Decomposing bulk signals to reveal hidden information in processive enzyme reactions: A case study in mRNA translation

Processive enzymes like polymerases or ribosomes are often studied in bulk experiments by monitoring time-dependent signals, such as fluorescence time traces. However, due to biomolecular process stochasticity, ensemble signals may lack the distinct features of single-molecule signals. Here, we demonstrate that, under certain conditions, bulk signals from processive reactions can be decomposed to unveil hidden information about individual reaction steps. Using mRNA translation as a case study, we show that decomposing a noisy ensemble signal generated by the translation of mRNAs with more than a few codons is an ill-posed problem, addressable through Tikhonov regularization. We apply our method to the fluorescence signatures of in-vitro translated LepB mRNA and determine codon-position dependent translation rates and corresponding state-specific fluorescence intensities. We find a significant change in fluorescence intensity after the fourth and the fifth peptide bond formation, and show that both codon position and encoded amino acid have an effect on the elongation rate. This demonstrates that our approach enhances the information content extracted from bulk experiments, thereby expanding the range of these time- and cost-efficient methods.

Figure A: Component-wise deviation between fitted (⃗ x out ) and given IFI vector (⃗ x in ) obtained from the analysis of the fluorescence signatures of an mRNA that consists of four identical codons.The shown data is an average for the decomposition of fluorescence signatures simulated from 1000 random input IFI vectors.The first entry of the IFI vector is always set to 1. IFI values related to the last states in Fig. 6 in the main text are fitted with higher reliability.

Determination of kinetic rate ω 45
We tested the robustness of our method by systematically using rates for the fit that are not the same as the rates used to generate the data and observing the effects on the fitted results.To perform this analysis, a for-loop is introduced into the fitting procedure that systematically changes the kinetic rate (in our case ω 45 ).For each iteration, the best theoretical fit in terms of least squares is saved.Once the for-loop is completed, the kinetic rate is found by identifying the minimum goodness-of-fit parameter χ 2 [1], which represents a measurement for the deviation of the simulated and calculated data curve: The standard deviation σ is used as an estimation for the error in the smoothed data curve.An increase in χ 2 by 1 results in an increase of 1 standard deviation σ for the parameter [1].We performed the simulation for uniform and non-uniform translation elongation rates.

4-codon-mRNA -uniform input rates
All transition rates apart from ω 45 of the Markov model description of the translation process are used as stated in Table 2 in the main paper.The rate ω 45 is set to ω 45 input = 14 s −1 for all codons and the fitting procedure to identify the unknown IFIs is performed as usual with the kinetic rate ω 45 as additional parameter.

4-codon-mRNA -non-uniform input rates
All transition rates apart from ω 45 of the Markov model description of the translation process are used as stated in Table 2 in the main paper.The rate ω 45 is set to different values for each codon, see Table A. The fitting procedure to identify the unknown IFIs is performed as usual with the kinetic rate ω 45 4 as additional parameter.

26-codon-mRNA -uniform input rates
All transition rates apart from ω 45 of the Markov model description of the translation process are used as stated in Table 2 in the main paper.The rate ω 45 is set to ω 45 input = 14 s −1 for all codons and the fitting procedure to identify the unknown IFIs is performed as usual with the kinetic rate ω 45 as additional parameter.Figure J shows the results of the decomposition.The best theoretical fit is found for ω 45 fit = 14s −1 (see Fig. K), which is in agreement with the input parameter.Table B: For each codon, a different rate for ω 45x is used in the Markov model description of the translation process resulting in non-uniform translation elongation rates.For selected codons, ω 45x is determined from the fluorescence signature of an mRNA that consists of 26 codons ('-' no simulation was performed for this position).It is assumed that all kinetic rates are known except for the fitted rate.A pronounced minimum in the χ 2 curve is necessary for a unique identification of the best fit.A distinct minimum can only be found for the first four codons.The * indicates that the χ 2 curve does not deploy an optimum but rather a minimal value for the kinetic rate, see the figures below.The codon-specific elongation rate ω elo is calculated by first step analysis [2] using the transition rates ω ij [3].

Figure B :
Figure B: Deviation between fitted (⃗ x out ) and given IFI vector (⃗ x in ) obtained from the analysis of the fluorescence signatures of an mRNA that consists of four identical codons.The shown data are averages of the decomposition of fluorescence signatures simulated from 1000 random input IFI vectors.The process was evaluated for different time spans.

