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### The mechanical stability of the helix-heterotetramer

We developed a single-molecule assay using our magnetic-tweezer setup^{50,51,52,53} to directly quantify the loading rate-dependent rupture forces and the force-dependent lifetime of the helix-heterotetramers (Supplementary Notes 1–2). Briefly, the single-molecule construct essentially contains the complementary components of the helix-heterotetramer (hereafter referred to as a and b), linked by a long flexible unstructured peptide chain (L) (Fig. 1a left panel, and Supplementary Fig. 1a). The components a and b, each containing two helices a1, a2 and b1, b2, respectively, are expected to form two helix-hairpins by design^{49}. The resulting complex by a and b is, therefore, a helix-heterotetramer. When a and b are separated under force, the linker L keeps them in close vicinity to allow reformation of the complex by dropping the force. In addition, the long linker gives rise to a large extension difference between the separated and associated states of the two components that can be unambiguous distinguished from domain unfolding. Hence, the linker increases both the experimental throughput and the detection accuracy. Importantly, as the peptide linker has a very low bending persistence length, it does not introduce significant mechanical perturbation to the helix-heterotetramers (Supplementary Note 3). The linked a and b (a-L-b) in the construct is spanned between two repeats of the well-characterized titin I27 domain^{27} at each side, which serve as molecular spacers and specificity control. The N- and C- termini of the construct contain a biotin-avi-tag and a spy-tag, respectively, which enables specific tethering for the single-molecule force spectroscopy experiments (Supplementary Notes 1–2).

The linked complementary a and b helix-hairpins can form a helix-heterotetramer at low forces, looping the linker (referred to as the looped state). Rupture of the helix-heterotetramer under force leads to release of the linker, accompanied with a stepwise extension increase (referred to as the unlooped state). As the linker has >200 residues, this stepwise extension can be easily differentiated from unfolding of the 89 residues of I27 domain in the handle (Supplementary Note 4)^{31}. The looped and unlooped states can be tuned by force (Fig. 1a, right panel). Since the a and b components in the construct are linked in such a way where the force is applied to the N-terminus of the a1 helix in the a component and the C-terminus of the b2 helix in the b component, we refer the protein construct to as a1a2Lb1b2 (left-to-right: N-to-C-termini). The way of force attaching to the construct results in an unzipping force geometry when the helix-heterotetramer forms (Fig. 1a left panel, and Supplementary Fig. 1a); therefore, we refer to the helix-heterotetramer in this construct as ab^{zip}.

In typical single-molecule magnetic-tweezer experiments, a molecule is either subjected to a time-varying force or a constant force, referred to as force-loading or force-clamping experiments, respectively. In the former, the force at which the structural transition of the molecule occurs is recorded, which depends on the force-loading rate. In the latter, the time taken to the transition is recorded, which depends on the level of the applied force. We investigated the force-response of the 13-ab^{zip} under both force-loading and constant force constraints, where the prefix 13 indicates the #13 helix-heterotetramer in the reported list^{49}.

For the force-loading experiments, we held the construct at sufficiently low force to allow the formation of the looped state and then linearly increased the force with a loading rate, and recorded the rupture force at which unlooping occurred. By repeating the force-loading procedure for sufficient number of cycles, we obtained the distribution of the rupture forces of the 13-ab^{zip} helix-heterotetramer at given force-loading rates. We observed characteristic stepwise unfolding signals from the a and b components for the unlooped 13-ab^{zip} construct at forces within 9 pN at a loading rate of 1 pN s^{−1}, associated with step sizes ~12 nm (Supplementary Fig. 2 and Supplementary Note 5). This result suggests that the helix-forming motifs in a and b indeed form stable helix-hairpins, which further interact with each other to form the helix-heterotetramer. For the quantification of the force-dependent lifetime, after formation of the 13-ab^{zip} helix-heterotetramer at low forces, we directly jumped to different levels of forces and measured the time duration until unlooping occurred. By repeating the force-jumping procedure for a sufficient number of cycles at multiple force levels, we obtained the statistics of the force-dependent lifetime of the 13-ab^{zip} helix-heterotetramer.

