Dipartimento di Matematica ed Informatica, Universitá degli Studi di Udine, Via delle Scienze 206, 33100 Udine, Italy

Division of Engineering and Applied Science, California Institute of Technology, 1200 E. California Blvd. Pasadena, CA 91125, USA

Abstract

Background

The molecular circuitry of living organisms performs remarkably robust regulatory tasks, despite the often intrinsic variability of its components. A large body of research has in fact highlighted that robustness is often a structural property of biological systems. However, there are few systematic methods to mathematically model and describe structural robustness. With a few exceptions, numerical studies are often the preferred approach to this type of investigation.

Results

In this paper, we propose a framework to analyze robust stability of equilibria in biological networks. We employ Lyapunov and invariant sets theory, focusing on the structure of ordinary differential equation models. Without resorting to extensive numerical simulations, often necessary to explore the behavior of a model in its parameter space, we provide rigorous proofs of robust stability of known bio-molecular networks. Our results are in line with existing literature.

Conclusions

The impact of our results is twofold: on the one hand, we highlight that classical and simple control theory methods are extremely useful to characterize the behavior of biological networks analytically. On the other hand, we are able to demonstrate that some biological networks are robust thanks to their structure and some qualitative properties of the interactions, regardless of the specific values of their parameters.

Background

The complex biochemistry of living organisms very often outperforms electrical and mechanical devices in terms of adaptability and robustness. Mapping such intricate reaction networks to high level design principles is the goal of systems biology, and it requires an immense collaborative effort among different disciplines, such as physics, mathematics and engineering

The most classical example of robust molecular circuitry is probably given by bacterial chemotaxis

In this work, we are going to ask a simple question: are there biological systems that present structurally stable equilibria, and preserve this property robustly with respect to their specific parameters? This question has been considered before in the literature. For instance, through extensive numerical analysis on three-node networks, the authors of

In this work, we provide a simple and general theoretical tool kit for the analysis bio-molecular systems. Such tools are suitable for the investigation of robust stability by means of Lyapunov and set-invariance methods. Provided that certain standard properties are verified, we demonstrate how a number of well known biological networks are asymptotically stable, robustly with respect to the model parameters. In some cases, we are also able to provide robust bounds on the system performance. Our approach does not require numerical simulation efforts. The contribution of the paper can be summarized as follows.

• The framework we suggest is easy and intuitive for

• The properties that can be derived from such modeling are, consequently, structurally robust because they are not inferred from mathematical formulas arbitrarily chosen to fit data.

• We suggest techniques based on set-invariance and Lyapunov theory, in particular piecewise-linear functions, to show that such models are amenable for robust investigation by

• We consider several models from the literature, reporting the original equations, and rephrasing them in our setup as case studies.

• We show how robust certifications can be given to important properties (some of which have been established based on specific models).

Methods

Robustness

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

where

**Definition 1 **

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

with positive and constant coefficients

If one is interested in the global system behavior, Lyapunov functions are a powerful tool providing sufficient conditions for stability. Given an equilibrium point

where

Non-smooth Lyapunov functions

The concept of Lyapunov derivative can be generalized when the function

where each _{i}

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

**Theorem 1 **

_{i }are smooth functions, and _{i }are given constants. Assume that

For instance, if our constraining functions are linear, ^{⊤}

Structural robustness investigation for biological networks

Let us begin with a simple biological example. Consider a protein _{1}, which represses the production of an RNA species _{2}. In turn, _{2 }can be the target of another RNA species _{2 }(and form an inactive complex to be degraded) or it can be translated into protein _{3}. A standard dynamical model

RNA species _{2 }determines the production rate of protein _{3 }by indexing the corresponding reaction rate as _{32}. Following the standard notation in control theory, we assume that the production rate of protein _{1 }is driven by some external signal or input _{1}, and that RNA _{2 }also acts as an external input on RNA _{2}. We assume that all the system parameters are positive and bounded scalars. Terms _{ij }
_{ih }
_{21}(_{1}) is a well known Hill function term _{1 }will converge to its equilibrium _{1}, however

