### A nonlinear state-space model of the GAL regulatory system

The state-space model derived in [4] is as follows. Let *R* and Gal be the states *x*_{1} and *x*_{2} of the system, and let *x* ≐ [*x*_{1}*x*_{2}]^{T}. Define *α*_{1} = -(*K*_{3}*S*_{0} + 1), *α*_{2} = -*K*_{3}*R*_{0}, *α*_{3} = - *K*_{3}*S*_{0} where *K*_{
i
}are the kinetic reaction constants, and the nonlinearity *f*(*ζ*_{1}, *ζ*_{2}) = (*K*_{3} - *K*_{2}) *ζ*_{1}*ζ*_{2}. Then, a state-space model of the GAL system is , where

**Remark 1** In [4], only the initial condition response, i.e., the response to *S*_{0} and *R*_{0}, is considered. The two inputs of interest are the galactose injected in the cell, and *R*; the first input can be varied using Gal2p, and the second input can be varied by transforming Gal4 deleted cells with a plasmid expressing Gal4.

**Remark 2** Arguing that *K*_{
i
}are all equal, Φ(*x*) is set to zero in [4], and the phase-plane method is applied on the linearized 2-state system to determine the conditions under which the system is stable and robust to the gene expression delays. In practice, however, the cells are not uniformly distributed whence *K*_{
i
}are not equal so that the nonlinearity Φ cannot be neglected. Further, as the following lemma shows, fails to exhibit bistability, a key property of the GAL regulatory network, even after Φ is accounted for.

**Lemma 1***has a unique steady state and does not exhibit a Hopf bifurcation*.

**Proof:** See Additional file 1.

**Remark 3** Lemma 1 implies that the GAL regulatory system model of [4] is not bistable. However, it is well known that the GAL regulatory system exhibits bistability (see [1]). This anomaly results because, in deriving , the nonlinear feedback loops of GAL3 and GAL80, one of which is positive whereas the other is negative, are overly simplified using a single negative feedback loop in [4]. We propose a correction by including more molecular reactions and, hence, more state variables in our model.

Let us choose the state variables *x*_{1} = , *x*_{2} = *Gal* 3/80, *x*_{3} = *R*, *x*_{4} = *BR*, *x*_{5} = *Gal*, and let *x* ≐ [*x*_{1}*x*_{2}*x*_{3}*x*_{4}*x*_{5}]^{T}. Then (1) can be expressed as = *Ax* + Φ(*x*) + *Bu*, where

where *a*_{
i
}and *b*_{
i
}are the kinetic reaction constants, *ζ*_{
i
}are the degradation rates, and *u* is the input galactose. This is our model of the GAL regulatory system. Note that the nonlinearity Φ(*x*) is quadratic and can be expressed as Φ(*x*) = *x*^{T}**N** *x* where **N** ≐ [*N*_{1}*N*_{2}... *N*_{5}]^{T}for some *N*_{
i
}∈ ℝ^{5×5}. Literature on the stability analysis of such systems is sparse although sufficiency conditions have been established in [6]. It appears that ℒ_{2} stability cannot be expected of multistable models due to the following reason.

**Lemma 2** *A bistable controllable state-space system is not* ℒ_{2}-*stable*.

**Proof:** Let *u*, *x* denote the input and output of the system. Since the system is bistable, there exists a time *τ* and control signals *u*_{1}, *u*_{2} ∈ *P*_{
τ
}ℒ_{2} that drive the system output to each of two distinct constant equilibrium output values, say *x*_{1o}and *x*_{2o}, at time *τ* such that *x*_{1}(*t*) = *x*_{1o}and *x*_{2}(*t*) = *x*_{2o}for all *t* ≥ *τ* . Hence, *u*_{1}, *u*_{2} ∈ ℒ_{2}, but *x*_{1} - *x*_{2} ∉ ℒ_{2}. Therefore, either *x*_{1} ∉ ℒ_{2} or *x*_{2} ∉ ℒ_{2} or both. QED.

