complex non-linear biodynamics in categories

Springer 2006

Axiomathes (2006) 16:65–122 DOI 10.1007/s10516-005-3973-8

I. C. BAIANU, R. BROWN, G. GEORGESCU and J. F. GLAZEBROOK

COMPLEX NON-LINEAR BIODYNAMICS IN CATEGORIES, HIGHER DIMENSIONAL ALGEBRA AND ŁUKASIEWICZ– MOISIL TOPOS: TRANSFORMATIONS OF NEURONAL, GENETIC AND NEOPLASTIC NETWORKS

ABSTRACT. A categorical, higher dimensional algebra and generalized topos framework for Łukasiewicz–Moisil Algebraic–Logic models of non-linear dynamics in complex functional genomes and cell interactomes is proposed. Łukasiewicz– Moisil Algebraic–Logic models of neural, genetic and neoplastic cell networks, as well as signaling pathways in cells are formulated in terms of non-linear dynamic systems with n-state components that allow for the generalization of previous logical models of both genetic activities and neural networks. An algebraic formulation of variable ‘next-state functions’ is extended to a Łukasiewicz–Moisil Topos with an nvalued Łukasiewicz–Moisil Algebraic Logic subobject classifier description that represents non-random and non-linear network activities as well as their transformations in developmental processes and carcinogenesis. The unification of the theories of organismic sets, molecular sets and Robert Rosen’s (M,R)-systems is also considered here in terms of natural transformations of organismal structures which generate higher dimensional algebras based on consistent axioms, thus avoiding well known logical paradoxes occurring with sets. Quantum bionetworks, such as quantum neural nets and quantum genetic networks, are also discussed and their underlying, non-commutative quantum logics are considered in the context of an emerging Quantum Relational Biology. KEY WORDS: adjoint functors and dynamically analogous systems, biogroupoids and organismal development, biological principles, nuclear equivalence and cell differentiation, categories, n-valued logics and higher dimensional algebra in neuroscience and genetics, cognitive and anticipatory processes, learning and quantum wave-pattern recognition, colimits, limits and adjointness relations in biology, generalized (M,R)-systems, neuro-categories and consciousness, quantum automata and relational biology, quantum bionetworks and their underlying quantum logics, quantum computers

1.

INTRODUCTION

The first published reports on applications of Category Theory to neural and metabolic networks date back to Robert Rosen’s seminal articles on (M,R)-systems (Rosen 1958a, b). These were

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followed over the next decades by a series of papers concerned with specific applications as well as non-linear dynamics of complex networks in biological systems (see for example Baianu, 1970, 1971a, b, 1972, 1973, 1977, 1987a, 1987b, 2004a, 2006). In his latest books, Rosen (1991, 2000) developed an idea previously shared with Elsasser that living systems are extremely complex in the sense of their intractability by any finite number of computer models/ automata/sequential machines, or even universal Turing machines. Although quite controversial, this inability to completely simulate biological systems through finite, numerical or mechanistic, models has important logical and mathematical underpinnings that need to be carefully considered in view of their possible fundamental importance to theoretical treatments of biology. This issue was also discussed in substantial detail from a categorical and dynamical standpoint in Baianu (1987a, b), and it is now further re-examined in a separate report (Baianu 2006). We shall begin in Section 2 with a concise consideration of basic mathematical concepts drawn from the Theory of Categories and Functors, presenting also their extension to Higher Dimensional Algebra. This will be carried out with a view to possible applications to Neurosciences and especially complex non-linear dynamics of neural networks which are essential to brain functions and cognition. The following brief account of category theory and higher dimensional algebra in Section 2 is largely as previously reported in recent articles (Brown and Porter 2003a, b) and Brown et al. (2004), albeit tailored for a different readership and with several important additions from previous literature.

2.

HIGHER DIMENSIONAL ALGEBRA AND CATEGORY THEORY IN NEUROSCIENCE

Since evolution is ‘concerned with’ efficiency, we must expect that the brain has evolved methods for dealing with structural information. It looks like a reasonable conjecture that category theory and higher dimensional category theory could be either necessary or even unavoidable for modeling this kind of behaviour. There remains, however, a wide range of options for modeling strategies (v. infra) that one can select from.

COMPLEX NON-LINEAR BIODYNAMICS IN CATEGORIES

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Higher Dimensional Algebra has already shown its use in models of information management, and in concurrency. Descriptions of systems by graphs are well known (see for example Rosen 1958a), with development described algebraically by paths in graphs. Interacting systems need, however, higher dimensional graphs, and a generalisation of the notion of path. Higher Dimensional Algebra is still young, and there are many new possibilities now being opened up, as a web search shows. The notion of local to global is a key issue in science: how do local conditions determine global behaviour? We can also see it as the problem of describing algebraic controls on a complex system in order to attempt to relate the total behaviour to the behaviour of its parts. We would like to understand such global behaviour, and in favourable circumstances make deductions and calculations. It is hoped that categorical concepts will help to shed light on ways to proceed in neuroscience. In this section we would like firstly to indicate the use of category theory and the particular notion of colimit to describe the gluing process or amalgamation of structures of the same type. Secondly, we would like to describe the new area of higher dimensional algebra, which allows new kinds of algebra suitable for encoding certain complex interactions. In particular, it is useful for describing processes of processes. 2.1. Relations in mathematics and neuroscience The Greeks devised the axiomatic method, but thought of it in a different manner to that we do today. One can imagine that the way Euclid’s Geometry evolved was simply through the delivering of a course covering the established facts of the time. In delivering such a course, it is natural to formalise the starting points, and so arranging a sensible structure. These starting points came to be called postulates, definitions and axioms, and they were thought to deal with real, or even ideal, objects, named points, lines, distance and so on. The modern view, initiated by the discovery of non Euclidean geometry, is that the words points, lines, etc. should be taken as undefined terms, and that axioms give the relations between these. This allows the axioms to apply to many other instances, and has led to the power of modern geometry and algebra. This suggests a task for the professionals in neuroscience, in order to help a trained mathematician struggling with the literature,

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namely to devise some kind of glossary with clear relations between these various words and their usages, in order to see what kind of axiomatic system is needed to describe their relationships. Clarifying, for instance, the meaning to be ascribed to ‘concept’, ‘percept’, ‘thought’, ‘emotion’, etc., and above all the relations between these words, is clearly a fundamental but difficult step. Although relations – in their turn – can be, and were, defined in terms of sets, their axiomatic/categorical introduction greatly expands their range of applicability. Ultimately, one deals with relations among relations and relations of higher order as discussed next. 2.2. Category theory and colimits: Gluing and structure One of the strong developments in mathematics of the 20th century has been that of category theory, with its power of describing the processes of mathematics, developing new logics, unifying different topics, and revealing underlying abstract processes which have turned out to have wide implications and uses (Maclean and Moerdijk 1992). Abstraction allows analogies, by encoding relations, and relations among relations – leading to logical structures of higher order. As an example, when we note that 2+3=3+2 and 2 3=3 2, and extend this to the abstract commutative law x y ¼ y x for a binary operation , we are making an analogy between addition and multiplication, and also make this law available in other situations. One may presume that the power of abstraction, in some sense of making maps, must be deeply encoded in evolutionary history as a technique for encouraging survival, since a map gives a small and manipulable model of the environment. Symbolic manipulation in mathematics often involves rewriting, an example of which is using the commutative law, i.e. replacing in a complicated formulae various instances of x y by y x. We do this rewriting for example in obtaining the equation 3 2 5 3 2 5 3 ¼ 22 33 52 :

ð2:1Þ

For more on rewriting, see Baader and Nipkow (1998), Brown and Heyworth (2000), Meseguer (1993) and Meseguer and Montanari (1998). A study of why and how mathematics works could be useful for making models for neurological functions involving maps of the


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environment. Mathematics may also possibly provide a comprehensible case study of the evolution of complex interacting structures, and so may yield analogies helpful for developing and evaluating models of brain activity, in order to derive better models, and so better understanding. We expect to need a new language, a new mathematics, for describing brain activity. To see what is involved in this search, it is reasonable to study the evolution of mathematics, and of particular branches such as category theory. Just as much as categories are used in the study of particles and atoms, complexification leads to their applications towards biological and neuronal systems. More specifically, complexification processes are features of evolutionary systems which can be based on quantum relational subsystems, particularly at the microscopic and ‘indeterministic’ levels of molecular biology. Such a change of states can be controlled by various Markovian systems and stochastic processes. In cell biology this is often exemplified by processes such as exocytosis and endocytosis. As pointed out in Ehresmann and Vanbremeersch (1987), the schematic representation is viewed as the complexification of a category with respect to a strategy. More specifically a strategy S on a category K consists of the data of external objects which are to be absorbed categorically. Partial functors represent transformations between successive states of the system. This is apparent in Memory Evolutive Systems (MES) and its subsystem the Archetypal Core (AC) where in a possible semantic framework, humans and other higher animals are able to assimilate, record and manage their experiences of their environment or their community, and this is most often achieved within the context of a hierarchical structure. Some description of category theory is given in Brown and Porter (2003b) and Brown et al. (2004). The relevance to biological development was recently described in a series of papers by Ehresmann and Vanbremeersch (2002, 2003). In particular, they see the notion of colimit in a category as describing a structure made up of inter-related parts, so that a category evolving with time can then allow evolving structures, the structures being given as colimits in the category Ct at time t. This notion of colimit gives a very general setting in which to describe the process of gluing or amalgamating complex structures, together with a description of the method of input to and output from these structures (Paton 1997, 2002). First we need to give the notion of category. This developed from a useful notation for a function: moving from the somewhat

