Journal of the Korean Physical Society, Vol. 53, No. 6, December 2008, pp. 34603467 Dynamic Analysis of Interlocked Positive Feedback Loops Cuong Nguyen and Seung Kee Han Physics Department, Chungbuk National University, Cheongju 361-763 (Received 3 September 2008) Positive feedback loops, which are responsible for reliable switches in biological systems, not only work alone but are also interlocked together. In this paper, we analyzed the possible dynamics of the interlocking of two dierent types of positive feedback loops, self-enhanced and self-recovering positive feedback loops. Using a bifurcation analysis, we showed that the interlocked positive feedback loop was able to generate diverse dynamical behaviors. They are classied as monostability, single bistability, non-overlapping, overlapping, nested and merged bistability with or without irreversibility. The diversity of dynamical behaviors implies the exibility of biological systems in response to external stimuli. PACS numbers: 87.17.Aa, 87.16.Yc, 87.16.Xa, 87.17.-d Keywords: Positive feedback loop, interlocked feedback loops, Bifurcation analysis, Dynamical diversity, Multistability I. INTRODUCTION Biological complex networks possess many simple modules with specic functions [1, 2]. These modules are built up from many building blocks with specic dynamical properties [3]. Positive feedback loops (PFLs), which are one of the building blocks, are repeatedly reported in many biological systems, such as cell cycles [3{6] and circadian rhythms [7]. PFLs have potential to exhibit bistability and irreversibility and play an important role in biological systems by converting a graded signal into a discontinuous response with hysteresis or reliable switches. The PFLs work in an interlocked manner, rather than alone. Interlocked PFLs can be found widely in many biological complex networks, such as mammalian cell cycle regulatory networks [6, 8, 9], transcriptional networks [10], gene regulation networks, etc. For example, in the budding yeast cell cycle model [4], the coupling of S and M phases is governed by interlocking of two PFLs of mutual inhibition. The inhibitor CKI mutually inhibits Clb5 6 and Clb1 2; Clb5 6 and Clb1 2 can, in turn, degrade CKI through uibiquitination [4]. Other interlocking modules can also be found in the ssion yeast cell cycle model [5], in which the whole cell cycle model is divided into three modules: G1=S, G2=M and Finish (M=G1). The cyclin-dependent kinase complex MP F is a unique regulator of those modules. Module G1=S is composed by interlocking two positive feedback loops of mutual inhibition (between M BF and its inhibitor Rum1) and mutual phosphorylation (between E-mail: skhan@chungbuk.ac.kr; Fax: +82-43-274-7811 MBF and Ste9); G2=M is composed by interlocking two PFLs, MP F with Cdc25 and MP F with W ee1. Other examples can be found in Ref. 9. Interlocking two or more PFLs together reinforces the properties of a single PFL and might bring new features that are not present in the single PFL case. In this study, we identied all possible dynamics of the interlocking of two types of PFL by performing a bifurcation analysis [11] in the one and the two parameters space of the stimulus amplitude and coupling strength. Finally, we constructed and compared phase diagrams in two coupling strengths. The diagrams show the dynamical structure of interlocking of the two types of PFLs. II. TWO DIFFERENT TYPES OF POSITIVE FEEDBACK LOOPS Here, we examined two types of positive feedback loop. One is the so-called self-enhancing PFL (epfl) in which the output P somehow increases its own synthesis rate; the other is the so-called self-recovering PFL (rpfl) in which the output P somehow removes its suppression. The two typical models of epfl and rpfl are auto-regulation (Figure 1(a)) and mutual phosphorylation (Figure 1(b)), respectively. er 1. Self-enhancing Positive Feedback: an Ampli- In the auto-regulation (Figure 1(a)), the output P positively regulates its own upstream regulator, T F, which, -3460-
