Symposium on Emergence; InterSymp '95 (Baden-Baden 1995)

Contrasting Two Representations of Emergence of Cellular Dynamics"

J.LR. Chandler, McLean, VA,

A.C. Ehresmann and J.-P. Vanbremeersch,

Faculté de Math. et Info.,33 rue Saint-Leu, F-80039 Amiens. France

Abstract

We contrast, in this paper, the organization of molecular and biological dynamics of a single cell in terms of two general theories: 1. Memory Evolutive Systems (MES) which present a mathematical model, based on category theory, for evolving self-organized hierarchical systems (Ehresmann and Vanbremeersch, 1987, 1991-94); 2. the C8 hypothesis which proposes specific methods of enumerating complexity (Chandler 1991, 1992).

Both theories give account of the hierarchical conformation of the cell and of its growth relying on a sequence of events during which manifestations of biological strategies emerge from exchanges with the environment and internal assemblies or disassemblies of higher complex structures. They indicate how a balanced cellular flow is achieved through the interactions among a net of overlapping cyclic internal communications channels. If a conflict arises, the biological coherence is restored by accelerating, delaying or inhibiting some of the cycles, thus explaining the simultaneous plasticity and bounded stability of a cell. The potential melding of these two theories to create applications describing the organization of evolutionary systems is being explored.

Introduction

The complex organisation of living systems is believed to have emerged from inanimate matter over geological time scales. Available geological and molecular biological evidence shows a highly specific chemical and dynamic relationship among all living species (Morowitz 1992). Such structural commonalities across species - polymers such as DNA, RNA, protein, and monomers such as sugars, lipids and amino acids - suggest an abstract set of rules of emergence may exist, revealing a deeper structure of biology and medicine than we currently recognize. Newer methods of analysis of mathematical dynamics and molecular biology may contribute to clarifying the co-emergence of new organisms with new ecologies.

Here, we contrast the organization of molecular and biological dynamics of a single cell in terms of two general theories - Memory Evolutive Systems (MES) based on Category theory (Ehresmann and Vanbremeersch, 1987, 1991-94), and a specific method of enumerating complexity - the C8 hypothesis (Chandler 1991, 1992). These theories were developed independently and presented at earlier Baden-Baden Symposia to which we refer for more details.

MES were introduced as a mathematical model for complex self-organized systems allowing for the representation of emergence. Their theory has been progressively developed with a view to applications, in particular to study higher cognitive processes in neural systems. The C8 hypothesis originated in an effort to enumerate the attributes of complexity in terms of a semantics and a syntax which transcended individual scientific languages and which was sufficient to describe the nature of human health and disease.

To distinguish the 2 theories, we adopt the form of a dialogue between the authors in order to give alternative descriptions and symbolisations of cellular dynamics in our separate languages.

1. How can a biological system be described?

JC. The closure (C1) distinguishes a system from its environment, and the conformation (C2) specifies the temporal form of a closure or of a closure and its environment. The cellular boundary may include projections (proteins and carbohydrates) which communicate with the interior which distinguishes closure from a simpler system's theory notion of a boundary. The closure and conformation of the cell at a given time are determined by the set of its components which include, in terms of classes of chemical structural diversity: simple substrates (organic acids, lipids, sugars, oxydative/reductive couples, organo-phosphate esters, and vitamins), monomers, polymers and organized organelles. For any arbitary short duration within a cell cycle, the composition and the dynamics of conformational changes describe the ever-changing relationships among the components of cells.

E+V. In the MES theory, the closure and conformation of the cell at a given time t is represented by a category, called the cell-category at t. We recall that a category is defined by objects and links between them (drawn as arrows) forming a directed graph (with possibly several arrows between two vertices). The graph structure is enriched by giving a law to combine 2 successive arrows f: A ® B and g: B ® C into a new arrow fg: A ® C, so that each path formed by successive links can be iteratively combined in a unique link (associativity).

The objects of the cell-category at t represent the components of the cell of any levels present at this time, and the links between them represent their biochemical affinities or energetical interactions. We add a special object 0 to model the loss of components before t (cf. below). To take into account the material constraints, we associate to each link a measure of its strength, as binding rates or propagation delays, and use these measures to define the combination law.