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obscure ‘a function is y=f(x) where y varies as x varies’ to the clearer ‘a function f: X fi Y assigns to each element x of the set X an element f(x) of the set Y’. This sees ‘function’ as being a ‘process’. The composition of functions then suggests the first step in the notion of a category C, which consists of a class ObC of ‘objects’, a set of ‘arrows’, or ‘morphisms’ f: X fi Y for any two objects X,Y, and a composition, giving, for instance, fg: X fi Z if also g: Y fi Z. This composite is represented by the following diagram f

g

X ! Y ! Z

ð2:2Þ

or even by ð2:3Þ

A category thus has not only a composition structure but also a ‘position’ structure given by its class of objects. The only rules are associativity f(gh)=(fg)h when both sides are defined, and the existence of identities 1X at each object X, so that with f as above, 1X f=f=f1Y. The notion of colimit in a category generalises the notion of forming the union X [ Y of two overlapping sets, with intersection X\Y. However, instead of concentrating on the sets X,Y themselves, we place them in context, and say that the utility of the union is that it allows us to construct functions f: X [ Y fi C for any C by specifying functions fX: X fi C, fY: Y fi C which agree on the intersection W=X \ Y. So we replace the specific construction of X [ Y by a property which describes, using functions, the relation of this construction to all other sets. That is, the emphasis is on the relation between input and output. A colimit has ‘input data’, a ‘cocone’. In the case of X [ Y, this cocone consist of the two functions iX: X \ Y fi X, iY: X \ Y fi Y. There are similar situations in other contexts. The numbers max (a,b), and 1cm (a,b) are all constructed from their ‘parts’, the numbers a,b. Here the ‘arrows’ of our previous ‘sets and functions’ example are replaced in the first case by the order relation ‘x 6 y’,


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‘x is less than or equal to y’, and in the second case by the divisibility relation ‘x 6 y’, ‘x divides y’. So we have the rules that: (i) if a 6 c and b 6 c then max (a,b) 6 c, (ii) if a 6 c and b 6 c then 1cm (a,b)6 c. Thus to make analogies between constructions for many different mathematical structures, we simply formulate a notion in a general category – that is all! In this way category theory has been a great unifying force in mathematics of the 20th century, and continues so to do. We also generalise now to more complex input data. So the ‘input data’ for a colimit is a diagram D, that is a collection of some objects in a category C and some arrows between them, such as:

ð2:4Þ

Next we need ‘functional controls’: this is a cocone with base D and vertex an object C.

ð2:5Þ

such that each of the triangular faces of this cocone is commutative. The output from such input data will be an object colimðDÞ in our category c defined by a special colimit cocone such that any cocone on D factors uniquely through the colimit cocone. The commutativity condition on the cocone in essence forces, in the colimit, an interaction of the images of different parts of the diagram D. The uniqueness condition makes the colimit the best possible solution to this factorisation problem. In the next picture the colimit is written

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r ¼ colimðDÞ; the dotted arrows represent new morphisms which combine to make the colimit cocone:

ð2:6Þ

and the broken arrow F is constructed from the other information. Again, all triangular faces of the combined picture are commutative. Now stripping away the ‘old’ cocone gives the factorisation of the cocone via the colimit:

ð2:7Þ

Intuitions: The object colimðDÞ is ‘put together’ from, or ‘composed of the parts of’, the constituent diagram D by means of the colimit cocone. From beyond (or above our diagrams) D, an object C ‘sees’ the diagram D ‘mediated’ through its colimit, i.e. if C tries to interact with the whole of D, it has to do so via colimðDÞ. The


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colimit cocone can be thought of as a kind of program: given any cocone on D with vertex C, the output will be a morphism U : colimðDÞ ! C constructed from the other data. How is this morphism realised, what are its values? To focus on a common example, consider the process of sending an email document, call it E. To send this we need a server S, which breaks down the document E into many parts Ei for i in some indexing set I, and labels each part Ei so that it becomes Ei¢. The labelled parts E¢i are then sent to various servers Si which then send these as messages E¢¢i to a server SC for the receiver C. The server SC combines the E¢¢i to produce the received message ME at C. Notice also that there is an arbitrariness in breaking the message down, and in how to route through the servers Si, but the system is designed so that the received message ME is independent of all the choices that have been made at each stage of the process. A description of the email system as a colimit may be difficult to realise precisely, but this analogy does suggest the emphasis on the amalgamation of many individual parts to give a working whole, which yields exact final output from initial input, despite choices at intermediate stages. Information is often ‘subdivided’ by the sensory organs and is reintegrated by the brain. To enable different parts of that information to be integrated, there must be some ‘glue’, some inter-relational information available. If we are given arrows a, b, c, d with no information on where they start or end, then we could form combinations which make no geometric sense. The colimit/composition process makes sense only where the inter-relations are also such as to enable the ‘integration’ to be well defined. Higher dimensional algebra allows more complex notions of ‘well formed composition’, and ones more adapted to geometry. Logical extentions (Lambek and Scott 1968) and computational models (Gadducci and Montanari 1995) of this new algebra are also being developed. Is the colimit notion useful to describe the way the brain (or a brain module) integrates structural information incoming in various forms to give a determined output? What other models are there? It is good to start with the assumption that the simplest idea works! Several biological examples of potential applications of the concept of colimit in Mathematical/Relational Biology arise in the context of the development of a complex multicellular organisms. Thus, the mature stage of the multicellular organism can be considered as

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the colimit of the dynamical state spaces of the developmental stages of a single cell – the fertilized ovum – which generates through its division and subsequent functional specialization all the intermediate stages of development as well as the mature stage. A similar example would arise during the development of the human brain of an individual, with the colimit corresponding to the most developed, or mature, stage of the brain architecture. Other biological examples of such possible colimit applications occur, or occurred, during biological evolution when the latter concept is understood in terms of the progression, or emergence, of species with increasing organizational complexity and anatomical structure (Bell and Holcombe 1996; Dioguardi 1995; Fisher et al. 2000). One question for neuroscientists is: does the brain use analogous processes for communication between its various structures? What we can say is that this general colimit notion represents a general mathematical process which is of fundamental importance in describing and calculating with many algebraic and other structures. To go back to our email analogy, the morphism F is constructed on some element z of colimðDÞ, by splitting z up somehow into pieces zi which come from parts z¢¢i in objects of D, mapping these z¢¢i over using the cocone on D, and combining them in C. This is how some proofs that certain cocones are colimits are actually carried out, see Brown (2004) and Cordier and Porter (1989). In quantum systems treated by the category of quantum objects and their interactions, a colimit can represent either a superposition, or an entanglement, as it will be further discussed in Section 8.

ð2:8Þ


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In such a hierarchical system as that represented in the above colimit diagram, the objects can be partitioned into a sequence of complexity levels 0,1,. . ., r, in which each object N at level n+1 is viewed as the colimit of at least one pattern of connected objects Ni of level n. In other words, N ¼ colimð pattern P of connected objects Ni at level nÞ where Ni ¼ colimð pattern pi Þ: In this respect the object N is a 2-iterated colimit of (P, pi) which is called a ramification of N of length 2 (Ehresmann and Vanbremeersch 1987). More generally, we can consider the k-iterated colimit and the k-ramification of an object. In this context, it is relevant to point out that Nicolas Rashevsky suggested as early as 1968 the presence of such multi-level, hierarchical systems in the organization of both biological and social organisms (Rashevsky 1968a, b; 1969); such systems were regarded by Nicolas Rashevsky as ‘organismic sets’ of various orders, with the lowest order, ‘zero’, being assigned to genetic networks, and the first complexity level being assigned to living cells. A neuron, for example, would be represented in this theory by an organismic set of the first order, in spite of its higher complexity for information processing than other less specialized cells of the same organism, such as the stem, or neuroglial cells. Such discreteness caused by the hierarchical organization in biological systems noted by Nicolas Rashevsky (loc.cit.) should also be considered in the context of his earlier topological approach to Relational Biology and Life which dealt primarily with the continuity of biological structures and processes in whole organisms (Rashevsky 1954). With this approach Rashevsky was able to derive in 1967 two important theorems concerning organismic sets: THEOREM 2.1. In any organismic sets of order higher than zero (i.e., higher than genetic networks), it is impossible to have all elements completely specialized (in the physiological, or functional sense, as stated in Rashevsky 1967.) THEOREM 2.2. Every specialized, or differentiated, organismic set (of any order) is mortal (Rashevsky 1967).

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A categorical formulation of Rashevsky’ s theory of organismic sets (Rashevsky 1968a, b) was then reported in terms of ‘organismic supercategories’ (Baianu and Marinescu 1968; Baianu 1970); the latter was presented as an axiomatic approach to organismic supercategories and organismic sets expressed in logical terms that utilize axioms of the Elementary Theory of Abstract Supercategories (ETAS, reported in Baianu 1970). One of the advantages of the ETAS axiomatic approach, which was inspired by the work of Lawvere (1963, 1966), is that ETAS avoids all the antimonies/paradoxes previously reported for sets (Russell and Whitehead 1925; Russell 1937). ETAS also provides an axiomatic approach to Higher Dimensional Algebra which is an alternative to the constructive approaches discussed next in Subsections 2.3 and 2.4. Subsequently, it was reported (Baianu 1980) that a unified theory of organismic sets, molecular sets and Metabolic–Replication, (M,R)-systems (Rosen 1958a, b) can be constructed in terms of natural transformations of functors between categories of organismic structures and their higher dimensional algebras (‘organismic supercategories’). Biologically relevant results in terms of functors and natural transformations of organismal structures were then reported using this ETAS-based approach (Baianu 1980, 1983, 1984; 1987a, b; 2004a). Several examples of such results will be given here in Sections 5 through 7 (Theorems 5.1 to 7.1). Thus, a conjecture as far as biological processes are concerned, is that the notion of colimit may give useful analogies to the way complex systems operate in a wide range of organisms, from Planaria to humans. More generally, it seems possible that this particular concept in category theory, seeing how a big object is built up of smaller related pieces, may be useful for the mathematics of complex processes. One may be inclined at first to consider such a view as being inconsistent with Robert Rosen’s non-reductionist view of complex systems biology; however, such an apparent inconsistency disappears if one does not associate directly these mathematical constructions with an actual physical, or material, construction of a living system, as in an attempt at physical reconstruction of a biological system through component mixing, for example, after it has been crudely decomposed into its molecular constituents. This leads directly to the important problem of realizability and entailment of mathematical constructions, and the related, non-trivial task of utilizing mathematical constructions to


