In biology, the processing of information at a cellular level also requires some digital capacity, because the categorisation of inputs and the structuration of an output behaviour require that decisions be made which may not be a compromise between different possibilities. For example, a cell should differentiate into a specific cell-type, and should not possess characteristics of different types in varying proportions. Positive feedback has for example been shown to be implicated in the reading out of morphogen gradients, with transcription factors responding differently to concentrations of morphogen, and being part of mutually-repressive positive feedback circuits [17]. Positive feedback circuits are crucial components of models of gradient interpretation, such as those presented in [18], [19], and [20] (interestingly, in one of the cases proposed in [20], positive feedback is necessary for stability, for reasons different from those discussed above). The role of feedback circuits in development has also been investigated in [21], based on a formalism developed in [22].
Possible elements to approach all-or-none responses are enzymes exhibiting allosteric kinetics, whose response curves are sigmoidal and not hyperbolic as in classical Michaelis-Menten kinetics, or modifying enzymes functioning in conditions of zero-order ultrasensitivity ([23], see [24] for an application to thresholds in development). However sigmoid functions can only come close to Heaviside step functions, and cannot ever be the source of a perfect, all-or-nothing response, as emphasised in [18]; the approximation of this kind of response can be enhanced, at the cost of complexification of the allosteric mechanism, but cannot be made perfect. An active mechanism based on positive feedback can however solve this problem [18], based on an underlying structural apparatus which needs not be sophisticated. The paradigm is that of a molecule controlled both by positive feedback and negative feedback, and for which negative (respectively positive) feedback predominates at low and high (respectively middle-valued) concentrations. Such a system is quite easy to design in such a way that there are only two stable stationary states, one with a high concentration of the molecule, and one with a low concentration. In such a system there is a third stationary state, which is unstable (and thus never observed in real-world conditions), and which plays the role of a threshold: starting at a concentration below that threshold, the concentration is brought back to its low stationary state, and starting above that threshold, the concentration is brought back to its high stationary state. Examples of biological situations where positive feedback circuits allow for threshold effects, as proposed in [18], are the activation of blood-coagulation enzymes [25], or the MAP kinase cascades (see below). The electronic equivalents of bi-stable systems would be "Schmitt triggers".
Positive feedback thus makes it possible, with a very simple mechanism, to constrain a system variable to a fixed set of possible values, and to have a behaviour close to ideal. In the case of discrete systems, the structure of feedback circuits, and especially their sign, has been shown to be an important determinant of the behaviour of the system [26]; it has been shown that the presence of a positive feedback circuit is necessary for a discrete system to have at least two stationary states [9,27]. In the continuous case, positive feedback is a necessary feature of a wide class of autonomous differential systems having 2 or more stationary states, if all stationary states are isolated. This was a conjecture made by R. Thomas [28], and proven independently in the case of constant signs of the feedback circuits in [7] and [29]. We recently provided a more general demonstration [30]. The demonstration we proposed shows that for such a multistationary system, there is at least one unstable stationary state which possesses a real, strictly positive eigenvalue in the Jacobian matrix; positive feedback being necessary for the existence of a real, positive eigenvalue, there is positive feedback in those systems, at least around this unstable stationary state.
Positive feedback is of course no miracle solution, and there are also energetical tradeoffs involved. For the evolution of the system to be quick around unstable stationary states (and thus less sensitive to stochastic fluctuations), the positive and negative forces exerted on the concentration of the relevant molecule must be strong, which is energetically costly.
It would be quite wasteful for a switch molecule to be under strong, simultaneous synthesis and degradation, as would be the case in the paradigmatic example described above. A way in which this can be avoided is by "filtering out small stimuli", as formulated in [31], so that competition between positive and negative feedback circuits is kept at a minimum. This filtering process can be provided by sigmoidal response curves; MAP kinase cascades, often involved in switch-like responses, exhibit such a type of response [31], and also have a positive feedback circuit [31,32]. The situation is thus the following: given a system which already exhibits some switch-like responses, even if quite imperfect, positive feedback circuits make it possible to give to the system an almost perfect switch-like behaviour; the better the original system, the lower the energetical cost to improve it.
An essential difference between positive and negative feedbacks is the existence of a real, positive eigenvalue allowed by positive feedback. Negative feedback can give rise to complex eigenvalues with positive real part; the complex component creates expanding oscillations in the dynamics around the stationary state. Such oscillations are most undesirable in a system trying to make a clear-cut decision.
Positive feedback is useful not only in making decisions, but also in memory, as emphasised in [18], and shown on a larger scale in [33]. First, positive feedback is a necessary condition for multistationarity, as shown in [30]; the greater the number of attractors of a system, the greater its storage capacity. Second, physical storage of information is a difficult task because of the natural tendency toward degradation; memory must be continuously self-renewed, a natural role of positive feedback. Computer memories make use of positive feedback, and so does the brain, if one goes with the hypothesis that memories are stored as strengths of connections between neurones, activated connections being reinforced (neurones also make use of positive feedback when their membrane depolarises [34]). A form of biological memory relying on positive feedback circuits, identified early on, is that of the lactose permease [35]. Different isoforms of the prion protein could also be a striking example of memory, as discussed in [36], which would have important implications both from the theoretical and therapeutical points of view. Positive feedback circuits could also provide memory at the level of ecosystems [37].
Finally, taking further the comparison between biological and computerised information processing, let us note that computers and brains both have highly oscillatory functioning, which, in the case of the computer, makes it possible to treat problems in a stepwise manner and break a complex computation into many elementary ones. It seems likely that intra-cellular regulation systems are not that sophisticated; a fundamental role for oscillations has not been proven so far in intra-cellular computations (but it has been shown that signals can be encoded by different frequencies of Ca
oscillations [38,39]). This however does not mean that temporal aspects are not important in cellular systems: cells can have qualitatively different responses to the same stimulus applied for differents lengths of time (p42 and p44 MAP-kinase pathways seem capable of such responses, as reported in [40]), and often prove capable of desensitisation overtime (which requires the use of integral feedback [41], which we won't discuss here). The progression through the cell cycle shows that temporal aspects are also important when it comes to arranging a set of tasks for physical, not informational, purposes. Also, cellular oscillations in the production of a messenger molecule makes it possible to create sharp peaks of that molecule, when a large cell population is temporarily synchronised.
These remarks bring us to the role of positive feedback circuits in the generation of oscillations.