1 Introduction

Despite tremendous advances in technology in recent years, modelling of gene regulatory networks is still hampered by the difficulty of determining rate constants, so that quantitative models are unreliable unless robust to parameter variation. Another issue is that the principles governing the behaviour of networks in general are not necessarily discoverable from any one particular network, no matter how well its fine details are described. These concerns have led to the development of qualitative models of gene networks, in which an attempt is made to assess qualitative behaviour somewhat independently of details, such as precise parameter values and biochemical kinetics. Thomas (1973), Glass and Kauffman (1973) and others began to explore this approach in the 1970s, using simplified model frameworks with the aim of capturing important qualitative features of the behaviour of gene networks and revealing fundamental principles of their design. Glass and Pasternack (1978), for example, showed that a class of networks that is broader than, but includes, negative feedback loops must show oscillatory behaviour, either damped or sustained according to a condition that could be calculated. A considerable literature has been built up over the intervening years, developing and elaborating this theory. Much of this work made some fairly drastic simplifying assumptions, including:

  • modelling only a single product of each gene, which is described as directly acting as a transcription factor in the regulation of other genes,

  • a step function limit of sigmoidal interactions, based loosely on the Michaelis–Menten and Hill function kinetics that arise in modelling enzyme-mediated or catalytic reaction kinetics, such as occurs in gene expression,

  • a single effective threshold of each gene product concentration, even when it regulates multiple other genes,

  • equal degradation rates of all gene products,

  • no autoregulation.

The class of systems in which at least the first two of these assumptions are made, without necessarily requiring the others, have been called ‘Glass networks’. These simplifying assumptions allowed some interesting mathematical work, which laid a foundation for the qualitative study of gene network dynamics. Of course, it is hoped that the insights gained still hold when some or all of these simplifying assumptions are relaxed, and more realistic models are used. Developments in synthetic biology has made this work all the more compelling and important. For instance, the groundbreaking papers by Elowitz and Leibler (2000) on the ‘Repressilator’ and by Gardner et al. (2000), on the genetic toggle switch (both published in the same issue of Nature), showed that simple networks built in the laboratory did, indeed, demonstrate the fundamental behaviours predicted by qualitative models. In more complex network structures, however, the correspondence is less clear-cut and considerable effort has gone into extending the analysis of the simplest class of qualitative models to more general classes in which one or more of the assumptions above are relaxed, though the mathematics becomes more difficult.

Multiple thresholds are the least problematic, and a number of studies have been able to effectively extend results to that situation (e.g. Farcot 2006). In fact, it has been shown that, mathematically, such networks can be embedded into a higher dimensional system with a single threshold per variable in such a way that the dynamics are essentially equivalent (Edwards et al. 2012).

Unequal degradation rates do not cause a real problem for local computation of solution trajectories, but make identification of stable periodic solutions more difficult, though some results along those lines have now been obtained (Farcot and Gouzé 2010; Edwards and Ironi 2014).

Allowing effective autoregulation means that singular situations have to be resolved when gene product concentrations become confined at or near a threshold value for an interval of time. This may occur because a single transcription factor simply inhibits its own production, but may also occur when pairs of variables have damped oscillations about a threshold intersection, such that in the step function limit, they converge to the threshold intersection in finite time (Edwards 2000). Such finite time convergence again requires autoregulation. These issues of singular behaviour seem, in one sense, to be mathematical artifacts, since we know that no real regulatory function has infinite slope. However, it has been shown that interesting behaviour of sigmoidal systems may be exposed by this approach (Machina et al. 2013b). There has been a controversy about how to handle these singular situations. Gouzé and colleagues used Filippov theory for discontinuous vector fields to produce set-valued solutions (Gouzé and Sari 2002; Casey et al. 2006), while Plahte and colleagues developed an approach based on singular perturbation theory (Plahte and Kjøglum 2005; Ironi et al. 2011). Both methods had their problems. The usual Filippov approach produces spurious solutions in some situations, while the classical singular perturbation theory can only be applied in a restricted class of problems. Recent work by Machina and colleagues (Machina and Ponosov 2011; Machina et al. 2013a, b) has clarified the relationships between these approaches and removed some of the difficulties. There exists a more appropriate definition of Filippov solution that avoids most of the spurious solutions and corresponds more closely to the singular perturbation solutions. Furthermore, there exists a generalization of the classical singular perturbation theory that avoids its restricted application. However, in the latter case, non-uniqueness can still arise in the limit of step function interactions, much like in the Filippov theory, though now it can be seen to arise from densely interwoven basins of attraction of different attractors and sensitive dependence on initial conditions in systems with steep sigmoidal interactions.

The singular perturbation approach to the handling of singular dynamics, in principle, obtains results about the behaviour of steep sigmoidal systems, using the step function limit only as a baseline from which to perturb. The work of Ironi et al. (2011), for example, allows a bound to be placed on the steepness parameter, above which the qualitative behaviour predicted must exist. Even here, though, the sigmoid interactions often have to be very steep, in biological terms, for the results to hold rigorously. There have also been numerical investigations of how behaviour changes as the steepness parameter is altered (Lewis and Glass 1991, 1992; Killough and Edwards 2005; Wilds and Glass 2009). Typically, if the step function network has complex behaviour, that behaviour is simplified by making the interactions less steep.

Very little work has been done in the qualitative model literature, however, on relaxing the first assumption above, and including in the model multiple interaction steps, a notable exception being a recent paper of Gedeon et al. (2012). Of course, in more realistic biochemical kinetic models, several steps are involved in producing a transcription factor that can play a regulatory role. The expression of a gene comprises transcription of copies of an mRNA molecule, characteristic of the gene, whose code is then translated into a protein, by building a corresponding string of amino acids, which are available in the cell. This protein may itself act as a transcription factor for another gene, but it may also need to undergo further modifications, such as dimerization, phosphorylation or other post-translational reactions, before it is effective as a regulator. In qualitative models, all of this is generally suppressed, and the situation is treated as if each gene product (protein), or combination of gene products, acts directly on the gene products of other genes by modifying their expression (by either repression or activation). The singular situations, including autoregulation, that arise in these simplified models, of course, hide the intermediate steps in every regulation process.

Thus, another way to approach the problem of autoregulation, or fast damped oscillation in negative feedback between two or more genes, which lead to the singularities in the step function limit of the simple qualitative models, is to expand the framework of the qualitative models to account for multiple steps in the regulatory process. For example, one can model the transcription of mRNA and then translation into protein as two separate steps, and one can also model post-translational processes as separate steps. Then, even when a gene regulates itself, the autoregulation is split into multiple steps. No variable appears in its own production term, and in that sense, no variable is directly self-regulating, and one might expect that the singularities, although present in the step function limit, should only be passed through transparently (transversally) or approached asymptotically. Although this makes the models more complicated, it may avoid the somewhat artificial mathematical difficulties associated with the singular dynamics and may still be mathematically tractable, while being truer to the biochemistry. One question that arises is whether the dynamics of these models converges to the dynamics of the simpler one-step models in the limit where all but one of the multiple steps are infinitely fast. If so, this provides at least some justification for the protein-only models. Another question that arises is how to analyse the new type of model. From observations, the timescales of the dynamics of mRNA and protein concentrations are typically different, but the magnitude of this difference depends on the gene and organism. For example, Bernstein et al. (2002) find mRNA half-lives of 3–8 min for most genes in E. coli, and Mosteller et al. (1980) found protein half-lives of 2–23 h or more, also in E. coli. However, in humans, mRNA half-lives can be on the order of 2–10 h and protein half-lives from 45 min to 28 h (see BioNumbers website). Thus, the ratio of decay rates of mRNA to protein is not at all clear and can range from at least 100 to much smaller values. Gedeon et al. (2012) use a value of 10 as biologically reasonable. Paetkau et al. (2006) cite particular examples of mRNAs and proteins in particular organisms where the ratio may be close to unity, even though such cases are recognized as exceptional. Note, furthermore, that the production rates of mRNA and protein are necessarily more closely linked than the decay rates, since the protein is produced from the mRNA by translation.

Here, we take the approach of accounting for two steps in each regulation, the transcription and translation, thus keeping track of both mRNA and protein concentrations. The transcription is still governed by sigmoid functions, while the translation is governed by a linear term in the differential equations. This is the model expansion considered by Gedeon and co-authors, but the conditions imposed on the general results in (Gedeon et al. 2012, Section 3.2) are restrictive. We discuss the relation between our results and theirs in Sect. 8.

Post-transcriptional modifications can take many forms. Dimerization, for example, requires a quadratic term in the model equations, while an enzyme-mediated or catalytic reaction requires a Michaelis–Menton or Hill Function (sigmoid) interaction term. Since these multiple possibilities make things more complicated, we do not consider them in detail here. However, in an Appendix we consider another mathematical possibility for avoiding the singular flow: a two-step system in which both steps are mediated by sigmoid functions, thus giving a larger sigmoidal network, but one in which no variable is directly self-regulating. This might be considered to model the expression of a protein as one step and the catalytic modification of that protein as a second step, though this is only one type of reaction that can occur in gene networks. We can also consider it as a mathematical method for dealing with the singular flow in a traditional gene network model by considering an infinitely fast limit of one of the variables in each pair.

