American Institute of Mathematical Sciences

Advanced
2011, 8(4): 1135-1168. doi: 10.3934/mbe.2011.8.1135

Mathematical analysis of steady-state solutions in compartment and continuum models of cell polarization

Zhenzhen Zheng 1, , Ching-Shan Chou 2, , Tau-Mu Yi 3, and  Qing Nie 4,

1. 

Department of Mathematics, Center for Complex Biological Systems & Center for Mathematical and Computational Biology, University of California, Irvine, CA 92697, United States
2. 

Department of Mathematics, Mathematica Biosciences Institute, The Ohio State University, Columbus, OH 43221, United States
3. 

Developmental and Cell Biology, Center for Complex Biological Systems & Center for Mathematical and Computational Biology, University of California, Irvina, CA 92697, United States
4. 

Department of Mathematics, Center for Complex Biological Systems & Center for Mathematical and Computational Biology, University of California, Irvine, California, 92697-3875

Received  February 2011 Revised  May 2011 Published  August 2011

Cell polarization, in which substances previously uniformly distributed become asymmetric due to external or/and internal stimulation, is a fundamental process underlying cell mobility, cell division, and other polarized functions. The yeast cell S. cerevisiae has been a model system to study cell polarization. During mating, yeast cells sense shallow external spatial gradients and respond by creating steeper internal gradients of protein aligned with the external cue. The complex spatial dynamics during yeast mating polarization consists of positive feedback, degradation, global negative feedback control, and cooperative effects in protein synthesis. Understanding such complex regulations and interactions is critical to studying many important characteristics in cell polarization including signal amplification, tracking dynamic signals, and potential trade-off between achieving both objectives in a robust fashion. In this paper, we study some of these questions by analyzing several models with different spatial complexity: two compartments, three compartments, and continuum in space. The step-wise approach allows detailed characterization of properties of the steady state of the system, providing more insights for biological regulations during cell polarization. For cases without membrane diffusion, our study reveals that increasing the number of spatial compartments results in an increase in the number of steady-state solutions, in particular, the number of stable steady-state solutions, with the continuum models possessing infinitely many steady-state solutions. Through both analysis and simulations, we find that stronger positive feedback, reduced diffusion, and a shallower ligand gradient all result in more steady-state solutions, although most of these are not optimally aligned with the gradient. We explore in the different settings the relationship between the number of steady-state solutions and the extent and accuracy of the polarization. Taken together these results furnish a detailed description of the factors that influence the tradeoff between a single correctly aligned but poorly polarized stable steady-state solution versus multiple more highly polarized stable steady-state solutions that may be incorrectly aligned with the external gradient.
Keywords: modeling., yeast, pheromone gradient, Cell polarization, stability.
Mathematics Subject Classification: 92C15, 92C37, 34D2.
Citation: Zhenzhen Zheng, Ching-Shan Chou, Tau-Mu Yi, Qing Nie. Mathematical analysis of steady-state solutions in compartment and continuum models of cell polarization. Mathematical Biosciences & Engineering, 2011, 8 (4) : 1135-1168. doi: 10.3934/mbe.2011.8.1135
References:
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G. L. Atkins, "Multicompartment Models for Biological Systems,", Willmer Brothers Limited, (1969). Google Scholar

[2]

D. M. Bryant and K. E. Mostov, From cells to organs: Building polarized tissue,, Nature Rev. Mol. Cell Biol., 9 (2008), 887. doi: 10.1038/nrm2523. Google Scholar

[3]

C.-S. Chou, Q. Nie and T. M. Yi, Modeling robustness tradeoffs in yeast cell polarization induced by spatial gradients,, PLoS One, 3 (2008). doi: 10.1371/journal.pone.0003103. Google Scholar

[4]

A. Dawes and L. Edelstein-Keshet, Phosphoinositides and Rho proteins spatially regulate actin polymerization to initiate and maintain directed movement in a 1D model of a motile cell,, Biophys. J., 92 (2007), 1. doi: 10.1529/biophysj.106.090514. Google Scholar

[5]

P. Devreotes and C. Janetopoulos, Eukaryotic chemotaxis: Distinctions between directional sensing and polarization,, J. Biol. Chem., 278 (2003), 20445. doi: 10.1074/jbc.R300010200. Google Scholar

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J. Dobbelaere and Y. Barral, Spatial coordination of cytokinetic events by compartmentalization of the cell cortex,, Science, 305 (2004), 393. doi: 10.1126/science.1099892. Google Scholar

