Advertisement

Bulletin of Mathematical Biology

, Volume 79, Issue 7, pp 1539–1563 | Cite as

Parametric Sensitivity Analysis of Oscillatory Delay Systems with an Application to Gene Regulation

  • Brian Ingalls
  • Maya Mincheva
  • Marc R. Roussel
Original Article

Abstract

A parametric sensitivity analysis for periodic solutions of delay-differential equations is developed. Because phase shifts cause the sensitivity coefficients of a periodic orbit to diverge, we focus on sensitivities of the extrema, from which amplitude sensitivities are computed, and of the period. Delay-differential equations are often used to model gene expression networks. In these models, the parametric sensitivities of a particular genotype define the local geometry of the evolutionary landscape. Thus, sensitivities can be used to investigate directions of gradual evolutionary change. An oscillatory protein synthesis model whose properties are modulated by RNA interference is used as an example. This model consists of a set of coupled delay-differential equations involving three delays. Sensitivity analyses are carried out at several operating points. Comments on the evolutionary implications of the results are offered.

Keywords

Sensitivity analysis Periodic solutions Delay-differential equations Gene expression RNA interference 

Notes

Acknowledgements

This research was supported by the Natural Sciences and Engineering Research Council of Canada.

References

  1. Acerenza L, Sauro HM, Kacser H (1989) Control analysis of time-dependent metabolic systems. J Theor Biol 137:423–444. doi: 10.1016/S0022-5193(89)80038-4 CrossRefGoogle Scholar
  2. Ahsen ME, Özbay H, Niculescu SI (2010) Analysis of deterministic cyclic gene regulatory network models with delays. Birkhäuser, Cham. doi: 10.1007/978-3-319-15606-4 zbMATHGoogle Scholar
  3. Baek D, Villén J, Shin C, Camargo FD, Gygi SP, Bartel DP (2008) The impact of microRNAs on protein output. Nature 455:64–71. doi: 10.1038/nature07242 CrossRefGoogle Scholar
  4. Baker CT, Rihan FA (1999) Sensitivity analysis of parameters in modelling with delay-differential equations. Tech. Rep. 349, Manchester Centre for Computational MathematicsGoogle Scholar
  5. Banks H, Robbins D, Sutton KL (2013a) Generalized sensitivity analysis for delay differential equations. In: Control and Optimization with PDE Constraints, Springer, pp 19–44Google Scholar
  6. Banks HT, Robbins D, Sutton KL (2013) Theoretical foundations for traditional and generalized sensitivity functions for nonlinear delay differential equations. Math Biosci Eng 10:1301–1333. doi: 10.3934/mbe.2013.10.1301 MathSciNetCrossRefzbMATHGoogle Scholar
  7. Bliss RD, Painter PR, Marr AG (1982) Role of feedback inhibition in stabilizing the classical operon. J Theor Biol 97:177–193. doi: 10.1016/0022-5193(82)90098-4 CrossRefGoogle Scholar
  8. Bocharov GA, Rihan FA (2000) Numerical modelling in biosciences using delay differential equations. J Comput Appl Math 125:183–199. doi: 10.1016/S0377-0427(00)00468-4 MathSciNetCrossRefzbMATHGoogle Scholar
  9. Boström K, Wettesten M, Borén J, Bondjers G, Wiklund O, Olofsson SO (1986) Pulse-chase studies of the synthesis and intracellular transport of apolipoprotein B-100 in Hep G2 cells. J Biol Chem 261:13,800–13,806Google Scholar
