Mathematical Stability Analysis of the Full SREBP-2 Pathway Model: Insights into Cholesterol Homeostasis

Abstract

We present a mathematical analysis of the sterol regulatory element-binding protein 2 (SREBP-2) pathway, a key regulator of intracellular cholesterol homeostasis. Using a compartment model formulated as a nonlinear system of ordinary differential equations, we investigate stability via M-matrix theory and norm-based criteria. We show that the Frobenius norm ${\parallel B\parallel }_F\ge 1$ cannot ensure stability, whereas the infinity norm condition ${\parallel B\parallel }_{\infty }<1$ provides a practical guarantee that the spectral radius $\rho \left(B\right)<1$. The spectral norm ${\parallel B\parallel }_2$ yields sharper intermediate bounds. Numerical simulations confirm these results, highlighting parameter regions of stability and showing that the dissociation rate $k_{-1}$ has the strongest influence on system behavior. These findings demonstrate the robustness of the ${\mathsf{\ell }}_{\infty }$ criterion, clarify the role of dissociation kinetics in cholesterol regulation, and provide a rigorous framework for assessing homeostatic control in the SREBP-2 pathway.

1. Introduction

In this study, we develop a comprehensive mathematical model of the cholesterol biosynthesis pathway that integrates genetic regulation, metabolic synthesis, and feedback control, building upon the framework originally proposed by [1]. The underlying dynamical system governing cholesterol metabolism is systematically derived and parameterized using biologically relevant data, thereby providing a rigorous foundation for the quantitative analysis of system dynamics. A primary objective of this work is to investigate the stability characteristics of the complete model through the application of M-matrix theory, following the methodological principles established in [2]. This stability analysis is essential for characterizing the long-term behavior of intracellular cholesterol concentrations under perturbations. By delineating the model’s stability properties, we aim to clarify the fundamental regulatory mechanisms governing cholesterol homeostasis and to evaluate the potential impact of external interventions, such as pharmacological treatments, on equilibrium states.

Cholesterol is a fundamental constituent of mammalian cell membranes, where it contributes to membrane fluidity, structural integrity, and selective permeability [3]. Maintaining appropriate intracellular cholesterol levels is vital, as excessive accumulation can lead to cytotoxic effects [4,5,6], whereas cholesterol deficiency may disrupt membrane stability and function. Beyond its structural role, cellular cholesterol metabolism is pivotal in regulating plasma cholesterol concentrations, which are strongly associated with cardiovascular risk [7]. Homeostatic control is achieved through the coordinated regulation of cholesterol influx, efflux, and endogenous biosynthesis, ensuring cholesterol levels remain within a physiologically optimal range. A central mechanism in cholesterol homeostasis is the low-density lipoprotein receptor (LDLR) pathway, which facilitates the removal of cholesterol from the bloodstream [8,9]. Endogenous cholesterol biosynthesis proceeds via a complex, multi-step enzymatic process, with the rate-limiting step catalyzed by 3-hydroxy-3-methylglutaryl coenzyme A reductase (HMGCR). The interplay between cholesterol uptake through LDLR and synthesis via HMGCR is regulated by a tightly regulated negative feedback loop. When intracellular cholesterol levels are low, the expression of both LDLR and HMGCR is upregulated, thereby promoting cholesterol influx and biosynthesis. Conversely, elevated intracellular cholesterol suppresses their expression, thereby limiting further accumulation and maintaining physiological balance.

Although numerous mathematical models have been developed to investigate lipoprotein metabolism and the dynamics of the low-density lipoprotein receptor (LDLR) pathway, many of these models omit an explicit representation of cholesterol biosynthesis and its regulatory feedback mechanisms [10,11,12,13,14]. This omission limits their capacity to accurately capture the complexity of cholesterol regulation and to predict the impact of pharmacological interventions, such as statins, which act as competitive inhibitors of HMG-CoA reductase (HMGCR). Furthermore, individual variability in statin efficacy is often attributed to genetic differences, which underscores the necessity for a more comprehensive modeling framework that integrates the genetic regulation of cholesterol metabolism [15,16].

To investigate the stability of the proposed system, we employ the theory of M-matrices, a powerful and well-established mathematical framework particularly suited for analyzing the stability of complex biological systems. As demonstrated in [17], this approach has been successfully applied to biological model equations, where the M-matrix structure naturally arises from compartmental formulations. It provides an efficient means to determine asymptotic stability properties, especially when traditional methods, such as direct eigenvalue computation, become infeasible in high-dimensional systems. For readers interested in a comprehensive treatment of the theory of M-matrices, detailed discussions can be found in [18,19]. In particular, Araki [2] demonstrated how this framework can be effectively applied to analyze the stability of large systems of differential equations, offering powerful tools for studying the collective behavior of interconnected dynamical subsystems. In contrast, classical stability analyses are often limited to systems of reduced dimensionality. For instance, in the work of Bhattacharya et al. [1], a seven-compartment model of the SREBP-2 cholesterol regulatory pathway was reduced to a three-compartment formulation, where the system’s stability and the onset of oscillations were studied via the Hopf bifurcation method. While this approach effectively characterizes local stability and the emergence of limit cycles, it becomes impractical when extended to large-scale interconnected systems with multiple feedback mechanisms.

An M-matrix is characterized by nonpositive off-diagonal entries and eigenvalues with positive real part properties that ensure desirable stability behavior in dynamical systems [20]. In our previous work, this framework has proven to be a powerful analytical tool for establishing norm-based stability criteria and bounding the spectral radius in complex biological models. Here, we combine Lyapunov-based analysis with spectral conditions derived from M-matrix theory to assess the stability of intracellular cholesterol regulation. In particular, we adopt the characterization developed by Araki [2,21], which provides sufficient conditions for the stability of large-scale interconnected systems through structured matrix analysis. This framework allows us to determine the parameter regimes under which intracellular cholesterol concentrations return to equilibrium after perturbations, thereby ensuring robust homeostatic regulation.

The remainder of this paper is organized as follows. Section 2 reviews the biological background of cholesterol biosynthesis and regulation, presents the derivation of the mathematical model together with the full system of equations, and summarizes parameter values obtained from the literature. In Section 3 and Section 4, we perform a detailed stability analysis based on M-matrix theory, demonstrating its effectiveness in characterizing system dynamics and interpreting the results within a biological framework. Finally, Section 5 highlights the main findings and discusses their broader implications.

2. Model Development

The mathematical model for the sterol regulatory element-binding protein 2 (SREBP-2) cholesterol biosynthesis pathway is developed from a comprehensive understanding of the biological processes involved in cholesterol homeostasis. Maintaining intracellular cholesterol levels within a physiological range is critical for cellular function and survival. The SREBP-2 pathway is central to this regulation and is characterized by intricate interactions among cholesterol, SREBP-2, and molecular components such as 3-hydroxy-3-methylglutaryl-coenzyme A reductase (HMGCR). These interactions are illustrated in the diagram shown in Figure 1, and the model parameters are listed in

Table 1. Model parameters used in the mathematical formulation of Equations (1)–(7), as established in [1].

Parameter	Description	Dimensional Value
$\kappa$	Cooperative binding exponent for SREBP-2 to DNA	3
$k_1$	Association rate constant of SREBP-2 binding to DNA, requiring $\kappa$ molecules to bind simultaneously	molecules−κ mLκ s−1
$k_{-1}$	Dissociation rate constant of the SREBP-2–DNA complex	s−1
$\iota$	Cooperative binding exponent for cholesterol–SREBP-2 formation	4
$k_2$	Association rate constant of cholesterol binding to SREBP-2, requiring $\iota$ molecules to bind simultaneously	molecules−ι mLι s−1
$k_{-2}$	Dissociation rate constant of the cholesterol–SREBP-2 complex	s−1
$K_m$	SREBP-2–HMGCR gene dissociation constant	$3.26\times {10}^{13}$ molecules mL−1
$K_c$	SREBP-2–Cholesterol dissociation constant	$6.02\times {10}^{13}$ molecules mL−1
$K_d$	Dissociation constant for the reaction between SREBP-2 and DNA (S and G), dependent on $\kappa$ binding sites	$3.46\times {10}^{40}$ moleculesκ mL−κ
$K_s$	Dissociation constant for the reaction between SREBP-2 and cholesterol (S and C), dependent on $\iota$ binding sites	$1.31\times {10}^{55}$ moleculesι mL−ι
${\mu }_m$	mRNA transcription rate	$5.17\times {10}^5$ molecules mL−1 s−1
${\delta }_m$	mRNA degradation rate	$4.48\times {10}^{-5}$ s−1
${\mu }_H$	HMGCR translation rate	$3.33\times {10}^{-2}$ s−1
${\delta }_H$	HMGCR degradation rate	$6.42\times {10}^{-5}$ s−1
${\mu }_C$	Cholesterol synthesis rate	$4.33\times {10}^{-2}$ s−1
${\delta }_C$	Cholesterol degradation rate	$1.20\times {10}^{-4}$ s−1

, see [1] for further details.

The regulatory mechanism relies on SREBP-2, a transcription factor that activates the expression of genes required for cholesterol biosynthesis. When intracellular cholesterol levels are low, SREBP-2 is transported from the endoplasmic reticulum (ER) to the Golgi apparatus for proteolytic activation, where it undergoes cleavage to release its active form. This active SREBP-2 translocates to the nucleus, binds to sterol regulatory elements (SREs) in the promoter region of target genes, and initiates the transcription of HMGCR messenger RNA (mRNA). The translated HMGCR enzyme catalyzes cholesterol production, increasing intracellular cholesterol levels and restoring homeostasis.

Conversely, when intracellular cholesterol levels are elevated, the transport of SREBP-2 from the endoplasmic reticulum to the Golgi apparatus is blocked, keeping it in an inactive state and preventing the transcription of cholesterol biosynthetic genes. This feedback mechanism ensures that cholesterol synthesis is tightly regulated, protecting cells from the cytotoxic effects of both cholesterol excess and deficiency.

The compartment mathematical model illustrated in the schematic diagram Figure 1 characterizes the biochemical interactions underlying cholesterol biosynthesis and its feedback regulation via the SREBP-2 pathway. As previously mentioned, the model was developed by [1] and integrates key mechanisms including transcription, translation, cholesterol synthesis, and feedback inhibition to comprehensively capture the dynamics of this biosynthetic process. These interactions are mathematically formulated as a system of nonlinear ordinary differential equations (ODEs), derived using mass-action kinetics to represent the reaction rates.

The genetic regulation of cholesterol biosynthesis is primarily regulated by the dynamic binding and dissociation of the transcription factor SREBP-2. The transcription factor SREBP-2, denoted by S, acts as a transcriptional activator by binding to the HMGCR gene G, forming an active transcriptional complex $S_b$. This interaction promotes the transcription of HMGCR mRNA M at a rate ${\mu }_m$. Subsequently, the mRNA M is translated into the HMG-CoA reductase enzyme H at a rate ${\mu }_h$, which catalyzes the synthesis of cholesterol C at a rate ${\mu }_c$.

The binding of SREBP-2 S to the gene G follows an association rate constant $k_1$ and a dissociation rate constant $k_{-1}$. The dissociation of the active complex regenerates unbound DNA promoters, ensuring dynamic responsiveness to intracellular cholesterol fluctuations. Elevated cholesterol levels facilitate the binding of cholesterol C to SREBP-2 S, forming an inactive complex and thus suppressing the transcription of HMGCR, providing a robust negative feedback mechanism to regulate cholesterol biosynthesis. Additionally, the degradation of HMGCR mRNA, the HMGCR enzyme, and cholesterol occurs at rates ${\delta }_m$, ${\delta }_h$, and ${\delta }_c$, respectively, which maintains cellular cholesterol homeostasis.

$${\displaystyle \frac{dG\left(t\right)}{dt}}=k_{-1}{S}_b\left(t\right)-k_1{S}^{\kappa }\left(t\right)G\left(t\right),$$

(1)

Here, $G\left(t\right)$ represents the concentration of free (unbound) HMGCR gene promoters (in molecules mL⁻¹), indicating the fraction available for binding by the transcription factor SREBP-2 at time t, with initial condition $G\left(0\right)=G_0$, reflecting the total gene copy number present in the cell before any binding has occurred. The variable $S\left(t\right)$ denotes the concentration of active, unbound SREBP-2 transcription factors not currently bound to gene promoters or cholesterol molecules (in molecules mL⁻¹), with initial condition $S\left(0\right)=S_0$, representing the baseline level of transcription factors in the nucleus. Finally, $S_b\left(t\right)$ corresponds to the concentration of the active gene complex formed when SREBP-2 binds to the HMGCR promoter, thereby promoting transcription of the gene (in molecules mL⁻¹). The initial condition $S_b\left(0\right)=0$ reflects that no gene-promoter complexes have yet formed at the initial time.

The parameter $k_{-1}$ represents the dissociation rate constant (in s⁻¹), whereas $k_1$ is the association rate constant (in molecules^−κ mL^κ s⁻¹). The exponent $\kappa$ accounts for the cooperative binding nature of SREBP-2 to DNA, such that the dissociation of one active DNA complex releases $\kappa$ molecules of unbound transcription factor, and the formation of an active DNA complex requires up to $\kappa$ DNA binding sites.

The mathematical model of the sterol regulatory element-binding protein 2 (SREBP-2) cholesterol biosynthesis pathway is developed based on a comprehensive understanding of the biological processes governing cholesterol homeostasis. Maintaining intracellular cholesterol levels within a physiological range is essential for cellular functionality and survival. The SREBP-2 pathway plays a central role in this regulation and is characterized by intricate interactions between cholesterol, SREBP-2, and molecular components such as 3-hydroxy-3-methylglutaryl-coenzyme A (HMG-CoA) reductase (HMGCR). Experimental findings by [1,22] indicate that the HMGCR gene contains three binding sites for SREBP-2, suggesting that cooperative binding is essential for transcriptional activation. Consequently, in our model, we adopt $\kappa =3$, reflecting the requirement of three SREBP-2 molecules to effectively activate HMGCR transcription.

The dynamics of active SREBP-2 S are determined by its binding to the HMGCR gene G, leading to the formation of the active complex $S_b$, as well as its interaction with cholesterol C, resulting in the formation of the inactive complex $C_b$:

$$\begin{array}{c}\hfill {\displaystyle \frac{dS\left(t\right)}{dt}}=\kappa k_{-1}{S}_b\left(t\right)-\kappa k_1{S}^{\kappa }\left(t\right)G\left(t\right)-k_2{C}^{\iota }\left(t\right)S\left(t\right)+k_{-2}{C}_b\left(t\right),\end{array}$$

(2)

Here, $C\left(t\right)$ represents the concentration of free (unbound) intracellular cholesterol (in molecules mL⁻¹), which is available to participate in regulatory and metabolic processes. The initial condition $C\left(0\right)=C_0$ reflects the baseline cholesterol level within the cell at the beginning of observation or simulation. The variable $C_b\left(t\right)$ denotes the concentration of the inactive complex formed when cholesterol binds to SREBP-2, thereby preventing it from activating gene transcription (in molecules mL⁻¹). The initial condition $C_b\left(0\right)=0$ indicates that no such complexes are present at time zero, representing a system in which cholesterol has not yet exerted feedback inhibition. The parameters $k_2$ and $k_{-2}$ are the association and dissociation rate constants for the interaction between S and C, with units molecules^−ι mL^ι s⁻¹ and s⁻¹, respectively, (see Figure 2).

The exponent $\iota$ characterizes the cooperative binding of cholesterol to SREBP-2, where each inactive complex requires up to $\iota$ cholesterol molecules to form. Moreover, experimental investigations on the sterol-sensing domain (SSD) of SCAP have demonstrated its tetrameric structure, indicating that four cholesterol molecules bind to a SCAP-SREBP complex to promote inactivation. Consequently, we set $\iota =4$ in our model [1,23,24]. The variables $S_b$ and $C_b$, play a crucial role in the regulation of cholesterol biosynthesis. These terms are defined as follows:

2.1. Active Complex Concentration ($S_b$)

The variable $S_b$ represents the concentration of the active DNA complex, where the transcription factor SREBP-2 (S) binds cooperatively to the HMGCR gene (G). This complex is denoted as $[G:\kappa S]$ and is responsible for activating transcription. The formation and dissociation of this complex are determined by the association rate constant $k_1$ and the dissociation rate constant $k_{-1}$: $G+\kappa S\rightleftharpoons G:\kappa S.$ The active complex $S_b$ facilitates the transcription of mRNA (M) at a rate ${\mu }_d$, without depleting the bound DNA complex. The dynamics of the active complex $S_b$ are determined by:

$$\begin{array}{c}\hfill {\displaystyle \frac{d{S}_b\left(t\right)}{dt}}=-k_{-1}{S}_b\left(t\right)+k_1{S}^{\kappa }\left(t\right)G\left(t\right).\end{array}$$

(3)

The exponent $\kappa$, as given above, represents the cooperative binding of SREBP-2 to the HMGCR gene, where each active complex formation requires up to $\kappa$ binding sites.

2.2. Inactive Complex Concentration ($C_b$)

The variable $C_b$ represents the concentration of the inactive complex formed between SREBP-2 (S) and cholesterol (C), denoted as $[S:\iota C]$. This complex inhibits the activation of the HMGCR gene and plays a key role in cholesterol regulation. The formation and dissociation of this complex are determined by the association rate constant $k_2$ and the dissociation rate constant $k_{-2}$: $S+\iota C\rightleftharpoons S:\iota C.$ The presence of $C_b$ sequesters SREBP-2, preventing its binding to the HMGCR gene. This inhibition suppresses transcription and plays a crucial role in the negative feedback regulation of cholesterol homeostasis. The dynamics of inactive complex $C_b$ are described by:

$$\begin{array}{c}\hfill {\displaystyle \frac{d{C}_b\left(t\right)}{dt}}=k_{2}S\left(t\right)C^{\iota }\left(t\right)-k_{-2}{C}_b\left(t\right).\end{array}$$

(4)

The parameters $k_2$ and $k_{-2}$ represent the association and dissociation rate constants, with units molecules^−ι mL^ι s⁻¹ and s⁻¹, respectively. The exponent $\iota$ represents the stoichiometric coefficient, indicating the number of cholesterol molecules required to bind to S to form the inactive complex. The transcription of HMGCR mRNA M occurs at a rate proportional to the active complex $S_b$ and is balanced by degradation:

$$\begin{array}{c}\hfill {\displaystyle \frac{dM\left(t\right)}{dt}}={\mu }_m{S}_b\left(t\right)-{\delta }_{m}M\left(t\right).\end{array}$$

(5)

Here, ${\mu }_m$ is the transcription rate constant (in s⁻¹), $M\left(t\right)$ represents the concentration of HMGCR mRNA (in molecules mL⁻¹), and ${\delta }_m$ is the degradation rate of mRNA (in s⁻¹). The initial condition $M\left(0\right)=0$ reflects that at time zero, no mRNA transcripts have been synthesized yet, consistent with a system where gene expression has not yet been initiated.

