Gut microbiota ecology: Biodiversity estimated from hybrid neutral-niche model increases with health status and aging

Volkov treats this model in the framework of the Chemical Master Equation and proves that under these assumptions the probability distribution of the RSA is expected to be the zero-truncated Negative Binomial PRSA n ð Þ ¼ Nobs 1 ð1 b=dÞS=b ð1 b=dÞS=b GðS=bÞ ðb=dÞn n! G n þ S=b ð Þ ð8Þ where Nobs refers to the total number of observed species and Γ is the gamma function. Biodi- versity can be estimated using the Hubbell biodiversity index [30, 31], defined as y ¼ Nobs ½ð1 b=dÞ S=b 1� � GðS=bÞ ð9Þ We modelled the empirical RSA derived from 16S rRNA data considering three possible scenarios. First, we tested pure neutrality by fitting the data with the 1NB model (Eq (8)). Then, we relaxed the hypothesis of species equivalence considering a hybrid niche-neutral model (2NB model) that assumes the existence of two non-interacting neutral niches (the evo- lutionary dynamics of each niche is neutral). Finally, we further relaxed the neutral hypothesis contemplating a hybrid niche-neutral model with three niches (3NB model). The mathematical aspects of the 2NB model are detailed in Bazzani et al. [38] and show that the stationary RSA distribution is a mixture of two zero-truncated Negative Binomials PRSA n ð Þ ¼ a � y1 ðb1=d1Þn n! G n þ S1=b1 ð Þ þ 1 a ð Þ � y2 ðb2=d2Þn n! G n þ S2=b2 ð Þ ð10Þ where α is the mixture coefficient, bi, di and Si are the birth, death, and influx rates of the i-th niche (i = 1,2), and θi is the biodiversity number relative to niche i, and is equivalent to the one in Eq (9).

Analogously, we assume the stationary RSA distribution for the 3NB model to be a mixture of three zero-truncated Negative Binomials (Eq (11)) to which 3 biodiversity numbers are associated: θ1, θ2 and θ3. PRSA n ð Þ ¼ a � y1 ðb1=d1Þn n! G n þ S1=b1 ð Þ þ b � y2 ðb2=d2Þn n! G n þ S2=b2 ð Þ þ 1 a b ð Þ � y3 ðb3=d3Þn n! G n þ S3=b3 ð Þ ð11Þ Model fitting and model selection Empirical RSA distributions were computed by counting the number of OTUs with a certain number of individuals. The 1NB, 2NB and 3NB models were fitted to the data using a custom implementation of the Approximate Bayesian Computation (ABC) rejection algorithm. Assuming a degree of similarity across samples, the goal was to implement a hierarchical model to estimate the parameters of the model, allowing the estimate from the general popula- tion to inform about the plausible values of each individual sample.To facilitate the algorithm convergence, we implemented this method with a two-step approach. In the first step, we assigned uninformative prior distributions to the model parameters and we fitted the model for each sample using the ABC algorithm detailed below. Then, for each data set, we con- structed the posterior distributions of the model parameters pooling the accepted parameters of all samples belonging to that data set and fitting them with either a Beta distribution (mix- ture coefficients) or a Gamma distribution (all the other parameters). Notice that, in general, PLOS ONE Gut microbiota biodiversity and healthy aging

PLOS ONE | https://doi.org/10.1371/journal.pone.0237207 October 30, 2020 15 / 23