Introduction

Bistability and multistability are common dynamic features of a wide variety of natural and synthetic systems1. A multistable system can switch between alternative stable states in response to external inputs. Multistability plays a key role in diverse biological processes such as cell differentiation2,3 and epithelial to mesenchymal transition (EMT)4,5,6,7, allowing for cell-fate determination among multiple phenotypes. Alternative differentiated states in development may correspond to multiple stable steady states of regulatory networks, while the hybrid phenotype in EMT can be interpreted as an intermediate state of a tristable circuit.

Positive feedback is necessary for bistability and multistability1,8. A positive feedback loop (PFL) with ultrasensitivity is able to exhibit bistability9; a single PFL seems insufficient to admit multistability10, but coupling a PFL with additional feedback loops is possible to realize multistability11,12,13,14. Indeed, tristability or tetrastability can be induced in interlinked PFLs, which are at the core of signaling pathways underlying the EMT4,5,6,7,15. Notably, these systems involve transcriptional coregulation of a microRNA (miRNA) and its targets, representing a recurrent motif in gene regulatory networks. Typical interlinked dual-PFLs include a mutual inhibitory loop (MIL) combined with a self-activation loop4 (Fig. 1A), two MILs sharing one component7 (Fig. 1B), and one component in an MIL inhibiting the other through a feed-forward loop4,5,6,7 (Fig. 1C). The first two structures have been investigated in detail15,16, whereas the last one was less probed. It is worth exploring since similar architectures are widely involved in cell signaling17,18,19, such as that responsible for the differentiation of human embryonic stem cells18 (Fig. 1D). Moreover, whether coupling a PFL with a negative feedback loop (NFL) can induce tristability is an open issue. Rather than exploring the underlying mechanism for realizing tristability in interlinked dual loops on a case-by-case basis, it is necessary to establish a general principle. It is essential to reveal how the features of individual feedback loops and their coupling manner affect the tristability.

Figure 1: Interlinked feedback loops.
figure 1

(A–D) Examples of miRNA-containing circuits where two loops share one (A,B) or two (C,D) components. (E) The model. The system of coupled feedback loops can be decomposed into two loops. The arrow- and bar-headed lines denote activation and inhibition, respectively, while circle-headed lines denote either activation or inhibition.

Based on the circuits in Fig. 1C,D, here we construct a circuit model comprising two transcription factors (TFs) and one miRNA. Mathematical equations characterizing the temporal evolution of circuit components take canonical forms. First, we explore whether the circuits of individual or coupled feedback loops can exhibit tristability over a wide range of parameter values by bifurcation analysis. Second, we introduce the measure of logarithmic gain and formulate the necessary condition for tristability in terms of the gain. Third, we investigate six representative systems (i.e., the circuits with different parameter values) and illustrate the requirements for tristability by presenting both analytical and numerical results. Finally, we discuss our methods, main results, and their relevance to cell signaling. This work reveals the essential mechanism for realizing tristability in multiplicatively coupled feedback loops.

Model and Method

The circuit model comprises three nodes termed X, Y and Z (Fig. 1E). X and Y are two TFs, while Z is a miRNA (M). Without loss of generality, we assume that two TFs mutually repress each other’s transcription, forming a TF-TF PFL. Y regulates the expression of Z and Z represses the production of X, constituting a TF-M-TF loop. This loop is either a PFL or an NFL, depending on whether Y activates or inhibits the transcription of Z. Furthermore, the sign of the TF-M-TF loop determines whether the coupled system is a PP (PFL-PFL) or PN (PFL-NFL) circuit (note that a dual-NFL circuit cannot admit multistability). The time evolution of the system is governed by the following ordinary differential equations:

where , , , and . All variables and parameters are normalized to be dimensionless (see SI Text 1 for original equations and the derivation).

