Synchrony in reaction-diffusion models of morphogenesis: Applications to curvature-dependent proliferation and zero-diffusion front waves

Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences

Volumn 367, Issue 1908, 2009, Pages 4829-4862

  Abbas, Lamia ^a   Demongeot, Jacques ^b   Glade, Nicolas ^b  

^a UNIVERSITÉ ET INSA DE ROUEN   (France)

^b UNIV GRENOBLE ALPES   (France)

Author keywords

Feather morphogenesis; Model of morphogenesis; Reaction diffusion curvature dependent proliferation; Zero diffusion front waves

Indexed keywords

AMPLIFICATION; CELL PROLIFERATION; DIFFUSION IN LIQUIDS;

AGE-DEPENDENT APPROACH; CELL DENSITY; CONTOUR LINE; GAUSSIAN CURVATURES; LOCAL CURVATURE; MORPHOGENS; MULTI-AGENTS SYSTEMS; REACTION DIFFUSION; REACTION-DIFFUSION CURVATURE-DEPENDENT PROLIFERATION; REACTION-DIFFUSION MODELS; ZERO-DIFFUSION FRONT WAVES;

MORPHOLOGY;

ANIMAL; ARTICLE; BIOLOGICAL MODEL; BIRD; CELL CYCLE; FEATHER; GASTRULATION; MORPHOGENESIS; PHYSIOLOGY; THERMODYNAMICS;

ANIMALS; BIRDS; CELL CYCLE; FEATHERS; GASTRULATION; MODELS, BIOLOGICAL; MORPHOGENESIS; THERMODYNAMICS;

EID: 73349130438     PISSN: 1364503X     EISSN: None     Source Type: Journal    
DOI: 10.1098/rsta.2009.0170     Document Type: Article

References (118)

