Abstract
The preceding chapter had some pretty heavy analysis, and the next has even more. In what may be welcome relief, the present chapter pushes in an orthogonal direction: it focuses on nondimensionalization and scaling, which are techniques for simplifying ODEs that arise in applications.
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Notes
- 1.
Even with this restriction, we are forced to omit many interesting problems; just to cite one example, reference [96] studies ODEs derived from neural models that do not belong to any of our three categories.
- 2.
Regarding the notation \(d^{2}x/d\hat{t}^{2}\): At this point, we introduce the convention of putting hats over variables—but not parameters—that have dimensions. Time has dimensions, but since x is measured in radians, it does not. Although this may seem obscure at the moment, we hope that the discussion in Section 5.2 will clarify the reasons for this convention. In the meantime, we suggest that you simply ignore the hats.
- 3.
Sel’kov’s model is represented more literally as a bathtub model in the cartoon on the frontispiece.
- 4.
Would you prefer light-years per year? Unity is such a nice velocity.
- 5.
Thus, this equation differs from Duffing’s equation (1.28) in that the linear part of the force law, \(k_{1}\hat{x}\), is attracting. This sign difference has no effect on scaling the equation; it just makes interpretation of the time scale \(T = \sqrt{m/k_{1}}\), which we introduce below, slightly more transparent.
- 6.
What about choosing T = m∕b to simplify the first coefficient? In Section 5.9, we argue that provided friction is not too large, the spring constant k 1 provides a more useful basis for scaling than b. But such choices are rarely cut-and-dried.
- 7.
However, to support scientific literacy we explain molarity in the Pearls. One point is relevant here: concentration in molarity is based on counting molecules per unit volume, not mass per unit volume. This choice allows one to readily compare the concentrations of the different substances X and Y, and it is responsible for the pleasant feature that the coefficients of the terms \(B\hat{x}\) and \(C\hat{x}\hat{y}^{2}\) have equal magnitudes in both equations (5.14).
- 8.
A reaction rate of the form \(\hat{x}^{n}/(K^{n} +\hat{ x}^{n})\), as appears in (5.23), is called a Hill function. If n = 1, this rate reduces to Michaelis–Menten kinetics (see Section 5.7). These rates arise from enzymatic reactions; when cooperative effects are important, n > 1. See Chapter 1 of Keener and Sneyd [47].
- 9.
Do you find this figure useful? If not, feel free to ignore it. We find the schematic diagrams less informative as they become more complicated.
- 10.
Let’s admit it—we needed several tries to get this right. After the fact, we can motivate focusing on the second equation, since this equation contains coefficients that appear in the largest parameter.
- 11.
Why are there only three parameters in (5.26), even though there are four dimensionless combinations of the parameters in (5.23)? The reason is that even though both \(\hat{x}\) and \(\hat{y}\) have units of molarity, we chose different x- and y-scales, and this gave us an extra degree of freedom in simplifying the equations. By contrast, this option was not attractive to us in scaling the equations (5.14) for Sel’kov’s model, because if we had chosen different scales for x and y in that problem, the terms \(B\hat{x}\) and \(C\hat{x}\hat{y}^{2}\) would then have had unequal coefficients in the two equations. The difference is that in Sel’kov’s model, single reaction terms change both concentrations, subtracting from X and adding to Y.
- 12.
Reaction (5.28) might seem to violate conservation of mass; otherwise, how could P be different from R? Typically, the product P is what’s called an isomer of R, a substance with the same chemical composition but with different spatial configuration. Incidentally, the second reaction [RE] → P+E is treated as irreversible; this assumption is appropriate if, for example, the product is produced in the cell nucleus but is transported out of the nucleus to the cell body before the reverse reaction can occur.
- 13.
Note that the same reaction constants appear in different equations. As in Sel’kov’s model, this simplification arises because we measure all concentrations in molarity, which is based on counting molecules. (Molarity is explained in the Pearls.) In contrast to Sel’kov’s model, when we simplify the present problem, we choose different scales for \(\hat{r}\) and \(\hat{c}\), scales that reflect typical orders of magnitude for the variables.
- 14.
These are not the parameters for any specific reaction; they are arbitrarily chosen values in the midrange of what is observed experimentally in reactions of this type.
- 15.
Incidentally, if you like jargon: chemists call the quantity k −1∕k +1 the dissociation constant of the reaction (5.33).
- 16.
- 17.
The angle x is already dimensionless. We prefer not to scale x, because that would mess up the trig functions.
- 18.
If (5.49) were written as a first-order system, it would be a fast–slow system, and neglecting this second-order derivative would be the approximation of letting the fast equation go to equilibrium.
- 19.
For the most part, we use Greek letters for dimensionless constants, but K deviates from this convention. However, invoking the pseudo-justification that too much consistency can be oppressive, we stick with K.
- 20.
Because of heavier isotopes, the molecular weight of carbon is a little greater than 12, and likewise for hydrogen and oxygen. Thus the molecular weight of glucose is also a little greater than 180, but this integer approximation is adequate for our purposes.
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Schaeffer, D.G., Cain, J.W. (2016). Nondimensionalization and Scaling. In: Ordinary Differential Equations: Basics and Beyond. Texts in Applied Mathematics, vol 65. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-6389-8_5
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