Single-Site Binding


Scheme is for binding of titrant \(T\) to a stationary species \(S\):

\[S + T \rightleftharpoons TS\]

To describe this, we use the following equilibrium constant:

\[K = \frac{[ST]}{[S]_{free}[T]_{free}}\]


parameter variable parameter name class
association constant \(K\) K thermodynamic
binding enthalpy \(\Delta H\) dH thermodynamic
fraction competent fx_competent nuisance
slope of heat of dilution dilution_heat nuisance
intercept of heat of dilution dilution_intercept nuisance


We can only manipulate \([T]_{total}\) and \([S]_{total}\) experimentally, so our first goal is to determine the concentration of \([ST]\), which we cannot manipulate or directly observe.

\[K = \frac{[ST]}{([S]_{total} - [ST])([T]_{total}-[ST])},\]
\[K \Big ([S]_{total}[T]_{total} - [ST]([S]_{total} + [T]_{total}) + [ST]^2 \Big ) = [ST],\]
\[[S]_{total}[T]_{total} - [ST](S_{total} + T_{total}) + [ST]^{2} - [ST]/K = 0,\]
\[[S]_{total}[T]_{total} - [ST]([S]_{total} + [T]_{total} + 1/K) + [ST]^2 = 0.\]

The real root of this equation describes \([ST]\) in terms of \(K\) and the total concentrations of \([S]\) and \([T]\):

\[[ST] = \frac{[S]_{total} + [T]_{total} + 1/K - \sqrt{([S]_{total} + [T]_{total} + 1/K)^2 -4[S]_{total}[T]_{total}}}{2}\]

The mole fraction \(ST\) is:

\[x_{ST} = \frac{[ST]}{[S]_{total}}\]


The heat for each shot \(i\) (\(q_{i}\)) is:

\[q_{i} = V_{0}[S]_{total,i}(\Delta H(x_{ST,i} - x_{ST,i-1})) + q_{dilution,i},\]

where \(V_{0}\) is the volume of the cell (fixed) and \(\Delta H\) is the enthalpy of binding. Note that we do not deal with dilution here, as pytc calculates \(x_{ST,i}\) for the entire titration, accouting for dilution at each step. \(V_{0}\) is held constant as the total cell volume (not the volume of solution including the neck) as only the cell, not the neck, is detected in the signal.