It is well known that auto-activation can lead to bistability [4], [12]–[15]. When an auto-activating circuit is bistable, it has two stable steady states: one in which the TF concentration is high and stays high because the TF activates the transcription of its own gene, and one in which the TF concentration is low and stays low because the TF's gene is not activated.

Fig. 2 summarizes which parameter values yield bistability in our model. As long as the fold change and the Hill coefficient of auto-activation are low, bistability cannot occur (regime I in the diagram). More precisely, bistability does not occur as long as (see Supporting Text S1)(3)( is the solid, blue line in Fig. 2.) In this regime, for any input , the circuit has a unique steady-state TF level (see the insets in Fig. 2). When exceeds the critical value , bistability sets in, but only for an intermediate range of values. For high values of and , this regime extends all the way up to (regime III, to the right of the dotted line). In this regime, the circuit is still bistable even when the signal is saturating.

Regime III seems inappropriate for a signaling circuit. Even at saturating signal levels (), the circuit will not be induced, because the low-expression state remains stable. (In stochastic models the system will eventually turn on by a random fluctuation, but the induction will be very slow, as demonstrated below.) In regime II the circuit is bistable for a range of values. There, the output is not uniquely determined by the input, because two output levels are compatible with a given . This behavior leads to hysteresis [16]. Bistability and hysteresis can presumably be beneficial in some systems [2], in particular if the circuit has to be cautious in turning on or off, or in the context of bet hedging [17]. However, in signaling systems, assuming that there is an optimal expression level for any given signal level, bistability and hysteresis will tend to trap the circuits in a non-optimal state. We argue therefore that a bistable response is often not desired. Based on these considerations, we expect that the circuit parameters should usually be in regime I (also see the discussion in Ref. [18]).

An advantage of positive feedback is that it can increase the sensitivity of a circuit [5], [6]. In the context of dose-response curves, a high sensitivity can be beneficial. In particular, highly sensitive signaling circuits allow the cell to ignore low-level signals, below a certain threshold value, which may be due to noise or cross-talk. We here quantify how much the sensitivity of signal-response circuits can benefit from auto-activation.

The sensitivity of response curves can be defined in various ways [19]. A common approach is to define the sensitivity of a response function as the maximum slope of that function in a log–log plot, that is, as . This definition has many desirable properties: a high indeed indicates that increases rapidly in some domain, the measure is invariant under scaling (), and convenient for mathematical analysis. For those reasons, we will use this measure below. However, a pitfall is that, in order for to be deemed sensitive, a high log–log slope in a single point is sufficient. As a result, a high does not guarantee that the circuit behavior resembles a binary switch [19]. Therefore, we also report results for the measure , defined as the average slope of in a log–log plot, calculated over the domain in which it switches from low to high. (In other words: in the switching domain.) The switching domain is defined heuristically as the domain in which increases from 10% to 90% of its maximum value.

Ultimately, the most relevant quantity is the sensitivity with which the expression of the target genes responds to changes in the signal level. This sensitivity is shaped by each step in the response network: the detection of the signal, the signal transduction, the modification of the TF, and the promoters of the target genes. Consequently, a sensitive response can be implemented at different places in the response network. Here we study the sensitivity that is contributed by the auto-activation circuit and therefore focus on the sensitivity of .

That auto-activation can strongly improve the sensitivity can be understood by revisiting Fig. 2. If the parameters are chosen near the border between regions I and II, the response is almost bistable, leading to a high log–log slope (see inset in Fig. 2). Indeed, for the model in Eq. 1 and 2, can be calculated exactly (see derivation in Supporting Text S1):(4)( was defined in Eq. 3.) This confirms that the maximal log–log derivative diverges when approaches the critical value . We conclude that an arbitrarily high can be obtained for any Hill coefficient by properly choosing the fold change, and vice versa.

In particular, if the auto-activation is non-cooperative (), Eq. 4 reduces to(5)proving that, even in the absence of molecular cooperativity, increases without bound when is increased. Indeed, in the limit of large a strict threshold response is obtained. We illustrate this in Fig. 3A and B. Assuming is large, the expression of the TF, , is fully inhibited when is below the threshold ; for it takes the form of a translated (shifted) Michaelis–Menten curve (Fig. 3A). In the same limit of high , the concentration of activated TF is threshold-linear (Fig. 3B). (Mathematical derivations are provided in the Supplementary Text S1.)

Fig. 3C graphically explains the origin of this behavior. It plots the rate of production of TFs, , as a function of the TF concentration, for four signal levels; the degradation rate is shown too. The steady state value of is the value at which production and degradation balance, i.e., where the production and degradation lines cross (indicated with dots). If is large, the steady state level is large. But if is reduced enough, the production line shifts below the degradation line and the steady state becomes . The threshold value is and can be varied biochemically by tuning .

