Dynamic linear models guide design and analysis of microbiota studies within artificial human guts

Weight, habitual fibre intake, and microbiome composition predict tolerance to fructan supplementation

Fructans are commonly used as dietary fibre supplements for their ability to promote the growth of beneficial gut microbes. However, fructan consumption has been associated with various dosage-dependent side effects. We characterised side effects in an exploratory analysis of a randomised trial in healthy adults (n = 40) who consumed 18 g/day inulin or placebo. We found that individuals weighing more or habitually consuming higher fibre exhibited the best tolerance. Furthermore, we identified associations between gut microbiome composition and host tolerance. Specifically, higher levels of Christensenellaceae R-7 group were associated with gastrointestinal discomfort, and a machine-learning-based approach successfully predicted high levels of flatulence, with [Ruminococcus] torques group and (Oscillospiraceae) UCG-002 sp. identified as key predictive taxa. These data reveal trends that can help guide personalised recommendations for initial inulin dosage. Our results support prior ecological findings indicating that fibre supplementation has the greatest impact on individuals whose baseline fibre intake is lowest.

miRglmm: a generalized linear mixed model of isomiR-level counts improves estimation of miRNA-level differential expression and uncovers variable differential expression between isomiRs

MicroRNA-seq data is produced by aligning small RNA sequencing reads of different miRNA transcript isoforms, called isomiRs, to known microRNAs. Aggregation to microRNA-level counts discards information and violates core assumptions of differential expression (DE) methods developed for mRNA-seq data. We establish miRglmm, a DE method for microRNA-seq data, that uses a generalized linear mixed model of isomiR-level counts, facilitating detection of miRNA with differential expression or differential isomiR usage. We demonstrate that miRglmm outperforms current DE methods in estimating DE for miRNA, whether or not there is significant isomiR variability, and simultaneously provides estimates of isomiR-level DE.

Impact of cryoprotective agents on human gut microbes and in vitro stabilized artificial gut microbiota communities

The availability of microbial biobanks for the storage of individual gut microbiota members or their derived and artificially assembled consortia has become fundamental for in vitro investigation of the molecular mechanisms behind microbe-microbe and/or microbe-host interactions. However, to preserve bacterial viability, adequate storage and processing technologies are required. In this study, the effects on cell viability of seven different combinations of cryoprotective agents were evaluated by flow cytometry for 53 bacterial species representing key members of the human gut microbiota after one and 3 months of cryopreservation at -80°C. The obtained results highlighted that no universal cryoprotectant was identified capable of guaranteeing effective recovery of intact cells after cryopreservation for all tested bacteria. However, the presence of inulin or skimmed milk provided high levels of viability protection during cryoexposure. These results were further corroborated by cryopreserving 10 artificial gut microbiota produced through in vitro continuous fermentation system technology. Indeed, in this case, the inclusion of inulin or skimmed milk resulted in a high recovery of viable cells, while also allowing consistent and reliable preservation of the artificial gut microbiota biodiversity. Overall, these results suggest that, although the efficacy of various cryoprotective agents is species-specific, some cryoprotectants based on glycerol and the addition of inulin or skimmed milk are preferable to retain viability and biodiversity for both single bacterial species and artificial gut microbiota.

Differential transcript usage analysis incorporating quantification uncertainty via compositional measurement error regression modeling

Differential transcript usage (DTU) occurs when the relative expression of multiple transcripts arising from the same gene changes between different conditions. Existing approaches to detect DTU often rely on computational procedures that can have speed and scalability issues as the number of samples increases. Here we propose a new method, CompDTU, that uses compositional regression to model the relative abundance proportions of each transcript that are of interest in DTU analyses. This procedure leverages fast matrix-based computations that make it ideally suited for DTU analysis with larger sample sizes. This method also allows for the testing of and adjustment for multiple categorical or continuous covariates. Additionally, many existing approaches for DTU ignore quantification uncertainty in the expression estimates for each transcript in RNA-seq data. We extend our CompDTU method to incorporate quantification uncertainty leveraging common output from RNA-seq expression quantification tool in a novel method CompDTUme. Through several power analyses, we show that CompDTU has excellent sensitivity and reduces false positive results relative to existing methods. Additionally, CompDTUme results in further improvements in performance over CompDTU with sufficient sample size for genes with high levels of quantification uncertainty, while also maintaining favorable speed and scalability. We motivate our methods using data from the Cancer Genome Atlas Breast Invasive Carcinoma data set, specifically using RNA-seq data from primary tumors for 740 patients with breast cancer. We show greatly reduced computation time from our new methods as well as the ability to detect several novel genes with significant DTU across different breast cancer subtypes.

Modeling Microbial Community Networks: Methods and Tools for Studying Microbial Interactions

Microbial interactions function as a fundamental unit in complex ecosystems. By characterizing the type of interaction (positive, negative, neutral) occurring in these dynamic systems, one can begin to unravel the role played by the microbial species. Towards this, various methods have been developed to decipher the function of the microbial communities. The current review focuses on the various qualitative and quantitative methods that currently exist to study microbial interactions. Qualitative methods such as co-culturing experiments are visualized using microscopy-based techniques and are combined with data obtained from multi-omics technologies (metagenomics, metabolomics, metatranscriptomics). Quantitative methods include the construction of networks and network inference, computational models, and development of synthetic microbial consortia. These methods provide a valuable clue on various roles played by interacting partners, as well as possible solutions to overcome pathogenic microbes that can cause life-threatening infections in susceptible hosts. Studying the microbial interactions will further our understanding of complex less-studied ecosystems and enable design of effective frameworks for treatment of infectious diseases.

Interplay between particle size and microbial ecology in the gut microbiome

Physical particles can serve as critical abiotic factors that structure the ecology of microbial communities. For non-human vertebrate gut microbiomes, fecal particle size (FPS) has been known to be shaped by chewing efficiency and diet. However, little is known about what drives FPS in the human gut. Here, we analyzed FPS by laser diffraction across a total of 76 individuals and found FPS to be strongly individualized. Contrary to our initial hypothesis, a behavioral intervention with 41 volunteers designed to increase chewing efficiency did not impact FPS. Dietary patterns could also not be associated with FPS. Instead, we found evidence that human and mouse gut microbiomes shaped FPS. Fecal samples from germ-free and antibiotic-treated mice exhibited increased FPS relative to colonized mice. In humans, markers of longer transit time were correlated with smaller FPS. Gut microbiota diversity and composition were also associated with FPS. Finally, ex vivo culture experiments using human fecal microbiota from distinct donors showed that differences in microbiota community composition can drive variation in particle size. Together, our results support an ecological model in which the human gut microbiome plays a key role in reducing the size of food particles during digestion. This finding has important implications for our understanding of energy extraction and subsequent uptake in gastrointestinal tract. FPS may therefore be viewed as an informative functional readout, providing new insights into the metabolic state of the gut microbiome.

