Open Access

Structurally robust biological networks

BMC Systems Biology20115:74

DOI: 10.1186/1752-0509-5-74

Received: 10 November 2010

Accepted: 17 May 2011

Published: 17 May 2011

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 [1].

The most classical example of robust molecular circuitry is probably given by bacterial chemotaxis [2, 3]. The action of the flagellar motor of E. coli cells is driven by a cascade of signaling proteins, whose active or inactive state is determined by the presence of nutrient in the environment. Both analysis on a simplified ordinary differential equation (ODE) model [2] and experiments [3] showed how the flagellar motion of E. coli presents a robustly stable steady state: steps in the nutrient concentration only temporarily alter the motor equilibrium. Cells are therefore sensitive to nutrient gradients, but always return to their stable motion mode (such property is also referred to as adaptability). Such stable steady state is only a function of the concentrations of the signaling cascade protein components and a few binding rates, and is therefore independent of external inputs. Further analysis also demonstrated how integral feedback is present in the chemotaxis network, and guarantees robustness (perfect adaptation) of the equilibrium [4].

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 [5] have shown that adaptability of these systems can be investigated solely based on their structure, regardless of the chosen reaction parameters. In [6], through numerical exploration of the Jacobian eigenvalues for two, three and four node networks, the authors isolated a series of interconnections which are stable, robustly with respect to the specific parameters. Such structures also turned out to be the most frequent topologies in existing biological networks databases. Numerical simulation has arguably been the most popular tool to investigate robustness of biological networks [712]. Analytical approaches to the study of robustness have been proposed in specific contexts. A series of recent papers [13, 14] focused on input/output robustness of ODE models for phosphorylation cascades. In particular, the theory of chemical reaction networks is used in [14] as a powerful tool to demonstrate the property of absolute concentration robustness. Indeed, the so-called deficiency theorems [15] are to date some of the most general results to establish robust stability of a chemical reaction network. Monotonicity is also a structural property that is useful to demonstrate robust dynamic behaviors of a class of biological models [16, 17]. Robustness has also been investigated in the context of compartmental models, which are often encountered in biology and chemistry [18].

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 biologists to formulate qualitative models without the need of exact mathematical expressions and parameters. We will propose analytical methods that only rely on qualitative interactions between network components.

  • 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 engineers and mathematicians. Such techniques are believed to be quite effective and promising in dealing with biological robustness [19], [20].

  • 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:
https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_Equ1_HTML.gif
(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 https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq1_HTML.gif be a class of systems and https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq2_HTML.gif be a property pertaining such a class. Given a family https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq3_HTML.gif we say that https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq2_HTML.gif is robustly verified by , in short robust, if it is satisfied by each element of .

Countless examples can be brought about families 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 https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq1_HTML.gif be the class of linear differential systems and the family of second order systems described by
https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_Equa_HTML.gif

with positive and constant coefficients a, b, c, d. Assume https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq4_HTML.gif . Then we can say that https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq2_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq5_HTML.gif , any convex function https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq6_HTML.gif for https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq7_HTML.gif 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:
https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_Equb_HTML.gif

where https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq8_HTML.gif 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:
https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_Equc_HTML.gif
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: https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq9_HTML.gif . If we define the generalized Lyapunov derivative as:
https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_Equd_HTML.gif
then the condition for stability becomes:
https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_Eque_HTML.gif

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 https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq10_HTML.gif of the state space is positively invariant if https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq11_HTML.gif implies that also https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq12_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq8_HTML.gif ) admits a unique solution. Consider the set:
https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_Equf_HTML.gif
where s i are smooth functions, and σ i are given constants. Assume that https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq13_HTML.gif . The set of active constraints is https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq14_HTML.gif , and is non-empty only on the boundary of https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq10_HTML.gif . Then the set https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq10_HTML.gif is positively invariant if and only if
https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_Equg_HTML.gif

