Positive feedback can be stabilising

On the other hand, positive feedback can be necessary for a system to preserve its stability. Again we will illustrate this point by two examples, a linear one, for which we do not have generating biological kinetic equations at hand, and one with a more straightforward biological interpretation.

Let us first consider the linear system with three variables $x,y,z$, whose interaction graph is constant, and is the one shown in Figure 2 (again, to this graph corresponds a unique linear system with 0 as initial condition). Such a system could arise from the linearisation of biological kinetic equations around a stationary state. The steady state (0,0,0) of this system is stable with the proposed coefficients; however, if the positive feedback on variable $z$ is diminished by a quantity equal at least to 1 (or replaced by negative feedback), other coefficients remaining unmodified, the system becomes unstable. The salient feature of this system is that there is one variable, $x$, which is under control of a negative feedback circuit of length greater than 2, while other variables are under control of negative feedback circuits of length equal to or less than 2. Intuitively, if the positive feedback on variable $z$ is diminished, the propagation of the "error reading" on $x$ will be too damped for the correction to be effective. Figures 3 and 4 show parametric timeplots of variables $x$ and $z$ after a perturbation on $x$, for two different values of $z$ autocatalysis. When the autocatalysis is high enough (Figure 3), the system goes back to its steady state $(0,0,0)$ in an oscillatory fashion. But when the autocatalysis coefficient is lowered by 1 unit, $z$ does not stay sufficiently negative to bring $x$ back to 0, and the system diverges (Figure 4): when $z$ has become sufficiently negative, $x$ starts decreasing (first vertical tangent on the plot, starting from the origin), but $z$ then starts going back to 0 (horizontal tangent), and $x$ finally sets on an ever-increasing course (starting from the second vertical tangent). In contrast, sufficient auto-catalysis of $z$ allows a more persistantly negative value of $z$: even though a decrease in $x$ also causes $z$ to go back toward zero after some time, this time interval is sufficiently great for $x$ to have decreased by a sufficient amount by the time it increases again (and thus causes $z$ to go back toward negative values), for the process to converge.

Another example is the following 3-dimensional system. Species and their concentrations are denoted in the same way, by $x,y,z$. A nefarious biochemical species $x$ is supposed to promote its own synthesis by an enzyme normally serving other purposes, following Michaelis kinetics. The presence of $x$ induces the transcription of an mRNA species $y$, following the same kinetics. Finally, protein $z$, translated from mRNA $y$, can competitively inhibit the auto-promoted creation of $x$, as well as the expression of its own mRNA $y$. An important feature of protein $z$ is that it activates its own translation from mRNA $y$, with a maximal speed $a\ \in \ \Re^{*}_{+}$. All three entities are supposed to undergo exponential decay, and translation as well as transcription are supposed to be leaky. The corresponding equations are the following, for a generic choice of parameters (leakage in mRNA and protein synthesis is chosen to be small compared to induced synthesis, and $x$ undergoes decay that is slow compared to its synthesis):

$$\begin{array}{lll} \frac{\mathrm{d}x}{\mathrm{dt}}=-0.1x+\fr... ...{\mathrm{d}z}{\mathrm{dt}}=-z+y(0.1+a\frac{z}{1+z}) \end{array}$$ (2)

Were there no leakage in $y$ transcription, the only steady state for which $x=0$ would be $x=y=z=0$, and would be unstable.

This system possesses a steady state for $x=0$ and $y=0.1$, and $z$ given by a quadratic equation which always has a positive root. A bifurcation study using the xppaut program [10] shows that the steady-state with $x=0$ and $z>0$ is stable if and only if $a>97.4$, i.e. if and only if the positive autocatalysis of $z$ is sufficiently strong. If not, small quantities of $x$ are amplified before $z$ can exert its repressing effect. Figure 5 shows the response of the system to a small perturbation in $x$, for $a=100$, starting from the steady state with $x=0$ and $z>0$: $z$ increases sharply, and the system then returns to $0$. Figure 6 shows the response of the system for $a=50$, also starting from the steady state with $x=0$ and $z>0$: the rise in $z$ is not as sharp, and the system does not go back to $0$. There is an important difference between the two responses around the region of the plots where $t=0$: while for $a=100$ the evolution of $z$ is markedly quicker than the evolution of $x$ (the parametric curve has a vertical tangent for $t=0$), when $a=50$ the opposite is true (the parametric curve has a flat tangent for $t=0$). Intuitively, in the case of a stable steady state, $z$ rises quickly enough to repress $x$ "before it is too late". Note that, in general, increasing parameters involved in positive feedback circuits does not necessarily lead to more positive weights of feedback circuits at stationary states, but to more positive weights at precise points of the state space (because the location of the stationary states is generally altered when the parameters of the system are changed).

It could make sense biologically to have an inducible expression of $z$ if $z$ was costly to produce, or if it interfered not only with the production of $x$ but also with physiologically desirable reactions. Delays between triggering of a signal and mRNA transcription, and then mRNA translation, are very noticeable in cells, especially eukaryotic cells; it could therefore be that positive feedback has a role in stabilising systems which involve long-range negative feedback loops, that length being bounded from below by structural, biological constraints.

Systems in which a variable is under positive auto-catalysis, and under control of a long negative feedback circuit, could also be relevant to the modelling of the immunological system. A virus replicates auto-catalytically, and the acquired immunological response requires many successive steps eventually leading to the prevention of its replication. The presence of memory cells allows subsequent infections to occur with different initial conditions (in the above system, leakage allows $z$ to be non-zero at rest, and makes it possible for the steady state with $x=0$ to be stable), but there are positive feedback circuits at play in immunological responses, for example between interleukin 12 and interferon $\gamma$ [11].

Figure 2: Example of a graph of a Jacobian matrix in which, other things being equal, a positive autocatalysis coefficient (highlighted in the figure) cannot be diminished by a quantity equal at least to 1, without the associated system becoming unstable.

Figure 3: Trajectory of the linear system whose matrix is given by the interaction graph in Figure 2, with initial conditions x=1,y=0,z=0. Arrows mark the direction of evolution of the system. The system converges to 0.

Figure 4: Trajectory of the linear system whose matrix is given by the interaction graph in Figure 2, the highlighted coefficient being diminished by one, with initial conditions x=1,y=0,z=0. The arrow marks the direction of evolution of the system. The system diverges.

Figure 5: Trajectory of system 2, for a=100, with initial conditions x=0.1,y=0.1,z=9.1 (the value of x being thus slightly increased from the steady state). The arrow marks the direction of evolution of the system. After a sharp increase in z, the system converges back to its steady state where x=0.

Figure 6: Trajectory of system 2, for a=50, with initial conditions x=0.1,y=0.1,z=4 (the value of x being thus slightly increased from the steady state). The arrow marks the direction of evolution of the system. The system does not return to the steady state where x=0.