A mathematical model of the biochemical network underlying left-right asymmetry establishment in mammals

The expression of the TGF-β protein Nodal on the left side of vertebrate embryos is a determining event in the development of internal-organ asymmetry. We present a mathematical model for the control of the expression of Nodal and its antagonist Lefty consisting entirely of realistic elementary reactions. We analyze the model in the absence of Lefty and find a wide range of parameters over which bistability (two stable steady states) is observed, with one stable steady state a low-Nodal state corresponding to the right-hand developmental fate, and the other a high-Nodal state corresponding to the left. We find that bistability requires a transcription factor containing two molecules of phosphorylated Smad2. A numerical survey of the full model, including Lefty, shows the effects of Lefty on the potential for bistability, and on the conditions that lead to the system reaching one or the other steady state.

[1]  B. Kendall.  Nonlinear Dynamics and Chaos , 2001 .

[2]  R. Thomas,et al.  Multistationarity, the basis of cell differentiation and memory. I. Structural conditions of multistationarity and other nontrivial behavior. , 2001, Chaos.

[3]  L Wolpert,et al.  Thresholds in development. , 1977, Journal of theoretical biology.

[4]  H. Moses,et al.  A tale of two proteins: Differential roles and regulation of Smad2 and Smad3 in TGF‐β signaling , 2007, Journal of cellular biochemistry.

[5]  S. Noselli,et al.  Strategies to establish left/right asymmetry in vertebrates and invertebrates. , 2007, Current opinion in genetics & development.

[6]  B. B. Edelstein.  Biochemical model with multiple steady states and hysteresis. , 1970, Journal of theoretical biology.

[7]  W. E. Briggs.  Zeros and factors of polynomials with positive coefficients and protein-ligand binding , 1985 .

[8]  Toshihiko Ogura,et al.  Inhibition of Nodal signalling by Lefty mediated through interaction with common receptors and efficient diffusion , 2002, Genes to cells : devoted to molecular & cellular mechanisms.

[9]  Paul A. Bates,et al.  Mathematical modeling identifies Smad nucleocytoplasmic shuttling as a dynamic signal-interpreting system , 2008, Proceedings of the National Academy of Sciences.

[10]  C. Hill.  Transcriptional Control by the SMADs. , 2016, Cold Spring Harbor perspectives in biology.

[11]  Drew N. Robson,et al.  Supplementary Materials for Differential Diffusivity of Nodal and Lefty Underlies a Reaction-Diffusion Patterning System , 2012 .

[12]  Michael M Shen,et al.  Two Modes by which Lefty Proteins Inhibit Nodal Signaling , 2004, Current Biology.

[13]  Alexandra Jilkine,et al.  Wave-pinning and cell polarity from a bistable reaction-diffusion system. , 2008, Biophysical journal.

[14]  Alexander F Schier,et al.  Lefty Blocks a Subset of TGFβ Signals by Antagonizing EGF-CFC Coreceptors , 2004, PLoS biology.

[15]  D. Constam.  Running the gauntlet: an overview of the modalities of travel employed by the putative morphogen Nodal. , 2009, Current opinion in genetics & development.

[16]  S. Ross,et al.  How the Smads regulate transcription. , 2008, The international journal of biochemistry & cell biology.

[17]  R. Serra,et al.  Unique and redundant roles of Smad3 in TGF‐β–mediated regulation of long bone development in organ culture , 2004, Developmental dynamics : an official publication of the American Association of Anatomists.

[18]  H. Yost,et al.  Nodal Signaling: CrypticLefty Mechanism of Antagonism Decoded , 2004, Current Biology.

[19]  A. Schier.  Nodal signaling in vertebrate development. , 2003, Annual review of cell and developmental biology.

[20]  Bruce L. Clarke.  Graph theoretic approach to the stability analysis of steady state chemical reaction networks , 1974 .

[21]  A. M. Turing,et al.  The chemical basis of morphogenesis , 1952, Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences.

[22]  H. Hamada,et al.  TGFβ signaling in establishing left-right asymmetry. , 2014, Seminars in cell & developmental biology.

