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November  2019, 24(11): 5989-6004. doi: 10.3934/dcdsb.2019117

Dynamic transitions and stability for the acetabularia whorl formation

1. 

Department of Mathematics, Indiana University, Bloomington, IN 47405, USA

2. 

School of Data Sciences, Zhejiang University of Finance & Economics, Hangzhou, 310018, China

* Corresponding author: Yiqiu Mao

Received  October 2018 Published  June 2019

Dynamical transitions of the Acetabularia whorl formation caused by outside calcium concentration is carefully analyzed using a chemical reaction diffusion model on a thin annulus. Restricting ourselves with Turing instabilities, we found all three types of transition, continuous, catastrophic and random can occur under different parameter regimes. Detailed linear analysis and numerical investigations are also provided. The main tool used in the transition analysis is Ma & Wang's dynamical transition theory including the center manifold reduction.

Citation: Yiqiu Mao, Dongming Yan, ChunHsien Lu. Dynamic transitions and stability for the acetabularia whorl formation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 5989-6004. doi: 10.3934/dcdsb.2019117
References:
[1]

A. M. Ashu, Some Properties of Bessel Functions with Applications to Neumann Eigenvalues in the Unit Disc, Bachelor's Theses in Mathematical Sciences, Lund University, 2013.Google Scholar

[2]

J. Dumais and L. G. Harrison, Whorl morphogenesis in the dasycladalean algae: The pattern formation viewpoint, Philosophical Transactions of the Royal Society of London B: Biological Sciences, 355 (2000), 281-305. doi: 10.1098/rstb.2000.0565. Google Scholar

[3]

J. DumaisK. Serikawa and D. F. Mandoli, Acetabularia: A unicellular model for understanding subcellular localization and morphogenesis during development, Journal of Plant Growth Regulation, 19 (2000), 253-264. doi: 10.1007/s003440000035. Google Scholar

[4] B. Goodwin, How the Leopard Changed Its Spots: The Evolution of Complexity, Princeton University Press, 2001. Google Scholar
[5]

B. Goodwin, J. Murray and D. Baldwin, Calcium: The elusive morphogen in acetabularia, in Proc. 6th Intern. Symp. on Acetabularia. Belgian Nuclear Center, CEN-SCK Mol, Belgium, (1984), 101–108.Google Scholar

[6]

B. C. Goodwin and L. Trainor, Tip and whorl morphogenesis in acetabularia by calcium-regulated strain fields, Journal of theoretical biology, 117 (1985), 79-106. doi: 10.1016/S0022-5193(85)80165-X. Google Scholar

[7]

L. G. Harrison, Reaction-diffusion theory and intracellular differentiation, International Journal of Plant Sciences, 153 (1992), S76–S85. doi: 10.1086/297065. Google Scholar

[8]

L. G. HarrisonJ. SnellR. VerdiD. VogtG. Zeiss and B. R. Green, Hair morphogenesis inacetabularia mediterranea: Temperature-dependent spacing and models of morphogen waves, Protoplasma, 106 (1981), 211-221. doi: 10.1007/BF01275553. Google Scholar

[9]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840. Springer-Verlag, Berlin-New York, 1981. Google Scholar

[10]

T. Ma and S. Wang, Bifurcation Theory and Applications, vol. 53, World Scientific, 2005. doi: 10.1142/9789812701152. Google Scholar

[11]

T. Ma and S. Wang, Phase Transition Dynamics, Springer, 2014. doi: 10.1007/978-1-4614-8963-4. Google Scholar

[12]

L. Martynov, A morphogenetic mechanism involving instability of initial forth, Journal of Theoretical Biology, 52 (1975), 471-480. doi: 10.1016/0022-5193(75)90013-2. Google Scholar

[13]

J. D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications, Third edition. Interdisciplinary Applied Mathematics, 18. Springer-Verlag, New York, 2003. Google Scholar

[14]

C. L. Siegel, Über einige anwendungen diophantischer approximationen, in On Some Applications of Diophantine Approximations, Springer, 2 (2014), 81–138. Google Scholar

