Molecular Dynamics and Potentials

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Molecular Dynamics and Potentials
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1. Classical Mechanics
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1.a General Solution
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1.b Damped System
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1.b.1 Overdamping
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1.b.2 Underdamping
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1.b.3 Critical damping
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2. Molecular Dynamics

• Simulate the motion of particles in a system as
  they react to forces caused by interactions with
  other particles.

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2.1 Proteins

• Consist of a chain of amino acid residues.
• Some features

• Bond Length
• Bond Angle
• Dihedral Angle

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2.2 Newtonian Dynamics(ND)

• No analytic solution due to the complicate form
  of

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2.3 Langevin Dynamics(LD)

• M mass matrix
• C Damping matrix, usually taken as
• U potential function
• D by fluctuation dissipation
  theorem
• W(t) Wiener process

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2.3.1 Purpose of LD

• Model the influence of heat bath by adding to the
  velocity of each particle a random and frictional
  force.

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2.3.2 Why Random Force?

• Reduce the degree of freedom dramatically.
• Missing forces are replaced by averaged and
  stochastic forces.

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2.3.3 Wiener Process W(t)
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2.4 Integrator

• Basic idea Turn a differential equation into a
  difference equation that can be solved
  iteratively.

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2.4.1 Leapfrog Method for ND

• ND in another form

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Step 1 half-kick
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Step 2 fluctuate(or drift)
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Step 3 half-kick
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2.4.2 Leapfrog Method for LD

• Rewrite the Langevin equation

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Step 1 half-kick
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Step 2 fluctuate
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Step 3 half-kick
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where
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2.4.3 Why Leapfrog?

• Leapfrog method is symplectic
• Leapfrog method preserves the conservation laws
  for linear and angular momentum and the property
  of time reversibility
• (it doesnt maintain conservation of energy)

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3 Potential Energy Function
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3.1 Bonded Energy
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3.2 Nonbonded Energy
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3.3 Energy Minimization

• Atoms move so as to reduce the net force
• Most methods can reach only local minimum
• MD imparts kinetic energy into system, it can be
  sufficient to progress over barriers

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References

• Molecular Modeling and Simulation An
  Interdisciplinary Guide, Tarmar Schlick
• Integration Schemes for Molecular Dynamics and
  Related Applications, Robert Skeel
• Molecular Modeling of Proteins and Mathematical
  Prediction of Protein Structure, Arnold Neumaier