The variables x, y and z denote the concentrations of X, Y and Z, respectively. The term ηf(y)g(z) describes the production rate of X; f(y) denotes the repressive regulation by Y at the transcriptional level, while g(z) characterizes the inhibition by Z at the translational level. The product of f(y) and g(z) characterizes the coregulation of X by Y and Z, representing a multiplicative coupling of two loops. η is the maximum production rate, taken as a control parameter here, and δ in g(z) reflects the strength of inhibition by Z. Similarly, h(x) describes the production rate of Y; κ is the maximum production rate. u(y) describes the regulation of Z transcription by Y; K relates to the dissociation constant for Y binding to the promoter of Z, while α > 1 and α < 1 represent the degree to which transcription is enhanced or weakened, respectively, by Y bound to the promoter. All the negative terms describe the process of degradation. n1, n2 and n3 are Hill coefficients, τ is the time in units of the reciprocal of the degradation rate of X, and 1/θy and 1/θz are the degradation rates of Y and Z normalized to that of X, respectively.

If we eliminate the indirect regulation of X by Y via Z, we get the equations for the TF-TF loop directly from the above equations:

and Eq. 2. If we ignore the direct regulation of X by Y, we get the equations for the TF-M-TF loop:

and Eqs 2 and 3. The parameter values are not fixed; instead, we explore the properties of these systems over a wide parameter range. Numerical simulations are performed using MATLAB 2008b. Bifurcation diagrams are plotted using Oscill8.

Results

Analysis of individual and coupled feedback loops

We first demonstrate that only the circuits of coupled feedback loops can admit tristability via bifurcation analysis. 50000 parameter sets are sampled for the PP (α > 1) and PN (α < 1) circuits, respectively, using the Latin hypercube sampling method (MATLAB built-in function lhsdesign). The ranges for parameter sampling are listed in Table S1. For each parameter set, we plot the steady-state value of x versus η.

For individual PFLs and PP circuits, all the bifurcation diagrams are categorized into seven types (Fig. 2). For the TF-TF or TF-M-TF PFL, only type I and type II are allowable, involving monostability and bistability (Fig. 2A,B). Thus, individual PFLs in our work cannot exhibit tristability. For convenience, we denote the PFL with the bifurcation diagram of type I or II by PI and PII, respectively. Therefore, PP circuits are grouped into four categories: PIPI, PIPII, PIIPI and PIIPII (the first item denotes the type of the TF-TF loop and the second denotes that of the TF-M-TF loop). Notably, PP circuits can admit five more types of bifurcation diagram besides types I and II: type III involves two separate bistability regions (Fig. 2C), while types IV–VII each have one tristability region, which is adjoined to two bistability regions, with different bifurcation points (Fig. 2D–G). As seen in Table S2, PIIPII is the most probable one to achieve tristability among PP circuits (with a probability of 0.37).

Figure 2: Schematic description of seven types of bifurcation diagrams for both the PP and PN circuits.
figure 2

Types I and II and type I are allowable for individual PFLs and NFLs, respectively. The solid and dashed lines denote stable and unstable steady states, respectively. The blue lines indicate possible transitions between steady states via SN bifurcation. Both axes are in log scale.

For individual NFLs and PN circuits, there appear more types of bifurcation diagram owing to the presence of Hopf bifurcation, via which some stable steady states become unstable. Fortunately, these unstable steady states can always be restabilized by adjusting parameters θy and θz to eliminate Hopf bifurcation (see SI Text 2). Consequently, all the bifurcation diagrams can be converted to the seven types illustrated in Fig. 2. This greatly simplifies the search for PN circuits with tristability. In such a setting, individual NFLs admit only the bifurcation diagram of type I, and PN circuits are divided into PIN and PIIN (‘N’ denotes the NFL). Only type I is allowed for PIN circuits, whereas types I–VII are observable for PIIN ones. In the following, we focus on how to realize tristability in the PIIPII and PIIN circuits.

Necessary condition for tristability in terms of gain

To quantify the essential condition for tristability, we tried multiple measures and found that logarithmic gain of circuits is an ideal candidate. Generally, the gain of a system not only reflects the effect of feedback like amplification or attenuation, but also tightly associates with the elements of the system’s Jacobian matrix, which govern the stability of steady states20,21. For individual loops, the logarithmic gain (denoted by G and called gain for short thereafter) is defined as the derivative of a transfer function (T) with respect to x in log-log coordinates. The transfer function is the composition of functions, each describing one of regulations in a feedback loop (see SI Text 3). Thus, the gain depends on all regulatory interactions in one loop. For the TF-TF loop, the transfer function and the gain are separately T1(x) = (ηfh)(x) and . For the TF-M-TF loop, the transfer function and the gain are T2(x) = (ηguh)(x) and , respectively.

