Abstract
Regulatory gene circuit motifs play crucial roles in performing and maintaining vital cellular functions. Frequently, theoretical studies of gene circuits focus on steadystate behaviors and do not include time delays. In this study, the inclusion of time delays is shown to entirely change the timedependent dynamics for even the simplest possible circuits with one and two gene elements with self and cross regulations. These elements can give rise to rich behaviors including periodic, quasiperiodic, weak chaotic, strong chaotic and intermittent dynamics. We introduce a special powerspectrumbased method to characterize and discriminate these dynamical modes quantitatively. Our simulation results suggest that, while a single negative feedback loop of either one or twogene element can only have periodic dynamics, the elements with two positive/negative feedback loops are the minimalist elements to have chaotic dynamics. These elements typically have one negative feedback loop that generates oscillations and another unit that allows frequent switches among multiple steady states or between oscillatory and nonoscillatory dynamics. Possible dynamical features of several simple one and twogene elements are presented in details. Discussion is presented for possible roles of the chaotic behavior in the robustness of cellular functions and diseases, for example, in the context of cancer.
Introduction
One of the challenges in molecular cell biology is to understand how cells fulfill their functions through specific gene regulations^{1}. Various stateofart experimental techniques, such as highthroughput DNA/RNA sequencing^{2} and wholecell genomic/proteomic profiling^{3}, have been developed to enable the mapping or the inference of gene regulatory network^{4,5}. Yet, it remains unclear how cells utilize the gene network to perform their specific tasks and to do so efficiently despite the innate high intracellular noise^{6,7,8,9}.
Many theoretical and synthetic biology studies have suggested that a specific set of topological links among a few genes, namely circuit motifs, may individually perform certain functions^{10,11,12,13}. For instance, a toggle switch (two genes mutualinhibiting each other) and its variations allow the coexistence of multiple stable steady states^{14,15}, which is essential to decisionmaking between cellular fates during cell differentiation^{16,17} and certain cell phenotypic transitions, such as epithelialtomensenchymal transitions^{18,19}. Motifs such as flipflop circuit^{20} and repressilator circuit^{21} have been shown to allow periodic oscillations in gene expression levels. Moreover, a feed forward loop may generate pulses, detect foldchanges or make adaptation in response to different external and internal signals^{22,23}.
It is commonly assumed that these circuit motifs are the building block modules of a larger modular network that can perform several elaborated tasks as needed. For the modular network to function efficiently, the individual circuit modules should have the following properties. First, each circuit should have sufficient functional flexibility to perform its specific function while it receives various inputs from the other modules in the network. Second, the module dynamical behavior should be stable in the presence of internal and external noise. Third, the module function should be robust to changes in the circuit parameters that vary from cell to cell. The current study is motivated by the need to investigate the sensitivity to the circuit parameters. We seek to check under which circumstances the circuit dynamics can be dramatically different, or even become chaotic for a certain range of parameters.
In general, the dynamics of gene expression for a gene circuit can be modeled by coupled nonlinear ordinary differential equations. Thus, chaotic behavior could theoretically exist in motifs comprised of three or more components (e.g. three genes) or whose dynamics is described by three or more equations^{24}. Chaotic behavior in gene circuits has been studied before^{20,25,26,27,28,29,30}, but it has gained limited attention in systems biology for the following reasons. First, it is commonly assumed that the selected gene networks, during the course of evolution, are those that are robust to noise and changes in the circuit dynamics. Second, it is very hard to quantitatively measure chaotic dynamics of gene expression due to the limited availability of temporal gene expression data and due to the presence of gene expression noises in circuit dynamics. Third, it has been shown in computational studies that chaos motifs are rare and the parameter range to observe the chaotic dynamics for such motifs is extremely narrow^{25}.
