Molecular Dynamics and the Rouse Model - An Introduction to the ...

Introduction
Methods
Results
Summary
Appendix
Molecular Dynamics and the Rouse Model
An Introduction to the Physics of Polymers
Daniel Bridges 1 Dr. Aniket Bhattacharya 2
1 Department of Physics & Astronomy
Middle Tennessee State University
2 Department of Physics
University of Central Florida
July 28, 2009
Daniel Bridges, Dr. Aniket Bhattacharya
Molecular Dynamics and the Rouse Model

Introduction
Methods
Results
Summary
Appendix
Polymers are essential to life.
Why am I here?
Polymer physics
Daniel Bridges, Dr. Aniket Bhattacharya
Molecular Dynamics and the Rouse Model

Introduction
Methods
Results
Summary
Appendix
Why am I here?
Polymer physics
Don’t sneeze; you might lose some polymers.
We can model a simplified DNA
strand as a beaded necklace.
Daniel Bridges, Dr. Aniket Bhattacharya
Molecular Dynamics and the Rouse Model

Introduction
Methods
Results
Summary
Appendix
Why am I here?
Polymer physics
Don’t sneeze; you might lose some polymers.
We can model a simplified DNA
strand as a beaded necklace.
Rouse model: ”Bead-spring
model”
Daniel Bridges, Dr. Aniket Bhattacharya
Molecular Dynamics and the Rouse Model

Polymers sit in solvents!
Introduction
Methods
Results
Summary
Appendix
Why am I here?
Polymer physics
Polymer/solvent interaction (varies with temp.) dictates three
situations:
Daniel Bridges, Dr. Aniket Bhattacharya
Molecular Dynamics and the Rouse Model

Polymers sit in solvents!
Introduction
Methods
Results
Summary
Appendix
Why am I here?
Polymer physics
Polymer/solvent interaction (varies with temp.) dictates three
situations:
1 polymer ”stretched” ⇒ ”good solvent”
Daniel Bridges, Dr. Aniket Bhattacharya
Molecular Dynamics and the Rouse Model

Polymers sit in solvents!
Introduction
Methods
Results
Summary
Appendix
Why am I here?
Polymer physics
Polymer/solvent interaction (varies with temp.) dictates three
situations:
1 polymer ”stretched” ⇒ ”good solvent”
2 ideal random walk ⇒ ”theta condition”
Daniel Bridges, Dr. Aniket Bhattacharya
Molecular Dynamics and the Rouse Model

Polymers sit in solvents!
Introduction
Methods
Results
Summary
Appendix
Why am I here?
Polymer physics
Polymer/solvent interaction (varies with temp.) dictates three
situations:
1 polymer ”stretched” ⇒ ”good solvent”
2 ideal random walk ⇒ ”theta condition”
3 collapses to form compact globule
Daniel Bridges, Dr. Aniket Bhattacharya
Molecular Dynamics and the Rouse Model

Polymers sit in solvents!
Introduction
Methods
Results
Summary
Appendix
Why am I here?
Polymer physics
Polymer/solvent interaction (varies with temp.) dictates three
situations:
1 polymer ”stretched” ⇒ ”good solvent”
2 ideal random walk ⇒ ”theta condition”
3 collapses to form compact globule
modeling this behavior ...
⃗F = − ⃗ ∇U
Daniel Bridges, Dr. Aniket Bhattacharya
Molecular Dynamics and the Rouse Model

Polymers sit in solvents!
Introduction
Methods
Results
Summary
Appendix
Why am I here?
Polymer physics
Polymer/solvent interaction (varies with temp.) dictates three
situations:
1 polymer ”stretched” ⇒ ”good solvent”
2 ideal random walk ⇒ ”theta condition”
3 collapses to form compact globule
modeling this behavior ...
⃗F = − ⃗ ∇U − ⃗ F R (t)
Daniel Bridges, Dr. Aniket Bhattacharya
Molecular Dynamics and the Rouse Model

