METABOLIC ‘CONTROL' IN MATURE ERYTHROCYTES

C. A. Hilgartner

This construct, termed directive correlation, comprises a rigorously

defined form of joint-causation involving a temporally

ordered sequence composed of coenetic variables (‘initial conditions’), and responses of environment and of organism to the coenetic variables, which in turn interact so as to achieve Ü¥e

METABOLIC ‘CONTROL’ IN MATURE ERYTHROCYTES

C. A. Hilgartner

Institute of General Semantics

Lakeville, Connecticut 06039, USA

INTRODUCTION

By about 1950, the main outlines of the major metabolic pathways of plant and animal organisms seemed well known. Since then, those mechanisms which determine the steady-state concentrations of metabolic intermediates or the overall rates of the various metabolic pathways have received increasing attention. But although our understanding of the minute details which make up living systems has steadily increased, our comprehension of what constitutes a ‘living system’ has not.

As I have pointed out elsewhere, the usual terminology of ‘adaptation’, ‘coordination’, ‘integration’, ‘control’, ‘regulation’, etc., impliespensable: If for any reason I try to deal with a ‘system’ less comprehensive than ‘an-organism-as-a-whole-in-its-environment-at-a-date’, by that act I annihilate the very phenomena of interest, viz. this ‘interacting at the boundary’, the ‘relatedness’ of ‘living systems’. For an ‘isolated living system’ comprises a contradiction in terms: an ‘organism’ ‘isolated’ from its ‘environment’ (e.g. a mammal cut off from air) cannot survive.

AN ERYTHROCYTE AS AN ‘ORGANISM’

In order to mathematize this fundamental ‘relatedness’, I have set up a Cartesian product space, (O ( E) , "in which to represent the interactions at the boundary in the (organism ( environment) field"9. Every element of this space, then, comprises an (o,e) couple, an interaction of organism with environment.

The array of set-theory sentences in which I describe the activities of an erythrocyte shows strict analogy to the first 24 set

sentences of Hilgartner & Randolph1. Thus I compare the activities of a single

celled organism to the transactional structure of ‘perception’, as summarized by Cantril & Livingston10. One particular red-blood-cell-in-its-environment-at-a-date comprises my ‘organism’ (O) ; its ‘environment’ I designate as (E) ; and I must specify the relations between them. As a beginning, I can accomplish this by specifying our set-space, and three mappings on this space, section (W) and projection with respect to the first and second variables (Pj1 and Pj2).

1) F = 0 ( E

2) W(oi) = {ei|(o,e) i ( F}

3) Pj1 (o,e) i = oi ( O

4) Pj2(o,e) i = ei ( E

The use of time

indices implies that the relations between O and E prove dynamic: any interaction progresses to an outcome (Oc).

5) (o,e) i ( ocj ( Oc

With multicellular (e.g. human) organisms, it seems obviously appropriate to specify some kind of ‘psychology of "perception"’. Although the context of a single

celled organism differs from that of a human organism, a one

celled organism also responds to ‘external’ dynamic influences, and thus in a sense does show a ‘psychology of "perception"’. These influences include mainly physical

chemical

biological factors, e.g. temperature, osmotic pressure, ionic strength, ionic composition, pH, the concentrations of various potential nutrients or other chemical substances, etc.

In order to specify these conditions, I define the physical boundary (B) as a subset of

O ( E ; I specify physical functions (f) which map elements or subsets of E into the set of stimuli (St) ; and I specify the process of ‘integration’ from elements of (o,e) interactions to sets of (o,e) interactions.

a) Those components which make up the boundary include a mechanical barrier or integument (Ig) ; a motor (M) system which includes contractile mechanisms involved in maintaining the normal biconcave shape, and also an array of mechanisms involved in "maintaining the difference between ‘inside’ and ‘outside’," e.g. the so

called "ion pump or ferry’’13.

Those mechanisms of our organism which undergo dynamic alterations as a function of environmental changes also belong to the boundary, occupying the role of ‘sensory receptors’, (S) (e.g. enzymes which undergo alterations of activity as a function of some environmental factor, such as the concentration of inorganic phosphorus |Pi| in the medium, where I indicate concentration by enclosing the symbol between vertical lines). Furthermore, of necessity the boundary must include some mechanism(s) for registering various aspects of the ‘state’ of the organism, analogous to the ‘proprioceptors’ (P) of a higher organism (e.g. enzymes which undergo alterations of activity as a function of the concentrations of key metabolic intermediates, or of the ratio of the intra-cellular concentrations of various coenzymes, such as |ATP| / |ADP|, or |NADH| / |NAD| ).

b) In principle, boundary elements analogous to ‘sensory receptors’ (or ‘proprioceptors’) must show some kind of characteristic thresholds (L) related to the absolute-magnitude-and-duration of the stimulus. Superliminal stimuli L, subliminal stimuli

EMBED Equation.2

.

c) If there exists the couple of a ‘sensory element’ with a supraliminal stimulus, at time t0, then at time t1 the ‘sensory element’ exists in an activated state, designated by an asterisk.

