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PLoS One. 2011; 6(10): e26302.
Published online Oct 14, 2011. doi:  10.1371/journal.pone.0026302
PMCID: PMC3194799

MicroRNA-Mediated Positive Feedback Loop and Optimized Bistable Switch in a Cancer Network Involving miR-17-92

Yichen Li,1,2 Yumin Li,1,2,* Hui Zhang,3 and Yong Chen2,3,*
Richard James Morris, Editor

Abstract

MicroRNAs (miRNAs) are small, noncoding RNAs that play an important role in many key biological processes, including development, cell differentiation, the cell cycle and apoptosis, as central post-transcriptional regulators of gene expression. Recent studies have shown that miRNAs can act as oncogenes and tumor suppressors depending on the context. The present work focuses on the physiological significance of miRNAs and their role in regulating the switching behavior. We illustrate an abstract model of the Myc/E2F/miR-17-92 network presented by Aguda et al. (2008), which is composed of coupling between the E2F/Myc positive feedback loops and the E2F/Myc/miR-17-92 negative feedback loop. By systematically analyzing the network in close association with plausible experimental parameters, we show that, in the presence of miRNAs, the system bistability emerges from the system, with a bistable switch and a one-way switch presented by Aguda et al. instead of a single one-way switch. Moreover, the miRNAs can optimize the switching process. The model produces a diverse array of response-signal behaviors in response to various potential regulating scenarios. The model predicts that this transition exists, one from cell death or the cancerous phenotype directly to cell quiescence, due to the existence of miRNAs. It was also found that the network involving miR-17-92 exhibits high noise sensitivity due to a positive feedback loop and also maintains resistance to noise from a negative feedback loop.

Introduction

MicroRNAs (miRNAs) are small, endogenous non-coding RNA molecules, typically An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e001.jpg nucleotides (nt) in length. Traditionally, miRNAs were thought to be an undesirable class of small RNAs that only served a relevant function in nonmammalian species. In 1993, Ambros and colleagues found that the lin-4 gene does not encode a protein product, but gives rise to a 61-nt precursor gene that matures to a more abundant 22-nt transcript in the model organism Caenorhabditis elegans [1]. The Ruvkun laboratory observed that lin-14 protein synthesis is regulated post-transcriptionally and that lin-14 levels are inversely proportional to those of lin-4 RNA [2]. Thus, they revealed the first miRNA and mRNA target interaction, where the lin-4 RNA has sequence complementarity to the An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e002.jpg untranslated region of the lin-14 gene. Currently, there have been over An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e003.jpg miRNAs identified and published in public databases (miRBase database, http://microrna.sanger.ac.uk). These miRNAs have been identified in animals, plants, and viruses and are involved in the regulation of a variety of biological processes.

By post-transcriptionally down-regulating gene expression, it is currently predicted that miRNAs regulate up to An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e004.jpg of human genes and their targets include signaling proteins, enzymes, and transcription factors [3]. The diversity and abundance of miRNA targets result in miRNAs playing important roles in nearly all fundamental cellular processes, such as developmental timing, cell proliferation, apoptosis, stem cell maintenance, differentiation, signaling pathways, and pathogenesis including carcinogenesis [4][6]. It is remarkable that approximately half of the known miRNAs are located inside or close to fragile sites and in minimal regions of loss of heterozygosity, minimal regions of amplifications, and common breakpoints associated with cancer [7]. Recent studies have shown that miRNAs are involved in the initiation and progression of a variety of cancers and can act as oncogenes or tumor suppressors depending on the tissue and the expression level of their targets [8], [9].

