In this section we will analyze five biological networks as case studies. Three of such examples, the L-arabinose, the sRNA and the Lac Operon networks, model the interaction and control of expression of a set of genes. The cAMP and the MAPK pathways are instead signaling networks, namely they represent sets of chemical species interacting for transmission and processing of upstream input signals. These networks are all well-known in the literature, and have been characterized mainly through experimental and numerical methods, although the MAPK pathway, for instance, has been thoroughly analyzed using the theory of monotone systems [17].

We will provide rigorous proofs that these networks are either mono or multi-stable in a robust manner. Such demonstrations rely on Lyapunov functions and invariant sets theory, according to our proposed methodology. In some cases, we are also able to provide bounds on their speed of convergence.

### The L-arabinose network

The arabinose network is analyzed in [28] as an example of feedforward loop. Two genes *araBAD* and *araFGH* are regulated by two transcription factors, AraC and CRP. AraC is a repressor, but turns into an activator when bound to the sugar L-arabinose. CRP is an activator when bound to the inducer cyclic AMP (cAMP), which is produced when cells are starving upon glucose (not produced during growth on glucose). CRP also binds to the *araC* promoter and enhances transcription of AraC, which has a significant basal rate of expression (i.e. it is produced by the cell also in absence of inducer CRP). A very simple model for this network can be derived by defining the state variables *x*_{1} and *x*_{2}, respectively the concentrations of the transcription factor AraC and of the output protein araBAD. The concentration of the transcription factor CRP is considered an external input *u*:

where *α*_{1}, *α*_{2} are the degradation and dilution rates of *x*_{1}, *x*_{2} respectively. The basal production rate of *x*_{1} (AraC) is *p*_{1}. The activation pathways are modeled by Hill functions *f* (*u*, *K*) = *u*^{H} /(*K*^{H} + *u*^{H} ), where *H* is the Hill coefficient and *K*_{
ij
} are the activation thresholds. The model can be recast into the general structure (4), which includes model (5) as special case::

where *u* is *nonnegative*-*constant*, *c*_{1}, *b*_{11} and *b*_{22} are *positive-constant*, while *c*_{1u}(*u*) and *c*_{2u 1}(*u*) are *sigmoidal* with respect to *u*, the latter increasing with respect to *x*_{1}. The graph representation of this network is in Figure 3A.

For this elementary network the analysis is straightforward. Variable *x*_{1} is not affected by *x*_{2}. Since *c*_{1u}(*u*) is bounded, *x*_{1} is also bounded and converges to an equilibrium point which is monotonically increasing in *u*. In turn, *x*_{2} is also positive and bounded for any value of *u* and stably converges to a unique equilibrium point , which is a monotonically increasing function of *u* (partially activated by ). The positive term *c*_{1} prevents *x*_{1}(*t*) and *x*_{2}(*t*) from staying at zero. It is worth remarking that the hierarchical structure of this network greatly facilitates the analysis; equilibria can in fact be iteratively found and their stability properties characterized.

### The sRNA pathway

Small regulatory RNAs (sRNA) play a fundamental role in the stress response of many bacteria and eukaryotes. In short, when the organism is subject to a stimulus that threatens the cell survival, certain sRNA species are transcribed and can down-regulate the expression of several other genes. For example, when *E. coli* cells are lacking a source of iron, the sRNA RyhB (normally repressed by the ferric uptake regulator Fur) is expressed and rapidly induces the degradation of at least other 18 RNA species encoding for non-essential proteins which use up Fe molecules. This allows essential iron-dependent pathways to use the low amount of available iron. Quantitative studies of the sRNA pathways have been carried out in [29–31]. Let us define *x*_{1} as the RNA concentration of the species which is targeted by the sRNA and *x*_{2} as the concentration of sRNA. The model often proposed in the literature is:

where *α*_{1}, *α*_{2} are the transcription rates of *x*_{1} and *x*_{2} respectively, *β*_{1}, *β*_{2} are their degradation rates (turnover), and *k* is the binding rate of the species *x*_{1} and *x*_{2}. The formation of the inactive complex *x*_{1} · *x*_{2} corresponds to a depletion of both free molecules of *x*_{1} and *x*_{2}. If *α*_{1} < *α*_{2} the pathway successfully suppresses the expression of the non-essential gene encoded by *x*_{1}. This model can be embedded in the general family:

by setting *b*_{12} = *kx*_{1} and *b*_{21} = *kx*_{2} (note that *b*_{12}(0) = *b*_{21}(0) as required). From our list of properties: *c*_{1}, *c*_{2}, *b*_{11} and *b*_{22} are *positive-constant*; *b*_{12}(*x*_{1}, *x*_{2}) and *b*_{21}(*x*_{1}, *x*_{2}) are *increasing-asymptotically-unbounded* in both variables; and *b*_{12}(*x*_{1}, *x*_{2})*x*_{2} = *b*_{21}(*x*_{1}, *x*_{2})*x*_{1} at all times. This network can be represented with the graph in Figure 2A.

The sRNA system is positive, because the nonnegativity Assumptions 1 and 4 are satisfied. The preliminary screening of this network tells us that each variable produce an inhibition control on the other, which increases with the variable itself. In other words *x*_{1} is "less tolerant" to an increase of *x*_{2} if the latter is present in a large amount. This means that the sum *x*_{1} + *x*_{2} is strongly kept under control. Also the mismatch between the two variables is controlled. ^{1} To prove stability of the (unique) equilibrium , we will use the 1-norm as Lyapunov function (see Figure 2B). This choice has a remarkable interpretation: denoting by and the sum and the difference of the two variables (referred to the equilibrium) we have

thus the function represents the worst case between the sum and the mismatch.

The following proposition shows that the sRNA pathway is a typical system in which robustness is structurally assured. We report the full demonstration of this proposition, because its steps and the techniques used are a model for the subsequent proofs in this paper.

**Proposition 2** *The variables of system (8) are bounded for any initial condition x*_{1}(0), *x*_{2}(0) ≥ 0. *The system admits a unique asymptotically stable equilibrium point**and the convergence is exponential*:

*for some β* > 0 *and any x*_{1}(0) ≥ 0, *x*_{2}(0) ≥ 0. *Moreover, no oscillations are possible around the equilibrium, in the sense that the condition**or**occurs at most once*.

**Proof** To prove boundedness of the variables we need to show the existence of an invariant set

Proposition 1 guarantees that the positivity constraints are respected. Then we just need to show that the constraint *x*_{1} + *x*_{2} ≤ *κ* cannot be violated for sufficiently large *κ* > 0. The derivative of function *s* (*x*_{1}, *x*_{2}) = *x*_{1} + *x*_{2} is

Define *κ* = (*c*_{1} + *c*_{2})/min {*b*_{11}, *b*_{22}} then for *s*(*x*_{1}, *x*_{2}) > *κ* the derivative becomes negative so *s*(*x*_{1}, *x*_{2}) cannot exceed *κ* (See Theorem 1).

Boundedness of the solution inside a compact set assures the existence of an equilibrium point. Let be any point in which the following equilibrium conditions holds:

The behavior of the candidate Lyapunov function:

will be examined in the different sectors represented in Figure 2B. Let us start by considering the sector defined by and (APB in Figure 2B) for which . In such a sector the Lyapunov derivative is:

where we have subtracted the null terms (10) and where we have exploited the fact that *b*_{12}(*x*_{1}, *x*_{2})*x*_{1} = *b*_{21}(*x*_{1}, *x*_{2})*x*_{2} is increasing in both variables. The inequality (CPD in Figure 2B) can be similarly proved to hold in the sector and .

Consider the sector defined by and (DPA in Figure 2B) for which in such a sector the Lyapunov derivative is:

Note that in the last step we have added and subtracted the null terms (10). In the opposite sector (BPC in Figure 2B) and , we can prove that . We just proved that

with *β* = min{*b*_{11}, *b*_{22}}. This implies (9) and the uniqueness of the equilibrium point.

We finally need to show that there are no oscillations. To this aim we notice that the sectors DPA, and , and its opposite

CPB, and , are both positive invariant sets.

