American Institute of Mathematical Sciences

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December  2011, 4(6): 1429-1441. doi: 10.3934/dcdss.2011.4.1429

Equilibrium submanifold for a biological system

Hongyu He 1, and  Naohiro Kato 2,

1. 

Department of Mathematics, Louisiana State University, Baton Rouge, LA 7080, United States
2. 

Department of Biology, Louisiana State University, Baton Rouge, LA 70803, United States

Received  March 2009 Revised  October 2009 Published  December 2010

The complexity in a biological system may be caused by both the number of variables involved and the number of system constants that can vary. A biological system in the subcellular level often stabilizes after a certain period of time. Its asymptote can then be described as an equilibrium under certain continuity assumptions. The biological quantities at the equilibrium can be detected by experiments and they observe some mathematical equations. The purpose of this paper is to study the equilibrium submanifold of vesicle trafficking in a two-compartment system. We compute the equilibrium submanifold under some fairly general assumption on the system constants. The disconnectedness of the equilibrium submanifold may have biological implications. We show that, unlike many other systems, the equilibrium is determined largely by system constants rather than the initial state. In particular, the equilibrium submanifold is locally a real algebraic variety, with small generic dimension and large degenerate dimension. Our result suggests that some biological system may be studied by algebraic or geometric methods.
Keywords: Vesicle Trafficking, Biological Systems., Equilibrium, Nonlinear ODE.
Mathematics Subject Classification: Primary: 92C; Secondary: 92.
Citation: Hongyu He, Naohiro Kato. Equilibrium submanifold for a biological system. Discrete & Continuous Dynamical Systems - S, 2011, 4 (6) : 1429-1441. doi: 10.3934/dcdss.2011.4.1429
References:
[1]

B. Alberts, J. Lewins, M. Raff, K. Roberts and P. Walter, editor., Intracellular vesicular traffic, in "Molecular Biology of the Cell," Fifth ed. New York, Garland Science, (2008), 749-812. Google Scholar

[2]

V. I. Arnold, "Ordinary Differential Equations," Springer, New York, 2006.  Google Scholar

[3]

B. Aulbach, "Continuous and Discrete Dynamics Near Manifolds of Equilibria," Lecture Notes in Mathematics, 1058. Springer-Verlag, Berlin, 1984.  Google Scholar

[4]

J. Bonifacino and B. Glick, The mechanisms of vesicle budding and fusion, Cell, 116 (2004), 153-166. Google Scholar

[5]

H. Cai, K. Reinisch and S. Ferro-Novick, Coats, tethers, Rabs, and SNAREs work together to mediate the intracellular destination of a transport vesicle, Dev. Cell, 12 (2007), 671-682. doi: 10.1016/j.devcel.2007.04.005.  Google Scholar

[6]

T. Chen, H. He and G. Church, Modeling gene expression with differential equations, "Pacific Symposium on Biocomputing' 99," World Scientific (2000), 29-40. Google Scholar

[7]

N. C. Collins, H. Thordal-Christensen, V. Lipka, S. Bau, E. Kombrink, J. L Qiu, R. Huckelhoven, M. Stein, A. Freialdenhoven, S. C. Somerville and P. Schulze-Lefert, SNARE-protein-mediated disease resistance at the plant cell wall, Nature, 425 (2003), 973-977. doi: 10.1038/nature02076.  Google Scholar

[8]

N. Kato, H. He and A. Steger, A systems model of vesicle trafficking in Arabidopsis pollen tubes, Plant Physiol., 152 (2010), 590-601. doi: 10.1104/pp.109.148700.  Google Scholar

[9]

M. de Maio, Therapeutic uses of botulinum toxin: From facial palsy to autonomic disorders, Expert. Opin. Biol. Ther., 8 (2008), 791-798. doi: 10.1517/14712598.8.6.791.  Google Scholar

[10]

H. Gong, D. Sengupta, A. Linstedt and R. Schwartz, Simulated de novo assembly of Golgi compartments by selective cargo capture during vesicle budding and targeted vesicle fusion, Biophysical Journal, 95 (2008), 1674-1688. doi: 10.1529/biophysj.107.127498.  Google Scholar

[11]

R. Heinrich and T. Rapoport, Generation of nonidentical compartments in vesicular transport systems, Journal of Cell Biology, 168 (2005), 271-280. doi: 10.1083/jcb.200409087.  Google Scholar

[12]

