The Shannon, Simpson, Nei, and other indices permit to
summarize quantitative parameters describing community
structure. As described above for the Shannon index, they
all have the same weakness, in that they do not take into
account the identity of the organisms in a community.
Because diversity indices compile information obtained from
a set of several species in a single value for each observation
before comparison, it is not surprising that complex changes in
patterns of diversity are not well converted into changes in these
indices. For example, Widmer and collaborators (2006) have not
observed significant changes in soil bacterial communities
subjected to various treatments using diversity indices, while
molecular analyses did show significant changes in bacterial
fingerprint, as well as in the structure of communities, through
multivariate analyses. Two strategies have been proposed to
ensure consistent results between diversity indices and multivariate analysis techniques: the strategy “CA-species-richness” suitable for data where rare species are important and the strategy
“nonsymmetric CA–Simpson” which is more appropriate for
data where a species dominates (Pelissier et al. 2003).
Interestingly, indices described above can be integrated
into a more general family that will allow to propose a
theoretical framework for estimating biodiversity. The
methods used to get the data on which the indices are
calculated may be taken into account in the choice of these
indices. This would enhance their relevance as it will be
discussed briefly. Here is thus an example describing how
to determine theoretically the choice of an index. The example is based on a counting method for studying biodiversity,
based on the reassociation kinetics of single-stranded DNA.
This example is taken from the article by Haegeman et al.
(2008). All indices described in this section have been proposed in contexts other than microbial ecology. However, as
already mentioned, the methods of observation and study of
microbial diversity induce bias in the estimation of biodiversity through the previous indices. To summarize, each index
corresponds to a few facets of biodiversity (specific, structural,
functional, etc.), and its expression involves the number of
individuals of a species, which cannot be achieved in practice
by methods that skew their estimates. For example, the index of
specific richness does not distinguish the relative abundance of
species and includes all in the same way. But in microbial
ecology, there is a large number of rare species. Simpson or
Shannon indices seem therefore better suited to this context, in
that they attribute gradually the relative importance of the
different species according to their abundance. The few indices
mentioned above does not constitute an exhaustive list and can
be imbedded into the general family of diversity indices of
Re ´nyi (1961), defined by
R α ¼
1
1 À α
ln
X S
s¼1
p
α
s
!
ð8:4Þ
where α is a positive number and where R 1 must be
understood as the limit of R α as α tends toward 1. This results
in the following relationships between Re ´nyi indices and the
indices presented previously (specific richness S, Shannon
index H
0 , and Simpson index D):
R 0 ¼ ln S
ð Þ
R 1 ¼ H
0
R 2 ¼ ln D
ð Þ
This approach (now α can be any positive number, which
defines the family of indices) allows us to determine a priori
which index should be used when a method is chosen.
The principle is to try to get a relationship between the
experimental methodology and the biodiversity indices considered. This process is easier if the choice of indices is
large, which is the case with infinite number of indices
provided by the family of Re ´nyi.
Making assumptions about the distribution of specific
abundances, we can more easily treat the case of rare species. Indeed, a rare species may be poorly sampled, and
estimated abundances by molecular methods usually add
uncertainty about the value of this estimate. However,
estimates of diversity using indices depend heavily (sometimes several orders of magnitude) on the assumptions about
the distribution of specific abundances.
Regarding the usual methods of microbial enumeration,
they are based on the analysis of 16S rRNA and require
amplification methods or cloning. The RNA strands are
then amplified by PCR and analyzed by “fingerprint”-type
methods, for instance. Other types of counting methods are
also available, such as methods of reassociation kinetics of
single-stranded DNA.
A simple model, based on the mass action law, to represent the reassociation kinetics is:
dC
dt
¼ kC
2
s
ð8:5Þ
where C s is the concentration of single-stranded DNA
corresponding to species s and k is a nonspecific parameter
characterizing the reactivity of the DNA strands. The solution of this equation is
C s t
ð Þ ¼
C s 0
ð Þ
1 þ kC s 0
ð Þt
ð8:6Þ
Consider the total concentration of the single strands of
DNA:
C t
ð Þ ¼
X S
s¼1
C s t
ð Þ ¼
X S
s¼1
C s 0
ð Þ
1 þ kC s 0
ð Þt
ð8:7Þ
8 Biodiversity and Microbial Ecosystems Functioning
277
summarize quantitative parameters describing community
structure. As described above for the Shannon index, they
all have the same weakness, in that they do not take into
account the identity of the organisms in a community.
