Bulletin of Mathematical Biology

, Volume 81, Issue 5, pp 1268–1302 | Cite as

Comparison of Deterministic and Stochastic Regime in a Model for Cdc42 Oscillations in Fission Yeast

  • Bin XuEmail author
  • Hye-Won Kang
  • Alexandra JilkineEmail author


Oscillations occur in a wide variety of essential cellular processes, such as cell cycle progression, circadian clocks and calcium signaling in response to stimuli. It remains unclear how intrinsic stochasticity can influence these oscillatory systems. Here, we focus on oscillations of Cdc42 GTPase in fission yeast. We extend our previous deterministic model by Xu and Jilkine to construct a stochastic model, focusing on the fast diffusion case. We use SSA (Gillespie’s algorithm) to numerically explore the low copy number regime in this model, and use analytical techniques to study the long-time behavior of the stochastic model and compare it to the equilibria of its deterministic counterpart. Numerical solutions suggest noisy limit cycles exist in the parameter regime in which the deterministic system converges to a stable limit cycle, and quasi-cycles exist in the parameter regime where the deterministic model has a damped oscillation. Near an infinite period bifurcation point, the deterministic model has a sustained oscillation, while stochastic trajectories start with an oscillatory mode and tend to approach deterministic steady states. In the low copy number regime, metastable transitions from oscillatory to steady behavior occur in the stochastic model. Our work contributes to the understanding of how stochastic chemical kinetics can affect a finite-dimensional dynamical system, and destabilize a deterministic steady state leading to oscillations.


Biochemical oscillations Stochastic model Cell polarity Noise-induced phenomena 



BX is supported by the Robert and Sara Lumpkins Endowment for Postdoctoral Fellows in Applied and Computational Math and Statistics at the University of Notre Dame. HWK is supported by NSF Grant DMS-1620403. AJ is supported by NSF Grant DMS-1615800. AJ and BX acknowledge the assistance of the Notre Dame Center for Research Computing (CRC).