The HMGCR enzyme H is synthesized at a rate ${\mu }_H$ from M and is degraded at a rate ${\delta }_H$:

$$\begin{array}{c}\hfill {\displaystyle \frac{dH\left(t\right)}{dt}}={\mu }_{H}M\left(t\right)-{\delta }_{H}H\left(t\right).\end{array}$$

(6)

Here, ${\mu }_H$ denotes the translation rate constant (in s⁻¹), $H\left(t\right)$ represents the concentration of the HMGCR enzyme (in molecules mL⁻¹), and ${\delta }_H$ corresponds to the degradation rate of HMGCR (in s⁻¹). The initial condition $H\left(0\right)=0$ reflects that no enzyme molecules are present at the initial time, consistent with a system in which translation has not yet been initiated.

Cholesterol C is produced by the HMGCR enzyme at a rate ${\mu }_C$ and is degraded at a rate ${\delta }_C$. Additionally, cholesterol forms inactive complexes with S, contributing to feedback inhibition:

$$\begin{array}{c}\hfill {\displaystyle \frac{dC\left(t\right)}{dt}}={\mu }_{C}H\left(t\right)+\iota k_{-2}{C}_b\left(t\right)-{\delta }_{C}C\left(t\right)-\iota k_2{C}^{\iota }\left(t\right)S\left(t\right).\end{array}$$

(7)

Here, ${\mu }_C$ denotes the cholesterol synthesis rate constant (in s⁻¹), ${\delta }_C$ is the degradation rate of cholesterol (in s⁻¹), and $\iota$ is the stoichiometric coefficient indicating the number of cholesterol molecules required to bind a single S molecule to form the inactive complex.

The right-hand sides of the system (1)–(7) are polynomial functions of the state variables and are therefore continuously differentiable on ${\mathbb{R}}^7$. Consequently, we consider the space $C^1({\mathbb{R}}_{\ge 0},{\mathbb{R}}^7)$ of all continuously differentiable functions that map time $t\in {\mathbb{R}}_{\ge 0}$ into ${\mathbb{R}}^7$.

Define the vector field $F:{\mathbb{R}}^7\to {\mathbb{R}}^7$ by

$${\displaystyle \frac{d\mathbf{x}}{dt}}=F\left(\mathbf{x}\right),\mathbf{x}={(G,S,S_b,M,H,C,C_b)}^{\top }.$$

(8)

where

$$F\left(\mathbf{x}\right)=\left(\begin{array}{c}{k}_{-1}{S}_b-k_1{S}^{\kappa }G\\ \kappa k_{-1}{S}_b-\kappa k_1{S}^{\kappa }G-k_2{C}^{\iota }S+k_{-2}{C}_b\\ -k_{-1}{S}_b+k_1{S}^{\kappa }G\\ {\mu }_m{S}_b-{\delta }_{m}M\\ {\mu }_{H}M-{\delta }_{H}H\\ {\mu }_{C}H+\iota k_{-2}{C}_b-{\delta }_{C}C-\iota k_2{C}^{\iota }S\\ k_{2}S{C}^{\iota }-k_{-2}{C}_b\end{array}\right).$$

(9)

In this context, all state variables are assumed to remain continuous and differentiable over time, reflecting the smooth evolution of the biochemical system under physiological conditions.

The initial concentration of unbound DNA, $G_0$, can be estimated using three different approaches:

• Direct Assumption ($S_0/\kappa$): This method assumes that the total transcription factors are evenly distributed among the available DNA binding sites. Since each DNA binding site requires $\kappa$ transcription factors, the total number of available binding sites is approximated as:

  $$G_0\approx {\displaystyle \frac{S_0}{\kappa }}$$
  This approach provides a straightforward estimate but does not account for the dynamic equilibrium conditions in transcriptional regulation.
• DNA Copy Number Approach: Building on this biological parallel, Milo and Phillips [25], p. 13 affirm that human somatic cells, including hepatocytes, are diploid and thus contain two copies of each gene. Accordingly, we assume $N_{\mathrm{gene}}=2$ for the HMGCR gene per nucleus. To convert gene copy number into a concentration (in molecules/mL), we divide by the volume of a single cell.
  Experimental work by Masyuk et al. [26] measured total liver volume in normal rats to be $10.22\pm 3.12$ mL. With an approximate hepatocyte count of $3\times {10}^9$ cells per liver [27], the mean volume of a single liver cell is estimated as:

  $$V_{\mathrm{cell}}={\displaystyle \frac{10.22\mathrm{mL}}{3\times {10}^9}}=(3.3\pm 1.0)\times {10}^{-12}\mathrm{mL}.$$
  The corresponding concentration of DNA binding sites per unit volume (in molecules/mL) is thus:

  $$G_0={\displaystyle \frac{N_{\mathrm{gene}}}{V_{\mathrm{cell}}}},$$

  where $N_{\mathrm{gene}}=2$ and $V_{\mathrm{cell}}=(3.3\pm 1.0)\times {10}^{-12}$ mL. Substituting these values yields:

  $$G_0=(6.06\pm 1.84)\times {10}^{11}\mathrm{molecules}/\mathrm{mL}.$$
  This range provides a conservative upper- and lower-bound estimate for $G_0$, offering a tractable method for model initialization when detailed equilibrium data are unavailable.
• Binding Equilibrium Approach: This method accounts for cooperative binding dynamics of transcription factors to DNA by incorporating the dissociation constant, given by $K_d={\displaystyle \frac{k_{-1}}{k_1}}$. The dynamics of the bound complex $S_b\left(t\right)$ are regulated by a nonlinear Equation (3) that is assumed to reach equilibrium rapidly (quasi-steady-state approximation). Therefore, we set ${\displaystyle \frac{d{\overline{S}}_b}{dt}}=0$, which leads to the steady-state relation:

  $${\overline{S}}_b={\displaystyle \frac{{\overline{S}}^{\kappa }\overline{G}}{K_d}}.$$
  Assuming $\overline{S}=S_0$, the expression becomes

  $${\overline{S}}_b={\displaystyle \frac{S_0^{\kappa }\overline{G}}{K_d}}.$$
  Using conservation of gene copy number and the initial condition $S_b\left(0\right)=0$, we impose the constraint

  $$G\left(0\right)+S_b\left(0\right)=G_0,\mathrm{so}\mathrm{that}\overline{G}=G_0-{\overline{S}}_b>0.$$
  Substituting this into the expression above yields

  $${\overline{S}}_b={\displaystyle \frac{S_0^{\kappa }(G_0-{\overline{S}}_b)}{K_d}},$$

  which can be rearranged to estimate:

  $$G_0={\overline{S}}_b\left({\displaystyle \frac{K_d+S_0^{\kappa }}{S_0^{\kappa }}}\right).$$
  This provides a biophysically realistic estimate of $G_0$, accounting for both cooperative binding and equilibrium dynamics.

The

Table 2. Comparison of three methods for estimating $G_0$.

Method	Formula	Estimated ${\mathit{G}}_0$ (Molecules mL−1)
Direct Assumption	$G_0\approx {\displaystyle \frac{S_0}{\kappa }}$	depends on $S_0$
DNA Copy Number Approach	$G_0={\displaystyle \frac{N_{\mathrm{gene}}}{V_{\mathrm{cell}}}}$	$(6.06\pm 1.84)\times {10}^{11}$
Binding Equilibrium Approach	$G_0={\overline{S}}_b{\displaystyle \frac{K_d+S_0^{\kappa }}{S_0^{\kappa }}}$	depends on ${\overline{S}}_b$, $S_0$, $K_d$

summarizes the three methods used to estimate $G_0$, highlighting their different assumptions and approaches. The Direct Assumption ($S_0/\kappa$) provides a quick estimation by assuming transcription factors distribute uniformly among available DNA binding sites. However, it does not account for biochemical binding kinetics.

The DNA Copy Number Approach estimates $G_0$ from the number of gene copies per cell, converted into a concentration by dividing by the average liver cell volume. This provides a biologically grounded estimate, assuming that all gene copies are equally accessible for binding. A limitation of this approach is that it may overestimate or underestimate the effective concentration of binding sites, since factors such as chromatin accessibility, transcriptional state, and cell-to-cell variability can decrease or increase the actual availability of gene copies for regulation.

Finally, the Binding Equilibrium Approach incorporates cooperative binding effects by considering the dissociation constant $K_d$, making it the most dynamically realistic estimation method. Since this method considers equilibrium binding kinetics, it provides a more accurate representation of the actual number of available unbound DNA sites in a real biological system.

3. Stability Analysis

The system (1)–(7) is an autonomous set of nonlinear ODEs for intracellular cholesterol regulation, with nonnegative initial data

$$G\left(0\right)=G_0,S\left(0\right)=S_0,S_b\left(0\right)=0,C_b\left(0\right)=0,M\left(0\right)=0,H\left(0\right)=0,C\left(0\right)=C_0.$$

(10)

It is important to note that the total concentration of the dynamic variables

$$T_C\left(t\right)=G\left(t\right)+S\left(t\right)+S_b\left(t\right)+M\left(t\right)+H\left(t\right)+C\left(t\right)+C_b\left(t\right)$$

(11)

is not conserved, because degradation acts as a sink and synthesis draws from upstream sources. Introduce nonnegative cumulative loss compartments

$${\dot{M}}_{\mathrm{los}}={\delta }_{m}M,{\dot{H}}_{\mathrm{los}}={\delta }_{H}H,{\dot{C}}_{\mathrm{los}}={\delta }_{C}C,M_{\mathrm{los}}\left(0\right)=H_{\mathrm{los}}\left(0\right)=C_{\mathrm{los}}\left(0\right)=0,$$

(12)

and nonnegative cumulative source production compartments

$${\dot{M}}_{\mathrm{src}}={\mu }_m{S}_b,{\dot{H}}_{\mathrm{src}}={\mu }_{H}M,{\dot{C}}_{\mathrm{src}}={\mu }_{C}H,M_{\mathrm{src}}\left(0\right)=H_{\mathrm{src}}\left(0\right)=C_{\mathrm{src}}\left(0\right)=0.$$

(13)

Define the augmented (closed) total

$$\begin{array}{cc}\hfill T_C^{\mathrm{closed}}\left(t\right)& :=G+S+(\kappa +1)S_b+C+(\iota +1)C_b+M+H\hfill \\ \hfill & +M_{\mathrm{los}}+H_{\mathrm{los}}+C_{\mathrm{los}}-\left(M_{\mathrm{src}}+H_{\mathrm{src}}+C_{\mathrm{src}}\right).\hfill \end{array}$$

(14)

Summing (1)–(7) with (12)–(13) cancels all internal exchanges ($G\leftrightarrow S_b$, $S\leftrightarrow C_b$) and balances each synthesis term by its source counter and each degradation term by its loss counter. Hence

$${\displaystyle \frac{d}{dt}}{T}_C^{\mathrm{closed}}\left(t\right)=0,T_C^{\mathrm{closed}}\left(t\right)\equiv T_C^{\mathrm{closed}}\left(0\right)=G_0+S_0+C_0.$$

(15)

Theorem 1.

Assume all parameters in (1)–(7) are nonnegative and the initial condition (10) satisfies $\mathbf{x}\left(0\right)\in {\mathbb{R}}_{\ge 0}^7$. Then the solution

$$\mathbf{x}\left(t\right)={(G,S,S_b,M,H,C,C_b)}^{\top }$$

exists and is unique on $[0,\infty )$, remains in ${\mathbb{R}}_{\ge 0}^7$ for all $t\ge 0$, and is uniformly bounded. In particular,

$$T_C\left(t\right)\le {\mathcal{T}}_{max}:=G_0+S_0+M^{★}+H^{★}+C^{★},\forall t\ge 0,$$

(16)

where the comparison-based constants are given by

$$M^{★}={\displaystyle \frac{{\mu }_m{G}_0}{{\delta }_m}},H^{★}={\displaystyle \frac{{\mu }_H{M}^{★}}{{\delta }_H}},$$

(17)

and

$$C^{★}=max\left\{C\left(0\right),{\displaystyle \frac{{\mu }_C{H}^{★}+\iota k_{-2}{S}_0}{{\delta }_C}}\right\}.$$

(18)

Proof. 

As previously noted, the right-hand sides of the system (1)–(7) are polynomial functions of the state variables and hence belong to $C^1([0,T_{max}),{\mathbb{R}}^7)$. Therefore, the vector field $F:{\mathbb{R}}^7\to {\mathbb{R}}^7$ defined in (9) is continuously differentiable and locally Lipschitz on ${\mathbb{R}}^7$. By the Picard–Lindelöf theorem (see, e.g., [28]), the initial value problem

$$\dot{\mathbf{x}}=F\left(\mathbf{x}\right),\mathbf{x}\left(0\right)={\mathbf{x}}_0,$$

(19)

with ${\mathbf{x}}_0\in {\mathbb{R}}_{\ge 0}^7$ admits a unique local solution $\mathbf{x}\left(t\right)\in C^1([0,T_{max}),{\mathbb{R}}^7)$.

We next show that the solution remains biologically meaningful, i.e., $\mathbf{x}\left(t\right)\in {\mathbb{R}}_{\ge 0}^7$ for all $t\ge 0$. Let the positive cone

$$K:={\mathbb{R}}_{\ge 0}^7=\{\mathbf{x}\in {\mathbb{R}}^7:x_i\ge 0,i=1,\dots ,7\}$$

denote the set of admissible (nonnegative) concentrations. To prove that K is forward invariant under (19), we employ a boundary first-exit argument.

Assume, for contradiction, that a trajectory starting in K leaves K. Define the first exit time

$$\tau :=inf\{t>0:\mathbf{x}(t)\notin K\}.$$

By continuity of $\mathbf{x}(·)$, we have $\mathbf{x}\left(\tau \right)\in K$, and there exists an index k such that $x_k\left(\tau \right)=0$ and $x_j\left(\tau \right)\ge 0$ for all $j\ne k$. Since $\mathbf{x}$ is continuously differentiable,

$${\dot{x}}_k\left(\tau \right)=F_k\left(\mathbf{x}\left(\tau \right)\right).$$

Evaluating $F_k$ at $\mathbf{x}\left(\tau \right)$ gives

$$\begin{array}{cccc}& G\left(\tau \right)=0\hfill & & \Rightarrow \dot{G}\left(\tau \right)=k_{-1}{S}_b\left(\tau \right)\ge 0,\hfill \\ & S\left(\tau \right)=0\hfill & & \Rightarrow \dot{S}\left(\tau \right)=\kappa k_{-1}{S}_b\left(\tau \right)+k_{-2}{C}_b\left(\tau \right)\ge 0,\hfill \\ & S_b\left(\tau \right)=0\hfill & & \Rightarrow {\dot{S}}_b\left(\tau \right)=k_1{S}^{\kappa }\left(\tau \right)G\left(\tau \right)\ge 0,\hfill \\ & C_b\left(\tau \right)=0\hfill & & \Rightarrow {\dot{C}}_b\left(\tau \right)=k_{2}S\left(\tau \right)C^{\iota }\left(\tau \right)\ge 0,\hfill \\ & M\left(\tau \right)=0\hfill & & \Rightarrow \dot{M}\left(\tau \right)={\mu }_m{S}_b\left(\tau \right)\ge 0,\hfill \\ & H\left(\tau \right)=0\hfill & & \Rightarrow \dot{H}\left(\tau \right)={\mu }_{H}M\left(\tau \right)\ge 0,\hfill \\ & C\left(\tau \right)=0\hfill & & \Rightarrow \dot{C}\left(\tau \right)={\mu }_{C}H\left(\tau \right)+\iota k_{-2}{C}_b\left(\tau \right)\ge 0.\hfill \end{array}$$

Hence, for every boundary point $\mathbf{x}\in \partial K$, the vector field satisfies $F\left(\mathbf{x}\right)\in T_K\left(\mathbf{x}\right)$, where

$$T_K\left(\mathbf{x}\right)=\{\mathbf{v}\in {\mathbb{R}}^7:v_i\ge 0\mathrm{whenever}{x}_i=0\}$$

is the tangent cone of K at $\mathbf{x}$. In particular, ${\dot{x}}_k\left(\tau \right)=F_k\left(\mathbf{x}\left(\tau \right)\right)\ge 0$. For sufficiently small $h>0$,

$$x_k(\tau +h)=x_k\left(\tau \right)+h{\dot{x}}_k\left(\tau \right)+o\left(h\right)\ge 0,$$

and all other components remain nonnegative by the definition of $\tau$. This contradicts the minimality of $\tau$. Therefore, no trajectory can leave K, and the positive cone is forward invariant:

$$\mathbf{x}\left(t\right)\in K\mathrm{for}\mathrm{all}t\in [0,T_{max}).$$

We have the conservation relations

$$G\left(t\right)+S_b\left(t\right)\equiv G_0,S\left(t\right)+\kappa S_b\left(t\right)+C_b\left(t\right)\equiv S_0,$$

which imply $0\le S_b\left(t\right)\le G_0$ and $0\le C_b\left(t\right)\le S_0$ for all $t\ge 0$. Taking into account the initial conditions $M\left(0\right)=0$, $H\left(0\right)=0$, and $C\left(0\right)=C_0$, we obtain the comparison bounds

$$\dot{M}\le {\mu }_m{G}_0-{\delta }_{m}M\Rightarrow M\left(t\right)\le M^{★}:={\displaystyle \frac{{\mu }_m{G}_0}{{\delta }_m}},$$

$$\dot{H}\le {\mu }_H{M}^{★}-{\delta }_{H}H\Rightarrow H\left(t\right)\le H^{★}:={\displaystyle \frac{{\mu }_H{M}^{★}}{{\delta }_H}},$$

$$\dot{C}\le {\mu }_C{H}^{★}+\iota k_{-2}{S}_0-{\delta }_{C}C\Rightarrow C\left(t\right)\le C^{★}:=max\left\{C_0,{\displaystyle \frac{{\mu }_C{H}^{★}+\iota k_{-2}{S}_0}{{\delta }_C}}\right\}.$$

Therefore,

$$\begin{array}{cc}\hfill T_C\left(t\right)& =G\left(t\right)+S\left(t\right)+S_b\left(t\right)+M\left(t\right)+H\left(t\right)+C\left(t\right)+C_b\left(t\right)\hfill \\ & =\left(G\left(t\right)+S_b\left(t\right)\right)+\left(S\left(t\right)+\kappa S_b\left(t\right)+C_b\left(t\right)\right)-\kappa S_b\left(t\right)-C_b\left(t\right)+M\left(t\right)+H\left(t\right)+C\left(t\right)\hfill \\ & =G_0+S_0-\kappa S_b\left(t\right)-C_b\left(t\right)+M\left(t\right)+H\left(t\right)+C\left(t\right)\hfill \\ & \le G_0+S_0+M^{★}+H^{★}+C^{★}=:{\mathcal{T}}_{max},\forall t\ge 0.\hfill \end{array}$$

Finally, the trajectory remains in the compact, positively invariant set

$$\mathcal{K}=\left\{\mathbf{x}\in {\mathbb{R}}_{\ge 0}^7:T_C\left(t\right)\le {\mathcal{T}}_{max}\right\},$$

on which the locally Lipschitz vector field F is bounded. By standard continuation arguments (cf. [28]), the solution extends globally, i.e., $T_{max}=\infty$. Hence, the system (1)–(7) admits a unique, globally defined solution that remains positive and bounded for all $t\ge 0$, thereby ensuring both mathematical well-posedness and biological feasibility. □

Definition 1.

