Lets say I measure kinetics of an enzyme under three toxic chemical exposure conditions. After fitting the enzyme velocity data using Michaelis Menten model, I am able to get the kinetic parameters (Km and Vmax) under each exposure conditions. Lets assume the sample size for each substrate concentration was 6 (n = 6). In order to determine if the Vmax and Km statistically differ from each other under the three different toxic chemical exposure conditions, is it acceptable to use n=6, and mean and standard error of vmax and km to apply one way ANOVA?
I know ANOVA will be the method of choice. Also I can calculate the Standard error easily. My major confusion is the sample size. Since, Vmax and Km values are not experimentally determined and are simply estimated by fitting a statistical model. Hence, I am not sure, if it is valid to use simply the sample size of 6 and proceed just like a normal statistical procedure. The problem is, how do we test the normality assumption if we go with this approach (because of the lack of individual Vmax and Km values)?
Thanks
Hi, Dear Kamran Shekh
As far as I know and understood about statistical analysis, your research sample size too small, may invalid to proceed just like a normal statistical procedure, if possible make your sample size bigger,
Regards,
Hi, thats just a hypothetical situation. I am just trying to understand the underlying principle.
I'm not a statistician, but here is my opinion. Each measurement of Km and Vmax comes from fitting a set of rate versus substrate concentration measurements using nonlinear least squares and the Michaelis-Menten equation. (Do not use linear regression and a linearization of the M-M equation.) This analysis provides an estimate of the parameter and a standard error for the estimate. I believe it is possible to obtain the standard error because it is assumed that the replicate rate measurements at each substrate concentration are normally distributed (which is not true for the linearization, hence the reason it should not be used). Therefore, multiple measurements of Km and Vmax made this way should also be normally distributed, allowing you to use normal statistics for your analysis.
Thanks Adam. Exactly, thats what I was looking for. Thanks for detailed explanation. As you mentioned, i did not use lineweaver-burk normalization because it disrupts normality assumption. Instead I fitted the data to michaelis menten kinetics using DRC package in R, which directly gives vmax and km with standard error. Thanks again for explaining.
Hi, I guess sample of size 6 is too small. There exists non-parametric alternative of ANOVA which does not require the assumption of normality of the data.
You may possibly try that.
Hope this helps.
Non Parametric alternative of ANOVA one way classification is Kruskal wallis ANOVA...
Hello Kamran,
The best way could be to do a common fit of all results with a covariate being the condition on Km and Vmax, and test if the fit with the covariate is significantly better than the fit without them (using nested models theory for instance). This avoid assumptions about estimation of the parameters distribution, also avoid the apparently low n=6 which corresponds in fact to a lot of data since for each curve you have several points. This method « only » assume that errors are normal, independant and have the same magnitude for all conditions (at least for the simplest model). The exact model to fit will depend on the way you do your replicate, and of course it would be better to have the same concentrations (x values) for each conditions.
The second option is to fit to each individual situation its own Km and Vmax value, to have for each condition 6 Km and Vmax values, then use usual statistical methods to compare these samples of Km and Vmax values. Compared to the previous method, it assumes additionally that the uncertainty of the estimation of (Km, Vmax) for a single curve is much lower than the between-curve variability, or at least not bigger, which may not be true especially if the model fits badly or if you have not enough data points for a single curve.
The third option is the one proposed by Adam Shapiro, using a fit for a condition on the average of the 6 curves and using the fit uncertainties on Km/Vmax as estimation of the residual error. It adds assumptions about the distribution of the parameters (which may well not be Gaussian because of the non-linearity of the model), and you'll probably have to do computations "by hand".
A mix between the second and third option is of course possible, to include (Km, Vmax) uncertainties, with the same limitations/assumptions.
Thanks Emmanuel for a very detailed explanation of the possible approaches.
Thanks
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