Clearly, miRNAs cannot independently perform a single task in cells. Instead, miRNAs regulate cellular networks as network components in many cellular functions. Indeed, it is thought that miRNAs lead to more effective noise buffering [10], [11]. In addition, Aguda et al. found that miR-17-92 plays an oncogenic role in one setting but suppresses tumor formation in a different scenario [12]. Here, there are two determinative factors, the definition of the postulated cancer zone and the switch behaviors of the system dynamics [12]. Note that the Myc/E2F/miR-17-92 network is composed of two feedback loops: a positive self-feedback loop for protein module (Myc/E2F) and a negative loop between the miRNA and the protein (Figure 2 in [12] or Figure S1). This network appears a typical bistable switch behavior and a one-way switch corresponding to the bistability and monostablility, respectively [12]. In fact, a single positive feedback loop without miRNA is enough to realize a bistable switch. Consequently, it is interesting to investigate the physiological significance of miRNAs or why cancer networks require miRNA regulation and the contributions from miRNAs.

Figure 2
Steady-state bifurcation diagrams of the dimensionless protein concentration

To examine this issue, we focus here on an abstract model of the Myc/E2F/miR-17-92 network described by Aguda et al. [12] and present simulations with experimental parameters. Our results show that the existence of miRNA improves the ability of the bistable switches in the systems. For the single-loop switch, there is a so-called fast/slow loop to describe the fast/slow response kinetics for the activation and inactivation [13]. Normally, the single fast-loop switch achieves more rapid responses. Moreover, we also found that miRNA can mediate the system between a fast loop and a slow loop due to the different rate coefficients of degradation for miR-17-92 and E2Fs/Myc (in general, miRNAs is more stable than protein), and can diversify the response behavior of the system to the input stimulus. Especially, the undamped relaxation oscillation behavior of the system indicates a possible digital regulation mode. Furthermore, the switching behaviors of the network involving miR-17-92 is both sensitive to stimuli and resistant to fluctuations in stimulus. Our finding show that miRNAs play a key role in the possibility to achieve a sensitive robustness in biological systems.

Results

Model Formulation

By following the mammalian G1-S regulatory network, the essential abstract structure of the Myc/E2F/miR-17-92 network is illustrated in Figure 1. An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e005.jpg denotes the protein module (Myc and E2Fs), and An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e006.jpg is the miRNA cluster module (see Ref. [12] or Figure S1 for the detailed reduction process). The positive feedback loop in module An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e007.jpg represents an autocatalytic process, which is also inhibited by module An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e008.jpg. At the same time, An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e009.jpg induces the transcription of An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e010.jpg.

Figure 1
Schematic illustration of the cancer network involving miR-17-92, E2F, and Myc.

The dynamics of the respective concentrations of An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e013.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e014.jpg are described by the following ordinary differential equations [12]:

equation image
(1)
equation image
(2)

where An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e017.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e018.jpg denote the concentrations of An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e019.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e020.jpg, respectively. An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e021.jpg describes the constitutive protein expression from the signal transduction pathway in the extracellular medium and is experimentally controlled in the cell culture medium. An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e022.jpg depicts the An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e023.jpg-independent constitutive transcription of An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e024.jpg. An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e025.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e026.jpg are the rate coefficients of degradation. An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e027.jpg is the constant of protein expression, and An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e028.jpg is the rate constant. An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e029.jpg is the coefficient of protein expression, and An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e030.jpg is a measure of the miRNA inhibition of protein expression.

Under a series of nondimensionalizing processes [12], Eqs. (1) and (2) can be rewritten as:

equation image
(3)
equation image
(4)

Here, An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e033.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e034.jpg, and An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e035.jpg. In general, An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e036.jpg is less than An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e037.jpg because miRNA is more stable than protein. An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e038.jpg is allowed to vary in the range of An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e039.jpg. The experimentally controllable parameters An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e040.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e041.jpg vary from An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e042.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e043.jpg, respectively [12], [14], [15]. The last parameter An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e044.jpg is set as An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e045.jpg. The process to deduce the dimensionless parameters is presented in Text S1.

Steady States, One-way Switches, and bistable Switches

Setting Eqs. (3) and (4) equal to zero and solving for the roots of the algebraic equations from the right-hand sides, we obtain the steady states of the model. As shown in [12], there exists the following relationship:

equation image
(5)

where An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e047.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e048.jpg represent the steady states of An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e049.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e050.jpg, respectively (see the explicit solutions in Text S2). Note that there is only a constant difference between An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e051.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e052.jpg. Moreover, An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e053.jpg denotes the dimensionless concentration of protein E2Fs and Myc which is directly correlated with the oncogene or tumor suppression. Thus, we present only the results of An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e054.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e055.jpg in the following context.

