1. Large Time Scale Molecular Paths Using Least Action.
Benjamin Gladwin, Thomas Huber.
What we are talking about is computational modeling of molecular processes.
Australian Research Council Centre of Excellence for Mathematics and 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.
How do we model molecules in computer. We represent all atoms in molecule in terms of their coordinates and these points interact in some fashion through an interaction function.

4. Molecular Interaction Types – Non-bonded Energy Terms.
Lennard-Jones Energy.
Coloumb Energy.
Coloumb law (rough detail)
All pairs of atoms
LJ eve for the noncharge interactions.

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:
Emphasis the potential as a function of the coordinates.

7. Molecular Representation and Potential Energy.
Energy surfaces and conformation.
With this function defined, can represent the dynamics as a point moving across this HD energy surface.

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.
Both exactly specify system, just use different data.
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. The Error and the Action.
The errors can be expressed as:
Force from potential
Force from path
Evaluating the probability that a particular step is correct
Check why there is a product over all i

13. The Error and the Action.
The errors can be expressed as:
Evaluating the probability that a particular step is correct
Check why there is a product over all i
Assuming the errors are independent and Gaussian around the correct path.

14. The Error and the Action.
Using Boltzmann’s Principle
Combining these ideas
GET ENTROPY INTO THAT LAST EQUATION!!!??

15. Least Action Approach. - (summary).
Reformulate Least action Principle:
Specify a path in terms of a set of parameters bi
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
Derivation Of Action in terms of errors
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.
Where are we going in the future?

23. In the Future. Practical Improvements: Theoretical 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. Acknowledgements. Thomas Huber Phil Pollett Benjamin Cairns
Support from the ARC Centre of Excellence for Mathematics and Statistics of Complex Systems
Department of
Mathematics.
Australian Research Council
Centre of Excellence for Mathematics
and Statistics of Complex Systems