Past exam of the biological physics course of the University of Oxford / 2018 / 5 / c / Solution

To find the optimal Förster distance for two-state contrast, take positive distinct distances $A,B$ and put $x=R_0^6>0$. The magnitude of the efficiency change isIts derivative has the sign of $A^6{B}^6-x^2$. It rises from zero, has a single maximum at $x=(AB{)}^3$, then falls back to zero. HenceThe printed form is equivalent, sinceAt this optimum the two efficiencies sum to one and the maximal contrast is $|B^3-A^3|/(B^3+A^3)$. If $A=B$, there is no distance change and the contrast is zero for every $R_0$; the source's unsimplified expression is then $0/0$ and does not specify a unique optimum. For example $A=B=5\mathrm{nm}$ yields zero contrast regardless of the fluorophore pair. The geometrical-mean result is a design criterion for distinct distances within the assumed fixed-orientation/photophysics FRET law, not a correction for dye-orientation changes.