A Spatio-Temporal Model of Notch Signalling in the Zebrafish Segmentation Clock: Conditions for Synchronised Oscillatory Dynamics

In the vertebrate embryo, tissue blocks called somites are laid down in head-to-tail succession, a process known as somitogenesis. Research into somitogenesis has been both experimental and mathematical. For zebrafish, there is experimental evidence for oscillatory gene expression in cells in the presomitic mesoderm (PSM) as well as evidence that Notch signalling synchronises the oscillations in neighbouring PSM cells. A biological mechanism has previously been proposed to explain these phenomena. Here we have converted this mechanism into a mathematical model of partial differential equations in which the nuclear and cytoplasmic diffusion of protein and mRNA molecules is explictly considered. By performing simulations, we have found ranges of values for the model parameters (such as diffusion and degradation rates) that yield oscillatory dynamics within PSM cells and that enable Notch signalling to synchronise the oscillations in two touching cells. Our model contains a Hill coefficient that measures the co-operativity between two proteins (Her1, Her7) and three genes (her1, her7, deltaC) which they inhibit. This coefficient appears to be bounded below by the requirement for oscillations in individual cells and bounded above by the requirement for synchronisation. Consistent with experimental data and a previous spatially non-explicit mathematical model, we have found that signalling can increase the average level of Her1 protein. Biological pattern formation would be impossible without a certain robustness to variety in cell shape and size; our results possess such robustness. Our spatially-explicit modelling approach, together with new imaging technologies that can measure intracellular protein diffusion rates, is likely to yield significant new insight into somitogenesis and other biological processes.

[1]  H. Takeda,et al.  Emergence of traveling waves in the zebrafish segmentation clock , 2010, Development.

[2]  D. A. Baxter,et al.  Effects of macromolecular transport and stochastic fluctuations on dynamics of genetic regulatory systems. , 1999, The American journal of physiology.

[3]  Daniel J. Muller,et al.  Movement Directionality in Collective Migration of Germ Layer Progenitors , 2010, Current Biology.

[4]  O. Pourquié,et al.  Vertebrate somitogenesis. , 2001, Annual review of cell and developmental biology.

[5]  Marek Kimmel,et al.  Mathematical model of NF- κB regulatory module , 2004 .

[6]  H. Meinhardt Models of biological pattern formation , 1982 .

[7]  P K Maini,et al.  A clock and wavefront mechanism for somite formation. , 2006, Developmental biology.

[8]  N. Monk Oscillatory Expression of Hes1, p53, and NF-κB Driven by Transcriptional Time Delays , 2003, Current Biology.

[9]  J. Kim Dale,et al.  Notch Is a Critical Component of the Mouse Somitogenesis Oscillator and Is Essential for the Formation of the Somites , 2009, PLoS genetics.

[10]  Ingmar H Riedel-Kruse,et al.  Synchrony Dynamics During Initiation, Failure, and Rescue of the Segmentation Clock , 2007, Science.

[11]  Wolfgang Wurst,et al.  Cell-based simulation of dynamic expression patterns in the presomitic mesoderm. , 2007, Journal of theoretical biology.

[12]  Anja Hanisch,et al.  Notch signaling, the segmentation clock, and the patterning of vertebrate somites , 2009, Journal of biology.

[13]  Andrew C Oates,et al.  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 , 2002 .

[14]  Olivier Pourquié,et al.  Segmental patterning of the vertebrate embryonic axis , 2008, Nature Reviews Genetics.

[15]  Frank Jülicher,et al.  Intercellular Coupling Regulates the Period of the Segmentation Clock , 2010, Current Biology.

[16]  N. Hirokawa The mechanisms of fast and slow transport in neurons: identification and characterization of the new kinesin superfamily motors , 1997, Current Opinion in Neurobiology.

[17]  A. Chitnis Why is delta endocytosis required for effective activation of notch? , 2006, Developmental dynamics : an official publication of the American Association of Anatomists.

[18]  T. Hays,et al.  Drosophila cytoplasmic dynein, a microtubule motor that is asymmetrically localized in the oocyte , 1994, The Journal of cell biology.

[19]  Olivier Pourquié,et al.  Control of segment number in vertebrate embryos , 2008, Nature.

[20]  Olivier Cinquin,et al.  Repressor Dimerization in the Zebrafish Somitogenesis Clock , 2007, PLoS Comput. Biol..