1. Aracena, J. & Demongeot, J. 2004 Mathematical methods for inferring regulatory networks interactions: application to genetic regulation. Acta Biotheor. 52, 391-400. (doi:10.1023/B:ACBI.0000046605.48037.7d)
2. Aracena, J., Ben Lamine, S., Mermet, M. A., Cohen, O. & Demongeot, J. 2003 Mathematical modelling in genetic networks: relationships between the genetic expression and both chromosomic breakage and positive circuits. IEEE Trans. Syst. Man Cybern. 33, 825-834. (doi:10.1109/TSMCB.2003.816928)
3. Aracena, J., Demongeot, J. & Goles, E. 2004a Mathematical modelling in genetic networks. IEEE Trans. Neural Netw. 15, 77-83. (doi:10.1109/TNN.2003.821555)
4. Aracena, J., Demongeot, J. & Goles, E. 2004b On limit cycles of monotone functions with symmetric connection graphs. Theor. Comp. Sci. 322, 237-244. (doi:10.1016/j.tcs.2004.03.010)
5. Aracena, J., Demongeot, J. & Goles, E. 2004c Fixed points and maximal independent sets on AND-OR networks. Discr. Appl. Math. 138, 277-288. (doi:10.1016/S0166-218X(03)00461-X)
6. Barnett-Hall, D. G. & Waugh, W. A. 1967 Indices of synchrony in cellular cultures. Biometrics 23, 693-716. (doi:10.2307/2528423)
7. Baserga, R. 1980 The biology of cell reproduction. Cambridge, MA: Harvard University Press.
8. Basse, B., Baguley, B. C., Marshall, E. S., Joseph, W. R., van Brunt, B., Wake, G. & Wall, D. J. N. 2003 A mathematical model for analysis of the cell cycle in human tumours. J. Math. Biol. 47, 295-312. (doi:10.1007/s00285-003-0203-0)
9. Baum, T. P., Pasqual, N., Thuderoz, F., Hierle, V., Chaume, D., Lefranc, M. P., Jouvin-Marche, E., Marche, P. & Demongeot, J. 2004 IMGT/Genelnfo: enhancing V(D)J recombination database accessibility. Nucleic Acids Res. 32, 51-54. (doi:10.1093/nar/gkh041)
10. Baum, T. P., Pasqual, N., Hierle, V., Bellahcene, F., Chaume, D., Lefranc, M. P., Jouvin-Marche, E., Marche, P. & Demongeot, J. 2006 IMGT/Genelnfo: new gamma and delta chains for database V(D)J recombination. BMC Bioinformatics 7, 224-228. (doi:10.1186/1471-2105-7-224)
11. Bertuzzi, A., Gandolfi, A. & Glovenco, M. A. 1981 Mathematical models of the cell cycle with a view to tumor studies. Math. Biosci. 53, 159-188. (doi:10.1016/0025-5564(81)90017-l)
12. Bieniawski, S. R., Kroo, I. M. & Wolpert, D. H. 2004 Discrete, continuous and constrained optimization using collectives. In 10th Multidisciplinary Analysis and Optimization Conf. 04, pp. l-9. Albany, NY: AIAA/ISSMO.
13. Blaschko, A. 1901 Die Neven-verteilung in der Haut in ihrer Beziehung zu den Enkrankungen der Haut. Wien, Austria: Wilhelm Braumuller.
14. Blumenthal, L. K. & Zahler, S. A. 1962 Index for measurement of synchronization of cell populations. Science 135, 724. (doi:10.ll26/science.l35.3505.724)
15. Bolognia, J. L., Orlow, S. J. & Glick, S. A. 1994 Lines of Blaschko. J. Am. Acad. Dermatol. 31, 157-190. (doi:10.1016/S0l90-9622(94) 70l43-l)
16. Borges, A. F. 1984 Relaxed skin tension lines (RSTL) versus other skin lines. Plast. Reconstr. Surg. 73, 44-50. (doi:10.1097/00006534- 198401000-00036)
17. Brunie, L., Leitner, F., Berthommier, F., Cinquin, P. & Demongeot, J. 1995 Interpretation of multimodal medical images using connectionist and variational methods. Technol. Health Care 3, 91-100.
18. Burns, V. W. 1962 Cell division synchronization. Prog. Biophys. Biophys. Chem. 12, l-20.
19. Burns, V. W. 1964 Synchrony in cell division and growth. New York, NY: Wiley.