Due to the positive feedback in module An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e056.jpg, this system exhibits a switching behavior, and it has two stable fixed points in the appropriate parameter regime [16]. Steady-state An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e057.jpg bifurcation diagrams as a function of the parameter An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e058.jpg for different values of An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e059.jpg are presented in the top panels of Figure 2. Considering the physiological constraints, the horizontal axis An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e060.jpg should be greater than An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e061.jpg and terminates at a maximal value of An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e062.jpg. Clearly, it also shows the oncogenic and tumor suppressor properties of miR-17-92 using the concept of the cancer zone [12] (see Figure 2A–D). Note that An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e063.jpg decreases with decreasing An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e064.jpg for fixed An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e065.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e066.jpg. Therefore, the positive feedback loop has a similar reversely regulating function of miRNA.

In the case of a bistable switch, the system exhibits hysteresis, which is a property of bistable systems. For convenience, we denote the lower/upper steady state (lower/higher protein concentration) as the off/on state. As an example, for An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e078.jpg in Figure 2A, as An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e079.jpg increases along the lower stable branch, An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e080.jpg remains in the off-state until An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e081.jpg. When An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e082.jpg is increased further, the off-state vanishes, and the switch moves towards the upper stable branch, corresponding to the on-state. Then, if we decrease An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e083.jpg, the switch moves along the upper stable branch until An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e084.jpg, and then a transition brings the switch back to the off-state. However, the system also acts as an extreme manifestation of hysteresis, a one-way switch, where the on-state branches into the negative domain but is actually eliminated due to a physically meaningful restriction (e.g., An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e085.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e086.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e087.jpg in Figure 2A and An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e088.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e089.jpg in Figure 2B).

It is well-known a bistable region is enclosed by two saddle-node bifurcation points (the left and right knees of the curves in the top panels of Figure 2). For any An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e090.jpg values in this region, the system has two stable solutions and one unstable one. The bottom panels of Figure 2 provide whole phase diagrams of switch behaviors including bistable and one-way switches and a monostable region for different An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e091.jpg values. By increasing the strength of the positive feedback An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e092.jpg, the system undergoes a transition from monostability to a bistable switch and then to a one-way switch. The parameter An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e093.jpg exhibits a similar influence. When we increase An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e094.jpg, the transition begins with a one-way switch to a bistable switch and then to monostablility. As for the dimensionless parameter of miRNA inhibition An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e095.jpg, it reduces the switch region enclosed in the effective domain of An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e096.jpg (from An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e097.jpg to An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e098.jpg, denoted by two red dashed lines). However, it should be noted that the system only has one-way switching behavior without An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e099.jpg (see Figure 2E). The effective region of the bistable switch is expanded by increasing the strength of An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e100.jpg (Figures 2E–H). Otherwise, for higher values of An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e101.jpg, the system is always monostable, but in the case of lower An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e102.jpg values, the system produces diverse dynamical behaviors with the existence of An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e103.jpg.

Note that there exists a cancer zone between levels associated with normal cell cycles and apoptosis, a range of Myc and E2F levels with increased probability of inducing cancer (see Figure S2 or Figures 3, ,66 in [12]). In general, the off-state denotes cell quiescence and the on-states represent other processes (cell cycles, the cancer zone, and cell apoptosis). In all cases, the regulation imposed by miRNA or the positive feedback loop is confined to the upper state, from cell cycles to the cancer zone and to cell apoptosis or the inverse order (see Figure 2 and Figure 6 in [12]). Considering a perturbation of An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e104.jpg along the on-state of a single bistable switch, An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e105.jpg changes slightly. For An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e106.jpg, the system almost stays in the on-state because there only exists the one-way switch (In fact, the probability of transferring to another cell status is also very low. See Figure 2A.) however, for An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e107.jpg (the existence of miRNA), it is possible to return to cell quiescence due to the bistable switch (Figures 2B–D). This means that the miRNAs lead to the recurrent process from the on-state to the off-state. Thus, the miRNA provides two possible regulation pathways, from cell apoptosis or the cancer zone to cell quiescence (Figure S3).

