The tree and the ring. Hierarchical and acentered structures in biology
The obscurantist movement against understanding what life is, promoting the funny idea of an "intelligent design" claims that macroevolution is more than unlikely. The present text shows that the concept of closure has the remarkable property to create, from a very simple event, a completely new organisation. This has been known in Eastern civilisations from antiquity. This is well known in mathematics as a surface which closes on itself defines suddenly an inside and an outside. Many more volumes, such as a torus, form in this way, after minute changes.
While creation of information is fairly straightforward in the data/program section of a Turing Machine, it has not been generally explored in the machine part of the Turing Machine. I show here that closure creates a remarkable, and sudden, change in the global information content of an entity which involves compartmentalisation, as expected to have happened during the reproduction time of the surface metabolism that developed at the origin of life.
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|Theory and Hypothesis||
IN GENERAL, living beings are perceived as complex hierarchies. This is the more so as one goes from unicellular organisms to multicellular entities, especially animals, where a series of levels organised by inclusion can easily be defined:
animal < organ < cell < (macro)molecule < atom
It is also implicitly assumed that such hierarchical organisation is the only means to achieve the overall behaviour of the complex individual. My purpose in the following lines is not to go against this generally accepted (and most often true) assumption, but to try and challenge its universality. I shall show that, in specific cases, an alternative global structure, be it static or dynamic, can be achieved in biological systems by means that are entirely foreign to hierarchies. It seems to me that it might be of interest to consider cancer processes in this new light.
Because of the abstractness (contrary to what is usually thought of biological concepts, I shall endeavour to give definitions, as simple and limited as possible, of a few recurrent words (and concepts)). A system is a collection of material objects (which can be isolated as such: cells or molecules for instance) that are linked by relationships.
Such relationships can be synchronous (i.e. they exist at a given time, and, for instance may persist after death such as the system of bones constituting a skeleton), or diachronous (i.e. they are brought up to date through time, such as the firing pattern of a neuronal network). The main object of biology, contrary to what is often perceived (mainly by those who are ignorant of its goals and achievements), is not the study of objects (although they are necessary prerequisites) but the study of the synchronous and diachronous relationships which link them. In this respect, questions of control or regulation are central issues for modern biology (*). Here, an interaction is a specific instance of relationship, when two objects are in physical contact with each other.
After having given a sketchy picture of hierarchical systems, I shall show that there might also exist accentered systems. which may be important in the formation of global biological entities. The consequences of the existence of the latter class of systems, if they indeed exist, are by no means obvious. It is the purpose of this article to prime a reassessment of old biological problems with this new view, even if this will lead to its rejection.
In his description of animal physiology, Claude Bernard, after having established the proper means to investigate the nature of inanimate matter organisation, at the empirical level, discusses the specific features of life:
“Il est très vrai que la vie n'introduit absolument aucune différence dans la méthode scientifique expérimentale qui doit être appliquée à l'étude des phénomènes physiologiques et que, sous ce rapport, les sciences physiologiques et les sciences physico-chimiques reposent exactement sur les mêmes principes d'investigation. Mais cependant il faut reconnaître que le déterminisme dans les phénomènes de la vie est non seulement un déterminisme très complexe mais que c'est en même temps un déterminisme qui est harmonieusement hiérarchisé”
Claude Bernard thus emphasizes two rules, specific to life: one is perceived as linked to aesthetical principles, harmony, and the second one is derived from organisation principles, characteristic to life, hierarchy. Indeed, the organisation as different, interwoven, levels is a feature that strikes all those who study living material. There are individual (partially,) autonomous systems that can be divided into subunits, having similar autonomy like Russian dolls:
organism < organs < tissues < cells < macromolecule, <...
This is what is immediately perceived when considering hierarchies. Furthermore, of course, there are many refinements of this order, which can be universalised for instance by adding entities such as societies, which would include organisms. However, in biological systems hierarchies are much richer than sets of inclusions: the organs which constitute the organism are not equally placed but finely interwoven. In addition there are links not only between subsystems at a given level, but between subsystems present at different levels.
Many questions may be posed about such organisations. Why and how do subunits aggregate (we shall see below that the answer is not necessarily a hierarchical structure)? Are they trying to satisfy some optimisation principle? It is clear that many emergent properties stem front hierarchies, and I shall emphasize one of them: stability. It it necessary however to remark at this point that the links which are established between subsystems (at a given level or between different levels are not entirely abstract relationships but are the consequence of the paterial nature of the elements constituting each level).
The adequacy of a prosthesis, for instance, comes from its ability to substitute for the organ at a precise level, by simulating in an appropriate way the surface, the contacts, with the other subsystems with which it interacts. In a way the shape of the missing element is fixed by the general shape of all the elements that interact with it. This kind of complementarity permits specification of local structures, by the whole of the system.