Dynamic Analysis of Interlocked Positive Feedback Loops { Cuong Nguyen and Seung Kee Han -3461- Fig. 1. (a) Autoregulation: output P positively regulates its own upstream regulator, T F, which, in turn, increases the synthesis rate of output P. (b) Mutual phosphorylation: the output P (red) removes its inhibited form P p (grey) by activating an enzyme E. Here, [P ] and [T F ] denote the concentrations of the output P and the regulator T F, respectively. The output P synthesis is proportional to stimulus S and is regulated by a transcription factor T F. Its degradation is proportional to itself. The activation and the deactivation of the regulator T F follow the Michaelis-Menten kinetic. The k a [P ] and the k itf are the maximal rates of activation and deactivation, respectively. The J atf and the J itf are Michaelis constants. The activation of the T F is induced by P with a coupling strength constant k a, which is the eect of P on T F. The chosen parameters are k s1 = 0.4, k s2 = 1, k d = 2, k a = 1, k itf = 0.5 and J atf = J itf = 0.01. The parameters are dimensionless. The small Michaelis constants, J's, are to ensure that the autoregulation possesses non-linearity, which is required to exhibit bistability [13]. Taking the stimulus S as the primary parameter, we created a one-parameter bifurcation diagram and steadystate values of the output P as a function of the stimulus S for dierent positive coupling strengths k a as in Figure 2(a)-(c). Fig. 2. Auto-regulation bifurcation diagram of a oneparameter bifurcation diagram: (a) monostability (k a = 0.01), (b) bistability (k a = 1), (c) irreversible bistability (k a = 1.5), (d) two-parameter bifurcation diagram of input S and coupling strength k a. Solid and dashed lines denote stable and unstable steady states, respectively. Dashed-dotted lines are a bifurcation diagram of auto-regulation without a feedback loop (k a = 0). in turn, increases the synthesis rate of the output P. Using the law of mass reaction and Michaelis-Menten kinetic [12], we can write the dierential equations governing the output P and the transcription factor T F of P as d[p ] = k s1 S + k s2 [T F ] k d [P ]; (1) d[t F ] = k a[p ](1 [T F ]) k itf [T F ] J atf + 1 [T F ] J itf + 1 [T F ] : (2) A. Monostability for weak coupling strength With a very small coupling strength k a, autoregulation exhibits monostability, as depicted by a monoincreasing line because there is no feedback eect in Figure 2(a). B. Bistability for strong coupling strength As the coupling strength k a gets stronger, the autoregulation exhibits bistability, which is the coexistence of two stable xed points (solid lines) separated by an unstable one (dashed line, Figure 2(b)). As long as the input S increases, the output P increases and reaches the threshold P 1 and then turns on the helper T F. The helper T F, in turn, increases the output level abruptly by increasing the output synthesis rate to a higher state (upper branch). If the input S decreases, the output P then decreases and falls below another threshold P 2 at which the output P is not strong enough to maintain the T F at the ON state. Therefore, the output P dramatically drops back to a lower state (lower branch). The two transitions occur at two dierent input S, called two saddle node bifurcation points, SN1 auto and SN2 auto, where the feedback loop turns on and o, respectively. To turn the feedback loop on, activating
-3462- Journal of the Korean Physical Society, Vol. 53, No. 6, December 2008 transcription factor T F, the output P has to exceed the threshold P 1. Similarly, P 2 is the threshold for turning the feedback o. The thresholds P 1 and P 2 are called feedback on/o levels. Also, the region of SN2 auto < S < SN1 auto is called the bistable region in which there are two stable and one unstable steady states. Here, the bistability of auto-regulation is said to be reversible because the output P can be ipped up or down by driving the stimulus S beyond SN1 auto or below SN2 auto. C. Irreversibility for very strong coupling strength As the coupling strength gets extremely large, the auto-regulation still exhibits bistability, but