A typical transfer function T(x) for the TF-TF or TF-M-TF PFL is a sigmoid function characterizing the involved cooperativity and saturation22,23 (Fig. 3A), while that for the TF-M-TF NFL is a reverse sigmoid function (Fig. S1A). The corresponding G(x) curve for the PFL is bell shaped with one positive maximum (Gmax) at xm (Fig. 3B), while that for the NFL is inverted bell shaped with one negative minimum (Gmin) at xm (Fig. S1B). Three features of the gain are used to represent individual loops, i.e., Gmax (Gmin), xm, and the full width W at half extremum. Since the (reverse) sigmoid function has the steepest slope at xm, xm is called activation threshold, especially when T(x) is close to a threshold-like function. W is defined as ln(w2/w1) with w1 and w2 denoting the x values at half extremum points. Notably, feedback loops with a bell-shaped G(x) seem to have a ‘bandpass’ property: only within a limited concentration range is the gain significantly nonzero; otherwise, it decays to zero rapidly. This ‘bandpass’ property has a critical role in shaping the feature of the system of coupled feedback loops as seen later.

Figure 3: Characteristics of individual PFLs in terms of gain.
figure 3

(A) Transfer function T is characterized by a sigmoid function. (B) Logarithmic gain G. If Gmax > 1, G equals 1 at b1 and b2. (C) Bifurcation diagram. Two SN bifurcations (marked by SN1 and SN2) appear at η1 and η2. (D) Scatter diagram of WB versus WG. Each dot represents one TF-TF loop. From blue to red, more dots are distributed locally. All axes except the G-axis are in log scale in (A–C).

Using the standard bifurcation analysis, we obtain a necessary condition for saddle-node (SN) bifurcation (see SI Text 3): if a dynamic system undergoes an SN bifurcation at a steady state denoted by x, the gain at x must be 1, i.e., x is a solution to the equation G(x) = 1. For the system to exhibit bistability, it must undergo two SN bifurcations, i.e., there should be two distinct real solutions to G(x) = 1. Similarly, the necessary condition for tristability is that there should be four distinct real solutions to G(x) = 1. For general multistability, G(x) = 1 should have at least four different real solutions. Of note, here we omit other types of bifurcation like pitchfork bifurcation because the appearance of bistability and tristability in the above systems all originates from SN bifurcation (Fig. 2), which is typical of cell-fate decision5,15,24.

For any PFL with Gmax > 1, there are at most two solutions to G(x) = 1 due to the bell-shaped G(x) curve; thus, the PFL can undergo at most two SN bifurcations and display bistability in some range of η, never admitting tristability (Fig. 3C). The width of the bistability region is defined as the logarithmic difference between its upper and lower thresholds, WB = ln(η1/η2). Notably, WB can be approximately expressed as follows (using an isosceles triangle to approximate the gain curve; see SI Text 4.1 for details):

To verify this, we separately sample 6200 and 5260 parameter sets for the TF-TF and TF-M-TF PFLs with bistability and calculate WB, Gmax and W for each case. The points in the (WG, WB) plane indeed scatter along the diagonal, with the Pearson’s correlation coefficient being 1.0 (Figs 3D and S2).

Difference between activation thresholds affects the presence of tristability in PIIPII circuits

If we apply the definitions of transfer function and gain to the system of coupled feedback loops, we have TC(x) = [η(fh)(guh)](x) and GC(x) = G1(x) + G2(x) (see SI Text 3), i.e., the gain of the coupled system is the sum of the gains of individual feedback loops. This additivity results from the product of functions when describing the regulation of X by Y and Z, i.e., the multiplicative coupling. Given the GC(x) curve is a superposition of G1(x) and G2(x) curves, the difference between x1m and x2m markedly affects the shape of GC(x) and presence of tristability. Similar to the definitions of W and WB, a logarithmic difference is defined, i.e., R = ln(x2m/x1m). To focus on the influence of R on tristability, we merely vary R while keeping other factors unchanged. Without loss of generality, we develop a numerical method in which three parameters n3, α, and K are adjusted simultaneously such that only x2m is changed (see SI Text 5). In the following, we present only the results for the case of , i.e., (we also investigated the case of R < 0, and similar conclusions are drawn).

