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PLoS One. 2012; 7(7): e40806.
Published online Jul 26, 2012. doi:  10.1371/journal.pone.0040806
PMCID: PMC3406091
Retroactive Signaling in Short Signaling Pathways
Jacques-Alexandre Sepulchre,1* Sofía D. Merajver,2 and Alejandra C. Ventura3
1Institut Non Linéaire de Nice, UMR 7335 CNRS - University of Nice Sophia Antipolis, Valbonne, France
2Department of Internal Medicine, and Comprehensive Cancer Center, University of Michigan, Ann Arbor, Michigan, United States of America
3Instituto de Fisiología, Biología Molecular y Neurociencias, CONICET and Departamento de Fisiología, Biología Molecular y Celular, Universidad de Buenos Aires, Buenos Aires, Argentina
Masaru Katoh, Editor
National Cancer Center, Japan
* E-mail: jacques-alexandre.sepulchre/at/inln.cnrs.fr
Conceived and designed the experiments: JAS ACV. Performed the experiments: JAS ACV. Analyzed the data: JAS ACV. Wrote the paper: JAS SDM ACV.
Received March 21, 2012; Accepted June 13, 2012.
In biochemical signaling pathways without explicit feedback connections, the core signal transduction is usually described as a one-way communication, going from upstream to downstream in a feedforward chain or network of covalent modification cycles. In this paper we explore the possibility of a new type of signaling called retroactive signaling, offered by the recently demonstrated property of retroactivity in signaling cascades. The possibility of retroactive signaling is analysed in the simplest case of the stationary states of a bicyclic cascade of signaling cycles. In this case, we work out the conditions for which variables of the upstream cycle are affected by a change of the total amount of protein in the downstream cycle, or by a variation of the phosphatase deactivating the same protein. Particularly, we predict the characteristic ranges of the downstream protein, or of the downstream phosphatase, for which a retroactive effect can be observed on the upstream cycle variables. Next, we extend the possibility of retroactive signaling in short but nonlinear signaling pathways involving a few covalent modification cycles.
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 [1]. 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 [2], [3]. 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 [4]. 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 [3], [5][7].
The mathematical modeling of signaling pathways often considers a simplified set of equations in which each cycle is described by a single variable [5]. 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 [8]. 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. [8], [9] this effect was described independently by two groups for the first time. The main focus in ref. [8] 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 [9]) was done mostly numerically in [8], 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 [9], [10]. In the context of engineering, and specifically in synthetic biology where modularity is required [11][13], 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 [9], [14][16]. Retroactivity tends to be attenuated in long signaling cascades [7], [10]. However, ref. [10] 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 [17]. 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 [18]. In short, retroactivity can be experimentally demonstrated in signaling pathways. In the recent paper by Wynn et al [16], 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. Fig. 1). 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.
Figure 1
Figure 1
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.
The Main Question
Figure 1 depicts simple motifs of 2-cycle and 3-cycle pathways. The goal is to study the conditions under which a signal, or a perturbation, that modifies the state of a downstream cycle, can be transmitted upstream, to another cycle in the context of these short pathways. We will focus most of our studies on what happens to the upstream cycle in a 2-cycle system, when control parameters of the downstream cycle are modified, as for instance its total available protein or its total phosphatase.
The mathematical equations describing these systems are discussed in the Methods section. To summarize our main notations, we name each cycle in a given signaling pathway by an index i(An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e001.jpg). We take the convention to call cycle 1 the starting cycle of a retroactive signaling scheme, and to increment the number of the other cycles following their position in the signaling network until the last cycle in the pathway has been reached. Figure 1(B) shows a simple example of retroactive signaling in the pathway 1An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e002.jpg2An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e003.jpg3 where cycle 2 is an enzyme for both cycles 1 and 3. For notational convenience we will use variable names to denote both a chemical species and its concentration. For instance, the instantaneous state of each cycle is described by the variables An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e004.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e005.jpg, denoting respectively the concentrations of the inactivated and of the activated protein An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e006.jpg, whose total amount is denoted by An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e007.jpg. The enzymatic activations of a given stage of the cascade on the next stages are indicated by vertical top-down arrows on Fig. 1, except for the activation of the uppermost stage for which the activating enzyme is a parameter, e.g. An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e008.jpg denoting the total concentration of the enzyme converting An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e009.jpg into An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e010.jpg. In all cases, the enzyme deactivating cycle An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e011.jpg has a total concentration denoted by An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e012.jpg.
In most signaling systems, the activated form of protein i corresponds to its phosphorylated form, in which case the converting enzymes are called kinase and phosphatase, respectively for the phosphorylation and the de-phosphorylation of the protein. Since this situation is the most frequently present in intracellular signaling modules, in what follows we will often name An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e013.jpg the kinase and An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e014.jpg the phosphatase of cycle 2, just for brevity. Moreover, the activating covalent modification will be referred to as phosphorylation. In fact, all the formalism used in this study can equally well apply to other covalent modifications like adenlylation, methylation, GTP-ase modifications.
Varying the Available Protein in a Signaling Cycle
In order to describe the 2-cycle cascade (cf. Fig. 1(A)) from the point of view of retroactive signaling, let us start by suppressing the phosphatase in the upstream cycle, i.e. set An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e015.jpg in cycle 2. Then, cycle 1 behaves like a single signaling cycle with kinase An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e016.jpg and with phosphatase An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e017.jpg. Let us analyse what happens to the activated and the non-activated proteins in cycle 1, when the total available amount of this protein, denoted by An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e018.jpg, is varied between An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e019.jpg and an arbitrarily large value. In what follows, we will see that answering this question will provide a way to analyse simple instances of retroactive signaling.
