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Simulation of bimolecular reactions: Numerical challenges with the graph Laplacian
Get PDFAn important framework for modelling and simulation of chemical reactions is a Markov process sometimes known as a master equation. Explicit solutions of master equations are rare; in general the explicit solution of the governing master equation for a bimolecular reaction remains an open question. We show that a solution is possible in special cases. One method of solution is diagonalization. The crucial class of matrices that describe this family of models are non-symmetric graph Laplacians. We illustrate how standard numerical algorithms for finding eigenvalues fail for the non-symmetric graph Laplacians that arise in master equations for models of chemical kinetics. We propose a novel way to explore the pseudospectra of the non-symmetric graph Laplacians that arise in this class of applications, and illustrate our proposal by Monte Carlo. Finally, we apply the Magnus expansion, which provides a method of simulation when rates change in time. Again the graph Laplacian structure presents some unique issues: standard numerical methods of more than second-order fail to preserve positivity. We therefore propose a method that achieves fourth-order accuracy, and maintain positivity.
References
A. Basak, E. Paquette, and O. Zeitouni. Regularization of non-normal matrices by gaussian noise–-the banded toeplitz and twisted toeplitz cases. In Forum of Mathematics, Sigma, volume 7. Cambridge University Press, 2019. doi:10.1017/fms.2018.29.
S. Blanes, F. Casas, J. A. Oteo, and J. Ros. The magnus expansion and some of its applications. Phys. Rep., 470(5-6):151–238, 2009. doi:10.1016/j.physrep.2008.11.001.
B. A. Earnshaw and J. P. Keener. Invariant manifolds of binomial-like nonautonomous master equations. SIAM J. Appl. Dyn. Sys., 9(2):568–588, 2010. doi10.1137/090759689.
J. Gunawardena. A linear framework for time-scale separation in nonlinear biochemical systems. PloS One, 7(5):e36321, 2012. doi:10.1371/journal.pone.0036321.
A. Iserles and S. MacNamara. Applications of magnus expansions and pseudospectra to markov processes. Euro. J. Appl. Math., 30(2):400–425, 2019. doi:10.1017/S0956792518000177.
S. MacNamara. Cauchy integrals for computational solutions of master equations. ANZIAM Journal, 56:32–51, 2015. doi:10.21914/anziamj.v56i0.9345.
S. MacNamara, A. M. Bersani, K. Burrage, and R. B. Sidje. Stochastic chemical kinetics and the total quasi-steady-state assumption: Application to the stochastic simulation algorithm and chemical master equation. J. Chem. Phys., 129:095105, 2008. doi:10.1063/1.2971036.
S. MacNamara and K. Burrage. Stochastic modeling of naive T cell homeostasis for competing clonotypes via the master equation. SIAM Multiscale Model. Sim., 8(4):1325–1347, 2010.
S. MacNamara, K. Burrage, and R. B. Sidje. Multiscale modeling of chemical kinetics via the master equation. SIAM Multiscale Model. and Sim., 6(4):1146–1168, 2008. doi:10.1137/060678154.
S. MacNamara, Wi. McLean, and K. Burrage. Wider contours and adaptive contours, pages 79–98. Springer International Publishing, 2019. doi:10.1007/978-3-030-04161-8_7.
M. J. Shon. Trapping and manipulating single molecules of DNA. PhD thesis, Harvard University, 2014. http://nrs.harvard.edu/urn-3:HUL.InstRepos:11744428.
M. J. Shon and A. E. Cohen. Mass action at the single-molecule level. J. Am. Chem. Soc., 134(35):14618–14623, 2012. doi:10.1021/ja3062425.
C. Timm. Random transition-rate matrices for the master equation. Phys. Rev. E, 80(2):021140, 2009. doi:10.1103/PhysRevE.80.021140.
L. N. Trefethen and M. Embree. Spectra and pseudospectra: The behavior of nonnormal matrices and operators. Princeton University Press, 2005. https://press.princeton.edu/books/hardcover/9780691119465/spectra-and-pseudospectra
On the hyperstability of the drygas functional equation on a restricted domain
No full texthttp://dx.doi.org/10.1017/S000497271900109
Character graphs with non-bipartite Hamiltonian complement
No full texthttp://dx.doi.org/10.1017/S000497271900116
Uniform asymptotic formulas of restricted bipartite partitions
No full texthttp://dx.doi.org/10.1017/S000497272000006
Power partitions and semi--Fibonacci partitions
No full texthttp://dx.doi.org/10.1017/S000497272000002
Modelling of hydrogen diffusion in the retina
Get PDFA simple mathematical model for diffusion of hydrogen within the retina has been developed. The model consists of three, well-mixed, one dimensional layers that exchange hydrogen via a diffusion process. A Fourier series method is applied to compute the hydrogen concentration. The effect of important parameters is examined and discussed. The results may contribute to an understanding of the hydrogen clearance technique to estimate blood flow. A two dimensional numerical method for the hydrogen diffusion is also presented. It is shown that the predominant features of the process are captured quite well by the simpler model.
