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α-‐diversity metrics

Marker-‐based metagenomic tutorial 1

Alpha Diversity: within sample diversity

Sample 1 Sample 2 Sample 3 Sample 4


Alpha Diversity: richness (R)


SPECIES RICHNESS (S) ESTIMATORS:

•  OTU richness – count of different species/OTUs

•  Observed Species – count of unique OTUs in each sample

•  Chao1 index – esGmate diversity from abundance data (importance of rare OTUs)

•  ACE index …


R = 5 R = 2 R = 4 R = 5

Let S0 be the number of taxa observed at least once in a sample,

a0 the unknown number of species present in the community but not observed.

1. OTU richness: R= S0 (no correcGon for taxa not observed);

2. Chao-‐1 index: assumes that the number of observaGons for a taxa has a Poisson distribuGon and corrects for variance;

R= S0 + a0 i.e. 3. ACE (abundance-‐based coverage esGmators): involves an arbitrary abundance threshold to label Sabun as the number of abundant taxa, Srare as the number of rare taxa; The expression basically inflates the number of rare taxa and inflates again the number of taxa with abundance 1.

R= S0 + a0 i.e.


Species richness indices

100

150

200

250

300

350

400

Richness

RU-BOL1-1

RU-KOR1

-1

GB-EL75-69

FI-N-47

FI-Xinb-3

FI-FHS

2-11

FI-FUT

1-2

SE-G1-9

FI-FSP

1-16

BE-OM-2

RU-RM1-9

DE-K35-Mu11

BY-G1-9

CH-H-149

DE-K35-Iinb

KE-1-1

RU-YAK

1-1

FR-C1-1

IT-IS

R1-8

IL-M1-8

IR-GG1-7


Species richness: example of use

Daphnia clones

Richne

ss (o

bserved species) Influence of temperature on

Daphnia microbiota:

20°C

28°C

SPECIES RICHNESS ESTIMATORS RICHNESS and EVENNESS ESTIMATORS: the calculated value of diversity increases both when the number of species increases and when evenness increases.

•  InformaSon staSsScs: Shannon-‐Wiener, Shannon-‐Weaver, Shannon entropy

•  Dominance indices: Inverse Simpson, Gini–Simpson, Berger–Parker index

Alpha Diversity: within sample diversity


Sample 1 Sample 3 Sample 4

R = 5

Sample 2

R = 2 R = 4 R = 5

Refers to how common or rare a species is relaGve to other species in a community.

Alpha Diversity: relaGve abundances



RelaGve species abundance distribuGons are graphed as rank-‐abundance diagrams.

Alpha Diversity: species evenness


Species evenness refers to how equally abundant species in an environment are.

Increasing richness and abundance


Alpha Diversity: indices’ performance

Sample size sensitivity!

Influenced by rare OTUs Influenced by dominance/abundance of OTUs


Shannon"

Species richness"

Diversity Index Value!

Inverse Simpson"

Berger-Parker"

Sequencing depth!

SPECIES RICHNESS ESTIMATORS RICHNESS and EVENNESS ESTIMATORS PHYLOGENETIC RICHNESS ESTIMATOR:

•  PhylogeneSc diversity (PD) – takes into consideraGon the phylogeny of microbes to esGmate diversity across a tree

Alpha Diversity: phylogeneGc diversity


R = 5


R = 2 R = 4 R = 5



Sample 2

R = 2

OTU 2 OTU 1



Sample 2

R = 2

OTU 2 OTU 1

Phylogene/c diversity is high



Sample 2

R = 2

OTU 2 OTU 1



Sample 2

R = 2

OTU 2 OTU 1

Phylogene/c diversity is low

»  What you want to know determines how you analyze your data

»  How important is each aspect of diversity? ˃  Richness? ˃  Evenness? ˃  Dominance? ˃  Abundance? ˃  Per-‐species (relaGve) abundance? ˃  Taxon diversity?


VariaGon among samples


Shannon"

Species richness"

Diversity Index Value!

Inverse Simpson"

Berger-Parker"

Sequencing depth!

Diversity: influence of sequencing effort



VariaSon in reads counts

Sample Addition Sequence!R

ichn

ess!

Samples: Accumulation"

Samples: Rarefaction"

Taxa: Accumulation" Taxa: Rarefaction"

RarefacSon


Rarefying was first recommended for microbiome counts in order to moderate differences in the presence of rare OTUs

(Lozupone et al. 2011 ISME J) Goal:

•  Standardize unequal sequencing effort. •  Enable similarity comparisons along a

range of samples or a gradient. •  Enable comparison of different runs or

replicates.

Procedure:

•  Determine the minimum sequencing depth. •  Subsample without replacement sequences from the larger libraries so that all

have the same smallest size. •  Note that the term is a bit misleading as this step should really be called

"subsampling to a given depth".