To illustrate how remarkable the values of are that can be achieved using auto-activation, we reexamine the open-loop case (see Fig. 1). The open-loop circuit itself is insensitive (), but this can be compensated by choosing a sensitive promoter for the target genes [20]. This requires that the TF binds cooperatively to multiple binding sites on the target promoters. If the TF binds fully cooperatively to binding sites and achieves a fold change (again assuming the Hill form of Eq. 2), the expression of the target gene responds with the following sensitivity to changes in :(6)(We provide a derivation in Supplementary Text S1.) Because , this shows that in open-loop circuits the sensitivity at best equals the number of cooperative binding sites at the promoter. However, this maximal is obtained only if the fold change is large, so that . The following numerical example illustrates this point. If the fold change in the non-cooperative auto-activation circuit is , the resulting is 5.5. To obtain the same value in the open-loop circuit, at , more than 7 cooperative TF binding sites are required at the target promoter. Clearly, if a threshold response is desired, auto-activation can be an excellent tool.

We repeat, however, that even though for the non-cooperative auto-activation circuit diverges in the limit of large , the response function does not converge to a step function (as Hill functions do in the limit of high ), but to the threshold-linear response in Fig. 3B. While this is obviously an excellent threshold response, its quality as a switch is better represented by the measure . Unlike , does not diverge in the large- limit, but it nevertheless acquires large values. For instance, in the example above (, , and ), we find . For comparison, for Hill functions,(7)where converges to 1 from below as the fold change increases (see derivation in Supplementary Text S1). From this it follows that, to obtain the value in the open-loop circuit (again assuming ), the promoter of the target gene should have a Hill coefficient . This shows that non-cooperative auto-activation circuits can constitute an excellent switch.

Fig. 3D summarizes the regions of parameter space that lead to a high sensitivity. The figure is based on Eq. 4 for . The line where diverges is shown–the same line that marks the boundary between mono- and bistable regimes in Fig. 2. If the circuit is to be sensitive, the parameters have to be close to that line, within the shaded region (where we chose a somewhat arbitrary cutoff ). An analogous figure based on is presented in the Supplementary Text S1 and leads to very similar conclusions.

Another important characteristic of a signaling circuit is its induction speed. A circuit that can be induced rapidly in a changing environment is expected to have a fitness advantage [7], [9]. We therefore study the induction time of the circuit.

As we explain below, the deterministic model is not adequate to describe the induction time of the circuit. We therefore introduce a stochastic model. This model is based on the following Master equation, describing the evolution of the probability distribution for the TF copy number at time :(8)All parameters are analogous to the deterministic model. As in the deterministic case, the mRNA level is not included explicitly but TFs are produced in bursts of size . We do not model the binding and unbinding of the TF to the DNA explicitly, but assume that the binding kinetics of the TF at the promoter are fast compared to the time scale of transcription initiation [21]. These assumptions simplify the analysis but are not critical: similar results can be obtained with other models previously presented in the literature [21]–[27].

To introduce the induction time, we imagine that the signal has been absent () for a period long enough to ensure that the circuit is in steady state. Then, at time , the signal is introduced at a saturating level (). We then define the induction time as the waiting time before the expression of the TF arrives at 50% of its steady state level. In the deterministic model, this can be calculated by solving the differential equation 1. In the stochastic model, the waiting time is actually a random variable; therefore, we report the mean waiting time, which can be calculated exactly from the Master equation 8. In the Supporting Text S1 we describe the method used; there we also discuss the full induction time probability distributions.

Fig. 4A shows results from both the deterministic and the stochastic model. Plotted is the induction time as a function of the fold change. We assume that the maximal expression level of the circuit is prescribed by the functional context of the circuit; therefore, we vary but keep fixed. The deterministic model predicts that the induction time increases mildly–less than two-fold–as is increased from 10 to 1000. Based on such results, one might conclude that the effect of auto-activation on the induction time is mild. The stochastic model, however, reveals a different picture. When is low, both models are in agreement. But when (as we explain below), the stochastic model deviates dramatically from the trend predicted by the deterministic theory, demonstrating that the induction time is much more strongly affected by the auto-activation than expected from deterministic rate equations [2], [3], [7], [9].

What causes the discrepancy between the deterministic model and its stochastic counterpart? When the fold change is increased while the maximal transcription rate is kept fixed, the basal transcription rate is reduced. As a result, the expected number of TFs present in the cell at time –just before the signal arrives–becomes small. This means that, when the signal is introduced, the TF concentration is initially too low to activate the TF's transcription significantly, so that the transcription rate remains of the order . Crucially, this means that the expected waiting time before the first transcription event occurs is close to . Using realistic parameters for bacteria, in which the maximal transcription rate is of the order , this waiting time can easily become large. Indeed, Fig. 4A shows that for large the induction time is of order , indicating that the first transcription events become the limiting step of the induction. This effect is not accounted for by the deterministic model, which disregards the discreteness of the molecular events.

