methods exist for refining the positions of transition states [26,27,49–52]. Once the position of the transition state is refined, a steepest descent pathway can be constructed [53,54].

VIII. HEURISTIC METHODS

An alternative approach to the construction of a smooth reaction pathway with a wellrefined transition state is the ‘‘conjugate peak refinement’’ (CPR) method of Fischer and Karplus [55]. As in the global methods, the path is optimized as a whole and self-consis- tently. However, all points along the path are not treated equally. The computational effort is always directed at bringing the highest energy segment of the path closer to the valley of the energy surface. Starting from some initial guess at the path, a simple set of rules known as a heuristic is applied in each cycle of CPR, when one path point is either added, improved, or removed. This is repeated until the only remaining high energy path points are the actual saddle points of the transition pathway.

For example, the method may proceed as follows:

1.Start from a straight line path as the first guess at the transition pathway connecting known reactant and product structures.

2.Search along that pathway to isolate the highest energy point.

3.From that highest energy point, called the ‘‘peak,’’ search conjugate to the direction of the reaction pathway and find the lowest energy point.

4.Make that lowest energy point a permanent intermediate point on the transition pathway.

5.Return to step 1 and refine the two intermediate segments in the same manner, and so on, until the desired level of detail is obtained.

The result of such a series of steps is depicted in Figure 9. This method has proved effective in isolating reaction pathways for conformational transitions involving localized torsional transitions including those involving a subtle isomerization mechanism [56].

On high dimensional energy surfaces, a newly added path point does not always reach the bottom of the valley in a single CPR cycle. Such a point is improved during a later CPR cycle, because eventually it will itself become the energy ‘‘peak’’ along the path, rk. In such a case, each intermediate structure rk is connected to two nearest neighbor structures rk 1 and rk 1, which define the unit displacement vectors uˆk (rk rk 1)/ |rk rk 1| and uˆk 1 (rk 1 rk)/|rk 1 rk|. We can then define the tangent vector at point rk as tk uˆk 1 uˆk. The path point rk is improved by performing an energy minimization conjugate to the tangent vector tk, which first involves maximizing the energy locally along tk. Sometimes, when this local maximization is not possible, the intermediate point rk is simply removed and the path is rebuilt from rk 1 to rk 1. Following this process, the intermediate segments can be continually refined by following the protocol described above until the desired level of detail is achieved. This process of local refinement of a globally defined reaction pathway leads to the optimal use of each algorithm.

CPR can be used to find continuous paths for complex transitions that might have hundreds of saddle points and need to be described by thousands of path points. Examples of such transitions include the quaternary transition between the R and T states of hemoglobin [57] and the reorganization of the retinoic acid receptor upon substrate entry [58]. Because CPR yields the exact saddle points as part of the path, it can also be used in conjunction with normal mode analysis to estimate the vibrational entropy of activation

Figure 9 The refinement of an initial straight line path to a smooth transition pathway using the conjugate peak refinement algorithm. The initial guess is a straight line path. That path is refined by the addition of an intermediate point (the long-stemmed arrow). Two additional intermediates are added to create a path of three intermediates before four more intermediates are inserted. The process can be continued until the desired level of smoothness in the transition pathway is obtained.

[59] and to analyze enzymatic mechanisms [60]. The efficiency and robustness of the CPR method make it an efficient means of mapping the topology of complex energy surfaces [61].

IX. SUMMARY

In the study of long-time-scale processes in macromolecular systems, the greatest challenge remains the isolation of one or more characteristic reaction pathways. These pathways define the reaction mechanism. They are also the starting points for the computation of rates of reaction. Once the potential of mean force along the reaction coordinate is known, the reaction rate constant can be computed by using transition state theory. Subsequently, the system dynamics can be followed to compute the transmission coefficient. It is currently possible to carry out such a series of steps not only on a modest chemical system [62] but also on a complex biomolecular system [31]. The greatest uncertainty in the computed value of the rate constant is due to the uncertainty in the activation energy. Potentially, the greatest uncertainty in the activation energy lies in the choice of the reaction coordinate.

Significant advances are being made in the development of effective methods for determining reaction pathways in complex systems. These methods appear to be the most

promising techniques for the study of long-time processes such as protein folding that continue to stand outside the range of direct molecular dynamics simulation.

ACKNOWLEDGMENT

I am grateful to Vio Buchete, Ron Elber, Stefan Fischer, and the editors for helpful comments.

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