All Nonlinear Phenomena (Nonlinear by Nonlinear) with a Total of Three Asymptotes

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Nonlinearity in Life and Biomedical Sciences

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All Nonlinear Phenomena (nonlinear by nonlinear) with a total of Three Asymptotes

Proportionality Law of the Sixth Kind (PL6)

In some natural phenomena, there may be a need to compare changes between two nonlinear numbers, with one nonlinear numbers having only a lower asymptote, and the other with both upper and lower asymptotes, adding up to a total of three asymptotes in the system. Phenomena that can be described by this type of equation are said to follow the proportionality law of the sixth kind (PL6). Examples of the phenomena following this law include: dose-response relationship in drug research and Hill’s equation.

Let us rewrite the equation of the proportionality law of the sixth kind in Table 2.1 as Eq. (8.0a):

d(qα21) = Kd(qβ11) (8.0a)

Subscripts 21 in α indicate that the α has two (2) asymptotes, and we are using the first (1) form of measurement for nonlinear numbers Y to account for both the upper and lower asymptotes. Subscript 11 in β indicates that the β has a single (1) asymptote and is using the first (1) form of measurementto measure the nonlinear numbers X relative to itslower asymptote. Substituting 21 with (Y – Ys)/(Yu – Y)and 11 with (X – Xs), we get Eq. (8.0b).

(8.0b)

In general, Eq. (8.0a) or Eq. (8.0b) states that thenonlinear change of the nonlinear face-value  is proportional to the nonlinear change of the nonlinear face-value . Alternatively, it states that the change of nonlinear true-value q((Y – Ys)/(Yu – Y)) is proportional to the changeof nonlinear true-value q(X – Xs). Integration of the above equations yields Eq. (8.0c).

(8.0c)

The log with superscript 1, q1, means “please plot the following nonlinear face-values on the nonlinear scale”. Logarithms of nonlinear face-values are nonlinear true-values.The above equations are the general forms for the proportionality law of the sixth kind. For graphical expression, the nonlinear face-values in the equations are plotted on a log-log graph to obtain a straight line. C is an integral constant or position constant. It is a point on the straight line. The nonlinear face-values (Y – Ys)/(Yu – Y) and (X – Xs) are plotted on logarithmic scale.

By stripping off the GVP notion, Eq. (8.0c)can be written into a conventional mathematical form, Eq. (8.0d).

(8.0d)

8.0 Characteristics of the Proportional Law of the Sixth Kind

Characteristics of the proportional law of the sixth kind are:

•In a nonlinear (of Y numbers) by nonlinear (of X numbers) phenomenon of the PL6, there are a total of three asymptotes.

•Both nonlinear numbers Y and nonlinear numbers X cannot attain their asymptotes.

•The nonlinear by nonlinear phenomenon can be expressed with a combination of three types of graphs: A Cartesian graph, a semi-log and a log-log graph.

8.1Dose Response Relationship - Contractile Effects of ACh in Isolated Guinea Pig Ileum

Dose response curves commonly generated in pharmacology, physiology, and many biological systems can be described and analyzed by the PL6 equations. In this section we will use the dose response data of contractile effects of acetylcholine (ACh) in isolated guinea pig ileum by Cheng for analysis [8.1].

Table 8.1A gives the data on contractile effect of acetylcholine (ACh) in isolated guinea pig ileum [8.1]. X is expressed as the molar concentration and the contractile effect Y is expressed as % of maximum response to the same agonist. y is the increment between Y for corresponding increase in the X.

Table 8.1A Contractile Effect of Acetylcholine (ACh) in Isolated Guinea Pig

Figure 8.1a is a linear by linear plot of the acetylcholine dose response curve, this is a primary graph; it is also a leading graph. In the graph, we see the Y is approaching an upper asymptote; however, the lower asymptote is not obvious to us. When plotting the same data in a linear (of Y) by log (of X) scale, it showsas a sigmoid curve indicating the existence of an upper asymptote Yu and a lower asymptote Ys, as shown in Fig. 8.1b. Figure 8.1b_1 is a log (of Y) by log (of X) plot of the same response data. Figures 8.1b and 8.1b_1 are two of the reference graphs without much usefulness in analysis.

Fig. 8.1c is a plot of y versus X in linear by linear scale; while Fig. 8.1c_1 is a plot of y versus X in linear by log scale. These are the primitive graphs.The curves in the above figures indicate that the Y and X numbers are nonlinear numbers. A nonlinear phenomenon needs to account for their asymptotes to obtain a proportionality relationship between the two numbers and describe them according tothe law of nature. Let us use Eq. (8.0c) of the PL6 phenomena to describe the above experimental data.

First, let us use an exploded graph of Fig. 8.1a to examine the nonlinear face-values in Eq. (8.0b), as shown in Fig. 8.1a_1 where the maximum of X scale is 1.0E-6. When referring to the upper asymptote Yu, the nonlinear change of the upper vertical solid arrow is negatively proportional to the nonlinear change of horizontal dashed arrow, the higher the vertical arrow the smaller the horizontal arrow, or vice visa.