We can apply Nagumo's theorem: consider the half-line PA originating in P, where and . Therefore we have that (again by adding the null term in (10)):

Similarly, on half-line *PD* where and , by considering (10) we derive

hence the claimed invariance of sector DPA. The proof of the invariance of sector CPB is identical.

**Remark 2** *Note that the constructed Lyapunov function**does not depend on the system parameters. This fact can be used to prove that if the transcription rates c*_{1}*(t) and c*_{2}*(t) are time-varying, but bounded, we have convergence to a neighborhood whose amplitude, obviously, depends on the bounds of c*_{1}*(t) and c*_{2}*(t). It is realistic to assume that the transcription rates vary over time: for instance, if the environmental conditions change, the cell may need to down or up-regulate entire groups of transcripts and therefore increase or decrease c*_{2}*(t)*.

The following corollary evidences the positive influence of *c*_{2}, which is positive over *x*_{2} and negative over *x*_{1}.

**Corollary 1** *Assume that* *x*_{1}(0), *x*_{2}(0) *is at the steady state corresponding* *to**and*. *Consider the new input**(keeping**). Then the system converges to a new equilibrium with**and*. *There is no undershoot, respectively, overshoot*.

**Proof** The steady state values and are respectively monotonically decreasing and increasing functions of *c*_{2}. Indeed, consider the steady-state condition

and regard it as a differentiable map (*x*_{1}, *x*_{2}) → (*c*_{1}, *c*_{2}). By the uniqueness proved in Proposition 2 the map is invertible. The Jacobian of the inverse map is the inverse of the Jacobian

where (keep in mind that *b*_{21}(*x*_{1}, *x*_{2})*x*_{1} = *b*_{12}(*x*_{1}, *x*_{2})*x*_{2}). The sign of the entries in the second column are negative and positive respectively, therefore, the steady-state values and are decreasing and increasing functions of *c*_{2}.

The absence of overshoot and undershoot is an immediate consequence of the invariance of the sector and previously proved.

Obviously, decreasing *c*_{2} increases *x*_{1} and decreases *x*_{2} and the same holds if we commute 1 and 2. It is worth noting that the same conclusions regarding the lack of multistability and oscillations for the sRNA pathway may be reached by qualitative analysis of the system's nullclines.

### The cAMP dependent pathway

The cyclic adenosine monophosphate (cAMP) pathway can activate enzymes and regulate gene expression based on sensing of extracellular signals. Such signals are sensed by the G protein-coupled receptors on the cell membrane. When a receptor is activated by its extracellular ligand, a series of conformational changes are induced in the receptor and in the attached intracellular G protein complex; the latter activates adenylyl cyclase, which catalyzes the conversion of ATP in cAMP. In yeast, cAMP causes the activation of the protein kinase A (PKA), which in turn regulates the cell growth, metabolism and stress response. Stochastic models are usually proposed for numerical analysis of the cAMP pathway. However, the cAMP pathway components in yeast are present in high numbers and a deterministic modeling approach is adopted in [31]. In such paper, the pathway is decomposed in several modules, here we consider the simplified cAMP Model A, which focuses on the parts of the pathway best characterized in the literature:

where *x*_{1} is the concentration of active G protein, *x*_{2} is the concentration of active PKA protein, *x*_{3} is the concentration of cAMP and *u* is the concentration of glucose input to the network. The parameters and model the "feedback" effect introduced by two phosphodiesterases (Pde1p and Pde2p) on the cAMP concentration. The numerical analysis in [31] typically shows that the cAMP concentration (*x*_{3}) responds with a large overshoot to steps in the glucose (*u*, input) concentration. We will analytically explore the dynamic behavior of *x*_{3}, showing that under certain assumptions, a bounded overshoot is a robust characteristic in the system. The parameters *k*_{
F
} and *k*_{
R
} are forward and reverse reaction rates for the formation of active *x*_{1} and *x*_{2}. Mass conservation allows to express the active protein amounts as a function of the total number of molecules, . The nonlinear expressions in equation *x*_{3} are derived by Michaelis-Menten enzymatic steps. We can re-write the above equations according to the general model (4):

where *u* is the external signal and where *b*_{23} = 0 for *x*_{2} = 0 and *b*_{32} = 0 for *x*_{3} = 0. A qualitative graphical representation of this network is in Figure 3B.