Supplementary material


  1. Altschuler SJ, Angenent SB, Wang Y, Wu LF (2008) On the spontaneous emergence of cell polarity. Nature 454(7206):886–889Google Scholar
  2. Amiranashvili A, Schnellbächer ND, Schwarz US (2016) Stochastic switching between multistable oscillation patterns of the Min-system. New J Phys 18(9):093049Google Scholar
  3. Anderson DF, Enciso GA, Johnston MD (2014) Stochastic analysis of biochemical reaction networks with absolute concentration robustness. J R Soc Interface 11(93):20130943Google Scholar
  4. Anderson DF, Cappelletti D, Kurtz TG (2017) Finite time distributions of stochastically modeled chemical systems with absolute concentration robustness. SIAM J Appl Dyn Syst 16(3):1309–1339MathSciNetzbMATHGoogle Scholar
  5. Ashkenazi M, Othmer HG (1978) Spatial patterns in coupled biochemical oscillators. J Math Biol 5(4):305–350MathSciNetzbMATHGoogle Scholar
  6. Barik D, Paul MR, Baumann WT, Cao Y, Tyson JJ (2008) Stochastic simulation of enzyme-catalyzed reactions with disparate timescales. Biophys J 95(8):3563–3574Google Scholar
  7. Barik D, Ball DA, Peccoud J, Tyson JJ (2016) A stochastic model of the yeast cell cycle reveals roles for feedback regulation in limiting cellular variability. PLoS Comput Biol 12(12):e1005230Google Scholar
  8. Bendezú FO, Vincenzetti V, Vavylonis D, Wyss R, Vogel H, Martin SG (2015) Spontaneous Cdc42 polarization independent of GDI-mediated extraction and actin-based trafficking. PLoS Biol 13(4):e1002097Google Scholar
  9. Benzi R, Sutera A, Vulpiani A (1981) The mechanism of stochastic resonance. J Phys A Math Gen 14(11):L453MathSciNetGoogle Scholar
  10. Bonazzi D, Haupt A, Tanimoto H, Delacour D, Salort D, Minc N (2015) Actin-based transport adapts polarity domain size to local cellular curvature. Curr Biol 25(20):2677–2683Google Scholar
  11. Bressloff PC (2010) Metastable states and quasicycles in a stochastic Wilson-Cowan model of neuronal population dynamics. Phys Rev E 82(5):051903Google Scholar
  12. Chang F, Martin SG (2009) Shaping fission yeast with microtubules. Cold Spring Harbor Perspect Biol 1(1):a001347Google Scholar
  13. Chiang H-D, Thorp JS (1989) Stability regions of nonlinear dynamical systems: a constructive methodology. IEEE Trans Autom Control 34(12):1229–1241MathSciNetzbMATHGoogle Scholar
  14. Das M, Drake T, Wiley DJ, Buchwald P, Vavylonis D, Verde F (2012) Oscillatory dynamics of Cdc42 GTPase in the control of polarized growth. Science 337(6091):239–243Google Scholar
  15. Dauxois T, Di Patti F, Fanelli D, McKane AJ (2009) Enhanced stochastic oscillations in autocatalytic reactions. Phys Rev E 79(3):036112Google Scholar
  16. Enciso GA (2016) Transient absolute robustness in stochastic biochemical networks. J R Soc Interface 13(121):20160475Google Scholar
  17. Endo M, Shirouzu M, Yokoyama S (2003) The Cdc42 binding and scaffolding activities of the fission yeast adaptor protein Scd2. J Biol Chem 278(2):843–852Google Scholar
  18. Erban R, Chapman SJ, Kevrekidis IG, Vejchodskỳ T (2009) Analysis of a stochastic chemical system close to a SNIPER bifurcation of its mean-field model. SIAM J Appl Math 70(3):984–1016MathSciNetzbMATHGoogle Scholar
  19. Etienne-Manneville S, Hall A (2002) Rho GTPases in cell biology. Nature 420(6916):629Google Scholar
  20. Forger DB, Peskin CS (2005) Stochastic simulation of the mammalian circadian clock. Proc Natl Acad Sci 102(2):321–324Google Scholar
  21. Freisinger T, Klünder B, Johnson J, Müller N, Pichler G, Beck G, Costanzo M, Boone C, Cerione RA, Frey E et al (2013) Establishment of a robust single axis of cell polarity by coupling multiple positive feedback loops. Nat Commun 4:1807Google Scholar
  22. Gammaitoni L, Hänggi P, Jung P, Marchesoni F (1998) Stochastic resonance. Rev Mod Phys 70(1):223Google Scholar
  23. Gang H, Ditzinger T, Ning CZ, Haken H (1993) Stochastic resonance without external periodic force. Phys Rev Lett 71(6):807Google Scholar
  24. Gardiner C (2009) Stochastic methods, vol 4. Springer, BerlinzbMATHGoogle Scholar
  25. Geffert PM (2015) Stochastic non-excitable systems with time delay: modulation of noise effects by time-delayed feedback. Springer, BerlinzbMATHGoogle Scholar
  26. Geva-Zatorsky N, Rosenfeld N, Itzkovitz S, Milo R, Sigal A, Dekel E, Yarnitzky T, Liron Y, Polak P, Lahav G et al (2006) Oscillations and variability in the p53 system. Mol Syst Biol 2:2006.0033Google Scholar