We remark that the knowledge of functions _{ij}x_{ih}x _{21 }is strictly decreasing and asymptotically converging to 0, while _{11}
_{1}, _{22}
_{2}, _{32}
_{2 }and _{33}
_{3 }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

where _{i}
_{i}
_{ij}
_{ih }

**A1 **
_{ij }
_{ih }
_{is }
_{il }

**A2 **
_{ij }
_{i}
_{ih }
_{i}

**A3 **
_{ij }
_{i}
_{j}
_{ih }
_{i}
_{h}
_{j }and x_{h }respectively

**A4 **
_{is}
_{s}
_{il}
_{l}
_{is}
_{il}

**A5 **
_{ih}
_{ih}
_{h}
_{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 **
_{i }
**≥ **0

Given the above assumptions, we can write equation (3) in an equivalent form. First of all, in view of A1-A3, we can write: _{ij}
_{i}
_{j}
_{i}
_{j}
_{j}
_{ih}
_{i}
_{h}
_{i}
_{h}
_{h}

The above expression is always valid due to the smoothness assumption A1 (see

Additionally, assumption A5 requires that _{ih}
_{h}
_{h}

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 _{i}
_{j}
_{k}
_{i}
_{j}
_{k}
_{i}
_{j}
_{k}
_{i}
_{j}
_{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

**P1 **
**≥ **0

**P2 **

**P3 **
**∞ **

**P4 **

**P5 **

**P6 **

**P7 **
**∞ **

**P8 **

**P9 **

**P10 **

As an example, the terms

Graphical representation of biological networks

**Graphical representation of biological networks**. A. The arcs associated with the functions _{31 }= _{31}(_{2}). B. The graph associated with equations (2); external inputs are represented as orange nodes. C. Examples of sigmoidal functions. D. Examples of complementary sigmoidal functions. In our general model (4), functions

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,

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

**Remark 1 **

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:

• Preliminary screening: establish essential information on the network structure, recognizing which properties (such as P1-P10) pertain to each link.

• Analytical investigation: infer robustness properties based on dynamical systems tools such as Lyapunov theory, set invariance and linearization.

Results and Discussion

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

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 _{1 }and _{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

where _{1}, _{2 }are the degradation and dilution rates of _{1}, _{2 }respectively. The basal production rate of _{1 }(AraC) is _{1}. The activation pathways are modeled by Hill functions ^{H }
^{H }
^{H }
_{ij }

where _{1}, _{11 }and _{22 }are _{1u
}(_{2u1}(_{1}. The graph representation of this network is in Figure

For this elementary network the analysis is straightforward. Variable _{1 }is not affected by _{2}. Since _{1u
}(_{1 }is also bounded and converges to an equilibrium point _{2 }is also positive and bounded for any value of _{1 }prevents _{1}(_{2}(

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 _{1 }as the RNA concentration of the species which is targeted by the sRNA and _{2 }as the concentration of sRNA. The model often proposed in the literature is:

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

by setting _{12 }= _{1 }and _{21 }= _{2 }(note that _{12}(0) = _{21}(0) as required). From our list of properties: _{1}, _{2}, _{11 }and _{22 }are _{12}(_{1}, _{2}) and _{21}(_{1}, _{2}) are _{12}(_{1}, _{2})_{2 }= _{21}(_{1}, _{2})_{1 }at all times. This network can be represented with the graph in Figure

The sRNA network

**The sRNA network**. A. The graph associated with the sRNA network B. Sectors, Lyapunov function level curves (orange) and qualitative behavior of the trajectories (green) for the sRNA system

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 _{1 }is "less tolerant" to an increase of _{2 }if the latter is present in a large amount. This means that the sum _{1 }+ _{2 }is strongly kept under control. Also the mismatch between the two variables is controlled. ^{1 }To prove stability of the (unique) equilibrium

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 **
_{1}(0), _{2}(0) ≥ 0.

_{1}(0) ≥ 0, _{2}(0) ≥ 0.

**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 _{1 }+ _{2 }≤ _{1}, _{2}) = _{1 }+ _{2 }is

Define _{1 }+ _{2})/min {_{11}, _{22}} then for _{1}, _{2}) > _{1}, _{2}) cannot exceed

Boundedness of the solution inside a compact set assures the existence of an equilibrium point. Let

The behavior of the candidate Lyapunov function:

will be examined in the different sectors represented in Figure

where we have subtracted the null terms (10) and where we have exploited the fact that _{12}(_{1}, _{2})_{1 }= _{21}(_{1}, _{2})_{2 }is increasing in both variables. The inequality (CPD in Figure

