Biochemical Problems, Mathematical Solutions

Marc R. Roussel; Moisés Santillán; Marc R. Roussel; Moisés Santillán

doi:10.3934/math.2022313

AIMS Mathematics

2022, Volume 7, Issue 4: 5662-5669. doi: 10.3934/math.2022313

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Editorial Special Issues

Biochemical Problems, Mathematical Solutions

Marc R. Roussel ^{1
,
,},
Moisés Santillán ²

1.
Alberta RNA Research and Training Institute, Department of Chemistry and Biochemistry, University of Lethbridge, Lethbridge, Alberta, Canada, T1K 3M4
2.
Centro de Investigación y de Estudios Avanzados del IPN, Unidad Monterrey, Vía del Conocimiento 201, Parque PIIT, 66628 Apodaca NL, MÉXICO

Received: 15 November 2021 Revised: 15 November 2021 Accepted: 31 December 2021 Published: 10 January 2022

Citation: Marc R. Roussel, Moisés Santillán. Biochemical Problems, Mathematical Solutions[J]. AIMS Mathematics, 2022, 7(4): 5662-5669. doi: 10.3934/math.2022313

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References

[1]	Md. R. Amin, M. R. Roussel, Graph-theoretic analysis of a model for the coupling between photosynthesis and photorespiration, Can. J. Chem., 92 (2014), 85–93. https://doi.org/10.1139/cjc-2013-0315 doi: 10.1139/cjc-2013-0315
[2]	S. Bakshi, A. Siryaporn, M. Goulian, J. C. Weisshaar, Superresolution imaging of ribosomes and RNA polymerase in live Escherichia coli cells, Mol. Microbiol., 85 (2012), 21–38. https://doi.org/10.1111/j.1365-2958.2012.08081.x doi: 10.1111/j.1365-2958.2012.08081.x
[3]	K. Ball, T. G. Kurtz, L. Popovic, G. Rampala, Asymptotic analysis of multiscale approximations to reaction networks, Ann. Appl. Probab., 16 (2006), 1925–1961. https://doi.org/10.1214/105051606000000420 doi: 10.1214/105051606000000420
[4]	A. F. Bartholomay, Enzymatic reaction-rate theory: A stochastic approach, Ann. N. Y. Acad. Sci., 96 (1962), 897–912. https://doi.org/10.1111/j.1749-6632.1962.tb54110.x doi: 10.1111/j.1749-6632.1962.tb54110.x
[5]	A. F. Bartholomay, A stochastic approach to statistical kinetics with application to enzyme kinetics, Biochemistry, 1 (1962), 223–230. https://doi.org/10.1021/bi00908a005 doi: 10.1021/bi00908a005
[6]	R. D. Bliss, Analysis of the Dynamic Behavior of the Tryptophan Operon of Escherichia coli: The Functional Significance of Feedback Inhibition, PhD thesis, University of California Riverside, 1979.
[7]	G. E. Briggs, J. B. S. Haldane, A note on the kinetics of enzyme action, Biochem. J., 19 (1925), 338–339. https://doi.org/10.1042/bj0190338 doi: 10.1042/bj0190338
[8]	S. Busenberg, J. Mahaffy, Interaction of spatial diffusion and delays in models of genetic control by repression, J. Math. Biol., 22 (1985), 313–333. https://doi.org/10.1007/BF00276489 doi: 10.1007/BF00276489
[9]	T. R. Caine Boyde, Foundation Stones of Biochemistry, Voile et Aviron, Hong Kong, 1980.
[10]	K. J. Dale, O. Pourquié, A clock-work somite, BioEssays, 22 (2000), 72–83. https://doi.org/10.1002/(SICI)1521-1878(200001)22:1%3C72::AID-BIES12%3E3.0.CO;2-S doi: 10.1002/(SICI)1521-1878(200001)22:1%3C72::AID-BIES12%3E3.0.CO;2-S
[11]	A. Ecoffet, F. Poitevin, K. D. Duc, MorphOT: Transport-based interpolation between EM maps with UCSF ChimeraX, Bioinformatics, 36 (2020), 5528–5529. https://doi.org/10.1093/bioinformatics/btaa1019 doi: 10.1093/bioinformatics/btaa1019
[12]	A. Ecoffet, G. Woollard, A. Kushner, F. Poitevin, K. D. Duc, Application of transport-based metric for continuous interpolation between cryo-EM density maps, AIMS Math., 7 (2022), 986–999. https://doi.org/10.3934/math.2022059 doi: 10.3934/math.2022059
[13]	B. B. Edelstein, Biochemical model with multiple steady states and hysteresis, J. Theor. Biol., 29 (1970), 57–62. https://doi.org/10.1016/0022-5193(70)90118-9 doi: 10.1016/0022-5193(70)90118-9