is irreversible (Figure 2(c)). This means that, once P ips up at S > SN1 auto, it never ips down, not even when all stimuli S are washed out (S = 0). The epfl could exhibit dierent dynamical behaviors depending on the coupling strength k a. As shown in the Figure 2(d), along the coupling axis, k a, two saddle node bifurcation points SN1 auto and SN2 auto emerge at k a = ka 1 (ka 1 = 0.0784); then move along to the left, the SN2 auto crosses zero at k a = ka 2 (ka 2 = 1.2103). Thus, there are three regions along the k a -axis: monostability k a < ka 1, bistability ka 1 < k a < ka 2 and irreversible bistability ka 2 < k a. We note that, once the helper T F turns on, it remarkably enhances the output P (solid line) in comparison with that of the simple regulation (dashed dotted line). This kind of positive feedback loop is, therefore, called a self-enhancing PFL (epfl). The epfl works as an amplier that amplies the output P. 2. Self-recovering Positive Feedback: the Buer In the PFL of mutual phosphorylation, the output P (red) removes the inhibited form P p (gray) by activating the enzyme E (Figure 1(b)). The active output P and its inactive form P p are thus antagonistic. The mathematical model of mutual phosphorylation is written in the following equations: d[p ] T = k s S k d [P ] T ; (3) d[p ] = k s S + k ap[e]([p ] T [P ]) J ap + [P ] T [P ] k ip [P ] k d [P ]; (4) J ip + [P ] d[e] = k b[p ](1 [E]) J ae + 1 [E] k ie [E] J ie + [E] : (5) Here, [P ] T, [P ] and [E] denote the total output P, the output P and the enzyme E, respectively. We note that output P can be in a free form P or in a phosphorylated Fig. 3. Mutual phosphorylation: one-parameter bifurcation diagram of P (solid and dashed lines): (a) monobistability, (b) bistability and (c) two-parameter bifurcation diagram of input S and coupling strength kb. Solid and dashed lines denote stable and unstable steady states, respectively. Dashdotted lines denote a one-parameter bifurcation diagram of total P. form P p ([P ] + [P ] p = [P ] T, where P T denotes the total P ). Here, the coupling strength k b is the aectivity of P on activation of the enzyme E. The parameters are k s = 0.4, k d = 2, k ap = 1, k ip = 0.5, k b = 1.3, k ie = 0.25 and J ap = J ip = J ae = J ie = 0.01. The parameters are dimensionless. The two-parameter bifurcation diagram of mutual phosphorylation (Figure 3(c)) indicates that, for a coupling strength k b smaller than a certain value, k b < kb1 (kb1 = 0.0638), mutual phosphorylation exhibits monostability (Figure 3(a)). It exhibits bistability (Figure 3(b)) when k b gets stronger, kb1 < k b. It does not, however, exhibit irreversibility, as shown in the twoparameter bifurcation diagram. Instead of crossing the vertical axis, the SN2 m phos line approaches the vertical axis while the stimulus S reaches 0. It is remarkable to note that, with strong coupling strength, k b > kb1 (Figure 3(b)), the output (solid line) is initially suppressed from simple regulation (dashed dotted line) by an inactive form of P p until the input S is strong enough, S > SN1 m phos, to activate the enzyme E, which recovers the output P by removing the inactive form. Therefore, the inactive form P p works as a buer to \store" the output P until the buer is fullled and then broken by the output P. This kind of PFL is called a self-recovering PFL (rpfl). III. INTERLOCKED POSITIVE FEEDBACK LOOPS (IPFLS) In a biological complex network, two or more PFLs could share common components to work together [3{
