On the numerical study of periodic solutions to delay equations in biological models

S. I. Fadeev ; V. V. Kogai ; T. M. Khlebodarova ; V. A. Likhoshvai

Sibirskij žurnal industrialʹnoj matematiki, Tome 19 (2016) no. 1, pp. 94-105.

Résumé

We present the results of a numerical study of periodic solutions of to a nonlinear delay equation in connection with mathematical models having a real biological prototype. The problem is formulated as a boundary value problem for a delay equation with periodicity and transversality conditions. We propose a spline-collocation finite-difference scheme of the boundary value problem using the Hermite interpolation cubic spline of class $C^1$ with fourth-order error. For the numerical study of the system of nonlinear equations of the difference scheme, the parameter-extension method is used, which allows us to identify the possible nonuniqueness of a solution to the boundary value problem and hence the nonuniqueness of periodic solutions regardless of their stability. It is shown by examples that periodic oscillations arise for values of the parameters typical for real molecular-genetic systems of higher organisms, for which the principle of “delay” is rather easy to implement.

Détail

Keywords: ordinary differential equation, delay, continuation method, boundary value problem
Mots-clés : oscillation.

Bibliographie

[1] Likhoshvai V. A., Fadeev S. I., Demidenko G. V., Matushkin Yu. G., “Modelirovanie uravneniem s zapazdyvayuschim argumentom mnogostadiinogo sinteza bez vetvleniya”, Sib. zhurn. industr. matematiki, 7:1(17) (2004), 73–94 | MR | Zbl

[2] Fadeev S. I., Likhoshvai V. A., Shtokalo D. N., “Issledovanie modeli sinteza lineinykh biomolekul s uchetom obratimosti protsessov”, Sib. zhurn. industr. matematiki, 8:3(23) (2005), 149–162 | MR | Zbl

[3] McManus C. J., Graveley B. R., “RNA structure and the mechanisms of alternative splicing”, Curr. Opin. Genet. Develop., 21:4 (2011), 37337–37339 | DOI

[4] Kelemen O., Convertini P., Zhang Z., Wen Y., Shen M., Falaleeva M., Stamm S., “Function of alternative splicing”, Gene, 514:1 (2013), 1–30 | DOI

[5] Likhoshvai V. A., Kogai V. V., Fadeev S. I., Khlebodarova T. M., “Alternative splicing can lead to chaos”, J. Bioinform. Comput. Biol., 13:1 (2015), 1540003 | DOI

[6] Likhoshvai V., Ratushny A., “Generalized Hill function method for modeling molecular processes”, J. Bioinform. Comput. Biol., 5:2B (2007), 521–531 | DOI

[7] Kogai V. V., Khlebodarova T. M., Fadeev S. I., Likhoshvai V. A., “Slozhnaya dinamika v sistemakh alternativnogo splaisinga mRNK: matematicheskaya model”, Vychisl. tekhnologii, 20:1 (2015), 38–52

[8] Hamburger V., Hamilton H. L., “A series of normal stages in the development of the chick embryo”, Develop. Dynam., 195:4 (1992), 231–272 | DOI

[9] Golubyatnikov V. P., Likhoshvai V. A., “Odnomernaya modelrazvitiya populyatsii zemnovodnykh”, Sib. zhurn. industr. matematiki, 5:2(10) (2002), 53–60 | MR | Zbl

[10] Dent P., Yacoub A., Contessa J., Caron R., Amorino G., Valerie K., Hagan M. P., Grant S., Schmidt-Ullrich R., “Stress and radiation–induced activation of multiple intracellular signaling pathways”, Radiat. Res., 159 (2003), 283–300 | DOI

[11] Keshet Y., Seger R., “The MAP kinase signaling cascades: a system of hundreds of components regulates a diverse array of physiological functions”, Meth. Mol. Biol., 661 (2010), 3–38 | DOI

[12] Munshi A., Ramesh R., “Mitogen-activated protein kinases and their role in radiation response”, Genes Cancer, 4 (2013), 401–408 | DOI

[13] Xu J., Zhang S., “Mitogen-activated protein kinase cascades in signaling plant growth and development”, Trends Plant Sci., 20 (2015), 56–64 | DOI

[14] Richardson M. K., Allen S. P., Wright G. M., Raynaud A., Hanken J., “Somite number and vertebrate evolution”, Development, 125 (1998), 151–160

[15] Gomez C., Ozbudak E. M., Wunderlich J., Baumann D., Lewis J., Pourquié O., “Control of segment number in vertebrate embryos”, Nature, 454:7202 (2008), 335–339 | DOI

[16] Dequéant M. L., Glynn E., Gaudenz K., Wahl M., Chen J., Mushegian A., Pourquié O., “A complex oscillating network of signaling genes underlies the mouse segmentation clock”, Science, 314:5805 (2006), 1595–1598 | DOI

