Architecture‐dependent robustness in a class of multiple positive feedback loops

Abstract

Many types of multiple positive feedbacks with each having potentials to generate bistability exist extensively in natural, raising the question of why a particular architecture is present in a cell. In this study, the authors investigate multiple positive feedback loops across three classes: one‐loop class, two‐loop class and three‐loop class, where each class is composed of double positive feedback loop (DPFL) or double negative feedback loop (DNFL) or both. Through large‐scale sampling and robustness analysis, the authors find that for a given class, the homogeneous DPFL circuit (i.e. the coupled circuit that is composed of only DPFLs) is more robust than all the other circuits in generating bistable behaviour. In addition, stochastic simulation shows that the low stable state is more robust than the high stable state in homogeneous DPFL whereas the high‐stable state is more robust than the low‐stable state in homogeneous DNFL circuits. It was argued that this investigation provides insight into the relationship between robustness and network architecture.

1 Introduction

Cellular networks are often required to carry out particular functions in changeable intracellular and extracellular environments [1, 2]. On the one hand, since these networks are composed of interacting chemical species with each present in low copy numbers, stochastic fluctuations in concentrations are inevitable. On the other hand, changes in the extracellular environment can affect the performance of the networks. Since evolution often selects traits that enhance the robustness of a biological system [1], a challenge theme in system biology is how to understand the relationship between robustness and network structure.

Bistability is one of the important dynamical behaviours observed in many biological systems [3, 4, 5], which allows a cell to switch between two distinct gene‐expression states in response to environmental stimuli or internal signals or both. Bistability plays a critical role in a variety of cellular processes such as cell‐cycle progression in budding yeast [6], B‐cell fate specification [7] and EGF receptor signalling [8]. It has been shown that the positive feedback loop is a key regulatory motif in creating bistability [9, 10]. In particular, a double positive feedback loop (DPFL) circuit (i.e. component A activates component B and vice versa) or a double negative feedback loop (DNFL) circuit (i.e. A inhibits B that in turn inhibits A) can, under certain circumstances, generate bistability. However, many biological systems are more complex, possibly containing multiple positive feedback loops rather than a single one [4, 11, 12], for example, 14 circuits with such a structure have been noted by Brandman et al. [4]. Also, for example, in the gene regulatory system that controls the progression through cell cycle, the mitotic trigger protein Cdc2 participates in three positive feedback loops; for the transversal of the start of cell cycle in budding yeast, the Cdc28 participates in two positive feedback loops. Thus, questions naturally arise: why are multiple positive feedback loops present in cellular networks? Furthermore, what advantages do they have in generating bistability?

Recently, more and more experimental evidences are showing that multiple positive feedback loops can make bistability more robust in contrast to a single positive feedback loop [11, 13, 14, 15]. For example, Chang et al. [13] developed an experimental multi‐loop system by combining two well‐characterised positive feedback loop. The interplay of both positive feedback loops results in bistability over a broad range of inducer concentration in contrast to the single positive feedback loop. This indicates that robust bistability can be achieved by the coherent linkage of distinct positive feedback loops. Other related studies include the work of Yao et al. [15], wherein they analysed all 768 gene circuits comprising all possible link combinations in a simplified Rb‐E2F network containing three nodes and ten links (refer to Fig. 1 b in Yao et al. [15] for more detail), and identified the network topologies that can generate robust, resettable bistability. As a result, they found that the most robust circuit to generate bistablity contains all the four positive‐feedback loops in the simplified Rb‐E2F network, indicating that multiple positive feedback loops can indeed enhance bistable robustness.

Fig. 1.

Schematic for three topologically equivalent classes to be studied, where the core modules are DPFL and DNFL

However, multiple positive feedback loops may exhibit different network architectures (patterns of regulatory interactions) in natural and synthetic networks. Two typical single positive feedback loop circuits that are often encountered in biological systems are DPFL and DNFL. Since they have the same loop gain (the loop gain is defined as the product of all the regulation signs in a loop), we define them as a topologically equivalent class. In addition to these simple network classes, there are more complex topologically equivalent classes, where each class is possibly composed of several DPFLs and DNFLs, for example, DPDP that is coupled by two single DPFLs with one common node and DNDN that is coupled by two single DNFLs with one common node. Henceforth, we will make the similar interpretation for other brief notations appearing in this paper. For each topologically equivalent network class, there is a so‐called network connection number, which is defined as the sum of DPFL and DNFL numbers, that is, the connection number of DPDP is 2. In this paper, taking DPFL and DNFL as two basic building blocks, we will consider several topologically equivalent network classes composed of them at the transcriptional level, such as the topologically equivalent class composed of two feedback loops (two‐loop class), which has three possible linking patterns: DPDP (i.e. one DPFL + one DPFL), DNDN (i.e. one DNFL + ne DNFL) and DNDP (i.e. one DNFL + one DPFL). We point out that all the three patterns can be found in natural systems and play an important role in many decision‐making processes (Table 1).