RarefacSon curves

RarefacGon curves represent the diversity as a funcGon of sequencing depth.


RarefacSon curves

Number of reads

Species Richness PhylogeneSc diversity

Host species: D. pulex D. longispina D. magna

RarefacSon


This approach simultaneously addresses problems when (1)  DNA sequencing libraries are of widely different sizes (also loss of informaGon in the

larger sample) (2) OTU count proporGons vary more than expected under a Poisson model. (3) power and accuracy is too low in the detecGon of differenGally abundant OTUs NB: A species/OTU is considered differenGally abundant if its mean proporGon is significantly different between two or more sample classes in the experimental design.

“In the case of differenGal abundance detecGon, it seems unlikely that the cost of rarefying is ever acceptable.”

McMurdie 2014 PLoS Comput Bio 10-‐e1003531

AlternaSves to rarefacSon


All that is leX aXer rarefacSon is the expected number of species per sample, not a real value or real data. DisGncGon between subsampling curves and normalizaGon. Randomly select evenly-‐sized samples from the larger sample •  Could be done iteraGvely to provide a normalized distribuGon of the expected

number of species •  Akin to the Jackknife value

Kempton & Wedderburn (1978) •  Produce equal sized samples aoer fipng species abundances to gamma

distribuGon •  Not commonly used Procedure: use normalizaSon tools or transform count data (e.g. use a log2(x + 1) transformaGon on count data to miGgate the impact of 0 and very high counts).

NormalizaSon •  Equalize depths by simply scaling OTU counts to a common depth in all samples •  Transform counts into relaGve abundances for each sample. This lead the counts

not to be integers any more. each number in the OTU table will represent the proporGon of sequences from that samples belonging to that OTU.

•  Note that the methods specifically tailored to integers will not apply but it will not change methods based on presence/absence or proporGons, such as UniFrac, Bray-‐CurGs, etc. . .

•  Other normalizaGon: normalize data based on 16S copy number. •  DESeq (Anders and Huber 2010), DESeq2 (Love et al. Genome Biology 2014) •  MetagenomeSeq’s CumulaGve Sum Scaling (CSS) (Paulson et al. Nature Methods

2013).

FiltraSon •  Remove taxa with 0 count (prune_taxa) •  Remove OTUs that appear less than n Gmes or in less than n samples

(genefilter_sample, filterfun_sample)


AlternaSves to rarefacSon

CH-Daphnia

CH-Water

Controls

FI-Daphnia


Ge]ng controls


Stock

IsolaGon

Daphnia, algae and ADaM substrates

76 sequences x max.607 nuc

Mock 76 strains

Clustering threshold

Number of OTUs

99% 58

98% 40

97% 35

96% 31

95% 30

90% 14

β-‐diversity metrics


Gamma Diversity: total species diversity


Sample 1 Sample 2

Sample 3 Sample 4

R = 7

Main concepts of beta-‐diversity: QuesSon – What is the influence of treatments A & B? What is the species diversity along transects & gradients? NoSon of similarity: Species differences among samples are posiGvely correlated to β-‐diversity values and inversely correlated to similarity.


Beta Diversity: between sample diversity


Treatment A Treatment B


R = 7

Sample 1 Sample 2

Sample 3 Sample 4

β = γ / α R = 5

Treatment A R = 6

Treatment B

Beta Diversity stricto sensu

Mean diversity per treatment



Presence/Absence Abundance

Without a phylogeneSc tree

•  Jaccard •  Bray-‐CurGs (PCoA)

•  Euclidean (PCA)

With a phylogeneSc tree


Beta Diversity: phylogeneGc diversity Sample 3

Sample 4

Treatm

ent B



Sample 4

Treatm

ent B



Sample 4

Treatm

ent B

Distance between samples is reduced if phylogeneSc approach is used


Beta Diversity: phylogeny-‐based assessment of differences in overall bacterial community composiGon.

UniFrac distances are based on the fracGon of branch length shared between two communiGes within a phylogeneGc tree constructed from the 16S rRNA gene sequences from all communiGes being compared.

•  With unweighted UniFrac, only the presence or absence of lineages are considered (community membership).

•  With weighted UniFrac, branch lengths are weighted based on the relaGve abundances of lineages within communiGes (community structure).

