^{1}

^{2}

^{3}

^{*}

Conceived and designed the experiments: RG JFP. Performed the experiments: RG JFP. Analyzed the data: RG JFP. Contributed reagents/materials/analysis tools: RG JFP. Wrote the paper: RG JFP.

The authors have declared that no competing interests exist.

It is now recognized that molecular circuits with positive feedback can induce two different gene expression states (bistability) under the very same cellular conditions. Whether, and how, cells make use of the coexistence of a larger number of stable states (multistability) is however largely unknown. Here, we first examine how autoregulation, a common attribute of genetic master regulators, facilitates multistability in two-component circuits. A systematic exploration of these modules' parameter space reveals two classes of molecular switches, involving transitions in bistable (progression switches) or multistable (decision switches) regimes. We demonstrate the potential of decision switches for multifaceted stimulus processing, including strength, duration, and flexible discrimination. These tasks enhance response specificity, help to store short-term memories of recent signaling events, stabilize transient gene expression, and enable stochastic fate commitment. The relevance of these circuits is further supported by biological data, because we find them in numerous developmental scenarios. Indeed, many of the presented information-processing features of decision switches could ultimately demonstrate a more flexible control of epigenetic differentiation.

An essential attribute of living cells is the capacity to select among various alternatives when confronted with external or internal cues. These decisions can be directly linked to survival, as the sporulation/competence choice in the bacterium

The capability of cells to present different stable expression states while maintaining identical genetic content plays a significant role in differentiation, signal transduction and molecular decision-making. These epigenetic phenotypes are partly associated to changes in genomic structural features, including several types of chromatin and DNA modifications

A positive feedback topology is nevertheless not sufficient to generate distinct epigenetic states. In addition, the circuit should display some degree of nonlinearity, i.e., sigmoidality, on its constituent interactions

How does the particular structure of a given positive feedback influence its function? Considering multistability as the most prominent attribute of these architectures, one could argue that genetic design does not really matter, as soon as sigmoidal interactions are achieved in some effective way. Careful analysis of some of the recent experimental reports seems to indicate the contrary. Two general patterns can be suggested. First, positive feedback loops at the core of more complex regulatory networks generally consists of simple structures controlling cell fate decisions. This is normally associated to two complementary expression states, i.e., bistability (like the p42–Cdc2 system involved in

The proposed patterns lead to a set of interesting questions. When is multistability, understood as at least three possible expression states, relevant in differentiation as opposed to bistability? Which simple feedback loop architectures can produce it, and what biologically relevant parameters do we need to quantify in order to predict these behaviors? Moreover, we can also ask to what extent complex feedback topologies are necessarily required for multifaceted information-processing and for the execution of elaborated developmental programs.

To address some of these issues, we first investigate the number of available expression states of two minimal complementary systems—a two-component mutual-activation and mutual-inhibition circuits—whose constituents are autoregulated (

Mutual-activation (A) or mutual-inhibition (B) circuits constituted by two master regulators (denoted 1 or 2). Both modules exhibit autoregulation of their components. (C,D) Differentiation as a progression or decision. In (C) a cell changes its expression state to a new one as a result, for instance, of a signalling event (wiggle arrow). This is implemented by a progression switch only requiring two expression states. Alternatively, a decision switch can drive the initial expression state to two different ones and thus needs three expression states, corresponding to the states before and after the decision. Different cell colors correspond to distinct expression states. See main text for details.

To describe dynamical aspects of genetic regulatory networks one can generally adopt two contrasting strategies. The first one is to collect all available molecular information about the regulatory interactions of a specific system. This allows to present the putative regulatory network involved in the mechanism under study, which can then be quantitatively described by using a particular mathematical formalism, e.g., a set of ordinary differential equations. While this method can be helpful to describe the dynamics of a very specific system, it usually incorporates a degree of complexity that can sometimes hide the key dynamical aspects, and molecular players, determining network behavior (with the additional drawback that new molecular agents could always be discovered and thus modify both network topology and dynamics). An alternative approach is to propose simplified mathematical models based on a number of realistic assumptions. Simple models help in the identification of basic design principles, might act as effective descriptions of more complex circuits and, as we see below, can actually correspond to extant regulatory modules found in several biological scenarios. These models also circumvent the lack of quantitative molecular details required in the more specific studies. We follow here this second approach.

