Download Logical depth and biological complexity

L O GI CA L D EPT H AS A ME AS U RE O F
B I OL O GI CA L O RGA NI ZA TI O N A ND IT S
R EL AT I ON TO AL GO RI T HMI C C OMPL E XI T Y
J . D . C O LLIE R
Department of Philosophy, University of Newcastle
Callaghan, NSW 2308 Australia
Although the complexity of biological systems and subsystems like DNA and various
transcription and translation pathways is of interest in itself, organization is of fundamental
importance to understanding biological systems. It would be convenient to have a general
definition of organization applicable to biological systems. I propose that C.H. Bennett’s notion
of logical depth is a suitable candidate. I discuss the problems with using complexity measures
alone, and then the relations between logical depth and algorithmic complexity. Last, I give some
examples in which depth gives a better measure of what might naively be taken to be complexity
in biological systems by many biologists,,and then argue that this must be augmented by
consideration of dynamical processes.
1. Algorithmic Complexity vs. Logical Depth
1.1 Biological Complexity
When biologists refer to “complexity”, they generally mean the nature
of interactions within biological systems rather than simply the measure
of information required to specify the system, such as Kolmogorov
complexity, or the complexity as determined by MML or MDL
methods.1,2,3 Biological interactions are characterized by their
functionality, which requires feedback and, in more advanced systems,
anticipation. Thus spatial and temporal closure of biological processes,
and overall interconnectedness imply organization. Function itself is
often characterized as a consequence of adaptation,4,5 but it is better
characterized by its role in maintaining system autonomy, itself an
organizational trait.6 A system is (relatively) autonomous if its functional
and interaction closure works so as to actively maintain system integrity
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against internal and external changes. Biological complexity is thus
organized complexity, and we need some measure of organization.
Complexity methods start with an isomorphic mapping of features
onto a string that is then amenable to MML or MDL methods of
compression to determine overall complexity. A simple case in which
the resulting measure might be misleading is found in eukaryotes, in
which the “reading frame”, a segment of DNA to be transcribed to
mRNA is controlled by transcription factors that can be influenced by
distant regulatory regions. The complexity of the reading frame gives
information about the complexity of the mRNA transcript, but the
regulatory function is not purely local, unlike in prokaryotes, so the
complexity of the local regulatory factors will give a misleading estimate
of the complexity of the regulatory factors. The organization as well as
the complexity of the genome is relevant to understanding gene
regulation. A further complexity is the editing phase of mRNA, in which
introns are deleted and the mRNA is modified to produce the final
functional mRNA transcript. The introns are typically random, or at least
chancy, and so increase the complexity of the reading frame of the DNA
without contributing to function. Furthermore, in many cases polypeptide
chains formed from more than one frame, often from disparate parts of
the genome, fold together to form functional proteins, again leading to
a divergence between local complexity measures of DNA and their
mRNA transcripts and the complexity of functional proteins. Each
element can be studied independently by complexity measures, but the
overall organization is likely to be overlooked by these methods alone.
I will give some further examples later. I should note now, though, that
translation itself is not a simple process, either in the complexity or
organizational sense. The connection to phenotypic traits is even more
complicated in most cases, with the notable exception of some genetic
defects. As I will show later, however, even genetic defects are not
organizationally simple in at least some cases. The relevant complexity
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for biological systems is organised complexity, not complexity
simplicitur.
1.2 Logical Depth
Logical depth was developed by C.H. Bennett, who hypothesized that it
was a suitable measure of organization.7 Formally, logical depth is a
measure of the minimal computation time (in number of computational
steps) required to compute an uncompressed string from its maximally
compressed form. Some adjustments are required to the definition to get
a reasonable value of depth for finite strings. We want to rule out cases
in which the most compressed program to produce a string is slow, but
a slightly longer program can produce the string much more quickly. To
accommodate this problem, the depth is defined relative to a significance
level s, so that the depth of a string at significance level s is the time
required to compute the string by a program no more than s bits longer
than the minimal program. Physically, the logical depth of a system
places a lower limit on how quickly the system can form from
disassembled resources.
Bennett has proposed that logical depth is a suitable measure of the
organization in a system. However, while adding more components to a
system at the will not increase the system organization, only the size of
the system organised, it will increase its depth because the sheer length
of the sequence to be computed has increased. All sequences of n
identical entries are intuitively equally trivial, however the depth of each
string depends on the depth of n itself. This effect can be made
negligible if we consider only relative depth: The depth of a sequence
relative to the depth of the length of the sequence. The relative depth
itself of a sequence of n identical entries is no more than the depth
required to specify the entry itself (and negligible if the entry is 0 or 1).
In the case of adding identical components to a system the relative depth
does not increase since the depth of a component is already included in
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the original system relative depth. It is not transparent whether relative
depth deals satisfactorily with all possible cases of this kind, but it is a
reasonable, and plausibly sufficient, refinement of logical depth
simplicitur to adopt.
