An energetic model for macromolecules unfolding in stretching experiments

Abstract

We propose a simple approach, based on the minimization of the total (entropic plus unfolding) energy of a two-state system, to describe the unfolding of multi-domain macromolecules (proteins, silks, polysaccharides, nanopolymers). The model is fully analytical and enlightens the role of the different energetic components regulating the unfolding evolution. As an explicit example, we compare the analytical results with a titin atomic force microscopy stretch-induced unfolding experiment showing the ability of the model to quantitatively reproduce the experimental behaviour. In the thermodynamic limit, the sawtooth force-elongation unfolding curve degenerates to a constant force unfolding plateau.

Keywords: biopolymers; macromolecule mechanics; macromolecules unfolding; protein stability; titin.

Figure 1.

Scheme of the energetic decomposition of the external work into unfolding (dissipated) energy Q_i, i = 1,2,3,4 and elastic stored energy Φ_e (dashed region) for a typical unfolding force–elongation curve (continuous line) reproduced from [14]. Dashed lines represent approximating worm-like chain curves each characterized by a different contour length L_c(i), i = 1, … , 4; l_c represents the (fixed) contour length increase at each unfolding event.

Figure 2.

Force–strain curve (log-scale evidences the differences in the low force regime) for the WLC compared with the usual Marko & Siggia [49] approximation (M&S) and with the simplified model in (3.5) for l_p = 0.42 nm. Observe that the proposed approximation keeps the same asymptotic behaviour as l→l_c of (3.3) and that for F > 10⁻¹ pN the approximation is of the same order as that of [49].

Figure 3.

Scheme of the energy minimization: with bold line we represent stable (global energy minimum) solutions.

Figure 4.

Unfolding behaviour for a system of n = 6 initial folded domains. Here, we considered the parameters: l_o = 58 nm, l_c = 28.43 nm, l_p = 0.36 nm, Q = 770k_BT, ΔQ = 420k_BT. Each equilibrium path is labelled by the number n_u of unfolded domains.

Figure 5.

Unfolding energies as a function of n_u deduced from the following experiments: (a) AFM experiment on titin from [14]; (b) AFM experiment on TNfnAll protein from [60]; (c) AFM experiment on titin from [48]; (d) AFM experiment on tenascin-C from [54]; (e) AFM experiment on titin from [59].

Figure 6.

Comparison between the AFM experiment for the titin protein reproduced from [14] (continuous curves) and the force versus elongation curves deduced by (3.5), (3.7), (4.3) and (4.2) (bold lines). Here, we considered the parameters: l_o = 58 nm, l_c = 28.43 nm, l_p = 0.36 nm, Q = 770k_BT, ΔQ = 420k_BT.

Figure 7.

Unfolding behaviour of the lattice in the thermodynamic limit. Bold lines represent the loading path, whereas continuous lines the unloading paths at different values of the unfolded fraction ν_u and maximum end-to-end length. Here, we used the same parameters as figure 4.