We thus introduce a two-component mathematical model to analyze the dynamical behavior of the mutual-activation/mutual-inhibition topologies (see equations in

System | Components | Mediators | Interaction Type | Output Fates | |

Mutual activation topology | Embryonic stem cells |
(Oct4/Sox2,Nanog) | Direct |
Transcriptional | (low,low)→differentiation |

(high,high)→self-renewal | |||||

Neurogenic network |
(Ac,Sc) | Da | Transcriptional | (low,low)→epidermal | |

(high,high)→neural | |||||

Myogenic differentiation |
(MyoG,Mef2C) | myogenic bHLH factors | Transcriptional | (low,low)→precursor cells | |

(high,high)→muscle cells | |||||

Pancreatic development |
(Sox9,FoxA2) | Direct |
Transcriptional | (low,low)→self-renewal | |

(high,high)→endocrine diff. | |||||

(p42,Cdc2) | Mos, Myt1 | Post-translational | (low,low)→immature oocyte | ||

(high,high)→mature oocyte | |||||

Apoptosis |
(Casp3, Casp9) | Direct | Post-translational | (low,low)→cell survival | |

(high,high)→apoptosis | |||||

Mutual inhibition topology | Mammalian embryogenesis |
(Cdx2,Oct3/4) | Direct | Transcriptional cooperative | (high,high)→precursor cells |

(high,low)→trophectoderm | |||||

(low,high)→inner cell mass | |||||

Hematopoietic development |
(GATA1,PU.1) | Direct | Transcriptional cooperative | (low,low)→priming state | |

(high,low)→erythroid/megakaryocytic | |||||

(low,high)→myelomonocytic | |||||

T-cell differentiation |
(T-bet,Gata3) | ITK | Post-translational | (high,high)→pluripotent state | |

(high,low)→Th-1 cells | |||||

(low,high)→Th-2 cells | |||||

Visual system specification |
(Pax6,Pax2) | Direct | Transcriptional | (high,high)→early eye epithelium | |

(high,low)→Optic cup | |||||

(low,high)→Optic stalk | |||||

(Wts,Melt) | Unknown | Transcriptional | (high,low)→‘Yellow’ photoreceptor | ||

(low,high)→‘Pale’ photoreceptor | |||||

( |
miRNAs | Transcriptional | (high,high)→equipotent | ||

(high,low)→ASEL neuron | |||||

(low,high)→ASER neuron |

What specific biological features determine multistability? We address this question by identifying a minimal set of biological determinants able to characterize circuit behavior. This analysis also helps us to highlight some unexpected features of the relation between module structure and epigenetics, and to introduce two main types of switches associated to the circuit dynamics.

The epigenetic profile exhibited by a particular positive feedback topology depends on the activation (inactivation) of the expression of its molecular constituents. This expression pattern is determined by the number of available equilibrium states of the system, which in turn depends on the specific values of its parameters. Thus, by changing the parameters, we can ultimately predict all the potential epigenetic regimes that a circuit can present. Such parameter space, commonly used in the study of dynamical systems

We show several phenotypic maps in _{H}, _{L} and II_{A}, _{H} and III_{L}→II_{A}, respectively). We further observe a qualitatively similar phenotypic map for both mutual inhibition (

This map shows the areas of coexistence of several expression states (multistability) in a _{{L,H}}; one (low,low)/(high,high) expression state, II_{{S,A}}; coexistence of (low,low)–(high,high), symmetric, and (low,high)–(high,low), antisymmetric, expression states, III_{{L,H}}; tristability with two antisymmetric states and one symmetric state, low or high, IV; coexistence of four expression states. The phenotypic map for mutual-inhibition (A,

The previous stereotypic behavior changes for strong crossactivation (_{S}). A different behavior arises when autoactivation strength dominates crossactivation, _{L} in

How can the epigenetic state of these circuits be modified? A change of gene expression is induced by external factors, e.g., due to a signalling event, which effectively modify the parameters of the circuits, and thus their location on the phenotypic map. This can generally happen in two ways. In some situations, the initial expression level progresses to a new one, this being the only possible steady state of the system (_{S}→I_{L} in _{H}→II_{A} transition in

Accordingly, we suggest that molecular switches driving these transitions can be classified as progression or decision switches. A progression switch drives the system to a final monostable epigenetic regime, while a decision switch takes the circuit towards a bi- or multistable regime. The phenotypic map shows then different parameter areas between which a particular topology could enable these switches.

