Discrete-State Stochastic Modeling of Morphogen Gradient Formation

  • Hamid Teimouri
  • Anatoly B. KolomeiskyEmail author
Part of the Methods in Molecular Biology book series (MIMB, volume 1863)


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.

Key words

Morphogen gradient Local accumulation time Reaction–diffusion processes Spatially varying degradation rate Discrete-state stochastic modeling Nonlinear degradation mechanism Direct-delivery mechanism 



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


  1. 1.
    Alaynick WA, Jessell TM, Pfaff SL (2011) SnapShot: spinal cord development. Cell 146:178–178.e1CrossRefPubMedPubMedCentralGoogle Scholar
  2. 2.
    Berezhkovskii AM (2011) Renewal theory for single-molecule systems with multiple reaction channels. J Chem Phys 134:074114CrossRefPubMedPubMedCentralGoogle Scholar
  3. 3.
    Berezhkovskii AM, Shvartsman SY (2011) Physical interpretation of mean local accumulation time of morphogen gradient formation. J Chem Phys 135:154115CrossRefPubMedPubMedCentralGoogle Scholar
  4. 4.
    Berezhkovskii AM, Shvartsman SY (2013) Kinetics of receptor occupancy during morphogen gradient formation. J Chem Phys 138:244105CrossRefPubMedPubMedCentralGoogle Scholar
  5. 5.
    Berezhkovskii AM, Sample C, Shvartsman SY (2010) How long does it take to establish a morphogen gradient? Biophys J 99:L59–L61CrossRefPubMedPubMedCentralGoogle Scholar
  6. 6.
    Berezhkovskii AM, Sample C, Shvartsman SY (2011) Formation of morphogen gradients: local accumulation time. Phys Rev E 83:051906CrossRefGoogle Scholar
  7. 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:e46CrossRefPubMedPubMedCentralGoogle Scholar
  8. 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 15:1269–1281CrossRefPubMedGoogle Scholar
  9. 9.
    Bozorgui B, Teimouri H, Kolomeisky AB (2015) Theoretical analysis of degradation mechanisms in the formation of morphogen gradients. J Chem Phys 143:025102CrossRefPubMedGoogle Scholar
  10. 10.
    Bressloff PC, Hyunjoong K (2018) Bidirectional transport model of morphogen gradient formation via cytonemes. Phys Biol 15:026010CrossRefPubMedGoogle Scholar
  11. 11.
    Briscoe J (2009) Making a grade: Sonic Hedgehog signalling and the control of neural cell fate. EMBO J 28:457–465CrossRefPubMedPubMedCentralGoogle Scholar
  12. 12.
    Briscoe J, Small S (2015) Morphogen rules: design principles of gradient-mediated embryo patterning. Development 142:3996–4009CrossRefPubMedPubMedCentralGoogle Scholar
  13. 13.
    Castle BT, Howard SA, Odde DJ (2011) Assessment of transport mechanisms underlying the bicoid morphogen gradient. Cell Mol Bioeng 4:116–121CrossRefPubMedPubMedCentralGoogle Scholar
  14. 14.
    Chen Y, Struhl G (1996) Dual roles for patched in sequestering and transducing Hedgehog. Cell 87:553–563CrossRefPubMedGoogle Scholar
  15. 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 141:124–135CrossRefPubMedPubMedCentralGoogle Scholar
  16. 16.
    Chou T, Mallick K, Zia RKP (2011) Non-equilibrium statistical mechanics: from a paradigmatic model to biological transport. Rep Prog Phys 74:116601CrossRefGoogle Scholar
  17. 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 19:78–89CrossRefPubMedGoogle Scholar
  18. 18.
    Crick FH (1970) Diffusion in embryogenesis. Nature 225:420–421CrossRefPubMedGoogle Scholar
  19. 19.
    Dalessi S, Neves A, Bergmann S (2012) Modeling morphogen gradient formation from arbitrary realistically shaped sources. J Theor Biol 294:130–138CrossRefPubMedGoogle Scholar
  20. 20.
    Deng J, Wang W, Lu LJ, Ma J (2010) A Two-dimensional simulation model of the bicoid gradient in Drosophila. PLoS Biol 5:e10275Google Scholar
  21. 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 26:1493–1517CrossRefGoogle Scholar
  22. 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 450:717–720CrossRefPubMedGoogle Scholar
  23. 23.
    Dilao R, Muraro D (2010) mRNA diffusion explains protein gradients in Drosophila early development. J Theor Biol 264:847–853CrossRefPubMedGoogle Scholar
  24. 24.
    Drocco JA, Grimm O, Tank DW, Wieschaus E (2011) Measurement and perturbation of morphogen lifetime: effects on gradient shape. Biophys J 101:1807–1815CrossRefPubMedPubMedCentralGoogle Scholar
  25. 25.
    Drocco JA, Wieschaus EF, Tank DW (2012) The synthesis-diffusion-degradation model explains Bicoid gradient formation in unfertilized eggs. Phys Biol 9:055004CrossRefPubMedPubMedCentralGoogle Scholar
  26. 26.