Let $x\left(t\right)$ denote a solution of the system (1)–(7) with nonnegative initial conditions. The system is said to be uniformly persistent (or permanent) with respect to a subset of coordinates $\mathcal{I}\subseteq \{1,\dots ,7\}$ if there exist positive constants ε and ϖ such that every corresponding component satisfies

$$\underset{t\to \infty }{lim\; inf}{x}_i\left(t\right)\ge \epsilon and\underset{t\ge 0}{sup}{x}_i\left(t\right)\le \varpi ,\forall i\in \mathcal{I}.$$

In biological terms, the variables in $\mathcal{I}$ remain permanently active: they neither diverge nor decay to zero as time increases.

To prove the uniform persistence (permanence) of the model Equations (1)–(7), we make use of the following Fluctuation Lemma [29] and of a standard calculus lemma, for which we include a detailed proof for the reader’s convenience.

Lemma 1

([29]). Let $f:{\mathbb{R}}_{\ge 0}\to \mathbb{R}$ be continuously differentiable, and define

$$\mathsf{\ell }:=\underset{t\to \infty }{lim\; inf}f\left(t\right),L:=\underset{t\to \infty }{lim\; sup}f\left(t\right),with\mathsf{\ell }<L.$$

Then there exist sequences $\left\{t_k\right\}$ and $\left\{s_k\right\}$ tending to infinity such that

$$f^{\prime }\left(t_k\right)=0,f^{\prime }\left(s_k\right)=0,f\left(t_k\right)\to \mathsf{\ell },f\left(s_k\right)\to L.$$

Lemma 2.

Let $f\in C^1\left([0,\infty )\right)$ be a bounded function such that ${\displaystyle \underset{t\to \infty }{lim}f\left(t\right)=\mathsf{\ell }}$ for some $\mathsf{\ell }\in \mathbb{R}$. Then there exists a sequence $t_k\to \infty$ such that

$$f\left(t_k\right)\to \mathsf{\ell },f^{\prime }\left(t_k\right)\to 0ask\to \infty .$$

Proof. 

Since f has a finite limit ℓ as $t\to \infty$, it follows in particular that f is bounded on $[0,\infty )$, i.e., there exists $M>0$ such that $\left|f\right(t\left)\right|\le M$ for all $t\ge 0$. We first claim that for every $n\in \mathbb{N}$ there exists $t_n\ge n$ such that

$$\left|f^{\prime }\left(t_n\right)\right|\le {\displaystyle \frac{1}{n}}.$$

Suppose, towards a contradiction, that this claim is false. Then there exists some $n_0\in \mathbb{N}$ such that

$$|f^{\prime }\left(t\right)|>{\displaystyle \frac{1}{n_0}}\mathrm{for}\mathrm{all}t\ge n_0.$$

Define $\epsilon :=1/n_0>0$ and $T:=n_0$. Then

$$|f^{\prime }\left(t\right)|\ge \epsilon \mathrm{for}\mathrm{all}t\ge T.$$

(20)

We now show that $f^{\prime }$ cannot change sign on $[T,\infty )$. Indeed, if there existed $t_1,t_2\ge T$ with $f^{\prime }\left(t_1\right)>0$ and $f^{\prime }\left(t_2\right)<0$, then by continuity of $f^{\prime }$ there would exist some $t^{\ast }\in (t_1,t_2)$ such that $f^{\prime }\left(t^{\ast }\right)=0$. This contradicts (20), which states that $|f^{\prime }\left(t\right)|\ge \epsilon >0$ for all $t\ge T$. Hence the sign of $f^{\prime }$ is constant on $[T,\infty )$. There are now two cases.

• Case 1: $f^{\prime }\left(t\right)\ge \epsilon$ for all $t\ge T$. Then f is strictly increasing on $[T,\infty )$. For any $t\ge T$ we have

$$f\left(t\right)-f\left(T\right)={\int }_T^t{f}^{\prime }\left(s\right)ds\ge {\int }_T^t\epsilon ds=\epsilon (t-T).$$

Therefore,

$$f\left(t\right)\ge f\left(T\right)+\epsilon (t-T)\mathrm{for}\mathrm{all}t\ge T.$$

Letting $t\to \infty$ shows that $f\left(t\right)\to +\infty$, which contradicts the boundedness of f on $[0,\infty )$.

• Case 2: $f^{\prime }\left(t\right)\le -\epsilon$ for all $t\ge T$. Then f is strictly decreasing on $[T,\infty )$. For any $t\ge T$ we have

$$f\left(t\right)-f\left(T\right)={\int }_T^t{f}^{\prime }\left(s\right)ds\le {\int }_T^t(-\epsilon )ds=-\epsilon (t-T),$$

so

$$f\left(t\right)\le f\left(T\right)-\epsilon (t-T)\mathrm{for}\mathrm{all}t\ge T.$$

Letting $t\to \infty$ shows that $f\left(t\right)\to -\infty$, again contradicting the boundedness of f on $[0,\infty )$. Therefore, for each $n\in \mathbb{N}$ there exists $t_n\ge n$ such that

$$|f^{\prime }\left(t_n\right)|\le {\displaystyle \frac{1}{n}}.$$

Finally, relabelling the sequence $\left(t_n\right)$ as $\left(t_k\right)$ gives the desired sequence $t_k\to \infty$ with

$$f\left(t_k\right)\to \mathsf{\ell },f^{\prime }\left(t_k\right)\to 0,$$

which completes the proof. □

Theorem 2.

Assume the admissible initial conditions (10) with $G_0>0$, $S_0\ge 0$, $C_0>0$, and all parameters in (1)–(7) are nonnegative. Then every solution is uniformly bounded and

$$\underset{t\to \infty }{lim\; inf}M\left(t\right)>0,\underset{t\to \infty }{lim\; inf}H\left(t\right)>0,\underset{t\to \infty }{lim\; inf}C\left(t\right)>0.$$

Hence the model is uniformly persistent (permanent) in the compartments M, H, and C.

Proof. 

We first establish the uniform boundedness of all state variables in the system (1)–(7), and then show that each biologically relevant component remains uniformly positive for all time. By the total concentration identity for the dynamic variables, established in (16), we conclude that all state variables are uniformly bounded on $[0,\infty )$. We next prove the positivity of cholesterol $C\left(t\right)$. Suppose, toward a contradiction, that ${lim\; inf}_{t\to \infty }C\left(t\right)=0$. There are two cases.

Case A: ${lim\; inf}_{t\to \infty }C\left(t\right)=:c_{\mathsf{\ell }}<c_L:={lim\; sup}_{t\to \infty }C\left(t\right)$. In this case we consider the subcase $c_{\mathsf{\ell }}=0$. Applying the fluctuation Lemma 1, there exist sequences $t_k,s_k\to \infty$ with

$$\dot{C}\left(t_k\right)=0,C\left(t_k\right)\to c_{\mathsf{\ell }}=0,\dot{C}\left(s_k\right)=0,C\left(s_k\right)\to c_L.$$

Passing to a subsequence, we may assume that

$$H\left(t_k\right)\to H_{\ast },C_b\left(t_k\right)\to C_{b,\ast },S_b\left(t_k\right)\to S_{b,\ast },G\left(t_k\right)\to G_{\ast },S\left(t_k\right)\to S_{\ast }$$

as $k\to \infty$, where $H_{\ast },C_{b,\ast },S_{b,\ast },G_{\ast },S_{\ast }\ge 0$. Evaluating (7) at $t_k$ and letting $k\to \infty$ yields

$$0={\mu }_{C}H\left(t_k\right)+\iota k_{-2}{C}_b\left(t_k\right)-{\delta }_{C}C\left(t_k\right)-\iota k_2{C}^{\iota }\left(t_k\right)S\left(t_k\right)⟶{\mu }_C{H}_{\ast }+\iota k_{-2}{C}_{b,\ast },$$

hence $H_{\ast }=C_{b,\ast }=0$.

Using

$$\dot{M}={\mu }_m{S}_b-{\delta }_{m}M,\dot{H}={\mu }_{H}M-{\delta }_{H}H,$$

the variation in constant formulas implies that if $S_{b,\ast }>0$ then necessarily $H_{\ast }>0$, a contradiction; thus $S_{b,\ast }=0$. From the conservation $G+S_b\equiv G_0$ we obtain $G_{\ast }=G_0>0$. Moreover, by stationarity of $S_b$ at its local minima (we can choose $t_k$ so that $S_b$ attains a local minimum near $t_k$; boundedness ensures the existence of such minima with ${\dot{S}}_b=0$) we have

$$0={\dot{S}}_b=-k_{-1}{S}_b+k_1{S}^{\kappa }G\Rightarrow S_{\ast }=0.$$

Therefore, near $t_k$, Equation (7) reduces to $\dot{C}\approx -{\delta }_{C}C$, which is strictly negative for $C>0$, contradicting $\dot{C}\left(t_k\right)=0$. Hence Case A is impossible.

Case B: ${\displaystyle \underset{t\to \infty }{lim}C\left(t\right)=0}$. Since all state variables remain bounded and the right-hand side of the system is smooth, the solution $\mathbf{x}\left(t\right)=(G,S,S_b,M,H,C,C_b)\left(t\right)$ is continuously differentiable on $[0,\infty )$ and all its components are bounded. Moreover, C is bounded and continuously differentiable with $C\left(t\right)\to 0$ as $t\to \infty$. Thus, by Lemma 2 (applied with $f\left(t\right)=C\left(t\right)$ and $\mathsf{\ell }=0$), there exists a sequence ${\sigma }_k\to \infty$ such that

$$C\left({\sigma }_k\right)\to 0,\dot{C}\left({\sigma }_k\right)\to 0\mathrm{as}k\to \infty .$$

Since H, $C_b$ and $S_b$ are bounded, we may, after passing to a subsequence (not relabelled), assume that

$$H\left({\sigma }_k\right)\to H_{\ast },C_b\left({\sigma }_k\right)\to C_{b,\ast },S_b\left({\sigma }_k\right)\to S_{b,\ast },$$

for some limits $H_{\ast },C_{b,\ast },S_{b,\ast }\ge 0$. From the conservation law $G\left(t\right)+S_b\left(t\right)\equiv G_0$ it follows that

$$G\left({\sigma }_k\right)=G_0-S_b\left({\sigma }_k\right)⟶G_{\ast }:=G_0-S_{b,\ast }\ge 0.$$

Next, evaluate the cholesterol Equation (7) at $t={\sigma }_k$:

$$\dot{C}\left({\sigma }_k\right)={\mu }_{C}H\left({\sigma }_k\right)+\iota k_{-2}{C}_b\left({\sigma }_k\right)-{\delta }_{C}C\left({\sigma }_k\right)-\iota k_2{C}^{\iota }\left({\sigma }_k\right)S\left({\sigma }_k\right).$$

Rewriting, we obtain

$${\mu }_{C}H\left({\sigma }_k\right)+\iota k_{-2}{C}_b\left({\sigma }_k\right)={\delta }_{C}C\left({\sigma }_k\right)+\iota k_2{C}^{\iota }\left({\sigma }_k\right)S\left({\sigma }_k\right)+\dot{C}\left({\sigma }_k\right).$$

Letting $k\to \infty$ and using $C\left({\sigma }_k\right)\to 0$, $\dot{C}\left({\sigma }_k\right)\to 0$ and boundedness of S, we obtain

$${\mu }_C{H}_{\ast }+\iota k_{-2}{C}_{b,\ast }=\underset{k\to \infty }{lim}\left({\delta }_{C}C\left({\sigma }_k\right)+\iota k_2{C}^{\iota }\left({\sigma }_k\right)S\left({\sigma }_k\right)+\dot{C}\left({\sigma }_k\right)\right)=0,$$

which, together with nonnegativity, implies

$$H_{\ast }=0,C_{b,\ast }=0.$$

Moreover, from the $C_b$ equation

$${\dot{C}}_b\left(t\right)=k_{2}S\left(t\right)C^{\iota }\left(t\right)-k_{-2}{C}_b\left(t\right),$$

and the facts that $C\left(t\right)\to 0$ while S is bounded, the inhomogeneous term $k_{2}S\left(t\right)C^{\iota }\left(t\right)$ tends to 0 as $t\to \infty$. By the variation-of-constants formula for this linear ODE with constant negative coefficient $-k_{-2}<0$, we obtain

$$C_b\left(t\right)\to 0\mathrm{as}t\to \infty .$$

Now consider the positive linear chain of equations

$$\dot{M}={\mu }_m{S}_b-{\delta }_{m}M,\dot{H}={\mu }_{H}M-{\delta }_{H}H.$$

Using the variation-of-constants formula, if ${lim\; inf}_{t\to \infty }{S}_b\left(t\right)>0$ then $H\left(t\right)$ would admit a positive lower bound as $t\to \infty$, contradicting the fact that $H\left({\sigma }_k\right)\to 0$. Hence

$$\underset{t\to \infty }{lim\; inf}{S}_b\left(t\right)=0.$$

We now distinguish two possibilities for the asymptotic behaviour of $S_b$.

• Subcase B1: ${\displaystyle \underset{t\to \infty }{lim\; sup}{S}_b\left(t\right)>0}$.

In this case, set

$$s_{\mathsf{\ell }}:=\underset{t\to \infty }{lim\; inf}{S}_b\left(t\right)=0,s_L:=\underset{t\to \infty }{lim\; sup}{S}_b\left(t\right)>0.$$

Applying the Fluctuation Lemma 1 to the function $t\mapsto S_b\left(t\right)$ yields a sequence $t_k\to \infty$ such that

$${\dot{S}}_b\left(t_k\right)=0,S_b\left(t_k\right)\to s_{\mathsf{\ell }}=0\mathrm{as}k\to \infty .$$

Since all state variables are bounded, we may, after passing to a subsequence (not relabelled), assume that

$$S\left(t_k\right)\to S_{\infty },G\left(t_k\right)\to G_{\infty }$$

for some $S_{\infty },G_{\infty }\ge 0$. From the conservation law $G\left(t\right)+S_b\left(t\right)\equiv G_0$ and $S_b\left(t_k\right)\to 0$ we obtain

$$G_{\infty }=G_0.$$

Evaluating the $S_b$ equation

$${\dot{S}}_b=-k_{-1}{S}_b+k_1{S}^{\kappa }G$$

at $t=t_k$ and using ${\dot{S}}_b\left(t_k\right)=0$ gives

$$0=-k_{-1}{S}_b\left(t_k\right)+k_1{S}^{\kappa }\left(t_k\right)G\left(t_k\right).$$

Passing to the limit $k\to \infty$ yields

$$0=k_1{S}_{\infty }^{\kappa }{G}_0.$$

Since $k_1>0$ and $G_0>0$, we conclude that $S_{\infty }=0.$ On the other hand, from the second conservation relation

$$S\left(t\right)+\kappa S_b\left(t\right)+C_b\left(t\right)\equiv S_0$$

and the already established facts $S_b\left(t_k\right)\to 0$ and $C_b\left(t_k\right)\to 0$, we obtain $S_0=S_{\infty }=0$, which contradicts the admissible initial condition $S_0>0$. Hence Subcase B1 cannot occur.

• Subcase B2: ${\displaystyle \underset{t\to \infty }{lim\; sup}{S}_b\left(t\right)=0}$.

