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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
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
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
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
Baker CT, Rihan FA (1999) Sensitivity analysis of parameters in modelling with delay-differential equations. Tech. Rep. 349, Manchester Centre for Computational Mathematics
Banks H, Robbins D, Sutton KL (2013a) Generalized sensitivity analysis for delay differential equations. In: Control and Optimization with PDE Constraints, Springer, pp 19–44
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
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
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
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 261:13,800–13,806
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
Bueler E, Butcher E (2002) Stability of periodic linear delay-differential equations and the Chebyshev approximation of fundamental solutions, preprint
Bure E, Rozenvasser E (1974) The study of the sensitivity of oscillatory systems. Autom Remote Control 35:1045–1052
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
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 59:895–922
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
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
Cornish-Bowden A, Cárdenas ML (eds) (1990) Control of metabolic processes. Plenum, New York
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
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
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
Driver RD (1962) Existence and stability of solutions of a delay-differential system. Arch Ration Mech Anal 10:401–426. doi:10.1007/BF00281203
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
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
Edmunds LN Jr (1988) Cellular and molecular bases of biological clocks. Springer, New York
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
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
Ermentrout B (2002) Simulating, analyzing, and animating dynamical systems. SIAM, Philadelphia
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
Fell DA (1992) Metabolic control analysis: a survey of its theoretical and experimental development. Biochem J 286:313–330. doi:10.1042/bj2860313
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
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
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
Goldbeter A (1996) Biochemical oscillations and cellular rhythms. Cambridge University Press, Cambridge
Goodwin BC (1963) Temporal organization in cells. Academic Press, London
Halaney A (1966) Differential Equations. Academic Press, New York, Stability, Oscillations, Time Lags
Hale JK, Ladeira LAC (1991) Differentiability with respect to delays. J Differ Equ 92:14–26. doi:10.1016/0022-0396(91)90061-D
Hale JK, Lunel SMV (1993) Introduction to functional differential equations. Springer, New York
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
Heinrich R, Schuster S (1996) The regulation of cellular systems. Chapman & Hall, New York
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
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.)
Hillman WS (1956) Injury of tomato plants by continuous light and unfavorable photoperiodic cycles. Am J Bot 43:89–96
Ingalls B (2008) Sensitivity analysis: from model parameters to system behaviour. Essays Biochem 45:177–193. doi:10.1042/bse0450177
Ingalls BP (2004) Autonomously oscillating biochemical systems: parametric sensitivity of extrema and period. Syst Biol 1:62–70. doi:10.1049/sb:20045005
Ingalls BP (2013) Mathematical modeling in systems biology. MIT Press, Cambridge
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
Izaurralde E (2015) Breakers and blockers–miRNAs at work. Science 349:380–382. doi:10.1126/science.1260969
Johnson CH (2001) Endogenous timekeepers in photosynthetic organisms. Annu Rev Physiol 63:695–728. doi:10.1146/annurev.physiol.63.1.695
Kacser H, Burns JA (1973) The control of flux. Symp Soc Exp Biol 27:65–104
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
Klarsfeld A, Rouyer F (1998) Effects of circadian mutations and LD periodicity on the life span of Drosophila melanogaster. J Biol Rhythms 13:471–478
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
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
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
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
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
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 131–153
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
MacDonald N (1987) An interference effect of independent delays. IEE Proc D 134:38–42
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
Mello CC, Conte D Jr (2004) Revealing the world of RNA interference. Nature 431:338–342. doi:10.1038/nature02872
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
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
Ouyang Y, Andersson CR, Kondo T, Golden SS, Johnson CH (1998) Resonating circadian clocks enhance fitness in cyanobacteria. Proc Natl Acad Sci USA 95:8660–8664
Pittendrigh CS, Minis DH (1972) Circadian systems: Longevity as a function of circadian resonance in Drosophila melanogaster. Proc Natl Acad Sci USA 69:1537–1539
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
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
Roussel CJ, Roussel MR (2001) Delay-differential equations and the model equivalence problem in chemical kinetics. Phys Can 57:114–120
Roussel MR (1996) The use of delay differential equations in chemical kinetics. J Phys Chem 100:8323–8330. doi:10.1021/jp9600672
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
Rozenwasser E, Yusupov R (1999) Sensitivity of automatic control systems. CRC Press, Boca Raton
Saltelli A, Ratto M, Andres T, Campolongo F, Cariboni J, Gatelli D, Saisana M, Tarantola S (2008) Global sensitivity analysis: the primer. Wiley, Chichester
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
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
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
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
Smith H (2011) An introduction to delay differential equations with applications to the life sciences, texts in applied mathematics, vol 57. Springer, New York
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
Stokes A (1962) A Floquet theory for functional differential equation. Proc Natl Acad Sci USA 48:1330–1334
Sweeney BM (1987) Rhythmic phenomena in plants, 2nd edn. Academic Press, San Diego
Taylor SR, Campbell SA (2007) Approximating chaotic saddles for delay differential equations. Phys Rev E 75:046215. doi:10.1103/PhysRevE.75.046215
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)
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
Tigges M, Marquez-Lago TT, Stelling J, Fussenegger M (2009) A tunable synthetic mammalian oscillator. Nature 457:309–312. doi:10.1038/nature07616
Tyson JJ (1975) Classification of instabilities in chemical reaction systems. J Chem Phys 62:1010–1015. doi:10.1063/1.430567
Tyson JJ, Csikasz-Nagy A, Novak B (2002) The dynamics of cell cycle regulation. BioEssays 24:1095–1109. doi:10.1002/bies.10191
Varma A, Morbidelli M, Wu H (2005) Parametric sensitivity in chemical systems. Cambridge University Press, Cambridge
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
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
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
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
Yanchuk S, Perlikowski P (2009) Delay and periodicity. Phys Rev E 79(046):221. doi:10.1103/PhysRevE.79.046221
Zak DE, Stelling J, Doyle FJ III (2005) Sensitivity analysis of oscillatory (bio)chemical systems. Comput Chem Eng 29:663–673
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
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
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
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
Zi Z (2011) Sensitivity analysis approaches applied to systems biology models. IET Syst Biol 5:336–346. doi:10.1049/iet-syb.2011.0015
Acknowledgements
This research was supported by the Natural Sciences and Engineering Research Council of Canada.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ingalls, B., Mincheva, M. & Roussel, M.R. Parametric Sensitivity Analysis of Oscillatory Delay Systems with an Application to Gene Regulation. Bull Math Biol 79, 1539–1563 (2017). https://doi.org/10.1007/s11538-017-0298-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11538-017-0298-x