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. 2012 Sep 4;109(36):14410-5.
doi: 10.1073/pnas.1202952109. Epub 2012 Aug 20.

Weak temporal signals can synchronize and accelerate the transition dynamics of biopolymers under tension

Won Kyu Kim  1 , Changbong Hyeon, Wokyung Sung
Affiliations

Weak temporal signals can synchronize and accelerate the transition dynamics of biopolymers under tension

Won Kyu Kim et al. Proc Natl Acad Sci U S A. .

Abstract

In addition to thermal noise, which is essential to promote conformational transitions in biopolymers, the cellular environment is replete with a spectrum of athermal fluctuations that are produced from a plethora of active processes. To understand the effect of athermal noise on biological processes, we studied how a small oscillatory force affects the thermally induced folding and unfolding transition of an RNA hairpin, whose response to constant tension had been investigated extensively in both theory and experiments. Strikingly, our molecular simulations performed under overdamped condition show that even at a high (low) tension that renders the hairpin (un)folding improbable, a weak external oscillatory force at a certain frequency can synchronously enhance the transition dynamics of RNA hairpin and increase the mean transition rate. Furthermore, the RNA dynamics can still discriminate a signal with resonance frequency even when the signal is mixed among other signals with nonresonant frequencies. In fact, our computational demonstration of thermally induced resonance in RNA hairpin dynamics is a direct realization of the phenomena called stochastic resonance and resonant activation. Our study, amenable to experimental tests using optical tweezers, is of great significance to the folding of biopolymers in vivo that are subject to the broad spectrum of cellular noises.

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Conflict of interest statement

The authors declare no conflict of interest.

Figures

Fig. 1.
Fig. 1.
Thermodynamics and kinetics of P5GA hairpin under tension. (A) Free energy profile as a function of molecular extension z under tension f, obtained from the molecular simulation using SOP model. Shown are the three free energy profiles at f = 13.0, 14.7, and 17.0 pN. At f = fm = 14.7 pN, the probability of being in UBA and NBA are equal. The profile is tilted towards UBA (or NBA) when f = 17.0 > fm (or f = 13.0 < fm). (B) Unfolding and folding probability PU(f) and PF(f)[ = 1 - PU(f)] of P5GA against tension f. Results from the SOP model [PF(f) and PU(f)] and two state model (solid curves) are in good agreement. formula image [or formula image] calculated from the tilted 1D free energy profile interpolated from results of the SOP model displays a good agreement with other results. (C) Mean folding time [τF(f)] and unfolding time [τU(f)] at varying tensions, obtained from the SOP simulation (circles),formula image and formula image obtained from the 1D simulation on the tilted free energy profile (squares), and mean first passage times (formula image, formula image calculated using the tilted free energy profile (solidcurves) are in a good agreement. The transition midforce determined at the intersection between τF(f) and τU(f) and is fm = 14.7 pN.
Fig. 2.
Fig. 2.
Demonstration of SR using simulation of P5GA hairpin under an oscillating force. (A) Structural ensemble of P5GA hairpin at NBA and UBA shown on the free energy profile F(z; f = 17 pN). Time trajectories of molecular extension z(t) at constant tension f = 17 pN are shown from the simulation of SOP model when a small time dependent oscillatory force δf sin(2πt/TΩ) (δf = 1.4 pN) is added with (B) TΩ( = 2π/Ω) = 1.3 ms, (C) 10.2 ms, and (D) 82 ms. Coherent hopping transition between folding and unfolding is found with TΩ = TSR ≈ 10.2 ms, which clearly shows the SR. The oscillatory forces with TΩ = 10.2 ms and 82 ms are overlaid on the time trajectory in C and D, respectively, to emphasize the effect of synchronization. See Fig. S1 for the SR of P5GA hairpin in the presence of handles.
Fig. 3.
Fig. 3.
Resonant activation (RA) in folding and unfolding transitions of P5GA. (A) Mean folding time τF of P5GA under f = 17 pN as a function of TΩ with δf = 1.4 pN (filled squares) and δf = 2.0 pN (filled circles). The transition times are minimized at TΩ = TRA ≃ 2.56 ms. (B) Mean unfolding time τU of P5GA under f = 13 pN as a function of TΩ with δf = 1.0 pN (empty squares) and δf = 1.4 pN (empty circles). The transition times are minimized at TΩ = TRA ≃ 2.56 ms.
Fig. 4.
Fig. 4.
SR in the folding and unfolding transitions of P5GA under time dependent tension f + δf sin(Ωt). (A) Time trajectories and (B) folding time distributions pfold(t) with f = 17 pN and δf = 1.4 pN at TΩ = 0.16 ms, 10.24 ms, and 40.96 ms are shown. The synchronization (SR condition) occurs around formula image, as shown in (A- 2). The inset in (B- 2) depicts the power spectrum S(ν), showing the sharpest peak at ν = 1/TSR ≈ 0.1 ms-1. (C) Mean folding time τF and measure of coherence P1 as a function of TΩ. (D) Mean unfolding time τU and measure of coherence P1 as a function of TΩ with f = 13 pN and δf = 1.4 pN. Note that the SR and RA conditions are made at different TΩs in C and D.
Fig. 5.
Fig. 5.
Filtering of SR condition. Shown are the pfold(t) of P5GA under formula image at varying combinations of δfi with Ω1 = 2π/0.1 ms-1, Ω2 = 2π/10 ms-1, Ω3 = 2π/100 ms-1, and f = 17 pN. For filled symbols, we fix δf2 = 1.4 pN, corresponding to the amplitude for the optimal frequency Ω2 for SR, and vary the other amplitudes to be δf1 = δf3 = 1.4 pN (filled squares), 1.7 pN (filled circles), and 2.0 pN (filled triangles), respectively. The starred symbols represent the cases where δf1, δf2 and δf3 are taken randomly between 0 and 2 pN every time step to make a nonequilibrium noise δf(t). Notably, the folding transition filters the optimal driving period despite other drivings with larger or transient amplitudes.
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