Dynamic Analysis of Interlocked Positive Feedback Loops { Cuong Nguyen and Seung Kee Han -3463- Fig. 4. Interlocking of two positive feedback loops: autoregulation and mutual phosphorylation. The output P exists in two forms, active P (left) and inactive form (right, phosphorylated form). The active output P could activate its helper T F and enzyme E. In turn, the helper T F enhances the synthesis rate of P, while the enzyme E dephosphorylates the inactive form of the output P to become an active form. k a and k b are the coupling strengths of autoregulation and mutual phosphorylation, respectively. 7,10,13]. As shown in Figure 4, an epfl and an rpfl share a common output P composed of interlocked PFLs. In this ipfl, the output P is cooperatively regulated by auto-regulation and mutual phosphorylation. The stimulus S and the transcription factor T F regulate P together; and the output P can be dephosphorylated or phosphorylated with the help of enzyme E. Both T F and E are activated by P with coupling strengths k a and k b, respectively (Figure 4). The mathematical model and parameters are inherited from individual PFLs as in the following equations: d[p ] T = k s S k d [P ] T ; (6) d[p ] = k s S + k ap[e]([p ] T [P ]) k ip [P ] k d [P ]; J ap + [P ] T [P ] J ip + [P ] (7) d[t F ] d[e] = k a[p ](1 [T F ]) J atf + 1 [T F ] = k b[p ](1 [E]) J ae + 1 [E] k itf [T F ] J itf + [T F ] ; (8) k ie [E] J ie + [E] : (9) All the parameter values here are the same as those for single PFLs. 1. Bifurcation Analysis in Two-dimensional Parameter Space Let us start with the case in which the epfl coupling strength k a is xed in a bistable region, ka 1 < k a < ka 2 Fig. 5. Two-parameter bifurcation diagram of the interlocked PFLs (black) on (S, k b ) parameter space with xed k a (k a = 1), the auto-regulation (dashed blue lines) with coupling strength k a = 1 and mutual phosphorylation (dashed red lines). The value of the k's are k 1 = 0.063, k 2 = 0.221, k 3 = 0.224, k 4 = 0.315, k 5 = 0.393, k 6 = 0.582, k 7 = 1.031; k 8 = 2.8 and k 9 = 24.361. (Figure 2(d)) while the rpfl coupling strength k b is varying. We performed a two-parameter bifurcation analysis of input S and rpfl coupling strength k b. The two-parameter bifurcation diagram of ipfl (black) is shown in Figure 5. Along the k b axis, two saddle node bifurcation points, SN1e and SN2e, which come from a single epfl, always exist. Meanwhile, other two saddle node bifurcation points, SN2m and SN1m, which correspond to those of a single rpfl, emerge at k b = k 1 and then move to the left and interfere with the SN1e and SN2e. The SN2m crosses the SN1e and SN2e at k b = k 2 and k b = k 4, respectively. The SN1m crosses the SN1e and then collapses with SN2e at k b = k 3 and k b = k 5, respectively. In the region of k 5 < k b < k 6, SN2m turns to be SN2e and SN1e turns to be SN1m due to the on/o level swapping between the epfl and the rpfl (refer to the next section for more details). The creation of SN2m and SN1e occurs at k b = k 6. Then, the SN1m crosses SN1e and SN2e at k b = k 7 and k b = k 9, respectively. The SN2m crosses SN2e at k b = k 8. Based on the interference between those saddle node bifurcation points, the k b axis could be divided into ten regions of dierent dynamical behaviors of ipfls as specied by the k's on the right axis. The detailed dynamical behavior of each region is explained in the next section. 2. Bifurcation Analysis in One-dimensional Parameter Space The ten regions along the k b axis are represented by one-parameter bifurcation diagrams, as shown in Figure 6(a)-(j), respectively. In Figure 6(a)-(j), the epfl coupling strength k a is xed so that the single epfl could
-3464- Journal of the Korean Physical Society, Vol. 53, No. 6, December 2008 Fig. 6. One-parameter bifurcation diagrams of epfl (blue), rpfl (red) and ipfl (black) with k a xed (k a = 1) for various values of k b : 0.05, 0.2, 0.23, 0.3, 0.35, 0.5, 0.8, 2, 5, 26 in correspondence to (a), (b), (j). yield bistability; meanwhile. the rpfl coupling strength k b is chosen according to the regions specied by k's in Figure 5. SN2e < S < SN1e and SN2m < S < SN1m, are separated from each other. Consequently, this conguration of the ipfl bifurcation diagram is called non-overlapping bistability (NoB). A. k b < k 1 In this region, a single rpfl exhibits suppressed monostability (red line in Figure 6(a)). Meanwhile, a single epfl could yield bistability with bistable regions of SN2 auto < S < SN1 auto (blue line in Figure 6(a)). Therefore, the ipfls exhibits a single bistable region of SN1e < S < SN2e (Figure 6(a)). However, the bistable region of ipfl is at larger stimulus S than that of epfl, SN2 