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PLoS Comput Biol. 2010 April; 6(4): e1000739.
Published online 2010 April 15. doi:  10.1371/journal.pcbi.1000739
PMCID: PMC2855316

Global Entrainment of Transcriptional Systems to Periodic Inputs

Giovanni Russo, 1 Mario di Bernardo, 1 , 2 , * and Eduardo D. Sontag 3 , ¶ , *
Herbert M. Sauro, Editor

Abstract

This paper addresses the problem of providing mathematical conditions that allow one to ensure that biological networks, such as transcriptional systems, can be globally entrained to external periodic inputs. Despite appearing obvious at first, this is by no means a generic property of nonlinear dynamical systems. Through the use of contraction theory, a powerful tool from dynamical systems theory, it is shown that certain systems driven by external periodic signals have the property that all their solutions converge to a fixed limit cycle. General results are proved, and the properties are verified in the specific cases of models of transcriptional systems as well as constructs of interest in synthetic biology. A self-contained exposition of all needed results is given in the paper.

Author Summary

The activities of living organisms are governed by complex sets of biochemical reactions. Often, entrainment to certain external signals helps control the timing and sequencing of reactions. An important open problem is to understand the onset of entrainment and under what conditions it can be ensured in the presence of uncertainties, noise, and environmental variations. In this paper, we focus mainly on transcriptional systems, modeled by Ordinary Differential Equations. These are basic building blocks for more complex biochemical systems. However, the results that we obtain are of more generality. To illustrate this generality, and to emphasize the use of our techniques in synthetic biology, we discuss the entrainment of a Repressilator circuit and the synchronization of a network of Repressilators. We answer the following two questions: 1) What are the dynamical mechanisms that ensure the entrainment to periodic inputs in transcriptional modules? 2) Starting from natural systems, what properties can be used to design novel synthetic biological circuits that can be entrained? For some biological systems which are always “in contact” with a continuously changing environment, entrainment may be a “desired” property. Thus, answering the above two questions is of fundamental importance. While entrainment may appear obvious at first thought, it is not a generic property of nonlinear dynamical systems. The main result of our paper shows that, even if the transcriptional modules are modeled by nonlinear ODEs, they can be entrained by any (positive) periodic signal. Surprisingly, such a property is preserved if the system parameters are varied: entrainment is obtained independently of the particular biochemical conditions. We prove that combinations of the above transcriptional module also show the same property. Finally, we show how the developed tools can be applied to design synthetic biochemical systems guaranteed to exhibit entrainment.

Introduction

Periodic, clock-like rhythms pervade nature and regulate the function of all living organisms. For instance, circadian rhythms are regulated by an endogenous biological clock entrained by the light signals from the environment that then acts as a pacemaker [1]. Moreover, such an entrainment can be obtained even if daily variations are present, like e.g. temperature and light variations. Another important example of entrainment in biological systems is at the molecular level, where the synchronization of several cellular processes is regulated by the cell cycle [2].

An important question in mathematical and computational biology is that of finding conditions ensuring that entrainment occurs. The objective is to identify classes of biological systems that can be entrained by an exogenous signal. To solve this problem, modelers often resort to simulations in order to show the existence of periodic solutions in the system of interest. Simulations, however, can never prove that solutions will exist for all parameter values, and they are subject to numerical errors. Moreover, robustness of entrained solutions needs to be checked in the presence of noise and uncertainties, which cannot be avoided experimentally.

From a mathematical viewpoint, the problem of formally showing that entrainment takes place is known to be extremely difficult. Indeed, if a stable linear time-invariant model is used to represent the system of interest, then entrainment is usually expected, when the system is driven by an external periodic input, with the system response being a filtered, shifted version of the external driving signal. However, in general, as is often the case in biology, models are nonlinear. The response of nonlinear systems to periodic inputs is the subject of much current systems biology experimentation; for example, in [3], the case of a cell signaling system driven by a periodic square-wave input is considered. From measurements of a periodic output, the authors fit a transfer function to the system, implicitly modeling the system as linear even though (as stated in the Supplemental Materials to [3]) there are saturation effects so the true system is nonlinear. For nonlinear systems, driving the system by an external periodic signal does not guarantee the system response to also be a periodic solution, as nonlinear systems can exhibit harmonic generation or suppression and complex behavior such as chaos or quasi-periodic solutions [4]. This may happen even if the system is well-behaved with respect to constant inputs; for example, there are systems which converge to a fixed steady state no matter what is the input excitation, so long as this input signal is constant, yet respond chaotically to the simplest oscillatory input; we outline such an example in the Materials and Methods Section, see also [5]. Thus, a most interesting open problem is that of finding conditions for the entrainment to external inputs of biological systems modeled by sets of nonlinear differential equations.

One approach to analyzing the convergence behavior of nonlinear dynamical systems is to use Lyapunov functions. However, in biological applications, the appropriate Lyapunov functions are not always easy to find and, moreover, convergence is not guaranteed in general in the presence of noise and/or uncertainties. Also, such an approach can be hard to apply to the case of non-autonomous systems (that is, dynamical systems directly dependent on time), as is the case when dealing with periodically forced systems.

The above limitations can be overcome if the convergence problem is interpreted as a property of all trajectories, asking that all solutions converge towards one another (contraction). This is the viewpoint of contraction theory, [6], [7], and more generally incremental stability methods [8]. Global results are possible, and these are robust to noise, in the sense that, if a system satisfies a contraction property then trajectories remain bounded in the phase space [9]. Contraction theory has a long history. Contractions in metric functional spaces can be traced back to the work of Banach and Caccioppoli [10] and, in the field of dynamical systems, to [11] and even to [12] (see also [13], [8], and e.g. [14] for a more exhaustive list of related references). Contraction theory has been successfully applied to both nonlinear control and observer problems, [7], [15] and, more recently, to synchronization and consensus problems in complex networks [16], [17],[18]. In [19] it was proposed that contraction can be particularly useful when dealing with the analysis and characterization of biological networks. In particular, it was found that using non Euclidean norms, as also suggested in [6] (Sec. 3.7ii), can be particularly effective in this context [19], [20].

One of the objectives of this paper is to give a self-contained exposition, with all proofs included, of results in contraction theory as applied to entrainment of periodic signals, and, moreover, to show their applicability to problems of biological interest. We believe that contraction analysis should be recognized as an important component of the “toolkit” of systems biology, and this paper should be useful to other researchers contemplating the use of these tools.

For concreteness, we focus mainly on transcriptional systems, as well as related biochemical systems, which are basic building blocks for more complex biochemical systems. However, the results that we obtain are of more generality. To illustrate this generality, and to emphasize the use of our techniques in synthetic biology design, we discuss as well the entrainment of a Repressilator circuit in a parameter regime in which endogenous oscillations do not occur, as well as the synchronization of a network of Repressilators. A surprising fact is that, for these applications, and contrary to many engineering applications, norms other than Euclidean, and associated matrix measures, must be considered.

