Nonexponential kinetics captured in sequential unfolding of polyproteins over a range of loads

Abstract

While performing under mechanical loads in vivo, polyproteins are vitally involved in cellular mechanisms such as regulation of tissue elasticity and mechano-transduction by unfolding their comprising domains and extending them. It is widely thought that the process of sequential unfolding of polyproteins follows an exponential kinetics as the individual unfolding events exhibit identical and identically distributed (iid) Poisson behavior. However, it was shown that under high loads, the sequential unfolding kinetics displays nonexponential kinetics that alludes to aging by a subdiffusion process. Statistical order analysis of this kinetics indicated that the individual unfolding events are not iid, and cannot be defined as a Poisson (memoryless) process. Based on numerical simulations it was argued that this behavior becomes less pronounced with lowering the load, therefore it is to be expected that polyproteins unfolding under lower forces will follow a Poisson behavior. This expectation serves as the motivation of the current study, in which we investigate the effect of force lowering on the unfolding kinetics of Poly-L₈ under varying loads, specifically high (150, 100 ​pN) and moderate-low (45, 30, 20 ​pN) forces. We found that a hierarchy among the unfolding events still exists even under low loads, again resulting in nonexponential behavior. We observe that analyzing the dwell-time distributions with stretched-exponentials and power laws give rise to different phenomenological trends. Using statistical order analysis, we demonstrated that even under the lowest load, the sequential unfolding cannot be considered as iid, in accord with the power law distribution. Additional free energy analysis revealed the contribution of the unfolded segments elasticity that scales with the force on the overall one-dimensional contour of the energy landscape, but more importantly, it discloses the hierarchy within the activation barriers during sequential unfolding that account for the observed nonexponentiality.

Keywords: Correlations; Energy landscape; Nonexponential kinetics; Polyprotein; Single-molecule force-spectroscopy.

Graphical abstract

Fig. 1

Unfolding of Poly-L₈ under constant stretching forces. The upper left panel illustrates the unfolding traces measured in Force-Clamp AFM (left-side, blue frame) and MT (right-side, red frame) settings, with seven folded domains of the Poly-L₈ constructs portrayed by dark gray spheres, and an unfolded domain chain marked with its Δx extension. Exemplary unfolding traces displaying eight events under loads of 150 ​pN (dark blue), with arrows marking the unfolding dwell-time of the 8th event (Δt₈) and its corresponding extension (Δx₈), 100 ​pN (blue), 45 ​pN (dark red), 30 ​pN (red), and 20 ​pN (light red). (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)

Fig. 2

Unfolding dwell-time CDFs for (a) 150 ​pN, (b) 100 ​pN, (c) 45 ​pN, (d) 30 ​pN, (e) 20 ​pN, and (f) all the CDFs on the same timescale, demonstrating their stretch over several decades.

Fig. 3

Poly-L₈ unfolding dwell-time PDFs and their TPL fits for (a) 150 ​pN, (b) 100 ​pN, (c) 45 ​pN, (d) 30 ​pN, (e) 20 ​pN, and (f) all the PDFs on the same timescale, demonstrating their stretch over several decades (the PDFs are given by the empty symbols, and the fits with the lines).

Fig. 4

Force variation of the unfolding dwell-time parametrization obtained from the SE (from CDFs, empty circles) and the TPL (from PDF, empty triangles) approaches. (a) Characteristic times obtained from CDFs, PDFs, and dwell-time medians (purple diamonds). The error bars to the fitted τ are given by the standard deviation of the fitting numerical error, and the medians error bars are given by the IQRs according to the distribution of the data at each force. (b) Characteristic-time percent relative errors of the parametric estimations of τ with respect to the medians for the CDFs (SE, empty circles; Single Exponential, filled circles), and for the PDFs (TPL, empty triangles; Single Exponential, filled triangles). (c) Characteristic exponents obtained from the fittings of the SE (β_SE, empty circles) and TPL (α_TPL, empty triangles). (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)

Fig. 5

Individual unfolding events within poly-L₈ at different forces. (a) Schematics of the unfolding dwell-times by event. (b) Characteristic time-intervals τ(j, F) fitted for each event j-PDF (empty symbols) and for all-events PDFs, τ(F) (horizontal thick lines), with medians of each event j-PDF (colored symbols). (c) Fitted values of the SE exponents, β(j, F), obtained for each event j-CDF (empty symbols) and for all-events CDFs, β(F) (horizontal thick lines). (d) Fitted values of the TPL exponent, α(j, F), obtained for each event j-PDF (filled symbols) and for all-events PDFs, α(F) (horizontal thick lines).

Fig. 6

Spearman rank correlation coefficient matrices between unfolding dwell-times at each event at (a) 150 ​pN, (b) 100 ​pN, (c) 45 ​pN, (d) 30 ​pN, and (e) 20 ​pN. (f) Determinants of the correlation matrices for three measures of correlation. Here, unlike in the correlation matrices shown in (a)–(e), lower determinant value indicates higher correlation.

Fig. 7

Q–Q plots of forces unfolding dwell-times at separate event combinations (rows) with the applied forces (columns). The dwell-time quantiles of the j event, Q_j(Δt), versus the dwell-time quantiles of their previous event, Q_j-1(Δt) are plotted along the first row, versus preceding two events, Q_j-2(Δt) (second row), and versus preceding four events, Q_j-4(Δt) (third row).

Fig. 8

The effect of the applied force on Poly-L₈ free energy reflected by its PMF along its extension coordinate. (a) Reconstructed PMFs of Poly-L₈ under varying loads of 150, 100, 45, 30 and 20 ​pN. (b) Separated PMFs of Poly-L₈ at low forces. (c) Activation barrier heights, calculated from the unfolding dwell-times.