Our preliminary analysis allows us to assume: *a*_{1u}, *a*_{23}: *decreasing-exactly-null* with threshold values and ; *d*_{32}, *a*_{31}: *decreasing-asymptotically-null*, *b*_{32} and *g*_{33} = *b*_{33}(*x*_{3})*x*_{3}: *increasing-asymptotically-constant; b*_{11}, *b*_{22}*are positive-constant*.

It is immediate to notice that for constant *u*, *x*_{1} robustly converges asymptotically to its equilibrium value such that

The solution of the previous equation is uniquely defined for each *u* since the function *ξ*^{-1}(*x*_{1}) on the right is strictly increasing and grows to infinity, precisely . Biologically, this means that external glucose signals are mapped to internal active G-protein concentration with a bijection, before saturating.

Also we have to note that the model is consistent with mass conservation: since *a*_{1u}(*x*_{1}) and *a*_{23}(*x*_{2}) are zero above the thresholds and , we have that and for and , respectively; therefore we assume , , for all *t* ≥ 0.

**Proposition 3** *There exists an equilibrium for system* (12) *if and only if*

*where**as previously defined* . *All the equilibrium values*, *and**are increasing functions of u. If condition (13) is satisfied* , *the equilibrium is unique and locally stable*.

The previous proposition assures only local stability, but this result can be extended to global stability. To this aim, we will assume that *x*_{1} is at its equilibrium value . Furthermore, under a suitable condition a performance bound on the transient values of *x*_{3}(*t*) can be given.

**Proposition 4** *Assume that x*_{1}*has reached its steady state*. *Then, the unique equilibrium point is globally attractive for any initial condition x*_{2}(0), *x*_{3}(0) ≥ 0. *Moreover, assume that*

*then we can give the following bound for the transient of x*_{3}(*t*)

The proof can be found in Section S{II of the Additional File 1.

**Remark 3** *The condition (14) has the following interpretation. It basically states that the inhibiting term b*_{33}(*x*_{3})*x*_{3}*at "full force" (x*_{3}*suitably large) dominates the activating term d*_{32}(*x*_{2}) + *a*_{31}(*x*_{2})*ξ when x*_{2}*is small. Note that, indeed, the feedback terms modulated by the two phosphodiesterases act in a complementary manner*, *in order to maintain a bounded concentration of cAMP in the cell*.

**Remark 4** *The system, even if initialized with small values x*_{
2
} (0) *and x*_{3}(0), *may exhibit a spike of cAMP x*_{3}*which is bounded by (15), if condition (14) is satisfied. If x*_{
3
}*(0) is small, then the bound is d*_{32}(0) + *a*_{31}(0)*ξ* (*u*): *the amplitude of the spike is, in general, an increasing function of the glucose concentration u. If condition (14) fails, then (see Figure S2 in the Additional File) the spike of x*_{3}*(t) can be arbitrarily large; thus condition (14) can be seen as a threshold*.

### The *Lac* operon

This genetic network was originally studied by Monod and Jacob [33]. The natural nutrient for *E. coli* bacterial cells is glucose, which is metabolized by enzymes normally produced by the bacteria. When glucose is absent, but the allolactose inducer is present in their environment, *E. coli* activates a set of genes that will regulate the lactose intake and breakdown. In particular, the cells start producing a permease protein, which binds to the cell membrane and increases the inflow of lactose; and cells also start producing the *β*-galactosidase protein, which converts lactose in allolactose.

In this paper we will consider the deterministic model proposed in [34]. This simple model does not capture the stochasticity of this genetic circuit, but it does explain the bimodal behavior of the system. Such behavior is observable experimentally: within the same population, the operon can be either induced or uninduced. Our analysis shows that for low or high intracellular inducer concentrations, the system is monostable and respectively reaches an uninduced or induced equilibrium; however, at intermediate inducer concentrations the system becomes multi-stable.

