Here we describe our method of back-predicting the FUS1 mRNA profile from the fluorescence data of the GFP* reporter. Our approach is similar to that of Wang et al. (2008), except that we assume that oxidation of GFP* is rate-limiting, as has been empirically determined by measurements on several GFP variants (Miyawaki et al., 2003). This assumption allows us to use a quasi-equilibrium approach to reduce the number of equations and parameters needed for fitting. We justify the simplification based on comparison of galactose induction profiles for fluorescence and protein (see results).
The temporal profile of the GFP reporter is related to mRNA profile by:
where k is the combined rate of translation and maturation and kdeg is the GFP* protein degradation rate. In all fittings, we held kdeg fixed at the measured value corresponding to a half-life of 7 min. Any arbitrary system of ordinary differential equations with linear reaction rates can be solved by a weighted sum of exponentials, whose arguments are the eigenvalues. Of course it is possible that the system contains highly non-linear reaction rates; however, this method is a good first approximation. Thus, the equation for the mRNA is of the form:
where the λn and bn parameters are similar to the eigenvalues and the eigenvalue weights of the mRNA induction, respectively. The parameter b0 is set by the initial condition. With this functional form for the mRNA profile, equation ((1)) has a straightforward analytical solution. We then estimated the free parameters by performing a non-linear least squares fit to this analytical solution of the GFP* profile to our data on the YΔkGFP* (short-lived reporter) pheromone induction profile driven by the FUS1 promoter. We found that the free parameters were constrained best when we used, at most, two exponentials to describe the mRNA profile. Our estimate of the parameters was b0 = –808, b1 = 818, b2 = –8.8, λ1 = 2.9E-5 min–1, λ2 = 0.1 min–1 (note that the parameters bn are unit-less). We also estimated the uncertainty in back-predicting the mRNA profile that is due to the variability of the observed GFP* reporter profile. To define the range, we added and subtracted the standard error of the GFP* reporter profile to the mean of the same profile. We then fitted our model to the result, and estimated the profile and parameters of the ± 1 standard error (max/min) mRNA induction time course. For the maximum profile the estimated parameters were b0 = –579, b1 = 589, b2 = –6, λ1 = 3.9E-5 min–1, λ2 = 0.1 min–1. For the minimum profile the estimated parameters were b0 = –930, b1 = 938, b2 = –10, λ1 = 2.4E-5 min–1, λ2 =0.1 min–1. For all fittings we used the Matlab (MathWorks) built-in function ‘lsqnonlin’.