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 & 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, (2008), 749.   Google Scholar

[2]

V. I. Arnold, "Ordinary Differential Equations,", Springer, (2006).   Google Scholar

[3]

B. Aulbach, "Continuous and Discrete Dynamics Near Manifolds of Equilibria,", Lecture Notes in Mathematics, (1058).   Google Scholar

[4]

J. Bonifacino and B. Glick, The mechanisms of vesicle budding and fusion,, Cell, 116 (2004), 153.   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.  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, (2000), 29.   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.  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.  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.  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.  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.  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.  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.  doi: 10.1016/j.tplants.2006.10.002.  Google Scholar

[14]

C. Taubes, "Modeling Differential Equations in Biology,", Second edition. Cambridge University Press, (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.  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, (2008), 749.   Google Scholar

[2]

V. I. Arnold, "Ordinary Differential Equations,", Springer, (2006).   Google Scholar

[3]

B. Aulbach, "Continuous and Discrete Dynamics Near Manifolds of Equilibria,", Lecture Notes in Mathematics, (1058).   Google Scholar

[4]

J. Bonifacino and B. Glick, The mechanisms of vesicle budding and fusion,, Cell, 116 (2004), 153.   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.  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, (2000), 29.   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.  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.  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.  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.  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.  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.  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.  doi: 10.1016/j.tplants.2006.10.002.  Google Scholar

[14]

C. Taubes, "Modeling Differential Equations in Biology,", Second edition. Cambridge University Press, (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.  doi: 10.1073/pnas.0800036105.  Google Scholar

[1]

Susanna Terracini, Juncheng Wei. DCDS-A Special Volume Qualitative properties of solutions of nonlinear elliptic equations and systems. Preface. Discrete & Continuous Dynamical Systems - A, 2014, 34 (6) : i-ii. doi: 10.3934/dcds.2014.34.6i

[2]

P. Adda, J. L. Dimi, A. Iggidir, J. C. Kamgang, G. Sallet, J. J. Tewa. General models of host-parasite systems. Global analysis. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 1-17. doi: 10.3934/dcdsb.2007.8.1

[3]

Franziska Hinkelmann, Reinhard Laubenbacher. Boolean models of bistable biological systems. Discrete & Continuous Dynamical Systems - S, 2011, 4 (6) : 1443-1456. doi: 10.3934/dcdss.2011.4.1443

[4]

Denis de Carvalho Braga, Luis Fernando Mello, Carmen Rocşoreanu, Mihaela Sterpu. Lyapunov coefficients for non-symmetrically coupled identical dynamical systems. Application to coupled advertising models. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 785-803. doi: 10.3934/dcdsb.2009.11.785

[5]

Anna Kostianko, Sergey Zelik. Inertial manifolds for 1D reaction-diffusion-advection systems. Part Ⅰ: Dirichlet and Neumann boundary conditions. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2357-2376. doi: 10.3934/cpaa.2017116

[6]

Anna Kostianko, Sergey Zelik. Inertial manifolds for 1D reaction-diffusion-advection systems. Part Ⅱ: periodic boundary conditions. Communications on Pure & Applied Analysis, 2018, 17 (1) : 285-317. doi: 10.3934/cpaa.2018017

[7]

Takanobu Okazaki. Large time behaviour of solutions of nonlinear ode describing hysteresis. Conference Publications, 2007, 2007 (Special) : 804-813. doi: 10.3934/proc.2007.2007.804

[8]

Junyong Eom, Kazuhiro Ishige. Large time behavior of ODE type solutions to nonlinear diffusion equations. Discrete & Continuous Dynamical Systems - A, 2019, 0 (0) : 1-15. doi: 10.3934/dcds.2019229

[9]

Martin Brokate, Pavel Krejčí. Optimal control of ODE systems involving a rate independent variational inequality. Discrete & Continuous Dynamical Systems - B, 2013, 18 (2) : 331-348. doi: 10.3934/dcdsb.2013.18.331

[10]

Wandi Ding. Optimal control on hybrid ODE Systems with application to a tick disease model. Mathematical Biosciences & Engineering, 2007, 4 (4) : 633-659. doi: 10.3934/mbe.2007.4.633

[11]

Ruyuan Zhang. Hopf bifurcations of ODE systems along the singular direction in the parameter plane. Communications on Pure & Applied Analysis, 2005, 4 (2) : 445-461. doi: 10.3934/cpaa.2005.4.445

[12]

John Bissell, Brian Straughan. Discontinuity waves as tipping points: Applications to biological & sociological systems. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 1911-1934. doi: 10.3934/dcdsb.2014.19.1911

[13]

Liping Zhang. A nonlinear complementarity model for supply chain network equilibrium. Journal of Industrial & Management Optimization, 2007, 3 (4) : 727-737. doi: 10.3934/jimo.2007.3.727

[14]

Youshan Tao, J. Ignacio Tello. Nonlinear stability of a heterogeneous state in a PDE-ODE model for acid-mediated tumor invasion. Mathematical Biosciences & Engineering, 2016, 13 (1) : 193-207. doi: 10.3934/mbe.2016.13.193

[15]

Mikhaël Balabane, Mustapha Jazar, Philippe Souplet. Oscillatory blow-up in nonlinear second order ODE's: The critical case. Discrete & Continuous Dynamical Systems - A, 2003, 9 (3) : 577-584. doi: 10.3934/dcds.2003.9.577

[16]

Karl Kunisch, Sérgio S. Rodrigues. Oblique projection based stabilizing feedback for nonautonomous coupled parabolic-ode systems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6355-6389. doi: 10.3934/dcds.2019276

[17]

MirosŁaw Lachowicz, Tatiana Ryabukha. Equilibrium solutions for microscopic stochastic systems in population dynamics. Mathematical Biosciences & Engineering, 2013, 10 (3) : 777-786. doi: 10.3934/mbe.2013.10.777

[18]

Zaki Chbani, Hassan Riahi. Existence and asymptotic behaviour for solutions of dynamical equilibrium systems. Evolution Equations & Control Theory, 2014, 3 (1) : 1-14. doi: 10.3934/eect.2014.3.1

[19]

Ivan Werner. Equilibrium states and invariant measures for random dynamical systems. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 1285-1326. doi: 10.3934/dcds.2015.35.1285

[20]

Lei Wang, Jinlong Yuan, Yingfang Li, Enmin Feng, Zhilong Xiu. Parameter identification of nonlinear delayed dynamical system in microbial fermentation based on biological robustness. Numerical Algebra, Control & Optimization, 2014, 4 (2) : 103-113. doi: 10.3934/naco.2014.4.103

2018 Impact Factor: 0.545

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]