BULLETIN OF MATHEMATICALBIOLOGY
VOLUME37 1975
A T H E O R Y FOR ENVIRONMENTAL SYSTEMS
9 CARLOSA. LEGUIZAM6N Gereneia de Investigaciones Comisi6n Nacional de Energia At6mica Av. del Libertador 8250 Buenos Aires, Argentina
A theory for environmental systems is defined on the basis of two elements, termed 'environmental unity' and 'behavior'. Environmental systems are regarded as non-living systems, each one related with only one biological system. We construct a materialenergetic environmental diagram, which is introduced in terms of the theory of categories, thereby giving rise to a new category E. By moans of two biological conditions, and the definition of static property of the biological system (related to its own environment), a set of theorems is obtained, exhibiting mathematical consequences for the represented theory.
1. Fundamental Concepts. The manifestation of life comes from organizational properties which we formalize as a biological system M. This organization provides its own continuous change, such that the whole life of this system involves a total period of time with a succession of different structures corresponding to consecutive short lifetime periods. The organization of the biological system is concerned with the relationships between it and its surrounding medium (Levins, 1968; Odum, 1971), such that these relationships will define the different conditions to be imposed on the environmental system E. This system E will be considered as formed by a non-hying system, such that the relationships between M and E will provide the new elements defined for the theory. As a starting point of the present theory, the following general postulate is assumed: For any period of time a given biological system M, there exists a unique 675
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CAI:CLOSA. LEGUIZAMON
system directly related to M, such that this system is its associated environmental system E. I f we define "exterior world" (W), as the totality of systems different from M and E; then, because of this postulate, W will not have any direct relationship with the biological system M. Thus, a simple diagram will be:
ME W where the arrows are relationships between the systems. Thus, the structure of the environmental system E is defined by the presence of a given biological system M and to the relationships between them (Lotka, 1956; Rashevsky, 1960). A limit must be specified in E in order to obtain the separation between the systems E and W. This environmental limit will be given by general characteristics adopted by the analysis, so that these characteristics m a y have geographical, energetical, chemical, theoretical, etc. properties. With respect to the limit between the systems M and E, this will be given by the respective biological and non-biological characteristic of each system. In order to obtain our theory, we start by defining the following two elements: (a) Environmental unity: a set of elements of identical material physical nature, such that these elements are associated with a determined quantity of energy. (b) Behavior: an oriented action between two environmental unities, such that a transformation (and/or an elimination) to other unities, m a y or m a y not be produced by their interaction. Let us make a few remarks about the definitions. The environmental unity requires a well defined quantity of energy to be j oined to the respective material physical nature, both being required to specify the unity. Thus, more than one unity must be defined if we have a different quantity of energy joined to the same material physical nature. This energy must be considered as of extrinsic characteristic to the material physical nature. An intrinsic energy is also existent, but it is undefinable from a specification of the respective material physical nature. For instance, an environmental unity (defined by a given material physical nature and its respective quantity of extrinsic energy) could directly be transformed in energy. This energy will be recognized by either as a new partial or total extrinsic energy linked to the same or different material physical nature as the original (but these concepts now defining another new unity); or as going to the biological system and/or to the exterior world. Thus, every form of intrinsic energy is not explicitly individualized. With respect to behavior, we consider this to be an oriented action from one
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unity to another unity. This action can be chemical, electrical, mechanical, etc.; that is, of diverse physical characteristics. (Later, our theory will provide theoretical characteristics emerging from the abstract mathematical elements of our representation.) Moreover, the definition assumes the simple possibility of change (transformation) in the material physical nature and/or in the quantity of energy of the elements defining one or both unifies concerned with the behavior. Similarly, a possible elimination is considered when, by this behavior, one or both unities are transferred to M and/or W. With respect to the selection of elements of each unity, this must be consistent with the environmental objectives that we are trying to obtain. Thus, adoption of a certain material physical standpoint does not, in general, preclude the possibility of other characterizations than those used for the definition of elements in the correct environmental unity. In a first case, a more elemental conception will give new behaviors following the intrinsic (in certain eases, physical) laws inherent to the material physical nature of the elements of the real environmental unity that must be defined. In other cases, when the structure is not too elemental, the behaviors will follow an intricate arrangement of unities which are not necessary for the environmental representation. In a second case (i.e. for a selection of elements of more coarse structures than the real one), we shall have a loss of information, because certain behaviors that are not representable within this coarse selection are abstracted out. This occurs in general, when the real material physical nature of elements has not been recognized in the environmental process.
2. Diagrams. In order to specify the arrangement of environmental unities and behaviors we have to define the necessary conditions for a first diagram which we shall call environmental diagram (E.D.). In this diagram, the environmental unities will be considered as simple sets, and the behaviors as simple mappings between sets. We shall call d(f) and r(f) the domain and range of the behavior represented by the mapping f. The E.D. for a given environmental system E, will be obtained from the following assumptions: (1) The environmental unities in E.D. representing the system E will be: (a) The unities B~ defined by the domain or the range of transfers between the environmental system E and the biological system M. (b) The unities B b defined by the domain or the range of behaviors connecting directly or through other unities, with the unities defined by (a). (c) The unities Bc defined by elements of identical material physical nature,
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CARLOS A. LEGUIZAMON
and quantity of energy, to the elements of unities defined by (a) or (b), but when in the system E there exist other elements which are not defined by means of the domain or the range of those transfers or behaviors giving the unities of the form (a) or (b). (d) The unities Bd defined by the union of the unities of the form Ba, B~ and B~; such that the material physical nature and quantity of energy defining B~ are the same as those giving the definition of B a, B~ and B~. (e) The unities Be defined for each unity of the form B~; such that each Be is defined by an element of theoretically empty material physical nature and quantity of energy.