Measuring$Microbial$Communi+es:$UniFrac$Metric$

hep://bmf.colorado.edu/unifrac/$

Used$to$analyzed$shared$branch$length$of$mul+ple$microbial$communi+es$

Beta Diversity: phylogeneGc diversity UniFrac:



Presence/Absence Abundance

Without a phylogeneSc tree

•  Jaccard •  Bray-‐CurGs (PCoA)

•  Euclidean (PCA)

With a phylogeneSc tree

•  Unweighted UniFrac •  Comdist

•  Weighted UniFrac •  Comdist

38

2

3

1 4

0.2

0.3

0.4

0.5

0.6

0.7

Cluster Dendrogram

hclust (*, "average")tutorial.data.bray

Hei

ght

s4

s3

s1

s2

Hierarchical clustering

Sample1 Sample2 Sample3 Sample4

OTU 1 2 0 1 2

OTU 2 4 1 2 4

OTU 3 1 0 0 2

OTU 4 1 0 3 2

OTU 5 1 9 0 0

OTU 6 0 0 2 0

OTU 7 0 0 0 2

#SampleID Treatment

Sample1 A

Sample2 A

Sample3 B

Sample4 B

Sample 3 Sample 4 Sample 1 Sample 2 Treatment A Treatment B

Marker-‐based metagenomic tutorial

OTU_table

MapFile


-0.4 -0.2 0.0 0.2

-0.3

-0.2

-0.1

0.0

0.1

0.2

PCoA 1: 79.5%

PC

oA 2

: 18.

9 %

site1

site2

site3

site4s4

s2

s1

s3

Beta Diversity: visualizaGon / ordinaGon

Sample 1 Sample 2

Sample 3 Sample 4

Treatment A

Treatment B

»  What you want to know determines how you analyze your data

»  How important is each aspect of diversity? ˃  Richness? ˃  Evenness? ˃  Dominance? ˃  Abundance? ˃  Per-‐species (relaGve) abundance? ˃  Taxon diversity?


Comparison of samples or group of samples:

How similar are communi4es?



Local specificity Community overlap

Dispersion sta4s4cs

VISUALIZATION

Differen4ally abundant taxa

Ordina4on plots Cluster analysis

Framework

Raw count data

Distance matrix Transformed data

STATISTICAL TESTS


Beta Diversity: similarity coefficients Based only on the number of species present in each sample All species are counted & weighted equally

•  Jaccard CJ = j / (a + b – j) a = richness in first site, b = richness in second site j = shared species

•  Sorensen CS = 2j / (a + b)

makes an effort to weight shared species by their rela4ve abundance •  Sorensen Quan4ta4ve CN = 2(jN) / (aN + bN) jN = sum of the lower of the two abundances recorded for species found in each site •  Morisita-‐Horn CmH is not influenced by sample size & richness but highly sensi4ve to

the abundance of the most abundant OTUs

•  Cluster Analyses use a similarity matrix of all samples o  Group Average clustering o  Centroid Clustering


Beta Diversity: phylogene4c distance

•  Unique Frac4on (UniFrac) metric

•  Qualita4ve phylogene4c β-‐diversity = Unweighted UniFrac

•  Distance = frac4on of the total branch length that is unique to any par4cular

environment

•  Quan4ta4ve phylogene4c β-‐diversity = Weighted UniFrac Lozupone and Knight (2005) Appl. Environ. Microbiol. Lozupone et al. (2007) Appl. Environ. Microbiol.


Distances To be used when…

Euclidean Manhagan

variables are expected to have equal variance

Bray–Cur4s Canberra

between-‐group differences in average absolute differenceare matched by propor4onate changes in average abundance

Gower variables are physical or chemical data

Jaccard Sorensen Ochiai

face presence/absence data

Whigaker Hellinger

analyze quan4ta4ve assemblage composi4on

Mean

Varia

nce

Mean [log-‐scale]

Varia

nce

[log-‐scale]

Distance metrics overview



Dispersion sta4s4cs

VISUALIZATION



Framework

Raw count data


STATISTICAL TESTS

OrdinaGon techniques plot_ordination / plot_samples 1. PCoA (Principal Coordinates Analyses) also called MDS (Metric Dimensional Scaling) relies on a dissimilarity or distance matrix. A non-‐metric variant of PCoA is called NMDS (Non Metric Dimensional Scaling). 2. DPCoA (Double Principal Coordinate Analysis) is a two step PCoA. The procedure first computes a distance matrix for taxa using the patris4c distance (length of the shortest path on a tree) over the taxa phylogeny. The posi4on of the communi4es in coordinate space is then the average posi4on (centroids) of their cons4tuent taxa, weighted by rela4ve abundances. The common space for taxa and communi4es allows for easier interpreta4on of the community. It also highlights leverage taxa that drive the differences between communi4es. 3. PCA (Principal Components Analysis) preserves the variance of samples. In par4cular, edge PCA, is an hybrid method where taxa abundance is combined to a phylogeny to create contrasts that are used as input variables. Unlike PCoA based on UniFrac distances and DPCoA, it does not use branch lengths. However, and unlike most other ordina4on methods, the principal components can be mapped onto the tree for easy visualiza4on and interpreta4on.