Mathematical tools

We consider in this paper systems of ordinary differential equations, generally time-dependent:

equation image
(1)

defined for An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e002.jpg and An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e003.jpg, where An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e004.jpg is a subset of An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e005.jpg. It will be assumed that An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e006.jpg is differentiable on An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e007.jpg, and that An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e008.jpg, as well as the Jacobian of An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e009.jpg with respect to An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e010.jpg, denoted as An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e011.jpg, are both continuous in An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e012.jpg. In applications of the theory, it is often the case that An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e013.jpg will be a closed set, for example given by non-negativity constraints on variables as well as linear equalities representing mass-conservation laws. For a non-open set An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e014.jpg, differentiability in An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e015.jpg means that the vector field An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e016.jpg can be extended as a differentiable function to some open set which includes An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e017.jpg, and the continuity hypotheses with respect to An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e018.jpg hold on this open set.

We denote by An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e019.jpg the value of the solution An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e020.jpg at time An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e021.jpg of the differential equation (1) with initial value An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e022.jpg. It is implicit in the notation that An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e023.jpg (“forward invariance” of the state set An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e024.jpg). This solution is in principle defined only on some interval An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e025.jpg, but we will assume that An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e026.jpg is defined for all An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e027.jpg. Conditions which guarantee such a “forward-completeness” property are often satisfied in biological applications, for example whenever the set An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e028.jpg is closed and bounded, or whenever the vector field An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e029.jpg is bounded. (See Appendix C in [21] for more discussion, as well as [22] for a characterization of the forward completeness property.) Under the stated assumptions, the function An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e030.jpg is jointly differentiable in all its arguments (this is a standard fact on well-posedness of differential equations, see for example Appendix C in [21]).

We recall (see for instance [23]) that, given a vector norm on Euclidean space (An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e031.jpg), with its induced matrix norm An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e032.jpg, the associated matrix measure An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e033.jpg is defined as the directional derivative of the matrix norm, that is,

equation image

For example, if An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e035.jpg is the standard Euclidean 2-norm, then An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e036.jpg is the maximum eigenvalue of the symmetric part of An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e037.jpg. As we shall see, however, different norms will be useful for our applications. Matrix measures are also known as “logarithmic norms”, a concept independently introduced by Germund Dahlquist and Sergei Lozinskii in 1959, [24],[25]. The limit is known to exist, and the convergence is monotonic, see [24], [26].

We will say that system (1) is infinitesimally contracting on a convex set An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e038.jpg if there exists some norm in An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e039.jpg, with associated matrix measure An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e040.jpg such that, for some constant An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e041.jpg,

equation image
(2)

Let us discuss informally (rigorous proofs are given later) the motivation for this concept. Since by assumption An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e043.jpg is continuously differentiable, the following exact differential relation can be obtained from (1):

equation image
(3)

where, as before, An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e045.jpg denotes the Jacobian of the vector field An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e046.jpg, as a function of An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e047.jpg and An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e048.jpg, and where An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e049.jpg denotes a small change in states and “An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e050.jpg” means An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e051.jpg, evaluated along a trajectory. (In mechanics, as in [27], An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e052.jpg is called “virtual displacement”, and formally it may be thought of as a linear tangent differential form, differentiable with respect to time.) Consider now two neighboring trajectories of (1), evolving in An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e053.jpg, and the virtual displacements between them. Note that (3) can be thought of as a linear time-varying dynamical system of the form:

equation image

once that An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e055.jpg is thought of as a fixed function of time. Hence, an upper bound for the magnitude of its solutions can be obtained by means of the Coppel inequality [28], yielding:

equation image
(4)

where An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e057.jpg is the matrix measure of the system Jacobian induced by the norm being considered on the states and An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e058.jpg. Using (4) and (2), we have that

equation image

Thus, trajectories starting from infinitesimally close initial conditions converge exponentially towards each other. In what follows we will refer to An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e060.jpg as contraction (or convergence) rate.

The key theoretical result about contracting systems links infinitesimal and global contractivity, and is stated below. This result can be traced, under different technical assumptions, to e.g. [6], [13], [12], [11].

Theorem 1. Suppose that An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e061.jpg is a convex subset of An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e062.jpg and that An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e063.jpg is infinitesimally contracting with contraction rate An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e064.jpg . Then, for every two solutions An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e065.jpg and An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e066.jpg of (1), it holds that:

equation image
(5)

In other words, infinitesimal contractivity implies global contractivity. In the Materials and Methods section, we provide a self-contained proof of Theorem 1. In fact, the result is shown there in a generalized form, in which convexity is replaced by a weaker constraint on the geometry of the space.

In actual applications, often one is given a system which depends implicitly on the time, An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e068.jpg, by means of a continuous function An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e069.jpg, i.e. systems dynamics are represented by An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e070.jpg. In this case, An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e071.jpg (where An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e072.jpg is some subset of An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e073.jpg), represents an external input. It is important to observe that the contractivity property does not require any prior information about this external input. In fact, since An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e074.jpg does not depend on the system state variables, when checking the property, it may be viewed as a constant parameter, An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e075.jpg. Thus, if contractivity of An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e076.jpg holds uniformly An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e077.jpg, then it will also hold for An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e078.jpg.

Given a number An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e079.jpg, we will say that system (1) is An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e080.jpg -periodic if it holds that

equation image

Notice that the system An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e082.jpg is An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e083.jpg-periodic, if the external input, An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e084.jpg, is itself a periodic function of period An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e085.jpg.

The following is the basic theoretical result about periodic orbits that will be used in the paper. A proof may be found in [6], Sec. 3.7.vi.

Theorem 2. Suppose that:

  • An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e086.jpg is a closed convex subset of An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e087.jpg;
  • An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e088.jpg is infinitesimally contracting with contraction rate An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e089.jpg;
  • An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e090.jpg is An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e091.jpg -periodic.

Then, there is a unique periodic solution An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e092.jpg of (1) of period An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e093.jpg and, for every solution An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e094.jpg , it holds that An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e095.jpg as An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e096.jpg.

In the Materials and Methods section of this paper, we provide a self-contained proof of Theorem 2, in a generalized form which does not require convexity.

A simple example

As a first example to illustrate the application of the concepts introduced so far, we choose a simple bimolecular reaction, in which a molecule of An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e097.jpg and one of An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e098.jpg can reversibly combine to produce a molecule of An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e099.jpg.

This system can be modeled by the following set of differential equations:

equation image
(6)

where we are using An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e101.jpg to denote the concentration of An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e102.jpg and so forth. The system evolves in the positive orthant of An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e103.jpg. Solutions satisfy (stoichiometry) constraints:

equation image
(7)

for some constants An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e105.jpg and An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e106.jpg.

We will assume that one or both of the “kinetic constants” An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e107.jpg are time-varying, with period An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e108.jpg. Such a situation arises when the An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e109.jpg's depend on concentrations of additional enzymes, which are available in large amounts compared to the concentrations of An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e110.jpg, but whose concentrations are periodically varying. The only assumption will be that An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e111.jpg and An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e112.jpg for all An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e113.jpg.

Because of the conservation laws (7), we may restrict our study to the equation for An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e114.jpg. Once that all solutions of this equation are shown to globally converge to a periodic orbit, the same will follow for An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e115.jpg and An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e116.jpg. We have that:

equation image
(8)

Because An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e118.jpg and An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e119.jpg, this system is studied on the subset of An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e120.jpg defined by An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e121.jpg. The equation can be rewritten as:

equation image
(9)

Differentiation with respect to An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e123.jpg of the right-hand side in the above system yields this (An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e124.jpg) Jacobian:

equation image
(10)

Since we know that An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e126.jpg and An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e127.jpg, it follows that

equation image

for An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e129.jpg. Using any norm (this example is in dimension one) we have that An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e130.jpg. So (6) is contracting and, by means of Theorem 2, solutions will globally converge to a unique solution of period An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e131.jpg (notice that such a solution depends on system parameters).