We propose that when time delays are included, chaotic dynamics could be observed even in simple circuit elements with one or two elements and for a much wider range of circuit parameters than previously expected. Chaos is observed even for two coupled genes with time delayed mutual regulations or even a single gene with two time delayed selfregulations. Moreover, we reason that chaotic dynamics can be very relevant in abnormal physiological conditions and in some diseases, such as cancer, where gene regulations and circuit parameters significantly differ from the evolutionary selected ones. Hence, different from previous studies of chaotic dynamics in gene circuits, we now include the effects of time delays in the self and cross regulations. Time delays in gene regulations may arise from the recruitment of RNA polymerase, transportation of mRNAs and translational process through ribosome and several other sources in the cell. While time delays do not change the steady states solutions of the gene networks, they can change the stability of these states and the circuits’ dynamical behaviors^{31,32,33}. Therefore in this study we show that time delays can add singularity to the nonlinear dynamics of gene expression. We note for completeness that in a different context some previous studies in an homogeneous population of circulating white blood cells have shown that inclusion of time delays can give rise to chaotic dynamics^{34}.
As it is mentioned earlier, this paper investigates the dynamics of simple one and two gene circuit motifs when time delays are included. In the next section, we present a concise description of the circuit motifs and their possible dynamical behaviors. We show that these circuits give rise to a rich variety of dynamical modesperiodic (P), quasiperiodic (QP), weak chaotic (WC) and strong chaotic (SC) dynamics. Over the years, researchers have been characterizing the properties of chaotic behaviors and distinguishing between the different types of chaos by various methods and criteria^{24}, including power spectrum, Lorenz map^{35}, features of a trajectory in the phase space^{36} such as strange attractor^{37}, Poincare map^{38}, Lyapunov stability^{39,40} and GrassbergerProcaccia algorithm^{41}. Yet, different authors use different terminology to describe the various chaotic behaviors^{42}. For example, some use the terms chaos and hyperchaos to describe weak chaos and strong chaos^{43,44}. Here we introduce a special powerspectrumbased method to distinguish among the various dynamical modes in the gene circuits that we have studied.
As elaborated in details in the results section, we found that even the simple one and two gene motifs, that are frequently recurring in most biological networks, are capable of generating P, QP, WC and SC dynamics for a wide range of time delays and circuit parameters. Moreover, transitions from nonchaotic dynamics (presumably corresponding to normal cells) to more elaborated dynamics (presumably in abnormal cells) only require relatively small changes in the time delays. Interestingly, for some circuits that contain both positive and negative feedback loops, the circuits can give rise to intermittent dynamics between two modes (e.g. PSC and WCSC). At the end of the article, we discuss possible roles of the chaotic behavior in the robustness of cellular functions and their relevance to some diseases.
Roadmap
In this section, we discuss the circuit motifs whose dynamics are studied in this paper (Fig. 1). They are among the simplest possible circuit motifs, consisting of either one gene or two genes. Yet, when considering time delays in the regulation, we found that even such simple circuits exhibit rich dynamical behaviors. To classify these dynamics, especially those nonperiodic oscillatory ones, we introduce a special powerspectrumbased method that is used throughout the work. The detailed analyses for each motif is shown in the following sections and in Supplementary Information.
We begin with the simplest gene elements that can give rise to periodic oscillations, since such feature is a prerequisite to have nonperiodic oscillations. One of the simplest oscillatory circuit motifs is repressilator. This element, which was first studied as a synthetic circuit^{21} and was later shown to be existent in naturally occurred gene network^{45,46}, is comprised of three genes with sequential inhibitions (A inhibits B, B inhibits C and C inhibits A). A repressilator is able to generate stable oscillations without introducing time delays in the repressive regulations. In Supplementary section SI1, we show that the dynamics of a repressilator is comparable to that of a selfinhibitory single gene element (Fig. 1a) with time delay. Similar time dynamics could also be observed in a flipflop circuit with time delays^{20,31}. Here, the flipflop motif is a twogene element (genes denoted by A and B) where A activates B and B inhibits A.