20. Carlier, M. F., Melki, R., Pantaloni, D., Hill, T. L. & Chen, Y. 1987 Synchronous oscillations in microtubule polymerisation. Proc. Natl Acad. Sci. USA 84, 5257-5261. (doi:10.l073/pnas.84.15.5257)
21. Castets, V., Dulos, E., Boissonade, J. & De Kepper, P. 1990 Experimental evidence of a sustained standing Turing-type nonequilibrium chemical pattern. Phys. Rev. Lett.. 64, 2953-2956. (doi:10.1103/PhysRevLett.64.2953)
22. Chiorino, G., Metz, J. A. J., Tomasini, D. & Ubezio, P. 2001 Desynchronization rate in cell populations: mathematical modeling and experimental data. J. Theor. Biol. 208, 185-199. (doi:10.1006/jtbi.2000.2213)
23. Chung, K. T., Nilson, E. H., Case, M. J., Marr, A. J. & Hungate, R. E. 1973 Estimation of growth rate from the mitotic index. Appl. Environ. Microbiol. 25, 778-780.
24. Cinquin, O. & Demongeot, J. 2002a Positive and negative feedback: striking a balance between necessary antagonists. J. Theor. Biol. 216, 229-241. (doi:10.1006/jtbi.2002.2544)
25. Cinquin, O. & Demongeot, J. 2002b Positive and negative feedback: mending the ways of sloppy systems. C. R. Acad. Sci. Biol. 325, 1085-1095. (doi:10.1016/S1631-0691(02)01533-0)
26. Cinquin, O. & Demongeot, J. 2005 High-dimensional switches and the modeling of cell differentiation. J. Theor. Biol. 233, 391-411. (doi:10.1016/j.jtbi.2004.10.027)
27. Clairambault, J., Gaubert, S. & Lepoutre T. 2008 Comparison of Perron and Floquet eigenvalues in age structured cell division cycle models. (http://arxiv.org/abs/0812.0803v1)
28. Cohen, O., Mermet, M. A. & Demongeot, J. 2001 HC Forum: a web site based on an international human cytogenetic database. Nucleic Acids Res. 29, 305-307. (doi:10.1093/nar/29.1.305)
29. Cottet, G. H., Demongeot, J., El Ayyadi, M. & Leitner, F. 1997 Image segmentation using snake-splines and anisotropic diffusion operators. See http://www-lmc.imag;fr/lmc-edp/mohamed.El-Ayyadi/demos.html.
30. Daszynkiewiez, Z., Gressman, H., Iraganos, F., Stein, J. & Kiamp, J. 1980 Cell heterogeneity during the cell cycle. J. Cell. Physiol. 112, 465-474.
31. De Boer, R. J. & Perelson, A. S. 2005 Estimating division and death rates from CFSE data. J. Comput. Appl. Math. 184, 140-164. (doi:10.1016/j.cam.2004.08.020)
32. Demetrius, L. 1983 Statistical mechanics and population biology. J. Stat. Phys. 30, 709-753. (doi:10.1007/BF01009685)
33. Demetrius, L. & Demongeot, J. 1984 A thermodynamic approach in the modelling of the cellular cycle. Biometrics 40, 259-260.
34. Demongeot, J. 1981 Asymptotic inference for Markov random field on Zd. Springer Series in Synergetics, vol. 9, pp. 254-267. Berlin, Germany: Springer.
35. Demongeot, J. In press. Biological boundaries and biological age. Acta Biotheor. 57.(doi:10.1007/s10441-009-9087-8)
36. Demongeot, J. & Demetrius, L. 1989 La derive demographique et la selection naturelle: etude empirique de la France (1850-1965. Population 2, 231-248.
37. Demongeot, J. & Frangoise, J. P. 2006 Approximation for limit cycles and their isochrons. C. R. Biol. 329, 967-970. (doi:10.1016/j.crvi.2006.05.005)
38. Demongeot, J. & Laurent, M. 1983 Sigmoidicity in allosteric models. Math. Biosci. 67, 1-17. (doi:10.1016/0025-5564(83 90015-9
39. Demongeot, J., Benaouda, D. & Jezequel, C. 1995 Dynamical confinement in neural networks and cell cycle. Chaos 5, 167-173. (doi:10.1063/1.166064)
40. Demongeot, J., Aracena, J., Thuderoz, F., Baum, T. P. & Cohen, O. 2003a Genetic regulation networks: circuits, regulons and attractors. C. R. Acad. Sci. Biol. 326, 171-188. (doi:10.1016/S1631-0691(03 00069-6)
