# Single-Site Binding¶

## Scheme¶

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:

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

## Parameters¶

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

## Species¶

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:

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

## Heat¶

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.