Skip to main content
SpringerLink
Log in

Discrete-State Stochastic Modeling of Morphogen Gradient Formation

  • Protocol
  • First Online:
Morphogen Gradients

Part of the book series: Methods in Molecular Biology ((MIMB,volume 1863))

  • 910 Accesses

Abstract

In biological development, positional information required for pattern formation is carried by the gradients of special signaling molecules, which are called morphogens. It is well known that the establishment of the morphogen gradients is a result of complex physical-chemical processes that involve diffusion, degradation of locally produced signaling molecules, and other biochemical reactions. Here we describe a recently developed discrete-state stochastic theoretical method to explain the formation of morphogen gradients in complex cellular environment.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Protocol
USD 49.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Alaynick WA, Jessell TM, Pfaff SL (2011) SnapShot: spinal cord development. Cell 146:178–178.e1

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  2. Berezhkovskii AM (2011) Renewal theory for single-molecule systems with multiple reaction channels. J Chem Phys 134:074114

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  3. Berezhkovskii AM, Shvartsman SY (2011) Physical interpretation of mean local accumulation time of morphogen gradient formation. J Chem Phys 135:154115

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  4. Berezhkovskii AM, Shvartsman SY (2013) Kinetics of receptor occupancy during morphogen gradient formation. J Chem Phys 138:244105

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  5. Berezhkovskii AM, Sample C, Shvartsman SY (2010) How long does it take to establish a morphogen gradient? Biophys J 99:L59–L61

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  6. Berezhkovskii AM, Sample C, Shvartsman SY (2011) Formation of morphogen gradients: local accumulation time. Phys Rev E 83:051906

    Article  CAS  Google Scholar 

  7. Bergmann S, Sandler O, Sberro H, Shnider S, Schejter E, Shilo B-Z, Barkai N (2007) Pre-steady-state decoding of the bicoid morphogen gradient. PLoS Biol 5:e46

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  8. Bischoff M, Gradilla AC, Seijo I, Andrés G, Rodríguez-Navas C, González-Méndez L, Guerrero I (2013) Cytonemes are required for the establishment of a normal Hedgehog morphogen gradient in Drosophila epithelia. Nature Cell Biol 15:1269–1281

    Article  CAS  PubMed  Google Scholar 

  9. Bozorgui B, Teimouri H, Kolomeisky AB (2015) Theoretical analysis of degradation mechanisms in the formation of morphogen gradients. J Chem Phys 143:025102

    Article  CAS  PubMed  Google Scholar 

  10. Bressloff PC, Hyunjoong K (2018) Bidirectional transport model of morphogen gradient formation via cytonemes. Phys Biol 15:026010

    Article  PubMed  Google Scholar 

  11. Briscoe J (2009) Making a grade: Sonic Hedgehog signalling and the control of neural cell fate. EMBO J 28:457–465

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  12. Briscoe J, Small S (2015) Morphogen rules: design principles of gradient-mediated embryo patterning. Development 142:3996–4009

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  13. Castle BT, Howard SA, Odde DJ (2011) Assessment of transport mechanisms underlying the bicoid morphogen gradient. Cell Mol Bioeng 4:116–121

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  14. Chen Y, Struhl G (1996) Dual roles for patched in sequestering and transducing Hedgehog. Cell 87:553–563

    Article  CAS  PubMed  Google Scholar 

  15. Cheung D, Miles C, Kreitman M, Ma J (2014) Adaptation of the length scale and amplitude of the Bicoid gradient profile to achieve robust patterning in abnormally large Drosophila melanogaster embryos. Development 141:124–135

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  16. Chou T, Mallick K, Zia RKP (2011) Non-equilibrium statistical mechanics: from a paradigmatic model to biological transport. Rep Prog Phys 74:116601

    Article  CAS  Google Scholar 

  17. Cohen M, Georgiou M, Stevenson NL, Miodownik M, Baum B (2010) Dynamic filopodia transmit intermittent Delta-Notch signaling to drive pattern refinement during lateral inhibition. Dev Cell 19:78–89

    Article  CAS  PubMed  Google Scholar 

  18. Crick FH (1970) Diffusion in embryogenesis. Nature 225:420–421

    Article  CAS  PubMed  Google Scholar 

  19. Dalessi S, Neves A, Bergmann S (2012) Modeling morphogen gradient formation from arbitrary realistically shaped sources. J Theor Biol 294:130–138

    Article  CAS  PubMed  Google Scholar 

  20. Deng J, Wang W, Lu LJ, Ma J (2010) A Two-dimensional simulation model of the bicoid gradient in Drosophila. PLoS Biol 5:e10275

    Google Scholar 

  21. Derrida B, Evans MR, Hakim V, Pasquier V (1993) Exact solution of a ID asymmetric exclusion model using a matrix formulation. J Phys A 26:1493–1517