[14]	R. Edwards, M. Wood, Branch prioritization motifs in biochemical networks with sharp activation, AIMS Math., 7 (2022), 1115–1146. https://doi.org/10.3934/math.2022066 doi: 10.3934/math.2022066
[15]	J. Eilertsen, M. R. Roussel, S. Schnell, S. Walcher, On the quasi-steady-state approximation in an open Michaelis-Menten reaction mechanism, AIMS Math., 6 (2021), 6781–6814. https://doi.org/10.3934/math.2021398 doi: 10.3934/math.2021398
[16]	N. Fenichel, Persistence and smoothness of invariant manifolds for flows, Indiana U. Math. J., 21 (1971), 193–226. https://doi.org/10.1512/iumj.1972.21.21017 doi: 10.1512/iumj.1972.21.21017
[17]	N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Differential Equations, 31 (1979), 53–98. https://doi.org/10.1016/0022-0396(79)90152-9 doi: 10.1016/0022-0396(79)90152-9
[18]	Herbert C. Friedmann (ed.), Enzymes, Hutchinson Ross Publishing, Stroudsburg, Pa., 1981.
[19]	D. S. Glass, X. Jin, I. H. Riedel-Kruse, Nonlinear delay differential equations and their application to modeling biological network motifs, Nat. Commun., 12 (2021), 1788. https://doi.org/10.1038/s41467-021-21700-8 doi: 10.1038/s41467-021-21700-8
[20]	A. Goeke, S. Walcher, E. Zerz, Determining "small parameters" for quasi-steady state, J. Differential Equations, 259 (2015), 1149–1180. https://doi.org/10.1016/j.jde.2015.02.038 doi: 10.1016/j.jde.2015.02.038
[21]	A. Goldbeter, R. Lefever, Dissipative structures for an allosteric model, Biophys. J., 12 (1972), 1302–1315. https://doi.org/10.1016/S0006-3495(72)86164-2 doi: 10.1016/S0006-3495(72)86164-2
[22]	D. Gonze, W. Abou-Jaoudé, D. A. Ouattara, J. Halloy, How molecular should your molecular model be? On the level of molecular detail required to simulate biological networks in systems and synthetic biology, Meth. Enzymol., 487 (2011), 171–215. https://doi.org/10.1016/B978-0-12-381270-4.00007-X doi: 10.1016/B978-0-12-381270-4.00007-X
[23]	B. C. Goodwin, Temporal Organization in Cells, Academic Press, London, 1963.
[24]	J. S. Griffith, Mathematics of cellular control processes. Ⅰ. Negative feedback to one gene, J. Theor. Biol., 20 (1968), 202–208. https://doi.org/10.1016/0022-5193(68)90189-6 doi: 10.1016/0022-5193(68)90189-6
[25]	R. Grima, S. Schnell, Modelling reaction kinetics inside cells, Essays Biochem., 45 (2008), 41–56. https://doi.org/10.1042/bse0450041 doi: 10.1042/bse0450041
[26]	S. M. Hanson, S. Schnell, Reactant stationary approximation in enzyme kinetics, J. Phys. Chem. A, 112 (2008), 8654–8658. https://doi.org/10.1021/jp8026226 doi: 10.1021/jp8026226
[27]	L. A. Harris, J. S. Hogg, J.-J. Tapia, J. A. P. Sekar, S. Gupta, I. Korsunsky, et al., BioNetGen 2.2: Advances in rule-based modeling, Bioinformatics, 32 (2016), 3366–3368. https://doi.org/10.1093/bioinformatics/btw469 doi: 10.1093/bioinformatics/btw469
[28]	F. G. Heineken, H. M. Tsuchiya, R. Aris, On the mathematical status of the pseudo-steady state hypothesis of biochemical kinetics, Math. Biosci., 1 (1967), 95–113. https://doi.org/10.1016/0025-5564(67)90029-6 doi: 10.1016/0025-5564(67)90029-6
[29]	J. Higgins, A chemical mechanism for oscillation of glycolytic intermediates in yeast cells, Proc. Natl. Acad. Sci. U.S.A., 51 (1964), 989–994. https://doi.org/10.1073/pnas.51.6.989 doi: 10.1073/pnas.51.6.989
[30]	A. V. Hill, The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves, J. Physiol., 40 (1910), ⅳ–ⅶ.
[31]	W. S. Hlavacek, J. R. Faeder, M. L. Blinov, R. G. Posner, M. Hucka, W. Fontana, Rules for modeling signal-transduction systems, Sci. STKE, (2006), re6. https://doi.org/10.1126/stke.3442006re6 doi: 10.1126/stke.3442006re6