Before trying to delineate what complementarity means in hierarchies, we shall divide such systems into two different types, both of which are very often found in living systems. The first type corresponds to organisation of aggregation properties. The whole system is an aggregate of subsystems, linked for instance by physical contact, but in which some substance or process, superimposed by the higher level, gives self-consistency to the whole. The second type corresponds to branched structures where elements present at each level control elements present in sublevels, as in a tree structure, typical of the organisation of neuronal structures. A ribosome made of two subunits, each made of a variety of proteins and RNAs, is an instance of a hierarchy organised by aggregation, whereas a neuronal network is typical of a control hierarchy. In general hierarchical structures are a mixture of both types.
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But they have a common property related to their stability, as we shall now see.
A vault is in architecture a hierarchical system where stones are subsystems. Because of gravity the function of the different stones are different, and they may be considered as forming different levels of the hierarchy. In particular the keystone (as its name indicates) has a special function: it is not absolutely required for the structure, which can stay firmly in usual conditions, but it ensures stability of the whole. Another remarkable feature is that its shape is strongly determined by the rest of the vault. It is not the form of the keystone which determines the shape of the vault but, rather the converse. Thus, in a hierarchy elements of a given level are more or less interchangeable whereas elements from different levels are specified or controlled by the other levels, the main feature being that elements present at different levels are not interchangeable. Stability results from cooperation of the different levels.
Is it therefore necessary to think which all what is related to living systems must be related to hierarchies? Is stability requiring construction of hierarchical systems? Must one consider that all aggregation of a large number of individuals, forming a whole with a self-consistent behaviour be a consequence of a hierarchical organisation?
“To close strongly, is to close without bars or bolts preventing however anyone to open; To tie well is tie without rope nor string preventing however anyone to untie”
So speaks the Tao. This suggests that there exists some means of conceiving vaults without a keystone; armies without a general but able to fight.
It took a long time for this old wisdom to be put into formal terms. This is undoubtedly because living forms are more often derived from trees, lineages or heads commanding bodily actions, than to whirls or circles. But the so-called "firing-squad" theorem shows how a set of “soldiers” all identical finite automata, able to perform a small number of elementary actions, as a function of the state of the other soldiers to which they are locally connected, and to count a little, may constitute, after a finite period of time, a whole that exhibits a coherent behaviour (e.g. , fire ~, all at the same time), in the complete absence of orders coming from a general. A fortiori more complex automata should display more elaborate behaviour. But analysis of networks of such automata suggests (this is only conjecture!) that their aptitude to do well in performing global tasks would be extremely unstable when their number changes. In particular such a network would lose its properties if the number of local automata grows. If, then, one considers usual living systems at least when they have not attained adulthood, it seems that a network of cells growing in number would prevent formation of acentered network endowed with global properties. Each new incoming cell would perturb, in an unpredictable way, the global properties just achieved. This is where the stability property of hierarchies comes in: formation of distinct levels permits stabilisation, for it depends on the (abstract) presence of levels much more than on the actual number of elements present at each level. But, in multicellular beings, the adult stage is precisely a moment, often very long, when, by definition, the organism maintains itself without major changes in the number of elements (cells or molecules) which are present at each level. It is however a frequent accident to suffer from a lesion in one or another tissue after aggression by a microorganism, or by all kinds of objects: and it is easy to observe that the global behaviour of the organism permits repair of the damage with remarkable efficiency. One could think, as I have shown above, that this would correspond to a hierarchical control (which it is, in a way). But one finds, in the case of a cut for instance, that the skin is reconstituted through a healing process which does not seem to reflect involvement of a specific hierarchical level (as in the case of the vault and the keystone) but, rather to reconstitution of the missing individuals, whatever the place that has been damaged.
This healing process corresponds to a certain kind of closing of an autonomous circle that has been broken fortuitously.
Does not one see, in this image of a circle, the manifestation of structures fundamentally different from tree structures? There are many reasons to think that this is meaningful. Ring closing is founded upon the creation of a recognition process between two adjacent similar structures. It is possible to describe such an interaction by looking into what happens at the molecular level. When identical molecules, able to interact with each other, are involved in a process of association.
The idea which lies at the roots of this molecular approach is simple. Let us consider two identical objects A. Either they don't interact (or interact during very short periods of time) or they form stable associations. In the latter case we are immediately placed in front of a very special situation. Indeed, the new object A2, comprising two objects A, is either symmetrical (closed on itself. forming a ring of two A objects) or possesses borders that are, on each side, identical with the borders found on the isolated A object.
A2 is thus able to interact again with A, creating object A3 But we are now placed in a situation similar to the preceding one: there is no reason for the three A objects to stay in a plane. Therefore there are three situations. Either A3 forms a three-partner ring (this is a special case of the following state), or it stays in a plane (and will form. after addition of more As, a more or less circular structure), or it comes out of the plane, starting with A2 and begins to form a helical structure. In general therefore, association of identical subunits will follow one side of an alternative: constitute a ring structure or form an endless helix, broken and reconstituted by random breaks.
