×

Theoretical study of the one self-regulating gene in the modified Wagner model. (English) Zbl 1394.92083

Summary: Predicting how a genetic change affects a given character is a major challenge in biology, and being able to tackle this problem relies on our ability to develop realistic models of gene networks. However, such models are rarely tractable mathematically. In this paper, we propose a mathematical analysis of the sigmoid variant of the Wagner gene-network model. By considering the simplest case, that is, one unique self-regulating gene, we show that numerical simulations are not the only tool available to study such models: theoretical studies can be done too, by mathematical analysis of discrete dynamical systems. It is first shown that the particular sigmoid function can be theoretically investigated. Secondly, we provide an illustration of how to apply such investigations in the case of the dynamical system representing the one self-regulating gene. In this context, we focused on the composite function \(f_a(m.x)\) where \(f_a\) is the parametric sigmoid function and \(m\) is a scalar not in \(\{0,1 \}\) and we have proven that the number of fixed-point can be deduced theoretically, according to the values of \(a\) and \(m\).

MSC:

92D10 Genetics and epigenetics
92C40 Biochemistry, molecular biology
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bahi, J.M.; Guyeux, C.; Perasso, A.; Chaos in DNA evolution; Int. J. Biomath. (IJB): 2016; Volume 9 ,1650076. · Zbl 1346.37071
[2] Guyeux, C.; Nicod, J.M.; Philippe, L.; Bahi, J.M.; The study of unfoldable self-avoiding walks. Application to protein structure prediction software; JBCB J. Bioinform. Comput. Biol.: 2015; Volume 13 ,1550009.
[3] Pai, A.A.; Pritchard, J.K.; Gilad, Y.; The genetic and mechanistic basis for variation in gene regulation; PLoS Genet.: 2015; Volume 11 .
[4] Bornholdt, S.; Modeling genetic networks and their evolution: A complex dynamical systems perspective; Biol. Chem.: 2001; Volume 382 ,1289-1299.
[5] Barabási, A.L.; Oltvai, Z.N.; Network biology: Understanding the cell’s functional organization; Nat. Rev. Genet.: 2004; Volume 5 ,101-113.
[6] Polynikis, A.; Hogan, S.; di Bernardo, M.; Comparing different ODE modelling approaches for gene regulatory networks; J. Theor. Biol.: 2009; Volume 261 ,511-530. · Zbl 1403.92095
[7] Perc, M.; Stochastic resonance on paced genetic regulatory small-world networks: Effects of asymmetric potentials; Eur. Phys. J. B: 2009; Volume 69 ,147-153. · Zbl 1188.82051
[8] Gosak, M.; Markovič, R.; Dolenšek, J.; Rupnik, M.S.; Marhl, M.; Stožer, A.; Perc, M.; Network science of biological systems at different scales: A review; Phys. Life Rev.: 2018; Volume 24 ,118-135.
[9] Wagner, A.; Does evolutionary plasticity evolve?; Evolution: 1996; Volume 50 ,1008-1023.
[10] Fierst, J.L.; Phillips, P.C.; Modeling the evolution of complex genetic systems: The gene network family tree; J. Exp. Zool. Part B Mol. Dev. Evol.: 2015; Volume 324 ,1-12.
[11] Wagner, A.; Evolution of gene networks by gene duplications: A mathematical model and its implications on genome organization; Proc. Natl. Acad. Sci. USA: 1994; Volume 91 ,4387-4391. · Zbl 0795.92019
[12] Masel, J.; Genetic assimilation can occur in the absence of selection for the assimilating phenotype, suggesting a role for the canalization heuristic; J. Evolut. Biol.: 2004; Volume 17 ,1106-1110.
[13] Pinho, R.; Borenstein, E.; Feldman, M.W.; Most networks in Wagner’s model are cycling; PLoS ONE: 2012; Volume 7 .
[14] Siegal, M.L.; Bergman, A.; Waddington’s canalization revisited: Developmental stability and evolution; Proc. Natl. Acad. Sci. USA: 2002; Volume 99 ,10528-10532.
[15] Carneiro, M.O.; Taubes, C.H.; Hartl, D.L.; Model transcriptional networks with continuously varying expression levels; BMC Evolut. Biol.: 2011; Volume 11 .
[16] Rünneburger, E.; Le Rouzic, A.; Why and how genetic canalization evolves in gene regulatory networks; BMC Evolut. Biol.: 2016; Volume 16 .
[17] Huerta-Sanchez, E.; Durrett, R.; Wagner’s canalization model; Theor. Popul. Biol.: 2007; Volume 71 ,121-130. · Zbl 1118.92042
[18] Le Cunff, Y.; Pakdaman, K.; Phenotype-genotype relation in Wagner’s canalization model; J. Theor. Biol.: 2012; Volume 314 ,69-83. · Zbl 1397.92503
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.