[32]	K. J. Hughes, M. Fairweather, J. F. Griffiths, R. Porter, A. S. Tomlin, The application of the QSSA via reaction lumping for the reduction of complex hydrocarbon oxidation mechanisms, Proc. Combust. Inst., 32 (2009), 543–551. https://doi.org/10.1016/j.proci.2008.06.064 doi: 10.1016/j.proci.2008.06.064
[33]	H. Kacser, J. A. Burns, The control of flux, Symp. Soc. Exp. Biol., 27 (1973), 65–104.
[34]	A. Kicheva, T. Bollenbach, O. Wartlick, F. Jülicher, M. Gonzalez-Gaitan, Investigating the principles of morphogen gradient formation: from tissues to cells, Curr. Opin. Genet. Dev., 22 (2012), 527–532. https://doi.org/10.1016/j.gde.2012.08.004 doi: 10.1016/j.gde.2012.08.004
[35]	T. R. Kiehl, R. M. Mattheyses, M. K. Simmons, Hybrid simulation of cellular behavior, Bioinformatics, 20 (2004), 316–322. https://doi.org/10.1093/bioinformatics/btg409 doi: 10.1093/bioinformatics/btg409
[36]	B. I. Kurganov, A. I. Dorozhko, Z. S. Kagan, V. A. Yakovlev, The theoretical analysis of kinetic behaviour of "hysteretic" allosteric enzymes. Ⅰ. The kinetic manifestations of slow conformational change of an oligomeric enzyme in the Monod, Wyman and Changeux model, J. Theor. Biol., 60 (1976), 247–269. https://doi.org/10.1016/0022-5193(76)90059-X doi: 10.1016/0022-5193(76)90059-X
[37]	K. J. Laidler, Theory of the transient phase in kinetics, with special reference to enzyme systems, Can. J. Chem., 33 (1955), 1614–1624. https://doi.org/10.1139/v55-195 doi: 10.1139/v55-195
[38]	H. D. Landahl, Some conditions for sustained oscillations in biochemical chains with feedback inhibition, Bull. Math. Biophys., 31 (1969), 775–787. https://doi.org/10.1007/BF02477786 doi: 10.1007/BF02477786
[39]	J. Le Sauteur-Robitaille, Z. S. Yu, M. Craig, Impact of estrogen population pharmacokinetics on a QSP model of mammary stem cell differentiation into myoepithelial cells, AIMS Math., 6 (2021), 10861–10880. https://doi.org/10.3934/math.2021631 doi: 10.3934/math.2021631
[40]	D. Lee, A. Koul, N. Lubna, S. A. McKenna, S. Portet, Mathematical modelling of OAS2 activation by dsRNA and effects of dsRNA lengths, AIMS Math., 6 (2021), 5924–5941. https://doi.org/10.3934/math.2021351 doi: 10.3934/math.2021351
[41]	T. I. Lee, N. J. Rinaldi, F. Robert, D. T. Odom, Z. Bar-Joseph, G. K. Gerber, et al., Transcriptional regulatory networks in Saccharomyces cerevisiae, Science, 298 (2002), 799–804. https://doi.org/10.1126/science.1075090 doi: 10.1126/science.1075090
[42]	D. Lloyd, E. L. Rossi, M. R. Roussel, The temporal organization of living systems from molecule to mind, in Ultradian Rhythms from Molecules to Mind (eds. D. Lloyd and E. L. Rossi), Springer, (2008), 1–8.
[43]	C. Macdonald, Gap junctions and cell-cell communication, Essays Biochem., 21 (1985), 86–118.
[44]	P. Macheras, A. Iliadis, Modeling in Biopharmaceutics, Pharmacokinetics and Pharmacodynamics: Homogeneous and Heterogeneous Approaches, vol. 30 of Interdisciplinary Applied Mathematics, Springer, New York, 2005.
[45]	M. C. Mackey, L. Glass, Oscillation and chaos in physiological control systems, Science, 197 (1977), 287–289. https://doi.org/10.1126/science.267326 doi: 10.1126/science.267326
[46]	M. C. Mackey, M. Santillán, M. Tyran-Kamińska, E. S. Zeron, Simple Mathematical Models of Gene Regulatory Dynamics, Lecture Notes on Mathematical Modelling in the Life Sciences, Springer, Cham, Switzerland, 2016.
[47]	J. Maynard Smith, Mathematical Ideas in Biology, Cambridge University Press, Cambridge, 1968.
[48]	H. G. Othmer, A continuum model for coupled cells, J. Math. Biol., 17 (1983), 351–369. https://doi.org/10.1007/BF00276521 doi: 10.1007/BF00276521
[49]	L. Ouellet, K. J. Laidler, Theory of the transient phase in kinetics, with special reference to enzyme systems — Ⅱ. The case of two enzyme-substrate complexes, Can. J. Chem., 34 (1956), 146–150. https://doi.org/10.1139/v56-018 doi: 10.1139/v56-018