As a result, we focus only on establishing a domain of attraction for an equilibrium point of such models. Determination of the domain of attraction is useful since it determines the stability region for cellular memory that can be controlled using a linear feedback of the gene expression states. An extreme example is that of persistent memory, obtained by deleting the GAL80 feedback loop; this phenomenon is observed in mutant genes [1].

**Remark 4** Experimentally, we have observed that the input-output map of *Kluyveromyces lactis* with GAL80 as the output and galactose as the input is an aberration of friction nonlinearity. Multiplier theoretic stability analysis results (see [7–10], and [11]) can be applied to determine the finite-gain stability of such reduced order models as we demonstrate in the Results section.

### Stability and multipliers

We now formally introduce the notation and the notion of stability; a detailed description of these notions is available in [9] and [10]. Let (ℝ^{+}) ℝ denote the set of all (nonnegative) real numbers. Let (·)' (or (·)^{T}) denote the transpose of a vector or a matrix (·). Let the inner-product and let the norm . The vector space ℒ_{2} comprises all signals *x* for which ||*x*|| < ∞. The norm . The Dirac delta function is denoted *δ*(·). The time-truncation operator is denoted *P*_{
τ
}. In stability analysis, a given system is often decomposed into two interconnected subsystems -- a *linear time-invariant* (LTI) subsystem in the feedforward path and an otherwise subsystem in the feedback path (see Fig. 3(i)). Stability of is then deduced if there exists a quadratic functional that separates the graph of from the inverse graph of (see [12]). Certain classes of convolution operators, also called *stability multipliers* (see [7]), specify such functionals. The larger the class of the stability multipliers, the lower the conservatism in the stability analysis [13]. Stability multipliers for memoryless monotone nonlinearities are the Zames-Falb multipliers [8] and their limiting cases include Popov multipliers [11] and RL/RC multipliers [14]. A key property of such a multiplier *M* is that it preserves the positivity of a memoryless monotone nonlinearity *N* in the sense that the positivity of *N* implies the positivity of *MN*. Well known examples of positivity preserving multipliers include the Popov multipliers and the Zames-Falb multipliers (see [7, 8], and [[9], Chapter 3]).

**Definition 1** *A system**mapping u* ∈ ℒ_{2}*into y* ∈ ℒ_{2}*is said to be* finite gain stable *if there exists γ* ≥ 0 *such that*||(*u*)|| ≤ *γ* ||*u*|| *for all u* ∈ ℒ_{2}.

**Definition 2** *The class**of* monotone nonlinearities *consists of all memoryless mappings N* : ℝ^{n}↦ ℝ^{n}*such that:* (*i*)*N is the gradient of a convex real-valued function, and (ii) there exists C* ∈ ℝ^{+}*s.t*. ||*N* (*x*)|| ≤ *C*||*x*|| ∀ *x* ∈ ℒ_{2}. *The class*.

**Definition 3** *The class* ℳ_{
ZF
}*of Zames-Falb multipliers denotes the class of convolution operators, either continuous-time or discrete-time, such that the impulse response of an M* ∈ ℳ_{
ZF
}*is of the form m*(·) = *g δ* (·) + *h*(·) *with* ||*h*||_{1} <*g*, *h*(*t*) ≤ 0 ∀ *t*, *where g*, *h*(·) ∈ ℝ.

**Remark 5** The Nyquist plot of a Zames-Falb multiplier is constrained to lie inside a disc in the open right-half *s*-plane, as shown in Fig. 3(ii). In [15], aberrations of monotone nonlinearities, as shown in Fig. 3(iii), are considered and a class of positivity preserving multipliers for these nonlinearities is established. The results of [15] facilitate a class of stabilizing controllers for systems featuring such nonlinearities. It turns out that *Kluyveromyces lactis* exhibits such a nonlinearity when the input is galactose and the output of interest is the GAL4 expression (see Fig. 4).