February  2014, 7(1): 35-51. doi: 10.3934/dcdss.2014.7.35

Existence and linear stability of solutions of the ballistic VSC model

1. 

VU University Amsterdam, Faculty of Sciences, De Boelelaan 1081, 1081 HV Amsterdam, Netherlands

2. 

TU Eindhoven, Faculty of Mathematics and Computer Science, P.O. Box 513, 5600 MB Eindhoven, Netherlands, Netherlands

Received  February 2012 Revised  March 2013 Published  July 2013

An equation for the dynamics of the vesicle supply center model of tip growth in fungal hyphae is derived. For this we analytically prove the existence and uniqueness of a traveling wave solution which exhibits the experimentally observed behavior. The linearized dynamics around this solution is analyzed and we conclude that all eigenmodes decay in time. Numerical calculation of the first eigenvalue gives a timescale in which small perturbations will die out.
Citation: Joost Hulshof, Robert Nolet, Georg Prokert. Existence and linear stability of solutions of the ballistic VSC model. Discrete and Continuous Dynamical Systems - S, 2014, 7 (1) : 35-51. doi: 10.3934/dcdss.2014.7.35
References:
[1]

S. Bartnicki-Garcia, F. Hergert and G. Giertz, Computer simulation of fungal morphogenesis and the mathematical basis for hyphal (tip) growth, Protoplasma, 153 (1989), 46-57. doi: 10.1007/BF01322464.

[2]

S. Bartnicki-Garcia, C. E. Bracker and G. Giertz, Mapping the growth of fungal hyphae: Orthogonal cell wall expansion during tip growth and the role of turgor, Biophys. J., 79 (2000), 2382-2390. doi: 10.1016/S0006-3495(00)76483-6.

[3]

S. Bartnicki-Garcia and G. Giertz, A three-dimensional model of fungal morphogenesis based on the vesicle supply center concept, J. Theor. Biol., 208 (2001), 151-164.

[4]

N. G. de Bruijn, "Asymptotic Methods in Analysis," Third edition, North-Holland Publishing Company, 1994.

[5]

E. Eggen, "Self-Regulating Tip Growth, Modeling Cell Wall Ageing," Master's Thesis, Utrecht University, 2006.

[6]

C. Gerhardt, Flow of Nonconvex Hypersurfaces into Spheres, J. Diff Geom., 32 (1990), 299-314.

[7]

A. Goriely and M. Tabor, Self-similar tip growth in filamentary organisms, Phys. Rev. Lett., 90 (2003), 108101. doi: 10.1103/PhysRevLett.90.108101.

[8]

G. Huisken and T. Ilmanen, A note on the inverse mean curvature flow, in "Proc. Workshop on Nonl. Part. Diff. Equ.," Saitama University, Sept. 1997. Available from: http://www.math.ethz.ch/~ilmanen/papers/saitama.ps

[9]

G. Huisken and T. Ilmanen, Higher regularity of the inverse mean curvature flow, J. Diff Geom., 80 (2008), 433-451. Available from: http://www.math.ethz.ch/~ilmanen/papers/imcfharnack.ps

[10]

A. Koch, The problem of hyphal growth in streptomycetes and fungi, J. Theor. Biol., 171 (1994), 137-150. doi: 10.1006/jtbi.1994.1219.

[11]

S. Tindemans, "Modeling Tip Growth in Fungal Hyphae," Master's thesis, University of Amsterdam, 2004. doi: 10.1016/j.jtbi.2005.07.004.

[12]

S. Tindemans, N. Kern and B. Mulder, The diffusive vesicle supply center model for tip growth in fungal hyphae, J. Theor. Biol., 238 (2006), 937-948. doi: 10.1016/j.jtbi.2005.07.004.

[13]

H. Triebel, "Higher Analysis," Hochschulbücher für Mathematik [University Books for Mathematics], Johann Ambrosius Barth Verlag GmbH, Leipzig, 1992.

[14]

J. Urbas, On the expansion of starshaped hypersurfaces by symmetric function of their principal curvatures, Math. Z., 205 (1990), 355-372. doi: 10.1007/BF02571249.

[15]

J. Weidmann, "Linear Operators in Hilbert Spaces," Graduate Texts in Mathematics, 68, Springer-Verlag, New York-Berlin, 1980

show all references

References:
[1]

S. Bartnicki-Garcia, F. Hergert and G. Giertz, Computer simulation of fungal morphogenesis and the mathematical basis for hyphal (tip) growth, Protoplasma, 153 (1989), 46-57. doi: 10.1007/BF01322464.

[2]

S. Bartnicki-Garcia, C. E. Bracker and G. Giertz, Mapping the growth of fungal hyphae: Orthogonal cell wall expansion during tip growth and the role of turgor, Biophys. J., 79 (2000), 2382-2390. doi: 10.1016/S0006-3495(00)76483-6.

[3]

S. Bartnicki-Garcia and G. Giertz, A three-dimensional model of fungal morphogenesis based on the vesicle supply center concept, J. Theor. Biol., 208 (2001), 151-164.

[4]

N. G. de Bruijn, "Asymptotic Methods in Analysis," Third edition, North-Holland Publishing Company, 1994.

[5]

E. Eggen, "Self-Regulating Tip Growth, Modeling Cell Wall Ageing," Master's Thesis, Utrecht University, 2006.

[6]

C. Gerhardt, Flow of Nonconvex Hypersurfaces into Spheres, J. Diff Geom., 32 (1990), 299-314.

[7]

A. Goriely and M. Tabor, Self-similar tip growth in filamentary organisms, Phys. Rev. Lett., 90 (2003), 108101. doi: 10.1103/PhysRevLett.90.108101.

[8]

G. Huisken and T. Ilmanen, A note on the inverse mean curvature flow, in "Proc. Workshop on Nonl. Part. Diff. Equ.," Saitama University, Sept. 1997. Available from: http://www.math.ethz.ch/~ilmanen/papers/saitama.ps

[9]

G. Huisken and T. Ilmanen, Higher regularity of the inverse mean curvature flow, J. Diff Geom., 80 (2008), 433-451. Available from: http://www.math.ethz.ch/~ilmanen/papers/imcfharnack.ps

[10]

A. Koch, The problem of hyphal growth in streptomycetes and fungi, J. Theor. Biol., 171 (1994), 137-150. doi: 10.1006/jtbi.1994.1219.

[11]

S. Tindemans, "Modeling Tip Growth in Fungal Hyphae," Master's thesis, University of Amsterdam, 2004. doi: 10.1016/j.jtbi.2005.07.004.

[12]

S. Tindemans, N. Kern and B. Mulder, The diffusive vesicle supply center model for tip growth in fungal hyphae, J. Theor. Biol., 238 (2006), 937-948. doi: 10.1016/j.jtbi.2005.07.004.

[13]

H. Triebel, "Higher Analysis," Hochschulbücher für Mathematik [University Books for Mathematics], Johann Ambrosius Barth Verlag GmbH, Leipzig, 1992.

[14]

J. Urbas, On the expansion of starshaped hypersurfaces by symmetric function of their principal curvatures, Math. Z., 205 (1990), 355-372. doi: 10.1007/BF02571249.

[15]

J. Weidmann, "Linear Operators in Hilbert Spaces," Graduate Texts in Mathematics, 68, Springer-Verlag, New York-Berlin, 1980

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