For instance, if our constraining functions are linear, sxσ, the Nagumo conditions are https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq15_HTML.gif . 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:
https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_Equ2_HTML.gif
(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 https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq16_HTML.gif . Similarly, https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq17_HTML.gif , https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq18_HTML.gif . Regardless of the specific parameter values, and therefore robustly, the system is stable. The equilibrium https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq19_HTML.gif could grow unbounded with u1, however https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq20_HTML.gif 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, https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq21_HTML.gif , 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
https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_Equ3_HTML.gif
(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 https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq22_HTML.gif which are not affected by other states and have the desired dynamics. The sets https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq23_HTML.gif 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.
https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_Equh_HTML.gif

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
https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_Equi_HTML.gif

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 ih. Once we adopt this notation, we can rewrite model (3) as follows:
https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_Equ4_HTML.gif
(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, https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq24_HTML.gif for some https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq25_HTML.gif .

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 https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq26_HTML.gif and f(x) = 0 for https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq27_HTML.gif for some https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq28_HTML.gif .

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.
https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_Fig1_HTML.jpg
Figure 1

Graphical representation of biological networks. A. The arcs associated with the functions a, b, c and d. We will use dashed arcs, connecting to arcs of the type a and b to highlight that the associated function is nonlinearly dependent on a species of the network: in the example above, a31 = a31(x2). 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 d(·) and c(·) are naturally associated with Hill function terms.

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:

  • 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 [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 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:
https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_Equ5_HTML.gif
(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::
https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_Equ6_HTML.gif
(6)

where u is nonnegative-constant, c1, b11 and b22 are positive-constant, while c1u(u) and c2u 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 c1u(u) is bounded, x1 is also bounded and converges to an equilibrium point https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq29_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq20_HTML.gif , which is a monotonically increasing function of u (partially activated by https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq29_HTML.gif ). 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 [2931]. 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:
https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_Equ7_HTML.gif
(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:
https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_Equ8_HTML.gif
(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.
https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_Fig2_HTML.jpg
Figure 2

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 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 https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq5_HTML.gif , we will use the 1-norm as Lyapunov function https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq30_HTML.gif (see Figure 2B). This choice has a remarkable interpretation: denoting by https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq31_HTML.gif and https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq32_HTML.gif the sum and the difference of the two variables (referred to the equilibrium) we have
https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_Equj_HTML.gif

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 https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq33_HTML.gif and the convergence is exponential:
https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_Equ9_HTML.gif
(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 https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq34_HTML.gif or https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq35_HTML.gif occurs at most once.

Proof To prove boundedness of the variables we need to show the existence of an invariant set
https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_Equk_HTML.gif
Proposition 1 guarantees that the positivity constraints are respected. Then we just need to show that the constraint x1 + x2κ cannot be violated for sufficiently large κ > 0. The derivative of function s (x1, x2) = x1 + x2 is
https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_Equl_HTML.gif

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 https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq36_HTML.gif be any point in which the following equilibrium conditions holds:
https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_Equ10_HTML.gif
(10)
The behavior of the candidate Lyapunov function:
https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_Equm_HTML.gif
will be examined in the different sectors represented in Figure 2B. Let us start by considering the sector defined by https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq37_HTML.gif and https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq38_HTML.gif (APB in Figure 2B) for which https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq39_HTML.gif . In such a sector the Lyapunov derivative is:
https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_Equn_HTML.gif

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) https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq40_HTML.gif can be similarly proved to hold in the sector https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq41_HTML.gif and https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq42_HTML.gif .

Consider the sector defined by https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq37_HTML.gif and https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq42_HTML.gif (DPA in Figure 2B) for which https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq43_HTML.gif in such a sector the Lyapunov derivative is:
https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_Equo_HTML.gif
Note that in the last step we have added and subtracted the null terms (10). In the opposite sector (BPC in Figure 2B) https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq41_HTML.gif and https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq38_HTML.gif , we can prove that https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq44_HTML.gif . We just proved that
https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_Equp_HTML.gif

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, https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq37_HTML.gif and https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq42_HTML.gif , and its opposite

CPB, https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq41_HTML.gif and https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq38_HTML.gif , are both positive invariant sets.