A representative system is chosen with G1max = 1.73, x1m = 1.28, W1 = 1.46, G2max = 1.60, and W2 = 0.96 (for parameter values see Table S3, System 1). When R is relatively large (e.g., R = 2.62), the GC(x) curve has two local maxima and one local minimum, intersecting with the line G = 1 four times (Fig. 4A). Accordingly, four SN bifurcation points appear at ηi(i = 1 ~ 4) in the bifurcation diagram (Fig. 4B). However, no tristability region arises, but two separate bistability regions appear (i.e., the bifurcation diagram is of type III). Actually, the widths of two bistability regions, WBC1 = ln(η1/η2) and WBC2 = ln(η3/η4), approximately equal those for the TF-TF and TF-M-TF loops (i.e., WB1 and WB2), respectively (SI Text 4.2). Based on Eq. 6, we further have

Figure 4: Effects of R on GC and bifurcation diagrams.
figure 4

x1m is fixed at 1.28, while x2m varies with R (System 1). (A) The curves of G1 (green), G2 (blue) and GC (red); (B) the corresponding bifurcation diagrams, where the solid and dashed curves denote stable and unstable steady states, respectively. The four bifurcation points and the corresponding η values are marked. The x-axis and η-axis are in log scale.

with i = 1, 2. On the other hand, the width of the intermediate branch of stable steady states (bounded by SN2 and SN3), WI = ln(η3/η2), obeys (SI Text 4.3):

If R is so large that WI > WBC1 + WBC2, two bistability regions remain separate. In this case, the coupled system acts as a bistable switch within two distinct ranges of η, determined by the features of individual loops.

For a smaller R (e.g., R = 2.39, 2.04, or 1.81), the GC(x) curve still has two local maxima and one local minimum and intersects with G = 1 four times. While Eqs 7, 8 still hold true, the reduction in R leads to WI < WBC1 + WBC2. Consequently, two bistability regions overlap, resulting in a tristability region. Accordingly, the bifurcation diagrams are separately of type IV, V, and VII. If R further drops (e.g., R = 1.22), however, the local minimum of GC(x) between two local maxima is elevated above 1, thus leaving only two intersections with G = 1. Consequently, there exist only two SN bifurcations and tristability is no longer possible. For R = 0.73, there is even no local minimum, and the GC(x) curve is single-bell shaped. The coupled system acts as a bistable switch in a wider range of η. We conclude that only when R is moderate, the bistability properties of two PFLs can be combined to generate tristability.

To quantify the range of R for tristability, we plot a phase diagram of R versus η, where four loci of SN bifurcation points are traced out, dividing the (η, R) plane into several domains, corresponding to three dynamic behaviors: monostability, bistability and tristability (Fig. 5B). Tristability exists in a limited range of R: its lower limit R1 is 1.51, determined by the intersection of the loci of SN2 and SN3, while its upper limit R2 is 2.51, determined by that of SN1 and SN4. In contrast, only two bistability regions may appear for R > R2, or one bistability region for (besides two monostability regions). Thus, for the dual-PFL circuit to exhibit tristability, R should be kept within a proper range.

Figure 5: Dependence of SN bifurcation points on R and η.
figure 5

(A) The curves of G1 (green), G2 (blue) and GC (red). (B) Phase diagram of R versus η. The solid curves denote the loci of bifurcation points: SN1 (blue), SN2 (cyan), SN3 (red) and SN4 (magenta). The white, orange, and green domains correspond to monostability, bistability, and tristability, respectively. The x-axis and η-axis are in log scale. Scatter diagrams of R versus (C) or (D). Each blue, red or green dot represents one coupled system with the bifurcation diagram of type III, IV–VII and II, respectively.