Figure 1 shows the behavior of the dynamical system (9), using two different values of An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e132.jpg. Notice that the asymptotic behavior of the system depends on the particular choice of the biochemical parameters being used. Furthermore, it is worth noticing here that the higher the value of An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e133.jpg, the faster will be the convergence to the attractor.

Figure 1
Entrainment of (9) to An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e134.jpg.

Results

Mathematical model and problem statement

We study a general externally-driven transcriptional module. We assume that the rate of production of a transcription factor An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e142.jpg is proportional to the value of a time dependent input function An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e143.jpg, and An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e144.jpg is subject to degradation and/or dilution at a linear rate. (Later, we generalize the model to also allow nonlinear degradation as well.) The signal An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e145.jpg might be an external input, or it might represent the concentration of an enzyme or of a second messenger that activates An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e146.jpg. In turn, An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e147.jpg drives a downstream transcriptional module by binding to a promoter (or substrate), denoted by An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e148.jpg with concentration An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e149.jpg. The binding reaction of An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e150.jpg with An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e151.jpg is reversible and given by:

equation image

where An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e153.jpg is the complex protein-promoter, and the binding and dissociation rates are An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e154.jpg and An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e155.jpg respectively. As the promoter is not subject to decay, its total concentration, An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e156.jpg, is conserved, so that the following conservation relation holds:

equation image
(11)

We wish to study the behavior of solutions of the system that couples An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e158.jpg and An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e159.jpg, and specifically to show that, when the input An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e160.jpg is periodic with period An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e161.jpg, this coupled system has the property that all solutions converge to some globally attracting limit cycle whose period is also An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e162.jpg.

Such transcriptional modules are ubiquitous in biology, natural as well as synthetic, and their behavior was recently studied in [29] in the context of “retroactivity” (impedance or load) effects. If we think of An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e163.jpg as the concentration of a protein An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e164.jpg that is a transcription factor for An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e165.jpg, and we ignore fast mRNA dynamics, such a system can be schematically represented as in Figure 2, which is adapted from [29]. Notice that An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e166.jpg here does not need to be the concentration of a transcriptional activator of An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e167.jpg for our results to hold. The results will be valid for any mathematical model for the concentrations, An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e168.jpg, of An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e169.jpg and An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e170.jpg, of An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e171.jpg (the concentration of An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e172.jpg is conserved) of the form:

equation image
(12)

Figure 2
A schematic diagram of the transcriptional system modeled in (12).

An objective in this paper is, thus, to show that, when An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e177.jpg is a periodic input, all solutions of system (12) converge to a (unique) limit cycle (Figure 3). The key tool in this analysis is to show that uniform contractivity holds. Since in this example the input appears additively, uniform contractivity is simply the requirement that the unforced system (An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e178.jpg) is contractive. Thus, the main step will be to establish the following technical result, see the Material and Methods:

Figure 3
Entrainment of the transcriptional module (12).

Proposition 1. The system

equation image

where

equation image
(13)

for all An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e188.jpg , and An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e189.jpg , An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e190.jpg , An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e191.jpg , and An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e192.jpg are arbitrary positive constants, is contracting.

Appealing to Theorem 2, we then have the following immediate Corollary:

Proposition 2. For any given nonnegative periodic input An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e193.jpg of period An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e194.jpg , all solutions of system (12) converge exponentially to a periodic solution of period An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e195.jpg.

In the following sections, we introduce a matrix measure that will help establish contractivity, and we prove Proposition 1. We will also discuss several extensions of this result, allowing the consideration of multiple driven subsystems as well as more general nonlinear systems with a similar structure. (A graphical algorithm to prove contraction of generic networks of nonlinear systems can also be found in [18] where this transcriptional module is also studied.)

Proof of Proposition 1

We will use Theorem 2. The Jacobian matrix to be studied is:

equation image
(14)

As matrix measure, we will use the measure An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e197.jpg induced by the vector norm An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e198.jpg, where An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e199.jpg is a suitable nonsingular matrix. More specifically, we will pick An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e200.jpg diagonal:

equation image
(15)

where An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e202.jpg and An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e203.jpg are two positive numbers to be appropriately chosen depending on the parameters defining the system.

It follows from general facts about matrix measures that

equation image
(16)

where An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e205.jpg is the measure associated to the An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e206.jpg norm and is explicitly given by the following formula:

equation image
(17)

Observe that, if the entries of An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e208.jpg are negative, then asking that An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e209.jpg amounts to a column diagonal dominance condition. (The above formula is for real matrices. If complex matrices would be considered, then the term An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e210.jpg should be replaced by its real part An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e211.jpg.)

Thus, the first step in computing An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e212.jpg is to calculate An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e213.jpg:

equation image
(18)

Using (17), we obtain:

equation image
(19)

Note that we are not interested in calculating the exact value for the above measure, but just in ensuring that it is negative. To guarantee that An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e216.jpg, the following two conditions must hold:

equation image
(20)

equation image
(21)

Thus, the problem becomes that of checking if there exists an appropriate range of values for An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e219.jpg, An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e220.jpg that satisfy (20) and (21) simultaneously.

The left hand side of (21) can be written as:

equation image
(22)

which is negative if and only if An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e222.jpg. In particular, in this case we have:

equation image

The idea is now to ensure negativity of (20) by using appropriate values for An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e224.jpg and An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e225.jpg which fulfill the above constraint. Recall that the term An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e226.jpg because of the choice of the state space (this quantity represents a concentration). Thus, the left hand side of (20) becomes

equation image
(23)

The next step is to choose appropriately An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e228.jpg and An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e229.jpg (without violating the constraint An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e230.jpg). Imposing An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e231.jpg, An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e232.jpg, (23) becomes

equation image
(24)

Then, we have to choose an appropriate value for An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e234.jpg in order to make the above quantity uniformly negative. In particular, (24) is uniformly negative if and only if

equation image
(25)

We can now choose

equation image

with An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e237.jpg. In this case, (24) becomes

equation image

Thus, choosing An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e239.jpg and An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e240.jpg, with An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e241.jpg, we have An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e242.jpg. Furthermore, the contraction rate An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e243.jpg, is given by:

equation image

Notice that An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e245.jpg depends on both system parameters and on the elements An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e246.jpg, An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e247.jpg, i.e. it depends on the particular metric chosen to prove contraction. This completes the proof of the Proposition.

Generalizations

In this Section, we discuss various generalizations that use the same proof technique.