Methodological Considerations in Longitudinal Analyses of Microbiome Data: A Comprehensive Review

Biological processes underlying health and disease are inherently dynamic and are best understood when characterized in a time-informed manner. In this comprehensive review, we discuss challenges inherent in time-series microbiome data analyses and compare available approaches and methods to overcome them. Appropriate handling of longitudinal microbiome data can shed light on important roles, functions, patterns, and potential interactions between large numbers of microbial taxa or genes in the context of health, disease, or interventions. We present a comprehensive review and comparison of existing microbiome time-series analysis methods, for both preprocessing and downstream analyses, including differential analysis, clustering, network inference, and trait classification. We posit that the careful selection and appropriate utilization of computational tools for longitudinal microbiome analyses can help advance our understanding of the dynamic host-microbiome relationships that underlie health-maintaining homeostases, progressions to disease-promoting dysbioses, as well as phases of physiologic development like those encountered in childhood.

Dietary Zn deficiency, the current situation and potential solutions

Zinc (Zn) deficiency is a worldwide problem, and this review presents an overview of the magnitude of Zn deficiency with a particular emphasis on present global challenges, current recommendations for Zn intake, and factors that affect dietary requirements. The challenges of monitoring Zn status are clarified together with the discussion of relevant Zn bioaccessibility and bioavailability issues. Modern lifestyle factors that may exacerbate Zn deficiency and new strategies of reducing its effects are presented. Biofortification, as a potentially useful strategy for improving Zn status in sensitive populations, is discussed. The review proposes potential actions that could deliver promising results both in terms of monitoring dietary and physiological Zn status as well as in alleviating dietary Zn deficiency in affected populations.

Current Viewpoint on Female Urogenital Microbiome-The Cause or the Consequence?

An increasing amount of evidence implies that native microbiota is a constituent part of a healthy urinary tract (UT), making it an ecosystem on its own. What is still not clear is whether the origin of the urinary microbial community is the indirect consequence of the more abundant gut microbiota or a more distinct separation exists between these two systems. Another area of uncertainty is the existence of a link between the shifts in UT microbial composition and both the onset and persistence of cystitis symptoms. Cystitis is one of the most common reasons for antimicrobial drugs prescriptions in primary and secondary care and an important contributor to the problem of antimicrobial resistance. Despite this fact, we still have trouble distinguishing whether the primary cause of the majority of cystitis cases is a single pathogen overgrowth or a systemic disorder affecting the entire urinary microbiota. There is an increasing trend in studies monitoring changes and dynamics of UT microbiota, but this field of research is still in its infancy. Using NGS and bioinformatics, it is possible to obtain microbiota taxonomic profiles directly from urine samples, which can provide a window into microbial diversity (or the lack of) underlying each patient's cystitis symptoms. However, while microbiota refers to the living collection of microorganisms, an interchangeably used term microbiome referring to the genetic material of the microbiota is more often used in conjunction with sequencing data. It is this vast amount of sequences, which are truly "Big Data", that allow us to create models that describe interactions between different species contributing to an UT ecosystem, when coupled with machine-learning techniques. Although in a simplified predator-prey form these multi-species interaction models have the potential to further validate or disprove current beliefs; whether it is the presence or the absence of particular key players in a UT microbial ecosystem, the exact cause or consequence of the otherwise unknown etiology in the majority of cystitis cases. These insights might prove to be vital in our ongoing struggle against pathogen resistance and offer us new and promising clinical markers.

Effect of Operational Parameters on the Cultivation of the Gut Microbiome in Continuous Bioreactors Inoculated with Feces: A Systematic Review

Since the early 1980s, multiple researchers have contributed to the development of in vitro models of the human gastrointestinal system for the mechanistic interrogation of the gut microbiome ecology. Using a bioreactor for simulating all the features and conditions of the gastrointestinal system is a massive challenge. Some conditions, such as temperature and pH, are readily controlled, but a more challenging feature to simulate is that both may vary in different regions of the gastrointestinal tract. Promising solutions have been developed for simulating other functionalities, such as dialysis capabilities, peristaltic movements, and biofilm growth. This research field is under constant development, and further efforts are needed to drive these models closer to in vivo conditions, thereby increasing their usefulness for studying the gut microbiome impact on human health. Therefore, understanding the influence of key operational parameters is fundamental for the refinement of the current bioreactors and for guiding the development of more complex models. In this review, we performed a systematic search for operational parameters in 229 papers that used continuous bioreactors seeded with human feces. Despite the reporting of operational parameters for the various bioreactor models being variable, as a result of a lack of standardization, the impact of specific operational parameters on gut microbial ecology is discussed, highlighting the advantages and limitations of the current bioreactor systems.

coda4microbiome: compositional data analysis for microbiome cross-sectional and longitudinal studies

BACKGROUND

One of the main challenges of microbiome analysis is its compositional nature that if ignored can lead to spurious results. Addressing the compositional structure of microbiome data is particularly critical in longitudinal studies where abundances measured at different times can correspond to different sub-compositions.

RESULTS

We developed coda4microbiome, a new R package for analyzing microbiome data within the Compositional Data Analysis (CoDA) framework in both, cross-sectional and longitudinal studies. The aim of coda4microbiome is prediction, more specifically, the method is designed to identify a model (microbial signature) containing the minimum number of features with the maximum predictive power. The algorithm relies on the analysis of log-ratios between pairs of components and variable selection is addressed through penalized regression on the "all-pairs log-ratio model", the model containing all possible pairwise log-ratios. For longitudinal data, the algorithm infers dynamic microbial signatures by performing penalized regression over the summary of the log-ratio trajectories (the area under these trajectories). In both, cross-sectional and longitudinal studies, the inferred microbial signature is expressed as the (weighted) balance between two groups of taxa, those that contribute positively to the microbial signature and those that contribute negatively. The package provides several graphical representations that facilitate the interpretation of the analysis and the identified microbial signatures. We illustrate the new method with data from a Crohn's disease study (cross-sectional data) and on the developing microbiome of infants (longitudinal data).

