Conjugate Peak Refinement (CPR)

Stefan Fischer & Martin Karplus
Chem. Phys. Lett. 1992, vol. 194, p. 252.  (Abstract) (Full PDF)

The low energy reactant and product conformations of a reaction or conformational change can be thought to be located at the bottom of two "basins" on a surface representing the energy landscape of the macromolecule.  The path that connects these two basins by traveling along the "valleys" of the surface and going over the "passes" (which are saddle-points on the surface and correspond to the transition states) is commonly referred to as the minimum energy path (MEP), also known as the intrinsic path (because it is close to the average over the most probable paths).

Many methods for finding the minimum energy path have been proposed.  One method that is robust and fast enough to find paths in macromolecules is called Conjugate Peak Refinement (CPR).

Schematic description of the CPR algorithm.
The contour lines show the two-dimensional energy landscape of a hypothetical molecule (with minima at R, I, P).  An initial guess of the path is made, here the straight interpolation line (-.-.-) from the reactant (R) to the product (P) states (results do not depend on whether the transition is considered as R->P, or P->R).

Each CPR-cycle (cycles are numbered in blue) automatically adds a path intermediate (.) to the highest-energy region of the path (the 'peak') and optimizes it by a conjugate minimization (->) so that it lies closer to the bottom of the energy-valley (X).  This new point is then incorporated into the path by the two adjacent interpolation segments (- - -).   Repeating this heuristic procedure yields a final path (--) that follows the valley of the energy surface and goes through the exact saddle-points (S1 ,S2), close to the minimum energy path (shown in green).

The real energy surface of a protein is very high-dimensional and has many MEPs.  On such surfaces, an alternative outcome of a CPR-cycle is the removal of the highest path-intermediate (X) from the path, which is then rebuild between the adjacent intermediates (this is a very frequent outcome for a CPR-cycle, but cannot be shown here, since it does not happen on simple 2D-surfaces).  The advantage of removal/rebuilding is that high energy barriers are often circumvented early during the path-refinement, resulting in final paths with globally lower barriers.

CPR finds the minimum energy path without applying any constraint to drive the reaction.  It does so by starting from an initial guess of the path, to which it adds a sufficient number of intermediate conformations and optimizing them until they lie in the "valleys" and until the highest-energy conformers coincide with the saddle-points.

The resulting intrinsic path constitutes a smooth 'movie' of the process, even though every atom is allowed to move independently.  It shows only the motions that are essential to the transition, unlike trajectories created by other methods used to accelerate slow conformational changes, such as targeted molecular dynamics.

Because no artificial strain is created, CPR yields meaningful and easily interpretable energy barriers along the path.  The latter is what distinguishes CPR from methods that determine pathways by some form of coordinate-driving.  In these, the transition-states often are not identified reliably and thus the energy barriers along the paths are not known, making it difficult to assess the probability of a path.

To find MPE with low barriers for very complex conformational transitions in proteins, it is not always possible to use an initial path built by simple linear interpolation in Cartesian coordinates (as in the Figure above).  Using more sophisticated interpolations and other procedures to generate the initial path has allowed to solve that problem, as described for the conformational transition in Ras p21.

CPR is a deterministic method, in the sense that it always optimizes the same initial path into the same final path.  To find different and better paths is achieved by varying the initial path.  This has been described for the search of chloride-transfer paths in halorhodopsin.

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