A SHORT COURSE IN THE MODELING OF CHEMOTAXIS

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exercise for the reader11 The solution is given by b(z) = c2 s(z) = � s0 s k(δ − µ) s0 � −cz/µ s0 1 + e � δ/µ e −cz/µ 19 (2.15) � −µ δ−µ (2.16) From this solution we see something similar to what was observed for the system (2.6)–(2.7). Note that the solution (b, s) above is only bounded if δ − µ > 0 i.e if δ > µ. That is to say the chemotactic influence must prevail over the diffusive influence on bacteria for the solution (2.15)–(2.16) to be meaningful—for traveling waves to exist. The model here is further supported by matching the wave speed c with reported data. This is done by the authors in [13] where they showed that the wave speed determined by (2.15)–(2.16) is in reasonable agreement with available data. The speed c can be determined by substituting (2.16) into (2.15) and integrating over all z (since the number of bacteria is assumed constant). One finds that c = Nk as0 � ∞ where N = a b(z) dz −∞ and a is the cross sectional area of the capillary tube. 11 H<strong>IN</strong>T: Integrate (2.12) and use the condition at infinity to deal with the constant of integration. Then divide through by b and integrate again. The solution will look like b = Qs δ/µ exp(−cz/µ) with Q arbitrary (corresponding to horizontal translation in z). To get the solution to look like (2.15)–(2.16) choose Q = c2s0 k(δ − µ) s−δ/µ 0 .
2.4 Some conclusions In this chapter we have seen the modeling of chemotaxis approached from the consideration of an entire population of organisms and how they interact with their environment. The most notable result is the equation (2.8) which captures in a mathematical language the phenomenon of taxis of a living species in response to a nonliving species—some chemical in the environment. The form of the second equation in the Classical Keller-Segel model as stated in (2.9) is decidedly vague. The influences that drive the evolution of the chemical species can vary from system to system and must be specified in correspondence with the assumptions of a particular model. Hopefully the model examples discussed in sections 2.1 and 2.3 have given you some idea of what types of factors might go into determining the appropriate right hand side of equation (2.9). • Does the organism produce the chemical? • Does the organism deplete the chemical, for example by eating it? • Are there other sources of the chemical, perhaps it is produce by some chemical reaction? • Does the chemical naturally degrade? • Is there appreciable diffusion of the chemical? These are just a few questions the modeler might want to consider. We will see more examples of equations governing chemical species in the fourth chapter. Now, we turn our attention to the modeling of chemotaxis from the microscopic perspective. Given the uncertainty encountered when attempting to quantify the 20
Page 1 and 2: A SHORT COURSE IN THE MODELING OF C
Page 3 and 4: Contents List of Figures iii 1 Intr
Page 5 and 6: List of Figures 1.1 Life Cycle of D
Page 7 and 8: mobile species.” Referring back t
Page 9 and 10: this new form, as a plasmoid, the i
Page 11 and 12: the macroscopic (see Patlak [20]),
Page 13 and 14: η(x, t) The concentration of acras
Page 15 and 16: The only way for this result to hol
Page 17 and 18: Equation (2.2) is the identifying e
Page 19 and 20: 2.2 The Keller-Segel Model of Chemo
Page 21 and 22: Figure 2.1: The experimental config
Page 23: , b ′ → 0 and s → s0 as z →
Page 27 and 28: 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
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The system (4.22)-(4.24) was solved
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Bibliography [1] E. De Angelis and
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[15] I. R. Lapidus, ”Pseudochemot
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Appendix A Linear Chemical Reaction
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proportional to A”. So that there
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Appendix B Bachmann-Landau Order Sy
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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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