CONCLUSIONS

coda4microbiome is a new algorithm for identification of microbial signatures in both, cross-sectional and longitudinal studies. The algorithm is implemented as an R package that is available at CRAN ( https://cran.r-project.org/web/packages/coda4microbiome/ ) and is accompanied with a vignette with a detailed description of the functions. The website of the project contains several tutorials: https://malucalle.github.io/coda4microbiome/.

Revealing the importance of prenatal gut microbiome in offspring neurodevelopment in humans

BACKGROUND

It has been widely recognized that a critical time window for neurodevelopment occurs in early life and the host's gut microbiome plays an important role in neurodevelopment. Following recent demonstrations that the maternal prenatal gut microbiome influences offspring brain development in murine models, we aim to explore whether the critical time window for the association between the gut microbiome and neurodevelopment is prenatal or postnatal for human.

METHODS

Here we leverage a large-scale human study and compare the associations between the gut microbiota and metabolites from mothers during pregnancy and their children with the children's neurodevelopment. Specifically, using multinomial regression integrated in Songbird, we assessed the discriminating power of the maternal prenatal and child gut microbiome for children's neurodevelopment at early life as measured by the Ages & Stages Questionnaires (ASQ).

FINDINGS

We show that the maternal prenatal gut microbiome is more relevant than the children's gut microbiome to the children's neurodevelopment in the first year of life (maximum Q² = 0.212 and 0.096 separately using the taxa at the class level). Moreover, we found that Fusobacteriia is more associated with high fine motor skills in ASQ in the maternal prenatal gut microbiota but become more associated with low fine motor skills in the infant gut microbiota (rank = 0.084 and -0.047 separately), suggesting the roles of the same taxa with respect to neurodevelopment can be opposite at the two stages of fetal neurodevelopment.

INTERPRETATION

These findings shed light, especially in terms of timing, on potential therapeutic interventions to prevent neurodevelopmental disorders.

FUNDING

This work was supported by the National Institutes of Health (grant numbers: R01AI141529, R01HD093761, RF1AG067744, UH3OD023268, U19AI095219, U01HL089856, R01HL141826, K08HL148178, K01HL146980), and the Charles A. King Trust Postdoctoral Fellowship.

BIRDMAn: A Bayesian differential abundance framework that enables robust inference of host-microbe associations

Quantifying the differential abundance (DA) of specific taxa among experimental groups in microbiome studies is challenging due to data characteristics (e.g., compositionality, sparsity) and specific study designs (e.g., repeated measures, meta-analysis, cross-over). Here we present BIRDMAn (Bayesian Inferential Regression for Differential Microbiome Analysis), a flexible DA method that can account for microbiome data characteristics and diverse experimental designs. Simulations show that BIRDMAn models are robust to uneven sequencing depth and provide a >20-fold improvement in statistical power over existing methods. We then use BIRDMAn to identify antibiotic-mediated perturbations undetected by other DA methods due to subject-level heterogeneity. Finally, we demonstrate how BIRDMAn can construct state-of-the-art cancer-type classifiers using The Cancer Genome Atlas (TCGA) dataset, with substantial accuracy improvements over random forests and existing DA tools across multiple sequencing centers. Collectively, BIRDMAn extracts more informative biological signals while accounting for study-specific experimental conditions than existing approaches.

Impact of Data and Study Characteristics on Microbiome Volatility Estimates

The human microbiome is a dynamic community of bacteria, viruses, fungi, and other microorganisms. Both the composition of the microbiome (the microbes that are present and their relative abundances) and the temporal variability of the microbiome (the magnitude of changes in their composition across time, called volatility) has been associated with human health. However, the effect of unbalanced sampling intervals and differential read depth on the estimates of microbiome volatility has not been thoroughly assessed. Using four publicly available gut and vaginal microbiome time series, we subsampled the datasets to several sampling intervals and read depths and then compared additive, multiplicative, centered log ratio (CLR)-based, qualitative, and distance-based measures of microbiome volatility between the conditions. We find that longer sampling intervals are associated with larger quantitative measures of change (particularly for common taxa), but not with qualitative measures of change or distance-based volatility quantification. A lower sequencing read depth is associated with smaller multiplicative, CLR-based, and qualitative measures of change (particularly for less common taxa). Strategic subsampling may serve as a useful sensitivity analysis in unbalanced longitudinal studies investigating clinical associations with microbiome volatility.

Technical versus biological variability in a synthetic human gut community

Synthetic communities grown in well-controlled conditions are an important tool to decipher the mechanisms driving community dynamics. However, replicate time series of synthetic human gut communities in chemostats are rare, and it is thus still an open question to what extent stochasticity impacts gut community dynamics. Here, we address this question with a synthetic human gut bacterial community using an automated fermentation system that allows for a larger number of biological replicates. We collected six biological replicates for a community initially consisting of five common gut bacterial species that fill different metabolic niches. After an initial 12 hours in batch mode, we switched to chemostat mode and observed the community to stabilize after 2-3 days. Community profiling with 16S rRNA resulted in high variability across replicate vessels and high technical variability, while the variability across replicates was significantly lower for flow cytometric data. Both techniques agree on the decrease in the abundance of Bacteroides thetaiotaomicron, accompanied by an initial increase in Blautia hydrogenotrophica. These changes occurred together with reproducible metabolic shifts, namely a fast depletion of glucose and trehalose concentration in batch followed by a decrease in formic acid and pyruvic acid concentrations within the first 12 hours after the switch to chemostat mode. In conclusion, the observed variability in the synthetic bacterial human gut community, as assessed with 16S rRNA gene sequencing, is largely due to technical variability. The low variability seen in HPLC and flow cytometry data suggests a highly deterministic system.