(2) The behaviors in E.D. representing the system E will determine through their domains and ranges, the unities connected by each one of them, and these behaviors will be: (a) The possible behaviors connecting unities according to assumption
(l-a). (b) The behaviors defining and connecting unities according to assumption (l-b). (c) The oriented behaviors connecting from each one of unities B d to each unity of the form B a, Bb and Be, such that for each Ba, this unity and its respective B~, B b and B c have elements of the same material physical nature and quantity of energy. (d) The theoretical behaviors connecting each unity of the form Be with its respective unity of the form B e. At this moment we must call attention to the fact that these two previous assumptions embody a certain general idea about the distinct future possibilities of this theory, from the viewpoint of abstract mathematica~ representation and the objectives of Relational Biology, in general. With this in mind, we adopt assumptions (l-e) and (2-d), in such a way that certain unities Be may represent the possible expansion of the environment through the unities Be. In other cases, unities Be may be considered as determinative of different factors inherent to the represented environment; such that, these factors would permit (or not) in E, the realizability of a given unities performing theoretical behaviors with unities Be. It is necessary to stress the fact that the assumption (2-d) does not consider a defined orientation for theoretical behaviors. This orientation will come (it is supposed) from the character assigned to the respective unity B e. (In the following Figures 1, 2 and 4 the orientation will only be conventional.) At this
A T H E O R Y F O R E N V I R O N M E N T A L SYSTEMS
679
moment, it is necessary to stress the fact that other possible environmental theories can be developed on the basis of the two previously defined elements, using other sets of assumptions. In order to give an idea about the applicability of our assumptions, we have the following examples: 1. We consider a chemical reaction, such that its reactants Bbl and Bb2 give the product Ba3 going to the biological system. Moreover we consider certain elements forming the unities Be1 and Bc~ which do not undergo the reaction to give B~. In fact, Bc~ has elements of the same material physical nature and quantity of energy as Bb~, and the same is true for Bc~ with respect to Bb2, and Bbl and Bb~ are defined by the assumption (l-b) and Br and B~ are defined by the assumption (l-c).
Bd.•.fBc,
gt
'~
Be,
M
B~2 gz
Figure I
x12: Behavior representing the mechanical action which puts the elements of Bbl in chemical reaction with the elements of Bb2. x21: Similarly for B~2 with Bbl. y: Behavior giving the transformation of Bbl to B~8. z: Behavior giving the transformation of Bb~ to Ba3. gl and g2: Theoretical behaviors. fl, f*, f2 and f*: Behaviors defined by (2-e). 2. Electromagnetic radiation from the exterior world W interacts with a fluorescent substance. A certain quantity of this substance is consumed by a biological system M, which recognizes the fluorescent radiation.
680
CARLOS A. LEGUIZAMON W
I
B3 " ~ - ' - - - B 2
B8
B7 "~"
/
/
/
Figure 2 Let us say a few words about this diagram. The total fluorescent substance in E is represented by B o. Only B 1 receives the electromagnetic radiation from W. Bz is a fluorescent substance that does not receive radiation. B 3 is the theoretical unity. B 1 produces B 4, such that the excitation state of electrons in, B4 is greater than in B 1. Moreover, the extrinsic energy considered in B 4 is different from the energy which is exciting the electrons. B4 emits fluorescent light entering to M, while B 4 is transformed to B5 with the same state given by the substance in B1, b u t with different quantity of extrinsic energy. A certain quantity of B5 is transferred to M as B 6. The part which is not transferred to M is the unity BT, and B 8 is the theoretical unity. We may consider the environmental diagram as a simple graph, such that the nodes of this graph will represent environmental unities and the oriented edges joining two nodes will represent behaviors. We shall now try to develop a new diagram from the double character of material and energy for each environmental unity. Thus, the original environmental unity B~ will be represented in terms of unities of the form Y~ and E~ defined in the following manner:
YV Material environmental unity, which determines the exclusive material physical nature of B~.
E~: Energetieal environmental unity, which determines the exclusive extrinsical energy of B~. In a similar form, each behavior in the environmental diagram will be considered as behaviors of the form a~j and a~j, such that:
a~i: Material behavior, which represents the exclusive material part of the original behavior, so that its range determines a material environmental unity.
a~j: Energetical behavior, which represents the exclusive energetical part of the
A T H E O R Y FOR ENVIRONMENTAL SYSTEMS
681
original behavior, so that its range determines an energetical environmental unity. We must make a few remarks concerning these new elements. Y~ must be seen as a simple set in a real mathematical sense, such that its caxdinality will be the same as that of B~ in E.D. El' must be regarded as a simple set of only one element. This element will represent a well defined value of energy linked to the elements of Y~. I f more than one energetical value corresponds to each Y~, we shall have more than one unity E~. In a mathematical sense, the linkage between material and energy will be given b y the cartesian product operation, such that the original unity B~ will now be represented as: Y~
x
E~
x
E~
x...
x
EX_ 1
={(y~,el, e2. . . . . en_l):y~eY~
and
ejeE~,
(j = 1,2 . . . . . n -
1)}.
(1)
The expression (1) must be seen independent of the order of factors up to isomorphism, and it follows from associativity that:
Y, • I--[ E~, ~ 1--[ E~, • Y, "~ E ~ • Y, ieJ
ieJ
• ~
E~'~_ Y~ • E ~ x 1-[ E~,.
tell
(2)
jell
It is noted that:
Hu{k}-~J,
k~H.
