Rate limit of protein elastic response is tether dependent

Abstract

The elastic restoring force of tissues must be able to operate over the very wide range of loading rates experienced by living organisms. It is surprising that even the fastest events involving animal muscle tissues do not surpass a few hundred hertz. We propose that this limit is set in part by the elastic dynamics of tethered proteins extending and relaxing under a changing load. Here we study the elastic dynamics of tethered proteins using a fast force spectrometer with sub-millisecond time resolution, combined with Brownian and Molecular Dynamics simulations. We show that the act of tethering a polypeptide to an object, an inseparable part of protein elasticity in vivo and in experimental setups, greatly reduces the attempt frequency with which the protein samples its free energy. Indeed, our data shows that a tethered polypeptide can traverse its free-energy landscape with a surprisingly low effective diffusion coefficient D(eff) ~ 1,200 nm(2)/s. By contrast, our Molecular Dynamics simulations show that diffusion of an isolated protein under force occurs at D(eff) ~ 10(8) nm(2)/s. This discrepancy is attributed to the drag force caused by the tethering object. From the physiological time scales of tissue elasticity, we calculate that tethered elastic proteins equilibrate in vivo with D(eff) ~ 10(4)-10(6) nm(2)/s which is two to four orders magnitude smaller than the values measured for untethered proteins in bulk.

Fig. 1.

Experimental design for time resolved measurements of the recoil dynamics of an extended polypeptide. (A) Schematic diagram of the force-clamp apparatus. A single polypeptide is extended between a high-speed cantilever (Olympus biolever) and a piezoelectric actuator with a high resonant frequency (300 kHz, Physik Instrumente PL-055.30). A well-tuned PID amplifier adjusts the piezoelectric actuator so as to keep the force on the polypeptide at the set point value (F_sp). Under force clamp conditions, F_c = F_pol + F_D, where F_c is the force applied by the cantilever, F_pol is the elastic force produced by the molecule and F_D is the flow drag force (dashed lines) from the motion of the piezoelectric actuator. (B) Typical experimental trace of the end-to-end length corresponding to the force protocol on the lower box. The inset shows a staircase of ubiquitin unfolding events (stars) measured at 180 pN providing a strong fingerprint for a single polypeptide and measures its initial length. The extended polypeptide is then repeatedly cycled between 250 pN and 100 pN in order to collect many trajectories (7 < n < 30) until the molecule detaches. (C) Example of a recoil trajectory (blue) obtained by averaging eight consecutive recoil trajectories. The averaged force trace shows a step (250 pN down to 100 pN) with a half time of approximately 33 μs, which is more than 10 times shorter than the relaxation time constant of the polypeptide measured to be approximately 1.11 ms. After recoil, the corresponding extension (green trace) is much faster with a time constant of approximately 0.68 ms. The force traces are superimposable. (D) Potential of mean force of a 250 nm long polymer calculated at two pulling forces of 250 pN and 100 pN, using the WLC model of polymer elasticity (lower box). The figure highlights the physics of our experimental design. The polypeptide starts extended at 250 pN (1) and then it is abruptly relaxed to 100 pN (2) where it recoils down to its new minimum (3), then it is switched back to the high force (4) where it extends to the PMF minimum at 250 pN (1).

Fig. 2.

Time constants and diffusion coefficients from extension and recoil traces of single polypeptides. (A) A series of four different recoil (upper box) and extension (lower box) trajectories measured from extended polypeptides of different contour lengths L_C (as labeled; 113 nm, 263 nm, 344 nm, and 391 nm). The data were fit with a single exponential to measure the value of the relaxation time constant, τ, (dashed lines). (B) Relaxation (filled circles) and extension (filled triangles) time constants, τ, as a function of the contour length of the polypeptide L_c. The asymmetry of the potential shown in Fig. 1D, results into very different values of τ for the extension and recoil, as shown in Fig. 1B. The linear contour length dependency of τ is readily reproduced by Brownian dynamics simulations using the potential shown in Fig. 1D and the values of D_eff measured from these data (open circles and triangles). The linear dependency of D_eff on L_c results from that of the total travel (ΔL) on L_c (inset). (C) Histograms of values for D_eff measured from the recoil (top box) and extension (bottom box) traces shown in Fig. 2A. (D and E) MD simulations of the end-to-end length as a function of time for ubiquitin maintained at a constant force of 250 pN (D; black curve); relaxation after force is quenched from 250 pN to 100 pN (D; grey curves) and averaged over five such trajectories (D; blue dots). Even if each of the time-origins of each trajectory differs, we make them coincide in the plot for clarity. (E) Normalized recoil (green curve) and extension (blue curve) averaged over five trajectories. The inset shows the PMF along the end-to-end distance at 100 pN (gray curve) and 250 pN (black curve). Both curves are shifted so that their minimum is 0. (F) Plot of the measured values of D_eff (circles from recoils; triangles from extensions) as a function of the viscoelastic drag, μ_D, measured from each cantilever. The upper squares show the values of D_max = k_BT/μ_D, marking the upper limit of D measured for each cantilever. We attribute the difference between D_eff and D_max to the noninstantaneous force step of the force-clamp apparatus. The open circles show the values of D_eff measured from BD simulations that include the time course of the force-step for each experiment. The inset portrays a simplified mechanical representation of the principal contributions to the forces of an extending polypeptide during these experiments. The Voigt model representing the polypeptide (k_pol, μ_pol) is in parallel with the drag (μ_D) generated by the motion of the piezoelectric actuator, giving μ_eff = μ_pol + μ_D.

Fig. 3.

Bandwidth of an extended polypeptide. (A) Brownian dynamics computes the length of a 30 nm long polypeptide placed under a 10 pN sinusoidal load of varying frequency, f, using the measured value of D_pol = 1,300 nm²/s. The figure shows polypeptide responses at two different frequencies (25 Hz, and 250 Hz). (B) We use lock-in detection to measure the in-phase (ϕ₀, triangles) and the out-of phase (ϕ₉₀, circles) components of the elastic response of the polypeptide as a function of the load frequency, f. The solid lines (inset), correspond to the real (triangles) and imaginary (circles) parts of the mechanical impedance of a Kelvin-Voigt model (inset), fitted to the data. From ϕ₀ and ϕ₉₀ we calculate power as a function of frequency for the polypeptide (triangles). The solid line fits the Kelvin-Voigt model to the data returning values of D_pol = 1,313 nm²/s and k_pol = 1.31 pN/nm. The arrow shows the -3 dB point, giving a bandwidth of 66 Hz. These measurements show that the elasticity of a polypeptide-based material is band-limited to a few hundred Hertz, in good agreement with broadly observed animal behavior.

Fig. 4.

Effect of tethering on the dynamics of an extending polypeptide. The act of tethering a polypeptide to an object (top inset) is an essential step in constructing an elastic system and introduces drag (μ_D; bottom inset) in parallel with the molecule (k_pol, μ_pol). Polypeptide PMF’s calculated at F = 0 and F = 12 pN (25). Any motion of the polypeptide along its PMF will cause drag from the tethered objects, greatly affecting its kinetics both for simple diffusion (1) or for barrier crossing events that significantly change the end-to-end length of the molecule (2).