Extracting Physically Intuitive Reaction Coordinates from Transition Networks of a β-Sheet Miniprotein

Abstract

Simulations are important for understanding complex reactions, but their interpretation is challenging owing to the large number of degrees of freedom typically involved. To address this issue, various means for relating the dynamics of a stochastic system to its structural and energetic features have been introduced. Here, we show how two leading approaches can be combined to advantage. We use the network of transitions observed in a reversible folding/unfolding simulation of a 20-residue three-stranded antiparallel beta-sheet peptide (beta3s) to estimate the probabilities of committing to stable states (the native state and major nonnative states), and these then serve as the basis for an efficient statistical procedure for identifying physical variables that describe the dynamics. We find that a single coordinate that jointly characterizes the formation of the two native turns of beta3s can adequately describe the overall folding process, despite its complex nature. Additional features associated with major pathways leading from individual nonnative states are resolved; indeed, a key result is an improved understanding of the unfolded state. Connections to other methods for analyzing complex reactions are discussed.

New Insights into the Folding of a β-Sheet Miniprotein in a Reduced Space of Collective Hydrogen Bond Variables: Application to a Hydrodynamic Analysis of the Folding Flow

“…Since the reaction coordinates are not identical, we matched their left and right boundaries to compare the FEPs. It is seen that the profiles are in good agreement, confirming that the sum of the bond distances 31 and the first principal coordinate determined with the HB PCA method (Sect. 2.2) can both serve as reaction coordinates for the overall description of the folding process.…”

“…The bonds involved in g 1 (the upper panel) are exactly the bonds Qi et al have found most appropriate to describe folding of beta3s 31 , and Zheng et al have indicated as the bonds that make the major contribution to the first “diffusion” coordinate 43 . Moreover, the contributions of different bonds are approximately equal, as was assumed 31 and confirmed 43 previously. A nonzero projection of g 1 onto a bond indicates that g 1 changes as the length of the bond changes.…”

“…To calculate two of them, the pfoldf method suggested by Krivov and Karplus 17 was used. One profile (the blue curve) uses the sum of distances for the above eight bonds as the reaction coordinate (i.e., that used by Qi et al 31 ), and the other (the red curve) the collective variable g 1 ; the reaction coordinate was divided into bins of width 0.01 Å and 0.005 Å respectively. Since the reaction coordinates are not identical, we matched their left and right boundaries to compare the FEPs.…”

Equilibration of Protein States: A Time Dependent Free-Energy Disconnectivity Graph

“…The last conformations that come in equilibrium with the others are the Ch-curl conformations, which are less stable than the antiparallel β-strands and more difficult to form dynamically because of the distant N-and C-terminal strands. The observed hierarchy of the establishment of equilibrium between the characteristic conformations is in good agreement with the results of the previous studies 6,11,12,25,28,[30][31][32]45 and has allowed further insight into the process of beta3s folding. One issue of particular interest is the difference between the Cs-or and Ns-or pathways, which has been indicated in the early work 25 and confirmed in the later, the first-passage folding studies: 11,12 although Cs-or and Ns-or conformations are "symmetrical" in that each of them has one hairpin formed and the other unstructured, the Ns-or pathway prevails.…”

Recent developments in methods for identifying reaction coordinates

“…[52] Other applications of the method included the study of a nucleotide flipping facilitated by O 6 -alkylguanine-DNA alkyltransferase [38] and the study of the folding of a 20-residue antiparallel-sheet miniprotein. [71]…”