41. Demongeot, J., Thuderoz, F., Baum, T. P., Berger, F. & Cohen, O. 2003b Bio-array images processing and genetic networks modelling. C. R. Acad. Sci. Biol. 326, 487-500. (doi:10.1016/S1631-0691(03 00114-8)
42. Demongeot, J., Bezy-Wendling, J., Mattes, J., Haigron, P., Glade, N. & Coatrieux, J. L. 2003c Multiscale modeling and imaging: the challenges of biocomplexity. Proc. IEEE 91, 1723-1737. (doi:10.1109/JPROC.2003.817878)
43. Demongeot, J., Glade, N., Hansen, O. & Moreira, A. 2007a An open issue: the inner mitochondrial membrane (IMM as a free boundary problem. Biochimie 89, 1049-1057. (doi:10.1016/j.biochi.2007.04.009)
44. Demongeot, J., Glade, N. & Forest, L. 2007b Lienard systems and potential-Hamiltonian decomposition. I. Methodology. C. R. Math. 344, 121-126. (doi:10.1016/j.crma.2006.10.016)
45. Demongeot, J., Glade, N. & Forest, L. 2007c Lienard systems and potential-Hamiltonian decomposition. II. Algorithm. C. R. Math. 344, 191-194. (doi:10.1016/j.crma.2006.10.013)
46. Demongeot, J., Fouquet, Y., Tayyab, M. & Vuillerme, N. 2009 Understanding physiological and degenerative natural vision mechanisms to define contrast and contour operators. PLoS ONE 4, e6010. (doi:10.1371/journal.pone.0006010)
47. Doumic, M. 2007 Analysis of a population model structured by the cells molecular content. Math. Model. Nat. Phenom. 2, 121-153. (doi:10.1051/mmnp:2007006)
48. Engelberg, J. 1964 Measurement of degrees of synchrony in cell populations. In Synchrony in cell division and growth (ed. E. Zeuthen), pp. 497-508. New York, NY: Wiley.
49. Ermentrout, B. 1991 Stripes or spots? Nonlinear effects in bifurcation of reaction-diffusion equations on the square. Proc. R. Soc. A 434, 413-417. (doi:10.1098/rspa.1991.0100)
50. Forest, L. & Demongeot, J. 2006 Cellular modelling of secondary radial growth in conifer trees: application to Pinus radiata. Bull. Math. Biol. 68, 753-784. (doi:10.1007/s11538-005-9004-5)
51. Forest, L. & Demongeot, J. 2008 A general formalism for tissue morphogenesis based on cellular dynamics and control system interactions. Acta Biotheor. 56, 51-74. (doi:10.1007/s10441-008-9030-4)
52. Forest, L., San Martin, J., Padilla, F., Chassat, F., Giroud, F. & Demongeot, J. 2004 Morphogenetic processes: application to cambial growth dynamics. Acta Biotheor. 52, 415-438. (doi:10.1023/B:ACBI. 0000046607.17817.20)
53. Forest, L., Martinez, S., Padilla, F., Demongeot, J. & San Martin, J. 2006a Modelling of auxin transport affected by gravity and differential radial growth. J. Theor. Biol. 241, 241-251. (doi:10.1016/j.jtbi.2005.11. 029)
54. Forest, L., Michon, F., Cadau, S., Demongeot, J. & Dhouailly, D. 2006b What is the biological basis of pattern formation of skin lesions? Viewpoint 4. Skin patterns belong to three main types, determined at three steps of development. Exp. Dermatol. 12, 559-564.
55. Forest, L., Glade, N. & Demongeot, J. 2007 Liénard systems and potential-Hamiltonian decomposition. Applications. C. R. Acad. Sci. Biol. 330, 97-106. (doi:10.1016/j.crvi.2006.12.001)
56. Glade, N. 2008 Computing with the cytoskeleton: a problem of scale. Int. J. Unconvent. Comput. 4, 33-44.
57. Glade, N., Demongeot, J. & Tabony, J. 2002 Numerical simulations of microtubule self-organisation by reaction and diffusion. Acta Biotheor. 50, 239-268. (doi:10.1023/A:1022608400954)