Figure 1b shows the representative force-bead height curves of the a1a2Lb1b2-construct from ~1 to ~20 pN measured with a loading rate of 1 pN s^{−1}. Each colored curve represents one independent force-increase loading cycle. The abrupt stepwise bead height jump in each curve indicates the force-dependent rupturing of the 13-ab^{zip} helix-heterotetramer and the resulting unlooping. The rupturing of the 13-ab^{zip} helix-heterotetramer and the unfolding of the separated a and b helix-hairpins typically occurred concurrently because the unfolding forces of the a and b helix- hairpins (<9 pN) are smaller than the unlooping forces (Supplementary Fig. 2 and Supplementary Note 5). Here, we note that in such force-loading experiments, due to the small transition distance of I27 (~0.6 nm^{27}), I27 retains its low unfolding rate (~10^{−3} s^{−1}) over the scanned force range up to 50 pN^{31,54}. Hence, I27 unfolding was not observed in such force-loading experiments (Supplementary Figs. 3–5 and Supplementary Note 4).

Figure 1c shows force–step size graph of the force-dependent rupture transitions of the 13-ab^{zip} helix-heterotetramer at three loading rates of 0.2 pN s^{−1} (dark gray), 1 pN s^{−1} (blue) and 5 pN s^{−1} (red). The corresponding normalized distributions of the rupture forces shows Gaussian-like distributions with peak values of ~8, ~12, and ~15 pN, respectively (Fig. 1d). Figure 1e shows examples of the bead height time traces of the a1a2Lb1b2-construct at 9 pN during force-clamping experiments (more representative time traces are provided in Supplementary Fig. 6). The ~80 nm height increase steps are from the force-induced rupturing of 13-ab^{zip} helix-heterotetramer that causes unlooping and concurrent unfolding of the a and b helix-hairpins obtained from different force-clamping cycles. Figure 1f shows the force-dependent average lifetimes of the 13-ab^{zip} helix-heterotetramer at different forces obtained by fitting the histogram of the measured lifetime data to a single-exponential decay function (Supplementary Fig. 7). The data show that the 13-ab^{zip} helix-heterotetramer can withstand forces of 5–12 pN for seconds to minutes depending on the force level.

### Modulating the mechanical stability of the helix-bundles

Using the 13-ab^{zip} helix-heterotetramer as a starting building block, we explored the possibilities to modulate the mechanical stability of the helix-heterotetramer by manipulating the force geometry. Since molecular complexes under shear-force geometry typically have a stronger mechanical stability than that under the unzipping force geometry^{54}, we modified the design of the construct so that the force is applied to the N-terminus of the a2 helix in the a component and the C-terminus of the b2 helix in the b component, leading to a shear-force geometry on the resulting helix-heterotetramer (Fig. 2a and Supplementary Fig. 1b). We refer the modified construct to as 13-a2a1Lb1b2 and the resulting helix-heterotetramer to as 13-ab^{shear}.

Figure 2b shows the representative force–height curves of 13-ab^{shear} helix-heterotetramer during force-increase scans with a loading rate of 1 pN s^{−1}. Figure 2c shows the force–step size graph of the force-dependent rupture transitions of the 13-ab^{shear} helix-heterotetramer at three loading rates of 0.2 pN s^{−1} (dark gray), 1 pN s^{−1} (blue) and 5 pN s^{−1} (red). The corresponding normalized distributions of the rupture forces are shown in Fig. 2d. At all three loading rates, a two-peak distribution was obtained. We reason that the mechanically weaker species could be an intermediate, partially folded structure involving only two or three helices. A possible mechanism that may lead to such intermediate is that the more extended flexible linker in the helix-heterotatramer formed by 13-ab^{shear} than that by 13-ab^{zip} could sterically slow down the complete folding process, resulting in such partially folded intermediate. The mechanically stronger species, which is expected to be a helix-heterotetramer involving four helices, is the major form, occupying ~70 ± 5% of the events.

