Maxwell relations for single-DNA experiments: Monitoring protein binding and double-helix torque with force-extension measurements

Single-DNA stretching and twisting experiments provide a sensitive means to detect binding of proteins, via detection of their modification of DNA mechanical properties. However, it is often difficult or impossible to determine the numbers of proteins bound in such experiments, especially when the proteins interact nonspecifically (bind stably at any sequence position) with DNA. Here we discuss how analogs of the Maxwell relations of classical thermodynamics may be defined and used to determine changes in numbers of bound proteins, from measurements of extension as a function of bulk protein concentration. We include DNA twisting in our analysis, which allows us to show how changes in torque along single DNA molecules may be determined from measurements of extension as a function of DNA linking number. We focus on relations relevant to common experimental situations (e.g., magnetic and optical tweezers with or without controlled torque or linking number). The relation of our results to Gibbs adsorption is discussed.

Figure 1

Cartoon of DNA molecule held at constant force and torque (a). When a DNA-binding protein is introduced at some finite concentration (b), it will bind and change the average extension and linking number of the molecule.

Figure 2

Behavior of (a) extension versus force, (b) protein occupation versus force, and (c) the mixed derivatives for the simple model analyzed in Sec. 2C. Parameters are chosen for a protein that contracts its DNA substrate against applied force when it binds. Successively lower extensions in (a), larger values of protein occupancy in (b), or rightward-shifted peak functions in (c) correspond to successively larger values of $\mu$ (see text for details).