    Eldar A, Rosin D, Shilo B-Z, Barkai N (2003) Self-enhanced ligand degradation underlies robustness of morphogen gradients. Dev Cell 5:635–646CrossRefPubMedGoogle Scholar
  27. 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 139:017101CrossRefPubMedGoogle Scholar
  28. 28.
    England JL, Cardy J (2005) Morphogen gradient from a noisy source. Phys Rev Lett 94:078101CrossRefPubMedGoogle Scholar
  29. 29.
    Entchev EV, Schwabedissen A, Gonzales-Gaitan M (2000) Gradient formation of the TGF-beta homolog Dpp. Cell 103:981–991CrossRefPubMedGoogle Scholar
  30. 30.
    Fairchild CL, Barna M (2014) Specialized filopodia: at the ‘tip’ of morphogen transport and vertebrate tissue patterning. Curr Opin Genet Devel 27:67–73CrossRefGoogle Scholar
  31. 31.
    Fedotov S, Falconer S (2014) Nonlinear degradation-enhanced transport of morphogens performing subdiffusion. Phys Rev E 89:012107CrossRefGoogle Scholar
  32. 32.
    Gradilla A-C, Guerrero I (2013) Cytoneme-mediated cell-to-cell signaling during development. Cell Tissue Res 352:59–66CrossRefPubMedGoogle Scholar
  33. 33.
    Gregor T, Wieschaus EF, McGregor AP, Bialek W, Tank DW (2007) Stability and nuclear dynamics of the bicoid morphogen gradient. Cell 130:141–152CrossRefPubMedPubMedCentralGoogle Scholar
  34. 34.
    Grimm O, Coppy M, Wieschaus EF (2009) Modelling the bicoid gradient. Development 137:2253–2264CrossRefGoogle Scholar
  35. 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 7:767–780CrossRefGoogle Scholar
  36. 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 47:2903–2916CrossRefGoogle Scholar
  37. 37.
    Gordon PV, Sample C, Berezhkovskii AM, Muratov CB, Shvartsman SY (2011) Local kinetics of morphogen gradients. Proc Natl Acad Sci USA 108:6157–6162CrossRefPubMedGoogle Scholar
  38. 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 138:104121CrossRefPubMedGoogle Scholar
  39. 39.
    Guerrero I, Kornberg TB (2014) Hedgehog and its circuitous journey from producing to target cells. Seminars Cell Dev Biol 33:52–62CrossRefGoogle Scholar
  40. 40.
    Hecht I, Rappel W-J, Levine H (2009) Determining the scale of the Bicoid morphogen gradient. Proc Natl Acad Sci USA 106:1710–1715CrossRefPubMedGoogle Scholar
  41. 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 97:12044–12049CrossRefPubMedGoogle Scholar
  42. 42.
    Kornberg TB (2012) The imperatives of context and contour for morphogen dispersion. Biophys J 103:2252–2256CrossRefPubMedPubMedCentralGoogle Scholar
  43. 43.
    Kerszberg M, Wolpert L (1998) Mechanisms for positional signalling by morphogen transport: a theoretical study. J Theor Biol 191:103–114CrossRefPubMedGoogle Scholar
  44. 44.
    Kerszberg M, Wolpert L (2007) Specifying positional information in the embryo: looking beyond morphogens. Cell 130:205–209CrossRefPubMedGoogle Scholar
  45. 45.
    Kicheva A, Pantazis P, Bollenbach T, Kalaidzidis Y, Bittig T, Jülicher F, Gonzales-Gaitan M (2007) Kinetics of morphogen gradient formation. Science 315:521–525CrossRefPubMedGoogle Scholar
  46. 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 22:527–532CrossRefGoogle Scholar
  47. 47.
    Kolomeisky AB (2011) Formation of a morphogen gradient: acceleration by degradation. J Phys Chem Lett 2:1502–1505CrossRefGoogle Scholar
  48. 48.
    Kornberg TB, Roy S (2014) Communicating by touch neurons are not alone. Trends Cell Biol 24:370–376CrossRefPubMedPubMedCentralGoogle Scholar
  49. 49.
    Kornberg TB, Roy S (2014) Cytonemes as specialized signaling filopodia. Development 141:729–736CrossRefPubMedPubMedCentralGoogle Scholar
  50. 50.
    Krotov D, Dubuis JO, Gregor T, Bialek W (2014) Morphogenesis at criticality. Proc Natl Acad Sci USA 111:3683–3688CrossRefPubMedGoogle Scholar
  51. 51.
    Lander DA (2007) Morpheus unbound: reimagining the morphogen gradient. Cell 128:245–256CrossRefPubMedGoogle Scholar
  52. 52.
    Lipshitz HD (2009) Follow the mRNA: a new model for Bicoid gradient formation. Nature Rev Mol Cell Biol 10:509–512CrossRefGoogle Scholar
  53. 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:e1000596CrossRefPubMedPubMedCentralGoogle Scholar
  54. 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 YorkGoogle Scholar
  55. 55.
    Martinez-Arias A, Stewart A (2002) Molecular principles of animal development. Oxford University Press, New YorkGoogle Scholar
  56. 56.
    Medioni C, Mowry K, Bess F (2012) Principles and roles of mRNA localization in animal development. Development 139:3263–3276CrossRefPubMedPubMedCentralGoogle Scholar