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guide the generation, or ‘bioengineering’, of real biological systems (Rosen 1971, 1973, 1991, 2000; Baianu and Marinescu 1974; Baianu 1970, 1973, 1974,1980,1983,1984,1987a,b; 2004a). The realizability of genetic networks will be discussed in further detail in Section 6. The colimit’s dual concept, that of limit (or direct limit), also plays an important role in Category theory, and its significance for representations of (M,R)-systems in biology and organismic sets was pointed out in conjunction with the optimal design of biological organisms, differentiation and organismal development (ontogeny) (Baianu 1968, 1970, 1980, 1983, 1984, 1987a, b; Baianu and Scripcariu 1974). 2.3. Natural transformations and functorial constructions in categories Categorical constructions make use of functors between categories as well as the higher order ‘morphisms’ between such functors called natural transformations that belong to a ‘2-category’ (see for example Lawvere 1966). Such constructions also pave the way to Higher Dimensional Algebra which will be introduced in the next section. Especially effective are the functorial constructions which employ the ‘hom’ functors defined next; this construction will then allow one to prove a very useful categorical result – the Yoneda–Grothendieck Lemma. Let C be any category and let X be an object of C. We denote by hX : C ! Set the functor obtained as follows: for any Y 2 ObC and any f : X ! Y; hX ðYÞ ¼ HomC ðX; YÞ; if g : Y ! Y 0 is a morphism of C then hX ðf Þ : HomC ðX; YÞ ! HomC ðX; YÞ0 is the map hX(f)(g)=fg. One can also denote hX as HomC ðX; Þ. Let us define now the very important concept of natural transformation which was first introduced by Eilenberg and MacLane (1945). Let X 2 ObC and let F : C ! Set be a covariant functor. Also, let x ˛ F(X). We shall denote by gX : hX ! F the natural transformation (or functorial morphism) defined as follows: if Y 2 ObC then ðgx ÞY : hX ðYÞ ! FðYÞ is the mapping defined by the equality ðgx ÞY ðf Þ ¼ Fðf ÞðxÞ; furthermore, one imposes the naturality (or commutativity) condition on the following diagram: ð2:9Þ

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LEMMA 2.1. (The Yoneda–Grothendieck Lemma). Let X 2 ObC and let F : C ! Set be a covariant functor. The assignment x 2 X!gX defines a bijection, or one-to-one correspondence, between the set F(X) and the set of natural transformations (or functorial morphisms) from hX to F. This important lemma can be interpreted as stating that any category can be realized as a category of family of ‘sets with structure’ and structure preserving families of functions between sets (see also Section 6 and the references cited therein for its applications to the construction of categories of genetic networks or (M,R)-systems). Note also that the Yoneda–Grothendieck Lemma was previously employed to construct generalized Metabolic–Replication, or (M,R)-systems (Baianu 1973; Baianu and Marinescu 1974), which are categorical representations of the simplest enzymatic (metabolic) and genetic networks (Rosen 1958a). We shall illustrate in subsequent Sections 4 to 7 several applications to bionetworks of another very important type of functorial construction which preserves colimits (and/or limits); this construction is only possible for those pairs of categories which exhibit certain important similarities represented by an adjointness relation. Therefore, adjoint functor pairs (Kan 1958) are here defined with the aim of utilizing their properties in representing similarities between categories of bionetworks, as well as preserving their limits and colimits. Previous attempts in this direction are those reported by Goguen (1999), Girault (1997), and Gauguen and Malcolm (2000). DEFINITION 2.1. Let us consider two covariant functors F and G between two categories C and C0 arranged as follows: F

G

C ! C0 ! C

ð2:10Þ

We shall define F to be a left adjoint functor of G, and we define G to be a right adjoint functor of F, if for any X an object of category C, and any object X¢ of C0 , there exists a bijection tðX; X0 Þ : HomC ðX; GðX0 ÞÞ ! Hom0C ðFðXÞ; X0 Þ; such that for any morphism f : X ! Y of C and morphism f 0 : X 0 ! Y 0 of C0 , the following diagrams of sets and canonically constructed mappings are natural (or commutative):


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ð2:11Þ

ð2:12Þ

In particular, we shall denote by gX : X ! GFðXÞ, the morphism t)1(X, F(X))(1F(X)). Also, we shall denote by eX0 : FGðX0 Þ ! X0 ; the morphism eðGðX0 Þ; X0 Þð1GðX0 Þ Þ, (N. Popescu 1975, p.11). One can easily verify that the following diagrams, which are canonically constructed, are also natural in C and C0 for any morphism f : X ! Y in C, and for any morphism f 0 : X 0 ! Y 0 in C0 , respectively, ð2:13Þ and ð2:14Þ Such adjoint functors commute, respectively, with either limits or colimits as specified by the following theorem (Theorem 5.4 on p.17 of N. Popescu 1975). THEOREM 2.3. Let F : C ! D be a covariant functor and let G : D ! C be its right adjoint functor. Then, one has: (1) F commutes with the limit in C of any functor; (2) G commutes with the colimit in D of any functor. One also has the following important theorem (N. Popescu 1975, Theorem 5.3, p. 13). THEOREM 2.4. Let F : C ! C0 be a covariant functor. The following assertions are equivalent:

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(1) F is full and faithful and any object X¢ of C0 is isomorphic to an object F(X), with X being an object of C; (2) F is full and faithful, and has a full and faithful left adjoint; (3) F is full and faithful, and has a full and faithful right adjoint. DEFINITION 2.2. Two categories C and C0 will be called equivalent if there is a covariant functor F : C ! C0 which satisfies any of the three assertions in Theorem 2.4. The functor F will be called an equivalence from C to C0 . 2.4. A simple introduction to higher dimensional algebra The aim here is to explain some mathematical ideas with which the first two authors have been preoccupied since the 1960s and 1970s, respectively. There was a lot of experimentation to produce the mathematics which would encompass some apparently simple intuitions. This experimentation can be viewed as ‘extraction of concepts’, and it also exemplifies mathematical concepts that might provide a model of inter-relationships that is much freer than the usual ones. Indeed, higher dimensional algebra is being used to model distributed systems and pattern recognition (Porter 1994a, b). A basic idea is that we may need to get away from ‘linear’ thinking in order to express intuitions clearly. Thus, the equation 2 ð5 þ 3Þ ¼ 2 5 þ 2 3

ð2:15Þ

is more clearly shown by the figure jjjjj jjj jjjjj jjj

ð2:16Þ

Indeed the number of conventions you need to understand Equation (2.15) make it seem barbaric compared with the picture (2.16). It is also interesting to see how you could express in a picture the general linear formula of the distributivity law a ðb þ cÞ ¼ a b þ a c:

ð2:17Þ

The importance of having simple comprehensible pictures instead of complex formulae is that the pictures help one to imagine theorems and their proofs. (The above exposition is borrowed from an account in week 53 of John Baez’s (1995) series ‘This week’s find in mathematical physics’).


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Those who have read Edwin Abbott’s famous book ‘Flatland’ (and those who have not, have a delight in store!) will recall the limited interactions available to the inhabitants of Lineland (Abbott 1884). It seems unreasonable to suppose that a purely linear mathematics can express reasonably the complex, highly non-linear interactions that occur in the brain. We often translate geometry into algebra. For example, a figure as follows: ð2:18Þ is easily translated into abcd and the proper language for expressing this is again that of category theory. It is useful to express this intuition as ‘composition is an algebraic inverse to subdivision’. The labelled subdivided line gives the composite word, abcd. But how do we express a diagram such as the following

where the squares are supposed filled and labelled? It seems that in doubling the number of dimensions from 1 to 2, you need to move from categories to double categories, or something similar, based on directed squares rather than on arrows. The extra richness involved is that a square can have more complicated relations to other squares than can happen in the linear situation. Such questions arose from a gluing or colimit problem in topology, namely to describe the behaviour of a big object in terms of the behaviour of its parts. When R. Brown started work on this, the particular problem in dimension 1 was well known and solved, but the question was to carry out similar methods in higher dimensions. For a survey of this work written for a mathematical audience, see Brown (2004) and Brown et al. (2005). In the topological situation from which these ideas arose, to obtain the uniqueness in the output of a colimit, as we always require

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for our emails, we had to go further than ‘algebraic inverses to subdivision’ and to use also the notion of ‘commutative cube’. A commutative square is very easy:

ð2:19Þ

because it is easily translated into mathematics as ab ¼ cd; or a ¼ cdb1 :

ð2:20Þ

The surprising thing is that to determine a commutative cube needs some new ideas. First we need to know how to compose the square faces of a cube. This can be done in two directions:

ð2:21Þ

So we get the notion of a double category, whose elements are squares, for which in the above diagram the composition x 1 z is defined if and only if the bottom edge of x is the same as the top edge of z, and similarly (but right and left edges) for x 2 y. Thus, the compositions are partially defined, under geometric conditions. It is important that one distinguishes between this concept of double category and that of 2-category (which was implicit in the functorial constructions of the previous Subsection 2.3). This new concept enables a close relation between the algebra and geometry. On these compositions, we have to impose all the obvious geometric rules. The rules specified above enable an easy description of multiple compositions. But there remains the question of how to define a commutative cube? A cube has six faces, which can be divided into two groups of three which glue on common edges as follows, where ¶ai, a=0,1, i=1,2,3, denotes for a=0,1, respectively, the back or front face in direction i.


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ð2:22Þ

But neither group of three has a well defined composition, because they do not, as required for a composition in a double category, form a rectangular array of little squares. So you need new elements to fill in the corners, and in fact you also need to expand out, in order to get the correct edges, as in the following diagrams.

ð2:23Þ

It is interesting that the extension from squares in dimension 2 to cubes in dimension 3 produces these gaps the handling of which require a new set of concepts. Thus 2-dimensional algebra needs some new basic constructions. This is not surprising. In dimension 1, you are limited to staying still, moving forward, or moving backward. In dimension 2, you can also turn left or right. This has been modelled formally with some special squares called thin squares, of which the first three are forms of horizontal and vertical identities corresponding to not moving in certain directions:

while our last two are known as connections, and correspond to turning left or right.

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Now we need laws on these operators, such as

The cancellation laws give expression to the idea that turning left and then right (or the other way round) is pointing in the same direction, and the transport laws give expression to the fact that turning with one’s arm outstretched is the same as turning. These laws lead to some amazing calculations and consequences when combined in large diagrams, in what we can call ‘2-dimensional rewriting’. Rewriting techniques in dimension 1 are well known and traditional: one uses say the associative, commutative, distributive, laws, or trigonometric identities, to change one expression in another in attempts to simplify, or to prove a theorem. Indeed, rewriting, or symbol manipulation, has proved a key element in the success of mathematics, in order to find the consequences of basic laws. There is, presumably, an evolutionary basis for this ability to replace ‘real objects’ by ‘symbols’ and so to make manipulation in the mind easier. Some substantial 2-dimensional rewriting has been carried using the above laws, and analogous rewriting has also been carried out in three dimensions in Al-Agl et al. (2002). As may be imagined, it is not easy to handle. As always, the development of a new mathematics solves some problems and then brings a range of new problems into view. Rewriting, changing one formula to another according to certain rules, is a basic facet of much mathematics. When multiple rewrites are occurring, for various operations, the study of this ‘distributed’


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or ‘concurrent’ rewriting can often be pictured as involving higher dimensional laws. Models of the complex message passing and distributed rewriting in the brain have yet to find the laws relevant to that context. Such laws could be key elements for the next level of understanding of cognition. We have so far considered the emergence of both algebraic (Bourbaki 1964) and topological structures, as well as the development of novel concepts in higher dimensional algebra from the standpoint of their potential applications to neurosciences. The question naturally arises here if such concepts might be extended – albeit with substantial rewriting of the underlying logical formalism(s) – to other major types of biological networks, such as the genetic and cellular networks. This approach might indeed pave the way towards a unified mathematical treatment of network biodynamics in complex biological systems as discussed in the next section. 3.