auto < S < SN1 auto, due to suppression in rpfl. B. k 1 < k b < k 2 In the region of k 1 < k b < k 2, the single rpfl exhibits bistability, SN2m phos < S < SN 1m phos, at very large stimulus S (Figure 6(b)). Therefore, ipfl exhibits an additional bistable region, SN 2m < S < SN 1m, coming from rpfl, beside the bistable region coming from epfl, SN2e < S < SN1e. However, the bistable region of ipfl, SN 1m < S < SN 2m is at smaller stimulus S than that of rpfl, SN2m phos < S < SN1m phos due to the enhancement of epfl, which is activated rst. Here, the two bistable regions of ipfl, C. k 2 < k b < k 3 By increasing k b, the two bifurcation points SN1m and SN2m move to the left and down and then interfere with SN1e and SN2e. In the region of k 2 < k b < k 3, the two bistable regions overlap, SN 2e < SN 2m < SN1e < SN1m (Figure 6(c)), creating a tristability region, SN2m < S < SN1e. Tristability means that three stable xed points coexist and are separated by two unstable xed points. Therefore, this conguration is called the overlapping bistable region (OLB). D. k 3 < k b < k 4 In this region, the bistable region of SN2m < S < SN1m is located inside the bistable region of SN2e < S < SN 1e, making a nesting bistability (NB) (Figure 6(d)). In this NB, switching from the lowest stable state to the highest one requires only one transition. Meanwhile, switching from the highest state to the lowest one requires two transitions. E. k 2 < k b < k 3
Dynamic Analysis of Interlocked Positive Feedback Loops { Cuong Nguyen and Seung Kee Han -3465- Here, two bistable regions merge together to create a large bistable region with a single transition between the two lowest and the highest stable states, SN2m < S < SN 1e (Figure 6(e)). Though inaccessible, there is a small region of tristability. Therefore, this is called merge bistability (MB'). F. k 5 < k b < k 6 In this region, two bistable regions emerge and the SN 1m and SN 2e collide and disappear, creating a bistable region without tristability inside (Figure 6(f)), the so-called merged bistability (MB). The merged bistable region is much larger than the two single bistable regions. This would enhance the reliability of biological toggle switches. G. k 6 < k b < k 7 Here, the ipfl exhibits another merged bistability with tristability (MB') again (Figure 6(g)). However, the rpfl on/o level is less than that of epfl in this conguration. Therefore, the bifurcation points of ipfl, SN1e, SN2e, SN1m and SN2m, occur at those of a single epfl and rpfl, respectively. H. k 7 < k b < k 8 Other nesting bistability (NB) is shown (Figure 6(h)). I. k 8 < k b < k 9 Other overlapping bistability (OLB) is shown (Figure 6(i)). J. k 8 < k b < k 9 Other non-overlapping bistability (NoB) is shown (Figure 6(j)). Thus, since k b > k 1, a single rpfl exhibits bistability and its two loci of saddle node bifurcations move to the left when k b is increasing (Figure 3(c)). As depicted in Figure 6(b)-(j), when k b is increased, the bistable region of a single rpfl, SN 2m-phos<S<SN 1mphos (red), moves to the left and down while those of a single epfl, SN1e and SN2e (blue), are xed. As a consequence, the ipfls exhibit an additional bistable region, SN2m < S < SN1m to SN2e < S < SN1e. Also, SN 2m < S < SN 1m moves to the left and down and then interferes with former bistable regions (SN2e < S < SN1e), creating diverse types of dynamical behaviors of ipfls. 3. Diversity of Dynamical Behaviors of ipfls To sum up, depending on the two coupling strengths k a and k b, interlocked system shows diverse dynamical behaviors. 1. Single bistability, SB, in which usually comes out when either auto-regulation or mutual activation exhibits bistability while the other does not (Figure 6(a)). The output turns on when the input exceeds the bifurcation point SN1e and the output turns o when the input S decreases below SN2e. 2. Non-overlapping bistability, NoB, in which bistable regions of each positive feedback loop are separated from each other (Figure 6(b) and (j)). To completely turn on the output P, the input S must reach SN1e to make the rst transition and then reach SN1m to make the second transition. As to turning o the output P, there are two transitions at SN2m and SN2e. 3. Overlapping bistability, OLB, in which two stable regions of an interlocked system overlap each other, creating tristability. Tristability means that three stable xed points coexist and are separated by two unstable xed points (Figure 6(c) and (i)). Turning on or o the output requires two transitions of output P at bifurcations. 