Next, we hypothesized that chaotic gene elements could be built on the basis of these minimalist oscillatory elements and demonstrate that this is indeed the case. We learned that to generate chaos in the gene expression, the elements must have at least two coupled feedback loops with different time delays – either the elements with at least two negative feedback loops (Fig. 1b), or the elements with at least one negative and one positive feedback loops (Fig. 1c,d). In addition, strong chaotic behavior (as explained and defined later) was found in elements with more than two coupled regulatory motifs. A typical example is the elements in Fig. 1c with two coupled negative feedback loops and one positive feedback loop. Note that in all these cases, we found similar nonperiodic dynamics for both singlegene and the corresponding twogene elements, suggesting that it is the coupled circuit motif rather than the number of genes that gives origin to the different chaotic behaviors.
We define the different chaotic behaviors by examining the power spectra of the time trajectories of the gene expression. We utilize two kinds of spectra – the spectrum of the whole time trajectories of the dynamics (termed full spectrum) and the spectrum of a corresponding discrete series that is composed of the maximum and the minimum expressions of every oscillation (termed maximumminimum spectrum). By using the combination of the two spectra, we were able to recognize different nonperiodic oscillatory dynamics. For example, Fig. 2 shows three different dynamics, representing the typical cases of quasiperiodic (QP, the left column), weak chaotic (WC, the middle column) and strong chaotic (SC, the right column) behaviors. The time trajectories for all these cases look very similar (Fig. 2, the first row). In terms of the full spectrum (Fig. 2, the third row), those for the QP and WC cases are similar since both have spikes at some discrete frequencies. Yet, the full spectrum of the SC case is distinct; it is marked by spikes through almost the whole frequency range. For the maximumminimum spectrum (Fig. 2, the fourth row), those for the WC and the SC cases are similar since both have many spikes through the whole frequency range. We also noted that the SC spectrum is dominant by the downward spikes, while the WC spectrum has balanced upward and downward spikes. The maximumminimum spectrum for the QP case is distinct; it is marked by the existence of only a few spikes and they are mostly upward. Another powerful tool is to visualize the time trajectory by projecting it onto a twodimensional phase space. Here we show the space by the gene expression level vs. the expression time derivative (Fig. 2, the second row). The QP map has a torus structure; the WC map covers the phases more but regularly, while the SC map is more irregular. Note that the combination of these tools also enables us to identify the intermittent PSC (the circuit in Fig. 1d) and the intermittent WCSC (Supplementary section SI5, the upper circuit in Fig. 1c) for some circuit motifs, as we explain in details in a later section.
Results
Oscillatory dynamics in onegene and twogene elements with time delay
As is mentioned in the previous section, a selfinhibitory single gene element with time delay (Fig. 3a) has nearly the same dynamics as a classical repressilator composed of three identical genes (see Supplementary section SI1 for details). More specifically, such singlegene element exhibits a Hopf bifurcation, as function of the time delay, from a steadystate dynamics into an oscillatory one at a threshold delay (Fig. 3a and 3b). The time dynamics of a selfinhibitory singlegene element with time delay τ is described by the following deterministic rate equation:
A represents the protein concentration measured in units of nM. Time t is measured in minutes. The protein production rate g_{A} is measured in units of nM/minute and the degradation rate k_{A} is measured in units of 1/minute. These units are used throughout the article.
H^{−}_{AA} is an inhibitory Hill function. In general, for gene X inhibited by gene Y, the inhibitory Hill function is defined by
Where Y_{0} is the threshold concentration of the Hill function and n is its rank. Note that for selfinhibitory gene A discussed here, X≡Y≡A.
We note that the Hopf bifurcation from the steadystate (black curve in Fig. 3a) to the oscillatory dynamics (red curves that mark the maximum and minimum levels of A) occurs at time delay threshold τ_{th} = 3.1 minutes that is the order of 1/k_{A}.