6) B ( O ( E : S ( P ( M ( Ig ( B

7) f:E ( St

7a) (xi ( E) ( f(xi) ( St

8) s ( S, e ( L, (s, e) 0 ( s*(e) 1 ( B ( O ( E

9) s*i ( S*i ( S ( B

CHEMICAL DYNAMICS OF ENZYMIC REACTIONS

In order to show how these relations apply at the level of a unicellular organism, I must devise ways of representing the chemical dynamics of enzyme

catalyzed reactions in this notation.

A. Chemical Reactions Without Catalysis

I can represent a generalized uni-directional chemical reaction as the interaction of elements in a simple conditional:

10) pi ( Pi, qi ( Qi, (p,q) i ( rj ( Rj ( sj ( Sj

I can represent a generalized reversible reaction by means of another sentence in conjunction with (10):

11) (r,s) i( pj ( qj

But if the reaction proves reversible, then the interacting configuration (p,q) i (which might correspond to the ‘activated state" of the reacting entity) must prove equivalent to the interacting configuration (r,s) i . Thus an exclusive union sign can serve to indicate the reversibility of this reaction:

12) (p,q) i ( (pj ( qj) ( (rj ( sj)

The notion of the concentration of a chemical substance concentration refers to the numbers of molecules of that substance per unit volume, Thus in this notation ‘concentration’ comprises an attribute of sets, not of elements (a measure of the ‘numbers of elements’ in the sets at any instant ti).

Likewise, the notion of reaction velocity comprises an attribute of sets, not of elements (a measure of the changes of ‘numbers of elements’ in the sets at successive instants).

Thus the familiar Law of Mass Action, which we ordinarily express in terms of the thermodynamic equilibrium constant, Keq , becomes a relation between the measures of the relevant sets, which I can indicate as:

13) [(p,q) i ( (pj( qj) ( (rj ( sj)]:

[|R| j] c ( [|S| j] d

[|P| j] a ( [|Q| j]b = Keq

(where a,b,c,d comprise small integers).

B. Enzymic Reactions

The simplest type of enzyme-catalyzed reaction involves the conversion of an element of the substrate A into an element of the product B in the presence of an element of the enzyme Z.

14) zi ( Zi, ai ( Ai,

[(z,a) i( za*i+1] ( z j ( b j ( B j

At the level of sets, I could write an equation in which a capital letter replaced each lower-case letter used in (16); in which case, I would avoid lengthy subscripts on the Z* term by writing it as Z*(A) i+1.

A large class of enzymic reactions involves the interaction of two or more substrates with an enzyme to produce two or more products, Here I do not intend to deal with details of enzymic mechanism, In set-theory terms, the general case becomes:

15) zi ( Zi, ai ( Ai, bi ( Bi, [(z,(a(b)) i ( zab*i+1] ( z j ( c j ( C j ( d j ( D j

Again, an exclusive union sign will serve to show an enzymic reaction as reversible.

In the notation, I represent reaction velocity, which constitutes a measure rather than a set, as a temporally-ordered mapping (for which I advance the time

indices): an operation which it takes a finite time to perform on the enzyme operator as it in turn operates on substrate(s), activator(s), inhibitor (s), etc. The context of measures of sets permits the use of ordinary algebraic operators, which would have no meaning with respect to the sets themselves.

MODIFER-SENSITIVE REACTIONS

Certain enzymic reactions prove sensitive to various types of modifying substances (Q), which in general serve to increase or to diminish the overall rate of reaction (inhibiting, Ib, and activating, Ac). As usual I do not deal here with subtleties of mechanism.

16) Z ( B ( O ( E : [[|Z| i, [|A| i> O], [|Q i ( O]

( vfzi = ( (Z*(A)) i+1],

[|Q| j > 0]( [QIb(Z*(A)) j+1 : Qvfz k < vfz i]]

where Z stands for the enzyme, A for the substrate, Q for the inhibiting modifier, and vfz for the reaction velocity in the ‘forward’ direction. Likewise,

17) Z ( B ( O ( E : [[|Z| i, [|A| i> 0], [|Q| i ( 0]

( vfzj = ((Z*(A)) i+1 ],

[|Q| j>0] ( [QAc(Z*(A)) k+1: Qvfz j > vfz i]]

As specific examples used below, red cell hexokinase (Hxk) undergoes strong inhibition in the presence of its product, glucose-6-phosphate (G6p) . Likewise, phosphofructokinase (Pfk) undergoes strong inhibition in the presence of an elevated ratio of concentrations of the coenzymes adenosine triphosphate ( |ATP| ) to adenosine diphosphate ( |ADP| ) . Both of these systems, then, qualify as ‘proprioceptors’. Contrariwise, both these enzymes in their inhibited states undergo activation ("de-inhibition") in the presence of elevated concentrations of inorganic phosphate ( |Pi| ) in the medium (‘environment’ ) of the cell, signified by a left subscript "e", e.g. ePi ).