  10. Buchholtz F, Schneider FW (1987) Computer simulation of T3/T7 phage infection using lag times. Biophys Chem 26:171–179. doi: 10.1016/0301-4622(87)80020-0 CrossRefGoogle Scholar
  11. Bueler E, Butcher E (2002) Stability of periodic linear delay-differential equations and the Chebyshev approximation of fundamental solutions, preprintGoogle Scholar
  12. Bure E, Rozenvasser E (1974) The study of the sensitivity of oscillatory systems. Autom Remote Control 35:1045–1052MathSciNetzbMATHGoogle Scholar
  13. Busenberg SN, Mahaffy JM (1988) The effects of dimension and size for a compartmental model of repression. SIAM J Appl Math 48:882–903. doi: 10.1137/0148049 MathSciNetCrossRefzbMATHGoogle Scholar
  14. Butcher EA, Ma H, Bueler E, Averina V, Szabo Z (2004) Stability of linear time-periodic delay-differential equations via Chebyshev polynomials. Int J Numer Methods Eng 59:895–922MathSciNetCrossRefzbMATHGoogle Scholar
  15. Cinquin O, Demongeot J (2002) Roles of positive and negative feedback in biological systems. C R Biol 325:1085–1095. doi: 10.1016/S1631-0691(02)01533-0 CrossRefGoogle Scholar
  16. Cooke KL, Grossman Z (1982) Discrete delay, distributed delay and stability switches. J Math Anal Appl 86:592–627. doi: 10.1016/0022-247X(82)90243-8 MathSciNetCrossRefzbMATHGoogle Scholar
  17. Cornish-Bowden A, Cárdenas ML (eds) (1990) Control of metabolic processes. Plenum, New YorkGoogle Scholar
  18. Danø S, Madsen MF, Sørensen PG (2005) Chemical interpretation of oscillatory modes at a Hopf point. Phys Chem Chem Phys 7:1674–1679. doi: 10.1039/B415437A CrossRefGoogle Scholar
  19. Darzacq X, Shav-Tal Y, de Turris V, Brody Y, Shenoy SM, Phair RD, Singer RH (2007) In vivo dynamics of RNA polymerase II transcription. Nat Struct Mol Biol 14:796–806. doi: 10.1038/nsmb1280 CrossRefGoogle Scholar
  20. Dill H, Linder B, Fehr A, Fischer U (2012) Intronic miR-26b controls neuronal differentiation by repressing its host transcript, ctdsp2. Genes Dev 26:25–30. doi: 10.1101/gad.177774.111 CrossRefGoogle Scholar
  21. Driver RD (1962) Existence and stability of solutions of a delay-differential system. Arch Ration Mech Anal 10:401–426. doi: 10.1007/BF00281203 MathSciNetCrossRefzbMATHGoogle Scholar
  22. Ebert MS, Sharp PA (2012) Roles for microRNAs in conferring robustness to biological processes. Cell 149:515–524. doi: 10.1016/j.cell.2012.04.005 CrossRefGoogle Scholar
  23. Eden E, Geva-Zatorsky N, Issaeva I, Cohen A, Dekel E, Danon T, Cohen L, Mayo A, Alon U (2011) Proteome half-life dynamics in living human cells. Science 331:764–768. doi: 10.1126/science.1199784 CrossRefGoogle Scholar
  24. Edmunds LN Jr (1988) Cellular and molecular bases of biological clocks. Springer, New YorkGoogle Scholar
  25. Engelborghs K, Luzyanina T, Roose D (2002) Numerical bifurcation analysis of delay differential equations using DDE-BIFTOOL. ACM Trans Math Softw 28:1–21. doi: 10.1145/513001.513002 MathSciNetCrossRefzbMATHGoogle Scholar
  26. Epstein IR (1990) Differential delay equations in chemical kinetics: some simple linear model systems. J Chem Phys 92:1702–1712. doi: 10.1063/1.458052 CrossRefGoogle Scholar
  27. Ermentrout B (2002) Simulating, analyzing, and animating dynamical systems. SIAM, PhiladelphiaCrossRefzbMATHGoogle Scholar