    Article  Google Scholar 

  22. Dessaud E, Yang LL, Hill K, Cox B, Ulloa F, Ribeiro A, Mynett A, Novitch BG, Briscoe J (2007) Interpretation of the sonic hedgehog morphogen gradient by a temporal adaptation mechanism. Nature 450:717–720

    Article  CAS  PubMed  Google Scholar 

  23. Dilao R, Muraro D (2010) mRNA diffusion explains protein gradients in Drosophila early development. J Theor Biol 264:847–853

    Article  CAS  PubMed  Google Scholar 

  24. Drocco JA, Grimm O, Tank DW, Wieschaus E (2011) Measurement and perturbation of morphogen lifetime: effects on gradient shape. Biophys J 101:1807–1815

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  25. Drocco JA, Wieschaus EF, Tank DW (2012) The synthesis-diffusion-degradation model explains Bicoid gradient formation in unfertilized eggs. Phys Biol 9:055004

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  26. Eldar A, Rosin D, Shilo B-Z, Barkai N (2003) Self-enhanced ligand degradation underlies robustness of morphogen gradients. Dev Cell 5:635–646

    Article  CAS  PubMed  Google Scholar 

  27. Ellery AJ, Simpson MJ, McCue SW (2013) Comment on local accumulation times for source, diffusion, and degradation models in two and three dimensions. J Chem Phys 139:017101

    Article  CAS  PubMed  Google Scholar 

  28. England JL, Cardy J (2005) Morphogen gradient from a noisy source. Phys Rev Lett 94:078101

    Article  CAS  PubMed  Google Scholar 

  29. Entchev EV, Schwabedissen A, Gonzales-Gaitan M (2000) Gradient formation of the TGF-beta homolog Dpp. Cell 103:981–991

    Article  CAS  PubMed  Google Scholar 

  30. Fairchild CL, Barna M (2014) Specialized filopodia: at the ‘tip’ of morphogen transport and vertebrate tissue patterning. Curr Opin Genet Devel 27:67–73

    Article  CAS  Google Scholar 

  31. Fedotov S, Falconer S (2014) Nonlinear degradation-enhanced transport of morphogens performing subdiffusion. Phys Rev E 89:012107

    Article  CAS  Google Scholar 

  32. Gradilla A-C, Guerrero I (2013) Cytoneme-mediated cell-to-cell signaling during development. Cell Tissue Res 352:59–66

    Article  CAS  PubMed  Google Scholar 

  33. Gregor T, Wieschaus EF, McGregor AP, Bialek W, Tank DW (2007) Stability and nuclear dynamics of the bicoid morphogen gradient. Cell 130:141–152

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  34. Grimm O, Coppy M, Wieschaus EF (2009) Modelling the bicoid gradient. Development 137:2253–2264

    Article  CAS  Google Scholar 

  35. Gordon PV, Muratov CB (2012) Self-similarity and long-time behavior of solutions of the diffusion equation with nonlinear absorption and a boundary source. Netw Heterog Media 7:767–780

    Article  Google Scholar 

  36. Gordon PV, Muratov CB (2015) Eventual self-similarity of solutions for the diffusion equation with nonlinear absorption and a point source. SIAM J Math Anal 47:2903–2916

    Article  Google Scholar 

  37. Gordon PV, Sample C, Berezhkovskii AM, Muratov CB, Shvartsman SY (2011) Local kinetics of morphogen gradients. Proc Natl Acad Sci USA 108:6157–6162

    Article  CAS  PubMed  Google Scholar 

  38. Gordon PV, Muratov CB, Shvartsman SY (2013) Local accumulation times for source, diffusion, and degradation models in two and three dimensions. J Chem Phys 138:104121

    Article  CAS  PubMed  Google Scholar 

  39. Guerrero I, Kornberg TB (2014) Hedgehog and its circuitous journey from producing to target cells. Seminars Cell Dev Biol 33:52–62

    Article  CAS  Google Scholar 

  40. Hecht I, Rappel W-J, Levine H (2009) Determining the scale of the Bicoid morphogen gradient. Proc Natl Acad Sci USA 106:1710–1715

    Article  CAS  PubMed  Google Scholar 

  41. Incardona JP, Lee JH, Robertson CP, Enga K, Kapur RP, Roelink H (2000) Receptor-mediated endocytosis of soluble and membrane-tethered Sonic hedgehog by Patched-1. Proc Natl Acad Sci USA 97:12044–12049

    Article  CAS  PubMed  Google Scholar 

  42. Kornberg TB (2012) The imperatives of context and contour for morphogen dispersion. Biophys J 103:2252–2256

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  43. Kerszberg M, Wolpert L (1998) Mechanisms for positional signalling by morphogen transport: a theoretical study. J Theor Biol 191:103–114