As the mechanically stronger major species could provide a more stable mechanical anchorage, we quantified the force-dependent lifetime of the 13-ab^{shear} helix-heterotetramer at forces greater than 10 pN (Fig. 2e and Supplementary Fig. 8). The helix-heterotetramer can withstand high forces (15–30 pN) over a time scale of seconds to minutes. For comparison, over the similar time scale the 13-ab^{zip} helix-heterotetramer can only withstand 5–12 pN. Altogether, these results indicate that greater mechanical stability of the helix-heterotetramer can be achieved by changing the geometry of the applied force from unzipping geometry to shearing geometry.

We also explored the possibility of modulating the mechanical stability by changing the number of interacting helices. This was done by creating a construct that contains only three out of the original four helices: the a2 helix in the a helix-hairpin and the two helices in the b helix-hairpin. This construct is referred to as 13-a2Lb1b2 (Fig. 2f and Supplementary Fig. 1c). In this construct, the b component forms a stable helix-hairpin structure (Supplementary Fig. 2 and Supplementary Note 5). We expect that the joining of the a2 peptide to the b helix-hairpin could lead to formation of a mechanically stable helix-heterotrimer, referred to as the 13-a^{half}b helix-heterotrimer, under a shear-force geometry.

From force-loading and force-clamping experiments, the three helices indeed form a stable complex, but with a weaker mechanical stability than either the 13-ab^{zip} or 13-ab^{shear} helix-heterotetramer (Fig. 2g–j and Supplementary Fig. 9). Its rupturing force distribution (Fig. 2i) is similar to the mechanically weaker species observed in Fig. 2d, consistent with the possibility that the mechanically weaker species of 13-ab^{shear} corresponds to an intermediate structure that involves three interacting helices. This result suggests that it is possible to modulate the mechanical stability via changing the number of interacting helices.

### The temperature-dependent mechanical stability

The ability of the helix-heterotetramers to retain their mechanical stability at higher temperature is important for its versatile applications at various temperatures. Hence, we quantified the mechanical stability of the ab^{zip} and ab^{shear} helix-heterotetramers at the human body temperature ~37 °C. For the 13-ab^{zip} and 13-ab^{shear} helix-heterotetramers, the force-loading rate-dependent rupture force distribution (Fig. 3a–b, d–e) and the force-dependent lifetimes (Fig. 3c, f and Supplementary Figs. 10 and 11) show moderate changes from the results obtained at 23 °C. Impressively, the helix-heterotetramer formed by 13-ab^{shear} retained significant mechanical stability at the highest temperature (~47 °C) tested in our experiments (Supplementary Fig. 12). These results suggest that the helix-heterotetramers still have sufficient mechanical stability at 37 °C which can be used over physiological temperature range.

### The formation of helix-heterotetramer is rapid and robust

The ability to rapidly form the helix-heterotetramer is also important for its various applications. Hence, we quantified the rate of the helix-heterotetramer formation. Briefly, we first kept the 13-a1a2Lb1b2-construct in the unlooped state at a sufficiently high force, followed by a force jump to a lower value of 1–3 pN, and then held the construct at the lower force for a time duration of (Delta t = 2,;5,;10, ldots) sec, to allow the potential formation of the 13-ab^{zip} helix-heterotetramer (Fig. 4a). If the 13-ab^{zip} helix-heterotetramer is formed during Δ*t*, the bead height will be shortened due to the looping of the linker (blue arrows in Fig. 4a). At forces >1 pN, the extension difference between the looped and the unlooped states is >10 nm. However, due to larger thermal motion of the bead at lower forces, the extension difference might not be visually clear enough at short duration of Δ*t* (Supplementary Fig. 13). Therefore, we added an additional force of ~9 pN to distinguish between the looped and the unlooped states of the construct. At this detecting force, the extension difference between two states is >70 nm (magenta arrow in Fig. 4a), which allows us to unambiguously determine the state of the construct prior to the force jump. By repeating the force-jumping assay over one hundred times from more than five tethers, we obtained the probability of the formation of the 13-ab^{zip} helix-heterotetramer at given lower forces over the waiting time Δ*t* (Fig. 4b).