The intermediate complex An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e020.jpg formed by enzyme An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e021.jpg and protein An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e022.jpg is a key chemical species in the coupling between cycle 2 and cycle 1. Thus it is relevant to study how An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e023.jpg grows when the total protein of cycle 1 is increased from the value An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e024.jpg. Figure 2(B) shows the case where cycle 1 is deactivated (i.e. An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e025.jpg). Then, An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e026.jpg first increases proportionally to An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e027.jpg, and reaches a plateau corresponding to its saturated value, An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e028.jpg, when An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e029.jpg. This saturating behavior suggests the definition of a characteristic range for the variation of An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e030.jpg, meaning that above this range a further increase of total protein in cycle 1 has not much effect on the sequestration of protein in cycle 2. For example, we can define the characteristic range for An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e031.jpg by extrapolating the initially linear growth of An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e032.jpg as a function of An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e033.jpg to its asymptotic value An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e034.jpg. This is indicated and denoted on Fig. 2 by An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e035.jpg. This characteristic range of An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e036.jpg can be analytically calculated as a function of the parameters of cycle 1. The result is:
A mathematical equation, expression, or formula.
 Object name is pone.0040806.e037.jpg
(1)
where An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e038.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e039.jpg are the maximal reaction rates defined in Eq.(19), and An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e040.jpg are the Michaelis-Menten coefficients of the cycle 1 (cf. section Methods). The quantity An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e041.jpg will be used in the following in order to non-dimensionalize the parameter An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e042.jpg by scaling it with An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e043.jpg whenever An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e044.jpg is plotted (e.g. in abscissa).
Figure 2
Figure 2
Behaviors of cycle 1 as a function of An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e045.jpg, the total protein in cycle 1.
Figure 2(C) shows the increase of An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e062.jpg when cycle 1 is activated (An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e063.jpg). It can be shown that in this case the maximum amount for An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e064.jpg is An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e065.jpg, with An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e066.jpg, meaning that the sequestration is lower than in the case where cycle 1 is deactivated. Therefore, we will see in the next Section that in order to optimize the retroactivity in a 2-cycle system, the downstream cycle should be deactivated, so that varying An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e067.jpg has a larger effect on An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e068.jpg and thereby a greater influence on the upstream cycle.
At the same time, two distinct behaviors are seen for variables (An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e069.jpg) as a function of total An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e070.jpg, according to whether cycle 1 is activated or not (cf. Fig. 2(D-E)). If cycle 1 is deactivated the asymptotic behavior is a linear increase of variable An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e071.jpg while An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e072.jpg tends to a constant. If cycle 1 is activated, the converse happens, namely An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e073.jpg grows linearly and An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e074.jpg reaches a constant value. Therefore, increasing the amount of substrate An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e075.jpg beyond the characteristic range An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e076.jpg in the covalent modification cycle 1 tends to an increase of either the activated or of the deactivated protein, but not of both, and the other variable tends to a constant. These latter values can be computed analytically as follows, if An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e077.jpg (cf. the section Methods):
  • if An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e078.jpg then
    A mathematical equation, expression, or formula.
 Object name is pone.0040806.e079.jpg
    (2)
  • if An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e080.jpg then
    A mathematical equation, expression, or formula.
 Object name is pone.0040806.e081.jpg
    (3)
Figures 2(D-E) illustrates also that the graphs of An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e082.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e083.jpg as a function of An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e084.jpg can be sketched by piecewise linear approximations. In particular, the initial slope of An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e085.jpg with respect to An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e086.jpg is found to be An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e087.jpg, whereas the initial slope of An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e088.jpg is An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e089.jpg (cf. section Methods).
The results of this section were obtained by assuming absence of phosphatase in cycle 2, so that cycle 1 behaved as an isolated cycle. In the general case of a 2-cycle system, with some phosphatase acting in the upstream cycle (An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e090.jpg), the obtained results can change, but the modifications are worked out in the Method section. Particularly, one shows that the characteristic range for An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e091.jpg, which are now denoted by An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e092.jpg, has a similar expression to the one defined by Eq.(1), but replacing in this equation An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e093.jpg by An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e094.jpg, where An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e095.jpg is the phosphorylated protein in cycle 2, in the limit of vanishing An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e096.jpg. Nevertheless, it appears that An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e097.jpg (Eq.1) is useful as an upper bound of the characteristic range An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e098.jpg, whose a lower bound is given by An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e099.jpg. Regarding the behavior of the cycle when An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e100.jpg, Eq.(2) still holds whatever the value of An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e101.jpg is, if An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e102.jpg. On the other hand, when An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e103.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e104.jpg, the limit (3) gives the final value of An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e105.jpg only approximately. The exact asymptotic behavior of An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e106.jpg, which cannot be formulated as a simple analytical expression, is given in the Method section (cf. Eq.(39)).
Retroactive Signaling in a 2-cycle Cascade
Having gained insight into how a covalent modification cycle behaves when its total protein An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e107.jpg is varied, we ask how the cycle 2, which is upstream with respect to cycle 1, can be influenced by varying parameters of the downstream cycle. In an experimental setup, the downstream cycle 1 can be characterized by 2 control parameters, namely the total protein An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e108.jpg as seen before, and the amount of phosphatase acting on the deactivation of cycle 1, i.e. An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e109.jpg. In this section the considered control parameters of the 2-cycle cascade will be An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e110.jpg or An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e111.jpg.
What kind of variables can we measure on the upstream cycle to observe the effect of varying the control parameters of the downstream cycle? One possibility is to measure the fraction of activated (e.g. phosphorylated) protein in cycle 2 [17]. The latter is defined by:
A mathematical equation, expression, or formula.
 Object name is pone.0040806.e112.jpg
(4)
Indeed the intermediate complexes An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e113.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e114.jpg both contain some fraction of the phosphorylated protein in cycle 2. In particular, An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e115.jpg represents the fraction of activated protein 2 that is sequestered in cycle 1. Thus this variable embodies the coupling between the two cycles and the source of retroactivity.