Table 1.

List for topologically equivalent natural circuits with three nodes: DPDP, DNDP and DNDN

	Systems	Interlinked positive feedback loops
DPDP	muscle cell fate specification [16]	CDO → MyoD → CDO Akt2 → MyoD → Akt2
	budding yeast polarisation [6]	Cdc24 → Cdc42 → Cdc24actin → Cdc42 → actin
	Kallikrein‐kinin system [16]	PLAT → PLG → PLATF12 → PLG → F12
DNDP	Mitotic trigger in Xenopus[8]	Cdc25 → cdc2→ Cdc25 Weel” cdc2 ” Weel
	EGF receptor signaling[17]	sheddases → EGFR → sheddases PTP ” EGFR ” PTP
	Start of cell cycle in budding yeast[18]	Cln → cdc28 → Cln Sicl ” cdc28 ” Sicl
DNDN	S.cerevisiae galactose regulation[I9]	GaI2 ” Gal80 ” Gal2 Gal3 ” Gal80 ” Gal3
	Motor neuron columnar fate specification [5]	Hox6 ” Hox9 ” Hox6 HoxIO” Hox9” HoxlO

Given a topologically equivalent class, we first computationally search for the bistable volume in the entire biologically reasonable parameter region by applying the large‐scale random sampling method, and then analyse global and local robustness. Interestingly, we find that the homogeneous DPFL circuit (i.e. the coupled circuit that is composed of only DPFLs) is more robust than all the other circuits in generating bistable behaviour, whereas the homogeneous DNFL circuit is the least robust circuit in generating bistability. In addition, stochastic simulation shows that the low stable state of the homogeneous DPFL circuit is easy to be destabilised by noise whereas the high stable state of the homogeneous DNFL circuit is easy to be destabilised by noise.

2 Models and methods

2.1 Models

First, for every network in a topologically equivalent class, we only consider transcriptional regulation. Second, when establishing a mathematical model, we integrate all biological process such as transcription, translation and promoter binding, into a single step and use the standard quasi‐steady‐state equilibrium assumption that the mRNA molecule dynamics and promoter binding are much faster than the protein dynamics. Third, we omit the basal expression rates of all species since they are generally very small [14, 20, 21]. Forth, we carry out non‐dimensionalisation of system parameters to reduce their number just as Ma et al. did in [15, 22].

Since DPFL and DNFL are two basic modules of more complex networks to be studied, we first establish the mathematical models for both. According to the above hypotheses and using mass‐action kinetics, the dimensionless differential equations governed these two circuits can be expressed as

$$\begin{array}{rl}\frac{\mathrm{d}x}{\mathrm{d}t}& =\frac{1}{{\tau }_1}\left[f_1(y)-x\right]\\ \frac{\mathrm{d}y}{\mathrm{d}t}& =\frac{1}{{\tau }_2}\left[f_2(x)-y\right]\end{array}$$ (1)

where each of functions f ₁ (z) and f ₂ (z) has the form of either (z /K_z ) ⁿ /[1 + (z /K_z ) ⁿ ] that corresponds to activation or 1/[1 + (z /K_z ) ⁿ ] that corresponds to repression. In (1), τ represents the half‐life time, K denotes the threshold for activation or inhibition and the parameter n is the Hill coefficient.

Then, we establish a mathematical model for each network in the two‐loop class that is composed of three topologically equivalent circuits with each composed of one DPFL and one DNFL, referring to the middle column of Fig. 1, where two double feedback loops share one common node (note: we always assume that the common node is x even for a more complex circuit). These circuits can be found in natural systems as listed in Table 1. For each circuit, we consider two typical ways of interaction between two feedback loops [13]: multiplicative way by which we mean that all input signals for a target gene are integrated on the cis‐regulatory module of the promoter in an AND‐type combinatorial manner; and additive way by which we mean that all input signals for a target gene are integrated in an OR‐type combinatorial manner. As examples, the mathematical models corresponding to these two kinds of interaction ways are described as

$$\begin{array}{rl}\frac{\mathrm{d}x}{\mathrm{d}t}& =\frac{1}{{\tau }_x}\left[g(y,z)-x\right]\\ \frac{\mathrm{d}y}{\mathrm{d}t}& =\frac{1}{{\tau }_y}\left[f_1(x)-y\right]\\ \frac{\mathrm{d}z}{\mathrm{d}t}& =\frac{1}{{\tau }_z}\left[f_2(x)-z\right]\end{array}$$ (2)

where g (y, z) = g ₁ (y) g ₂ (z) if the regulation is in the AND‐type combinatorial manner and g (y, z) = αg ₁ (y) + (1 − α) g ₂ (z)(0 < α < 1) if the regulation is in the OR‐type combinatorial manner. The meaning of functions f_k and g_j is similar to that in the caption of (1).

