Tutorials¶
TCSPC¶
Gaussian¶
Under our measurement conditions we have a mixture of monomeric and dimeric hGBP1. Thus all donor fluorescence decays FD(t) were fitted with the decays of at least two molecular species, FD(0)(t) for the Donor-only species and FD(A)(t) for the dimeric FRET species:
where xDOnly corresponds to the fraction of Donor-only molecules and c is a constant offset. Due to local quenching the fluorescence decay of the donor in the absence of FRET is tri-exponential with the individual species fractions \(x_{\rm{D}}^{(i)}\) and fluorescence lifetimes \(tau _{{\rm{D(0)}}}^{(i)}\) (see Table S3)
Thus, the time resolved fluorescence intensity decays of donor-/acceptor-labeled protein-complex (FRET-sample) were fitted globally with the decays of the donor-/unlabeled protein-complexes (donor only sample, DOnly). Generally it is reasonable to assume that the radiative lifetime of the donor is not affected by quenching. Hence, the FRET-rate constant (kFRET) is only determined by the donor-acceptor distance and their relative orientation [Ref: Kalinin, S.,& Johansson, L.B.-Å. Energy Migration and Transfer Rates Are Invariant to Modeling the Fluorescence Relaxation by Discrete and Continuous Distributions of Lifetimes. (2004) J. Phys. Chem. B 108, 3092-3097]. Expressing the FRET-rate constant in terms of distances the donor-fluorescence in presence of acceptor is given by:
Whereas p(RDA) is a FRET-rate distribution expressed as distance and R0 is the Forster-radius (in this case R0 =52 Å) and \(k_0=1/\tau_0\) is the radiative rate the unquenched dye. The flurophores are attached to the biomolecule by long flexible linkers. Hence, a donor-acceptor distance distribution is expected which is not averaged during the fluorescence lifetime of the dyes [Sindbert, S., Kalinin, S., Nguyen, H., Kienzler, A., Clima, L., Bannwarth, W., Appel, B., Müller, S. & Seidel, C.A.M. (2011) J. Am. Chem. Soc. 33, 2463-2480.] and the fluorescence decay FD(A) has to be expressed as by an donor-acceptor distance distribution p(RDA) with a non-zero width. Here the experimental time-resolved fluorescence intensities were either fitted by a Gaussian distribution of donor-acceptor distances (p(RDA)) with a mean inter dye distance RDA and a width wDA (Eq. 4) or, analog to the Tikhonov regularization, p(RDA) was determined by deconvolution of the fluorescence intensity decays by using the maximum-entropy method (MEM) [Livesey, A. K.; Skilling J. Maximum Entropy Theory. Acta Crystallogr. Sect. A 1985, 41, 113-122. Ref. Brochon, J. C. Methods Enzymol. 1994, 240, 262-311.].
The width of the Gaussian donor-acceptor distance distribution wDA should not be misinterpreted as the experimental/statistical-error but it describes a real physical property of the donor-acceptor pair. The experimental fluorescence decays presented below are described by combining the above formulas and were fitted by custom software written in Python.
\({F_{{\rm{D(A)}}}}(t) = {F_{D(0)}} \cdot \int\limits_{{R_{{\rm{DA}}}}} {\frac{1}{{{w_{{\rm{DA}}}}\sqrt {{\pi \mathord{\left/ {\vphantom {\pi 2}} \right. \kern-\nulldelimiterspace} 2}} \,}}\exp \left( { - 2\,{{\left[ {\frac{{{R_{{\rm{DA}}}} - \left\langle {{R_{{\rm{DA}}}}} \right\rangle }}{{{w_{DA}}}}} \right]}^2}} \right)\exp \left( { - t \cdot {k_0}\left[ {1 + {{({R_0}/{R_{{\rm{DA}}}})}^6}} \right]} \right)\,d{R_{{\rm{DA}}}}}\)
Stopped Flow¶
In the reaction system model a system of reactions has to be defined by the user. Given the user-defined reaction system and initial values of the reacting species the time-evolution of the species is calculated by numerical integration of the differential equations. Below the theoretical frame-work is described followed by a description how to define custom reaction models and fit experimental data. Given a set of species and a number of elementary reactions with associated rate constants the reaction system can be described by the following chemical reactions:
Using the law of mass action the flux of molecules per time and unit-volume is given by: