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MicroRNA regulation of a cancer network: Consequences of the feedback loops involving miR1792, E2F, and Myc

Contributed by Avner Friedman, November 4, 2008 (received for review September 3, 2008)
Abstract
The transcription factors E2F and Myc participate in the control of cell proliferation and apoptosis, and can act as oncogenes or tumor suppressors depending on their levels of expression. Positive feedback loops in the regulation of these factors are predicted—and recently shown experimentally—to lead to bistability, which is a phenomenon characterized by the existence of low and high protein levels (“off” and “on” levels, respectively), with sharp transitions between levels being inducible by, for example, changes in growth factor concentrations. E2F and Myc are inhibited at the posttranscriptional step by members of a cluster of microRNAs (miRs) called miR1792. In return, E2F and Myc induce the transcription of miR1792, thus forming a negative feedback loop in the interaction network. The consequences of the coupling between the E2F/Myc positive feedback loops and the E2F/Myc/miR1792 negative feedback loop are analyzed using a mathematical model. The model predicts that miR1792 plays a critical role in regulating the position of the off–on switch in E2F/Myc protein levels, and in determining the on levels of these proteins. The model also predicts largeamplitude protein oscillations that coexist with the off steady state levels. Using the concept and model prediction of a “cancer zone,” the oncogenic and tumor suppressor properties of miR1792 is demonstrated to parallel the same properties of E2F and Myc.
MicroRNAs (miRs) are small noncoding RNAs, 18–24 nt in length, that are predicted to regulate the expression of approximately onethird of all human genes (1, 2). This regulation occurs posttranscriptionally through miR binding to mRNA targets leading to target degradation or inhibition of translation. Current targetprediction computer programs (3, 4) often predict that a miR could target tens to hundreds of genes, and that a gene can be targeted by many miRs—thus, the expectation that miRs play important roles in coordinating many cellular processes, particularly those involved in development and disease (5). Indeed, miRs have been implicated in various cancers, acting either as oncogenes or tumor suppressor genes (6). In this article, we investigate the role of a set of miRs in the important “cancer network” shown in Fig. 1. A cancer network is a molecular or gene interaction network involving oncogenes or tumor suppressor genes. The network shown is associated with the control of the G1S transition in the mammalian cell cycle (7, 8).
The extent to which miRs change the levels of their target mRNAs is marginal compared with the effect of other regulators such as transcription factors and posttranslational protein modifiers (10). Thus, it is thought that the primary role of miRs is to modulate or finetune the dynamics of regulatory networks (10–13). The significance of this role is now increasingly recognized as there are now many reported cases in which abnormal miR expressions correlate with cancer development (reviewed in ref. 6). Here, we focus on miR1792, which behaves as an oncogene or a tumor suppressor in different situations (2, 14).
The miR1792 cluster is a polycistronic gene located in human chromosome 13 ORF 25 (C13orf25) located at 13q31q32. The cluster is composed of 7 mature miRs, namely, miR17–5p, miR17–3p, miR18a, miR19a, miR20a, miR19b, and miR92–1 (Fig. 2A). The nucleotide sequences and organization of this cluster is highly conserved in vertebrates (reviewed in ref. 2). He et al. (15) reported the first evidence of the oncogenic activity of miR1792. Gene expression data show overexpression of miR1792 in several tumors, including cancers of the breast, lung, colon, stomach, pancreas, and prostate (16, 17). Although the tumorsuppressor property of miR1792 remains to be demonstrated directly in vivo, the following observations are quite suggestive: This cluster is deleted in 16.5% of ovarian cancers, 21.9% of breast cancers, and 20% of melanomas (14, 15).
Among the experimentally validated targets of some miR1792 cluster members are the transcription factors Myc, E2F1, E2F2, and E2F3; interestingly, these same factors have been shown to induce the transcription of miR1792 (reviewed in ref. 14). The negative feedback loops thus formed are depicted in Fig. 2A. In the modeling that follows, we will focus on E2F1, which possesses both oncogenic and tumor suppressor properties (18). Earlier, we analyzed a model that views E2F1 and Myc as members of a control node in a regulatory network coordinating cell cycle entry and apoptosis (9, 19). Among the predictions of the model is that increasing levels of E2F or Myc drives the sequence of cellular states, namely, quiescence, cell proliferation, and apoptosis. Here, we postulate that in between levels associated with normal cell cycles and apoptosis, there exists a range of Myc and E2F1 levels with increased probability of inducing cancer. We call this range the cancer zone. We summarize in Fig. 3 our hypothesis on how the oncogenic and tumor suppressive properties of miR1792 arise in relation to the cancer zone. Using a mathematical model abstracted from the complex network (Fig. 2), we illustrate the mechanisms and conditions under which these miR1792 properties operate.
Formulation of the Model
Dimensionless Equations.
Fig. 2 summarizes how the complex regulatory network is reduced to a model with 2 components representing the protein module p (Myc and the E2Fs) and the miR cluster m. Step 1 in Fig. 2C represents the autocatalytic (positive feedback) growth of p, which is inhibited by m. Step 2 in the same figure depicts the pinduced transcription of the miR cluster. The dynamics of the respective concentrations of these modules, p and m, are described by Eqs. 1 and 2. The rate of step 1 is given by the second term on the righthand side of Eq. 1 where the Hill exponent of Eq. 2 on p assumes a threshold kinetics; furthermore, the presence of m in the denominator accounts for the miRdependent downregulation of protein expression. The aforementioned threshold on the rate of protein expression is determined by the term (Γ_{1} + Γ_{2}m) in the denominator. The value of the parameter Γ_{2} is a measure of the efficiency of miR inhibition of protein expression, and lumps all factors that could affect the binding of the members of the miR1792 cluster to their targets and the inhibition of protein translation.
The constant term α in Eq. 1 stands for constitutive protein expression due to signal transduction pathways stimulated by growth factors present in the extracellular medium. The parameter α therefore corresponds to an experimentally controllable condition such as the concentration of nutrients in the cell culture medium. The rightmost term in Eq. 1 is a firstorder protein degradation term with fixed rate coefficient of δ. The constant term β in Eq. 2 represents pindependent constitutive transcription of m. The second term in Eq. 2 is the rate of pinduced transcription of m (assumed to be first order in p for simplicity), and the last term is a degradation term with rate coefficient γ.
Eqs. 1 and 2 can be nondimensionalized as follows where the dimensionless variables and parameters are
Delay Differential Equation.
Step 1 in Fig. 2C is an abstraction of all of the steps involved in protein expression—from transcription factor binding to DNA, gene transcription, to translation in ribosomes. Thus, the rate of synthesis of the protein is not a function of its instantaneous concentration (as assumed in Eq. 3), but rather of its concentration at some time Δ in the past. In a second set of computer simulations presented in the Results Section, this time delay Δ is considered in the second term on the righthand side of Eq. 3, which is rewritten explicitly in Eq. 6.
Solving for the Steady States.
The steady states of the system of Eqs. 3 and 4 are determined by equating the righthand sides to zero. After eliminating μ in the steady equations, we obtain the following cubic polynomial whose nonnegative roots give the steady states of φ (symbolized by φ_{s}): where The steady state of μ (symbolized by μ_{s}) is given by We are interested in threshold or switching behavior of the system, and, therefore, the conditions on the parameters for the existence of multiple steady states are relevant. From ref. 20, the set T of parameters that guarantee existence of 3 positive real roots of Eq. 7 is: where Thus, the necessary (but not sufficient) conditions for the existence of 3 steady states of the model are:
Parameter Values and Numerical Solution of the Differential Equations.
The parameter ε is expected to be less than unity because miRs are typically more stable than proteins; for example, δ for E2F1 and Myc are ≈0.25 h^{−1} and ≈0.7 h^{−1}, respectively (21), and γ ≈ 0.02 h^{−1} (22). The value of ε = 0.02 is used in our computer simulations (noting that δ for Myc is of order unity, and making allowances for the other E2Fs besides E2F1). An estimate of k_{1} for E2F1 is ≈0.4 μM h^{−1} and Γ_{1} ≈0.1 μM^{2} (21). We arbitrarily set (k_{2}/β) ≈3 μM^{−1} so that Γ′_{1} ≈1 and κ ≈5 (the parameter β is assumed to be manipulated experimentally via gene transfection, for example). The dimensionless parameters α′ and Γ′_{2}—whose values can be tuned experimentally—are allowed to vary in the ranges 0–0.4 and 0–2.5, respectively, to explore the effect of increasing rate of growth factorinduced protein synthesis and inhibition efficiency of the miRs. The differential equations of the model are solved using the computer software described in Methods.
Results and Discussion
Steady States of the Model and Significance of the Parameter α′.
According to Eq. 8, the steady states of m and p increase or decrease in the same direction. This model prediction agrees with observations in various tumors that levels of Myc and miR1792 are both increased (23, 24). The model also clarifies the interpretation of Hayashita et al. (23) that members of the miR1792 cluster promote proliferation—this is because increase in the miR level correlates with increase in the levels of Myc or E2F, which are both proliferative.
The steady state φ_{s} as a function of the parameter α′ for different values of Γ′_{2} is shown in Fig. 4. These diagrams are referred to as a “steadystate bifurcation diagrams,” and α′ is referred to as a bifurcation parameter. Because of physiological constraints, the horizontal axis in Fig. 4 does not extend to infinity but terminates at some maximum value of α′. To generate experimental curves similar to those shown in this figure, one can envisage a laboratory experiment in which cells are grown at different nutrient concentrations and then measuring the longterm protein levels.
With parameters satisfying the relationships given in Eq. 10, the model predicts that there is a range of α′ in which the system has 3 coexisting steady states (e.g., those with Γ′_{2} = 0, 1, 1.5, 1.8, 2). For example, for the curve with Γ′_{2} = 1.8, values of α′ from 0.05 (corresponding to the left knee of the curve) to 0.18 (right knee) give 3 steady states. We interpret the “right knee” of a steadystate diagram as a “switchon” point at which a sharp irrevocable increase in protein level occurs until the upper steady state (the “on” position) is attained. In other words, the model predicts that there exists a threshold in growthfactor requirement for cells to “turn on” the protein synthesis. (We interpret the low protein steady states in Fig. 4 as the “off” positions.) Note that one of the hallmarks of cancer is a decreased growth factor requirement for proliferation. As this switchon value of α′ (growthfactor requirement) decreases as the parameter Γ′_{2} decreases, the significance of this parameter is discussed next.
Significance of the Parameter Γ′_{2} and miR Regulation of Protein Levels.
The dimensionless parameter Γ′_{2} represents the inhibition efficiency of miR1792 against its target proteins. The expression of Γ′_{2} in terms of the 4 parameters Γ_{2}, k_{2}, β, and γ (Eq. 5) suggests the ways to manipulate the miR inhibition efficiency experimentally. For example, Γ′_{2} can be increased by increasing Γ_{2} or k_{2}, or by decreasing β or γ. The dependence of Γ′_{2} on β seems counterintuitive because Eq. 5 states that an increase in the constitutive or pindependent expression of miR1792 leads to a decrease in the miR inhibition efficiency.
The case of Γ′_{2} = 0 represents any of the following situations: deletion of the miR1792 cluster; members of the cluster do not bind the transcripts of the target proteins (p) perhaps due to mutations; p does not induce expression of miR1792 (case of k_{2} = 0). Although p is no longer coupled to m, the 1dimensional model of the autocatalytic variable p is still capable of exhibiting 3 steady states for α′ between 0 and ≈0.05 (see Fig. 4). Standard linear stability analysis shows that the system is bistable in this range of α values—that is, the bottom and upper branches of steady states are stable, whereas the middle branch of steady states are unstable. Very interestingly, bistability involving E2F1 and Myc have recently been demonstrated experimentally by Yao and colleagues (21). In addition to the E2F1 loop, these authors invoked another source of positive feedback loop—specifically, the E2FCdk2pRbE2F loop shown in Fig. 1—in their model of the system.
Two key observations can be made from Fig. 4 with regards to the role of miR1792: (i) as Γ′_{2} is increased, the switchon values of α′ (corresponding to the right knees of the curves) increase; (ii) as Γ′_{2} is increased, the upper branch of steady states (the on states) are lowered. The first observation suggests that the miRs counteract the cancerassociated decreased growth factor requirement for cell proliferation. The second observation agrees with the current thinking about the role of miR1792 in preventing a runaway E2F/Myc positive feedback loop that may induce uncontrolled cell proliferation.
The model also predicts 3 qualitatively different types of steadystate bifurcation diagrams as illustrated in Fig. 4: (i) 2 disconnected curves with just the right knee (e.g., Γ′_{2} = 0, 1, 1.5), (ii) a continuous curve with left and right knees (e.g., Γ′_{2} = 1.8 and 2), and (iii) a continuous curve with no knees (e.g., Γ′_{2} = 2.5). Type (i) is an irreversible switch to the upper branch of steady states, whereas type (ii) allows a transition from the upper branch to the lower branch by decreasing α′ below the value corresponding to the left knee of the curve.
NonSteadyState Behavior and Sensitivity of Protein Levels to miRs.