Then $S_b\left(t\right)\to 0$ as $t\to \infty$. Since $S_b$ is bounded and continuously differentiable, we can apply Lemma 2 with $f\left(t\right)=S_b\left(t\right)$ and $\mathsf{\ell }=0$ to find a sequence $t_k\to \infty$ such that

$$S_b\left(t_k\right)\to 0,{\dot{S}}_b\left(t_k\right)\to 0\mathrm{as}k\to \infty .$$

As before, boundedness allows us to assume (passing to a subsequence if necessary) that

$$S\left(t_k\right)\to S_{\infty },G\left(t_k\right)\to G_{\infty }.$$

From $G\left(t\right)+S_b\left(t\right)\equiv G_0$ and $S_b\left(t_k\right)\to 0$ we again obtain $G_{\infty }=G_0$. Evaluating the $S_b$ equation at $t=t_k$,

$${\dot{S}}_b\left(t_k\right)=-k_{-1}{S}_b\left(t_k\right)+k_1{S}^{\kappa }\left(t_k\right)G\left(t_k\right),$$

and letting $k\to \infty$ gives

$$0=k_1{S}_{\infty }^{\kappa }{G}_0,$$

so that $S_{\infty }=0$. On the other hand, using

$$S\left(t\right)+\kappa S_b\left(t\right)+C_b\left(t\right)\equiv S_0$$

and the convergence $S_b\left(t_k\right)\to 0$, $C_b\left(t_k\right)\to 0$, we obtain again $S_{\infty }=S_0=0$, and giving the same contradiction. Since both Subcase B1 and Subcase B2 are impossible, Case B cannot occur. Therefore

$$\underset{t\to \infty }{lim\; inf}C\left(t\right)>0.$$

We now show that $H\left(t\right)$ and $M\left(t\right)$ also remain permanently positive. Set

$${\epsilon }_C:=\frac{1}{2}\underset{t\to \infty }{lim\; inf}C\left(t\right)>0.$$

By the definition of the liminf, there exists ${\sigma }_C>0$ such that

$$C\left(t\right)\ge {\epsilon }_C\mathrm{for}\mathrm{all}t\ge {\sigma }_C.$$

(21)

Since C is bounded and continuously differentiable on $[0,\infty )$, there exists $L_C>0$ with

$$|\dot{C}\left(t\right)|\le L_C\mathrm{for}\mathrm{all}t\ge 0.$$

We use the following elementary selection fact: for every bounded $C^1$ function $f:{\mathbb{R}}_{\ge 0}\to \mathbb{R}$ there exists a sequence $t_k\to \infty$ such that $f^{\prime }\left(t_k\right)\to 0$ as $k\to \infty$. For completeness, we recall the argument. For each integer $n\ge 0$, the mean value theorem applied to $[n,n+1]$ yields a point ${\tau }_n\in [n,n+1]$ with

$$f^{\prime }\left({\tau }_n\right)={\displaystyle \frac{f(n+1)-f\left(n\right)}{1}}=f(n+1)-f\left(n\right).$$

Since the sequence ${\left\{f\left(n\right)\right\}}_{n\in \mathbb{N}}$ is bounded, it has a Cauchy subsequence $f\left(n_j\right)$. For this subsequence we have $f(n_j+1)-f\left(n_j\right)\to 0$, and hence $f^{\prime }\left({\tau }_{n_j}\right)\to 0$ as $j\to \infty$. Setting $t_k:={\tau }_{n_k}$ gives the desired sequence. Applying this fact with $f\left(t\right)=C\left(t\right)$, we obtain a sequence $t_k\to \infty$ such that

$$\dot{C}\left(t_k\right)\to 0\mathrm{as}k\to \infty .$$

We may assume $t_k\ge {\sigma }_C$ for all k, so by (21) we have

$$C\left(t_k\right)\ge {\epsilon }_C\mathrm{for}\mathrm{all}k.$$

Since all state variables are bounded, we may, after passing to a subsequence (not relabelled), assume that

$$C\left(t_k\right)\to C_{\ast },S\left(t_k\right)\to S_{\ast },H\left(t_k\right)\to H_{\ast },C_b\left(t_k\right)\to C_{b,\ast }$$

for some limits $C_{\ast },S_{\ast },H_{\ast },C_{b,\ast }\ge 0$. In particular, $C_{\ast }\ge {\epsilon }_C$.

Evaluating the cholesterol Equation (7) at $t=t_k$ gives

$${\mu }_{C}H\left(t_k\right)+\iota k_{-2}{C}_b\left(t_k\right)={\delta }_{C}C\left(t_k\right)+\iota k_2{C}^{\iota }\left(t_k\right)S\left(t_k\right)+\dot{C}\left(t_k\right).$$

Letting $k\to \infty$ and using $\dot{C}\left(t_k\right)\to 0$, we obtain

$$\underset{k\to \infty }{lim}\left({\mu }_{C}H\left(t_k\right)+\iota k_{-2}{C}_b\left(t_k\right)\right)={\delta }_C{C}_{\ast }+\iota k_2{C}_{\ast }^{\iota }{S}_{\ast }\ge {\delta }_C{C}_{\ast }\ge {\delta }_C{\epsilon }_C.$$

Hence we can choose

$${\gamma }_0:=\frac{1}{2}{\delta }_C{\epsilon }_C>0$$

and find $k_0$ such that

$${\mu }_{C}H\left(t_k\right)+\iota k_{-2}{C}_b\left(t_k\right)\ge {\gamma }_0\mathrm{for}\mathrm{all}k\ge k_0.$$

(22)

Next we extend this pointwise lower bound to short time intervals. Since H and $C_b$ are bounded and continuously differentiable, their derivatives are bounded: there exists $L_{HC}>0$ such that

$$|\dot{H}\left(t\right)|\le L_{HC},\left|{\dot{C}}_b\left(t\right)\right|\le L_{HC}\mathrm{for}\mathrm{all}t\ge 0.$$

Thus the combined input

$$I\left(t\right):={\mu }_{C}H\left(t\right)+\iota k_{-2}{C}_b\left(t\right)$$

satisfies

$$|\dot{I}\left(t\right)|\le L_I\mathrm{for}\mathrm{all}t\ge 0$$

for some $L_I>0$. Choose

$$\tau :={\displaystyle \frac{{\gamma }_0}{2L_I}}>0.$$

For any $k\ge k_0$ and any $t\in [t_k-\tau ,t_k+\tau ]$ we have

$$I\left(t\right)-I\left(t_k\right)={\int }_{t_k}^t\dot{I}\left(\xi \right)d\xi ,$$

and therefore

$$|I\left(t\right)-I\left(t_k\right)|\le {\int }_{t_k}^t|\dot{I}\left(\xi \right)|d\xi \le L_I|t-t_k|\le L_I\tau ={\displaystyle \frac{{\gamma }_0}{2}}.$$

Combining this with (22), which gives $I\left(t_k\right)\ge {\gamma }_0$ for all $k\ge k_0$, we obtain

$$I\left(t\right)\ge I\left(t_k\right)-|I\left(t\right)-I\left(t_k\right)|\ge {\gamma }_0-{\displaystyle \frac{{\gamma }_0}{2}}={\displaystyle \frac{{\gamma }_0}{2}},t\in [t_k-\tau ,t_k+\tau ],k\ge k_0.$$

In other words, we have constructed an unbounded sequence of intervals $[t_k-\tau ,t_k+\tau ]$ such that

$${\mu }_{C}H\left(t\right)+\iota k_{-2}{C}_b\left(t\right)=I\left(t\right)\ge {\displaystyle \frac{{\gamma }_0}{2}}\mathrm{for}\mathrm{all}t\in [t_k-\tau ,t_k+\tau ],k\ge k_0.$$

We now use this to obtain a recurrent positive forcing from $S_b$ into the $(M,H)$ chain of equations

$$\dot{M}={\mu }_m{S}_b-{\delta }_{m}M,\dot{H}={\mu }_{H}M-{\delta }_{H}H.$$

Suppose, toward a contradiction, that at least one of M or H is not permanent, that is,

$$\underset{t\to \infty }{lim\; inf}M\left(t\right)=0\mathrm{or}\underset{t\to \infty }{lim\; inf}H\left(t\right)=0.$$

If ${lim\; inf}_{t\to \infty }M\left(t\right)>0$ then the variation-of-constants formula for H implies ${lim\; inf}_{t\to \infty }H\left(t\right)>0$ as well. Hence non-permanence of M or H implies

$$\underset{t\to \infty }{lim\; inf}M\left(t\right)=0\mathrm{and}\underset{t\to \infty }{lim\; inf}H\left(t\right)=0.$$

Therefore there exist sequences $\left\{r_j\right\}$ and $\left\{{\rho }_j\right\}$ with $r_j\to \infty$, ${\rho }_j\to \infty$ such that

$$M\left(r_j\right)\to 0,H\left({\rho }_j\right)\to 0\mathrm{as}j\to \infty .$$

Since M and H are bounded and continuously differentiable, their derivatives are bounded: there exists $L_{MH}>0$ such that

$$|\dot{M}\left(t\right)|\le L_{MH},\left|\dot{H}\left(t\right)\right|\le L_{MH}\mathrm{for}\mathrm{all}t\ge 0.$$

Using $r_j\to \infty$ and ${\rho }_j\to \infty$, we can, after passing to subsequences and relabelling, arrange that

$$|r_j-{\rho }_j|\le {\delta }_0$$

for some fixed ${\delta }_0>0$ and all j. Then

$$|M\left({\rho }_j\right)-M\left(r_j\right)|\le L_{MH}|r_j-{\rho }_j|\le L_{MH}{\delta }_0,|H\left(r_j\right)-H\left({\rho }_j\right)|\le L_{MH}|r_j-{\rho }_j|\le L_{MH}{\delta }_0.$$

Choosing ${\delta }_0$ so small that $L_{MH}{\delta }_0<1$, and using $M\left(r_j\right)\to 0$, $H\left({\rho }_j\right)\to 0$, we obtain (after a further subsequence if necessary) a single sequence, still denoted $r_j$, such that

$$M\left(r_j\right)\to 0,H\left(r_j\right)\to 0\mathrm{as}j\to \infty .$$

On the other hand, assume that there exists a sequence $s_j\to \infty$ such that $S_b\left(s_j\right)\to 0$ and $S_b\left(t\right)$ remains small on whole neighbourhoods of each $s_j$. Since $S_b$ is bounded and continuously differentiable, there is $L_S>0$ such that $|{\dot{S}}_b\left(t\right)|\le L_S$ for all $t\ge 0$. Hence, for any fixed $\eta >0$ we can find ${\tau }_S>0$ such that

$$S_b\left(t\right)\le \eta \mathrm{for}\mathrm{all}t\in [s_j,s_j+{\tau }_S]$$

whenever j is large enough. In particular, on $[s_j,s_j+{\tau }_S]$ the forcing term ${\mu }_m{S}_b$ in the M equation is bounded by ${\mu }_m\eta$.

Because M, H, and $S_b$ have bounded derivatives, there exists $L>0$ such that

$$|\dot{M}\left(t\right)|,|\dot{H}\left(t\right)|,|{\dot{S}}_b\left(t\right)|\le L\mathrm{for}\mathrm{all}t\ge 0.$$

Since $r_j\to \infty$ and $s_j\to \infty$, we can, after passing to subsequences and relabelling, arrange that

$$|r_j-s_j|\le \delta$$

for some fixed $\delta >0$ and all j, where $\delta$ is chosen so small that $L\delta <\eta$. Then

$$|M\left(s_j\right)-M\left(r_j\right)|\le L|s_j-r_j|\le L\delta <\eta ,|H\left(s_j\right)-H\left(r_j\right)|\le L|s_j-r_j|\le L\delta <\eta .$$

Since $M\left(r_j\right)\to 0$ and $H\left(r_j\right)\to 0$, it follows that

$$M\left(s_j\right)\to 0,H\left(s_j\right)\to 0\mathrm{as}j\to \infty .$$

On such an interval $[s_j,s_j+{\tau }_S]$ the equation

$$\dot{M}={\mu }_m{S}_b-{\delta }_{m}M$$

together with the variation-of-constants formula

$$M\left(t\right)=M\left(s_j\right)e^{-{\delta }_m(t-s_j)}+{\mu }_m{\int }_{s_j}^t{e}^{-{\delta }_m(t-\xi )}{S}_b\left(\xi \right)d\xi$$

shows that

$$0\le M\left(t\right)\le M\left(s_j\right)e^{-{\delta }_m(t-s_j)}+{\mu }_m\eta {\int }_{s_j}^t{e}^{-{\delta }_m(t-\xi )}d\xi \le M\left(s_j\right)+{\displaystyle \frac{{\mu }_m}{{\delta }_m}}\eta$$

for all $t\in [s_j,s_j+{\tau }_S]$. Since $M\left(s_j\right)\to 0$ and $\eta >0$ is arbitrary, we may choose $\eta$ small and then j large so that $M\left(t\right)$ is as small as we wish on $[s_j,s_j+{\tau }_S]$.

In turn, from

$$\dot{H}={\mu }_{H}M-{\delta }_{H}H$$

and the corresponding variation-of-constants formula,

$$H\left(t\right)=H\left(s_j\right)e^{-{\delta }_H(t-s_j)}+{\mu }_H{\int }_{s_j}^t{e}^{-{\delta }_H(t-\xi )}M\left(\xi \right)d\xi ,$$

we obtain that $H\left(t\right)$ can also be made arbitrarily small on $[s_j,s_j+{\tau }_S]$ for large j, since both $H\left(s_j\right)\to 0$ and $M\left(t\right)$ is small there.

Moreover, from the conservation law

$$S\left(t\right)+\kappa S_b\left(t\right)+C_b\left(t\right)\equiv S_0$$

and the fact that $S_b$ is small on $[s_j,s_j+{\tau }_S]$, we deduce that $C_b\left(t\right)\le S_0$ on these intervals. Hence, on $[s_j,s_j+{\tau }_S]$ and for large j, both $H\left(t\right)$ and $C_b\left(t\right)$ are uniformly small. Consequently, the combined input

$$I\left(t\right)={\mu }_{C}H\left(t\right)+\iota k_{-2}{C}_b\left(t\right)$$

is uniformly small on $[s_j,s_j+{\tau }_S]$. However, the intervals $[t_k-\tau ,t_k+\tau ]$ on which $I\left(t\right)\ge {\gamma }_0/2$ are unbounded in time. For j sufficiently large, the intervals $[s_j,s_j+{\tau }_S]$ and $[t_k-\tau ,t_k+\tau ]$ must overlap for infinitely many k. This yields a contradiction, since on the overlap we cannot have simultaneously $I\left(t\right)$ arbitrarily small (by the behaviour of $S_b$, M, and H) and $I\left(t\right)\ge {\gamma }_0/2$. Therefore our assumption was false: $S_b$ cannot remain arbitrarily small on arbitrarily long time intervals for large t. In particular, there exists ${\epsilon }_b>0$ and an unbounded sequence of times ${\left({\theta }_k\right)}_{k\in \mathbb{N}}$ such that

$$S_b\left({\theta }_k\right)\ge {\epsilon }_b>0\mathrm{for}\mathrm{all}k.$$

(23)

Thus $S_b$ provides a recurrent, strictly positive forcing into the linear equations

$$\dot{M}={\mu }_m{S}_b-{\delta }_{m}M,\dot{H}={\mu }_{H}M-{\delta }_{H}H.$$

By comparison for this linear system, the recurrent positive forcing from $S_b$ implies strictly positive lower bounds for $M\left(t\right)$ and $H\left(t\right)$ as $t\to \infty$. Indeed, from the recurrent lower bound (23) and the variation-of-constants formulas, one can find constants $m_{\ast }>0$ and $h_{\ast }>0$ such that

$$\underset{t\to \infty }{lim\; inf}M\left(t\right)\ge m_{\ast },\underset{t\to \infty }{lim\; inf}H\left(t\right)\ge h_{\ast }.$$

Together with ${lim\; inf}_{t\to \infty }C\left(t\right)>0$, this shows that M, H, and C are uniformly persistent, and completes the proof of Theorem 2. □

Remark 1

(Non-persistent states). The states G, S, $S_b$, and $C_b$ are nonnegative and uniformly bounded but need not be uniformly persistent. Indeed,

$$G\left(t\right)+S_b\left(t\right)\equiv G_0,S\left(t\right)+\kappa S_b\left(t\right)+C_b\left(t\right)\equiv S_0,$$

so the total mass in each group is fixed while its distribution may shift. Consequently, trajectories with $S_b\left(t\right)\to 0$ (hence $G\left(t\right)\to G_0$) or with $C_b\left(t\right)\to 0$ (hence $S\left(t\right)\to S_0$) are compatible with boundedness, and thus ${lim\; inf}_{t\to \infty }G\left(t\right)$, ${lim\; inf}_{t\to \infty }S\left(t\right)$, ${lim\; inf}_{t\to \infty }{S}_b\left(t\right)$, or ${lim\; inf}_{t\to \infty }{C}_b\left(t\right)$ can be zero.

Note that the positivity of $S_b$ is not part of the persistence statement; it is used only as an intermediate estimate to ensure a strictly positive forcing in the linear chain

$$\dot{M}={\mu }_m{S}_b-{\delta }_{m}M,\dot{H}={\mu }_{H}M-{\delta }_{H}H,$$

which is essential to conclude ${lim\; inf}_{t\to \infty }M\left(t\right)>0$ and ${lim\; inf}_{t\to \infty }H\left(t\right)>0$.

The model Equations (1)–(7) can be written in vector form as

$$\begin{array}{ccc}\hfill {\displaystyle \frac{d\mathbf{x}\left(t\right)}{dt}}& =& \mathbf{J}\mathbf{x}\left(t\right)+f\left(\mathbf{x}\right),\hfill \end{array}$$

(24)

where

$$\mathbf{J}=\left(\begin{array}{ccccccc}0& 0& k_{-1}& 0& 0& 0& 0\\ 0& 0& \kappa k_{-1}& 0& 0& 0& k_{-2}\\ 0& 0& -k_{-1}& 0& 0& 0& 0\\ 0& 0& {\mu }_m& -{\delta }_m& 0& 0& 0\\ 0& 0& 0& {\mu }_H& -{\delta }_H& 0& 0\\ 0& 0& 0& 0& {\mu }_C& -{\delta }_C& \iota k_{-2}\\ 0& 0& 0& 0& 0& 0& -k_{-2}\end{array}\right),f\left(\mathbf{x}\right)=\left(\begin{array}{c}-k_1{S}^{\kappa }G\\ -\kappa k_1{S}^{\kappa }G-k_2{C}^{\iota }S\\ k_1{S}^{\kappa }G\\ 0\\ 0\\ -\iota k_2{C}^{\iota }S\\ k_2{C}^{\iota }S\end{array}\right).$$

To study stability via M-matrix techniques, we first characterize the biologically relevant equilibrium of (1)–(7), and then apply a general theorem on M-matrices. Let

$$\overline{\mathbf{x}}={\left(\overline{G},\overline{S},{\overline{S}}_b,\overline{M},\overline{H},\overline{C},{\overline{C}}_b\right)}^{\top }$$

denote an equilibrium of (1)–(7), i.e.,

$${\displaystyle \frac{d\overline{\mathbf{x}}}{dt}}=0.$$

(25)

In particular, for the active complex (3) we have at equilibrium

$${\displaystyle \frac{d{\overline{S}}_b}{dt}}=0.$$

(26)

Using the gene conservation law

$$\overline{G}+{\overline{S}}_b=G_0⟹\overline{G}=G_0-{\overline{S}}_b>0,$$

(27)

and the steady–state relation from (3),

$$0=-k_{-1}{\overline{S}}_b+k_1{\overline{S}}^{\kappa }\overline{G},$$

we obtain

$${\overline{S}}_b={\displaystyle \frac{(k_1/k_{-1}){\overline{S}}^{\kappa }{G}_0}{1+(k_1/k_{-1}){\overline{S}}^{\kappa }}}.$$

(28)

The downstream balances in (5)–(7) give

$$\overline{M}={\displaystyle \frac{{\mu }_m}{{\delta }_m}}{\overline{S}}_b,\overline{H}={\displaystyle \frac{{\mu }_H}{{\delta }_H}}\overline{M},\overline{C}={\displaystyle \frac{{\mu }_C}{{\delta }_C}}\overline{H}=\gamma {\overline{S}}_b,\gamma :={\displaystyle \frac{{\mu }_m{\mu }_H{\mu }_C}{{\delta }_m{\delta }_H{\delta }_C}}>0.$$

(29)

From the inactive–complex Equation (4) we likewise obtain

$$0=k_2{\overline{C}}^{\iota }\overline{S}-k_{-2}{\overline{C}}_b⟹{\overline{C}}_b={\displaystyle \frac{k_2}{k_{-2}}}{\overline{C}}^{\iota }\overline{S}={\displaystyle \frac{k_2}{k_{-2}}}{\left(\gamma {\overline{S}}_b\right)}^{\iota }\overline{S}.$$

(30)

Crucially, ${\overline{C}}_b$ closes the steady state via the SREBP-2 pool conservation

$$\overline{S}+\kappa {\overline{S}}_b+{\overline{C}}_b=S_0,$$

(31)

which, in view of (28)–(30), reduces the equilibrium conditions to a single scalar equation for $\overline{S}$. Writing S for the unknown equilibrium value of $\overline{S}$, we define

$$\Phi \left(S\right):=S+\kappa S_b\left(S\right)+C_b\left(S_b\left(S\right),S\right)-S_0,$$

(32)

where

$$S_b\left(S\right)={\displaystyle \frac{(k_1/k_{-1})S^{\kappa }{G}_0}{1+(k_1/k_{-1})S^{\kappa }}},C_b\left(S\right)={\displaystyle \frac{k_2}{k_{-2}}}{\left(\gamma S_b\left(S\right)\right)}^{\iota }S.$$

Lemma 3.