18) Hxk ( P ( B: [ |G6p| i > 0 ( G6pIb(Hxk*(Glu,ATP)) i+1]

19) Hxk ( S ( B : [ | ePi| i > 0 ( PiAc(G6pIb(Hxk*(Glu,ATP)) i+1]

20) Pfk ( P ( B : [[ |ATP| / |ADP| ] i > 1 ( ATPIb(Pfk*(F6p,ATP))) i+1]

21) Pfk ( S ( B: [ | ePi| i > 0 ( PiAc(ATPIb(Pfk*(F6p,ATP))) i+1]

ENZYMIC REACTIONS AS DIRECTIVELY CORRELATED

Thus, I can represent enzymic reactions as a special case of generalized chemical reactions. But in the most fundamental sense, enzymic reactions comprise biologically ‘purposive’ activities, and thus they should meet the criteria for classification as directively correlated processes. As example, I shall consider a well-known reversible single-substrate reaction, the phosphohexose isomerase (Phi) system, which ‘serves’ to inter-convert glucose-6-phosphate (G6p) and

fructose-6-phosphate (F6p).

The coenetic variables in this reaction include conditions of pH, ionic strength, temperature, etc., compatible with enzymic activity; the presence of a suitable concentration of the enzyme, and of one substrate (or both); etc. The existence of an enzymic reaction in vivo indicates that the organism has "chosen" a thermodynamically possible reaction, which in the absence of perturbing conditions takes place to an extent indicated by the thermodynamic equilibrium constant.

Almost certainly, in sterile aqueous solution at room temperatures, G6p will spontaneously change into F6p at some very slow rate; but the half-life of a given sample (as determined from thermodynamic considerations) might amount to thousands or tens of thousands of years. For biological systems, this reaction must undergo enough rate increase to give half-lives somewhere on the order of hours to tiny fractions of a second. Thus I can express the focal condition of this presumed directive correlation as the requirement that the in vivo ratios of |F6p|i to |G6p|i approximate the value of the equilibrium constant, with the corollary requirement that the rate of this reaction remain greater than vL , some limit intrinsic to the organism, e.g. great enough that this reaction not comprise the rate-limiting step in the metabolic pathway(s) in which it plays a part.

Then the directively correlated variables in this reaction comprise precisely those relations subsumed in the set-theory representation of the reaction:

22) Phi ( O: [|Phi|i, |G6p|i > 0, |F6p| i ( 0,

[(Phi, G6p) i ( Phi*(G6p) i+1] ( [|Phij| ( [|F6pj| > |F6pi| ( |G6pj| < |G6pi|]]:

[[|F6p| j / |G6p| j ] ( Keq , vfPhiq > vL q]

Clearly, this enzymic reaction meets the criteria for classification as a directively correlated activity; and so would any other. Thus the topic of the ‘control’ of metabolism in red blood cells does provide a suitable test case against which to examine the question of the adequacy of the construct of ‘directive correlation’ as a model or ‘map’ of the structure of biologically ‘purposive’ activities.

METABOLITE INHIBITING OF THE HEXOKINASE SYSTEM

As the next step in this test, I examine a well-known phenomenon, the inhibiting of the irreversible Hxk reaction by its own product. As a bridge from the specific to the general, I shall use data from an experiment published by Rose, et al. 11 obtained with a suspension of red blood cells from a normal donor used as a control sample in an experiment:

OBSERVED:

Total rate of glucose utilization

vtGlu0 = 1.132 (M/hr/ml packed cells

Rate of CO2 product production

vCO2o = 0.142 (M/hr/ml packed cells (12.5% of total)

Phosphofructokinase rate

vPfk0 = 0.99 (M/hr/ml packed cells (87.5% of total)

Concentration of glucose-6-phosphate

|G6p|0 = 0.0447 (M/ml packed cells

Under these conditions, the vtGlu remains roughly constant, as does the |G6p| . Since no functional phosphatase exists in erythrocytes12, and since erythrocytes do not form glycogen, the vtGlu must equal the sum of the vCO2 and the vPfk . Since the |G6p| remains roughly constant, therefore the vHxk likewise must equal the sum of the other two rates. But the Hxk system undergoes inhibition by its own product, G6p . Thus

23) O0: [G6pIb(Hxk*(Glu,ATP)) 0 : [ |G6p|0 = 0.0447 (M/ml packed cells]

([ vHxk = (vCO2 + vPfk)0 = 1.132 (M/hr/ml packed cells]],

[[[(0.0447

|G6p| i ) > LHxki ] ( vHxk i+1 > (vCO2 + vPfk) i+1]

( [ |G6p| i+2 > |G6p| i ]

( [(G6pIb(Hxk*(Glu,ATP)) i+3 : vHxk i+4 < vHxk i] ( ...