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D. G. Drubin and W. J. Nelson, Origins of cell polarity,, Cell, 84 (1996), 335. doi: 10.1016/S0092-8674(00)81278-7. Google Scholar

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A. B. Goryachev and A. V. Pokhilko, Dynamics of cdc42 network embodies a turing-type mechanism of yeast cell polarity,, FEBS Lett., 582 (2008), 1437. doi: 10.1016/j.febslet.2008.03.029. Google Scholar

[9]

J. Haugh and I. Schneider, Spatial analysis of 3' phosphoinositide signaling in living fibroblasts: I. Uniform stimulation model and bounds on dimensionless groups,, Biophys. J., 86 (2004), 589. doi: 10.1016/S0006-3495(04)74137-5. Google Scholar

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P. A. Iglesias and A. Levchenko, Modeling the cell's guidance system,, Sci STKE, 2002 (2002). doi: 10.1126/stke.2002.148.re12. Google Scholar

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A. Jilkine, A. F. M. Marée and L. Edelstein-Keshet, Mathematical model for spatial segregation of the Rho-family GTPases based on inhibitory crosstalk,, Bull. Math. Biol., 69 (2007), 1943. doi: 10.1007/s11538-007-9200-6. Google Scholar

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J. Krishnan and P. A. Iglesias, Uncovering directional sensing: Where are we headed?,, Syst. Biol. (Stevenage), 1 (2004), 54. doi: 10.1049/sb:20045001. Google Scholar

[13]

J. Krishnan and P. Iglesias, A modeling framework describing the enzyme regulation of membrane lipids underlying gradient perception in Dictyostelium cells,, J. Theor. Biol., 229 (2004), 85. doi: 10.1016/j.jtbi.2004.03.005. Google Scholar

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A. Levchenko and P. A. Iglesias, Models of eukaryotic gradient sensing: Application to chemotaxis of amoebae and neutrophils,, Biophys. J., 82 (2002), 50. doi: 10.1016/S0006-3495(02)75373-3. Google Scholar

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I. Maly, H. Wiley and D. Lauffenburger, Self-organization of polarized cell signaling via autocrine circuits: Computational model analysis,, Biophys. J., 86 (2004), 10. doi: 10.1016/S0006-3495(04)74079-5. Google Scholar

[16]

A. Marée, A. Jilkine, A. Dawes, V. Grieneisen and L. Edelstein-Keshet, Polarization and movement of keratocytes: A multiscale modeling approach,, Bull. Math. Biol., 68 (2006), 1169. Google Scholar

[17]

F. R. Maxfield, Plasma membrane microdomains,, Curr Opin Cell Biol, 14 (2002), 483. doi: 10.1016/S0955-0674(02)00351-4. Google Scholar

[18]

H. Meinhardt, "Models of Biological Pattern Formation,", Academic Press, (1982). Google Scholar

[19]

H. Meinhardt, Orientation of chemotactic cells and growth cones: Models and mechanisms,, J. Cell Sci., 112 (1999), 2867. Google Scholar

[20]

I. Mellman and W. J. Nelson, Coordinated protein sorting, targeting and distribution in polarized cells,, Nature Rev. Mol. Cell Biol., 9 (2008), 833. doi: 10.1038/nrm2525. Google Scholar

[21]

Y. Mori, A. Jilkine and L. Edelstein-Keshet, Wave-pinning and cell polarity from a bistable reaction-diffusion system,, Biophys J., 94 (2008), 3684. doi: 10.1529/biophysj.107.120824. Google Scholar

[22]

A. Narang, Spontaneous polarization in eukaryotic gradient sensing: A mathematical model based on mutual inhibition of frontness and backness pathways,, J. Theor. Biol., 240 (2006), 538. doi: 10.1016/j.jtbi.2005.10.022. Google Scholar

[23]

M. Onsum and C. V. Rao, A mathematical model for neutrophil gradient sensing and polarization,, PLoS Comput. Biol., 3 (2007), 436. doi: 10.1371/journal.pcbi.0030036. Google Scholar

[24]

M. Otsuji, S. Ishihara, C. Co, K. Kaibuchi, A. Mochizuki and S. Kuroda, A mass conserved reaction-diffusion system captures properties of cell polarity,, PLoS Comput. Biol., 3 (2007), 1040. doi: 10.1371/journal.pcbi.0030108. Google Scholar

[25]

D. Pruyne and A. Bretscher, Polarization of cell growth in yeast I. Establishment and maintenance of polarity states,, J. Cell Sci., 113 (2000), 365. Google Scholar

[26]