It is also important to distinguish how the state previous to the expression change is abandoned. In a progression switch, this equilibrium is no longer available after a given stimulus strength, and the system necessarily jumps toward a new expression state. In comparison, the initial state of a decision switch does not disappear but becomes unstable. While both situations seem equivalent experimentally, their implications for switch function are totally different. The unstable state acts effectively as a boundary splitting mutually exclusive domains of expression (see below and also next section). This qualitative reasoning can be formally described using the language of Dynamical Systems Theory, where the previous transitions correspond to steady state bifurcations

The intersection between the circuit response curves or nullclines (lines in the

In

What type of signal processing enables the presented switches? In the following, we show the potential of decision switches to robustly discriminate several characteristics of biochemical stimuli, e.g., strength, duration, timing, etc. These capacities offer flexible control of epigenetic expression, far beyond that attributed to bistable (progression) switches.

We first imagine a situation in which a decision switch, initially in a symmetric state of high expression—(

How reliable is this discrimination? For instance, how big should the duration difference be to correctly recognize this distinction? We quantify this ability in

(A) A signal pulse inducing degradation of

Discrimination of differences in stimulus strength rather than duration works in a similar vein as before, but it additionally displays other features. A particularly interesting one is linked to the phenomenon of stochastic fate commitment. Cells appear in some occasions to choose among two different phenotypes (cell classes) in a random manner, e.g.,

To examine this, we start as in the previous section with a population of cells in the symmetric high expression state, i.e., we consider noisy gene expression. The signal is acting in both components but this time with the very same duration. As a result of this stimulus the circuit remains in the symmetric state, as with duration discrimination, while signal strength is below a particular value. However, when this value crosses a threshold, the symmetric states becomes unstable (insets

Several features of this process are of interest. First note that, as compared with differential duration, it lacks any threshold. Why is this? Examination of the way the symmetric state is abandoned provides the answer. For strength discrimination this state does not disappear, as in the previous section, but becomes unstable. This distinction is relevant from a biological point of view, although it might not seem to be so

In the examples above, we considered deterministic pulses acting post-translationally (fast time scale). However, realistic signaling events fluctuate in several ways and may also act in slower scales, e.g., by transcriptional regulation. We now briefly study these aspects in connection with stimulus strength and duration computations.

Signal fluctuations originate opposite effects when considering duration and strength detection. To dissect the role of this noise, we consider again a post-translational (fast) signal and introduce stimulus stochastic dynamics as a birth-death process (see

The switch parameters are the same as those in

What about the discrimination of signal strength? Signal noise modifies this task (stochastic fate determination) in a completely opposite way. In this case, we obtain that correct discrimination gets worse with the presence of stimulus fluctuations (

Cells need to discriminate not only relatively simultaneous stimuli, as discussed previously, but also stimuli received with certain time delays. This task is generally linked to the cellular capacity to maintain a memory of recent signaling events, i.e., bistability and positive feedback regulation

We consider two signal pulses separated by a particular time delay, which are operating alternatively on the

Stochastic time evolution (A,C) of the concentration of the

The preceding section discussed several new computational features associated to the presence of multistatibility in these systems. We also argued before how multistatibility is linked to autoregulation, a connection that we now further elaborate. Autoregulation favors multistability by either amplifying or compensating transient differences in expression between the circuit constituents. This modifies the type of steady states usually found in two-component switches; (low,low) or (high,high) expression for mutual-activation and (low,high) or (high,low) expression for mutual-inhibition, but see also

How does autoregulation-based amplification work? We consider a mutual-activation circuit in an initial (transient) asymmetric expression state, i.e., _{0}≠_{0}, with _{0}, _{0}, being the concentration of each circuit component in non-dimensional units. We plot the time evolution of the species concentration (

Deterministic trajectories (A,B) and time evolution of different promoter occupancies (C,D) for a mutual-activation switch with strong autoregulation (_{0} = 1, _{0} = 3 (non-dimensional units). Insets correspond to a population distribution for the _{0}, _{0})—evolve towards an asymmetric (low,high) state.