Ecological memory of prior nutrient exposure in the human gut microbiome

Many ecosystems have been shown to retain a memory of past conditions, which in turn affects how they respond to future stimuli. In microbial ecosystems, community disturbance has been associated with lasting impacts on microbiome structure. However, whether microbial communities alter their response to repeated stimulus remains incompletely understood. Using the human gut microbiome as a model, we show that bacterial communities retain an "ecological memory" of past carbohydrate exposures. Memory of the prebiotic inulin was encoded within a day of supplementation among a cohort of human study participants. Using in vitro gut microbial models, we demonstrated that the strength of ecological memory scales with nutrient dose and persists for days. We found evidence that memory is seeded by transcriptional changes among primary degraders of inulin within hours of nutrient exposure, and that subsequent changes in the activity and abundance of these taxa are sufficient to enhance overall community nutrient metabolism. We also observed that ecological memory of one carbohydrate species impacts microbiome response to other carbohydrates, and that an individual's habitual exposure to dietary fiber was associated with their gut microbiome's efficiency at digesting inulin. Together, these findings suggest that the human gut microbiome's metabolic potential reflects dietary exposures over preceding days and changes within hours of exposure to a novel nutrient. The dynamics of this ecological memory also highlight the potential for intra-individual microbiome variation to affect the design and interpretation of interventions involving the gut microbiome.

Nano-Al2O3 particles affect gut microbiome and resistome in an in vitro simulator of the human colon microbial ecosystem

Nano-Al₂O₃ has been widely used in various consumer products and water treatment processes because of its unique physicochemical properties. The probability of human exposure to nano-Al₂O₃ increases significantly, of which oral ingestion is an important route. However, effects and underlying mechanisms of nano-Al₂O₃ on gut microbiota and resistome are still not well delineated. Here, we systematically investigated the effects of nano-Al₂O₃ on the human gut microbiome by an in vitro simulator of human colon microbial ecosystem. Results indicated that nano-Al₂O₃ interfered with the gut microbiota, and significantly suppressed the short-chain fatty acids metabolism, which might pose adverse effects on the host. More seriously, high level of nano-Al₂O₃ (50 mg/L) was more destructive to the gut flora, though the damage might be temporary. In addition, sub-inhibitory low-dose of nano-Al₂O₃ (0.1 mg/L) significantly enhanced the abundance of antibiotic resistance genes (ARGs) after 7-day exposure. This is attributed to that low concentration of nano-Al₂O₃ can promote horizontal transfer of ARGs by increasing cell membrane permeability and relative abundance of transposase (e.g. tnpA, IS613, and Tp614). Our findings confirmed the adverse effects of nano-Al₂O₃ on the human gut resistome and emphasized the necessity to assess potential risks of nanomaterials on the human gut health.

Dynamic metabolic interactions and trophic roles of human gut microbes identified using a minimal microbiome exhibiting ecological properties

Microbe–microbe interactions in the human gut are influenced by host-derived glycans and diet. The high complexity of the gut microbiome poses a major challenge for unraveling the metabolic interactions and trophic roles of key microbes. Synthetic minimal microbiomes provide a pragmatic approach to investigate their ecology including metabolic interactions. Here, we rationally designed a synthetic microbiome termed Mucin and Diet based Minimal Microbiome (MDb-MM) by taking into account known physiological features of 16 key bacteria. We combined 16S rRNA gene-based composition analysis, metabolite measurements and metatranscriptomics to investigate community dynamics, stability, inter-species metabolic interactions and their trophic roles. The 16 species co-existed in the in vitro gut ecosystems containing a mixture of complex substrates representing dietary fibers and mucin. The triplicate MDb-MM’s followed the Taylor’s power law and exhibited strikingly similar ecological and metabolic patterns. The MDb-MM exhibited resistance and resilience to temporal perturbations as evidenced by the abundance and metabolic end products. Microbe-specific temporal dynamics in transcriptional niche overlap and trophic interaction network explained the observed co-existence in a competitive minimal microbiome. Overall, the present study provides crucial insights into the co-existence, metabolic niches and trophic roles of key intestinal microbes in a highly dynamic and competitive in vitro ecosystem.

Microbiota responses to different prebiotics are conserved within individuals and associated with habitual fiber intake

BACKGROUND

Short-chain fatty acids (SCFAs) derived from gut bacteria are associated with protective roles in diseases ranging from obesity to colorectal cancers. Intake of microbially accessible dietary fibers (prebiotics) lead to varying effects on SCFA production in human studies, and gut microbial responses to nutritional interventions vary by individual. It is therefore possible that prebiotic therapies will require customizing to individuals.

RESULTS

Here, we explored prebiotic personalization by conducting a three-way crossover study of three prebiotic treatments in healthy adults. We found that within individuals, metabolic responses were correlated across the three prebiotics. Individual identity, rather than prebiotic choice, was also the major determinant of SCFA response. Across individuals, prebiotic response was inversely related to basal fecal SCFA concentration, which, in turn, was associated with habitual fiber intake. Experimental measures of gut microbial SCFA production for each participant also negatively correlated with fiber consumption, supporting a model in which individuals' gut microbiota are limited in their overall capacity to produce fecal SCFAs from fiber.

CONCLUSIONS

Our findings support developing personalized prebiotic regimens that focus on selecting individuals who stand to benefit, and that such individuals are likely to be deficient in fiber intake. Video Abstract.

Mouse Subcutaneous BCG Vaccination and Mycobacterium tuberculosis Infection Alter the Lung and Gut Microbiota

The objective of this study was to characterize the effect of Bacillus Calmette-Guérin (BCG) vaccination and M. tuberculosis infection on gut and lung microbiota of C57BL/6 mice, a well-characterized mouse model of tuberculosis. BCG vaccination and infection with M. tuberculosis altered the relative abundance of Firmicutes and Bacteroidetes phyla in the lung compared with control group. Vaccination and infection changed the alpha- and beta-diversity in both the gut and the lung. However, lung diversity was the most affected organ after BCG vaccination and M. tuberculosis infection. Focusing on the gut-lung axis, a multivariate regression approach was used to compare profile evolution of gut and lung microbiota. More genera have modified relative abundances associated with BCG vaccination status at gut level compared with lung. Conversely, genera with modified relative abundances associated with M. tuberculosis infection were numerous at lung level. These results indicated that the host local response against infection impacted the whole microbial flora, while the immune response after vaccination modified mainly the gut microbiota. This study showed that a subcutaneous vaccination with a live attenuated microorganism induced both gut and lung dysbiosis that may play a key role in the immunopathogenesis of tuberculosis. IMPORTANCE The microbial communities in gut and lung are important players that may modulate the immunity against tuberculosis or other infections as well as impact the vaccine efficacy. We discovered that vaccination through the subcutaneous route affect the composition of gut and lung bacteria, and this might influence susceptibility and defense mechanisms against tuberculosis. Through these studies, we can identify microbial communities that can be manipulated to improve vaccine response and develop treatment adjuvants.