In Fig. 3c,d we demonstrate that a twogene flipflop element can give rise to oscillatory dynamics when time delay is included. This element also exhibits a Hopf bifurcation, as function of the time delay τ, from a steadystate dynamics into an oscillatory one at a threshold delay τ = τ_{th}. The model equations for this twogene motif are included in Supplementary section SI1. We note that the origin of the correspondence between the selfinhibitory onegene element and the twogene flipflop element can be understood as follows. Since gene A activates its inhibitor gene B, it acts effectively as a selfinhibitory gene. Since each of the two regulatory paths (AtoB and BtoA) has a time delay τ, the flipflop element corresponds to a single selfinhibitory gene with a time delay that is equal to twice τ plus a time delay associated with the dynamics of gene B. This is why the threshold time delay to induce oscillation (the bifurcation point) for the flipflop element τ_{th} is 1.05 minutes, which is smaller than half of the time delay for the case of the single gene element.
Weak chaotic dynamics in elements with two negative feedback loops
In the previous section, we showed that a negative feedback loop, either in a singlegene or in a twogene element, can give rise to periodic dynamics when there are time delays in the inhibitory regulation. Thus, it can be anticipated that circuit motifs with more than one negative feedback loops can exhibit more complex dynamics. Here, we investigate on two circuit motifs; one gene with two selfinhibitions (Fig. 4a) and a flipflop element with one of the genes having a selfinhibition (Fig. 4d). Compared to the elements studied in the previous section (Fig. 3), each of the elements in this section contains an additional motif of timedelayed selfinhibition.
The time dynamics of a single gene element with two selfinhibitions with time delays τ_{1} and τ_{2}, respectively, are described by the following deterministic rate equation:
H^{−}_{1AA} and H^{−}_{2AA} are inhibitory Hill functions as defined by equation (2), representing the two selfinhibitions. Note that the time delays τ_{1} and τ_{2} are not necessarily the same. And in such a situation, the corresponding circuit can exhibit nontrivial dynamics.
To demonstrate this, we investigate the circuit dynamics while fixing the time delay for the first selfinhibition τ_{1} to be 18 minutes and varying the second time delay τ_{2}. The rest circuit parameters are listed in Supplementary section SI2. In this specific case, the circuit dynamics always exhibit oscillations, regardless of the values of τ_{2}. Thus, we plot a bifurcation diagram of the maximum levels of protein A for each oscillation with respect to the values of time delay τ_{2} (Fig. 4a). When τ_{2} varies from 4 minutes to 5 minutes, the circuit exhibits transitions from P to QP/WC dynamics and back to P again. As an example, when τ_{2} = 4.65 minutes (navy arrow in Fig. 4a), the circuit exhibits a nonperiodic oscillatory dynamics (Fig. 4b, Supplementary Fig. SI2.1). The corresponding full spectrum (Fig. 4c) and the maximumminimum spectrum (Supplementary Fig. SI2.1) indicate this to be a weak chaotic dynamics (WC). We also found that WC/QP dynamics are observed when 3.09 minutes <τ_{2} < 3.113 minutes (Supplementary Fig. SI2.2).
Next we study the twogene flipflop element, where the first gene (A) has an additional selfinhibition (Fig. 4d). This twogene element exhibits similar dynamics to the abovediscussed onegene element with two selfinhibitions. We considered a specific case in which gene A selfinhibition has time delay τ_{1}, while the gene A activation of gene B and the backward inhibition of gene A by gene B have the same time delay τ_{2}. One can also choose different time delays for the regulations in the flipflop element, but similar dynamical behaviors will be observed. The model equations for this motif are included in Supplementary section SI2. For this circuit we considered the dynamics for τ_{1} equal to 5.3 minutes and the rest parameters are listed in Supplementary section SI2. Similar to the previous circuit, the current motif also exhibits nonperiodic dynamics when τ_{2} is approximately from 8 to 9.18 minutes. For example, when τ_{2} = 8.2 minutes (navy arrow in Fig. 4d), the circuit exhibits WC dynamics (Fig. 4e,f, Supplementary Fig. SI2.1).
