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IntroductionMethods
ResultsSummaryAppendix
Molecular Dynamics and the Rouse ModelAn Introduction to the Physics of Polymers
Daniel Bridges1 Dr. Aniket Bhattacharya2
1Department of Physics & AstronomyMiddle Tennessee State University
2Department of PhysicsUniversity of Central Florida
July 28, 2009
Daniel Bridges, Dr. Aniket Bhattacharya Molecular Dynamics and the Rouse Model
IntroductionMethods
ResultsSummaryAppendix
Why am I here?Polymer physics
Polymers are essential to life.
Daniel Bridges, Dr. Aniket Bhattacharya Molecular Dynamics and the Rouse Model
IntroductionMethods
ResultsSummaryAppendix
Why am I here?Polymer physics
Don’t sneeze; you might lose some polymers.
We can model a simplified DNAstrand as a beaded necklace.
Rouse model: ”Bead-springmodel”
Daniel Bridges, Dr. Aniket Bhattacharya Molecular Dynamics and the Rouse Model
IntroductionMethods
ResultsSummaryAppendix
Why am I here?Polymer physics
Don’t sneeze; you might lose some polymers.
We can model a simplified DNAstrand as a beaded necklace.
Rouse model: ”Bead-springmodel”
Daniel Bridges, Dr. Aniket Bhattacharya Molecular Dynamics and the Rouse Model
IntroductionMethods
ResultsSummaryAppendix
Why am I here?Polymer physics
Polymers sit in solvents!
Polymer/solvent interaction (varies with temp.) dictates threesituations:
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
IntroductionMethods
ResultsSummaryAppendix
Why am I here?Polymer physics
Polymers sit in solvents!
Polymer/solvent interaction (varies with temp.) dictates threesituations:
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
IntroductionMethods
ResultsSummaryAppendix
Why am I here?Polymer physics
Polymers sit in solvents!
Polymer/solvent interaction (varies with temp.) dictates threesituations:
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
IntroductionMethods
ResultsSummaryAppendix
Why am I here?Polymer physics
Polymers sit in solvents!
Polymer/solvent interaction (varies with temp.) dictates threesituations:
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
IntroductionMethods
ResultsSummaryAppendix
Why am I here?Polymer physics
Polymers sit in solvents!
Polymer/solvent interaction (varies with temp.) dictates threesituations:
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
IntroductionMethods
ResultsSummaryAppendix
Why am I here?Polymer physics
Polymers sit in solvents!
Polymer/solvent interaction (varies with temp.) dictates threesituations:
1 polymer ”stretched” ⇒ ”good solvent”2 ideal random walk ⇒ ”theta condition”3 collapses to form compact globule
modeling this behavior ...
~F = −~∇U − ~FR(t)
Daniel Bridges, Dr. Aniket Bhattacharya Molecular Dynamics and the Rouse Model
IntroductionMethods
ResultsSummaryAppendix
Why am I here?Polymer physics
Polymers sit in solvents!
Polymer/solvent interaction (varies with temp.) dictates threesituations:
1 polymer ”stretched” ⇒ ”good solvent”2 ideal random walk ⇒ ”theta condition”3 collapses to form compact globule
modeling this behavior ...
~F = −~∇U − ~FR(t)− ζ d~rdt
Daniel Bridges, Dr. Aniket Bhattacharya Molecular Dynamics and the Rouse Model
IntroductionMethods
ResultsSummaryAppendix
Why am I here?Polymer physics
Polymers sit in solvents!
Polymer/solvent interaction (varies with temp.) dictates threesituations:
1 polymer ”stretched” ⇒ ”good solvent”2 ideal random walk ⇒ ”theta condition”3 collapses to form compact globule
Langevin dynamics
~F = −~∇U − ~FR(t)− ζ d~rdt = m d2~r
dt2
Daniel Bridges, Dr. Aniket Bhattacharya Molecular Dynamics and the Rouse Model
IntroductionMethods
ResultsSummaryAppendix
Why am I here?Polymer physics
Polymers sit in solvents!
