^{1}and Olga K. Dudko

^{1,a)}

### Abstract

The unfolding of a biomolecule by stretching force is commonly treated theoretically as one-dimensional dynamics along the reaction coordinate coincident with the direction of pulling. Here we explore a situation, particularly relevant to complex biomolecules, when the pulling direction alone is not an adequate reaction coordinate for the unfolding or rupture process. We show that in this case the system can respond to pulling force in unusual ways. Our theory points out a remarkably simple, but largely overlooked, mechanism of the complex responses of biomolecules to force. The mechanism originates from the basic property of the transition state to change its structure under applied force. A relationship is established between a key experimental observable—force-dependent lifetime—and the microscopic properties of the biomolecule in the form of an analytical solution to the problem of a force-induced molecular transition in two dimensions. The theory is applicable to biological contexts ranging from protein folding to ligand–receptor interactions.

We are grateful to Attila Szabo for insightful discussions. This work was supported by the NSF CAREER Award No. MCB-0845099 (to O.K.D.) and the NSF Center for Theoretical Biological Physics Grant No. PHY-0822283. O.K.D. is the Hellman Family Foundation Faculty Fellow.

I. INTRODUCTION

II. MODEL

III. FORCE-DEPENDENT LIFETIME ON THE 2D LANDSCAPE: ANALYTICAL SOLUTION

IV. UNUSUAL LIFETIME VERSUS FORCE DEPENDENCE: WHAT IS THE MECHANISM?

V. FORCE-DEPENDENT LIFETIME ON THE 2D LANDSCAPE: A 1D DESCRIPTION

VI. WHEN CAN THE 2D RUPTURE DYNAMICS BE ADEQUATELY VIEWED AS A 1D PROCESS ALONG Q?

VII. CONCLUSIONS

### Key Topics

- Biomolecules
- 20.0
- Free energy
- 15.0
- Diffusion
- 12.0
- Dissociation
- 10.0
- Molecular dissociation
- 9.0

## Figures

Single-molecule unfolding/rupture as the diffusive crossing of a barrier on a free energy landscape in the space of two coordinates: the pulling coordinate *x* and a slow degree of freedom *Q*. Left: Minimal 2D landscape *G* _{0}(*x*,*Q*) with a single dissociation path that connects the bound state and the free state via a single barrier. Right: Several representative scenarios for the force-dependent lifetime that can be realized on the landscape depicted on the left. Symbols: lifetimes from Brownian dynamics simulations; lines: lifetimes from the 2D analytical theory in Eq. (2) with parameter values listed in Table I.

Single-molecule unfolding/rupture as the diffusive crossing of a barrier on a free energy landscape in the space of two coordinates: the pulling coordinate *x* and a slow degree of freedom *Q*. Left: Minimal 2D landscape *G* _{0}(*x*,*Q*) with a single dissociation path that connects the bound state and the free state via a single barrier. Right: Several representative scenarios for the force-dependent lifetime that can be realized on the landscape depicted on the left. Symbols: lifetimes from Brownian dynamics simulations; lines: lifetimes from the 2D analytical theory in Eq. (2) with parameter values listed in Table I.

The rollover scenario (also shown in Fig. 1, curve B) as an illustration of the simple mechanism of the complex response of a biomolecule to force. (A) Lifetime as a function of stretching force from the analytical solutions to the 2D and 1D problems, Eqs. (2) and (3), respectively. Inset: Activation barrier as a function of force. The barrier grows with the force initially, resulting in a prolonged lifetime. High enough force lowers the barrier, shortening the lifetime. A crossover between the two regimes occurs at a force of 48 pN. (B) Comparison of the 2D and 1D solutions for the lifetime, Eqs. (2) and (3), respectively, reported in the form of the ratio *τ* _{1D}/*τ* _{2D}. Inset: topological features of the landscape that determine the difference between *τ* _{2D} and *τ* _{1D}; see also Eq. (4) and Fig. 4. (C**)** Snapshots of the landscape at four values of force corresponding to the open symbols in Fig. 2(a). Deformation of the reaction pathway (black curve) in the *x*-*Q* plane is caused by a shift of the barrier (■) with respect to the native state (•) as force is increased. Cartoons of a protein (depicted as a coil) and a ligand-receptor complex (depicted as a pair of hooks) provide a molecular interpretation of the mechanism of the rollover. Because the end-to-end distance of the molecule in the transition state is initially shorter than that in the folded state, a weak stretching force counteracts the intrinsic mechanism of unfolding. At a force past the rollover force (48 pN), the situation reverses.

The rollover scenario (also shown in Fig. 1, curve B) as an illustration of the simple mechanism of the complex response of a biomolecule to force. (A) Lifetime as a function of stretching force from the analytical solutions to the 2D and 1D problems, Eqs. (2) and (3), respectively. Inset: Activation barrier as a function of force. The barrier grows with the force initially, resulting in a prolonged lifetime. High enough force lowers the barrier, shortening the lifetime. A crossover between the two regimes occurs at a force of 48 pN. (B) Comparison of the 2D and 1D solutions for the lifetime, Eqs. (2) and (3), respectively, reported in the form of the ratio *τ* _{1D}/*τ* _{2D}. Inset: topological features of the landscape that determine the difference between *τ* _{2D} and *τ* _{1D}; see also Eq. (4) and Fig. 4. (C**)** Snapshots of the landscape at four values of force corresponding to the open symbols in Fig. 2(a). Deformation of the reaction pathway (black curve) in the *x*-*Q* plane is caused by a shift of the barrier (■) with respect to the native state (•) as force is increased. Cartoons of a protein (depicted as a coil) and a ligand-receptor complex (depicted as a pair of hooks) provide a molecular interpretation of the mechanism of the rollover. Because the end-to-end distance of the molecule in the transition state is initially shorter than that in the folded state, a weak stretching force counteracts the intrinsic mechanism of unfolding. At a force past the rollover force (48 pN), the situation reverses.

