# can the parameters of this nonlinear equation be reliably estimated using a limited set of experimental data?

A chemical system at equilibrium is described by this equation: $$D_{tot} = D + D \cdot \frac {P \cdot X}{D+K_P} + D \cdot \frac {E}{D+K_E}$$ The only parameters that the experimenter can control (by setting up the experiment with different concentrations of certain substances) are $$X$$ and $$D_{tot}$$.
$$X$$ is a real number in $$(0,1]$$ and $$D_{tot}$$ is a positive real number.
Usually, $$X$$ is varied, whereas $$D_{tot}$$ is kept constant, but it's possible to vary $$D_{tot}$$ if needed. $$P, K_P, E, K_E$$ are positive real constants whose values are not known.

The main goal is estimating $$P$$ and $$K_P$$.

The experimenter can't measure $$D$$ directly. However, by appropriate sampling of the system, a quantity $$f_u$$ can be measured, for which it is known that:

$$\frac 1 {f_u} = {1+\frac {P \cdot X}{D+K_P}}$$ The 'usual' approach is to assume (quite arbitrarily) that $$E \approx 0$$, which also implies $$D = D_{tot} \cdot f_u$$, thus:

$$D_{tot} \approx D_{tot} \cdot f_u + D_{tot} \cdot f_u \cdot \frac {P \cdot X}{D_{tot} \cdot f_u+K_P}$$

(...)

$$\frac X {1 - f_u} \approx \frac {D_{tot}} P + \frac 1 {f_u} \cdot \frac {K_P}{P}$$

Auxiliary variables are defined:

$$y = \frac X {1 - f_u}$$ $$x = \frac 1 {f_u}$$

thus:

$$y \approx \frac {D_{tot}} P + x \cdot \frac {K_P}{P}$$

By linear regression, using data where $$X$$ was varied and the corresponding values of $$f_u$$ were measured, the slope and intercept of this equation are found, and knowing the experimental value of $$D_{tot}$$ in theory one can obtain the desired parameter estimates.

Example (in R):

    HSA_data <- data.frame(X=c(1, 0.4, 0.1, 0.01), fu=c(0.003,
0.011, 0.028, 0.224))
HSA_data["x"] <- with(HSA_data, 1/fu)
HSA_data["y"] <- with(HSA_data, X/(1-fu))
HSA_lm <- lm(y~x, HSA_data)
summary(HSA_lm)

Call:
lm(formula = y ~ x, data = HSA_data)

Residuals:
1        2        3        4
-0.02211  0.09838 -0.03948 -0.03679

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.036432   0.054364   0.670    0.572
x           0.002966   0.000313   9.476    0.011 *
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.08087 on 2 degrees of freedom
Multiple R-squared:  0.9782,    Adjusted R-squared:  0.9673
F-statistic:  89.8 on 1 and 2 DF,  p-value: 0.01095


My first question is: do you think that this approach is valid, given that 'linearizing' nonlinear equations has long been criticized?

[BTW, while the slope is generally OK, the intercept is often negative, which is nonsensical given the theory from which the equation is derived. That for me points to a fundamental problem with this approach.]

The second, even more important question is: given that the assumption $$E \approx 0$$ is really arbitrary and often wrong, do you think the data in the above example could be somehow used with the original equation, i.e. without making this assumption?
I can see that there would be 4 parameters to estimate with only 4 data points.
Would you suggest collecting more data? Manipulating the equation in some clever way, considering that $$f_u$$ is independent from $$E, K_E$$? Any other suggestions?