Nonlinearity arising from noncooperative transcription factor binding enhances negative feedback and promotes genetic oscillations

Nonlinearity arising from noncooperative transcription factor binding enhances negative feedback and promotes genetic oscillations

Iván M. Lengyel,[1] Daniele Soroldoni,[2, 3] Andrew C. Oates,[2, 3] Luis G. Morelli[1] Email:


    • [1] Departamento de Física, FCEyN UBA and IFIBA, CONICET, Pabellón 1, Ciudad Universitaria, 1428 Buenos Aires, Argentina.
    • [2] MRC-National Institute for Medical Research, The Ridgeway, Mill Hill, NW7 1AA London, UK.
    • [3] Department of Cell and Developmental Biology, University College London, Gower Street, WC1E 6BT London, UK.

I. Introduction

Cells can generate temporal patterns of activity by means of genetic oscillations [4, 5]. Genetic oscillations are biochemical oscillations in the levels of gene products [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]. They can be produced by negative feedback regulation of gene expression, in which a gene product inhibits its own production directly or indirectly [18]. Such autoinhibition is often performed by transcriptional repressors, proteins or protein complexes that bind the promoter of a gene and inhibit the transcription of new mRNA molecules [19, 20], see Fig. 1. A theoretical description of biochemical oscillations requires a delayed negative feedback, together with sufficient nonlinearity and a balance of the timescales of the different processes involved [21]. Delays may occur naturally in transcriptional regulation, since the assembly of gene products involves intermediate steps [19, 22, 23]. Nonlinearity refers to the presence of nonlinear terms in the equations describing the dynamics. Such nonlinear terms may occur in the equations due to the presence of cooperative biochemical processes, where cooperativity is understood as a phenomenon in which several components act together to orchestrate some collective behavior [24]. Although some processes giving rise to nonlinear terms are known, in general it is still an open question how nonlinearity is built into genetic oscillators.

Figure 1: Delayed autoinhibition can produce genetic oscillations. (A) The gene (light blue box) is transcribed and translated into gene products x, with a delay . In this example, gene products form dimers that act as transcriptional repressors, inhibiting transcription of the gene (blunted arrow), and decay at a rate . (B) and (C) Numerical solutions of Eq. (III.) describing the oscillator in A. (B) Phase space: monomer concentration vs. the delayed concentration settled to a limit cycle. (C) Dimer concentration oscillates as a function of time. Parameters in B, C: , , , , , .

A compelling model system for genetic oscillations is the vertebrate segmentation clock [25, 26, 27, 28]. This is a tissue-level pattern generator that controls the formation of vertebrate segments during embryonic development [28, 29]. The spatiotemporal patterns generated by the segmentation clock are thought to be initiated by a genetic oscillator at the single cell level [10, 30, 31, 32, 12, 13]. In this genetic oscillator, negative feedback is provided by genes of the Her family, which code proteins that form dimers and can act as transcriptional repressors [33, 34, 35, 36, 37]. The time taken from transcription, translation and splicing may introduce the necessary feedback delays in zebrafish and mouse [22, 38, 39, 37, 40]. One source of nonlinearity in the segmentation clock oscillator is the dimerization of gene products that bind the promoter of cyclic genes [35]. However, this may be insufficient to generate the observed oscillations in the levels of gene products. One way to increase nonlinearity would be cooperative binding of repressors to regulatory binding sites at the promoter [41, 24, 42]. Cooperative binding to multiple binding sites can make the negative feedback steeper [43]. A similar effect occurs with ultrasensitivity in phosphorylation cascades [44]. The presence of clusters of binding sites for transcription factors may be a common motif in gene regulation [45], and cooperativity in transcription factor binding has been reported for some systems [46].

In zebrafish, multiple binding sites for Her dimers have been identified in the promoter region of Her1, Her7, and other genes of the Notch pathway [47, 35, 48]. However, there is no evidence for cooperative binding of transcriptional regulators in the case of the zebrafish segmentation clock. Therefore, although cooperative binding is not ruled out, this lack of evidence raises the general question of what contribution could be expected from the multiple binding sites reported in the promoter of segmentation clock cyclic genes. Here we use theory to study how multiple binding sites affect nonlinearity and biochemical oscillations in a generic description of a genetic oscillator.