The dynamics of the cell is modelled by the Evolutive System consisting of the successive cell-categories during its life-time, and of the 'transition functors' between them. The transition functor from t to t' models the global change of conformation of the cell between these 2 dates. It is a mapping from the cell-category at t to the cell-category at t' respecting the combination of links. This mapping is not 1-1 nor onto (to account for the emergence of new objects, e.g. by synthesis or endocytosis). And it maps on the special object 0 those components at t which disappear (e.g., by decomposition or exocytosis) at some date between t and t'. The transition functor from t to t' measures the global change which results of the different flows from t to t', without describing the intermediary processes nor giving information on the time-scales of the various components. It just specifies that a particular polymer has subsisted from t to t', or that another has been formed or decomposed at an unspecified time between t and t'.

2. How are more complex objects formed?

JC. To link, chain, or associate simpler "elements" together is defined as concatenation (C3); to separate elements as deconcatenation. From a chemical perspective, the cell has an organizational hierarchy in which each component above the simple substrates is the specific concatenation of lower level components. More complex components are formed from less complex components by uniquely specified reactions. Following concatenation of monomers to form a complex polymer, it may change among two or more conformations by spontaneously "folding" into a form with radically different chemical properties. For example, its affinities for other chemicals may shift, resulting in a new flow of conformational patterns In the C8 hypothesis, the term concatenation denotes the construction or destruction of an a object, that is, a change in the description of the composition. The number of interactions resulting in the association of two or more objects within a cell ranges from simple covalent bond formation between two carbon atoms to highly articulated associations of thousands of hydrogen, oxygen, carbon, nitrogen, sulfur and divalent metal atoms at the interface of DNA - protein interactions.

The organization and function of a cell is critically dependent on the latter types of interactions in which the causal actions are many in number, geometrically specific, spatially diffuse and energetically cooperative. Both macromolecular associations as well as organelle formation tend to emerge from such polyglotic discourses. The term ligature can be used to distinguish this form of dynamical association from traditional chemical bonding.

E+V. The concatenation is modelled in a category by the colimit operation. Given a pattern (or sub-category) in the category, formed by objects and some distinguished links between them, we say that an object of the category is the colimit of the pattern if its links to any other object are in 1-1 correspondence with the collective links of the pattern. These collective links represent the coordinated interactions the objects of the pattern may have when cooperating through their specific links, and which could not be performed by the objects acting separately. The colimit of a pattern must not be confused with the simple aggregate of its objects, for it accounts for the constraints imposed by their distinguished links. Analytically, a colimit can be represented as an attractor for the dynamics.

A category is defined as hierarchical if its objects are partitioned into levels so that each object of a level is the colimit of a pattern of linked objects of the preceding levels. Then the passage from a pattern to its colimit entails a change of code. The cell-category at t is hierarchical, the levels being defined by the increasing orders of structural complexity of its components. For example, a specific macromolecule of RNA is an object of the macromolecular level, which is the colimit of a pattern of the lower molecular level representing its geometric conformation.

Let us develop this example of a colimit. The pattern is the sub-category of the cell-category at t described as follows: The objects are the bases of the RNA. We first describe a graph by considering as links the phospho-diester covalent bonds between successive bases and the non-covalent H-bonds between complementary bases; each of these links has a particular length. To obtain a category, we need a combination of links, so that we add paths fg formed by successive links; the length of fg determines the angles between the bonds f and g. The spatial conformation of the macromolecule is entirely entailed by this pattern, though not yet realized in it. The colimit operation models the folding which leads from this pattern to the RNA with its tertiary structure. The properties of the colimit determine the interactions of the RNA with other components of different levels that emerge from the folding which uncovers specific sites of the tertiary structure, such as the anti-codon. Hence they explain how the RNA can participate in the information processing of the cell (e.g., bind to a ribosome) in a way that the bases before being binded together could not.

The colimit process may be iterated (and in fact the RNA is formed as an iterated colimit, cf. below). A ligature would correspond to an iterated colimit.

.3. How does complexification emerge?

JC. Convolution (C5) is the mutual relationship between the closure and the surroundings - the set of continuous interactions between the system and its environment. The conformation of a closure depends on the flow of concatenating or deconcatenating processes within the closure that depend on the flow with the ecology. Growth can be represented as a cumulative process in which a cell acquires external materials and constructs "self" via iterative cycles of concatenations.

E+V.. The change of states is modelled by the complexification process for a strategy. This process takes into account concatenations (assembly of new components, e.g. synthesis of proteins,...) and deconcatenations (dis-assembly of some complexes), but also the iteractions with the environment (convolution), consisting in the ingestion of extracellular products (endocytosis) and the rejection of some products. Here the environment is not considered by itself, but only through its repercussions on the system.