Furthermore, we derive the estimates of R1 and R2, and , given G1max > 1 and G2max > 1 (SI Text 4.4):

The width of R’s range for tristability thus obeys . Therefore, the wider the bistability regions for two PFLs, the broader the range of R for tristability. To validate Eq. 9, we sample 3200 parameter sets for the coupled system with G1max > 1, G2max > 1, and , and calculate , , and R for each case. For the systems with the bifurcation diagram of type II, nearly all points scatter in the region of (Fig. 5C), while for those of type III, the points spread in the region of (Fig. 5D); for those of type IV–VII, the points are mainly distributed in the region of . Thus, the range of R for tristability is indeed approximately determined by the features of individual PFLs. This has important implications for inducing tristability in coupled feedback loops.

WB of individual PFLs significantly affects the bifurcation type of the coupled system

It is worthy to note that WB1 and WB2 affect the bifurcation type of the coupled system. For example, if WB1 is decreased by setting G1max and W1 to 1.49 and 1.44 and WB2 is increased by setting G2max and W2 to 2.00 and 0.99 (see Table S3, System 2), then WB1 is smaller than WB2, leading to WBC1 < WBC2. Consequently, the bifurcation diagram of type V, which requires WBC1 > WBC2, no longer occurs given (Fig. S3). Instead, type VI can be observed for some values of R (e.g. R = 2.14). That is, type V and type VI are mutually exclusive, and which type occurs depends on WB1 and WB2. Together, the width of the bistability region for individual PFLs greatly influences the type of bifurcation diagram that the coupled system can admit.

W, R and Gmin affect the tristability of PIIN circuits

Here, we probe how to produce tristability in PIIN circuits with G1max + G2min < 1. Intuitively, an NFL may go against bistability since its negative gain leads to GC < G1. For a representative PIIN circuit with G1max = 2.59, x1m = 1.81, W1 = 1.62, G2min = −3.70, x1m = 1.77 and W2 = 1.37 (see Table S3, System 3), GC is less than 1 (Fig. S4A), and bistability is eliminated (Fig. S4C). In fact, this occurs when W1 and W2 are comparable or W2 is much larger than W1. If W2 is much less than W1, e.g., W2 = 0.39 (for parameter values see Table S3, System 4), the modulation of G1 by G2 can be much localized (Fig. S4B). Only the part around x2m is quenched by G2, and the others nearly remain unchanged. Consequently, GC(x) can have two local maxima (>1) and one local minimum (<1), and a tristability region arises in the bifurcation diagram (Fig. S4D). Thus, the G(x) curve for the NFL should be much narrower than that for the PFL so as to realize tristability.

Take System 4 as an example to explore how R affects the tristability. With R = 0.10, −0.02, or −0.29, G1 is modulated locally in a similar way (Fig. 6A). As expected, a tristability region exists in the corresponding bifurcation diagrams (Fig. 6B). However, if R rises to 0.35 or drops to −0.61, one local maximum of GC becomes less than 1, and there are only two solutions to GC(x) = 1. Thus, no tristability can be realized. An overview of R’s influence is presented in the (R, η) phase diagram (Fig. 7A,D). The tristability arises within a limited range of R. Its upper limit is 0.28, determined by the intersection of the loci of SN3 and SN4; the lower limit is −0.45, determined by that of SN1 and SN2. For R > 0.28 or R < −0.45, only one bistability and two monostability regions are observable. Therefore, a tristability region appears when R approaches 0. This is in contrast to the condition for tristability in PIIPII circuits; instead, two PFLs with a small |R| seem to merge into one PFL, unable to admit tristability. Collectively, for PIIN with G1max + G2min < 1 to exhibit tristability, W2 needs to be much less than W1, and |R| should be sufficiently small.

Figure 6: Effects of R on GC and bifurcation diagrams of PIIN (System 4).
figure 6

(A) The curves of G1 (green), G2 (blue) and GC (red) for different values of R; (B) the corresponding bifurcation diagrams, where the solid and dashed lines denote stable and unstable steady states, respectively. The x-axis and η-axis are in log scale.

Figure 7: Dependence of bifurcation points on R and η for PIIN.
figure 7

(A–C) The curves of G1 (green), G2 (blue) and GC (red); (D–F) the (R, η) phase diagrams. The solid curves denote the loci of SN bifurcation points: SN1 (blue), SN2 (cyan), SN3 (red) and SN4 (magenta). The white, orange, and green domains correspond to monostability, bistability, and tristability, respectively. (A,D), (B,E) and (C,F) correspond to Systems 4, 5, 6, respectively. The x-axis and η-axis are in log scale.