  28. Falaleeva M, Stamm S (2013) Processing of snoRNAs as a new source of regulatory non-coding RNAs. BioEssays 35:46–54. doi: 10.1002/bies.201200117 CrossRefGoogle Scholar
  29. Fell DA (1992) Metabolic control analysis: a survey of its theoretical and experimental development. Biochem J 286:313–330. doi: 10.1042/bj2860313 CrossRefGoogle Scholar
  30. Feng J, Sevier SA, Huang B, Jia D, Levine H (2016) Modeling delayed processes in biological systems. Phys Rev E 94:032408. doi: 10.1103/PhysRevE.94.032408 CrossRefGoogle Scholar
  31. Ferrell JE Jr (1996) Tripping the switch fantastic: how a protein kinase can convert graded inputs into switch-like outputs. Trends Biochem Sci 21:460–466. doi: 10.1016/S0968-0004(96)20026-X CrossRefGoogle Scholar
  32. Ferrell JE Jr, Xiong W (2001) Bistability in cell signaling: how to make continuous processes discontinuous, and reversible processes irreversible. Chaos 11:227–236. doi: 10.1063/1.1349894 CrossRefzbMATHGoogle Scholar
  33. Goldbeter A (1996) Biochemical oscillations and cellular rhythms. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
  34. Goodwin BC (1963) Temporal organization in cells. Academic Press, LondonGoogle Scholar
  35. Halaney A (1966) Differential Equations. Academic Press, New York, Stability, Oscillations, Time LagsGoogle Scholar
  36. Hale JK, Ladeira LAC (1991) Differentiability with respect to delays. J Differ Equ 92:14–26. doi: 10.1016/0022-0396(91)90061-D MathSciNetCrossRefzbMATHGoogle Scholar
  37. Hale JK, Lunel SMV (1993) Introduction to functional differential equations. Springer, New YorkCrossRefzbMATHGoogle Scholar
  38. Heinrich R, Reder C (1991) Metabolic control analysis of relaxation processes. J Theor Biol 151:343–350. doi: 10.1016/S0022-5193(05)80383-2 CrossRefGoogle Scholar
  39. Heinrich R, Schuster S (1996) The regulation of cellular systems. Chapman & Hall, New YorkCrossRefzbMATHGoogle Scholar
  40. Highkin HR, Hanson JB (1954) Possible interaction between light-dark cycles and endogeneous daily rhythms on the growth of tomato plants. Plant Physiol 29:301–302. doi: 10.1104/pp.29.3.301 CrossRefGoogle Scholar
  41. Hill AV (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol (Lond) 40:4–7. doi: 10.1113/jphysiol.1910.sp001386 (Suppl.)
  42. Hillman WS (1956) Injury of tomato plants by continuous light and unfavorable photoperiodic cycles. Am J Bot 43:89–96CrossRefGoogle Scholar
  43. Ingalls B (2008) Sensitivity analysis: from model parameters to system behaviour. Essays Biochem 45:177–193. doi: 10.1042/bse0450177 CrossRefGoogle Scholar
  44. Ingalls BP (2004) Autonomously oscillating biochemical systems: parametric sensitivity of extrema and period. Syst Biol 1:62–70. doi: 10.1049/sb:20045005 CrossRefGoogle Scholar
  45. Ingalls BP (2013) Mathematical modeling in systems biology. MIT Press, CambridgezbMATHGoogle Scholar
  46. Ingalls BP, Sauro HM (2003) Sensitivity analysis of stoichiometric networks: an extension of metabolic control analysis to non-steady state trajectories. J Theor Biol 222:23–36. doi: 10.1016/S0022-5193(03)00011-0 MathSciNetCrossRefGoogle Scholar
  47. Izaurralde E (2015) Breakers and blockers–miRNAs at work. Science 349:380–382. doi: 10.1126/science.1260969 CrossRefGoogle Scholar
  48. Johnson CH (2001) Endogenous timekeepers in photosynthetic organisms. Annu Rev Physiol 63:695–728. doi: 10.1146/annurev.physiol.63.1.695 CrossRefGoogle Scholar