Multistability is however generally not expected in circuits exhibiting relatively strong mutual activation, as this might imply unrealistically strong autoregulation (

How biologically relevant is the general framework that we have proposed? We investigated how these topologies enable different epigenetic regimes (the phenotypic map), characterized the molecular switches driving transitions between them, and discussed several information-processing features exhibited by these switches. To put these ideas in a biological context, we now first identify the presence of these regulatory architectures in specific scenarios, and then discuss several signal discrimination features that have been analyzed in these and related settings.

Indeed, we found a number of differentiation programs controlled by circuits constituted by two molecular regulators in a mutual-activation or mutual-inhibition configuration (

Similarly, various stages of hematopoietic lineage specification are driven by modules exhibiting these architectures. In this situation, the presence of a third expression state, or priming state, is currently under inspection

What sort of signal discrimination is found in these contexts? The influence of signal attributes in various developmental scenarios hints at the possibility that more elaborated signal processing could be at work. One example of this influence is the role of signal strength in thymocite differentiation

Moreover, the precise temporal expression programs exhibited by genes involved in cell differentiation suggests that discrimination between signals at different times could be also important in these situations. For instance, during pancreas development (

Signal processing ultimately leads to a fate decision. Two models, not necessarily exclusive, have been proposed. In a first model (sometimes termed ‘instructive’ or ‘selective’ regulation

What specific molecule is critical for this particular physiological response? This question is usually asked when a given cellular behavior comes under scrutiny. The search for such master regulators is specially relevant in the context of differentiation, where they become lineage specification factors, whose expression, or the lack of it, is associated to distinct cell fates. This approach, however, does not seem to be sufficient anymore. Indeed, an increasing number of studies confirm the view that regulators do not work in isolation, and that we need to study them as parts of genetic control circuits to properly recognize their function

Even though the molecular components of such control circuits are obviously diverse, their architectures do exhibit two main unifying attributes. First, they represent a relatively simple positive loop structure, and second, this structure is constituted by interactions with a degree of sigmoidality (threshold-like action) that enables circuits to exhibit bistability

To analyze these issues, we characterized the function of two-component circuits with the use of mathematical models. An additional property in these systems is that their main constituents are autoregulated. We made use of the phenotypic map, a parameter space characterizing the patterns of expression associated to these modules. Transitions between expression states were then considered to be induced by two major switch classes. A progression switch corresponds to a transition in which at most two expressions states should be available. Alternatively, a decision switch needs of three expression states, one before and two after the decision. Both types correspond to distinct bifurcations of the system equilibria

We examined a number of scenarios where master regulators and their interactions have been experimentally uncovered. We identified several architectures corresponding to the analyzed circuits, i.e., constituted by two principal autoregulated molecular agents in a mutual-activation/mutual-inhibition topology. Our study also provided an elaborated rationale of why master regulators largely exhibit autoregulation. In addition, we correlated mutual-activation/mutual-inhibition switches with differentiation as a progression or decision, respectively. These theoretical arguments helped thus to unify a wide range of biological data, and present progression and decision switches as fundamental design principles in the control of epigenetic differentiation.