In Sect. 2, we summarize the traditional protein-only model using the description of Plahte and Kjøglum, and we briefly summarize the singular perturbation analysis (Plahte and Kjøglum 2005). Then in Sect. 3 we introduce the two-step extension to make it an mRNA-protein model and give a result on the singular stationary points (SSPs) in this expanded model. In Sect. 4, we show that solutions of the original system are recovered if we take an infinitely fast limit in the rate of the protein equations. In Sect. 5, we do the same for infinitely fast mRNA rates. In the more sensitive case of SSPs in domains where two variables are switching simultaneously, we show in Sect. 6 that such points are generally unstable. Then in Sect. 7, we determine the flow from one switching point to the next, to allow integration of trajectories in regular domains. In the Discussion, we deal with the relationship between our results and those of Gedeon et al. (2012), as well as the relationship between our model and the differential delay systems of Ponosov (2005) and Shlykova et al. (2008). We also summarize the results obtained and questions remaining unanswered. In an Appendix, we consider an alternate two-step model where both steps involve sigmoidal regulation.

2 Preliminaries

We are concerned here with a class of qualitative protein-only gene network models as discussed in the introduction, which has been formalized in a number of ways. Using the notation of Plahte and Kjøglum (2005), the network equations are

$$\begin{aligned} \dot{y}_i=F_i(Z)-\gamma _i y_i,\qquad i=1,\ldots ,n, \end{aligned}$$
(1)

with \(y_i\) the concentration of the protein product of gene \(i,\) \(\gamma _i>0\) and \(Z=(Z_1,\ldots ,Z_n)\) an \(n\)-dimensional vector of sigmoid functions \(Z_i=\mathcal {S}(y_i,\theta _i,q)\). In general, \(\mathcal {S}(y_i,\theta _i,q),\) \(i=1,\ldots ,n,\) must satisfy a number of conditions listed in (Plahte and Kjøglum 2005). For example, these conditions are satisfied by the Hill function given by

$$\begin{aligned} Z_i=H(y_i,\theta _i,q)=\frac{y_i^{1/q}}{y_i^{1/q}+\theta _i^{1/q}}, \qquad i=1,\ldots ,n, \qquad 0<q\le 1. \end{aligned}$$
(2)

Here, \(q\) is the steepness parameter (smaller values of \(q\) correspond to greater steepness of the sigmoid function) and \(\theta _i\) is called the threshold, because \(H(y_i,\theta _i,q)\in (0,1)\) and \(H(\theta _i,\theta _i,q) =\frac{1}{2}\). We take \(\mathcal {S}=H\) in the main body of the paper, but a different sigmoid function arises in the Appendix. The production rate function \(F_i\ge 0\) is a multilinear polynomial, i.e., an affine function with respect to each \(Z_i\). Production rates are inherently bounded, so that \(0\le F_i(Z)\le \overline{F}_i,\) \( i=1,\ldots ,n,\) where each \(\overline{F}_i,\) is a constant. Plahte and Kjøglum allow the degradation rate to also depend on concentrations of transcription factors so that the model equations become

$$\begin{aligned} \dot{y}_i=F_i(Z)-G_i(Z) y_i,\qquad i=1,\ldots ,n. \end{aligned}$$

Although this does not make mathematical analysis more difficult, in practice degradation rates can normally be considered constant, and \(G_i(Z)=\gamma _i\) makes our exposition more straightforward. The regulatory effect of the transcription factors occurs only through the production rates.

An alternative to the model given by (1) and (2) is to approximate steep sigmoidal functions \(Z_i\) by step functions, also denoted by \(Z_{i}\), with the jump at \(\theta _i\). System (1) then becomes a switched system with the phase space partitioned into regular domains by the threshold hyperplanes (walls) \(y_{i}=\theta _i\). Within each of the regular domains, (1) is linear, since \(Z\) is a binary vector that has a constant value in the domain. System (1) has a unique ‘focal point’ for the value of \(Z\) appropriate to the domain, the point at which the right-hand side functions are all \(0\). If this focal point lies in the domain itself, it is an asymptotically stable node to which solutions converge, but if not then solution trajectories hit a threshold hyperplane or intersection of hyperplanes before reaching the focal point. On the other side of this threshold or thresholds, the value of \(Z\) is different and the focal point is therefore different in general.

Whenever we deal with the functions \(Z_i\) as step functions (whether as limits of sigmoids or as an approximation), we will make the following assumption to avoid technical difficulties that only arise for very special sets of parameters (on a set of measure 0 in parameter space).

Assumption 1

No focal point, \(\Phi \), where \(\Phi _i=F_i(Z)/\gamma _i\) for any binary vector \(Z\), lies on a threshold hyperplane. Thus, \(\Phi _i\ne \theta _i\) for any \(i\).

Instead of simply replacing the sigmoid with a step function, we can attempt to deal with steep sigmoids by means of a singular perturbation from the step function limit. We will refer to a singularly perturbed system several times in the rest of this paper, not only in relation to system (1). In general, such a system involves coupled slow and fast motion, say

$$\begin{aligned} \begin{array}{l} \frac{\hbox {d}X_i}{\hbox {d}t}=f_i(X,Y),\\ \varepsilon \frac{\hbox {d}Y_i}{\hbox {d}t}=g_i(X,Y), \end{array} \qquad i=1,\ldots ,n, \end{aligned}$$
(3)

where \(X=(X_1,\ldots ,X_n),\) \(Y=(Y_1,\ldots ,Y_n),\) and \(\varepsilon >0\) is a small parameter. The initial value problem is determined by initial conditions \(X(0)=X_0,\) \(Y(0)=Y_0\). The variables \(X\) and \(Y\) are referred to as the slow and fast variables, respectively. The solution to the initial value problem (3) is denoted by \((X^\varepsilon (\cdot ),Y^\varepsilon (\cdot ))\). The standard analysis based on Tikhonov’s theorem (see e.g. Artstein 2002) guarantees under certain conditions that the solution \((X^\varepsilon (\cdot ),Y^\varepsilon (\cdot ))\) converges as \(\varepsilon \rightarrow 0\) to the solution of the differential-algebraic system

$$\begin{aligned} \begin{array}{l} \frac{\hbox {d}X_i}{\hbox {d}t}=f_i(X,Y), \\ 0=g_i(X,Y), \end{array}\qquad i=1,\ldots ,n, \end{aligned}$$

obtained from (3) when \(\varepsilon =0\). In particular, a crucial condition is that for each fixed \(X,\) solutions of the differential equation

$$\begin{aligned} \frac{\hbox {d}Y_i}{\hbox {d}\tau }=g_i(X,Y), \qquad i=1,\ldots ,n, \end{aligned}$$

representing fast flow in a new time variable, \(\tau =t/\varepsilon ,\) should converge as \(\tau \rightarrow \infty \) to a solution of the algebraic equation \(0=g_i(X,Y)\). If the fast system does not converge to a fixed point, then the Tikhonov theorem does not apply and we have to resort to a more general theory.

It is useful to have a relatively simple example of our class of gene networks to which we can refer in later sections.

Example 1

(Plahte and Kjøglum 2005)

$$\begin{aligned} \begin{array}{l} \dot{y}_1=Z_1+Z_2-2Z_1Z_2-\gamma _1y_1,\\ \dot{y}_2=1-Z_1Z_2-\gamma _2y_2,\\ \end{array} \end{aligned}$$
(4)

with \(Z_1,Z_2\) given by (2). Example parameter values chosen by Plahte and Kjøglum (2005, Fig. 2) are \(\gamma _1=0.6,\gamma _2=0.9,\theta _1=\theta _2=1\). Solutions with small \(q>0\) (i.e., steep Hill functions) are studied by means of singular perturbation analysis with reference to the limiting system in which \(q\rightarrow 0\). However, the limit system in the vicinity of threshold values is studied only in a transformed system of differential equations, where the variables near threshold (switching variables) are described by their \(Z_i\) values and time is rescaled by \(\tau =t/q\). In this transformed system, there is no discontinuity.

In the limit \(q\rightarrow 0\), the switching domain \(\{y_1=\theta _1, y_2<\theta _2 \}\) is a ‘white wall’, i.e. the solutions depart from it. The switching domain \(\{y_1<\theta _1, y_2=\theta _2 \}\) is a ‘transparent wall’, i.e. the solutions pass through it. The switching domains \(\{y_1>\theta _1, y_2=\theta _2 \}\) and \(\{y_1=\theta _1, y_2>\theta _2 \}\) are ‘black walls’, i.e. the solutions are attracted to these domains. In the first of these black walls, there exists a unique singular stationary point (SSP), which is at \((1.5,1)\) with the parameter values chosen by Plahte and Kjøglum and is asymptotically stable. For some parameter values, \(\gamma _1,\gamma _2,\theta _1,\theta _2\), including those chosen above, and small enough \(q\), there is also a stable SSP near the threshold intersection, \((\theta _1,\theta _2)\). This behaviour is depicted by Plahte and Kjøglum (2005, Fig. 2). \(\square \)

3 The mRNA-protein Model

We now expand the \(n\)-dimensional model (1), which accounts only for the protein transcription factors that regulate other genes, to obtain a model of dimension \(2n\) that includes both mRNA and proteins as separate variables:

$$\begin{aligned} \begin{array}{l} \dot{x}_i=F_i(Z)-\beta _i x_i,\\ \dot{y}_{i}=\kappa _{i}x_i-\alpha _{i}y_{i}, \\ \end{array}\qquad i=1,\ldots ,n, \end{aligned}$$
(5)

where \(y_i\) is the concentration of the protein product of gene \(i\), \(x_i\) is the concentration of the \(i\)th mRNA, \(Z=(Z_{1},\ldots ,Z_{n})\) with \(Z_{i}=H(y_{i}, \theta _i,q)\), and \(0<q<<1\). (In Gedeon et al. 2012, \(\alpha _i\) is taken to be \(\gamma _i\), the constant protein degradation rate from the protein-only model.) The functions \(F_i(Z)\) satisfy the same assumptions as in the protein-only model (1) and \(\beta _i>0\), \(\alpha _i>0\), \(\kappa _i\ge 0\). Note that since \(\dot{y}_i\) is independent of \(Z_i,\) all switching hyperplanes \(y_i=\theta _i\) are ‘transparent’ (solutions just pass through switching hyperplanes).

