Fig 1.
Schematic representation of the decomposition of ensemble time traces.
a): The action of an individual processive enzyme is monitored over time. Changes in the signal are interpreted to obtain a detailed picture of step-by-step transitions and related kinetics of this molecule. b): Ensemble time trace for a large number of enzymes in parallel. The detected signal is a superposition of all individual signals. Due to the stochasticity of biochemical processes, these individual signals differ from each other to some extend. Consequently, the features of the individual signals are blurred or even lost during superposition. c): The ensemble time trace needs to be computationally decomposed to reveal the hidden information on the transitions of a single molecule.
Fig 2.
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.
Fig 3.
Fitting fluorescence signatures of longer mRNAs reveals signs of an ill-posed problem.
a): Fluorescence signature of an mRNA that consists of 26 identical codons for a random IFI input vector. The simulated fluorescence signature is compared to the best theoretical fit in terms of least squares. No regularization is applied. b): Fitted IFIs obtained from the decomposition of the fluorescence signature in a) compared to the given IFI input vectors. No regularization is applied.
Fig 4.
Characteristics of the occupancy probabilities matrix P.
a): Condition number κ for translation of mRNAs consisting of n identical codons with elongation rates ωelo = 10s−1 and ωelo = 20s−1. b): Condition number κ for translation of an mRNA that consists of 26 identical codons for various elongation rates ωelo.
Fig 5.
Decomposition of fluorescence signatures of mRNAs consisting of 26 identical codons with Tikhonov regularization.
a): For different values of α, the norm of the regularized solution is plotted versus the residual norm
. The log-log plot shows the characteristic L-shaped curve. The optimal regularization parameter in terms of trade-off between regularization and perturbation error is α = 0.01 and is found by locating the corner of the curve. b): Input and fitted IFI vectors for three different regularization parameter values.
Fig 6.
Decomposition of fluorescence signatures of LepB mRNAs.
Measured fluorescence signatures of the in-vitro translation of LepB3 and LepB11 mRNA (colored lines) and best fits in terms of least squares (black lines).
Fig 7.
Fitted IFIs obtained from the analysis of the fluorescence signatures of truncated LepB mRNAs of different lengths.
The IFIs correspond to the different states of the translation process (see Fig 8 and [8]). The lowest IFI (IFI5) is associated to the states after binding of Phe and before binding of the second Ala to the nascent peptide chain. The IFIs associated to the artificial state of a ribosome at the end of a truncated mRNA are not shown.
Fig 8.
Representation of mRNA translation elongation as a Markov process and assigned IFIs for a mRNA that consists of four codons.
The Markov model for translation of a short mRNA consisting of n = 4 codons includes 6n + 2 = 26 states and n + 3 = 7 assigned IFIs. Each state in the process corresponds to one biochemically resolved step: State 01 represents the initiation complex with the start codon in the ribosomal A site. States 01-41 describe initial selection, including ternary complex binding and recognition, GTPase activation, GTP hydrolysis and rearrangement of EF-Tu. This is followed by tRNA accommodation and peptide bond formation (51). Afterwards, the ribosome translocates to the second codon (02). The elongation cycle is repeated for the next three codons before the ribosome reaches the end of the truncated mRNA (E4) where the P site is occupied by the fourth codon while the A site remains empty. See refs. [39–45] for more details on the translation process. Possible transitions between the states are indicated by arrows and occur with transitions rates ωij, which is exemplified by the ternary complex binding rate ω01 and the unbinding rate ω10. The average rate of translation for one codon is denoted by ωelo. For the first codon, changes in the state-specific IFIs are assumed to occur both after peptide bond formation and translocation, and states assigned with identical IFI values have the same color. Note that due to initial synchronization of ribosomes, the two IFI changes on the first codon become visible in the ensemble fluorescence signature. For all following codons only one change in fluorescence intensity after peptide bond formation (after transition from state (4k) to state (5k)) is considered to avoid overfitting. The final intrinsic fluorescence intensity E4 corresponds to the artificial state of the ribosome at the end of the truncated mRNA. Previously published as the 6n2/n3 model in [8]. Nomenclature of IFIs for a): a generic mRNA, and b): the specific case of LepB5 mRNA.
Table 1.
Codon position-specific translation rates calculated from fitted in-vitro rates.
Table 2.
Transition rates.
Fig 9.
a): Simulated ensemble fluorescence signature constructed from the random IFI input vector
weighted by the time-dependent state occupancy probabilities P of the Markov process. b): Superimposition of Gaussian noise to imitate thermal fluctuations. c): The simulated data curve
is smoothed using a moving average. Note: To make differences in
and
visible, a Gaussian noise with a standard deviation of 0.01 and a moving average with a non-optimal subset size of 200 steps were used in this figure.