For biochemical reactions, stochastic fluctuation (or molecule noise) are inevitable [23, 24, 25], as some molecule species are often present in low copy numbers. To capture the effect of noise on the steady states of each circuit, we also construct stochastic models by introducing additive noise to each equation in deterministic models, since it has been shown that a stochastic term in the rate equation can capture fluctuations in the gene expression [24]. For example, the stochastic model for the DNDP circuit is described as follow

$$\begin{array}{rl}{\tau }_1\frac{\mathrm{d}x}{\mathrm{d}t}& =\frac{1}{1+{\left(y/K_1\right)}^{n_1}}\frac{{\left(z/K_2\right)}^{n_2}}{1+{\left(z/K_2\right)}^{n_2}}-x+{\xi }_x(t)\\ {\tau }_2\frac{\mathrm{d}y}{\mathrm{d}t}& =\frac{1}{1+{\left(x/K_3\right)}^{n_3}}-y+{\xi }_y(t)\\ {\tau }_3\frac{\mathrm{d}z}{\mathrm{d}t}& =\frac{{\left(x/K_4\right)}^{n_4}}{1+{\left(x/K_4\right)}^{n_4}}-z+{\xi }_z(t)\end{array}$$ (3)

where 〈ξ_i (t)〉 = 0, 〈ξ_i (t) ξ_i (t′)〉 = Dδ (t − t′), 〈ξ_i (t) ξ_j (t′)〉 = 0(i ≠ j); 〈• 〉 denotes the average; D represents the noise intensity. We set D = 0.01 in simulation.

Similarly, we can establish deterministic and stochastic models for all circuits in a more complex class such as in the three‐loop class. For all the circuits to be studied, K and τ are uniformly sampled on the log scale over the ranges K ∈[0.001, 1], τ ∈[0.2, 20] and n is uniformly sampled on the linear scale over the range n ∈[2, 10]. All the parameter ranges are estimated according to the published parameter values for fundamental processes in gene expression [22, 15].

2.2 Methods

As our interest is in the robustness of bistability, we first define some criteria to identify bistable behaviour. Since all the circuits to be studied are monotone systems [3, 26], we can use the monotonicity‐based graphic method introduced by Angeli et al. [3] and its extension to determine whether a circuit exhibits bistability. The key to this approach is to view a circuit as a closure of an open‐loop system, which contains one input signal and one output signal, denoted by ω and η (ω), respectively, thus corresponding to an incidence graph. For example, for DNDP circuit shown in Fig. 2 a, the incidence graph is shown in Fig. 2 b. Note that ω can be viewed as a stimulus whereas η (ω) as a response. Thus, we have an I /O curve for every circuit in a topologically equivalent class: η = η (ω). Now, we define the following two criteria to identify bistability based on the I /O curve:

1. There must be three intersecting points between the I /O curve η = η (ω) and the diagonal curve η = ω in the (ω, η) plane, denoted by X _min, X _mid and X _max, which correspond, respectively, to I, II and III in Fig. 2 c. Moreover, the system's state corresponding to X _mid is unstable since the slope rate of the I /O curve at this state is more than one [4], whereas the states X _min and X _max are stable and are called low and high stable states, respectively.

2. The condition [η]_max − [η]_min > ε is satisfied, where [η]_max and [η]_min denote the maximum and the minimum of η evaluated at the two stable‐steady states, respectively, and ε is a pre‐given small number. We set ε = 0.1 in our numerical simulation.

Fig. 2.

Framework of robustness analysis for circuits in a topologically‐equivalent class

a Schematic view of the coupled positive feedback loop circuit after breaking one feedback loop

b Incidence graph for this circuit, where ω represents the input and η = x represents the output

c Flowchart for robustness analysis, where the arrows represent the flow direction

Then, we use the random parameter sampling method combined with the mentioned above graphical method to assess robustness of each circuit as was done in the previous literature [27, 28, 29, 30], referring the flowchart shown in Fig. 2. The first step of our approach involves the random sampling over a large set, which can span the biologically feasible parameter space, denoted by V. Specifically, up to M = 10⁵ parameter sets are sampled using the Latin hypercube sampling method [31] and assigned to the kinetic constants of every circuit in a topologically equivalent class. Then, we find a viable subset V ₀ ⊆ V that allows the circuit system to exhibit bistability. Thus, we define the global robustness as R_T = |V ₀ |/|V |, where |V | denotes the size of the V set.

