^{1,a)}, A. Imparato

^{2,b)}and A. Pelizzola

^{1,c)}

### Abstract

We study the mechanical unfolding pathways of the domain of fibronectin by means of an Ising-like model, using both constant force and constant velocity protocols. At high forces and high velocities our results are consistent with experiments and previous computational studies. Moreover, the simplicity of the model allows us to probe the biologically relevant low force regime, where we predict the existence of two intermediates with very close elongations. The unfolding pathway is characterized by stochastic transitions between these two intermediates.

A.I. gratefully acknowledges financial support from Lundbeck Fonden and from Danish Centre for Scientific Computing (DCSC). A.I. is grateful to Anders Irbäck and to Simon Mitternacht for helpful and stimulating discussions.

I. INTRODUCTION

II. MODEL AND METHODS

A. Model

B. Model parameters, simulation, and analysis

III. EQUILIBRIUM PROPERTIES

IV. UNFOLDING PATHWAYS

A. Force clamp

B. Constant velocity

V. CONCLUSIONS

### Key Topics

- Atomic force microscopy
- 12.0
- Proteins
- 12.0
- Monte Carlo methods
- 11.0
- Acids
- 7.0
- Free energy
- 5.0

## Figures

Sketch of the native structure of (Protein Data Bank ID 1ttf) with -strands labeled A–G in sequence order. Figure generated by PyMOL.

Sketch of the native structure of (Protein Data Bank ID 1ttf) with -strands labeled A–G in sequence order. Figure generated by PyMOL.

Free energy landscape at temperature and for forces (red line), (green line), and (blue line). .

Free energy landscape at temperature and for forces (red line), (green line), and (blue line). .

Mean unfolding time as a function of the force applied to the molecule (average over 100 different trajectories). The red line is a fit to Arrhenius’ law in the range of forces from 25 to 60 pN. In this range we find from the fit . The green line is a fit from 60 to 115 pN, .

Mean unfolding time as a function of the force applied to the molecule (average over 100 different trajectories). The red line is a fit to Arrhenius’ law in the range of forces from 25 to 60 pN. In this range we find from the fit . The green line is a fit from 60 to 115 pN, .

Unfolding pathways scheme of pulled by a constant force. Transitions denoted by red arrows have been observed only at low forces (40, 36, and 28 pN). Oblique red arrows represent refolding transitions.

Unfolding pathways scheme of pulled by a constant force. Transitions denoted by red arrows have been observed only at low forces (40, 36, and 28 pN). Oblique red arrows represent refolding transitions.

Typical MC trajectories: end-to-end length (red line) and a few order parameters as functions of time, with a . Green line: fraction of native contacts, whole . Blue line: fraction of native contacts between strands G and F. Purple line: fraction of native contacts between strands A and B. Cyan line: fraction of native contacts between strands C and F.

Typical MC trajectories: end-to-end length (red line) and a few order parameters as functions of time, with a . Green line: fraction of native contacts, whole . Blue line: fraction of native contacts between strands G and F. Purple line: fraction of native contacts between strands A and B. Cyan line: fraction of native contacts between strands C and F.

Mixed AG-GF trajectory: MC time evolution of the end-to-end length (red line) and of some order parameters with a constant force of 28 pN. Colors as in Fig. 5

Mixed AG-GF trajectory: MC time evolution of the end-to-end length (red line) and of some order parameters with a constant force of 28 pN. Colors as in Fig. 5

Histograms of the intermediate life times for AG pathway (red line) and GF pathway (green line) at force . Data obtained from 3600 different trajectories. The lines are exponential fits.

Histograms of the intermediate life times for AG pathway (red line) and GF pathway (green line) at force . Data obtained from 3600 different trajectories. The lines are exponential fits.

Histograms of the unfolding times at force . Data obtained from 5000 different trajectories and the bin size of the histogram is 1. The fit is to a lognormal distribution.

Histograms of the unfolding times at force . Data obtained from 5000 different trajectories and the bin size of the histogram is 1. The fit is to a lognormal distribution.

Unfolding pathways scheme of pulled at constant velocity. Intermediate states in the full square boxes have a rupture force remarkably higher than those in dashed boxes.

Unfolding pathways scheme of pulled at constant velocity. Intermediate states in the full square boxes have a rupture force remarkably higher than those in dashed boxes.

MC time evolution of the end-to-end length (red line) and of a few order parameters with a constant velocity of . Green line: weighted fraction of native contacts, whole . Blue line: weighted fraction of native contacts between strands G and F. Purple line: weighted fraction of native contacts between strands A and B. Cyan line: weighted fraction of native contacts between strands C and F.

MC time evolution of the end-to-end length (red line) and of a few order parameters with a constant velocity of . Green line: weighted fraction of native contacts, whole . Blue line: weighted fraction of native contacts between strands G and F. Purple line: weighted fraction of native contacts between strands A and B. Cyan line: weighted fraction of native contacts between strands C and F.

MC time evolution of end-to-end length (red line), force (yellow line), and of some order parameters with a constant velocity of for a mixed AG-GF trajectory. Green line: weighted fraction of native contacts, whole . Blue line: weighted fraction of native contacts between strands G and F. Purple line: weighted fraction of native contacts between strands A and B. Cyan line: weighted fraction of native contacts between strands C and F. Bins of 2000 MC steps have been used to reduce fluctuations in the plot.

MC time evolution of end-to-end length (red line), force (yellow line), and of some order parameters with a constant velocity of for a mixed AG-GF trajectory. Green line: weighted fraction of native contacts, whole . Blue line: weighted fraction of native contacts between strands G and F. Purple line: weighted fraction of native contacts between strands A and B. Cyan line: weighted fraction of native contacts between strands C and F. Bins of 2000 MC steps have been used to reduce fluctuations in the plot.

Distributions of the rupture forces of the native state at pulling velocity . Data obtained from 500 different trajectories; bin size of the histogram is 2. The fit is to Eq. (9).

Distributions of the rupture forces of the native state at pulling velocity . Data obtained from 500 different trajectories; bin size of the histogram is 2. The fit is to Eq. (9).

## Tables

Unfolding time , waiting phase life time , AG intermediate life time , and GF intermediate life time at different constant forces. Values are in MC steps and are approximated averages on 100 different trajectories at each force.

Unfolding time , waiting phase life time , AG intermediate life time , and GF intermediate life time at different constant forces. Values are in MC steps and are approximated averages on 100 different trajectories at each force.

Relative frequencies of unfolding pathways at constant force. 100 trajectories for each value of the force.

Relative frequencies of unfolding pathways at constant force. 100 trajectories for each value of the force.

Relative frequencies of unfolding pathways. 100 trajectories for each value of the velocity.

Relative frequencies of unfolding pathways. 100 trajectories for each value of the velocity.

Average rupture forces.

Average rupture forces.

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