Meanwhile, when referring to the lower asymptote Ys, the nonlinear change of lower vertical solid arrow is directly proportional to the nonlinear change of horizontal dashed arrow, the higher the vertical arrow the larger the horizontal arrow, or vice visa. Since both the vertical arrows need to change with the horizontal arrow simultaneously, we can use an additive (or subtraction) form to account for both change simultaneously.

When the nonlinear change of the upper vertical solid arrow is negatively proportional to the nonlinear change of horizontal dashed arrow, its equation is d(q(Yu – Y)) = -Kd(q(X – Xs)). When the nonlinear change of lower vertical solid arrow is directly proportional to the nonlinear change of horizontal dashed arrow, its equation is d(q(Y – Ys)) = Kd(q(X – Xs)). For simultaneous change of two vertical arrows with the horizontal arrow, the equation is a combination of the above two sub equations, i.e., d(q(Y – Ys) –q(Yu – Y)) = Kd(q(X – Xs), or d(q((Y – Ys)/(Yu – Y))) = Kd(q(X – Xs)). This is equation Eq. (8.0b).

Table 8.1B lists the data calculation and the equation parameters from final analysis. We will illustrate how to obtain optimal upper and lower asymptotes as well asthe unique proportionality equation later in this section.

Table 8.1B Data calculation and equation parameters

Column D of Table 8.1B is the calculation of face-value (Y – Ys)/(Yu – Y), using the parameters Yu = 102 and Ys = -0.4. The plot of nonlinear face-value (Y – Ys)/(Yu – Y) vs. nonlinear face-value (X – Xs) in a linear by linear scale is given in Fig. 8.1d. This is a pre-proportionality graph. By converting both axes into nonlinear logarithmic scale and inserting the trendline equation and coefficient of determination, we obtain Fig. 8.1e. This is the proportionality plot.

In the following, let us use the data of ACh to demonstrate how to locate the upper asymptote. Let us start with Table 8.1Cwhere the data on the contractile effect of acetylcholine is given. When plotting Column A (X) versus Column B (Y)in a linear by linear scale, we obtain a curve as shown previously in Fig. 8.1a.

Next, we need to assign a cell each for Yu and Ys to calculate (Y – Ys)/(Yu – Y) in Column C. Let us assign Cell E1 for Yu and Cell E2 for Ys. By examining Y data in Column B, we notice that the largest number is 99.83; we need to assume a value slightly larger than this value, say 99.9, and input this number into Cell E1; let us assume Ys = 0 for the time being and input 0 into Cell E2. Then calculate the value of (Y – Ys)/ (Yu – Y) in Column C.

Formula for Cell C2 is shown in formula bar as “= (B2 - $E$2)/($E$1 – B2)”. Next, we need to copy Cell C2 to Cell C3 through Cell C10 to complete Column C.

Next, by plotting Column C vs. Column A in a linear by linear scale, we obtain Fig. 8.1d_1; the next step is converting both axes into nonlinear logarithmic scale. This is done by right clicking the mouse on vertical axis, followed by selecting “Format Axis”, and selecting logarithmic scale. By repeating the same procedure for the horizontal axis, we obtain Fig. 8.1.e_1; notice that when converting X axis from a linear to nonlinear logarithmic scale, we have a new situation with a nonlinear lower asymptote Xs = 0. The next step is to generate trendline equation and R2.

Table 8.1C Searching for upper asymptote Yu

In the next step, we click on data series followed by right click and select “add trendline”. Then in the Trendline Option manual, select regression type “Power”, and select “display equation on chart”, and also select “display R-squared value on chart”, then click O.K., we obtain Fig. 8.1e_2, where the data line curved upward because the assumed Yu value is too small. We record the R2 = 0.9347 for Yu = 99.9. By changing Yu value in Cell E2 from 99.9 to 100, the R2 changes to 0.9600 (not shown as graph). By changing Yu value in Cell E2 from 100 to 101, the R2 changes to 0.9957, as shown in Fig. 8.1e_3. By changing Yu value in Cell E2 to 106, the R2 changes to 0.9899, as shown in Fig. 8.1e_4, where the data line curved downward because the assumed Yu value is too large.

We can systematically increase the Yu value in Cell E2 and collect all Yu for the corresponding R2, as shown in the right lower corner of Table 8.1D.This is followed by plotting collected Yu vs. R2, as shown in Fig. 8.1f, where an arrow is pointing at the optimal Yu. From this plot we identify the unique asymptote from the maximum of R2 (R2 = 0.9981)as Yu = 102.

We notice from Table 8.1D that when the Yu value is less than unique Yu (e.g., Yu = 101) the R2 is smaller than the maximum (e.g., 0.9957) and the data line curved upward, as shown in Fig. 8.1e_2. When the Yu value is larger than unique Yu (e.g., Yu = 106), the R2 is also smaller the maximum (e.g., 0.9899) and the data line curved downward, as shown in Fig. 8.1e_4. At Yu = 102, the R2 is at its maximum, as shown in Fig. 8.1e, and Table 8.1D.

Once the optimal upper asymptote Yu has been resolved, we can fine tune the lower asymptote Ys. To fine tune Ys, let us start with Table 8.1E, where the Yu is fixed at Yu = 102 and Ys will vary.