With respect to material and energetical behaviors, let us consider the following E.D.: B1 ~12> B2. This diagram will be transformed into a new diagram, which we shall call material-energetic environmental diagram (M.E.E.D.), according to the rules given above: QY
12
(2e'
Yz
Yl
X
E) ,-E
Figure 3
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CARLOS A. LEGUIZAMON
The domain of each behavior will be defined b y the product:
• 1-I
(3)
teJ
and the range of each behavior defines the material unity (Y2), or the energetical unity ( E ~ J , j = 1, 2 . . . . . n - 1). At this point we recall that similar developments were considered in a previous paper (Leguizam6n, 1975), in the context of the representation of biological systems. From the mathematical properties of the cartesian product, we can define some special kinds of material and energetical unities and material and energetical behaviors. Thus, we shall define the one-point sets of the form: lr' = {Yo} and
1~ = {%}
such that Yo and eo will be represented b y the ordinal O. In fact, I r, and 1E~ will be the theoretical material unity of zero material physical nature and the theoretical energetical unity of zero energetical value. These new unities will be considered as identities under the cartesian product operation, with the bijections: 1r, x Y , - - ~
Y, D,
Y~ x 1r,
1E~ • E~ -~> E~ 2) with behaviors of the form: either (4-b), (4-c) and (4-d); or (4-c); or (4-c) and (4-d); or (4-d); or (4-b) and (4-c); or (4-b) and (4-d) (b) one behavior of the form (4-b) can give a n-tuple (n /> 2) with behaviors of the form: either (4-e) and (4-d); or (4-c); or (4-d). (6) The environmental unities defined by the range of behaviors of the same n-tuple, perform the cartesian product. From these assumptions we note the following: (a) the same general assumptions given for E.D., (b) cardinality of each Y~ equal to the cardinality of B~, (c) the one-point sets E~, (d) the one-point sets I r, and 1s~, (e) the theoretical unity lr, • 1-IsGs ls?J representing in M.E.E.D., Be of E.D., (f) theoretical material and theoretical energetical behaviors and their different arrangements. In order to have a material-energetic environmental diagram, we consider the diagram corresponding to example 2 above: w I I
(t
~y ~,Y~I
2 X
eo
X Y
~OI
o;X\ fx N ,,, ,-
,Y,
ltE~ I
8
re
rE?
\
1
Eg Figure 4
/
-.
\
/"
A THEORY FOR ENVIRONMENTAL SYSTEMS
685
The incident energy from the exterior world W defines E~. We suppose that Yo, y eo and Y1 and Y2 have the same original energy, and b y this, we have (ao~aol) eo (a~2, aDs). The behaviors a~l a~2, a~s and a~v may be considered as mechanical separation from Yo x E~ and Y5 x E~. We do not consider the one-point sets lr, and 1E~ when they are not required.
3. Representation by the Theory of Categories. In order to have a representation for the environmental theory, we shall give a set of axioms on the collections of objects {G} and morphisms {g} in a category (MacLane, 1971; Mitchell, 1965), that will comprise a material-energetic environmental diagram. These are: (1) For each object G~ of the collection {G}, there exists an object la~ = {0} of the same collection {G}, performing: 1a, x G~_ G ~
G~ • lat
(2) In the collection of objects (G) may exist objects le~ (i = 1, 2 , . . . , n), performing: 1G1 • 162 • 2 1 5
la. _- 1Go,
such that objects of the form G of the collection {G} may not produce the cartesian product with 1%. (3) Each range of each morphism gt e {g} of each n-tuple (n t> 2), defines an object G~ or 1v, of the collection {G}, so that if n morphisms form the n-tuple (gl, g2. . . . , gn), then each range will define: either (a) objects G~ = r(g~), for (i = 1, 2 , . . . , n), such that these objects perform: G1 x G2 x . . .
x 6/.
o r ( b ) objects Gt =r(g~), for i = 1 , 2 , . . . , / % and objects tej for j = k + 1, k + 2 , . . . , n; such that in the collection {G} there exists one or more than one objects Gn+ l, performing: G 1 X G2...
X G k X 1G1r
la~+~ x . . .
X
x le- x G,+ 1
or (c) objects la, = r(.q~), for i = 1, 2, . . . , n: such that in the collection {G} there exists objects G~ for the same i = 1, 2 , . . . , n; performing the axiom 1, such that: la~
x
Gi
x
I~2
x
Gg.
x...
x
I~
x
G~
or (d) objects 1~, = r(g~), for i = l, 2 . . . . , n; performing the axiom 2, but when the domain of gl, g 2 , . . . , gn does not perform this identical arrangement.
686
CARLOSA. LEGUIZAMON
(4) The domain of morphisms g,, g j , . . . , gm of one or more than one n-tuple (n I> 2) is defined by similar objects and arrangements as those were given by eases a, b and c of axiom 3. The case (3-d) m a y be performed as a domain of g~, gj . . . . , gin, when the ranges of these morphisms do not perform identical arrangement. (5) In the collection of objects (G}, objects of the form (7, (i = 1, 2 , . . . , n) m a y or m a y not exist, performing G1 x G2 x . . . x G, for n /> 2, such that each object G, and its 1G,given by the axiom 1, are not defined by any domain or range of morphism of the collection (g} different from the identity morphism. A category satisfying the above axioms, will be called an environmental category E. The first axiom introduces two kinds of objects in E. Thus, objects of the form G, correspond to unities Y, and E~ in M.E.E.D. Objects 1c, correspond to one-point sets 1r, and 1sf whose unique element was defined by the ordinal 0. The previous defined identity is given by the expression: 1Q, • G, ~ G, ~_ G, x le,.
Thus, the assumptions (2), (3-a) and (3-b) defining M.E.E.D., are, in a general form, given by this axiom. The second axiom expresses the possibility of existence of objects I e, (i = 1, 2 , . . . , n) related by a cartesian product. In fact, this axiom introduces the original unity Be of E.D., such that, this unity was represented by assumption (3.c-4) in M.E.E.D. We must stress the fact that the axiom introduces only the "possibility" of I e, (i = 1, 2, . . . , n) because for certain representations of environmental systems, the original unities B e of E.D. cannot exist (B e are due to the presence of unities Be). Moreover, as cartesian products of the form G x 1% m a y occur in E, the axiom introduces the possibility of non existence of this previous arrangement, if the product leo = lal • la2 x -. 9 x la~ must be adopted alone. The axioms (3) and (4) incorporate similar characteristics in the arrangement between objects orE. It is necessary to stress the fact, that by (3-a), (3-b) and (3-c), it is possible to define an n-tuple of ranges, which cartesian product is only a subproduct of a greater number of factors. For instance, for each object G, of (3-a), there will be defined another object 1G, (axiom 1), and these objects were not defined by ranges of n-tuples. Also, by (3-b), it is possible to have one or more than one object of the form G~+I (for instance E~ in figure 4) and this object is not defined by ranges. Moreover, with respect to (3-c), we have the same for each G~because the n-tuple will follow similar arrangement of only one theoretical material behavior together to n - 1 theoretical energetical behaviors of M.E.E.D.