Using Community Ecology Theory and Computational Microbiome Methods To Study Human Milk as a Biological System

Human milk is a complex and dynamic biological system that has evolved to optimally nourish and protect human infants. Yet, according to a recent priority-setting review, “our current understanding of human milk composition and its individual components and their functions fails to fully recognize the importance of the chronobiology and systems biology of human milk in the context of milk synthesis, optimal timing and duration of feeding, and period of lactation” (P. Christian et al., Am J Clin Nutr 113:1063–1072, 2021, https://doi.org/10.1093/ajcn/nqab075). We attribute this critical knowledge gap to three major reasons as follows. (i) Studies have typically examined each subsystem of the mother-milk-infant “triad” in isolation and often focus on a single element or component (e.g., maternal lactation physiology or milk microbiome or milk oligosaccharides or infant microbiome or infant gut physiology). This undermines our ability to develop comprehensive representations of the interactions between these elements and study their response to external perturbations. (ii) Multiomics studies are often cross-sectional, presenting a snapshot of milk composition, largely ignoring the temporal variability during lactation. The lack of temporal resolution precludes the characterization and inference of robust interactions between the dynamic subsystems of the triad. (iii) We lack computational methods to represent and decipher the complex ecosystem of the mother-milk-infant triad and its environment. In this review, we advocate for longitudinal multiomics data collection and demonstrate how incorporating knowledge gleaned from microbial community ecology and computational methods developed for microbiome research can serve as an anchor to advance the study of human milk and its many components as a “system within a system.”

Modeling approaches for probing cross-feeding interactions in the human gut microbiome

Microbial communities perform emergent activities that are essentially different from those carried by their individual members. The gut microbiome and its metabolites have a significant impact on the host, contributing to homeostasis or disease. Food molecules shape this community, being fermented through cross-feeding interactions of metabolites such as lactate, acetate, and amino acids, or products derived from macromolecule degradation. Mathematical and experimental approaches have been applied to understand and predict the interactions between microorganisms in complex communities such as the gut microbiota. Rational and mechanistic understanding of microbial interactions is essential to exploit their metabolic activities and identify keystone taxa and metabolites. The latter could be used in turn to modulate or replicate the metabolic behavior of the community in different contexts. This review aims to highlight recent experimental and modeling approaches for studying cross-feeding interactions within the gut microbiome. We focus on short-chain fatty acid production and fiber fermentation, which are fundamental processes in human health and disease. Special attention is paid to modeling approaches, particularly kinetic and genome-scale stoichiometric models of metabolism, to integrate experimental data under different diet and health conditions. Finally, we discuss limitations and challenges for the broad application of these modeling approaches and their experimental verification for improving our understanding of the mechanisms of microbial interactions.

BEEM-Static: Accurate inference of ecological interactions from cross-sectional microbiome data

The structure and function of diverse microbial communities is underpinned by ecological interactions that remain uncharacterized. With rapid adoption of next-generation sequencing for studying microbiomes, data-driven inference of microbial interactions based on abundance correlations is widely used, but with the drawback that ecological interpretations may not be possible. Leveraging cross-sectional microbiome datasets for unravelling ecological structure in a scalable manner thus remains an open problem. We present an expectation-maximization algorithm (BEEM-Static) that can be applied to cross-sectional datasets to infer interaction networks based on an ecological model (generalized Lotka-Volterra). The method exhibits robustness to violations in model assumptions by using statistical filters to identify and remove corresponding samples. Benchmarking against 10 state-of-the-art correlation based methods showed that BEEM-Static can infer presence and directionality of ecological interactions even with relative abundance data (AUC-ROC>0.85), a task that other methods struggle with (AUC-ROC<0.63). In addition, BEEM-Static can tolerate a high fraction of samples (up to 40%) being not at steady state or coming from an alternate model. Applying BEEM-Static to a large public dataset of human gut microbiomes (n = 4,617) identified multiple stable equilibria that better reflect ecological enterotypes with distinct carrying capacities and interactions for key species.

Characterizing the ecological interactions among microbial members is an important step towards understanding the structure and function of diverse microbial communities. Widely used correlation based approaches for inferring interactions from cross-sectional microbiome sequencing data are not able to predict the directionality of interactions, and their results may not be interpretable. We developed an expectation-maximization algorithm (BEEM-Static) that can infer directed interaction networks from cross-sectional data based on an ecological model. Our benchmarking results showed that BEEM-Static inferred presence and directionality of interactions accurately, while correlation based methods had performance slightly better than random guesses. In addition, BEEM-Static was robust to various types of noises using statistical filters to identify and remove data points violating its assumptions. Applying BEEM-Static to a large public dataset of human gut microbiomes, we were able to identify multiple stable equilibria with distinct ecological properties.

Technological tools and strategies for culturing human gut microbiota in engineered in vitro models

The gut microbiota directly impacts the pathophysiology of different human body districts. Consequently, microbiota investigation is an hot topic of research and its in vitro culture has gained extreme interest in different fields. However, the high sensitivity of microbiota to external stimuli, such as sampling procedure, and the physicochemical complexity of the gut environment make its in vitro culture a challenging task. New engineered microfluidic gut-on-a-chip devices have the potential to model some important features of the intestinal structure, but they are usually unable to sustain culture of microbiota over an extended period of time. The integration of gut-on-a-chip devices with bioreactors for continuous bacterial culture would lead to fast advances in the study of microbiota-host crosstalk. In this review, we summarize the main technologies for the continuous culture of microbiota as upstream systems to be coupled with microfluidic devices to study bacteria-host cells communication. The engineering of integrated microfluidic platforms, capable of sustaining both anaerobic and aerobic cultures, would be the starting point to unveil complex biological phenomena proper of the microbiota-host crosstalks, paving to way to multiple research and technological applications.

Measuring and mitigating PCR bias in microbiota datasets

PCR amplification plays an integral role in the measurement of mixed microbial communities via high-throughput DNA sequencing of the 16S ribosomal RNA (rRNA) gene. Yet PCR is also known to introduce multiple forms of bias in 16S rRNA studies. Here we present a paired modeling and experimental approach to characterize and mitigate PCR NPM-bias (PCR bias from non-primer-mismatch sources) in microbiota surveys. We use experimental data from mock bacterial communities to validate our approach and human gut microbiota samples to characterize PCR NPM-bias under real-world conditions. Our results suggest that PCR NPM-bias can skew estimates of microbial relative abundances by a factor of 4 or more, but that this bias can be mitigated using log-ratio linear models.

