Granger Causality Analysis of Chignolin Folding

Abstract

Constantly advancing computer simulations of biomolecules provide huge amounts of data that are difficult to interpret. In particular, obtaining insights into functional aspects of macromolecular dynamics, often related to cascades of transient events, calls for methodologies that depart from the well-grounded framework of equilibrium statistical physics. One of the approaches toward the analysis of complex temporal data which has found applications in the fields of neuroscience and econometrics is Granger causality analysis. It allows determining which components of multidimensional time series are most influential for the evolution of the entire system, thus providing insights into causal relations within the dynamic structure of interest. In this work, we apply Granger analysis to a long molecular dynamics trajectory depicting repetitive folding and unfolding of a mini β-hairpin protein, CLN025. We find objective, quantitative evidence indicating that rearrangements within the hairpin turn region are determinant for protein folding and unfolding. On the contrary, interactions between hairpin arms score low on the causality scale. Taken together, these findings clearly favor the concept of zipperlike folding, which is one of two postulated β-hairpin folding mechanisms. More importantly, the results demonstrate the possibility of a conclusive application of Granger causality analysis to a biomolecular system.

Figure 1

(A) Free energy map as a function of two dominant TICA ICs. Red dots indicate the location of representative frames for six (un)folding steps. (B–D) Descriptors of the folding process as functions of the reaction coordinate. Shaded areas indicate estimation errors (see text for details). (E) Transition matrix between six folding steps, with circle areas proportional to corresponding populations of simulation frames (transitions with probability < 0.03 not shown).

Figure 2

Representative structures for subsequent folding steps and schematic depiction of inter-residue contact frequencies. Shown are major hydrogen bonds captured in particular structures. Gray stars indicate turn locations in polypeptide backbone. Detailed frequencies of contact formation and the distribution of turn angles are provided in the Supporting Information.

Figure 3

Fractions of hydrogen bonds formed in subsequent CLN025 (un)folding steps. Bonds formed in <5% of respective simulation frames are not shown.

Figure 4

Contact-based descriptors of Granger causality. Color codes for contact groups: magenta, turn; blue, arms; green, ladder; black, direct.

Figure 5

Degree of distance overlap with the native ensemble, Ω(ξ), for four contacts with highest (left plot) and lowest (right plot) G° values. CF, commitor function. Hairpin scheme: G° values for all contacts.