Because diversity indices compile information obtained from
a set of several species in a single value for each observation
before comparison, it is not surprising that complex changes in
patterns of diversity are not well converted into changes in these
indices. For example, Widmer and collaborators (2006) have not
observed significant changes in soil bacterial communities
subjected to various treatments using diversity indices, while
molecular analyses did show significant changes in bacterial
fingerprint, as well as in the structure of communities, through
multivariate analyses. Two strategies have been proposed to
ensure consistent results between diversity indices and multivariate analysis techniques: the strategy “CA-species-richness” suitable for data where rare species are important and the strategy
“nonsymmetric CA–Simpson” which is more appropriate for
data where a species dominates (Pelissier et al. 2003).
Interestingly, indices described above can be integrated
into a more general family that will allow to propose a
theoretical framework for estimating biodiversity. The
methods used to get the data on which the indices are
calculated may be taken into account in the choice of these
indices. This would enhance their relevance as it will be
discussed briefly. Here is thus an example describing how
to determine theoretically the choice of an index. The example is based on a counting method for studying biodiversity,
based on the reassociation kinetics of single-stranded DNA.
This example is taken from the article by Haegeman et al.
(2008). All indices described in this section have been proposed in contexts other than microbial ecology. However, as
already mentioned, the methods of observation and study of
microbial diversity induce bias in the estimation of biodiversity through the previous indices. To summarize, each index
corresponds to a few facets of biodiversity (specific, structural,
functional, etc.), and its expression involves the number of
individuals of a species, which cannot be achieved in practice
by methods that skew their estimates. For example, the index of
specific richness does not distinguish the relative abundance of
species and includes all in the same way. But in microbial
ecology, there is a large number of rare species. Simpson or
Shannon indices seem therefore better suited to this context, in
that they attribute gradually the relative importance of the
different species according to their abundance. The few indices
mentioned above does not constitute an exhaustive list and can
be imbedded into the general family of diversity indices of
Re ´nyi (1961), defined by
R α ¼
1
1 À α
ln
X S
s¼1
p
α
s
!
ð8:4Þ
where α is a positive number and where R 1 must be
understood as the limit of R α as α tends toward 1. This results
in the following relationships between Re ´nyi indices and the
indices presented previously (specific richness S, Shannon
index H
0 , and Simpson index D):
R 0 ¼ ln S
ð Þ
R 1 ¼ H
0
R 2 ¼ ln D
ð Þ
This approach (now α can be any positive number, which
defines the family of indices) allows us to determine a priori
which index should be used when a method is chosen.
The principle is to try to get a relationship between the
experimental methodology and the biodiversity indices considered. This process is easier if the choice of indices is
large, which is the case with infinite number of indices
provided by the family of Re ´nyi.
Making assumptions about the distribution of specific
abundances, we can more easily treat the case of rare species. Indeed, a rare species may be poorly sampled, and
estimated abundances by molecular methods usually add
uncertainty about the value of this estimate. However,
estimates of diversity using indices depend heavily (sometimes several orders of magnitude) on the assumptions about
the distribution of specific abundances.
Regarding the usual methods of microbial enumeration,
they are based on the analysis of 16S rRNA and require
amplification methods or cloning. The RNA strands are
then amplified by PCR and analyzed by “fingerprint”-type
methods, for instance. Other types of counting methods are
also available, such as methods of reassociation kinetics of
single-stranded DNA.
A simple model, based on the mass action law, to represent the reassociation kinetics is:
dC
dt
¼ kC
2
s
ð8:5Þ
where C s is the concentration of single-stranded DNA
corresponding to species s and k is a nonspecific parameter
characterizing the reactivity of the DNA strands. The solution of this equation is
C s t
ð Þ ¼
C s 0
ð Þ
1 þ kC s 0
ð Þt
ð8:6Þ
Consider the total concentration of the single strands of
DNA:
C t
ð Þ ¼
X S
s¼1
C s t
ð Þ ¼
X S
s¼1
C s 0
ð Þ
1 þ kC s 0
ð Þt
ð8:7Þ
8 Biodiversity and Microbial Ecosystems Functioning
277