Figure 3 shows the variations of the activated fraction An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e116.jpg as a function of parameters An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e117.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e118.jpg under several conditions, depending on cycle 2 is activated or not. As will become clearer in the next sections, the main message of Fig. 3 is that varying the downstream parameters, the retroactivity on the phosphorylated fraction An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e119.jpg is significant only when the upstream cycle starts in deactivated state (left column). It is relatively negligible however, when the upstream cycle starts out activated.
Figure 3
Figure 3
Phosphorylated fraction of protein 2 as a function of 2 control parameters of the downstream cycle 1, namely An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e120.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e121.jpg.
Varying the available protein of the downstream cycle
Let us consider in detail the effect of varying the total protein An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e137.jpg in cycle 1. In practice, this can be achieved in various ways, e.g. by overexpressing the gene coding for protein 1, or by interfering with this quantity by adding a drug able to inhibit this protein [16], or by sequestration of An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e138.jpg resulting from modifying its substrates [19]. Since the retroactive control of cycle 1 on cycle 2 depends crucially on the complex An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e139.jpg, the relevant range of variation for An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e140.jpg can be estimated by An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e141.jpg given by Eq.(1). Therefore, the graphs presented in Figs. 3(B-C) show variations of An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e142.jpg over a range of An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e143.jpg, which is adequate to capture the significant variations of the activated fraction of protein 2 induced by varying An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e144.jpg. Figure 3(B) shows that when cycle 2 is deactivated, the variation of An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e145.jpg can pass from a value close to An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e146.jpg to a value close to An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e147.jpg. Moreover the amplitude variation of An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e148.jpg is maximum when cycle 1 is deactivated. In the latter case, we have seen in the previous section that the non-activated protein An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e149.jpg grows proportionally to An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e150.jpg (Fig. 2(D)). This arbitrarily large increase of the substrate of An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e151.jpg causes the saturation of enzyme 1 for cycle 1 and the complex An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e152.jpg increases towards its maximal allowed value An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e153.jpg like in Fig. 2(B). Therefore, by increasing An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e154.jpg, the phosphorylated fraction An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e155.jpg tends to its maximal value An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e156.jpg; in this case we have a phenomenon of total sequestration of protein 2 in cycle 1.
On the other hand, if cycle 1 is activated and cycle 2 is still deactivated, the results of the previous section show that An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e157.jpg reaches only a fraction of total protein 2, namely An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e158.jpg (Fig. 2(C)). Here we observe a phenomenon of partial sequestration of species 2 by cycle 1. Once this partial sequestration has occurred, a further increase of An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e159.jpg has no longer an effect on the upstream cycle 2. The latter behaves then as a single covalent modification cycle with a reduced amount of protein 2, equal to An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e160.jpg. Therefore, the fraction An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e161.jpg saturates sooner than before and remains inferior to An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e162.jpg. It is seen on Fig. 3(B) (thin red lines) that a piecewise-linear sketch for the variations of An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e163.jpg is sufficient to describe the behavior of An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e164.jpg as a function of An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e165.jpg.
Finally, the case where cycle 2 starts out activated is depicted on Fig. 3(C). In this situation, the phosphorylated fraction An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e166.jpg hardly varies whatever the value of An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e167.jpg is, especially if cycle 1 starts out also activated. If it is deactivated, the variation of An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e168.jpg is non zero, but very weak. In conclusion, in order to enhance the retroactive control of cycle 1 on cycle 2, that is to get the larger possible increase of the fraction of phosphorylated protein in cycle 2, and this as a function of parameter An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e169.jpg of cycle 1, one should start from a situation where both cycles 1 and 2 are deactivated.
Varying the phosphatase of the downstream cycle
We turn now to the retroactive effect of varying the phosphatase of the downstream cycle, An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e170.jpg, on the fraction of phosphorylated protein in cycle 2. Here the total protein An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e171.jpg is fixed. Figures 3(D-E) show the variation of the phosphorylated fraction An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e172.jpg as a function of An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e173.jpg, that is a non-dimensionalized parameter proportional to An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e174.jpg (Eq.(19)). In the same manner as before, one observes that the phosphorylated fraction An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e175.jpg exhibits a significative variation only in the case where cycle 2 is deactivated (Fig. 3(D)). Moreover, the variation of An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e176.jpg is seen only when the control parameter An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e177.jpg varies in the interval An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e178.jpg, that is when cycle 1 passes from its activated to its deactivated state. Then, the level of An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e179.jpg increases proportionally to An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e180.jpg, until reaching a plateau depending on the chosen amount of An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e181.jpg. This plateau, that is the maximum fraction of upstream protein 2 that can be phosphorylated by increasing the phosphatase of the downstream cycle, can be predicted by the expression:
A mathematical equation, expression, or formula.
 Object name is pone.0040806.e182.jpg
(5)
This equation is derived below, in the section Methods. In this equation, An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e183.jpg is the maximum free protein 2 that is activated in the limit of arbitrarily large phosphatase An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e184.jpg. Thus it is unknown a priori but, as a first approximation, it can be replaced by An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e185.jpg (the value of An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e186.jpg in absence of cycle 1). To get a better estimate, the actual value of An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e187.jpg can be found by using an iterative process.
Equation (5) allows us to estimate the level of An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e188.jpg necessary to reach a given fraction An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e189.jpg in the limit of large phosphatase An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e190.jpg:
A mathematical equation, expression, or formula.
 Object name is pone.0040806.e191.jpg
(6)
In summary, in a 2-cycle cascade, in order to create conditions that may substantially modify the fraction of the activated protein in the upstream cycle by perturbing the parameters of the downstream cycle, it is recommended to deactivate the upstream cycle 2. Then, if the downstream cycle 1 is also maintained deactivated a substantial change in An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e192.jpg can be obtained by varying the total protein in the downstream cycle, within a range An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e193.jpg, where An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e194.jpg can be computed as a function of the system parameters (Eq.(1)). In the case where the downstream cycle is activated, it is also possible to change An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e195.jpg by varying the total protein, but in a smaller range than before, namely An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e196.jpg. Varying the phosphatase of the downstream cycle will not modify An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e197.jpg, if cycles 1 and 2 are both deactivated. If, on the other hand, the downstream cycle is activated, then a retroactive signaling in An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e198.jpg can be achieved by modifying the downstream phosphatase, provided that the total protein 1 is sufficiently abundant (cf. Eq.(6)).
