A SHORT COURSE IN THE MODELING OF CHEMOTAXIS

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[24] G. M. Saidel, E. D. Morris, and G. M. Chisolm, Transport of Macro- molecules in Arterial Wall in vivo:a Mathematical Model and Analytic Solutions, Bull. Math. Biol. (1987) 49, 153–169 [25] D. Steinberg, Low Density Lipoprotein Oxidation and its Pathobiological Sig- nificance, J. Bio. Chem. (1997) 272, 20963–20966 [26] A. Stevens, The Derivation of Chemotaxis Equations as Limit Dynamics of Moderately Interacting Stochastic Many-Particle Systems, SIAM J. Appl. Math. (2000), Vol. 61 no. 1 183–212 (electronic) 75
Appendix A Linear Chemical Reactions There are several ways that one can translate a chemical schematic into mathematical equations. The most common assumption is that the reactions are linear in the reactants. Here, I think it most instructive to explain this by going through a typical (albeit fictitious) example. But first a few words about notation. In the literature, particularly when composed by chemists, a given chemical species is denoted differently when it appears in a chemical schematic than when it appears in a differential equation. For example, suppose that the reaction involves the polymerization of methyl methacrylate. In the chemical schematic, this might be assigned a variable A (A + B → C) which is a symbol that denotes that methyl methacrylate is one of the reactants but that has no notion of dimension associ- ated with it. When translated into a mathematical equation, chemists will typically denote the concentration of methyl methacrylate, which would have specified dimensions of Moles/Liter or perhaps micro Moles/Liter, by the related variable [A] Thus the mathematical expression would look something like d[A] dt = −k[A][B]. 76
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A SHORT COURSE IN THE MODELING OF C
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Contents List of Figures iii 1 Intr
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List of Figures 1.1 Life Cycle of D
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mobile species.” Referring back t
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this new form, as a plasmoid, the i
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the macroscopic (see Patlak [20]),
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η(x, t) The concentration of acras
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The only way for this result to hol
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Equation (2.2) is the identifying e
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2.2 The Keller-Segel Model of Chemo
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Figure 2.1: The experimental config
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, b ′ → 0 and s → s0 as z →
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2.4 Some conclusions In this chapte
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Chapter 3 Modeling: The Microscopic
Page 29 and 30: master equation of our pdf we need
Page 31 and 32: This equation assumes that our walk
Page 33 and 34: A number of observations can be mad
Page 35 and 36: The notable characteristic of this
Page 37 and 38: efore, the corresponding continuous
Page 39 and 40: or in two or more dimensions 34 ∂
Page 41 and 42: Chapter 4 Applications of the Kelle
Page 43 and 44: (SMCs). The ECs are the main cellul
Page 45 and 46: the review [17] and the references
Page 47 and 48: The first two assumptions immediate
Page 49 and 50: arrive at the system of interest
Page 51 and 52: 1/40 th of an hour, and during this
Page 53 and 54: Figure 4.3: Time evolution of the e
Page 55 and 56: Figure 4.5: Evolution of the growth
Page 57 and 58: tree. More advanced atherosclerotic
Page 59 and 60: three distinctive layers-the intima
Page 61 and 62: to be characterized by a change in
Page 63 and 64: eaction with a free radical R at a
Page 65 and 66: Lesion progression Smooth muscle ce
Page 67 and 68: Modeling Assumptions We begin by id
Page 69 and 70: gradient. For the immune cells and
Page 71 and 72: Both production and consumption app
Page 73 and 74: assume that the threshold values of
Page 75 and 76: The system (4.22)-(4.24) was solved
Page 77 and 78: Bibliography [1] E. De Angelis and
Page 79: [15] I. R. Lapidus, ”Pseudochemot
Page 83 and 84: proportional to A”. So that there
Page 85 and 86: Appendix B Bachmann-Landau Order Sy
Page 87: 3. Let h(τ) = e −τ and p(τ) =
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A SHORT COURSE IN THE MODELING OF CHEMOTAXIS

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