^{1,a)}, A. Pelizzola

^{1,b)}and M. Zamparo

^{2,c)}

### Abstract

A simple model, recently introduced as a generalization of the Wako-Saitô model of protein folding, is used to investigate the properties of widely studied molecules under external forces. The equilibrium properties of the model proteins, together with their energy landscape, are studied on the basis of the exact solution of the model. Afterwards, the kinetic response of the molecules to a force is considered, discussing both force clamp and dynamic loading protocols and showing that theoretical expectations are verified. The kinetic parameters characterizing the protein unfolding are evaluated by using computer simulations and agree nicely with experimental results, when these are available. Finally, the extended Jarzynski equality is exploited to investigate the possibility of reconstructing the free energy landscape of proteins with pulling experiments.

The authors thank A. Szabo for interesting discussions and J. Klafter for his interest in our work.

I. INTRODUCTION

II. THE MODEL

III. EQUILIBRIUM PROPERTIES

IV. FORCE CLAMP

V. DYNAMIC LOADING

VI. EVALUATING THE FREE ENERGY LANDSCAPE FROM PULLING EXPERIMENTS

VII. CONCLUSIONS

## Figures

Cartoon of the model protein, with a force applied to one of the free ends. Dots denote amino acids and dashed lines denote contacts.

Cartoon of the model protein, with a force applied to one of the free ends. Dots denote amino acids and dashed lines denote contacts.

Panel (a): Average order parameter as a function of the external force , for different molecules: 1TIT (full line), 1COA (dotted line), 1I6C (dashed line), and 1BBL (dashed-dotted line). The temperature value is taken to be . Panel (b): Root mean square length of the same molecules as a function of the external force , with .

Panel (a): Average order parameter as a function of the external force , for different molecules: 1TIT (full line), 1COA (dotted line), 1I6C (dashed line), and 1BBL (dashed-dotted line). The temperature value is taken to be . Panel (b): Root mean square length of the same molecules as a function of the external force , with .

Phase diagram of 1TIT (full line), 1COA (dotted line), 1I6C (dashed line), and 1BBL (dashed-dotted line). For each molecule, the curves are defined by . The lower-left region of the phase diagram corresponds to folded molecules, and the upper-right region corresponds to unfolded molecules.

Phase diagram of 1TIT (full line), 1COA (dotted line), 1I6C (dashed line), and 1BBL (dashed-dotted line). For each molecule, the curves are defined by . The lower-left region of the phase diagram corresponds to folded molecules, and the upper-right region corresponds to unfolded molecules.

Panel (a): Free energy landscape as a function of the molecular elongation , at , for different molecules: 1TIT (full line), 1COA (dotted line), 1I6C (dashed line), and 1BBL (dashed-dotted line). Panel (b): Tilted free energy landscapes , with .

Panel (a): Free energy landscape as a function of the molecular elongation , at , for different molecules: 1TIT (full line), 1COA (dotted line), 1I6C (dashed line), and 1BBL (dashed-dotted line). Panel (b): Tilted free energy landscapes , with .

Mean unfolding time as a function of the force, at , for four different molecules: boxes, 1I6C; triangles, 1BBL; circles, 1COA; and inset, 1TIT.

Mean unfolding time as a function of the force, at , for four different molecules: boxes, 1I6C; triangles, 1BBL; circles, 1COA; and inset, 1TIT.

Histograms of the unfolding time for the 1TIT molecule in a force clamp, at with forces (main figure) and (inset). The lines in the main figure are fits of the data to a log-normal function (full line) and to the function (9) (line points). The line in the inset corresponds to a fit to a negative exponential function.

Histograms of the unfolding time for the 1TIT molecule in a force clamp, at with forces (main figure) and (inset). The lines in the main figure are fits of the data to a log-normal function (full line) and to the function (9) (line points). The line in the inset corresponds to a fit to a negative exponential function.

Panel (a): Mean first passage time at for the 1I6C molecule and for different temperatures (in K), as obtained by direct computer simulations (points) and by Eq. (10) (lines). Panel (b): Free energy landscape as a function of the molecule length , at , for different values of the force (in pN). Inset: Free energy landscape at .

Panel (a): Mean first passage time at for the 1I6C molecule and for different temperatures (in K), as obtained by direct computer simulations (points) and by Eq. (10) (lines). Panel (b): Free energy landscape as a function of the molecule length , at , for different values of the force (in pN). Inset: Free energy landscape at .

Panel (a): Plot of the typical unbinding force of the 1TIT molecule as a function of the pulling velocity , for the three values of the temperature. The lines are fits to the data in the linear regime defined by Eq. (11). From such fits one can obtain the characteristic unfolding length . We find for , for , and for . Panel (b): The lines are fits of the data to Eq. (12).

Panel (a): Plot of the typical unbinding force of the 1TIT molecule as a function of the pulling velocity , for the three values of the temperature. The lines are fits to the data in the linear regime defined by Eq. (11). From such fits one can obtain the characteristic unfolding length . We find for , for , and for . Panel (b): The lines are fits of the data to Eq. (12).

Distribution of unfolding force under dynamical loading for the 1TIT molecule for (a) and (b) , with . The dotted line is a fit to Eq. (13).

Distribution of unfolding force under dynamical loading for the 1TIT molecule for (a) and (b) , with . The dotted line is a fit to Eq. (13).

Unfolding length as a function of , as obtained from fits of the unfolding force probability distribution to Eq. (13), with , for the (a) 1COA and (b) 1TIT molecules.

Unfolding length as a function of , as obtained from fits of the unfolding force probability distribution to Eq. (13), with , for the (a) 1COA and (b) 1TIT molecules.

Reconstructed free energy landscape of the PIN1 (a) and the 1BBL (b) for . The lines correspond to the exact result. The pulling rate is expressed in units of .

Reconstructed free energy landscape of the PIN1 (a) and the 1BBL (b) for . The lines correspond to the exact result. The pulling rate is expressed in units of .

## Tables

Width and height of the energy barrier separating the two minima of the free energy , with and , for the molecules considered here.

Width and height of the energy barrier separating the two minima of the free energy , with and , for the molecules considered here.

Unfolding length , as given from linear fits to Eq. (8), for the molecules considered in this paper.

Unfolding length , as given from linear fits to Eq. (8), for the molecules considered in this paper.

Unfolding length , as given from linear fits to Eq. (11), for the molecules considered in this paper.

Unfolding length , as given from linear fits to Eq. (11), for the molecules considered in this paper.

Unfolding length , barrier height , and characteristic exponent , as given from fits of the unfolding data with dynamic loading technique to Eq. (12).

Unfolding length , barrier height , and characteristic exponent , as given from fits of the unfolding data with dynamic loading technique to Eq. (12).

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