Ii. A promoter with multiple binding sites

We first evaluate the regulatory function of a promoter that contains binding sites for a transcriptional repressor, see Fig. 2. We consider transcriptional repressors although the results in this section are more general and would apply to other types of transcription factors. We focus on a single transcriptional repressor for the sake of simplicity, and assume that all binding sites are identical. That is, binding and unbinding of repressors to the different sites occur at the same rates for all sites. Moreover, we assume that there is no cooperativity in repressor binding. This means that binding and unbinding rates are not affected by the presence of bound factors to any of the other sites.

Figure 2: Schematic representation of a promoter with multiple binding sites, numbered from to . Transcriptional repressors (orange squares) bind and unbind from binding sites (numbered platforms) at the promoter of a gene (light blue stretch). (A) and (B) illustrate two equivalent configurations of the promoter with identical number of bound transcriptional repressors having the same inhibiting strength.

The state of the promoter at any time can be characterized by the number of bound factors, which goes from to . With a rate , a free repressor binds to an empty site at the promoter, which steps from state to state ; with a rate , a bound repressor falls off the promoter, which steps from state to state :


Denoting by the promoter occupation probabilities which describe the fraction of time that the promoter spends at state in the thermodynamic limit [49], the kinetics of binding and unbinding of transcriptional repressors to the binding sites is given by the set of ordinary differential equations

together with the conservation law


where is the repressor concentration and is proportional to the number of gene copies.

We assume here that binding and unbinding of repressors to the promoter occur much faster than other processes like transcription, translation, transport and decay of molecules. This means that the promoter occupation probabilities quickly reach equilibrium with a given concentration of transcriptional repressors [49, 50]. This situation is described by for all in Eq. (II.), and the resulting algebraic equations can be solved by induction to obtain


where is the equilibrium constant for binding of factors. Using the constraint Eq. (II.) we express in terms of


Equation (II.) and Eq. (II.) describe the equilibrium occupation of the promoter in terms of the concentration of the transcriptional repressor.

Iii. Abrupt inhibition

In the previous section, we evaluated the kinetics of noncooperative binding to a promoter with multiple binding sites. How does the presence of bound transcriptional repressors affect the transcription rate of the gene downstream of the promoter? In general, the strength of inhibition will depend on the number of bound repressors, and the regulatory function will have the form


where is the basal transcription rate in the absence of bound repressors and is the transcription rate in the presence of bound repressors. Here we assume, for the sake of simplicity, that transcription proceeds at its basal rate while there are or less sites occupied by the repressors, and drops to zero when the number of occupied sites is larger than , see Fig. 3. We shall consider an alternative scenario below.

In this situation, Eq. (III.) becomes


Using Eq. (II.) and Eq. (II.), the resulting regulatory function for a promoter with binding sites is


The zebrafish segmentation clock may be an interesting system to evaluate these results. In the her1/her7 locus of zebrafish, an estimated number of about binding sites has been reported [35, 48]. The regulatory functions resulting from Eq. (III.) for are displayed in Fig. 3B. Although it is clear that inhibition of the promoter for a given level of repressors shifts to the right as increases, it is less obvious how the value of affects the steepness —that is the nonlinearity— of the negative feedback. We use Hill functions to parametrize the regulatory functions Eq. (III.) in a more transparent form, see Appendix. Hill functions are parametrized by a Hill coefficient characterizing the steepness of the curve, and an inhibition threshold that describes the concentration of repressors that halves the production rate. We fit Hill functions to and obtain an effective Hill coefficient and effective inhibition threshold , for each value of and , Fig. 4A,B. Increasing the number of binding sites while keeping fixed can increase the Hill coefficient. For fixed , increasing changes the Hill coefficient in a nonmonotonic way: there is an optimal value of that maximizes the Hill coefficient and therefore nonlinearity. The effective inhibition threshold changes in a simpler form, increasing both with and . In conclusion, multiple binding sites can effectively increase the nonlinearity of the feedback via the regulatory function.