A THEORY FOR ENVIRONMENTAL SYSTEMS
687
The part (3-d) introduces the general condition that between objects lao there are no morphisms. The same is given in axiom 4. Axiom (3) introduces in E, the assumption (6) of M.E.E.D., and this is given by the different defined cartesian products. The fifth axiom permits cartesian products in E, such that their factors (objects) are not defined by any domain or range of morphisms of E (different from the identity morphism). The presence of these objects is due either to transfers between M and E (without morphisms with another object of E), or to similar transfers to the previous one, but also going or coming from the system W. We must keep in mind that our objects will be simple sets, and the morphisms will be simple set theoretic mappings. Thus, our category E will be a subcategory S of all sets. Moreover, E will represent the environmental system, such that this system satisfies the general postulate and the assumptions to get YI.E.E.D. 4. Some Consequences of the Environmental Theory. We adopt the following conditions for biological systems: M.1--- Every biological system has, at least, one input and one output. M.2-- In the biological system the inputs are connected to the outputs. Similarly, we specify the condition of non-living systems for the system represented by the present environmental theory: E. 1 --No biological system is considered the environment of another biological system. THEOREM 1. Every environmental category E haz, at least, two objects G1 and G2, such that G1 x G2 r E. Proof. In order to have existence, every biological system transforms many of its inputs of a given material physical nature and quantity of energy, into outputs of other material physical natures or quantity of energy. (The connections between inputs and outputs are imposed by M.2.) Thus, the proof of this theorem is immediate. This comes from these previous remarks, and from the conditions given by: M. 1, the general postulate, the assumptions in order to obtain E.D. and M.E.E.D., and the axioms giving the category E. THEOR~ 2. I f an environmental category E has a set Gm of all the objects which are not defined by transfers between M and E, and which do not perform cartesian product operation with other objects defined by transfers; then, the category E will not be a discrete category. Proof. The set Gm is restrained to objects which do not perform a cartesian
688
CARLOS A. LEGLrlZAMON
product operation with other objects defined by transfers. In fact, this condition is imposed on Gm, because this possibility can be feasible. Thus, by means of this condition, and from the general construction of the environmental theory and its respective category E, each object of Gm will have morphisms connecting it (directly or through other objects of Gin) with objects which do not belong to Gm. Then, the present category E will not be a discrete category. It is necessary to stress the fact, that a given category E m a y not be discrete, but the set of objects Gm -- ~ . Definition. Given the whole life of a biological system M, where each consecutive time period determines each system M s, M 2 , . . . , M n respectively, and such that these systems are related with their respective environmental systems El, E2, . . . , E.; then the biological system M will be environmentally static if each system E s (j e I, I = 1, 2 , . . . , n) has the unities defined by transfers from M s to E~ for every i < j, i e I; and Ej has the unities which must be used by transfers from E k to M~ for every/c > j, k e I. THEOREM 3. Given a biological system M, with consecutive time periods defining the respective systems M1, M2, 9 9 M~ for the whole life of M, where these systems are related with their respective environmental systems E~, E2 . . . . , E~; and such that: (a) M is an environmentally static system, (b) the transfers between each pair M f - E j (j e I, I = l, 2 , . . . , n) define material environmental unities transferred from and to M~;
then, each category E s representing each system Ej, will not be a discrete category. Proof. We note some simple considerations with respect to the previous conditions. By means of the definition of M as environmentally static system, each system Ej will have the unities defined by transfers from M~ to E~ for every i < j, i e I; and Ej will have the unities which must be used by transfers from E~ to M~ for every ~ > j, k e I. But these transfers m a y be energetical, exclusively; in such a manner that they may be determined by energetical environmental unities which m a y have a cartesian product operation with material environmental unities which may be the same through the total chain of environmental systems. In this case, certain categories Ej may be discrete. These considerations are not valid when the condition (b) is included; such that, this condition together with the definition of an environmentally static system, will produce in each system Ej: other material environmental unities defined by transfers from Mt to E~ (i < j, i e I), and other material environmental unities which must be used as transfers from E~ to M~ (]r > j, k e I). Thus, by using
A THEORY FOR ENVIRONMENTAL SYSTEMS
689
Theorem 2, these last two sets of unities will belong to a set Gm of objects in each category Ej, such that b y the same theorem, each category E~ will not be a discrete category. As a remark about the consequences of the environmentally static property, we must say that the definition of different environmental unities in each system Ej depends on the environmental limit, which will be defined by the geographical, energetical, chemical, theoretical, etc., characteristics adopted. In fact, we may recognize the environmental static property in M, when, during the whole life of M, the limit is fixed, or when the change of limit does not affect unities and behaviors under the conditions of the above theorems. I wish to express my gratitude to Robert Rosen, who was my research director and from whom I received the best in order to obtain the present theory. This paper was prepared while the author was a visitor at the Center for Theoretical Biology, State University of New York at Buffalo, U.S.A.; whose hospitality is acknowledged. This work was made possible by a Fellowship from the Consejo Nacional de Investigaciones Cienttficas y T~cnieas of the Repfiblica Argentina. LITERATURE Leguizam6n, C.A. 1975. "Concept of Energy in Biological Systems." Bull..Math. Biology, 37, 565-572. Levins, R. 1968. Evolution in Changing Environments. Princeton: Princeton University Press. Lotka, A.J. 1956. Element~o/Mathematical Biology. New York: Dover Publications. MacLane, S. 1971. Categories/orthe Working Mathematician. New York: Springer. Mitchell, B. 1965. Theoryo/Categories. New York: Academic Press. Odum, E.P. 1971. Fundamentals of Ecology. Philadelphia: W. B. Saunders. Rashevsky, N. 1960. Mathematical Biophysics. New York: Dover Publications. RECEl~D 4-1-74 Rv.WS~D 3-12-75
VOLUME37 1975
A T H E O R Y FOR ENVIRONMENTAL SYSTEMS
9 CARLOSA. LEGUIZAM6N Gereneia de Investigaciones Comisi6n Nacional de Energia At6mica Av. del Libertador 8250 Buenos Aires, Argentina
A theory for environmental systems is defined on the basis of two elements, termed 'environmental unity' and 'behavior'. Environmental systems are regarded as non-living systems, each one related with only one biological system. We construct a materialenergetic environmental diagram, which is introduced in terms of the theory of categories, thereby giving rise to a new category E. By moans of two biological conditions, and the definition of static property of the biological system (related to its own environment), a set of theorems is obtained, exhibiting mathematical consequences for the represented theory.
