Fig 1.
Input parameters of the simulation based on Scheme 2.
(A) The dominating DADSs and (B) the corresponding macroscopic time constants. (C) Temporal and (D) spectral representation of the kinetic data calculated from the parameters presented in (A) and (B).
Fig 2.
Model selection by BO from the simulated data presented in Fig 1C and 1D at a fixed value of ω = 1.
Noise level: σrel = 10−3. (A) Mean MSE obtained without (cyan) and with 10-fold CV (blue and red). (C) Mean MSE obtained with RCV(nv). (B) and (D) Solution of the GENP calculated with the value of λ selected at the minimum presented in (A) and (C), respectively. A line in a particular color in panel (B) and (D) corresponds to a column x*,k in Eq (8), plotted against the grid of the time constants, while the colors represent different wavelengths. (The black dot beyond the upper limit of τ corresponds to the value of τ = ∞.) The sparsity of the solution is manifested in the low number of features in the form of narrow spikes, representing single exponentials, whose time constant and amplitude are indicated by their location and height, respectively.
Fig 3.
Model selection by BO based on RCV(nv) from the simulated data with different noise levels.
Results of the steps of Algorithm 1. (A) Step 1: BO in the joint space of λ and ω, explored by the BO process at the blue dots. (B) Step 2: BO in the space of ω with the fixed value of λ obtained in Step 1. The purple line (right axis) represents the average support size against ω. (C) Step 3: BO in the space of λ with the fixed value of ω obtained in Step 2. The purple line (right axis) represents the number of features against λ.
Fig 4.
Results of Algorithms 1 and 2 on the selected models presented in Fig 3.
(A) Representation over time constants (Algorithm 1, see captions for Fig 2B and 2D for details). The arrows point to the position of the true values (red) and those obtained by Algorithm 2 (valid ones blue, false positive black). (B) Representation by DADSs, neglecting components below 5% of the maximal one (Algorithm 1, dotted) compared to the true DADSs as presented in Fig 1A. (C) DADS obtained by Algorithm 2.
Table 1.
Kinetic parameters (τ and A) predicted at low and high noise levels on the simulated data by Algorithm 1 and Algorithm 2 (8 exponentials).
MSE refers to the mean square error of the fit.
Fig 5.
Experimental fluorescence kinetic data on FAD.
(A)Temporal and (B) spectral representation.
Fig 6.
Model selection from the fluorescence kinetic data presented in Fig 5.
For details see the legends and caption of Fig 3.
Fig 7.
Fluorescence kinetic parameters predicted by the selected models presented in Fig 6.
For details see the caption of Fig 4. In (B) and (C) the continuous lines are smoothing splines over the data plotted by dots.
Fig 8.
The kinetics corresponding to the distributions presented in S2 Fig (blue).
For comparison see the kinetics corresponding to the single discrete value at the maximum (200) of the activation energy distribution (red).
Table 2.
Kinetic parameters (τ and A) predicted from the fluorescence kinetic data by Algorithm 1 and Algorithm 2 (5 exponentials).
Fig 9.
Model selection based on RCV(nv) and 10-fold CV from the distributed kinetic data presented in Fig 8.
Noise level: σrel = 10−3. For details see the legends and caption of Fig 3.
Fig 10.
Solution of GENP on the selected models based on RCV(nv) and 10-fold CV presented in Fig 9.
(A) Distribution of time constants. (B) The residual of the fits calculated with the distributions presented in (A).
Fig 11.
Comparison of the true distribution (copied from S3B Fig) to those predicted by the selected models.
The model selection by BO based on 10-fold CV (Fig 9 right column) was carried out on the simulated data (Fig 8 blue) at different noise levels.
Fig 12.
Solution of the GENP obtained from data described by formula (14) at low noise.
For (A) and (C), model selection was carried out by RCV(nv), while for (B) and (D) by 10-fold CV. For (A) and (B), the value of A in the formula is 0, while for (C) and (D) its value is 20. τ1 = 101, τ2 = 10−1, τ3 = 103. Noise level: σrel = 10−7. λ and ω are the hyperparameters found by the BO process.
Fig 13.
Solution of the GENP obtained from data described by formula (15) at low noise.
For (A) and (C), the model selection was carried out by RCV(nv), while for (B) and (D) by 10-fold CV. For (A) and (B), the value of A in the formula is 0, while for (C) and (D) its value is 1. C = 8, τ1 = 101, τ2 = 10−1, τ3 = 103. Noise level: σrel = 10−7. λ and ω are the hyperparameters found by the BO process.