Figure 3: Abrupt inhibition. Full inhibition occurs when more than sites are bound by transcriptional repressors. (A) Transcription rate as a function of the number of bound transcriptional repressors. (B) Normalized regulatory function, Eq. (III.), as a function of repressor concentration , with and (dark blue to dark red).
Figure 4: Multiple binding sites can increase nonlinearity and enhance oscillations. (A) and (B) Effective Hill parameters for the regulatory function , Eq. (III.), with and . (A) Effective Hill coefficient . (B) Effective inhibition threshold . (C) and (D) Oscillations described by Eq. (III.) with , , and . (C) Amplitude of oscillations. (D) Period of oscillations. In C and D, the white region represents the nonoscillatory regime, in which the system settles to a fixed point. Color bar labels indicate values in each panel.

As discussed above, nonlinearity is an essential ingredient in a theory of biochemical oscillations [21]. We therefore ask how multiple binding sites in the promoter affect a biochemical oscillator. We use the regulatory function Eq. (III.) in a generic model for genetic oscillations. We consider a gene that encodes a protein that forms dimers, and these dimers can bind to multiple binding sites at the promoter to inhibit transcription, Fig. 1A. We introduce an explicit delay to account for transcription, translation, splicing, and other processes involved in the assembly of the gene product and its dimerization. We assume that dimerization is a fast reaction, with a separation of timescales from other processes. Therefore, at any time the dimer concentration can be approximated by that of the monomers squared. The dynamics of the product concentration is given by


where is the basal production rate, is the decay rate of products, relates to the number of gene copies, is the total delay for product assembly, and is the product concentration that halves the basal transcription rate. The regulatory function Eq. (III.) is parametrized by and . This genetic oscillator is a reduction of the models proposed by Lewis [22], and other authors [51, 52]. It describes the protein concentration, but does not include the mRNA; the duration of transcription and translation are both included in the delay . Furthermore, it does not describe effects present in theories that include more than one regulator [53, 35, 54], but here it is enough to illustrate the effects of multiple binding sites in a simpler context.

We integrate Eq. (III.) numerically and evaluate the resulting dynamics by calculating the amplitude and period of oscillations. In all numerical simulations, we use the function dde23 from MATLAB [55]. Scanning the values of and , we determine whether the system oscillates in steady state: when the difference in the maxima over the last ten cycles falls below 0.01, the simulation is stopped and we record the output.

The amplitude of oscillations grows with the number of binding sites , Fig. 4C. The range where the system oscillates grows with the number of binding sites . The amplitude is nonmonotonic in : for a fixed number of binding sites , there is an optimal value for that maximizes the amplitude of oscillations, Fig.4C. The period of oscillations grows with and decreases with . These results show how the change in nonlinearity observed in the regulatory functions is reflected in the oscillations as the number of binding sites change.

Iv. Gradual inhibition

There is strong evidence for the products of some cyclic genes of the zebrafish segmentation clock binding their own promoters and acting as transcriptional repressors [30, 38, 36, 47, 35, 26]. However, we do not have detailed knowledge of how these transcriptional repressors affect transcription rates when bound to the promoters of cyclic genes. There is evidence from transcriptional analysis of the Hes1 gene in mouse indicating that inhibition is gradual [33]. While the wildtype Hes1 promoter containing all three N Box elements that are bound by Hes1 proteins showed a 30-fold inhibition of transcription in the presence of Hes1, mutations in one, two, and three of the N Box elements showed impaired inhibition with 14-, 7-, and 2-fold inhibition, respectively [33].

Motivated by these results, we consider here a scenario where additional bound repressors gradually reduce transcription rate until it drops to zero, Fig. 5A. For the sake of simplicity, we assume that transcription rate drops linearly as a function of bound repressors, from a basal rate , to zero for occupied sites


see Fig. 5A. Using this gradual inhibition in Eq. (III.) together with Eq. (II.) and (II.), we obtain regulatory functions


see Fig. 5B. As in the previous case, it is clear that the effective inhibition threshold shifts to the right as increases, but it is not so clear if the steepness of the regulatory function changes and, if so, how. Performing fits to Hill functions, we find that the nonmonotonic behavior of the effective Hill exponent is observed again as increases, Fig. 6A,B. Oscillations are similarly affected by noncooperative binding with gradual inhibition, Fig. 6C,D. These results show that the prediction of an optimal value for the number of bound repressors that fully inhibits the promoter is robust with respect to the details of how multiple bound repressors reduce the transcription rate.