( [[Oj : (0.0447 - |G6p| j) < LHxkj ] ( vHxk j = (vCO2 + vPfk) j

= (vCO2 + vPfk) 0]],

[[[|G6p| i - 0.0447)> LHxk i]( vHxk i+1 < (vCO2 + vPfk) i+1 ]

( [|G6p| i+2 < |G6p| i ]

( [(G6pIb(Hxk*(Glu,ATP)) i+3 : vHxk i+4 > vHxk i ] ( ...

( [Oj : [|G6p| j

0.0447) < (LHxkj] ( vHxk j = (vCO2 + vPfk) j = (vCO2 + vPfk) 0]]

This sentence describes a directively correlated process, in which the coenetic variables comprise those aspects of environment and of the organism which determine the total rate of glucose utilization and which determine the state of the hexokinase system such that if the |G6p| stays close enough to some specific value (e.g. 0.0447 (M/ml packed cells), then vHxk = vtGlu ; and the focal condition comprises dynamically to maintain this equality of rates. Then any outcome in which the vHxk remains equal to the vtGlu constitutes a subset of this focal condition. If the vHxk either exceeded or fell short of the vtGlu , then the |G6p| would increase (perhaps without apparent limit) or would diminish markedly (perhaps to vanishingly small values). Neither of these outcomes would constitute a subset of this focal condition.

INTERACTIONS OF SEVERAL ENZYMIC REACTIONS

As the next step in this test, I have examined the complicated interactions between two irreversible, magnesium-dependent, modifier

sensitive enzymic reaction systems, Hxk and Pfk , and the reversible Phi system which connects them. In order to show these, I must first touch on several topics: A) The dynamics of reaction velocity as a function of a concentration ratio; B) The dynamics of ATP utilization and production; C) The complexing of adenosine phosphates with divalent cations (mainly Mg++ ), and its effects on modifier-sensitive reactions; and finally, D) The structure of directive correlations which involve more than one enzymic system.

A. Reaction Velocity as a Function of a Concentration Ratio

Although on a long

term basis erythrocytes may turn over and replace their store of adenosine phosphates13 (and perhaps also their supplies of the other coenzymes), on a short

term basis the total concentration of the adenosine phosphates within an erythrocyte appears fixed. The enzyme adenylate kinase brings about such rapid interconversion of AMP , ADP , and ATP that the value of the ratio [ |ATP| ][|ADP| ] / [ |ADP| ] 2 remains close to the relevant value of the Keq 14. But the rates of utilization of these three adenosine phosphates turns over a large enough fraction of the fixed total pool per unit time so that relatively small changes in these utilization rates can result in rather prompt shifts in the proportions of these three coenzymes. Consequently, the kinetics of any enzymic system which uses an adenosine phosphate as one substrate depends on at least one concentration value which can change rapidly.

If in addition the enzymic reaction proves sensitive to a different adenosine phosphate as inhibiting or activating modifier, the reaction system may show marked changes of velocity as a function of variations in the ratios of the two adenosine phosphates involved15,16.

B. Dynamics of ATP Utilization and Production

1. Utilization

As noted elsewhere17, an erythrocyte utilizes ATP to maintain various trans-membrane gradients and to run the sodium-potassium "pump", to run the ‘motor’ system involved in maintaining the normal biconcave disc shape, to perform various other metabolic syntheses (e.g. purine "salvage" pathway), and to "prime" the Embden-Meyerhof pathway. In order to specify the relations between the production-and-utilization of ATP and various metabolic possibilities concerning the Embden-Meyerhof pathway, I lump all the uses of ATP except the priming of the Embden-Meyerhof pathway into one term, and speak of a rate of utilization of ATP, uATPi, which has a positive value as long as the cell contains any ATP. On simple kinetic grounds, it seems reasonable to assume that uATPi may increase with increasing |ATP| / |ADP| .

2. Production

In the Embden-Meyerhof pathway, each glucose molecule catabolized requires expenditure of one molecule of ATP in the Hxk reaction, and one molecule of ATP in the Pfk reaction. For each molecule of glucose which becomes converted to pyruvate (or lactate), two molecules of ATP result from the diphosphoglyerate kinase (Dgk) reaction and two molecules of ATP result from the pyruvate kinase (Pvk) reaction: an investment of 2 ATP’s for a yield of 4 ATP’s.

The starting material for the hexose monophosphate shunt pathway consists of G6p . In overall stoichiometry, for each three molecules of G6p which enter the "shunt" pathway, one molecule of glyceraldehyde-3-phosphate (Ga3) , two molecules of F6p , and three molecules of carbon dioxide result. Further processing of those products through the Embden-Meyerhof pathway yields a total of 10 molecules of ATP : an investment of 5 ATP’s for a yield of 10 ATP’s .