The most frequent case is the case of helical structures, because they do not ask for constraints limiting associations to co-planar architectures. This is the reason for the universality of the helical form in biology. But a helix has borders that are ill-defined: in principle it is an infinite structure. The extremities of the helix have different properties from the main body of the structure, and one can easily understand why such structures, when they are long, are able to bind their axis, forming a circular helix (this is a means to achieve infinity). Another means is to associate two helices, having the same axis (internal to each other), but differing in pitch: this defines a vernier, and gives the association a very precise length. But a very efficient way to solve the problem is to construct a ring structure, by forcing the pitch of the helix to zero (i.e. making a coplanar association of identical subunits). It is therefore quite understandable that, during evolution, helical structures may have become ring structures. The evident advantage of such a situation is that all subunits have strictly equivalent positions, by symmetry. One finds there a case of a global structure (the ring derived from strictly local properties).
A important consequence of such a situation is that any lesion is easily repaired, just by inserting a new subunit in place of the missing one. In the case of rings of cells, the simple duplication of cells at the border of the lesion will fill the gap. Ring closing is an intrinsic local property of the structure. But this rests in the quality of the subunit that permits co-planar association. Any departure from the constraints permitting co-planarity, any change in form, will result in formation of an endless helix. As new mutant subunits are added, during the process which should normally lead to ring closing, the helix is growing more, always trying to close an impossible ring.
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A minute departure from a well formed architecture results in uncontrollable growth. The very process which is meant to permit closing results in irrepressible growth. In the case of cells, interacting by their surface, one is led to think that their cytoplasmic membrane harbours specific objects that have a self-aggregating property by mutual recognition. If formation of a regular paving is a prerequisite for closing the system, its healing after a wound will result, as described, by a simple multiplication of cells surrounding the wound, but this requires a specific interlock between two adjacent cells membranes and places stringent constraints on the structure of the macromolecules responsible for the interaction. I suggest that both specific receptors and molecules similar in architecture to immunoglobulins assume such a function (and, indeed, molecules of the histocompatibility complex are related to immunoglobulins), as we shall now see.
The model which has been roughly outlined above. in the case of planar structures, could be illustrated by association of macromolecules, or epithelial cells. One of its predictions is that one should find ring structures in epithelia. But it can be generalized to other, more abstract, structures, especially if one enters the time dimension in the construction of rings. A ring structure can very well exist as a system of permanentlv circulating objects. This is now the probability of encounter (and the duration of the contact) that fixes closing conditions. It only requires the contacts be frequent enough to control multiplication of the objects forming the dynamic ring. If there are not enough objects (equivalent of the epithelial wound they must multiply), until they are concentrated enough to trigger closing of the dynamic ring. One can easily visualise the immune network as such an entity, comprising in fact several interlocked rings (B and T cells, as well as immunoglobulin itself). The role of the antigen, the foreign agent in the system, is to trap some of the elements which control ring closing. This starts up a process of multiplication, analogous to healing, until the network of the perturbing agent, thus forming a dual image, is restored by an appropriate closing.
Since during this process it is highly likely that variants that would be inapt for allowing closing will be produced, evolution has selected a system that corrects its own mistakes, incorporating errors as parts of the network. This is the raison d'être of the generation of immune diversity. In fact, the response to a foreign antigen is but a side-effect of a general mechanism meant to produce a stable, dynamic image of the self as a closed structure.
The general ideas that have a been presented above are just indications of lines of thought that might be useful in exploring new aspects of living systems. It is clear that, usually, hierarchies are at work. But it seems to me worth considering closed static or dynamic networks (as in epithelia or in the blood system), which would permit construction of global entities without asking for specific hierarchies. Obviously, however, such structures should be stringently coupled to the hierarchies which permit control of gene expression or cell multiplication. I think that the immune system is a paragon of such systems having the structure of a ring, so that most of acentered systems might be derived in biology from a set of molecules similar in architecture to the immunoglobulin, placed in different contexts thanks to a modular construction, which permits introducing immunoglobulin-like molecules in different places.
If such a system is at work it needs not only that the molecules forming the ring (be it static or dynamic) be of a type permitting closing (and avoiding formation of "helical", static or dynamic structures) but also that closing controls expression of these very molecules (either from a control operating inside a given class of cells or through appropriate cell multiplication). This requires a very large number of genes products, and may be reflected in the ever growing number of oncogenes.
Note (*) This is not well perceived unfortunately by many molecular biologists who are still fascinated by the discovery of new objects. Let us remember the cries of victory when the first oncogene was discovered and the disappointment when new oncogenes almost every month are added to the list This is is not perceived either by the quasi-religious adversaries of molecular biology who are often repeating the trite nonsense “a living being is more than the sum of its parts”! (back to text)
1. Balzer RM. Studies concerning minimal time solutions to the firing squad synchronization problem. Carnegie Institute of technology, Pittsburgh, 1966.
2. Balzer RM. An 8-state minimal olution in the firing squad synchronization problem, in Information and Control, X, pp. 22- 42, 1967.
3. Waksmann A, An optimum solution to the Firing squad synchronization problem, in Information and Control, IX, pp. 66-78, 1966.