BIOLOGICAL AND COGNITIVE INTERACTIVE NETWORKS

Protein synthesis as a channel of information is transcribed into the protein amino acid sequence which acknowledging the genetic code, DNA ¢ mRNA

translation

!

Amino acid sequence of protein

and if error free, contains a true replica of the information contained in sense codons that are transcribed into mRNA anti-codons.1 Biological interactive networks as a class of complex networks consist of local cellular communities (or ‘organismic sets’) organized and managed by their characteristic selection procedures. Thus, in any partitioning of the structure, it is necessary to regulate the local properties of the organism rather than the global mechanism, which explains an organism’s need for ‘modular constructions’. The colimit concept seems here appropriate because genetic information affords a hierarchical structure and genetic switches operate as transcription factors encoding and switching on other genes within this hierarchy. Moreover, one can include systems which by their intrinsic structure interact via noncommutative relationships. In this way, categorical techniques employed in genetic networks as regulatory systems, may be specialized to the study of equivalence relations, involving groupoids, colimits and other various techniques of higher dimensional algebra (Lambek and Scott 1986; Krsti et al. 2001; Gadducci and Montanari 1995).

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More specifically, inter-regulatory systems of genetic networks via activation or inhibition of DNA transcription can be modeled at several differing levels where various factors influence distinct states usually by some embryonic process or by the actual network structure itself. For each gene network it is important to understand the dynamics of inter-regulatory genetic groups which of themselves create hierarchical systems with their own characteristics. A gene positively (or negatively) regulates another when the protein coding of the former activates (respectively, inhibits) the properties of the latter. In this way, genetic networks are comprised of interconnecting positive and negative feedback loops. The DNA binding protein is encoded by a gene at a vertex i say, activating a target gene j where the transcription rate of i is realized in terms of a function of the concentration xj of the regulatory protein. Acting towards a given gene, the regulatory genes are protein coded and induce a transcription factor. Recent modeling techniques draw from a variety of mathematical sources: graph theory, stochastic differential equations, and Boolean networks are just a few examples (specific approaches were recently realized in de Jong et al. (2000, 2003), and also reviewed in de Jong (2005)). From our viewpoint, the graph–theoretic procedures can be generalized to categories, colimits and, in particular, groupoids. The traditional use of comparatively rigid Boolean networks can be extended to flexible, multi-valued logics with non-commutative features, a procedure which is central to this paper. In particular, the globalization of the dynamics of local groups of organisms (such as certain ‘organismic sets’ of higher order) that are representable in the groupoid context involves certain geometric-topological constructions arising from the iterates of such local procedures towards the concept of holonomy. To obtain an intuitive picture of this concept, imagine you are tracing a path that is representative of a flow which is generated – in a certain specific sense – by an equivalence relation linking local objects in the system. You may observe that the neighbouring flow paths tend to veer off; but as you progress steadily further, other flow paths appear to approach asymptotically. The holonomy concept embraces the idea of a phase transition throughout the physical and biological sciences. Perhaps some of the most notable examples include the Berry phase whereby a slowly evolving quantum system in returning to its original state retains a memory of its motion via a geometric phase in the wavefunction, a phase as given by


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R expði c AÞ, where A is a suitable potential and c is the path in question. Likewise, in the Born–Oppenheimer approximation as nuclei describe a closed path in a certain parameter space, the electronic wave function acquires such a phase (Anandan 1992). 3.1. Neuronal groups, re-entry and the dynamic core hypothesis Suppose we consider two surfaces consisting of neurons and receptor cells denoted by X and Y, respectively, and also let f : X ! Y be a mapping of points of X to assigned points of Y under f. In the maps/re-entry model (Edelman 1989, 1992; Edelman and Tononi 2000) such a mapping should be considered as a component of the cerebral anatomy which is equipped and genetically coded with such mapping networks, as for instance, the operational part of the visual cortex. Re-entry is a selective process whereby a multitude of neuronal groups interact rapidly by two-way signaling (reciprocity) where parallel signals are inter-relayed between maps; take for instance the field of signaling cycles active within the thalamocortical meshwork. A priori, such a process is not a feedback system since there are many parallel streams operating simultaneously and re-entry channels serve to link, in a certain sense, the compositions f1 f2 f3 of distinct maps. In general, these mappings are defined locally throughout, where a global mapping can be considered as defining a perceptual category. The maps/re-entry processes comprise a representational schemata for external stimuli on the nervous system, ensuring the context dependence of local synaptic dynamics and, at the same time, mediating conflicting signals. Impediments and general malfunctioning in the re-entry processes in the presence of a certain biochemical imbalance, may be equated with a demise in consciousness, or representative of certain mental illnesses such as schizophrenia. In any event, when we talk about ‘neuronal groups’, what do we really mean? We have the reciprocity in relay of signaling (invertibility), but we hardly consider these as groups in the mathematical sense, and in practice there is no real reason why we should since the efficiency of re-entry is dependent upon the widespread variation in strength of connection, orientation and the potential convergenece/divergence of paths. Suggestive of the Jamesian idea of consciousness through processes (or processes of processes), the dynamic core hypothesis

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(Edelman and Tononi 2000) concerns the strength-framework of neural interactions within a functional cluster, mainly prevalent in the thalamocortical meshwork. A point here is that the dynamic core defines a neuronal state space (space of objects) and paths connecting points in this space represent a sequence of conscious states over time. Consequently, classifying the paths in the above groupoid sense amounts to classifying these states (over time) where the invertibility/ reciprocity property is already built into the categorical structure of the groupoid. (We are reminded that mathematically a ‘group’ is just a special case of a groupoid admitting a single object – its identity). Also, the local property of ‘maps’ is suggestive of the approach to groupoids considered in terms of sheaves of local equivalence relations. Consciousness loops (Edelman 1989) and the neuronal workspace (Baars 1988) are among an assortment of models that seem to have some tentative interpretation in the context of cat-neurons (‘categorical neurons’ or ‘neuro-categories’ as introduced by Baianu 1972, 1987a, and also discussed by Ehresmann and Vanbremeersch 1987). Analogous to how neurons communicate mainly through synaptic networks, cat-neurons interact in accordance with certain linking procedures and can be extended to higher order categorical structures. Among other things, there are proposed several criteria for studying the binding problem via the overall integration of neuronal assemblies and concepts such as the archetypal core: the cat-neuron resonates as an echo propagates to target concepts through series of thalamocortical loops. Intuitively, the groupoid concept is implicit here because we have observed that groupoids are indeed certain types of categories. Nevertheless, the holonomy groupoid is an essential global concept since it pieces-up and classifies ‘paths’ (in whatever prevailing sense) up to holonomy, or phase-transition; such globalization procedures are treated in Aof and Brown (1992). The relevance of these latter ideas has been discussed in the context of interactive cognitive modules and information theory in Wallace (2005). Thus in a broader approach one might propose a sequence of inter-relationships (denoted ‘‘ ()’’): • The theory of neuronal group experiential selection and stimuli between maps () morphisms of groupoids, and groupoid actions on the corresponding configuration spaces.


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• The perceptual category, interaction and synchronization of differing neuronal groups () the holonomy groupoid, groupoid actions, products of groupoids and crossed modules/complexes of groupoids. • Collective assemblies of neuronal groups () stacks in groupoids. • Complex-interactive neuronal groups with reciprocity () multiple groupoids (groupoid atlases, Bak et al. 2004). The results obtained with a similar approach designed for nonrandom genetic networks will be reviewed next in Sections 4 to 6, and several new consequences will be also derived in Section 7 for developing cellular networks and neoplastic growth structures. Furthermore, we shall be able to relate and compare here our results to those obtained through Robert Rosen’s (M,R)-systems approach to life itself (Rosen 1991, 2000). Thus, we shall proceed in Sections 4 through 6 with more specific applications involving non-linear dynamics and transformations of genetic and neoplastic networks. Such considerations lead to Łukasiewicz–Moisil Algebras and their transformations as explained next. Previously, the assumption was made (Baianu 1977) that certain genetic activities have n levels of intensity, and this assumption is justified both by the existence of epigenetic controls, as well as by the coupling of the genome to the rest of the cell through specific signaling pathways that are involved in the modulation of both translation and transcription control processes. This model is a description of genetic activities in terms of n-valued Łukasiewicz–Moisil logics. For operational reasons the model is directly formulated in an algebraic form by means of Łukasiewicz–Moisil Logic algebras. Łukasiewicz–Moisil algebras were introduced by Moisil in 1941 (cited in Georgescu 2006) as algebraic models of n-valued logics: further improvements were then made by utilizing categorical constructions of Łukasiewicz–Moisil Logic algebras (Georgescu and Popescu 1968; Georgescu and Vraciu 1970). 4.

