The main contribution of this work is to propose a new one-variable per cycle model for signaling cascades of covalent modification cycles, consistent with a mechanistic complete description. Our model reveals new and biologically relevant properties of such cascades. These properties are characterized completely for the case of single-phosphorylated cascades. Furthermore, single and doubly-phosphorylated cases are compared regarding their stimulus-response curves, while a more exhaustive characterization of the scheme involving double phosphorylation will be presented in a future article.
The scheme in , which has been employed by many groups, is suggestive of the concept of a “cascade”. From a systemic point of view, a cascade is a system composed of units, the output of which is successively an input to the next unit. Based on this structure, powerful concepts from control theory can be applied successfully to the study of signaling cascades 
. Although these concepts have proven its utility in many contexts, this kind of schematic representation implicitly conveys the idea that a signaling cascade is only a feed-forward chain in which signal transmission is analogous to a domino effect 
. Our study sheds a different light on this system, showing that this schematic representation can be misleading, since it turns out that each unit is actually coupled not only to the following one but also to the previous one, and interesting dynamics can arise from these interactions.
Our initial motivation for developing a new one-variable description of signaling cascades, was the following observation. The main assumption underlying the GK description of a single cycle is that the concentration of the target protein is in large excess compared to those of the converting enzymes. Holding the same assumption over a cascade of units would mean that the target proteins are in higher and higher concentration as the cascade progresses, since they act as the transforming enzyme for the following cycle. To our knowledge, this important issue has not been remarked upon in the literature, except for a brief comment in the work of Millat et. al. (
, page 11).
In order to get more insight into this point, we have sought special limiting cases for which the mechanistic and the GK-like model are in good agreement. However, it turns out that the dynamics of the signaling cascade described by the mechanistic and the GK-like models cannot be compared consistently. The fundamental reason for this mismatch is that a careful perturbation analysis applied to the mechanistic model provides a different set of equations.
We note that in search for an adequate set of hypothesis leading from the mechanistic equations to the model given by Equations 4, we have studied an alternative scheme in which the modified protein Yi*
is not directly the kinase of the next reaction. Instead, we studied the case where Yi*
activates that kinase. This scheme was suggested by the work of Goldbeter 
. The resulting equation (see Text S7
) is fundamentally different from the GK-like model. In reality, no set of assumptions can give rise to the GK-like model as a limiting case of our model.
Our mathematical method relies on the standard quasi-steady state assumption (QSSA), which can be applied under well defined conditions to elicit a clear separation between the slow and fast dynamics of the mechanistic model. Under this standard QSSA framework, our analysis shows that a good slow variable for which evolution equations can be written is the sum of the free activated enzyme which is available in the ith
cycle plus the amount of this protein which is captured by the next inter-converting cycle. The idea of working with a mixed variable xi
can be further generalized by considering the “total” variable corresponding to the total amount of activated enzyme found not only as free molecules or bound to the next substrate, but also complexed with the reverse enzyme E′i
. In fact, this choice is the key ingredient of the method called the “total” quasi-steady state approximation (tQSSA) which has been proved to be a simple but most efficient extension of the standard QSSA 
. The application of this extended framework to the description of the signaling cascade of is concerned with our current research. In the same context, other authors have recently applied the tQSSA method to the study of small networks of GK cycles 
. These systems do not form cascades, but involve a more complicated coupling between the units. Nevertheless, their results show that indeed the tQSSA method is successful in obtaining a reduced set of equations, with one variable per cycle, which faithfully reproduces the dynamics of the network for a large range of system parameters.
Even in the less extended QSSA framework, the conditions under which the model is valid are made clear. Under such conditions, our new model is indeed in perfect agreement with the complete mechanistic model (). Those conditions are expressed in terms of three key parameters (Equation 5) we have defined to simplify the study. Even though the phenomenological equations, Equation 4, are appealing because of their simpler form and modular nature, we could not find any set of assumptions that would enable us to recover those descriptions. Our simplified model reveals properties of signaling cascades that were either hidden by the complex structure of the complete mechanistic model or lost in the simplified phenomenological descriptions.
It was stated that the reduced mechanistic model is valid whenever these two conditions are satisfied: εi«
1 and µi ηi~εi
. The study of the performance of the new approximation ( and the corresponding computed errors) makes it clear that even when those conditions are satisfied only moderately, the new model is still robust in approximating the complete description. As an example, we have computed a 5% error for ε
1, and µ
0.5 (meaning µη~
). Moreover, we have observed that the steady state predictions of the reduced model are highly accurate. Therefore the properties of signaling cascades we are unveiling thanks to the new reduced model, are not restricted by a tight relationship between concentrations and reaction rates hard to achieve in in vivo
or in vitro
All the novel properties of a signaling cascade reported in this paper are linked, as previously mentioned, to the negative feedback from each unit to the previous one. This backward negative feedback can produce damped temporal oscillations in the chain, or it can create amplified “pathway” oscillations in the steady states of the cascade. Interestingly, it can also transduce a signal both forward and backwards. Given the multi-branched complex nature of many signal transduction pathways, this finding could have wide implications and can help focus further experimental investigation.
It has recently been reported that the 3-level MAPK cascade has autonomous oscillations without any kind of added explicit feedback 
. Following a systematic numerical exploration of the corresponding mechanistic model 
, the authors provide a qualitative description of the mechanism responsible for these sustained oscillations. Their explanation strongly suggests the necessity of a bistable behavior at the second or third levels of the cascade, thus requiring double-phosphorylation at these stages 
. Consistent with their findings, we have observed only damped oscillations in the dynamics of the single-phosphorylated cascade (Equations 6–7), which has been the main focus of the present work. Interestingly, preliminary numerical simulations of our reduced doubly-phosphorylated cascade model (Text S3
), indicates that these autonomous oscillations are recovered in the simplified description.
The stimulus-response curves of the new model were also investigated (). They have the usual sigmoidal shape characteristic of ultrasensitive responses; however, they exhibit lower steepness when compared with the output of the GK-like model. This result corroborates the conclusions stated in the work of Blüthgen et al. 
, where an analysis of the effect of sequestration was conducted. This effect is partially mitigated by double-phosphorylation (), as expected from the literature 
To further characterize the new model within realistic conditions, we have studied it subject to different sets of published parameters corresponding to a well-known signaling pathway, such as the MAPK one ( and , and Text S6
). We have found that the ability of the model to transduce a signal both forward and backwards is widespread and that the effect is of enough magnitude to allow experimental verification.
Finally, we have applied a modular response analysis to determine the network architecture of the cascade described by the new model equations (). This well-known approach enables not only to test the bidirectional structure of the cascade, but also to estimate the relative strength of the backward interaction.
In summary, our findings do not at all weaken the importance of previous models like the GK-like models and those with linear rates. To the contrary, the results of our model provide a different approach to deal with a simple one-variable per cycle model to describe signaling cascades. We hope that our contribution will help in the understanding of existing models for signaling cascades, will improve the description of available data, and will inspire both theoretical and experimental investigation.