1. Fundamental Concepts. The manifestation of life comes from organizational properties which we formalize as a biological system M. This organization provides its own continuous change, such that the whole life of this system involves a total period of time with a succession of different structures corresponding to consecutive short lifetime periods. The organization of the biological system is concerned with the relationships between it and its surrounding medium (Levins, 1968; Odum, 1971), such that these relationships will define the different conditions to be imposed on the environmental system E. This system E will be considered as formed by a non-hying system, such that the relationships between M and E will provide the new elements defined for the theory. As a starting point of the present theory, the following general postulate is assumed: For any period of time a given biological system M, there exists a unique 675
676
CAI:CLOSA. LEGUIZAMON
system directly related to M, such that this system is its associated environmental system E. I f we define "exterior world" (W), as the totality of systems different from M and E; then, because of this postulate, W will not have any direct relationship with the biological system M. Thus, a simple diagram will be:
ME W where the arrows are relationships between the systems. Thus, the structure of the environmental system E is defined by the presence of a given biological system M and to the relationships between them (Lotka, 1956; Rashevsky, 1960). A limit must be specified in E in order to obtain the separation between the systems E and W. This environmental limit will be given by general characteristics adopted by the analysis, so that these characteristics m a y have geographical, energetical, chemical, theoretical, etc. properties. With respect to the limit between the systems M and E, this will be given by the respective biological and non-biological characteristic of each system. In order to obtain our theory, we start by defining the following two elements: (a) Environmental unity: a set of elements of identical material physical nature, such that these elements are associated with a determined quantity of energy. (b) Behavior: an oriented action between two environmental unities, such that a transformation (and/or an elimination) to other unities, m a y or m a y not be produced by their interaction. Let us make a few remarks about the definitions. The environmental unity requires a well defined quantity of energy to be j oined to the respective material physical nature, both being required to specify the unity. Thus, more than one unity must be defined if we have a different quantity of energy joined to the same material physical nature. This energy must be considered as of extrinsic characteristic to the material physical nature. An intrinsic energy is also existent, but it is undefinable from a specification of the respective material physical nature. For instance, an environmental unity (defined by a given material physical nature and its respective quantity of extrinsic energy) could directly be transformed in energy. This energy will be recognized by either as a new partial or total extrinsic energy linked to the same or different material physical nature as the original (but these concepts now defining another new unity); or as going to the biological system and/or to the exterior world. Thus, every form of intrinsic energy is not explicitly individualized. With respect to behavior, we consider this to be an oriented action from one
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677
unity to another unity. This action can be chemical, electrical, mechanical, etc.; that is, of diverse physical characteristics. (Later, our theory will provide theoretical characteristics emerging from the abstract mathematical elements of our representation.) Moreover, the definition assumes the simple possibility of change (transformation) in the material physical nature and/or in the quantity of energy of the elements defining one or both unifies concerned with the behavior. Similarly, a possible elimination is considered when, by this behavior, one or both unities are transferred to M and/or W. With respect to the selection of elements of each unity, this must be consistent with the environmental objectives that we are trying to obtain. Thus, adoption of a certain material physical standpoint does not, in general, preclude the possibility of other characterizations than those used for the definition of elements in the correct environmental unity. In a first case, a more elemental conception will give new behaviors following the intrinsic (in certain eases, physical) laws inherent to the material physical nature of the elements of the real environmental unity that must be defined. In other cases, when the structure is not too elemental, the behaviors will follow an intricate arrangement of unities which are not necessary for the environmental representation. In a second case (i.e. for a selection of elements of more coarse structures than the real one), we shall have a loss of information, because certain behaviors that are not representable within this coarse selection are abstracted out. This occurs in general, when the real material physical nature of elements has not been recognized in the environmental process.
2. Diagrams. In order to specify the arrangement of environmental unities and behaviors we have to define the necessary conditions for a first diagram which we shall call environmental diagram (E.D.). In this diagram, the environmental unities will be considered as simple sets, and the behaviors as simple mappings between sets. We shall call d(f) and r(f) the domain and range of the behavior represented by the mapping f. The E.D. for a given environmental system E, will be obtained from the following assumptions: (1) The environmental unities in E.D. representing the system E will be: (a) The unities B~ defined by the domain or the range of transfers between the environmental system E and the biological system M. (b) The unities B b defined by the domain or the range of behaviors connecting directly or through other unities, with the unities defined by (a). (c) The unities Bc defined by elements of identical material physical nature,
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CARLOS A. LEGUIZAMON
and quantity of energy, to the elements of unities defined by (a) or (b), but when in the system E there exist other elements which are not defined by means of the domain or the range of those transfers or behaviors giving the unities of the form (a) or (b). (d) The unities Bd defined by the union of the unities of the form Ba, B~ and B~; such that the material physical nature and quantity of energy defining B~ are the same as those giving the definition of B a, B~ and B~. (e) The unities Be defined for each unity of the form B~; such that each Be is defined by an element of theoretically empty material physical nature and quantity of energy.
