Figure 1.
Experimental mRNA decay patterns.
The relative mRNA number defined in Eq. (1) can be measured at different time points after the interruption of transcription for S. cerevisiae as adapted from [23]. In this semi-log plot we show only those decay patterns that are monotonically decreasing and satisfy the convexity properties according to the general condition derived in Eq. (18). From the 51 decay patterns shown here, 21 curves show a cross-over from fast to slow decay (red) while 4 curves show a cross-over from slow to fast decay (blue). This indicates that the purely exponential decay is only one of several possible decay patterns.
Figure 2.
Prototypical pathways of mRNA degradation.
In panel A degradation is depicted as a relatively simple process determined by only a single step, e.g. by unspecific and fast endonucleolytic decay, such as the degradation pathway mediated by RNase E in prokaryotic cells. In panel B, instead, we show a schematic representation of the degradation pathway known as decapping which is one of the main degradation mechanisms in eukaryotic cells. The decapping mechanism consists of several biochemical steps, possibly triggered by a specific miRNA, which contribute to destabilize the mRNA until complete degradation takes place. This mechanism can be considered as a prototype of multi-step degradation.
Figure 3.
mRNA decay patterns and lifetime distributions.
Relative mRNA number as a function of the time
after the interruption of transcription. (A) Two different experimental decay patterns are reproduced from [23] (circles) corresponding to the genes MGS1 (red) and RPS16B (blue) of S. cerevisiae. The solid lines represent decay patterns as calculated by the Markov chain model and the corresponding rates are estimated from a non-linear regression analysis (see Models and Methods for details about the fit parameters). For comparison, we also show a fit with a simple exponential function (dashed lines) which is clearly not suitable to capture the entire information of the degradation process. In particular, the exponential fit for the red data points suggests a half-life (intersection with the horizontal line) that is almost twice as large as the true half-life. (B) The corresponding lifetime densities
are derived using the rates obtained via the fit of Eq. (1) with the data. Evidently, both densities differ strongly from an exponential distribution indicated by the dashed lines. While the red line shows a rapidly decaying lifetime distribution, the blue line is broadly distributed, with a clear maximum at an intermediate time.
Figure 4.
Effective degradation rate as a function of the age a of an mRNA.
The lifetime distribution of an mRNA can be translated into an age-dependent degradation rate via Eq. (4). Here, we illustrate the change of the degradation rate during the lifetime of an mRNA for the two decay patterns shown in Fig. 3. For the mRNA encoding MGS1 (red), the degradation rate is high for young mRNAs and decreases to a constant value after some transient time. In contrast, for RPS16B mRNA (blue), the degradation rate is close to zero upon birth of the mRNA and increases gradually to a constant value. For comparison, the constant rates corresponding to a fit of the decay data with purely exponential functions (dashed lines) are also included.
Figure 5.
mRNA decay patterns in S. cerevisiae.
The patterns are obtained from a systematic fitting procedure applying Eqs. (1) and (9) or (10) to the experimental data from Ref. [23]. The curves show the theoretical decay patterns that minimize the deviation between theory and experiment. Note that the experimental data points are omitted here for better legibility. The left panel shows 21 fitted curves that decay exponentially in good approximation (the best fit was either an exponential function or fitting by another function leads to an error reduction of less than 10 per cent). Conversely, 94 curves show a moderate decay followed by a fast decay (central panel, the best fit was obtained by Eq. (10) with ) and 309 curves decay rapidly at short times and then level off (right panel, the best fit was obtained by Eq. (9) with
). Thus, the majority of decay patterns does not follow a single-exponential decay law. For this visualization of the different categories, we considered data that were nearly bona-fide (see text) resulting in 424 genes. Moreover, in the central and right panel we highlighted a number of decay patterns that display a strong contrast to an exponential decay.
Figure 6.
Average residual lifetime and residual protein synthesis capacity.
(A) Average residual lifetime as function of time
, as defined in Eq. (20), after the interruption of transcription for MGS1 (red) and RPS16B mRNA (blue). Under steady state conditions, i.e.
, both have similar residual lifetimes. However, if transcription is stopped, the remaining mRNA population ages. The average residual lifetime of MGS1 mRNA still present in the cell at time
increases with
because only old mRNAs with a low degradation rate are still in the sample (see Fig. 4). In contrast, for RPS16B mRNA the average residual lifetime decreases. Only for exponentially distributed lifetimes (dashed lines) the average residual lifetime stays constant, which reflects the memoryless property of the exponential distribution; (B) Residual protein synthesis capacity
versus
as defined in Eq. (7). The capacity
is proportional to the amount of proteins that will be produced by an average mRNA from the sample. The residual protein synthesis capacity decays exponentially if the mRNA has an exponential lifetime distribution (dashed lines) but follows a different pattern if the process of degradation is more complex. The small differences between the exponential and the true decay patterns indicate that the non-exponential character of the lifetime distributions may be difficult to deduce from measurements of the residual protein synthesis capacity
.
Figure 7.
Maturation and degradation of mRNA viewed as a Markov chain.
During its lifetime, each mRNA undergoes biochemical modifications described by transitions from state i to state with a rate
. These alterations may result in a change of the degradation rate
that governs the transition from state i to the absorbing state 0. The probability density of the time to absorption provides the distribution
for the lifetime of the mRNA. Many known degradation pathways can be described in this manner and thus provide the mRNA lifetime distribution if quantitative information about the rates is available. This model can also be used to fit experimental data, as was done in Figure 3, in order to derive the empirical lifetime distribution.
Figure 8.
The plot shows the residual sum of squares (RSS) after fitting the exponential model (abscissa) and the multistep model (ordinate) to the experiment data from Ref. [23]. Clearly, the multistep model leads to a considerable improvement of the fitting procedure, resulting in an average error reduction by almost one order of magnitude. Moreover, we also display the errors corresponding to the different categories in Fig. 5 as black, blue and red dots, respectively. The latter two represent the non-exponential patterns and typically imply a strong reduction of the fitting error. Additionally, we have highlighted the two representative cases RPS16B and MGS1 belonging to the two non-exponential categories, as given in Fig. 3A. One may notice that there are some black dots, corresponding to the exponential decay patterns in Fig. 5. For these decay patterns, the fitting with a multi-step model does not provide a significant improvement of the fit compared to the exponential function.