One of the most vital processes in biology is the transduction of signals along biochemical pathways, enabling the living cell to elicit appropriate responses to chemical and physical stimuli 
. In this context, the concept of signaling cascade is used as a paradigm or a model of signaling pathways. It consists of a chain of enzymatic reactions wherein a protein is interconverted reversibly between two forms. At each stage in the cascade, the activated form of the protein, which usually is a covalently modified derivative of the native protein, serves as the enzyme to activate the protein in the next stage in the chain and so on. Thus, a signaling cascade consists of a succession of covalent modification cycles, whose classical representative example is the phosphorylation/dephosphorylation cycle, but the general concept is broadly applicable. In some important cases, such as the well-studied MAPK cascades, the stages are in fact composed of double phosphorylations 
. In all cases, the concept of cascade clearly indicates a notion of flow oriented unidirectionally.
A general intracellular signaling network may consist of several interconnected cascades 
. Its topology can then be described as an oriented graph whose nodes represent stages of the cascades and the arrows serve to relate the activated proteins at a given stage to other covalent modification cycles or to a substrate targeted by the network. Associated with such a graph one may define several signaling pathways, namely several paths in the oriented graph, going from a top vertex, representing a biochemical entry of the system, e.g. a ligand, towards the bottom stage of one of the cascades, e.g. a transcription factor for some genes. A simple type of signal that can be transmitted in this system is a step increase of the enzyme activating the top cycle of one signaling pathway. Several studies have been devoted to the modeling of the propagation of such signal in signaling chains, and on the transmission properties as a function of most of the parameters of the cascade 
The mathematical modeling of signaling pathways often considers a simplified set of equations in which each cycle is described by a single variable 
. In a previous study, we highlighted that these simplified models overlooked the property of retroactivity between two successive stages of the cascades, and we proposed a new type of simplified modeling for cascades to account for this important signaling property 
. The concept of retroactivity means that the response property of a well-characterized input/output isolated device can change dramatically when this device is coupled to a downstream load. In the context of signaling pathways, retroactivity is a phenomenon that arises due to enzyme sequestration in the intermediate complex enzyme-next protein in the cascade. Its main consequence is that a downstream perturbation -e.g. of the protein- can produce a response in a component upstream of the perturbation without the need for explicit feedback connections. In refs. 
this effect was described independently by two groups for the first time. The main focus in ref. 
was to derive a simplified description of signaling cascades with one variable per cycle while keeping the retroactive property, after noticing that the standard simplifications on modeling cascades were explicitly avoiding such effects. The study of the effect (referred to as retroactivity in 
) was done mostly numerically in 
, introducing the notion of “reverse stimulus response curve”. Now, we study in detail reverse stimulus response curves, by characterizing both analytically and numerically when to expect a measurable upstream effect due to a downstream change in a control parameter. This work provides a roadmap for planning experiments that carefully account for this phenomena.
The absence of retroactivity for a signaling module implies that the state variables of this module do not change when its output is used as the input of another device. Special conditions are to be met in the design of a network unit in order to minimize the retroactivity 
. In the context of engineering, and specifically in synthetic biology where modularity is required 
, retroactivity is usually considered as a nuisance, often preventing the proper functioning of devices that consists of assemblies. The property of pathway retroactivity started to gain interest in the systems biology community 
. Retroactivity tends to be attenuated in long signaling cascades 
. However, ref. 
also shows that the probability that a 3-stage cascade exhibits retroactivity is around 0.5, so under many commonly encountered conditions, retroactivity occurs. Indeed, recent experiments demonstrate that retroactivity can be set in evidence and measured in vivo
in the MAPK cascade controlling the early development of drosophila embryos 
. An in vitro
study shows that retroactivity effects can be easily induced at one stage of the signaling system regulating the nitrogen assimilation in E. coli
. In short, retroactivity can be experimentally demonstrated in signaling pathways. In the recent paper by Wynn et al 
, it is shown that an important consequence of retroactivity is its role in the cellular response to a targeted therapy. In particular, we characterized the fact that kinase inhibitors can produce off-target effects as a consequence of retroactivity. In this numerical study, a statistical methodology based on a random sampling of the parameter space was utilized. In particular, that study considered a signaling topology with 3 single cycles, where one of them activates the other two in parallel. This system is also analysed in the present paper which is based on a numerical and analytical study of the nonlinear equations. In that sense, both articles complement each other.
Moreover, in the present work, we make use of the property of retroactivity in order to extend, theoretically, the standard view of signaling to a new type of intracellular signaling. Indeed, the existence of retroactivity in signaling pathways turns the usually one-way oriented graphs mentioned above, into two-way oriented graphs, with arrows going now from downstream to upstream. We call retroactive signaling the design of a pathway that exploits this possibility, that is to say, an extended signaling pathway which comprises a connected path of upstream signaling from output to input (cf. ). Since retroactivity is a secondary effect, when compared with the usual activation in signaling cascades, a retroactive signaling pathway would typically include only one or a few upstream arrows combined with the usual downstream arrows. Nevertheless, the possibility of retroactive steps in a signaling pathway opens up previously unexplored possibilities for signal transduction.
Motifs of short signaling pathways illustrating the concept of retroactive signaling in (A) a 2-cycle cascade and (B) in a 3-cycle cascade.
In this paper we explore this concept for the first time in short signaling pathways like the basic case of a 2-cycle cascade and simple extensions of it. The 2-cycle cascade, or the bi-cyclic cascade, is usually described as a motif comprising 2 cycles and a single arrow linking the activated protein of the first onto the second cycle. In this article, retroactive signaling in this system will be dealt with by analysing how a variation of the parameters affecting the downstream cycle, e.g. varying the total protein concentration in this cycle, or its phosphatase, can induce a response in variables of the upstream cycle. The upstream response can be computed numerically and estimated analytically. We will illustrate the theoretical work with examples of retroactive signaling in short multi-cycle pathways.