Tyson et al., (2003). Sniffers, buzzers, toggles and blinkers: dynamics of regulatory and signaling pathways in the cell.
Biological systems often are of a staggering complexity, not only in the number of components and interactions, but also in the behaviours they can exhibit, and unravelling the underlying regulatory networks is a daunting task. One way lies in the creation of mechanistic mathematical models of a molecular interaction network, that capture the dynamics of the biological system, and allow in depth analysis of its properties and potential behaviours. Sometimes it is possible to identify certain parts of such a network, that can be linked to distinct functions, allowing bigger systems to be decomposed into smaller modules, akin to electrical circuits.
In their article  Tyson et al., first introduce a number of simple functional motifs which produce a certain response in dependence on an incoming signal, and then show how some of these can be identified as underlying modules of a more complex network, the regulation of the eukaryotic cell cycle, and how similar motifs can underlie pattern formation in spatially inhomogeneous systems. For the mathematical formulation of the systems only simple mass action and Michelis-Menten kinetics are used, keeping the applicability of the modules very general. The functional motifs are classified as sniffers, buzzers, toggles and blinkers.
A sniffer, similar to the human sense of smell, is a system that shows a response to changing signal strength and then quickly falls back to a basal level. This is realised in the article by a direct positive influence of the signal S on the response R and an indirect negative influence over the intermediate regulator X (see figure 1 and BIOMD0000000312). With increasing S, production of R is increased, leading to a rise in R. At the same time S also increases production of X, which enhances the degradation of R, thereby lowering its concentration again, giving only a short overall response of R to changes of S (figure 1). This kind of regulation with of R by S is also known as feed-forward, and has been found as a common motif in gene regulation .
Figure 1: Network diagram of the sniffer or perfectly adapted system and time courses of R and X under stepwise increasing signal strength S. In the diagram rounded rectangles are entities, crossed circles stand for sinks or sources, and arrows with filled heads and boxes denote reactions. Connections between an entity node and a reaction denote either activatory (empty arrowhead) or inhibitory (blunt arrowhead) regulations. In the time course the signal strength S is increased by 1 unit every 4 time units. (after fig. 1d in )
Figure 2: Network diagram and signal-response curve for the toggle switch with mutual inhibition (after fig. 1f in ). In the signal-response curve the steady states of R are plotted over a range of values of signal strength S. Thick lines stand for stable, thin ones for unstable steady states. The two points SN5 and SN9 stand for the saddle-node bifurcations at which the system switches between mono- and bistability. Starting from low values of S, R stays low until S reaches the value at SN5, at which R switches to the upper stable state. If S is subsequently reduced, R stays in the upper stable branch until S falls below the value at SN9, at which R switches down again.
The buzzer is a system that, at a certain threshold signal strength, switches reversibly from a level of low to high response. The form of the signal-response curve is also known as sigmoidal, due to the likeness of the response curve to the letter S. In the article this form is achieved using a reversible covalent modification cycle, in which the signal S acts as for example as a kinase, phosphorylating a protein R. Koshland and Goldbeter  first described, how cooperativity could arise in such systems, if at least the one of the modifying enzymes operates close to saturation. Their derivation of a the steady state ratio of the modified to the unmodified form of the modified protein is also known as the Goldbeter- Koshland function.
More complex regulatory interactions are needed for the last two categories, toggle switches and blinkers or oscillators. To create systems showing these kinds of behaviour, components of the system have to show a direct or indirect influence on their levels. These kinds of interactions are also known as feedbacks.  gives a concise introduction to feedback loops in biology and some of the dynamics emerging from them.
While Buzzers and toggles are similar both show a sudden, non-linear change in response to crossing of a threshold value of signal, toggles show a different kind of behaviour, when the signal strength goes back again. A toggle switch has two critical values of signal strength, between which it exhibits two stable states of response, or bistability. The system stays in the off state until the signal strength surpasses the upper critical value, and only switches back from the on to the off state, if it falls beyond the lower critical value. The phenomenon that the systems response does not only depend on the signal strength, but also on the state the system is in, is also known as hysteresis. If the lower critical value is smaller than zero, the switch becomes irreversible.
Toggle switches can occur in systems with positive feedbacks. The authors achieve this positive feedback via two divergent strategies. Once via a cycle of two activating interactions and once via two negative interactions. In the mutually activating case (BIOMD0000000311), the response R activates the phosphorylation of an intermediate E to Ep, which in turn activates expression of R. With the right parameters, this loop can lead to irreversible switches. For the mutually inhibitory cycle (BIOMD0000000310), R is negatively regulated by E, which enhances the degradation of R. By phosphorylating E to EP, R relieves this inhibition, leading to an overall positive feedback and formation of an reversible switch (figure 2).
Oscillatory systems, in contrast to switches, generally need a negative feedback in combination with some sort of a temporal delay. One way of achieving this delay is using an intermediary species in the feedback loop, meaning that the loop hast to consist of at least three components. Three fundamentally different oscillators are presented in the article.
The simple negative feedback oscillator (BIOMD0000000308) contains a linear signalling cascade activating the response R by phosphorylation to Rp, which in turn inhibits the signalling cascade. This system can exhibit sustained oscillations in its response Rp over a range of values of signal strength S (figure 3). Between a lower and an upper critical value of S, the system's steady state becomes unstable, while outside this region it possesses a single steady state. At the critical values the system undergoes a so-called super-critical Hopf bifurcation, which means that the amplitudes of the oscillations slowly increase from the bifurcation point onwards.
Figure 3: Network diagram and signal-response curve for the negative feedback oscillator (after fig. 2a in ). In the signal-response curve, the steady states of Rp are plotted in black, the minimal and maximal amplitude of the sustained oscillations in red. Thick and thin and dotted lines indicate stable and unstable steady states or limit cycles. The blue dots at the transition from stable to unstable steady states indicate the Hopf bifurcations.
Figure 4: Network diagram and signal-response curve for the negative feedback oscillator (after fig. 2c in ).
Another possibility for oscillations to arise are coupled negative and positive feedbacks, as exemplified in the activator-inhibitor oscillator (BIOMD0000000306). It consists of the mutually activating switch with an added component X, which is expressed by R and enhances its degradation, connecting R and X in a small, two component negative feedback loop. In this system the positive feedback loop switches R on in the bistable region and the delayed negative feedback via X switches R off to the lower level again.
A way of achieving oscillations without explicit negative feedback is shown in the substrate-depletion oscillator (figure 4 and BIOMD0000000307). Here R again exerts a positive feedback via the architecture of the mutually activating switch, only this time the signal S first activates production of an intermediate X, that is then converted into R. Above a critical value of S, the positive feedback of R now can lead to a depletion of X, which in turn leads to a decrease in R, allowing building up of X again (figure 4). In the latter two cases of oscillations, the system switches from a stable state to an unstable one with sustained oscillations at sub-critical Hopf bifurcations. This manifests itself in the sudden appearance of oscillations with big and not gradually increasing amplitudes.
In their article Tyson et al. show in clear and very elegant examples how different kinds of basic regulatory networks can give rise to varying dynamics and qualitative behaviours. The modules they introduce can be found at the base of a variety of important biological phenomena, and the authors exemplify this by dissecting the eukaryotic cell cycle. A general understanding of which kind of regulatory interactions can produce certain behaviours and finding these motifs in bigger networks is paramount to making sense of the complexity that living systems posses. This article is both a wonderful introduction to computational modelling in biology for both biologists and mathematicians and a interesting read for scientists already involved in it.
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