Figure 3
The dynamical behaviors of the system when increasing the inhibition of miRNA
Figure 6
Time course of fluctuation-induced escape from the on-state (upper state) to the off-state (lower state).

Diversification of the Signal-response Behaviors

As shown in Figure 2, the emergence of the inhibition of miRNA An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e123.jpg induces diversification of the system dynamics in the physiological regions of An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e124.jpg. In fact, the repression of An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e125.jpg by An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e126.jpg yields negative feedback (see Figure 1). This system, composed of positive and negative feedback, is a flexible motif that can exhibit various behaviors [17], [18]. It has already been shown that the negative feedback makes oscillation possible [19], [20]. Here, we show that the system can exhibit diverse signal-response behaviors corresponding to the different regimes of Figures 2F–H.

For any fixed strength of the positive feedback An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e127.jpg, the behavior of the system is different for varying An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e128.jpg (see the bottom panels of Figure 2). With An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e129.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e130.jpg ranging from 1.0 to 1.2 and 1.6, Figure 3 shows that the response of the system to the input signal is tuned by An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e131.jpg. The top row shows the time evolution of An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e132.jpg under the input stimulus, and the bottom row depicts the corresponding bifurcation diagram of the system. The solid and dashed lines denote stable and unstable steady states, respectively. An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e133.jpg/An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e134.jpg denotes a saddle-node bifurcation and An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e135.jpg/An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e136.jpg represents a Hopf bifurcation. The input stimulus is a pulse with An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e137.jpg except for when it is 0.16 from time 20 to 25 in Figures 3A and 3B, but it always is constant at An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e138.jpg in Figure 3C. The initial values used were An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e139.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e140.jpg.

First, in the case of An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e141.jpg, the system behaves irreversibly. Under the pulse input, the system settles to the on-state and cannot return to the initial off-state (Figure 3A). This behavior is typical of a one-way switch (Figure 3D). As An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e142.jpg is increased to 1.2, the strength of the pulse input is in the range between An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e143.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e144.jpg (Figure 3E). In this case, there are three steady states, but only the lower state (off-state) is stable while the others (the upper and middle ones) are unstable. The system exhibits excitability where An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e145.jpg is first driven to the on-state due to the instability from the pulse input and then completely recovers to the original off-state after a short shift (Figure 3B). Thus, the model system also generates a large-amplitude transient pulse to respond to the pulse input.

When An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e146.jpg is further increased and the strength of the constant stimulus is between two Hopf bifurcation points An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e147.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e148.jpg, the system enters into limit-cycle oscillations (see Figures 3C and 3F, and the respective 3-dimension bifurcation in Figure S4). An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e149.jpg rises gradually and then drops rapidly after reaching a maximum, and the cycle repeats. As we all know, the undamped relaxation oscillation is a periodic process in which slow smooth change of the state of an object over a finite interval of time is alternated with rapid irregular change of the state during an infinitely short time. Since van der Pol presented the classic example of a one-dimensional system having relaxation oscillations [21], such oscillatory processes are observed in many real mechanical, radiotechnical, biological, laser physical etc., objects. So, such an oscillation is a type of undamped relaxation oscillation. Notably there is no time delay in the recent limit-cycle oscillation system. This means that this oscillation resulted from the hysteresis induced by the positive feedback [22], [23].

