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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 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.

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  1. 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.

  2. Biologically Interesting Processes. • Chemical Processes • Reaction Kinetics • Thermodynamic properties. • Reaction Intermediates • Biological Processes • Cellular mechanics. • Physical ion pumps. • Docking Mechanisms. • Protein Folding Pathways.

  3. Outline. • Molecular Representation. • Our Approach. • Example. • Discussion and Comments.

  4. Molecular Interaction Types – Non-bonded Energy Terms. • Lennard-Jones Energy. • Coloumb Energy.

  5. Molecular Interaction Types – Bonded Energy Terms. • Bond energy: • Bond Angle Energy:

  6. Molecular Interaction Types – Bonded Energy Terms. • Improper Dihedral Angle Energy: • Dihedral Angle Energy:

  7. Molecular Representation and Potential Energy. • Potential energy. • Energy surfaces and conformation.

  8. 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).

  9. Outline. • Molecular Representation. • Our Approach. • Example. • Discussion and Comments.

  10. Boundary Value Reformulation. • Using information from the start and end points ) Fill in intermediate points. • Only Positional Information needed. • Directs the path.

  11. 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.

  12. 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

  13. 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.

  14. Combining these ideas The Error and the Action. • Using Boltzmann’s Principle

  15. 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

  16. Outline. • Molecular Representation. • Our Approach. • Example. • Discussion and Comments.

  17. Dihedral Angle Potential. • Four Bodied interaction term • Experimental Setup:

  18. Results.

  19. Seven Particles. • Lennard-Jones Potential. • Rearrangement of only three particles. • Starting at Equilibrium separations.

  20. Outline. • Molecular Representation. • Our Approach. • Example. • Discussion and Comments.

  21. 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.

  22. 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.

  23. 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.

  24. 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.

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