NON-LINEAR DYNAMICS IN NON-RANDOM GENETIC NETWORKS CONSIDERED AS ŁUKASIEWICZ–MOISIL LOGIC ALGEBRAS

Rashevsky has pointed out as early as 1968 that the interactions among the genes of an operon may be relationally analogous to interactions among the neurons of a certain neural net (Rashevsky

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1968b). Thus, it would be natural to define any assembly, or aggregate, of interacting genes as a genetic network, without considering the ‘clustering’ of genes as a necessary condition for all possible biological organisms. Jacob and Monod (1961a, b) have shown, that in E. Coli the ‘‘regulatory gene’’ and three ‘‘structural genes’’ concerned with lactose metabolism lie near one another in the same region of the chromosome. Another special region near one of the structural genes has the capacity of responding to the regulatory gene, and it is called the ‘‘operator gene’’. The three structural genes are under the control of the same operator and the entire aggregate of genes represents a functional unit or ‘‘operon’’. The presence of this ‘‘clustering’’ of genes seems to be doubtful in the case of higher organisms although in certain eukaryotes, such as yeast, there is also evidence of such gene clustering and of its significant consequences for the dynamic structure of the cell interactome which is neither random nor linear. Furthermore, the recent experimental findings of ‘jumping’ genes, which were predicted earlier theoretically (Baianu 1971a; 1983), and represented in the form of linked diagrams of genetic, quantum observables/algebraic operators (Baianu 1971a), bring in the requirement of a higher algebraic dimension to genetic network dynamics and epigenetic controls that has been so far ignored by many theoretical approaches to embryogenesis/biological development and evolution. Had the structural genes presented an ‘‘all-or-none’’ type of response to the action of regulatory genes, the neural nets might be considered to be dynamically analogous to the corresponding genetic networks, especially since the former also have coupled, intra-neuronal signaling pathways resembling – but distinct – from those of other types of cells in higher organisms. In a broad sense, both types of network could be considered as two distinct realizations of a network which is built up of two-factor elements (Rashevsky 1954; Rosen, 1971; Baianu 1987a). This allows for detailed, both qualitative and quantitative, dynamica1 ‘analyses’ of their actions (Rosen 1971; Baianu 1987a, b; Baianu 2004a). However, the case that was considered first as being the more suitable alternative (Baianu 1977) is the one in which the activities of the genes are not necessarily of the ‘‘all-or-none’’ type. Nevertheless, the representation of the elements of a network (in our case these are genes, operons, or groups of


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genes), as black boxes is convenient, and is here retained to keep the presentation both simple and intuitive (see Figure 1). The formalization of genetic networks that was previously introduced (Baianu 1977, 1987a, b) in terms of Łukasiewicz–Moisil Logics, and the appropriate definitions are here recalled in order to maintain a self-contained presentation. (Further algebraic logic/ categorical details and fundamental properties of Łukasiewicz– Moisil Logic algebras can also be found in this volume in the accompanying paper by Georgescu 2006). The genetic network presented in Figure 1 is a discriminating network (for a pertinent review see also Baianu, 1987a). Consider Figure 1 and apply to it a type of formalization similar to that of McCulloch and Pitts (1943). The level (or chemical concentration) of P1 is zero when the operon A is inactive, and it will take some definite non-zero values on levels ‘1’, ‘2’, and ‘(n)1)’, otherwise. The first level of A is obtained for a threshold value ‘i ’ of P2 – which corresponds to a certain level ‘j’ of B. Similarly, the other corresponding thresholds for levels ‘1’, ‘2’, . . ., ‘(n)1)’ are, respectively, lA1 :; lA2 :; . . . ; lAðn1Þ :. The thresholds are indicated inside the black boxes, in a sequential order, as shown in Figure 2. Thus, if A is inactive (that is, on the zeroth level), then B will be active on the k-th level which is characterized by certain concentration of P2. Symbolically, we write: Aðt; 0Þ: ¼ :Bðt þ d; kÞ;

ð4:1Þ

where t denotes time and d is the ‘time lag’ or delay after which the inactivity of A is reflected into the activity of B, on the kth level. Similarly, one has:

Figure 1. The simplest control unit in a genetic network and its corresponding black-box images.

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Aðt0 þ e; n 1Þ: ¼ :Bðt0 ; 0Þ:

Figure 2.

ð4:2Þ

Black-boxes with n-levels of activity.

The levels of A and B, as well as the time lags d and e, need not be the same. More complicated situations arise when there are many concomitant actions on the same gene. These situations are analogous to a neuron with alterable synapses. Such complex situations could thus arise through interactions which belong to distinct metabolic pathways. In order to be able to deal with any particular situation of this type one needs the symbols of n-valued logics. Relabel the last (n ) 1) level of a gene by 1. An intermediate level of the same gene should be then relabeled by a lower case letter, x or y. The zero level will be labeled by ‘0’, as before. Assume that the levels of all other genes can be represented by intermediate levels. (It is only a convenient convention and it does not impose any further restriction on the number of situations which could arise). With all assertions of the type ‘‘gene A is active on the ith level and gene B is active on the jth level’’ one can form a distributive lattice L. The composition laws for the lattice will be denoted by and . The symbol will stand for the logical non-exclusive ‘or’, and will stand for the logical conjunction ‘and’. Another symbol ‘‘:’’ allows for the ordering of the levels and is the canonical ordering of the lattice. Then, one is able to give a symbolic characterization of the dynamics of a gene in a network with respect to each level i. This is achieved by means of the maps /i_t: L fi L and N: L fi L, (with N being the logical ‘negation’) as specified below. The necessary logical restrictions on the actions of these maps lead to the definition of an n-valued Łukasiewicz– Moisil algebra as specified next.


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DEFINITION 4.1. (First reported by Moisil, 1941, cited in Georgescu 2006). An n-valued Łukasiewicz–Moisil algebra, (LMn -algebra) is a structure of the form (L,,,N,(/)i˛{1,...,n)1},0,1) such that: (L1) (L, , , N, 0, 1) is a de Morgan algebra, that is, a bounded distributed lattice with a decreasing involution N satisfying the de Morgan property N(x y)=Nx Ny; (L2) For each i ˛{1,...,n)1},/i: L fi L is a lattice endomorphism;2 (L3) For each i ˛{1,...,n)1}, x˛L, / i(x) N/i (x)=1 and /i (x) N/ i(x)=0; (L4) For any i, j ˛{1,...,n)1}, one has /i /j ¼ /j ; (L5) For each i, j ˛{1,...,n)1},i £ j implies /i £ /j; (L6) For each i ˛{1,...,n)1} and x˛L, /i(N x)=N/n)i(x). (L7) (Moisil’s determination principle) ["i˛{1,...,n)1}, /i(x)= /i (y)] implies x=y, EXAMPLE 4.1. Let Ln={0,1/(n)1),...,(n)2)/(n)1),1}. This set can be naturally endowed with a LMn-algebra structure as follows: – the bounded lattice operations are those induced by the usual order on rational numbers; – for each j˛{0,...,n)1}, N(j/(n)1))=(n)j)/(n)1); – for each i˛{1,...,n)1} and j ˛{0,...,n)1}, /i(j/(n)1))=0 if jk. The commutativity of this diagram is compatible with conditions (M1), (M2) and (M3) that define the morphisms of algebraic logic lattices. Thus, G j is constructed as the Cartesian product PGi, and furthermore G1 is taken to be ‘0’, the initial object of Net, or – in the context of biological evolution – the genetic net of the ‘primordial’ organism (the very first living ‘cell’!). This result leads also to the statement that ‘‘the genetic network corresponding to a fertilized ovum is the unique colimit (or ‘projective limit’) of all the subsequent genetic networks – corresponding to the later stages of development of that organism’’. Such an important algebraic property represents the ‘potentialities for development of a fertilized ovum’. The two theorems presented above also reveal a dominant feature of the category of genetic nets. The algebraic properties of Net are similar to those exhibited by the category of all automata (sequential machines), and by its subcategory of (M,R)-systems, MR (for details see Theorems 1 and 2, Baianu 1973). Whereas these categories share an important global property it would also be important to determine their local properties that are dissimilar. Thus, Theorems 5.1 and 5.2 hint at a more fundamental conjecture which may be stated as follows: ‘‘There exist adjoint functors between the category of genetic networks described here and the category of (M,R)-systems which was previously characterized’’ (Theorems 1 and 2 of Baianu 1977, 1973, respectively). One may then consider as a strong possibility the statement that such adjoint functors are not both full and faithful. There may also exist certain Kan extensions (Kan 1958) of the (M,R)-system subcategory in the Net- and CLukn-categories which could be constructed explicitly for specific equivalence classes of (M,R)-systems and their underlying, adjoint genetic networks. Such Kan extensions may be restricted to the subcategory of centered Łukasiewicz–Moisil Logic Algebras, CLukn, defined by the Boolean–compatible dynamic transformations of the simplest (M,R)-systems (as defined by Rosen 1971, 1973). 6.

REALIZABILITY OF GENETIC NETWORKS

The genes in a given network G will be relabeled in this section by g1, g2, g3,..., gN. The peripheral genes of G are defined as the genes

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of G which are not influenced by the activity of other genes, and that in their turn do not influence more than one gene by their activity. Such genes have connectivities that are very similar to those present in random genetic networks, and could be presumably studied in Łukasiewicz–Moisil Logic extensions of random genetic networks, rather than in strictly Boolean logic nets. The intermediate case of centered Łukasiewicz–Moisil Algebra models of random genetic networks will thus provide a seamless link between various type of logic-based random networks, and also to the Bayesian analysis of simpler organism genomes, such as that of yeast, and also possibly to other interesting, unicellular organisms such as Archeas. The assertion A(t; 0) in Equation (4.1) is called the action of a gene, gA. The predicates which define the activities of genes comprise their syntactical class. As in the formalization of McCullouch and Pitts (1943), a solution of G will be a class of sentences of the form: St : Apþ1 ðz1 Þ: ¼ :Pri ðA; B; . . . ; Np ; zn Þ;

ð6:1Þ

with ‘‘Pri’’ being a predicate expression which contains no free variable save z1, and such that St has one of the values of the n-valued logic, except zero. The functor S is defined by the two equalities: SðPÞðt; kÞ: ¼ :Pðkx; tÞ ¼ kxS2 Pr ¼ SðSðPrÞÞ; . . . ; Sk ðPrÞ ¼ SðSð. . . ðSðPrÞÞÞktimes : Given a predicate expression Sm(Pr1)(P1,...,Pp,z1), with m a natural number and s a constant sequence, then it is said to be realizable if there exists a genetic, or a neural, network, G and a series of activities such that: A1(z1)=Pr1(A1,A2,...,z1,sa1) has a non-zero logical value for sa1 =A(0). Here the realizing gene will be denoted by gp1. Two laws concerning the activities of the genes, which are such that every S which is realizable for one of them is also realizable for the other, will be called equivalent. The equivalence classes formed by such ‘equivalent’ genes would thus lead to groupoid structures (Ehresmann 1965, 1966, 1967; Brown, 1987). A genetic network will be called cyclic if each gene of the net is arranged in a functional chain with the same beginning and end. In a cyclic net each gene acts on its next neighbour and is influenced by its preceding neighbour. If a set of genes g1, g2, g3, ..., gp of the