(2) The behaviors in E.D. representing the system E will determine through their domains and ranges, the unities connected by each one of them, and these behaviors will be: (a) The possible behaviors connecting unities according to assumption
(l-a). (b) The behaviors defining and connecting unities according to assumption (l-b). (c) The oriented behaviors connecting from each one of unities B d to each unity of the form B a, Bb and Be, such that for each Ba, this unity and its respective B~, B b and B c have elements of the same material physical nature and quantity of energy. (d) The theoretical behaviors connecting each unity of the form Be with its respective unity of the form B e. At this moment we must call attention to the fact that these two previous assumptions embody a certain general idea about the distinct future possibilities of this theory, from the viewpoint of abstract mathematica~ representation and the objectives of Relational Biology, in general. With this in mind, we adopt assumptions (l-e) and (2-d), in such a way that certain unities Be may represent the possible expansion of the environment through the unities Be. In other cases, unities Be may be considered as determinative of different factors inherent to the represented environment; such that, these factors would permit (or not) in E, the realizability of a given unities performing theoretical behaviors with unities Be. It is necessary to stress the fact that the assumption (2-d) does not consider a defined orientation for theoretical behaviors. This orientation will come (it is supposed) from the character assigned to the respective unity B e. (In the following Figures 1, 2 and 4 the orientation will only be conventional.) At this
A T H E O R Y F O R E N V I R O N M E N T A L SYSTEMS
679
moment, it is necessary to stress the fact that other possible environmental theories can be developed on the basis of the two previously defined elements, using other sets of assumptions. In order to give an idea about the applicability of our assumptions, we have the following examples: 1. We consider a chemical reaction, such that its reactants Bbl and Bb2 give the product Ba3 going to the biological system. Moreover we consider certain elements forming the unities Be1 and Bc~ which do not undergo the reaction to give B~. In fact, Bc~ has elements of the same material physical nature and quantity of energy as Bb~, and the same is true for Bc~ with respect to Bb2, and Bbl and Bb~ are defined by the assumption (l-b) and Br and B~ are defined by the assumption (l-c).
Bd.•.fBc,
gt
'~
Be,
M
B~2 gz
Figure I
x12: Behavior representing the mechanical action which puts the elements of Bbl in chemical reaction with the elements of Bb2. x21: Similarly for B~2 with Bbl. y: Behavior giving the transformation of Bbl to B~8. z: Behavior giving the transformation of Bb~ to Ba3. gl and g2: Theoretical behaviors. fl, f*, f2 and f*: Behaviors defined by (2-e). 2. Electromagnetic radiation from the exterior world W interacts with a fluorescent substance. A certain quantity of this substance is consumed by a biological system M, which recognizes the fluorescent radiation.
680
CARLOS A. LEGUIZAMON W
I
B3 " ~ - ' - - - B 2
B8
B7 "~"
/
/
/
Figure 2 Let us say a few words about this diagram. The total fluorescent substance in E is represented by B o. Only B 1 receives the electromagnetic radiation from W. Bz is a fluorescent substance that does not receive radiation. B 3 is the theoretical unity. B 1 produces B 4, such that the excitation state of electrons in, B4 is greater than in B 1. Moreover, the extrinsic energy considered in B 4 is different from the energy which is exciting the electrons. B4 emits fluorescent light entering to M, while B 4 is transformed to B5 with the same state given by the substance in B1, b u t with different quantity of extrinsic energy. A certain quantity of B5 is transferred to M as B 6. The part which is not transferred to M is the unity BT, and B 8 is the theoretical unity. We may consider the environmental diagram as a simple graph, such that the nodes of this graph will represent environmental unities and the oriented edges joining two nodes will represent behaviors. We shall now try to develop a new diagram from the double character of material and energy for each environmental unity. Thus, the original environmental unity B~ will be represented in terms of unities of the form Y~ and E~ defined in the following manner:
YV Material environmental unity, which determines the exclusive material physical nature of B~.
E~: Energetieal environmental unity, which determines the exclusive extrinsical energy of B~. In a similar form, each behavior in the environmental diagram will be considered as behaviors of the form a~j and a~j, such that:
a~i: Material behavior, which represents the exclusive material part of the original behavior, so that its range determines a material environmental unity.
a~j: Energetical behavior, which represents the exclusive energetical part of the
A T H E O R Y FOR ENVIRONMENTAL SYSTEMS
681
original behavior, so that its range determines an energetical environmental unity. We must make a few remarks concerning these new elements. Y~ must be seen as a simple set in a real mathematical sense, such that its caxdinality will be the same as that of B~ in E.D. El' must be regarded as a simple set of only one element. This element will represent a well defined value of energy linked to the elements of Y~. I f more than one energetical value corresponds to each Y~, we shall have more than one unity E~. In a mathematical sense, the linkage between material and energy will be given b y the cartesian product operation, such that the original unity B~ will now be represented as: Y~
x
E~
x
E~
x...
x
EX_ 1
={(y~,el, e2. . . . . en_l):y~eY~
and
ejeE~,
(j = 1,2 . . . . . n -
1)}.
(1)
The expression (1) must be seen independent of the order of factors up to isomorphism, and it follows from associativity that:
Y, • I--[ E~, ~ 1--[ E~, • Y, "~ E ~ • Y, ieJ
ieJ
• ~
E~'~_ Y~ • E ~ x 1-[ E~,.
tell
(2)
jell
It is noted that:
Hu{k}-~J,
k~H.
