Structurally robust biological networks

Robustness

We will consider biological dynamical systems which are successfully modeled with ODEs and can be written as:

(1)

where x is the system state, u models external inputs, and both are vectors of appropriate dimensions. Such class of models is appropriate for biological systems where stochasticity and anisotropy can be neglected. We define robustness as follows:

Definition 1 Let be a class of systems and be a property pertaining such a class. Given a family we say that is robustly verified by ℱ, in short robust, if it is satisfied by each element of ℱ.

Countless examples can be brought about families ℱ and candidate properties. In this work, we will focus on the property of stability, which is an important feature for the equilibria of biological networks [1, 6, 17]. If we take into account a linear or linearized dynamical system, we can immediately provide an example that clarifies our definition of robustness [21]. Let be the class of linear differential systems and ℱ the family of second order systems described by

with positive and constant coefficients a, b, c, d. Assume . Then we can say that is robust. The situation is different if we admit that a(t), b(t), c(t), d(t) can vary with time, yielding a system which is possibly unstable.

If one is interested in the global system behavior, Lyapunov functions are a powerful tool providing sufficient conditions for stability. Given an equilibrium point , any convex function for and zero at the origin is a candidate Lyapunov function. If f(x, u) is continuous, and V (·) is smooth, then V (·) is a Lyapunov function if:

where is fixed and κ (·) is a negative definite function (i.e. κ (·) < 0 on all its domain, except for κ (0) = 0).

Non-smooth Lyapunov functions

The concept of Lyapunov derivative can be generalized when the function V (·) is non-smooth. For instance, consider the convex function:

where each V _i (·) is smooth and convex, and assume that V (·) is positive definite. The set of active functions is never empty and is defined as: . If we define the generalized Lyapunov derivative as:

then the condition for stability becomes:

Positively invariant sets

We are interested in cases where the trajectories of system (1) remain trapped in bounded sets at all times, therefore behaving consistently with respect to some desired criterion.

We say that a subset of the state space is positively invariant if implies that also for all t > 0. The following theorem (which relies on the concept of Lyapunov function) provides a general necessary and sufficient condition for a set to be invariant.

Theorem 1 (Nagumo, 1943) Assume that system (1) (for a fixed constant input ) admits a unique solution. Consider the set:

where s _i are smooth functions, and σ _i are given constants. Assume that . The set of active constraints is , and is non-empty only on the boundary of . Then the set is positively invariant if and only if

For instance, if our constraining functions are linear, s^⊤x ≤ σ, the Nagumo conditions are . We refer the reader to [22] for further details on positively invariant sets; more recent works on this topic are [23] and [24].

Structural robustness investigation for biological networks

Let us begin with a simple biological example. Consider a protein x₁, which represses the production of an RNA species x₂. In turn, x₂ can be the target of another RNA species u₂ (and form an inactive complex to be degraded) or it can be translated into protein x₃. A standard dynamical model [25] of this process is:

(2)

RNA species x₂ determines the production rate of protein x₃ by indexing the corresponding reaction rate as a₃₂. Following the standard notation in control theory, we assume that the production rate of protein x₁ is driven by some external signal or input u₁, and that RNA u₂ also acts as an external input on RNA x₂. We assume that all the system parameters are positive and bounded scalars. Terms a _ij are first order production rates: species i is produced at a rate which is linear in species j; b _ih denote in this case first order degradation rates. The term d₂₁(x₁) is a well known Hill function term [26]. The stability properties of this small network can be immediately assessed: x₁ will converge to its equilibrium . Similarly, , . Regardless of the specific parameter values, and therefore robustly, the system is stable. The equilibrium could grow unbounded with u₁, however is always bounded.

We remark that the knowledge of functions a _ij x, b _ih x and d(·) is not necessary at all: the previous conclusions can be easily derived by the qualitative information that d₂₁ is strictly decreasing and asymptotically converging to 0, while b₁₁x₁, b₂₂x₂, , a₃₂x₂ and b₃₃x₃ are increasing.