3. Balance

Then the ratio of |ATP| / |ADP| will remain constant if and only it the rate of production of ATP, pATPi , exactly equals the rate of utilization of ATP , uATPi , over the period of measurement. As I show below, under "usual" conditions the total rate of glucose utilization will tend to become stabilized so as to maintain approximately constant |ATP| / |ADP|.

Under certain circumstances the total rate of glucose utilization may increase above this "steady-state" value. Then, other things remaining unchanged, as long as the sum of one-third the rate of carbon dioxide production plus the total rate of lactate production equals twice the total rate of glucose utilization, then the |ATP| / |ADP| ratio will continue to rise above its starting value. This would hold even if the uATPi increases somewhat with this increasing ratio.

However, a sufficient increase in |ATP| / |ADP| will bring into play at least two further mechanisms to counteract its further increase: a decrease in the concentration of free Mg++ ion (discussed below) and the accumulation of metabolic intermediates. Although the main mechanism now under consideration does not produce an increase of |ATP| / |ADP| sufficient to invoke these two further mechanisms, I must briefly present them in order to define the limits of the present main mechanism.

If the various metabolic intermediates which figure in the Embden-Meyerhof pathway subsequent to the investment of ATP’s and prior to the production of any ATP’s (e.g.

fructose-1,6-diphosphate (Fdp) , dihydroxyacetone phosphate (Dap) , Ga3 , and

1,3-diphosphoglycerate (Dpg)) should accumulate within the cell, the rate of production of ATP will diminish and therefore, depending on the proportion of metabolites accumulating to those undergoing catabolism, the rate of increase of the |ATP| / |ADP| ratio might slow, the ratio might become constant, or the ratio might even decrease. A lesser modification of the rise in this ratio at high glycolytic rates would result from accumulation of metabolic intermediates which figure in the Embden-Meyerhof pathway after the Dgk but prior to the Pvk reaction (e.g.

3-phosphoglycerate (P3g) , 2,3-diphospho-glycerate (G23) , 2-phosphoglycerate (P2g) , or phosphoenolpyruvate (Pep)) .

C. Divalent Cations and the Adenosine Phosphates

The adenosine phosphates, like other nucleotides, readily form complexes with divalent cations, especially Ca++ and Mg++ . Since in erythrocytes a "physiological pump" keeps the concentrations of calcium small,18 the main cation involved in this complexing comprises Mg++.

Table I 19 shows the dissociation constants for Mg++ , H+ , and K+ complexes of AMP , ADP , and ATP (determined at 35(C and 0.1 ionic strength). A comparison of these values shows that ATP binds Mg++ much more firmly than do the other two coenzymes. Thus ATP comprises a major intra-cellular ligand of Mg++ 20; and the adenylate kinase system (discussed above) not only makes up an important test system which allows the study of the state of magnesium in red blood cells, but also comprises an important factor in establishing the intra-cellular concentration of free Mg++ ions21. But the adenylate kinase system and the system composed or the adenosine phosphates and their magnesium complexes affect each other in a reciprocal fashion: the value of the equilibrium constant for the adenylate kinase system comprises a biphasic function of the total magnesium concentration22. From Rose’s measurements, it appears that something on the order of 70% of the ATP and 15% of the ADP in his samples exist in the complexed form under his glycolizing conditions23.

Then increases in the intra-cellular concentration of ATP would have the effect of diminishing the intra-cellular concentration of free Mg++ , or decreases in |ATP| would result in increased |Mg++| .

"Magnesium functions in many enzymic reactions as a cofactor and in complex with nucleotides acting as substrates." 24 At least six of the enzymes of the Embden-Meyerhof pathway (Hxk, Pfk, Dgk, Pgm, Enl, Pvk) require Mg++ as a cofactor. Thus increases in concentration of ATP , with the consequent decreased |Mg++| , could have the effect of diminishing the rates of reaction of these enzymic systems.

Furthermore, the magnesium complexes of the adenosine phosphates may interact with enzymes in ways quite different from those of the free coenzymes25:

a) With Hxk (and fumarase, which figures in the citric acid cycle of Krebs, absent from red blood cells), ATP---- proves inhibitory while ATPMg-- does not.

b) Pfk proves tenfold more sensitive to inhibition by ATP---- than by ATPMg-- .

c) The kinases which use ADP for the generation of ATP , such as Pgk and Pvk , behave as if ADPMg- comprised their substrate. In cells with Mg++ concentrations in the lower ranges (such as red cells), only a small part of their total |ADP| will exist in this form; hence these low concentrations of ADPMg- may serve to limit the rates of these reactions. But the concentrations of this complex will remain sensitive to changes in concentration of Mg++ and of total |ADP| , both of which change reciprocally with changes in total |ATP| .

D. Directive Correlations Involving More Than One Enzymic System

Pfk comprises one example of the class of enzymes which shows marked changes in reaction velocity as a function of changes in a ratio : it undergoes inhibition in the presence of high ratios of |ATP| to |ADP| (sentence (20)). Thus by making suitable substitutions in sentence (23), I can demonstrate that the dynamics of the ratio-inhibited Pfk reaction will increase or decrease the total rate of glucose utilization vtGlu so as to maintain approximately constant |ATP | / |ADP| .