[15] G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge university press, 1995. Google Scholar
[16]

Y. You, Global dynamics of the brusselator equations, Dynamics of Partial Differential Equations, 4 (2007), 167-196. doi: 10.4310/DPDE.2007.v4.n2.a4. Google Scholar

show all references

References:
[1]

A. M. Ashu, Some Properties of Bessel Functions with Applications to Neumann Eigenvalues in the Unit Disc, Bachelor's Theses in Mathematical Sciences, Lund University, 2013.Google Scholar

[2]

J. Dumais and L. G. Harrison, Whorl morphogenesis in the dasycladalean algae: The pattern formation viewpoint, Philosophical Transactions of the Royal Society of London B: Biological Sciences, 355 (2000), 281-305. doi: 10.1098/rstb.2000.0565. Google Scholar

[3]

J. DumaisK. Serikawa and D. F. Mandoli, Acetabularia: A unicellular model for understanding subcellular localization and morphogenesis during development, Journal of Plant Growth Regulation, 19 (2000), 253-264. doi: 10.1007/s003440000035. Google Scholar

[4] B. Goodwin, How the Leopard Changed Its Spots: The Evolution of Complexity, Princeton University Press, 2001. Google Scholar
[5]

B. Goodwin, J. Murray and D. Baldwin, Calcium: The elusive morphogen in acetabularia, in Proc. 6th Intern. Symp. on Acetabularia. Belgian Nuclear Center, CEN-SCK Mol, Belgium, (1984), 101–108.Google Scholar

[6]

B. C. Goodwin and L. Trainor, Tip and whorl morphogenesis in acetabularia by calcium-regulated strain fields, Journal of theoretical biology, 117 (1985), 79-106. doi: 10.1016/S0022-5193(85)80165-X. Google Scholar

[7]

L. G. Harrison, Reaction-diffusion theory and intracellular differentiation, International Journal of Plant Sciences, 153 (1992), S76–S85. doi: 10.1086/297065. Google Scholar

[8]

L. G. HarrisonJ. SnellR. VerdiD. VogtG. Zeiss and B. R. Green, Hair morphogenesis inacetabularia mediterranea: Temperature-dependent spacing and models of morphogen waves, Protoplasma, 106 (1981), 211-221. doi: 10.1007/BF01275553. Google Scholar

[9]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840. Springer-Verlag, Berlin-New York, 1981. Google Scholar

[10]

T. Ma and S. Wang, Bifurcation Theory and Applications, vol. 53, World Scientific, 2005. doi: 10.1142/9789812701152. Google Scholar

[11]

T. Ma and S. Wang, Phase Transition Dynamics, Springer, 2014. doi: 10.1007/978-1-4614-8963-4. Google Scholar

[12]

L. Martynov, A morphogenetic mechanism involving instability of initial forth, Journal of Theoretical Biology, 52 (1975), 471-480. doi: 10.1016/0022-5193(75)90013-2. Google Scholar

[13]

J. D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications, Third edition. Interdisciplinary Applied Mathematics, 18. Springer-Verlag, New York, 2003. Google Scholar

[14]

C. L. Siegel, Über einige anwendungen diophantischer approximationen, in On Some Applications of Diophantine Approximations, Springer, 2 (2014), 81–138. Google Scholar

[15] G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge university press, 1995. Google Scholar
[16]

Y. You, Global dynamics of the brusselator equations, Dynamics of Partial Differential Equations, 4 (2007), 167-196. doi: 10.4310/DPDE.2007.v4.n2.a4. Google Scholar