Assume $G_0>0$, $S_0>0$, and that all parameters in (1)–(7) are positive. Then the scalar Equation (32) admits a unique root $S^{★}>0$. Consequently, the equilibrium

$${\overline{\mathbf{x}}}^{★}=\left(G_0-S_b\left(S^{★}\right),S^{★},S_b\left(S^{★}\right),{\overline{M}}^{★},{\overline{H}}^{★},{\overline{C}}^{★},{\overline{C}}_b^{★}\right)$$

determined by (28)–(30) is componentwise positive and unique.

Proof. 

For $S>0$ we recall

$$S_b\left(S\right)={\displaystyle \frac{(k_1/k_{-1})S^{\kappa }{G}_0}{1+(k_1/k_{-1})S^{\kappa }}},C_b\left(S\right)={\displaystyle \frac{k_2}{k_{-2}}}{\left(\gamma S_b\left(S\right)\right)}^{\iota }S,$$

and

$$\Phi \left(S\right)=S+\kappa S_b\left(S\right)+C_b\left(S\right)-S_0.$$

We first verify continuity of $\Phi$ and its limits at 0 and $+\infty$. Since $S_b\left(S\right)$ and $C_b\left(S\right)$ are obtained by algebraic operations and compositions involving $S>0$, both functions are continuous on $(0,\infty )$, and hence $\Phi$ is continuous on $(0,\infty )$.

From (28) and (30), we have

$$\underset{S\downarrow 0}{lim}\Phi \left(S\right)=-S_0<0.$$

By continuity we may extend $\Phi$ continuously to $S=0$ by setting $\Phi \left(0\right):=-S_0$.

Next, we consider the behavior as $S\to \infty$. From (28) we obtain

$$S_b\left(S\right)={\displaystyle \frac{(k_1/k_{-1})S^{\kappa }{G}_0}{1+(k_1/k_{-1})S^{\kappa }}}\nearrow G_0\mathrm{as}S\to \infty .$$

Hence,

$$C_b\left(S\right)={\displaystyle \frac{k_2}{k_{-2}}}{\left(\gamma S_b\left(S\right)\right)}^{\iota }S\sim {\displaystyle \frac{k_2}{k_{-2}}}{\left(\gamma G_0\right)}^{\iota }S\mathrm{as}S\to \infty .$$

It follows that

$$\Phi \left(S\right)=S+\kappa S_b\left(S\right)+C_b\left(S\right)-S_0⟶+\infty \mathrm{as}S\to \infty .$$

We now show that $\Phi$ is strictly increasing on $(0,\infty )$. Introduce the positive constants

$$a_0:={\displaystyle \frac{k_1}{k_{-1}}}{G}_0>0,b_0:={\displaystyle \frac{k_1}{k_{-1}}}>0,$$

so that

$$S_b\left(S\right)={\displaystyle \frac{a_0{S}^{\kappa }}{1+b_0{S}^{\kappa }}},S>0.$$

Differentiating gives

$$\begin{array}{cc}\hfill S_b^{\prime }\left(S\right)& ={\displaystyle \frac{a_0\kappa S^{\kappa -1}(1+b_0{S}^{\kappa })-a_0{S}^{\kappa }\left(b_0\kappa S^{\kappa -1}\right)}{{(1+b_0{S}^{\kappa })}^2}}\hfill \\ & ={\displaystyle \frac{a_0\kappa S^{\kappa -1}}{{(1+b_0{S}^{\kappa })}^2}}.\hfill \end{array}$$

Since $a_0>0$, $\kappa \ge 1$ and $S>0$, we have

$$S_b^{\prime }\left(S\right)>0\mathrm{for}\mathrm{all}S>0,$$

so $S_b$ is strictly increasing. It is convenient to rewrite $C_b$ as

$$C_b\left(S\right)=d_{0}S{\left[S_b\left(S\right)\right]}^{\iota },d_0:={\displaystyle \frac{k_2}{k_{-2}}}{\gamma }^{\iota }>0.$$

Differentiating and applying the product and chain rules yields

$$\begin{array}{cc}\hfill C_b^{\prime }\left(S\right)& =d_0\left({\left[S_b\left(S\right)\right]}^{\iota }+S\iota {\left[S_b\left(S\right)\right]}^{\iota -1}{S}_b^{\prime }\left(S\right)\right).\hfill \end{array}$$

Thus $C_b$ is strictly increasing on $(0,\infty )$. Since $S_b^{\prime }\left(S\right)>0$ and $C_b^{\prime }\left(S\right)>0$ for all $S>0$, it follows that

$${\Phi }^{\prime }\left(S\right)>1\mathrm{for}\mathrm{all}S>0,$$

and therefore $\Phi$ is strictly increasing on $(0,\infty )$. Combining continuity of $\Phi$ on $(0,\infty )$ with

$$\Phi \left(0\right)=-S_0<0,\underset{S\to \infty }{lim}\Phi \left(S\right)=+\infty ,$$

and strict monotonicity, the intermediate value theorem yields a unique $S^{★}>0$ such that $\Phi \left(S^{★}\right)=0$.

Finally, the remaining equilibrium components are determined by (28)–(30). Since $G_0>0$, $S_0>0$ and all rate parameters are positive, we obtain

$${\overline{S}}_b=S_b\left(S^{★}\right)>0,\overline{M}={\displaystyle \frac{{\mu }_m}{{\delta }_m}}{\overline{S}}_b>0,\overline{H}={\displaystyle \frac{{\mu }_H}{{\delta }_H}}\overline{M}>0,$$

$$\overline{C}={\displaystyle \frac{{\mu }_C}{{\delta }_C}}\overline{H}>0,{\overline{C}}_b=C_b\left(S^{★}\right)>0,\overline{G}=G_0-{\overline{S}}_b>0.$$

Thus the equilibrium

$${\overline{\mathbf{x}}}^{★}=\left(G_0-S_b\left(S^{★}\right),S^{★},S_b\left(S^{★}\right),{\overline{M}}^{★},{\overline{H}}^{★},{\overline{C}}^{★},{\overline{C}}_b^{★}\right)$$

is componentwise positive. Uniqueness of $S^{★}$ immediately implies uniqueness of the full equilibrium vector ${\overline{\mathbf{x}}}^{★}$. □

To analyze the stability of the equilibrium, we introduce perturbations

$$\tilde{\mathbf{x}}=\mathbf{x}-\overline{\mathbf{x}}={(\tilde{G},\tilde{S},{\tilde{S}}_b,\tilde{M},\tilde{H},\tilde{C},{\tilde{C}}_b)}^T,$$

where

$$\tilde{G}=G-\overline{G},\tilde{S}=S-\overline{S},{\tilde{S}}_b=S_b-{\overline{S}}_b,\tilde{M}=M-\overline{M},$$

$$\tilde{H}=H-\overline{H},\tilde{C}=C-\overline{C},{\tilde{C}}_b=C_b-{\overline{C}}_b.$$

We begin by analyzing the stability of a perturbed nonlinear system around a nontrivial equilibrium. Substituting the perturbed variables into the original system and using the equilibrium condition

$$\mathbf{J}\overline{\mathbf{x}}+f\left(\overline{\mathbf{x}}\right)=0,$$

we obtain the perturbed system

$${\displaystyle \frac{d\tilde{\mathbf{x}}}{dt}}=\mathbf{J}\tilde{\mathbf{x}}+f(\overline{\mathbf{x}}+\tilde{\mathbf{x}})-f\left(\overline{\mathbf{x}}\right).$$

Expanding the nonlinear term, the system becomes and collecting terms, the final expression for the perturbed system reads

$${\displaystyle \frac{d\tilde{\mathbf{x}}}{dt}}=\left(\begin{array}{c}{k}_{-1}{\tilde{S}}_b-k_1{(\overline{S}+\tilde{S})}^{\kappa }(\overline{G}+\tilde{G})+k_1{\overline{S}}^{\kappa }\overline{G}\\ \kappa k_{-1}{\tilde{S}}_b+k_{-2}{\tilde{C}}_b-\kappa k_1{(\overline{S}+\tilde{S})}^{\kappa }(\overline{G}+\tilde{G})-k_2{(\overline{C}+\tilde{C})}^{\iota }(\overline{S}+\tilde{S})+\kappa k_1{\overline{S}}^{\kappa }\overline{G}+k_2{\overline{C}}^{\iota }\overline{S}\\ -k_{-1}{\tilde{S}}_b+k_1{(\overline{S}+\tilde{S})}^{\kappa }(\overline{G}+\tilde{G})-k_1{\overline{S}}^{\kappa }\overline{G}\\ {\mu }_m{\tilde{S}}_b-{\delta }_m\tilde{M}\\ {\mu }_H\tilde{M}-{\delta }_H\tilde{H}\\ {\mu }_C\tilde{H}+\iota k_{-2}{\tilde{C}}_b-{\delta }_C\tilde{C}-\iota k_2{(\overline{C}+\tilde{C})}^{\iota }(\overline{S}+\tilde{S})+\iota k_2{\overline{C}}^{\iota }\overline{S}\\ -k_{-2}{\tilde{C}}_b+k_2{(\overline{C}+\tilde{C})}^{\iota }(\overline{S}+\tilde{S})-k_2{\overline{C}}^{\iota }\overline{S}\end{array}\right).$$

To facilitate the stability analysis, we rewrite the perturbed system in a decomposed form as

$${\displaystyle \frac{d{\tilde{\mathbf{x}}}_i}{dt}}={\mathfrak{F}}_i\left({\tilde{\mathbf{x}}}_i\right)+{\mathfrak{G}}_i\left(\tilde{\mathbf{x}}\right),i=1,\dots ,7,$$

(33)

where the vector fields satisfy ${\mathfrak{F}}_i\left(0\right)=0$ and ${\mathfrak{G}}_i\left(0\right)=0$ for all i. The functions ${\mathfrak{F}}_i$ describe the internal (self) dynamics of the isolated subsystems,

$${\displaystyle \frac{d{\tilde{\mathbf{x}}}_i}{dt}}={\mathfrak{F}}_i\left({\tilde{\mathbf{x}}}_i\right),i=1,\dots ,7,$$

(34)

whereas the functions ${\mathfrak{G}}_i$ represent the nonlinear interaction terms that couple the subsystems. This decomposition isolates the intrinsic dynamics of each compartment from the nonlinear interconnections, thus allowing the use of the composite–system framework for large-scale stability analysis.

Explicitly, the isolated dynamics are given by $\mathfrak{F}\left(\tilde{\mathbf{x}}\right)={\left({\mathfrak{F}}_i\left({\tilde{\mathbf{x}}}_i\right)\right)}_{i=1,\dots ,7}$, taking into account the perturbation $\tilde{G}=\tilde{S}={\tilde{S}}_b=\tilde{M}=\tilde{H}=\tilde{C}={\tilde{C}}_b\approx 0$; that is,

$$\begin{array}{cc}\hfill {\mathfrak{F}}_1\left(\tilde{G}\right)& =-\tilde{G},{\mathfrak{F}}_2\left(\tilde{S}\right)=-\tilde{S},{\mathfrak{F}}_3\left({\tilde{S}}_b\right)=-k_{-1}{\tilde{S}}_b,{\mathfrak{F}}_4\left(\tilde{M}\right)=-{\delta }_m\tilde{M},\hfill \\ \hfill {\mathfrak{F}}_5\left(\tilde{H}\right)& =-{\delta }_H\tilde{H},{\mathfrak{F}}_6\left(\tilde{C}\right)=-{\delta }_C\tilde{C},{\mathfrak{F}}_7\left({\tilde{C}}_b\right)=-k_{-2}{\tilde{C}}_b.\hfill \end{array}$$

The nonlinear interconnection terms are collected into the vector-valued function $\mathfrak{G}\left(\tilde{\mathbf{x}}\right)={\left({\mathfrak{G}}_i\left(\tilde{\mathbf{x}}\right)\right)}_{i=1,\dots ,7},$ where:

$$\mathfrak{G}\left(\tilde{\mathbf{x}}\right)=\left(\begin{array}{c}{k}_{-1}{\tilde{S}}_b-k_1{(\overline{S}+\tilde{S})}^{\kappa }(\overline{G}+\tilde{G})+k_1{\overline{S}}^{\kappa }\overline{G}\hfill \\ \kappa k_{-1}{\tilde{S}}_b+k_{-2}{\tilde{C}}_b-\kappa k_1{(\overline{S}+\tilde{S})}^{\kappa }(\overline{G}+\tilde{G})\hfill \\ -k_2{(\overline{C}+\tilde{C})}^{\iota }(\overline{S}+\tilde{S})+\kappa k_1{\overline{S}}^{\kappa }\overline{G}+k_2{\overline{C}}^{\iota }\overline{S}\hfill \\ k_1{(\overline{S}+\tilde{S})}^{\kappa }(\overline{G}+\tilde{G})-k_1{\overline{S}}^{\kappa }\overline{G}\hfill \\ {\mu }_m{\tilde{S}}_b\hfill \\ {\mu }_H\tilde{M}\hfill \\ {\mu }_C\tilde{H}+\iota k_{-2}{\tilde{C}}_b-\iota k_2{(\overline{C}+\tilde{C})}^{\iota }(\overline{S}+\tilde{S})+\iota k_2{\overline{C}}^{\iota }\overline{S}\hfill \\ k_2{(\overline{C}+\tilde{C})}^{\iota }(\overline{S}+\tilde{S})-k_2{\overline{C}}^{\iota }\overline{S}\hfill \end{array}\right).$$

This decomposition enables the application of the composite-system method to analyze the stability of interconnected nonlinear subsystems, where the functions $\mathfrak{F}$ and $\mathfrak{G}$ are continuously differentiable and satisfy $\mathfrak{F}\left(0\right)=0$ and $\mathfrak{G}\left(0\right)=0$, as established by the foundational result of Araki [2], stated in the following theorem.

Theorem 3

([2]). Consider the composite nonlinear system defined by the decomposition (33). This system is uniformly asymptotically stable in the large (uniformly a.s.i.l.) if there exist constants ${\alpha }_i$, functions $v_i\left({\tilde{\mathbf{x}}}_i\right)$, $u_i\left({\tilde{\mathbf{x}}}_i\right)$, and $w_i\left({\tilde{\mathbf{x}}}_i\right)$, for each subsystem $i=1,\dots ,7$, explicitly satisfying the following conditions:

(a)

For each subsystem i, the function $v_i\left({\tilde{\mathbf{x}}}_i\right)$ is positive definite, decrescent, radially unbounded, and continuously differentiable. Moreover, the isolated subsystem dynamics must satisfy:

$${\displaystyle \frac{d{v}_i}{dt}}{|}_{\mathit{isolated}}\le -{\alpha }_i{\left\{u_i\left({\tilde{\mathbf{x}}}_i\right)\right\}}^2-w_i\left({\tilde{\mathbf{x}}}_i\right),$$

where $u_i\left({\tilde{\mathbf{x}}}_i\right)$ is nonnegative-definite. The function $w_i\left({\tilde{\mathbf{x}}}_i\right)$ must be either strictly positive definite or identically zero. Specifically:

• If ${\alpha }_i>0$, then $w_i\left({\tilde{\mathbf{x}}}_i\right)$ may be identically zero or positive definite function.
• If ${\alpha }_i\le 0$, then $w_i\left({\tilde{\mathbf{x}}}_i\right)$ must be positive definite function.

(b)

There exist explicitly defined positive constants ${\beta }_{ik}$ for $i\ne k$, bounding the nonlinear interconnection terms as:

$${\nabla }_i{v}_i^{\top }\left({\tilde{\mathbf{x}}}_i\right){\mathfrak{G}}_i\left(\tilde{\mathbf{x}}\right)\le u_i\left({\tilde{\mathbf{x}}}_i\right)\sum _{k=1}^7{\beta }_{ik}{u}_k\left({\tilde{\mathbf{x}}}_k\right).$$

(c)

The $7\times 7$ matrix $A=\left(a_{ik}\right)$ defined explicitly by:

$$a_{ii}={\alpha }_i-{\beta }_{ii},a_{ik}=-{\beta }_{ik}\mathit{for}i\ne k,$$

must be an M-matrix.

Remark 2.

These explicit conditions accommodate both scenarios:

• When isolated subsystems possess inherent negative definite stability (${\alpha }_i>0$, $w_i=0$).
• When isolated subsystems lack inherent negative definite stability (${\alpha }_i\le 0$, $w_i\left({\tilde{\mathbf{x}}}_i\right)>0$ for all ${\tilde{\mathbf{x}}}_i\ne 0$), explicitly relying on stability through nonlinear interconnections and the M-matrix condition. Here, the term positive definite function means:

  $$f\left({\tilde{\mathbf{x}}}_i\right)=0\mathrm{if}{\tilde{\mathbf{x}}}_i=0,f\left({\tilde{\mathbf{x}}}_i\right)>0\mathit{otherwise}$$
  This distinguishes it from a merely nonnegative function which might vanish at multiple points.

