Fig 1.
Age distribution of molecules during pulse chase experiments.
(a) Depiction of molecules in a cell during a pulse chase experiment with pulse duration of 70 in arbitrary units (a.u.). For the purpose of illustration we show four snapshots of the experiment. In snapshot I (30 a.u.), pulse has just begun. The white dots depict the population of molecules already existing in the cell before the pulse. All newly synthesized molecules (red dots) are labeled by the pulse and measurable by the experimentalist. As the pulse continues in snapshot II (60 a.u.) we see more labeled molecules appear. Meanwhile, both labeled and unlabeled molecules degrade. In snapshot III (90 a.u.), the pulse has ended since some time. Newly synthesized molecules from this moment on are unlabeled. Again, both labeled and unlabeled molecules degrade. In snapshot IV (200 a.u.), all labeled molecules have degraded. Unlabeled molecules continue to be synthesized and degraded. (b) Age distribution of the labeled molecules, each curve corresponds to one phase in panel (a). In snapshot I, the pulse has just begun, and all molecules that are labeled by the pulse are no older than the time elapsed since the pulse has begun. In snapshot II, the pulse has been applied for some time; some labeled molecules may be quite old. In snapshot III the molecules cannot be younger than the time elapsed since pulse has been stopped. By snapshot IV, if there were molecules left, they would have that age distribution.
Fig 2.
Markov chain representation of a Markov process and 2-state model fit to a decay curve.
(a) Markov chain representation of a degradation process. Biochemical pathways (such as degradation) can be readily translated into Markov chain models: each biochemical entity is represented as a Markov state (circles) and the reaction speeds are represented as fluxes between the states (arrows). Here we show a possible Markov model of degradation containing two states. Newly synthesized molecules are in state 1. From state 1, there are two possible paths, either to state 2 with rate κ12 or degraded with rate κ10. For those molecules that reach state 2, they are degraded with rate κ20. The rates κ10 and κ20 are necessarily different, otherwise the model collapses into a 1 state model. (b) 2-state model fit (black line) and exponential fit (red line) to sample data with 1 minute pulse (data from [9], blue spots). Note that in the log(abundance)-linear(time) scale, the data does not resemble a straight line, thus necessitating a model more complicated than a single exponential. The best fit using Eq (11) with C(Δt) from Eq (16) gives the following parameters: κ10 = 0.0109 min − 1, κ20 = 0.002 min − 1, κ12 = 0.0189 min − 1, and pulse = 1 min. κexp = 0.0029. The decision in favor of the two-stage model is made on the basis of the AIC criterion thanks to its very small RSS.
Fig 3.
Model calibration with fabricated data.
(a) Verification of fitting procedure using simulated data separately. Using the parameters obtained from the best fit model to the data from [9] for pulse = 1 min, we fabricate sample data by calculating the abundance over time (dots) for different pulse lengths (1, 5, 30, 120, 1200 minutes) using the function that gives the decay pattern of the relative abundance C(Δt), Eq (16), as function of the measurement time Δt. We then fit resultant decay patterns with our fitting routine. We get back the same rates that were used to simulated the data for each experiment (Table 1). This shows that if the system in the background is unchanging, we can reliably extract the parameters of the system by fitting the decay patterns individually. κ10 = 0.0109 min − 1, κ20 = 0.002 min − 1, and κ12 = 0.0189 min − 1. (b) Simultaneous fit of pooled simulated data. Here the simulated data is augmented with a small amount of multiplicative noise, Eq (17). We fit the whole collection of data simultaneously (see Methods). Values very close to our original simulation parameters are obtained (Table 1 last row). This shows that under steady experimental conditions, we can reliably extract the parameters of the system by fitting the decay patterns simultaneously.
Table 1.
Model parameters for simulated data with different pulse lengths.
Parameters from the best fit to the data simulated with different pulse lengths and the 2-state model as found by Multistart (MATLAB®) with 1000 start points. Minimization by fmincon (bounds κ10, κ20, κ12 ∈ [0.000001,1]). Notice that the fits yield results identical to those used to generate the data. This proves the self-consistency of the procedure. Conversely in Table 2, the parameter values are not stable; suggesting different degradation system dynamics in each experiment.
Fig 4.
Model calibration with data from Ref. [9].
(a) Decay patterns from Ref. [9] fit individually. Each decay pattern from Ref. [9] is fit with the 2-state model using Eq (11) with C(Δt) from Eq (16). We find that for some of the decay patterns, the parameters obtained from the fitting are different from the others (Table 2). This implies that the underlying system has changed in the different experiments. Possibly the labeling procedure has affected the cells and contributed to a change in the internal environment. (b) Simultaneous fit of pooled real data with Eq (12). Here we pool the decay patterns and fit them simultaneously with the 2-state model. No good fits were found, despite using global and multistart techniques in the parameter search process. This implies that the underlying systems across the experiments can not be described by one unified model, at least not the two state model that we have considered. Possibly the labeling procedure has affected the cells and contributed to a change in the internal environment.
Table 2.
Model parameters for data from Ref. [9].
Parameters from the best fit to the 2-state model as found by Multistart (MATLAB®) with 1000 start points. Minimization by fmincon (bounds κ10, κ20, κ12 ∈ [0.000001,1] min − 1). Notice that each fit yields different parameters. This suggests that the degradation system dynamics in each experiment is different. In contrast Table 1 shows that the fits to simulated data with consistent parameters return the same rates. Data are reported in Table A in S1 Supporting Information
Table 3.
Model parameters for simulated pulse no chase data.
Parameters from the best fit to the 2-state model as found by Multistart (MATLAB®) with 100 start points. Minimization by fmincon (bounds κ10, κ20, κ12 ∈ [0.000001,1] min − 1). The first row is the best fit for the data as if the experimentalist only took 3 measurements at t = 1,2,3 minutes. The 2nd row takes one more measurement (t = 10 minutes) into consideration.
Fig 5.
Simulation of pulse no chase experiments.
Simulation of pulse no chase experiments, where the pulse is applied up until the time of measurement. Data is produced with rates κ10 = 0.0109 min−1, κ20 = 0.0002 min−1 and κ12 = 0.0189 min−1 using Eq (19). Traces show the results of several fits, each fitting taking into consideration one additional data point (Table 3).
Fig 6.
Sensitivity of P(Δt) to the parameters κ10, κ20 and κ12.
We plot the sensitivities of the measurements in a pulse no chase experiment assuming a 2-state model with κ10 = 0.0109 min − 1, κ20 = 0.0002 min − 1 and κ12 = 0.0189 min − 1. (a) Output is only sensitive to small values of κ10. (b) Output is sensitive to a range of κ20. (c) Output is only sensitive to small values of κ12. For all parameters, taking more measurements at later timepoints does not help the parameter estimation because the output is not sensitive to deviations in the parameters at late times.