We specially studied the abilities of these modules to respond and monitor stimuli, with special emphasis on decision switches. This revealed a series of findings. First, decision switches are able to elicit richer responses to differential signal parameters, like strength or duration, enhancing signal specificity

To derive the mathematical models used in this study, we consider all biochemical reactions involved in transcription regulation and expression of two interacting genes (dimerization reactions, binding/unbinding of transcription factors to promoters, transcription, translation and degradation, see _{i}_{i}_{i}_{i}_{i}_{x}_{y}_{xy}

Phenotypic maps are obtained numerically by sampling the parameter space with different initial conditions and letting the system to reach all available steady states. Moreover, stochastic gene expression is simulated by using the Gillespie's algorithm in most cases (taking into account mRNA dynamics, see Equation 7 in

Deterministic dynamics for the reduced two-variable model (black lines) compared to the four-variable model with different ratios of mRNA and protein degradation (Δ). Red line: Δ = 10. Blue line: Δ = 1. Solid lines correspond to the time evolution of the x component, and dashed lines to the y component. The system is in the tri-stability regime with ρ_{x} = ρ_{y} = 10, ν_{x} = ν_{y} = 0 (mutual inhibition), σ_{x} = σ_{y} = 0.2 and α_{x} = α_{y} = 1.

(0.88 MB EPS)

Comparison of different algorithms and intrinsic noise sources. Probability distribution of the x component concentration for a population of cells in a symmetric high expression state. Solid lines: simulations with Gillespie's algorithm. Dashed lines: solution of the chemical Langevin equations. Black lines correspond to a burst parameter b = 1 and a volume factor Ω = 10 (V = V_{0}·Ω). Blue lines: effect of translational bursting (b = 10, Ω = 10). Red lines: Effect of finite size noise (b = 1, Ω = 1). Other parameters of the model for the stochastic simulations are ρ_{x} = ρ_{y} = 10, ν_{x} = ν_{y} = 0, σ_{x} = σ_{y} = 0.2 (mutual inhibition, tri-stability regime), α_{x} = α_{y} = 1 and k^{x}_{x} = k^{y}_{y} = 0.001 nM^{−2}.

(0.79 MB EPS)

(A,B). Phenotypic map of the circuit with average production rate a = 1 and different cross-interaction strengths. (A) ν = 2, (B) ν = 20 (the cross-inhibition case ν = 0 is shown in _{xy} = 0). (C,D) A possible role played by cooperativity among species. Here we plot the phenotypic map for a = 1, as a function of the autoregulation and the joint interaction strength parameter μ, Eq. (1) main text, for slightly non-overlapping promoters (σ_{xy} = 0.001) and cross-interaction strengths ν = 2 (C) and ν = 20 (D). In the case of total competition for the same promoter site, panels (A,B), positive cross-interaction is not able to generate bistability of symmetric expression states (0,0), (1,1), since at an average production rate a = 1 the lower (0,0) state is not stabilized. Strong cooperativity (recall that μ = ρ×ν for independent regulation) together with competition for the same binding sites favors the appearance of a low (0,0) expression state and bistability (stability regions correspond to the areas inside cusps).

(1.34 MB EPS)

Reversible (graded) deci-switch. The intersection between the circuit response curves or nullclines (lines in the x-y planes) identifies the system steady states, being these either stable (filled circles), or unstable (empty circles). In this way, a range of different initial concentrations of the circuit components (basin of attraction; light and dark grey areas) ends up in the same expression state. A reversible deci-switch is associated to a transition in which the initial expression state (0,0) becomes unstable (A). Two new asymmetric states appear in a graded fashion (B). This is a supercritical pitchfork bifurcation, insets (A–B), where the magnitude and types of available equilibria are plotted as a given parameter changes in the x-axis (solid line; steady state, dotted line; unstable state). Note that in this case there exist no hysteresis. The transition is reversible, which means that the appearance of new expression states strongly depends on the presence of a external factor (acting as bifurcation parameter). This could represent, for instance, a primary master regulator.

(0.78 MB EPS)

(A) Increased autoregulation enhances duration detection. Here we examine how the response of a decision switch to stimulus duration depends on autoregulation strength. The response for an autoregulation strength ρ = 10 (red line and filled circles, the same as in

(1.10 MB EPS)

Effect of fast and slow signals on strength discrimination. A mutual inhibition switch is placed in a regime (ρ = 30, ν = 0, σ = 0.2, a = 0.1) where a symmetric (high,high) expression state becomes unstable with similar amplitudes for: A. fast and B. slow degradation signals. Red lines and circles show the performance using deterministic signal pulses, and blue lines (squares) adding noise to the signals such that the mean number of signal molecules is the same in both cases. Lines are fits to Weibull or stretched exponential functions.

