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Large Time Scale Molecular Paths Using Least Action. Benjamin Gladwin, Thomas Huber. gladwin@maths.uq.edu.au Australian Research Council Centre of Excellence for Mathematics and Statistics of Complex Systems and the University of Queensland, Department of Mathematics.

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large time scale molecular paths using least action

Large Time Scale Molecular Paths Using Least Action.

Benjamin Gladwin, Thomas Huber.

gladwin@maths.uq.edu.au

Australian Research CouncilCentre of Excellence forMathematicsand Statistics of Complex Systems and the University of Queensland, Department of Mathematics.

biologically interesting processes
Biologically Interesting Processes.
  • Chemical Processes
    • Reaction Kinetics
    • Thermodynamic properties.
    • Reaction Intermediates
  • Biological Processes
    • Cellular mechanics.
      • Physical ion pumps.
      • Docking Mechanisms.
    • Protein Folding Pathways.
outline
Outline.
  • Molecular Representation.
  • Our Approach.
  • Example.
  • Discussion and Comments.
molecular interaction types bonded energy terms6
Molecular Interaction Types – Bonded Energy Terms.
  • Improper Dihedral Angle Energy:
  • Dihedral Angle Energy:
molecular representation and potential energy
Molecular Representation and Potential Energy.
  • Potential energy.
  • Energy surfaces and conformation.
molecular dynamics
Molecular Dynamics.
  • Traditionally initial value approach.
  • Small time scales:
  • Disadvantages:
  • Initial conditions specified positions and velocities.
  • Stepwise numerically integrated in time.
  • Integration step ¼ 1 fs (10-15s)
  • Protein folding timescale:

1 ms ! tens of seconds.

  • >109 steps for even the fastest folding protein.
  • Current Simulations ¼ tens of nanoseconds
  • No guarantee to find final state (in finite time).
outline9
Outline.
  • Molecular Representation.
  • Our Approach.
  • Example.
  • Discussion and Comments.
boundary value reformulation
Boundary Value Reformulation.
  • Using information from the start and end points

) Fill in intermediate points.

  • Only Positional Information needed.
  • Directs the path.
the idea behind the action
The Idea behind the Action.
  • Hamilton’s Least action Principle says
  • Force from potential:
  • Force from the path:
  • Balancing these forces means that the path moves along the potential.
  • Numerical simulations will contain errors.
the error and the action

Force from potential

Force from path

The Error and the Action.
  • The errors can be expressed as:
  • Evaluating the probability that a particular step is correct
the error and the action13
The Error and the Action.
  • The errors can be expressed as:
  • Evaluating the probability that a particular step is correct
  • Assuming the errors are independent and Gaussian around the correct path.
least action approach summary

Specify a path in terms of a set of parameters bi

S=0 path

Least Action Approach. - (summary).
  • Reformulate Least action Principle:
  • Action measures error from a Real dynamical path.
  • Current path in energy space is a point on an ‘Action Surface’.
  • Calculate the gradient of the Action w.r.t parameters bi.
  • Minimise Action using this gradient by adjusting bi’s.

S>0 path

outline16
Outline.
  • Molecular Representation.
  • Our Approach.
  • Example.
  • Discussion and Comments.
dihedral angle potential
Dihedral Angle Potential.
  • Four Bodied interaction term
  • Experimental Setup:
seven particles
Seven Particles.
  • Lennard-Jones Potential.
  • Rearrangement of only three particles.
  • Starting at Equilibrium separations.
outline20
Outline.
  • Molecular Representation.
  • Our Approach.
  • Example.
  • Discussion and Comments.
advantages of approach
Advantages of Approach.
  • Conceptual
    • Smaller step sizes increases time resolution.
    • More expansions increases path accuracy.
    • No step size limitation.
    • Always have a stable solution (trajectories).
  • Computational:
    • Allows hierarchical optimization (unlike Molecular Dynamics).
    • Well suited to parallel processing.
    • Minimises search space by directing transition.
disadvantages of approach
Disadvantages of Approach.
  • Computationally:
    • Calculation of the second derivative.
  • Conceptually:
    • Artificial force imposed by time constraint ! Naturally inaccessible regions of energy surface.
    • Possible avoidance of key events from misplaced sample points.
in the future
In the Future.
  • Practical Improvements:
    • Improve program accuracy by redistribution of time slices.
    • Further code development
  • Theoretical Improvements:
    • Applying a different interpretation of the path in terms of angles instead of positions.
  • Real System Tests:
    • Organic charge-transfer complex b-(BEDT-TTF)2I3 in cooperation with Physics Dept. University of Queensland.
acknowledgements

Australian Research Council

Centre of Excellence for Mathematics

and Statistics of Complex Systems

Acknowledgements.
  • Thomas Huber
  • Phil Pollett
  • Benjamin Cairns
  • Support from the ARC Centre of Excellence for Mathematics and Statistics of Complex Systems

Department of

Mathematics.