Historically, the M-matrix criterion used in this work originates from the eigenvalue localization principle introduced by Geršgorin [30] in his classical disk theorem, presented as Theorem 7.6 on page 223 of [31]. This principle was further developed by Ostrowski [32,33,34], who established determinant inequalities and diagonal dominance conditions characterizing M-matrices. A systematic account of these results was later given by Poole and Boullion [35], where the spectral properties of M-matrices positivity of principal minors, nonpositive off-diagonal entries, and eigenvalues lying in the open left half-plane were unified into an algebraic framework. The theory was subsequently extended by Araki [2] to nonlinear interconnected systems, linking the classical M-matrix structure with the stability analysis of large-scale nonlinear differential systems.

• We aim to establish uniform asymptotic stability in the large (uniformly a.s.i.l.) of the composite system by verifying conditions (a) and (b) of Theorem 3.

Consider the Lyapunov candidate

$$v_1\left(\tilde{G}\right)=\frac{1}{2}{\tilde{G}}^2,$$

which is positive definite, radially unbounded, and continuously differentiable. Differentiating $v_1$ along the isolated subsystem dynamics, given explicitly by

$$\dot{\tilde{G}}{|}_{\mathrm{iso}}={\mathfrak{F}}_1\left(\tilde{G}\right)=-\tilde{G},$$

yields

$${\displaystyle \frac{d{v}_1}{dt}}{|}_{\mathrm{iso}}=\tilde{G}·(-\tilde{G})=-{\tilde{G}}^2.$$

Thus condition (a) of Theorem 3 holds with

$${\alpha }_1=1,u_1\left(\tilde{G}\right)=\left|\tilde{G}\right|,w_1\left(\tilde{G}\right)=0.$$

We write

$$\dot{\tilde{G}}={\mathfrak{F}}_1\left(\tilde{G}\right)+{\mathfrak{G}}_1\left(\tilde{\mathbf{x}}\right),\nabla v_1\left(\tilde{G}\right)=\tilde{G}.$$

From the perturbed full vector field,

$$\dot{\tilde{G}}=k_{-1}{\tilde{S}}_b-k_1\left({(\overline{S}+\tilde{S})}^{\kappa }(\overline{G}+\tilde{G})-{\overline{S}}^{\kappa }\overline{G}\right),$$

so we set the interconnection part, starting from the perturbed full vector field

$${\mathfrak{G}}_1\left(\tilde{\mathbf{x}}\right):=k_{-1}{\tilde{S}}_b-k_1\left({(\overline{S}+\tilde{S})}^{\kappa }(\overline{G}+\tilde{G})-{\overline{S}}^{\kappa }\overline{G}\right).$$

A first-order Taylor expansion around $(\overline{S},\overline{G})$ gives

$${(\overline{S}+\tilde{S})}^{\kappa }(\overline{G}+\tilde{G})-{\overline{S}}^{\kappa }\overline{G}=\kappa {\overline{S}}^{\kappa -1}\overline{G}\tilde{S}+{\overline{S}}^{\kappa }\tilde{G}+R,\parallel R\parallel =O\left(\right|\tilde{S}{|}^2+|\tilde{G}{|}^2),$$

and hence

$${\mathfrak{G}}_1\left(\tilde{\mathbf{x}}\right)=-k_1{\overline{S}}^{\kappa }\tilde{G}-k_1\kappa {\overline{S}}^{\kappa -1}\overline{G}\tilde{S}+k_{-1}{\tilde{S}}_b+O\left({\epsilon }^2\right),$$

for $|\tilde{S}|,|\tilde{G}|\le \epsilon$ small. Therefore,

$$\nabla v_1^{\top }{\mathfrak{G}}_1\left(\tilde{\mathbf{x}}\right)=\tilde{G}{\mathfrak{G}}_1\left(\tilde{\mathbf{x}}\right)=-k_1{\overline{S}}^{\kappa }|\tilde{G}{|}^2-k_1\kappa {\overline{S}}^{\kappa -1}\overline{G}|\tilde{G}||\tilde{S}|+k_{-1}|\tilde{G}||{\tilde{S}}_b|+O\left({\epsilon }^3\right).$$

Taking absolute values and using the triangle inequality,

$$|\nabla v_1^{\top }{\mathfrak{G}}_1\left(\tilde{\mathbf{x}}\right)|\le k_1{\overline{S}}^{\kappa }|\tilde{G}{|}^2+k_1\kappa {\overline{S}}^{\kappa -1}\overline{G}|\tilde{G}||\tilde{S}|+k_{-1}|\tilde{G}||{\tilde{S}}_b|+O\left({\epsilon }^3\right).$$

This matches condition (b) in the small-gain form

$$|\nabla v_1^{\top }{\mathfrak{G}}_1\left(\tilde{\mathbf{x}}\right)|\le u_1\left(\tilde{G}\right)\sum _{k=1}^7{\beta }_{1k}{u}_k\left({\tilde{x}}_k\right),$$

by taking

$$u_1\left(\tilde{G}\right)=|\tilde{G}|,u_2\left(\tilde{S}\right)=|\tilde{S}|,u_3\left({\tilde{S}}_b\right)=\left|{\tilde{S}}_b\right|,$$

and setting the (nonzero) gains explicitly as

$${\beta }_{11}=k_1{\overline{S}}^{\kappa },{\beta }_{12}=k_1\kappa {\overline{S}}^{\kappa -1}\overline{G},{\beta }_{13}=k_{-1},$$

with ${\beta }_{1k}=0$ for $k\ge 4$, and the $O\left({\epsilon }^3\right)$ term negligible as $\epsilon \to 0$.

Therefore, conditions (a) and (b) of Theorem 3 are explicitly verified for the subsystem corresponding to $\tilde{G}$.

We now verify conditions (a) and (b) for the subsystem associated with $\tilde{S}$. Let $v_2\left(\tilde{S}\right)=\frac{1}{2}{\tilde{S}}^2$. Then $\nabla v_2\left(\tilde{S}\right)=\tilde{S}$, and, under the isolated dynamics $\dot{\tilde{S}}=-\tilde{S}$,

$${\displaystyle \frac{d{v}_2}{dt}}{|}_{\mathrm{iso}}=\nabla v_2\left(\tilde{S}\right)\dot{\tilde{S}}=\tilde{S}(-\tilde{S})=-{\tilde{S}}^2,$$

which is negative definite in $\tilde{S}$, so condition (a) of Theorem 3 holds with

$${\alpha }_2=1,u_2\left(\tilde{S}\right)=\left|\tilde{S}\right|,w_2\equiv 0.$$

From the interconnection part, starting from the perturbed full vector field, we define

$${\mathfrak{G}}_2\left(\tilde{\mathbf{x}}\right)=\kappa k_{-1}{\tilde{S}}_b+k_{-2}{\tilde{C}}_b-\kappa k_1\left[{(\overline{S}+\tilde{S})}^{\kappa }(\overline{G}+\tilde{G})-{\overline{S}}^{\kappa }\overline{G}\right]-k_2\left[{(\overline{C}+\tilde{C})}^{\iota }(\overline{S}+\tilde{S})-{\overline{C}}^{\iota }\overline{S}\right].$$

First–order Taylor expansions around $(\overline{S},\overline{G},\overline{C})$ yield

$${(\overline{S}+\tilde{S})}^{\kappa }(\overline{G}+\tilde{G})-{\overline{S}}^{\kappa }\overline{G}=\kappa {\overline{S}}^{\kappa -1}\overline{G}\tilde{S}+{\overline{S}}^{\kappa }\tilde{G}+R_1,\parallel R_1\parallel =O(|\tilde{S}{|}^2+|\tilde{G}{|}^2),$$

$${(\overline{C}+\tilde{C})}^{\iota }(\overline{S}+\tilde{S})-{\overline{C}}^{\iota }\overline{S}={\overline{C}}^{\iota }\tilde{S}+\iota {\overline{C}}^{\iota -1}\overline{S}\tilde{C}+R_2,\parallel R_2\parallel =O(|\tilde{S}{|}^2+|\tilde{C}{|}^2).$$

Hence, for $|\tilde{S}|,|\tilde{G}|,|\tilde{C}|\le \epsilon$,

$${\mathfrak{G}}_2\left(\tilde{\mathbf{x}}\right)=-\underset{=:A_S}{\underbrace{\left({\kappa }^2{k}_1{\overline{S}}^{\kappa -1}\overline{G}+k_2{\overline{C}}^{\iota }\right)}}\tilde{S}-\underset{=:A_G}{\underbrace{\kappa k_1{\overline{S}}^{\kappa }}}\tilde{G}-\underset{=:A_C}{\underbrace{k_2\iota {\overline{C}}^{\iota -1}\overline{S}}}\tilde{C}+\kappa k_{-1}{\tilde{S}}_b+k_{-2}{\tilde{C}}_b+O\left({\epsilon }^2\right).$$

Therefore,

$$\nabla v_2^{\top }{\mathfrak{G}}_2=\tilde{S}{\mathfrak{G}}_2=-A_S|\tilde{S}{|}^2-A_G|\tilde{S}||\tilde{G}|-A_C|\tilde{S}||\tilde{C}|+\kappa k_{-1}|\tilde{S}||{\tilde{S}}_b|+k_{-2}|\tilde{S}||{\tilde{C}}_b|+O\left({\epsilon }^3\right),$$

and hence, by the triangle inequality,

$$|\nabla v_2^{\top }{\mathfrak{G}}_2\left(\tilde{\mathbf{x}}\right)|\le A_S|\tilde{S}{|}^2+A_G|\tilde{S}||\tilde{G}|+A_C|\tilde{S}||\tilde{C}|+\kappa k_{-1}|\tilde{S}||{\tilde{S}}_b|+k_{-2}|\tilde{S}||{\tilde{C}}_b|+O\left({\epsilon }^3\right).$$

This matches condition (b) in the small-gain form

$$|\nabla v_2^{\top }{\mathfrak{G}}_2\left(\tilde{\mathbf{x}}\right)|\le u_2\left(\tilde{S}\right)\sum _{k=1}^7{\beta }_{2k}{u}_k\left({\tilde{x}}_k\right)+O\left({\epsilon }^3\right),$$

by taking

$$u_1\left(\tilde{G}\right)=|\tilde{G}|,u_2\left(\tilde{S}\right)=|\tilde{S}|,u_3\left({\tilde{S}}_b\right)=|{\tilde{S}}_b|,u_6\left(\tilde{C}\right)=|\tilde{C}|,u_7\left({\tilde{C}}_b\right)=\left|{\tilde{C}}_b\right|,$$

and the nonzero gains

$${\beta }_{22}={\kappa }^2{k}_1{\overline{S}}^{\kappa -1}\overline{G}+k_2{\overline{C}}^{\iota },{\beta }_{21}=\kappa k_1{\overline{S}}^{\kappa },{\beta }_{23}=\kappa k_{-1},{\beta }_{26}=k_2\iota {\overline{C}}^{\iota -1}\overline{S},{\beta }_{27}=k_{-2}$$

with all other ${\beta }_{2k}=0$. Thus, conditions (a) and (b) of Theorem 3 are explicitly verified for the subsystem associated with $\tilde{S}$.

Similarly to the subsystems associated with $\tilde{G}$ and $\tilde{S}$, we now consider the remaining subsystems corresponding to the state variables

$${\tilde{S}}_b,\tilde{M},\tilde{H},\tilde{C},{\tilde{C}}_b.$$

For each of these, we define the Lyapunov candidate

$$v_i\left({\tilde{x}}_i\right)=\frac{1}{2}{\tilde{x}}_i^2,i=3,\dots ,7,$$

which is positive definite, radially unbounded, and continuously differentiable. The corresponding isolated dynamics take the linear form

$${\dot{\tilde{x}}}_i={\mathfrak{F}}_i\left({\tilde{x}}_i\right)=-{\alpha }_i{\tilde{x}}_i,$$

with strictly positive decay rates

$${\alpha }_3=k_{-1},{\alpha }_4={\delta }_m,{\alpha }_5={\delta }_H,{\alpha }_6={\delta }_C,{\alpha }_7=k_{-2}.$$

Differentiating $v_i$ along these isolated dynamics gives

$${\displaystyle \frac{d{v}_i}{dt}}{|}_{\mathrm{iso}}=-{\alpha }_i{\tilde{x}}_i^2=-{\alpha }_i{u}_i^2\left({\tilde{x}}_i\right),u_i\left({\tilde{x}}_i\right):=\left|{\tilde{x}}_i\right|,$$

so condition (a) of Theorem 3 is satisfied with $w_i\left({\tilde{x}}_i\right)\equiv 0$.

The nonlinear interconnection terms ${\mathfrak{G}}_i\left(\tilde{\mathbf{x}}\right)$ are obtained from the full system by isolating each component’s coupling. Expanding these terms about the equilibrium $\overline{\mathbf{x}}$ yields Taylor remainders satisfying

$$\parallel R_i\parallel =O\left({\epsilon }^2\right)\mathrm{for}\parallel \tilde{\mathbf{x}}\parallel <\epsilon ,\epsilon \to 0.$$

Hence, the derivative of each Lyapunov function satisfies

$$|\nabla v_i^{\top }{\mathfrak{G}}_i\left(\tilde{\mathbf{x}}\right)|\le u_i\left({\tilde{x}}_i\right)\sum _{k=1}^7{\beta }_{ik}{u}_k\left({\tilde{x}}_k\right)+O\left({\epsilon }^3\right),$$

with explicitly computable, nonnegative interconnection gains ${\beta }_{ik}$ that depend on model parameters and equilibrium values.

The nonzero gains are:

$$\begin{array}{cc}& {\beta }_{21}=\kappa k_1{\overline{S}}^{\kappa },{\beta }_{22}={\kappa }^2{k}_1{\overline{S}}^{\kappa -1}\overline{G}+k_2{\overline{C}}^{\iota },{\beta }_{23}=\kappa k_{-1},{\beta }_{26}=k_2\iota {\overline{C}}^{\iota -1}\overline{S},{\beta }_{27}=k_{-2},\hfill \\ & {\beta }_{31}=k_1{\overline{S}}^{\kappa },{\beta }_{32}=k_1\kappa {\overline{S}}^{\kappa -1}\overline{G},\hfill \\ & {\beta }_{43}={\mu }_m,{\beta }_{54}={\mu }_H,\hfill \\ & {\beta }_{62}=\iota k_2{\overline{C}}^{\iota },{\beta }_{65}={\mu }_C,{\beta }_{66}={\iota }^2{k}_2{\overline{C}}^{\iota -1}\overline{S},\hfill \\ & {\beta }_{72}=k_2{\overline{C}}^{\iota },{\beta }_{76}=k_2\iota {\overline{C}}^{\iota -1}\overline{S}.\hfill \end{array}$$

All other ${\beta }_{ik}=0$. Thus, conditions (a) and (b) of Theorem 3 are verified for all subsystems $i=3,\dots ,7$.

Remark 3.

The use of big-O terms in the Lyapunov expansion makes it explicit that the argument is local: the Lyapunov functions guarantee asymptotic stability in a neighborhood of the equilibrium, with decay along the full nonlinear trajectories up to higher-order terms. In contrast to a purely linear eigenvalue test via the Jacobian which guarantees only linearized stability, the Lyapunov framework provides scalar energy-like functionals that decrease along solutions and thus establish a nonlinear stability mechanism. Moreover, the M-matrix formulation translates these differential properties into explicit algebraic inequalities on parameters (e.g., diagonal dominance and positivity conditions), which are verifiable without computing spectra. This is particularly advantageous in large-scale systems, where direct eigenvalue computations are often impractical, and structural properties (sparsity, monotonicity) can still be exploited to guarantee robustness of the equilibrium.

To complete the application of Theorem 3, we now verify that the matrix $A=\left(a_{ik}\right)\in {\mathbb{R}}^{7\times 7}$, defined by

$$a_{ii}={\alpha }_i-{\beta }_{ii},a_{ik}=-{\beta }_{ik}(i\ne k),$$

is an M-matrix; that is, all off-diagonal entries are nonpositive and all principal minors are positive.

Using the identified parameters ${\alpha }_i,{\beta }_{ik}$, the explicit form of A is:

$$A=\left(\begin{array}{ccccccc}1-k_1{\overline{S}}^{\kappa }& -k_1\kappa {\overline{S}}^{\kappa -1}\overline{G}& -k_{-1}& 0& 0& 0& 0\\ -\kappa k_1{\overline{S}}^{\kappa }& 1-\left({\kappa }^2{k}_1{\overline{S}}^{\kappa -1}\overline{G}+k_2{\overline{C}}^{\iota }\right)& -\kappa k_{-1}& 0& 0& -k_2\iota {\overline{C}}^{\iota -1}\overline{S}& -k_{-2}\\ -k_1{\overline{S}}^{\kappa }& -k_1\kappa {\overline{S}}^{\kappa -1}\overline{G}& k_{-1}& 0& 0& 0& 0\\ 0& 0& -{\mu }_m& {\delta }_m& 0& 0& 0\\ 0& 0& 0& -{\mu }_H& {\delta }_H& 0& 0\\ 0& -\iota k_2{\overline{C}}^{\iota }& 0& 0& -{\mu }_C& {\delta }_C-{\iota }^2{k}_2{\overline{C}}^{\iota -1}\overline{S}& 0\\ 0& -k_2{\overline{C}}^{\iota }& 0& 0& 0& -k_2\iota {\overline{C}}^{\iota -1}\overline{S}& k_{-2}\end{array}\right).$$

Thus, provided suitable parameter conditions ensuring A is an M-matrix are explicitly satisfied, Theorem 3 guarantees uniform asymptotic stability in the large for the entire composite system. An M-matrix is a special class of matrices with widespread applications in stability analysis, numerical methods, and mathematical modeling of interconnected systems. Formally, an M-matrix is a real square matrix whose off-diagonal elements are non-positive and whose eigenvalues have positive real parts [18,20]. Equivalently, M-matrices can be characterized by positive principal minors or diagonal dominance conditions [2,19,21]. Due to their beneficial spectral and structural properties, M-matrices naturally arise in the study of stability for composite dynamical systems and iterative numerical methods [18,19]. To demonstrate that the matrix A is an M-matrix, we utilize well-established properties from the literature [18,35,36]. Specifically, if $A\in {\mathbb{R}}^{n\times n}$ can be written as $A=sI-B$, where $s>0$ and $B={\left(b_{ij}\right)}_{1\le i,j\le n}$ is a non-negative matrix ($B\ge 0$, meaning $b_{ij}\ge 0$ for all $1\le i,j\le n$), and $s\ge \rho \left(B\right)$, then A is termed an M-matrix. In this context, $\rho \left(A\right)=max\left\{\right|\lambda |:det(\lambda I-A)=0\}$ denotes the spectral radius of A, representing the largest absolute value of its eigenvalues.