The state variables of the ODE model we will study are the concentration of nonfunctional permease protein *x*_{1}; the concentration of functional permease protein *x*_{2}; the concentration of inducer (allolactose) inside the cell *x*_{3}, and the concentration of *β*-galactosidase *x*_{4}, a quantity that can be experimentally measured. The concentration of inducer external to the cell is here denoted as an input function *u*.

where *β*_{1}, *β*_{2}, *δ*_{1}, *δ*_{2}, *δ*_{3} and *γ* are constants and *f*_{
i
} are functions that are experimentally measurable. In particular, at low inducer concentrations, , where *k*_{
i
} 's are constant; at high *x*_{3} concentrations *f*_{1} saturates. The functions *f*_{2} and *f*_{3} are assumed to depend hyperbolically on their arguments. According to the proposed setup, the previous equations can rewritten as follows:

where *c*_{13}(*x*_{3}) = *f*_{1}(*x*_{3}), *b*_{11} = *δ*_{1}, *a*_{21} = *β*_{1}, *b*_{22} = *δ*_{2}, *a*_{32}(*u*) = *f*_{2}(*u*) =, *b*_{32}(*x*_{3}) = *f*_{3}(*x*_{3}), *c*_{3u}= *β*_{2}, *b*_{33} = *δ*_{3}, *c*_{43}(*x*_{3}) = *γ f*_{1}(*x*_{3}) and *b*_{44} = *δ*_{4}. This corresponds to the network in Figure 3C.

From our preliminary analysis step: *c*_{13} is *constant-sigmoidal*, *a*_{32}(*u*) and *b*_{32}(*x*_{3}) are *increasing-asymptotically-constant*, and the remaining functions *a*_{21}, *b*_{11}, *b*_{22} and b_{33} are *positive-constant*.

We can start to study this network without any specific knowledge of the parameters in equations (17). First of all, as evident in Figure 3C, note that the *β*-galactosidase concentration *x*_{4} does not affect any other chemical species: therefore, the fourth equation can be considered separately. As long as the inducer concentration of *x*_{3} within the cell reaches an equilibrium , *x*_{4} converges to . Therefore, we can restrict our attention to the first three equations; this is consistent with the model proposed in [35, 36]. From now on we will consider this reduced model (see Section S-III of the Additional File), neglecting the linear term *c*_{3u}*u* as in [35, 36]. We will not introduce delays in our model, as done in [37]. Our preliminary screening also shows that the evolution of this system is necessarily bounded. Indeed *x*_{1} receives a bounded signal from *x*_{3} and the degradation term -*b*_{11}*x*_{1} keeps *x*_{1} bounded. In turn, *x*_{2} remains bounded. The inducer concentration *x*_{3} receives a bounded signal form *u* and *x*_{2}; therefore *x*_{3} stays bounded as well, being both *a*_{32}(*u*) and *b*_{32}(*x*_{3}) bounded.

The following proposition evidences that fundamental results can be established starting from our general framework. These results are consistent with the findings in [36], whose analysis relies on assuming Hill-type functions in the model.

**Proposition 5** *For any functional terms in Equations* 17, satisfying the general assumptions formulated above, the system admits a unique equilibrium for large u > 0 *or small u* > 0.

*For some chioces of such functional terms, the system may have multiple positive equilibria x*^{A} , *x*^{B} , *x*^{C} ,... ∈ IR^{3}*(typically three) for intermediate values of u. If multiple equilibria exist, then they are ordered in the sense that x*^{A} ≤ *x*^{B} ≤ *x*^{C} ... *where the inequality has to be considered componentwise. If the equilibria are all distinct, then they are alternatively stable and unstable. In the case of three equilibria, x*^{A} , *x*^{B} , *x*^{C} *they are stable, unstable and stable, respectively. Finally, given any equilibrium point, the positive and negative cones x* ≤ *x** *and ×* ≥ *x** *are positively invariant*.

The proof is given in Section S-III of the Additional File. The cone-invariance property implies that the state variables cannot exhibit oscillations around their equilibria. For instance, if *x*^{A} is the first (hence stable) equilibrium, given any initial condition upper bounded by *x*^{A} (*x*(0) *x*^{A} ) in the domain of attraction, the convergence to *x*^{A} has no overshoot (and if *x*(0) ≥ *x*^{A} there is no undershoot).