4. Nesting bistability, NB, in which one bistable region nests in the other (Figure 6(d) and (h)). In this type, the transition number is asymmetric. For example, turning on the output P requires only one transition while two transitions are required to turn o the output P (Figure 6(d)). 5. Merging bistability, MB and MB', in which two individual bistable regions become a large merged one (Figure 6(e)-(g)). In this region, two single PFLs, epfl and rpfl, cooperate with each other to create a large bistable region. We also searched for all possible dynamical behaviors of an ipfl for a rpfl coupling strength k b when the epfl coupling strength k a is in the irreversible bistable region, k a > ka 2 (Figure 2(d)). All possible one-parameter bifurcation diagrams of a single epfl (blue), a single rpfl (red) and an ipfl (black) are shown in Figure 7(a)-(j). Because the epfl bistable region (blue) is irreversible, the ipfl bistable region (black) is irreversible. Despite that, the dynamical behavior of the ipfl is similar to that described in the case of ka 1 < k a < ka 2 (Figure 6(a)-(j)), except that their bistability is irreversible. Therefore, these dynamical behaviors could be classied into dierent groups as follows: 1. Irreversible single bistability (isb), Figure 7(a). There is one irreversible bistable region.
-3466- Journal of the Korean Physical Society, Vol. 53, No. 6, December 2008 Fig. 7. One-parameter bifurcation diagrams of epfl with strong coupling strength (blue), rpfl (red) and ipfl (black). The epfl coupling strength is k a = 3.5. The rpfl coupling strength is k b = [0.05, 0.29, 0.34, 0.5, 0.58, 0.7, 0.9, 2, 10.0, 30.0] in correspondence to (a), (b), (j). The ipfl coupling strengths k a and k b are same as those of a single epfl and rpfl. one of them is irreversible. 3. Irreversible overlapping bistability (iolb), Figure 7(c). Two bistable regions overlap creating tristability; one of them is irreversible. 4. Irreversible nesting bistability (inb). One bistable region nests in the other; one of them is irreversible (Figure 7(d) and (j)). 5. Double irreversible nesting bistability (i 2 NB). One bistable region nests in the other; both of them are irreversible (Figure 7(e)). Fig. 8. Phase diagram on the coupling-strength plane of (k a, k b ). Mono: monostability; SB: single bistability; NoB: non-overlapped bistability; OLB: overlapped bistability; NB: nested bistability; and MB: merged bistability. \i" and \i 2 " denote irreversibility and double irreversibility. Regions with the same color are of the same type. See the text for more details. 2. Irreversible non-overlapping bistability (inob), Figure 7(b). Two bistable regions are separated; 6. Irreversible merged bistability with or without tristability inside (imb and imb'), Figure 7(f)- (i). Two individual bistable regions become a large merged one, which is irreversible. and k b 4. Phase Diagram of the coupling strength k a The ipfls are composed of two PFLs with two different positive coupling strengths, k a and k b. To understand more how ipfls depend on their two coupling strengths, we built up a parameter-phase diagram in the (k a, k b ) plane (Figure 8). The whole phase plane
