Unveiling the influence of device stiffness in single macromolecule unfolding

Abstract

Single-molecule stretching experiments on DNA, RNA, and other biological macromolecules opened up the possibility of an impressive progress in many fields of life and medical sciences. The reliability of such experiments may be crucially limited by the possibility of determining the influence of the apparatus on the experimental outputs. Here we deduce a model that let us analytically evaluate such influence, fundamental for the interpretation of Single Molecule Force Spectroscopy experiments and intermolecular interactions phenomena. As we show, our model is coherent with previous numerical results and quantitively reproduce AFM experimental tests on titin macromolecules and P-selectin with variable probe stiffnesses.

Figure 1

Scheme of an SFMS experiment for the analysis of the unfolding of a protein molecule constituted by modules undergoing a folded/unfolded transition here modeled through a two wells potential energy assumption.

Figure 2

Plot of $\tilde{V}$ as a function of δ and of $\tilde{F}$ as a function of both δ and $\overline{\epsilon }$ at zero temperature for a chain with N = 5 elements and ε_u = 1: γ = 0.8 (a–c) and γ = 0.3 (d–f).

Figure 3

Plot of hysteresis cycles at zero temperature for a chain with N = 5 elements and ε_u = 1: γ = 0.8 (a) and γ = 0.3 (b). See the text for details about the sequence of loading-unloading path.

Figure 4

(a) Force-strain curves for different values of γ (and k_d). Here N = 5, ε_u = 1, T = 300 K, l = 30 nm and ${\overline{k}}_{\exp }=k_p/Nl=4\text{pN}/\text{nm}$. The curves are obtained for ${\overline{k}}_d=k_d/(\alpha L)=1,5,10,20\mathrm{pN}/\mathrm{nm}$ corresponding to $\gamma \simeq 0.2,0.56,0.71,0.83$, respectively (see SM). (b) $\langle \overline{\epsilon }\rangle$–δ curves for the same parameters of (a).

Figure 5

Force-strain curve with variable number of elements N = 5,20,100. Left: fixed element length l = 30 nm and ${\overline{k}}_p=k_p/l=4\mathrm{pN}/\mathrm{nm}$. Right: fixed total length L = Nl = 150 nm and ${\overline{k}}_{\exp }=k_p/Nl=4\text{pN}/\text{nm}$. Larger values of N correspond to smaller amplitude of the oscillations. Here ε_u = 1, T = 300 K and ${\overline{k}}_d=k_d/(\alpha L)=20\mathrm{pN}/\mathrm{nm}$.

Figure 6

Force-strain curve showing the hardening effect of local maxima of F increasing with $\langle \overline{\epsilon }\rangle$. Here N = 10, ε_u = 1, T = 300 K (left), T = 600 K (right), l = 30 nm and ${\overline{k}}_{\exp }=k_p/Nl=0.4\mathrm{pN}/\mathrm{nm},{\overline{k}}_d=k_d/(\alpha L)=0.1\mathrm{pN}/\mathrm{nm}$ (giving γ = 0.2).

Figure 7

Comparison of the force-strain curves obtained in the thermodynamical limit (monotonic curve) and for N = 100 (chainsaw curve). Here ε_u = 1, T = 300 K, l = 30 nm, ${\overline{k}}_p=k_p/l=4\mathrm{pN}/\mathrm{nm},\gamma =0.7$.

Figure 8

Comparison of the the force-strain curves obtained from the model with the experimental results (see). We have fixed N = 5, ε_u = 1, T = 300 K, l = 24 nm and ${\overline{k}}_{\exp }=k_p/Nl=4\mathrm{pN}/\mathrm{nm},{\overline{k}}_d=k_d/(\alpha L)=20\mathrm{pN}/\mathrm{nm}$ (corresponding to $\gamma \simeq 0.86$).

Figure 9

Dependence of the rupture force on the spring constant of a (very soft) transducer compared with experimental results (see). We have fixed N = 1, ε_u = 35, T = 300 K, l = 40 nm and ${\overline{k}}_p=1\mathrm{pN}/\mathrm{nm},{\overline{k}}_d=k_d/(\alpha L)$ varying from 1 × 10⁻³ pN/nm to 40 × 10⁻³ pN/nm. Different symbols correspond to different loading rates.

Figure 10

Comparison of the the force-strain curves obtained the wlc model (dashed lines) with experimental results (top, see and Fig. 8) and the results form our model (bottom, same parameters of Fig. 8). The parameters of the WLC are consistent with those reported in: persistence length l_p = 0.4 nm, increase of the contour length l_c = 28 nm and device stiffness ${\overline{k}}_d\mathrm{=20}\mathrm{pN}/\mathrm{nm}$.