[23]  A. Ciechanover,et al.  Ubiquitin‐mediated proteolysis: biological regulation via destruction , 2000, BioEssays : news and reviews in molecular, cellular and developmental biology.

[24]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[25]  A. Goldbeter,et al.  Biochemical Oscillations And Cellular Rhythms: Contents , 1996 .

[26]  BARRY B. EDELSTEN,et al.  Biochemical Model with Multiple Steady States and Hysteresis , 2003 .

[27]  Y. Saijoh,et al.  Left–right asymmetric expression of the TGFβ-family member lefty in mouse embryos , 1996, Nature.

[28]  C. Hill,et al.  Stoichiometry of Active Smad-Transcription Factor Complexes on DNA* , 2002, The Journal of Biological Chemistry.

[29]  D. Reinberg,et al.  Inhibition of Excess Nodal Signaling During Mouse Gastrulation by the Transcriptional Corepressor DRAP1 , 2002, Science.

[30]  Avalanche of bifurcations and hysteresis in a model of cellular differentiation. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[31]  Michael C. Mackey,et al.  Simple Mathematical Models of Gene Regulatory Dynamics , 2016 .

[32]  James P. Quirk,et al.  Qualitative Economics and the Stability of Equilibrium , 1965 .

[33]  Maya Mincheva,et al.  Graph-theoretic methods for the analysis of chemical and biochemical networks. I. Multistability and oscillations in ordinary differential equation models , 2007, Journal of mathematical biology.

[34]  A. Schier,et al.  Nodal Stability Determines Signaling Range , 2005, Current Biology.

[35]  C. Wernstedt,et al.  Phosphorylation of Ser465 and Ser467 in the C Terminus of Smad2 Mediates Interaction with Smad4 and Is Required for Transforming Growth Factor-β Signaling* , 1997, The Journal of Biological Chemistry.

[36]  A. Schier.  Nodal morphogens. , 2009, Cold Spring Harbor perspectives in biology.

[37]  H. Meinhardt,et al.  Pattern formation by local self-activation and lateral inhibition. , 2000, BioEssays : news and reviews in molecular, cellular and developmental biology.

[38]  A. Mochizuki,et al.  Fluid flow and interlinked feedback loops establish left–right asymmetric decay of Cerl2 mRNA , 2012, Nature Communications.

[39]  J. King,et al.  Wave pinning and spatial patterning in a mathematical model of Antivin/Lefty–Nodal signalling , 2013, Journal of mathematical biology.

[40]  Wave pinning in strips , 2006, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[41]  H. Hamada,et al.  The left-right axis in the mouse: from origin to morphology , 2006, Development.

[42]  H. Hamada,et al.  Roles of nodal‐lefty regulatory loops in embryonic patterning of vertebrates , 2001, Genes to cells : devoted to molecular & cellular mechanisms.

[43]  James D. Murray.  Mathematical Biology: I. An Introduction , 2007 .

[44]  G. L. Ermakov,et al.  Simplest Kinetic Schemes for Biochemical Oscillators , 2004, Biochemistry (Moscow).

[45]  Y. Saijoh,et al.  lefty-1 Is Required for Left-Right Determination as a Regulator of lefty-2 and nodal , 1998, Cell.

[46]  W. Talbot,et al.  Mouse Lefty2 and zebrafish antivin are feedback inhibitors of nodal signaling during vertebrate gastrulation. , 1999, Molecular cell.

[47]  Bard Ermentrout,et al.  Simulating, analyzing, and animating dynamical systems - a guide to XPPAUT for researchers and students , 2002, Software, environments, tools.

[48]  James A. Gagnon,et al.  Nodal patterning without Lefty inhibitory feedback is functional but fragile , 2017, eLife.

[49]  J. Massagué,et al.  Smad transcription factors. , 2005, Genes & development.

[50]  Harm Hinrich Rotermund,et al.  Reaction diffusion patterns in the catalytic CO‐oxidation on Pt(110): Front propagation and spiral waves , 1993 .

[51]  Robert Rosen,et al.  Structural stability and morphogenesis , 1977 .