As the global robustness analysis does not show the persistence of bistable behaviour when the extracellular environment is changed, the next step of our method makes use of the identified viable parameter set V ₀ to carry our local robustness analysis, including robustness to parameter perturbations and robustness to molecular noise. For this, for every point in the set V ₀, we randomly perturb every component of the parameter vector N times (N = 1000 is set in simulation) by a Gaussian generator with mean 1 and standard deviation 0.4 [28], leading to a small region V _L (see Fig. 2). Then, we let R _p compute the fraction of the N parameter sets (i.e. the ratio of the number of parameter sets that yield bistable behaviour over N), and ${\overline{R}}_{\mathrm{p}}$ compute the mean fraction for all the points in V ₀. The index ${\overline{R}}_{\mathrm{p}}$ describes a system's mean robustness to its parameter perturbations, called ‘mean parameter perturbation fraction’. Note that a common role of noise is to destabilise the stable steady state of a biological system [32]. To quantify the noise‐induced destabilisation, we estimate the robustness of two stable states for each bistable circuit in the identified viable parameter set V ₀. For each bistable circuit, we first set the initial state at the low stable state and perform 100 independent stochastic simulations with different random seeds for a long period of time. Then, we define the robustness of the low stable state (R _l) as the fraction of transition (the ratio of number of the cells in the transition state over the number of all the cells). Such a definition implies that the smaller the R _l, the more robust is the low stable state against noise. Similarly, we set the initial state at the high stable state and compute the robustness of the high stable state R _h. Furthermore, we use ${\overline{R}}_{\mathrm{l}}\text{and}{\overline{R}}_{\mathrm{h}}$ to compute the average of R _l and R _h for all the points in V ₀. ${\overline{R}}_{\mathrm{l}}\text{and}{\overline{R}}_{\mathrm{h}}$ will be called as the mean robustness of the low stable state and the high stable state, respectively.

3 Results

To assess the extent to which the network architectures influence robustness in generating bistability, we perform global analysis and local analysis for all circuits across three topologically equivalent class: one‐loop class, two‐loop class and three‐loop class. Since the DPFL and DNFL circuits are basic modules of all the circuits in a topologically‐equivalent class, we perform more detailed analysis and demonstrate more results for them than for coupled circuits.

3.1 In the one‐loop case

3.1.1 Global robustness

In this section, we combine global analysis with local analysis to show the robustness of bistability for DPFL and DNFL circuits. We randomly sample M = 10⁵ parameter vectors, which cover a range of several orders of magnitude for each parameter. The numerical results are shown in Figs. 3 a and b. Specifically, Fig. 3 a shows the global robustness score R_T for both circuits. From this figure, one can observe that R_T for the DPFL circuit is significantly higher than that for the DNFL circuit. In fact, R_T ≃ 0.74 for the former whereas R_T ≃ 0.18 for the latter. The relationship between the distribution of bistable systems in the (K ₁, K ₂) phase plane for DPFL and DNFL systems are shown in Fig. 3 b; as shown, a broader range of parameters resulted in bistable systems for the DPFL circuits. The combination of Figs. 3 a and b indicates that the DPFL circuit has much better global robustness than the DNFL circuit. To better understand it, here we give an intuitive interpretation. Fix some parameter values τ ₁ = τ ₂ = 1, n ₁ = n ₂ = 2 and investigate the effect of variation in parameter K ₁ and K ₂ on the bistability of both circuits. A codimension‐two bifurcation diagram is shown in Fig. 3 c, where regions marked by I and II are the bistable regions of DPFL and DNFL, respectively. We observe from this figure that both bistable regions can vary with two control parameters K ₁ and K ₂. In particular, when they are small, both systems can display bistable behaviour, but when K ₁ or K ₂ increases, the DNFL system will become monostable earlier than the DPFL system. Therefore the latter is more efficient in generating bistability than the former, which is consistent with our above conclusion.

Fig. 3.

Robustness analysis for DPFL and DNFL: (A and B) Global robustness analysis, where

a Global robustness score R_T , and

b Parameter region that allows the circuit system to generate bistability in the (K ₂, K ₁) phase plane with the grey corresponding to DPFL and the black to DNFL

c Codimension two bifurcation diagram for DNFL and DPFL, where some parameter values are set as τ ₁ = τ ₂ = 1, n ₁ = n ₂ = 2. The regions, denoted by I and II, correspond to the bistable regions of DPFL and DNFL, respectively

d Local robustness against parameter perturbations (${\overline{R}}_{\mathrm{p}}$)

e Standard variation (λ_i ) along each principal axes in the viable parameter space (see the main text)

f Components of the normalized eigenvector along the last principal axis (also see the main text), where parameter 1–6 denote τ ₁, τ ₂, K ₁, K ₂, n ₁ and n ₂, respectively

To explain why the DNFL has the poorer global robustness than the DPFL, we performed the principal component analysis (PCA) [28], which can show sensitivity of a system to parameters. Note that associated with the identified viable parameter sets is the variance–covariance matrix Σ with eigenvalues λ_i and eigenvectors k _i = (k_i1 , k_i2 , …,k_ip ) for both circuit systems. The i th PCA axis is then defined as ${\upsilon }_i={\sum }_j{k}_{ij}{\mathit{\theta }}_j$, where θ _j represents the unit vector of the j th axis in the original six‐dimensional parameter space (τ ₁, τ ₂, K ₁, K ₂, n ₁, n ₂), and the variation along this axis is determined by λ_i . The smaller the eigenvalue λ_i , the more sensitive is the system to the parameter combination corresponding to the v_i axis [33]. Thus, the system is most sensitive to the parameter corresponding to the last PCA axis (${\upsilon }_6$) that corresponds to the last eigenvalue (λ ₆) and most insensitive to the parameter corresponding to the first PCA axis (${\upsilon }_1$) that corresponds to the first eigenvalue (λ ₁).

