A theoretical model for the mechanical unfolding of repeat proteins

Abstract

We consider the mechanical stretching of a polypeptide chain formed by multiple interacting repeats. The folding thermodynamics and the interactions among the repeats are described by the Ising model. Unfolded repeats act as soft entropic springs, whereas folded repeats respond to a force as stiffer springs. We show that the resulting force-extension curve may exhibit a pronounced force maximum corresponding to the unfolding of the first repeat. This event is followed by the unfolding of the remaining repeats, which takes place at a lower force. As the protein extension is increased, the force-extension curve of a sufficiently long repeat protein displays a plateau, where the force remains nearly constant and the protein unfolds sequentially so that the number of unfolded repeats is proportional to the extension. Such a sequential mechanical unfolding mechanism is displayed even by the repeat proteins whose thermal denaturation is highly cooperative, provided that they are long enough. By contrast, the unfolding of short repeat progressions can be cooperative.

Figure 1

Illustration of the model used. The protein is described as a sequence of folded ($s_i=+1$) and unfolded ($s_i=0$) repeat units. The folded and unfolded repeats behave as Hookean springs with stiffnesses $k_{\mathrm{f}}$ and $k_{\mathrm{u}}$, respectively.

Figure 2

Force-extension curve of a repeat protein computed numerically and estimated using the approximations described in the first part of the Results section. The parameters of the model used are: H = 4, J = 8, N = 10, and $b_{\mathrm{u}}=10b_{\mathrm{f}}$. (a) Numerical data (solid line) versus the approximation using Eq. 20. Two horizontal lines show two different values of the force, F. For $F=F_1$, the equation $G\prime \left(z\right)=F$ has three solutions (three crossings of the horizontal line), and for $F=F_2$, there are two solutions. (b) Numerical data (solid line) compared with the force-extension curve obtained directly from Eq. 11 without invoking a continuous approximation for the number of folded repeats (dashed line).

Figure 3

Force-extension curve of a repeat protein with the same J + H but different J and H. Solid line, J = H = 6; dashed line, J = 0, H = 12; points, H = −15, J = 27. The number of repeats is N = 10 and $b_{\mathrm{u}}=10b_{\mathrm{f}}$.