  49. Kacser H, Burns JA (1973) The control of flux. Symp Soc Exp Biol 27:65–104Google Scholar
  50. Kholodenko BN, Demin OV, Westerhoff HV (1997) Control analysis of periodic phenomena in biological systems. J Phys Chem B 101:2070–2081. doi: 10.1021/jp962336u CrossRefGoogle Scholar
  51. Klarsfeld A, Rouyer F (1998) Effects of circadian mutations and LD periodicity on the life span of Drosophila melanogaster. J Biol Rhythms 13:471–478CrossRefGoogle Scholar
  52. Lander ES, Linton LM, Birren B, Nusbaum C et al (2001) Initial sequencing and analysis of the human genome. Nature 409:860–921. doi: 10.1038/35057062, errata: Nature 412, 565–566
  53. Lapidot M, Pilpel Y (2006) Genome-wide natural antisense transcription: coupling its regulation to its different regulatory mechanisms. EMBO Rep 7:1216–1222. doi: 10.1038/sj.embor.7400857 CrossRefGoogle Scholar
  54. Larter R (1983) Sensitivity analysis of autonomous oscillators. Separation of secular terms and determination of structural stability. J Phys Chem 87:3114–3121. doi: 10.1021/j100239a032 CrossRefGoogle Scholar
  55. Lenz SM, Schlöder JP, Bock HG (2014) Numerical computation of derivatives in systems of delay differential equations. Math Comput Simul 96:124–156. doi: 10.1016/j.matcom.2013.08.003 MathSciNetCrossRefGoogle Scholar
  56. Lewis J (2003) Autoinhibition with transcriptional delay: a simple mechanism for the zebrafish somitogenesis oscillator. Curr Biol 13:1398–1408. doi: 10.1016/S0960-9822(03)00534-7 CrossRefGoogle Scholar
  57. Long X, Insperger T, Balachandran B (2009) Systems with periodic coefficients and periodically varying delays: semidiscretization-based stability analysis. In: Delay Differential Equations, Springer, pp 131–153Google Scholar
  58. MacDonald N (1977) Time lag in a model of a biochemical reaction sequence with end product inhibition. J Theor Biol 67:549–556. doi: 10.1016/0022-5193(77)90056-X MathSciNetCrossRefGoogle Scholar
  59. MacDonald N (1987) An interference effect of independent delays. IEE Proc D 134:38–42CrossRefzbMATHGoogle Scholar
  60. Meijer HA, Kong YW, Lu WT, Wilczynska A, Spriggs RV, Robinson SW, Godfrey JD, Willis AE, Bushell M (2013) Translational repression and eIF4A2 activity are critical for microRNA-mediated gene regulation. Science 340:82–85. doi: 10.1126/science.1231197 CrossRefGoogle Scholar
  61. Mello CC, Conte D Jr (2004) Revealing the world of RNA interference. Nature 431:338–342. doi: 10.1038/nature02872 CrossRefGoogle Scholar
  62. Mier-y-Terán-Romero L, Silber M, Hatzimanikatis V (2010) The origins of time-delay in template biopolymerization processes. PLoS Comput Biol 6:e1000726. doi: 10.1371/journal.pcbi.1000726 MathSciNetCrossRefGoogle Scholar
  63. Monk NAM (2003) Oscillatory expression of Hes1, p53, and NF-\(\kappa \)B driven by transcriptional time delays. Curr Biol 13:1409–1413. doi: 10.1016/S0960-9822(03)00494-9 CrossRefGoogle Scholar
  64. Ouyang Y, Andersson CR, Kondo T, Golden SS, Johnson CH (1998) Resonating circadian clocks enhance fitness in cyanobacteria. Proc Natl Acad Sci USA 95:8660–8664CrossRefGoogle Scholar
  65. Pittendrigh CS, Minis DH (1972) Circadian systems: Longevity as a function of circadian resonance in Drosophila melanogaster. Proc Natl Acad Sci USA 69:1537–1539CrossRefGoogle Scholar