Assuming An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e248.jpg activation by enzyme kinetics

The previous model assumed that An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e249.jpg was created in proportion to the amount of external signal An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e250.jpg. While this may be a natural assumption if An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e251.jpg is a transcription factor that controls the expression of An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e252.jpg, a different model applies if, instead, the “active” form An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e253.jpg is obtained from an “inactive” form An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e254.jpg, for example through a phosphorylation reaction which is catalyzed by a kinase whose abundance is represented by An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e255.jpg. Suppose that An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e256.jpg can also be constitutively deactivated. Thus, the complete system of reactions consists of

equation image

together with

equation image

where the forward reaction depends on An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e259.jpg. Since the concentrations of An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e260.jpg must remain constant, let us say at a value An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e261.jpg, we eliminate An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e262.jpg and have:

equation image
(26)

We will prove that if An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e264.jpg is periodic and positive, i.e. An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e265.jpg, then a globally attracting limit cycle exists. Namely, it will be shown, after having performed a linear coordinate transformation, that there exists a negative matrix measure for the system of interest.

Consider, indeed, the following change of the state variables:

equation image
(27)

The system dynamics then becomes:

equation image
(28)

As matrix measure, we will now use the measure An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e268.jpg induced by the vector norm An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e269.jpg. (Notice that this time, the matrix An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e270.jpg is the identity matrix).

Given a real matrix An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e271.jpg, the matrix measure An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e272.jpg is explicitly given by the following formula (see e.g. [23]):

equation image
(29)

(Observe that this is a row-dominance condition, in contrast to the dual column-dominance condition used for An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e274.jpg.)

Differentiation of (28) yields the Jacobian matrix:

equation image

Thus, it immediately follow from (29) that An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e276.jpg is negative if and only if:

equation image
(30)

equation image
(31)

The first inequality is clearly satisfied since by hypotheses both system parameters and the periodic input An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e279.jpg are positive. In particular, we have:

equation image

By using (27) (recall that An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e281.jpg), the right hand side of the second inequality can be written as:

equation image

Since all system parameters are positive and An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e283.jpg, the above quantity is negative and upper bounded by An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e284.jpg.

Thus, we have that An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e285.jpg, where:

equation image

The contraction property for the system is then proved. By means of Theorem 2, we can then conclude that the system can be entrained by any periodic input.

Simulation results are presented in Figure 4, where the presence of a stable limit cycle having the same period as An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e287.jpg is shown.

Figure 4
Entrainment of the transcriptional module (26).

Multiple driven systems

We may also treat the case in which the species An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e297.jpg regulates multiple downstream transcriptional modules which act independently from each other, as shown in Figure 5. The biochemical parameters defining the different downstream modules may be different from each other, representing a situation in which the transcription factor An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e298.jpg regulates different species. After proving a general result on oscillations, and assuming that parameters satisfy the retroactivity estimates discussed in [29], one may in this fashion design a single input-multi output module in which e.g. the outputs are periodic functions with different mean values, settling times, and so forth.

Figure 5
Multiple driven transcriptional modules.

We denote by An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e299.jpg the various promoters, and use An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e300.jpg to denote the concentrations of the respective promoters complexed with An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e301.jpg. The resulting mathematical model becomes:

equation image
(32)

We consider the corresponding system with no input first, assuming that the states satisfy An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e303.jpg and An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e304.jpg for all An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e305.jpg.

Our generalization can be stated as follows:

Proposition 3. System (32) with no input (i.e. An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e306.jpg ) is contracting. Hence, if An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e307.jpg is a non-zero periodic input, its solutions exponentially converge towards a periodic orbit of the same period as An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e308.jpg.

Proof. We only outline the proof, since it is similar to the proof of Proposition 2. We employ the following matrix measure:

equation image
(33)

where

equation image
(34)

and the scalars An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e311.jpg have to be chosen appropriately (An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e312.jpg).

In this case,

equation image
(35)

and

equation image
(36)

Hence, the An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e315.jpg inequalities to be satisfied are:

equation image
(37)

and

equation image
(38)

Clearly, the set of inequalities above admits a solution. Indeed, the left hand side of (38) can be recast as

equation image

which is negative definite if and only if An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e319.jpg for all An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e320.jpg. Specifically, in this case we have

equation image

Also, from (37), as An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e322.jpg for all An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e323.jpg, we have that (37) can be rewritten as:

equation image

Since An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e325.jpg, we can impose An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e326.jpg (with An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e327.jpg) and the above inequality becomes

equation image

Clearly, such inequality is satisfied if we choose An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e329.jpg sufficiently small; namely:

equation image

Following a similar derivation to that of the previous Section, we can choose

equation image

with An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e332.jpg. In this case, we have:

equation image

Thus, An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e334.jpg, where

equation image

The second part of the Proposition is then proved by applying Theorem 2.

In Figure 6 the behavior of two-driven downstream transcriptional modules is shown. Notice that both the downstream modules are entrained by the periodic input An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e336.jpg, but their steady state behavior is different.

Figure 6
Entrainment of two-driven transcriptional modules.

Notice that, by the same arguments used above, it can be proven that

equation image
(39)

is contracting.

Transcriptional cascades

A cascade of (infinitesimally) contracting systems is also (infinitesimally) contracting [6], [30] (see Materials and Methods for an alternative proof). This implies that any transcriptional cascade, will also give rise to a contracting system, and, in particular, will entrain to periodic inputs. By a transcriptional cascade we mean a system as shown in Figure 7. In this figure, we interpret the intermediate variables An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e351.jpg as transcription factors, making the simplifying assumption that TF concentration is proportional to active promoter for the corresponding gene. (More complex models, incorporating transcription, translation, and post-translational modifications could themselves, in turn, be modeled as cascades of contracting systems.)

Figure 7
Transcriptional cascade discussed in the text.

More abstract systems

We can extend our results even further, to a larger class of nonlinear systems, as long as the same general structure is present. This can be useful for example to design new synthetic transcription modules or to analyze the entrainment properties of general biological systems. We start with a discussion of a two dimensional system of the form:

equation image
(40)

In molecular biology, An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e353.jpg would typically represent a nonlinear degradation, for instance in Michaelis-Menten form, while the function An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e354.jpg represents the interaction between An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e355.jpg and An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e356.jpg. The aim of this Section is to find conditions on the degradation and interaction terms that allow one to show contractivity of the unforced (no input An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e357.jpg) system, and hence existence of globally attracting limit cycles.

We assume that the state space An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e358.jpg is compact (closed and bounded) as well as convex. Since the input appears additively, we must prove contractivity of the unforced system.

Theorem 3. System (40), without inputs An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e359.jpg , evolving on a convex compact subset of phase space is contracting, provided that the following conditions are all satisfied, for each An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e360.jpg :

  • An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e361.jpg;
  • An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e362.jpg;
  • An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e363.jpg does not change sign;
  • An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e364.jpg.

Notice that the last condition is automatically satisfied if An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e365.jpg, because An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e366.jpg.

As before, we prove contraction by constructing an appropriate negative measure for the Jacobian of the vector field. In this case, the Jacobian matrix is:

equation image
(41)

Once again, as matrix measure we will use:

equation image
(42)

with

equation image
(43)

and An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e370.jpg appropriately chosen.

Using (42) we have

equation image
(44)

Following the same steps as the proof of Proposition 1, we have to show that:

equation image
(45)

equation image
(46)

Clearly, if An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e374.jpg for every An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e375.jpg and An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e376.jpg, the first inequality is satisfied, with

equation image

To prove the theorem we need to show that there exists An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e378.jpg and An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e379.jpg satisfying (46). For such inequality, since An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e380.jpg does not change sign in An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e381.jpg by hypothesis, we have two possibilities:

  1. An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e382.jpg, An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e383.jpg;
  2. An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e384.jpg, An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e385.jpg.

