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Diffusion with attrition. (English) Zbl 1113.92027

Summary: This article treats the problem of the sharp front observed when a diffusing substance interacts irreversibly with binding sites within the medium. The model consists of two simultaneous partial differential equations that are nonlinear and cannot be solved in closed form. The parameters are the diffusion coefficient \(D\) in the direction under consideration \((x)\), the interaction constant \(k\), the binding-site concentration \(\mu\) and the boundary concentration of the diffusing ion \(c_0\). Our aim is to develop methods to enable the estimation of these parameters from the experimental data. An analytical solution for the case \(k\to\infty\), as found by others, is given first and then a finite element analysis package is used to obtain numerical solutions for the general case.
Graphs are presented to illustrate the effects of the various parameters. Simple graphical procedures are described to compute \(\mu\) and \(c_0\). The position of the advancing front \(\xi\) then provides, together with \(\mu\), a way to estimate \(D\). A mathematical identity relating \(D\) and \(x\) and a second one involving \(D\), \(k\) and \(t\) help to reduce the complexity of the problem. A new, measurable quantity \(S(t)\) is defined as \(S(t)=\max_x (-df(x,t)/dx)\), where \(f\) is the total concentration (free + bound) of the diffusing ion at time \(t\), and detailed plots are furnished that permit the computation of \(k\) directly from \(S(t)\), \(\mu\) and \(D\). The accuracy with which such methods can be expected to determine the various parameters of the model is considered at some length. Finally, in a concluding section, we simulate typical experimental data, examine the validity of our methods, and see how their accuracy is affected by controlled amounts of various kinds of noise.

MSC:

92C40 Biochemistry, molecular biology
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
92C45 Kinetics in biochemical problems (pharmacokinetics, enzyme kinetics, etc.)
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[1] Ben-Naim E., Redner S.,Weiss G.H. (1993) Partial absorption and ”virtual” traps. J. Stat. Phys. 71, 75–88 · Zbl 0920.60088 · doi:10.1007/BF01048089
[2] Crank J. (1975) The Mathematics of Diffusion, 2nd ed, pp. 349–350. Oxford University Press, Oxford
[3] Crank J., Nicolson P. (1947) A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type. Proc. Camb. Philos. Soc. 43, 50–67 · Zbl 0029.05901 · doi:10.1017/S0305004100023197
[4] Edelstein-Keshet L., Ermentrout G.B. (2000) Models for spatial polymerization dynamics of rod-like polymers. J. Math. Biol. 40, 64–96 · Zbl 0998.92015 · doi:10.1007/s002850050005
[5] FlexPDE, Professional Edition: PDE Solutions. Antioch, CA, Version 5.0.10(2006)
[6] Grover N.B. (1966) Anisometric transport of ions and particles in anisotropic tissue spaces. Biophys. J. 6, 71–85 · doi:10.1016/S0006-3495(66)86640-7
[7] Hermans J.J. (1947) Diffusion with discontinuous boundary. J. Colloid Sci. 2, 387–398 · doi:10.1016/0095-8522(47)90042-1
[8] Hill A.V. (1929) The diffusion of oxygen and lactic acid through tissues. Proc. Roy. Soc. B104, 39–96 · doi:10.1098/rspb.1928.0064
[9] IMSL (ZBREN): Math and Stat Libraries. In: Fortran PowerStation, Professional Edition, Microsoft, Redmond, CA, Version 4.0(1995)
[10] Katz S.M., Kubu E.T., Wakelin J.H. (1950) The chemical attack on polymeric materials as modified by diffusion. Text. Res. J. 20, 754–760 · doi:10.1177/004051755002001102
[11] Marino M.A. (1974) Numerical and analytical solutions of dispersion in a finite, adsorbing porous medium. Water Resour. Bull. 10, 81–90
[12] Nicolson P., Roughton F.J.W. (1951) A theoretical study of the influence of diffusion and chemical reaction velocity on the rate of exchange of carbon monoxide and oxygen between the red blood corpuscle and the surrounding fluid. Proc. Roy. Soc. B138, 241–264 · doi:10.1098/rspb.1951.0020
[13] Olbris D.J., Herzfeld J. (1999) An analysis of actin delivery in the acrosomal process of Thyone. Biophys. J. 77, 3407–3423 · doi:10.1016/S0006-3495(99)77172-9
[14] Oster G., Perelson A.S.(1994) Cell protrusions. In: Levin S.A.(eds) Frontiers in Mathematical Biology, Lecture Notes in Biomathematics. Springer, Berlin Hedelberg New York, pp 53–78 · Zbl 0821.92012
[15] Oster G.F., Perelson A.S., Tilney L.G. (1982) A mechanical model for elongation of the acrosomal process in Thyone sperm. J. Math. Biol. 15, 259–265 · Zbl 0492.92004 · doi:10.1007/BF00275078
[16] Perelson A.S., Coutsias E.A. (1986) A moving boundary model of acrosomal elongation. J. Math. Biol. 23, 361–379 · Zbl 0585.92027 · doi:10.1007/BF00275254
[17] Perelson A.S., Segel L.A. (1978) A singular perturbation approach to diffusion reaction equations containing a point source, with application to hemolytic plaque assay. J. Math. Biol. 6, 75–85 · Zbl 0387.92004 · doi:10.1007/BF02478519
[18] Petropoulos J.H., Roussis P.P. (1969) Diffusion of penetrants in organic solids accompanied by other rate processes. In: Adler G., (eds) Organic Solid State Chemistry. Gordon & Breach, New York, pp. 343–357
[19] Putnam D.D., Burns M.A. (1997) Predicting the filtration of noncoagulating particles in depth filters. Chem. Eng. Sci. 52, 93–105 · doi:10.1016/S0009-2509(96)00367-3
[20] Reese C.E., Eyring H. (1950) Mechanical properties and the structure of hair. Text. Res. J. 20, 743–753 · doi:10.1177/004051755002001101
[21] Roughton F.J.W. (1959) Diffusion and simultaneous chemical reaction velocity in haemoglobin solutions and red cell suspensions. Prog. Biophys. Biophys. Chem. 9, 55–104
[22] Tilney L.G., Kallenbach N. (1979) Polymerization of actin. VI. The polarity of the actin filaments in the acrosomal process and how it might be determined. J. Cell Biol. 81, 608–623 · doi:10.1083/jcb.81.3.608
[23] Zhu C., Skalak R. (1988) A continuum model of protrusion of pseudopod in leukocytes. Biophys. J. 54, 1115–1137 · doi:10.1016/S0006-3495(88)83047-9
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