Fig. 3 e shows the standard variation of the parameters along the last PCA axes of DNFL and DPFL. We observe that the standard variation along the last PCA axis for the DNFL circuit is substantially smaller than that along the last PCA axis for the DPFL circuit, but the standard variation along the other PCA axes are basically similar for both circuits. This implies that a strong association exists between individual parameters in the viable parameter space for the DNFL circuit. Fig. 3 f shows the components of the normalised eigenvector along the last PCA axis for DNFL and DPFL, respectively. We observe from the figure that the linear combination (K ₁ − K ₂) dominates the last PCA axis of DNFL but the linear combination (K ₁ + K ₂) dominates the last PCA axis of DPFL (K_i denotes the threshold for activation or inhibition). Furthermore, by computing the correlation (denoted by r) between parameters K ₁ and K ₂ for both circuits, we find that the two parameters are positively correlated with r = 0.6 for DNFL but negatively correlated with r = − 0.1 for DPFL. Thus, it is the high association between parameters K ₁ and K ₂ that contributes to the lack of global robustness that we observed in DNFL circuit. When examining (1) for the DNFL circuit, we find that if K ₁ is too large and K ₂ is too small, then the concentration of protein Y is very high, which will, in turn, prevent the persisting existence of the high stable state of protein X; if K ₁ is too small and K ₂ is too large, then the concentration of protein Y is very low, which will in turn prevent the persisting existence of the high stable state of protein X. Therefore the parameters K ₁ and K ₂ are required to be delicately balanced to generate bistable behaviour.

To test whether the above conclusion depends on sampling size, we vary the sample size from 1000 to 10 000 and compute the global robustness index for both DNFL and DPFL (Fig. 4 a). We find that this index changes in a small range for both circuits, indicating that the sampling size does not influence the conclusion. Furthermore, to test the whether the qualitative result depends on the choice of parameter range, we perform the global robustness analysis when the range of K is extended from the interval (0.001, 1.0) to the interval (0.0001, 10), and find that the global robustness of DPFL is also much better than that of DNFL (Fig. 4 b).

Fig. 4.

Effect of sample size and parameter range on global robustness

A Global robustness scores for DNFL and DPFL when the sample size varies from 2000 to 10 000

B Global robustness scores for DPFL and DNFL with an extended range [K ∈ (0.0001, 10)]

3.1.2 Local robustness

First, we analyse local robustness of both circuits to parameter perturbations. Fig. 3 d shows the numerical results for ${\overline{R}}_{\mathrm{p}}$, from which we observe that there is a small difference in ${\overline{R}}_{\mathrm{p}}$ between two circuits, but ${\overline{R}}_p$ for DPFL is still larger than that for DNFL, indicating that the former has better local robustness than the latter. Comparing Figs. 3 a and 3d, we find an interesting fact, that is, the high R_T corresponds to the high average ${\overline{R}}_{\mathrm{p}}$ and the low R_T to the low average ${\overline{R}}_{\mathrm{p}}$. In short, such an investigation on local robustness further supplements the conclusion obtained by global robustness analysis.

Then, we analyse local robustness of both circuits to noise. Previous experiments [34] and theoretic analysis [23, 35] have shown that noise can induce spontaneous switch between two distinct stable states. Our major focus in this paper is on investigating the difference between the stability of two stable states. Fig. 5 a shows the robustness of low stable states (${\overline{R}}_{\mathrm{l}}$) for both DPFL and DNFL. From the figure, we observe that only 2% for the DNFL system transits to the high stable state, but as high as 56% for the DPFL system transits to the high stable state, indicating that the low stable state of the DNFL system is more robust to noise whereas the low stable state of the DPFL system is more fragile to noise. The robustness of the high stable state (${\overline{R}}_{\mathrm{h}}$) of both circuits is shown in Fig. 5 b. As shown in this figure, the transition proportion of DNFL (30%) is much higher than that of DPFL (2%), indicating that the high stable states of the DNFL system is more fragile to noise while the high stable state of the DPFL system is more robust to noise. In conclusion, for the DNFL circuit, the low stable state is more robust to noise and the high stable state is more fragile to noise; but for DPFL circuits, the low stable state is more fragile to noise and the high stable state is more robust to noise. Figs. 5 c and d show the robustness of low and high stable steady states for the DPFL circuit against noise intensity. If the noise is absent, there exists no transition between two stable states and all cells are trapped in the initial states. With the increasing of the noise intensity, the frequency of transition between the two stable states is enhanced and the stable state is easier to be destabilised by noise. Comparing Figs. 5 c and d, we find that the frequency of transition from the low stable state to the high stable state (${\overline{R}}_{\mathrm{l}}$) increases more fast than the frequency of transition from the high stable state to the low stable state (${\overline{R}}_{\mathrm{h}}$), which further verify our conclusion that the low stable state of DPFL circuits is more fragile to noise and the high stable state.

