Transient-state binding kinetics

Experiments designed to study ligand association and dissociation processes in the (often brief) period between introducing the ligand to the population of target proteins and attainment of equilibrium binding are referred to as transient-state kinetics studies. While it can be relatively straightforward to quantify the degree of occupation of a population of target proteins by a ligand under equilibrium binding conditions by employing steady-state kinetic approaches, quantifying changes occurring in a more dynamic system pre-steady-state can be much more challenging, particularly when equilibrium binding is achieved within extremely short time periods that may be shorter than one second. The example below illustrates how k_on and k_off may be determined in a system that reaches equilibrium extremely rapidly, making use of specialised equipment and an approach known as stopped-flow spectrophotometry. The calculation and graphing procedures are appropriate regardless of the experimental approach taken to obtain data.

It is necessary that formation of a complex between a ligand molecule and a target protein creates a measurable signal that is not present when the target is unoccupied. In this example, binding of a ligand to a target protein results in a shift in an absorbance maximum for the protein, from 450 nm to around 490 nm. It is preferable that the signal results from of a change in the protein’s absorbance rather than in the ligand’s absorbance, as the ligand will generally be present at much higher concentrations than the protein and it may be difficult to observe a change in absorbance characteristics of a tiny fraction of ligand bound to protein over what may be a substantial background signal in relative terms. It is also preferable to measure a gain in signal over a low protein background absorbance rather than a loss of signal from a high initial absorbance measurement. In this example, measuring a decrease in absorbance at 450 nm or an increase in absorbance at 490 nm following addition of ligand to the protein target will provide information about the rate and extent of formation of the complex, and measuring the increase at 490 nm is preferred. The target protein in this example is a solubilised protein that has been over-expressed in a cell culture system and purified prior to experimentation.

The figure below shows a stopped-flow spectrophotometer manufactured by Applied Photophysics, similar to an instrument in the author’s laboratory.

The protein and ligand are loaded into two separate gas-tight syringes, and on command from the control software, a hydraulic ram pushes equal volumes from the two syringes through a mixing chamber and into an optical cell, and measurement of absorbance begins within a millisecond or so of the ligand and target protein being mixed together. A rapid-scanning optical unit is able to measure a full absorbance spectrum of the mixture every few milliseconds. The figures below illustrate several absorbance spectra (left) obtained over a 1 second time period, with the data being shown instead as difference spectra relative to the initial spectrum of the protein with no ligand present (right). It is apparent that the changes in absorbance are virtually complete within 50 milliseconds, and thus that equilibrium binding is reached within a little over 50 milliseconds.

Note that spectra are shown to illustrate the changes in absorbance that are taking place when ligand binds to the target protein. The information that will be replotted to obtain k_obs values are the absorbance values at 450 nm, or those at 490 nm, versus time. However it would be sufficient simply to measure the decrease in absorbance that occurs at 450 nm, or the increase in absorbance that occurs at 490 nm, over the 1 second measurement period. The plot obtained would show an exponential curve with a rate constant of k_obs. The figure below (left) shows three exponential curves illustrating the increase in absorbance at 490 nm when three different ligand concentrations were mixed with the buffer solution containing the protein targets. Each of the exponential curves yields a rate constant, k_obs, and the values are indicated on the figure. Note that as the ligand concentration increases, the values for k_obs also increase, meaning that the half-lives associated with these exponential curves (equal to 0.693/k_obs) will become progressively shorter as ligand concentration increases. The height of the plateau – the fraction of targets occupied when equilibrium binding is reached – also increases as ligand concentration increases. Plotted versus [ligand], these plateau values for equilibrium binding would yield a binding hyperbola from which the equilibrium dissociation constant (K_D) could be obtained.

The figure above (right) shows the three values for k_obs obtained from the raw data on the left, plotted against the concentration of ligand at which the data were obtained. While the values for k_obs shown were obtained by fitting of the data to an equation for exponential association by nonlinear regression, manual determination of the half-life for each of the curves would allow estimation of k_obs from k_obs = 0.693/t_½. The observed rate constant, k_obs, is impacted both by k_on and by k_off, with the relationship, k_obs = k_on × [L] + k_off, representing an equation for a straight line. It is then a straightforward matter to determine the slope and Y-intercept to obtain values for k_on and k_off, respectively.

This indirect approach to obtaining values for k_on and k_off is necessary because since neither association nor dissociation can exist in isolation, it is extremely difficult to measure either association rate or dissociation rate directly in the absence of some interference from the opposing process.

Stopped-flow measurements are commonly used to establish k_on and k_off, and thus K_D, for an enzyme substrate. A functional enzyme assay which measures the rate of formation of product as substrate is metabolised yields a hyperbola from which the substrate’s K_M can be determined in a manner analogous to the determination of K_D from a binding curve. However, contrary to popular opinion, the K_M value does not generally provide information about the substrate’s binding affinity for the enzyme. Rather, it is the substrate concentration at which product is formed at half of the maximum possible rate, and this is impacted by k_cat, the catalytic rate constant, which indicates how many product molecules can be generated by a single enzyme molecule working at V_max in unit time.