With respect to material and energetical behaviors, let us consider the following E.D.: B1 ~12> B2. This diagram will be transformed into a new diagram, which we shall call material-energetic environmental diagram (M.E.E.D.), according to the rules given above: QY
12
(2e'
Yz
Yl
X
E) ,-E
Figure 3
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CARLOS A. LEGUIZAMON
The domain of each behavior will be defined b y the product:
• 1-I
(3)
teJ
and the range of each behavior defines the material unity (Y2), or the energetical unity ( E ~ J , j = 1, 2 . . . . . n - 1). At this point we recall that similar developments were considered in a previous paper (Leguizam6n, 1975), in the context of the representation of biological systems. From the mathematical properties of the cartesian product, we can define some special kinds of material and energetical unities and material and energetical behaviors. Thus, we shall define the one-point sets of the form: lr' = {Yo} and
1~ = {%}
such that Yo and eo will be represented b y the ordinal O. In fact, I r, and 1E~ will be the theoretical material unity of zero material physical nature and the theoretical energetical unity of zero energetical value. These new unities will be considered as identities under the cartesian product operation, with the bijections: 1r, x Y , - - ~
Y, D,
Y~ x 1r,
1E~ • E~ -~> E~ 2) with behaviors of the form: either (4-b), (4-c) and (4-d); or (4-c); or (4-c) and (4-d); or (4-d); or (4-b) and (4-c); or (4-b) and (4-d) (b) one behavior of the form (4-b) can give a n-tuple (n /> 2) with behaviors of the form: either (4-e) and (4-d); or (4-c); or (4-d). (6) The environmental unities defined by the range of behaviors of the same n-tuple, perform the cartesian product. From these assumptions we note the following: (a) the same general assumptions given for E.D., (b) cardinality of each Y~ equal to the cardinality of B~, (c) the one-point sets E~, (d) the one-point sets I r, and 1s~, (e) the theoretical unity lr, • 1-IsGs ls?J representing in M.E.E.D., Be of E.D., (f) theoretical material and theoretical energetical behaviors and their different arrangements. In order to have a material-energetic environmental diagram, we consider the diagram corresponding to example 2 above: w I I
(t
~y ~,Y~I
2 X
eo
X Y
~OI
o;X\ fx N ,,, ,-
,Y,
ltE~ I
8
re
rE?
\
1
Eg Figure 4
/
-.
\
/"
A THEORY FOR ENVIRONMENTAL SYSTEMS
685
The incident energy from the exterior world W defines E~. We suppose that Yo, y eo and Y1 and Y2 have the same original energy, and b y this, we have (ao~aol) eo (a~2, aDs). The behaviors a~l a~2, a~s and a~v may be considered as mechanical separation from Yo x E~ and Y5 x E~. We do not consider the one-point sets lr, and 1E~ when they are not required.
3. Representation by the Theory of Categories. In order to have a representation for the environmental theory, we shall give a set of axioms on the collections of objects {G} and morphisms {g} in a category (MacLane, 1971; Mitchell, 1965), that will comprise a material-energetic environmental diagram. These are: (1) For each object G~ of the collection {G}, there exists an object la~ = {0} of the same collection {G}, performing: 1a, x G~_ G ~
G~ • lat
(2) In the collection of objects (G) may exist objects le~ (i = 1, 2 , . . . , n), performing: 1G1 • 162 • 2 1 5
la. _- 1Go,
such that objects of the form G of the collection {G} may not produce the cartesian product with 1%. (3) Each range of each morphism gt e {g} of each n-tuple (n t> 2), defines an object G~ or 1v, of the collection {G}, so that if n morphisms form the n-tuple (gl, g2. . . . , gn), then each range will define: either (a) objects G~ = r(g~), for (i = 1, 2 , . . . , n), such that these objects perform: G1 x G2 x . . .
x 6/.
o r ( b ) objects Gt =r(g~), for i = 1 , 2 , . . . , / % and objects tej for j = k + 1, k + 2 , . . . , n; such that in the collection {G} there exists one or more than one objects Gn+ l, performing: G 1 X G2...
X G k X 1G1r
la~+~ x . . .
X
x le- x G,+ 1
or (c) objects la, = r(.q~), for i = 1, 2, . . . , n: such that in the collection {G} there exists objects G~ for the same i = 1, 2 , . . . , n; performing the axiom 1, such that: la~
x
Gi
x
I~2
x
Gg.
x...
x
I~
x
G~
or (d) objects 1~, = r(g~), for i = l, 2 . . . . , n; performing the axiom 2, but when the domain of gl, g 2 , . . . , gn does not perform this identical arrangement.
686
CARLOSA. LEGUIZAMON
(4) The domain of morphisms g,, g j , . . . , gm of one or more than one n-tuple (n I> 2) is defined by similar objects and arrangements as those were given by eases a, b and c of axiom 3. The case (3-d) m a y be performed as a domain of g~, gj . . . . , gin, when the ranges of these morphisms do not perform identical arrangement. (5) In the collection of objects (G}, objects of the form (7, (i = 1, 2 , . . . , n) m a y or m a y not exist, performing G1 x G2 x . . . x G, for n /> 2, such that each object G, and its 1G,given by the axiom 1, are not defined by any domain or range of morphism of the collection (g} different from the identity morphism. A category satisfying the above axioms, will be called an environmental category E. The first axiom introduces two kinds of objects in E. Thus, objects of the form G, correspond to unities Y, and E~ in M.E.E.D. Objects 1c, correspond to one-point sets 1r, and 1sf whose unique element was defined by the ordinal 0. The previous defined identity is given by the expression: 1Q, • G, ~ G, ~_ G, x le,.
Thus, the assumptions (2), (3-a) and (3-b) defining M.E.E.D., are, in a general form, given by this axiom. The second axiom expresses the possibility of existence of objects I e, (i = 1, 2 , . . . , n) related by a cartesian product. In fact, this axiom introduces the original unity Be of E.D., such that, this unity was represented by assumption (3.c-4) in M.E.E.D. We must stress the fact that the axiom introduces only the "possibility" of I e, (i = 1, 2, . . . , n) because for certain representations of environmental systems, the original unities B e of E.D. cannot exist (B e are due to the presence of unities Be). Moreover, as cartesian products of the form G x 1% m a y occur in E, the axiom introduces the possibility of non existence of this previous arrangement, if the product leo = lal • la2 x -. 9 x la~ must be adopted alone. The axioms (3) and (4) incorporate similar characteristics in the arrangement between objects orE. It is necessary to stress the fact, that by (3-a), (3-b) and (3-c), it is possible to define an n-tuple of ranges, which cartesian product is only a subproduct of a greater number of factors. For instance, for each object G, of (3-a), there will be defined another object 1G, (axiom 1), and these objects were not defined by ranges of n-tuples. Also, by (3-b), it is possible to have one or more than one object of the form G~+I (for instance E~ in figure 4) and this object is not defined by ranges. Moreover, with respect to (3-c), we have the same for each G~because the n-tuple will follow similar arrangement of only one theoretical material behavior together to n - 1 theoretical energetical behaviors of M.E.E.D.
