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Conceived and designed the experiments: SW JM. Performed the experiments: SW JM. Analyzed the data: SW JM. Wrote the paper: SW JM RVS.

The authors have declared that no competing interests exist.

Genetic toggle switches are widespread in gene regulatory networks (GRN). Bistability, namely the ability to choose among two different stable states, is an essential feature of switching and memory devices. Cells have many regulatory circuits able to provide bistability that endow a cell with efficient and reliable switching between different physiological modes of operation. It is often assumed that negative feedbacks with cooperative binding (i.e. the formation of dimers or multimers) are a prerequisite for bistability. Here we analyze the relation between bistability in GRN under monomeric regulation and the role of autoloops under a deterministic setting. Using a simple geometric argument, we show analytically that bistability can also emerge without multimeric regulation, provided that at least one regulatory autoloop is present.

Bistability is known to pervade key relevant biological phenomena

A general consensus indicates that such switches are based on a mutual regulation of two transcription factors (

In (a) a genetic circuit with monomeric autoloops and cross-regulation involving two genes (G_{A}, G_{B}) coding for two proteins (A, B) acting as transcription factors. Under certain conditions, this type of genetic circuit can show bistability. Here all possible regulatory modes are shown (+/−). (b) Simplified diagram summarizing the logic of this system.

Focusing on two-components genetic circuits, their regulatory proteins are known to form homodimers (or multimers) to be effective transcription factors allowing to turn ON or OFF the state of target genes

For general systems without any specific assumptions, multimeric regulation was assumed to be essential to obtain bistable behaviour

In deterministic dynamics, bistability requires the existence of three fixed points. In this paper we demonstrate, to our knowledge for the first time, that deterministic bistability can emerge for two-component gene circuits by considering solely auto-regulatory loops. This is unlike the previously briefly mentioned cases

In order to perform a general analysis of the nullclines, as introduced in _{+}>0 and ξ_{−}<0 ) with the horizontal axis given by

(a) Graphical representation of the nullcline's components. The numerator is the parabolic curve and the denominator the straight line. Two feasible scenarios are shown: the solid line denotes ξ^{+}>φ, the dashed line corresponds to ξ_{+}<φ. (b),(c) Qualitative behaviour of the nullclines applying the two possible conditions.

The denominator is a lineal function crossing the horizontal axis in φ = γ_{A}α^{B}_{c}/d_{A}. The points φ and ξ_{+} are the upper and lower bound of the protein concentrations of the system within the biological meaningful region. Combining the two components, two different scenarios are feasible, φ<ξ_{+} or φ>ξ_{+}, comprising different geometrical features. In both cases we find two crossing points with the horizontal axis in ξ_{±}, no inflection points, and the nullclines tending towards their oblique asymptotes with an identical slope ^{A}_{l}/ω^{B}_{c} for both settings A→±∞. From this expression we see that the autoloop is related with certain geometrical features. Systems without auto-regulatory loops (ω^{A}_{l} = 0) do not exhibit oblique asymptotes, but horizontal. As shown later the existence of oblique asymptotes is closely related with the number of possible fixed points and bistability.

In the first case, ξ_{+}>φ, we obtain a vertical asymptote in φ with its lateral behaviour given by lim_{A→φ±}(B)_{dA/dt = 0} = ±∞ For the second case, ξ_{+}<φ, we find similar asymptotes with opposite lateral behaviour according to lim_{A→φ±}(B)_{dA/dt = 0} = ∓∞. In order to determine possible extrema of the nullcline (dB/dA = 0), we find, after some algebra, that the inequality_{+} and φ<ξ_{+} by using the previous expressions for φ and ξ_{+} we conclude that only φ>ξ_{+} satisfies condition (5) and hence provides extrema. However, according with the vertical asymptotic behaviour and the existence of only one crossing point (ξ_{+}) within the positive domain, we conclude that the extrema are located within B<0. Hence, no extrema can be obtained within the biologically meaningful domains, i.e. by imposing the biological constraint that the levels of proteins must be positive (A>0, B>0), for either scenario. In

Using the previous geometrical approach, we are in the position to reassemble both nullclines within the biological meaningful region determining how many crossing points between both nullclines can arise under different regulatory conditions. The crossings between nullclines define the so called fixed points, i.e. the levels of proteins A and B such that dA/dt = 0 and dB/dt = 0 simultaneously, thus no changes in protein concentration will take place Four possible cases are obtained based on the symmetry of the expressions for nullcline dA/dt = 0 and dB/dt = 0. They are shown in _{+}>φ]_{dA/dt = 0} ∧ [ξ_{+}<φ]_{dB/dt = 0} and [ξ_{+}<φ]_{dA/dt = 0} ∧ [ξ_{+}>φ]_{dB/dt = 0} (3(a) and 3(b), respectively), equal geometrical arguments apply. In both cases the nullclines exhibit opposite monotonies and opposite curvatures within the entire domain due to the absence of extrema and inflexion points. These conditions solely allow for a single crossing, hence monostability. In the case [ξ_{+}<φ]_{dA/dt = 0} ∧ [ξ_{+}<φ]_{dB/dt = 0}, depicted in _{+}]_{dA/dt = 0} = [ξ_{+}]_{dB/dt = 0} = 0 two crossing point arise. In accordance with expression (4), these conditions can be satisfied, if 4d_{i}_{i}^{i}_{l}_{i}_{i}^{i}_{l}_{A}_{B}^{A}_{c}^{A}_{c}_{+}>φ]_{dA/dt = 0} ∧ [ξ_{+}>φ]_{dB/dt = 0} both nullclines show the same type of curvature and monotony. Due to the oblique asymptote, introduced by the autoloop, no analytical constraints prevent the existence of three crossing points. In

