  2 s = -1 if discriminant < 0, miss rjorjo rjtrjt rjt+rjt+ riorio

17 Collision Mechanics r 12 c  y v2’v2’ x v 12 v1’v1’   v1’v1’ v2’v2’ Momentum conserved Kinetic energy conserved Must evaluate r ij c and b c = r ij cv ij at collision Collision force acts in direction vector between point of contact and center of site, r 12 c /  for 1 and - r 12 c /  for 2.

18 Event Tracking Start particles with desired E KE Calculate next collision. Move all particles to time of next collision. Calculate new velocities for particles that collide. Loop.

19 DMD and square well (a)(a) (b)(b) (c)(c)(d)(d) Will accelerate when fall into well. Will bounce on cores. Approaching, but cores will miss. Will decelerate if escape well, or will bounce inside well. Departing. Will decelerate if escape well, or will bounce inside well. 10 different types of events (this is a subset)

20 Time and velocity changes d ij = distance of event s = +1 or -1 depending on event where b c = r ij cv ij b = r ijv ij s = +1 or -1 depending on event

21 Bonded Sites bond pseudobond (1) Sites can not escape from wells. (2) Bond distances < site diameters.

Flow sheet 22

23 Applications of DMD Protein Fibril formation (Hall) Square-well attractive forces promote helix fibril formation. Vapor Pressure prediction (Elliott) DMD provides the reference state from which a perturbation is applied.