Dynamic Analysis of Interlocked Positive Feedback Loops { Cuong Nguyen and Seung Kee Han -3467- of (k a, k b ) is divided into seven regions with dierent colors, as shown in Figure 8. The monostability region (Mono) at small k a and k b (k a < ka 1, k b < k 1 ) is where both epfl and rpfl exhibit monostability; so the ipfls also exhibit monostability. The single bistability regions with or without irreversibility (SB and isb) are where either epfl or rpfl exhibits bistability, (ka 1 < k a and k b < k 1 ) or (k a < ka 1 and k 1 < k b ). The region of (ka 1 < k a, k 1 < k b ) is divided into four small regions and show a diversity of dynamical behaviors of ipfl, such as non-overlapping bistability with or without irreversibility (NoB and inob), overlapping bistability with or without irreversibility (OLB and iolb), nesting bistability with or without irreversibility (NB, inb and i 2 NB) and merged bistability with or without irreversibility (MB, MB', imb and imb'). IV. CONCLUSION In complex biological systems, epfl and rpfl cooperate and interlock together rather than working alone [9]. Our dynamical analysis indicated that, by interlocking two PFLs, the ipfl would show a diversity of dynamical behaviors, such as single bistability (SB), non-overlapping bistability (NoB), overlapping bistability (OLB), nesting bistability (NB) and merging bistability (MB) (Figure 6), or irreversible single bistability (isb), irreversible non-overlapping bistability (inob), double irreversible non-overlapping bistability (i 2 NoB), irreversible overlapping bistability (iolb), irreversible nesting bistability (inb), double irreversible Nesting bistability (i 2 NB) and irreversible Merging bistability (imb) (Figure 7). Those dynamical behaviors could be easily transformed to any one of the others by changing one of the coupling strengths, k a or k b. This provides exibility for biological systems in response to various external stimuli. Remarkable enough, the bistable region of ipfl in the MB type is greatly enhanced in comparison to that of a single one (Figure 6(e)-(g) and Figure 7(f)-(i)). In this region, the epfl and the rpfl cooperate to greatly improve the reliability of the decisive switch. The cooperation of the two PFLs creates a more reliable toggle switch, which is important in biological decisive systems. In this work, positive feedback loops were classied into two groups, self-enhancing (epfl) and self-recovery (rpfl) positive feedback loops. Then the epfl and rpfl were interlocked together, the interlocked positive feedback loop (ipfl) showed diverse dynamical behaviors with respect to the relative coupling strengths of the feedback loops. Interestingly, when the epfl and the rpfl cooperate with comparable coupling strengths, the two bistable regions of the single epfl and rpfl merge together, resulting in a much larger bistable region. Therefore, cooperativity of the epfl and the rpfl greatly enhances the reliability of switches. ACKNOWLEDGMENTS This work was supported by a Korea Science and Engineering Foundation grant (R15-2004-033-007001-0) funded by the Ministry of Science of Technology, Korea and also by the BK21 program at Chungbuk National University. REFERENCES [1] L. H. Hartwell, J. J. Hopeld, S. Leibler and A. W. Murray, Nature 402, C47 (1999). [2] C. Nguyen, C. N. Yoon, S. K. Han, D. L. Pham and H. Y. Kim, J. Korean Phys. Soc. 50, 332 (2007). [3] J. J. Tyson, K. C. Chen and B. Novak, Curr. Opin. Cell. Biol. 15, 221 (2003). [4] K. C. Chen, A. Csikasz-Nagy, B. Gyory, J. Val, B. Novak and J. J. Tyson, Mol. Biol. Cell. 11, 369 (2000); C. N. Yoon, S. K. Han and H. Y. Kim, J. Korean Phys. Soc. 44, 638 (2004). [5] B. Novak, Z. Pataki, A. Ciliberto and J. J. Tyson, Chaos 11, 277 (2001). [6] Z. Qu, W. R. MacLellan and J. N. Weiss, Biophys. J. 85, 3600 (2003). [7] P. Smolen, D. A. Baxter and J. H. Byrne, J. Neurosci. 21, 6644 (2001). [8] M. Swat, A. Kel and H. Herzel, Bioinformatics 20, 1506 (2004). [9] O. Brandman, J. E. Ferrell, Jr., R. Li and T. Meyer, Science 310, 496 (2005). [10] T. I. Lee, N. J. Rinaldi, F. Robert, D. T. Odom, Z. Bar- Joseph, G. K. Gerber, N. M. Hannett, C. T. Harbison, C. M. Thompson, I. Simon, J. Zeitlinger, E. G. Jennings, H. L. Murray, D. B. Gordon, B. Ren, J. J. Wyrick, J. B. Tagne, T. L. Volkert, E. Fraenkel, D. K. Giord and R. A. Young, Science 298, 799 (2002). [11] S. G. Lee, J. Korean Phys. Soc. 52, 11 (2008). [12] G. E. Briggs and J. B. Haldane, Biochem J. 19, 338 (1925). [13] J. E. Ferrell and W. Xiong, Chaos 11, 227 (2001).