[52]  Y. Saijoh,et al.  Left-right asymmetry in the level of active Nodal protein produced in the node is translated into left-right asymmetry in the lateral plate of mouse embryos. , 2011, Developmental biology.

[53]  S. Marcellini.  When Brachyury meets Smad1: the evolution of bilateral symmetry during gastrulation , 2006, BioEssays : news and reviews in molecular, cellular and developmental biology.

[54]  Maya Mincheva,et al.  A graph-theoretic method for detecting potential Turing bifurcations. , 2006, The Journal of chemical physics.

[55]  David Terman,et al.  Propagation Phenomena in a Bistable Reaction-Diffusion System , 1982 .

[56]  J. Creemers,et al.  Lefty Proteins Exhibit Unique Processing and Activate the MAPK Pathway* , 2001, The Journal of Biological Chemistry.

[57]  Hiroshi Hamada,et al.  Generation of robust left-right asymmetry in the mouse embryo requires a self-enhancement and lateral-inhibition system. , 2006, Developmental cell.

[58]  James P. Keener,et al.  Propagation and its failure in coupled systems of discrete excitable cells , 1987 .

[59]  Qiang Wang,et al.  Molecular regulation of Nodal signaling during mesendoderm formation , 2018, Acta biochimica et biophysica Sinica.

[60]  A. Brivanlou,et al.  The orphan receptor ALK7 and the Activin receptor ALK4 mediate signaling by Nodal proteins during vertebrate development. , 2001, Genes & development.

[61]  Laura N. Vandenberg,et al.  A unified model for left-right asymmetry? Comparison and synthesis of molecular models of embryonic laterality. , 2013, Developmental biology.

[62]  M. Shen.  Nodal signaling: developmental roles and regulation , 2007, Development.

[63]  From Clocks to Chaos: The Rhythms of Life , 1988 .

[64]  U. Alon,et al.  Negative autoregulation speeds the response times of transcription networks. , 2002, Journal of molecular biology.

[65]  H. Yost,et al.  Molecular mechanisms of vertebrate left-right development. , 1998, Trends in genetics : TIG.

[66]  C. Tabin,et al.  Mechanisms of Left–Right Determination in Vertebrates , 2000, Cell.

[67]  Hans V Westerhoff,et al.  What makes biochemical networks tick? , 2004, European journal of biochemistry.

[68]  M. Roussel,et al.  A new class of biochemical oscillator models based on competitive binding. , 1997, European journal of biochemistry.

[69]  M. Feinberg.  Chemical reaction network structure and the stability of complex isothermal reactors—II. Multiple steady states for networks of deficiency one , 1988 .

[70]  S. Tabibzadeh,et al.  Lefty Inhibits Receptor-regulated Smad Phosphorylation Induced by the Activated Transforming Growth Factor-β Receptor* , 2001, The Journal of Biological Chemistry.

[71]  Y. Saijoh,et al.  Role of asymmetric signals in left-right patterning in the mouse. , 2001, American journal of medical genetics.

[72]  W. Zhou,et al.  Single-Molecule Imaging Reveals the Activation Dynamics of Intracellular Protein Smad3 on Cell Membrane , 2016, Scientific Reports.

[73]  Roeland M. H. Merks,et al.  Nodal signaling range is regulated by proprotein convertase-mediated maturation. , 2015, Developmental cell.

[74]  Markus Bär,et al.  Spiral waves in a surface reaction: Model calculations , 1994 .

[75]  D. Besser.  Expression of Nodal, Lefty-A, and Lefty-B in Undifferentiated Human Embryonic Stem Cells Requires Activation of Smad2/3* , 2004, Journal of Biological Chemistry.

[76]  D. Constam,et al.  Cripto recruits Furin and PACE4 and controls Nodal trafficking during proteolytic maturation , 2008, The EMBO journal.

[77]  H. Hamada,et al.  Left-right patterning: conserved and divergent mechanisms , 2012, Development.

[78]  Matthew Hartley,et al.  GraTeLPy: graph-theoretic linear stability analysis , 2014, BMC Systems Biology.