In the first case, the right hand side of (46) becomes

equation image
(47)

Choosing An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e387.jpg, with An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e388.jpg, we have:

equation image

Specifically, if we now pick

equation image

where An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e391.jpg and An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e392.jpg, we have that the above quantity is uniformly negative definite, i.e.

equation image

In the second case, the right hand side of (46) becomes

equation image
(48)

Again, by choosing An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e395.jpg, with An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e396.jpg, we have the following upper bound for the expression in (48):

equation image
(49)

Thus, it follows that An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e398.jpg provided that the above quantity is uniformly negative definite. Since, by hypotheses,

equation image
(50)

then An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e400.jpg. The proof of the Theorem is now complete.

From a biological viewpoint, the hardest hypothesis to satisfy in Theorem 3 might be that on the derivatives of An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e401.jpg. However, it is possible to relax the hypothesis on An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e402.jpg if the rate of change of An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e403.jpg with respect to An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e404.jpg, i.e. An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e405.jpg, is sufficiently larger than An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e406.jpg. In particular, the following result can be proved.

Theorem 4. System (40), without inputs An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e407.jpg , evolving on a convex compact set, is contractive provided that:

  • An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e408.jpg, An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e409.jpg;
  • An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e410.jpg, An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e411.jpg;
  • An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e412.jpg.

Proof. The proof is similar to that of Theorem 3. In particular, we can repeat the same derivation to obtain again inequality (46). Thence, as no hypothesis is made on the sign of An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e413.jpg, choosing An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e414.jpg we have

equation image
(51)

Thus, it follows that, if An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e416.jpg, then An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e417.jpg An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e418.jpg such that An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e419.jpg, implying contractivity. The above condition is satisfied by hypotheses, hence the theorem is proved.

Remarks

Theorems 3 and 4 show the possibility of designing with high flexibility the self-degradation and interaction functions for an input-output module.

This flexibility can be further increased, for example in the following ways:

  • Results similar to that of the above Theorems can be derived (and also extended) if some self degradation rate for An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e420.jpg is present in (40), i.e.
    equation image
    (52)
    with An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e422.jpg.
  • Theorem 3 and Theorem 4 can also be extended to the case in which the An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e423.jpg-module drives more than one downstream transcriptional modules.

Applications to synthetic biology

We introduced above a methodology for checking if a given transcriptional module can be entrained to some periodic input. The aim of this section is to show that our methodology can serve as an effective tool for designing synthetic biological circuits that are entrained to some desired external input.

In particular, we will consider the synthetic biological oscillator known as the Repressilator [31], for which an additional coupling module has been recently proposed in [32]. A numerical investigation of the synchronization of a network of non-identical Repressilators was independently reported in [33].

We will show that our results can be used to isolate a set of biochemical parameters for which one can guarantee the entrainment to any external periodic signal of this synthetic biological circuit. In what follows, we will use the equations presented in [32] to model the Repressilator and the additional coupling model.

Entrainment using an intra-cellular auto-inducer

The Repressilator is a synthetic biological circuit that consists of three genes that inhibit each other in a cyclic way [31]. As shown in Figure 8, gene lacI (associated to the state variable An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e424.jpg in the model) expresses protein LacI (An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e425.jpg), which inhibits the transcription of gene tetR (An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e426.jpg). This translates into protein TetR (An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e427.jpg), which inhibits transcription of gene cI (An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e428.jpg). Finally, the protein CI (An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e429.jpg), translated from cI, inhibits expression of lacI, completing the cycle.

Figure 8
The Repressilator circuit.

In Figure 9 a modular addition to the three-genes circuit is presented. The module was first presented in [32] and makes the Repressilator circuit sensitive to the concentration of the auto-inducer (labeled as An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e430.jpg in the model) which is a small molecule that can pass through the cell membrane. Specifically, the module makes use of two proteins: (i) LuxI, which synthesizes the auto-inducer; (ii) LuxR, with which the auto-inducer synthesized by LuxI forms a complex that activates the transcription of various genes.

Figure 9
Modular addition to the Repressilator circuit.

We model the above circuit with the simplified set of differential equations proposed in [32]. Specifically, the dynamics of the mRNAs are

equation image
(53)

Notice that the above equations are dimensionless. This is done by: (i) measuring time in units of mRNA lifetime (which is assumed equal for the three genes), and (ii) expressing the protein levels in units of their Michaelis constant. The parameter An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e432.jpg represents the dimensionless transcription rate in the absence of self-repression, while An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e433.jpg denotes the maximum contribution of the auto-inducer to the expression of lacI.

The dynamics of the proteins are described by

equation image
(54)

The parameters An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e435.jpg, An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e436.jpg, An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e437.jpg represent the ratios between the mRNAs and the respective proteins' lifetimes and An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e438.jpg, An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e439.jpg, An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e440.jpg represent the protein decay rate.

The last differential equation of the model from [32] keeps track of the evolution of the intra-cellular auto-inducer. It is assumed that the proteins TetR and LuxI have equal lifetimes. This in turn implies that the dynamics of such proteins are identical, and hence one uses the same variable to describe both protein concentrations. Thus, the dynamics of the auto-inducer are given by:

equation image

where An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e442.jpg is the rate of degradation of An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e443.jpg.

We now model the forcing on the intracellular auto-inducer concentration by adding an external input An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e444.jpg to the above dynamical equation. The equation for An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e445.jpg becomes:

equation image
(55)

where An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e447.jpg can be thought as a diffusion rate.

We will now use the analytical methodology developed in the previous sections, to properly tune the biochemical parameters of the Repressilator circuit, whose mathematical model consists of the set of differential equations (53), (54), (55), so that it shows entrainment to the periodic input An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e448.jpg. That is, the measured output (e.g. An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e449.jpg), oscillates asymptotically with a period equal to that of An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e450.jpg. Of course, the periodic orbit of the output will depend on the particular choice of the parameters.

In what follows, we assume that all the system parameters can be varied except for the self-degradations that we assume to be fixed as, in practice, they are difficult to modify.

In this case, the Jacobian matrix to be studied is

equation image
(56)

The matrix measure that we will use to prove contraction is

equation image

where An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e453.jpg is a An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e454.jpg diagonal matrix having on the main diagonal the positive arbitrary scalars An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e455.jpg. Computation of An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e456.jpg yields

equation image
(57)

Thus, from the definition of An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e458.jpg given in (29), we have that there exists some An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e459.jpg such that An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e460.jpg, An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e461.jpg if and only if there exists a set of scalars An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e462.jpg, An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e463.jpg, such that

equation image
(58a)

equation image
(58b)

equation image
(58c)

equation image
(58d)

equation image
(58e)

equation image
(58f)

equation image
(58g)

It is easy to check that the nonlinear terms in the above equations satisfy the following inequalities:

equation image

and

equation image

for all An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e473.jpg. Hence, the system of inequalities (58a)–(58g) are satisfied, if the following set is fulfilled:

equation image
(59a)

equation image
(59b)

equation image
(59c)

equation image
(59d)

equation image
(59e)

equation image
(59f)

equation image
(59g)

The system can then be proved to be contracting for a given set of biochemical parameters, if there exists a set of scalars An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e481.jpg, An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e482.jpg satisfying the above inequalities. For example, if the repressilator parameters are chosen so that

equation image
(60)

then it is trivial to prove that, for any constant value An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e484.jpg, the set of scalars An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e485.jpg, for An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e486.jpg, satisfies (59a)–(59g). Indeed, in Figure 10 we provide a set of biochemical parameters for which the circuit is contracting and shows entrainment to the periodic input An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e487.jpg. (These parameters, except for the maximal transcription rate An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e488.jpg, are in the same ranges as those used in [31], [32]. These parameters are also close to those used in [33] and [34]. The reason for picking an An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e489.jpg much smaller than in [32], is that we need to slow down transcription so as to eliminate intrinsic oscillations; otherwise the entrainment effect cannot be shown. This lowering of An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e490.jpg by two orders of magnitude is also found in other works, for example in [35], where the same model is studied, with An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e491.jpg somewhat larger but of the same order of magnitude as here.)