Dashed line corresponds to nullcline dA/dt = 0, solid line to dB/dt = 0, φ^{A} and φ^{B} denote the location of the asymptote for dA/dt = 0 and dB/dt = 0, respectively. Due to the symmetry of the nullclines' expressions, the vertical asymptote of dB/dt = 0 corresponds to the horizontal of dA/dt = 0. Analogously, ξ^{A}_{+} and ξ^{B}_{+} are the crossing points with the axis. (d) The geometrical features of the nullclines allow for two possible cases. Three crossing points (depicted) or a single crossing (not depicted).

In order to determine the impact of the number of autoloops on bistability, we have numerically analyzed the effect of downsizing the system from two to one autoloop (ω^{i}_{l}^{j}_{l}

In (a) circuit with two autoloops and in (b) circuit with one autoloop are shown. Circle denotes a stable, square an unstable fixed point. The basins of attraction are shown in grey and white. The following sets of parameters have been used: (a) γ_{A} = 1, d_{A} = 1, α^{A}_{l} = 10, ω_{l}^{A} = 1, ω^{B}_{c} = 1, α^{B}_{c} = 0, γ_{B} = 1.1, d_{B} = 0.1, α^{B}l = 2.1, ω^{B}_{l} = 0.1, ω^{A}_{c} = 1.1, α^{A}_{c} = 0 and (b) γ_{A} = 5, d_{A} = 8, α^{A}_{l} = 9, ω_{l}^{A} = 1, ω^{B}_{c} = 1, α^{B}_{c} = 0, γ_{B} = 8.5, d_{B} = 1, α^{B}_{l} = 0, ω^{B}_{l} = 0, ω_{c}^{A} = 1, α^{A}_{c} = 0.

In the previous sections the type of regulatory interactions, given by α^{i}_{l} and α^{i}_{c} was handled generally. However, the individual regulatory interactions, i.e. activation or inhibition, introduce additional constraints for the emergence of bistability. Applying some algebra to condition φ<ξ_{+} (bistability), we obtain an equivalent expression as in (5) with the opposite inequality. Focusing on the type of regulation, it can be rewritten as

This leads us to two different instances: (a) if α^{B}_{c}>1, then α^{A}_{l}>α^{B}_{c} and (b) if α^{B}_{c}<1, then α^{A}_{l}>α^{B}_{c} ∨ α^{A}_{l}<α^{B}_{c}. As a consequence systems with inhibitory regulation in the autoloop and activatory cross-regulation can not exhibit bistability. In all the other cases no geometric impediments are present.

In other circuit topologies the emergence of bistability is possible but conditioned to the specific parameters of the system.

To summarize, a general, analytic set of conditions for bistability in simple two-element genetic circuits has been derived for monomeric regulation. Although previous work suggested that such kind of mechanism would be unlikely to be observed, here a simple geometric argument reveals that wide parameter spaces allow monomeric regulation to generate multiple stable states. These results permit to predict the expected scenarios where a reliable switch could be obtained. Current efforts in engineering cellular systems

Finally, further work should explore how noise can act on these types of dynamical systems. In eucaryotic cells, dimerization has been shown to provide a source of noise reduction at least at the level of simple GRNs

We focus our analysis on the most general system formed by two genes. Gene A is expressed under the constrains of two different monomeric regulatory modes. Protein A exhibits an auto-regulatory loop by binding to its own promoter, as well as a cross-regulation mediated by protein B. Gene B expression is analogously regulated (see

We are assuming basal transcription, the standard rapid equilibrium approximations supposing that binding and unbinding processes are faster than synthesis and degradation, and constancy of the total number of promoter sites. Furthermore, the concentration of the other biochemical elements involved remains constant during time and can be subsumed in the kinetic constant γ_{i}. The binding equilibrium of the autoloop and the cross-regulators are denoted by ω^{i}_{l} and ω^{i}_{c}, respectively. Furthermore α ^{i}_{l} and α ^{i}_{c} denote the regulatory rates with respect to the basal transcription, for the autoloop and cross-regulation respectively. Values<1 correspond to inhibitory regulation, whereas >1 accounts for activation. Finally, d_{i}

In order to analyze the system's dynamics we obtain the following expressions for the nullclines imposing dA/dt = 0 and dB/dt = 0 considering monomeric regulation:

The number of crossing points between (2) and (3) defines the number of different fixed points within the system. Both nullclines have mathematically symmetric expressions, tunable by the set of parameters. This symmetry facilitates their analysis due to interchangeability of the characteristic features. Hence, the problem can be evaluated by reducing the analysis to one expression. Here (2) is analyzed.

We thank the members of the CSL for useful discussions. We also thank an anonymous referee for valuable comments, particularly the potential implementation of our proposed mechanism.