Fig. 5.

Local robustness of both circuits to noise

a and b Mean robustness of the low stable state and the high stable state for two circuits

c and d Mean robustness of the low and high stable states versus noise intensity for DPFL

e Distribution of quasi‐potential corresponding to low and high stable states of the DNFL circuit

f Distribution of quasi‐potential corresponding to low and high stable states of the DPFL circuit, where blue bar represents the distribution of the low stable state and red bar represents the distribution of the high stable state

Furthermore, we use the deterministic quasi‐potential method [36] to quantify the stability of the low and high stable states. The quasi‐potential to be defined is actually a Liapunov function that is minimised along the trajectory from an initial state to the attractor in the phase space. For example, the quasi‐potential for the DNFL circuit (1) can be written as

$$\frac{\mathrm{d}{V}_{\mathrm{q}}}{\mathrm{d}t}=-\left[{\left(\frac{\mathrm{d}x}{\mathrm{d}t}\right)}^2+{\left(\frac{\mathrm{d}y}{\mathrm{d}t}\right)}^2\right]$$ (4)

Sudin et al. [36] have shown that the valley of low elevation of the quasi‐potential corresponds to the stable state, with the deeper valley associated with higher probability of occurrence than the shallow valley.

Here, we set the quasi‐potential level corresponding to the unstable point (X _mid) as zero. The trajectory originating from the adjacent area of the unstable point would converge to the low stable state (X _min) or high stable point (X _max). Integrating (4) along these trajectories, we obtain the quasi‐potential corresponding to the low stable state and high stable state. Figs. 5 e and f show the distribution of quasi‐potential corresponding to the low stable state and high stable state of DNFL and DPFL. From the figure, we observe that, for DNFL, the median quasi‐potential of the low stable state is lower (V _q = − 0.33) than that of the high stable state (V _q = − 0.13). However, for DPFL, the median quasi‐potential of the high stable state (V _q = − 0.67) is lower than that of the low stable state (V _q = − 0.03). These results further verify our conclusion that the low stable state is more robust than the high stable state in the DNFL circuit, and the high stable state is more robust than the low stable state in the DPFL circuit.

3.2 In multi‐loop cases

Here, we only consider two topologically equivalent classes: two coupled positive feedback loops, which institute three kinds of circuits: DNDN, DNDP, DPDP, and three coupled positive feedback loops, which institute four kinds of circuits: DNDNDN, DNDNDP, DNDPDP, DPDPDP, and show some results obtained by (global and local) robustness analysis. For more complex topologically‐equivalent classes, the results are similar and hence omitted.

First, we perform global robustness analysis. For three topologically‐equivalent circuits with two feedback loops, that is, DNDN, DNDP and DPDP, we randomly sample M = 10⁵ parameter vectors and assign these parameter vectors to every circuit in this class. The numerical results are shown in Fig. 6 a. Compared with DNDN and DNDP circuits, the DPDP circuit exhibits bistability with a significantly larger proportion of parameter sets with the global robustness score as high as 92%; in contrast, the global robustness scores for DNDN and DNDP are 30 and 71%, respectively. This indicates that DPDP is the most robust circuit among the three topologically‐equivalent circuits, whereas the DNDN circuit is the most fragile one. The similar result is also held for four topologically‐equivalent circuits: DNDNDN, DNDNDP, DNDPDP, DPDPDP, that is, the DPDPDP circuit is most robust in generating bistability. We can see from the above analysis that for a topological‐equivalent class, increasing the number of DPFLs and decreasing the number of DNFLs will improve robustness of bistability. In more complex cases of coupled feedback loops, this qualitative result is still kept (data are not shown here).

Fig. 6.