Furthermore, Rose and his co-workers have implicated Pfk in a general metabolic effect, in which the total rate of glucose utilization (and of lactate production) increases as a function of increasing concentration of inorganic phosphorus (Pi) in the medium (the environment of our erythrocyte) 26. Their evidence indicates that the Pi counteracts the inhibitors of the first two irreversible steps of the Embden-Meyerhof pathway: G6p inhibiting Hxk , and |ATP| / |ADP| ratio inhibiting Pfk . Since the intra-cellular |G6p| and |ATP| remain approximately constant in the face of these increasing rates of glucose utilization (Figure 1) 27, the Pi effect must change the rates of these two reactions by exactly the same amount. And since this Pi effect does not alter the steady-state |G6p| (and since presumably |Pi| has no direct effect on the hexose monophosphate shunt nor (in cells which show this reaction) on glycogen synthesis), it does not alter the rates of other metabolic pathways.

This observed relation between three enzymic reactions ( Hxk , Phi , and Pfk ), metabolite inhibitors of two of the reactions, an environmental activator which has the effect of "de-inhibiting" these two reactions, and the overall rate of glucose utilization, comprises another type of directive correlation.

Here, the "dramatic situation " involves those metabolic changes which occur after our erythrocyte in a steady-state condition under a given set of circumstances encounters an abrupt increase in |ePi| . The initial coenetic variables comprise those aspects of environment and of the organism which determine the total rate of glucose utilization; which determine the state of the Hxk system such that if the |G6p| equals some specific value then the rate of the Hxk reaction will equal the total rate of glucose utilization; and which determine the state of the Pfk system such that if the ratio of |ATP| / |ADP| equals some specific value (and the |F6p| remains in equilibrium with the |G6p| , via the Phi system) then the rate of the Pfk reaction will equal the difference term (vHxki - vCO2i) . The initial focal condition comprises dynamically to maintain this equality of rates.

Following the increase in environmental |ePi| the components of this dynamically interacting system undergo a series of alterations of state and mutual interactions which soon result in the attainment of a new steady state. For the moment, I shall assume that all reactions beyond the Pfk system remain in equilibrium and the sole results of this perturbation comprise a set of equivalent increases in the rates of each of these subsequent reactions, and thus an increase in the rate of lactate production and of ATP production (and consequently a moderate increase in the rate of ATP utilization). (This seems not a bad assumption, for moderate increases of |ePi|, judging from the data in Table I of reference 28.)

Then the interactions of the Hxk - Phi - Pfk system might go something like this:

24) O0: CV0 ( |ePi| 0 = j) ( [ |G6p| 0 = l ] ( [vtGlu 0 = m]

( [[ |ATP| / |ADP| ] 0 = n] ,

FC0 ( [[vHxk 0+1 = ( vCO2 + vPfk) 0+1 = vtGlu 0+1 = m] : [[ |ATP| / |ADP| ] 0+1 = n]]

25) O1 : G6pIb(Hxk*(Glu, ATP)) 1 : [[ |G6p| 1 =1 ] ( [vHxk 2 = (vCO2 + vPfk) 2

= vtGlu = m] ,

Phi*(G6p, F6p) 1 : [ vPhi2 >> [ vHxk 2 = ( vCO2 + vPfk ) 2] ( [[ |F6p| / |G6p| ] 2 ( Phi Keq]],

ATPIb (Pfk*(F6p,ATP)) 1 : [[[[ |ATP| / |ADP| ] 1 = n]

( [[ |F6p| / |G6p| ] 1 ( Phi Keq]] ( [ vPfk 2 = ( vHxk - vCO2 ) 2]] ( FC0

26) O3 : CV3 ( [ |ePi| 3 = ( j + (j ) ], FC3 ( [[ vt Glu i > vt Glu 0 ]

( [ |G6p| i = |G6p| 0 ]]

27) O4: PiAc(G6pIb(Hxk*(Glu, ATP))) 4 : vHxk5 = (vHxk 0 + (vHxk 5),

PiAc(ATPIb(Pfk*(G6p, ATP))) 4 : vPfk 5 = (vPfk 0 + (vPfk 5)

In order to increase the speed of his vehicle, a driver usually depresses the accelerator quite a bit at first; as the vehicle approaches the desired speed, he will "let up" on the accelerator until the vehicle detectably loses speed, and then will depress the accelerator just enough to maintain the desired speed. Likewise, it seems reasonable to assume that in coming to a new steady state, the reaction velocities, etc., of a red cell will go through a series of "oscillations" or "closer approximations" in which the variables deviate enough from the steady state condition of the moment to exceed the system’s ‘sensory’ or ‘proprioceptive’ threshold.