Consider the sector defined by

Note that in the last step we have added and subtracted the null terms (10). In the opposite sector (BPC in Figure

with _{11}, _{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,

CPB,

We can apply Nagumo's theorem: consider the half-line PA originating in P, where

Similarly, on half-line

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

**Remark 2 **
_{1}
_{2}
_{1}
_{2}
_{2}

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

**Corollary 1 **
_{1}(0), _{2}(0)

**Proof **The steady state values _{2}. Indeed, consider the steady-state condition

and regard it as a differentiable map (_{1}, _{2}) → (_{1}, _{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 _{21}(_{1}, _{2})_{1 }= _{12}(_{1}, _{2})_{2}). The sign of the entries in the second column are negative and positive respectively, therefore, the steady-state values _{2}.

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

Obviously, decreasing _{2 }increases _{1 }and decreases _{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

where _{1 }is the concentration of active G protein, _{2 }is the concentration of active PKA protein, _{3 }is the concentration of cAMP and _{3}) responds with a large overshoot to steps in the glucose (_{3}, showing that under certain assumptions, a bounded overshoot is a robust characteristic in the system. The parameters _{F }
_{R }
_{1 }and _{2}. Mass conservation allows to express the active protein amounts as a function of the total number of molecules, _{3 }are derived by Michaelis-Menten enzymatic steps. We can re-write the above equations according to the general model (4):

where _{23 }= 0 for _{2 }= 0 and _{32 }= 0 for _{3 }= 0. A qualitative graphical representation of this network is in Figure

Graphs associated with case studies

**Graphs associated with case studies**. A. The graph associated with the L-arabinose network, external inputs are represented as orange nodes. B. The graph associated with the cAMP pathway. C. The graph associated with the lac Operon network. D. The graph associated with the MAPK signaling pathway.

Our preliminary analysis allows us to assume: _{1u
}, _{23}: _{32}, _{31}: _{32 }and _{33 }= _{33}(_{3})_{3}: _{11}, _{22 }

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

The solution ^{-1}(_{1}) on the right is strictly increasing and grows to infinity, precisely

Also we have to note that the model is consistent with mass conservation: since _{1u
}(_{1}) and _{23}(_{2}) are zero above the thresholds

**Proposition 3 **

The previous proposition assures only local stability, but this result can be extended to global stability. To this aim, we will assume that _{1 }is at its equilibrium value _{3}(

**Proposition 4 **
_{1 }
_{2}(0), _{3}(0) ≥ 0.

_{3}(

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

**One additional file includes the proofs for Propositions 3, 4, 5 and 6 in the main paper**.

Click here for file

**Remark 3 **
_{33}(_{3})_{3 }
_{3 }
_{32}(_{2}) + _{31}(_{2})_{2 }

**Remark 4 **
_{2}
_{3}(0), _{3 }
_{3}(0) is small, then the bound is d_{32}(0) + _{31}(0)_{3}

The

This genetic network was originally studied by Monod and Jacob

In this paper we will consider the deterministic model proposed in

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

where _{1}, _{2}, _{1}, _{2}, _{3 }and _{i }
_{i}
_{3 }concentrations _{1 }saturates. The functions _{2 }and _{3 }are assumed to depend hyperbolically on their arguments. According to the proposed setup, the previous equations can rewritten as follows:

where _{13}(_{3}) = _{1}(_{3}), _{11 }= _{1}, _{21 }= _{1}, _{22 }= _{2}, _{32}(_{2}(_{32}(_{3}) = _{3}(_{3}), _{3u
}= _{2}, _{33 }= _{3}, _{43}(_{3}) = _{1}(_{3}) and _{44 }= _{4}. This corresponds to the network in Figure

From our preliminary analysis step: _{13 }is _{32}(_{32}(_{3}) are _{21}, _{11}, _{22 }and b_{33 }are

We can start to study this network without any specific knowledge of the parameters in equations (17). First of all, as evident in Figure _{4 }does not affect any other chemical species: therefore, the fourth equation can be considered separately. As long as the inducer concentration of _{3 }within the cell reaches an equilibrium _{4 }converges to _{3u
}
_{1 }receives a bounded signal from _{3 }and the degradation term -_{11}
_{1 }keeps _{1 }bounded. In turn, _{2 }remains bounded. The inducer concentration _{3 }receives a bounded signal form _{2}; therefore _{3 }stays bounded as well, being both _{32}(_{32}(_{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

