a pinch of verlet-velocity algorithm, a dash of langevin dynamics: a recipe for classical molecular...
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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
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Today’s Outline
Who am I and How did I get here?
Systems and Questions
Examples of Mathematics Used
Summary
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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
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Systems and Questions
Chlorine dioxide relaxation*
P-selectin/PSGL-1 conformation*
Carbon nanotubes
Crystal kinetics
*Dissertation subject
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Systems and QuestionsChlorofluorocarbon’s (CFC’s) Role in Ozone
Depletion
08/05/03 Austral Winter
11/04/03Austral Spring
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Systems and Questions
1.4698 Å1.6270Å
117.41º106.20º
hν
What are the relaxation dynamics of the photoexcitation of chlorine dioxide?
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Systems and Questions
Leukocyte Extravasation
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Systems and Questions
Are there conformational changes in the protein and/or ligand in response to an external force?
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????????
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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
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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
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Mathematical Ingredients
Velocity Verlet Algorithm
mtf
tvtxtx)(
21
)()()( 2
mtftf
tvtv2
)()()()(
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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.
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Mathematical Ingredients
Langevin (stochastic) Equation
2( ) ( )
k Tdm F r R tdt m
Dissipative force Fluctuating force
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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)
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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.
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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
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Summary
Computational chemistry relies heavily on algorithms to do simulation of molecular systems.
Velocity Verlet AlgorithmLangevin DynamicsBBK Method
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SummaryProcessor speed is a limiting factor for size of system and time-scale of simulation.
Faster processor speedsParallel ComputingHigh speed cluster
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Summary
Optimization of algorithms is a possible solution to the limits of computer technology
Future Dream: Quantum Computers
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Summary
Classical Molecular Dynamics is a powerful tool used by computational chemists to study molecular systems.
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Thanks!!!
Major Resource: Phillips, et.al. Scalable Molecular Dynamics with NAMD. J Comput Chem. (26) 1781-1802. 2005
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