A Pinch of Verlet-Velocity Algorithm, A Dash of Langevin Dynamics: A Recipe for Classical Molecular...

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