Robustness analysis for coupled circuits with two feedback loops: DNDN, DNDP and DPDP and three feedback loops: DNDNDN, DNDNDP, DNDPDP and DPDPDP

a and e Global robustness score (R_T )

b and f Mean parameter perturbation fraction (${\overline{R}}_{\mathrm{p}}$)

c and g Mean robustness of the low stable state (${\overline{R}}_{\mathrm{l}}$)

d and h Mean robustness of the low stable state (${\overline{R}}_{\mathrm{h}}$)

Then, we analyse the local robustness to parameters in V ₀ for every bistable circuit. For this, we perturb each set of parameters in V ₀ 1000 times, and compute the corresponding mean parameter perturbation fraction ${\overline{R}}_{\mathrm{p}}$. The numerical result is shown in Figs. 6 b and f. We observe that even though the difference in ${\overline{R}}_{\mathrm{p}}$ between the circuits is not large, the DPDP (or DPDPDP) circuit has the largest ${\overline{R}}_{\mathrm{p}}$, whereas the DNDN (or DNDNDN) circuit has the lowest ${\overline{R}}_{\mathrm{p}}$, indicating that the DPDP (or DPDPDP) circuit has better local robustness than the other two circuits. Now, we analyse local robustness of circuits to noise for each circuit in V ₀. To show the effect of the noise, we compute the mean robustness of the low stable state, ${\overline{R}}_{\mathrm{l}}$, and the high stable state, ${\overline{R}}_{\mathrm{h}}$. The result is shown in Figs. 6 c and 6d or Figs. 6 g and h. We find that there is a large difference between the circuits in each of two topologically equivalent classes. Specifically, DNDN (or DNDNDN) has the lowest ${\overline{R}}_{\mathrm{l}}$ and the highest ${\overline{R}}_{\mathrm{h}}$; DPDP (or DPDPDP) has the highest ${\overline{R}}_{\mathrm{l}}$ and the lowest ${\overline{R}}_{\mathrm{h}}$; and DNDP has moderate ${\overline{R}}_{\mathrm{l}}$ and ${\overline{R}}_{\mathrm{h}}$. Thus, in the noisy environment, the DNDN (or DNDNDN) circuit has a robust low stable state and a fragile high stable state; and the DPDP (or DPDPDP) circuit has a fragile low stable state and a robust high stable state. However, for DNDP, we find that two stable‐steady states seem to have the similar attracting basin since ${\overline{R}}_{\mathrm{l}}$ and ${\overline{R}}_{\mathrm{h}}$ are almost the same.

Similar to the single‐loop case, the PCA of the viable parameter vectors can provide us some insight into the topologically equivalent circuits. Fig. 7 a shows the standard deviation of the viable parameters along the principal axes of three circuits with two loops. We observe that the standard deviation along most of the principal axes are basically similar in quantity for all three circuits, but the last two principal components for DNDN are significantly smaller than those for DNDP and DPDP, indicating that the smaller standard deviations in these axes contribute substantially to the lack of robustness of the DNDN circuit. Figs. 7 b –d show the last PCA axes for DNDN, DNDP and DPDP circuits. From the subfigures, we find that the linear combination (K ₃ − K ₁) + (K ₄ − K ₂) dominates the first PCA axis of DNDN (where K_i denotes the threshold for activation or inhibition). Similarly, the combination (K ₃ − K ₁) dominates the first PCA axis of DNDP; the combination (K ₃ + K ₁) + (K ₄ + K ₂) dominates the first PCA axis of DNDP. Note that K ₃ and K ₁ (or K ₄ and K ₂) in the DNDN circuit [see (2)] represent the threshold of inhibition in two DNFLs, so they are required to be delicately balanced to generate bistability.

Fig. 7.

Results by PCA analysis for three topologically equivalent circuits with two loops

a Standard deviation along the principal axes of viable parameters

b –d Last principal axes for DNDN, DNDP and DPDP, where parameter 1–11 denotes τ ₁, τ ₁, τ ₁, K ₁, K ₂, K ₃, K ₄, n ₁, n ₂, n ₃ and n ₄, respectively

We also performed the PCA for four topologically‐equivalent circuits with three loops, and found that the results in this case are similar to those in the case of two loops (data are not shown here).

To test whether our conclusion depends on the interaction mechanism between two positive feedback loops, here we perform the global robustness analysis only for the circuits with two loops coupled in an additive way. The results are shown in Fig. 8. We observe from Fig. 8 a that the DPDP circuit is also most robust among three topologically equivalent circuits. Here for DPFL and DNFL, we assume that multimeric regulation (n > 1) is essential to generate bistability (note: for interlinked positive feedback loops, monomeric regulation (n = 1) may induce bistability via some topologies of the interaction between feedback loops, referring to [37]). Fig. 8 b shows the global robustness for monomeric circuits with two loops coupled in a multiplicative way. We find that all robustness scores for three types of circuits are very small compared with the multimeric circuits, but the DPDP circuit can still outperform the other two types of circuits in generating bistability. In addition, to test whether our conclusion depends on the choice of the parameter range, we also perform the global robustness analysis when the range of K is extended to the interval (0.0001, 10) and the range of n is extended to (1, 10). The result shows that the global robustness of DPDP also outperforms that of the other two circuits in generating bistability, referring to Fig. 8 c.

Fig. 8.