For smooth sigmoids, the point \((\frac{\alpha _1}{\kappa _1}y_1^*,\ldots ,\frac{\alpha _n}{\kappa _n}y_n^*,\; y_1^*,\ldots ,y_n^*)\) is a stationary solution of (5) if and only if \((y_1^*,\ldots ,y_n^*)\) is a stationary solution of

$$\begin{aligned} \begin{array}{l} \dot{y}_i=\frac{\kappa _i}{\alpha _i}F_i({Z}_1,\ldots ,{Z}_n)-\beta _i y_i,\qquad i=1,\ldots ,n.\\ \end{array} \end{aligned}$$
(6)

For switched systems, stationary points are traditionally defined as the limit as \(q\rightarrow 0\) of the corresponding stationary points of the continuous systems with smooth sigmoids. Thus, the same relation holds between the stationary solutions of the switched systems (5) and (6) with the step regulatory functions (\(q\rightarrow 0\)). The problem of locating the stationary solutions of switched systems (5) is then reduced to locating the stationary solutions of (6), which is well studied in the literature (Plahte and Kjøglum 2005; Casey et al. 2006). Stationary points in regular domains are straightforward and are always asymptotically stable.

Below we provide a stability result for the system (5) in case of only one variable \(y_i\) assuming its threshold value. For notational simplicity, we assume that \(y_1=\theta _1\) but the result is valid for any fixed \(y_i=\theta _i\).

Theorem 1

Suppose that \(y^*=(\theta _1,y_2^*,\ldots ,y_n^*),\) \(y_i^*\ne \theta _i,\) \(i=2,\ldots ,n,\) is a SSP of (6) in the limit as \(q\rightarrow 0\). Then \((x^*,y^*)\) with \(x^*=(\frac{\alpha _1}{\kappa _1}\theta _1,\frac{\alpha _2}{\kappa _2}y_2^*,\ldots ,\frac{\alpha _n}{\kappa _n}y_n^*)\) is an asymptotically stable (respectively, unstable) SSP of (5) in the limit as \(q\rightarrow 0\) if and only if \(y^*_1=\theta _1\) is an asymptotically stable (respectively, unstable) solution of (6) with \(i=1\).

Proof

We separate the singular variables \(x_1,\) \(y_1\) and rewrite (5) as the following \(2n\) dimensional system

$$\begin{aligned} \begin{array}{l} \dot{x}_1=F_1(Z_1,Z_2,\ldots ,Z_n)-\beta _1 x_1=f_1(x_1,y),\\ \dot{y}_{1}=\kappa _{1}x_1-\alpha _{1}y_{1}=f_2(x_1,y_1), \\ \dot{x}_r=F_r(Z_1,Z_2,\ldots ,Z_n)-\beta _r x_r=f_{r+1}(x_r,y),\\ \dot{y}_{r}=\kappa _{r}x_r-\alpha _{r}y_{r}=f_{r+n}(x_r,y_r), \qquad r=2,\ldots ,n.\\ \end{array} \end{aligned}$$
(7)

We look at the Jacobian matrix \(J(q)\) of the system (7) at \((x^*,y^*)\) for a small \(q>0\), written by columns \((\frac{\partial f}{\partial x_1}, \frac{\partial f}{\partial y_1},\frac{\partial f}{\partial x_2},\ldots ,\frac{\partial f}{\partial x_n},\frac{\partial f}{\partial y_2},\ldots ,\frac{\partial f}{\partial y_n})\)

$$\begin{aligned} J(q)=\left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} \frac{\partial f_1}{\partial x_1}&{}\frac{\partial f_1}{\partial y_1}&{}0&{}{\cdots }&{}0&{}\frac{\partial f_1}{\partial y_2}&{}{\cdots }&{}\frac{\partial f_1}{\partial y_n}\\ \frac{\partial f_2}{\partial x_1}&{}\frac{\partial f_2}{\partial y_1}&{}0&{}{\cdots }&{}0&{}0&{}{\cdots }&{}0\\ 0&{}\frac{\partial f_3}{\partial y_1}&{}{-\beta _2}&{}{\cdots }&{}0&{}\frac{\partial f_3}{\partial y_2}&{}{\cdots }&{}\frac{\partial f_3}{\partial y_n}\\ {\cdots }&{}{\cdots }&{}{\cdots }&{}{\cdots }&{}{\cdots }&{}{\cdots }&{}{\cdots }&{}{\cdots }\\ 0&{}\frac{\partial f_{n+1}}{\partial y_1}&{}0&{}{\cdots }&{}-\beta _n&{}\frac{\partial f_{n+1}}{\partial y_2}&{}{\cdots }&{}\frac{\partial f_{n+1}}{\partial y_n}\\ 0&{}0&{}\kappa _2&{}{\cdots }&{}0&{}{-\alpha _2}&{}{\cdots }&{}{0}\\ {\cdots }&{}{\cdots }&{}{\cdots }&{}{\cdots }&{}{\cdots }&{}{\cdots }&{}{\cdots }&{}{\cdots }\\ 0&{}0&{}0&{}{\cdots }&{}\kappa _n&{}0&{}{\cdots }&{}{-\alpha _n} \end{array}\right) . \end{aligned}$$
(8)

Note that since \(y_r\) is away from its threshold, \(\frac{\hbox {d}Z_r}{\hbox {d}y_r},\) \(r=2,\ldots ,n,\) vanishes when \(q\rightarrow 0\), and, in fact, does so exponentially fast, so the entries in the last \(n-1\) columns of the first \(n+1\) rows of the matrix are all negligible. The characteristic equation of (8) is thus

$$\begin{aligned} \left| \begin{array}{cc} \frac{\partial f_1}{\partial x_1}-\lambda &{}\frac{\partial f_1}{\partial y_1}\\ \frac{\partial f_2}{\partial x_1}&{}\frac{\partial f_2}{\partial y_1}-\lambda \end{array} \right| (-\beta _2-\lambda )\cdots (-\beta _n-\lambda )(-\alpha _2-\lambda ) \cdots (-\alpha _n-\lambda )+\zeta (q)=0 \end{aligned}$$
(9)

where \(\lim _{q\rightarrow 0}\zeta (q)/q=0\), since \(\frac{\hbox {d}Z_r}{\hbox {d}y_r}\rightarrow 0\) as \(q\rightarrow 0\) exponentially fast. The terms of \(J(q)\) that approach \(\infty \) as \(q\rightarrow 0\) are those in the second column. These take the form \(\frac{\partial f_i}{\partial y_1}=\frac{\partial F_i}{\partial Z_1}\frac{\hbox {d}Z_1}{\hbox {d}y_1}\), with \(\frac{\hbox {d}Z_1}{\hbox {d}y_1}\big |_{y_1^*=\theta _1}=\frac{1}{4q\theta _1}\), so these grow only proportionally to \(\frac{1}{q}\). Thus, the terms of the characteristic equation involving \(\zeta (q)\) become insignificant as \(q\rightarrow 0\). Also, \(\beta _i>0\) and \(\alpha _i>0\) for all \(i=1,\ldots , n\). Thus, the stability analysis at \((x^*,y^*)\) reduces to studying the \(2\times 2\) Jacobian

$$\begin{aligned} \widehat{J}(q)=\left( \begin{array}{cc} \frac{\partial f_1}{\partial x_1}&{}\frac{\partial f_1}{\partial y_1}\\ \frac{\partial f_2}{\partial x_1}&{}\frac{\partial f_2}{\partial y_1} \end{array} \right) =\left( \begin{array}{cc} -\beta _1&{}\frac{\partial f_1}{\partial y_1}\\ \kappa _1&{}-\alpha _1 \end{array} \right) \end{aligned}$$
(10)

with \(\text{ trace }(\widehat{J}(q))=-\beta _1-\alpha _1<0\) and \(\text{ det }(\widehat{J}(q))=\beta _1\alpha _1-\kappa _1 \frac{\partial f_1}{\partial y_1},\) where \(\frac{\partial f_1}{\partial y_1}=\frac{\partial F_1}{\partial Z_1}\frac{d Z_1}{d y_1}\) at \((x^*,y^*)\). Since \(\frac{d Z_1}{d y_1}\big |_ { y_1^*=\theta _1} =\frac{1}{4q\theta _1}\rightarrow \infty \) as \(q\rightarrow 0,\) the sign of \(\text{ det }(\widehat{J}(q))\) is determined by the sign of \(\frac{\partial F_1}{\partial Z_1}\). Thus, \((x^*,y^*)\) is asymptotically stable (respectively, unstable) if and only if \(\frac{\partial F_1}{\partial Z_1}<0\) (respectively, \(>0\)), which is equivalent to \((\theta _1,y_2^*,\ldots ,y_n^*)\) being an asymptotically stable (respectively, unstable) stationary point of (6) in the limit \(q\rightarrow 0\). \(\square \)

Remark 1

There exists a sufficiently small \(\bar{q}>0\) such that for all \(0<q<\bar{q}\), the result of Theorem 1 holds qualitatively with an asymptotically stable stationary point, \((x^*,y^*)\) with \(y_1^*\) near \(\theta _1\).

