From: Pedro C. Marijuan <[email protected]>

Date: Thu 26 Feb 1998 - 09:38:00 CET

Date: Thu 26 Feb 1998 - 09:38:00 CET

THE LANGUAGE OF CELLS

Pedro C. Marijuan & Josi Pastor

Department of Electronics and Communications Engineering

Universidad de Zaragoza, Zaragoza 50015, Spain

1. Introduction: Positional versus Compositional Information

The notion of a "language" of cells does not look consistent in relation

with the standard views of Information Theory applied to biology. Although

Shannon and Wiener (1948) distinguished between discrete, continuous, and

mixed information sources, the standard application of their ideas to the

biology of the cell has been heavily influenced by the DNA and RNA

sequential structure, and only the discrete-positional case has been

traditionally considered (e.g., Gatlin, 1972). As a consequence, the lack

of distinction between "positional" and "compositional" forms of

information and the subsequent neglect of the latter--we are going to argue

here--have implied an analytical dead-end concerning the possibilities of

elucidating formal mechanisms of cellular languages.

In biology and in human to human communication, the assumed preconditions

for information transmission, and for any workable language, refer to

sequences of messages containing combinations of symbols which will be

accepted (or emitted, or read, or transmitted, or deciphered) always

following a positional order. Therefore, Shannon's formula appears as the

natural way of measuring the average combinatory content of these messages,

and of establishing their relative index of surprise, in order to design

appropriate channels, codes, etc. A workable language can be created,

subsequently, as a series of grammatical (Markovian) rules to abide by when

connecting successive positional messages comprised within the dictionary

scope of the language.

However, one can point at a number of instances in natural and social

communication where symbols are used in a rather different way. Instead of

a "positional" context, a "compositional" one is the case. In this

alternative context, messages are exchanged as presences or absences of

symbols which have been accumulated upon predetermined sets of objects. No

meaningful positional relationships among the objects within the set or

among the symbols accumulated upon the objects are assumed. For example,

several glasses on a tray may contain a variable number of different

symbolic items (ice cubes, soda, vermouth, olives, cherries). The set of

glasses on the tray become the message, each glass being an individual

object that accumulates several symbols which precisely configure it as a

distinguishable object. Communication among two subjects could be

established by the exchange of trays, with variable number of glasses and

variable contents within them. Theoretically, that messages might be

reliably distinguished and transmitted by this method "of concurrent

processing of discrete states of media" has been postulated by Karl

Javorszky (1996). A whole body of partitional calculus (or granularity

algebra) has been envisioned by this author (Javorszky, 1995).

More reasonable examples of the compositional way of information exchanges

may be found in the communicational use of colors, odors and tastes by

individual organisms; also in social insects' pheromones; and anecdotally

in the etiquette "language of flowers"; and perhaps within musical

compositions and within the formative frequencies of vowels and consonants

of our own spoken languages as well. The "language of cells", we will

discuss here, may be one of the most interesting instances of communication

by means of such compositional tools; and it has been the forerunner of any

further means of biological communication.

Marshall Mc Luhan's famous dictum "the medium is the message" and the

particular disdain this author showed about Shannon's information theory

(Mc Luhan, 1968) are worth be remembered when considering this fundamental

distinction between positional and compositional forms of information

exchange.

2. Analyzing a Compositional Message

When receiving a compositional message encrusted upon a set of N elements,

there is very little a subject can do: just counting the presence of the

different symbols on each element of the set. Hence the elements can be

grouped in homogeneous classes of overlapping or non-overlapping nature

(each class is defined by the presence of a specific symbol, and it

actually demarcates a partition of the set N). After the classes defined by

single symbols, the more complex coincidences of combinations of symbols

(class overlaps) among the elements can be counted. It can be easily proved

that all the possible countings of symbolic presences among the N elements

of the set, in the first case of linear or one-dimensional partitions for

single symbols, conduce to the whole set of partitions of N, known as E(N)

(Javorszky, 1995). Whereas the coincidences or overlaps of successive

combinations of two, three, four symbols, etc., can be considered as second

order partitions, third order, fourth order ones, etc.--E2(N), E3(N),

E4(N)...

Mathematically, partitions are a very straight concept: the additive

decompositions of natural numbers. For instance, the set { (5), (4,1),

(3,2), (3,1,1), (2,2,1), (2,1,1,1), (1,1,1,1,1) } represents the whole

onedimensional partitions of 5. By adhering to this mathematical treatment,

one can use the well-known partitional properties of numbers to discuss the

most probable logical states of a compositional message encrusted upon the

elements of the set N.

The set of partitions E(N) can be transformed right away into a

probability body. The probability of any state of the set to exist as is

described by a specific partition is given by the relative frequency of

this partition among all partitions. For instance, on E(5) the probability

is

1/7 - for states (5), (2,1,1,1) and (1,1,1,1,1) each,

2/7 - for states with either 2 or 3 summands each,

15/20 - for any summand to be an odd number.