As stated in Figure 3A and 3D, the system exhibits a fundamental phenomenon in nature, the so-called bistability. A bistable system is able to rest in two states, which need not be symmetric. The defining characteristic of bistability is simply that two stable states (minima) are separated by a barrier (local maximum). For example, for an ensemble of particles, the bistability comes from that free energy has three critical points. Two of them are minima and the last is a maximum. However, Figure 3B and 3E display the excitability of the system. Common to all excitable systems is the existence of a rest state, an excited (or firing) state, and a refractory (or recovery) state, such as action potential in neural systems. The system is in the presence of one stable and one or more unstable fixed points. If unperturbed, the system resides in the rest state; small perturbations result only in a small-amplitude response of the system. For a sufficiently strong stimulus (for example, larger than the An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e150.jpg value of An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e151.jpg in Figure 3E), the system can leave the rest state, going through the firing and refractory states and then comes back to rest again [24].

As illustrated above, with increasing An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e152.jpg, the system undergoes a transition from bistability to excitability and to undamped relaxation oscillation without a time delay. We suggest that there is a feasible way to produce diverse signal-response behaviors by combining the inhibition of miRNA An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e153.jpg and the experimentally tunable parameter An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e154.jpg. Figure 4 presents an overview of the tunability using the phase diagram of the system dynamics in An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e155.jpg plane with An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e156.jpg. The bulk diagram is composed of four kinds of dynamics: monostability, bistability, excitability, and undamped relaxation oscillation where the borderlines between these dynamics (solid lines in Figure 4) are saddle-node and Hopf bifurcation points, respectively. We cannot observe the codimension 2 or other higher codimension bifurcations in our studied parameter range. It is obvious that the values of An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e157.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e158.jpg can be cooperatively tuned in the corresponding regions to achieve desirable behaviors and functions. In fact, when we fix any one parameter in An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e159.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e160.jpg, and An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e161.jpg, the others can perform similar synergetic function to achieve the diverse response behaviors (see Figure S5).

Figure 4
Bifurcation diagram in the space spanned by

Indeed, a single positive feedback loop with ultrasensitivity is able to act as a bistable switch and a single negative feedback loop with a time delay can produce sustained oscillations [25][27]. However a motif assembled by a positive loop and a negative loop not only performs the both functions without changing the topological structure, but also presents excitable behavior. Note that the last two behaviors resulted from the negative feedback loop of miR-17-92 because there can only be one-way switches without miRNAs (Figure 2A). These signal-response behaviors offer diverse regulating options. The oscillation (pulses) could provide potential precise regulation, such as the digital response of p53 to DNA damage [28], [29].

Optimized Bistable Switch

According to Eqs. (3–4), the dimensionless parameter An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e166.jpg is a time constant for the activation and inactivation of An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e167.jpg and determines whether the switch is fast or slow. Note that An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e168.jpg is the ratio of the degradation rate for E2Fs and Myc for miR-17-92. Consequently, the switching behavior of the protein concentration from the positive feedback loop can also be represented as interlinked dual-time feedback loops [13], [30], [31].

Figure 5 illustrates the switch responses of the system for two different time constants An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e169.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e170.jpg to a step stimulus with or without fluctuation. The basal strength of the stimulus strength is set at An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e171.jpg, where the system is initially in the off-state, and then jumps to An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e172.jpg at time An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e173.jpg, corresponding to the on-state of the system. The panels in the left column denote the stimulus input, where the left bottom panel corresponds to a fluctuating environment described by Gaussian white noise with mean An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e174.jpg and variance An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e175.jpg (Figures 5A, 5D). The middle and right columns show the response of the switch for An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e176.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e177.jpg, respectively. In all of the cases, the system with a smaller An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e178.jpg increases rapidly at the initial stage and responds much faster than that of the slower loop. The fluctuation amplitude of An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e179.jpg for larger An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e180.jpg values is much smaller than that of the faster loop (Figures 5E, 5F). That is, the fast loop is critical for the switching sensitivity, but the slow loop increases the switching stability to resist stimulus fluctuations. Furthermore, the inhibition of miRNA An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e181.jpg also slows down the switching process, especially near the on-state, and effectively represses fluctuations (Figures 5B and 5C, Figures 5E and 5F).

Figure 5
Responses of switches to a step stimulus input.

