Interplay between Viscoelasticity and Force Rate Affects Sequential Unfolding in Polyproteins Pulled at Constant Velocity

Polyproteins are unique constructs, comprised of folded protein domains in tandem and polymeric linkers. These macromolecules perform under biological stresses by modulating their response through partial unfolding and extending. Although these unfolding events are considered independent, a history dependence of forced unfolding within polyproteins was reported. Here we measure the unfolding of single poly(I91) octamers, complemented with Brownian dynamics simulations, displaying increasing hierarchy in unfolding-foces, accompanied by a decrease in the effective stiffness. This counters the existing understanding that relates stiffness with variations in domain size and probe stiffness, which is expected to reduce the unfolding forces with every consecutive unfolding event. We utilize a simple mechanistic viscoelastic model to show that two effects are combined within a sequential forced unfolding process: the viscoelastic properties of the growing linker chain lead to a hierarchy of the unfolding events, and force-rate application governs the unfolding kinetics.


Brownian dynamics (BD) simulations
Based on the procedure given in ref. [1], we performed BD simulations for a polyprotein comprised of a short linker and 8 folded domains in line with a cantilever. The evolution of the time dependent position for each one of the eight protein domains in the polyprotein is given by their length xp,N, where the index, N = 1 -8, denotes the domain number:  , D = 1500 nm/s is the value of the diffusion coefficient taken here, representing the drag inflicted by the viscous hydrodynamic forces acting on the cantilever tip, 2-4 kB = 1.38 . 10 -2 pN . nm/K is Boltzmann's constant, T = 300 K is the temperature, and KS = 25 pN/nm taken as the spring constant of the cantilever.
The thermal random force resulting from the fluctuations of each element with its surroundings, (t), is described by the fluctuation-dissipation theorem, with (t) = 0, and (t)(t + t) = (2/D)(kBT) 2 (t), where (t) is a delta function. The contribution of the linker between the protein domains is accounted by the time evolution of its length, xl: For a polyprotein of eight domains and a linker, the eight equations given by eq. (S1) and another by eq.
(S2), are coupled by a constant velocity constraint: is the initial extension of the linker, and xp,0 = Rc [nm] is the size of a folded protein domain. The polymeric elastic extensibility forces experienced by the linker (l), and each one of the protein domains (p) (once unfolding occurred), are approximated by the worm like chain (WLC) interpolation formula, 5,6 (S4) Here lp ~ 0.4 nm is taken as the persistence length of the polypeptide polymeric linker, LC,l0 = 14 nm is the contour length of the linker, and LC,l = 28 nm is the contour length of the unfolded protein domain.
Finally, the forces required to unfold each one of the protein domains are approximated with 1 4 where UP is the one-dimensional representation of a protein domain energy profile with U0 = 115 pN . nm ~ 28 kT is the free energy barrier height of a folded domain, Rc = 4 nm, is the size of a folded protein domain, and b = 12, is a curvature parameter in the folded domain potential. The total produces the recorded force restored by the polyprotein molecule: The time interval taken was Δt = 2 . 10 -7 s. The solution of the expressions above guaranteed that there is no particular hierarchy in the unfolding events, such that when the domains are similar (as the case here), the unfolding sequence will be random. 7 BD simulations were performed as specified above with pulling velocities varying between 40 -4,000 nm/s, spanning over three velocity decades that are still within characteristic experimental range. n = 100 traces were collected for each pulling velocity. Figure S1 shows three exemplary BD simulation FX traces, each comprised of eight "unfolding" events, at three different velocities, V = 40, 400, and 4,000 nm/s colored with light blue, purple, and black, respectively. This color coding will remain with the all the relevant calculations and parameters concerning these velocities throughout the manuscript.