  66. Purcell O, Savery NJ, Grierson CS, di Bernardo M (2010) A comparative analysis of synthetic genetic oscillators. J R Soc Interface 7:1503–1524. doi: 10.1098/rsif.2010.0183 CrossRefGoogle Scholar
  67. Rihan FA (2003) Sensitivity analysis for dynamic systems with time-lags. J Comput Appl Math 151:445–462. doi: 10.1016/S0377-0427(02)00659-3 MathSciNetCrossRefzbMATHGoogle Scholar
  68. Roussel CJ, Roussel MR (2001) Delay-differential equations and the model equivalence problem in chemical kinetics. Phys Can 57:114–120Google Scholar
  69. Roussel MR (1996) The use of delay differential equations in chemical kinetics. J Phys Chem 100:8323–8330. doi: 10.1021/jp9600672 CrossRefGoogle Scholar
  70. Roussel MR, Zhu R (2006) Validation of an algorithm for delay stochastic simulation of transcription and translation in prokaryotic gene expression. Phys Biol 3:274–284. doi: 10.1088/1478-3975/3/4/005 CrossRefGoogle Scholar
  71. Rozenwasser E, Yusupov R (1999) Sensitivity of automatic control systems. CRC Press, Boca RatonCrossRefGoogle Scholar
  72. Saltelli A, Ratto M, Andres T, Campolongo F, Cariboni J, Gatelli D, Saisana M, Tarantola S (2008) Global sensitivity analysis: the primer. Wiley, ChichesterzbMATHGoogle Scholar
  73. Schmiedel JM, Klemm SL, Zheng Y, Sahay A, Blüthgen N, Marks DS, van Oudenaarden A (2015) MicroRNA control of protein expression noise. Science 348:128–132. doi: 10.1126/science.aaa1738 CrossRefGoogle Scholar
  74. Shen J, Liu Z, Zheng W, Xu F, Chen L (2009) Oscillatory dynamics in a simple gene regulatory network mediated by small RNAs. Phys A 388:2995–3000. doi: 10.1016/j.physa.2009.03.032 MathSciNetCrossRefGoogle Scholar
  75. Shimoni Y, Friedlander G, Hetzroni G, Niv G, Altuvia S, Biham O, Margalit H (2007) Regulation of gene expression by small non-coding RNAs: a quantitative view. Mol Syst Biol 3:138. doi: 10.1038/msb4100181 CrossRefGoogle Scholar
  76. Singh J, Padgett RA (2009) Rates of in situ transcription and splicing in large human genes. Nat Struct Mol Biol 16:1128–1133. doi: 10.1038/nsmb.1666 CrossRefGoogle Scholar
  77. Smith H (2011) An introduction to delay differential equations with applications to the life sciences, texts in applied mathematics, vol 57. Springer, New YorkCrossRefGoogle Scholar
  78. Smolen P, Baxter DA, Byrne JH (2000) Modeling transcriptional control in gene networks-methods, recent results, and future directions. Bull Math Biol 62:247–292. doi: 10.1006/bulm.1999.0155 CrossRefzbMATHGoogle Scholar
  79. Stokes A (1962) A Floquet theory for functional differential equation. Proc Natl Acad Sci USA 48:1330–1334MathSciNetCrossRefzbMATHGoogle Scholar
  80. Sweeney BM (1987) Rhythmic phenomena in plants, 2nd edn. Academic Press, San DiegoGoogle Scholar
  81. Taylor SR, Campbell SA (2007) Approximating chaotic saddles for delay differential equations. Phys Rev E 75:046215. doi: 10.1103/PhysRevE.75.046215 MathSciNetCrossRefGoogle Scholar
  82. Taylor SR, Gunawan R, Petzold LR, Doyle FJ (2008) Sensitivity measures for oscillating systems: application to mammalian circadian gene network. IEEE Trans Automat Control 53:177–188 (Special Issue)MathSciNetCrossRefGoogle Scholar