Next, we present an important result from spectral analysis regarding the upper bound of the spectral radius. This theorem, which is essential for understanding the influence of model parameters on stability, is stated in [37]. The spectral radius plays a key role in determining the stability of the system, and this theorem provides bounds on the spectral radius based on the matrix norm.

Theorem 4

([37]). Let $\parallel ·\parallel$ be an operator norm on $M_n$ induced by a vector norm, and let $A\in M_n$. Then the following inequalities hold:

(a)

Upper bound on the spectral radius:

$$\rho \left(A\right)\le \parallel A\parallel .$$

(b)

Lower bound on the spectral radius (for nonsingular matrices): If A is nonsingular, then

$$\rho \left(A\right)\ge {\displaystyle \frac{1}{\parallel A^{-1}\parallel }}.$$

Moreover, for any eigenvalue λ of A, one always has $\left|\lambda \right|\le \rho \left(A\right)$.

To complement Theorem 4, we recall several important relationships between common matrix norms, which will be used later to estimate the spectral radius and compare different stability criteria.

Remark 4.

The bounds established in Theorem 4 apply only to operator norms induced by vector norms (such as the ${\mathsf{\ell }}_1$, ${\mathsf{\ell }}_2$, or ${\mathsf{\ell }}_{\infty }$ norms), and not necessarily to non-operator norms such as the Frobenius norm. For any matrix $B\in M_n$, the following standard norm inequalities hold [37]:

$$\rho \left(B\right)\le {\parallel B\parallel }_2\le {\parallel B\parallel }_F,{\parallel B\parallel }_2\le \sqrt{{\parallel B\parallel }_1{\parallel B\parallel }_{\infty }}.$$

These relations are frequently used to estimate the spectral radius when only computable norm bounds are available and provide a direct connection between different matrix norms in spectral stability analysis.

We now apply spectral theory to analyze the structure of the matrix A. Recall that the small-gain condition can be interpreted through the spectral radius of the associated nonnegative matrix B, obtained from the decomposition

$$\begin{array}{cc}\hfill A& =I-B,\hfill \\ \hfill B& =\left(\begin{array}{ccccccc}{k}_1{\overline{S}}^{\kappa }& k_1\kappa {\overline{S}}^{\kappa -1}\overline{G}& k_{-1}& 0& 0& 0& 0\\ \kappa k_1{\overline{S}}^{\kappa }& {\kappa }^2{k}_1{\overline{S}}^{\kappa -1}\overline{G}+k_2{\overline{C}}^{\iota }& \kappa k_{-1}& 0& 0& k_2\iota {\overline{C}}^{\iota -1}\overline{S}& k_{-2}\\ k_1{\overline{S}}^{\kappa }& k_1\kappa {\overline{S}}^{\kappa -1}\overline{G}& 1-k_{-1}& 0& 0& 0& 0\\ 0& 0& {\mu }_m& 1-{\delta }_m& 0& 0& 0\\ 0& 0& 0& {\mu }_H& 1-{\delta }_H& 0& 0\\ 0& \iota k_2{\overline{C}}^{\iota }& 0& 0& {\mu }_C& 1-{\delta }_C+{\iota }^2{k}_2{\overline{C}}^{\iota -1}\overline{S}& 0\\ 0& k_2{\overline{C}}^{\iota }& 0& 0& 0& k_2\iota {\overline{C}}^{\iota -1}\overline{S}& 1-k_{-2}\end{array}\right).\hfill \end{array}$$

(35)

Here B is a nonnegative matrix. From the model Equations (1)–(7) and Theorem 1, the equilibrium values and parameters satisfy

$$\overline{S}\ge 0,\overline{G}\ge 0,\overline{C}\ge 0,k_1,k_{-1},k_2,k_{-2},{\mu }_m,{\mu }_H,{\mu }_C,{\delta }_m,{\delta }_H,{\delta }_C,\kappa ,\iota \ge 0.$$

Under these assumptions, all off-diagonal entries of B are nonnegative. For the diagonal entries, nonnegativity holds if and only if

$$\begin{array}{cc}\hfill & 1-k_{-1}\ge 0,1-{\delta }_m\ge 0,1-{\delta }_H\ge 0,\hfill \\ \hfill & 1-{\delta }_C+{\kappa }^2{k}_2{\overline{C}}^{\kappa -1}\overline{S}\ge 0,1-k_{-2}\ge 0.\hfill \end{array}$$

(36)

If the inequalities in (36) are satisfied, then $B\ge 0$ entrywise. Moreover, if the spectral radius of B satisfies $\rho \left(B\right)<1$, then $A=I-B$ is a nonsingular M-matrix. Thus, the ${\mathsf{\ell }}_1$-norm of the matrix B, defined as the maximum column sum, is

$${\parallel B\parallel }_1=max\left\{\begin{array}{cc}& (\kappa +2)k_1{\overline{S}}^{\kappa },\hfill \\ & k_1\kappa (\kappa +2){\overline{S}}^{\kappa -1}\overline{G}+(\iota +2)k_2{\overline{C}}^{\iota },\hfill \\ & 1+\kappa k_{-1}+{\mu }_m,\hfill \\ & 1-{\delta }_m+{\mu }_H,\hfill \\ & 1-{\delta }_H+{\mu }_C,\hfill \\ & (1-{\delta }_C)+\left({\iota }^2+2\iota \right)k_2{\overline{C}}^{\iota -1}\overline{S},\hfill \\ & 1\hfill \end{array}\right\}.$$

In order for $A=I-B$ to be a nonsingular M-matrix, it is sufficient that ${\parallel B\parallel }_1<1$. However, from the column sums computed above we have

$${\parallel B\parallel }_1\ge max\{\kappa k_{-1}+1+{\mu }_m,1\}\ge 1,$$

since the third column sum equals $\kappa k_{-1}+1+{\mu }_m>1$ and the seventh column sum is exactly 1. Hence, the sufficient condition ${\parallel B\parallel }_1<1$ is not satisfied. This does not imply that $\rho \left(B\right)\ge 1$; it only shows that the crude ${\mathsf{\ell }}_1$-norm estimate is too conservative to conclude $\rho \left(B\right)<1$.

Consequently, we turn to alternative methods for establishing $\rho \left(B\right)<1$, such as estimates based on the ${\mathsf{\ell }}_{\infty }$-norm or the Frobenius norm of the matrix B. The ${\mathsf{\ell }}_{\infty }$-norm of B corresponds to the maximum absolute row sum:

$${\parallel B\parallel }_{\infty }=max\left\{\begin{array}{cc}& k_1{\overline{S}}^{\kappa }+k_1\kappa {\overline{S}}^{\kappa -1}\overline{G}+k_{-1},\hfill \\ & \kappa k_1{\overline{S}}^{\kappa }+{\kappa }^2{k}_1{\overline{S}}^{\kappa -1}\overline{G}+k_2{\overline{C}}^{\iota }+\kappa k_{-1}+\iota k_2{\overline{C}}^{\iota -1}\overline{S}+k_{-2},\hfill \\ & k_1{\overline{S}}^{\kappa }+k_1\kappa {\overline{S}}^{\kappa -1}\overline{G}+1-k_{-1},\hfill \\ & {\mu }_m+1-{\delta }_m,\hfill \\ & {\mu }_H+1-{\delta }_H,\hfill \\ & \kappa k_2{\overline{C}}^{\iota }+{\mu }_C+1-{\delta }_C+{\kappa }^2{k}_2{\overline{C}}^{\iota -1}\overline{S},\hfill \\ & k_2{\overline{C}}^{\iota }+\iota k_2{\overline{C}}^{\iota -1}\overline{S}+1-k_{-2}\hfill \end{array}\right\}.$$

To ensure that $A=I-B$ is a nonsingular M-matrix, we require:

$${\parallel B\parallel }_{\infty }<1.$$

Finally, the Frobenius norm of the matrix B, denoted ${\parallel B\parallel }_F$, is defined by

$${\parallel B\parallel }_F=\sqrt{tr\left(B^{\top }B\right)}.$$

We obtain the symbolic expression

$${\parallel B\parallel }_F=\sqrt{\begin{array}{cc}& {(1-{\delta }_H)}^2+{(1-{\delta }_m)}^2+{(1-k_{-1})}^2+{(1-k_{-2})}^2+{\mu }_C^2+{\mu }_H^2+{\mu }_m^2\hfill \\ & +{(1-{\kappa }^2{k}_1{\overline{S}}^{\kappa -1}\overline{G}-k_2{\overline{C}}^{\iota })}^2+{(1-k_1{\overline{S}}^{\kappa })}^2+{(1-{\delta }_C+{\iota }^2{k}_2{\overline{C}}^{\iota -1}\overline{S})}^2\hfill \\ & +k_{-1}^2+{\kappa }^2{k}_{-1}^2+k_{-2}^2+{\iota }^2{k}_{-2}^2\hfill \\ & +{\overline{C}}^{2\iota }{k}_2^2(1+{\iota }^2)+{\overline{S}}^{2\kappa }{k}_1^2(1+{\kappa }^2)+2{\overline{C}}^{2\iota -2}{\overline{S}}^2{\iota }^2{k}_2^2+2{\overline{G}}^2{\overline{S}}^{2\kappa -2}{k}_1^2{\kappa }^2\hfill \end{array}}.$$

Although the Frobenius norm ${\parallel ·\parallel }_F$ is not an operator norm, it satisfies

$$\rho \left(B\right)\le {\parallel B\parallel }_2\le {\parallel B\parallel }_F,$$

and therefore the sufficient condition

$${\parallel B\parallel }_F<1$$

implies that $\rho \left(B\right)<1$, and hence $A=I-B$ is a nonsingular M-matrix.

Theorem 5

(Norm-based M-matrix criterion and stability for the SREBP-2 model). Let $A=I-B$ denote the matrix obtained from the interconnection analysis of the perturbed system (1)–(7), as defined in (35). Here, $\overline{S},\overline{G},\overline{C}\ge 0$ denote the equilibrium values, and the parameters of (1)–(7) satisfy

$$k_1,k_{-1},k_2,k_{-2},{\mu }_d,{\mu }_h,{\mu }_C,{\delta }_m,{\delta }_h,{\delta }_C\ge 0,\nu ,\kappa \ge 0,$$

together with the diagonal nonnegativity conditions

$$\begin{array}{cc}\hfill & 1-k_{-1}\ge 0,1-{\delta }_m\ge 0,1-{\delta }_H\ge 0,\hfill \\ \hfill & 1-{\delta }_C+{\iota }^2{k}_2{\overline{C}}^{\iota -1}\overline{S}\ge 0,1-k_{-2}\ge 0.\hfill \end{array}$$

(37)

so that $B\ge 0$ entrywise.

If either of the following norm bounds holds:

$${\parallel B\parallel }_{\infty }<1\mathrm{or}{\parallel B\parallel }_F<1,$$

(38)

then $\rho \left(B\right)<1$, and hence $A=I-B$ is a nonsingular M-matrix. Consequently, condition (c) of Theorem 3 holds, and the equilibrium of the full SREBP-2 composite system is uniformly asymptotically stable in the large.

Proof. 

Under (37), $B\ge 0$. For the ${\mathsf{\ell }}_{\infty }$ operator norm, $\rho \left(B\right)\le {\parallel B\parallel }_{\infty }$, so ${\parallel B\parallel }_{\infty }<1$ implies $\rho \left(B\right)<1$. For the Frobenius norm, use $\rho \left(B\right)\le {\parallel B\parallel }_2\le {\parallel B\parallel }_F$ (see, e.g., [37]); thus ${\parallel B\parallel }_F<1$ also implies $\rho \left(B\right)<1$. Hence $A=I-B$ is a nonsingular M-matrix [18,19,20,37]. By construction, A is precisely the matrix in condition (c) of Theorem 3; with (a)–(b) already verified for the subsystems, Theorem 3 yields uniform asymptotic stability in the large for the equilibrium of the composite SREBP-2 model. □

4. Spectral Norm Analysis and Simulation

The condition ${\parallel B\parallel }_1<1$ is often too restrictive and fails to hold for realistic parameter combinations. In such cases, the ${\mathsf{\ell }}_{\infty }$- and Frobenius-norm bounds in (38) provide alternative sufficient conditions ensuring $\rho \left(B\right)<1$, and thus the M-matrix property of A. For any matrix B, the following standard inequalities hold [37]:

$$\rho \left(B\right)\le {\parallel B\parallel }_2\le {\parallel B\parallel }_F,{\parallel B\parallel }_2\le \sqrt{{\parallel B\parallel }_1{\parallel B\parallel }_{\infty }}.$$

Therefore, a sufficient (though not necessary) approach to ensure ${\parallel B\parallel }_2<1$ is either ${\parallel B\parallel }_F<1$, or the simultaneous conditions ${\parallel B\parallel }_1<1$ and ${\parallel B\parallel }_{\infty }<1$. As will be shown below, ${\parallel B\parallel }_F<1$ is unattainable for the present interconnection, and ${\parallel B\parallel }_1<1$ is typically too conservative in practice. Consequently, we establish stability via the small-gain condition ${\parallel B\parallel }_{\infty }<1$, which directly implies $\rho \left(B\right)<1$. Since $\rho \left(B\right)\le {\parallel B\parallel }_2$, the spectral norm ${\parallel B\parallel }_2$ may, but need not, lie below 1.

Lemma 4

(Frobenius–norm obstruction). For the matrix B of the SREBP-2 interconnection, one always has ${\parallel B\parallel }_F\ge 1$. Hence, the condition ${\parallel B\parallel }_F<1$ can never guarantee $\rho \left(B\right)<1$ for this model.

Proof. 

Even if all entries of B except

$$B_{1,3}=k_{-1},B_{3,3}=1-k_{-1},B_{2,7}=k_{-2},B_{7,7}=1-k_{-2}$$

were (hypothetically) set to zero, we would still have

$${\parallel B\parallel }_F^2\ge k_{-1}^2+{(1-k_{-1})}^2+k_{-2}^2+{(1-k_{-2})}^2.$$

Each quadratic term $x^2+{(1-x)}^2$ attains its minimum at $x=\frac{1}{2}$ with value $\frac{1}{2}$, so the right-hand side is minimized when $k_{-1}=k_{-2}=\frac{1}{2}$, yielding ${\parallel B\parallel }_F^2\ge \frac{1}{2}+\frac{1}{2}=1$, and therefore ${\parallel B\parallel }_F\ge 1$.

In the full model, B contains many additional nonnegative entries, such as the degradation and coupling coefficients ${\mu }_m,{\mu }_H,{\mu }_C>0$ and the feedback terms $k_1{\overline{S}}^{\kappa },k_1\kappa {\overline{S}}^{\kappa -1}\overline{G},k_2{\overline{C}}^{\iota }$. Each of these strictly increases the Frobenius norm. Hence, for all admissible parameter values, one has ${\parallel B\parallel }_F>1$. □

Let $r_i:={\sum }_j{B}_{ij}$ denote the row sums of B. Using the steady-state relations and introducing the grouped rate parameters

$${\alpha }_1:=k_1{\overline{S}}^{\kappa },{\alpha }_2:=\kappa k_1{\overline{S}}^{\kappa -1}={\alpha }_1{\displaystyle \frac{\kappa }{\overline{S}}},\beta :=k_2{\overline{S}}^{\iota },$$

(39)

we can rewrite the cholesterol feedback terms as

$$k_2{\overline{C}}^{\iota }=\beta {\left(\frac{\overline{C}}{\overline{S}}\right)}^{\iota },k_2{\overline{C}}^{\iota -1}\overline{S}=\beta {\left(\frac{\overline{C}}{\overline{S}}\right)}^{\iota -1}.$$

(40)

From the explicit structure of B, the row sums are then

$$\begin{array}{cc}\hfill r_1& ={\alpha }_1\left(1+\kappa \frac{\overline{G}}{\overline{S}}\right)+k_{-1},\hfill \\ \hfill r_2& =2\kappa {\alpha }_1\frac{\overline{G}}{\overline{S}}+\beta {\left(\frac{\overline{C}}{\overline{S}}\right)}^{\iota }+\kappa k_{-1}+k_{-2},\hfill \\ \hfill r_3& ={\alpha }_1\left(1+\kappa \frac{\overline{G}}{\overline{S}}\right)+(1-k_{-1}),\hfill \\ \hfill r_4& ={\mu }_m+(1-{\delta }_m),r_5={\mu }_H+(1-{\delta }_H),\hfill \\ \hfill r_6& =\iota \beta {\left(\frac{\overline{C}}{\overline{S}}\right)}^{\iota }+{\mu }_C+\left(1-{\delta }_C+{\iota }^2\beta {\left(\frac{\overline{C}}{\overline{S}}\right)}^{\iota -1}\right),\hfill \\ \hfill r_7& =\beta {\left(\frac{\overline{C}}{\overline{S}}\right)}^{\iota }+\iota \beta {\left(\frac{\overline{C}}{\overline{S}}\right)}^{\iota -1}+(1-k_{-2}).\hfill \end{array}$$

(41)

For the DNA–binding block (rows 1–3), introduce the shorthand

$$L:={\alpha }_1\left(1+\kappa \frac{\overline{G}}{\overline{S}}\right)>0.$$

(42)

The inequalities $r_1<1$ and $r_3<1$ are equivalent to

$$r_1<1\iff L+k_{-1}<1\iff k_{-1}<1-L,$$

and

$$r_3<1\iff L+(1-k_{-1})<1\iff L<k_{-1},$$

so that

$$r_1<1\mathrm{and}{r}_3<1⟺k_{-1}\in (L,1-L),$$

(43)

which is only possible if $L<\frac{1}{2}$.

In addition, we will later require $\kappa k_{-1}<1$ in order to select $k_{-2}>0$ with $\kappa k_{-1}+k_{-2}<1$. To encode both requirements, we strengthen the feasibility conditions to

$$k_{-1}\in \left(L,min\{1-L,1/\kappa \}\right),L<min\left\{\frac{1}{2},\frac{1}{\kappa }\right\}.$$

(44)

Under (44), we can then choose

$$k_{-2}\in \left(0,1-\kappa k_{-1}\right),\mathrm{for}\mathrm{example}{k}_{-2}:=\frac{1}{2}(1-\kappa k_{-1}),$$

(45)

which guarantees $\kappa k_{-1}+k_{-2}<1$.

Finally, we collect the degradation–dominance inequalities

$${\delta }_m>{\mu }_m,{\delta }_H>{\mu }_H,{\delta }_C>{\mu }_C,$$

(46)

which ensure that the purely metabolic and hormonal rows $r_4,r_5,r_6$ can be made strictly less than one.