Figure C :
Figure C: Singular values σ i of the occupancy probabilities matrix P for translation of an mRNA consisting of 26 codons.The singular values decay gradually to zero without jumps between consecutive singular values.

Figure D :Figure E :
Figure D: Component-wise deviation between fitted (⃗ x out ) and given IFI vectors (⃗ x in ) obtained from the analysis of fluorescence signatures of mRNAs that consist of 26 identical codons.The shown data are averages of the decomposition (with Tikhonov regularization) of fluorescence signatures simulated from 1000 random input IFI vectors.IFI values corresponding to the first and last states of the translation process are fitted with higher accuracy.
Figure F shows the results of the decomposition.The best theoretical fit is found for ω 45 fit = 14s −1 (see Fig. G), which is in agreement with the input parameter.

Figure F :
Figure F: Fluorescence signature of an mRNA that consists of four identical codons for random IFI input vector.a): The simulated fluorescence signature is compared to the best theoretical fit in terms of least squares.The theoretical model and simulated data curves are in perfect agreement.b): Fitted IFIs obtained from the analysis of the fluorescence signature compared to the given IFI input vector.

Figure G :
Figure G: Determination of kinetic rate ω 45 from fluorescence signature of an mRNA that consists of four identical codons for random IFI input vector.a): The goodness-of-fit parameter χ 2 is calculated for different values of the unknown kinetic rate ω 45 .The kinetic rate with the minimal χ 2 is identified.b): Enhanced depiction of the local minimum which is found for ω 45 fit = 14s −1 , which is in agreement with the input parameter ω 45 input = 14 s −1 .
Figure H shows the results of the decomposition.The best theoretical fit is found for ω 45 4 ,fit = 14s −1 (see Fig. I), which is in agreement with the input parameter.

Figure H :
Figure H: Fluorescence signature of an mRNA that consists of four codons with non-uniform translation rates for random IFI input vector.a): The simulated fluorescence signature is compared to the best theoretical fit in terms of least squares.The theoretical model and simulated data curves are in perfect agreement.b): Fitted IFIs obtained from the analysis of the fluorescence signature compared to the given IFI input vector.

Figure I :
Figure I: Determination of kinetic rate ω 45 4 from fluorescence signature of an mRNA that consists of four codons with non-uniform translation rates for random IFI input vector.a): The goodness-of-fit parameter χ 2 is calculated for different values of the unknown kinetic rate ω 45 4 .The kinetic rate with the minimal χ 2 is identified.b): Enhanced depiction of the local minimum which is found for ω 45 4 ,fit = 14s −1 , which is in agreement with the input parameter ω 45 4 ,input = 14s −1 .

Figure J :
Figure J: Fluorescence signature of an mRNA that consists of 26 identical codons for random IFI input vector.a): The simulated fluorescence signature is compared to the best theoretical fit in terms of least squares.The theoretical model and simulated data curves are in perfect agreement.b): Fitted IFIs obtained from the analysis of the fluorescence signature compared to the given IFI input vector.

Figure K :
Figure K: Determination of kinetic rate ω 45 from fluorescence signature of an mRNA that consists of 26 identical codons for random IFI input vector.a): The goodness-of-fit parameter χ 2 is calculated for different values of the unknown kinetic rate ω 45 .The kinetic rate with the minimal χ 2 is identified.b): Enhanced depiction of the local minimum which is found for ω 45 fit = 14s −1 , which is in agreement with the input parameter ω 45 input = 14 s −1 .

Figure S :
Figure S: Fluorescence signatures of mRNAs that consist of 26 codons for random IFI input vectors.a, c): The simulated fluorescence signature is compared to the best theoretical fit in terms of least squares.The theoretical model and simulated data curves are in perfect agreement.b, d): Fitted IFIs obtained from the analysis of the fluorescence signature compared to the given IFI input vector.The rates ω 45 1 (a, b) and ω 45 18 (c, d) were treated as unknown fitting parameters.

Table A :
For each codon, a different rate for ω 45x is used in the Markov model description of the translation process resulting in a non-uniform translation elongation rate.*ω 45 4 is assumed to be the unknown kinetic rate to be fitted.