The load dependence of rate constants

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.