To clearly investigate the switching robustness to fluctuations in the stimulus, we use the definition of the fraction of transition An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e196.jpg [17], [30]. The system is driven by the same noisy stimulus An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e197.jpg with different seeds, where An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e198.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e199.jpg is Gaussian white noise. Initially, we settled on a population of An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e200.jpg cells in the on-state of the system, with some cells that may flip to the off-state due to a noise-induced switch. An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e201.jpg is defined as the ratio of the number of cells in the transition state to the number of all cells at each time point. Moreover, one can define the time required for An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e202.jpg to reach the midpoint between its initial and steady-state values as the so-called response time An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e203.jpg.

Figure 6 displays the time courses of An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e204.jpg for An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e205.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e206.jpg, and An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e207.jpg. For smaller An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e208.jpg values, it takes a relatively long time for An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e209.jpg to reach a smaller steady state. We denote An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e210.jpg as the value of the steady state of An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e211.jpg. For example, An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e212.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e213.jpg for An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e214.jpg, which means that most of cells are still trapped in the on-state for a long time. In the case of An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e215.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e216.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e217.jpg indicating that a significant number of cells (An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e218.jpg) have flipped to the off-state and are taking less time to reach the steady state of An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e219.jpg.

Furthermore, we show the fraction of transition An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e227.jpg and the response time An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e228.jpg as functions of An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e229.jpg for different An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e230.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e231.jpg values in Figure 7. In the simulations, all An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e232.jpg cells initially settled in the on-state. The fluctuation of An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e233.jpg follows a Gaussian white noise distribution with variance An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e234.jpg. With increasing An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e235.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e236.jpg decreases from An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e237.jpg to An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e238.jpg, but An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e239.jpg also increases quickly in the case of An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e240.jpg. Especially in the flipping region of An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e241.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e242.jpg decreases when An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e243.jpg increased, as can be similarly observed in Figure 6. These switching behaviors resulted from the binding between the on-state and the off-state decreasing when An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e244.jpg or An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e245.jpg increases in the region of the bistable switch (see the dynamic diagram of An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e246.jpg vs An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e247.jpg, Figure S6).

Figure 7
Noise tolerance and response features of the systems including the inhibition of miRNA.

Note that An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e259.jpg is An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e260.jpg in the left column of Figure 7 and is An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e261.jpg in the right one. Obviously, for smaller values of An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e262.jpg (fast loop) and larger values of An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e263.jpg, there exists a larger region of An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e264.jpg for which almost all cells are trapped in the on-state (An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e265.jpg). In the critical region of An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e266.jpg for An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e267.jpg from An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e268.jpg to An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e269.jpg, the flipping process with small An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e270.jpg values is less sensitive to An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e271.jpg than with larger ones (Figures 7A and 7B). In addition, An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e272.jpg for small An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e273.jpg values is significantly less than for larger ones (Figures 7C and 7D).

Cellular processes are essentially stochastic and occur in a fluctuating environment [32]. A small perturbation in the stimulus input could be amplified by positive feedback [33], [34]. (Sometimes, positive feedback can work as a noise-filtering device [35]). Additionally, noise-induced switching behavior may induce a false decision regarding cell fate (case An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e274.jpg in Figure 5E). By involving the negative feedback loop of miRNAs, the Myc/E2F/miR-17-92 cancer network operates like a dual-time switch interlinked by fast and slow positive loops. The entire system exhibits high noise sensitivity in the off-state due to the rapid responses of the positive loop, which is regulated by the ratio between the miRNA and protein degradation rates (note that An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e275.jpg is often much less than An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e276.jpg). At the same time, the system is resistant to noise when it is in the on-state as a result of the negative feedback.