1.3 Relation of Logical Depth to Complexity
The first thing to note is that logical depth implies redundancy. A
maximally complex string is not compressible, and necessarily has a
computation time equivalent to its uncompressed form. Secondly, logical
depth cannot be produced by low order redundancy, where redundancy
order is ½ the minimal length required to detect the redundancy. Simple
repetition, for example, requires only a relatively short computation time
and has low relative logical depth. On the contrary, relative logical depth
requires higher order redundancy, and relatively greater depth requires
relatively higher order redundancy.
MML and MDL methods are well adapted to detecting redundancies
of various orders.2 It would be convenient if relative logical depth was
implied by higher order redundancy, but this is not generally the case.
For example, a sequence of incompressible strings of length n will have
redundancy order n, but will not require a long computation time from
the compressed form of the complete string. Therefore, higher order
redundancy can be taken as a sign of possible organization, worth further
investigation. This example, however, implies that (relative) logically
deep strings will show redundancy of both higher and lower orders, the
discovery of which would be further evidence for organization.
Unfortunately, there has been little work on minimal computation time
for computing the uncompressed form from the compressed form of a
string. Most of the available work deals with the relative computational
difficulty of classes of problems rather than for particular problems.
Nonetheless, this work can give some indication of the class of
organization involved in a particular problem.
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Relative logical depth promises to capture the core of the intuitive
sense of organization discussed in the first part of this section. Crystals
are much easier to produce than are computers and, within the latter,
otherwise comparable memory chips are much easier to produce than
CPUs, which are much more complicatedly organized (as well as being
more complex in the algorithmic sense). However I do not know of a
theorem that requires a relation between ease of production and depth.
I will assume that higher redundancy order computationally associated
with lower order redundancy indicates higher logical depth, other things
being equal. Whether or not organization requires anything else is
somewhat unclear at present. It is, for example, unclear to me whether
relative logical depth can be used to satisfactorily distinguish the kind of
deep organization exhibited by a computer running a complex coherent
program and that exhibited by a living cell, which is autonomous,
defends an internal/external phase separation through a controlling
boundary membrane, and is self-reproducing, all organizational features
the computer lacks.
A further complication is that biological systems are organised
hierarchically through causal closure conditions (a paradigmatic example
is the cohesion of species as well as their members, but similar
considerations apply to cells and the organisms they comprise).8
Consider two systems constituted of the same basic components and
with equal depth, the first a single level system with subtly correlated
parts so that they mutually regulate and some control others, and the
second a multi-level hierarchy with internally simple levels but crosslevel correlations, none of which are control relations. If the notion of
organization is fundamentally concerned with just the extent of coordination in the sense of depth, then ex hypothesi these two systems
would have the same measure of organization. If, on the other hand, the
notion of organization also includes reference to the degree to which
correlations are hierarchically organised by levels, as it will for many,
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then, the second system would have the greater organization. Secondly,
there is the question of whether ordering of regulation or control should
play any intrinsic role in the concept of organization. If so, then the first
system will be judged to possess more organization than the second.
Finally, these two intuitions are often joined to emphasize the
importance of hierarchical order of regulation or control to organization.
In this case a system with coherent interactions at a high level used to
control lower levels will be judged to possess more organization than the
first, since the first system can show only first order hierarchical
regulation or control while the second system can show higher order
hierarchical regulation and/or control. Perhaps we need to introduce
concepts of ‘hierarchical organization’, ‘regulatory (control)
organization’ and ‘hierarchical regulatory (control) organization’ in
addition to organization simplicitur. Further development of this would
take me too far afield, but it is worth noting that the nature of the
interactions in a system rather than its logical structure alone is of some
importance to understanding and classifying the system as “organized”.
2. Some Examples of Organization that Seem to Defy Complexity
Analysis
Molecular genes are segments of DNA that can be transcribed into
functional mRNA or translated into polypeptides that are either
functional themselves or can be enfolded with other polypeptides to form
functional proteins. In prokaryotes, the regulatory DNA is adjacent to the
transcribed DNA, and the whole can be regarded as a functional
molecular gene. As described above, however, in eukaryotes regulatory
factors can arise at some distance from the transcribed “reading frame”;
the exact mechanism is at present unknown. In order to define a
functional molecular gene for eukaryotes, then, a more global
perspective is required to determine the computational relations that
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underlie the gene. It should be noted that replicators need not include
complete functional genes, nor need they contain only complete
functional genes. This means that even for molecular reductionists the
unit of replication (inheritance) is not a functional gene. In order to
understand inherited traits, it is necessary to understand how the DNA
interacts with itself and the cell metabolism to produce phenotypic traits.