The resulting time-dependent probability of forming the 13-ab^{zip} helix-heterotetramer at given forces show that at increased forces, the time taken for the looping probability to reach an equilibrium state increased. The looping probability can be fitted to ({mathrm{P}}left( {Delta t} right) = frac{{k_p}}{{k_p + k_u}}left( {1 – e^{ – (k_{mathrm{p}} + k_{mathrm{u}})Delta t}} right)), where *k*_{p} and *k*_{u} are the rates of looping (13-ab^{zip} helix-heterotetramer formation) and unlooping (13-ab^{zip} helix-heterotetramer rupturing), respectively, which can be determined by fitting to the experimentally measured time evolution of the looping probability (Fig. 4b and Supplementary Table 1).

The values show that *k*_{p} has a strong dependence on force, deceasing over 50 folds when force increased from 1.7 pN (~0.58 s^{−1}) to 2.9 pN (~0.01 s^{−1}). In contrast, *k*_{u} is much less sensitive to force over the tested force range, with a value of ~0.005–0.015 s^{−1} around the force range. The strong force dependence of the looping rate is consistent with a large transition distance as a result from the looping of the flexible linker to reach the transition state. Applying the Arrhenius rate equation, a zero-force looping rate is estimated to be ~15.7 s^{−1} based on the *k*_{p}(*F*) data points obtained at the four force values (Supplementary Table 1, Supplementary Notes 6–7, and Supplementary Fig. 14). Furthermore, the formation of the 13-ab^{zip} helix-heterotetramer is highly robust, as it can be formed over hundreds of cycles of looping and unlooping. Overall, these results collectively suggest that the 13-ab^{zip} helix-heterotetramer formation is rapid and robust.

To find whether other reported helix-heterotetramer systems have similar properties, we also quantified the #37 helix-heterotetramer under both unzipping (Supplementary Figs. 15a, 16–21) and shearing force (Supplementary Figs. 15b, 18, 19, 22–24) geometries. The results show that the mechanical stability of the #37 helix-heterotetramer, and its dependence on the force geometry and temperature, are similar to those of the #13 helix-heterotetramer. Therefore, the principle of modulating the mechanical stability of the complexes could be generally applied to all the helix-heteroteramers in the reported list^{49}.

### Helix-heterotetramer as a mechanically stable anchorage

We have shown that the #13 and #37 helix-heterotetramers can withstand a significant range of mechanical forces over physiological temperature range. The formation of the helix-heterotetramers is also rapid and robust. These properties make them appealing candidates to be used as mechanically stable anchorage/crosslinker over physiological temperature range. We demonstrate helix-heterotetramer system’s applications as a mechanically stable anchorage in single-molecule studies of force and temperature-dependent protein unfolding/refolding dynamics and protein stability.

In the first example, we used the #13 helix-heterotetramer to anchor a protein construct (bio-4I27-b) containing four repeats of titin I27 domains (Fig. 5a). In this construct, the I27 domains are spanned between an avi-biotin tag at N-terminus and the b helix-hairpin at the C-terminus. The C-terminus of bio-4I27-b construct is specifically tethered to the complementary a helix-hairpin immobilized on the bottom coverslip surface under the shear-force geometry (referred as split-13-ab^{shear}, Supplementary Note 2). The biotin-tagged N-terminus is tethered to a superparamagnetic bead via a 572 bp DNA handle (Fig. 5a, left). As previously mentioned, I27 has a low unfolding rate (~10^{−3} s^{−1}) within 50 pN^{27,54}. On the other hand, the 13-ab^{shear} helix-heterotetramer has a comparable unfolding rate at forces <10 pN at 23 °C (Fig. 2e). Therefore, at forces within 10 pN, the long lifetime of the 13-ab^{shear} helix-heterotetramer can potentially be utilized as a mechanically stable anchorage to investigate the mechanical stability of I27.

Figure 5b shows six representative time traces of the bead height obtained from six different tethers after a force jump from <1 pN, at which all domains are folded, to a constant force of ~8 pN at 23 °C. In each time trace, four stepwise increases of the bead height are observed, corresponding to unfolding of the four I27 repeats. The step sizes are distributed around 14.9 ± 1.0 nm (mean ± standard deviation), consistent with the release of ~89 residues of I27 into a disordered polypeptide (Supplementary Note 6 and Supplementary Figs. 4 and 5). Refolding of the unfolded domains was not observed at this force, because the force is greater than the equilibrium critical force of I27 (~5.4 pN) at 23^{ o}C as reported previously^{27,31}. At this force, the I27 domain has very slow unfolding rate, resulting in long experimental time duration of more than one and half hour for each experiment. This example demonstrates that the long lifetime of the split-13-ab^{shear} helix-heterotetramer under shear-force geometry over physiological force range, typically a few pN, provides a specific anchoring method to support long duration of single-molecule experiments.

