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. 2009 Dec 1;106(48):20288-93.
doi: 10.1073/pnas.0905764106. Epub 2009 Nov 13.

Refolding dynamics of stretched biopolymers upon force quench

Changbong Hyeon  1 , Greg Morrison, David L Pincus, D Thirumalai
Affiliations

Refolding dynamics of stretched biopolymers upon force quench

Changbong Hyeon et al. Proc Natl Acad Sci U S A. .

Abstract

Single-molecule force spectroscopy methods can be used to generate folding trajectories of biopolymers from arbitrary regions of the folding landscape. We illustrate the complexity of the folding kinetics and generic aspects of the collapse of RNA and proteins upon force quench by using simulations of an RNA hairpin and theory based on the de Gennes model for homopolymer collapse. The folding time, tau(F), depends asymmetrically on deltaf(S) = f (S) - f (m) and deltaf (Q) = f (m) - f (Q) where f (S) (f (Q)) is the stretch (quench) force and f (m) is the transition midforce of the RNA hairpin. In accord with experiments, the relaxation kinetics of the molecular extension, R(t), occurs in three stages: A rapid initial decrease in the extension is followed by a plateau and finally, an abrupt reduction in R(t) occurs as the native state is approached. The duration of the plateau increases as lambda = tau (Q)/tau (F) decreases (where tau (Q) is the time in which the force is reduced from f (S) to f (Q)). Variations in the mechanisms of force-quench relaxation as lambda is altered are reflected in the experimentally measurable time-dependent entropy, which is computed directly from the folding trajectories. An analytical solution of the de Gennes model under tension reproduces the multistage stage kinetics in R(t). The prediction that the initial stages of collapse should also be a generic feature of polymers is validated by simulation of the kinetics of toroid (globule) formation in semiflexible (flexible) homopolymers in poor solvents upon quenching the force from a fully stretched state. Our findings give a unified explanation for multiple disparate experimental observations of protein folding.

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Conflict of interest statement

The authors declare no conflict of interest.

Figures

Fig. 1.
Fig. 1.
Force-quench kinetics of P5GA hairpins. (A) Refolding from a fixed f S to varying f Q (left) or from varying f S to a fixed f Q (right). (B) τF versus f Q starting from different f S. A fit to the Bell equation gives ΔxU→N‡≈(0.6−0.7) nm. The extrapolated value τF o to zero quench force is plotted in C (* symbols). (C) τF versus f S for each value of f Q. The variations of τF with f S are fit to Eq. 1. From the fit, we find (Δx UN/nm, τ(f Q)/μs) = (1.5,89.7), (2.5, 126), (1.8, 150), (2.6, 224), (3.8, 357), (4.4, 622) for f Q = 0, 2, 3.5, 7, 10, 12 pN, respectively. (D) Time traces of the molecular extension, R(t), upon force quench. Multiple time traces are plotted in gray, and the averaged time trace is shown by the thick black line. A few representative time traces are in color. The ensemble of time traces are shown for f S = 70 pN and f Q = 3.5 pN.
Fig. 2.
Fig. 2.
Analysis of the folding trajectories associated with force quench for various values of (f S,f Q). (A) f S = 56 pN → pN. (B) f S = 14 pN → pN. (C) f S = 56 pN → pN. (D) f S = 14 pN → pN. (E) f S = 56 pN → pN. P ERE(R) and P TSE(R) are shown in the right-hand frames of A–E, and the corresponding structures are shown for the ERE (left) and TSE (right). Thermally denatured ensemble P TDE(R) for T ≫ T f is shown in A.
Fig. 3.
Fig. 3.
The time evolution of the entropy, measured by using Eq. 2. (A) S(t)/k B from f S = 56 pN to f Q = 12 pN (red) and 2 pN (blue). (B) S(t)/k B from f S = 14 pN to f Q = 12 pN (red) and 2 pN (blue). (C) The time evolution of S(t)/k B for varying f S with fixed f Q = 12 pN. In this frame, we take the running average of S(t)/k B every 4.7 μ s.
Fig. 4.
Fig. 4.
The expanding sausage model for force-quench dynamics. (A) The free energy profile as a function of R for increasing tension. The free energy profile F(R) is plotted in dimensionless form by using F―(R/L)=FτcηL3=2(RL)1/2+f―pR24L2(3−2R/L1−R/L)−f―Q(RL). Two stable, free energy minima occur at R = 0 and R = R 0(f Q) ≲ L when f―Q>f―Q*. The profiles show that R TS decreases and R 0 increases with increasing f. In B–D, the dynamics of R(t) (left) and the average refolding time 〈τ〉 = τF (right) are shown for varying f―Q, f―S, and chain flexibility (f―p=kBT/lp), respectively.
Fig. 5.
Fig. 5.
Polymer collapse in a poor solvent upon force quench. (A) The collapse of a WLC in a poor solvent. (B) Toroidal, single racquet, or multiple racquet structures are formed in various trajectories. (C) The entropy production using Eq. 2, displaying the low → high → low pathway from high f S to low f Q. (D) Linear reduction in force over a period, τQ, during which the free energy gradually changes from a profile with f > f m to one with f < f m. Depending on the time scale for hopping between the native and unfolded basins of attraction, τhop, in comparison with τQ, the pattern of relaxation dynamics upon force quench can be greatly affected.
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References

    1. Thirumalai D, Hyeon C. RNA and protein folding: Common themes and variations. Biochemistry. 2005;44:4957–4970. - PubMed
    1. Onuchic JN, Wolynes PG. Theory of protein folding. Curr Opin Struct Biol. 2004;14:70–75. - PubMed
    1. Dill KA, Ozkan SB, Shell MS, Weikl TR. The protein folding problem. Annu Rev Biophys. 2008;37:289–316. - PMC - PubMed
    1. Shakhnovich E. Protein folding thermodynamics and dynamics: Where physics, chemistry, and biology meet. Chem Rev. 2006;106:1559–1588. - PMC - PubMed
    1. Thirumalai D. From minimal models to real proteins: Time scales for protein folding kinetics. J Physique I. 1995;5:1457–1467.

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