[50]	J. Pantoja-Hernández, M. Santillán, Segmentation-clock synchronization in circular-lattice networks of embryonic presomitic-mesoderm cells, AIMS Math., 6 (2021), 5817–5836. https://doi.org/10.3934/math.2021344 doi: 10.3934/math.2021344
[51]	L. Petzold, W. Zhu, Model reduction for chemical kinetics: An optimization approach, AIChE J., 45 (1999), 869–886. https://doi.org/10.1002/aic.690450418 doi: 10.1002/aic.690450418
[52]	I. Prigogine, R. Lefever, A. Goldbeter, M. Herschkowitz-Kaufman, Symmetry breaking instabilities in biological systems, Nature, 223 (1969), 913–916. https://doi.org/10.1038/223913a0 doi: 10.1038/223913a0
[53]	M. Ptashnyk, Multiscale modelling and analysis of signalling processes in tissues with non-periodic distribution of cells, Vietnam J. Math., 45 (2017), 295–316. https://doi.org/10.1007/s10013-016-0232-9 doi: 10.1007/s10013-016-0232-9
[54]	O. Radulescu, A. N. Gorban, A. Zinovyev, V. Noel, Reduction of dynamical biochemical reactions networks in computational biology, Front. Genet., 3 (2012), 131. https://doi.org/10.3389/fgene.2012.00131 doi: 10.3389/fgene.2012.00131
[55]	O. Radulescu, A. N. Gorban, A. Zinovyev, A. Lilienbaum, Robust simplifications of multiscale biochemical networks, BMC Syst. Biol., 2 (2008), 86. https://doi.org/10.1186/1752-0509-2-86 doi: 10.1186/1752-0509-2-86
[56]	M. R. Roussel, The use of delay differential equations in chemical kinetics, J. Phys. Chem., 100 (1996), 8323–8330. https://doi.org/10.1021/jp9600672 doi: 10.1021/jp9600672
[57]	M. A. Savageau, The behavior of intact biochemical control systems, Curr. Topics Cell. Reg., 6 (1972), 63–130. https://doi.org/10.1016/B978-0-12-152806-5.50010-2 doi: 10.1016/B978-0-12-152806-5.50010-2
[58]	L. A. Segel, M. Slemrod, The quasi-steady-state assumption: A case study in perturbation, SIAM Rev., 31 (1989), 446–477. https://doi.org/10.1137/1031091 doi: 10.1137/1031091
[59]	P. K. Sorger, S. R. Allerheiligen, D. R. Abernethy, R. B. Altman, K. L. R. Brouwer, A. Califano, et al., Quantitative and Systems Pharmacology in the Post-genomic Era: New Approaches to Discovering Drugs and Understanding Therapeutic Mechanisms, Technical report, National Institutes of Health, 2011. Available from: https://www.nigms.nih.gov/training/documents/systemspharmawpsorger2011.pdf.
[60]	I. Stoleriu, F. A. Davidson, J. L. Liu, Quasi-steady state assumptions for non-isolated enzyme-catalyzed reactions, J. Math. Biol., 48 (2004), 82–104. https://doi.org/10.1007/s00285-003-0225-7 doi: 10.1007/s00285-003-0225-7
[61]	P. Thomas, A. V. Straube, R. Grima, Limitations of the stochastic quasi-steady-state approximation in open biochemical reaction networks, J. Chem. Phys., 135 (2011), 181103. https://doi.org/10.1063/1.3661156 doi: 10.1063/1.3661156
[62]	P. P. Thumfort, D. B. Layzell, C. A. Atkins, A simplified approach for modeling diffusion into cells, J. Theor. Biol., 204 (2000), 47–65. https://doi.org/10.1006/jtbi.2000.1071 doi: 10.1006/jtbi.2000.1071
[63]	T. Turányi, Reduction of large reaction mechanisms, New J. Chem., 14 (1990), 795–803.
[64]	C. Walter, Enzyme Kinetics: Open and Closed Systems, Ronald Press, New York, 1966.
[65]	L. E. Whitehouse, A. S. Tomlin, M. J. Pilling, Systematic reduction of complex tropospheric chemical mechanisms, Part Ⅰ: Sensitivity and time-scale analyses, Atmos. Chem. Phys., 4 (2004), 2025–2056. https://doi.org/10.5194/acp-4-2025-2004 doi: 10.5194/acp-4-2025-2004
[66]	D. M. Wolf, A. P. Arkin, Motifs, modules and games in bacteria, Curr. Opin. Microbiol., 6 (2003), 125–134. https://doi.org/10.1016/S1369-5274(03)00033-X doi: 10.1016/S1369-5274(03)00033-X

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