**Remark 5** *It is interesting to notice that, due to the competition between terms a*_{32}*and b*_{32}, *the considered Lac Operon model is not a monotone system according to the definition in*[16], *where a different model was considered*.

### MAPK signaling pathway

Mitogen-activated protein (MAP) kinases are proteins that respond to the binding of growth factors to cell surface receptors. The pathway consists of three enzymes, MAP kinase, MAP kinase kinase (MAP2K) and MAP kinase kinase kinase (MAP3K) that are activated in series. By activation or phosphorylation, we mean the addition of a phosphate group to the target protein. Extracellular signals can activate MAP3K, which in turn phosphorylates MAP2K at two different sites; in the last round, MAP2K phosphorylates MAPK at two different sites. The MAP kinase signaling cascade can transduce a variety of growth factor signals, and has been evolutionary conserved from yeast to mammals.

Several experimental studies have highlighted the presence of feedback loops in this pathway, which result in different dynamic properties. This work will focus on a specific positive-feedback topology, where doubly-phosphorylated MAPK has an activation effect on MAP3K. Such positive feedback has been extensively studied in the literature, since the biochemical analysis of Huang and Ferrell [37, 38] on the MAPK cascade found in *Xenopus* oocytes. In this type of cells, Mos (MAP3K) can activate MEK (MAP2K) through phosphorylation of two residues (converting unphosphorylated MEK to monophosphorylated MEK-P and then bisphosphorylated MEK-PP). Active MEK then phosphorylates p42 (MAPK) at two residues. Active p42 can then promote Mos synthesis, completing the closed positive-feedback loop.

The presence of such positive feedback in the MAPK cascade has been linked to a bistable behavior: the switch between two stable equilibria in *Xenopus* oocytes denotes the transition between the immature and mature state. A standard ODE model for the cascade is proposed in [17], where the authors demonstrate bi-stability of the system by applying the general theory of monotone systems. We adopt such model, which is reported below:

where *x* is concentration of Mos (MAP3K), *y*_{1} is the concentration of unphosphorylated MEK (MAP2K), *y*_{2} is the concentration of phosphorilated MEK-P, *y*_{3} is the concentration of MEK-PP, *z*_{1}, *z*_{2} and *z*_{3} are respectively the concentrations of unphosphorylated, phosphorylated and doubly-phosphorylated p42 (MAPK). Finally, *u* is the input to the system.

While bi-stability may occur due to other phenomena, such as multisite phosphorylation [39], rather than due to feedback loops, a large body of literature focuses on bi-stability induced by the positive-feedback in the Huang-Ferrel model in *Xenopus*[40, 41] reported above. In [37] the feedback *f* (*u*) was characterized, through *in vitro* studies, as an activating Hill-function with high cooperativity. In [17] instead, *f* (*u*) was assumed to be a first order linear term in the concentration of active MAP3K, *x*_{7}. In Proposition 6, we will explore the effects of different qualitative functional assumptions on the feedback loop dynamics *f* (*u*). We will highlight that the system loses its well-known bi-stability not only in the absence of feedback, but also when the feedback becomes unbounded. An unbounded positive feedback would be caused, for instance, by an autocatalytic process of MAP3K activation, mediated by active MAPK. We choose to rewrite the above model as follows:

