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Positive feedback is a ubiquitous signal transduction motif that allows systems to convert graded inputs into decisive, all-or-none outputs. Here we investigate why the positive feedback switches that regulate polarization of budding yeast, calcium signaling, Xenopus oocyte maturation, and various other processes use multiple interlinked loops rather than single positive feedback loops. Mathematical simulations revealed that linking fast and slow positive feedback loops creates a “dual-time” switch that is both rapidly inducible and resistant to noise in the upstream signaling system.
Studies in many biological systems have identified positive feedback as the key regulatory motif in the creation of switches with all-or-none “digital” output characteristics (1). Although a single positive feedback loop (A activates B and B activates A) or the equivalent double-negative feedback loop (A inhibits B and B inhibits A) can, under the proper circumstances, generate a bistable all-or-none switch (1–5), it is intriguing that many biological systems have not only a single but multiple positive feedback loops (Table 1). Three examples of positive feedback systems are shown in more detail in Fig. 1. Polarization in budding yeast depends on two positive feedback loops, a rapid loop involving activity cycling of the small guanosine triphosphatase Cdc42 and a slower loop that may involve actin-mediated transport of Cdc42 (Fig. 1A) (6). In many cell types, the induction of prolonged Ca2+ signals involves initial rapid positive feedback loops centered on Ca2+ release mediated by inositol 1,4,5-trisphosphate (IP3) combined with a much slower loop that induces Ca2+ influx mediated by the depletion of Ca2+ stores (7, 8) (Fig. 1B). Xenopus oocytes respond to maturation-inducing stimuli by activating a rapid phosphorylation/dephosphorylation–mediated positive feedback loop (between Cdc2, Myt1, and Cdc25) and a slower translational positive feedback loop [between Cdc2 and the the mitogen-activated protein kinase (MAPK or ERK) cascade, which includes Mos, MEK (MAPK kinase), and p42] (Fig. 1C).
The presence of multiple interlinked positive loops raises the question of the performance advantage of the multiple-loop design. One clue is provided by recent studies of budding yeast polarization. When the slow positive feedback loop is selectively compromised by treatment with the actin-depolymerizing agent latrunculin, the result is rapid but unstable cell polarization (6). In contrast, cells lacking a functional fast loop (by deletion of Bem1) form stable poles, but with reduced speed (6). These experimental observations led us to hypothesize that the slow positive feedback loop is crucial for the stability of the polarized “on” state, whereas the fast loop is critical for the speed of the transition between the unpolarized “off” state and polarized on state.
To test this hypothesis computationally, we created models of positive feedback switches containing either a single positive feedback loop (Fig. 2A) or two interlinked loops (Fig. 2B). For the single-loop switch, we assumed either fast or slow kinetics for the activation and inactivation of loop component A. For the dual-loop switch, we assumed either fast kinetics for both the A and B loops, slow kinetics for both loops, or fast kinetics for the A loop and slow for the B loop (9).
Each model switch responded to a noise-free stimulus (Fig. 2, C to G, left) and a noisy stimulus (Fig. 2, C to G, right) as shown. As expected, the single-slow-loop switch turned on and off slowly and filtered out noise (Fig. 2C). Adding a second slow loop produced a higher basal activity in the off state, a quicker switch from off to on, and a slower switch from on to off (Fig. 2D). The behavior of the two-slow-loop switch was exactly equivalent to that of a single-loop switch in which the concentration of B was doubled. Thus, adding a second loop with identical kinetic constants provides a backup in the event of gene deletion, but does not otherwise alter the behavior of the system beyond what could be achieved with a single loop.
The single-fast-loop switch turned on and off rapidly and was highly susceptible to noise in both the off and on states (Fig. 2E), and adding a second fast loop quickened the transition from off to on and delayed the transition from on to off (Fig. 2F). Thus, the fast-loop switch achieved more rapid responses, but at the cost of increased noise.
In contrast, the system in which a slow and a fast positive feedback loop are linked together introduces marked advantages over single-loop systems, as well as dual-loop systems with the same time constant. In this “dual-time” switch, the output turned on rapidly, as a consequence of the kinetic properties of the fast loop, and turned off slowly as a consequence of the kinetics of the slow loop (Fig. 2G). This allows for independent tuning of the activation and deactivation times. More important, although the dual-time switch exhibited high noise sensitivity when in the off state, as a result of the rapid responses of its fast loop, it became resistant to noise once it settled in its on state as a result of the properties of its slow loop. Thus, the dual-time switch provides the ability to transit rapidly from the off state to the on state together with robust stability of the on state (10).
These computational studies help understand the yeast phenotypes described above and provide a rationale for the existence of dual-time positive feedback systems in Ca2+ signaling, oocyte maturation, and other biological systems. In the case of Ca2+ signaling, the dual-time system enables rapid Ca2+ responses from IP3-induced Ca2+ release, while also enabling long-term robust Ca2+ signals once the store-operated Ca2+ influx is triggered. Although weak stimuli or noise have been shown to trigger IP3-mediated Ca2+ spikes, more persistent stimuli are needed to induce Ca2+ influx and prolonged Ca2+ responses (7). These long-term Ca2+ signals are required for T-cell activation and differentiation and many other cellular processes (7, 8). Xenopus oocyte maturation includes a period termed interkinesis, during which Cdc2 becomes partially deactivated (11). We conjecture that the slow positive feedback loop helps prevent a transition to the off state during this critical interkinesis period.
Our study suggests that many biological systems have evolved interlinked slow and fast positive feedback loops to create reliable all-or-none switches. These dual-time switches have separately adjustable activation and deactivation times. They combine the important features of a rapid response to stimuli and a marked resistance to noise in the upstream signaling pathway.
We thank R. Brandman, Y. Brandman, T. Galvez, R. S. Lewis, L. Milenkovic, D. Mochly-Rosen, M. P. Scott, P. M. Vitorino, and R. Wedlich-Soldner who provided helpful suggestions. This work was supported by an NSF predoctoral fellowship awarded to O.B., NIH grants GM46383 to J.E.F., GM057063 to R.L., and MH064801 and GM063702 to T.M.