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CRITICAL TIMESCALES AND TIME INTERVALS FOR COUPLED LINEAR PROCESSES

  • MATTHEW J. SIMPSON (a1), ADAM J. ELLERY (a1), SCOTT W. MCCUE (a1) and RUTH E. BAKER (a2)

Abstract

In 1991, McNabb introduced the concept of mean action time (MAT) as a finite measure of the time required for a diffusive process to effectively reach steady state. Although this concept was initially adopted by others within the Australian and New Zealand applied mathematics community, it appears to have had little use outside this region until very recently, when in 2010 Berezhkovskii and co-workers [A. M. Berezhkovskii, C. Sample and S. Y. Shvartsman, “How long does it take to establish a morphogen gradient?” Biophys. J. 99 (2010) L59–L61] rediscovered the concept of MAT in their study of morphogen gradient formation. All previous work in this area has been limited to studying single-species differential equations, such as the linear advection–diffusion–reaction equation. Here we generalize the concept of MAT by showing how the theory can be applied to coupled linear processes. We begin by studying coupled ordinary differential equations and extend our approach to coupled partial differential equations. Our new results have broad applications, for example the analysis of models describing coupled chemical decay and cell differentiation processes.

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References

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[1]Barenblatt, G. I., Scaling (Cambridge University Press, Cambridge, 2003).
[2]Berezhkovskii, A. M., Sample, C. and Shvartsman, S. Y., “How long does it take to establish a morphogen gradient?Biophys. J. 99 (2010) L59L61; doi:10.1016/j.bpj.2010.07.045.
[3]Berezhkovskii, A. M., Sample, C. and Shvartsman, S. Y., “Formation of morphogen gradients: Local accumulation time”, Phys. Rev. E 83 (2011) 051906; doi:10.1103/PhysRevE.83.051906.
[4]Berezhkovskii, A. M. and Shvartsman, S. Y., “Physical interpretation of mean local accumulation time of morphogen gradient formation”, J. Chem. Phys. 135 (2011) 154115; doi:10.1063/1.3654159.
[5]Cho, C. M., “Convective transport of ammonium with nitrification in soil”, Canad. J. Soil Sci. 51 (1970) 339350; doi:10.4141/cjss71-047.
[6]Clement, T. P., Sun, Y., Hooker, B. S. and Petersen, J. N., “Modelling multispecies reactive transport in ground water”, Ground Water Modeling and Remediation 18 (1998) 7992; doi:10.1111/j.1745-6592.1998.tb00618.x.
[7]Denman, P. K., McElwain, D. L. S., Harkin, D. G. and Upton, Z., “Mathematical modelling of aerosolised skin grafts incorporating keratinocyte clonal subtypes”, Bull. Math. Biol. 69 (2007) 157179; doi:10.1007/S11538-006-9082-z.
[8]Ellery, A. J., Simpson, M. J., McCue, S. W. and Baker, R. E., “Critical timescales for advection–diffusion–reaction processes”, Phys. Rev. E 85 (2012) 041135; doi:10.1103/PhysRevE.85.041135.
[9]Ellery, A. J., Simpson, M. J., McCue, S. W. and Baker, R. E., “Moments of action provide insight into critical times for advection–diffusion–reaction processes”, Phys. Rev. E 86 (2012) 031136; doi:10.1103/PhysRevE.86.031136.
[10]Fernando, A. E., Landman, K. A. and Simpson, M. J., “Nonlinear diffusion and exclusion processes with contact interactions”, Phys. Rev. E 81 (2010) 011903; doi:10.1103/PhysRevE.81.011903.
[11]Gordon, P. V., Sample, C., Berezhkovskii, A. M., Muratov, C. V. and Shvartsman, S. Y., “Local kinetics of morphogen gradients”, Proc. Natl Acad. Sci. 108 (2011) 61576162; doi:10.1073/pnas.1019245108.