    Article  CAS  PubMed  Google Scholar 

  44. Kerszberg M, Wolpert L (2007) Specifying positional information in the embryo: looking beyond morphogens. Cell 130:205–209

    Article  CAS  PubMed  Google Scholar 

  45. Kicheva A, Pantazis P, Bollenbach T, Kalaidzidis Y, Bittig T, Jülicher F, Gonzales-Gaitan M (2007) Kinetics of morphogen gradient formation. Science 315:521–525

    Article  CAS  PubMed  Google Scholar 

  46. Kicheva A, Bollenbach T, Wartlick O, Jülicher F, Gonzalez-Gaitan M (2012) Investigating the principles of morphogen gradient formation: from tissues to cells. Curr Opin Gen Dev 22:527–532

    Article  CAS  Google Scholar 

  47. Kolomeisky AB (2011) Formation of a morphogen gradient: acceleration by degradation. J Phys Chem Lett 2:1502–1505

    Article  CAS  Google Scholar 

  48. Kornberg TB, Roy S (2014) Communicating by touch neurons are not alone. Trends Cell Biol 24:370–376

    Article  PubMed  PubMed Central  Google Scholar 

  49. Kornberg TB, Roy S (2014) Cytonemes as specialized signaling filopodia. Development 141:729–736

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  50. Krotov D, Dubuis JO, Gregor T, Bialek W (2014) Morphogenesis at criticality. Proc Natl Acad Sci USA 111:3683–3688

    Article  CAS  PubMed  Google Scholar 

  51. Lander DA (2007) Morpheus unbound: reimagining the morphogen gradient. Cell 128:245–256

    Article  CAS  PubMed  Google Scholar 

  52. Lipshitz HD (2009) Follow the mRNA: a new model for Bicoid gradient formation. Nature Rev Mol Cell Biol 10:509–512

    Article  CAS  Google Scholar 

  53. Little SC, Tkacik G, Kneeland TB, Wieschaus EF, Gregor T (2011) The formation of the bicoid morphogen gradient requires protein movement from anteriorly localized mRNA. PLoS Biol 9:e1000596

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  54. Lodish H, Berk A, Kaiser C, Krieger M, Scott MP, Bretscher A, Ploegh H, Matsudaira P (2007) Molecular cell biology, 6th edn. W.H. Freeman, New York

    Google Scholar 

  55. Martinez-Arias A, Stewart A (2002) Molecular principles of animal development. Oxford University Press, New York

    Google Scholar 

  56. Medioni C, Mowry K, Bess F (2012) Principles and roles of mRNA localization in animal development. Development 139:3263–3276

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  57. Mogilner A, Odde D (2011) Modeling cellular processes in 3D. Trends Cell Biol 21:692–700

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  58. Müller P, Rogers KW, Jordan BM, Lee JS, Robson D, Ramanathan S, Schier AF (2012) Differential diffusivity of Nodal and Lefty underlies a reaction-diffusion patterning system. Science 336:721–724

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  59. Müller P, Rogers KW, Yu SR, Brand M, Schier AF (2013) Morphogen transport. Development 140:1621–1638

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  60. Porcher A, Dostatni N (2010) The Bicoid morphogen system. Curr Biol 20:R249–R254

    Article  CAS  PubMed  Google Scholar 

  61. Redner S (2001) A guide to first-passage processes. Cambridge University Press, New York

    Book  Google Scholar 

  62. Reingruber J, Holcman D (2014) Computational and mathematical methods for morphogenetic gradient analysis, boundary formation and axonal targeting. Seminars Cell Dev Biol 35:189–202

    Article  Google Scholar 

  63. Richards DM, Saunders TE (2015) Spatiotemporal analysis of different mechanisms for interpreting morphogen gradients. Biophys J 108:2061–2073

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  64. Rogers KW, Schier AF (2011) Morphogen gradients: from generation to interpretation. Annu Rev Cell Dev Biol 27:377–407

    Article  CAS  PubMed  Google Scholar 

  65. Rørth P (2014) Reach out and touch someone. Science 343:848–849

    Article  PubMed  Google Scholar 

  66. Roy S, Kornberg TB (2015) Paracrine signaling mediated at cell-cell contacts. Bioessays 37:25–33

    Article  CAS  PubMed  Google Scholar 

  67. Sample C, Shvartsman SY (2010) Multiscale modeling of diffusion in the early Drosophila embryo. Proc Natl Acad Sci USA 107:10092–10096

    Article  CAS  PubMed  Google Scholar 

  68. Sanders TA, Llagostera E, Barna M (2013) Specialized filopodia direct long-range transport of SHH during vertebrate tissue patterning. Nature 497:628–632

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  69. Saunders T, Howard M (2009) When it pays to rush: interpreting morphogen gradients prior to steady-state. Phys Biol 6:046020

