How to mathematically characterize a feedback loop in ODEs?

Question

I have a biological system that exhibits a feedback type of behavior. The diagram is a schematic of the system of ODEs. In this system, the total amount of $x_1, x_2, x_3$ is conserved; however, there are transitions between them at a rate of $m_{ij} > 0$ proportional to state $x_i$. During state $x_2$, the byproduct $y$ is produced at a rate of $\alpha > 0$ proportional to $x_2$. Additionally, $y$ degrades at a rate of $\gamma > 0$ proportional to itself. The amount of $y$ affects the transition rate from $x_2$ to $x_3$ positively. This is represented by a monotone function $f(y)$ that is bounded between $1$ and $m>1$.

Edit 2: the production of $y$ does not consume $x_2$, just the materials floating in the environment. $x_2$ acts like a machine that takes those materials to make $y$. The schematic is a simplified diagram of the transcription-translation processes for DNA-RNA-protein.

The standard biological description would categorize the process involved $y$ as a positive loop. In fact, the underlying biological process is known to exhibit a "positive feedback loop". But this does not make logical sense. My observation is that as $y$ increases, $f(y)$ increases, leading to a higher transition rate from $x_2$ to $x_3$; however, it is not clear that would lead to either an increase in $y$ or not. Furthermore, it seems that whether this is a positive or negative feedback loop relies on the relative ratios of the other transition rates. Numerically, I thought of characterizing the loop as the ratio between $\frac{dy/dt}{d^2y/dt^2}$, for instance, if $\frac{dy/dt}{d^2y/dt^2} > 0$, then I have a positive feedback loop and vice versa.

My question is: how do I characterize whether $y$ is involved in a negative or positive feedback loop?

I found a similar concept based on the Jacobian matrix that is used to characterize "qualitative stability", but I don't think it is quite the same. I have searched through many system biology/math-bio references, but I have not found such materials. If there is some standard method for such characterization, please help point me to the right references. Thanks in advance!

Edit 1: I believe the standard qualitative stability theory states that in general, stability will follow in a negative feedback loop (for instance, page 240 of Leah Edelstein-Keshet's Mathematical Models in Biology). Hence, I think it may be more precise for me to ask for a mathematical characterization of a "self-activating" or "self-inhibiting" loop involving the $y$ compartment.

Edit 3: As Alexandre Eremenko pointed out, I should explain the question and motivation from a mathematical point of view. The current theory of qualitative stability uses a linearized characterization of the feedback loop (either a self-inhibiting or a self-activating loop). In this example, this is equivalent to looking at $\partial Y'/\partial Y = -\gamma < 0$. Hence, $Y$ is concluded to be in a negative feedback loop. However, this is not the case (at least for some situations). A better description would involve the sign of $Y'/Y''$, which is better and allows for more possibilities but is still only an approximation. Thus I would like to ask if there is a way to mathematically characterize the feedback loop of $Y$ without the need for approximation.

Answer 1

This is a site for mathematical questions. Let me try to state your question in mathematical terms, and you tell us whether I translated it correctly or not. Let $\mathbf{x}=(x_1,x_2,x_3,y)^T$ be a time dependent vector in $R^4$, satisfying the differential equation $$\mathbf{x}'=A(\mathbf{x})\mathbf{x},$$ where $A$ is the matrix $$A=\left(\begin{array}{cccc}-m_{12}&m_{21}&0&0\\ m_{12}&-(m_{21}+m_{23}f(y)+\alpha)&m_{32}&0\\ 0&m_{23}f(y)&-m_{32}&0\\ 0&\alpha&0&-\gamma\end{array}\right),$$ and (I suppose) all $m_{ij}$ are positive and $f$ is some function $|f|\leq 1$. And you want to know whether $0$ is a stable equilibrium.

In fact the system decouples: the first 3 equations are independent of $y$.

Certainly the answer will depend on $m_{ij}$ and $k:=f(0)$, under some reasonable assumption about the behavior of $f$. If $f$ is continuous, the answer depends first of all on the eigenvalues of the Jacobi matrix $$J=\left(\begin{array}{cccc}-m_{12}&m_{21}&0&0\\ m_{12}&-(m_{21}+m_{23}k+\alpha)&m_{32}&0\\ 0&m_{23}k&-m_{32}&0\\ 0&\alpha&0&-\gamma \end{array}\right).$$ A sufficient condition of stability (negative feedback) is that all these eigenvalues have negative real part, and if one of them has positive real part then the system is unstable (positive feedback).

Since one negative root of the characteristic polynomial, namely $-\gamma$ factors out, the question is reduced to stability of a cubic polynomial.

The stability criterion for a polynomial of third degree $$\lambda^3+a_1\lambda^2+a_2\lambda+a_3$$ is $a_1>0,\; a_1a_2-a_3>0,\; a_3>0$. Computation with Maple (if I made no mistake) shows that your system is indeed stable for all positive values parameters $m_{ij},\alpha,\gamma,k$ are positive. If some of them are allowed to be $0$, the question may depend on other properties of $f$.

Edit. In fact no computaton is necessary: the 3 times 3 upper left submatrix is a Jacobi matrix, and always has negative eigenvalues.