A possible mechanism to generate chaos for elements with two coupled feedback loops
In the previous two sections, we showed that elements with one negative feedback loop allow oscillations and elements with two coupled negative feedback loops can generate chaos. To understand the mechanism of the chaotic dynamics, we further study the onegene element with two selfinhibitions (Fig. 4a). When τ_{1} = 18 minutes, the dynamics of the element have oscillations with a period of around 40 minutes for a large range of τ_{2} (Supplementary section SI2), including those of the P, QP and WC cases mentioned above. In addition, when τ_{1} = 3 or 4 minutes, we also observed bifurcation of the dynamics from a steadystate one to an oscillatory one with respect to the value of τ_{2}. The first bifurcation point τ_{2} ~ 3 minutes, which is very close to the range of τ_{2} to observe nonperiodic dynamics of the same element (Fig. 4a and Supplementary Fig. SI2.2). This observation suggests that the first negative feedback loop drives the underlying oscillations and the second element switches frequently from the oscillatory phase to the steadstate phase, which probably causes the coupled motif to have quasi periodic or chaotic behavior. This mechanism could be generalized to circuit motifs with a positive feedback loop, which allows phase transitions between two stable steady states (next section).
Strong chaotic dynamics in elements with one positive and two negative feedback loops
In this section, we study a singlegene element with two selfinhibitions and one selfactivation (inset of Fig. 5a) and a circuit with two selfinhibitory and mutually activating genes (Fig. 5d). Compared with those from the previous section, the gene elements here contain a positive feedback loop in addition to two negative feedback loops. We found that this combination of two negative feedback loops and one positive feedback loop can give rise to elaborate dynamics.
We first studied the singlegene element with three selfregulation loops with time delays τ_{1}, τ_{2} and τ_{3}, (Fig. 5a). The deterministic rate equation for this circuit is given by
Here H^{+}_{3AA} is an excitatory Hill function, representing the selfactivation of gene A. By definition, H^{+}_{3AA}(A) ≡ 1H^{−}_{3AA}(A).
We studied the dependence of the circuit dynamics on τ_{3} when setting τ_{1} to be equals to 18 minutes and τ_{2} to be equal to 8 minutes. The rest circuit parameters are listed in Supplementary section SI3. The corresponding bifurcation diagram is shown in Fig. 5a. By varying τ_{3} from roughly 10 to 20 minutes, we observed transitions among the P, QP, WC and SC dynamics. Interesting, the circuits exhibit nonperiodic dynamics for a much wider range of time delays, compared to the circuits from the previous section. More detailed results are shown in Supplementary sections SI3 (QP, SC) and SI5 (WC).
Next we studied the twogene circuit with two selfinhibitory and mutually activating genes (inset of Fig. 5d). Biologically, this circuit motif can correspond to single cell containing such circuit, or two positively interacting cells, each of which has a singlegene element with selfinhibition. Depending on the nature of the mutual activations, the circuit has slightly different nonperiodic dynamics. If the mutual activations are modeled by excitatory Hill functions, the circuit can give rise to P, QP and WC dynamics (Supplementary section SI3). On the other hand, if the mutual activations are modeled by linear functions (corresponding to two interacting cells), the two cells can give rise to P, QP, WC and SC dynamics (Fig. 5d–f, Supplementary Fig. SI3.3 and Fig. SI3.4).
We studied the dependence of the circuit dynamics on the time delay τ_{21} of the AtoB activation when setting the first selfinhibition τ_{1} to be equal to 6.0 minutes, the second one τ_{2} to be equal to 5.0 minutes and the BtoA activation time delay τ_{12} to be equal to 7.5 minutes where both activations are given by linear functions. The corresponding bifurcation diagram is shown in Fig. 5d. Similar to the singlegene element described above, this twogene element can also give rise to elaborated nonperiodic dynamics for a wide range of time delay τ_{21}. More detailed results are shown in the Supplementary section SI3.