Discussion

It has been reported that miR-17-92 behaves as an oncogene and a tumor suppressor depending on different situations [36], [37]. For the first time, Aguda et al. analyzed the reduced model of the coupling between the E2F/Myc positive feedback loops and the E2F/Myc/miR-17-92 negative feedback loop. They showed that miR-17-92 plays a critical role in regulating the protein levels (on/off). Most important, they demonstrated the parallel oncogenic and tumor suppressor properties of miR-17-92 using the concept, cancer zone (Figure S2 or Figure 3 in [12]). By considering the bistable switch behaviors, Aguda et al. [12] predicted that increasing miRNA level drives E2F/Myc level in normal cell cycle to enter the cancer zone (oncogene, case a), or drives protein levels to exit cancer zone and enter the cell apoptosis (tumor suppressor, case b), and vice versa (see Figure S2 or Figure 3 in [12]). The reduced abstract model of the Myc/E2F/miR-17-92 network is typically interlinked by positive and negative feedback loops (Figure 1). A bistable system with interlinked loops has been illustrated in the yeast galactose-utilization network [38], [39], the mitogen-activated protein kinase 1,2/protein kinase C signaling network [40], [41], circadian clocks [42], [43], the eukaryotic cell cycle [44], [45], the p53-Mdm2 network [29], and so on. It has been shown that a system with interlinked loops behaves as a tunable motif and performs diverse behaviors [17]. The essential dynamic in the Myc/E2F/miR-17-92 network is a bistable switch, which can be realized only by a positive feedback loop without miRNAs (Figure 2A). Thus, the physiological importance of miRNAs remains unclear. The present work is based on the hypothesis that the miRNAs are essential in optimizing the switching behavior of the Myc/E2F/miR-17-92 network, and we focus on the role of miR-17-92 on the response-signal behavior without or with noise.

In this paper, using simulation parameters that are biologically plausible, we have shown that the system represents various behaviors (monostability, a bistable switch, a one-way switch) instead of a simple one-way switch because of the existence of miR-17-92 (see Figures 2E–H. The bistable region is also larger with miRNA present). As a result, the system is capable of generating a diverse array of signal-response behaviors with suitable combined parameters (Figures 3A–C). Especially, we find that, due to the the existence of miR-17-92, the range (parameter An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e277.jpg) of normal cell cycles is enlarged and this transition (from cell death/cancer to quiescence) is probably realized by noise-induced switches. In addition, the response time constant of the protein module can be regulated by the miRNA degradation rate (An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e278.jpg in Figures 5B–C). The Myc/E2F/miR-17-92 network can run as a dual-time switch (interlinked fast and slow positive loops) and appears to be more sensitive to stimuli and resistant to stimulus fluctuations (Figures 5E–F and Figure 6). It means that miR-17-92 can perform efforts to optimize the bistable switch where miR-17-92 confers signaling robustness (to limit undesired signaling fluctuations, buffering effect) and achieves optimal signaling efficacy (balancing effect).

In addition, processes in gene regulatory systems are typically subject to considerable delays induced by underlying biochemical reactions. Time delays in combination with a positive/negative feedback loop can induce sustained oscillations and multistability [12]. The model of the Myc/E2F/miR-17-92 network should also account for multiple time delays. It has been shown that the inclusion of a long-time delay in a negative feedback loop can generate oscillations and that the addition of a positive feedback loop can increase the oscillation amplitude and widen the stimulus regime for the oscillation, thus promoting the robustness of the oscillations [17], [46]. Therefore, it is conjectured that the effect of a time delay would not qualitatively change our results.

Because an individual miRNA usually targets many genes that are involved in various cellular signaling pathways [11], modulations in the level of a single miRNA could eventually affect many pathways at the same time. The miR-17-92 cluster, which comprises six miRNAs (miR-17, miR-18a, miR-19a, miR-20a, miR-19b-1, and miR-92-1), plays an important role in the cell cycle (by targeting the E2F family) [47], apoptosis (by downmodulating the antiapoptotic protein Bim and tumor suppressors PTEN and p21) [48][50], and angiogenesis (activated by c-Myc and VEGF) [51]. In this study, the model is constrained in a network associated with the cell cycle, without considering other networks. It would be worthwhile to construct a large-scale gene regulation network including different biological functions of a small group of miRNAs and to develop potential strategies of miRNA-based therapeutic targets.