**Proposition 5 **

^{A}
^{B}
^{C }
^{3 }
^{A }
^{B }
^{C }
^{A}
^{B}
^{C }they are stable, unstable and stable, respectively. Finally, given any equilibrium point, the positive and negative cones x

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 ^{A }
^{A }
^{A}
^{A }
^{A }

**Remark 5 **
_{32 }
_{32},

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

The presence of such positive feedback in the MAPK cascade has been linked to a bistable behavior: the switch between two stable equilibria in

where _{1 }is the concentration of unphosphorylated MEK (MAP2K), _{2 }is the concentration of phosphorilated MEK-P, _{3 }is the concentration of MEK-PP, _{1}, _{2 }and _{3 }are respectively the concentrations of unphosphorylated, phosphorylated and doubly-phosphorylated p42 (MAPK). Finally,

While bi-stability may occur due to other phenomena, such as multisite phosphorylation _{7}. In Proposition 6, we will explore the effects of different qualitative functional assumptions on the feedback loop dynamics

The term _{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 _{11}(_{1})_{1}, functions _{23}(_{3}), _{21}(_{2}), _{41}(_{3}) and _{44}(_{4})_{4}, functions _{56}(_{6}), _{54}(_{5}), _{74}(_{6}) and _{77}(_{7})_{7 }are _{31}(_{2}) = _{21}(_{2}), _{34}(_{4}) = _{44}(_{4})_{4}, _{31}(_{3}) = _{41}(_{3}), _{33}(_{3})_{3 }= _{23}(_{3}) and _{64}(_{5}) = _{54}(_{5}), _{67}(_{7}) = _{77}(_{7})_{7}, _{64}(_{6}) = _{74}(_{6}), _{66}(_{6})_{6 }= _{56}(_{6}). We assume _{10 }to be a

The graph in Figure _{1}}, Σ_{234 }= (_{2}, _{3}, _{4}) and Σ _{567 }= {_{5}, _{6}, _{7}}. Signal _{1 }is the only input for Σ_{234}, signal _{4 }is the only input for Σ_{567}. Then _{7 }is fed back to the first subsystems by arc _{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

with _{2}(0) + _{3}(0) + _{4}(0) and _{5}(0) + _{6}(0) + _{7}(0). Since _{i }
_{1 }are bounded. The system can be studied by removing variables _{3 }= _{2 }- x_{4 }and _{6 }= _{5 }- _{7}. We must assume that

**Proposition 6 **

_{17}(_{1})

_{17}(_{1}) ^{2 }

_{17}(_{1}

The proof of this last proposition also shows that multiple equilibria ^{A}
^{B}

**Remark 6 **
_{17 }
^{3 }
_{17},

**Remark 7 **

Conclusions

A property is structurally robust if it is satisfied by a class of systems of a given structure, regardless the choice of specific expressions adopted and of the parameter values in the model. We have considered five relevant biological examples and proposed to capture their dynamics with parameter-free, qualitative models. We have shown that specific robust properties of such models can be assessed by means of solid theoretical tools based on Lyapunov methods, set-invariance theory and matrix theory. Robustness is often tested through simulations, at the price of exhaustive campaigns of numerical trials and, more importantly, with no theoretical guarantee of robustness. We are far from claiming that numerical simulation is useless. It it important, for instance, to falsify "robustness conjectures" by finding suitable numerical counterexamples. Furthermore, for very complex systems in which analytic tools can fail, simulation appears be the last resort. Indeed a limit of the considered theoretical investigation is that its systematic application to more complex cases is challenging. However, the set of techniques we employed can be successfully used to study a large class of simple systems, and are in general suitable for the analytical investigation of structural robustness of biological networks, complementary to simulations and experiments.

Authors' contributions

FB and EF performed research and wrote the paper.

Notes

^{1}The concentration mismatch is more "softly" controlled, since the derivative of the difference _{12}(_{1}, _{2})_{2 }= _{21}(_{1}, _{2})_{1}.

^{2}I.e. the nullclines have no common tangent lines.

^{3}Cf. the erratum: http://www.math.rutgers.edu/~sontag/FTPDIR/angeli-ferrell-sontag-pnas04-errata. txt and

Acknowledgements

The authors acknowledge financial support by the National Science Foundation (NSF) grant CCF-0832824 (The Molecular Programming Project). We are grateful to R. M. Murray, for helpful advise and discussions, and to the Reviewers for their constructive comments.