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genetic net G is such that its removal from G leaves G without cycles, and if no proper subset has this property, then the set is called cyclic. The cardinality of this set is an index of the complexity of its behaviour. It will be seen later that this index does not uniquely determine the complexity of behaviour of a genetic network. Furthermore, such cyclic subnetworks of the genome may have additional algebraic structure that can be characterized by a certain type of algebraic groups that will be called genetic groups, which will form a Category of Genetic Groups, GrG, with group transformations as group morphisms. GrG is obviously a subcategory of Net, the category of genetic networks, or genomes. In the general case, however, one would also expect, to encounter genetic networks endowed with both an algebraic and a compatible topological structure (Baianu 1971a, b, c; 1977), such as in the case of topological groupoids. The subcategory of these algebraic-topological genetic networks that comprises their representative topological/genetic groupoids as objects and their transformations represented here as groupoid morphisms will be denoted by TGd, whereas the general category of topological groupoids will be denoted here by TopGd. In its turn, the category Net is a subcategory of the higher order Cell Interactome (Baianu 2004a) category, IntC, that includes all signaling pathways coupled with the genetic networks, as well as their dynamic transformations and other metabolic processes essential to cell survival, division, differentiation, growth and development. There is, therefore, in terms of the organizational hierarchy and complexity indices of the various categories of networks the following partial, and/or strict, ordering: ASGMRCLuknGrGNetTopGdIntCLukn; ð6:3Þ where ASG is the Automata Semigroup Category and TGd TopGd is the associated sequence of categories of possible network structures. It is interesting that these related structures form a finite, organizational semi-lattice of subcategories of network models in Lukn. Their classification can be effectively carried out by selecting the Łukasiewicz–Moisil Logic Algebras as the subobject classifier in a Łukasiewicz–Moisil Logic Algebras Topos (Baianu et al. 2005) that includes the Cartesian closed category (Baianu 1973) of all complete and cocomplete networks (which have both limits and colimits). A particularly interesting example is that of the TGd


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category (which is a subcategory of TopGd) that is endowed with sheaves of genetic networks which have striking, ‘emerging’ properties such as ‘genetic memory’. Perhaps, ‘genetic memory’ reflects certain underlying holonomic quantum genetic processes; it would also involve the related concept of quantum automaton (Subsection 8.3) if endowed with special reversibility properties. Ubiquitous examples of such processes are provided by many enzyme reactions and/or quantum relational oscillations in genetic networks during cell cycling (Baianu 1971a), or certain quantum-level changes that can initiate the neoplastic transformations of cells during carcinogenesis (Baianu 1969, 1970, 1971a, c; 1977). Let us consider next a generalisation via many-valued logics of the important concept of time-dependent propositional functions introduced by McCullouch and Pitts (1943). (D3) An n-valued propositional expression (NTPE) designates a temporal propositional function (TPF) and is defined by the following recursion: (NT1). An lpl[z] is an NTPE iff P1 is a predicate variable with n-possible logical values; (NT2). If S1 and S2 are two NTPEs containing the same free individual variable, then so are: S1 S2, S1 S2, S1Æ S2, and S1~ S2. Note that these definitions have similar content as the corresponding ones of McCullouch and Pitts (1943), except for the presence of n-logical values. As a consequence, one can easily prove mutatis mutandi the following theorems. THEOREM 6.1. Every genetic net (of order zero) can be solved in terms of n-valued temporal propositional expressions (NTPE), (Baianu 1977). THEOREM 6.2. Every NTPE is realizable in terms of a genetic net of 0th order (Baianu, 1977). THEOREM 6.3. Any complex sentence S1 (built up in any manner out of elementary sentences of the form p(z1)zz), with zz being any numeral, by means of negation, conjunction, implication and logical equivalence), is an NTPE, (Baianu 1977).

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S1 acquires zero value only when all its constituents p(z1)zz0) have all the zero logical value (‘‘false’’). Let us recall that if two or more genes influence the activity of the same gene, then the influenced genes are said to be alterable. One readily obtains the following theorem concerning alterable genes: THEOREM 6.4. Alterable genes can be replaced by cycles. (See also Theorem VII and its proof in the original paper of McCullouch and Pitts 1943). For cyclic genetic nets of order p one can adopt the construction method introduced by McCullouch and Pitts. However, there will be no different sentences formed out of the pN1 by joining to the conjunction of some set the conjunctions of the ‘‘negated’’ forms of each level of the rest. Consequently, the logical expression which is a solution of G, will have the form: ðz4 Þðzi Þzzp : Pri ðzi ; z4 Þ: :ð9fÞðziþ1 Þ½zðiþ3Þ 1 f ðzðiþ1Þ Þ; ð6:4Þ with i=1, 2,..., (n ) 3) and zzn=n. (For further details see also Baianu 1977, p. 256). In the case of genetic networks with n-levels of activity, the realizability of a set of Si is not as simple as it was in the case of neural nets operating with just Boolean logic, such as digital computers. In this case, it involves n simultaneous conditions for the n distinct logical values, instead of just the two values from Boolean logic. As a consequence, it is possible that certain genetic networks will be able to ‘take into account’ the future of their peripheral genes in their switching sequence and levels of activities, thus effectively anticipating sudden threats to the cell survival, and also exhibiting multiple adaptation behaviors in response to exposure to several damaging chemicals, or mutagens, antibiotics, etc. Thus, another index of complexity of behavior of genetic networks is the number of future peripheral genes which are taken into account by a specific realization of a network. In contrast to a feedback system, this will be called a feedforward system. Furthermore, the fact that the number of active genes, or simply the number or genes, is not constant in an organism during its development, but increases until a fully-developed, organismal stage is reached, hinders somewhat the direct application of the ‘purely’ logical formalization introduced in this section. However, the categorical and


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Łukasiewicz–Moisil Logic Topos formalisms introduced in the previous Sections 2, 4 and 5 can now be readily applied to organismic developmental processes and may allow one to practically solve these realizability problems through effective categorical constructions such as: pre-sheaves, sheaves, colimits, limits, adjoint functors, Kan extensions and higher dimensional algebras. 7.

NATURAL TRANSFORMATIONS IN BIOLOGICAL DIFFERENTIATION AND DEVELOPMENT

We shall consider in this section the problem of modeling biological differentiation and development in terms of a small number of hypotheses based on certain empirical facts established through biological experimentation, such as a very large number of welldesigned nuclear transplantation experiments. The biomathematical consequences of such interesting biological experiments were previously considered and several theorems were derived in terms of categorical models of dynamics in developing biological organisms (Baianu and Scripcariu 1974). The important question arises: How similar are the differentiated stages of a fertilized ovum to the original germ cell considered from either a dynamic or a physiological/functional standpoint? Rosen (1968a, b) proposed formal definitions of similarity between dynamic systems which he called ‘analogous’, and represented such a similarity as a dynamic isomorphism between their state spaces, defined by him in a classical sense; his dynamic isomorphism also commutes with both internal transition functions for any pair of analogous systems. This commutative, classical notion of analogy was subsequently generalized to complex biological systems in terms of adjoint functors of bionetwork categories (Baianu and Scripcariu 1974). (As it will be shown here in Section 8, the representation of quantum analogous systems requires, on the other hand, non-commutative quantum logics that lead to much less restrictive conditions than those imposed by the dynamic isomorphism for both classical and relativistic mechanics definitions of analogous systems.) Dynamic systems which are only partially similar were called by Rosen (1968a, b) ‘simply analogous’. This latter concept was then generalized by employing surjective functors instead of adjoint functors; the partial dynamic similarity between developmental stages

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that exists in organisms during their development was thus represented as a ‘weak adjointness’ relation (Baianu and Scripcariu 1974), instead of the adjointness relation which exists between equivalent nuclei only during the initial stages of organismic development. 7.1. Modeling hypotheses We shall begin here with a deliberately, much simpler representation than those widely employed by ‘simulations’ of biological development, or embryogenesis, which evoke epigenesis and/or structural pattern formation through local interactions involving both diffusion and a finite number of postulated, coupled biochemical reactions. We shall argue that such alternative approaches which involve some form of Turing machines, or cellular automata/ tessallation models, are greatly underestimating the extreme complexity of a developing organism which cannot be modeled by any finite number of sequential machines, numerical computer models (Baianu 1987a), or universal Turing machines in the sense defined by von Neumann. The heterogeneous classes that are ubiquitously present in living systems should then be considered at the next stage in the form of a higher dimensional algebra of mathematical structures built upon the underlying, different structures. Clearly, the ‘linear’, deterministic, or causal models previously employed in ‘Complex Systems’ research to model morphogenesis are bypassed by such structures within structures..., within more, distinct structures, that are highly non-linear in a qualitative dynamic sense. They are also highly intractable and shall be called here ‘ultra-complex’ (living) systems in order to distinguish them sharply from those currently considered by current ‘Complex Systems’ research. Such ultra-complex systems were previously called ‘‘complex’’ by Rosen (1991), in a logical sense very close to ours, but very different from the conventional meaning in current ‘Complex Systems’ research. Instead of making complicated models with simple ‘physical’ components, we shall make ultra-complex, functional-modular models (in the sense defined above, and also that of Rosen’s (M,R)-systems), with specified mathematical structures that are algebraically and topologically more manageable than complicated models with simple mechanistic processes. Our ultra-complex models may also prove to be quite similar to the living systems in their dynamics and


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observed behaviors (albeit in a qualitative sense, rather than a purely numerical one). Such ultra-complex systems – although they are mathematically quite simple in their algebraic structure – they are also extremely intractable by the well-known, linear ‘engineering’ methods developed for simple system dynamics; they can thus be regarded only as ‘special’ systems with an extremely high degree of complexity. Their biological functionality seems to include ‘creativity’ and ‘re-creativity’. Measurements on ultra-complex biological systems usually yield a wide range of values for different systems and there is always a ‘quantum fuzziness’ present which is intrinsic to the ultra-complex living systems (Baianu and Marinescu 1968). This quantum fuzziness is quite distinct from the wellknown, ‘random-event’ fuzziness introduced earlier by Zadeh (1965). Possible reasons for its presence in ultra-complex, living systems will be discussed here in Section 8. Remarkably, such biological/quantum fuzziness is not simply quantitative but it also extends to the patterns of relations representing key biological functions, and indeed to the entire metabolism of an organism. This also contributes significantly to the extreme degree of complexity of living systems and is part-and-parcel of the observed biological variability. 7.1.1. Germ cells These are cells that have the highest transformation, or differentiation, ‘potential’. Biomathematically, this biological, empirical finding can be expressed as follows: germ cells have the highest cardinal, a, of the class of equivalent sets of descendant cells. The latter cells are derived from germ cells through specialization to certain physiological functions such as: carrying oxygen and/or CO2 by red blood cells, vision/sensitivity to light of retinal cells, immune response (by both leukocytes and T-cells), ‘conduction’, or propagation, of electrical impulses by neurons, etc. 7.1.2. Stem cells Such cells are at various intermediate stages of differentiation between germ cells and fully-differentiated, specialized cells, such as the ones specified above. Furthermore, stem cells are close in their differentiation ‘potential’ to germ cells, but usually their differentiation potential has decreased somewhat in comparison with that of the germ cells from which they have originated.

