Non-linear analysis of a model for yeast cell communication. (English) Zbl 1434.35251
Summary: We study the non-linear stability of a coupled system of two non-linear transport-diffusion equations set in two opposite half-lines. This system describes some aspects of yeast pairwise cellular communication, through the concentration of some protein in the cell bulk and at the cell boundary. We show that it is of bistable type, provided that the intensity of active molecular transport is large enough. We prove the non-linear stability of the most concentrated steady state, for large initial data, by entropy and comparison techniques. For small initial data we prove the self-similar decay of the molecular concentration towards zero. Informally speaking, the rise of a dialog between yeast cells requires enough active molecular transport in this model. Besides, if the cells do not invest enough in the communication with their partner, they do not respond to each other; but a sufficient initial input from each cell in the dialog leads to the establishment of a stable activated state in both cells.
MSC:
| 35Q92 | PDEs in connection with biology, chemistry and other natural sciences |
| 35B32 | Bifurcations in context of PDEs |
| 35B35 | Stability in context of PDEs |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35K40 | Second-order parabolic systems |
| 92B05 | General biology and biomathematics |
| 92C17 | Cell movement (chemotaxis, etc.) |
| 92C37 | Cell biology |
Keywords:
nonlinear stability; asymptotic convergence; logarithmic Sobolev inequality; HWI inequalityReferences:
| [1] | B. Alberts, A. Johnson, J. Lewis, M. Raff, K. Roberts and P. Walter, Molecular Biology of the Cell, 5th edition. Garland Science (2007). |
| [2] | A. Arnold, P. Markowich, G. Toscani and A. Unterreiter, On convex sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations. Commun. Partial Differ. Equ. 26 (2001) 43-100. · Zbl 0982.35113 · doi:10.1081/PDE-100002246 |
| [3] | M. Bagnat and K. Simons, Cell surface polarization during yeast mating. Proc. Nat. Acad. Sci. 99 (2002) 14183-14188. · doi:10.1073/pnas.172517799 |
| [4] | P. Biler, G. Karch, P. Laurençot and T. Nadzieja, The 8π-problem for radially symmetric solutions of a chemotaxis model in the plane. Math. Methods Appl. Sci. 29 (2006) 1563-1583. · Zbl 1105.35131 |
| [5] | A. Blanchet, On the parabolic-elliptic Patlak-Keller-Segel system in dimension 2 and higher, Séminaire Laurent Schwartz - EDP et applications (2011-2012). · Zbl 1326.35400 |
| [6] | M.J. Cáceres and R. Schneider, Blow-up, steady states and long time behaviour of excitatory-inhibitory nonlinear neuron models. Kinet. Relat. Models 10 (2017) 587-612. · Zbl 1358.35062 · doi:10.3934/krm.2017024 |
| [7] | M.J. Cáceres, J.A. Carrillo and B. Perthame, Analysis of nonlinear noisy integrate & fire neuron models: blow-up and steady states. J. Math. Neurosci. 1 (2011) 7. · Zbl 1259.35198 · doi:10.1186/2190-8567-1-7 |
| [8] | V. Calvez, N. Meunier and R. Voituriez, A one-dimensional Keller-Segel equation with a drift issued from the boundary. C. R. Math. Acad. Sci. Paris 348 (2010) 629-634. · Zbl 1197.35149 · doi:10.1016/j.crma.2010.04.009 |
| [9] | V. Calvez, R. Hawkins, N. Meunier and R. Voituriez, Analysis of a nonlocal model for spontaneous cell polarization. SIAM J. Appl. Math. 72 (2012) 594-622. · Zbl 1248.35024 |
| [10] | J.A. Carrillo, M.M. González, M.P. Gualdani and M.E. Schonbek, Classical solutions for a nonlinear Fokker-Planck equation arising in computational neuroscience. Commun. Partial Differ. Equ. 38 (2013) 385-409. · Zbl 1282.35382 · doi:10.1080/03605302.2012.747536 |
| [11] | J.A. Carrillo, B. Perthame, D. Salort and D. Smets, Qualitative properties of solutions for the noisy integrate and fire model in computational neuroscience. Nonlinearity 28 (2015) 3365. · Zbl 1336.35332 |
| [12] | W. Chen, Q. Nie, T.-M. Yi and C.-S. Chou, Modelling of yeast mating reveals robustness strategies for cell-cell interactions. PLoS Comput. Biol. 12 (2016) e1004988. |
| [13] | C.-S. Chou, Q. Nie and T.-M. Yi, Modeling robustness tradeoffs in yeast cell polarization induced by spatial gradients. PLoS One 3 (2008) e3103. · doi:10.1371/journal.pone.0003103 |
| [14] | C.-S. Chou, L. Bardwell, Q. Nie and T.-M. Yi, Noise filtering tradeoffs in spatial gradient sensing and cell polarization response. BMC Syst. Biol. 5 (2011) 196. |
| [15] | I. Csiszár, Information-type measures of difference of probability distributions and indirect observations. Studia Sci. Math. Hungar. 2 (1967) 299-318. · Zbl 0157.25802 |
| [16] | G. Dumont, J. Henry and C.O. Tarniceriu, Noisy threshold in neuronal models: connections with the noisy leaky integrate-and-fire model. J. Math. Biol. 73 (2016) 1413-1436. · Zbl 1358.35202 · doi:10.1007/s00285-016-1002-8 |