Kmax is that number (1..n < N) which generates the most numerous set of

partitions of N into k summands. In this most probable partitional state,

the set shows Kmax distinct summands with respect to a one-describing

dimension. In the case of E (5), there is a Kmax shared both by 2 and 3.

Heuristically, it appears that a compositional message can be univocally

described by its corresponding "trace" of unidimensional partitions

(Javorszky, 1996), if a few additional statistical measures that act as a

sort of context or shared background in the communicational process have

been previously established: most probable message length, ratio of

symbols/elements, structural depth, shallowness, etc. Then the use of

partitions of further order (second, third, etc.) becomes redundant--and

its inclusion would notably complicate the mathematical description of the

message.

Additionally, the Kmax. dimension of every property or symbolic presence

may be used as an origin or natural cannon to which the respective

deviations of successive messages can refer. This further simplifies the

partitional "trace" describing a specific message in the context of an

ongoing communication process.

Karl Javorszky (1996) has argued that an efficient communication procedure,

massively parallel, can be built around such minimized partitional traces

or message simplifications. It seems to work particularly well with data

sets of a moderate size, which are preferably prestructured and come in a

quasi continuous stream, so that the number of possible symbols is always

kept rather finite--although symbols might come from an infinite multitude,

there should be a relatively small collection of distinguishing items

employed at the communicational session, and their group relations should

not generate a cardinality overstatement symbols/elements above a certain

limit.

In the extent that Javorszky4s estimates are correct, the overall capacity

of a compositional channel making use of discrete states of media can be

generically expressed as

T(N) = E(N) exp ln E(N),

understanding T(N) as the number of different logical states which can be

distinguished by means of collections of symbols put on the elements of the

set N -only non-redundant states are counted, for redundant symbol groups

can always substitute by single symbols, coalescing into a unique logical

state. E (N) is the already mentioned number of partitions of the set N.

Calculating the respective probabilities and taking logarithms, one could

obtain an expression in bits for the compositional-channel capacity (or the

entropy of a compositional information-source) somehow paralleling the

famous Shannonian entropy.

It is also interesting the comparison between T(N) and the strictly

positional use of the same elements of the set N in a combinatory way

(which, in principle, should provide a total of N! different messages or

logical states). In this comparison, T(N) yields a larger number of logical

states than N! for values of N in between 31 and 95, with a maximum around

63-64. However, for N = 12, the number of combinations N! reaches a maximum

with respect to T(N).

Seemingly, several parameters of the genetic code would correspond with

such max./min. extremes that characterize the compositional-positional

interrelationship (see Javorszky, 1995, for detailed expression of all

these formulae and calculations).

3. A Partitional Approach to Cellular Communication

How can the above compositional considerations be applied to the analysis

of inter- and intra-cellular processes? Instead of DNA sequences, it seems

that the natural target of this new approach should be the "mysterious"

processing operations performed by the cellular signaling system.

The basic idea to play with is that any array (sample) of chemical

compositions detected by the set of cellular receptors may be recast as a

partitional state, for we may consider that it has been obtained by

distinguishing the presence of a series of specific chemical signs within

an hypothetical set of N elements or receptor complexes--it is a transient

message coming from the environment through an ongoing compositional

communication process.

Then, an immediate question arises. Could the system of receptors,

membrane-bound enzyme and protein complexes, second messengers, and the

dedicated kinase and phosphatase chains, be understood as an abstract

partitional processing-system capable of extracting the minimized "trace"

or relative information differences within the stream of incoming

compositional messages and physically transport these differences down to

final effectors at the nucleus, cytoplasm, or membrane? That's the basic

hypothesis these two authors are presently trying to explore.

If (and what a big "if") cells would make use of formal tools of

partitional nature in their management by means of the cellular signaling

system of the compositional messages they receive, then the notion of a

genuine cellular language, with specific dialects for every organismic

tissue, could be seriously argumented. And perhaps more interesting than

that, quite a few other bizarre aspects of the signaling system could

receive some more formal (and simpler) treatment: the cross-talk between

signaling pathways, the checkpoints relating signaling operations with

cell-cycle stages, and even the widespread formation of aggregates and

complexes among signaling components...

Partitions are very direct formal tools, but at the same time sophisticated

ones. For instance, if natural numbers are left to "oscillate" and

simultaneously their corresponding partitions are allowed to grow and

shrink (Javorszky, 1998), then there emerges a variety of self-organization

patterns and internally-driven numerical processes which seemingly parallel

the constructive/degradative aspects so prevalent in biological and social

occurrences (and even physical ones).

The studies by Caianiello (1987) on the partitional dynamics inherent of

monetary systems and the suggestion by one of us (Marijuan, 1998) about the

"currency" role actually played by the set of second messengers in the

internal measurement of cellular function might finally be stepping stones

pointing out in the same direction.

----------------

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Pedro C. Marijuan --FAX 34 976 761 861 --TEL 34 976 761 927

Dto. Ingenieria Electronica y Comunicaciones

CPS, Universidad de Zaragoza

Zaragoza 50015, SPAIN

Received on Thu Feb 26 10:00:31 1998

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