  27. Gillespie D (2007) Stochastic simulation of chemical kinetics. Annu Rev Phys Chem 58:35–55Google Scholar
  28. Gonze D, Halloy J, Goldbeter A (2002a) Deterministic versus stochastic models for circadian rhythms. J Biol Phys 28(4):637–653Google Scholar
  29. Gonze D, Halloy J, Goldbeter A (2002b) Robustness of circadian rhythms with respect to molecular noise. Proc Natl Acad Sci 99(2):673–678Google Scholar
  30. Goryachev AB, Leda M (2017) Many roads to symmetry breaking: molecular mechanisms and theoretical models of yeast cell polarity. Mol Biol Cell 28(3):370–380Google Scholar
  31. Hegemann B, Unger M, Lee SS, Stoffel-Studer I, van den Heuvel J, Pelet S, Koeppl H, Peter M (2015) A cellular system for spatial signal decoding in chemical gradients. Dev Cell 35(4):458–470Google Scholar
  32. Howard M, Rutenberg AD (2003) Pattern formation inside bacteria: fluctuations due to the low copy number of proteins. Phys Rev Lett 90(12):128102Google Scholar
  33. Hu J, Kang H-W, Othmer HG (2014) Stochastic analysis of reaction–diffusion processes. Bull Math Biol 76(4):854–894MathSciNetzbMATHGoogle Scholar
  34. Jilkine A, Angenent SB, Wu LF, Altschuler SJ (2011) A density-dependent switch drives stochastic clustering and polarization of signaling molecules. PLoS Comput Biol 7(11):e1002271Google Scholar
  35. Johnson JM, Jin M, Lew DJ (2011) Symmetry breaking and the establishment of cell polarity in budding yeast. Curr Opin Genet Dev 21(6):740–746Google Scholar
  36. Johnston MD, Anderson DF, Craciun G, Brijder R (2018) Conditions for extinction events in chemical reaction networks with discrete state spaces. J Math Biol 76(6):1535–1558MathSciNetzbMATHGoogle Scholar
  37. Kang H-W, Kurtz TG, Popovic L (2014) Central limit theorems and diffusion approximations for multiscale markov chain models. Ann Appl Probab 24(2):721–759MathSciNetzbMATHGoogle Scholar
  38. Kar S, Baumann WT, Paul MR, Tyson JJ (2009) Exploring the roles of noise in the eukaryotic cell cycle. Proc Natl Acad Sci 106(16):6471–6476Google Scholar
  39. Keizer J (1987) Statistical thermodynamics of nonequilibrium processes. Springer, BerlinGoogle Scholar
  40. Kerr RA, Levine H, Sejnowski TJ, Rappel W-J (2006) Division accuracy in a stochastic model of Min oscillations in Escherichia coli. Proc Natl Acad Sci USA 103(2):347–352Google Scholar
  41. Kim JK, Josić K, Bennett MR (2014) The validity of quasi-steady-state approximations in discrete stochastic simulations. Biophys J 107(3):783–793Google Scholar
  42. Klünder B, Freisinger T, Wedlich-Söldner R, Frey E (2013) GDI-mediated cell polarization in yeast provides precise spatial and temporal control of Cdc42 signaling. PLoS Comput Biol 9(12):e1003396Google Scholar
  43. Kuo C-C, Savage NS, Chen H, Wu C-F, Zyla TR, Lew DJ (2014) Inhibitory GEF phosphorylation provides negative feedback in the yeast polarity circuit. Curr Biol 24(7):753–759Google Scholar
  44. Kurtz TG (1971) Limit theorems for sequences of jump markov processes approximating ordinary differential processes. J Appl Probab 8(2):344–356MathSciNetzbMATHGoogle Scholar
  45. Kurtz TG (1972) The relationship between stochastic and deterministic models for chemical reactions. J Chem Phys 57(7):2976–2978Google Scholar
  46. Kuske R, Gordillo LF, Greenwood P (2007) Sustained oscillations via coherence resonance in SIR. J Theor Biol 245(3):459–469MathSciNetGoogle Scholar
  47. Lawson MJ, Drawert B, Khammash M, Petzold L, Yi T-M (2013) Spatial stochastic dynamics enable robust cell polarization. PLoS Comput Biol 9(7):e1003139Google Scholar
  48. Lipan O, Ferwerda C (2018) Hill functions for stochastic gene regulatory networks from master equations with split nodes and time-scale separation. Phys Rev E 97(2):022413Google Scholar
  49. Manninen T, Linne M-L, Ruohonen K (2006) Developing Itô stochastic differential equation models for neuronal signal transduction pathways. Comput Biol Chem 30(4):280–291zbMATHGoogle Scholar
  50. McKane AJ, Newman TJ (2005) Predator–prey cycles from resonant amplification of demographic stochasticity. Phys Rev Lett 94(21):218102Google Scholar
  51. McKane AJ, Nagy JD, Newman TJ, Stefanini MO (2007) Amplified biochemical oscillations in cellular systems. J Stat Phys 128(1–2):165–191MathSciNetzbMATHGoogle Scholar
  52. McKane AJ, Biancalani T, Rogers T (2014) Stochastic pattern formation and spontaneous polarisation: the linear noise approximation and beyond. Bull Math Biol 76(4):895–921MathSciNetzbMATHGoogle Scholar
