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A Pinch of Verlet-Velocity Algorithm, A Dash of Langevin Dynamics:A Recipe for Classical Molecular Dynamics
Kim GunnersonUndergraduate 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 QuestionsChlorofluorocarbon’s (CFC’s) Role in Ozone
Depletion
08/05/03 Austral Winter
11/04/03Austral Spring
Systems and Questions
1.4698 Å1.6270Å
117.41º106.20º
hν
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 IngredientsClassical Equations of Motion:
Newton leads the way
1 2, ,..., , 1,2...total Nm r U r r r Nr
BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB
2
2
d d rF ma m m
dt dt
F U
Mathematical Ingredients
Potential:
total bond angle dihedral vdw coulombU U U U U U
12 6
4 ij ijvdw ij
i j i ij ij
Ur r
Lennard-Jones 6-12 Potential
04i j
coulombi j i ij
q qU
r
Electrostatic Interaction
Force Fields: Contain values determined via a combination of empirical techniques and quantum mechanical calculations.
IntermolecularIntramolecular
Mathematical Ingredients
Velocity Verlet Algorithm
mtf
tvtxtx)(
21
)()()( 2
mtftf
tvtv2
)()()()(
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 Tdm F r R tdt m
Dissipative force Fluctuating force
Mathematical Ingredients
Now the integration of position and velocity changes as well…Brünger-Brooks-Karplus (BBK) method
1 1
1 / 2
1 / 2n n n n
tr r r r
t
2 1 21
( )1 / 2 n n
k Tt M F r Z
t M
Dissipative termFluctuating term
Potential force
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 IngredientsSteered Molecular Dynamics
There are two methods for adding an additional force to the system
F
ORSMD atom
ν
DUMMY ATOMSMD atom
Spring withConstant
k
Constant velocity:
Constant Force
20 ])([
21
nrrtkU
UF
Summary
Computational chemistry relies heavily on algorithms to do simulation of molecular systems.
Velocity Verlet AlgorithmLangevin DynamicsBBK Method
SummaryProcessor speed is a limiting factor for size of system and time-scale of simulation.
Faster processor speedsParallel ComputingHigh 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