Unfolding forces and stiffness from BD simulations
We collected the maximal (unfolding) forces form all the traces for each event, and calculated their probability distribution functions (pdfs). Figure S2A shows the force pdfs at the pulling velocities for the 1 st , 5 th and 8 th events, from which the first and second moments are evaluated. Figure S2B shows similar distributions calculated from poly-(I91) unfolding at V = 50, 100 and 1,000 nm/s for the 1 st , 3 rd and 7 th unfolding events.  Figure 3B shows the changes of stiffness with unfolding events. The stiffness values were estimated in as described in the main text, by taking the slope of the force with respect to the end-to-end extension in the linear regime before every unfolding event (as shown in red in Fig. S4A), and averaging these values with respect to the event number. Additionally, we estimated Keff using eq. (2) from the main text with respect to the parameters obtained from the simulations. Here, as with the experimental data shown in Fig. 1C in the main text, the stiffness decrease with every consecutive unfolding event, where, in general, Keff underestimates dF/dx at high velocities and overestimate it at the low velocity, for N > 4. The stiffness obtained from the BD simulation is extremely high for relatively short linkers (N < 4), and particularly high before the 1 st unfolding event. These values are very different from the ones observed experimentally, and result in part from the chosen functional form and parameters of the folded domains, and the short linker assumed in the BD simulations. Also, the experimental system experiences hydrodynamic drag effects that are highly prominent in the near vicinity of the surface to which the molecule is anchored 8-10 (and were not accounted for in the simulations). Comparing the behavior of dF/dx obtained from BD simulations to the one measured from poly-(I91) pulling experiments (Fig. 1C in the main text), a similar behavior is observed for N > 3, with the increase of the distance from the surface. Therefore, for the following, in our analysis of the BD simulations, we will disregard the 1 st event. can be intiuitively illustrated. Fig. S4A shows an unfodling FX trace of poly-(I91) measured at a pulling velocity of 800 nm/s (blue). The linker extensions prior to every unfolding event are fitted with the WLM moodel given by eq. (S4) (black curves). These fits are obtained at a constant separation distance, LC, between each unfolding event. For clarity we also colored in red the segment used for the evaluation of the experimental stiffness dF/dx on one of the peaks. In Fig. S4B we ploted the numerical derivatives of the specific WLC curves from Fig. S4A, and ploted red triangles at the extact position of the end-to-end extension at which every unfolding event occurred. This illustrate how constant increment that forms a longer linker chain becomes "softer", althouth the mean force required to unfold a domain somewhere linearly connected with it increase.

Variation of mean unfolding-forces stiffness and force-rates with pulling velocity at constant length
Maximal unfoldign forces were shown in the literature and in the main text to increase with the advancement of unfolding during the pulling experiments and simulatoins. 7,[11][12][13] In a typicall analysis of FX experiments, these values are avergaed per pulling velocity, and then plotted against the logarythm of the velocity. Here we are interested in the changes of the parameters monitored in this study (not only the mean unfolding forces) with the pulling velocity, with respect to their event number. While previously we dwelled on the efect of N, where V = const., here we monitor their trends where V varies and N is considered constant. Figure S6 shows an increase of the unfolding forces, averaged with respect to their unfolding event, with the applied pulling velocities for poly-(I91) (Fig. S6A)    with the applied pulling velocity, and the difference between (dF/dt)N and dF/dxVN that decreases as N grows. The behavior of the mean stiffness and force-rates, averaged over all the N events for both estimations is shown in Fig. 3 in the main text.

Zener (standard linear solid) model for the viscoelastic description of the linker
As mentioned above, and in the main text, the variation of the stiffness with the pulling velocity indicates the viscoelastic nature of the unfolding polyproteins. For this reason we chose to use the Zener model (illustrated in Fig. 4A in the main text) to describe the viscoelastic behavior of the extending linker as the polyprotein unfolds. The standard linear solid model, also known as the Zener model, is described in the Kelvin representation by two springs signifying the elastic moduli of the linker and probe (Kl and KS respectively), with a viscous term given by , the internal friction of the polymeric linker: where V = dx/dt, is the velocity in which the linker is pulled. Kl, describes the stiffness of the linker with its extension, 14 and it is expressed as a function of LC, and F. 15 We therefore take LC as the extension coordinate x, such that Kl(x, F), and continue under the high force approximation: where Kl = (/x)F 3/2 ,  = 4(lp/kBT) 1/2 according to eq. (2) in the main text. The overall stiffness of the system can be described by taking the derivative of the force with respect to the position: From eq.(S9) we can obtain an expression for at high extensions (x/LC → 1), which is described by the contour length of the chain, the stiffness of the probe and of the linker, but more importantly, with the overall stiffness of the system, dF/dx (which is a combination of KS and Kl, and can be directly measured by the slope of the linear regime in the FX curve before unfolding event), of the applied pulling velocity V and the force rare dF/dt: Figure S8 presents the relation between  and dF/dx as a function of N and as a function of V. For constant V,  shows an increase with dF/dx for poly-(I91) (Fig. S8A), and for the BD simulations (Fig.   S8C), while for constant N it exhibit the opposite trend (Figs. S8B and S8D). In both cases, the trend can be observed more explicitly in the BD simulations. When the polyprotein chain extends at constant pulling velocity, its stiffness reduces with every unfolding event, and with it the force-rate at which consecutive unfolding occurs. As the chain becomes more loose, it transfer momentum less efficiently, which is reflected through  (viscosity is a measure for momentum transfer, the higher the viscosity, them better momentum transfer is). This can also explain why higher forces are required to unfold the preceding domains. We see this in Figs S8A and S8C, as the internal friction (viscosity) increases with the overall stiffness of the polyprotein (or decrease when the stiffness decreases). When N is constant, i.e. a chain of fixed length is pulled at increasing velocities. The chain stiffness (mildly) grows with V, and with it the force rates (to larger extents) as shown in Fig. S7. The force-rates can be viewed as shearrates, who reduces viscosity in polymeric liquids (shear thinning). 16 Additionally, as discussed in the main text, the coupling with the term   V predicts the decrease of  as V increases (and the stiffness and force-rates with it).