DPCoA

PCoA/MDS with wUniFrac

PCoA/MDS with UniFrac

Most sensi4ve to outliers*, least sensi4ve to noise**, upweights deep differences, gives OTU loca4ons, long run4me (depends on the number of OTUs but not of samples)

* Outliers = highly abundant OTUs ** Noise = noise in detec4ng low-‐abundance OTUs (around 0 count)

Less sensi4ve to outliers* / more sensi4ve to noise** than DPCoA, linear run4me in the number of OTUs and samples

Puts more weight on shallow branches on the tree than either DPCoA or wUniFrac, sensi4ve to noise**, picks up shallower differences, linear run4me in the number of OTUs and samples

Rely on rela4ve abundance of taxa

Rely on presence/absence of taxa


OrdinaGon techniques

Linear

Unimodal


OrdinaGon techniques

Raw data (presence/absence or abundance data) Distance-‐based

(db)-‐data (distance matrix)

Response along a gradient

How you interprete results Linear Unimodal

Unconstrained by env. factors tb-‐PCA CA, DCA PCoA, NMDS Env. factors are used post hoc

Constrained by env. factors tb-‐RDA CCA db-‐RDA

Test of significance and par44oning of variance explained by env. factors

Hierarchical clustering


1.  Use a distance measure that is ecologically meaningful. A good analysis prac4ce is to repeat the analysis with several good distance measures and inves4gate whether all these analyses lead to the same conclusion.

2.  Inves4gate how well the distances in the ordina4on graph represent the total distances (e.g.


Distance-‐based analyses



Dispersion sta4s4cs

VISUALIZATION



Framework

Raw count data


STATISTICAL TESTS

Local Specicity: iden4fy indicator species


edgeR and DESeq based on raw counts and use nega4ve binomial distribu4ons to model count data. edgeR is usually more conserva4ve than DESeq. A two-sided t-test with unequal variances use the mt wrapper in phyloseq and based either on rank test or moderated taxa-‐wise ANOVA. Is usually more conserva4ve than the nega4ve binomial model implemented in edgeR and DESeq. Nevertheless, they are quite sufficient to detect large size effects and are easier to use (e.g. Fisher test).


DifferenGally Abundant OTUs

Number of reads

Num

ber o

f OTU

s

StaGsGcal tests for different membership •  UniFrac Significance: frac4on of random trees that have more Unique branch

length than the real tree.

•  Phylogene4c (P) Test: based on the number of changes between states (samples) required to explain the distribu4on of sequences on the tree (Fitch parsimony). Sensi4ve to tree topology but not to branch lengths. See Mar4n AP (2002) Appl. Environ. Microbiol.

•  LibShuff



The libshuff method is a generic test that describes whether two or more communi4es have the same structure using the Cramer-‐von Mises test sta4s4c. The significance of the sta4s4cal test indicates the probability that the communi4es have the same structure by chance. Because each pairwise comparison requires two significance tests, a correc4on for mul4ple comparisons (e.g. Bonferroni's correc4on) must be applied. The program calculates a homologous and a heterologous coverage curve for the libraries then calculates the distance between the two curves and use a Monte Carlo test procedure to compare them. NB: Monte Carlo simula4ons: randomly permute the data (environment assignments) and determine how oten the random data has a more extreme value than the real data. Singleton DR et al. (2001) Appl. Environ. Microbiol. Schloss PD et al. (2004) Appl. Environ. Microbiol. hgps://toolshed.g2.bx.psu.edu/repos/jjohnson/mothur_toolsuite

Libshuff (Library shuffling)

StaGsGcal tests for different membership •  UniFrac Significance: frac4on of random trees that have more Unique branch

length than the real tree.

•  Phylogene4c (P) Test: based on the number of changes between states (samples) required to explain the distribu4on of sequences on the tree (Fitch parsimony). Sensi4ve to tree topology but not to branch lengths. See Mar4n AP (2002) Appl. Environ. Microbiol.

•  LibShuff

•  ADONIS: Analysis of variance using distance matrices (vegan package in R). formal tes4ng of sample covariates is also done using a permuta4on MANOVA with the (squared) distances and covariates as response and linear predictors, respec4vely. See Anderson (2001) Austral Ecology

•  ANOSIM

•  Mantel test

Note that these mul4variate analyses can be heavily influenced by heterogeneity of dispersion across groups in an unbalanced design. See Anderson and Walsh (2013) Ecological Monographs


Dispersion •  Dispersion is defined as a change in mean–variance rela4onship

•  Permuta4on test of homogeneity of group dispersion. It is an analogue to homogeneity of variances.

•  Test if all groups share a common dispersion (i.e. if the varia4on between samples is similar to the varia4on between groups).

•  Dispersion of sequences in the tree

See Webb CO (2000) American Naturalist


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