R. Jahn and R. H. Scheller, SNAREs-engines for membrane fusion, Nat. Rev. Mol. Cell Biol., 7 (2006), 631-643. doi: 10.1038/nrm2002.  Google Scholar

[13]

J. Samaj, J. Muller, M. Beck, N. Bohm and D. Menzel, Vesicular trafficking, cytoskeleton and signalling in root hairs and pollen tubes, Trends Plant Sci., 11 ( 2006), 594-600. doi: 10.1016/j.tplants.2006.10.002.  Google Scholar

[14]

C. Taubes, "Modeling Differential Equations in Biology," Second edition. Cambridge University Press, Cambridge, 2008 Google Scholar

[15]

J. H. Williams, Novelties of the flowering plant pollen tube underlie diversification of a key life history stage, Proc. Natl. Acad. Sci. USA, 105 (2008), 11259-11263. doi: 10.1073/pnas.0800036105.  Google Scholar

show all references

References:
[1]

B. Alberts, J. Lewins, M. Raff, K. Roberts and P. Walter, editor., Intracellular vesicular traffic, in "Molecular Biology of the Cell," Fifth ed. New York, Garland Science, (2008), 749-812. Google Scholar

[2]

V. I. Arnold, "Ordinary Differential Equations," Springer, New York, 2006.  Google Scholar

[3]

B. Aulbach, "Continuous and Discrete Dynamics Near Manifolds of Equilibria," Lecture Notes in Mathematics, 1058. Springer-Verlag, Berlin, 1984.  Google Scholar

[4]

J. Bonifacino and B. Glick, The mechanisms of vesicle budding and fusion, Cell, 116 (2004), 153-166. Google Scholar

[5]

H. Cai, K. Reinisch and S. Ferro-Novick, Coats, tethers, Rabs, and SNAREs work together to mediate the intracellular destination of a transport vesicle, Dev. Cell, 12 (2007), 671-682. doi: 10.1016/j.devcel.2007.04.005.  Google Scholar

[6]

T. Chen, H. He and G. Church, Modeling gene expression with differential equations, "Pacific Symposium on Biocomputing' 99," World Scientific (2000), 29-40. Google Scholar

[7]

N. C. Collins, H. Thordal-Christensen, V. Lipka, S. Bau, E. Kombrink, J. L Qiu, R. Huckelhoven, M. Stein, A. Freialdenhoven, S. C. Somerville and P. Schulze-Lefert, SNARE-protein-mediated disease resistance at the plant cell wall, Nature, 425 (2003), 973-977. doi: 10.1038/nature02076.  Google Scholar

[8]

N. Kato, H. He and A. Steger, A systems model of vesicle trafficking in Arabidopsis pollen tubes, Plant Physiol., 152 (2010), 590-601. doi: 10.1104/pp.109.148700.  Google Scholar

[9]

M. de Maio, Therapeutic uses of botulinum toxin: From facial palsy to autonomic disorders, Expert. Opin. Biol. Ther., 8 (2008), 791-798. doi: 10.1517/14712598.8.6.791.  Google Scholar

[10]

H. Gong, D. Sengupta, A. Linstedt and R. Schwartz, Simulated de novo assembly of Golgi compartments by selective cargo capture during vesicle budding and targeted vesicle fusion, Biophysical Journal, 95 (2008), 1674-1688. doi: 10.1529/biophysj.107.127498.  Google Scholar

[11]

R. Heinrich and T. Rapoport, Generation of nonidentical compartments in vesicular transport systems, Journal of Cell Biology, 168 (2005), 271-280. doi: 10.1083/jcb.200409087.  Google Scholar

[12]

R. Jahn and R. H. Scheller, SNAREs-engines for membrane fusion, Nat. Rev. Mol. Cell Biol., 7 (2006), 631-643. doi: 10.1038/nrm2002.  Google Scholar

[13]

J. Samaj, J. Muller, M. Beck, N. Bohm and D. Menzel, Vesicular trafficking, cytoskeleton and signalling in root hairs and pollen tubes, Trends Plant Sci., 11 ( 2006), 594-600. doi: 10.1016/j.tplants.2006.10.002.  Google Scholar

[14]

C. Taubes, "Modeling Differential Equations in Biology," Second edition. Cambridge University Press, Cambridge, 2008 Google Scholar

[15]

J. H. Williams, Novelties of the flowering plant pollen tube underlie diversification of a key life history stage, Proc. Natl. Acad. Sci. USA, 105 (2008), 11259-11263. doi: 10.1073/pnas.0800036105.  Google Scholar

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