[21]  A. Parks,et al.  Ligand endocytosis drives receptor dissociation and activation in the Notch pathway. , 2000, Development.

[22]  Kevin J Painter,et al.  Adding Adhesion to a Chemical Signaling Model for Somite Formation , 2009, Bulletin of mathematical biology.

[23]  Leann Tilley,et al.  Fluorescence photobleaching analysis for the study of cellular dynamics , 2002, European Biophysics Journal.

[24]  Julian Lewis,et al.  Setting the Tempo in Development: An Investigation of the Zebrafish Somite Clock Mechanism , 2007, PLoS biology.

[25]  Richard H. Rand,et al.  Hopf bifurcation in a DDE model of gene expression , 2008 .

[26]  B C Goodwin,et al.  A cellular oscillator model for periodic pattern formation. , 2001, Journal of theoretical biology.

[27]  Hiroshi Momiji,et al.  Dissecting the dynamics of the Hes1 genetic oscillator. , 2008, Journal of theoretical biology.

[28]  L. Segel SIMPLIFICATION AND SCALING , 1972 .

[29]  Frank Jülicher,et al.  Delayed coupling theory of vertebrate segmentation , 2008, HFSP journal.

[30]  Julian Lewis,et al.  Notch Signalling Synchronizes the Zebrafish Segmentation Clock but Is Not Needed To Create Somite Boundaries , 2007, PLoS genetics.

[31]  Robert Geisler,et al.  her1 and the notch pathway function within the oscillator mechanism that regulates zebrafish somitogenesis. , 2002, Development.

[32]  M. Barresi,et al.  DEVELOPMENTAL DYNAMICS 219:287–303 (2000) REVIEWS A PEER REVIEWED FORUM Somite Development in Zebrafish , 2022 .

[33]  Paul Houston,et al.  Models for pattern formation in somitogenesis: a marriage of cellular and molecular biology. , 2002, Comptes rendus biologies.

[34]  Daniel St Hughes,et al.  Extensive molecular differences between anterior- and posterior-half-sclerotomes underlie somite polarity and spinal nerve segmentation , 2009, BMC Developmental Biology.

[35]  Yoshihiro Morishita,et al.  Traveling wave formation in vertebrate segmentation. , 2009, Journal of theoretical biology.

[36]  J. Cooke,et al.  A gene that resuscitates a theory--somitogenesis and a molecular oscillator. , 1998, Trends in genetics : TIG.

[37]  J. Campos-Ortega,et al.  her1, a zebrafish pair-rule like gene, acts downstream of notch signalling to control somite development. , 1999, Development.

[38]  Bruce P. Graham,et al.  Continuum model for tubulin-driven neurite elongation , 2004, Neurocomputing.

[39]  Michael C Mackey,et al.  The segmentation clock in mice: interaction between the Wnt and Notch signalling pathways. , 2007, Journal of theoretical biology.

[40]  Yang Cao,et al.  Stochastic simulation of enzyme-catalyzed reactions with disparate timescales. , 2008, Biophysical journal.

[41]  A. Kuroiwa,et al.  Fgf/MAPK signalling is a crucial positional cue in somite boundary formation. , 2001, Development.

[42]  Heino Prinz,et al.  Hill coefficients, dose–response curves and allosteric mechanisms , 2010, Journal of chemical biology.

[43]  Albert Goldbeter,et al.  Modeling the segmentation clock as a network of coupled oscillations in the Notch, Wnt and FGF signaling pathways. , 2008, Journal of theoretical biology.

[44]  Julian Lewis Autoinhibition with Transcriptional Delay A Simple Mechanism for the Zebrafish Somitogenesis Oscillator , 2003, Current Biology.

[45]  Joseph M. Mahaffy,et al.  Genetic control models with diffusion and delays , 1988 .

[46]  Johannes Müller,et al.  Modeling the Hes1 Oscillator , 2007, J. Comput. Biol..

[47]  M. Campanelli Multicellular mathematical models of somitogenesis , 2009 .

[48]  James N. Weiss The Hill equation revisited: uses and misuses , 1997, FASEB journal : official publication of the Federation of American Societies for Experimental Biology.

[49]  Albert Cardona,et al.  Dynamics of zebrafish somitogenesis , 2008, Developmental dynamics : an official publication of the American Association of Anatomists.