Remark 2

The result of Theorem 1 holds for system (5) even when the protein reactions (translation) are much faster than the mRNA reactions (transcription) and vice versa. The situation with infinitely fast reactions is considered below in Sects. 4 and 5.

In the next two sections, we assume different timescales for the protein and mRNA reactions. Our results in the extreme case of infinitely fast rates of one of the variable types (protein or mRNA) provide a partial justification for using the protein-only equations (1), but it should be noted that asymptotic behaviour may still be different. In practice, the actual ratio of mRNA to protein rates may not be extremely small or extremely large. Estimates of this ratio in different contexts vary (see, for example, Garcia-Ojalvon et al. 2004, p. 10,958, 2nd paragraph; Gedeon et al. 2012).

4 Infinitely Fast Protein Rates

In this section, we consider a situation where the protein translation rates are much faster than the mRNA transcription rates. Such infinitely fast protein dynamics are modelled by introducing a small parameter \(\varepsilon \rightarrow 0\). The singular perturbation theory is then applied with \(\varepsilon \) as the small parameter. We show that the limit dynamics of the expanded system after a short transient is represented by (1) with an appropriate choice of \(F_i\).

Consider the system of dimension \(2n\) as in (5) but with \(\frac{1}{\varepsilon }\) scaling the rate of the protein dynamics.

$$\begin{aligned} \begin{array}{l} \dot{x}_i=F_i(Z)-\beta _i x_i,\qquad i=1,\ldots ,n,\\ \dot{y}_{i}=\frac{1}{\varepsilon }(\kappa _ix_i-\alpha _{i}y_{i}), \\ \end{array} \end{aligned}$$
(11)

where \(Z=(Z_{1},\ldots ,Z_{n}),\) \(Z_{i}=H(y_{i}, \theta _i,q)\). We denote by \((x^{\varepsilon },y^{\varepsilon })\) a solution to (11) for \(\varepsilon >0\). The initial value problem is determined by the initial conditions \(x_i(0)=x_{i,0},\) \(y_i(0)=y_{i,0}\).

Let \(q>0\) be fixed. The system (11) represents a singular perturbation problem as \(\varepsilon \rightarrow 0\) with the fast flow \(y_{i}\) and the slow flow \(x_i\). The pseudo-stationary solution \(y_{i}^*=\kappa _ix_i/\alpha _{i}\) of the fast flow is obtained from \(\kappa _ix_i-\alpha _{i}y_{i}=0\). It is easy to check that the stationary solution \(y_{i}^*\) is asymptotically stable for each fixed \(x_i\). Thus, all assumptions of the Tikhonov theorem are fulfilled, and the slow flow is therefore governed by

$$\begin{aligned} \dot{x}_i=F_i(Z^*_{1},\ldots , Z^*_{n})-\beta _i x_i, \end{aligned}$$
(12)

where \(Z^*_{i}=H(y_{i}^*, \theta _i,q)=H(\kappa _ix_i/\alpha _{i}, \theta _i,q)\) is the Hill function. Note that due to the properties of the Hill function, \(H(\kappa _ix_i/\alpha _{i}, \theta _i,q)=H(x_i, \theta _i\alpha _{i}/\kappa _i,q)\). The system (12) can be rewritten as

$$\begin{aligned} \dot{x}_i=F_i(\widetilde{Z}_1,\ldots ,\widetilde{Z}_n)-\beta _i x_i,\qquad i=1,\ldots ,n, \end{aligned}$$
(13)

where \(\widetilde{Z}_i=H(x_i, \widetilde{\theta }_i,q),\) with \(\widetilde{\theta }_i=\theta _i\alpha _i/\kappa _i,\) or as

$$\begin{aligned} \begin{array}{l} \dot{y}_i^*=\frac{\kappa _i}{\alpha _i}F_i({Z}_1^*,\ldots ,{Z}_n^*)-\beta _i y_i^*,\qquad i=1,\ldots ,n.\\ \end{array} \end{aligned}$$
(14)

By the Tikhonov theorem, we have the following result (see, e.g., Artstein 2002, Theorem 2.1).

Theorem 2

Let \(q>0\) and an initial condition \((x_0,y_0)\) be fixed. The solutions \(x_i^{\varepsilon }(t),\) \(i=1,\ldots ,n,\) of (11) converge as \(\varepsilon \rightarrow 0\) to the solution \(\widetilde{x}_i(t)\) of the system (13) uniformly on intervals of the form \([0,T]\). The solutions \(y_i^{\varepsilon }(t),\) \(i=1,\ldots ,n,\) converge as \(\varepsilon \rightarrow 0\) to the solution \(y_{i}^*(t)\) of the system (14) uniformly on intervals of the form \([\delta ,T]\) for \(\delta >0\). On intervals \(\tau \in [0,S]\) with \(S>0\) fixed and \(\tau =t/\varepsilon \), the trajectories \(y_i^{\varepsilon }(\tau )\) converge uniformly as \(\varepsilon \rightarrow 0\) to

$$\begin{aligned} y_i(\tau )=\dfrac{\kappa _ix_{i,0}}{\alpha _i}+\left( y_{i,0}-\dfrac{\kappa _ix_{i,0}}{\alpha _i} \right) e^{-\alpha _i\tau }, \end{aligned}$$

the solution of \(\dfrac{\hbox {d}y_i(\tau )}{\hbox {d}\tau }=\kappa _ix_{i,0}-\alpha _{i}y_i(\tau )\) with \(y_i(0)=y_{i,0}\). Finally,

$$\begin{aligned} \lim _{S\rightarrow \infty }\lim _{\varepsilon \rightarrow 0}y_i^\varepsilon (\varepsilon S)=\kappa _ix_{i,0}/\alpha _{i}. \end{aligned}$$

Remark 3

Note that Theorem 2 only discusses the situation \(\epsilon \rightarrow 0\) and \(q>0,\) where the limit solution is given by (13) and (14). However, the systems (13) and (14), where \({Z}_i^*,\) \(\widetilde{Z}_i\) are Hill functions dependent on \(q,\) are well studied by several authors. For the analysis and calculations of the limit solutions of (13) and (14) as \(q\rightarrow 0\), we refer the reader to Plahte and Kjøglum 2005; Gouzé and Sari 2002, and Machina et al. 2013a.

Remark 4

Theorem 2 shows that \(y\) approaches its pseudo-equilibrium quickly and then for an arbitrarily long finite time interval stays close to the pseudo-equilibrium, which evolves slowly with \(x\). Solutions, \(x_i\), to system (11) converge as \(\epsilon \rightarrow 0\) to solutions of system (13), which is of the same form as the original system, (1), uniformly on finite time intervals of arbitrary length, but not necessarily on \(t\in (0,\infty )\). Thus, asymptotics may differ when \(\varepsilon >0\). If system (13) as \(q\rightarrow 0\) has an asymptotically stable fixed point, then the point may not be stable in system (11) for any finite \(\varepsilon \) (see Remark 6 and Example 4 in Sect. 6, where the system does not explicitly contain \(\epsilon \) but admits arbitrarily small parameters \(\alpha _i>0,\) \(\kappa _i>0,\) \(i=1,2\)).

Example 2

Consider the following system, which is an expansion of the system in Example 1.

$$\begin{aligned} \begin{array}{l} \dot{x}_1=Z_1+Z_2-2Z_1Z_2-\beta _1x_1,\\ \dot{x}_2=1-Z_1Z_2-\beta _2x_2,\\ \dot{y}_1=\frac{1}{\varepsilon }(\kappa _1x_1-\alpha _1y_1),\\ \dot{y}_2=\frac{1}{\varepsilon }(\kappa _2x_2-\alpha _2y_2). \end{array} \end{aligned}$$
(15)

where \(Z_i=H(y_i,\theta _i,q),\) \(i=1,2,\) and \(0<\varepsilon <<1\).

By Theorem 2, as \(\varepsilon \rightarrow 0\), the solutions \(y_i^\varepsilon \) of (15) converge to the solution of the system

$$\begin{aligned} \begin{array}{l} \dot{y}_1^*=\frac{\kappa _1}{\alpha _1}(Z_1^*+Z_2^*-2Z_1^*Z_2^*)-\beta _1y_1^*,\\ \dot{y}_2^*=\frac{\kappa _2}{\alpha _2}(1-Z_1^*Z_2^*)-\beta _2y_2^*.\\ \end{array} \end{aligned}$$
(16)

uniformly on intervals of the form \([\delta ,T]\) for \(\delta >0\). The system (16) can be studied following the analysis of Plahte and Kjøglum (2005) and in fact can be rescaled to put it into the form of our Example 1 but with different values of parameters. Thus, it has the same white, black and transparent walls, and an SSP in the wall as in Example 1, for appropriate parameter values (Fig. 1).