Figure 10
Simulation of the Repressilator model described by (53), (54), (55).

Note that using the set of inequalities (59a)–(59g) as a guideline, it is possible to find other parameter regions where the system is still contracting but exhibit some other desired properties. For instance, to tune (e.g. increase) the amplitude of the output oscillations shown in Figure 10, a possible approach can be that of increasing the biochemical parameter An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e503.jpg so as to make stronger the effect of the auto-inducer on the dynamics of the gene An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e504.jpg (variable An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e505.jpg in the model).

Again we can prove that the set of inequalities (59a)–(59g) is satisfied for An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e506.jpg arbitrarily large, if we set An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e507.jpg, for An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e508.jpg and choose An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e509.jpg such that

equation image

and

equation image

Now, due to biochemical constraints the parameter An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e512.jpg is considerably smaller than An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e513.jpg and An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e514.jpg (in our simulations the ratio is of about two orders of magnitude). Therefore, whatever the value of An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e515.jpg, it suffices to set An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e516.jpg and An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e517.jpg, with An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e518.jpg being a positive arbitrary constant, to get a solution to (59a)–(59g) and hence guarantee the system to be contracting.

Figure 11 shows the behavior of the system output with the modified parameters confirming that with this choice of parameters the oscillation amplitude is indeed larger as expected.

Figure 11
Increasing the amplitude of oscillations for the model described by (53), (54), (55).

Observe the nonlinear character of the oscillation depicted in Figure 11, which is reflected in the lack of symmetry in the behavior at minima and maxima of cI An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e523.jpg. Our theory predicts the existence (and uniqueness) of such a nonlinear oscillations. None of the usual techniques, based on linear analysis, can explain such behavior.

Entrainment using an extra-cellular auto-inducer

We now consider the case in which the extracellular auto-inducer can change due to an external signal as well as diffusion from intracellular auto-inducer, as represented in Figure 9. A new variable must be introduced, to keep track of the extracellular auto-inducer concentration. The only difference in the new model with respect to the previous one is that the differential equation for An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e524.jpg becomes:

equation image
(61)

Notice that the parameter An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e526.jpg measures the diffusion rate of the auto-inducer across the cell membrane, i.e. An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e527.jpg, with An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e528.jpg representing the membrane permeability, An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e529.jpg its surface area and An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e530.jpg the cell volume. In the above equation, An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e531.jpg denotes the concentration of the extra-cellular auto-inducer, whose dynamics are given by:

equation image
(62)

where An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e533.jpg, with An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e534.jpg denoting the total extracellular volume, while An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e535.jpg stands for the decay rate.

In analogy with the previous section, we will ensure entrainment of the dynamical system consisting of (53), (54), (61), (62), by tuning the biochemical parameters of this new circuit. Again, the guidelines for engineering the parameters will be provided by the tools developed in the previous sections.

Following the schematic of the previous section, we will prove that there exists An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e536.jpg and a An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e537.jpg constant diagonal matrix An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e538.jpg, such that An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e539.jpg, where An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e540.jpg is the system Jacobian.

If we denote with An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e541.jpg, An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e542.jpg the diagonal elements of An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e543.jpg, we obtain the following block-structure for the matrix An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e544.jpg:

equation image
(63)

where An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e546.jpg is given in (57) and:

equation image
(64)

Thus, we have that An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e548.jpg if and only if there exist some An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e549.jpg, An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e550.jpg such that inequalities (58a)–(58f) are all satisfied and additionally:

equation image
(65a)

equation image
(65b)

Again, we can find sets of biochemical parameters in order to satisfy the above inequalities and hence ensure global entrainment of the circuit to some external input. For example, if we set

equation image
(66)

then, as in the previous section, it is trivial to show that setting all An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e554.jpg to the same identical value satisfies the set of inequality required to prove contraction and hence guarantees entrainment. Notice that the last constraint in (66) is automatically satisfied by the physical (i.e. positivity) constraints on the system parameters.

In Figure 12, the behavior of the circuit is shown with the parameters chosen so as to satisfy the constraints given in (66).

Figure 12
Simulation of the Repressilator forced by some extra-cellular molecule.

Entraining a population of Repressilators

Consider, now, a population of An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e564.jpg Repressilator circuits, which are coupled by means of an auto-inducer molecule. We can think of such a network as having an all-to-all topology, with the coupling given by the concentration of the extracellular auto-inducer, An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e565.jpg. The aim of this section is to show that the methodology proposed here can also be used as an effective tool to guarantee the synchronization of an entire population of biochemical oscillators onto some entraining external periodic input.

We denote with the subscript An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e566.jpg the state variables of the An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e567.jpg-th circuit in the network, which is modelled using the equations reported in [32] as:

equation image
(67)

Figure 13 shows a simulation of a population of Repressilators modeled as in (67), with biochemical parameters tuned as in the previous Section: all the circuits composing the network evolve asymptotically towards the same synchronous evolution, which has period equal to that of the input signal An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e569.jpg. The interested reader is referred to the Materials and Methods for the proof.

Figure 13
Synchronization of Repressilators.

Materials and Methods

All simulations are performed in MATLAB (Simulink), Version 7.4, with variable step ODE solver ODE23t. Simulink models are available upon request. The proofs of the results are as follows.

An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e573.jpg-reachable sets

We will make use of the following definition:

Definition 1. Let An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e574.jpg be any positive real number. A subset An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e575.jpg is An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e576.jpg-reachable if, for any two points An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e577.jpg and An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e578.jpg in An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e579.jpg there is some continuously differentiable curve An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e580.jpg such that:

  1. An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e581.jpg,
  2. An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e582.jpg and
  3. An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e583.jpg, An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e584.jpg.

For convex sets An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e585.jpg, we may pick An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e586.jpg, so An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e587.jpg and we can take An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e588.jpg. Thus, convex sets are An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e589.jpg-reachable, and it is easy to show that the converse holds as well.

Notice that a set An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e590.jpg is An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e591.jpg-reachable for some An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e592.jpg if and only if the length of the geodesic (smooth) path (parametrized by arc length), connecting any two points An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e593.jpg and An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e594.jpg in An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e595.jpg, is bounded by some multiple An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e596.jpg of the Euclidean norm, An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e597.jpg. Indeed, re-parametrizing to a path An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e598.jpg defined on An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e599.jpg, we have:

equation image

Since in finite dimensional spaces all the norms are equivalent, then it is possible to obtain a suitable An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e601.jpg for Definition 1.