Figure 5: Gradual inhibition. Binding of multiple transcriptional repressors gradually inhibits transcription. (A) Transcription rate as a function of the number of bound transcriptional repressors. Transcription occurs at the basal rate in the absence of bound repressors, and decreases linearly to zero for more than bound repressors. (B) Normalized regulatory function Eq. (IV.) as a function of repressor concentration , with and (dark blue to dark red).
Figure 6: Multiple binding sites can increase nonlinearity and enhance oscillations in the presence of gradual inhibition. (A) and (B) Effective Hill parameters for the regulatory function , Eq. (IV.), with and . (A) Effective Hill coefficient . (B) Effective inhibition threshold . (C) and (D) Oscillations obtained using regulatory functions Eq. (IV.) with , , and . (C) Amplitude of oscillations. (D) Period of oscillations. In C and D the white region represents the nonoscillatory regime, in which the system settles to a fixed point. Color bar labels indicate values in each panel.

V. Discussion

We studied the effects of multiple noncooperative binding sites in the promoter of a genetic oscillator. We evaluated the behavior of a promoter with multiple binding sites when binding of transcriptional repressors is noncooperative, Fig. 2. We considered two different hypotheses for how bound transcriptional repressors affect transcription rates, Figs. 3 and 5. In both cases, we calculated how the number of binding sites and the number of bound repressors required to produce full inhibition affect the nonlinearity of regulatory functions. We showed that there is an optimal value of the number of repressors needed to fully inhibit transcription that maximizes nonlinearity, Figs. 4AB and 6AB. This increased nonlinearity is reflected in the behavior of a genetic oscillator controlled by such regulatory functions, Figs. 4CD and 6CD.

Cooperative binding is a well known means to increase the nonlinearity of a biological dynamical system [24, 42]. Here we show that the nonlinearity of a regulatory function can be increased by multiple binding sites, even if binding is noncooperative. This idea may have application in other biological control systems as well. Using the same formulation as we did here, one can also describe nonlinearity in the transcriptional activation of gene expression, thereby creating effective on-switches in developmental and physiological regulatory networks. A similar effect has been reported in a theoretical study of enzymes with multiple phosphorylation sites [56, 57, 58]. It was found that nonessential phosphorylation sites give rise to an increase in effective Hill coefficients, enhancing ultrasensitivity in signal transduction [58].

Previous work on the segmentation clock addressed the case of three binding sites [43], motivated by experimental observations of the Hes7 mouse promoter [34]. The authors assumed that a single bound dimer inhibits transcription completely, corresponding to the particular case of the theory developed here. They considered both noncooperative and cooperative binding, and showed that cooperativity increases the effective Hill coefficient as expected. In a follow-up [59], the authors addressed the case of Hes1 regulation based on the report of four binding sites in the Hes1 mouse promoter [33]. Again assuming that a single bound dimer inhibits transcription completely, they used the data from the transcriptional analysis of the Hes1 gene to estimate an effective Hill coefficient for the Mouse Hes1 oscillator, obtaining an upper bound of about . More recently, the effect of two binding sites as compared to a single binding site was discussed together with differential decay of the monomers [60].

Our theory predicts how changing the number of binding sites , and the number of bound repressors that produce full inhibition affects a single cell oscillator. Although an experiment that changes the value of may currently be challenging in a cellular system, the number of binding sites is more amenable to experimental manipulation. For example, binding sites could be mutated [33], or deleted from the promoter, or they could be interfered with using genome editing strategies such as TALEN [61] or CRISPR [62] to alter or delete specific binding sites. To assess the effects of these perturbations in experiments may also pose some challenges. Dropping the number of binding sites from to introduces a period change of about 2.5%, while amplitude halves over the same range, Fig. 4C,D. Experiments will require at least such precision to reliably detect changes.