Y. Sakumura, Y. Tsukada, N. Yamamoto and S. Ishii, A molecular model for axon guidance based on cross talk between Rho GTPases,, Biophys. J., 89 (2005), 812. doi: 10.1529/biophysj.104.055624. Google Scholar

[27]

R. Skupsky, W. Losert and R. Nossal, Distinguishing modes of eukaryotic gradient sensing,, Biophys. J., 89 (2005), 2806. doi: 10.1529/biophysj.105.061564. Google Scholar

[28]

K. Subramanian and A. Narang, A mechanistic model for eukaryotic gradient sensing: Spontaneous and induced phosphoinositide polarization,, J. Theor. Biol., 231 (2004), 49. doi: 10.1016/j.jtbi.2004.05.024. Google Scholar

[29]

D. W. Thompson, "On Growth and Form,", Dover, (1992). Google Scholar

[30]

M. Tomishige, Y. Sako and A. Kusumi, Regulation mechanism of the lateral diffusion of Band 3 in erythrocyte membranes by the membrane skeleton,, J. Cell Biol., 142 (1998), 989. doi: 10.1083/jcb.142.4.989. Google Scholar

[31]

A. M. Turing, The chemical basis of morphogenesis,, Phil. Trans. Roy. Soc. Lond. B, 237 (1952), 37. doi: 10.1098/rstb.1952.0012. Google Scholar

[32]

M. Vicente-Manzanares and F. Sánchez-Madrid, Cell polarization: A comparative cell biology and immunological view,, Clin. Dev. Immunol., 7 (2000), 51. Google Scholar

show all references

References:
[1]

G. L. Atkins, "Multicompartment Models for Biological Systems,", Willmer Brothers Limited, (1969). Google Scholar

[2]

D. M. Bryant and K. E. Mostov, From cells to organs: Building polarized tissue,, Nature Rev. Mol. Cell Biol., 9 (2008), 887. doi: 10.1038/nrm2523. Google Scholar

[3]

C.-S. Chou, Q. Nie and T. M. Yi, Modeling robustness tradeoffs in yeast cell polarization induced by spatial gradients,, PLoS One, 3 (2008). doi: 10.1371/journal.pone.0003103. Google Scholar

[4]

A. Dawes and L. Edelstein-Keshet, Phosphoinositides and Rho proteins spatially regulate actin polymerization to initiate and maintain directed movement in a 1D model of a motile cell,, Biophys. J., 92 (2007), 1. doi: 10.1529/biophysj.106.090514. Google Scholar

[5]

P. Devreotes and C. Janetopoulos, Eukaryotic chemotaxis: Distinctions between directional sensing and polarization,, J. Biol. Chem., 278 (2003), 20445. doi: 10.1074/jbc.R300010200. Google Scholar

[6]

J. Dobbelaere and Y. Barral, Spatial coordination of cytokinetic events by compartmentalization of the cell cortex,, Science, 305 (2004), 393. doi: 10.1126/science.1099892. Google Scholar

[7]

D. G. Drubin and W. J. Nelson, Origins of cell polarity,, Cell, 84 (1996), 335. doi: 10.1016/S0092-8674(00)81278-7. Google Scholar

[8]

A. B. Goryachev and A. V. Pokhilko, Dynamics of cdc42 network embodies a turing-type mechanism of yeast cell polarity,, FEBS Lett., 582 (2008), 1437. doi: 10.1016/j.febslet.2008.03.029. Google Scholar

[9]

J. Haugh and I. Schneider, Spatial analysis of 3' phosphoinositide signaling in living fibroblasts: I. Uniform stimulation model and bounds on dimensionless groups,, Biophys. J., 86 (2004), 589. doi: 10.1016/S0006-3495(04)74137-5. Google Scholar

[10]

P. A. Iglesias and A. Levchenko, Modeling the cell's guidance system,, Sci STKE, 2002 (2002). doi: 10.1126/stke.2002.148.re12. Google Scholar

[11]

A. Jilkine, A. F. M. Marée and L. Edelstein-Keshet, Mathematical model for spatial segregation of the Rho-family GTPases based on inhibitory crosstalk,, Bull. Math. Biol., 69 (2007), 1943. doi: 10.1007/s11538-007-9200-6. Google Scholar

[12]

J. Krishnan and P. A. Iglesias, Uncovering directional sensing: Where are we headed?,, Syst. Biol. (Stevenage), 1 (2004), 54. doi: 10.1049/sb:20045001. Google Scholar

[13]