Figure 1.  Classification in the parameter space
Figure 2.  $ n_c $ plot against $ a $ and $ d $, numbers inside indicates critical wave number $ n_c $ when $ \lambda $ crosses $ \lambda_c $
Figure 3.  Critical Eigenvector $ \cos{n_c\theta}R_{n_c, J_c}(r) $ with $ n_c = 6, j_c = 3 $ and $ \delta = 1.2 $, dark or bright regions indicate potential whorl hair growth, in this graph we have around 18 growth regions
Table 1.  Calculation for $ q(\lambda_c) $
$ \delta $ $ a $ $ d $ $ R $ $ (n_c, j_c) $ $ q(\lambda_c)(u_c-v_c) $
$ 1.05 $ $ 0.2 $ $ 13 $ $ 4 $ $ (2, 1) $ $ 73.5557 $
$ 1.05 $ $ 0.2 $ $ 15 $ $ 4 $ $ (2, 1) $ $ 147.17 $
$ 1.05 $ $ 0.2 $ $ 20 $ $ 4 $ $ (2, 1) $ $ 240.462 $
$ 1.05 $ $ 0.2 $ $ 80 $ $ 4 $ $ ( 1, 1) $ $ 82.1464 $
$ 1.05 $ $ 0.4 $ $ 65 $ $ 4 $ $ ( 2, 1) $ $ 79.4266 $
$ 1.05 $ $ 0.2 $ $ 13 $ $ 10 $ $ (5, 1) $ $ 459.715 $
$ 1.05 $ $ 0.4 $ $ 80 $ $ 10 $ $ ( 4, 1) $ $ 527.142 $
$ 1.2 $ $ 0.2 $ $ 15 $ $ 4 $ $ (2, 1 ) $ $ 31.0966 $
$ 1.2 $ $ 0.2 $ $ 15 $ $ 20 $ $ ( 9, 1) $ $ 523.768 $
$ 1.2 $ $ 0.4 $ $ 175 $ $ 4 $ $ ( 1, 1) $ $ -1.97781 $
$ 2 $ $ 0.2 $ $ 30 $ $ 4 $ $ ( 2, 1) $ $ 5.66053 $
$ 2 $ $ 0.4 $ $ 60 $ $ 4 $ $ ( 3, 1) $ $ 2.0822 $
$ 2 $ $ 0.4 $ $ 100 $ $ 4 $ $ ( 2, 1) $ $ 2.479171 $
$ 8 $ $ 0.4 $ $ 80 $ $ 10 $ $ ( 10, 5) $ $ 6.07011 $
$ \delta $ $ a $ $ d $ $ R $ $ (n_c, j_c) $ $ q(\lambda_c)(u_c-v_c) $
$ 1.05 $ $ 0.2 $ $ 13 $ $ 4 $ $ (2, 1) $ $ 73.5557 $
$ 1.05 $ $ 0.2 $ $ 15 $ $ 4 $ $ (2, 1) $ $ 147.17 $
$ 1.05 $ $ 0.2 $ $ 20 $ $ 4 $ $ (2, 1) $ $ 240.462 $
$ 1.05 $ $ 0.2 $ $ 80 $ $ 4 $ $ ( 1, 1) $ $ 82.1464 $
$ 1.05 $ $ 0.4 $ $ 65 $ $ 4 $ $ ( 2, 1) $ $ 79.4266 $
$ 1.05 $ $ 0.2 $ $ 13 $ $ 10 $ $ (5, 1) $ $ 459.715 $
$ 1.05 $ $ 0.4 $ $ 80 $ $ 10 $ $ ( 4, 1) $ $ 527.142 $
$ 1.2 $ $ 0.2 $ $ 15 $ $ 4 $ $ (2, 1 ) $ $ 31.0966 $
$ 1.2 $ $ 0.2 $ $ 15 $ $ 20 $ $ ( 9, 1) $ $ 523.768 $
$ 1.2 $ $ 0.4 $ $ 175 $ $ 4 $ $ ( 1, 1) $ $ -1.97781 $
$ 2 $ $ 0.2 $ $ 30 $ $ 4 $ $ ( 2, 1) $ $ 5.66053 $
$ 2 $ $ 0.4 $ $ 60 $ $ 4 $ $ ( 3, 1) $ $ 2.0822 $
$ 2 $ $ 0.4 $ $ 100 $ $ 4 $ $ ( 2, 1) $ $ 2.479171 $
$ 8 $ $ 0.4 $ $ 80 $ $ 10 $ $ ( 10, 5) $ $ 6.07011 $
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