It is appropriate at this point to outline a series of general assumptions that will be useful in the following analysis.

We will consider a class of biological network models consisting of n first order differential equations

(3)

where x _i , i = 1,..., n are the dynamic variables. For the sake of notation simplicity, we are not denoting external inputs with a different symbol. Inputs can be easily included as dynamic variables which are not affected by other states and have the desired dynamics. The sets denote the subsets of variables affecting x _i . The different terms in equation (3) are associated with a specific biological and physical meaning. The terms f _ij (· , ·) are associated with production rates of reagents, typically, these functions are assumed polynomial in their arguments; similarly, terms g _ih (· , ·) model degradation or conversion rates and are also likely to be polynomial in practical cases. Finally, terms c(·) and d(·) are associated with monotonic nonlinear terms, often given by Michaelis-Menten or Hill functions [26]. We assume that system (3) satisfies the following assumptions:

A1 (Smoothness) Functions f _ij (· , ·), g _ih (· , ·), c _is (·) and d _il (·) are unknown nonnegative continuously differentiable functions.

A2 f _ij (x _i , 0) = 0 and g _ih (x _i , 0) = 0, ∀x.

A3 Functions f _ij (x _i , x _j ) and g _ih (x _i , x _h ), are strictly increasing in x _j and x _h respectively.

A4 (Saturation) Functions c _is (x _s ) and d _il (x _l ) are nonnegative and, respectively, non-decreasing and non-increasing. Moreover c _is (∞) > 0 and d _il (0) > 0.

A5 Functions g _ih (· , ·) are null at the lower saturation levels : g _ih (0, x _h ) = 0, ∀x _h .

In view of the nonnegativity assumptions and Assumption 5, the general model (4) is a nonlinear positive system, according to the next proposition, and its investigation will be restricted to the positive orthant.

Proposition 1 The nonnegative orthant x _i ≥ 0 is positively invariant for system (4).

Given the above assumptions, we can write equation (3) in an equivalent form. First of all, in view of A1-A3, we can write: f _ij (x _i , x _j ) = a(x _i , x _j )x _j , g _ih (x _i , x _h ) = b(x _i , x _h )x _h , with

The above expression is always valid due to the smoothness assumption A1 (see [18], Section 2.1).

Additionally, assumption A5 requires that b _ih (0, x _h ) = 0, ∀x _h , for i ≠ h. Once we adopt this notation, we can rewrite model (3) as follows:

(4)

To simplify the notation, we have considered functions depending on two variables at most. However, we can straightforwardly extend assumptions A1-A5 to multivariate functional terms in equation (3). In turn, the model structure (4) can be easily generalized to include terms as a(x _i , x _j , x _k ,...), b(x _i , x _j , x _k ,...), c(x _i , x _j , x _k ,...), d(x _i , x _j , x _k ,...).

If we restrict our attention to the general class of models (4), under assumptions A1-A5, we can proceed to successfully analyzing the robust stability properties of several biological network examples.

The structural analysis of system (4) can be greatly facilitated whenever it is legitimate to assume that functions a, b, c d have certain properties. For the reader's convenience, a list of possible properties is given below. Given a general function f(x):

P1 f (x) = const ≥ 0 is nonnegative-constant.

P2 f(x) = const > 0 is positive-constant.

P3 f (x) is sigmoidal: it is non-decreasing, f(0) = f '(0) = 0, if 0 < f(∞) < ∞ and its derivative has a unique maximum point, for some .

P4 f (x) is complementary sigmoidal: it is non-increasing, 0 < f(0), f'(0) = 0, f(∞) = 0 and its derivative has a unique minimum point. In simple words, f is a CSM function iff f(0) - f(x) is a sigmoidal function.

P5 f (x) is constant-sigmoidal, the sum of a sigmoid and a positive constant.

P6 f (x) is constant-complementary-sigmoidal, the sum of a complementary sigmoid and a constant.

