KcsA channel. Two opposing monomers of the tetrameric channel are shown. Each monomer contains two TM helices (TM1, blue; TM2, green). The TM helices are connected to each other by the turret loop (gray), pore helix (red), and selectivity filter (yellow). Two K⁺ ions within the selectivity filter are shown as purple spheres. Locations of intracellular and extracellular regions are indicated.

Alignment between two pathways. Conformations along optimal pathway (i.e., pathway with the lowest energy barrier) are in cyan, and other conformations are in purple. (a) To align two pathways, we regard each pathway as a word with the constituting conformations as letters, which makes the problem analogous to that of the alignment of similar protein sequences. Thus, we can use dynamic programming for the aligning of two words, to align two pathways where two conformations from different pathways are matched, based on their rmsd. Zero or more gaps between conformations are allowed, to maximize the global alignment between two pathways. (b) In this example, the alignment can be used to shorten the optimal pathway (upper pathway) by removing the conformation that is aligned to a gap in the lower pathway. (c) The alignment can be used to hybridize pathways, to obtain an energetically favorable pathway by replacing subpathways. The first and last conformations along the two pathways are aligned with each other, and the other conformations are aligned with gaps. In other words, the two subpathways that are encircled by the ellipsoids cannot be matched. If the energy barrier of the purple subpathway is lower than that of the cyan subpathway, we may exchange between the two subpathways and obtain a better energy barrier for the optimal pathway.

Putative three-phase motion pathway between closed and open conformations of KcsA. (a) Intracellular perspective of the KcsA channel in closed conformation (gray), with location of the Cα atom of residue V¹⁵⁵ along the pathway marked in space-fill representation using different colors. The optimal pathway begins with Brownian motion (in gray). Phase I of the pathway, where the closed conformation is unlocked, is shown in cyan. Phase II, where the gate opens, is shown in purple, and phase III, where the channel is locked in its open conformation, is shown in red. (b) Energy profile of pathway. Arbitrary units of the Rosetta energy function were used (see Methods). The closed conformation (step 1) was assigned the lowest energy, and is the most stable. The first five steps are around the initial conformation, and are not an integral part of the opening pathway. The pathway from the closed conformation to the (model structure of the) open conformation (step 23) involves crossing an energy barrier (step 12). This is exactly the transition between phases I and II in a. (c and d) HOLE (45) analysis of changes in profile of the channel pore along the pathway. (c) Magnitude of the radius along the pore axis (Z). The green and red curves were obtained using the open and closed conformations, and the blue curves were obtained for intermediate conformations along the pathway between these two end-conformations. The gate is around Z = 20 Å, and the selectivity filter, which was held fixed, is around Z = 50–60 Å. (d) The green and red curves mark changes in the pore radius at Z = 35 Å and Z = 23 Å along the pathway, congruently. The black curve marks changes in pore radius at minimal constriction. The depth of the minimal constriction along the Z axis changes along the motion pathway.

Scheme of the overall algorithm: finding a motion pathway with a minimal energy barrier between open and closed conformations. (a) Two RRT conformation trees were grown simultaneously, starting from the initial (solid) and goal (open) conformations, until trees could be connected to create a full-motion pathway. The goal conformation (right) was based on the known x-ray structure of KvAP (10). (b) General outline of the methodology.

Alignment graph for two pathways P = {p₁, p₂, p₃} and Q = {q₁, q₂, q₃, q₄}. Two examples of alignments, as described in the text, are indicated by solid and dashed arrows. Each path in the graph suggests a pairwise alignment between two pathways.