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# Calculation of DNA binder concentrations

## Intention

• First approach was to determine the appropriate concentrations of DNA binders with a simple approximation:

$K_{D} = \frac{c_{B}c_{bp}}{c_{B bp}}$

with KD the dissociation constant of the DNA binder, cB the concentration of DNA binder, cbp the concentration of base pairs and cBbp the concentration of the occupied basepairs

However, this formula is not suited for such small concentrations, therefore we derived a more exact one.

## Derivation

• Take the dissociation constant KD and replace the concentrations with the total concentrations of the DNA binders and base pairs:

${c_{B}}^{T} = c_{B bp} + c_{B}$$c_{B}= {c_{B}}^{T} - c_{B bp}$

and

${c_{bp}}^{T} = c_{B bp} + c_{bp}$$c_{bp} = {c_{bp}}^{T} - c_{B bp}$

• Now assume that every n-th base a DNA binder should have bound in equilibrium

${c_{B bp}} = \frac{{c_{bp}}^T}{n}$

• With this, we get to a KD which is only dependent on the total concentrations of base pairs (c.f. structure) and DNA binders

$K_{D} = ({c_{B}}^{T} - \frac{{c_{bp}}^{T}}{n}) (n -1)$

## Result

• Now we are able to calculate the right concentration of the DNA binders in the sample to get an occupancy of n (i.e. one DNA binder each n-th base pair)

${c_{B}}^{T} = \frac{K_{D}}{n -1} + \frac{{c_{bp}}^{T}}{n}$