After unfolding, I27 domain exists in a highly disordered state that carries a larger conformational entropy^{27}. Therefore, increased temperature is expected to decrease the domain stability. In order to probe the thermodynamic properties of I27, we increased the temperature to 37 °C and measured the equilibrium unfolding and folding transitions, which were used to calculate the folding energy of I27. Figure 5c is a representative time trace of the bead height obtained at ~3.5 pN showing reversible unfolding and folding transitions. From the trace, we obtained the probability distributions of bead height during dynamic unfolding and refolding of the four I27 domains. The probability of having *n* unfolded I27 domains in four independent I27 repeats, *P*_{4}(*n*), can be directly read out from the bead height distribution peaks (Fig. 5c, right panel). It follows the binomial distribution: (P_4(n) = C_4^np^n(1 – p)^{4 – n}), where (C_4^n = frac{{4!}}{{n!left( {4 – n} right)!}}) is the binomial coefficient. The single-free parameter *p* denotes the probability of an I27 domain in the unfolded state at this force.

Fitting of the binomial distribution to the normalized probability of the number of unfolded I27 repeats, we determined that *p*(*F*) ~ 0.27 at 3.5 pN at 37^{ o}C (Fig. 5c). The value of *p*(*F*) is related to the force-dependent free energy difference between the unfolded and folded states, Δ*G*(*F*), by (Delta Gleft( F right) = Delta G_0 + Delta phi left( F right) = – k_{mathrm{B}}T{mathrm{ln}}frac{{1 – p(F)}}{{p(F)}}). Here, (Delta G_0) is the zero-force folding energy of the I27 domain, (Delta phi left( F right) = mathop {smallint }limits_0^F (x_{mathrm{u}}(f) – x_{mathrm{f}}(f)){mathrm{d}}f) is the force-dependent conformational free energy difference between the unfolded state and the folded state^{27,31}. Using the equation and based on the measured *p*(*F*) and force-extension curves of I27 in the folded and unfolded states (Supplementary Note 6), we obtained (Delta G_0 = – 4.9 pm 0.4;k_{mathrm{B}}T). The critical force (F_csim 3.9) pN at which the unfolded and folded states have equal probabilities, is also obtained by (Delta G_0 + Delta phi left( {F_c} right) = 0). Comparing these values with those obtained at 23 °C^{27,31} (Table 1), it clearly shows that when temperature increases from 23 to 37 °C, (Delta G_0) of I27 increases significantly by more than 3 *k*_{B}T (i.e., thermodynamic stability decreases) and correspondingly the equilibrium critical force decreases by ~1.5 pN.

In the above example, the I27 domain has a typical β-sheet structure. In order to probe the thermodynamic properties of protein domains with different ternary structures, such as α-helix bundles, we created another protein construct (bio-6SR-b) that contains six repeats of the fourth α-actinin spectrin-repeat (SR) domain^{55}, spanned between an avi-biotin tag at N-terminus and the b helix-hairpin of the #13 helix-heterotetramer system at the C-terminus. Each SR domain is formed with three α-helices bundled together. Figure 6a shows representative time traces of the bead height obtained from six independent tethers, each containing six stepwise domain unfolding steps. The step sizes are distributed around ~17.9 ± 1.0 nm, consistent with the release of ~104 residues of SR4 into a disordered polypeptide (Supplementary Note 6 and Supplementary Figs. 4 and 5). Using similar approach to quantify the temperature-dependent thermal stability of I27, we quantified the temperature-dependent thermal stability of the spectrin-repeat based on equilibrium two-state transitions over a temperature range from 23 °C to 31 °C (Fig. 6b–d and Table 1), and revealed that the stability of the spectrin-repeat is negatively influenced by increased temperatures.

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