Our results suggest a possible evolutionary mechanism to increase nonlinearity in gene regulatory systems. In this mechanism, point mutations in the promoter that increase the number of binding sites for transcription factors may increase the steepness of regulatory functions. If the resulting steeper regulation performs some function better, such mutations would have a good chance to be conserved by natural selection. In the case of the segmentation clock, the amplitude of oscillations could increase with the number of binding sites, possibly reducing the signal to noise ratio. Furthermore, the range for oscillations would be wider, making the oscillatory regime less sensitive to slow extrinsic fluctuations of parameter values. Remarkably, it may happen that after an increase in , an increase in also raises nonlinearity, see Fig. 4A. This means that weaker repressors would result in better oscillations. This evolutionary mechanism would provide a simple way to gradually increase the nonlinearity of a feedback or other regulatory function.

The theory for the zebrafish regulatory function could be refined using the experimentally measured relative affinities of the binding sites at the her1 and her7 promoters [35]. Apart from the effects reported here, the number of binding sites may have additional roles. For example, it could serve as a buffer for fluctuations in gene expression [63, 64, 65, 66, 67, 23], augmenting the precision of genetic oscillations. This will be the topic of future work.

Acknowledgements - We thank Saúl Ares, Luciana Bruno, Ariel Chernomoretz and Hernan Grecco for helpful comments on an early version of this work, and James Ferrell for calling our attention to Ref. [56]. Thanks to E. Aikau for inspiration. LGM acknowledges support from MINCyT Argentina (PICT 2012 1952). A.C.O. and D.S. were supported by the Medical Research Council UK (MC_UP_1202/3) and the Wellcome Trust (WT098025MA).

Figure 7: Hill functions are characterized by two parameters, the Hill coefficient and the inhibition threshold . (A) Hill functions with and (red), (green) and (blue). (B) Hill functions with and (red), (green) and (blue).

Appendix: Effective Hill functions

Hill functions are often used to describe the nonlinearities present in gene regulatory networks [20]. Hill functions are sigmoidal step functions defined by


where the steepnes of the step is characterized by the exponent , and the inhibition threshold is the concentration of repressor that halves the production rate, here scaled to unity, Fig. 7. Here we include an explicit factor in the exponent to account for the dimerization of the transcriptional repressors. One advantage of Hill functions is that they are very simply parametrized. In the main text, we fit Hill functions to the more complex regulatory functions Eqs. (III.) and (IV.). Some fits of Eq. (III.) in the case are displayed in Fig. 8 for illustration.

Figure 8: Effective Hill functions Eq. (Appendix: Effective Hill functions) can fit regulatory functions of the multiple binding site regulatory function, Eq. (III.) for , with and . (A-C) Examples of fits of Hill functions (solid lines) to regulatory functions (dashed lines) for (A) (green), (B) (light blue) and (C) (purple). (D) Effective Hill coefficient as a function of . (E) Effective inhibition threshold as a function of . Dots in D and E correspond to fits from panels A, B and C.