Remark 1. The notion of An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e602.jpg -reachable set is weaker than that of convex set. Nonetheless, in Theorem 5, we will prove that trajectories of a smooth system, evolving on a An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e603.jpg -reachable set, converge towards each other, even if An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e604.jpg is not convex. This additional generality allows one to establish contracting behavior for systems evolving on phase spaces exhibiting “obstacles”, as are frequently encountered in path-planning problems, for example. A mathematical example of a set with obstacles follows.

Example 1. Consider the two dimensional set, An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e605.jpg , defined by the following constraints:

equation image

Clearly, An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e607.jpg is a non-convex subset of An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e608.jpg . We claim that An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e609.jpg is An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e610.jpg -reachable, for any positive real number An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e611.jpg . Indeed, given any two points An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e612.jpg and An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e613.jpg in An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e614.jpg , there are two possibilities: either the segment connecting An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e615.jpg and An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e616.jpg is in An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e617.jpg , or it intersects the unit circle. In the first case, we can simply pick the segment as a curve ( An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e618.jpg ). In the second case, one can consider a straight segment that is modified by taking the shortest perimeter route around the circle; the length of the perimeter path is at most An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e619.jpg times the length of the omitted segment. (In order to obtain a differentiable, instead of merely a piecewise-differentiable, path, an arbitrarily small increase in An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e620.jpg is needed.)

Proof of Theorem 1

We now prove the main result on contracting systems, i.e. Theorem 1, under the hypotheses that the set An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e621.jpg, i.e. the set on which the system evolves, is An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e622.jpg-reachable.

Theorem 5. Suppose that An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e623.jpg is a An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e624.jpg -reachable subset of An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e625.jpg and that An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e626.jpg is infinitesimally contracting with contraction rate An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e627.jpg . Then, for every two solutions An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e628.jpg and An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e629.jpg it holds that:

equation image
(68)

Proof. Given any two points An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e631.jpg and An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e632.jpg in An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e633.jpg, pick a smooth curve An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e634.jpg, such that An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e635.jpg and An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e636.jpg. Let An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e637.jpg, that is, the solution of system (1) rooted in An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e638.jpg, An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e639.jpg. Since An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e640.jpg and An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e641.jpg are continuously differentiable, also An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e642.jpg is continuously differentiable in both arguments. We define

equation image

It follows that

equation image

Now,

equation image

so, we have:

equation image
(69)

where An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e647.jpg. Using Coppel's inequality [28], yields

equation image
(70)

An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e649.jpg, An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e650.jpg, and An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e651.jpg. Notice the Fundamental Theorem of Calculus, we can write

equation image

Hence, we obtain

equation image

Now, using (70), the above inequality becomes:

equation image

The Theorem is then proved.

Proof of Theorem 1. The proof follows trivially from Theorem 5, after having noticed that in the convex case, we may assume An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e655.jpg.

Proof of Theorem 2

In this Section we assume that the vector field An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e656.jpg is An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e657.jpg-periodic and prove Theorem 2.

Before starting with the proof of Theorem 2 we make the following:

Remark 2. Periodicity implies that the initial time is only relevant modulo An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e658.jpg . More precisely:

equation image
(71)

Indeed, let An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e660.jpg , An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e661.jpg , and consider the function An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e662.jpg , for An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e663.jpg . So,

equation image

where the last equality follows by An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e665.jpg -periodicity of An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e666.jpg . Since An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e667.jpg , it follows by uniqueness of solutions that An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e668.jpg , which is (71). As a corollary, we also have that

equation image
(72)

where the first equality follows from the semigroup property of solutions (see e.g. [21] ), and the second one from (71) applied to An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e670.jpg instead of An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e671.jpg.

Define now

equation image

where An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e673.jpg. The following Lemma will be useful in what follows.

Lemma 1. An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e674.jpg for all An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e675.jpg and An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e676.jpg.

Proof. We will prove the Lemma by recursion. In particular, the statement is true by definition when An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e677.jpg. Inductively, assuming it true for An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e678.jpg, we have:

equation image

as wanted.

Theorem 6. Suppose that:

  • An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e680.jpg is a closed An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e681.jpg -reachable subset of An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e682.jpg ;
  • An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e683.jpg is infinitesimally contracting with contraction rate An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e684.jpg ;
  • An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e685.jpg is An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e686.jpg -periodic;
  • An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e687.jpg.

Then, there is an unique periodic solution An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e688.jpg of (1) having period An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e689.jpg . Furthermore, every solution An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e690.jpg , such that An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e691.jpg , converges to An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e692.jpg , i.e. An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e693.jpg as An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e694.jpg.

Proof. Observe that An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e695.jpg is a contraction with factor An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e696.jpg: An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e697.jpg for all An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e698.jpg, as a consequence of Theorem 5. The set An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e699.jpg is a closed subset of An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e700.jpg and hence complete as a metric space with respect to the distance induced by the norm being considered. Thus, by the contraction mapping theorem, there is a (unique) fixed point An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e701.jpg of An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e702.jpg. Let An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e703.jpg. Since An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e704.jpg, An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e705.jpg is a periodic orbit of period An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e706.jpg. Moreover, again by Theorem 5, we have that An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e707.jpg. Uniqueness is clear, since two different periodic orbits would be disjoint compact subsets, and hence at positive distance from each other, contradicting convergence. This completes the proof.

Proof of Theorem 2. It will suffice to note that the assumption An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e708.jpg in Theorem 6 is automatically satisfied when the set An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e709.jpg is convex (i.e. An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e710.jpg) and the system is infinitesimally contracting.

Notice that, even in the non-convex case, the assumption An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e711.jpg can be ignored, if we are willing to assert only the existence (and global convergence to) a unique periodic orbit, with some period An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e712.jpg for some integer An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e713.jpg. Indeed, the vector field is also An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e714.jpg-periodic for any integer An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e715.jpg. Picking An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e716.jpg large enough so that An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e717.jpg, we have the conclusion that such an orbit exists, applying Theorem 6.

Cascades

In order to show that cascades of contracting systems remain contracting, it is enough to show this, inductively, for a cascade of two systems.

Consider a system of the following form:

equation image

equation image

where An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e720.jpg and An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e721.jpg for all An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e722.jpg (An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e723.jpg and An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e724.jpg are two An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e725.jpg-reachable sets). We write the Jacobian of An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e726.jpg with respect to An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e727.jpg as An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e728.jpg, the Jacobian of An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e729.jpg with respect to An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e730.jpg as An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e731.jpg, and the Jacobian of An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e732.jpg with respect to An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e733.jpg as An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e734.jpg,

We assume the following:

  1. The system An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e735.jpg is infinitesimally contracting with respect to some norm (generally indicated as An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e736.jpg), with some contraction rate An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e737.jpg, that is, An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e738.jpg for all An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e739.jpg and all An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e740.jpg, where An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e741.jpg is the matrix measure associated to An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e742.jpg.
  2. The system An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e743.jpg is infinitesimally contracting with respect to some norm (which is, in general different from An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e744.jpg, and is denoted by An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e745.jpg), with contraction rate An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e746.jpg, when An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e747.jpg is viewed a a parameter in the second system, that is, An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e748.jpg for all An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e749.jpg, An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e750.jpg and all An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e751.jpg, where An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e752.jpg is the matrix measure associated to An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e753.jpg.
  3. The mixed Jacobian An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e754.jpg is bounded: An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e755.jpg, for all An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e756.jpg, An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e757.jpg and all An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e758.jpg, for some real number An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e759.jpg, where “An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e760.jpg” is the operator norm induced by An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e761.jpg and An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e762.jpg on linear operators An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e763.jpg. (All norms in Euclidean space being equivalent, this can be verified in any norm.)