If the negative feedback is weakened by increasing G2min from −3.70 to −1.5 such that G1max + G2min > 1 (Fig. 7B; for parameter values see Table S3, System 5), the tristability can appear when |R| is moderate rather than close to 0 (Figs 7E and S5). However, the tristability regions are rather small, and it is difficult for such systems to exhibit robust tristability. Actually, if G2min further rises, there is no tristability region any more. Conversely, if the negative feedback is enhanced by decreasing G2min from −3.70 to −4.50 (Fig. 7C; for parameter values see Table S3, System 6), the bifurcation diagram is always of type II or III (Fig. S6), and no tristability appears at any R or η (Fig. 7F). Together, a proper Gmin is required for PIIN to realize tristability.

Timescale of the NFL markedly influences the tristability of PIIN circuits

It is well known that coupling an NFL with a PFL can also induce oscillation via Hopf bifurcation under some conditions25,26. For PIIN circuits, the necessary condition for Hopf bifurcation at a steady state represented by x is (see SI Text 3):

where θy and θz determine the timescales of Y and Z dynamics. Thus, both the gains and timescales have a role in governing the stability of steady states. It is worthy to explore how θy and θz affect the tristability of systems with a given gain. Take System 4 with R = −0.02 as an example.

We first fix θy at 1.0 and investigate the influence of θz. The system can admit tristability at θz = 0.7 (Fig. 8A). All the steady states in the intermediate branch between SN2 and SN3 are stable. When θz rises to 1.0, two Hopf bifurcations (HB1 and HB2) appear at the intermediate branch. Thus, the steady states bounded by HB1 and HB2 become unstable, and stable limit-cycle oscillations may arise (Fig. 8B). If θz further rises to 1.2, some limit cycles disappear after a supercritical saddle homoclinic orbit bifurcation, where a limit cycle collides with a saddle and then becomes a homoclinic orbit to the saddle (Fig. 8C). If θz is increased to 1.5, Hopf bifurcation points no longer exist after the collision with SN bifurcation points (via Bogdanov-Takens (BT) bifurcation); no limit cycles appear and all the steady states in the branch between SN2 and SN3 become unstable (Fig. 8D), leaving only two stable branches. Consequently, only monostability or bistability can be realized. This kind of bistability has been reported27, where a bistable system exhibits diverse and flexible switching behaviors.

Figure 8: Bifurcation diagrams of PIIN (System 4) with R = −0.02, θy = 1.0 and different θz.
figure 8

(A) All the steady states in the intermediate branch between SN2 and SN3 are stable at θz = 0.7. (B) The steady states between two Hopf bifurcation points HB1 and HB2 are unstable and accompanied by stable limit cycles at θz = 1.0. Red curves denote the maximum and minimum of x in limit cycles. (C) The η range allowing for limit cycles is shortend at θz = 1.2. Dashed lines denote the supercritical saddle homoclinic orbit bifurcations. (D) All the steady states between SN2 and SN3 are unstable, and no limit cycle occurs at θz = 1.5. Both axes are in log scale.

To systematically demonstrate how θz influences the stability of steady states in the intermediate branch between SN2 and SN3, we then construct a phase diagram of θz versus η, where four loci of SN2, SN3, HB1 and HB2 are traced out (Fig. 9A). The loci of HB1 and HB2 meet at θz = 0.84, the minimum of θz that allows for Hopf bifurcation (with η = 62.53). The domains below and above these two loci correspond to the stability and instability of steady states, respectively. Therefore, some steady states between SN2 and SN3 become unstable if θz > 0.84. For θz > 1.36 (the value at which HB2 and SN3 meet and a BT bifurcation occurs), all the steady states are unstable, and tristability is completely forbidden. Conversely, all the steady states between SN2 and SN3 are stable for θz < 0.84. Thus, the system admits tristability in a wide range of η. Together, decreasing θz facilitates the occurrence of tristability. In other words, when the PFL is coupled with a fast enough NFL, tristability is more easily produced.