The above analysis focussed on the changes of the fraction of phosphorylated protein in cycle 2 because the variable An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e199.jpg is experimentally accessible. However, it is also interesting to describe the behaviors of the 2-cycle cascade in terms of the free proteins in cycle 2, respectively An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e200.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e201.jpg, as will be covered in the next section. Indeed, as discussed below, An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e202.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e203.jpg are responsible for the possible crosstalk effects in cascades with more than 2 cycles.
Downregulation of the free proteins in the upstream cycle
When the upstream cycle 2 is deactivated, Figs. 3(B,D) demonstrate that the phosphorylated fraction An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e204.jpg can be raised by increasing An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e205.jpg or An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e206.jpg from An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e207.jpg. How does this growth affect the amount of free non-active and active proteins in the upstream cycle? It is seen on Fig. 3(F) and (H) that the growth of An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e208.jpg coincides with a decrease of the non-active protein An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e209.jpg. Conversely, the variation of the free activated protein An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e210.jpg is negligible (not shown). Moreover, if cycle 1 is deactivated, addition of the substrate An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e211.jpg in cycle 1 can lead to a complete depletion of protein An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e212.jpg in the upstream cycle. The decrease of An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e213.jpg is roughly linear in the range An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e214.jpg, and then beyond this range it is inversely proportional, An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e215.jpg. When the downstream cycle is activated, the decrease of An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e216.jpg occurs on the smaller range An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e217.jpg and then reaches a plateau that can be analytically predicted (cf. thin continuous lines on Fig. 3)(F)). This situation reflects the phenomenon of partial sequestration of protein of cycle 2 in the dynamics of cycle 1.
As illustrated on Fig. 3(H), the variation of phosphatase in the downstream cycle can also retroactively affects the amount of non-activated protein An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e218.jpg, provided that cycle 1 is activated and that the quantity An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e219.jpg is large enough. This figure also shows that the variation of An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e220.jpg is well approximated by a linear decrease as a function of An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e221.jpg or, equivalently, of An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e222.jpg.
When the upstream cycle 2 is activated, Figs. 3(C,E) showed that a variation of control parameters in cycle 1 entailed only minor changes in the fraction of phosphorylated protein in the upstream cycle 2. This result might convey the idea that when cycle 2 is activated no retroactivity can be observed on cycle 2. In reality, this view would be wrong, because in this case there can exist a large decrease of the free active enzyme An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e223.jpg, as illustrated on Figs. 3(G,I). Indeed, although the fraction An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e224.jpg stayed relatively constant on Figs. 3(C,E), these graphs showed also that the amount of protein 2 sequestrated by cycle 1 increased under a boost of the control parameters An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e225.jpg or An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e226.jpg. In fact, the growth of the intermediate complex An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e227.jpg is compensated by a corresponding decrease in An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e228.jpg, keeping a roughly constant total phosphorylated fraction An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e229.jpg. As before, to get a large variation of An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e230.jpg by making available more protein An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e231.jpg, cycle 1 should be deactivated, leading to the phenomenon of total sequestration in a range of An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e232.jpg (Fig. 3(G)). In contrast, if the control parameter is the phosphatase of the downstream cycle, then a retroactive response on cycle 2 is possible if the downstream cycle starts activated, while An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e233.jpg is large enough (cf. Fig. 3(I)).
Retroactive Signaling in Multi-cycle Pathways
The results obtained with a 2-cycle cascade can predict the effect of retroactivity in short signaling pathways with more than 2 cycles. We first consider a 3-cycle pathway where the activated protein in the cycle at the top of the pathway is an enzyme that activates two other cycles which are not directly linked together (Fig. 4(A)-(B)). In the last section we have demonstrated that a change in the parameters of a downstream cycle, for example the amount of phosphatase or the available protein of the cycle 1, can affect the state of the upstream cycle 2. More precisely, we anticipate that when the phosphatase is increased in cycle 1, it can augment the deactivated form of the protein An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e234.jpg. The latter then can bind to a greater amount of enzyme An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e235.jpg, which become less available for the activation of other substrates such as the protein in cycle 3. Therefore, to implement the scheme of retroactive signaling 1An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e236.jpg2An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e237.jpg3, we start by assuming that the upstream cycle 2 is activated and we consider a signal having the form of an increase in the phosphatase of the downstream cycle 1. We know from the above results (cf. Fig. 3)(I)) that to create a substantial variation in the upstream cycle 2, the phosphatase signal should switch the cycle 1 from an activated state to a deactivated state, considering at the same time a relatively large amount of available protein in cycle 1 (cf. Eq.(6)). Then Fig. 3(I) showed that the switching of the downstream cycle caused a complete decrease of the free phosphorylated enzyme An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e238.jpg in the upstream cycle 2. This behavior of An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e239.jpg can be considered as an output response of the pathway 1An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e240.jpg2 that can be used as the input of the conventional signaling pathway 2An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e241.jpg3. Therefore a retroactive signaling in the 3-cycle pathway 1An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e242.jpg2An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e243.jpg3 shown on Fig. 4 is promoted when there is a strong retroactivity on the segment 2An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e244.jpg1, but a weak retroactivity on the segment 2An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e245.jpg3 with respect to the considered input. Another condition is that, when the downstream cycle 1 is completely activated (i.e. when the phosphatase signal on cycle 1 is absent), cycle 3 should be activated by cycle 2. In this case only, it will feel the strong decay of the free phosphorylated enzyme in the upstream cycle 2 caused by its sequestration in the compounds of cycle 1. Figure 4(A) illustrates this type of signaling. One sees that cycle 3 can be switched on or off by varying the phosphatase regulating the input cycle 1.