  • [4] A Goldbeter, Biochemical Oscillations and Cellular Rhythms: The Molecular Bases of Periodic and Chaotic Behaviour, Cambridge University Press, Cambridge (1997).
  • [5] L Glass, M C Mackey, From Clocks to Chaos: The Rhythms of Life, Princeton University Press, Princeton (1988).
  • [6] Y Liu, N F Tsinoremas, C H Johnson, N V Lebedeva, S S G M Ishiura, T Kondo, Circadian orchestration of gene expression in cyanobacteria, Gene. Dev. 9, 1469 (1995).
  • [7] E Nagoshi, C Saini, C Bauer, T Laroche, F Naef, U Schibler, Circadian gene expression in individual fibroblasts: Cell-autonomous and self-sustained oscillators pass time to daughter cells, Cell 119, 693 (2004).
  • [8] I Mihalcescu, W Hsing, S Leibler, Resilient circadian oscillator revealed in individual cyanobacteria, Nature 430, 81 (2004).
  • [9] A Goldbeter, C Gérard, D Gonze, J C Leloup, G Dupont, Systems biology of cellular rhythms, FEBS Lett. 586, 2955 (2012).
  • [10] I Palmeirim, D Henrique, D Ish-Horowicz, O Pourquié, Avian hairy gene expression identifies a molecular clock linked to vertebrate segmentation and somitogenesis, Cell 91, 639 (1997).
  • [11] A Aulehla, W Wiegraebe, V Baubet, M B Wahl, C Deng, M Taketo, M Lewandoski, O Pourquié, A -catenin gradient links the clock and wavefront systems in mouse embryo segmentation, Nat. Cell Biol. 10, 186 (2008).
  • [12] Y Masamizu, T Ohtsuka, Y Takashima, H Nagahara, Y Takenaka, K Yoshikawa, H Okamura, R Kageyama, Real-time imaging of the segmentation clock: Revelation of unstable oscillators in the individual presomitic mesoderm cells, Proc. Natl. Acad. Sci. USA 103, 1313 (2006).
  • [13] A J Krol, D Roellig, M L Dequéante, O Tassy, E Glynn, G Hattem, A Mushegian, A C Oates, O Pourquié, Evolutionary plasticity of segmentation clock networks, Development 138, 2783 (2011).
  • [14] H Shimojo, T Ohtsuka, R Kageyama, Oscillations in notch signaling regulate maintenance of neural progenitors, Neuron 58, 52 (2008).
  • [15] N Geva-Zatorsky, N Rosenfeld, S Itzkovitz, R Milo, A Sigal, E Dekel, T Yarnitzky, Y Liron, P Polak, G Lahav, U Alon, Oscillations and variability in the p53 system, Mol. Syst. Biol. 2, 2006.0033 (2006).
  • [16] M B Elowitz, S Leibler, A synthetic oscillatory network of transcriptional regulators, Nature 403, 335 (2000).
  • [17] J Stricker, S Cookson, M R Bennett, W H Mather, L S Tsimring, J Hasty, A fast, robust and tunable synthetic gene oscillator, Nature 456, 516 (2008).
  • [18] J J Tyson, Computational cell biology, Chap. 9, Pag. 230, Springer, Berlin (2002).
  • [19] B Alberts, A Johnson, J Lewis, M Raff, K Roberts, P Walter, Molecular biology of the cell, 4 Ed., Garland Science, New York (2002).
  • [20] U Alon, An introduction to systems biology: Design principles of biological circuits, Chapman & Hall/CRC Press, Boca Raton, Florida (2006).
  • [21] B Novák, J J Tyson, Design principles of biochemical oscillators, Nat. Rev. Mol. Cell Biol. 9, 981 (2008).
  • [22] J Lewis, Autoinhibition with transcriptional delay: A simple mechanism for the zebrafish somitogenesis oscillator, Curr. Biol. 13, 1398 (2003).
  • [23] L G Morelli, F Jülicher, Precision of genetic oscillators and clocks, Phys. Rev. Lett. 98, 228101 (2007).
  • [24] J E Ferrell, Q&A: Cooperativity, J. Biol. 8, 53 (2009).
  • [25] O Pourquié, Vertebrate segmentation: From cyclic gene networks to scoliosis, Cell 145, 650 (2011).
  • [26] A C Oates, L G Morelli, S Ares, Patterning embryos with oscillations: Structure, function and dynamics of the vertebrate segmentation clock, Development 139, 625 (2012).
  • [27] Y Saga, The synchrony and cyclicity of developmental events, Cold Spring Harb. Perspect. Biol. 4, a008201 (2012).
  • [28] D Roellig, L G Morelli, S Ares, F Jülicher, A C Oates, Snapshot: The segmentation clock, Cell 145, 800 (2011).
