A SHORT COURSE IN THE MODELING OF CHEMOTAXIS

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(iii) The MMP degrades fibronectin to produce an intermediate product that in turn produces an MMP plus a degraded product. c + f k3 k4 −→ q −→ c + degraded product (The intermediate product q is not of interest.) This occurs by Michaelis- Menten kinetics 4 (as opposed to linear reactions). (iv) The ECs produce fibronectin at a rate of βf(fM − f) per EC; where fM is the amount of fibronectin in a normal capillary. (v) The decay of fibronectin and MMP is negligible. (vi) The ECs follow a reinforced random walk using a nearest neighbor model with transitional probabilities ˆ T (c, f) = ˆ T1(c) ˆ T2(f). (vii) The kinetics of the reactions in assumptions (i) and (ii) are in steady state prior to EC migration. (viii) The system is closed with no flux of EC into or out of the domain. (ix) The system is on a 1D lattice x = 0 to x = 1 4 For a linear reaction, we might interpret this schematic as (see appendix A) df = −k3cf. dt The assumption that the kinetics are of Michaelis-Menten type means that we modify this equation so that if f is small, the equation is essentially the same linear kinetics, but when f is large the equation is of zero order—df/dt = constant. This can be achieved by writing df −k3cf = dt 1 + ν2f . 41
The first two assumptions immediately give us preliminary differential equations for v, r, l, and c based on the condition of linear reactions. These are 42 ∂v ∂t = −k1rv + k−1l (4.1) ∂r ∂t = −k1rv + (k−1 + k2)l (4.2) ∂l ∂t = k1rv − (k−1 + k2)l (4.3) ∂c ∂t = k2l (4.4) Assumptions (iii) through (v) allow us to write the equation for fibronectin ∂f ∂t = −λ2cf 1 + ν2f + βf(fM − f)η (4.5) where the λ2 = k3, and ν2 = k4/k3. Finally, we can obtain the equation ∂η ∂t � � ∂ ∂η ∂ = D − η (ln τ) ∂x ∂x ∂x (4.6) which is an equation consistent with (3.12) and assumption (vi), namely η follows a reinforced random walk of a nearest neighbor model 5 . At this point, an equation for each of the species has been specified. But rather than consider the system (4.1)– (4.6) the authors of [16] note that while EC move on a time scale of days, the chemical kinetics described in the first two assumptions occur on the time scale of seconds. They thus argue that species r and l are in a pseudo steady state during EC movement. The phrase ”pseudo” steady state is used because the solution r, 5 The term τ in equation (4.6) is equivalent to the term ˆ T in (3.12) and should not be confused with the use of the character τ used in chapter 3 to discuss the gradient based model. While this notational change may be confusing, the use of the character τ in chapter 3 was consistent with the notation of Othmer and Stevens whose work chapter 3 was based on, and here the use of τ is consistent with the notation of Levine et al. the authors of the current model.
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 and 24: , b ′ → 0 and s → s0 as z →
Page 25 and 26: 2.4 Some conclusions In this chapte
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: the review [17] and the references
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 and 80: [15] I. R. Lapidus, ”Pseudochemot
Page 81 and 82: Appendix A Linear Chemical Reaction
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) =

A SHORT COURSE IN THE MODELING OF CHEMOTAXIS

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