First of all, the increased rates of the Hxk and Pfk systems (before the subsequent reactions have had time to become affected) will "burn up" ATP at an increased rate, lowering the |ATP| / |ADP| ratio.

28) [[[ vHxk 5 > vHxk 0 ] ( [ vPfk 5 > vPfk 0 ]]

( [[ |ATP| / |ADP| ] 6 < [ |ATP / |ADP| ] 0]]

( [((ATPIb(Pfk*(F6p, ATP))) 7 : vPfk 8 > vPfk 5 > vPfk 0]]

If the initial altered rates of the Hxk and Pfk systems turn our markedly unequal, the mutual interactions of this complex ‘sensory-and-propriceptive’ inhibiting-and-activating system come into play. For example, if the reaction velocity of the Pfk system markedly exceeds that of the Hxk system, the interactions might resemble:

29) [[ vPfk8 >> vHxk5 ] ( [ |G6p|9 < |G6p|0 ]] ( [vHxk10 > vHxk5 ]

30) [[[ vPfk 8 > vPfk 0 ] ( [pATP9 > pATP0 ]]

( [[ |ATP| / |ADP| ] 10 > [ |ATP| / |ADP| ] 0 > [ |ATP| / |ADP| ] 6 ]]

( [[ uATP 11 > uATP 0 ] ( [((ATPIb(Pfk*(F6p,ATP))) 11 : vPfk12 < vPfk5]] ( …

31) Oi : [[ |ePi| i = j + (j ] ( [(G6p)i = (G6p)0 = 1]

( [vHxk i = (vCO2 + vPfk)i = vtGlu i = m + (m]

( [[ |ATP| / |ADP| ] i = n + (n]] ( FC3

The size of the final increase in |ATP| / |ADP| will depend mainly on two factors: A) the magnitude of the initial disparity between vPfk and vHxk , and B) the slope of the change in vPfk as a function of changing |ATP| / |ADP| . In this context, a reexamination of Figure ( 1 ) shows that in the course of the almost threefold increase in vtGlu engendered by changing the |ePi| from 3 to 20 to 50 (M the intra-cellular |ATP| increases from about 1.16 to about 1.18 to about 1.26 (M — an overall change of about 10%. Not having the original data, I cannot test for statistical significance; but I hazard the guess that this exceeds the limits of the measurement errors of the methods used. And a 10% increase in |ATP| would represent a somewhat larger change in

|ATP| / |ADP| .

In a similar fashion, I could write a series of sentences describing the mechanisms which would come into play if the reaction velocity of the Pfk system started out markedly less than that of the Hxk system; but since this for the most part involves no more than a reversal of the direction of the inequality signs, I leave it as an exercise for readers to do.

I infer, then, that this three-enzyme ‘sensory-and-proprioceptive’ inhibiting-and-activating system qualifies for classification as a directively correlated system which directly depends on the two lower ‘orders’ of directive correlations I have discussed, and which has as its focal condition dynamically to maintain a constant steady-state |G6p| and an equality of reaction velocities such that vtGlu i = vHxk i = (vCO2 + vPfk) i, regardless of the value of the total glucose utilization rate vtGlu i . If these reaction velocities should become unequal, then the |G6p| would decrease markedly (perhaps to vanishingly small values) or would increase markedly (perhaps without apparent limit). Neither of these outcomes would constitute a subset of the focal condition of this directively correlated system. (In a subsequent publication, I shall consider one example of this type, in which there occurs a marked alteration of the |G6p| and the accumulation of other metabolic intermediates; and I predict that this will turn out to constitute an even more elaborate directive correlation.)

To summarize, I have shown that enzymic reactions qualify for classification as directively correlated processes. In notation I have shown the structure of one metabolite-inhibited (‘proprioceptive’) enzymic reaction system (Hxk) (and implicitly shown another, viz. Pfk ), and have shown that this class of reactions constitutes a set of ‘higher-order’ directive correlations which serves to keep constant the steady-state concentrations of the inhibiting metabolites. Finally, I have shown the structure of an even ‘higher-order’ directively correlated system which determines the overall rate of glucose utilization.

Sommerhoff29 defines a ‘living system’ as

a compact physical system of mechanically connected parts whose states and activities are related by an integrated set of directive correlations which, over and above any proximate focal condition, has the continued existence of the system as an ultimate focal condition.

Furthermore, he points out that

The concept of an integrated sequence of activities... stands for a relation between these activities which enables us to attribute an individual goal to each, and at the same time an ultimate goal to the whole sequence. 30

Finally, in his ‘integration theorem’, he defines this relation:

Theorem: If GA is the goal-event (focal condition) of a directive correlation A and if the occurrence of GA is a necessary condition for the occurrence of the goal-event GB of a directive correlation B, than GB is also a goal-event of A . 30

Then GA and GB constitute a proximate and an ulterior (or ultimate) goal of A respectively.