The complexification process describes what becomes of a category when there is applied on it a strategy aiming at the addition or suppression of certain objects and the formation or disassembly of certain colimits. In particular this construction describes the new interactions between the components after the strategy is realized. The links between two new objects N and N' obtained as colimits of already existing patterns P and P' are of two types: the simple links just assemble clusters of correlated lower level links between the objects of P and P'; the complex links are obtained as combinations of simple links made possible by the fact that an object can be the colimit of two different patterns (degeneracy principle, cf. below), and they represent emerging constraints.

For instance in the cell the synthesis of RNA requires an iteration of complexifications: the first one leads to the primary structure, the second forms the different domains (folding of the loops and helices), then the last complexification gives the complete folding of the tertiary structure. Simple transport processes mediated by substrate specific transport proteins also proceed from a sequence of complexifications.

The formation of a new (iterated) colimit such as the RNA synthesis relies both on local and global informations: locally it consists in the formation and strengthening of bonds between objects in agreement with energetical laws. Globally once the pattern has been concatenated into a new object, the colimit acquires functional properties of its own which determines its relationships with its environment.

4. What are the origins of the dynamical stability of biological systems?

JC. Cyclicity (C4) within the context of a cell means that the dynamics of C1, C2, and C3 are constrained within a specific region of space. Dynamic forms of cyclicity include oscillations, iterations, repetitions, and orbits. These processes may recur with regular or irregular (chaotic) frequencies which may be modulated by external factors (Convolution, C5) In a cell, the dynamics of normal function emerge from continuous interaction among chemical constituents. The genetic organization of a cell can be represented in terms of cyclic communications channels formed by affinities among the constituents. Collaboration among intertwined cyclic communications channels select energy flow strategies for the cell. Collaborative Configurations (CoCos) among the sets of channels organize the system to sustain internal and external flows consistent with biological function.

E+V. In a MES, the intertwined cyclic communications channels are modeled by a net of competitive internal Centers of Regulation (CR). A CR consists of a pattern of components of the evolutive system of the same level working together along specific links to communicate observations and constraints.

Each CR operates a stepwise cyclic process at its specific time-scale and period. A cycle from t to t' consists first in decoding the partial information the CR can gain from the system-category at t; this is modeled by a category, called the landscape of the CR; then as a response, a strategy is selected through this landscape, and finally the CR encodes commands to realize this strategy.

The landscape retains only those attributes of the system which are observable by the CR during its cycle, in accordance with the constraints coming from the complexity level and the propagation delays. Usual mathematical models represent a cycle of a particular CR.

An example of a lower CR in the cell is given by the promoter of a gene. The informations it can decode are the absence or presence of an activator protein on its operator (we think of a positive regulation). It has 2 possible strategies: if there is no activator, it is "do nothing". If there is an activator, the strategy consists in binding the RNA-polymerase to open the gene helix and begin the transcription of the gene.

Higher CRs, such as the nucleus, or the membrane which regulates a traffic essential for the dynamics, have more strategies available, and they can control some lower CRs and possibly evaluate the results of their strategies and memorize it.

5. What are the roots of biological organization?

JC. Coherence (C7) is defined as a specific coordinated pattern of behavior within a closure over a range of flows generated during convolutionary relationships. Biochemical cycles emerge from positive and negative feedback relationships and feedforward relationships which are driven by attractions among the molecular orbitals. Balanced cellular flow is achieved by adjusting concentrations of both simple and complex molecules. If imbalances occur, the organism seeks normalcy by accelerating, delaying or inhibiting some of the cyclic communication channels.

E+V. The result of the strategy of a CR has far reaching global repercussions; for the above cited promoter it entails the transcription of the gene, and the whole process of synthesis of a protein. It follows that the regulations encoded by the different CRs at a given time are overlapping and competitive.

There is no central regulation to coordinate the strategies of the different CRs. Each CR operates a cyclic process at its own period, and one cycle of a higher CR may encompass several cycles of a slower CR; moreover the realization of its selected strategy may depend on some external resources and require differing delays. When the various strategies of the CRs in progress at a given time entail non-coherent energetical or temporal constraints, the global result will be context-dependent, and some of the strategies will be inhibited as a result of the competition to sustain the stability of the global system. In this case, we say that there is a fracture for the CRs whose strategies are not realized in due time.