While the original system (4) in Example 1 has an asymptotically stable SSP at the threshold intersection, Theorem 2 does not allow us to conclude that system (15) has a corresponding asymptotically stable stationary point that converges to that of system (4) as \(\varepsilon \rightarrow 0\). We investigate this question further in Sect. 6. \(\square \)

Fig. 1
figure 1

a The phase space of (16) with a few trajectories plotted. The small parameter is \(q=0.01\), and the other parameter values are as follows: \(\alpha _1=1,\) \(\alpha _2=2,\) \(\beta _1=0.6,\) \(\beta _2=0.9,\) \(\kappa _1=3,\) \(\kappa _2=4,\) \(\theta _1=\theta _2=1\). b Evolution in the \(y_1,y_2\) plane of two trajectories of (15) with different initial values: \(x_0=(1, 2),\) \(y_0=(1, 0.5)\) for the solid line, and \(x_0=(1.3, 0.75),\) \(y_0=(0.5, 1.5)\) for the dashed line. The small parameter values are \(q=0.01\) and \(\varepsilon =0.02\), and other parameter values are as in (a). Initially, the trajectory in the \(y_1,y_2\) plane quickly reaches the vicinity of the point \((\kappa _1x_{1,0}/\alpha _{1},\kappa _2x_{2.0}/\alpha _{2}),\) which is \((3,4)\) for the solid line and \((3.9, 1.5)\) for the dashed line. Then it follows the dynamics of (16) while approaching the SSP near \((2.25,1)\)

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5 Infinitely Fast mRNA Rates

Now we consider the situation where the mRNA transcription rates are much faster than the protein translation rates. We show that the limit dynamics of the expanded system with a constant \({G}_i\) is given by (1), with an appropriate choice of \(F_i\) and \(G_i\) for that system.

Consider the system of dimension \(2n\) as in (5) but with \(\frac{1}{\varepsilon }\) scaling the rate of the mRNA dynamics.

$$\begin{aligned} \begin{array}{l} \dot{x}_i=\frac{1}{\varepsilon }(F_i(Z)-\beta _i x_i),\\ \dot{y}_{i}=\kappa _ix_i-\alpha _{i}y_{i}, \\ \end{array}\qquad i=1,\ldots ,n, \end{aligned}$$
(17)

where \(Z=(Z_{1},\ldots ,Z_{n}),\) \(Z_{i}=H(y_{i}, \theta _i,q)\).

Let \(q>0\) be fixed. The system (17) represents a singular perturbation problem as \(\varepsilon \rightarrow 0\) with the fast flow \(x_{i}\) and the slow flow \(y_i\). The pseudo-stationary solution \(x_{i}^*=F_i(Z)/\beta _i\) of the fast flow is obtained from \(F_i(Z)-\beta _i x_i=0\). It is easy to check that the stationary solution \(x_{i}^*\) is asymptotically stable for each fixed \(y_i\). Thus, all assumptions of the Tikhonov theorem are fulfilled, and the slow flow is therefore governed by

$$\begin{aligned} \dot{y}_i=\kappa _i\frac{F_i(Z)}{\beta _i}-\alpha _{i}y_{i}. \end{aligned}$$
(18)

The system (18) is of the form (1) with \(F_i(Z)\) replaced by \(\frac{\kappa _i}{\beta _i}F_i(Z)\).

By the Tikhonov theorem, the following result holds.

Theorem 3

Let \(q>0\) and an initial condition \((x_0,y_0)\) be fixed. The solutions \(y_i^{\varepsilon }(t),\) \(i=1,\ldots ,n,\) of (17) converge uniformly on any finite time interval as \(\varepsilon \rightarrow 0\) to \(y_{i},\) which is a solution to the system (18).

Remark 5

The same remarks apply here as Remarks 3 and  4 following Theorem 2. The limit solutions of system (18) as \(q\rightarrow 0\) are well studied by Plahte and Kjøglum 2005; Gouzé and Sari 2002; Machina et al. 2013a. Solutions to system (17) converge to solutions of system (18) on finite time intervals of arbitrary length, but the result does not tell us about the asymptotic behaviour for any \(\varepsilon >0\).

Example 3

Consider the following system, which is an expansion of the system in Example 1.

$$\begin{aligned} \begin{array}{l} \dot{x}_1=\frac{1}{\varepsilon }(Z_1+Z_2-2Z_1Z_2-\beta _1x_1),\\ \dot{x}_2=\frac{1}{\varepsilon }(1-Z_1Z_2-\beta _2x_2),\\ \dot{y}_1=\kappa _1x_1-\alpha _1y_1,\\ \dot{y}_2=\kappa _2x_2-\alpha _2y_2. \end{array} \end{aligned}$$
(19)

where \(Z_i=H(y_i,\theta _i,q),\) \(i=1,2,\) \(0<\varepsilon <<1\).

By Theorem 3, as \(\varepsilon \rightarrow 0\), the solutions \(y_i^\varepsilon \) of (19) converge to the solution of the system

$$\begin{aligned} \begin{array}{l} \dot{y}_1=\frac{\kappa _1}{\beta _1}(Z_1+Z_2-2Z_1Z_2)-\alpha _1y_1,\\ \dot{y}_2=\frac{\kappa _2}{\beta _2}(1-Z_1Z_2)-\alpha _2y_2.\\ \end{array} \end{aligned}$$
(20)

uniformly on intervals of the form \([0,T]\). The system (20) can again be studied following the analysis of Plahte and Kjøglum (2005) and can be rescaled to put it into the form of our Example 1 with different values of the parameters. Thus, again it has the same white, black and transparent walls, and a SSP in the wall, though not necessarily at the threshold intersection point (Fig. 2). \(\square \)

Fig. 2
figure 2

A few trajectories (black solid lines) of (20) are plotted in its phase space. Parameter values are as follows: \(\alpha _1=1,\) \(\alpha _2=2,\) \(\beta _1=0.6,\) \(\beta _2=0.9\), \(\kappa _1=3\), \(\kappa _2=4\), \(\theta _1=\theta _2=1\). The two bold dashed lines represent the \(y_1,y_2\) coordinates of the trajectories of (19) with the initial values at \(x_0=(1, 2),\) \(y_0=(1.5, 4)\) and \(x_0=(1, 2),\) \(y_0=(3, 0)\). All the trajectories approach the SSP near \((2.25,1)\). The small parameter values are \(q=0.01\) and \(\varepsilon =0.02\).

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6 SSPs in Co-dimension 2 Switching Domains

Switching domains in which more than one variable is switching simultaneously are potentially more complicated to analyse. The number of variables that are switching in a domain of a protein-only system is the co-dimension of the domain because if \(k\) of \(n\) variables are switching then the domain itself is of dimension \(n-k\) in the step function limit. In this section, we will discuss some stability results for general 4-dimensional mRNA-protein systems, in which both of the protein concentration variables are switching. The results will also apply to switching domains in \(2n\)-dimensional systems where exactly two of the variables are switching and the rest are regular. Thus, we consider

$$\begin{aligned} \begin{array}{l} \dot{x}_1=F_1(Z)-\beta _1 x_1\\ \dot{x}_2=F_2(Z)-\beta _2 x_2\\ \dot{y}_{1}=\kappa _1x_1-\alpha _{1}y_{1} \\ \dot{y}_{2}=\kappa _2x_2-\alpha _{2}y_{2},\\ \end{array} \end{aligned}$$
(21)

with \(Z_i=H(y_i,\theta _i,q)\) as before. Note that when \(\kappa _i>0,\) \(\alpha _i>0,\) \(i=1,2,\) are small, the system (21) represents the situation with different timescales for the dynamics of mRNA and protein concentrations. In particular, we are interested in stability of fixed points of (21) where \(y_1\) and \(y_2\) are simultaneously switching. Thus, we seek fixed points with the coordinates \((x^{*}_{1},x^{*}_{2},Z^{*}_{1},Z^{*}_{2})\), with \(Z^{*}_{i} \, \in \, (0,1)\). We therefore rewrite (21) as

$$\begin{aligned} \begin{array}{l} \dot{x}_1=F_1(Z)-\beta _1 x_1\\ \dot{x}_2=F_2(Z)-\beta _2 x_2\\ \dot{Z}_{1}=\frac{Z_{1}(1-Z_{1})}{q \theta _1\left( \frac{Z_{1}}{1-Z_{1}}\right) ^{q}}\left( \kappa _1x_1-\alpha _{1}\theta _1\left( \frac{Z_{1}}{1-Z_{1}}\right) ^{q}\right) \\ \dot{Z}_{2}=\frac{Z_{2}(1-Z_{2})}{q \theta _2\left( \frac{Z_{2}}{1-Z_{2}}\right) ^{q}}\left( \kappa _2x_2-\alpha _{i}\theta _2\left( \frac{Z_{2}}{1-Z_{2}}\right) ^{q}\right) ,\\ \end{array} \end{aligned}$$
(22)

with \(q>0\). Solving \(\dot{Z}_{i}=0\), we obtain \(x^{*}_{i}=\frac{\alpha _{i}\theta _i Z^{*^{q}}_{i}}{\kappa _{i}(1-Z^{*}_{i})^{q}}\), where \(Z^{*}_{i}\) are roots of the system

$$\begin{aligned} \begin{array}{l} F_1(Z)-\beta _1 x^{*}_1=0\\ F_2(Z)-\beta _2 x^{*}_2=0.\\ \end{array} \end{aligned}$$
(23)