Polymer/solvent interaction (varies with temp.) dictates threesituations:
1 polymer ”stretched” ⇒ ”good solvent”2 ideal random walk ⇒ ”theta condition”3 collapses to form compact globule
Langevin dynamics
~F = −~∇U − ~FR(t)− ζ d~rdt = m d2~r
dt2
Daniel Bridges, Dr. Aniket Bhattacharya Molecular Dynamics and the Rouse Model
IntroductionMethods
ResultsSummaryAppendix
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
IntroductionMethods
ResultsSummaryAppendix
Verlet AlgorithmThe Rouse ModelThe Zimm ModelSome 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) + δt~v(t) + 12δt2~a(t) + ...
~r(t − δt) = ~r(t)− δt~v(t) + 12δt2~a(t) + ...
The Verlet Algorithm
~r(t + δt) = 2~r(t)−~r(t − δt) + δt2~a(t) + ...
Daniel Bridges, Dr. Aniket Bhattacharya Molecular Dynamics and the Rouse Model
IntroductionMethods
ResultsSummaryAppendix
Verlet AlgorithmThe Rouse ModelThe Zimm ModelSome 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) + δt~v(t) + 12δt2~a(t) + ...
~r(t − δt) = ~r(t)− δt~v(t) + 12δt2~a(t) + ...
The Verlet Algorithm
~r(t + δt) = 2~r(t)−~r(t − δt) + δt2~a(t) + ...
Daniel Bridges, Dr. Aniket Bhattacharya Molecular Dynamics and the Rouse Model
IntroductionMethods
ResultsSummaryAppendix
Verlet AlgorithmThe Rouse ModelThe Zimm ModelSome 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) + δt~v(t) + 12δt2~a(t) + ...
~r(t − δt) = ~r(t)− δt~v(t) + 12δt2~a(t) + ...
The Verlet Algorithm
~r(t + δt) = 2~r(t)−~r(t − δt) + δt2~a(t) + ...
Daniel Bridges, Dr. Aniket Bhattacharya Molecular Dynamics and the Rouse Model
IntroductionMethods
ResultsSummaryAppendix
Verlet AlgorithmThe Rouse ModelThe Zimm ModelSome 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) + δt~v(t) + 12δt2~a(t) + ...
~r(t − δt) = ~r(t)− δt~v(t) + 12δt2~a(t) + ...
The Verlet Algorithm
~r(t + δt) = 2~r(t)−~r(t − δt) + δt2~a(t) + ...
Daniel Bridges, Dr. Aniket Bhattacharya Molecular Dynamics and the Rouse Model
IntroductionMethods
ResultsSummaryAppendix
Verlet AlgorithmThe Rouse ModelThe Zimm ModelSome Topics Considered
Basic assumptions of the the ”Bead-Spring” Model
N ”beads” (monomers are points) with
spring constant ksp = 3kBTb2
No hydrodynamic interactions betweenbeads
Proposed by P. E. Rouse, J Chem Phys 21,1272 (1953).
Drawbacks: Does not accurately express diffusion coefficient orrelaxation time.
Daniel Bridges, Dr. Aniket Bhattacharya Molecular Dynamics and the Rouse Model
IntroductionMethods
ResultsSummaryAppendix
Verlet AlgorithmThe Rouse ModelThe Zimm ModelSome Topics Considered
Basic assumptions of the the ”Bead-Spring” Model
N ”beads” (monomers are points) with
spring constant ksp = 3kBTb2
No hydrodynamic interactions betweenbeads
Proposed by P. E. Rouse, J Chem Phys 21,1272 (1953).
Drawbacks: Does not accurately express diffusion coefficient orrelaxation time.
Daniel Bridges, Dr. Aniket Bhattacharya Molecular Dynamics and the Rouse Model
IntroductionMethods
ResultsSummaryAppendix
Verlet AlgorithmThe Rouse ModelThe Zimm ModelSome Topics Considered
Basic assumptions of the the ”Bead-Spring” Model
N ”beads” (monomers are points) with
spring constant ksp = 3kBTb2
No hydrodynamic interactions betweenbeads
Proposed by P. E. Rouse, J Chem Phys 21,1272 (1953).
Drawbacks: Does not accurately express diffusion coefficient orrelaxation time.
Daniel Bridges, Dr. Aniket Bhattacharya Molecular Dynamics and the Rouse Model
IntroductionMethods
ResultsSummaryAppendix
Verlet AlgorithmThe Rouse ModelThe Zimm ModelSome Topics Considered
Building on Rouse: the Zimm model
addresses hydrodynamicinteractions between beads
the diffusion coefficient andrelaxation times agree withexperiment.