Materials and Methods

All numerical bifurcation analyses of the ordinary differential equations were performed with OSCILL 8.28 [52]. The ordinary differential equations and stochastic differential equations were numerically solved and separately integrated using the fourth-order Runge-Kutta scheme and the fourth-order stochastic Runge-Kutta scheme [53], [54] in Fortran 95 codes, respectively.

Supporting Information

Text S1

The deduction of the dimensionless parameter ranges.

(PDF)

Text S2

The explicit solutions for the steady states of protein and miRNA levels.

(PDF)

Figure S1

Reducing process of the Myc/E2F/miR-17-92 network to an abstract model. Solid arrows mean activation and hammerhead means inhibition. (A) Summary of the interactions among the transcription factors Myc, E2F, and miR-17-92 cluster. (B) The first step in the model reduction. (C) The final abstract model is composed of the protein module An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e279.jpg (Myc/E2F) and the miRNA module An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e280.jpg (miR-17-92). It is also presented in detail in Figures 1 and and22 by Aguda et al. [12].

(EPS)

Figure S2

The schematic diagram of cancer zone. Clearly, the miR-17-92 clusters act as an oncogene or as a tumor suppressor designed by Aguda et al. [12].

(EPS)

Figure S3

The switching behavior between cell statuses corresponding to the transcriptional activities of E2F or Myc. Note that the plausible experimental range of An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e281.jpg is An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e282.jpg. Without the inhibition of miRNAs (left panel), the switch is limited in on-states (cell cycle, cancer, and cell death). Moreover, most of cells are settled on cancer or apoptosis which return back to cell cycles only by decreasing the positive feedback An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e283.jpg. For example, the black dashed-line arrows from An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e284.jpg to An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e285.jpg denotes a regulation from cell death to cell cycles. In the presence of miRNAs (right panel), the protein concentration of steady states An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e286.jpg is significantly less than that in left panel. So, the range of normal cell cycles is enlarged and the regulation of An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e287.jpg is also at work effectively. Especially, it is possible that cells back to quiescence from cancer or apoptosis directly. As an example, for An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e288.jpg, there exists noise-induced switch from cancer zone to cell quiescence (the red dashed-line arrow). Similarly, the regulation of An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e289.jpg can also realize the same function (see the black dashed-line arrow, regulation from An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e290.jpg to one optional value between An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e291.jpg). Here parameter An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e292.jpg.

(EPS)

Figure S4

The 2-dimension phase portraits for An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e293.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e294.jpg. The systems come into steady states for An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e295.jpg and (A) An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e296.jpg, (C) An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e297.jpg. In the case of An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e298.jpg, the systems appear periodic oscillations for (B) An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e299.jpg and (D) An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e300.jpg. The circles and the solid square denote the initial status (An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e301.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e302.jpg) and the fixed steady states, respectively. The arrows indicate the time evolution of the systems.

(EPS)

Figure S5

The diverse signal-response behaviors tuned by the positive feedback An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e303.jpg and the miRNA inhibition An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e304.jpg for fixed An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e305.jpg. Note that the oscillation region appears for the high An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e306.jpg (normally, An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e307.jpg), but the bistable switch disappears. Parameter An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e308.jpg.

(EPS)

Figure S6

The bifurcation diagram spanned by An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e309.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e310.jpg for An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e311.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e312.jpg. Obviously, in the region of the bistable switch, the range of An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e313.jpg or An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e314.jpg between on-state and off-state decreases with increasing An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e315.jpg (panel A) or An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e316.jpg (panel B), respectively. The additional red dashed lines are the drawing borderlines between bistable and one-way switches under different preconditions (for a fixed value of An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e317.jpg/An external file that holds a picture, illustration, etc.
Object name is pone.0026302.e318.jpg in panel A/B).

(EPS)

Footnotes

Competing Interests: The authors have declared that no competing interests exist.

Funding: This work was supported by the National Natural Science Foundation of China (Grant No. 10975063, 30870364), the High Technology Research and Development Program of China (863 program, Grant No. 2007AA021807), and The Key Technology R&D Program of Gansu Province (Grant No. 0708NKCA129). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

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