In the limit \(q\rightarrow 0\), notice that \(x^{*}_{i}=\frac{\alpha _{i}\theta _i}{\kappa _{i}}\), which greatly simplifies the root finding problem in (23). Assuming that such a fixed point \((x^{*}_{1},x^{*}_{2},Z^{*}_{1},Z^{*}_{2})\) exists, we seek conditions on the parameters of the expanded system, \(\kappa _{i}\) and \(\alpha _{i}\), that guarantee stability or instability at the fixed point. We analyse the Jacobian at \(\mathcal {P}=(x^{*}_{1},x^{*}_{2},Z^{*}_{1},Z^{*}_{2})\),

$$\begin{aligned} J(\mathcal {P})= \begin{bmatrix} -\beta _1&0&\frac{\partial \dot{x}_1}{\partial Z_{1}}&\frac{\partial \dot{x}_1}{\partial Z_{2}}\\ 0&-\beta _2&\frac{\partial \dot{x}_2}{\partial Z_{1}}&\frac{\partial \dot{x}_2}{\partial Z_{2}}\\ \frac{A_{1}}{q}&0&{-\alpha _{1}}&0\\ 0&\frac{A_{2}}{q}&0&{-\alpha _{2}}, \end{bmatrix} \end{aligned}$$

where \(A_{i}=\frac{\kappa _{i}Z_{i}^*(1-Z_{i}^*)}{\theta _i\left( \frac{Z_{i}^*}{1-Z_{i}^*}\right) ^{q}}\). We are interested in stability when \(q\) is very small. If \(\lambda \) denotes an eigenvalue of \(J(\mathcal {P})\), we write the characteristic polynomial in terms of \(\mu =\lambda \sqrt{q}\), multiply through by \(q^2\) and take the limit as \(q\rightarrow 0\), to obtain

$$\begin{aligned} \mu ^{4} - \left( \frac{\partial \dot{x}_2}{\partial Z_{2}}A_{2}+\frac{\partial \dot{x}_1}{\partial Z_{1}}A_{1}\right) \mu ^{2} -A_{1}A_{2}\left( \frac{\partial \dot{x}_2}{\partial Z_{1}}\frac{\partial \dot{x}_1}{\partial Z_{2}}-\frac{\partial \dot{x}_2}{\partial Z_{2}}\frac{\partial \dot{x}_1}{\partial Z_{1}} \right) =0. \end{aligned}$$
(24)

Since the roots of a polynomial are continuous in their coefficients, we can make the roots of the characteristic polynomial in \(\mu \) as close to the roots of (24) as we wish by taking \(q\) small enough. This implies that whenever there is a root of (24) with a nonzero real part, we can find a \(q\) small enough that the characteristic polynomial in \(\mu \) will also have a root with a nonzero real part. More importantly, since if \(\lambda \) is a root of (24), then \(- \lambda \) is also a root of (24), we can find a \(q\) small enough such that the characteristic polynomial in \(\mu \) will have a root with a positive real part. Finally, since the roots of the characteristic polynomial in \(\lambda \) are positive scalar multiples of roots of the characteristic polynomial in \(\mu \), whenever the characteristic polynomial in \(\mu \) has a root with a positive real part, there is an eigenvalue of \(J(\mathcal {P})\) with a positive real part. Therefore, for a given fixed point \(\mathcal {P}\) of (22), with \(Z_{i} \in (0,1)\), we know that \(\mathcal {P}\) is unstable when \(q<\delta \) for some \(\delta >0\), if there exists a root of (24) with a nonzero real part. Thus, the only situation in which we do not immediately ascertain instability for small enough \(q\) is when all solutions of (24) have pure imaginary roots. In that case, we do not know the stability of the equilibria of (22).

Remark 6

We have shown that any possible SSPs of system (21) located in co-dimension 2 switching domains are generically unstable, except for possibly very specific cases when stability of the equilibria is unknown. Note that this is true for arbitrary positive values of \(\alpha _i, \kappa _i\), including cases when the timescales of mRNA and protein are very different. However, the corresponding SSPs of the protein-only model reduced from (21) may in general be stable, as in the Plahte–Kjøglum system (2005) and the example below. Our results agree with the analysis of Polynikis et al. (2009) for protein-only systems without self-regulation. Under the assumption that mRNA \(x_i\) is not regulated by \(Z_i,\) they show that for some values of \(\alpha _i> 0,\) \(\kappa _i > 0\) the system (21) exhibits oscillations around an unstable singular stationary point. The amplitude of these oscillations decreases as \(\alpha _i\) and \(\kappa _i\) decrease. In the limit \(\alpha _i\rightarrow 0,\) \(\kappa _i \rightarrow 0,\) the equilibria predicted by their reduced protein-only model are always stable; see Polynikis et al. (2009, Sections 5.2, 5.3). Similar results were found earlier by Tyson and Othmer (1978), also for systems without self-regulation.

Example 4

We now apply this result to the mRNA-protein expansion of the Plahte–Kjøglum system,

$$\begin{aligned} \begin{array}{l} \dot{x}_1=Z_1+Z_2-2Z_1Z_2-\beta _1x_1,\\ \dot{x}_2=1-Z_1Z_2-\beta _2x_2,\\ \dot{y}_1=\kappa _1x_1-\alpha _1y_1,\\ \dot{y}_2=\kappa _2x_2-\alpha _2y_2, \end{array} \end{aligned}$$
(25)

with the same parameters as in the original system: \(\beta _{1}=0.6, \, \beta _{2}=0.9\) and \(\theta _i=1\). We ask whether there exist values of \(\kappa _{i}\) and \(\alpha _{i}\) such that the dynamics of the expanded system differ from those in the original Plahte–Kjøglum system (4). First, we look for conditions on \(\kappa _{i}\) and \(\alpha _{i}\) ensuring the existence of fixed points of the type \((x^{*}_{1},x^{*}_{2},Z^{*}_{1},Z^{*}_{2})\), with \(Z^{*}_{i} \, \in \, (0,1)\).

Following the singular perturbation approach, making the fast time substitution \(\tau =\frac{t}{q}\) with \(x^{\prime } = \frac{\partial x}{\partial \tau }\) and taking \(q\rightarrow 0\), the system becomes

$$\begin{aligned} \begin{array}{l} x^{\prime }_{1}=0,\\ x^{\prime }_{2}=0,\\ Z^{\prime }_{1}=\frac{Z_{1}(1-Z_{1})}{y_{1}}(\kappa _1x_1-\alpha _1y_1),\\ Z^{\prime }_{2}=\frac{Z_{2}(1-Z_{2})}{y_{2}}(\kappa _2x_2-\alpha _2y_2). \end{array} \end{aligned}$$

Fixed points with \(Z^{*}_{i} \, \in \, (0,1)\) occur when \(y_{i}=\theta _i\), which yields \(x^{*}_{i}=\frac{\alpha _{i}\theta _i}{\kappa _{i}}\). Returning to slow time, putting these values for \(x^{*}_{i}\) into the first two equations above gives

$$\begin{aligned} \begin{array}{l} \dot{x}_1=Z_1+Z_2-2Z_1Z_2-\beta _1x^{*}_{1},\\ \dot{x}_2=1-Z_1Z_2-\beta _2x^{*}_{2},\\ \dot{Z}_1=0,\\ \dot{Z}_2=0. \end{array} \end{aligned}$$

At a stationary point, all these derivatives must be \(0\). From the second equation above,

$$\begin{aligned} Z^{*}_{2}=\frac{1-\beta _2x^{*}_{2}}{Z^{*}_{1}}, \end{aligned}$$
(26)

and then we obtain a quadratic in \(Z^{*}_{1}\),

$$\begin{aligned} \mathcal {F}(Z^{*}_{1})=Z_1^{*^{2}}-(c_{1}+2(1-c_{2}))Z_1^*+(1-c_{2})=0, \end{aligned}$$
(27)

where \(c_{i}=\beta _ix^{*}_{i}\). The roots of \(\mathcal {F}\), when combined with \(Z^{*}_{2}\) and \(x^{*}_{i}\), are fixed points of (25). Since we are only interested in roots in the open interval \((0,1)\), it is clear from (26) that in order to have a stationary point \(c_2<1\) and thus \(\mathcal {F}(0)=1-c_2>0\). In that case, if \(\mathcal {F}(1)=c_2-c_1<0\), then \(\mathcal {F}\) has exactly one root in \((0,1)\) by the intermediate value theorem. If \(\mathcal {F}(1)>0\), then the number of roots in \((0,1)\) depends on \(\mathcal {F}_{\text {min}}=(c_{2}-c_{1})(1-c_{2})-\frac{c^{2}_{1}}{4}\): If \(\mathcal {F}_{\text {min}}>0\) then there are no roots in \((0,1)\), while if \(\mathcal {F}_{\text {min}}<0\) there are 2 roots in \((0,1)\).

Thus, the conditions for existence of exactly one stationary point are \(c_{2}<1\) and \(c_{1}>c_{2}\). The conditions for the existence of two stationary points are \(c_{1}<c_{2}<1\) and \((c_{2}-c_{1})(1-c_{2})-\frac{c^{2}_{1}}{4}<0\). Figure 3 labels the regions in the \(c_1,c_2\) parameter plane where system (25) has \(0,1\) or \(2\) fixed points, \((Z_1^*,Z_2^*)\), in the open square \((0,1)^2\).