The term *μx*_{7} introduces the positive feedback loop and represents a key parameter for the analysis to follow. A preliminary screening of the system immediately highlights the following properties. Function *b*_{11}(*x*_{1})*x*_{1}, functions *c*_{23}(*x*_{3}), *b*_{21}(*x*_{2}), *a*_{41}(*x*_{3}) and *b*_{44}(*x*_{4})*x*_{4}, functions *c*_{56}(*x*_{6}), *b*_{54}(*x*_{5}), *a*_{74}(*x*_{6}) and *b*_{77}(*x*_{7})*x*_{7} are *increasing-asymptotically-constant*. Moreover, *a*_{31}(*x*_{2}) = *b*_{21}(*x*_{2}), *c*_{34}(*x*_{4}) = *b*_{44}(*x*_{4})*x*_{4}, *b*_{31}(*x*_{3}) = *a*_{41}(*x*_{3}), *b*_{33}(*x*_{3})*x*_{3} = *c*_{23}(*x*_{3}) and *a*_{64}(*x*_{5}) = *b*_{54}(*x*_{5}), *c*_{67}(*x*_{7}) = *b*_{77}(*x*_{7})*x*_{7}, *b*_{64}(*x*_{6}) = *a*_{74}(*x*_{6}), *b*_{66}(*x*_{6})*x*_{6} = *c*_{56}(*x*_{6}). We assume *c*_{10} to be a *positive-constant*.

The graph in Figure 3D can be partitioned considering three aggregates of variables, precisely {*x*_{1}}, Σ_{234} = (*x*_{2}, *x*_{3}, *x*_{4}) and Σ _{567} = {*x*_{5}, *x*_{6}, *x*_{7}}. Signal *x*_{1} is the only input for Σ_{234}, signal *x*_{4} is the only input for Σ_{567}. Then *x*_{7} is fed back to the first subsystems by arc *a*_{17}. Without the positive feedback loop, we will demonstrate that the system is a pure stable cascade. Note also that Σ_{234} and Σ_{567} can be reduced since and and therefore the following sums are constant

with *k* ≐ *x*_{2}(0) + *x*_{3}(0) + *x*_{4}(0) and *h* ≐ *x*_{5}(0) + *x*_{6}(0) + *x*_{7}(0). Since *x*_{
i
} ≥ 0, all the variables but *x*_{1} are bounded. The system can be studied by removing variables *x*_{3} = *k* - *x*_{2} - x_{4} and *x*_{6} = *h* - *x*_{5} - *x*_{7}. We must assume that otherwise no equilibrium is possible. The following result is proved in Section S-IV of the Additional File.

**Proposition 6** *For μ* = 0 *the system admits a unique globally asymptotically stable equilibrium*.

*For μ* > 0, *the system may have multiple equilibria, for specific choices of the involved functions a*, *b*, *c*.

*For μ* > 0 *suitably large and a*_{17}(*x*_{1}) *lower bounded by a positive number, then the system has no equilibria*.

*For μ* > 0 *suitably bounded and a*_{17}(*x*_{1}) *increasing, or non-decreasing, and bounded, if multiple simple*^{2}*equilibria exist, then such equilibria are alternatively stable and unstable. In the special case of three equilibria, then the system is bistable*.

*For μ* > 0 *suitably bounded and a*_{17}(*x*_{1}*) increasing asymptotically unbounded, then the number of equilibria is necessarily even (typically* 0 *or* 2*). Moreover, if we assume that there exists μ** > 0 *such that the system admits two distinct equilibria for any* 0 < *μ* ≤ *μ**, *then one is stable, while the other is unstable*.

The proof of this last proposition also shows that multiple equilibria *x*^{A} , *x*^{B} ,... have a partial order: while and have the reverse order and

**Remark 6** *The simplest case of constant a*_{17}*has been fully developed in*[17]^{3}*and*[16], *and it turns out that the system may exhibit bi-stability for suitable values of the feedback strength μ. Here we show that, for constant a*_{17}, *bi-stability is actually a robust property. Our results are consistent with the fact that the MAPK cascade is a monotone system and some of them could be demonstrated with the same tools used in*[16, 17]. *With respect to such literature, our contribution is that of inferring properties such as number of equilibria and mono or bi-stability starting from qualitative assumptions on the dynamics of the model, without invoking monotonicity*.

**Remark 7** *Finally, it is necessary to remark that our results on the MAPK pathway robust behaviors hold true given the model (19) and its structure. Other work in the literature shows that feedback loops are not required to achieve a bistable behavior in the MAPK cascade*[38], *when the dual phosphorylation and de-phosphorylation cycles are non-processive (i.e. sites can be phosphorylated/de-phosphorylation independently) and distributed (i.e. the enzyme responsible for phosphorylation/de-phosphorylation is competitively used in the two steps)*.