| [17] | J.M. Dyer, N.S. Savage, M. Jin, T.R. Zyla, T.C. Elston and D.J. Lew, Tracking shallow chemical gradients by actin-driven wandering of the polarization site. Curr. Biol. 23 (2013) 32-41. · doi:10.1016/j.cub.2012.11.014 |
| [18] | L.C. Evans, Partial differential equations, 2nd edition. In: Vol. 19 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI (2010). · Zbl 1194.35001 · doi:10.1090/gsm/019 |
| [19] | T. Freisinger, B. Klünder, J. Johnson, N. Müller, G. Pichler, G. Beck, M. Costanzo, C. Boone, R.A. Cerione, E. Frey and R. Wedlich-Soldner, Establishment of a robust single axis of cell polarity by coupling multiple positive feedback loops. Nat. Commun. 4 (2013) 1807. |
| [20] | H. Gajewski, On a variant of monotonicity and its application to differential equations. Nonlinear Anal.: Theory Methods App. 22 (1994) 73-80. · Zbl 0821.35002 · doi:10.1016/0362-546X(94)90006-X |
| [21] | R. Hawkins, O. Bénichou, M. Piel and R. Voituriez, Rebuilding cytoskeleton roads: active transport induced polarisation of cells. Phys. Rev. E 80 (2009) 040903. |
| [22] | C.L. Jackson and L.H. Hartwell, Courtship in s. cerevisiae: both cell types choose mating partners by responding to the strongest pheromone signal. cell 63 (1990) 1039-1051. |
| [23] | M. Jin, B. Errede, M. Behar, W. Mather, S. Nayak, J. Hasty, H.G. Dohlman and T.C. Elston, Yeast dynamically modify their environment to achieve better mating efficiency. Sci. Signal. 4 (2011) ra54-ra54. |
| [24] | I. Kim and Y. Yao, The Patlak-Keller-Segel model and its variations: properties of solutions via maximum principle. SIAM J. Math. Anal. 44 (2012) 568-602. · Zbl 1261.35080 · doi:10.1137/110823584 |
| [25] | S. Kullback, On the convergence of discrimination information. IEEE Trans. Inf. Theory IT-14 (1968) 765-766. |
| [26] | M.J. Lawson, B. Drawert, M. Khammash, L. Petzold and T.-M. Yi, Spatial stochastic dynamics enable robust cell polarization. PLoS Comput. Biol. 9 (2013) e1003139. |
| [27] | A.T. Layton, N.S. Savage, A.S. Howell, S.Y. Carroll, D.G. Drubin and D.J. Lew, Modeling vesicle traffic reveals unexpected consequences for Cdc42p-mediated polarity establishment. Curr. Biol. 21 (2011) 184-194. · doi:10.1016/j.cub.2011.01.012 |
| [28] | T. Lepoutre, N. Meunier and N. Muller, Cell polarisation model: the 1D case. J. Math. Pures Appl. 101 (2014) 152-171. · Zbl 1282.35196 |
| [29] | K. Madden and M. Snyder, Cell polarity and morphogenesis in budding yeast. Ann. Rev. Microbiol. 52 (1998) 687-744. PMID: 9891811. · doi:10.1146/annurev.micro.52.1.687 |
| [30] | T.I. Moore, C.-S. Chou, Q. Nie, N.L. Jeon and T.-M. Yi, Robust spatial sensing of mating pheromone gradients by yeast cells. PLoS One 3 (2008) e3865. · doi:10.1371/journal.pone.0003865 |
| [31] | T.I. Moore, H. Tanaka, H.J. Kim, N.L. Jeon and T.-M. Yi, Yeast G-proteins mediate directional sensing and polarization behaviors in response to changes in pheromone gradient direction. Mol. Biol. Cell 24 (2013) 521-34. · doi:10.1091/mbc.e12-10-0739 |
| [32] | N. Muller, M. Piel, V. Calvez, R. Voituriez, J. Gonçalves-Sá, C.-L. Guo, X. Jiang, A. Murray and N. Meunier, A predictive model for yeast cell polarization in pheromone gradients. PLoS Comput. Biol. 12 (2016) e1004795. |
| [33] | N.S. Savage, A.T. Layton and D.J. Lew, Mechanistic mathematical model of polarity in yeast. Mol. Biol. Cell 23 (2012) 1998-2013. · doi:10.1091/mbc.e11-10-0837 |
| [34] | M.-N. Simon, C. De Virgilio, B. Souza, J.R. Pringle, A. Abo and S.I. Reed, Role for the rho-family gtpase cdc42 in yeast mating-pheromone signal pathway. Nature 376 (1995) 702-705. |
| [35] | B.D. Slaughter, S.E. Smith and R. Li, Symmetry breaking in the life cycle of the budding yeast. Cold Spring Harbor Perspect. Biol. 1 (2009) a003384. · doi:10.1101/cshperspect.a003384 |
| [36] | B.D. Slaughter, J.R. Unruh, A. Das, S.E. Smith, B. Rubinstein and R. Li, Non-uniform membrane diffusion enables steady-state cell polarization via vesicular trafficking. Nat. Commun. 4 (2013) 1380. |
| [37] | C. Villani, Topics in optimal transportation. In: Vol. 58 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI (2003). · Zbl 1106.90001 · doi:10.1090/gsm/058 |
| [38] | R. Wedlich-Soldner, S. Altschuler, L. Wu and R. Li, Spontaneous cell polarization through actomyosin-based delivery of the cdc42 gtpase. Science 299 (2003) 1231-1235. |
| [39] | R. Wedlich-Soldner, S.C. Wai, T. Schmidt and R. Li, Robust cell polarity is a dynamic state established by coupling transport and gtpase signaling. J. Cell Biol. 166 (2004) 889-900. · doi:10.1083/jcb.200405061 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