  53. Othmer HG, Aldridge JA (1978) The effects of cell density and metabolite flux on cellular dynamics. J Math Biol 5(2):169–200MathSciNetzbMATHGoogle Scholar
  54. Pablo M, Ramirez SA, Elston TC (2018) Particle-based simulations of polarity establishment reveal stochastic promotion of Turing pattern formation. PLoS Comput Biol 14(3):e1006016Google Scholar
  55. Pavin N, Paljetak HČ, Krstić V (2006) Min-protein oscillations in Escherichia coli with spontaneous formation of two-stranded filaments in a three-dimensional stochastic reaction-diffusion model. Phys Rev E 73(2):021904Google Scholar
  56. Pikovsky AS, Kurths J (1997) Coherence resonance in a noise-driven excitable system. Phys Rev Lett 78(5):775MathSciNetzbMATHGoogle Scholar
  57. Press WH, Teukolsky SA, Vetterling WT, Flannery BP (1996) Numerical recipes in C, vol 2. Cambridge University Press, CambridgezbMATHGoogle Scholar
  58. Reichenbach T, Mobilia M, Frey E (2006) Coexistence versus extinction in the stochastic cyclic Lotka–Volterra model. Phys Rev E 74(5):051907MathSciNetGoogle Scholar
  59. Samoilov M, Plyasunov S, Arkin AP (2005) Stochastic amplification and signaling in enzymatic futile cycles through noise-induced bistability with oscillations. Proc Natl Acad Sci USA 102(7):2310–2315Google Scholar
  60. Schnoerr D, Sanguinetti G, Grima R (2017) Approximation and inference methods for stochastic biochemical kinetics—a tutorial review. J Phys A Math Theor 50(9):093001MathSciNetzbMATHGoogle Scholar
  61. Slaughter BD, Smith SE, Li R (2009) Symmetry breaking in the life cycle of the budding yeast. Cold Spring Harb Perspect Biol 1(3):a003384Google Scholar
  62. Thomas P, Straube AV, Grima R (2012) The slow-scale linear noise approximation: an accurate, reduced stochastic description of biochemical networks under timescale separation conditions. BMC Syst Biol 6(1):39Google Scholar
  63. Thomas P, Straube AV, Timmer J, Fleck C, Grima R (2013) Signatures of nonlinearity in single cell noise-induced oscillations. J Theor Biol 335:222–234MathSciNetzbMATHGoogle Scholar
  64. Toner DLK, Grima R (2013) Molecular noise induces concentration oscillations in chemical systems with stable node steady states. J Chem Phys 138(5):02B602Google Scholar
  65. Tostevin F, Howard M (2005) A stochastic model of Min oscillations in Escherichia coli and Min protein segregation during cell division. Phys Biol 3(1):1Google Scholar
  66. Ushakov OV, Wünsche H-J, Henneberger F, Khovanov IA, Schimansky-Geier L, Zaks MA (2005) Coherence resonance near a Hopf bifurcation. Phys Rev Lett 95(12):123903Google Scholar
  67. Van Kampen NG (1992) Stochastic processes in physics and chemistry, vol 1. Elsevier, AmsterdamzbMATHGoogle Scholar
  68. Vellela M, Qian H (2007) A quasistationary analysis of a stochastic chemical reaction: Keizer’s paradox. Bull Math Biol 69(5):1727–1746MathSciNetzbMATHGoogle Scholar
  69. Wheatley E, Rittinger K (2005) Interactions between Cdc42 and the scaffold protein Scd2: requirement of SH3 domains for GTPase binding. Biochem J 388(1):177–184Google Scholar
  70. Wilkie J, Wong YM (2008) Positivity preserving chemical langevin equations. Chem Phys 353(1–3):132–138Google Scholar
  71. Wu C-F, Lew DJ (2013) Beyond symmetry-breaking: competition and negative feedback in GTPase regulation. Trends Cell Biol 23(10):476–483Google Scholar
  72. Xu B, Bressloff PC (2016) A PDE–DDE model for cell polarization in fission yeast. SIAM J Appl Math 76(5):1844–1870MathSciNetzbMATHGoogle Scholar
  73. Xu B, Jilkine A (2018) Modeling Cdc42 oscillation in fission yeast. Biophys J 114(3):711–722Google Scholar
  74. Zakharova A, Vadivasova T, Anishchenko V, Koseska A, Kurths J (2010) Stochastic bifurcations and coherencelike resonance in a self-sustained bistable noisy oscillator. Phys Rev E 81(1):011106Google Scholar
  75. Zakharova A, Feoktistov A, Vadivasova T, Schöll E (2013) Coherence resonance and stochastic synchronization in a nonlinear circuit near a subcritical Hopf bifurcation. Eur Phys J Spec Top 222(10):2481–2495Google Scholar

Copyright information

© Society for Mathematical Biology 2019

Authors and Affiliations

  1. 1.Department of Applied and Computational Mathematics and StatisticsUniversity of Notre DameNotre DameUSA
  2. 2.Department of Mathematics and StatisticsUniversity of Maryland Baltimore CountyBaltimoreUSA

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