58. Glade, N., Forest, L. & Demongeot, J. 2007 Lienard systems and potential-Hamiltonian decomposition. III. Applications in biology. C. R. Math. 344, 253-258. (doi:10.1016/j.crma.2006.11.014)
59. Glade, N., Ben Amor, H., Bastien, O. 2009 Trail systems as fault tolerant wires and their use in bio-processors. In Modelling Complex Biological Systems in the Context of Genomics, Proc. of the 'La Colle sur Loup' Spring School (eds P. Amar, F. Kepes & V. Norris), pp. 85-119. France: EDP Sciences.
60. Goldbeter, A. 1991 A minimal cascade model for the mitotic oscillator involving cyclin and cdc2 kinase. Proc. Natl Acad. Sci. USA 88, 9107-9112. (doi:10.1073/pnas.88.20.9107)
61. Goldstein, S. 1981 Entropy increase in dynamical systems. Israel J. Math. 38, 241-256. (doi:10.1007/BF02760809)
62. Goldstone, R. L. & Roberts, M. E. 2006 Self-organized trail systems in groups of humans. Complexity 11, 43-50. (doi:10.1002/cplx. 20135)
63. Greiner, G. 1984 A typical Perron-Frobenius theorem with applications to an age-dependent population equation. In Infinite-dimensional systems. Lecture Notes in Mathematics, vol. 1076, pp. 86-100. Berlin, Germany: Springer. (doi:10.1007/BFb0072769)
64. Guardavaccaro, D. & Pagano, M. 2006 Stabilizers and destabilizers controlling cell cycle oscillators. Mol. Cell 22, 1-4. (doi:10.1016/j.molcel.2006.03.017)
65. Hahn, G. M. 1970 Mammalian cell populations. Math. Biosci. 6, 295-304. (doi:10.1016/0025-5564(70)90069-6)
66. Harrison, L. G. 1993 Kinetic theory of living pattern. Cambridge, UK: Cambridge University Press. Helbing, D., Keltsch, J. & Molnár, P. 1997 Modelling the evolution of human trail systems. Nature 388, 47-50 (doi:10.1038/40353)
67. Jagers, P. 1975 Branching processes with biological applications. New York, NY: Wiley.
68. Jiang, T. X., Widelitz, R. B., Shen, W. M., Will, P., Wu, D. Y., Lin, C. M., Jung, H. S. & Chuong, C. M. 2004 Integument pattern formation involves genetic and epigenetic controls: feather arrays simulated by digital hormone models. Int. J. Dev. Biol. 48, 117-136. (doi:10.1387/ijdb. 15272377)
69. Johnson, K. & Rossi, L. F. 2006 A mathematical and experimental study of ant foraging trail dynamics. J. Theor. Biol. 241, 360-369 (doi:10.1016/j.jtbi.2005.12.003)
70. Kimmel, M. 1980 Cellular population dynamics I and II. Math. Biosci. 48, 201-299.
71. Kolmogorov, A. N., Petrovsky, I. G. & Piskunov, N. S. 1937 Etude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique. Bull. Univ. d'Etat à Moscou, Sér. Intern. A 1, 1-25.
72. Kot, M. 2001 Elements ofmathematical ecology. Cambridge, UK: Cambridge University Press.
73. Kozubowski, T. J. & Podgorski, K. 2007 Invariance properties of the negative binomial Levy process and stochastic self-similarity. Int. Math. Forum 2, 1457-1468.
74. Kraissl, C. J. 1951 The selection of appropriate lines for elective surgical incisions. Plast. Reconstr. Surg. 8, 1-28. (doi:10.1097/00006534-195107000-00001)
75. Kroo, I. 2004 Innovations in aeronautics. In 42nd Aerospace Sciences Meeting 04, pp. 1-11. Reno, NV: AIAA.
76. Laforge, B., Guez, D., Martinez, M. & Kupiec, J. J. 2005 Modeling embryogenesis and cancer: an approach based on an equilibrium between the autostabilization of stochastic gene expression and the interdependence of cells for proliferation. Prog. Biophys. Mol. Biol. 89, 93-120. (doi:10.1016/j.pbiomolbio.2004.11.004)
77. Liu, Y., Guo, Y., Valles, Jr, J. M. & Tang, J. X. 2006 Microtubule bundling and nested buckling drive stripe formation in polymerizing tubulin solutions. Proc. Natl Acad. Sci. USA 103, 10 654-10 659. (doi:10.1073/pnas.0510381103)