  57. 57.
    Mogilner A, Odde D (2011) Modeling cellular processes in 3D. Trends Cell Biol 21:692–700CrossRefPubMedPubMedCentralGoogle Scholar
  58. 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 336:721–724CrossRefPubMedPubMedCentralGoogle Scholar
  59. 59.
    Müller P, Rogers KW, Yu SR, Brand M, Schier AF (2013) Morphogen transport. Development 140:1621–1638CrossRefPubMedPubMedCentralGoogle Scholar
  60. 60.
    Porcher A, Dostatni N (2010) The Bicoid morphogen system. Curr Biol 20:R249–R254CrossRefPubMedGoogle Scholar
  61. 61.
    Redner S (2001) A guide to first-passage processes. Cambridge University Press, New YorkCrossRefGoogle Scholar
  62. 62.
    Reingruber J, Holcman D (2014) Computational and mathematical methods for morphogenetic gradient analysis, boundary formation and axonal targeting. Seminars Cell Dev Biol 35:189–202CrossRefGoogle Scholar
  63. 63.
    Richards DM, Saunders TE (2015) Spatiotemporal analysis of different mechanisms for interpreting morphogen gradients. Biophys J 108:2061–2073CrossRefPubMedPubMedCentralGoogle Scholar
  64. 64.
    Rogers KW, Schier AF (2011) Morphogen gradients: from generation to interpretation. Annu Rev Cell Dev Biol 27:377–407CrossRefPubMedGoogle Scholar
  65. 65.
    Rørth P (2014) Reach out and touch someone. Science 343:848–849CrossRefPubMedGoogle Scholar
  66. 66.
    Roy S, Kornberg TB (2015) Paracrine signaling mediated at cell-cell contacts. Bioessays 37:25–33CrossRefPubMedGoogle Scholar
  67. 67.
    Sample C, Shvartsman SY (2010) Multiscale modeling of diffusion in the early Drosophila embryo. Proc Natl Acad Sci USA 107:10092–10096CrossRefPubMedGoogle Scholar
  68. 68.
    Sanders TA, Llagostera E, Barna M (2013) Specialized filopodia direct long-range transport of SHH during vertebrate tissue patterning. Nature 497:628–632CrossRefPubMedPubMedCentralGoogle Scholar
  69. 69.
    Saunders T, Howard M (2009) When it pays to rush: interpreting morphogen gradients prior to steady-state. Phys Biol 6:046020CrossRefPubMedGoogle Scholar
  70. 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:e1003629CrossRefGoogle Scholar
  71. 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 136:605–614CrossRefPubMedPubMedCentralGoogle Scholar
  72. 72.
    Tabata T, Takei Y (2004) Morphogens, their identification and regulation. Development 131:703–712CrossRefPubMedGoogle Scholar
  73. 73.
    Teimouri H, Kolomeisky AB (2014) Development of morphogen gradient: the role of dimension and discreteness. J Chem Phys 140:085102CrossRefPubMedGoogle Scholar
  74. 74.
    Teimouri H, Kolomeisky AB (2015) The role of source delocalization in the development of morphogen gradients. Phys Biol 12:026006CrossRefPubMedGoogle Scholar
  75. 75.
    Teimouri H, Kolomeisky AB (2016) New model for understanding mechanisms of biological signaling: direct transport via cytonemes. J Phys Chem Lett 7:180–185CrossRefPubMedGoogle Scholar
  76. 76.
    Teimouri H, Bozorgui B, Kolomeisky AB (2016) Development of morphogen gradients with spatially varying degradation rates. J Phys Chem B 120:2745–2750CrossRefPubMedGoogle Scholar
  77. 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 111:4397–4402CrossRefGoogle Scholar
  78. 78.
    Tufcea DE, Francois P (2015) Critical timing without a timer for embryonic development. Biophys J 109:1724–1734CrossRefPubMedPubMedCentralGoogle Scholar
  79. 79.
    Wartlick O, Kicheva A, Gonzales-Gaitan M (2009) Morphogen gradient formation. Cold Spring Harb Perspect Biol 1:a001255CrossRefPubMedPubMedCentralGoogle Scholar
  80. 80.
    Wolpert L (1969) Positional information and the spatial pattern of cellular differentiation. J Theor Biol 25:1–47CrossRefPubMedGoogle Scholar
  81. 81.
    Wolpert L (1998) Principles of development. Oxford University Press, New YorkGoogle Scholar
  82. 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 461:533–536CrossRefPubMedGoogle Scholar
  83. 83.
    Yuste SB, Abad E, Lindenberg K (2010) Reaction-subdiffusion model of morphogen gradient formation. Phys Rev E 82:061123CrossRefGoogle Scholar
  84. 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 22:668–675CrossRefPubMedPubMedCentralGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Department of Physics and FAS Center for Systems BiologyHarvard UniversityCambridgeUSA
  2. 2.Department of Chemistry and Center for Theoretical Biological PhysicsRice UniversityHoustonUSA

Personalised recommendations