  83. Tian T, Burrage K, Burrage PM, Carletti M (2007) Stochastic delay differential equations for genetic regulatory networks. J Comput Appl Math 205:696–707. doi: 10.1016/j.cam.2006.02.063 MathSciNetCrossRefzbMATHGoogle Scholar
  84. Tigges M, Marquez-Lago TT, Stelling J, Fussenegger M (2009) A tunable synthetic mammalian oscillator. Nature 457:309–312. doi: 10.1038/nature07616 CrossRefGoogle Scholar
  85. Tyson JJ (1975) Classification of instabilities in chemical reaction systems. J Chem Phys 62:1010–1015. doi: 10.1063/1.430567 CrossRefGoogle Scholar
  86. Tyson JJ, Csikasz-Nagy A, Novak B (2002) The dynamics of cell cycle regulation. BioEssays 24:1095–1109. doi: 10.1002/bies.10191 CrossRefGoogle Scholar
  87. Varma A, Morbidelli M, Wu H (2005) Parametric sensitivity in chemical systems. Cambridge University Press, CambridgeGoogle Scholar
  88. Wang Y, Liu CL, Storey JD, Tibshirani RJ, Herschlag D, Brown PO (2002) Precision and functional specificity in mRNA decay. Proc Natl Acad Sci USA 99:5860–5865. doi: 10.1073/pnas.092538799 CrossRefGoogle Scholar
  89. Wilkins AK, Tidor B, White J, Barton PI (2009) Sensitivity analysis for oscillating dynamical systems. SIAM J Sci Comput 31:2706–2732. doi: 10.1137/070707129 MathSciNetCrossRefzbMATHGoogle Scholar
  90. Woelfle MA, Ouyang Y, Phanvijhitsiri K, Johnson CH (2004) The adaptive value of circadian clocks: An experimental assessment in cyanobacteria. Curr Biol 14:1481–1486. doi: 10.1016/j.cub.2004.08.023 CrossRefGoogle Scholar
  91. Yan X, Hoek TA, Vale RD, Tanenbaum ME (2016) Dynamics of translation of single mRNA molecules in vivo. Cell 165:976–989. doi: 10.1016/j.cell.2016.04.034 CrossRefGoogle Scholar
  92. Yanchuk S, Perlikowski P (2009) Delay and periodicity. Phys Rev E 79(046):221. doi: 10.1103/PhysRevE.79.046221 MathSciNetGoogle Scholar
  93. Zak DE, Stelling J, Doyle FJ III (2005) Sensitivity analysis of oscillatory (bio)chemical systems. Comput Chem Eng 29:663–673CrossRefGoogle Scholar
  94. Zhang HM, Kuang S, Xiong X, Gao T, Liu C, Guo AY (2015) Transcription factor and microRNA co-regulatory loops: Important regulatory motifs in biological processes and diseases. Br Bioinform 16:45–58. doi: 10.1093/bib/bbt085 CrossRefGoogle Scholar
  95. Zhang Y, Liu H, Zhou J (2016) Oscillatory expression in Escherichia coli mediated by microRNAs with transcriptional and translational time delays. IET Syst Biol 10:203–209. doi: 10.1049/iet-syb.2016.0017 CrossRefGoogle Scholar
  96. Zhdanov VP (2009) Bistability in gene transcription: Interplay of messenger RNA, protein, and nonprotein coding RNA. Biosystems 95:75–81. doi: 10.1016/j.biosystems.2008.07.002 CrossRefGoogle Scholar
  97. Zhdanov VP (2011) Kinetic models of gene expression including non-coding RNAs. Phys Rep 500:1–42. doi: 10.1016/j.physrep.2010.12.002 CrossRefGoogle Scholar
  98. Zi Z (2011) Sensitivity analysis approaches applied to systems biology models. IET Syst Biol 5:336–346. doi: 10.1049/iet-syb.2011.0015 CrossRefGoogle Scholar

Copyright information

© Society for Mathematical Biology 2017

Authors and Affiliations

  1. 1.Department of Applied MathematicsUniversity of WaterlooWaterlooCanada
  2. 2.Department of Mathematical SciencesNorthern Illinois UniversityDeKalbUSA
  3. 3.Alberta RNA Research and Training Institute, Department of Chemistry and BiochemistryUniversity of LethbridgeLethbridgeCanada

Personalised recommendations