[12]Hickson, R. I., Barry, S. I. and Mercer, G. N., “Critical times in multilayer diffusion. Part 1: Exact solutions”, Int. J. Heat Mass Transfer 52 (2011) 57765783; doi:10.1016/j.ijheatmasstransfer.2009.08.013.
[13]Hickson, R. I., Barry, S. I. and Mercer, G. N., “Critical times in multilayer diffusion. Part 2: Approximate solutions”, Int. J. Heat Mass Transfer 52 (2011) 57845791; doi:10.1016/j.ijheatmasstransfer.2009.08.012.
[14]Hickson, R. I., Barry, S. I., Sidhu, H. S. and Mercer, G. N., “Critical times in single-layer reaction diffusion”, Int. J. Heat Mass Transfer 54 (2011) 26422650; doi:10.1016/j.ijheatmasstransfer.2011.01.019.
[15]Kolomeisky, A. B., “Formation of a morphogen gradient: acceleration by degradation”, J. Phys. Chem. Lett. 2 (2011) 15021505; doi:10.1021/jz2004914.
[16]Landman, K. A., Cai, A. Q. and Hughes, B. D., “Travelling waves of attached and detached cells in a wound-healing cell migration assay”, Bull. Math. Biol. 69 (2007) 21192138; doi:10.1007/S11538-007-9206-0.
[17]Landman, K. A. and McGuinness, M. J., “Mean action time for diffusive processes”, J. Appl. Math. Decision Sci. 4 (2000) 125141; doi:10.1155/S1173912600000092.
[18]Landman, K. A. and White, L. R., “Predicting filtration time and maximizing throughput in a pressure filter”, AIChE J. 43 (1997) 31473160; doi:10.1002/aic.690431204.
[19]Lunn, M., Lunn, R. J. and Mackay, R., “Determining analytic solutions of multiple species contaminant transport, with sorption and decay”, J. Hydrol. 180 (1996) 195210; doi:10.1016/0022-1694(95)02891-9.
[20]Montgomery, J. H., Groundwater chemicals desk reference, 4th edn. (CRC Taylor and Francis, Boca Raton, FL, 2007).
[21]McNabb, A., “Mean action times, time lags, and mean first passage times for some diffusion problems”, Math. Comput. Modell. 18 (1993) 123129; doi:10.1016/0895-7177(93)90221-J.
[22]McNabb, A. and Wake, G. C., “Heat conduction and finite measures for transition times between steady states”, IMA J. Appl. Math. 47 (1991) 193206; doi:10.1093/imamat/47.2.193.
[23]Simpson, M. J. and Landman, K. A., “Analysis of split operator methods applied to reactive transport with Monod kinetics”, Adv. Water Resour. 30 (2007) 20262033; doi:10.1016/j.advwatres.2007.04.005.
[24]Simpson, M. J., Landman, K. A. and Clement, T. P., “Assessment of a nontraditional operator split algorithm for simulation of reactive transport”, Math. Comput. Simul. 70 (2005) 4460; doi:10.1016/j.matcom.2005.03.019.
[25]Simpson, M. J., Landman, K. A. and Hughes, B. D., “Cell invasion with proliferation mechanisms motivated by time-lapse data”, Phys. A 389 (2010) 37793790; doi:10.1016/j.physa.2010.05.020.
[26]Simpson, M. J., Towne, C., McElwain, D. L. S. and Upton, Z., “Migration of breast cancer cells: Understanding the roles of volume exclusion and cell-to-cell adhesion”, Phys. Rev. E 82 (2010) 041901; doi:10.1103/PhysRevE.82.041901.
[27]van Genuchten, M. Th., “Convective-dispersive transport of solutes involved in sequential first-order decay reactions”, Comput. Geosci. 11 (1985) 129147; doi:10.1016/0098-3004(85)90003-2.
[28]Vogel, T. M. and McCarty, P. L., “Biotransformation of tetrachloroethylene to trichloroethylene, dichloroethylene, vinyl chloride, and carbon dioxide under methanogenic conditions”, Appl. Environ. Microbiol. 49 (1985) 10801083; http://www.ncbi.nlm.nih.gov/pmc/articles/PMC238509/.
[29]Zheng, C. Z. and Bennett, G. D., Applied contaminant transport modeling, 2nd edn. (John Wiley, New York, 2002).
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