Figure 4
Figure 4
Retroactive signaling in multi-cycle pathways.
A similar retroactive signaling in the same 3-cycle pathway can be achieved by modifying not the phosphatase but the available protein in the starting cycle 1. Keeping the same parameters as above, Fig. 4(B) shows that increasing the signaling protein 1 from a low value to four times the characteristic range An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e277.jpg entails a deactivation cycle 3. This happens because of the retroactive mechanism between cycles 1 and 2, as discussed in the previous section (cf. Fig. 3)(G)). In the latter case, the increase of the total protein available in the downstream cycle 1 downregulated the activated enzyme in the upstream cycle 2, assuming that the downstream cycle was deactivated. Here again, by combining a large retroactivity between cycles 1 and 2, but a low one between cycle 2 and 3, one achieves a retroactive signaling between cycle 1 and 3.
In some covalent modification cycles, the deactivated protein can serve also as an enzyme for another protein modification [18], [20]. For example a variation of the motif shown on Fig. 4(A) is a 3-cycle network consisting of one upstream cycle and 2 downstream cycles activated respectively by the phosphorylated and non-phosphorylated forms of protein in the upstream cycle. Then we checked that a change in the phosphatase of one downstream cycle can produce a transition in the other downstream cycle activated by the non-phosphorylated protein in the upstream cycle (not shown).
To extend the possibility of retroactive signaling to more complex situations than a 3-cycle pathways we now consider a motif of a 5-cycle network in which the activated protein in the top cycle acts as the enzyme regulating two 2-cycle cascades, as shown on Fig. 4(C). Can we produce in this case an example of retroactive signaling from one bottom cycle to the other bottom one, numbered respectively by 1 and 5, initiated for instance by a phosphatase variation in cycle 1? Here, the study of the 2-cycle and the 3-cycle systems reported above can also help to answer this question. In this 5-cycle pathway, the subnetwork formed by cycles 2-3-4 has the same topology than the 3-cycle pathway discussed previously. Therefore, since this latter subsystem is suitable for retroactive signaling, let us consider the subnetwork 2-3-4 with the same parameters as considered for the 3-cycle network of Fig. 4(B). Then we can link to this system the cycle 1 downstream to cycle 2, and the cycle 5 downstream to cycle 4. For recall, cycle 2 is deactivated. Now we use the result shown on Fig. 3(H), showing that increasing the phosphatase in cycle 1 is going to reduce the available protein in cycle 2 in such a way that the free activated enzyme in cycle 3 is strongly reduced. This, in turn, deactivates cycle 4, and then cycle 5 as for standard cascades. This example of retroactive signaling scenario is seen on Fig. 4(C) where the increase in the phosphatase in cycle 1 entails not only the deactivation of cycle 1 (not shown) but also the deactivation of the remote cycle 5. Let us remark that this crosstalk effect can propagate to possible downstream effectors activated by cycle 5.
Cell signaling is generally thought in terms of a series of reversible biochemical reactions that are chained together in a feedforward network where extra connections, called feedbacks, could regulate the information flow from bottom-up. In particular the expression “signaling cascade” was coined to suggest the idea of an upstream to downstream signal transmission. In the simplest scheme of a cascade of two covalent modification cycles, the input signal typically is a steep increase of the enzyme modifying the first protein. Then the latter acts as the enzyme activating the second protein whose concentration is interpreted as the output of this system. In this paper, however, we show that in such a cascade a retroactive signaling is also possible, i.e. transmitting an input signal from downstream to upstream, and we predict conditions for which this phenomenon can be observed. The input signal is now a variation of a biochemical species that can change the state of the downstream cycle. Two cases are considered, namely a change of the total amount of the downstream signaling protein, or a variation of the phosphatase deactivating the same protein. In both cases we work out characteristic ranges of the concentrations of the species for which a retroactive effect can be observed in the upstream cycle. Moreover we show that this potentiality can help to perform retroactive signaling in short multi-cycle pathways.
A covalent modification cycle is generally described as a two-state entity for which the total level of protein is fixed. However, like all the molecules inside the cell, this signaling protein is subjected to a turnover governed by several processes, including synthesis and degradation [21]. The changes in these processes alters the total level of proteins. For example the degradation of several signaling proteins is actively regulated by proteases, which has consequences on the signaling dynamics [22]. The present study shows that the variation of the total amount of available protein in a downstream signaling cycle can also affect the states of signaling modules upstream in the transduction cascade.
There are several ways to modify the available protein in the downstream cycle in a cascade of covalent modifications. One way is to change the amount of substrates to which the activated protein of the downstream cycle can bind. For example, in a recent study reported in [17], the authors perform experiments on the ERK/MAPK pathway associated with the syncytium state of the Drosophila embryo. They manage to modify the amount of substrates of the doubly phosphorylated form of ERK by constructing mutants missing the corresponding substrates. Another way to alter the available protein in the downstream cycle is to add in the medium a kinase inhibitor that can bind to the activated enzyme at the end stage of the pathway [16], [23]. Both ways can be modeled by considering an additional chemical reaction of the form:
A mathematical equation, expression, or formula.