P7 f (x) is increasing-asymptotically-constant: f'(x) > 0, 0 < f (∞) < ∞ and its derivative is decreasing.

P8 f (x) is decreasing-asymptotically-null: f'(x) < 0, f (∞) = 0 and its derivative is increasing.

P9 f (x) is decreasing-exactly-null: f'(x) < 0, for and f(x) = 0 for for some .

P10 f (x) is increasing-asymptotically-unbounded: f'(x) > 0, f (∞) = + ∞.

As an example, the terms d(·) and c(·) are associated with Hill functions, which are sigmoidal and complementary sigmoidal functions. A graphical sketch of their profile is in Figure 1C and 1D.

Network graphs

Building a dynamical model for a biological system is often a long and challenging process. For instance, to reveal dynamic interactions among a pool of genes of interest, biologists may need to selectively knockout genes, set up micro RNA assays, or integrate fluorescent reporters in the genome. The data derived from such experiments are often noisy and uncertain, which implies that also the estimated model parameters will be uncertain. However, qualitative trends can be reliably assessed in the dynamic or steady state correlation of biological quantities.

Graphical representations of such qualitative trends are often used by biologists, to provide intuition regarding the network main features. We believe that, indeed, such graphs may be useful even to immediately construct models analogous to (3). We propose a specific method to construct such graphs: the biochemical species of the network will be associated to the nodes in the graph, the qualitative relationships between the species will be instead associated with different types of arcs: in particular, the terms of a, b, c and d will be represented as arcs having different end-arrows, as shown in Figure 1A. These graphs can be immediately constructed, by knowing the correlation trends among the species of the network, and aid the construction of a dynamical model. For simple networks, this type of graph may provide intuition regarding their behavior and may facilitate their structural robustness analysis. For instance, the graph associated to equations (2) is shown in Figure 1B. Throughout the paper, we will consider similar case studies and use their graph representation as a visual support for the analysis.

Remark 1 In this work, properties such as positivity, monotonicity, boundedness and other functional characteristics are labeled as "qualitative and structural properties"[27]. Through such properties, we can draw conclusions on the dynamic behaviors of the considered systems without requiring specific knowledge of parameters and without numerical simulations. However, it is clear that our approach requires more information than other methods, such as boolean networks and other graph-based frameworks.

Investigation method

The main objective of this work is to show that, at least for reasonably simple networks, structural robust stability can be investigated with simple analytical methods, without the need for extensive numerical analysis. We will suggest a two stage approach:

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 x₁ and x₂, 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:

(5)

where α₁, α₂ are the degradation and dilution rates of x₁, x₂ respectively. The basal production rate of x₁ (AraC) is p₁. 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::

(6)

where u is nonnegative-constant, c₁, b₁₁ and b₂₂ are positive-constant, while c_1u(u) and c_{2u 1}(u) are sigmoidal with respect to u, the latter increasing with respect to x₁. The graph representation of this network is in Figure 3A.

For this elementary network the analysis is straightforward. Variable x₁ is not affected by x₂. Since c_1u(u) is bounded, x₁ is also bounded and converges to an equilibrium point which is monotonically increasing in u. In turn, x₂ 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 c₁ prevents x₁(t) and x₂(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 x₁ as the RNA concentration of the species which is targeted by the sRNA and x₂ as the concentration of sRNA. The model often proposed in the literature is:

(7)

where α₁, α₂ are the transcription rates of x₁ and x₂ respectively, β₁, β₂ are their degradation rates (turnover), and k is the binding rate of the species x₁ and x₂. The formation of the inactive complex x₁ · x₂ corresponds to a depletion of both free molecules of x₁ and x₂. If α₁ < α₂ the pathway successfully suppresses the expression of the non-essential gene encoded by x₁. This model can be embedded in the general family:

(8)

by setting b₁₂ = kx₁ and b₂₁ = kx₂ (note that b₁₂(0) = b₂₁(0) as required). From our list of properties: c₁, c₂, b₁₁ and b₂₂ are positive-constant; b₁₂(x₁, x₂) and b₂₁(x₁, x₂) are increasing-asymptotically-unbounded in both variables; and b₁₂(x₁, x₂)x₂ = b₂₁(x₁, x₂)x₁ 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 x₁ is "less tolerant" to an increase of x₂ if the latter is present in a large amount. This means that the sum x₁ + x₂ 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 x₁(0), x₂(0) ≥ 0. The system admits a unique asymptotically stable equilibrium point and the convergence is exponential:

(9)

for some β > 0 and any x₁(0) ≥ 0, x₂(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 x₁ + x₂ ≤ κ cannot be violated for sufficiently large κ > 0. The derivative of function s (x₁, x₂) = x₁ + x₂ is

Define κ = (c₁ + c₂)/min {b₁₁, b₂₂} then for s(x₁, x₂) > κ the derivative becomes negative so s(x₁, x₂) 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:

(10)

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 b₁₂(x₁, x₂)x₁ = b₂₁(x₁, x₂)x₂ 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{b₁₁, b₂₂}. 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 c₁(t) and c₂(t) are time-varying, but bounded, we have convergence to a neighborhood whose amplitude, obviously, depends on the bounds of c₁(t) and c₂(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 c₂(t).

The following corollary evidences the positive influence of c₂, which is positive over x₂ and negative over x₁.

Corollary 1 Assume that x₁(0), x₂(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 c₂. Indeed, consider the steady-state condition

and regard it as a differentiable map (x₁, x₂) → (c₁, c₂). 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 b₂₁(x₁, x₂)x₁ = b₁₂(x₁, x₂)x₂). 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 c₂.

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

Obviously, decreasing c₂ increases x₁ and decreases x₂ 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:

(11)

where x₁ is the concentration of active G protein, x₂ is the concentration of active PKA protein, x₃ 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 (x₃) responds with a large overshoot to steps in the glucose (u, input) concentration. We will analytically explore the dynamic behavior of x₃, 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 x₁ and x₂. Mass conservation allows to express the active protein amounts as a function of the total number of molecules, . The nonlinear expressions in equation x₃ are derived by Michaelis-Menten enzymatic steps. We can re-write the above equations according to the general model (4):

(12)

where u is the external signal and where b₂₃ = 0 for x₂ = 0 and b₃₂ = 0 for x₃ = 0. A qualitative graphical representation of this network is in Figure 3B.

Our preliminary analysis allows us to assume: a_1u, a₂₃: decreasing-exactly-null with threshold values and ; d₃₂, a₃₁: decreasing-asymptotically-null, b₃₂ and g₃₃ = b₃₃(x₃)x₃: increasing-asymptotically-constant; b₁₁, b₂₂are positive-constant.

It is immediate to notice that for constant u, x₁ 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(x₁) 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(x₁) and a₂₃(x₂) 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

(13)

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 x₁ is at its equilibrium value . Furthermore, under a suitable condition a performance bound on the transient values of x₃(t) can be given.

Proposition 4 Assume that x₁has reached its steady state . Then, the unique equilibrium point is globally attractive for any initial condition x₂(0), x₃(0) ≥ 0. Moreover, assume that

(14)

then we can give the following bound for the transient of x₃(t)

(15)

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 b₃₃(x₃)x₃at "full force" (x₃suitably large) dominates the activating term d₃₂(x₂) + a₃₁(x₂)ξ when x₂is 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 x₃(0), may exhibit a spike of cAMP x₃which is bounded by (15), if condition (14) is satisfied. If x ₃ (0) is small, then the bound is d₃₂(0) + a₃₁(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 x₃(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 x₁; the concentration of functional permease protein x₂; the concentration of inducer (allolactose) inside the cell x₃, and the concentration of β-galactosidase x₄, a quantity that can be experimentally measured. The concentration of inducer external to the cell is here denoted as an input function u.

