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In general, the notion of multiset can be considered at any level, because multisets
of second level can be considered as objects which can be aggregated by providing
multisets of third level, and the same kind of construction can be iterated. We re-
serve the word
population
for denoting multisets of the first level, while the term
hypermultisets
is used for multisets at a level greater than one. Therefore, a
pop-
ulation
over a set
A
is a multiset of elements which belong to
A
, and a multiset of
level
k
0, over a set
A
, is a multiset of elements which are multisets of levels lower
than
k
(over
A
).
Populations are ubiquitous in life phenomena, at any level, from populations of
molecules, to populations of organisms and species. However, the peculiarity of
membranes is that they provide a biochemical counterpart of parentheses of mathe-
matical constructs. In fact, when molecules are put inside membranes, these become
elements of other aggregation levels, where they are elements (possibly occurring
in many copies) of a second level multiset. We note the following statement empha-
sizing the structural role of membranes.
>
Membranes constitute the biochemical realization of hierarchical multiset ag-
gregation of biological components.
If we consider an atom as a multiset of subatomic particles, that is, protons, neu-
trons, and electrons, then a molecule such as
CO
2
is a multiset of two levels:
CO
2
=
C
+
2
O
=(
6
·
(
p
+
n
+
e
))+
2
(
8
·
(
p
+
n
+
e
))
.
Analogously to hypermultisets, we can consider hypersets and hypersequences
(however, this is not standard terminology). A
hyperset
is a set including sets as
its elements. For example,
which is the set
of elements
a
and
b
.A
hypersequence
is a sequence of sequences. For example,
(
{
a
,{
a
,
b
}}
includes as element
{
a
,
b
}
is a hypersequence. Any sequence can be represented by a hy-
persequence constituted by pairs (sequences of length 2). For example
a
,
(
a
,
b
)
,
(
a
,
(
a
,
b
)))
(
a
,
b
,
c
)
can
be represented by
. The notion of level for hypersets and hypersequences
can be defined as in the case of hypermultisets. It is easy to realize that hyperstruc-
tures are expressed, up to here, using parentheses. For example, a hypersequence of
level two has two levels of parentheses. This concept is the basis of hierarchies, and
trees are the mathematical concept behind any notion of hierarchy, as we show in
the next section.
Before considering trees and graphs in basic structures of life, let us conclude
regarding the intertwined roles of membranes and polymers. The basic mathemati-
cal structure underlying molecules and reactions are multisets. Efficient realizations
of chemical reactions require complex molecules facilitating and driving them (en-
zymes), and a compartmentalization of molecules in order to select, concentrate, and
protect all the elements involved in reactions. Membranes provide the biological so-
lution to compartmentalization, and polymers, which are sequences of monomers,
provide the enormous molecular variety and complexity within which molecular
((
a
,
b
)
,
c
)