The functional properties of the E2F/Myc/miR1792 network—in particular, the role of the miR cluster—can be further understood by studying its nonsteady state kinetics. For example, the dynamics of the system can be very sensitive to the initial levels of the miR cluster. Shown in Fig. 5 are computer simulations from various initial conditions of φ and μ. In Fig. 5A, 5 initial conditions located at the lower left corner of the box are very close to each other, with identical initial φ_{0} but with 5 close values of μ_{0}. All of the trajectories ultimately approach a stable steady state (shown as empty circle), but the initial conditions where μ_{0} = 0.340, 0.343, and 0.345 lead to trajectories with wide swings in protein levels that even surpass the upper steady state (see Fig. 4, α′ = 0.1); in contrast, for the initial conditions μ_{0} = 0.350 and 0.355 the system goes to the steady state directly. (The value of μ_{0} that delineates these two dynamics is between 0.345 and 0.350, as exhibited in Fig. 5B.) Thus, the model predicts that the system could be prone to large bursts of protein synthesis if the level of mir17–92 is below a certain threshold.
At α′ = 0.1 (as in Fig. 5 a and b) the system has 3 coexisting steady states, but only the lowest one is stable (shown as empty circle in Fig. 5A). When α′ is increased beyond the right knee of the curve in Fig. 4 (for Γ′_{2} = 1.8), only one steady state is available for the system; this steady state is asymptotically stable as shown by the phase plane trajectories plotted in Fig. 5C and the temporal course of φ in Fig. 5D.
The nonsteady state behavior of the system as shown in Fig. 5 could explain an experimental observation that seemingly contradicts the prediction of the model at steady state. Several groups (14, 25, 26) have shown that miR20a and miR17–5p (members of the miR1792 cluster) are antiapoptotic because the downregulation of these miRs leads to increased cell death and their overexpression decreases cell death. These observations were explained (25, 27) in terms of the downregulation of E2F1 protein levels by miR20a and miR17–5p, and E2F1's induction of apoptosis when overexpressed. However, our model suggests that, at steady state, increased levels of miR1792 are associated with increased levels of Myc and the E2Fs (Eq. 8 and Fig. 4) and therefore increased apoptosis. One way to resolve this dilemma is to view the model's transient dynamics instead of steady states, and to illustrate the possibility that reported experimental observations were made under nonsteady state conditions. In Fig. 5 A and C, the slow segments of the trajectories (from the maximum φ to the maximum μ) correspond to increasing miR (μ) and decreasing protein (φ). This decrease in the protein, which is associated with increased miR, could then lead to a decreased rate of apoptosis—thus, the reference to the miR as antiapoptotic.
miR1792 as an Oncogene and a Tumor Suppressor.
Viewed in terms of miR steady state levels, the oncogenic and tumor suppressor properties of miR1792—cases a and d in Fig. 3, respectively—parallel those of E2F1 or Myc because of Eq. 8. This idea is explained more explicitly in Fig. 6. (Note that in this figure, the range of φ_{s} that defines the cancer zone is chosen arbitrarily; one would expect that the range of the cancer zone would depend on the specific cellular system and on the perturbations of the system that drives it toward a cancerous state.) In the direction of the arrow in Fig. 6A, the on steady states increase to levels that enter the cancer zone. The corresponding increase in μ_{s} is interpreted as promoting cancer and therefore classifies the miR as an oncogene. However, in the direction of the arrow in Fig. 6B, both φ_{s} and μ_{s} increase—driving exit from the cancer zone and into apoptosis, thereby classifying the miR cluster as a tumor suppressor.
An alternative view of the oncogenic and tumor suppressor properties of miR1792—cases b and c in Fig. 3, respectively—arises when the focus is on the miR inhibition efficiency parameter, Γ′_{2}, rather than the miR levels as in the preceding paragraph. The inhibition of the exit from the cancer zone (case b in Fig. 3) is carried out by preventing a decrease of Γ′_{2}; in other words, the direction of increasing Γ′_{2} when miR levels are high would be oncogenic. As a tumor suppressor, the inhibition of the entry into the cancer zone (case c in Fig. 3) is done by preventing a decrease of Γ′_{2} when miR levels are low.
The discussion in this section illustrates the confusion that may arise in using the labels “oncogene” and “tumor suppressor” if the attribute of the miR (that is, the miR level, μ_{s}, or the inhibition efficiency parameter, Γ′_{2}) used to correlate with entry into or exit from the cancer zone is not clearly specified. However, determining Γ′_{2} experimentally would be more difficult in practice compared with measuring miR levels, and therefore classification using changes in miR levels is commonly used.
Time Delays and Oscillations.