In order to compare the \(x_i\) variables of the expanded system (25) to those of the original Plahte–Kjøglum system (4), note that if \(\kappa _{i}=\alpha _{i}\), then \(c_{i}=\beta _{i}\theta _i\). In this case \(c_{1}=0.6\) and \(c_{2}=0.9\), which satisfy the requirements for the existence of two fixed points and give the same \(Z^{*}_{i}\) values as in the original Plahte–Kjøglum system (4). With these values for \(Z^{*}_{i}\), the roots of the reduced characteristic equation (24) are \((0.28321i, -0.28321i, 0.31456, -0.31456)\) for the first fixed point, and \((0.17325-0.24305i, 0.17325+0.24305i, -0.17325-0.24305i, -0.17325+0.24305i)\) for the second fixed point (after rounding). In both cases, there is a root with positive real part, so we can find a \(q>0\) small enough such that both fixed points are unstable. Numerical solutions of (25) agree with this analysis, as shown in Fig. 4, with \(q=0.001\). This is not consistent with the behaviour of the original Plahte–Kjøglum system (4) in the vicinity of the threshold intersection (i.e., in the fast variables, \(Z_1,Z_2\)), where one fixed point is stable. The implication is that there are parameter values \(\kappa _i, \alpha _i\) (even if \(\beta _i=\gamma _i\) from the original system) for which the SSP in (25) at \(y_1=\theta _1, y_2=\theta _2\) is no longer stable (see Remark 4). \(\square \)

Fig. 3
figure 3

(Color Figure Online) Values of \(c_{1}\) and \(c_{2}\) where system (25) will have \(0,1\) or \(2\) fixed points, \((Z_1^*,Z_2^*)\), in the open square \((0,1)^2\)

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Fig. 4
figure 4

(Color Figure Online) Numerical simulations of system (25) with \(\kappa _1=\alpha _1=3.33, \beta _1=0.6, \kappa _2=\alpha _2=5.0, \beta _2=0.9\), and \(q=0.001\). a A projection of the 4-dimensional phase space onto the \(x_1,x_2\)-plane, showing four solutions, none of which converge to the threshold intersection. b A blow-up of the region around the fixed point near \((1.5,1)\). In both panes, the asterisk indicates the fixed point in the wall where \(y_1>1\), \(x_1>1\), and \(y_2\) and \(x_2\) are near 1

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7 Analysis of the Flow in Regular Domains

If timescales are of the same order (not orders of magnitude different), the protein dynamics of (5) cannot be analysed just by studying the simpler system (1). A natural question is whether the methods of analysis developed to determine the flow of system (1) through regular domains can be extended to an analysis for system (5). In this section, we take \(q\rightarrow 0\) and describe a mapping from one wall to another that allows an analytical construction of the solution of the switched system (5) for any finite number of transitions.

Given a Boolean vector \([B^{(k)}_1,\ldots ,B^{(k)}_n],\) we start in the regular domain \(\mathcal {B}^{(k)}\) where \(Z_{i}=B^{(k)}_{i},\) \(i=1,\ldots ,n,\) and denote by the constants \(\pi _i^{(k)}=F_i(B^{(k)}_{1},\ldots ,B^{(k)}_{n}),\) so that initially (5) is given by

$$\begin{aligned} \begin{array}{l} \dot{x}_i=\pi _i^{(k)}-\beta _ix_i,\\ \dot{y}_{i}=\kappa _{i}x_i-\alpha _{i}y_{i}, \\ \end{array}\qquad i=1,\ldots ,n, \end{aligned}$$
(28)

with \(y_i(0)=y_i^{(k)}\) as the initial value in the box \(\mathcal {B}^{(k)}\) and \(x_i(0)=x_i^{(k)}\).

The solution to the first equation in (28) is

$$\begin{aligned} {x}_i(t)=\frac{\pi _i^{(k)}}{\beta _i}-\left( \frac{\pi _i^{(k)}}{\beta _i} -x_i^{(k)}\right) e^{-\beta _it},\qquad i=1,\ldots ,n. \end{aligned}$$
(29)

Inserting this into the second equation in (28) gives

$$\begin{aligned} \dot{y}_{i}=\kappa _i\frac{\pi _i^{(k)}}{\beta _i}+\kappa _i \left( x_i^{(k)}-\frac{\pi _i^{(k)}}{\beta _i}\right) e^{-\beta _i t} -\alpha _{i}y_{i}, \end{aligned}$$

which has solution

$$\begin{aligned} {y}_{i}(t)= & {} \frac{\kappa _i\pi _i^{(k)}}{\beta _i\alpha _{i}} +\frac{\kappa _i}{\alpha _i-\beta _i}\left( x_i^{(k)}- \frac{\pi _i^{(k)}}{\beta _i}\right) e^{-\beta _i t}\nonumber \\&+\left( y_i^{(k)}-\frac{\kappa _i\pi _i^{(k)}}{\beta _i\alpha _{i}}-\frac{\kappa _i}{\alpha _i-\beta _i}\left( x_i^{(k)}- \frac{\pi _i^{(k)}}{\beta _i}\right) \right) e^{-\alpha _i t}, \end{aligned}$$
(30)

under the assumption (which we make here and in the remainder of this section) that \({\beta _i\ne \alpha _{i}}\).

Thus in a particular box \(\mathcal {B}^{(k)}\) given by a Boolean vector \([B^{(k)}_{1},\ldots ,B^{(k)}_{n}],\) the solution to (28) is given by (29) and (30). Since all walls in the model (5) are transparent (\(\dot{y}_i\) does not depend on \(Z_i\)), the solution can be continuously extended from one box to a neighbouring box using these formulas.

For example, suppose that at some time \(T^{(k)}=T_{s_{k}}\), the solution passes through the wall \(y_{s_k}=\theta _{s_k}\) at the point \(y_{s_k}(T^{(k)})=y^{(k)}_{s_k}=\theta _{s_k},\) \(x_{i}(T^{(k)})=x_{i}^{(k)},\) and \(y_{i}(T^{(k)})=y^{(k)}_{i},\) \(i=1,\ldots ,n,\) and proceeds into the box \(\mathcal {B}^{(k)}\). The index \((k)\) is a counter for the intersections of the solution and walls. Consider all \(j\) such that \(y_j=\theta _j\) are \(n\) walls adjacent to the box \(\mathcal {B}^{(k)}\). For each \(j\), we have, from (30),

$$\begin{aligned} \theta _j= & {} \frac{\kappa _j\pi _j^{(k)}}{\beta _j\alpha _j}+ \frac{\kappa _j}{\alpha _j-\beta _j}\left( x_j^{(k)} -\frac{\pi _j^{(k)}}{\beta _j}\right) e^{-\beta _jT_j}\nonumber \\&+\left( y_j^{(k)}-\frac{\kappa _j\pi _j^{(k)}}{\beta _j\alpha _j}- \frac{\kappa _j}{\alpha _j-\beta _j}\left( x_j^{(k)} -\frac{\pi _j^{(k)}}{\beta _j}\right) \right) e^{-\alpha _jT_j}. \end{aligned}$$
(31)

With the substitutions \(u=e^{-\beta _jT_j},\) \(a=\frac{\kappa _j\pi _j^{(k)}}{\beta _j\alpha _j}-\theta _j,\) \(b=\frac{\kappa _j}{\alpha _j-\beta _j}\left( x_j^{(k)} -\frac{\pi _j^{(k)}}{\beta _j}\right) ,\) and \(c=y_j^{(k)}-\frac{\kappa _j\pi _j^{(k)}}{\beta _j\alpha _j}- \frac{\kappa _j}{\alpha _j-\beta _j}\left( x_j^{(k)} -\frac{\pi _j^{(k)}}{\beta _j}\right) \), the equality (31) can be rewritten as

$$\begin{aligned} 0=a+bu+cu^{\frac{\alpha _{j}}{\beta _j}}, \end{aligned}$$
(32)

which is potentially a transcendental equation (if \(\frac{\alpha _{j}}{\beta _j}\) is an irrational number). If for a particular \(j\), there exist \(N\ge 1\) positive solutions of (32), say \(\bar{u}_{\ell }\) for \(\ell =1,\ldots , N\), then \(T_j=\min \limits _{\ell }\left\{ -\frac{\ln \bar{u}_\ell }{\beta _j}\right\} \) (the first moment of hitting a wall).

If there exists a positive solution of (32) for some \(j,\) then the index \(s_{k+1}\), specified by \(T_{s_{k+1}}=\min \limits _{j}\{ T_{j} \}\), indicates the wall \(y_{s_{k+1}}=\theta _{s_{k+1}}\) that the solution crosses at step \((k+1)\). Otherwise, nothing switches and the solution stays in the same box \(\mathcal {B}^{(k)}\) as \(t\rightarrow \infty \) with \(x_i(t)\rightarrow \frac{\pi _i^{(k)}}{\beta _i},\) and \({y}_{i}(t)\rightarrow \frac{\kappa _{i}\pi _i^{(k)}}{\beta _i\alpha _{{i}}}>0,\) monotonically.

When a solution \(\bar{u}=\bar{u}(x_{s_{k+1}}^{(k)},y_{s_{k+1}}^{(k)},\pi _{s_{k+1}}^{(k)}, \beta _{s_{k+1}},\kappa _{s_{k+1}},\alpha _{{s_{k+1}}},\theta _{s_{k+1}})\) of (32) is inserted into (29) and (30) for \(e^{-\beta _it},\) the remaining coordinates \(x_{i}\) and \(y_i,\) \(i=1,\ldots ,n,\) of the intersection point \((x^{(k+1)},y^{(k+1)})\) with the wall \(y_{s_k}=\theta _{s_k}\) are obtained.

We thus have described a mapping \(M: \mathbb {R}^{2n}_+\rightarrow \mathbb {R}^{2n}_ +\) from one wall to another, i.e., \((x^{(k+1)},y^{(k+1)})=M(x^{(k)},y^{(k)})\).