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7.1.3. The differentiation potential This can be defined biomathematically as the cardinal of the class of equivalent sets of its descendant cells of various degree(s) of functional/physiological specialization. (A drawback of this definition is that the differentiation potential is not known experimentally until after the occurrence of the development, whereas a predictive definition based on the existing experimental observations is currently unavailable). With these three basic concept definitions available one is in a position to derive several new results based on the formalism and definitions presented in a previous publication (Baianu and Scripcariu 1977). 7.1.4. Biogroupoids and development Such biological equivalence classes can lead to a biogroupoid structure (see for completeness several relevant publications on groupoids: Brown 1987; Buneci 2003; Ehresmann 1966, 1967). Assuming, in addition, the Darwinian classification hypothesis that biological species are uniquely defined and comprise genetically distinct classes of ‘equivalent’ organisms, thus one obtains a unique biogroupoid (up to an isomorphism) representing such classes. In this case, genetic ‘equivalence’ could be defined, for example, in terms of an equal number of chromosomes of the same type. Such species equivalence can also be defined in terms of either overall (‘global’) morphological structures or specific, molecular structures of proteins/enzymes and their underlying genome maps. Such equivalence relations lead to biologically unique groupoid structures (that is ‘biogroupoids’): two elements (s,t) and (u,v) in the graph of an equivalence relation are composable if and only if s=v and (u,v)(v,t)=(u,t) and (u,v))1=(v, u). These considerations lead us also to postulate the following: RELATIONAL EQUIVALENCE PRINCIPLE:. The equivalence classes of descendant cells of either germ or stem cells lead to unique biogroupoid structures that are characteristic for a specific organism or biological species. The caveat for such concepts is the inter-species exchange of genes that is known to occur between several species not only of bacteria but also of some higher plants, as distant as wheat and


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rye! Furthermore, we note here that the groupoid really contains information not just about the equivalence classes but about how things are, or are seen to be, equivalent. COROLLARY 7.1. The germ cells of any multi-cellular organism of a given biological species are naturally equivalent. (The proof of this corollary utilizes Theorems 2.3 and 2.4, as well as Definitions 2.1 and 2.2.) THEOREM 7.1. The nucleus of a developing germ cell is dynamically adjoint with all of its descendants only up to the blastula stage of development (Baianu and Scripcariu 1974). Simply stated, this theorem expresses the developmental ‘equivalence’ of nuclei from different stages of development of the same organism, or ‘nuclear equivalence’, only up to the blastula stage of development of an organism, (also observed through nuclear transplant experiments, as explained in further detail in loc.cit.). After the blastula stage, the nuclei of post-blastula stage cells when transplanted into cells of the earlier stages no longer result in a normal growth and development of the entire organism (‘defective cloning’). The results in this section lead us therefore to propose the following related principle, or postulate, governing the course of development of individual higher organisms capable of sexual reproduction. THE POSTULATE OF EPIMORPHIC ORGANISMAL DEVELOPMENT:. The nuclei of germ and stem cells in a fully developed organism are weakly adjoint dynamical systems with those of all somatic cells of the same organism. The Principle of Biological Epimorphism in its first general formulation for comparing different organisms that developed through biological evolution goes back to Rashevsky’s seminal paper in 1961, Rashevsky (1961), and its final formulations by Rashevsky (1967, 1972). Its roots, however, can be traced even further back to D’Arcy Thompson’s (1900) book ‘‘On Growth and Form’’ (notwithstanding Thompson’s apparently unfounded criticism of the evolutionary theory first reported by Charles Darwin). Our postulate above represents the extension of this bio-epimorphism principle to Ontogeny (i.e., the development of an individual organism from a

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single germ cell or fertilized egg). When it was first stated, it had the strong, incorrect form (which is now known not to be supported at all by observations): ‘‘Ontogeny partially repeats Phylogeny’’, (or in the original formulation of this idea by Ernst Haeckel ‘‘ontogeny recapitulates phylogeny’’). Broadly ‘similar’ to symmetry breaking governing transitions in the ‘simpler’ physical processes, a relationship of dynamic weak adjointness reflects the partial breakdown of ‘nuclear equivalence’ at certain stages of development in higher organisms, which is implicit in an embryogenetic ‘principle’ that still needs a precise formulation in terms of the underlying genetic networks and related epigenetic controls; this introduces limitations to the complete cloning of organisms which undergo complex embryogenesis and also to the maintenance of ‘immortal’ cell lines in culture (see also Theorem 2.2 of Rashevsky 1967, briefly discussed here in Section 2.2). CONJECTURE. Cancer cells, or neoplastically transformed cells, have only weak dynamic adjointness relations with somatic cells from which they descended, on the one hand, and with stem cells from the same tissue, on the other hand. This could also be stated differently as follows: ‘‘there exist inequivalent embedding functors from the dynamic state space of a cancer cell and those of stem, or specialized/somatic, cells from whose neoplastic transformation the cancer cell originated ’’. If this conjecture could be proven one would be then able to introduce detailed topological biogroupoid models of cancer cell dynamics, thus generating cellular state spaces whose specific mathematical structures would allow one to represent different types of cancer cells and their dynamic state space transformations. Future developments of the results and concepts presented in this section may lead to definitive tests of this conjecture in the context of categories of genetic networks, their associated Łukasiewicz–Moisil algebras and the natural transformations of genetic network functors that represent genetic mutations and biological evolution. 8.

QUANTUM NEURAL NETWORKS, QUANTUM GENETICS AND THEIR UNDERLYING NON-COMMUTATIVE QUANTUM LOGICS

In this section we shall attempt to consider non-linear dynamic bionetworks at the microscopic, quantum level. Then we shall


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proceed to enquire into the nature of the underlying quantum logics that determines the non-linear dynamics of such quantum bionetworks. Quantum automata were originally introduced (Baianu 1971a) as a quantum, dynamic representation of simple genetic networks and their linked, microscopic observables (loc.cit.). This concept was then generalised to quantum computers (Baianu 1971b) and its current development is accelerated by the technological goal of making quantum nanocomputers (Baianu 2004b). Such recent developments are also motivated by the pressing need for a deeper understanding and improved mathematical formulation of Quantum Algebraic Topology underlying Quantum Field and Quantum Gravity theories (Baianu et al. 2005). 8.1. Quantum neural networks Analogous to genetic networks, neural networks traditionally represent information via a complex of synaptic connections/weights regulated by the Hebb memory/learning matrix J={Jij}. Learning processes follow essentially in accordance with the gradient flow of the corresponding Hamiltonian of the system, and are manifestly the processes of interaction between systems of neurons and elements of J, as is the usual phase difference woutput)winput over states {wk}. The traditional models possessing a ‘discrete’ character may be considered as too rigid by their discrete character and perhaps are only meaningful for AI. The introduction of their quantum analogs, obtained by replacing the Hebbian memory-storage by quantum holography, may prove to be particularly useful in high resolution imaging, wave-function reconstruction and tomography. Such models exhibited by Perusˇ et al. (2003, 2004) implement neuro-quantum isomorphisms with a numbers-to-waves translation. Typically, the data consists of: • A quantum wave function Y acting as the net’s state vector q. • Eigen-wave functions wk (k=1, ..., P) act as Hopfields’s patternbearing eigenvectors (attractors) vk. • The quantum Green’s function propagator G (replacing the Hebbian matrix J). • Sum of self-interferences wk wk of quantum waves implementing the sum of auto-correlations of input pattern configurations vk vk (namely, the content-addressable memory J).

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Following Perusˇ et al. (2004), the matrix representation for G is given by Ghj ¼

P X

wkh ðwkJ Þ ;

ð8:1Þ

k¼1

where h and j denote the unit/pixel/neuron/quantum point at locations r1 and r2 at time t and 1 £ h,j £ N (and N can be arbitrarily large). Relative to the k0th stored state, the input–output system is given by " # N N X P X X output input k k

Wh ¼ Ghj Wj ¼ wh ðwj Þ Winput j j¼1

¼

" P N X X k¼1

j¼1

#

k¼1

ð8:2Þ

wkh ðwkj Þ Winput : j

j¼1

Alternatively, in the Dirac notation the latter can be read as ! X k k output input jW i ¼ GjW i¼ jw ihw j jWinput k ð8:3Þ X k k k0 input ¼ hw jW ijw i ¼ w ; k

b when and so defines another obvious equivalence relation WRW b states Y and W are related by the above equality. In other words, a quantum neuro-groupoid structure is realizable via such an equivalence relation but notably without any explicit account of ‘paths’, where mention of the latter concept would otherwise be deemed as inherently problematic (cf. the Heisenberg Uncertainty Principle). 8.2. Quantum effects Let H be a (complex) Hilbert space (with inner product denoted Æ,æ) and L(H) the bounded linear operators on on H. We place a natural partial ordering ‘‘6’’ on L(H) by S 6 T if hSw; wi hTw; wi;

for all w 2 H:

ð8:4Þ


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In the terminology of Gudder (2004), an operator A ˛H is said to represent a quantum effect if 0 6 A 6 I. Let eðHÞ denote the set of quantum effects on H. Next, let PðHÞ ¼ fP 2 LðHÞ : P2 ¼ P; P ¼ P g;

