Spontaneous Rate Constant fcj and Unfolding Distance

While the kinetics of a reaction is pathway-dependent, thermodynamic properties such as the free energy of unfolding only depend on the initial and final stages of the process, ttus, the question arises as whether the "mechanical stability" of a protein is a thermodynamic or a kinetic property. If mechanical stability were a property of the equilibrium it should depend only on the state variable that describes the system, i.e., the end-to-end extension of the molecule, tte dependence of the force on the pulling speed reveals that mechanical unfolding is a kinetic process and it holds the key to extracting kinetic information from these experiments.

What then is the effect of a mechanical force on the kinetics of the protein unfolding reaction? tte effect of force on the lifetime of a bond was first addressed for intermolecular bonds by Bell (1978); Evans and Ritchie then provided a more detailed interpretation in 1997. ttis formulation is directly applicable to the case of intramolecular bonds since the forces stabilizing both types of interactions are the same (Sect. 8.2.2). Still, there is an important difference between both processes as forced unfolding of a protein can be reversed and, therefore, refolding cannot be neglected as it is rebinding, in the case of force unbinding.

From basic classical mechanics, these authors developed a phenomenological procedure to obtain intrinsic rate constants from pulling experiments. Based on conventional transition state theory, the mechanical unfolding reaction is described by a simple two-state kinetics model in which the protein adopts only a native (folded) or a denatured (unfolded) state (Fig. 8.6c, panel 1), with a single high-energy transition (or activation) state ($). In this model the process is dominated by the thermal fluctuations of the protein and it is characterized by the relative height of the activation energy barrier (AG*) and the distance to the barrier, i.e., the position of the transition state along the mechanical reaction coordinate, also dubbed the width of the folding potential (Axu; Fig. 8.6c, panel 2). As with any denaturant, proteins unfold under force because the free-energy barrier (transition state) to unfolding is lowered, in comparison to the native state, tte force-dependent rate constants (fcu, unfolding rate, and fcf, folding rate) are given in this treatment by

where A is the frequency factor and Axu and Axf are the distances along the reaction coordinate (end-to-end length) from the folded state to the transition state and from the unfolded state to the transition state, respectively; fcg is the Boltzmann constant and T the absolute temperature, ttus, the application of a force Fu on a protein increases its unfolding rate (by reducing the activation energy of unfolding by FuAxu) but decreases its refolding rate. A force applied at an angle 8 to the reaction coordinate x adds a mechanical potential -(Fcos6)x, which tilts the reaction landscape and lowers the barrier (Fig. 8.6c, panel 2). tte rate constant of the unfolding process therefore depends exponentially on the force applied and the strain induced (deformation). Ms constant reflects the stochastic transition from the folded to unfolded states and thus it determines the time required to cross the barrier.

We can calculate a probability density for unfolding, which predicts the most likely force of unfolding in terms of the spontaneous unfolding rate constant as follows:

Using an analytical solution of this type, we should in principle be able to calculate the kinetic parameters for the process, fc° (spontaneous rate of unfolding) and Axu (width of the activation energy barrier, i.e., the distance on the reaction coordinate over which the force must be applied to reach the transition state). However, although pulling at a fixed loading rate (i.e., constant force, Eq. 8.5) is experimentally feasible (see later), most protein unfolding experiments have been carried out at a constant length. In these experiments the loading rate varies as the molecule is stretched because the stiffness of the molecule depends on the applied force; hence, this equation cannot be used directly. Instead, fc° and Axu are typically determined using Monte Carlo simulations that mimic the stochastic nature of the thermally driven unfolding of a protein molecule stretched at a constant rate (Oberhauser et al. 1998; Rief et al. 1998a; Carrion-Vazquez et al. 1999a). ttis procedure models the extensibility of the protein with an entropic elasticity, such as that described by the WLC, and a force-dependent all-or-none (two-state model) unfolding of its individual modules, tte unfolding of the modules is described by Eyring rate theory with a simple two-state Markovian model and a force-dependent rate constant. In this description the probability of observing unfolding of any module at each step is where N is the number of folded modules, ku is the unfolding rate constant, and Af is the polling interval (Rief et al. 1998a). tte simulation is performed for a number of pulling speeds using different combinations of fc° and Axu until they replicate the observed speed dependence (Fig. 8.6b; Carrion-Vazquez et al. 1999a). To make this method more accurate, the values of fc° and Axu must also fit the force histogram simultaneously (Fig. 8.5b). An alternative method has recentlybeen proposed that compares mutant and wild-type data to calculate these parameters (Best et al. 2002).

P=NkuAt

tte mechanical kinetics parameters (i.e., fc° and Axu) offer a useful description of the energy landscape of a protein under force that, although simplistic, allows us to compare the mechanical strength between different proteins (Sect. 8.4).

Finally, it has recently been suggested that the Eyring kinetic model may not be the most appropriate theory to describe this process (Howard 2001; Bustamante et al. 2004). In the Eyring model, the transition state is considered similar to the initial state but, in principle, this is only applicable when covalent bonds are made or broken. In contrast, Kramers rate theory may better apply to the typical protein conformational changes, in which a large number of bonds are made or broken (reviewed by Bieri and Kiefaaber 2000).

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