The Michealis Menten Hyperbola of xlab = Substrate Concentration and ylab = Velocity usually looks like a rising hyperbola to a maximum. The parameters which you are measuring look like an exponential decay as the substrate concentration increases. I'm not sure the Michealis Menten Equation works so well here. Also you shouldn't be using the lines function. It doesn't give you a curve. You should be using the curve function.
x <- c(0, 2.5, 5.0, 10.0, 25.0)
y <- c(4.91, 1.32, 1.18, 1.12, 1.09)
mm <- data.frame(x, y)
I think you should make the dataframe before you call it into a function.
model.mm <- nls(y ~ Vmax*x / (Km + x), data = mm, start = list(Km = max(mm$y)/2, Vmax = max(mm$y)))
plot(y ~ x, type = "p", lwd = 2, xlab = "Lopinavir Concentrations (uM)", ylab = "Efflux Ratio", pch = 16, main = "Lopinavir Transport in MDCK-MDR1 Cells")
summary(model.mm)
Formula: y ~ Vmax * x/(Km + x)
Parameters:
Estimate Std. Error t value Pr(>|t|)
Km -0.4772 6.6246 -0.072 0.947
Vmax 1.0678 2.1382 0.499 0.652
Residual standard error: 2.835 on 3 degrees of freedom
Number of iterations to convergence: 4
Achieved convergence tolerance: 6.634e-06
Now I think the curve function is pretty self explanatory:
?curve
curve(x * 1.0678 / (x + -0.4772), col = "red", lwd = 2, add = TRUE)
fx <- function(x){x * 1.0678 / (x + -0.4772)}
range(x)
1 0 25
We can integrate this Michealis-Menten FUnction and calculate area under curve:
require(pracma)
integrate(fx, lower = 0, upper = 25)
Error in integrate(fx, lower = 0, upper = 25) :
the integral is probably divergent
This divergence is because you plot looks a lot like y = (1 / x) which is divergent.
If you push x a little bit away from zero, where y tends to infinity we can get a finite answer.
integrate(fx, lower = 0.5, upper = 25)
29.71809 with absolute error < 0.00069
but this integral is questionable for divergence reasons.
You can estimate the area under your scatterplot points using the trapezoidal approximation for integration:
trapz(mm$x, mm$y)
1 33.2375
Note: I tried fitting exponential functions to your plot but that doesn't work. The curve drops too fast missing most of the points.
I think I figured out what's wrong with your function.
Type: y = (x * v) / (x + K)
into https://www.desmos.com/calculator
and see what happens when you make K negative and K - positive, and when you make both K and v negative, etc.