  • [29] Y Saga, The mechanism of somite formation in mice, Curr. Opin. Genet. Dev. 22, 331 (2012).
  • [30] A C Oates, R K Ho, Hairy/E(spl)-related (Her) genes are central components of the segmentation oscillator and display redundancy with the Delta/Notch signaling pathway in the formation of anterior segmental boundaries in the zebrafish, Development 129, 2929 (2002).
  • [31] H Hirata, S Yoshiura, T Ohtsuka, Y Bessho, T Harada, K Yoshikawa, R Kageyama, Oscillatory expression of the bHLH factor Hes1 regulated by a negative feedback loop, Science 298, 840 (2002).
  • [32] S A Holley, D Jülich, G J Rauch, R Geisler, C Nüsslein-Volhard, her1 and the notch pathway function within the oscillator mechanism that regulates zebrafish somitogenesis, Development 129, 1175 (2002).
  • [33] K Takebayashi, Y Sasai, Y Sakai, T Watanabe, S Nakanishi, R Kageyama, Structure, chromosomal locus, and promoter analysis of the gene encoding the mouse helix-loop-helix factor hes-1. negative autoregulation through the multiple n box elements, J. Biol. Chem. 269, 5150 (1994).
  • [34] Y Bessho, G Miyoshi, R Sakata, R Kageyama, Hes7: A bhlh-type repressor gene regulated by notch and expressed in the presomitic mesoderm, Genes Cells 6, 175 (2001).
  • [35] C Schröter, S Ares, L G Morelli, A Isakova, K Hens, D Soroldoni, M Gajewski, F Jülicher, S J Maerkl, B Deplancke, A C Oates, Topology and dynamics of the zebrafish segmentation clock core circuit, PLoS Biol. 10, e1001364 (2012).
  • [36] A Trofka, J Schwendinger-Schreck, T Brend, W Pontius, T Emonet, S A Holley, The Her7 node modulates the network topology of the zebrafish segmentation clock via sequestration of the Hes6 hub, Development 139, 940 (2012).
  • [37] A Hanisch, M V Holder, S Choorapoikayil, M Gajewski, E M Özbudak, J Lewis, The elongation rate of RNA polymerase ii in zebrafish and its significance in the somite segmentation clock, Development 140, 444 (2013).
  • [38] F Giudicelli, E M Özbudak, G J Wright, J Lewis, Setting the tempo in development: An investigation of the zebrafish somite clock mechanism, PLoS Biol. 5, 1309 (2007).
  • [39] E M Özbudak, J Lewis, Notch signalling synchronizes the zebrafish segmentation clock but is not needed to create somite boundaries, PLoS Genet. 4(2), e15 (2008).
  • [40] Y Harima, Y Takashima, Y Ueda, T Ohtsuka, R Kageyama, Accelerating the tempo of the segmentation clock by reducing the number of introns in the hes7 gene, Cell Rep. 3, 1 (2013).
  • [41] J Keener, J Sneyd, Mathematical physiology I: Cellular physiology, 2 Ed. Springer, Berlin (2008).
  • [42] H Qian, Cooperativity in cellular biochemical processes: Noise-enhanced sensitivity, fluctuating enzyme, bistability with nonlinear feedback, and other mechanisms for sigmoidal responses, Ann. Rev. Biophys. 41, 179 (2012).
  • [43] S Zeiser, H V Liebscher, H Tiedemann, I Rubio-Aliaga, G K H Przemeck, M H de Angelis, G Winkler, Number of active transcription factor binding sites is essential for the hes7 oscillator, Theor. Biol. Med. Model. 3, 11 (2006).
  • [44] J Gunawardena, Multisite protein phosphorylation makes a good threshold but can be a poor switch, P. Natl. Acad. Sci. USA 102, 14617 (2005).
  • [45] V Gotea, A Visel, J M Westlund, M A Nobrega, L A Pennacchio, I Ovcharenko, Homotypic clusters of transcription factor binding sites are a key component of human promoters and enhancers, Genome Res. 20, 565 (2010).
  • [46] D S Burz, R Rivera-Pomar, H Jäckle, S D Hanes, Cooperative dna-binding by bicoid provides a mechanism for threshold-dependent gene activation in the drosophila embryo, EMBO J. 17, 5998 (1998).
  • [47] T Brend, S A Holley, Expression of the oscillating gene her1 is directly regulated by hairy/enhancer of split, t-box, and suppressor of hairless proteins in the zebrafish segmentation clock, Dev. Dynam. 238, 2745 (2009).
