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Figure 4 | Theoretical Biology and Medical Modelling

Figure 4

From: The quantitation of buffering action I. A formal & general approach

Figure 4

Proportions in Two-Partitioned Systems. The buffering measures are dimensionless proportions between two parts of a whole, or between one particular part and the whole. A, Bisection of a straight line. An oriented line of length σ (row 1) can be divided in several ways into two parts of lengths τ (red) and β (blue), respectively (rows 2–5). Dividing the line at a point that is lying on the line itself gives rise to an "inner division" (rows 2 and 3), whereas dividing the line outside the interval yields an "outer divison" (rows 4 and 5). Proportions between the two parts can be expressed in various ways (Figure 4D). For inner divisions, proportions are positive-valued. For outer divisions, "negative proportions" and fractional lengths greater than 1 or smaller than 0 are obtained. B, Bisection of a function The principle of dividing a quantity into two is also applicable to the values of a function of x at a given value of x. Thus, there are multiple ways to split an entire function σ into two functions τ and β such that the sum of their values τ(x) and β(x) equals the value σ(x) for every x. C, Bisection of a slope, or rate of change. The quantity to be bisected may as well be the slope σ' of a function σ. Again, there are multiple ways to split a function σ into two functions τ and β such that the sum of their first derivatives τ'(x) and β'(x) equals σ'(x) for every x. For functions and derivatives of functions alike, the proportions between the two parts into which they were split can be expressed by the four measures indicated in Figure 4D. "Buffering" relates to the proportion between such partial rates of change of two complementary processes. D, Measures of proportionality between the parts of a bisected slope. Proportions among two partial rates of change τ' and β' that result from bisection of a whole rate σ' can be expressed either as fractions of a part with respect to the whole (τ'/σ' and β'/σ') or as as ratio of one part over the other (τ'/β' and β'/τ'). The following terminology is suggested: t, "transfer coefficient"; b, "buffering coefficient"; T, "transfer ratio"; B, "buffering ratio". These parameters serve to quantitate buffering action. E, Relation between the four measures of proportion. Any single one of the four measures (t,b,T,B) fully determines the other three. The plot shows b, T, and B as functions of t. F, Trigonometric measure of proportionality between parts of a bisected slope. The two partial rates of change τ'(x) and β'(x) may be interpreted as two perpendicular vectors. Their resultant τ'(x) + β'(x) encloses an "buffering angle α" with τ'(x). The buffering angle and the four buffering parameters (t, b, T, B) are related by four bijections with the buffering angle . A buffering angle α = 0 is equivalent to zero buffering, a buffering angle of 90° to perfect buffering. This representation of buffering behavior does not have discontinuities at "infinite" transfer or buffering odds, and is able to reflect the full range of buffering withing half a unit circle (-45° to 135°). (See Supplement 5 for further details)

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