[17] Dequéant M. L., Pourquié O., “Segmental patterning of the vertebrate embryonic axis”, Nat. Rev. Genet., 9:5 (2008), 370–382 | DOI

[18] Kageyama R., Masamizu Y., Niwa Y., “Oscillator mechanism of Notch pathway in the segmentation clock”, Develop. Dynam., 236:6 (2007), 1403–1409 | DOI | MR

[19] Ferjentsik Z., Hayashi S., Dale J. K., Bessho Y., Herreman A., De Strooper B., del Monte G., de la Pompa J. L., Maroto M., “Notch is a critical component of the mouse somitogenesis oscillator and is essential for the formation of the somites”, PLoS Genet., 5:9 (2009), e1000662 | DOI

[20] Ozbudak E. M., Pourquié O., “The vertebrate segmentation clock: the tip of the iceberg”, Curr. Opin. Genet. Develop., 18:4 (2008), 317–323 | DOI

[21] Schróter C., Herrgen L., Cardona A., Brouhard G. J., Feldman B., Oates A. C., “Dynamics of zebrafish somitogenesis”, Develop. Dynam., 237:3 (2008), 545–553 | DOI

[22] Gibb S., Maroto M., Dale J. K., “The segmentation clock mechanism moves up a notch”, Trends Cell. Biol., 20:10 (2010), 593–600 | DOI

[23] Gennerich A., Vale R. D., “Walking the walk: how kinesin and dynein coordinate their steps”, Curr. Opin. Cell. Biol., 21:1 (2009), 59–67 | DOI

[24] Kolomeisky A. B., “Motor proteins and molecular motors: how to operate machines at the nanoscale”, J. Phys. Condens. Matter., 25:46 (2013), 463101 | DOI

[25] Shiroguchi K., Kinosita K. (Jr.), “Myosin V walks by lever action and Brownian motion”, Science, 316:5828 (2007), 1208–1212 | DOI

[26] Fadeev S. I., “Programma chislennogo resheniya nelineinykh kraevykh zadach dlya sistem obyknovennykh differentsialnykh uravnenii s parametrom”, Vychisl. metody lineinoi algebry, Nauka, Novosibirsk, 1990, 104–200 | MR

[27] Fadeev S. I., Kogai V. V., Kraevye zadachi dlya sistem obyknovennykh differentsialnykh uravnenii, Ucheb. posobie, izd. NGU, Novosibirsk, 2012

[28] Kogai V. V., Fadeev S. I., Likhoshvai V. A., “O chislennom issledovanii avtokolebanii v gipoteticheskikh gennykh setyakh”, Vychisl. tekhnologii, 10:3 (2005), 56–71

[29] Likhoshvai V. A., Fadeev S. I., “Modeli tsiklicheskikh gennykh setei”, Sistemnaya kompyuternaya biologiya, Integratsionnye proekty SO RAN, 14, Izd-vo SO RAN, Novosibirsk, 2008, 410–420

[30] Likhoshvai V. A., Khlebodarova T. M., Bazhan S. I., Gainova I. A., Chereshnev V. A., Bocharov G. A., “Mathematical model of the Tat-Rev regulation of HIV-1 replication in an activated cell predicts the existence of oscillatory dynamics in the synthesis of viral components”, BMC Genomics, 15, Suppl. 12 (2014), S1 | DOI

[31] Pigolotti S., Krishna S., Jensen M. H., “Oscillation patterns in negative feedback loops”, Proc. Nat. Acad. Sci. USA, 104:16 (2007), 6533–6537 | DOI | MR | Zbl

[32] Goldbeter A., Pourquié O., “Modeling the segmentation clock as a network of coupled oscillations in the Notch, Wnt and FGF signaling pathways”, J. Theor. Biol., 252:3 (2008), 574–585 | DOI | MR

[33] Wang Y., Paszek P., Horton C. A., Kell D. B., White M. R., Broomhead D. S., Muldoon M. R., “Interactions among oscillatory pathways in NF-kappa B signaling”, BMC Syst. Biol., 5 (2011), 23 | DOI | Zbl

[34] Jensen P. B., Pedersen L., Krishna S., Jensen M. H., “A Wnt oscillator model for somitogenesis”, Biophys. J., 98:6 (2010), 943–950 | DOI

[35] Schrøder T. D., Özalp V. C., Lunding A., Jernshøj K. D., Olsen L. F., “An experimental study of the regulation of glycolytic oscillations in yeast”, FEBS J., 280:23 (2013), 6033–6044 | DOI

[36] Klevecz R. R., Li C. M., Marcus I., Frankel P. H., “Collective behavior in gene regulation: the cell is an oscillator, the cell cycle a developmental process”, FEBS J., 275 (2008), 2372–2384 | DOI