 Object name is pone.0040806.e278.jpg
(7)
where An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e279.jpg represents a substrate or a kinase inhibitor of the downstream protein An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e280.jpg. Then it can be shown that the set of stationary state equations of the signaling pathway is affected only in the conservation equation for the total protein An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e281.jpg. More precisely this latter quantity is replaced by An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e282.jpg, where 2 additional parameters characterize respectively the total amount An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e283.jpg of binding chemical species and the dissociation constant An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e284.jpg. Thus, the effect of varying An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e285.jpg is qualitatively analog to changing the amount of available protein An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e286.jpg. In particular, when the affinity of An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e287.jpg for protein 1 is high (i.e., An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e288.jpg small), the available downstream protein is approximately reduced by An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e289.jpg. Therefore under this hypothesis the upstream response in a 2-cycle cascade to a variation of An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e290.jpg can straightforwardly be inferred from the curves shown on Figs. 3. For instance, from Fig. 3(B) one predicts that in a 2-stage cascade increasing An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e291.jpg can decrease the phosphorylated fraction An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e292.jpg of the upstream protein, especially if the upstream cycle is in a deactivated state. This phenomenon may be the source of undesirable off-target effects in targeted therapies based on kinase inhibitors [16].
In Ossareh et al, the authors performed mathematical analysis of retroactivity in a signaling cascade with an arbitrary number of stages. They achieved necessary and sufficient conditions for which retroactivity exists in such chains. Their analysis is based on the linearization of the steady state equations in order to predict how a small downstream perturbation is amplified in the upstream response of an arbitrarily long signaling chain. Those results are complementary to the ones presented in the present paper, in the sense that here we consider short signaling pathways but our analysis is based on the resolution of the full nonlinear equations, and not only on the linearized system. So, it is concerned with arbitrarily large perturbations of the parameters. In fact we show that retroactive signaling is meant to work only for a characteristic range of parameter variations that we analytically estimate by working on the asymptotic behaviors of the system for small and large parameter perturbations.
Signaling pathways are regulated by several mechanisms, like positive or negative feedback loops linking the output of the cascades and some upstream stages. This requires the existence of specific chemical interactions between the output protein of the cascade and the upstream proteins that are involved in the feedback loop. Our study shows that the property of retroactive signaling can be another way to regulate the functioning of signaling cascades in branched pathways, without explicit feedbacks. In fact, we can further speculate that in natural signaling pathways with possibly several branches, some of the latter would be sensitive to retroactivity and be devoted to the regulation of the usual branches, where signals go in the top-down direction. These results prompt new experiments concerning signaling cascades and possibly new ways to interpret previous results.
Our theoretical study is performed in the framework of coupled nonlinear equations describing the rate of changes of protein concentrations in signaling cascades formed of covalent modification cycles. The model equations are deterministic and based on the law of mass action. Only stationary states of these equations are analysed and thus the mathematical method amounts to solving sets of algebraic nonlinear equations. Thus the issue of how the biochemical species reach the equilibrium is not discussed here, as it has been addressed in some previous studies [8], [9], [24]. In this respect our analysis is independent of questions related to possible time-scale differences between the kinetics of enzyme/substrate. For example, the usual quasi-steady state approximations are not to be considered since all the variables are at equilibrium.
Let us note that we assume that the studied signaling pathways possess a stable equilibrium. Although in this paper we will not explicitly discuss the generality of this assumption by performing the linear stability analysis of the equation set, the hypothesis of a stable equilibrium is consistent with the current knowledge. In the literature, published results indicate that non steady behaviors (e.g. sustained oscillations) can arise in signaling cascades only with the concomitant occurence of bistability in the signaling modules [6], [25]. However, this situation was only met with signaling modules described by double-phosphorylations cycles, like in the MAPK cascade. Here the considered signaling pathways do not include double-phosphorylation. Therefore this paper will not consider retroactive signaling in oscillating systems.
Steady States in Basic Models of Signaling Cascades
Let us introduce the notations used for writing the equations in the case of the simple 2-cycle cascade as depicted on Fig. 1(A). Assuming that this system is isolated from other biochemical reactions, the chemical equations describing the transformations of these species can be written as follows:
A mathematical equation, expression, or formula.
 Object name is pone.0040806.e293.jpg
A mathematical equation, expression, or formula.
 Object name is pone.0040806.e294.jpg
A mathematical equation, expression, or formula.
 Object name is pone.0040806.e295.jpg
A mathematical equation, expression, or formula.
 Object name is pone.0040806.e296.jpg
(8)
where An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e297.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e298.jpg denote enzyme concentrations, whereas An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e299.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e300.jpg (An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e301.jpg) are intermediate enzyme-substrate complexes. These chemical equations readily generalize to the other motifs, e.g. the one shown on Fig. 1(B). The kinetic equations of the state variables of the cascades are written using the law of mass actions.
A mathematical equation, expression, or formula.
 Object name is pone.0040806.e302.jpg
(9)
A mathematical equation, expression, or formula.
 Object name is pone.0040806.e303.jpg
A mathematical equation, expression, or formula.
 Object name is pone.0040806.e304.jpg
A mathematical equation, expression, or formula.
 Object name is pone.0040806.e305.jpg
(10)
A mathematical equation, expression, or formula.
 Object name is pone.0040806.e306.jpg
A mathematical equation, expression, or formula.
 Object name is pone.0040806.e307.jpg
with the conservation laws for the total proteins An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e308.jpg and total enzyme concentrations An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e309.jpg:
A mathematical equation, expression, or formula.
 Object name is pone.0040806.e310.jpg
(11)
A mathematical equation, expression, or formula.
 Object name is pone.0040806.e311.jpg
(12)
A mathematical equation, expression, or formula.
 Object name is pone.0040806.e312.jpg
A mathematical equation, expression, or formula.
 Object name is pone.0040806.e313.jpg
A mathematical equation, expression, or formula.
 Object name is pone.0040806.e314.jpg
Since we focus only on the stationary states of the system, the time-derivatives of the concentrations can be equaled to zero. This enables to express the variables An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e315.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e316.jpg (An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e317.jpg) in terms of the protein concentrations as follows:
A mathematical equation, expression, or formula.
 Object name is pone.0040806.e318.jpg
A mathematical equation, expression, or formula.
 Object name is pone.0040806.e319.jpg
A mathematical equation, expression, or formula.