Global robustness scores for three circuits

a Coupled in an additive way

b With monomeric regulation (n = 1)

c With extended sample ranges [K∈ (0.001, 10) and n∈ (1, 10)]

3.3 Inferring the robustness from the shape of I/O curve

The previous theoretical analysis [3, 13] has shown that robustness of bistability seems to depend on the ultrasensitivity of the system. In particular, Chang et al. [13] revealed that two characteristics of the I /O curve (see the definition in method) are closely related to the robustness of bistability. The first factor is the steepness of the I /O curve, that is, the steeper the I /O curve is, the more robust does the system keep bistability against parameter perturbations; The second factor is the absolute height of the I /O curve, that is, the higher the absolute magnitude is, the more robustly does the system generate bistable behaviour. In this section, we will use these two characteristics of I /O curves to analyse topologically equivalent circuits across three classes.

To simultaneously incorporate the above two factors to our analysis, we first convert each non‐linear model into a linear curve by using a three‐section piecewise linearisation method [38] (Fig. 9 a), and then compute the maximal output η _max and the minimal output η _min of the I /O curve. Furthermore, we introduce two new quantities

$$\begin{array}{rl}{\eta }_{0.1}\left({\omega }_{0.1}\right)& ={\eta }_{\min }+0.1\left({\eta }_{\max }-{\eta }_{\min }\right)\\ {\eta }_{0.9}\left({\omega }_{0.9}\right)& ={\eta }_{\min }+0.9\left({\eta }_{\max }-{\eta }_{\min }\right)\end{array}$$ (5)

where ω _0.9 is the concentration of input for generating 90% of the maximum response, and ω _0.1 is the concentration of input for generating 10% of the maximum response. Then, the slope of the simplified linear I /O curve is S = [η _0.9 (ω _0.9)− η _0.1 (ω _0.1)]/(ω _0.9 − ω _0.1). Furthermore, we have the mean slope when randomly sampling parameter values, denoted by $\overline{S}$. Clearly, the larger the $\overline{S}$ is, the higher is the mean steepness or the absolute magnitude, implying that the robustness of bistability is better.

Fig. 9.

Mean slope of linear I/O curves for three topologically equivalent classes

a Linearisation of the I /O curve

b Single‐loop case

c Two‐loop case

d Three‐loop cases

Fig. 9 b shows the mean slope for DNFL and DPFL. We observe that the $\overline{S}$ for DPFL is larger than that for DNFL, so we can accordingly infer that the former is more robust than the latter, which is in accord with the above numerical simulation (see Fig. 3 a). Such an agreement indicates that the robustness of a circuit can be estimated from the shape of the I /O curve without recourse to extensive numerical simulation. We further estimate the $\overline{S}$ for the coupled circuits with two feedback loops and three feedback loops, with results being shown in the Figs. 9 c and d. From these subfigures, we observe that with the increase of the number of DPFLs in each class, the mean slope of the corresponding circuit is increased, implying that the robustness becomes better. The similar conclusion is also held for other more complex circuits (data are not shown here). In a word, we obtain the same results as obtained through large‐scale random sampling. Such an agreement further verifies the correctness of our conclusion.

4 Conclusion and discussion

It has been shown that non‐linear interaction between simply coupled feedback loops can contribute to the robustness of bistability [12, 13]. However, natural or synthetic circuits often exist in topologically‐equivalent classes with each containing several circuits. For instance, coupled positive feedback circuits with two loops (i.e. topologically‐equivalent two‐loop class), which can be divided into DNDN, DNDP and DPDP patterns, are typical examples (see Table 1). In this paper, by using large‐scale parameter sampling method to assess the robustness of topologically‐equivalent circuits, we have shown that the interlinked pattern of feedback loops can affect the robustness of the circuit systems in generating bistability. Specifically, for a given class, the homogeneous DPFL circuit is more robust than all the other circuits in generating bistable behaviour. In particular, increasing the number of DPFL will improve the robustness of the corresponding circuit but increasing the number of DNFL will weaken the robustness of the circuit. In addition, stochastic simulation shows that the low state of homogeneous DPFL circuits is easy to be destabilised by noise while the high state of homogeneous DNFL circuits is easy to be destabilised by noise.

In natural networks, a bistable circuit that performs a particular biological function is usually required to be sufficiently robust against perturbations. An efficient way to enhance bistablity is to interlink multiple feedback–positive feedback loops [39, 40, 41]. Thus, our study is helpful for understanding design principles by which a robust bistable switch is generated. That is, in addition to an appropriate cooperation between feedbacks loops as shown previously, the interlinking patterns are also an important factor affecting the robustness of bistability. However, we point out that our robustness analysis cannot exclude the possibility that the least robust (in the sense of this paper) circuits exist in real biological systems (see the examples listed in Table 1), implying that these circuits possibly play other specific roles. In summary, the combination of global and local analysis is a reliable method for identifying the structure of networks with a particular biological function.