(16)

where β₁, β₂, δ₁, δ₂, δ₃ 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 x₃ concentrations f₁ saturates. The functions f₂ and f₃ are assumed to depend hyperbolically on their arguments. According to the proposed setup, the previous equations can rewritten as follows:

(17)

where c₁₃(x₃) = f₁(x₃), b₁₁ = δ₁, a₂₁ = β₁, b₂₂ = δ₂, a₃₂(u) = f₂(u) =, b₃₂(x₃) = f₃(x₃), c_3u= β₂, b₃₃ = δ₃, c₄₃(x₃) = γ f₁(x₃) and b₄₄ = δ₄. This corresponds to the network in Figure 3C.

From our preliminary analysis step: c₁₃ is constant-sigmoidal, a₃₂(u) and b₃₂(x₃) are increasing-asymptotically-constant, and the remaining functions a₂₁, b₁₁, b₂₂ and b₃₃ 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 x₄ does not affect any other chemical species: therefore, the fourth equation can be considered separately. As long as the inducer concentration of x₃ within the cell reaches an equilibrium , x₄ 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_3uu 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 x₁ receives a bounded signal from x₃ and the degradation term -b₁₁x₁ keeps x₁ bounded. In turn, x₂ remains bounded. The inducer concentration x₃ receives a bounded signal form u and x₂; therefore x₃ stays bounded as well, being both a₃₂(u) and b₃₂(x₃) 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 ,... ∈ IR³(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 a₃₂and b₃₂, 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:

(18)

where x is concentration of Mos (MAP3K), y₁ is the concentration of unphosphorylated MEK (MAP2K), y₂ is the concentration of phosphorilated MEK-P, y₃ is the concentration of MEK-PP, z₁, z₂ and z₃ 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, x₇. 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:

(19)

The term μx₇ 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 b₁₁(x₁)x₁, functions c₂₃(x₃), b₂₁(x₂), a₄₁(x₃) and b₄₄(x₄)x₄, functions c₅₆(x₆), b₅₄(x₅), a₇₄(x₆) and b₇₇(x₇)x₇ are increasing-asymptotically-constant. Moreover, a₃₁(x₂) = b₂₁(x₂), c₃₄(x₄) = b₄₄(x₄)x₄, b₃₁(x₃) = a₄₁(x₃), b₃₃(x₃)x₃ = c₂₃(x₃) and a₆₄(x₅) = b₅₄(x₅), c₆₇(x₇) = b₇₇(x₇)x₇, b₆₄(x₆) = a₇₄(x₆), b₆₆(x₆)x₆ = c₅₆(x₆). We assume c₁₀ to be a positive-constant.

The graph in Figure 3D can be partitioned considering three aggregates of variables, precisely {x₁}, Σ₂₃₄ = (x₂, x₃, x₄) and Σ ₅₆₇ = {x₅, x₆, x₇}. Signal x₁ is the only input for Σ₂₃₄, signal x₄ is the only input for Σ₅₆₇. Then x₇ is fed back to the first subsystems by arc a₁₇. Without the positive feedback loop, we will demonstrate that the system is a pure stable cascade. Note also that Σ₂₃₄ and Σ₅₆₇ can be reduced since and and therefore the following sums are constant

(20)

with k ≐ x₂(0) + x₃(0) + x₄(0) and h ≐ x₅(0) + x₆(0) + x₇(0). Since x _i ≥ 0, all the variables but x₁ are bounded. The system can be studied by removing variables x₃ = k - x₂ - x₄ and x₆ = h - x₅ - x₇. 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 a₁₇(x₁) lower bounded by a positive number, then the system has no equilibria.

For μ > 0 suitably bounded and a₁₇(x₁) increasing, or non-decreasing, and bounded, if multiple simple²equilibria 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 a₁₇(x₁) 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 a₁₇has been fully developed in[17]³and[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 a₁₇, 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).