References
V. A. Alder, D. Y. Yu, S. J. Cringle and E. N. Su. Experimental approaches to diabetic retinopathy. Asia-Pac. J. Ophthalmol. 4:20–25, 1992.
J. C. Arciero, P. Causin and F. Malgoroli. Mathematical methods for modeling the microcirculation. AIMS Biophys. 4:362–399, 2017. doi:10.3934/biophy.2017.3.362
D. E. Farrow, G. C. Hocking, S. J. Cringle and D.-Y. Yu. Modeling Hydrogen clearance from the retina. ANZIAM J. 59:281–292, 2018. doi:10.1017/S1446181117000426
A. B. Friedland. A mathematical model of transmural transport of oxygen to the retina. Bull. Math. Biol. 40:823–837, 2018; doi:10.1007/BF02460609
D. Goldman. Theoretical models of microvascular oxygen transport to tissue. Microcirculation 15:795–811, 2008. doi:10.1080/10739680801938289
A. C. Hindmarsh. ODEPACK, A Systematized Collection of ODE Solvers. In Scientific Computing, R. S. Stepleman, et al., Eds., pp. 55-64. North-Holland, Amsterdam, 1983.
S. S. Kety. The theory and applications of the exchange of inert gas at the lungs and tissues. Pharmacol. Rev. 3:1–41, 1951. http://pharmrev.aspetjournals.org/content/3/1/1
B. P. Leonard. A stable and accurate convective modelling procedure based on quadratic upstream interpolation. Comput. Methods Appl. Mech. Eng. 19:59–98, 1979. doi:10.1016/0045-7825(79) 90034-3
S. L. Mitchell. Coupling transport and chemistry: numerics, analysis and applications. PhD thesis, University of Bath, UK, 2003. https://researchportal.bath.ac.uk/en/studentTheses/coupling-transport-and-chemistry-numerics-analysis-and-applicatio
G. A. Winchell. Mathematical model of inert gas washout from the retina: evaluation of hydrogen washout as a means of determining retinal blood flow in the cat. Master\textquoteright s Thesis, Northwestern University, Evanston, USA, 1983. https://search.library.northwestern.edu/permalink/f/5c25nc/01NWU_ALMA21563278530002441
D. Y. Yu, V. A. Alder and S. J. Cringle. Measurement of blood flow in rat eyes by hydrogen clearance. Am. J. Physiol. (Heart Circ. Physiol.) 261:H960–H968, 1991. doi:10.1152/ajpheart.1991.261.3.H960
D. Y. Yu, S. J. Cringle, V. A. Alder, E. N. Su, and P. K. Yu, Intraretinal oxygen distribution and choroidal regulation in the avascular retina of guinea pigs. Am. J. Physiol. (Heart Circ. Physiol.) 270:H965-H973, 1996. doi:10.1152/ajpheart.1996.270.3.H965
S. Cringle, D.-Y. Yu, V. Alder, E.-N. Su, and P. Yu. Choroidal regulation of oxygen supply to the guinea pig retina. In A. G. Hudetz, and D. F. Bruley (Eds.), Oxygen Transport to Tissue XX, pp. 385–389. Springer, 1998. doi:10.1007/978-1-4615-4863-
Strategic customers in Markovian queues with vacations and synchronized abandonment
No full textWe study impatient customers’ joining strategies in a single-server Markovian queue with synchronized abandonment and multiple vacations. Customers receive the system information upon arrival, and decide whether to join or balk, based on a linear reward-cost structure under the acquired information. Waiting customers are served in a first-come-first-serve discipline, and no service is rendered during vacation. Server’s vacation becomes the cause of impatience for the waiting customers, which leads to synchronous abandonment at the end of vacation. That is, customers consider simultaneously but independent of others, whether to renege the system or to remain. We are interested to study the effect of both information and reneging choice on the balking strategies of impatient customers. We examine the customers’ equilibrium and socially optimal balking strategies under four cases of information: fully/almost observable and fully/almost unobservable cases, assuming the linear reward-cost structure. We compare the social benefits under all the information policies.
doi: 10.1017/S144618112000011
Critical length for the spreading-vanishing dichotomy in higher dimensions
No full textWe consider an extension of the classical Fisher–Kolmogorov equation, called the “Fisher–Stefan” model, which is a moving boundary problem on 0<x<L(t) . A key property of the Fisher–Stefan model is the “spreading–vanishing dichotomy”, where solutions with L(t) > L_{\textrm{c}} will eventually spread as , whereas solutions where will vanish as . In one dimension it is well known that the critical length is . In this work, we re-formulate the Fisher–Stefan model in higher dimensions and calculate as a function of spatial dimensions in a radially symmetric coordinate system. Our results show how depends upon the dimension of the problem, and numerical solutions of the governing partial differential equation are consistent with our calculations.
doi: 10.1017/S144618112000010
Semilinear Calderon problem on Stein manifolds with Kahler metric
No full texthttp://dx.doi.org/10.1017/S000497271200033
Sums of partial theta functions through an extended Bailey transform
No full texthttp://dx.doi.org/10.1017/S000497271200033