To visualize the process, Fig. 4B shows five representative time traces of the induction, obtained by kinetic Monte Carlo simulations, for a circuit with a large fold change . Before the signal is switched on, the copy number fluctuates around the average value . The signal is introduced at time . The traces clearly show that the induction time is dominated by the waiting time before the first transcription event; next, the circuit usually switches on rapidly. This has another important consequence: because transcription is (modeled as) a Poisson process, the probability distribution of induction times is approximately exponential (see Supplementary Text S1). The standard deviation of the induction times is therefore as large as the mean. This means that, in this parameter regime, the induction process is not only slow, but also unpredictable.

The anomalous induction time ultimately results from a low basal transcription rate . However, assuming that the maximal expression level of the TF is set by its biological function, so that can be considered given, this directly leads to constraints on . Fig. 4C is a contour plot of the induction time according to the stochastic model. It clearly shows that large fold changes lead to long induction times. Also, unless the fold change is small, increasing the Hill coefficient strongly slows down induction; comparison to Fig. 2 shows that this is because the circuit then approaches and eventually enters the (deterministically) bistable regime. If we arbitrarily decide that the induction time should not exceed 50 min, the admissible values of and are limited to the small shaded region.

To be precise, we distinguish between bistability and bimodality. We call a circuit bistable if the deterministic model predicts two stable steady states, and bimodal if the stochastic model predicts a steady state probability distribution with two peaks. Naively, one might expect that, in order for a circuit to be bimodal, it should be bistable. Theoretical work has shown, however, that this correspondence does not always hold [24], [26], [28], [29]. In particular, stochastic models of auto-activation circuits can produce bimodality even if the auto-regulation is non-cooperative (), in which case the circuit cannot be bistable (see Fig. 2) [24], [26]. Recently, this theoretical result was verified experimentally using a synthetic auto-activation circuit in Saccharomyces cerevisiae [27]. As we will explain, this phenomenon is directly related to the slow induction time we just described.

We explained that the slow induction was due to the fact that, if the circuit is prepared in a state with no TFs (), it on average lingers in that state for a time , which becomes large if is large. Obviously, the same lingering time holds if the circuit arrives in a state with by a rare random fluctuation. Therefore, as is increased, the stochastic circuit will spend an increasing fraction of its time in the state, which for large enough results in a peak in the steady-state distribution at . (In fact, in the limit the state becomes an absorbing state, so that the steady-state probability distribution becomes concentrated entirely on .) If another peak is present at non-zero expression levels, a bimodal distribution results. In other words, the anomalous bimodality and the slow induction are different manifestations of the same underlying characteristics of the circuit.

It can readily be derived from the Master equation 8 that the bimodality for non-cooperative auto-activating circuits requires that (see Supporting Text S1). In other words, in the non-cooperative circuit, bimodality is obtained only if the basal transcription rate is smaller than the protein degradation/dilution rate . This holds independent of , or ; moreover, more detailed stochastic models yield the same result (see supplementary text of Ref. [27]). Given the relation between the anomalous induction time and bimodality, this explains why the induction time of the stochastic model deviates noticeably from the deterministic one when ≳ α/β, as we indeed observed in Fig. 4A.

The above interpretation also indicates that a large burst size is not required to obtain bimodality at [27]. Bimodality can occur at any burst size, provided the fold change is large enough, such that (or, equivalently, provided the basal expression level is low enough such that ). Yet, the burst size can be important in an indirect way: to maintain a fixed steady state expression level, an increased burst size has to be compensated by a decreased or an increased . In both cases the requirement is relaxed.

In models that explicitly treat the binding of the TF to its own promoter, bimodality can also occur due to a different mechanism [21], [25]. If the binding kinetics of the TF are slow, the steady state distribution can have two peaks: one corresponding to the expression when a TF is bound to the promoter, and one corresponding to the expression when the promoter is unbound. For this type of bimodality the duality with an anomalous induction does not necessarily hold.

We have assumed that signaling circuits generally care about speed and sensitivity, and should avoid bistability and hysteresis. Each of these properties imposes constraints on the auto-regulation by the TF, as shown in the phase diagrams Figs. 2, 3D and 4C. In Fig. 5 we combine these results to analyze which parameters are compatible with all these constraints. To eliminate bistability, the operating point of the circuit should be below the black solid line. To obtain sensitivity, it should be close to this line. To avoid a slow induction, the fold change should not be too large; yet, to have any benefit from the auto-activation, it should not be too small. These constraints restrict the system to the small parameter region indicated in the plot. In this region, the fold change is at most moderate (≲ 40), and the Hill coefficient is roughly in the range 1 to 2. Such a low Hill coefficient can be achieved using auto-regulation by a single TF dimer.