For instance, in the cell cycle, mitosis is induced by the strategy at the protein-kinase level leading to the formation of a complex with the G2 cyclins. However if the DNA strategy of replication is not achieved in due time, the unreplicated DNA sends a message to inhibit the formation of this complex (by the activation of Wee1) up to the time the DNA replication is achieved.

Thus at each time there is an interplay between the strategies that the different CRs try to enforce. Higher CRs have more latency because of their longer periods, but they have more possibilities to impose strategies on lower CRs,

The repair of a fracture may come from the same CR, or from other CRs which impose a new strategy, with the risk of causing new fractures later on. Finally the global (continuous) dynamics of the cell is modulated by this dialectics between CRs which are heterogeneous with respect to their complexity levels and their (discrete) time-scales.

This description explains that an "information theory" which is centered on the message (as Shannon theory) is not well-suited to the study of a biological system as the cell. An adequate biological information theory should be multi-centered, and evaluate the constraints and results of the decoding/encoding processes of the different CRs.

6. What are the origins of the complex dynamics of a cell?

JC. Contractability (C6) represents the capacity of a system to represent itself in some skeletal form. (A contract represents a mutual agreement between two entities. Thus, the dynamics of a cell emerges from the agreement between the ecology and the cell to sustain energy flows. The concept of a contract, when it comes into being, is to agree to future patterns of behavior which are deemed mutually desirable. Thus, contractability is closely associated with the activities of anticipation, design, planning and feedforward processes.) For a cell, it represents an opportunity for a set of dynamic trajectories to emerge in the system. The biological principle of "no cell, without a cell" (keine Zell ohne Zell) emphasizes the birthing process in biological systems.

At birth, a cell receives an initial endowment of its parent(s). This endowment includes a set of molecular orbitals for its chemical components; these are called the "connate" set of orbitals (L. Con = With, nat = born). From observations on inheritance, the connate set is hypothesized to provide guidance for the future dynamics of the cell.

From a broad perspective, the universe continues to be creative as the rate of complexification increases. Within the terminology of the C8 hypothesis, the growth processes of a specific cell is not included under the concept of creativity. Rather, growth and development are the consequences of the contract. In a broad sense, creativity of new cells, new biological organisms, depends on the genesis of a new contract with new terms of agreement between the organism and its ecology. Cyclic communications channels constitute portions of the terms of the contract.

E+V. In a MES, there is a memory which is developed from an innate (or connate) state, through the memorization by the CRs of their strategies and their results, leading to a procedural memory. In the higher levels of the memory, a unit represents a complex strategy under a contracted form, since its realization entails the progressive decomposition of the iterated colimit.

In the cell, the innate memory allows for the correct unfolding of the different cyclic processes, which explains the robustness of the cell. And the dialectics between heterogeneous CRs, which emerges from the interlevel relationships and differential periods of the CRs, explains the plasticity of the system, allowing it to adapt in variable environments.

The plasticity relies in particular on the Degeneracy Principle (also called the Multiplicity Principle) for the colimit operation. This principle (singled out by Edelman for neural groups) models 2 properties:

1. an economy of means due to the fact that a same object can participate in different patterns, and so have different functions; for instance the same protein-kinase cdc2 regulates the cell-cycle through association with different classes of cyclins depending on the cycle phase.

2. a stability of higher scales which comes from the fact that several patterns may have the same colimit; more precisely we give measures of the extent of modifications a pattern may tolerate while its overall functions remain the same, and in that way, it is related to noise in information theory. It is exemplified by the fact that different genotypes lead to the same phenotype. A biological system is error-prone, but it is able to do well in spite of these errors.

Creativity could also be represented as a consequence of the degeneracy principle, for this principle is at the root of the possibility of an extension of a hierarchy by successive complexifications leading to the emergence of more and more complex objects. The hierarchy of the cell is fixed; but it is not fixed in more complex systems. In particular in neural systems we have shown that the development of higher cognitive processes relies on the formation and memorization of iterated colimits representing assemblies of assemblies... of coherent Hebb assemblies of synchronous neurons; and this MES representation gives some philosophical insights on the brain/mind problem (Ehresmann and Vanbremeersch, 1994).

The possibility to use the complexification process for emergence of new structures could also have applications in Evolution Theory.