Proposition 1

(Sufficient conditions for ${\parallel B\parallel }_{\infty }<1$). Fix $\kappa ,\iota \ge 1$ and let ${\mu }_m,{\mu }_H,{\mu }_C\ge 0$. Let $\overline{S}>0$, $\overline{G}\ge 0$, $\overline{C}\ge 0$ denote the equilibrium values, and introduce the grouped rates ${\alpha }_1,\beta$ as in (39). Assume that the degradation–dominance conditions (46) hold, and that the quantity L in (42) satisfies

$$L<min\left\{\frac{1}{2},\frac{1}{\kappa }\right\}.$$

(47)

Choose $k_{-1},k_{-2}$ so that the feasibility conditions (44) and the dissociation constraint (45) are satisfied. Then there exists ${\epsilon }_0>0$ such that, for all $0\le {\alpha }_1,\beta \le {\epsilon }_0$, the row sums $r_i$ of B given in (41) satisfy $r_i<1$ for all $i=1,\dots ,7$. Consequently,

$${\parallel B\parallel }_{\infty }=\underset{1\le i\le 7}{max}{r}_i<1,$$

which implies $\rho \left(B\right)<1$, and therefore $A=I-B$ is a nonsingular M-matrix.

Proof. 

From (41) and (42) we have

$$r_1=L+k_{-1},r_3=L+(1-k_{-1}).$$

The choice (44) yields $L<k_{-1}<1-L$, hence

$$r_1=L+k_{-1}<L+(1-L)=1,r_3=L+(1-k_{-1})<L+(1-L)=1.$$

Thus $r_1<1$ and $r_3<1$.

By (45) we have $0<k_{-2}<1-\kappa k_{-1}$, so

$$\kappa k_{-1}+k_{-2}<1.$$

The remaining contributions in $r_2$,

$$2\kappa {\alpha }_1\frac{\overline{G}}{\overline{S}}\mathrm{and}\beta {\left(\frac{\overline{C}}{\overline{S}}\right)}^{\iota },$$

depend linearly on ${\alpha }_1$ and $\beta$ and vanish as $({\alpha }_1,\beta )\to (0,0)$. Hence there exists ${\epsilon }_0>0$ such that, for all $0\le {\alpha }_1,\beta \le {\epsilon }_0$,

$$2\kappa {\alpha }_1\frac{\overline{G}}{\overline{S}}+\beta {\left(\frac{\overline{C}}{\overline{S}}\right)}^{\iota }<1-(\kappa k_{-1}+k_{-2}),$$

and therefore $r_2<1$.

The rows $r_4$ and $r_5$ are independent of ${\alpha }_1,\beta$ and satisfy

$$r_4={\mu }_m+(1-{\delta }_m)<1,r_5={\mu }_H+(1-{\delta }_H)<1$$

by (46). For $r_6$, rewrite

$$r_6={\mu }_C+(1-{\delta }_C)+\beta \left[\iota {\left(\frac{\overline{C}}{\overline{S}}\right)}^{\iota }+{\iota }^2{\left(\frac{\overline{C}}{\overline{S}}\right)}^{\iota -1}\right].$$

The first bracket ${\mu }_C+(1-{\delta }_C)$ is strictly less than one by (46), and the $\beta$-dependent term tends to zero as $\beta \to 0$. Shrinking ${\epsilon }_0$ if necessary, we ensure $r_6<1$ for all $0\le \beta \le {\epsilon }_0$.

Similarly,

$$r_7=(1-k_{-2})+\beta \left[{\left(\frac{\overline{C}}{\overline{S}}\right)}^{\iota }+\iota {\left(\frac{\overline{C}}{\overline{S}}\right)}^{\iota -1}\right].$$

Since $0<k_{-2}<1$, we have $1-k_{-2}<1$, and the $\beta$-term again vanishes as $\beta \to 0$. Reducing ${\epsilon }_0$ once more if needed, we obtain $r_7<1$ for all $0\le \beta \le {\epsilon }_0$.

Thus, for all $0\le {\alpha }_1,\beta \le {\epsilon }_0$,

$$r_i<1\mathrm{for}\mathrm{all}i=1,\dots ,7,$$

so that

$${\parallel B\parallel }_{\infty }=\underset{1\le i\le 7}{max}{r}_i<1.$$

It follows that $\rho \left(B\right)\le {\parallel B\parallel }_{\infty }<1$, hence $A=I-B$ is a nonsingular M-matrix. □

To solve the model numerically without encountering extreme magnitudes, we nondimensionalize all compartments by the initial SREBP-2 pool $S_0>0$:

$$g={\displaystyle \frac{G}{S_0}},s={\displaystyle \frac{S}{S_0}},s_b={\displaystyle \frac{S_b}{S_0}},m={\displaystyle \frac{M}{S_0}},h={\displaystyle \frac{H}{S_0}},c={\displaystyle \frac{C}{S_0}},c_b={\displaystyle \frac{C_b}{S_0}},$$

while keeping time t measured in seconds.

Collecting all $S_0$-powers into grouped rate parameters, we define

$${\alpha }_1:=k_1{S}_0^{\kappa },{\alpha }_2:=\kappa k_1{S}_0^{\kappa -1},\beta :=k_2{S}_0^{\iota },$$

so that the terms $k_1{S}^{\kappa }$, $\kappa k_1{S}^{\kappa -1}G$, and $k_2{C}^{\iota }S$ become ${\alpha }_1{s}^{\kappa }$, ${\alpha }_{2}g{s}^{\kappa -1}$, and $\beta c^{\iota }s$, respectively.

The scaled ODE system (for $\kappa ,\iota \ge 1$) then reads

$$\begin{array}{cc}\hfill \dot{g}& =k_{-1}{s}_b-{\alpha }_{1}g{s}^{\kappa },\hfill \\ \hfill \dot{s}& =\kappa k_{-1}{s}_b-{\alpha }_{2}g{s}^{\kappa -1}-\beta c^{\iota }s+k_{-2}{c}_b,\hfill \\ \hfill {\dot{s}}_b& ={\alpha }_{1}g{s}^{\kappa }-k_{-1}{s}_b,\hfill \\ \hfill \dot{m}& ={\mu }_m{s}_b-{\delta }_{m}m,\hfill \\ \hfill \dot{h}& ={\mu }_{H}m-{\delta }_{H}h,\hfill \\ \hfill \dot{c}& ={\mu }_{C}h-{\delta }_{C}c-\iota \beta c^{\iota }s+\iota k_{-2}{c}_b,\hfill \\ \hfill {\dot{c}}_b& =\beta c^{\iota }s-k_{-2}{c}_b,\hfill \end{array}$$

with dimensionless initial conditions

$$g\left(0\right)={\displaystyle \frac{G_0}{S_0}},s\left(0\right)=1,s_b\left(0\right)=0,m\left(0\right)={\displaystyle \frac{M_0}{S_0}},h\left(0\right)={\displaystyle \frac{H_0}{S_0}},c\left(0\right)={\displaystyle \frac{C_0}{S_0}},c_b\left(0\right)=0.$$

We then computed heatmaps over logarithmic grids in the parameter plane $({\alpha }_1,\beta )$ to visualize the behavior of the reduced Jacobian B through its induced matrix norms. Two representative presets are presented below to illustrate the theoretical small-gain predictions and the associated stability regions.

• Stable regime. We fix the grouped rates ${\alpha }_{1,\mathrm{base}}={10}^{-3}{\mathrm{s}}^{-1}$ and ${\beta }_{\mathrm{base}}={10}^{-8}{\mathrm{s}}^{-1}$, stoichiometric exponents $\kappa =3$ and $\iota =4$, and choose

  $$k_{-1}=0.10,k_{-2}=0.10,{\delta }_m={\delta }_H={\delta }_C=0.995(\mathrm{all}\mathrm{in}{\mathrm{s}}^{-1}).$$
  The fixed kinetic and scaling constants are

  $$S_0=8.21\times {10}^{16},G_0=1.00\times {10}^{13},K_m=3.26\times {10}^{13}(\mathrm{all}\mathrm{in}\mathrm{molecules}{\mathrm{mL}}^{-1}),$$

  $${\mu }_m=5.17\times {10}^5(\mathrm{molecules}{\mathrm{mL}}^{-1}{\mathrm{s}}^{-1}),$$

  $${\mu }_H=3.33\times {10}^{-2},{\mu }_C=4.33\times {10}^{-2}(\mathrm{both}\mathrm{in}{\mathrm{s}}^{-1}).$$
  These parameters satisfy $\kappa k_{-1}+k_{-2}=3·0.10+0.10=0.40<1$, so the system operates close to the theoretical small-gain stability threshold.
  On the $({\alpha }_1,\beta )$-multiplier grid, the heatmaps in Figure 3 show that both the spectral norm ${\parallel B\parallel }_2$ and the infinity norm ${\parallel B\parallel }_{\infty }$ remain strictly below 1 across the entire domain, while the Frobenius norm ${\parallel B\parallel }_F$ stays consistently above 1. Hence, stability can be certified by the ${\mathsf{\ell }}_{\infty }$- and spectral-norm criteria, whereas the Frobenius criterion fails to do so (cf. Lemma 4).
  To assess parametric sensitivity, Figure 4 shows stability fractions under pairwise variation of $(k_{-1},{\delta }_{m,H,C})$. The system is found to be most sensitive to the dissociation rate $k_{-1}$, with only about $29\%$ of the sampled domain satisfying ${\parallel B\parallel }_{\infty }<1$, while the degradation fractions ${\delta }_m,{\delta }_H,{\delta }_C$ exert comparatively minor influence, yielding stability fractions above $90\%$. The contour plots delineate the stability region and illustrate how the infinity-norm criterion quantitatively partitions stable versus unstable configurations.
  For any parameter tuple $\theta$ (e.g., grouped-rate multipliers $({\alpha }_1,\beta )$ or a pairwise scan $(x,y)\in {[0,1]}^2$), let $B\left(\theta \right)\in {\mathbb{R}}^{7\times 7}$ denote the associated interconnection matrix. We define the stability fraction over a finite grid $\mathcal{G}=\{{\theta }_1,\dots ,{\theta }_N\}$ by

  $${\mathrm{frac}}_{\infty }\left(\mathcal{G}\right)={\displaystyle \frac{1}{N}}\sum _{k=1}^N{\mathbf{1}}_{\{\parallel B\left({\theta }_k\right){\parallel }_{\infty }<1\}},$$

  where ${\mathbf{1}}_A$ is the indicator function of the event A:

  $${\mathbf{1}}_A=\left\{\begin{array}{cc}1,\hfill & \mathrm{if}A\mathrm{is}\mathrm{true},\hfill \\ 0,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$
  Thus each grid point ${\theta }_k$ contributes 1 if $\parallel B\left({\theta }_k\right){\parallel }_{\infty }<1$, and 0 otherwise. Dividing by N gives the proportion of the sampled domain in which the small-gain stability condition is satisfied.
• Critical boundary. We fix the grouped multipliers ${\alpha }_{1,\mathrm{base}}={10}^{-3}{\mathrm{s}}^{-1}$ and ${\beta }_{\mathrm{base}}={10}^{-8}{\mathrm{s}}^{-1}$, and use the model’s stoichiometric exponents $\kappa =3$ (DNA binding) and $\iota =4$ (cholesterol binding). Choose the following rate parameters (all in ${\mathrm{s}}^{-1}$):

  $$\begin{array}{cc}\hfill k_{-1}& =0.18,k_{-2}=0.40,{\delta }_m={\delta }_H={\delta }_C=0.990,\hfill \\ \hfill {\mu }_H& =3.33\times {10}^{-2},{\mu }_C=4.33\times {10}^{-2}.\hfill \end{array}$$

  $$S_0=8.21\times {10}^{16},G_0=1.00\times {10}^{13},K_m=3.26\times {10}^{13}(\mathrm{all}\mathrm{in}\mathrm{molecules}{\mathrm{mL}}^{-1}),$$

  and

  $${\mu }_m=5.17\times {10}^5(\mathrm{molecules}{\mathrm{mL}}^{-1}{\mathrm{s}}^{-1}).$$
  With these values we obtain

  $$\kappa k_{-1}+k_{-2}=3·0.18+0.40=0.94<1,$$

  placing the system near the small–gain stability threshold. On the $({\alpha }_1,\beta )$ multiplier grid, the heatmaps in Figure 5 show that the contour ${\parallel B\parallel }_{\infty }=1$ intersects the domain, producing both stable regions (${\parallel B\parallel }_{\infty }<1$) and unstable regions (${\parallel B\parallel }_{\infty }>1$). In contrast, ${\parallel B\parallel }_2>1$ and ${\parallel B\parallel }_F>1$ throughout, so neither the spectral nor the Frobenius norm can certify stability in this regime.
  For the pairwise scans in Figure 6, we vary $k_{-1}$ against a common degradation rate $\delta :={\delta }_m={\delta }_H={\delta }_C$. The dissociation rate $k_{-1}$ is the most influential: only about $18\%$ of the $(k_{-1},\delta )$ plane satisfies ${\parallel B\parallel }_{\infty }<1$, whereas the degradation rates have a comparatively minor effect, with stability fractions exceeding $90\%$. The resulting contours delineate the stability boundary and illustrate how the ${\mathsf{\ell }}_{\infty }$-criterion partitions parameter space into stable and unstable regions.

These simulations are consistent with the analytical results. The Frobenius norm cannot be used to guarantee stability, whereas the infinity norm ${\parallel B\parallel }_{\infty }$ can. Moreover, the spectral norm ${\parallel B\parallel }_2$ lies between the spectral radius $\rho \left(B\right)$ and the Frobenius norm ${\parallel B\parallel }_F$, and therefore yields a sharper, yet still sufficient, stability test whenever ${\parallel B\parallel }_2<1$.

5. Discussion

In this work, we developed and analyzed a comprehensive mathematical model of the SREBP-2 cholesterol biosynthesis pathway, integrating transcriptional regulation, metabolic synthesis, and feedback inhibition. Using M-matrix theory, we established sufficient norm-based conditions that guarantee stability of the equilibrium. A central outcome is the identification of the ${\mathsf{\ell }}_{\infty }$-criterion ${\parallel B\parallel }_{\infty }<1$ as a reliable and biologically meaningful stability test, while showing that the Frobenius norm ${\parallel B\parallel }_F\ge 1$ does not guarantee stability. The transition at ${\parallel B\parallel }_{\infty }=1$ marks a loss of diagonal dominance and is suggestive of a bifurcation threshold (by analogy with Hopf bifurcation), a phenomenon to be investigated in detail for this large-scale model in future work. The spectral norm ${\parallel B\parallel }_2$, lying between $\rho \left(B\right)$ and ${\parallel B\parallel }_F$, provides a sharper yet still sufficient condition whenever ${\parallel B\parallel }_2<1$.

Our simulations reinforce these theoretical findings. Heatmaps across $({\alpha }_1,\beta )$-multiplier grids confirm that stability is robustly captured by ${\parallel B\parallel }_{\infty }$, whereas ${\parallel B\parallel }_F$ remains uninformative. Sensitivity analyses reveal that the dissociation rate $k_{-1}$ is the dominant parameter influencing stability, while degradation fractions $({\delta }_m,{\delta }_h,{\delta }_c)$ exert comparatively weaker effects. These results provide mechanistic insight into cholesterol regulation: while degradation processes ensure baseline clearance, the dynamics of transcription-factor dissociation chiefly determine whether the system remains in a homeostatic regime. Biologically, this highlights that the decision point for HMGCR transcription (active vs. silenced) depends more strongly on SREBP-2 promoter dissociation dynamics than on downstream turnover processes. In practice, even modest alterations in $k_{-1}$, whether via mutations, diet-induced signaling, or drug action, can tip the balance between stable cholesterol control and pathological dysregulation.

From a translational perspective, these findings emphasize the importance of controlling dissociation dynamics in therapeutic interventions targeting cholesterol metabolism. Pharmacological strategies that alter SREBP-2 binding affinities or modulate degradation rates may shift the system across the stability boundary. Our results therefore suggest that $k_{-1}$ could serve as a critical “control knob” for homeostasis, and that drugs designed to stabilize the SREBP-2-DNA interaction might buffer against instability more effectively than those acting solely on degradation pathways. The mathematical framework developed here thus offers a predictive tool for assessing how biochemical perturbations impact homeostatic balance. More broadly, the robustness of the ${\mathsf{\ell }}_{\infty }$ criterion admits a clear biological interpretation: the pathway remains stable as long as no single interaction (binding, dissociation, or synthesis) overwhelms the others. This mirrors how cells maintain cholesterol within a narrow range despite large environmental fluctuations, a property essential for membrane integrity and signaling.

Several limitations should be noted. First, the model assumes a well-mixed cellular environment and neglects stochastic fluctuations, both of which are known to affect transcriptional regulation. Second, parameter estimates are drawn from available experimental data and may vary across organisms and conditions. Future work should integrate parameter uncertainty, explore stochastic formulations, and extend the analysis to multiscale models that couple intracellular regulation with systemic lipid transport. It is also important to note that current funding support was dedicated to the full mathematical analysis of the SREBP-2 pathway. To fully validate these findings, experimental data should be collected under physiologically realistic conditions, enabling direct comparison between predicted and observed homeostatic responses. Our framework provides a solid theoretical basis for such studies, and the next step will be to perform targeted experiments to confirm and refine the model’s predictions.

Overall, our study demonstrates the utility of M-matrix theory and norm-based stability analysis in clarifying the regulatory principles of complex biochemical networks. By connecting rigorous mathematical results with biologically interpretable conclusions, we provide a framework for future studies of cholesterol regulation and related metabolic pathways. In particular, by linking mathematical stability criteria directly to biological control points, our analysis reveals which molecular parameters govern whether cells maintain cholesterol balance or drift toward disease states. Numerical simulations and stability maps presented in Section 4 (Figure 3, Figure 4, Figure 5 and Figure 6) illustrate stability and instability regions across parameter variations, as well as sensitivity with respect to $k_{-1}$ and degradation fractions. These results provide strong support for the proposed M-matrix framework and norm-based criteria. An important direction for future work is the application of bifurcation theory [38], which offers a complementary viewpoint for the analysis of this comprehensive model. Although a detailed bifurcation study lies beyond the present scope, our results already indicate that suitable parameter combinations can drive the system into instability, with the selection of appropriate norm criteria playing a central role. We therefore conclude that the present framework delivers rigorous and quantitative support for stability analysis, while bifurcation methods represent a natural extension to further characterize the dynamics of this large system.