Polymers sit in solvents!
Introduction
Methods
Results
Summary
Appendix
Why am I here?
Polymer physics
Polymer/solvent interaction (varies with temp.) dictates three
situations:
1 polymer ”stretched” ⇒ ”good solvent”
2 ideal random walk ⇒ ”theta condition”
3 collapses to form compact globule
modeling this behavior ...
⃗F = − ⃗ ∇U − ⃗ F R (t) − ζ d⃗r
dt
Daniel Bridges, Dr. Aniket Bhattacharya
Molecular Dynamics and the Rouse Model

Polymers sit in solvents!
Introduction
Methods
Results
Summary
Appendix
Why am I here?
Polymer physics
Polymer/solvent interaction (varies with temp.) dictates three
situations:
1 polymer ”stretched” ⇒ ”good solvent”
2 ideal random walk ⇒ ”theta condition”
3 collapses to form compact globule
Langevin dynamics
⃗F = − ⃗ ∇U − ⃗ F R (t) − ζ d⃗r
dt = m d2 ⃗r
dt 2
Daniel Bridges, Dr. Aniket Bhattacharya
Molecular Dynamics and the Rouse Model

Polymers sit in solvents!
Introduction
Methods
Results
Summary
Appendix
Why am I here?
Polymer physics
Polymer/solvent interaction (varies with temp.) dictates three
situations:
1 polymer ”stretched” ⇒ ”good solvent”
2 ideal random walk ⇒ ”theta condition”
3 collapses to form compact globule
Langevin dynamics
⃗F = − ⃗ ∇U − ⃗ F R (t) − ζ d⃗r
dt = m d2 ⃗r
dt 2
Daniel Bridges, Dr. Aniket Bhattacharya
Molecular Dynamics and the Rouse Model

Introduction
Methods
Results
Summary
Appendix
Why am I here?
Polymer physics
Relaxation time = MD equilibriation
Want polymer to be ”at rest”.
Important for MD accuracy and clarity
Daniel Bridges, Dr. Aniket Bhattacharya
Molecular Dynamics and the Rouse Model

Introduction
Methods
Results
Summary
Appendix
Verlet Algorithm
The Rouse Model
The Zimm Model
Some Topics Considered
Verlet Algorithm is ”numerical integration” of motion eqns.
Next position ⃗r(t + δt) ”future” determined by
current position ⃗r(t) and acceleration ⃗a(t): t ⇒ ”now”
previous position ⃗r(t − δt): t − δt ⇒ ”past”
⃗r(t + δt) = ⃗r(t) + δt⃗v(t) + 1 2 δt2 ⃗a(t) + ...
⃗r(t − δt) = ⃗r(t) − δt⃗v(t) + 1 2 δt2 ⃗a(t) + ...
The Verlet Algorithm
⃗r(t + δt) = 2⃗r(t) −⃗r(t − δt) + δt 2 ⃗a(t) + ...
Daniel Bridges, Dr. Aniket Bhattacharya
Molecular Dynamics and the Rouse Model

Introduction
Methods
Results
Summary
Appendix
Verlet Algorithm
The Rouse Model
The Zimm Model
Some Topics Considered
Verlet Algorithm is ”numerical integration” of motion eqns.
Next position ⃗r(t + δt) ”future” determined by
current position ⃗r(t) and acceleration ⃗a(t): t ⇒ ”now”
previous position ⃗r(t − δt): t − δt ⇒ ”past”
⃗r(t + δt) = ⃗r(t) + δt⃗v(t) + 1 2 δt2 ⃗a(t) + ...
⃗r(t − δt) = ⃗r(t) − δt⃗v(t) + 1 2 δt2 ⃗a(t) + ...
The Verlet Algorithm
⃗r(t + δt) = 2⃗r(t) −⃗r(t − δt) + δt 2 ⃗a(t) + ...
Daniel Bridges, Dr. Aniket Bhattacharya
Molecular Dynamics and the Rouse Model