Three different models intended to explain a nonmonotonic dependence of the lifetime on force, τ(*F*). Models A (Refs. 17–19) and B (Refs. 17, 20, and 21, and 22) assume a discrete switch between two competing pathways, one favored at low force (upper panel) and the other at high force (lower panel). Model C (this work, Refs. 7 and 19) reveals an alternative, implicitly simpler mechanism of the nonmonotonic τ(*F*): a single pathway is continuously distorted by the force such that the molecular transition (unfolding or rupture) is counteracted by low force (upper panel) and favored by high force (lower panel). Because Model C accounts for the natural movement of the transition state with force, it allows nonmonotonic scenarios of τ(*F*) to be realized for dynamics as simple as that on a landscape with a single transition state.

Three different models intended to explain a nonmonotonic dependence of the lifetime on force, τ(*F*). Models A (Refs. 17–19) and B (Refs. 17, 20, and 21, and 22) assume a discrete switch between two competing pathways, one favored at low force (upper panel) and the other at high force (lower panel). Model C (this work, Refs. 7 and 19) reveals an alternative, implicitly simpler mechanism of the nonmonotonic τ(*F*): a single pathway is continuously distorted by the force such that the molecular transition (unfolding or rupture) is counteracted by low force (upper panel) and favored by high force (lower panel). Because Model C accounts for the natural movement of the transition state with force, it allows nonmonotonic scenarios of τ(*F*) to be realized for dynamics as simple as that on a landscape with a single transition state.

The nature of the response of a biomolecule to force is determined both by the topological features of the landscape (A and B) and by the diffusive properties of motion on this landscape. (A) Slope of the dissociation pathway with respect to the *Q*-direction in the saddle region of the landscape; upper: small slope, lower: large slope. (B) Ratio |λ_{∩}(*F*)|/*k*(*Q* _{∩}(*F*)) of the curvatures of the landscape in the *Q*- and *x*-directions, in the saddle region (see text for the precise definition of λ_{∩}). Figure 2(b), inset, shows how these features evolve with the force in the rollover scenario. (C) “Phase diagram” summarizing how the landscape topology (see A and B) and diffusion anisotropy *D* _{ Q }/*D* _{ x } affect the rupture kinetics on the 2D landscape depicted in Fig. 1. When the ratio τ_{1D}/*τ* _{2D} approaches 1, force-induced molecular transition on the 2D landscape can be adequately described as a 1D diffusion process on the potential of mean force along the coordinate *Q*.

The nature of the response of a biomolecule to force is determined both by the topological features of the landscape (A and B) and by the diffusive properties of motion on this landscape. (A) Slope of the dissociation pathway with respect to the *Q*-direction in the saddle region of the landscape; upper: small slope, lower: large slope. (B) Ratio |λ_{∩}(*F*)|/*k*(*Q* _{∩}(*F*)) of the curvatures of the landscape in the *Q*- and *x*-directions, in the saddle region (see text for the precise definition of λ_{∩}). Figure 2(b), inset, shows how these features evolve with the force in the rollover scenario. (C) “Phase diagram” summarizing how the landscape topology (see A and B) and diffusion anisotropy *D* _{ Q }/*D* _{ x } affect the rupture kinetics on the 2D landscape depicted in Fig. 1. When the ratio τ_{1D}/*τ* _{2D} approaches 1, force-induced molecular transition on the 2D landscape can be adequately described as a 1D diffusion process on the potential of mean force along the coordinate *Q*.

## Tables

Parameter values used in simulations and theory in Figs. 1 and 2. Diffusion coefficients *D* _{ Q } and *D* _{ x } = 10*D* _{ Q } were chosen so that the lifetime at zero force calculated from Langer theory, **,** was 10^{3} s, with Δ*Q*‡ = 0.5. For concreteness, the *Q*-coordinate of the transition state at zero force, *Q* _{∩}(*F* = 0), was taken to be 0.5, which corresponds to a shift of the landscape by 0.5 in the *Q*-direction; however, the resulting dynamics does not depend on a particular choice of *Q* _{∩}(*F* = 0).

Parameter values used in simulations and theory in Figs. 1 and 2. Diffusion coefficients *D* _{ Q } and *D* _{ x } = 10*D* _{ Q } were chosen so that the lifetime at zero force calculated from Langer theory, **,** was 10^{3} s, with Δ*Q*‡ = 0.5. For concreteness, the *Q*-coordinate of the transition state at zero force, *Q* _{∩}(*F* = 0), was taken to be 0.5, which corresponds to a shift of the landscape by 0.5 in the *Q*-direction; however, the resulting dynamics does not depend on a particular choice of *Q* _{∩}(*F* = 0).

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