# skbio.math.diversity.alpha.michaelis_menten_fit¶

skbio.math.diversity.alpha.michaelis_menten_fit(counts, num_repeats=1, params_guess=None)[source]

Calculate Michaelis-Menten fit to rarefaction curve of observed OTUs.

The Michaelis-Menten equation is defined as

$S=\frac{nS_{max}}{n+B}$

where $$n$$ is the number of individuals and $$S$$ is the number of OTUs. This function estimates the $$S_{max}$$ parameter.

The fit is made to datapoints for $$n=1,2,...,N$$, where $$N$$ is the total number of individuals (sum of abundances for all OTUs). $$S$$ is the number of OTUs represented in a random sample of $$n$$ individuals.

Parameters: counts : 1-D array_like, int Vector of counts. num_repeats : int, optional The number of times to perform rarefaction (subsampling without replacement) at each value of $$n$$. params_guess : tuple, optional Initial guess of $$S_{max}$$ and $$B$$. If None, default guess for $$S_{max}$$ is $$S$$ (as $$S_{max}$$ should be >= $$S$$) and default guess for $$B$$ is round(N / 2). S_max : double Estimate of the $$S_{max}$$ parameter in the Michaelis-Menten equation.

Notes

There is some controversy about how to do the fitting. The ML model given in [R57] is based on the assumption that error is roughly proportional to magnitude of observation, reasonable for enzyme kinetics but not reasonable for rarefaction data. Here we just do a nonlinear curve fit for the parameters using least-squares.

References

 [R57] (1, 2) Raaijmakers, J. G. W. 1987 Statistical analysis of the Michaelis-Menten equation. Biometrics 43, 793-803.