Intermittency between periodic and strong chaotic mode in a circuit element with both selfinhibition and selfactivation
While the classification previously shown is a good characterization of the different dynamical regimes, some other circuits may show an even more complex dynamics that could not be either one of the dynamic regimes. During time evolution, they may switch among different dynamics, for example, in a singlegene element with both selfactivation and selfinhibition (inset of Fig. 6a). The circuit is special in that it exhibits intermittency between periodic and chaotic behaviors in a single time trajectory.
Consider that the time delays for the selfinhibition and the selfactivation are τ_{1} and τ_{2,} respectively, the deterministic rate equation for the circuit is
We studied the dependence of the circuit dynamics on the selfactivation time delay τ_{2} when setting the selfinhibition time delay τ_{1} to be equal to 26 minutes. The rest circuit parameters are listed in Supplementary section SI4. The corresponding bifurcation diagram is shown in Fig. 6a. We found that this circuit exhibits periodic dynamics for majority of the τ_{2} values we have tested, but it has also nonperiodic oscillations when τ_{2} varies between approximately 22 to 26 minutes and 28 to 29 minutes. Close inspection of the dynamical behavior, especially for cases with τ_{2} that are close to the bifurcation point (e.g. around 26 minutes), we observed transitions from SC (τ_{2} = 26.0 minutes, Fig. 6f,g) to P (τ_{2} = 26.3 minutes, Fig. 6b,c) dynamics. Interestingly, for cases where the values of τ_{2} are in between, we observed intermittent dynamics between periodic and strong chaotic behaviors (i.e. PSC intermittency, τ_{2} = 26.29 minutes, Fig. 6d,e). Here we refer the intermittent chaotic dynamics to the cases where substantially long durations of both periodic and chaotic dynamics are observed in a single trajectory, while transitions between these two modes are not periodic ^{43,47,48}. We also noticed that, in the intermittent chaotic dynamics, the amplitudes of the oscillations in both modes decay slightly when time advances, until the mode is switched to the other one (Fig. 6d). Interestingly, the closer the τ_{2} to the bifurcation point around 26 minutes, the longer the time intervals of the periodic dynamics in each intermittent cycle, while the duration of the SC dynamics remains the same. Note that the full spectrum for the SC dynamics has mostly downward spikes (Fig. 7e), while that for the P dynamics has mostly upward and discrete spikes (Fig. 7a). As expected, that for intermittent dynamics has a mixture of both up and downward spikes (Fig. 7c). Moreover, the 2D map (AdA/dt) for an intermittent dynamics (Fig. 6e) has both the trajectory of the periodic dynamics (green) and that of the chaotic dynamics (purple), indicating the coexistence of both modes. As shown in Supplementary section SI5, the onegene element with two selfinhibitions and one selfactivation (Fig. 5a) is another example of circuit that can generates intermittent dynamics, but in this case it is between the WC and the SC dynamics.
Discussion
In this study, we showed that simple onegene and twogene elements can give rise to elaborate dynamics when time delays are included for the self and cross regulation loops. The observed time behaviors include periodic (P), quasiperiodic (QP), weak chaotic (WC) and strong chaotic (SC) dynamics as well as intermittent dynamics between periodic and chaotic behaviors. To quantitatively distinguish between the different dynamical modes, we developed a dedicated spectrumbased method that is essentially a combination of both the powerspectrum of the full dynamics and the powerspectrum of a series of discrete maximum/minimum expressions for each oscillation. Note that this combination of both spectra is needed to distinguish among the P, the QP, the WC and the SC dynamics. Either spectrum alone is insufficient to achieve this goal. We have also tried to describe the nonperiodic dynamics by the Lyapunov exponents. However, the Lyapunov exponents obtained from a time series are sensitive to some parameters (e.g. embedding dimension and time lag for embedding) and it is hard to estimate these parameters for a timedelay nonlinear system^{49}.
