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A Pinch of Verlet-Velocity
Algorithm, A Dash of
Langevin Dynamics:
A Recipe for Classical
Molecular Dynamics
Kim Gunnerson
Undergraduate Mathematics Seminar
Wed Oct. 25, 2006
Today’s Outline
Who am I and How did I get here?
Systems and Questions
Examples of Mathematics Used
Summary
Who am I? How did I get here?
1987-BS Chemistry from PLU
1985-1994:Laboratory Chemist
1994-2002:HS Chemistry and MS
science teacher including stint as
science department chair
2002-today:Physical Chemistry grad
student and GK-12 Project Manager
Systems and Questions
Chlorine dioxide relaxation*
P-selectin/PSGL-1 conformation*
Carbon nanotubes
Crystal kinetics
*Dissertation subject
Systems and Questions
Chlorofluorocarbon’s (CFC’s) Role in Ozone Depletion
08/05/03
Austral Winter
11/04/03
Austral Spring
Systems and Questions
hν
1.4698 Å
117.41º
1.6270Å
106.20º
What are the relaxation dynamics of the
photoexcitation of chlorine dioxide?
Systems and Questions
Leukocyte Extravasation
Systems and Questions
Are there conformational changes in the protein and/or
ligand in response to an external force?
????????
Mathematical Ingredients
Classical Equations of Motion:
Newton leads the way
d
d 2r
F  ma  m
m 2
dt
dt

m r  
U total r1 ,r2 ,..., rN ,   1,2... N
 r

F  U

Mathematical Ingredients
Potential:
U total  U bond  U angle  U dihedral  U vdw  U coulomb
Intermolecular
Intramolecular
Lennard-Jones 6-12 Potential
U coulomb
qi q j
 
i j i 4 0 rij
U vdw
  12   6 
  4 ij  ij    ij  
i j i
 rij 
 rij  
Electrostatic Interaction
Force Fields: Contain values determined via a combination of
empirical techniques and quantum mechanical calculations.
Mathematical Ingredients
Velocity Verlet Algorithm
1 f (t )
x(t   )  x(t )  v(t )   2
2
m
f (t )  f (t   )
v (t   )  v (t ) 

2m
Mathematical Ingredients
Problem: Need to represent the
correct ensemble distribution for the
specified temperature and pressure.
Solution: Slightly modify the
Newtonian equations in order to take
into account the temperature (and
pressure) dependency of the molecular
system.
Mathematical Ingredients
Langevin (stochastic) Equation
2 k  T
d
m
 F ( r )   
R (t )
dt
m
Dissipative force
Fluctuating force
Mathematical Ingredients
Now the integration of position and
velocity changes as well…BrüngerBrooks-Karplus (BBK) method
rn 1
 1
2 k  T 
1  t / 2
1
2
 rn 
t  M F ( rn ) 
Zn 
 rn  rn 1  
1  t / 2
1  t / 2
M


Dissipative term
Potential force
Fluctuating term
Note: This will reduce to the Verlet algorithm as γ→0 (limit of
Newtonian dynamics)
Mathematical Ingredients
One final ingredient for the bio-chemical
system, the hemodynamic force
experienced by the leukocyte is modeled
by adding an additional force…this
method is called steered molecular
dynamics.
Mathematical Ingredients
Steered Molecular Dynamics
There are two methods for adding an additional force to the system
SMD atom
DUMMY ATOM
Spring with
Constant
ν
k
Constant velocity:
F  U
1
U  k [t  ( r  r 0 )  n ]2
2
OR
SMD atom
F
Constant Force
Summary
Computational chemistry relies
heavily on algorithms to do
simulation of molecular systems.
Velocity Verlet Algorithm
Langevin Dynamics
BBK Method
Summary
Processor speed is a limiting factor
for size of system and time-scale of
simulation.
Faster processor speeds
Parallel Computing
High speed cluster
Summary
Optimization of algorithms is a
possible solution to the limits of
computer technology
Future Dream: Quantum Computers
Summary
Classical Molecular Dynamics is a
powerful tool used by computational
chemists to study molecular systems.
Thanks!!!
Major Resource: Phillips, et.al. Scalable
Molecular Dynamics with NAMD. J Comput
Chem. (26) 1781-1802. 2005