  • [48] D Soroldoni, personal communication (2014).
  • [49] L Bintu, N E Buchler, H G Garcia, U Gerland, T Hwa, J Kondev, R Phillips, Transcriptional regulation by the numbers: Models, Curr. Opin. Genet. Dev. 15, 116 (2005).
  • [50] H G Garcia, A Sanchez, T Kuhlman, J Kondev, R Phillips, Transcription by the numbers redux: experiments and calculations that surprise, Trends Cell Biol. 20, 723 (2010).
  • [51] N A M Monk, Oscillatory expression of Hes1, p53, and NF-B driven by transcriptional time delays, Curr. Biol. 13, 1409 (2003).
  • [52] M H Jensen, K Sneppen, G Tiana, Sustained oscillations and time delays in gene expression of protein Hes1, FEBS Lett. 541, 176 (2003).
  • [53] O Cinquin, Repressor dimerization in the zebrafish somitogenesis clock, PLoS Comp. Biol. 3, e32 (2007).
  • [54] A Ay, S Knierer, A Sperlea, J Holland, E M Özbudak, Short-lived her proteins drive robust synchronized oscillations in the zebrafish segmentation clock, Development 140, 3244 (2013).
  • [55] L F Shampine, S Thompson, Solving ddes in Matlab, Appl. Numer. Math. 37, 441 (2001).
  • [56] L Wang, Q Nie, G Enciso, Nonessential sites improve phosphorylation switch, Biophys. J. 99, L41 (2010).
  • [57] S Ryerson, G Enciso, Ultrasensitivity in independent multisite systems, J. Math. Biol. 69, 977 (2014).
  • [58] G Enciso, Nonautonomous and random dynamical systems in life sciences Lecture Notes in Mathematics (Mathematical Biosciences Subseries) No 2102, Springer Verlag, Berlin (2013).
  • [59] S Zeiser, J Müller, V Liebscher, Modeling the hes1 oscillator, J. Comput. Biol. 14, 984 (2007).
  • [60] M Campanelli, T Gedeon, Somitogenesis clock-wave initiation requires differential decay and multiple binding sites for clock protein, PLoS Comp. Biol. 6, e1000728 (2010).
  • [61] V M Bedell, Y Wang, J M Campbell, T L Poshusta, C G Starker, R G K II, W Tan, S G Penheiter, A C Ma, A Y H Leung, S C Fahrenkrug, D F Carlson, D F Voytas, K J Clark, J J Essner, S C Ekker, In vivo genome editing using a high-efficiency talen system, Nature 491, 114 (2012).
  • [62] E Pennisi, The crispr craze, Science 341, 833 (2013).
  • [63] N Barkai, S Leibler, Biological rhythms: Circadian clocks limited by noise, Nature 403, 267 (2000).
  • [64] M B Elowitz, A J Levine, E D Siggia, P S Swain, Stochastic gene expression in a single cell, Science 297, 1183 (2002).
  • [65] J Raser, E O’Shea, Noise in gene expression: Origins, consequences, and control, Science 309, 2010 (2005).
  • [66] B Munsky, G Neuert, A V Oudenaarden, Using gene expression noise to understand gene regulation, Science 336, 183 (2012).
  • [67] L S Tsimring, Noise in biology, Rep. Prog. Phys. 77, 026601 (2014).
Comments 0
Request Comment
You are adding the first comment!
How to quickly get a good reply:
  • Give credit where it’s due by listing out the positive aspects of a paper before getting into which changes should be made.
  • Be specific in your critique, and provide supporting evidence with appropriate references to substantiate general statements.
  • Your comment should inspire ideas to flow and help the author improves the paper.

The better we are at sharing our knowledge with each other, the faster we move forward.
The feedback must be of minimum 40 characters and the title a minimum of 5 characters
Add comment
Loading ...
This is a comment super asjknd jkasnjk adsnkj
The feedback must be of minumum 40 characters
The feedback must be of minumum 40 characters

You are asking your first question!
How to quickly get a good answer:
  • Keep your question short and to the point
  • Check for grammar or spelling errors.
  • Phrase it like a question
Test description