    Article  CAS  PubMed  Google Scholar 

  70. Sigaut L, Pearson JE, Colman-Lerner A, Dawson SP (2014) Messages do diffuse faster than messengers: Reconciling disparate estimates of the morphogen bicoid diffusion coefficient. PLoS Comp Biol 10:e1003629

    Article  CAS  Google Scholar 

  71. Spirov A, Fahmy K, Schneider M, Frei E, Noll M, Baumgartner S (2009) Formation of the bicoid morphogen gradient: an mRNA gradient dictates the protein gradient. Development 136:605–614

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  72. Tabata T, Takei Y (2004) Morphogens, their identification and regulation. Development 131:703–712

    Article  CAS  PubMed  Google Scholar 

  73. Teimouri H, Kolomeisky AB (2014) Development of morphogen gradient: the role of dimension and discreteness. J Chem Phys 140:085102

    Article  CAS  PubMed  Google Scholar 

  74. Teimouri H, Kolomeisky AB (2015) The role of source delocalization in the development of morphogen gradients. Phys Biol 12:026006

    Article  PubMed  Google Scholar 

  75. Teimouri H, Kolomeisky AB (2016) New model for understanding mechanisms of biological signaling: direct transport via cytonemes. J Phys Chem Lett 7:180–185

    Article  CAS  PubMed  Google Scholar 

  76. Teimouri H, Bozorgui B, Kolomeisky AB (2016) Development of morphogen gradients with spatially varying degradation rates. J Phys Chem B 120:2745–2750

    Article  CAS  PubMed  Google Scholar 

  77. Tompkins N, Li N, Girabawe C, Heymann M, Ermentrout GB, Epstein IR, Fraden S (2013) Testing Turing’s theory of morphogenesis in chemical cells. Proc Natl Acad Sci USA 111:4397–4402

    Article  CAS  Google Scholar 

  78. Tufcea DE, Francois P (2015) Critical timing without a timer for embryonic development. Biophys J 109:1724–1734

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  79. Wartlick O, Kicheva A, Gonzales-Gaitan M (2009) Morphogen gradient formation. Cold Spring Harb Perspect Biol 1:a001255

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  80. Wolpert L (1969) Positional information and the spatial pattern of cellular differentiation. J Theor Biol 25:1–47

    Article  CAS  PubMed  Google Scholar 

  81. Wolpert L (1998) Principles of development. Oxford University Press, New York

    Google Scholar 

  82. Yu SR, Burkhardt M, Nowak M, Ries J, Petrasek Z, Scholpp S, Schwille P, Brand M (2009) Fgf8 morphogen gradient forms by a source-sink mechanism with freely diffusing molecules. Nature 461:533–536

    Article  CAS  PubMed  Google Scholar 

  83. Yuste SB, Abad E, Lindenberg K (2010) Reaction-subdiffusion model of morphogen gradient formation. Phys Rev E 82:061123

    Article  CAS  Google Scholar 

  84. Zhou S, Lo WC, Suhalim JL, Digman MA, Grattom E, Nie Q, Lander AD (2012) Free extracellular diffusion creates the Dpp morphogen gradient of the drosophila wing disc. Curr Biol 22:668–675

    Article  CAS  PubMed  PubMed Central  Google Scholar 

Download references

Acknowledgements

A.B.K. acknowledges the support from the Center for Theoretical Biological Physics (NSF Grant PHY-1427654).

Author information

Authors and Affiliations

  1. Department of Physics and FAS Center for Systems Biology, Harvard University, Cambridge, MA, 02138, USA

    Hamid Teimouri

  2. Department of Chemistry and Center for Theoretical Biological Physics, Rice University, Houston, Texas, 77005-1892, USA

    Anatoly B. Kolomeisky

Authors
  1. Hamid Teimouri
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Anatoly B. Kolomeisky
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Anatoly B. Kolomeisky .

Editor information

Editors and Affiliations

  1. Integrated Microscopy Core, Baylor College of Medicine, Houston, TX, USA

    Julien Dubrulle

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer Science+Business Media, LLC, part of Springer Nature

About this protocol

Check for updates. Verify currency and authenticity via CrossMark

Cite this protocol

Teimouri, H., Kolomeisky, A.B. (2018). Discrete-State Stochastic Modeling of Morphogen Gradient Formation. In: Dubrulle, J. (eds) Morphogen Gradients. Methods in Molecular Biology, vol 1863. Humana Press, New York, NY. https://doi.org/10.1007/978-1-4939-8772-6_12

Download citation

  • DOI: https://doi.org/10.1007/978-1-4939-8772-6_12

  • Published:

  • Publisher Name: Humana Press, New York, NY

  • Print ISBN: 978-1-4939-8771-9

  • Online ISBN: 978-1-4939-8772-6

  • eBook Packages: Springer Protocols

Publish with us

Policies and ethics

Search

Navigation