We claim that, under these assumptions, the complete system is infinitesimally contracting. More precisely, pick any two positive numbers An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e764.jpg and An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e765.jpg such that

equation image

and let

equation image

We will show that An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e768.jpg, where An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e769.jpg is the full Jacobian:

equation image
(73)

with respect to the matrix measure An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e771.jpg induced by the following norm in An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e772.jpg:

equation image

Since

equation image

for all An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e775.jpg and An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e776.jpg, we have that, for all An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e777.jpg and An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e778.jpg:

equation image

equation image

where from now on we drop subscripts for norms. Pick now any An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e781.jpg and a unit vector An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e782.jpg (which depends on An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e783.jpg) such that An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e784.jpg. Such a vector An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e785.jpg exists by the definition of induced matrix norm, and we note that An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e786.jpg, by the definition of the norm in the product space. Therefore:

equation image

where the last inequality is a consequence of the fact that An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e788.jpg for any nonnegative numbers with An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e789.jpg (convex combination of the An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e790.jpg's). Now taking limits as An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e791.jpg, we conclude that

equation image

as desired.

Entraining a population of Repressilators: proof

The general principle that we apply to prove entrainment of a population of Repressilators is as follows.

Assume that the cascade system

equation image
(74)

with An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e794.jpg being an exogenous input, satisfies the contractivity assumptions of the above Section. Then, consider the interconnection of An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e795.jpg identical systems which interact through the variable An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e796.jpg as follows:

equation image
(75)

Suppose that An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e798.jpg is a solution of (75) defined for all An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e799.jpg, for some input An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e800.jpg. Then, we have the synchronization condition: An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e801.jpg, as An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e802.jpg.

Indeed, we only need to observe that every pair An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e803.jpg is a solution of (74) with the same input

equation image

Furthermore, if An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e805.jpg is a An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e806.jpg-periodic function, the An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e807.jpg interconnected dynamical systems synchronize onto a An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e808.jpg-periodic trajectory.

The above principle can be immediately applied to prove that synchronization onto a An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e809.jpg-periodic orbit is attained for the Repressilator circuits composing network (67) (see also [19]).

Specifically, let An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e810.jpg and An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e811.jpg; we have that An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e812.jpg is a solution of (67). We notice that any pair An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e813.jpg is a solution of the following cascade system

equation image
(76)

Thus, as shown above, contraction of (76) implies synchronization of (67). Differentiation of (76) yields the Jacobian matrix

equation image
(77)

where An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e816.jpg and An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e817.jpg denote the partial derivatives of decreasing and increasing Hill functions with respect to the state variable of interest and An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e818.jpg, An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e819.jpg.

Note that the Jacobian matrix An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e820.jpg has the structure of a cascade, i.e.

equation image

with:

equation image

An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e823.jpg, An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e824.jpg. Thus, to prove contraction of the virtual system (76) it suffices to prove that there exist two matrix measures, An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e825.jpg and An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e826.jpg such that:

  1. An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e827.jpg;
  2. An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e828.jpg;

where An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e829.jpg. Clearly, since An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e830.jpg is a positive real parameter, the second condition above is satisfied (with An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e831.jpg being any matrix measure). Now, notice that matrix An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e832.jpg has the same form as the Jacobian matrix of the Repressilator circuit (56). Hence, if the parameters of the Repressilator are chosen so that they satisfy (66), then there exist a set of positive real parameters An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e833.jpg, An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e834.jpg, such that An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e835.jpg (that is, the first condition above is also satisfied with An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e836.jpg).

Thus, we can conclude that (76) is contracting. Furthermore, all the trajectories of the virtual system converge towards a An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e837.jpg-periodic solution (see Theorem 6). This in turn implies that all the trajectories of network (67) converge towards the same An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e838.jpg-periodic solution. That is, all the nodes of (67) synchronize onto a periodic orbit of period An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e839.jpg.

A counterexample to entrainment

In [5] there is given an example of a system with the following property: when the external signal An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e840.jpg is constant, all solutions converge to a steady state; however, when An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e841.jpg, solutions become chaotic. (Obviously, this system is not contracting.) The equations are as follows:

equation image

where An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e843.jpg and An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e844.jpg. Figure 14 shows typical solutions of this system with a periodic and constant input respectively. The function “rand” was used in MATLAB to produce random values in the range An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e845.jpg.

Figure 14
Simulation of counter-example.

Discussion

We have presented a systematic methodology to derive conditions for various types of biochemical systems to be globally entrained to periodic inputs. For concreteness, we focused mainly on transcriptional systems, which constitute basic building blocks for more complex biochemical systems. However, the results that we obtained are of more generality. To illustrate this generality, and to emphasize the use of our techniques in synthetic biology design, we discussed as well the entrainment of a Repressilator circuit in a parameter regime in which endogenous oscillations to not occur, as well as the synchronization of a network of Repressilators. These latter examples serve to illustrate the power of the tools even when a large amount of feedback is present.

Our key tool is the use of contraction theory, which we believe should be recognized as an important component of the “toolkit” of systems biology. In all cases conditions are derived by proving that the module of interest is contracting under appropriate generic assumptions on its parameters. A surprising fact is that, for these applications, and contrary to many engineering applications, norms other than Euclidean, and associated matrix measures, must be considered. Of course, more than one norm may be appropriate for a given problem: for example we can pick different An external file that holds a picture, illustration, etc.
Object name is pcbi.1000739.e855.jpg's in our weighted norms, and each such choice gives rise to a different estimate of convergence rates. This is entirely analogous to the use of Lyapunov functions in classical stability analysis: different Lyapunov functions provide different estimates.

Ultimately, and as with any other method for the analysis of nonlinear systems, such as the classical tool of Lyapunov functions, finding the “right” norm is more of an art than a science. A substantial amount of trial and error, intuition, and numerical experimentation may be needed in order to come up with an appropriate norm, and experience with a set of already-studied systems (such as the ones studied here) should prove invaluable in guiding the search.

Acknowledgments

The authors wish to thank Dr Diego di Bernardo, Telethon Institute of Genetics and Medicine (TIGEM), Naples (Italy) for suggesting entrainment of Repressilators as a possible suitable application of the tools presented in this paper and Prof J J Slotine, MIT, Boston (USA) for all the insightful discussions about contraction theory.

Footnotes

The authors have declared that no competing interests exist.

GR and MdB acknowledge support from the European Union Project ‘Engineering Complexity in Biological Systems (COBIOS)’, VI Framework program (Grant no. 043379). EDS acknowledges support from grants NSF-DMS-0614371, NIH-1R01GM086881, and AFOSR-FA9550-08. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

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