ð8:5Þ

denote the space of projection operators on H. The space PðHÞ EðHÞ constitute the sharp quantum effects on H. Likewise a natural partial ordering ‘‘|’’ can be placed on P(H) by defining P £ Q if PQ=P. A quantum state is specified in terms of a probability measure m : PðHÞ P![0,1],Pwhere m(I)=1 and if Pi are mutually orthogonal, then m( Pi)= m(Pi). The corresponding quantum probabilities and stochastic processes, may be either ‘‘sharp’’ or ‘‘fuzzy’’. A brief mathematical formulation following Gudder (2004) accounts for these distinctions as it will be explained next. Let AðHÞ be a r-algebra generated by open sets and consider the pure states as denoted by XðHÞ ¼ fx 2 H : kxk ¼ 1g. We have then relative to the latter an effects space eðXðHÞ; AðHÞÞ less ‘‘sharp’’ than the space of projections P(H) and thus comprising an entity which is ‘‘fuzzy’’ in nature. Now, for a given unitary operator U :H fi H, a sharp observable XU is expressed abstractly by a map XU : AðHÞ ! EðXðHÞ; AðHÞÞ;

ð8:6Þ

for which XU ðAÞ ¼ IU1 ðAÞ . Suppose then we have a dynamical group ( t 2 R) satisfying U(s +t)=U(s) U(t), such as in the case U(t)=exp()it H) where H denotes the energy operator of Schro¨dinger’s equation. Such a group of operators extends XU as above to a fuzzy (quantum) stochastic process X~UðtÞ : AðHÞ ! EðXðHÞ; AðHÞÞ:

ð8:7Þ

One can thus define classes of analogous quantum processes with ‘similar’ dynamic behaviours (see also our discussion in the previous Section 7) by employing dynamical group isomorphisms, whereas comparisons between dissimilar quantum processes could be represented by dynamical group homomorphisms. To what degree the visual and auditory processes are ‘‘sharp’’ or ‘‘fuzzy’’ remains open to further research. Nevertheless, it is conceivable that certain membrane-interactive neurophysiological

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phenomena occur via a fuzzy, a semi-classical or a quantum stochastic process. For a discussion of quantum measurements see Krips (1999). From the ‘‘sharp’’ point of view, Stapp (1999) has described a dynamic/body/brain/mind schemata as a quantum system complete with an observer on the basis of the von Neumann–Wigner theory involving projection operators P as above. The intentional viewpoint interprets ‘‘Yes’’=P and in the complementary case, ‘‘No’’=I ) P, where I is the identity operator. The projection P is said to act on the degrees of freedom of the brain of the observer and reduces the latter as well as a universal state to one that is compatible with ‘‘Yes’’ or ‘‘No’’ reduced states~: ð‘‘Yes’’ÞS 7! PSP ð‘‘No’’ÞS 7! ðI PÞSðI PÞ: The actualization of a single thought creates a chain of subsequent thoughts and conscious action which might be realized by projection into the future of a component of the thought to which the body/world scheme itself becomes actualized. In turn, the neuronal processes that result from this associated body/world scheme eventually achieve the actual intention itself. As this process unfolds, consciousness is sustained through the continued interplay of fundamental neuro-cognitive processes (such as, recognition, sensorymotor responses, information management, and also especially through logical inferences, learning, and so on). It is another part of the picture. 8.3. Quantum genetics Following Schro¨dinger’s attempt (Schro¨dinger 1945), Robert Rosen’s report in 1960 is perhaps one of the earliest quantum-theoretical approaches to genetic problems that utilized explicitly the properties of von Neumann algebras and spectral measures/ self-adjoint operators (Rosen 1960). A subsequent approach considered genetic networks as quantum automata (Baianu 1971a) and genetic reduplication processes as quantum relational oscillations of such bionetworks (loc.cit.). This approach was also utilized in subsequent reports to introduce representations of genetic changes that occur during differentiation, biological development, or oncogenesis (Baianu 1971c) in terms of natural transformations of organismal (or organismic) structures (Baianu 1980, 1983, 1984,


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1987a, b; 2004a, 2004b; Baianu and Prisecaru, 2004), thus paving the way to a Quantum Relational Biology (Baianu 1971a, 2004a). The significance of these previous results for quantum bionetworks will be considered next from both a logical and an axiomatic viewpoint. 8.4. The non-commutative logics underlying quantum bionetworks Quantum bionetworks, be they neuronal or genetic, are likely to have underlying, non-commutative quantum logics. Clearly, such quantum logics are distinct from the n-valued logics considered above in Sections 4 to 6. On the other hand, the Łukasiewicz–Moisil logic algebras could be modified to yield quantum logics if their distributivity axioms, such as L1 (and also L6), are either removed or appropriately modified for both the logical conjunction and negation operations. Furthermore, one expects on the basis of our results in Sections 4 and 5, above, as well as those previously reported (Baianu 1977; 1987a, b), that the specific quantum logics underlying quantum genetic networks would be distinct from those underlying quantum neuronal networks. The quantum effects considered above in Subsection 8.1 and 8.2 define special types of quantum logics that are mostly ‘‘fuzzy’’ rather than sharp, and may not therefore possess any underlying lattice structure, in sharp contrast with what one might reasonably expect for the specific quantum logics underlying quantum genetic networks. According to our novel results presented in Sections 4 through 6, the latter quantum logics should still retain a non-commutative, non-distributive and also nonassociative lattice structure capable of leading to ‘sharp’ responses of the associated quantum genetic networks, but still subject to the ‘limited fuzziness’ specified by Heisenberg’s Uncertainty Principle in Quantum Theory. Such ‘sharp’ responses are observed with definite – but non-commutative – probabilities for the occurrence of specific quantum events (Gudder 2004), such as genetic mutations induced by radiation in microscopic genetic networks considered as quantum systems. This interesting dichotomy of the underlying quantum logics between neuronal and genetic networks is most likely to be a significant part of the answer to Erwin Schro¨dinger’s, Nicolas Rashevsky’s and Robert Rosen’s very important questions leading to ‘‘What is Life Itself? ’’.

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9.

DISCUSSION AND CONCLUSIONS

One of the first successful applications of Logics to Biology was the use of predicate calculus (Hilbert and Ackerman 1927; Carnap 1938) for a dynamical description of activities in neural nets (McCulloch and Pitts 1943), that was subsequently developed by several neural network theorists. Another significant application related to Boolean Logic was the calculus of predicates which was applied by Nicolas Rashevsky (1965) to more general situations in Relational Biology and Organismic Set theory. Only a few years later, Lo¨fgren (1968) introduced a novel, non-Boolean logical approach to the problem of complete self-reproduction that complemented Robert Rosen’s own ideas and approach to Quantum Genetics (personal communication to the first author from Robert Rosen in 1969). Lo¨fgren’s approach (loc.cit.) solved some of the paradoxes previously uncovered by Rosen (1960) in Quantum Genetics who approached this field from the standpoint of the von Neumann’s algebraic and logical lattice (Birkhoff 1948) formalism. The characterization of genetic activities in terms of Łukasiewicz–Moisil Logic Algebras that was here presented has only certain broad similarities to the well known method of McCulloch and Pitts (1943). There are major differences arising in genetic networks both from the fact that the genes are considered to act in a multiple, step-wise manner, as well as from the coupling of the genetic network to the cell interactomics through intracellular signaling pathways. The ‘‘all-or-none’’ type of activity often considered in conjunction with genes is recovered as a particular case of our generalized description for n=2 in centered Łukasiewicz–Moisil logic algebras. Thus, the new concept of a Łukasiewicz–Moisil Topos expands the application range of such models of genetic activities to whole genomes, cell interactomics, neoplastic transformations and morphogenetic, or evolutionary, processes. The approach of genetic activities from the standpoint of Łukasiewicz–Moisil Logic algebras, categories and Topoi that was presented in Section 6 leads to the conclusion that the use of n-valued logics for the description of genetic activities reveals new algebraic and transformation properties that are in agreement with several lines of experimental evidence (such as adaptability of genetic nets and feed-forward/anticipatory processes and observations in evolutionary biology), as well as a wide array of cell genomic and interactomic data for the simpler


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organisms, such as yeast and a nematode (C. elegans) species. In principle – and hopefully soon in practice – such categorical- and Topos-based applications to cell genomes and interactomes will not be limited to the simpler organisms but will also include the much more complex, higher organisms such as Homo sapiens sapiens. Non-linear dynamics of non-random genetic and cell networks can be thus formulated explicitely through categorical constructions enabled by the Łukasiewicz–Moisil Logic algebras that are in principle computable through symbolic programming on existing high performance workstations and supercomputers even for modeling networks composed of huge numbers of interacting ‘biomolecular’ species (Baianu 1983, 1984, 1987a, b; Baianu and Prisecaru 2004). Strategies for meaningful measurements and observations in real, complex biological systems (Baianu 2004b; Baianu et al. 2005), such as individual human organisms, may thus be combined with genomic and proteomic testing on individuals, and may very well lead to optimized, individualized therapies for life-threatening diseases such as cancer and cardiovascular diseases. On the other hand, one has to consider the fact that the problem of computability, or solvability, of complex models increases in difficulty with the number n of ‘truth’ values present in n-valued logics. The categorical notion of representable functor may thus provide an adequate description of the computability concept for genetic nets. The results presented in Sections 4 and 5 also strongly indicate that the genetic nets are not generally equivalent to Turing machines as the simpler neural nets are. Furthermore, the results that were reviewed in Sections 3 to 6 show that only those particular genetic networks which are characterized completely by centered Łukasiewicz–Moisil algebras may indeed possess equivalent Turing machines, with the categorical equivalence being expressed in terms of adjoint functors that are both full and faithful (see Theorem 4.1 in Section 4). The formalization introduced in this paper in terms of categories, functors, higher dimensional algebra and Łukasiewicz–Moisil Topoi does allow one to obtain several additional results that we intend to present in subsequent papers.

ACKNOWLEDGMENTS

This work has been supported by a Leverhulme Emeritus Fellowship, 2002–2004, for R. Brown and an EPRSC Visiting Fellowship

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for J. F. Glazebrook. I. C. Baianu gratefully acknowledges several personal communications from the late Professor Robert Rosen over two decades (1969–1989) that are pertinent to the categorical developments presented here concerning Relational Biology and the Metabolic-Replication processes entailed by Robert Rosen’s settheoretical approach to (M,R)-Systems.

NOTES 1

Recall that DNA stores information in the neucleotide bases A (Adenine), C (Cytosin), G (Guanine), T (Thymine); a triplet of such nucleotides in the DNA sequence is called a ‘codon’ and it may encode the information necessary to specify a single amino acid but the genetic code is a redundant one, without any overlap, quasi-universal and also capable of ‘reverse transcription’ from certain types of RNA back into DNA. Furthermore, not all nucleotide or codon sequences present in DNA are transcribed in vivo. 2 The /i’s are called the Chrysippian endomorphisms of L.

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