J. Krishnan and P. Iglesias, A modeling framework describing the enzyme regulation of membrane lipids underlying gradient perception in Dictyostelium cells,, J. Theor. Biol., 229 (2004), 85. doi: 10.1016/j.jtbi.2004.03.005. Google Scholar

[14]

A. Levchenko and P. A. Iglesias, Models of eukaryotic gradient sensing: Application to chemotaxis of amoebae and neutrophils,, Biophys. J., 82 (2002), 50. doi: 10.1016/S0006-3495(02)75373-3. Google Scholar

[15]

I. Maly, H. Wiley and D. Lauffenburger, Self-organization of polarized cell signaling via autocrine circuits: Computational model analysis,, Biophys. J., 86 (2004), 10. doi: 10.1016/S0006-3495(04)74079-5. Google Scholar

[16]

A. Marée, A. Jilkine, A. Dawes, V. Grieneisen and L. Edelstein-Keshet, Polarization and movement of keratocytes: A multiscale modeling approach,, Bull. Math. Biol., 68 (2006), 1169. Google Scholar

[17]

F. R. Maxfield, Plasma membrane microdomains,, Curr Opin Cell Biol, 14 (2002), 483. doi: 10.1016/S0955-0674(02)00351-4. Google Scholar

[18]

H. Meinhardt, "Models of Biological Pattern Formation,", Academic Press, (1982). Google Scholar

[19]

H. Meinhardt, Orientation of chemotactic cells and growth cones: Models and mechanisms,, J. Cell Sci., 112 (1999), 2867. Google Scholar

[20]

I. Mellman and W. J. Nelson, Coordinated protein sorting, targeting and distribution in polarized cells,, Nature Rev. Mol. Cell Biol., 9 (2008), 833. doi: 10.1038/nrm2525. Google Scholar

[21]

Y. Mori, A. Jilkine and L. Edelstein-Keshet, Wave-pinning and cell polarity from a bistable reaction-diffusion system,, Biophys J., 94 (2008), 3684. doi: 10.1529/biophysj.107.120824. Google Scholar

[22]

A. Narang, Spontaneous polarization in eukaryotic gradient sensing: A mathematical model based on mutual inhibition of frontness and backness pathways,, J. Theor. Biol., 240 (2006), 538. doi: 10.1016/j.jtbi.2005.10.022. Google Scholar

[23]

M. Onsum and C. V. Rao, A mathematical model for neutrophil gradient sensing and polarization,, PLoS Comput. Biol., 3 (2007), 436. doi: 10.1371/journal.pcbi.0030036. Google Scholar

[24]

M. Otsuji, S. Ishihara, C. Co, K. Kaibuchi, A. Mochizuki and S. Kuroda, A mass conserved reaction-diffusion system captures properties of cell polarity,, PLoS Comput. Biol., 3 (2007), 1040. doi: 10.1371/journal.pcbi.0030108. Google Scholar

[25]

D. Pruyne and A. Bretscher, Polarization of cell growth in yeast I. Establishment and maintenance of polarity states,, J. Cell Sci., 113 (2000), 365. Google Scholar

[26]

Y. Sakumura, Y. Tsukada, N. Yamamoto and S. Ishii, A molecular model for axon guidance based on cross talk between Rho GTPases,, Biophys. J., 89 (2005), 812. doi: 10.1529/biophysj.104.055624. Google Scholar

[27]

R. Skupsky, W. Losert and R. Nossal, Distinguishing modes of eukaryotic gradient sensing,, Biophys. J., 89 (2005), 2806. doi: 10.1529/biophysj.105.061564. Google Scholar

[28]

K. Subramanian and A. Narang, A mechanistic model for eukaryotic gradient sensing: Spontaneous and induced phosphoinositide polarization,, J. Theor. Biol., 231 (2004), 49. doi: 10.1016/j.jtbi.2004.05.024. Google Scholar

[29]

D. W. Thompson, "On Growth and Form,", Dover, (1992). Google Scholar

[30]

M. Tomishige, Y. Sako and A. Kusumi, Regulation mechanism of the lateral diffusion of Band 3 in erythrocyte membranes by the membrane skeleton,, J. Cell Biol., 142 (1998), 989. doi: 10.1083/jcb.142.4.989. Google Scholar

[31]

A. M. Turing, The chemical basis of morphogenesis,, Phil. Trans. Roy. Soc. Lond. B, 237 (1952), 37. doi: 10.1098/rstb.1952.0012. Google Scholar

[32]

M. Vicente-Manzanares and F. Sánchez-Madrid, Cell polarization: A comparative cell biology and immunological view,, Clin. Dev. Immunol., 7 (2000), 51. Google Scholar

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