A THEORY FOR ENVIRONMENTAL SYSTEMS
687
The part (3-d) introduces the general condition that between objects lao there are no morphisms. The same is given in axiom 4. Axiom (3) introduces in E, the assumption (6) of M.E.E.D., and this is given by the different defined cartesian products. The fifth axiom permits cartesian products in E, such that their factors (objects) are not defined by any domain or range of morphisms of E (different from the identity morphism). The presence of these objects is due either to transfers between M and E (without morphisms with another object of E), or to similar transfers to the previous one, but also going or coming from the system W. We must keep in mind that our objects will be simple sets, and the morphisms will be simple set theoretic mappings. Thus, our category E will be a subcategory S of all sets. Moreover, E will represent the environmental system, such that this system satisfies the general postulate and the assumptions to get YI.E.E.D. 4. Some Consequences of the Environmental Theory. We adopt the following conditions for biological systems: M.1--- Every biological system has, at least, one input and one output. M.2-- In the biological system the inputs are connected to the outputs. Similarly, we specify the condition of non-living systems for the system represented by the present environmental theory: E. 1 --No biological system is considered the environment of another biological system. THEOREM 1. Every environmental category E haz, at least, two objects G1 and G2, such that G1 x G2 r E. Proof. In order to have existence, every biological system transforms many of its inputs of a given material physical nature and quantity of energy, into outputs of other material physical natures or quantity of energy. (The connections between inputs and outputs are imposed by M.2.) Thus, the proof of this theorem is immediate. This comes from these previous remarks, and from the conditions given by: M. 1, the general postulate, the assumptions in order to obtain E.D. and M.E.E.D., and the axioms giving the category E. THEOR~ 2. I f an environmental category E has a set Gm of all the objects which are not defined by transfers between M and E, and which do not perform cartesian product operation with other objects defined by transfers; then, the category E will not be a discrete category. Proof. The set Gm is restrained to objects which do not perform a cartesian
688
CARLOS A. LEGLrlZAMON
product operation with other objects defined by transfers. In fact, this condition is imposed on Gm, because this possibility can be feasible. Thus, by means of this condition, and from the general construction of the environmental theory and its respective category E, each object of Gm will have morphisms connecting it (directly or through other objects of Gin) with objects which do not belong to Gm. Then, the present category E will not be a discrete category. It is necessary to stress the fact, that a given category E m a y not be discrete, but the set of objects Gm -- ~ . Definition. Given the whole life of a biological system M, where each consecutive time period determines each system M s, M 2 , . . . , M n respectively, and such that these systems are related with their respective environmental systems El, E2, . . . , E.; then the biological system M will be environmentally static if each system E s (j e I, I = 1, 2 , . . . , n) has the unities defined by transfers from M s to E~ for every i < j, i e I; and Ej has the unities which must be used by transfers from E k to M~ for every/c > j, k e I. THEOREM 3. Given a biological system M, with consecutive time periods defining the respective systems M1, M2, 9 9 M~ for the whole life of M, where these systems are related with their respective environmental systems E~, E2 . . . . , E~; and such that: (a) M is an environmentally static system, (b) the transfers between each pair M f - E j (j e I, I = l, 2 , . . . , n) define material environmental unities transferred from and to M~;
then, each category E s representing each system Ej, will not be a discrete category. Proof. We note some simple considerations with respect to the previous conditions. By means of the definition of M as environmentally static system, each system Ej will have the unities defined by transfers from M~ to E~ for every i < j, i e I; and Ej will have the unities which must be used by transfers from E~ to M~ for every ~ > j, k e I. But these transfers m a y be energetical, exclusively; in such a manner that they may be determined by energetical environmental unities which m a y have a cartesian product operation with material environmental unities which may be the same through the total chain of environmental systems. In this case, certain categories Ej may be discrete. These considerations are not valid when the condition (b) is included; such that, this condition together with the definition of an environmentally static system, will produce in each system Ej: other material environmental unities defined by transfers from Mt to E~ (i < j, i e I), and other material environmental unities which must be used as transfers from E~ to M~ (]r > j, k e I). Thus, by using
A THEORY FOR ENVIRONMENTAL SYSTEMS
689
Theorem 2, these last two sets of unities will belong to a set Gm of objects in each category Ej, such that b y the same theorem, each category E~ will not be a discrete category. As a remark about the consequences of the environmentally static property, we must say that the definition of different environmental unities in each system Ej depends on the environmental limit, which will be defined by the geographical, energetical, chemical, theoretical, etc., characteristics adopted. In fact, we may recognize the environmental static property in M, when, during the whole life of M, the limit is fixed, or when the change of limit does not affect unities and behaviors under the conditions of the above theorems. I wish to express my gratitude to Robert Rosen, who was my research director and from whom I received the best in order to obtain the present theory. This paper was prepared while the author was a visitor at the Center for Theoretical Biology, State University of New York at Buffalo, U.S.A.; whose hospitality is acknowledged. This work was made possible by a Fellowship from the Consejo Nacional de Investigaciones Cienttficas y T~cnieas of the Repfiblica Argentina. LITERATURE Leguizam6n, C.A. 1975. "Concept of Energy in Biological Systems." Bull..Math. Biology, 37, 565-572. Levins, R. 1968. Evolution in Changing Environments. Princeton: Princeton University Press. Lotka, A.J. 1956. Element~o/Mathematical Biology. New York: Dover Publications. MacLane, S. 1971. Categories/orthe Working Mathematician. New York: Springer. Mitchell, B. 1965. Theoryo/Categories. New York: Academic Press. Odum, E.P. 1971. Fundamentals of Ecology. Philadelphia: W. B. Saunders. Rashevsky, N. 1960. Mathematical Biophysics. New York: Dover Publications. RECEl~D 4-1-74 Rv.WS~D 3-12-75