Figure 9: Phase diagrams of θz versus η.
figure 9

(A) Stability of steady states in the intermediate branch between SN2 and SN3 shown in Fig. 8 for different θz. The solid curves denote the loci of Hopf bifurcation points HB1 (magenta) and HB2 (blue), and SN bifurcation points SN2 (cyan) and SN3 (red). Bogdanov-Takens bifurcations occur at the intersections between the HB and SN loci, BT1 and BT2. Four dashed lines denote the values of θz corresponding to the bifurcation diagrams in Fig. 8. (B) The loci of Hopf bifurcation points for differnt θy. Both axes are in log scale.

Finally, we investigate the effect of θy on the tristability. We change θy and determine the loci of Hopf bifurcation points. When increasing or decreasing θy, the loci have similar shapes (Fig. 9B), and thus the above conclusion qualitatively holds true for different θy. Nevertheless, when θy drops (e.g., to 0.7), the minimum of θz that allows Hopf bifurcation becomes smaller, meaning that to widen the tristability region, the NFL dynamics need to be faster. Conversely, if θy rises (e.g., to 1.4), the requirement for a fast NFL is loosened. Thus, θy can influence the timescale range of the NFL required for tristability.

Conditions for different bifurcation diagrams of simplified PIIN systems

To further interpret the above results, we approximate the gain functions by piecewise linear functions and derive the conditions for different types of bifurcation diagrams (see SI Text 6). Set , , , , , and . According to Table 1, we can explain how the features of individual feedback loops affect the tristability in PIIN circuits. First, the bifurcation diagrams of type IV-VII appear only when W1/W2 > r1 for G1max + G2min > 1 or W1/W2 > r3 for G1max + G2min < 1. Obviously, r1 > 1 for G1max + G2min > 1 and r3 > r2 > 1 for G1max + G2min < 1 always hold. Thus, W2 < W1 is necessary for tristability (Fig. S4). Second, a limited range of |R| is also crucial for tristability. For G1max + G2min < 1, tristability can be effectively induced when R~0; for G1max + G2min > 1, a non-zero but moderate R is required (Fig. 7). Third, G1max or G2min influences the critical values of W1/W2 and |R| for tristability. With G1max, W1 and W2 fixed, for example, decreasing G2min can increase r3 but decrease r1. Given G1max + G2min > 1, a decrease in G2min can lead to a change from W1/W2 < r1 to W1/W2 > r1 and the occurrence of tristability. Given G1max + G2min < 1, however, a reduction in G2min can cause a change from W1/W2 > r3 to W1/W2 < r3, thus eliminating the tristability at any R. This accounts for why a proper G2min is required for inducing tristability. Finally, it is worthy to note that the estimates of critical thresholds in Table 1 would be in more agreement with numerical results when a more accurate approximation to the gain functions is acquired.

Table 1 Conditions for different types of bifurcation diagrams that simplified PIIN systems (with G1max > 1) can admit.

Dynamics in tristable systems

It is worth noting that the tristable PIIN and PIIPII circuits may exhibit distinct dynamics in approaching a steady state. Take System 4 with R = −0.02 and System 1 with R = 2.04 as examples of the PIIN and PIIPII circuits, respectively; they exhibit tristability separately at η = 66 (Fig. S7A) and η = 250 (Fig. S8A). We plot the phase portraits to demonstrate their dynamics around each steady state. When approaching the intermediate stable steady state, the PIIN system shows damped oscillation (Fig. S7B), whereas the PIIPII system does not (Fig. S8B). This difference is related to distinct features of eigenvalues for that state. For the PIIN system, the real part of a pair of conjugate complex eigenvalues is greater than the real eigenvalue (Fig. S7C,D); thus, the oscillatory mode decays more slowly, resulting in damped oscillation. In contrast, a real eigenvalue has the maximal negative real part for the PIIPII system (Fig. S8C,D). Taken together, such a dynamic difference may be used to distinguish between the two types of tristable systems.

Discussion

We revealed how the features of individual feedback loops affect the presence of tristability in the coupled system and determined various conditions for tristability to arise in terms of logarithmic gain. For two coupled bistable PFLs, there should be a moderate difference between their activation thresholds. When coupled with a bistable PFL, an NFL should have a narrower gain curve and a comparable activation threshold compared with the PFL, a proper minimum gain and fast dynamics. These results have important implications for realizing tristability in interlinked feedback loops.