[50]  Stefan Zeiser,et al.  Number of active transcription factor binding sites is essential for the Hes7 oscillator , 2006, Theoretical Biology and Medical Modelling.

[51]  C. Pao,et al.  Models of genetic control by repression with time delays and spatial effects , 1984, Journal of mathematical biology.

[52]  David Ish-Horowicz,et al.  Notch signalling and the synchronization of the somite segmentation clock , 2000, Nature.

[53]  S. Busenberg,et al.  Interaction of spatial diffusion and delays in models of genetic control by repression , 1985, Journal of Mathematical Biology.

[54]  Olivier Cinquin,et al.  Is the somitogenesis clock really cell-autonomous? A coupled-oscillator model of segmentation. , 2003, Journal of theoretical biology.

[55]  Hans Meinhardt,et al.  Models of Segmentation , 1986 .

[56]  H. Hirata,et al.  Oscillatory Expression of the bHLH Factor Hes1 Regulated by a Negative Feedback Loop , 2002, Science.

[57]  D. S. Broomhead,et al.  Pulsatile Stimulation Determines Timing and Specificity of NF-κB-Dependent Transcription , 2009, Science.

[58]  A. Verkman,et al.  Translational Diffusion of Macromolecule-sized Solutes in Cytoplasm and Nucleus , 1997, The Journal of cell biology.

[59]  J. Campos-Ortega,et al.  SUMMARY her 1 , a zebrafish pair-rule like gene , acts downstream of notch signalling to control somite development , 2022 .

[60]  E. C. Zeeman,et al.  A clock and wavefront model for control of the number of repeated structures during animal morphogenesis. , 1976, Journal of theoretical biology.

[61]  Ruth E Baker,et al.  Mathematical models for somite formation. , 2008, Current topics in developmental biology.

[62]  O. Pourquié,et al.  Avian hairy Gene Expression Identifies a Molecular Clock Linked to Vertebrate Segmentation and Somitogenesis , 1997, Cell.

[63]  D. A. Ede,et al.  Somites in Developing Embryos , 1986, NATO ASI Series.

[64]  P. Swain,et al.  Intrinsic and extrinsic contributions to stochasticity in gene expression , 2002, Proceedings of the National Academy of Sciences of the United States of America.

[65]  John J Tyson,et al.  A model of yeast cell-cycle regulation based on multisite phosphorylation , 2010, Molecular systems biology.

[66]  A. Oates,et al.  Segment Number and Axial Identity in a Segmentation Clock Period Mutant , 2010, Current Biology.

[67]  Takeharu Nagai,et al.  Direct measurement of protein dynamics inside cells using a rationally designed photoconvertible protein , 2008, Nature Methods.

[68]  R. Natalini,et al.  A spatial model of cellular molecular trafficking including active transport along microtubules. , 2010, Journal of theoretical biology.

[69]  R. Weinberg,et al.  The Biology of Cancer , 2006 .

[70]  R. Lasser,et al.  Oscillations of Hes7 caused by negative autoregulation and ubiquitination , 2008, Comput. Biol. Chem..

[71]  John J Tyson,et al.  Exploring the roles of noise in the eukaryotic cell cycle , 2009, Proceedings of the National Academy of Sciences.

[72]  Shigeru Kondo,et al.  Noise-resistant and synchronized oscillation of the segmentation clock , 2006, Nature.

[73]  M. Maroto,et al.  The chick somitogenesis oscillator is arrested before all paraxial mesoderm is segmented into somites , 2010, BMC Developmental Biology.

[74]  V. Zhdanov Interplay of master regulatory proteins and mRNA in gene expression: 3D Monte Carlo simulations , 2008 .

[75]  Mads Kærn,et al.  Segmentation and somitogenesis derived from phase dynamics in growing oscillatory media. , 2000, Journal of theoretical biology.

[76]  Thierry Soussi,et al.  Shaping genetic alterations in human cancer: the p53 mutation paradigm. , 2007, Cancer cell.

[77]  Michael C. Mackey,et al.  A Proposed Mechanism for the Interaction of the Segmentation Clock and the Determination Front in Somitogenesis , 2008, PloS one.

[78]  Scott A Holley,et al.  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 , 2009, Developmental dynamics : an official publication of the American Association of Anatomists.