Introduction
Methods
Results
Summary
Appendix
Verlet Algorithm
The Rouse Model
The Zimm Model
Some Topics Considered
Verlet Algorithm is ”numerical integration” of motion eqns.
Next position ⃗r(t + δt) ”future” determined by
current position ⃗r(t) and acceleration ⃗a(t): t ⇒ ”now”
previous position ⃗r(t − δt): t − δt ⇒ ”past”
⃗r(t + δt) = ⃗r(t) + δt⃗v(t) + 1 2 δt2 ⃗a(t) + ...
⃗r(t − δt) = ⃗r(t) − δt⃗v(t) + 1 2 δt2 ⃗a(t) + ...
The Verlet Algorithm
⃗r(t + δt) = 2⃗r(t) −⃗r(t − δt) + δt 2 ⃗a(t) + ...
Daniel Bridges, Dr. Aniket Bhattacharya
Molecular Dynamics and the Rouse Model

Introduction
Methods
Results
Summary
Appendix
Verlet Algorithm
The Rouse Model
The Zimm Model
Some Topics Considered
Verlet Algorithm is ”numerical integration” of motion eqns.
Next position ⃗r(t + δt) ”future” determined by
current position ⃗r(t) and acceleration ⃗a(t): t ⇒ ”now”
previous position ⃗r(t − δt): t − δt ⇒ ”past”
⃗r(t + δt) = ⃗r(t) + δt⃗v(t) + 1 2 δt2 ⃗a(t) + ...
⃗r(t − δt) = ⃗r(t) − δt⃗v(t) + 1 2 δt2 ⃗a(t) + ...
The Verlet Algorithm
⃗r(t + δt) = 2⃗r(t) −⃗r(t − δt) + δt 2 ⃗a(t) + ...
Daniel Bridges, Dr. Aniket Bhattacharya
Molecular Dynamics and the Rouse Model

Introduction
Methods
Results
Summary
Appendix
Verlet Algorithm
The Rouse Model
The Zimm Model
Some Topics Considered
Basic assumptions of the the ”Bead-Spring” Model
N ”beads” (monomers are points) with
spring constant k sp = 3k BT
b 2
No hydrodynamic interactions between
beads
Daniel Bridges, Dr. Aniket Bhattacharya
Molecular Dynamics and the Rouse Model

Introduction
Methods
Results
Summary
Appendix
Verlet Algorithm
The Rouse Model
The Zimm Model
Some Topics Considered
Basic assumptions of the the ”Bead-Spring” Model
N ”beads” (monomers are points) with
spring constant k sp = 3k BT
b 2
No hydrodynamic interactions between
beads
Proposed by P. E. Rouse, J Chem Phys 21,
1272 (1953).
Daniel Bridges, Dr. Aniket Bhattacharya
Molecular Dynamics and the Rouse Model

Introduction
Methods
Results
Summary
Appendix
Verlet Algorithm
The Rouse Model
The Zimm Model
Some Topics Considered
Basic assumptions of the the ”Bead-Spring” Model
N ”beads” (monomers are points) with
spring constant k sp = 3k BT
b 2
No hydrodynamic interactions between
beads
Proposed by P. E. Rouse, J Chem Phys 21,
1272 (1953).
Drawbacks: Does not accurately express diffusion coefficient or
relaxation time.
Daniel Bridges, Dr. Aniket Bhattacharya
Molecular Dynamics and the Rouse Model

Introduction
Methods
Results
Summary
Appendix
Verlet Algorithm
The Rouse Model
The Zimm Model
Some Topics Considered
Building on Rouse: the Zimm model
addresses hydrodynamic
interactions between beads
Daniel Bridges, Dr. Aniket Bhattacharya
Molecular Dynamics and the Rouse Model

Introduction
Methods
Results
Summary
Appendix
Verlet Algorithm
The Rouse Model
The Zimm Model
Some Topics Considered
Building on Rouse: the Zimm model
addresses hydrodynamic
interactions between beads
the diffusion coefficient and
relaxation times agree with
experiment.
Daniel Bridges, Dr. Aniket Bhattacharya
Molecular Dynamics and the Rouse Model

