The load dependence of rate constants

Abstract

As experimental techniques in biophysics have progressed at the single molecule level, there has been considerable interest in understanding how external mechanical influences (such as load) affect chemical reactions. The majority of biophysical studies investigating load-dependent kinetics use an equation where the rate constant exponentially depends on force, which is sometimes called Bell's equation. This equation requires the determination of two parameters that describe the potential energy-strain function: k(0), which is the reaction rate in the absence of load, and x(c), which is the difference in strain between the reactant and transition states. However, there have been theoretical studies based on Kramers' theory suggesting that the rate constant should have load-dependent pre-exponential terms and nonlinear load-dependent terms in the exponential. Kramers' theory requires an exact knowledge of the potential energy-strain function, which is in general not known for an experimental system. Here, we derive a general approximation of Kramers' theory where the potential energy-strain function is described by five parameters, which can, for small loads, be reduced to four-, three-, and finally to two parameters (Bell's equation). We then use an idealized physical system to validate our approximations to Kramers' theory and show how they can predict parameters of interest (such as k(0) and x(c)) better than Bell's equation. Finally, we show previously published experimental data that are not well fitted by Bell's equation but are adequately fitted by these more exact equations.

Observing the osmophobic effect in action at the single molecule level

“…Not only is the size of the activation energy of the transition state measurable by SMFS experiments, but also a direct measure of the position of the transition state along the pulling coordinate can be obtained 16, 17. Consequently, from such experiments it is possible to gain geometric information about the transition state and appreciate its putative movements along the reaction coordinate in response to different conditions 16, 17–28…”

“…The unfolding kinetics at different velocities were well described by Bell's equation which expresses the dependence of the unfolding rate on the force applied $k_{\rm u} \left( F \right) = k_{\rm u} ^0 {\rm exp}\left[ {F{\rm }\Delta x_{\rm u} /\left( {K_{\rm b} T} \right)} \right]$ as an approximation to Kramers' reaction rate theory25 where k is the spontaneous unfolding rate in absence of applied force, Δ x u is the unfolding distance separating the folded state and the unfolding transition barrier measured along the reaction coordinate, T is the temperature in Kelvin and K b is Boltzmann's constant. Note that the spontaneous dissociation rate can be expressed as where A is a prefactor and Δ G u is the height of the unfolding activation energy barrier.…”

Condensation of actin filaments pushing against a barrier

“…Note that γ itself could be a function of the force; however, in the following we assume that it is just a constant. More elaborate treatments of the load dependence of the transition rates can be found in, for instance, [25]. An essential feature of this model is that although multiple filaments interact with the barrier, when a monomer is added to one of the filaments in contact, it must do work against the entire load.…”

Phosphate and ADP Differently Inhibit Coordinated Smooth Muscle Myosin Groups

“…The rate of the z2 nm step is increased or lowered by assisting or retarding mechanical loads, respectively (18,21). We assumed that the rate of the z4 nm mechanical step depended on external load in the same way (22)(23)(24). Note that myosin was thiophosphorylated; dephosphorylation and phosphorylation were not treated in our model.…”