Replicating Robert Rosen's (M,R) Systems
Ron Cottam, Willy Ranson & Roger Vounckx
Abstract
Much of Robert Rosen’s professional work targeted the
development of relational biology and the way in which ‘efficient cause’ could
be internalized in an organism. His book ‘Life Itself’ [1] focuses on precisely
this aspect of living systems. Unfortunately, editorial errors in the book lead
to a degree of confusion, most particularly in the case of Figure 10C.5, where
Rosen’s two different relational arrows have been interchanged. Careful
assessment also prompts a number of questions as to the validity and
comprehensiveness of the book’s arguments.
Aloisius Louie [2] has correctly pointed out that there is a degree of
inconsistency in the notation which is used in ‘Life Itself’, but if we replace
its ‘morphism: domain→codomain’ mapping by his ‘element-chasing’ version [2,
Figure (11)] we apparently end up with an ‘organism’ which depends on no more
than one gene [3, p.265]: the majority of life’s complexity, to some extent
hinted at by Rosen’s notation, disappears in the formalization. Recent research
indicates that complete knowledge of the human genome is insufficient to
determine human fabrication [4, 5]. Is the mathematical mapping of sets, with
its restriction to one-to-one and many-to-one relationships, sufficient to
describe biological processes, most particularly those relating to the
complexity of genetic networks, to protein gene-switching, and to non-protein
RNA-based catalysis?
Our own studies of natural hierarchy are persuasive that both organisms and
rationality itself constitute hyperscalar systems, and that they always operate
within birational frameworks of ‘entity and ecosystem’. How does this relate to
Rosen’s scheme of internalized ‘efficient cause’, which is apparently
mono-scalar and mono-rational? Rosen intentionally eliminates environmental
influences from his model: is this feasible in a multi-scalar system, let alone
a birational one? His relational description in terms of “entailment without
states” rejects the implications of specific embodiment, whereas a birational
hierarchy depends on the ‘assimilation’ of both interrelations and embodiment.
We accept that Rosen’s relational model has provided a useful stepping stone to
understanding the nature of life, but also suggest that it induces potentially
digressive conclusions. Rosen indicates [e.g. in 6] that in a specific manner
metabolism, repair and replication are interchangeable. Is this conjecture valid
outside the confines of his formal mathematics?
This paper presents conclusions drawn from a comparison between Rosen’s
relational model of an organism and that of a birational complementary natural
hierarchy. Rosen’s model is ‘replicated’ in a number of different ways which
lend credence to the argument that birationality sheds new light on the nature
of life and the usefulness of his accomplishments.
Rosen’s scheme of a sequence of relational arrows can be ‘replicated’ as a
nested abstract association between the hand-written code for a computer
program, its compiler, and a resulting program. Both yield the same
graph-theoretic description. It is noteworthy that the relevant characterization
of an organism is in terms of three components – metabolism, repair and
replication. Further ‘replication’ of Rosen’s scheme is as a circulation, in a
triangular loop of three independent components, where on each circuit the flow
is boosted by a kick from the environment; a two-component system of this kind
would necessarily fall victim to dissipation. Rosen’s final well-known Figure
10C.6 appears in ‘Life Itself’ as an asymmetric diagram. When ‘replicated’
centro-symmetrically it looks quite different; the two outlying sides of a
‘figure-of-eight’ (∞) circulation are functional complements of each other; the
central region ‘assimilates’ both their outcomes.
In a natural hierarchy we cannot successfully fractionate {functor and function}
– it consequently makes no sense to talk about f, b/B, or φ/Ф in isolation, and
we must look at Rosen’s solid- and hollow-headed arrows as related pairs, as
Rosen himself recognized – as ‘functors/functions’ or ‘operators/operations’.
Recognition in general of a single object implies the existence of not two but
three separate domains: the object, its ecosystemic environment and their
interface: the bifurcating categorization of nature into the complement of
mechanism and organism is insufficient. It is important in this context to note
that a mechanism can ‘contain’ an organism, but more to the point that an
organism can ‘contain’ a mechanism [e.g. 1, Figure 10C.6]. An organism is not
‘the complement of a mechanism’: the complement of a mechanism is its ecosystem:
an organism is the ‘complex interface’ between mechanism and ecosystem.
In our ‘figure-of-eight’ replication of Rosen’s scheme, metabolism is now the
outlying f→a→b, repair is the outlying f→φ→b, and replication is now the central
assimilation of both their outcomes b→f. Operator/operation f→a→b falls into
Rosen’s category of mechanisms, where solid-headed arrow f→a is the induction of
software flow a→b. Operator/operation f→φ→b, however, is very different: it is
the opposite or complement of a mechanism: hollow-headed arrow f→φ is the
induction of hardware flow φ→b. Operator/operation b→f is the intimate
association of induction of software flow, induction of hardware flow, and both
software and hardware flows themselves! This intimate four-fold association of
birational causes and effects only exists in an organism. An organism is an
intimate (complementary) coupling between a mechanism and its ecosystem.
Rosen’s model ‘does what he wanted it to do’; it (almost) ‘internalizes
efficient cause’. But it is also hyperscalar, because everything is! His
notational mixing of sets & elements is in fact superficially useful, in that it
maintains at least an impression of implicit complexity. The next most useful
step in validating Rosen’s work will be to reformulate the multi-scalar
generality of a hyperscalar natural hierarchy in terms of notional mappings, to
see what happens when the formally-mathematical relationships are extended to
include not only one-to-one and many-to-one relationships but also the
one-to-many relationships which Rosen excluded. In the absence of mutation,
one-to-one and many-to-one mappings preclude evolution: one-to-many does not.
Will this crude ‘injection’ of Rosennean-style complexity into a
notionally-mapped self-correlating natural hierarchy convert a merely
complicated mechanistic network into an organism?
Keywords: Rosen; life; organisms; birationality; hyperscale
References
[1] Rosen, R. Life Itself. New York: Columbia UP, 1992.
[2] Louie, A.H. “A Series of Unfortunate Misprints.” July 2005; http://www.panmere.com/rosen/Louie_LI_typos.pdf
(23.02.2006).
[3] Wolkenhauer, O. “Systems Biology: the Reincarnation of Systems Theory
Applied in Biology?” Briefings in Bioinformatics 2(3), 258-270, 2001.
[4] Mattick, J.S. “Challenging the Dogma: the Hidden Layer of Non-Protein-Coding
RNAs in Complex Organisms.” BioEssays 25, 930-939, 2003.
[5] Gravely, B.R. “Alternative Splicing: Increasing Diversity in the Proteomic
World.” Trends in Genetics 17(2), 100-107, 2001.
[6] Rosen, R. “Some Relational Cell Models: the Metabolism-Repair Systems.” In
Rosen, R. (ed.) Foundations of Mathematical Biology II, p. 236. London: Academic
Press, 1972.
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