Our investigation reveals that, in order to generate chaotic dynamics, the circuit motifs need to have one negative feedback loop, which gives rise to oscillations and another feedback loop (either positive, negative, or both) that creates two dynamical phases (e.g. two stable states, or a stable state and an oscillatory state). Chaos might take place when the first element generates sustainable oscillations while the second one allows hopping between the two dynamical modes. It is our expectation that this might be also true for several other chaotic motifs.
We also observed periodic/chaotic and weak/strong chaotic intermittency in certain circuit motifs. More efforts should be made to investigate the mechanism of the intermittent dynamics, in particular the dependence of the ratio between the time intervals of the two modes as function of the time delays. Another interesting future direction is to examine the effective landscape of the circuit ^{16,50,51,52,53} in the presence of time delays and to obtain an understanding of the difference between the effective landscape for a chaotic system^{54} and a nonchaotic one.
Moreover, we demonstrated that the dynamics for certain circuit motifs can be converted from periodic oscillations to nonperiodic, chaotic dynamics by just adjusting the values of time delays in a gene regulation. Note that time delays are common in genegene interactions. There are multiple mechanisms to adjust this time delays such as the effects of molecular crowding and the reduced efficacy of recruiting essential molecular components for gene regulation. Therefore, the delayinduced chaos could potentially affects normal cellular functions that typically require proper stable dynamics. Unlike some diseases which are caused by the malfunction of a gene, time delays in gene regulation provide an additional source for abnormal behavior.
The delayinduced chaos could also be relevant to diseases that are associated with cellular functions that require oscillatory gene expressions. For example, some cancer patients are known to suffer from an irregular circadian clock^{55,56}, which in turn facilitates the advance of tumorigenesis by interfering with the immune system^{57}. We hypothesize that cancer could induce additional time delays to the regulations of circadian genes, therefore generating nonperiodic dynamics which disrupt the usual process. Indeed, the core circadian gene regulatory circuit could potentially have delayinduced chaos because the circuit motif has a negative feedback loop and a positive feedback loop (Fig. 1e)^{58}. Chaotic behavior has also recently been studied in the context of interactions between cancer and the immune system^{59}. A good example is the NFκB genetic circuit, which plays important roles in both immune response and proliferation signaling. Again, the NFκB circuit (Fig.1f) is composed of two coupled negative feedback loops^{60}. The normal functions of NFκB could be potentially affected by nonperiodic expression dynamics of the circuit. Since time delays are inherent in every gene network, it is our expectation that several additional biological processes will be affected by delay induced chaos.
Additional Information
How to cite this article: Suzuki, Y. et al. Periodic, Quasiperiodic and Chaotic Dynamics in Simple Gene Elements with Time Delays. Sci. Rep. 6, 21037; doi: 10.1038/srep21037 (2016).
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Acknowledgements
This work was supported by the Center for Theoretical Biological Physics sponsored by the NSF (Grant PHY 1427654), by NSF MCB 1214457, by the Cancer Prevention and Research Institute of Texas (CPRIT) and by the Tauber Family Funds and the MagutGlass Chair in Physics of Complex Systems at TelAviv University. JNO is a CPRIT Scholar in Cancer Research sponsored by CPRIT. ML is supported by a training fellowship from the Keck Center for Interdisciplinary Bioscience Training of the Gulf Coast Consortia (CPRIT Grand No. RP140113).
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Y.S., M.L. and E.B.J. developed the theoretical framework. Y.S. performed the simulations. Y.S., M.L., E.B.J. and J.N.O. contributed to the analysis of the results and the generation of the manuscript.
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Suzuki, Y., Lu, M., BenJacob, E. et al. Periodic, Quasiperiodic and Chaotic Dynamics in Simple Gene Elements with Time Delays. Sci Rep 6, 21037 (2016). https://doi.org/10.1038/srep21037
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DOI: https://doi.org/10.1038/srep21037
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