7. How can the constraints on cellular dynamics be enumerated?

JC. Necessary and sufficient conditions for the organization of a living cell have been described as an interdependency between the necessities and the sufficiencies to sustain a specific genome in a specific ecology (Chandler, 1992). To analyze these constraints, it is important to enumerate both the structural complexity of the cell and the dynamical constraints on its behavior. The C8 theory was designed with the specific objective of enumerability (Chandler, 1991).

In particular, the closure is enumerated by the volume of the organism and the number of exchanges with the ecology. Its complexity depends on the number of the spatially distinctive internal units it is partioned into. The flows within a closure can be represented by comparing successive conformations of the cell.

The fate of discrete units and the structural reorganizations can be enumerated by the number of concatenations or sequences of concatenations. Observations in all species show that the internal dynamics of living systems are constrained by structural relationships among components. That is, the connate set generates cyclic communications channels which confine the dynamics to an infinitestimal portion of the usual chemical expectations. Cyclicity and the distinction between homogeneous and heterogeneous cycles is also important in the analysis of complex systems. Convolution may be enumerable in terms of the time-dependent relationships among the components participating in generation of the closure. Further work is needed to crisply define the remaining conceptual classes of observation.

E+V. All these enumerations are adaptable in a MES, though up to now the geometric aspect of the theory has been more thoroughly developed. On a conceptual level, MES theory emphasizes time over matter and space, and it insists on the variable durations of concurrent processes.

In particular, an important measure for a complex object is its stability span which is related to its lifetime and to its rates of synthesis and of degradation. It intervenes to enumerate the structural and temporal constraints which must be respected for the cycle of a CR to be completed in due time. These constraints are formulated in the form of inequalities correlating the period of the CR with the time-lags of the decoding and encoding processes, and with the stability spans of the objects with which it interacts. If these constraints cannot be realized during a long enough time, there will be a de-resynchronization consisting in a change of period for the CR. In particular, we have proposed a theory of aging based on such a cascade of de-resynchronisations at increasing levels (Ehresmann and Vanbremeersch, 1993); this theory seems to unify the various physiological theories of aging.

Conclusion

Biological systems such as a cell have many characteristics that differentiate them from other material systems, such as their organization in a hierarchy of interacting components of different scales, their dynamics by assembly and disassembly, directed by overlapping internal regulations at various complexity levels and time-scales, and their plasticity. These are taken into account both in the conceptual taxonomy of C8 and in the mathematical model of MES, based on Category theory, though in a different perspective.

The C8 hypothesis was formed to classify the common features of scientific observations from the viewpoint of complex systems theory. The initial goal was to focus on enumerable observations (attributes) such that the relative complexity of systems of different behaviors could be compared. The first four observational concepts can be roughly associated with constraints on space, matter, change and stability and may be sufficient to enumerate physical systems. The second four terms are essential to describe observations on biological systems. Efforts to enumerate the relationships between these concepts and the physical concepts have proved to be a challenging task because of the tightly coupled dynamic relationships between the genome, internal energy flows and external energy exchanges with an ecology. A critical focus of the C8 hypothesis is the supra-local relationships between the semantics of biochemical descriptions and the syntax of quantitative relationships which can be related to scientific principles.

MES has been developed as a mathematical model for complex self-organized systems, such as bio-sociological or neural systems. The complexification process allows to describe the formation of complex objects and of their interactions, and to explain the emergence of higher order processes. Their dynamics is regulated by the competitive interactions of a net of internal Centers of Regulation, each of which operates a cyclic process at a specific complexity level and with its own time-scale. The stability and plasticity of the system result from the dialectics between these CRs which modulates the evolution.of the system and gives it its specificity.

The mutual recognition of the substantial similarity of the two hypotheses, developed independently and from different intellectual perspectives, has encouraged this collaborative effort. We are working toward bridging our semantic, syntactical and cultural differences with the goal of providing a unified description of emergence within natural systems.

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Table 1. A possible semantic comparison between the representations of C8, MES and a Cell.

(C8) ----- MES ------------------------------- Cell

C1 ------ category-------------------- Components

C2 ------ evolutive system ------------ Configuration

C3 ------ colimit ---------------------- Binding and folding

C4 ------ CR ------------------------- Cyclic processes

C5 ------ complexification ------------ Metabolism

C6 ------ memory ------------------- Genome

C7 ------ dialectics between CRs ---- Overlapping regulations

C8 ------ degeneracy principle ------- Plasticity

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