This is a problem of organization, not of complexity. The information
transferred is organizational information, not the information of
complexity theory.
This suggests that the unit of inheritance is a trait, not a molecular
gene. It is well known that the fitness of Mendelian (or evolutionary)
alleles is highly context dependent. E.O. Wilson describes a trait as
genetically determined if it makes a difference in some environment.9
For example, eye color is genetically determined because there are
environments in which genetic differences make a difference to eye
color, even though the environment and other factors can affect eye
color. This definition of genetic determination has limited interest to
molecular geneticists, since a genetically determined trait need not
depend on a single stretch of DNA, and may have a large environmental
influence. However it is of interest to evolutionary biologists because
genes are selected because of the phenotypic traits they express.
Typically, for complex traits, a number of segments of DNA, perhaps not
even on the same chromosome, may be involved. Wilson’s definition of
genetic determinism is not entirely satisfactory even for his own
purposes (not only because it diverges from popular conceptions of
genetic determinism, however useful it is for evolutionary biology), since
it allows any environment. It would be better to restrict the environments
to the evolutionarily effective ones, and to consider a norm of reaction
across these environments to determine the strength of genetic
determination for a particular trait across a range of environments. It may
well turn out that for a particular trait the genetic determination
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(difference attributable to genetic causes) is low in one environment and
high in another. Intelligence (or more operationally, IQ), for example,
may be such a trait.
As I mentioned above, genetic defects are more likely to be traceable
to individual segments of DNA, but even this is not a simple matter.
Consider for example the evolutionary gene for Myotonic Dystrophy.
The molecular difference between sufferers of this genetic disease is that
they possess any one of a number of sequences of between 50 and 200
repeats, whereas normal people in this respect have any of several
sequences of between 5 and 27 repeats. In other words, there is a
threshold effect beyond which the translated proteins are sufficient in
number to cause the disease. This cannot be determined by genetics
alone, at least for the local regions, and requires understanding the
metabolic processes. A full genetic understanding requires considering
the genetic basis for the organization of these processes as well as the
specific threshold level. This likely involves non-additive (non-linear)
interactions among a number of molecular genes through their products.
This organization should show up in higher order redundancies in the
genome involving the relevant repeated sequences, but whether or not
this would be a productive line of research is questionable.
Many normal traits in most species show considerable variability
across a species. The evolutionary definition of genetic determination
cannot be isolated from environmental interaction; as noted above, there
is a norm of reaction. This is in line with the requirement of interaction
closure for autonomy and the possibility of functionality. This presents
further problems for complexity analysis, since environmental
interactions must be taken into consideration. It is possible to compare
the complexity of the environment with the complexity of the functional
parts of the genome to get some idea of the complexity of adaptation.10
Unfortunately, this correlation tells only a small part of the story. Again,
a major factor is the organization of environmental interaction and
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adaptive strategies, which again involve logical depth. A deep strategy
is not necessarily more flexible, though it will involve higher order
redundancy. It may represent a deeply entrenched and complicated
relationship between an organism and a particular environment to which
it has specialized. If the depth is associated with higher cohesive levels,
however, it is more likely that the depth is associated with adaptability
and less specialization. Similar considerations apply to ecological
resiliency. Mutual complexity tells us something about degree of
adaptation, but depth can tell us more, and the type of depth can tell us
about the kind of adaptive strategy in use. Similarly, ecological
complexity, especially the sum of mutual complexity of interactions, can
tell us a good deal about the nature of an ecological system, but this may
indicate a brittle ecological community or one with considerable
resiliency, depending on how the interactions are organized.11 In this
case too great a depth may indicate an ecological community that has
lost its resilience, much as a specialized adaptation is not an effective
strategy in a varying environment. Organization as well as complexity
is required to understand such systems, but as pointed out in the previous
section, the type of organization is also relevant. Complexity and
organization studies can place limits on the possibilities for biological
theories of specific systems, but the dynamical character of the system
processes is required for full understanding.
3. Conclusions
Biological complexity is organized complexity. Even relatively basic and
supposedly simply biological subsystems like DNA, when regarded as
functional, show complex organization. To some extent biological
organization can be understood in terms of logical depth. Depth, in turn,
can be recognized, but not reliably, through higher order redundancy
bound together with lower order redundancies. Algorithmic complexity
places limits on possible biological processes, and the logical depth of
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a biological system or subsystem places further limits. However, even
these constraints leave open a wide range of strategies. Full
understanding requires knowing the dynamical character of the processes
of the system.
Acknowledgments
I would like to thank Professor C.A. Hooker, who supported this work
by granting me a Research Associateship under his Australian Research
Council Large Grant. Several of the ideas in this paper were developed
together with him and Wayne Christensen. Several of the examples I
used were derived from a talk by Paul Griffiths based on his unpublished
work.
References
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