Table 8.1D

Table 8.1E Fine tuning the Ys

We start with inputting Cell E3 with value -0.9, and calculate Column C. The formula for Cell C2 is “=(B2-$E$3)/($E$2-B2)”; then copy Cell C2 to Cell C3 through Cell C10. By plotting Column A vs. Column C on a linear by linear scale followed by converting the scale into log-log scale,followed by adding trend line equation and R2, we obtain Fig. 8.1g.

The R2 for Yu = 102 and Ys = -0.9 is 0.9984. We can systematically increase the Ys value and obtain the corresponding R2 values as shown in lower right side of Table 8.1E. By plotting collected Ys vs. R2 values, we obtain Fig. 8.1f_1 where the maximum value of 0.9988 occurred when Ys is -0.4. The proportionality plot with Yu = 102 and Ys = -0.4 is shown in Fig. 8.1e.

Table 8.1B-1 lists theoretical calculation of Y in column E, along with all the parameters including Yu, Ys, C, and K for the equation. Formula for calculating theoretical Y is given in formula bar. Calculated theoretical Y from this equation is plotted as Open Square in Fig. 8.1b.

Table 8.1B-1 All data calculation and equation parameters

8.2Comments on Historical Misunderstandings (100 years mystery solved) – Hill’s Equation

In the literature of pharmacology, physiology, and biological systems, it has been shown that in a Hill’s plot, the line drown according to the Hill’s equation does not fit the points exactly except in the middle of the range [8.2, 8.3]. The fact is that all the biomedical phenomena are nonlinear phenomena, and that all nonlinear numbers need to be measured relative to the asymptotes; however, the Hill’s equation doesn’t address the asymptote, hence its equation does not fit the data points.

In the experimental world,the realistic nonlinear asymptotes, not the numbers 0, 1, or 100, dictate the behavior of the curves according to the physical law. In data analysis, we need to obey the law of nature to obtain meaningful analysis rather thandoing analysis based on data fitting. Any analysis without the input of the law of nature has no physical meaning.

Hill’s equation can be viewed as a special case of PL6 equation shown in Eq. (8.0b) and Eq. (8.0c). When replacing 1 or 100 for Yu, setting Ys = 0 and Xs = 0, Eq. (8.0c) yields Hill's equation in conventional mathematical form, Eq. (8.2a).

(8.2a)

When applying the full equation, Eq. (8.0c), without assuming Ys = 0 and Yu = 100, but resort to resolving for Yu and Ys, we obtain a realistic proportionality graph, as shown in Fig. 8.1e, where the trend line exhibiting a good fit with R2 = 0.9988.

Hill's equation is a special case of Eq. (8.0c) when Ys = 0 and Yu = 1 or 100 [8.1, 8.2]. In reality, experimental data tend to deviate from the upper asymptote Yu and lower asymptote Ys; therefore, the data in Hill's plot tend to deviate from the straight line. By assuming Ys = 0 and Yu = 100 in Eq. (8.0c), we obtain the second form of Eq. (8.2a).

The proportionality plot of ACh data according to this equation is shown in Fig. 8.2a, where the coefficient of determination deteriorates from R2 = 0.9988 (in Fig. 8.1e) to R2 = 0.96 and the data points are not on a straight line.

One another way of writing the Hill’s equation is as Eq. (A)below [8.2]. The Hill’s plot from the literature is shown in the accompanied graph.

Hill’s equation: (A)

Hill’s plot: The line is drawn according to the Hill equation (Eq. A), and does not fit the points exactly except in the middle of the range. It is often difficult to make measurements outside the range -1 to +1 on the ordinate, which corresponds to a range 0.09 – 0.91 in the value of Y [8.2].

It has been stated: “The above equation (A) cannot account for the curve and from the time of Hill (1910) onwards, much effort has been devoted to the search for a plausible physical model that can” [8.2].

From the figures in Section 8.1 and 8.2, we can easily identify that the basic problem associated with Hill’s plot lies in the fact that the traditional equation doesn’t account for the nonlinearity of the data and doesn’t account for nonlinear asymptotes.

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Note I: Equations for Y and ED50 can also be written as conventional math forms Eq. (8.2b) and Eq. (8.2c).

(8.2b)

(8.2c)

Note II: The ratio of (Y – Ys)/(Yu – Y) can also be described physically as the fraction affected over the fraction unaffected, as shown in Fig. 8.2b. The physical interpretation of the figure is that the nonlinear change of (Y – Ys) is proportional to the nonlinear change of X, and at the same time, the nonlinear change of (Yu – Y) is proportional to the nonlinear change of X.

References

8.1Hsien C. Cheng and Ralph W. Lai, "Use of Proportionality Equations for Analyses of Dose-Response Curves," Pharmacological Research, Vol. 47, 2003, pp. 163 -173.

8.2Athel Cornish-Bowden, Fundamentals of Enzyme Kinetics, Butterworths & Co. Ltd, London, 1979, pp.149-153

8.3Lubert Stryer, Biochemistry, W. H. Freeman and Company, New York, NY, 1988, pp.151-157.