78. Mandelkow, E., Mandelkow, E.M., Hotani, H., Hess, B. & Muller, S. C. 1989 Spatial patterns from oscillating microtubules. Science 246, 1291-1293. (doi:10.1126/science.2588005)
79. McCallum, H. 2000 Population parameters: estimation for ecological models. Oxford, UK: Blackwell. McKendrick, A. G. 1926 Applications of mathematics to medical problems. Proc. Edinb. Math. Soc. 44, 98-130. (doi:10.1017/S0013091500034428)
80. Michel, P. 2006 Existence of a solution to the cell division eigenproblem. Math. Models Methods Appl. Sci. 16, 1125-1153. (doi:10.1142/S0218202506001480)
81. Michel, P., Mischler, S. & Perthame, B. 2004 General entropy equations for structured population models and scattering. C.R. Acad. Sci. Serie I 338, 697-702.
82. Michon, F., Forest, L., Collomb, E., Demongeot, J. & Dhouailly, D. 2008 BMP-2 and BMP-7 play antagonistic roles in feather induction. Development 135, 2797-2805. (doi:10.1242/dev.018341)
83. Murray, J. D. 1981 A pre-pattern formation mechanism for animal coat markings. J. Theor. Biol. 88, 161-199. (doi:10.1016/0022-5193(81) 90334-9)
84. Murray, J. D. 1989 Mathematical biology. Berlin, Germany: Springer.
85. Namy, P., Ohayon, J. & Tracqui, P. 2004 Critical conditions for pattern formation and in vitro tubulogenesis driven by cellular traction fields. J. Theor. Biol. 227, 103-120. (doi:10.1016/j.jtbi.2003.10. 015)
86. Park, S. W., Lee, K. M., Lim, Y. H., Kim, H. J., Woo, S. H., Paek, S. H. & Jeon, B. S. 2006 Correlation between phase synchronization of neural activity in patients with Parkinson's disease and the clinical outcome of deep brain stimulation. In Dynamics Day 2006, pp. 52. Bethesda, MD: University of Maryland.
87. Piazzera, S. & Tonetto, L. 2005 Asynchronous exponential growth for an age dependent population equation with delayed birth process. J. Evol. Equ. 5, 61-77. (doi:10.1007/s00028-004-0159-6)
88. Pirollet, F., Job, D., Margolis, R. L. & Garel, J. R. 1987 An oscillatory mode for microtubule assembly. EMBO J. 6, 3247-3252.
89. Prescott, D. M. 1976 The cell cycle and the control of cellular reproduction. Adv. Genet. 18, 99-177. (doi:10.1016/S0065-2660(08) 60438-1)
90. Rashevsky, N. 1938 Mathematical biophysics. Chicago, IL: University of Chicago Press.
91. Ridge, M. D. & Wright, V. 1996 The directional effects of skin. A bio-engineering study of skin with particular reference to Langer's lines. J. Invest. Dermatol. 46, 341-346.
92. Ronot, X., Doisy, A. & Tracqui, P. 2000 Dynamic behaviour of cell monolayers during in vitro wound healing: a quantitative study by optical flow analysis. Cytometry 41, 19-30. (doi:10.1002/1097-0320(20000901)41: 1<19:AID-CYTO3>3.0.CO;2-X)
93. Rueckert, R. R. & Mueller, G. C. 1960 Studies on unbalanced growth in tissue culture. I. Indication and consequences of thymidine deficiency. Cancer Res. 20, 1584-1591.
94. Scherbaum, O. H. 1957 The division index and multiplication in a mass culture of Tetrahymena following inoculation. J. Protozool. 4, 257-259.
95. Scherbaum, O. H. I960 Synchronous division of microorganisms. Ann. Rev. Microbiol. 14, 283-310. (doi:10.ll46/annurev.mi.l4.l00l60. 00l435)
96. Sept, D. 1999 Model for spatial microtubule oscillations. Phys. Rev. E 60, 838-841. (doi:10.1103/PhysRevE.60.838)
97. Sgibnev, M. S. 2001 An asymptotic expansion for the expectation of an age-dependent branching process with a submultiplicative estimate of the remainder. J. Appl. Prob. 38, 807-814. (doi:10.1239/jap/ 1005091046)
