Variate-covariate models

Consider a titration experiment to determine the dissociation constant of binding partners A and B. The total concentration of B, \(c_\mathrm{B}^0\) is held constant while the total concentration of A, \(c_\mathrm{A}^0\), is varied. The equilibrium concentration of the duplex AB, \(c_\mathrm{AB}\), is measured for each value of \(c_\mathrm{A}^0\). For the reaction AB ⇌ A + B with dissociation constant \(K_d\), we can solve for the concentration of the AB duplex using the equation

\[\begin{align} K_d = \frac{c_\mathrm{A}\,c_\mathrm{B}}{c_\mathrm{AB}} = \frac{(c_\mathrm{A}^0 - c_\mathrm{AB})\,(c_\mathrm{A}^0 - c_\mathrm{AB})}{c_\mathrm{AB}}, \end{align} \]

which is solved to give

\[\begin{align} c_\mathrm{AB} = \frac{1}{2}\left(K_d + c_\mathrm{A}^0 + c_\mathrm{B}^0 - \sqrt{\left(K_d + c_\mathrm{A}^0 + c_\mathrm{B}^0\right)^2 - 4c_\mathrm{A}^0\,c_\mathrm{B}^0}\right). \end{align} \]

So, we have an equation for our measured quantity \(c_\mathrm{AB}\) in terms of our manipulated quantity \(c_\mathrm{A}^0\) (with \(c_\mathrm{B}^0\) held constant). This is an example of a variate-covariate model, where the measured quantity is called the variate and manipulated quantities are called covariates. This constitutes an important class of models that are routinely encountered in science.

Note that analysis of variate-covariate models is often referred to as “curve fitting.” I eschew this term because I prefer to focus on the generative model, as opposed to whatever technique we may use to obtain estimates of its parameters.