 Object name is pone.0040806.e320.jpg
(13)
A mathematical equation, expression, or formula.
 Object name is pone.0040806.e321.jpg
(14)
with the coefficients An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e322.jpg (An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e323.jpg) defined as a function of he kinetic parameters An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e324.jpg. One thus recognizes the usual Michaelis-Menten form for the substrate-enzyme complexes. The substitution of these expressions in Eqs.(9)–(10) and in the conservation laws given Eqs.(11)–(12) leads finally to 4 algebraic equations in the unknowns An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e325.jpg. Therefore a reduced set of equations (9–14) can be written as:
A mathematical equation, expression, or formula.
 Object name is pone.0040806.e326.jpg
(15)
A mathematical equation, expression, or formula.
 Object name is pone.0040806.e327.jpg
(16)
A mathematical equation, expression, or formula.
 Object name is pone.0040806.e328.jpg
(17)
A mathematical equation, expression, or formula.
 Object name is pone.0040806.e329.jpg
(18)
A 2-cycle cascade involves 4 enzymatic reactions. Each of those can be characterized also by their maximum reaction rates (An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e330.jpg). We denote the latter as follows:
A mathematical equation, expression, or formula.
 Object name is pone.0040806.e331.jpg
(19)
The upper bound of the velocity An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e332.jpg, which describes the activation of An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e333.jpg, will depend on the total protein in cycle 2. In the following section we will seek the conditions under which the variations of parameters of cycle 1 produce a significant effect in cycle 2 due to retroactivity. As will be discussed, this property will depend on the states of the variables of both, upstream and downstream cycles. We will use the following terminology: cycle An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e334.jpg is said to be activated if An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e335.jpg. Otherwise, it is said to be deactivated. This property is easily related to the ratio An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e336.jpg in the symmetric case An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e337.jpg. Then cycle An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e338.jpg is activated if and only if An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e339.jpg [5].
The following sections give details on the derivation of equations (1)–(3) and (5) used in the section Results.
Variation of the Total Downstream Protein in a 2-cycle Cascade
Let us consider a 2-cycle cascade as drawn on Fig. 2(A), with total upstream protein An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e340.jpg, total downstream protein An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e341.jpg, and total deactivating enzyme An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e342.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e343.jpg, respectively for the upstream and downstream cycles. We wish to determine a suitable value of An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e344.jpg that can be used as a characteristic dose of downstream protein inducing a retroactive effect on the upstream cycle. The steady state of this system is given by the solution of Eqs.(15)–(18). As motivated above, we focus on the behavior of An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e345.jpg, i.e. the intermediate substrate-kinase complex, which at equilibrium is given by An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e346.jpg. The change of An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e347.jpg as a function of the total protein An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e348.jpg is illustrated on Fig. 2(B)–(C) in the case where An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e349.jpg, but the behavior is the same if An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e350.jpg. It can be sketched by an increase of An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e351.jpg proportional to An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e352.jpg followed by a saturation to a constant value, that is An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e353.jpg when cycle 1 is deactivated (i.e. An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e354.jpg). Therefore the quantity
A mathematical equation, expression, or formula.
 Object name is pone.0040806.e355.jpg
(20)
defines a proper characteristic range of An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e356.jpg for the variation of An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e357.jpg. The upper index of An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e358.jpg reminds that the result of the right-hand side of this equality depends on the value of An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e359.jpg. In particular, we will be interested to the case An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e360.jpg which corresponds to the situation of the isolated signaling cycle 1 with kinase An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e361.jpg and with phosphatase An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e362.jpg. To simplify the notations, we will denote in the following:
A mathematical equation, expression, or formula.
 Object name is pone.0040806.e363.jpg
(21)
and we will show that Eq.(1) holds with this definition. Since An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e364.jpg, one deduces that
A mathematical equation, expression, or formula.
 Object name is pone.0040806.e365.jpg
(22)
Now, it suffices to compute the derivative of An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e366.jpg w.r.t. An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e367.jpg and evaluate it at An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e368.jpg. This can be analytically performed by differenciating each equation of the system (15)–(18) with respect to An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e369.jpg. This calculation provides a system of linear equations in the coupled variables An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e370.jpg. Solving this linear system we find that the solution can be written as:
A mathematical equation, expression, or formula.
 Object name is pone.0040806.e371.jpg
(23)
A mathematical equation, expression, or formula.
 Object name is pone.0040806.e372.jpg
(24)
A mathematical equation, expression, or formula.
 Object name is pone.0040806.e373.jpg
(25)
A mathematical equation, expression, or formula.
 Object name is pone.0040806.e374.jpg
(26)
where An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e375.jpg is the activated upstream enzyme when An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e376.jpg, and An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e377.jpg. Let us remark that An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e378.jpg or An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e379.jpg means respectively that the upstream cycle is highly activated or strongly deactivated.
By combining Eqs.(20), (22) and (25), one obtains the characteristic range for An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e380.jpg, as defined by Eq.(20):
A mathematical equation, expression, or formula.
 Object name is pone.0040806.e381.jpg
(27)
In the case where An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e382.jpg, the upstream cycle is such that there is no phosphatase to deactivate it, so that An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e383.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e384.jpg. In this case, using the definition An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e385.jpg, Eq.(27) becomes the sought relation Eq.(1), i.e.:
A mathematical equation, expression, or formula.
 Object name is pone.0040806.e386.jpg
(28)
One easily shows that An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e387.jpg (because An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e388.jpg). Therefore An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e389.jpg can be used as an upper bound of the characteristic range for An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e390.jpg. Particularly, if the downstream cycle is strongly activated, then An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e391.jpg and then An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e392.jpg is an excellent approximation of An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e393.jpg. On the other hand, if the downstream cycle is strongly deactivated, so that An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e394.jpg, one can use An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e395.jpg, that is the lower value reached by An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e396.jpg in the limit An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e397.jpg.
