In each cell, the information encoded in the DNA is transcribed into messenger RNAs (mRNAs), which then, in turn, are translated into proteins. From a single-molecule point of view, each mRNA has a finite lifetime and its decay arises from the action of a variety of degrading enzymes that break down the mRNA into its constituents, the nucleotides. The encounters between mRNA and degrading proteins are largely dominated by stochastic effects.
Given the relevance of mRNA concentration on protein abundance 
, much effort has been dedicated to improve our understanding of mRNA degradation 
. In the past decades, a number of different mechanisms responsible for the degradation of the mRNA have been identified 
. Some mechanisms of degradation are known to affect the decay of all mRNA species and are thus unspecific. In contrast, other mechanisms are known to affect certain mRNAs more than others depending on different physical and chemical properties of the nucleotide chain. For example, micro-RNAs mediate the docking of degrading enzymes specifically to each mRNA and contribute thus to the large variation of the stability between mRNA species 
One widely studied degradation pathway in bacteria is known as endonucleolytic
. This degradation process is initiated by cleavage within the nucleotide chain by the action of a single protein or protein complex. For instance in E. coli
, RNAse E and its homologues are proteins responsible to initiate endonucleolytic decay. Once the degradation process has been initiated, it leads to a rapid decay of the attacked mRNA with a sudden interruption of translation. In this case, the time scale related to the random encounter between the degradation complex and the mRNA primarily determines the lifetime of the mRNAs. It is commonly believed that eukaryotic mRNAs are affected to a lesser extent by endonucleolytic degradation than prokaryotic mRNAs 
. In eukaryotic cells, the most common mechanisms of degradation are those that lead to decapping
. This mechanism requires deadenylation at the 3′ region and the destabilization of the 5′ cap structure before degradation occurs in the 5′ to 3′ direction behind the last translating ribosome 
. Different exonucleolytic degradation pathways exist also in bacteria. In E. coli
, for instance, modification of the 3′ stem-loop is a prerequisite of exonucleolytic degradation initiation 
. Moreover, in B. subtilis
a 5′ exonuclease has been discovered recently 
. Furthermore, a variety of miRNA and small-RNA mediated degradation mechanisms have been identified 
. These mechanisms require several biochemical steps for complete degradation or complete loss of functionality.
Irrespective of the degradation pathway, the lifetime of a single mRNA is a random variable that will depend on the diffusion time of the degrading complexes and on the time scale of enzymatic activity at the various steps of degradation. Moreover, the particular form of the lifetime distribution for each species of mRNA depends on the specific mechanisms that are responsible for its degradation. A species of mRNA that is mostly degraded by the action of an endonuclease, for instance, will have an exponential lifetime distribution. The same holds also for degradation processes which involve only one relatively slow, rate-limiting step.
In contrast, during the decapping process of degradation or during the degradation process triggered by miRNA, the mRNAs undergo a series of biochemical modifications 
, several of which are characterized by relatively slow rates, which implies that their lifetime distribution cannot be described by a single exponential function.
This simple observation has dramatic consequences. Indeed, a basic result in probability theory states that the exponential distribution is memoryless, i.e. the life expectancy does not depend on the age of the process, while any other probability distribution carries a memory of the past. For the mRNAs, this memory is encoded in the biochemical transformations or in other transient phenomena that characterize the aging of the mRNA.
In this paper, we show that complex mRNA degradation processes necessarily lead to lifetime distributions that are not exponential. Our study addresses the relationship between the mRNA lifetime distribution and the experimentally observed mRNA decay patterns. The diverse degradation processes described above call for a general theory of mRNA degradation. The theory that we present here provides a robust mathematical framework, into which one can incorporate additional molecular details about specific degradation mechanisms.
Experimentally, the decay patterns are often determined by the measurement of the decaying average amount of each mRNA species at steady state expression, in a cell culture after the interruption of transcription. This method proceeds by taking several samples at different time points, as described in 
. The data points are then fit by an exponential function in order to compute the half-life of the mRNAs. However, this fitting procedure has many shortcomings. In Ref. 
a considerable amount of data have been eliminated because they could not be fit by an exponential function and in Refs. 
many rather distinct decay patterns were observed so that the idea of a fit with a simple exponential has been rejected for the majority of the mRNAs.
reproduces some of the measured decay curves for S. cerevisiae
from Ref. 
. We highlighted those patterns that strongly deviate from an exponential decay. The red patterns show a marked cross-over from a quick decay at short time scales to a slow decay at larger time scales. The blue patterns, instead, show the opposite behavior with a cross-over from a slower decay at short time scales to a quicker decay at large time scales. In the background, the gray lines show those decay patterns that are approximately exponential. In particular, the analysis of the data shows that short-lived mRNA species tend to belong to the set of the red decay patterns while long-lived mRNAs tend to belong to the set of the blue patterns in .
Experimental mRNA decay patterns.
The inhibition of transcription is likely to strongly stress the cells and possibly leads to undesired side effects. Therefore, several laboratories have developed an alternative method and applied the non-perturbing pulse-chase technique to assess mRNA stability 
. This method is based on the labeling of the mRNAs with a heavy nucleotide, which is added to the cell culture over a period of time. Again, a measurement of the relative amount of labeled mRNAs over time reveals the time scale of degradation for the observed mRNA species. However, also here the pattern may or may not be exponential and the quality of the data relies on the number of recorded time points and on a good choice of the fitting curve, which reflects the assumptions about the biochemical mechanisms for the degradation process.
In this article, we will first derive a general relationship between the mRNA lifetime distribution and the decay patterns for the amount of mRNA, starting from a steady state expression level. Furthermore, we will introduce the concept of mRNA aging and show that the residual lifetime distribution of the mRNAs as well as their potential protein yield change during the experimental determination of the half-life. Finally, we will develop a model for degradation by multiple steps and apply it to data of S. cerevisiae in order to gain insight into the form of the associated lifetime distributions and a possible classification of the decay patterns of the entire mRNA pool.