The current work indicates that the logarithmic gain is an important measure, greatly facilitating the identification of critical features of feedback loops that contribute to the tristability. This is because the gain is the derivative of the transfer function, providing sufficient information about its nonlinearity. With this gain, it is easy to judge whether a system satisfies the essential condition for tristability. One feasible method to experimentally measure the gain is to interrupt a feedback loop and characterize the input-output relationship of the open-loop system21,23. Our theoretical framework can also be applicable to the other two types of dual-loop circuits shown in Fig. 1A,B, given two loops are multiplicatively coupled (see SI Text 7). In this scenario, the additivity of logarithmic gains and the necessary condition for saddle-node bifurcation still hold true. If the gain curves of individual loops are also bell shaped, the conditions for tristability in the coupled loops can be analyzed similarly.

Analyzing the gain of individual feedback loops may gain insight into the feature of coupled feedback loops. For example, the system comprising the NR2F2/OCT4 and OCT4/miR-302/NR2F2 PFLs governs a one-step transition from the undifferentiated to differentiated state18 (Fig. 1D). Intuitively, this system may not exhibit tristability. As seen in Fig. 2F, however, the bifurcation diagram of type VII, admitting a one-step transition (given that there is no critical slowing down), also contains a tristability region. Thus, the system may possess a hidden intermediate state that is hard to observe. This awaits experimental verification, which may begin with testing whether both the two PFLs exhibit bistability (equivalent to checking whether G1max and G2max exceed 1.0).

The epithelial-mesenchymal transition (EMT) has been widely studied because it is involved in various processes such as cancer metastasis, wound repair and embryonic development28. Two theoretical models revealed how the tristability and two-step transitions underlie the EMT4,5. Despite differences between the two models in modeling details, the tristability is realized in a dual-PFL module, and the two-step transitions are governed by the sequential activation of two PFLs. Moreover, different parameter values lead to distinct transition routes5,6. Indeed, here we identify four types of bifurcation diagrams with a tristability region, and they admit distinct routes of transitions among multiple stable states (Fig. 2). More importantly, distinct transitions can be guaranteed by adjusting the features of individual feedback loops. This provides an effective method for interlinking two PFLs to accomplish the expected transition, which may be applicable to controllable cell differentiation and reprogramming.

Very recently, it was shown that the circuit comprising the Ovol2/Zeb1, Snail1/miR-34a and Zeb1/miR-200 PFLs can realize tetra-stability in the EMT7. Of note, there exists another PFL involving an indirect regulation, Ovol2/TGF-β/Snail1/Zeb1, which can be coupled with the Ovol2/Zeb1 loop in a similar manner to that in Fig. 1C. According to our results, this coupled system may also exhibit tristability over some range of parameters even when the Snail1/miR-34a and Zeb1/miR-200 PFLs are blocked, which awaits further justification.

Whereas it was widely shown that coupling a PFL with an NFL can act as a bistable switch, an oscillator or excitable device16,29,30,31,32,33 and also improve the robustness of those functional motifs34,35,36,37, whether it can generate tristability remains open. Here, we found that the coupled system can admit tristability when the NFL quenches the PFL locally in terms of the gain: the single bell in the G(x) curve is split into two, equivalent to production of a dual-PFL system. The timescale of the NFL should be fast enough; otherwise, the coupled system may exhibit oscillation or bistability26,27. As far as we know, there is no experimental observation supporting such a novel tristable switch; it would be interesting to experimentally validate this two-step transition.

The current work is limited to deterministic analysis. Given the inherent stochasticity in biological systems, the robustness of multiple stable states against intrinsic fluctuation is important for their existence in a noisy environment. Moreover, the robustness to extrinsic signal fluctuation is also critical for multi-stable switches to respond accurately. Thus, it is worth exploring the influence of intrinsic and extrinsic noise on the presence of and transition between multiple steady states.

Additional Information

How to cite this article: Huang, B. et al. Realization of tristability in a multiplicatively coupled dual-loop genetic network. Sci. Rep. 6, 28096; doi: 10.1038/srep28096 (2016).