98. Spirov, A. V. 1993 Changes of initial symmetry in the pattern-form interaction model of sea urchin early development. J. Theor. Biol. 161, 491-504. (doi:10.1006/jtbi.1993.1069)
99. Tabony, J. 1994 Morphological bifurcations involving reaction-diffusion processes during microtubule formation. Science 264, 245-248. (doi:10.1126/science.8146654)
100. Takahashi, M. 1966 Theoretical basis for cell cycle analysis I. J. Theor. Biol. 13, 202-211. (doi:10.1016/0022-5193(66)90017-8)
101. Takahashi, M. 1968 Theoretical basis for cell cycle analysis II. J. Theor. Biol. 18, 195-209. (doi:10.1016/0022-5193(68)90161-6)
102. Thellier, M., Vincent, J. C., Alexandre, S., Lassalles, J. P., Deschrevel, B., Norris, V. & Ripoll, C. 2003 Biological processes in organised media. C. R. Biol. 326, 149-159. (doi:10.1016/S1631- 0691(03)00062-3)
103. Thellier, M., Demongeot, J., Guespin, J., Ripoll, C., Norris, V. & Thomas, R. 2004 A logical (discrete) formulation model for the storage and recall of environmental signals in plants. Plant Biol. 10, 1055-1075.
104. Theraulaz, G. et al. 2002 Spatial patterns in ant colonies. Proc. Natl Acad. Sci. USA 99, 9645-9649. (doi:10.1073/pnas. 152302199)
105. Thom, R. 1972 Stabilité structurelle et morphogenèse. Essai d'une thérie générale des modèles. New York, NY: Benjamin.
106. Tlüiler, K. 2002 Spatial and temporal distribution of coat patterns of Eurasian Lynx (Lynx lynx) in two re-introduced populations in Switzerland. Zürich, Switzerland: Kora Berichte 13.
107. Tracqui, P. 2006 Mechanical instabilities as a central issue for in silico analysis of cell dynamics. Proc. IEEE Soc. 94, 710-724.
108. Tranquillo, R. T. & Murray, J. D. 1992 Continuum model of fibroblast-driven wound contraction: Inflammation-mediation. J. Theor. Biol. 158, 135-172 (doi:10.1016/S0022-5193(05)80715-5)
109. Tuljapurkar, S. 1982 Why use population entropy? It determines the rate of convergence. J. Math. Biol. 13, 325-337. (doi:10.1007/ BF00276067)
110. Tuljapurkar, S. 2004 Entropy and convergence in dynamics and demography. J. Math. Biol. 31, 253-271. (doi:10.1007/BF00166145)
111. Turing, A. M. 1952 The chemical basis of morphogenesis. Phil. Trans. R. Soc. Lond. B 237, 37-72. (doi:10.1098/rstb.1952.0012)
112. Tyson, J. J. & Novak, B. 2001 Regulation of the eukaryotic cell cycle: molecular antagonism, hysteresis, and irreversible transitions. J. Theor. Biol. 210, 249-263. (doi:10.1006/jtbi.2001.2293)
113. Vincent, J. C. & Thellier, M. 1983 Theoretical analysis of the significance of whether or not enzyme or transport systems in structured media follow Michaelis-Menten kinetics. Biophys. J. 41, 23-28 (doi:10.1016/S0006-3495(83)8440l-4)
114. von Foerster, H. 1959 Some remarks on changing populations. In The kinetics of cellular proliferation (ed. F. Stohlman Jr), pp. 382-407. New York, NY: Grune and Stratton.
115. Williams, N. E. & Scherbaum O. H. 1959 Morphogenetics events in normal and synchronously dividing Tetrahymena. J. Embryol. Exp. Morphol. 7, 241-256.
116. Zeng, W., Thomas, G. L. & Glazier, J. A. 2004 Non-Turing stripes and spots: a novel mechanism for biological cell clustering. Physica A 341, 482-494. (doi:10.1016/j.physa.2004.03.089)
117. Zeuthen, E. 1958 Artificial and induced periodicity in living cells. Adv. Biol. Med. Phys. 6, 37-73.
118. Zeuthen, E. 1964 Synchrony in cell division and growth. New York, NY: Intersciences Publishers Inc.