In this section, Eqs. 6 and 4 are numerically solved. Fig. 7 shows the dynamics of the system as the parameter α′ is increased or decreased. The rate of change of α′ is slow enough so that the system keeps close to the locally stable steady states. As shown by the diagrams on the top row of Fig. 7, for both cases without (Δ = 0) and with time delay (Δ = 0.2), increasing α′ leads to a sharp transition to the upper steady states; this transition occurs at the α′ corresponding to the right knee of the diagram shown in Fig. 4 for Γ′_{2} = 1.8. Once at the top branch of steady states, oscillations are initially observed with rapidly decreasing amplitudes as α′ is increased further. When the direction of change of α′ is reversed (bottom diagrams of Fig. 7), new interesting dynamics appear. For the case without time delay (Δ = 0), the system initially traces the curve that was made in the direction of increasing α′, but then goes a little bit beyond (to the left) of the right knee before undergoing a sharp drop to the lower branch of steady states. Note that the system does not switch down at the left knee (see Fig. 4 for Γ′_{2} = 1.8); this is because of the emergence of largeamplitude oscillations (data not shown) that exist in a narrow range of α′ values just to the left of the right knee; these largeamplitude oscillations suddenly disappear and the system gets trapped by the lower branch of stable steady states. In contrast, for the case with time delay of Δ = 0.2, the system exhibits largeamplitude protein oscillations for wider ranges of α′s as indicated by the black region in the bottom right diagram (the accompanying smallamplitude oscillations in μ are also shown in gray). In the same range of α′ values where this oscillatory region is located, the system could get trapped in the lower branch of steady states depending on the initial conditions.
Thus, the model with time delay predicts the coexistence between largeamplitude oscillations and low steadystate protein levels. To try to understand the physiological significance of these largeamplitude protein oscillations, we checked the stability of the lower branch of steady states (the off states) and found that these states are quite robust against perturbations—for example, at α′ = 0.1, it takes a perturbation of ≈370% above the value of φ_{s} to switch the system to the largeamplitude oscillations (simulations not shown); also, large amounts of perturbation above μ_{s} do not induce switching, but, as shown in Fig. 5A (no delay), perturbations that decrease the miR level below a certain threshold induce largeamplitude swings in protein concentrations. The robustness of the off states and of the switchon values of α′ is required for proteins that control important cellular processes (such as entry into Sphase of the cell cycle, which E2F1 and Myc regulate). However, the model predicts that large bursts in protein levels (the largeamplitude oscillatory states) are possible and can be used by the system to obtain apoptotic levels quickly to avert any danger that may be caused by large perturbations.
Conclusions
We proposed and analyzed a simple model of the interactions between miR1792 and the transcription factors E2F and Myc. Our goal is to explore the broad consequences of the structure of the network on the levels, steady states and dynamics of the miR and the group of proteins that the miR targets. The simplicity of the 2variable model precludes it from capturing the different properties of Myc, E2F1, E2F2, and E2F3 with respect to their proliferative or apoptotic effects or nature of repression by miR1792 members. The model couples the positive feedback loops involving E2F and Myc—generating bistability and the concomitant offon switching behavior of the system—and the negative feedback loop between these proteins and members of the miR1792 cluster. The model predicts that the steady states of these proteins and the miRs change in the same direction, although slow nonsteady state or transient dynamics are possible where the changes could be in opposite directions. We have illustrated how changes in the miR inhibition efficiency—the parameter Γ′_{2} in Eq. 3—controls the value of the offon switch in growthfactor requirement and how it attenuates the on levels of the proteins. An important prediction of the model is that decreasing Γ′_{2} leads to decreasing growthfactor requirement for switching the protein on, and that the on levels increase with decreasing Γ′_{2}. Possible experimental means of manipulating the value of Γ′_{2} are discussed. Due to the negative feedback loop in the network, largeamplitude protein oscillations are predicted to coexist with the off steady state levels, allowing the system to respond through apoptosis to dangerously large perturbations. Finally, using the postulate of a cancer zone, we have shown that the oncogenic and tumor suppressor properties of miR1792 parallel those of E2F and Myc.
Methods
For the first set of computer simulations, the system composed of Eqs. 3 and 4 is solved using the routine ode23 in MATLAB (The Mathworks). For the second set of simulations involving time delay, the system composed of Eqs. 6 and 4 is solved using the routine dde23 in MATLAB.
Acknowledgments
This work was supported by U.S. National Science Foundation Agreement 0112050 and National Institutes of Health Grant RO1HL67176.
Footnotes
 ^{1}To whom correspondence should be addressed. Email: afriedman{at}mbi.osu.edu

Author contributions: B.D.A. designed research; B.D.A. and Y.K. performed research; B.D.A., M.G.P.H., A.F., and C.B.M. analyzed data; and B.D.A. and C.B.M. wrote the paper.

The authors declare no conflict of interest.

Freely available online through the PNAS open access option.
 © 2008 by The National Academy of Sciences of the USA
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