We can apply Nagumo's theorem: consider the half-line PA originating in P, where https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq45_HTML.gif and https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq37_HTML.gif . Therefore we have that (again by adding the null term in (10)):
https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_Equq_HTML.gif
Similarly, on half-line PD where https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq46_HTML.gif and https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq42_HTML.gif , by considering (10) we derive
https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_Equr_HTML.gif

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 https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq47_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq48_HTML.gif and https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq49_HTML.gif . Consider the new input https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq50_HTML.gif (keeping https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq51_HTML.gif ). Then the system converges to a new equilibrium with https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq52_HTML.gif and https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq53_HTML.gif . There is no undershoot, respectively, overshoot.

Proof The steady state values https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq19_HTML.gif and https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq20_HTML.gif are respectively monotonically decreasing and increasing functions of c2. Indeed, consider the steady-state condition
https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_Equs_HTML.gif
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
https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_Equt_HTML.gif

where https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq54_HTML.gif (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 https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq19_HTML.gif and https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq20_HTML.gif are decreasing and increasing functions of c2.

The absence of overshoot and undershoot is an immediate consequence of the invariance of the sector https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq37_HTML.gif and https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq42_HTML.gif 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:
https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_Equ11_HTML.gif
(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 https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq55_HTML.gif and https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq56_HTML.gif 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, https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq57_HTML.gif . 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):
https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_Equ12_HTML.gif
(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.
https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_Fig3_HTML.jpg
Figure 3

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: a1u, a23: decreasing-exactly-null with threshold values https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq58_HTML.gif and https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq59_HTML.gif ; 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
https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_Equu_HTML.gif

The solution https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq60_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq61_HTML.gif . 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 a1u(x1) and a23(x2) are zero above the thresholds https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq58_HTML.gif and https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq59_HTML.gif , we have that https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq62_HTML.gif and https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq63_HTML.gif for https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq64_HTML.gif and https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq65_HTML.gif , respectively; therefore we assume https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq66_HTML.gif , https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq67_HTML.gif , for all t ≥ 0.

Proposition 3 There exists an equilibrium for system (12) if and only if
https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_Equ13_HTML.gif
(13)

where https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq60_HTML.gif as previously defined . All the equilibrium values https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq60_HTML.gif , https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq20_HTML.gif and https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq68_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq19_HTML.gif . 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 https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq19_HTML.gif . Then, the unique equilibrium point is globally attractive for any initial condition x2(0), x3(0) ≥ 0. Moreover, assume that
https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_Equ14_HTML.gif
(14)
then we can give the following bound for the transient of x3(t)
https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_Equ15_HTML.gif
(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.
https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_Equ16_HTML.gif
(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, https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq69_HTML.gif , 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:
https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_Equ17_HTML.gif
(17)

where c13(x3) = f1(x3), b11 = δ1, a21 = β1, b22 = δ2, a32(u) = f2(u) =, b32(x3) = f3(x3), c3u= 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 https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq68_HTML.gif , x4 converges to https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq70_HTML.gif . 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 c3uu 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 xx* 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:
https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_Equ18_HTML.gif
(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:
https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_Equ19_HTML.gif
(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 https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq71_HTML.gif and https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq72_HTML.gif and therefore the following sums are constant
https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_Equ20_HTML.gif
(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 https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq73_HTML.gif 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 < μμ*, 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: https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq74_HTML.gif while https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq20_HTML.gif and https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq75_HTML.gif have the reverse order https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq76_HTML.gif and https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq77_HTML.gif

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).

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.

Notes

1The concentration mismatch is more "softly" controlled, since the derivative of the difference https://static-content.springer.com/image/art%3A10.1186%2F1752-0509-5-74/MediaObjects/12918_2010_Article_691_IEq78_HTML.gif is not influenced by the nonlinear term b12(x1, x2)x2 = b21(x1, x2)x1.

2I.e. the nullclines have no common tangent lines.

3Cf. the erratum: http://www.math.rutgers.edu/~sontag/FTPDIR/angeli-ferrell-sontag-pnas04-errata. txt and [42].

Declarations

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.

Authors’ Affiliations

(1)
Dipartimento di Matematica ed Informatica, Universitá degli Studi di Udine
(2)
Division of Engineering and Applied Science, California Institute of Technology

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This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.