High-throughput DNA sequencing is often used to profile the species composition of host-associated microbial communities (microbiota). One important step in DNA sequencing is amplification where DNA from many different bacteria are repeatedly copied using a technique called Polymerase Chain Reaction (PCR). However, PCR is known to introduce multiple forms of bias as DNA from some bacteria are more efficiently copied than others. Here we introduce experimental and computational procedures that allows PCR NPM-bias (PCR bias from non-primer-mismatch sources) to be measured and mitigated in studies of microbial communities.

Impact of Temporal pH Fluctuations on the Coexistence of Nasal Bacteria in an in silico Community

To manipulate nasal microbiota for respiratory health, we need to better understand how this microbial community is assembled and maintained. Previous work has demonstrated that the pH in the nasal passage experiences temporal fluctuations. Yet, the impact of such pH fluctuations on nasal microbiota is not fully understood. Here, we examine how temporal fluctuations in pH might affect the coexistence of nasal bacteria in in silico communities. We take advantage of the cultivability of nasal bacteria to experimentally assess their responses to pH and the presence of other species. Based on experimentally observed responses, we formulate a mathematical model to numerically investigate the impact of temporal pH fluctuations on species coexistence. We assemble in silico nasal communities using up to 20 strains that resemble the isolates that we have experimentally characterized. We then subject these in silico communities to pH fluctuations and assess how the community composition and coexistence is impacted. Using this model, we then simulate pH fluctuations-varying in amplitude or frequency-to identify conditions that best support species coexistence. We find that the composition of nasal communities is generally robust against pH fluctuations within the expected range of amplitudes and frequencies. Our results also show that cooperative communities and communities with lower niche overlap have significantly lower composition deviations when exposed to temporal pH fluctuations. Overall, our data suggest that nasal microbiota could be robust against environmental fluctuations.

Context-aware dimensionality reduction deconvolutes gut microbial community dynamics

The translational power of human microbiome studies is limited by high interindividual variation. We describe a dimensionality reduction tool, compositional tensor factorization (CTF), that incorporates information from the same host across multiple samples to reveal patterns driving differences in microbial composition across phenotypes. CTF identifies robust patterns in sparse compositional datasets, allowing for the detection of microbial changes associated with specific phenotypes that are reproducible across datasets.

Application of OU processes to modelling temporal dynamics of the human microbiome, and calculating optimal sampling schemes

BACKGROUND

The vast majority of microbiome research so far has focused on the structure of the microbiome at a single time-point. There have been several studies that measure the microbiome from a particular environment over time. A few models have been developed by extending time series models to accomodate specific features in microbiome data to address questions of stability and interactions of the microbime time series. Most research has observed the stability and mean reversion for some microbiomes. However, little has been done to study the mean reversion rates of these stable microbes and how sampling frequencies are related to such conclusions. In this paper, we begin to rectify this situation. We analyse two widely studied microbial time series data sets on four healthy individuals. We choose to study healthy individuals because we are interested in the baseline temporal dynamics of the microbiome.

RESULTS

For this analysis, we focus on the temporal dynamics of individual genera, absorbing all interactions in a stochastic term. We use a simple stochastic differential equation model to assess the following three questions. (1) Does the microbiome exhibit temporal continuity? (2) Does the microbiome have a stable state? (3) To better understand the temporal dynamics, how frequently should data be sampled in future studies? We find that a simple Ornstein-Uhlenbeck model which incorporates both temporal continuity and reversion to a stable state fits the data for almost every genus better than a Brownian motion model that contains only temporal continuity. The Ornstein-Uhlenbeck model also fits the data better than modelling separate time points as independent. Under the Ornstein-Uhlenbeck model, we calculate the variance of the estimated mean reversion rate (the speed with which each genus returns to its stable state). Based on this calculation, we are able to determine the optimal sample schemes for studying temporal dynamics.

CONCLUSIONS

There is evidence of temporal continuity for most genera; there is clear evidence of a stable state; and the optimal sampling frequency for studying temporal dynamics is in the range of one sample every 0.8-3.2 days.

Naught all zeros in sequence count data are the same

Genomic studies feature multivariate count data from high-throughput DNA sequencing experiments, which often contain many zero values. These zeros can cause artifacts for statistical analyses and multiple modeling approaches have been developed in response. Here, we apply different zero-handling models to gene-expression and microbiome datasets and show models can disagree substantially in terms of identifying the most differentially expressed sequences. Next, to rationally examine how different zero handling models behave, we developed a conceptual framework outlining four types of processes that may give rise to zero values in sequence count data. Last, we performed simulations to test how zero handling models behave in the presence of these different zero generating processes. Our simulations showed that simple count models are sufficient across multiple processes, even when the true underlying process is unknown. On the other hand, a common zero handling technique known as "zero-inflation" was only suitable under a zero generating process associated with an unlikely set of biological and experimental conditions. In concert, our work here suggests several specific guidelines for developing and choosing state-of-the-art models for analyzing sparse sequence count data.

Short-Chain Fatty Acid Production by Gut Microbiota from Children with Obesity Differs According to Prebiotic Choice and Bacterial Community Composition

Pediatric obesity remains a public health burden and continues to increase in prevalence. The gut microbiota plays a causal role in obesity and is a promising therapeutic target. Specifically, the microbial production of short-chain fatty acids (SCFA) from the fermentation of otherwise indigestible dietary carbohydrates may protect against pediatric obesity and metabolic syndrome. Still, it has not been demonstrated that therapies involving microbiota-targeting carbohydrates, known as prebiotics, will enhance gut bacterial SCFA production in children and adolescents with obesity (age, 10 to 18 years old). Here, we used an in vitro system to examine the SCFA production by fecal microbiota from 17 children with obesity when exposed to five different commercially available over-the-counter (OTC) prebiotic supplements. We found microbiota from all 17 patients actively metabolized most prebiotics. Still, supplements varied in their acidogenic potential. Significant interdonor variation also existed in SCFA production, which 16S rRNA sequencing supported as being associated with differences in the host microbiota composition. Last, we found that neither fecal SCFA concentration, microbiota SCFA production capacity, nor markers of obesity positively correlated with one another. Together, these in vitro findings suggest the hypothesis that OTC prebiotic supplements may be unequal in their ability to stimulate SCFA production in children and adolescents with obesity and that the most acidogenic prebiotic may differ across individuals.IMPORTANCE Pediatric obesity remains a major public health problem in the United States, where 17% of children and adolescents are obese, and rates of pediatric "severe obesity" are increasing. Children and adolescents with obesity face higher health risks, and noninvasive therapies for pediatric obesity often have limited success. The human gut microbiome has been implicated in adult obesity, and microbiota-directed therapies can aid weight loss in adults with obesity. However, less is known about the microbiome in pediatric obesity, and microbiota-directed therapies are understudied in children and adolescents. Our research has two important findings: (i) dietary prebiotics (fiber) result in the microbiota from adolescents with obesity producing more SCFA, and (ii) the effectiveness of each prebiotic is donor dependent. Together, these findings suggest that prebiotic supplements could help children and adolescents with obesity, but that these therapies may not be "one size fits all."

