Chemical reactions inside cells have long been correctly described as both discrete and stochastic [1–3], often entailing acute spatial patterns or dependencies [4–6]. Despite the intrinsic uncertainty in the occurrence of these chemical events, and basically against all odds, cells prevail as efficient decision makers. Not only are their fate decisions influenced by stochastic events and embedded within widely fluctuating environments, but they are stochastic themselves , the underlying mechanisms of which remain widely unknown.
So, one cannot help but wonder: how do cells process widely varying information from their environment, control their own chemical 'noise', and still manage to produce appropriate responses? The key to this question lies in signal transduction pathways, a series of interconnected chemical events that lead to highly specific cell responses. One such mechanism is adaptation, a common term used to represent sets of chemical reactions that generate a transient response in the presence of a sustained stimulus . These transient responses have been shown to affect gene expression and regulatory processes, where the cell decision is determined by the strength and duration of the input signal .
Adaptive behavior can result from three basic signaling motifs: integral control, negative feedback, and feed-forward regulation . The first is an abstraction of an engineering principle, where regulation is achieved by integrating the differences between a desired response and the state of the system. A cellular system may proceed in a similar fashion, by comparing 'actual' to 'desired' conditions, as has been found to be the case in bacterial chemotaxis [10–13] or calcium homeostasis .
Integral control can be achieved through appropriate combination of negative feedback loops, the latter of which are ubiquitous elements of signaling pathways, allowing for myriads of types of physiological homeostasis. In a self-regulating gene, a transcriptional repressor negatively regulates its own expression and, within certain network architectures and ranges of feedback strength, noise can be effectively reduced. In this sense, negative feedback allows a system to respond by decreasing the magnitude of any input perturbation, generally resulting in stabilization of the input signal. However, while the latter is true in a deterministic setting, several types of non-classic behavior can be observed once considering discrete signals and stochasticity .
In contrast, feed-forward architectures let the system respond to known cues (input signals) in a predetermined way, independently of the system's response. This is the essential difference from feedback mechanisms, where the output influences ('feeds back') the system to create a new response. For feed-forward to produce adaptation, two signal-dependent pathways must affect a third component, in opposite ways, otherwise known as 'incoherent' feed-forward loops .
Several exhaustive studies have shown that negative feedback regulation rarely yields perfect adaptation, whereas integral control and feed-forward regularly do so [8, 10, 11, 17]. Nevertheless, it should be noted that negative feedback can produce adaptation states close to 'perfect', and basically indistinguishable in terms of biological functions . By perfect adaptation it is generally understood that the system will return to the exact state where it was before the input signal was introduced, provided the system was already in equilibrium.
Furthermore, in order to consider a system adaptive, certain eligibility criteria in terms of amplitude and duration of the system response have to be met. It should be noted that no homogeneous criteria exist in the literature, and comparison between different adaptation models can become a daunting task. Quite generally, though, amplitude has been assessed in terms of sensitivity and precision, namely, the difference between maximal response and pre-stimulation values, and the difference between equilibrium values before and after stimulation, respectively [18, 19].
Recently, some types of adaptive systems (such as the incoherent feed-forward loop) have been shown to display fold-change detection (FCD) properties. Namely, that the system generates a response to fold-changes in the input signal, rather than absolute levels [20, 21]. The latter is related to Weber's law, which describes the relationship between a stimulus and its perceived intensity, a widely used concept in perception studies.
In this respect, some experimental studies have shown how important transduction mechanisms (such as ERK2 translocation  or Wnt signaling ) display robust fold-change responses. From these studies, several hypotheses have already arisen, such as whether cells detect and process information in relative rather than absolute terms, or whether fold-change detection facilitates the production of adjustable noise filters. Proving such hypotheses would greatly aid our understanding of cell signaling pathways, as FCD could rescale meaningful signal changes with respect to the background noise.
With all these points in mind, and in response to some of the open questions posed in , we study the effects of stochasticity in a minimalistic adaptation architecture, a 'two-state protein' scheme [24, 25]. For such, we wanted to analyze how stochastic profiles in a single-cell system propagate to population behavior, and what this actually entails in terms of system predictability. Surprisingly, our preliminary simulations highlighted how single cell and population behavior can be completely different, adaptation largely being an emergent property of a large ensemble. This led us to analyze adaptation in an exact stochastic setting, and understand why one should think of adaptation processes in probability space, rather than in numbers of molecules.
Until now, no one had noticed how ergodicity breaks down in simple linear scenarios devoid of cell growth and replication properties. Hence, our results provide key novel insights that need to be considered in any future study of adaptation, as well as any study where biological ergodicity is readily assumed. An example of the latter is linear and nonlinear signaling pathway studies.
Additionally, we also respond to some of the open questions in  and show how the simple linear 'two-state protein' scheme in a stochastic setting displays fold-change detection properties, both for consecutive stimulation inputs and separate fold-stimulations. This is the first study of FCD under stochasticity, the importance of which extrapolates to any cell signaling study.
Lastly, we discuss how extensions of the 'two-state protein' scheme (by considering discrete mediators, e.g. kinases and phosphatases) retain many of the properties observed in the purely linear system, including rupture of ergodicity.