(1.15 MB EPS)

Multistability domains as a function of relative interaction strength (a = 1). For moderate to large average production rates and autoregulation strengths, the boundaries between monostable and multistable domains follow a linear relation, ρ/ν∝1/σ. For instance, ρ/ν>20 indicates a tri-stable domain at σ = 0.2. Notice that for high σ values the symmetric expression state (1,1) is no longer available and only two asymmetric equilibria coexist.

(0.84 MB EPS)

Autoregulation as a compensation mechanism. For mutual inhibition (ν = 0) and moderate autoactivation (ρ = 5), the ratio of binding affinities (σ parameter) determines if the circuit behaves as a toggle switch (A,C) or generates tri-stability (B,D). (A) With similar binding affinities (σ = 0.6), the autoregulation is acting at the same time than cross-interaction. Then mutual inhibition dominates, amplifying the expression of the ‘winner’ species in detriment of the ‘looser’ one. In this regime, only two asymmetric states exist [(low,high), (high,low)]. This is illustrated in the inset by the probability distribution of the x component, obtained by solving the stochastic system. (C) The probability of promoter occupation for autoactivation of the looser species (in this case, x-auto, black solid line) never reaches the necessary level for effective activation. (B) If relative binding affinity is strongly favored for autoactivation (σ = 0.2), the species with smaller initial expression is rapidly increased, compensating the initial difference. Here a new (high,high) expression state is available compared to the previous case (see inset). (D) Probability of occupation for autoregulation is increased faster in the less abundant species (black solid line).

(1.14 MB EPS)

Role of positive autoregulation in mutual-activation switches. (A) Response to a signal increasing the degradation of both components as a function of cross-interaction strength, for a switch without autoregulation (a = 1). The system is initially (no stimulus) in a monostable high expression regime. For ν<12, the signal decreases gradually the expression. For higher ν values, a low expression state is also stabilized and a progression (1,1)→(0,0) can take place depending on signal strength. (B) For the same stimulus, we take a cross-interaction strength of ν = 10 (no response regime) and examine the response as a function of autoregulation. For ρ>10 a bistability regime, and eventually a progre-switch, transition can take place. Thus, the presence of autoregulation enables a circuit to work as a switch in a signaling environment where it would not work as such otherwise, i.t., without autoactivation. Other parameters in (B) are a = 1, σ = 0.2.

(1.65 MB EPS)

Autoregulation favors flexibility in signal processing. Normalized response after a sustained degradation signal with different intensities in x and y components. (A) A switch without autoregulation in a symmetric bistable regime (a = 0.1, ν = 20) responds to signal asymmetries. In a color code, we show the concentration of the x species normalized by the initial equilibrium value (no signal). Note that for signal larger than {similar, tilde operator } 0.1 in one component a transition (1,1)→(0,0) is attained, irrespective of the strength of the signal in the other component. (B) For the same value of crossinteraction (ν = 20) but strong autoregulation (ρ = 20, σ = 0.1, see

(1.50 MB EPS)

Influence of Hill coefficients on phenotypic maps. This map shows the areas of coexistence of several expression states (multistability) in a σ-ρ parameter space for a Hill coefficient of n = 4, e.g., x, y species acting as tetramers. These regions are: I_{L}; one (low,low) expression state, II_{A}; coexistence of (low,high)-(high,low), antisymmetric, expression states, III_{{L,H}}; tri-stability with two antisymmetric states and one symmetric state, low or high, IV; coexistence of four expression states. (A) Phenotypic map for mutual-inhibition with low basal production rate (ν = 0, a = 0.1) corresponding to _{L} if the initial expression state is (high,high), Fig. S11.A.

(1.21 MB EPS)

Positive autoregulation of the factors involved in mutual activation/inhibition architectures in

(0.05 MB PDF)

Further discussions on the mathematical models and the approximations considered.

(0.12 MB PDF)