Stochastic logistic models reproduce experimental time series of microbial communities

We analyze properties of experimental microbial time series, from plankton and the human microbiome, and investigate whether stochastic generalized Lotka-Volterra models could reproduce those properties. We show that this is the case when the noise term is large and a linear function of the species abundance, while the strength of the self-interactions varies over multiple orders of magnitude. We stress the fact that all the observed stochastic properties can be obtained from a logistic model, that is, without interactions, even the niche character of the experimental time series. Linear noise is associated with growth rate stochasticity, which is related to changes in the environment. This suggests that fluctuations in the sparsely sampled experimental time series may be caused by extrinsic sources.

Interindividual Variation in Dietary Carbohydrate Metabolism by Gut Bacteria Revealed with Droplet Microfluidic Culture

Bacterial culture and assay are components of basic microbiological research, drug development, and diagnostic screening. However, community diversity can make it challenging to comprehensively perform experiments involving individual microbiota members. Here, we present a new microfluidic culture platform that makes it feasible to measure the growth and function of microbiota constituents in a single set of experiments. As a proof of concept, we demonstrate how the platform can be used to measure how hundreds of gut bacterial taxa drawn from different people metabolize dietary carbohydrates. Going forward, we expect this microfluidic technique to be adaptable to a range of other microbial assay needs.

Culture and screening of gut bacteria enable testing of microbial function and therapeutic potential. However, the diversity of human gut microbial communities (microbiota) impedes comprehensive experimental studies of individual bacterial taxa. Here, we combine advances in droplet microfluidics and high-throughput DNA sequencing to develop a platform for separating and assaying growth of microbiota members in picoliter droplets (MicDrop). MicDrop enabled us to cultivate 2.8 times more bacterial taxa than typical batch culture methods. We then used MicDrop to test whether individuals possess similar abundances of carbohydrate-degrading gut bacteria, using an approach which had previously not been possible due to throughput limitations of traditional bacterial culture techniques. Single MicDrop experiments allowed us to characterize carbohydrate utilization among dozens of gut bacterial taxa from distinct human stool samples. Our aggregate data across nine healthy stool donors revealed that all of the individuals harbored gut bacterial species capable of degrading common dietary polysaccharides. However, the levels of richness and abundance of polysaccharide-degrading species relative to monosaccharide-consuming taxa differed by up to 2.6-fold and 24.7-fold, respectively. Additionally, our unique dataset suggested that gut bacterial taxa may be broadly categorized by whether they can grow on single or multiple polysaccharides, and we found that this lifestyle trait is correlated with how broadly bacterial taxa can be found across individuals. This demonstration shows that it is feasible to measure the function of hundreds of bacterial taxa across multiple fecal samples from different people, which should in turn enable future efforts to design microbiota-directed therapies and yield new insights into microbiota ecology and evolution.

IMPORTANCE Bacterial culture and assay are components of basic microbiological research, drug development, and diagnostic screening. However, community diversity can make it challenging to comprehensively perform experiments involving individual microbiota members. Here, we present a new microfluidic culture platform that makes it feasible to measure the growth and function of microbiota constituents in a single set of experiments. As a proof of concept, we demonstrate how the platform can be used to measure how hundreds of gut bacterial taxa drawn from different people metabolize dietary carbohydrates. Going forward, we expect this microfluidic technique to be adaptable to a range of other microbial assay needs.

Efficient and Accurate Inference of Mixed Microbial Population Trajectories from Longitudinal Count Data

The recently completed second phase of the Human Microbiome Project has highlighted the relationship between dynamic changes in the microbiome and disease, motivating new microbiome study designs based on longitudinal sampling. Yet, analysis of such data is hindered by presence of technical noise, high dimensionality, and data sparsity. Here, we introduce LUMINATE (longitudinal microbiome inference and zero detection), a fast and accurate method for inferring relative abundances from noisy read count data. We demonstrate that LUMINATE is orders of magnitude faster than current approaches, with better or similar accuracy. We further show that LUMINATE can accurately distinguish biological zeros, when a taxon is absent from the community, from technical zeros, when a taxon is below the detection threshold. We conclude by demonstrating the utility of LUMINATE on a real dataset, showing that LUMINATE smooths trajectories observed from noisy data. LUMINATE is freely available from https://github.com/tyjo/luminate.

Counts: an outstanding challenge for log-ratio analysis of compositional data in the molecular biosciences

Thanks to sequencing technology, modern molecular bioscience datasets are often compositions of counts, e.g. counts of amplicons, mRNAs, etc. While there is growing appreciation that compositional data need special analysis and interpretation, less well understood is the discrete nature of these count compositions (or, as we call them, lattice compositions) and the impact this has on statistical analysis, particularly log-ratio analysis (LRA) of pairwise association. While LRA methods are scale-invariant, count compositional data are not; consequently, the conclusions we draw from LRA of lattice compositions depend on the scale of counts involved. We know that additive variation affects the relative abundance of small counts more than large counts; here we show that additive (quantization) variation comes from the discrete nature of count data itself, as well as (biological) variation in the system under study and (technical) variation from measurement and analysis processes. Variation due to quantization is inevitable, but its impact on conclusions depends on the underlying scale and distribution of counts. We illustrate the different distributions of real molecular bioscience data from different experimental settings to show why it is vital to understand the distributional characteristics of count data before applying and drawing conclusions from compositional data analysis methods.