Proposed by B. H. Zimm, J ChemPhys 24, 269 (1956).
Daniel Bridges, Dr. Aniket Bhattacharya Molecular Dynamics and the Rouse Model
IntroductionMethods
ResultsSummaryAppendix
Verlet AlgorithmThe Rouse ModelThe Zimm ModelSome Topics Considered
Building on Rouse: the Zimm model
addresses hydrodynamicinteractions between beads
the diffusion coefficient andrelaxation times agree withexperiment.
Proposed by B. H. Zimm, J ChemPhys 24, 269 (1956).
Daniel Bridges, Dr. Aniket Bhattacharya Molecular Dynamics and the Rouse Model
IntroductionMethods
ResultsSummaryAppendix
Verlet AlgorithmThe Rouse ModelThe Zimm ModelSome Topics Considered
Building on Rouse: the Zimm model
addresses hydrodynamicinteractions between beads
the diffusion coefficient andrelaxation times agree withexperiment.
Proposed by B. H. Zimm, J ChemPhys 24, 269 (1956).
Daniel Bridges, Dr. Aniket Bhattacharya Molecular Dynamics and the Rouse Model
IntroductionMethods
ResultsSummaryAppendix
Verlet AlgorithmThe Rouse ModelThe Zimm ModelSome Topics Considered
Interesting Quantities (Rouse)
Average Radius of Gyration:
R2g = 1
npart
∑npartn=1
⟨(~Rn − ~RG )2
⟩End-to-end Distance:
∥∥∥~R1N
∥∥∥ =∥∥∥∑N
n=1~rn
∥∥∥Diffusion Constant:DG = kBT
Nζ = 16t
⟨‖~ri (t)−~ri (0)‖2
⟩Relaxation Time: τr ' Nb2
DG
ζ, (viscous) frictioncoefficient
N, number ofmonomers
kB , Boltzmannconstant
T , temperature
τ , relaxation time
Daniel Bridges, Dr. Aniket Bhattacharya Molecular Dynamics and the Rouse Model
IntroductionMethods
ResultsSummaryAppendix
Diffusion Constant
Diffusion Constant
DG = kBTNζ = 1
6t
⟨‖~ri (t)−~ri (0)‖2
⟩Rahman: 2.43× 10−5 cm2
sMine: 2.47× 10−5 cm2
s
Daniel Bridges, Dr. Aniket Bhattacharya Molecular Dynamics and the Rouse Model
IntroductionMethods
ResultsSummaryAppendix
Future Outlook
Future goals:
understanding forced translocation
manipulate polymer movement
gene therapyvirus injectionprotein sequencing
Daniel Bridges, Dr. Aniket Bhattacharya Molecular Dynamics and the Rouse Model
IntroductionMethods
ResultsSummaryAppendix
Future Outlook
Future goals:
understanding forced translocation
manipulate polymer movement
gene therapyvirus injectionprotein sequencing
Daniel Bridges, Dr. Aniket Bhattacharya Molecular Dynamics and the Rouse Model
IntroductionMethods
ResultsSummaryAppendix
Future Outlook
Future goals:
understanding forced translocation
manipulate polymer movement
gene therapy
virus injectionprotein sequencing
Daniel Bridges, Dr. Aniket Bhattacharya Molecular Dynamics and the Rouse Model
IntroductionMethods
ResultsSummaryAppendix
Future Outlook
Future goals:
understanding forced translocation
manipulate polymer movement
gene therapyvirus injection
protein sequencing
Daniel Bridges, Dr. Aniket Bhattacharya Molecular Dynamics and the Rouse Model
IntroductionMethods
ResultsSummaryAppendix
Future Outlook
Future goals:
understanding forced translocation
manipulate polymer movement
gene therapyvirus injectionprotein sequencing
Daniel Bridges, Dr. Aniket Bhattacharya Molecular Dynamics and the Rouse Model
IntroductionMethods
ResultsSummaryAppendix
Future Outlook
Thank you!
Daniel Bridges, Dr. Aniket Bhattacharya Molecular Dynamics and the Rouse Model
IntroductionMethods
ResultsSummaryAppendix
For Further Reading
For Further Reading I
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 PropertiesJohn Wiley and Sons, Inc.
Daniel Bridges, Dr. Aniket Bhattacharya Molecular Dynamics and the Rouse Model