Example 5

Consider the system

$$\begin{aligned} \begin{array}{l} \dot{x}=1-Z-\beta x,\\ \dot{y}=\kappa x-\alpha y, \end{array} \end{aligned}$$
(33)

where \(Z=H(y,\theta ,q),\) \(\alpha =1,\) \(\beta =0.6,\) \(\kappa =3,\) \(\theta =1\). These equations would be obtained by considering the original Plahte–Kjøglum example in the vicinity of either of its black walls and expanding to the \(4\)-dimensional mRNA-protein system. Some trajectories, calculated according to the above analysis, are plotted in Fig. 5. In this example, there is only one threshold, dividing the phase space into two boxes, and we calculated repeated transitions across the same threshold at \(y=1\).

The stationary solution of (33) is \((\frac{\alpha }{\kappa }y^*, y^*)\) (see Sect. 3), where \(y^*\) is the stationary solution of

$$\begin{aligned} \dot{y}=\frac{\kappa }{\alpha }(1-Z)-\beta y. \end{aligned}$$
(34)

The system (34) has a unique asymptotically stable singular stationary solution at \(y^*=\theta =1\). Note that in this case, the stability of the SSP in the original protein-only model is preserved in the expanded mRNA-protein model. \(\square \)

Fig. 5
figure 5

A few trajectories of (33) are plotted in its phase space. All the trajectories spiral into the singular stationary point near \((1/3,1)\), which is asymptotically stable by Theorem 1. The small parameter value is \(q=0.01\)

Full size image

Figure 4b shows similar behaviour in a four-dimensional system with different parameters, where \(x_2\) and \(y_2\) spiral in to a fixed point as above.

8 Discussion

We have obtained results regarding the more realistic model framework for gene regulatory networks that keeps track of both mRNA and protein concentrations and thus models both transcription and translation as separate steps. We have also drawn some comparisons with the simpler modelling framework, studied at length in the past, in which the only variables are protein concentrations and they are taken to regulate each other’s production directly.

We have shown that in the limit of infinitely fast rates for either the mRNA or the protein, solutions of the expanded system converge to solutions of the protein-only model, but for less-than-infinite rates this conclusion only holds for a finite time interval. Thus, asymptotic behaviour can differ in the expanded system and stability properties of fixed points can be changed. Our Theorem 1, however, guarantees that stability of a SSP in a co-dimension 1 switching domain of the protein-only model is preserved in the expanded model, at least for steep enough sigmoids. Thus, it is only in higher co-dimension switching domains that stability can be changed in the expanded model. In fact, our analysis of SSPs in co-dimension 2 switching domains shows that they are generally unstable, although there are special situations where the linearized system in the limit \(q\rightarrow 0\) is neutrally stable (Sect. 6, in the discussion following (24)). In these cases, we have not determined the effect on stability of the SSP of the nonlinear terms nor of the perturbation \(q>0\). This question remains for future work. It is clear, though, that the stability of a fixed point in a co-dimension 2 domain of the protein-only model can be lost in the expanded model, depending on the values of the parameters. This is true for similar as well as different timescales between the protein and mRNA dynamics.

The overall picture is that the switching domains are more benign in the expanded model than in the protein-only model. In the infinitely steep limit (\(q\rightarrow 0\)) case of the protein-only model, solutions can reach the switching threshold of a single variable (a wall) in finite time and remain stuck there for an interval of time (infinite if approaching an SSP within the switching domain). In the infinitely steep limit of the mRNA-protein model, there may still be fixed points in walls, but these are approached only asymptotically after spiralling in the corresponding mRNA and protein variables for an infinite period of time (Sect. 7, Example 5). All walls are transparent, so there is no sliding motion in switching domains, even in the limit \(q\rightarrow 0\). Co-dimension 2 switching domains are generally unstable, so solutions can normally not be trapped there, though there are special cases where we are not yet sure what happens, as mentioned above. We conjecture, though, that any stable fixed points in higher co-dimension switching domains are also approached only asymptotically, after an infinite period of spiralling.

Because of this transparency of switching domains, and the lack of singular flow, the most mathematically difficult and sensitive situations that arise in the protein-only models are avoided in the mRNA-protein models. This promises to make analysis easier, though the flow in regular domains (the mapping from entry wall to exit wall, which we derived in Sect. 7) is now somewhat more complicated. For example, the flow may enter and leave such a domain across the same wall.

We now discuss briefly the relationship between our work and that of two other groups. One group reduced a delay system to an ordinary differential equation system with additional variables such that if only one additional variable is introduced, our mRNA-protein system is obtained. The other tackled a similar model expansion to that of the current paper, but focussed on different questions, such as the behaviour of feedback loops and an oscillating example. There is some overlap, however, and that is discussed next.

8.1 Relation to the Analysis of Gedeon, Cummins and Heys

In general, our conclusions regarding stability of SSPs are consistent with those Gedeon et al. (2012), who deal with a similar expansion to include both mRNA and protein dynamics, but only with smooth sigmoids (never the infinitely steep limit). Much of their paper deals with specific example networks, but their Section 3.2 gives a general result about the stability of fixed points in the protein-only model compared to the mRNA-protein model. We agree, in particular, with their statement that the inclusion of the mRNA variables generally has a destabilizing effect on equilibria of the protein-only system or else preserves their stability. However, they also find a small parameter region in which it is possible for an unstable equilibrium to be stabilized by the inclusion of the mRNA variables (their Theorem 3.3, part 3), which we have not found. The conflict is only apparent, however, as their result does not apply to our model, except possibly in some rare special cases.

Their Theorem 3.3 can be satisfied, for example, by expanding the equation

$$\begin{aligned} \dot{x}=\kappa Z(x)-\hbox {d}x,\quad \text{ with } \quad Z(x)=\frac{x}{\theta +x} \end{aligned}$$
(35)

(having the Hill function with \(q=1\)) and \(d,\kappa ,\theta >0\). The equilibrium at \(x=0\) is unstable if \(\frac{\kappa }{\theta }>d\). The expanded system is then

$$\begin{aligned} \begin{array}{l} \dot{x}= cy-\hbox {d}x,\\ \dot{y}=\kappa Z(x) - by, \\ \end{array} \end{aligned}$$
(36)

where \(x\) and \(y\) here are the concentrations of protein and mRNA, respectively (as in Gedeon et al. 2012). The Jacobian matrix at the origin is

$$\begin{aligned} J(0,0)=\left[ \begin{array}{cc} -d &{} c\\ \frac{\kappa }{\theta } &{} -b\end{array}\right] . \end{aligned}$$

The origin is stable if det\((J(0,0))>0\), i.e., if \(\frac{\kappa }{\theta }<\frac{bd}{c}\). So if \(d< \frac{\kappa }{\theta }<\frac{bd}{c}\), the origin is stabilized by the expansion. However, this cannot occur with Hill functions with exponent \(>1\) (i.e., \(q<1\)) in this example since then the equilibrium at \(x=0\) of (35) is stable as is the equilibrium of the expanded system (36). The condition in their Theorem 3.3, resulting from the assumption that the equilibrium point is in the identical location in the protein-only and expanded systems, means that it does not apply in almost all cases with which we are concerned.

Gedeon and co-authors also consider the infinitely fast limits, in a slightly different way, but do not deal with the steep sigmoid limit and focus on a particular example network with periodic behaviour.

8.2 Relation to Delay Systems

Our model (5) fits the delay differential model considered by Ponosov (2005) and Shlykova et al. (2008). Indeed, after the variable substitution, \(\alpha _iX_i=\kappa _ix_i\), and allowing a variable degradation term for the mRNA, system (5) becomes

$$\begin{aligned} \begin{array}{l} \dot{X}_i=\widetilde{F}_i(Z)-G_i(Z)X_i,\qquad i=1,\ldots ,n,\\ \dot{y}_{i}=\alpha _iX_{i}-\alpha _{i}y_{i}, \\ \end{array} \end{aligned}$$
(37)

with \(\widetilde{F}_i(Z)=\frac{\kappa _i}{\alpha _i}F_i(Z),\) \(Z=(Z_1,\ldots ,Z_n)\) and \(Z_{i}=H(y_{i}, \theta _i,q)\).

Ponosov and co-authors considered the delay system

$$\begin{aligned} \begin{array}{l} \dot{X}_i=\widetilde{F}_i(Z)-G_i(Z)X_i,\qquad i=1,\ldots ,n\\ \dot{y}_{i}=(\mathfrak {R}X_{i})(t) \end{array} \end{aligned}$$
(38)

with the delay operator

$$\begin{aligned} (\mathfrak {R}X_{i})(t)=\int \limits _{-\infty }^tG(t-s)X_i(s)ds, \end{aligned}$$

where the function \(G\) is a linear combination of gamma distributions (Ponosov 2005, Eqs. (4.4)–(4.6)). Our system (37) is a particular case of (38), where \(G(t-s)=\alpha _ie^{-\alpha _i(t-s)},\) also known in the literature as the weak generic delay kernel (see, e.g., Ponosov 2005).

The delay differential models (38) were studied by Ponosov (2005) and Shlykova et al. (2008), who considered (solved) the problem of singular stationary points (see Ponosov (2005), Theorem 8.2 and Conclusion). Theorem 6.6 (localization principle) in Shlykova et al. 2008, Theorem 7.2) reduced the stability problem to singular coordinates only (their analysis is done under the assumption that sigmoids are logoid functions).