Compositional Lotka-Volterra describes microbial dynamics in the simplex

Dynamic changes in microbial communities play an important role in human health and disease. Specifically, deciphering how microbial species in a community interact with each other and their environment can elucidate mechanisms of disease, a problem typically investigated using tools from community ecology. Yet, such methods require measurements of absolute densities, whereas typical datasets only provide estimates of relative abundances. Here, we systematically investigate models of microbial dynamics in the simplex of relative abundances. We derive a new nonlinear dynamical system for microbial dynamics, termed “compositional” Lotka-Volterra (cLV), unifying approaches using generalized Lotka-Volterra (gLV) equations from community ecology and compositional data analysis. On three real datasets, we demonstrate that cLV recapitulates interactions between relative abundances implied by gLV. Moreover, we show that cLV is as accurate as gLV in forecasting microbial trajectories in terms of relative abundances. We further compare cLV to two other models of relative abundance dynamics motivated by common assumptions in the literature—a linear model in a log-ratio transformed space, and a linear model in the space of relative abundances—and provide evidence that cLV more accurately describes community trajectories over time. Finally, we investigate when information about direct effects can be recovered from relative data that naively provide information about only indirect effects. Our results suggest that strong effects may be recoverable from relative data, but more subtle effects are challenging to identify.

Dynamic changes in microbial communities play an important role in human health and disease. Specifically, deciphering how microbial species in a community interact with each other and their environment can elucidate mechanisms of disease, a problem typically investigated using tools from community ecology. Yet, such methods require measurements of absolute densities, whereas typical only provide estimates of relative abundances. We investigate methods for describing microbial dynamics in terms of relative abundances using approaches from machine learning and dynamical systems. Across three real datasets, we show that relative abundances are sufficient to describe compositional dynamics. Additionally, we show that models trained on relative abundances alone predict future compositions as well models trained on absolute abundances. Finally, we provide criteria for when direct effects, which typically can only be learned from absolute abundances, are recoverable for relative data. As a proof of concept, we recapitulate a previously proposed interaction network for C. difficile colonization.

Primate microbiomes over time: Longitudinal answers to standing questions in microbiome research

To date, most insights into the processes shaping vertebrate gut microbiomes have emerged from studies with cross-sectional designs. While this approach has been valuable, emerging time series analyses on vertebrate gut microbiomes show that gut microbial composition can change rapidly from 1 day to the next, with consequences for host physical functioning, health, and fitness. Hence, the next frontier of microbiome research will require longitudinal perspectives. Here we argue that primatologists, with their traditional focus on tracking the lives of individual animals and familiarity with longitudinal fecal sampling, are well positioned to conduct research at the forefront of gut microbiome dynamics. We begin by reviewing some of the most important ecological processes governing microbiome change over time, and briefly summarizing statistical challenges and approaches to microbiome time series analysis. We then introduce five questions of general interest to microbiome science where we think field-based primate studies are especially well positioned to fill major gaps: (a) Do early life events shape gut microbiome composition in adulthood? (b) Do shifting social landscapes cause gut microbial change? (c) Are gut microbiome phenotypes heritable across variable environments? (d) Does the gut microbiome show signs of host aging? And (e) do gut microbiome composition and dynamics predict host health and fitness? For all of these questions, we highlight areas where primatologists are uniquely positioned to make substantial contributions. We review preliminary evidence, discuss possible study designs, and suggest future directions.

An expectation-maximization algorithm enables accurate ecological modeling using longitudinal microbiome sequencing data

BACKGROUND

The dynamics of microbial communities is driven by a range of interactions from symbiosis to predator-prey relationships, the majority of which are poorly understood. With the increasing availability of high-throughput microbiome taxonomic profiling data, it is now conceivable to directly learn the ecological models that explicitly define microbial interactions and explain community dynamics. The applicability of these approaches is severely limited by the lack of accurate absolute cell density measurements (biomass).

METHODS

We present a new computational approach that resolves this key limitation in the inference of generalized Lotka-Volterra models (gLVMs) by coupling biomass estimation and model inference with an expectation-maximization algorithm (BEEM).

RESULTS

BEEM outperforms the state-of-the-art methods for inferring gLVMs, while simultaneously eliminating the need for additional experimental biomass data as input. BEEM's application to previously inaccessible public datasets (due to the lack of biomass data) allowed us to construct ecological models of microbial communities in the human gut on a per-individual basis, revealing personalized dynamics and keystone species.

CONCLUSIONS

BEEM addresses a key bottleneck in "systems analysis" of microbiomes by enabling accurate inference of ecological models from high throughput sequencing data without the need for experimental biomass measurements.

Timescales of gut microbiome dynamics

Vast communities of microorganisms inhabit the gastrointestinal tracts of humans and other animals. Understanding their initial development, fluctuations in composition, stability over long times, and responses to transient perturbations - in other words their dynamics - is important both for gaining basic insights into these ecosystems and for rationally manipulating them for therapeutic ends. Gut microbiome dynamics, however, remain poorly understood. We review here studies of gut microbiome dynamics in the presence and absence of external perturbations, noting especially the long timescales associated with overall stability and the short timescales associated with various underlying biological processes. Integrating these disparate timescales, we suggest, is an important goal for future work and is necessary for developing a predictive understanding of microbiome dynamics.

Establishing microbial composition measurement standards with reference frames

Differential abundance analysis is controversial throughout microbiome research. Gold standard approaches require laborious measurements of total microbial load, or absolute number of microorganisms, to accurately determine taxonomic shifts. Therefore, most studies rely on relative abundance data. Here, we demonstrate common pitfalls in comparing relative abundance across samples and identify two solutions that reveal microbial changes without the need to estimate total microbial load. We define the notion of "reference frames", which provide deep intuition about the compositional nature of microbiome data. In an oral time series experiment, reference frames alleviate false positives and produce consistent results on both raw and cell-count normalized data. Furthermore, reference frames identify consistent, differentially abundant microbes previously undetected in two independent published datasets from subjects with atopic dermatitis. These methods allow reassessment of published relative abundance data to reveal reproducible microbial changes from standard sequencing output without the need for new assays.

Community ecology as a framework for human microbiome research

The field of human microbiome research has revealed the intimate co-association of humans with diverse communities of microbes in various habitats in the human body, and the necessity of these microbes for the maintenance of human health. Microbial heterogeneity between humans and across spatial and temporal gradients requires multidimensional datasets and a unifying set of theories and statistical tools to analyze the human microbiome and fully realize the potential of this field. Here we consider the utility of community ecology as a framework for the interrogation and interpretation of the human microbiome.

Correction to: Dynamic linear models guide design and analysis of microbiota studies within artificial human guts

AbstractFollowing publication of the original article [1], the authors noticed an error in the presentation of equations in the PDF version.