Introduction
Methods
Results
Summary
Appendix
Verlet Algorithm
The Rouse Model
The Zimm Model
Some Topics Considered
Building on Rouse: the Zimm model
addresses hydrodynamic
interactions between beads
the diffusion coefficient and
relaxation times agree with
experiment.
Proposed by B. H. Zimm, J Chem
Phys 24, 269 (1956).
Daniel Bridges, Dr. Aniket Bhattacharya
Molecular Dynamics and the Rouse Model

Introduction
Methods
Results
Summary
Appendix
Interesting Quantities (Rouse)
Verlet Algorithm
The Rouse Model
The Zimm Model
Some Topics Considered
Average Radius of Gyration:
Rg 2 = 1 ∑ npart
npart n=1
〈( R ⃗ n − R ⃗ 〉
G ) 2 End-to-end Distance: ∥R ⃗ ∥ ∥ ∥∥ ∥∥ ∑ 1N = N
n=1 ⃗r n∥
Diffusion Constant:
D G = k BT
Nζ
= 1 6t
〈‖⃗r i (t) −⃗r i (0)‖ 2〉
Relaxation Time: τ r ≃ Nb2
D G
ζ, (viscous) friction
coefficient
N, number of
monomers
k B , Boltzmann
constant
T , temperature
τ, relaxation time
Daniel Bridges, Dr. Aniket Bhattacharya
Molecular Dynamics and the Rouse Model

Diffusion Constant
D G = k BT
Nζ
= 1 6t
Rahman: 2.43 × 10−5 cm2
s
Introduction
Methods
Results
Summary
Appendix
〈‖⃗r i (t) −⃗r i (0)‖ 2〉
Diffusion Constant
Mine: 2.47 × 10−5 cm2
s
Daniel Bridges, Dr. Aniket Bhattacharya
Molecular Dynamics and the Rouse Model

Introduction
Methods
Results
Summary
Appendix
Future Outlook
Future goals:
understanding forced translocation
Daniel Bridges, Dr. Aniket Bhattacharya
Molecular Dynamics and the Rouse Model

Introduction
Methods
Results
Summary
Appendix
Future Outlook
Future goals:
understanding forced translocation
manipulate polymer movement
Daniel Bridges, Dr. Aniket Bhattacharya
Molecular Dynamics and the Rouse Model

Introduction
Methods
Results
Summary
Appendix
Future Outlook
Future goals:
understanding forced translocation
manipulate polymer movement
gene therapy
Daniel Bridges, Dr. Aniket Bhattacharya
Molecular Dynamics and the Rouse Model

Introduction
Methods
Results
Summary
Appendix
Future Outlook
Future goals:
understanding forced translocation
manipulate polymer movement
gene therapy
virus injection
Daniel Bridges, Dr. Aniket Bhattacharya
Molecular Dynamics and the Rouse Model

Introduction
Methods
Results
Summary
Appendix
Future Outlook
Future goals:
understanding forced translocation
manipulate polymer movement
gene therapy
virus injection
protein sequencing
Daniel Bridges, Dr. Aniket Bhattacharya
Molecular Dynamics and the Rouse Model

Thank you!
Introduction
Methods
Results
Summary
Appendix
Future Outlook
Daniel Bridges, Dr. Aniket Bhattacharya
Molecular Dynamics and the Rouse Model

For Further Reading I
Introduction
Methods
Results
Summary
Appendix
For Further Reading
Allen, M.P., Tildesley, D.J. (1987).
Computer Simulation of Liquids.
Oxford: Clarendon Press.
Doi, M. (1995).
Introduction to Polymer Physics.
Oxford: Clarendon Press
Jones, R. A. L. (2002).
Soft Condensed Matter.
Oxford: Oxford University Press.
Teraoka, I. (2002).
Polymer Solutions: An Introduction to Physical Properties
John Wiley and Sons, Inc.
Daniel Bridges, Dr. Aniket Bhattacharya
Molecular Dynamics and the Rouse Model

For Further Reading II
Introduction
Methods
Results
Summary
Appendix
For Further Reading
Rahman, A. (1964).
Correlations in the Motion of Atoms in Liquid Argon.
Physical Review, 136, Number 2A
Daniel Bridges, Dr. Aniket Bhattacharya
Molecular Dynamics and the Rouse Model