Avenues for future research based on the results of this thesis may be opened by trying to answer the following questions:
As a number of alternative double-ended approaches have been developed in the past few years, such as, for example, the string method [287,283,284,285,286], the growing string method [288,109], and a super-linear NEB based on adopted basis Newton-Raphson minimiser [289], it would be interesting to make a detailed comparison of these methods on a set of problems we are likely to be solving in the future.
The connection algorithm and the algorithm for sampling the fastest path presented in Chapter 2 and Appendix E, respectively, are far from optimum because they operate on evolving databases but use the static Dijkstra algorithm to build the shortest path tree. When applying these methods to databases larger than these discussed in this thesis the use of dynamic graph algorithms may be of benefit.
It would be exciting to see more applications of the GT method that would allow us to come to a more detailed understanding of its strengths and weaknesses. Comparisons with sparse-optimised numerical approaches for solving the master equation and iterative solvers based on first-step analysis are also desirable.
From a theoretical point of view, it would be interesting to extend the approach of Section 4.3 to obtain higher moments of the escape time distribution function and maybe even the distribution function itself.
Although the DNEB method and connection algorithm presented in this thesis have seen a number of successful applications already the scope for further applications and development is ample. Potential areas include the design of better initial pathway guessing strategies for proteins, reduction of memory requirements of the connection algorithm, better understanding of the relationship of the optimal edge weight function and the form of the potential, and parallelisation of these methods for distributed-memory computing, to name but a few. Much ongoing work is now focused on attempts to construct an initial folding pathway for a medium-sized protein, barnase. Preliminary results showed that further improvements (with emphasis on the large number of degrees of freedom) to both double-ended and single-ended transition state searching methods are required to complete this task.