# American Institute of Mathematical Sciences

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

## Equilibrium submanifold for a biological system

 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.
Citation: Hongyu He, Naohiro Kato. Equilibrium submanifold for a biological system. Discrete and 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. [2] V. I. Arnold, "Ordinary Differential Equations," Springer, New York, 2006. [3] B. Aulbach, "Continuous and Discrete Dynamics Near Manifolds of Equilibria," Lecture Notes in Mathematics, 1058. Springer-Verlag, Berlin, 1984. [4] J. Bonifacino and B. Glick, The mechanisms of vesicle budding and fusion, Cell, 116 (2004), 153-166. [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. [6] T. Chen, H. He and G. Church, Modeling gene expression with differential equations, "Pacific Symposium on Biocomputing' 99," World Scientific (2000), 29-40. [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. [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. [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. [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. [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. [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. [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. [14] C. Taubes, "Modeling Differential Equations in Biology," Second edition. Cambridge University Press, Cambridge, 2008 [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.

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##### 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. [2] V. I. Arnold, "Ordinary Differential Equations," Springer, New York, 2006. [3] B. Aulbach, "Continuous and Discrete Dynamics Near Manifolds of Equilibria," Lecture Notes in Mathematics, 1058. Springer-Verlag, Berlin, 1984. [4] J. Bonifacino and B. Glick, The mechanisms of vesicle budding and fusion, Cell, 116 (2004), 153-166. [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. [6] T. Chen, H. He and G. Church, Modeling gene expression with differential equations, "Pacific Symposium on Biocomputing' 99," World Scientific (2000), 29-40. [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. [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. [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. [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. [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. [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. [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. [14] C. Taubes, "Modeling Differential Equations in Biology," Second edition. Cambridge University Press, Cambridge, 2008 [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.
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