A ligand, L, binds with an indicator /: Z + / ^ IL
The Law o f Mass Action states that
A+i (Z . 7 ) = k.\ IL
Z, I and IL are the concentrations o f free ligand, free indicator and bound indicator respectively.
Æd is the apparent equilibrium dissociation constant.
■ u - ¥
... ®
IL * and I* are the fractions o f indicator in the bound and unbound state
I*
= - f ... (2)
IfIL* ... (3) It
And can be re-written in terms o f /* and IL *
I I *
^ - - n y ...
The fluorescence intensity F is related to the indicator concentration / by an efficiency term S dependent on excitation intensity, extinction coefficient, pathlength, quantum yield and instrument efficiency (Grynkiewicz et a l , 1985).
In the unbound state ( / = /t) : So = (5)
It
F
In the bound state ( / Z = / t ) : Si = (6)
T
fluorescence emitted by the indicator in the absence o f ligand
fluorescence emitted by the indicator in the presence o f saturating concentrations o f the ligand
In the presence o f non-saturating concentrations o f the ligand, the fluorescence emitted by the dye is the sum o f the products o f the concentrations o f the dye in the bound and unbound form with their respective S-factors:
F — I So IL Si
Substituting with (5) and (6) and using (2) and (3) gives:
F = I* Fmm + IL* Fmax...(7)
Since I* = I - IL*... (8)
Equation (7) may be re-written and rearranged:
F - Fmin = IL* (F m B X ' Fmin )...(9)
L I * And from equation (4) IL* = ---
Which may be rearranged to
K , { F - F ^ )
L = - J * ^m in + ^m ax Using equation (8) F , ( F - F ^ ) L =And (9) to get the final equation
K , ( F - F ^ ) F ^ - F
This equation permits the calculation o f the concentration o f free ligand (L) when the fluorescence emitted by the indicator at that concentration (F), Fmin, ^max and the K i are known.
The assumptions made are that the indicator forms a 1:1 complex with the ligand and that the indicator is sufficiently dilute for the fluorescence intensity to be
proportional to the concentration o f fluorescent species. In addition, it is assumed that F, Fniin and Fmax are determined at the same instrument sensitivity, optical pathlength and indicator concentration.
Appendix 2: Calcium buffering by calcium indicators
[Ca] is the free calcium concentration; [D] is the free dye concentration; [DCa] is the bound dye concentration; Kc& is the dissociation constant
[Ca] + [D] [DCa]
^ [Ca] [D] [DCa] fraction bound =
[Ca]
[ D ]t o tis the total dye concentration
[DCa] + [Z)] = [D]jqj
Using [D ] = [ D ] j x ) t ~ [DCa] to substitute for [Z)]:
fraction bound = [D]jot - [DCa]
K c a [DCa] = fraction bound = [Ca] [D ]joj ^Ca + V^^TOT ^Ca +
At [Ca] = 0.1 mM, fluo-4 (500 mM; Kca = 0.35 mM) buffers calcium at a ratio o f 2500 bound: 1 free. The ratio for fura-2FF (500 mM; Kcn = 29.8 mM) is 15:1.
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