Let us note that using the definition of An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e398.jpg in the simple situation An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e399.jpg, the derivatives An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e400.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e401.jpg in eqs.(25)–(26) can be written in a compact form, namely:
A mathematical equation, expression, or formula.
 Object name is pone.0040806.e402.jpg
(29)
Incidently, these expressions give the initial slope of the curves drawn on Figs. 2(D-E).
Now, to justify Eqs.(2)–(3) given in the Results, we wish to compute the asymptotic values of An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e403.jpg in the limit of large An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e404.jpg. As suggested by the numerical computations, we first suppose that the asymptotic behavior of these variables are described by:
A mathematical equation, expression, or formula.
 Object name is pone.0040806.e405.jpg
(30)
A mathematical equation, expression, or formula.
 Object name is pone.0040806.e406.jpg
(31)
A mathematical equation, expression, or formula.
 Object name is pone.0040806.e407.jpg
(32)
A mathematical equation, expression, or formula.
 Object name is pone.0040806.e408.jpg
(33)
where An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e409.jpg are unknown constants to be worked out. Substitution of these relations in Eqs.(15)–(18) with An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e410.jpg determines An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e411.jpg. Since An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e412.jpg must be positive, this case is only consistent with the hypothesis An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e413.jpg, that is equivalent to Eq. (2) given in the Result section. Let us notice that here the result is independent on considering the case An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e414.jpg or not. The values of the other unknowns are found to be An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e415.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e416.jpg, and
A mathematical equation, expression, or formula.
 Object name is pone.0040806.e417.jpg
(34)
Secondly, in order to justify Eq.(3), we suppose another asymptotic behavior for the system variables in the limit of large An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e418.jpg:
A mathematical equation, expression, or formula.
 Object name is pone.0040806.e419.jpg
(35)
A mathematical equation, expression, or formula.
 Object name is pone.0040806.e420.jpg
(36)
A mathematical equation, expression, or formula.
 Object name is pone.0040806.e421.jpg
(37)
A mathematical equation, expression, or formula.
 Object name is pone.0040806.e422.jpg
(38)
where An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e423.jpg are new unknown constants to be determined. The calculation can be done in 2 steps. First An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e424.jpg can be calculated by solving Eqs.(15)–(16) which here becomes:
A mathematical equation, expression, or formula.
 Object name is pone.0040806.e425.jpg
A mathematical equation, expression, or formula.
 Object name is pone.0040806.e426.jpg
This system can be interpreted as finding the activated and deactivated proteins in the upstream cycle with the reduced amount of total protein An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e427.jpg. The latter must be positive, that is equivalent to the condition An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e428.jpg related to Eq.(3). The solution of this system is hard to write explicitly, except in the case An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e429.jpg where An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e430.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e431.jpg.
The second step is to solve Eqs.(17)–(18) in the limit An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e432.jpg. Then one easily finds that An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e433.jpg, and therefore
A mathematical equation, expression, or formula.
 Object name is pone.0040806.e434.jpg
(39)
where An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e435.jpg has been found in the first step. The latter equation generalizes Eq. (3), which holds in the case where An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e436.jpg. Then the simple expression of An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e437.jpg leads to the equality An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e438.jpg which is Eq. (3). Finally the value of An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e439.jpg is the same expression as Eq.(34), but swapping the “primed” and “not primed” parameters.
In conclusion, by using Eqs.(29)–(38), let us note that we can sketch the behavior of An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e440.jpg and of An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e441.jpg as a function of An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e442.jpg as piecewise linear graphs (see red lines on Figs. 2(D)–(E)).
Variation of the Downstream Phosphatase in a 2-cycle Cascade
Let us consider a 2-cycle cascade as drawn on Fig. 1(A) and suppose now that the control parameter is the quantity of phosphatase An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e443.jpg in the downstream cycle 1. We wish to prove the result of Eq.(5) giving the phosphorylated fraction An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e444.jpg of protein in cycle 2 in the limit of large An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e445.jpg.
First recall that An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e446.jpg is defined by the chemical compounds containing An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e447.jpg, namely (cf. Eq.(4)):
A mathematical equation, expression, or formula.
 Object name is pone.0040806.e448.jpg
Thus, by using the steady expression for the complexes An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e449.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e450.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e451.jpg is also expressed as:
A mathematical equation, expression, or formula.
 Object name is pone.0040806.e452.jpg
(40)
We wish to remove the dependency in An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e453.jpg of this expression. The steady state equations of cycle 1 can be written as follows:
A mathematical equation, expression, or formula.
 Object name is pone.0040806.e454.jpg
(41)
A mathematical equation, expression, or formula.
 Object name is pone.0040806.e455.jpg
Since An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e456.jpg is an enzyme, in the limit An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e457.jpg, none of the biochemical variables should diverge. Therefore the second equation in the above system implies that in this limit one has An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e458.jpg. Thus the Eq.(41) can be simplified into the form:
A mathematical equation, expression, or formula.
 Object name is pone.0040806.e459.jpg
(42)
This enables to write An external file that holds a picture, illustration, etc.
Object name is pone.0040806.e460.jpg as:
A mathematical equation, expression, or formula.
 Object name is pone.0040806.e461.jpg
(43)
And by using this expression in Eq.(40), one finds Eq.(5), or:
A mathematical equation, expression, or formula.
 Object name is pone.0040806.e462.jpg
(44)
Footnotes
Competing Interests: The authors have declared that no competing interests exist.
Funding: ACV is a member of the Carrera del Investigador Científico (CONICET) and was supported by the Department of Defense Breast Cancer Research Program, the Center for Computational Medicine and Bioinformatics (University of Michigan), and the Agencia Nacional de Promoción Científica y Tecnológica (Argentina). SDM receives funding from the Breast Cancer Research Foundation. The international program of scientific collaboration PICS 05922 between CNRS (France) and CONICET (Argentina) supported this work. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
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