But the directively correlated system which determines vtGlu can achieve its focal condition if and only if the Hxk and Pfk (and Phi ) systems achieve their respective focal conditions, viz. to maintain |G6p| and |ATP| / |ADP| at steady-state values; and these systems in turn can achieve their focal conditions if and only if the enzymic reaction systems which underlie them achieve their focal conditions, viz. sufficiently to increase the spontaneous reaction rates of their substrates to reach values within the limits intrinsic to the organism.

Thus these mechanisms qualify for classification as a hierarchically ordered set of ‘integrated’ directive correlations.

Therefore the results of this study already suffice to demonstrate that an erythrocyte qualifies as an element of the set of ‘living systems’.

But the ability to make this decision on formal grounds means that Sommerhoff’s definition of the term ‘living system’, as incorporated into this theoretical system, constitutes an acceptable set-theory operator. This comprises a degree of rigor in the definition of ‘living systems’ not previously attained, to the best of my knowledge.

And this achievement, in turn, does not cast doubt upon the adequacy of the construct of ‘directive correlation’ as a model for the apparently ‘purposive’ activities of living systems.

But this study only reinterprets a theory of the structure of human behaving-and- experiencing, extending it to deal with biological and biochemical systems, Therefore these results fail to cast doubt upon the biological presuppositions of the existing "general theory of the structure of human psycho

dynamics," and indeed make that theory appear more general and generally valid than we at first supposed.

These results, taken together, seem to imply that further extensions of this theoretical system which succeed in encompassing other biological and biochemical systems might prove rewarding indeed. And certain hints and suggestions in the present study, e.g. the successful translation of certain aspects of physical chemistry into our notation, suggest the possibility that further study of this theoretical system might serve to improve our understanding not only of the social and biological sciences but of the physical sciences as well.

REFERENCES

1. C. A. Hilgartner & J. F. Randolph, J. Theo. Biol. 23, 285-336 (1969a), pp. 297-9.

2. G. Sommerhoff, Analytical Biology. London: Oxford University Press, 1950.

3. C. A. Hilgartner & J. F. Randolph, J. Theo. Biol. 23, 247-274 (1969b ).

4. C. A. Hilgartner & J. F. Randolph, J. Theo. Biol. 24, 1-29 (1969c).

5. C. A. Hilgartner & J. F. Randolph, (submitted for publication) (1969d).

6. C. A. Hilgartner, General Semantics, Psychotherapy, and the Logic of Science,

unpublished manuscript 1963, revised 1967. Abbreviated version, ETC.: Rev.

Gen. Sem. 25, 315-324 (1968).

7. C. A. Hilgartner, Feelings, Orientation, and Survival: The Psychological

Dimension of the Current Human Crisis, presented at the Ninth International

Conference on General Semantics, San Francisco State College, August 1965.

8. F. S. Perls, R. Hefferline & P. Goodman, Gestalt Therapy: Excitement & Growth in

the Human Personality. New York: Julian Press, 1951, p. 227. (Cited in

Hilgartner & Randolph (1969a), p. 303.)

9. C. A. Hilgartner & J. F. Randolph (1969a) op. cit., p. 295, also p. 303 ff.

10. H. Cantril & W. K. Livingston, J. indiv. Psychol. 19, 3 (1963). Reprinted in Gen.

Sem. Bull. no 30-31, pp. 32-38, 1963-64.

11. I. A. Rose & E. L. O’Connell, J. Biol. Chem. 239, 12-17 (1964), Table I.

12. I. A. Rose & E. L. O’Connell, ibid., p. 14.

13. J. W. Harris, The Red Cell. Cambridge, Mass.: Harvard University Press, 1963,

p. 214.

14. I. A. Rose, Proc. Natl. Acad. Sci. 61, 1079

1086 (1968), pp. 1079-80.

15. D. E. Atkinson & D. U. Walton, J. Biol. Chem. 242, 3239 (1967); D. E.

Atkinson & L. Fall, J. Biol. Chem. 242, 3241 (1967).

16. John Westley, Enzymic Catalysis. New York: Harper & now, 1969, p. 177.

17. J. W. Harris, op. cit., p. 214.

18. I. A. Rose (1968), op. cit., p. 1083.

19. Ibid., p. 1080.

20. Ibid., p. 1084.

21. Ibid., p. 1084.

22. Ibid., p. 1080.

23. Ibid., p. 1083.

24. Ibid., p. 1079.

25. Ibid., p. 1085.

26. I. A. Rose, J. V. B. Warms, & E. L. O’Connell, Biochem. Biophys.

Res. Comm. 15, 33-37, (1964); I. A. Rose & E. L. O’Connell (1964), op. cit.

27. I. A. Rose, J. V. B. Warms, & E. L. O’Connell, (1964), op. cit., Fig. 1.

28. I. A. Rose & J. V. B. Warms (1966), op. cit., Table I.

29. G. Sommerhoff, (1950), op. cit., p. 195.

30. G. Sommerhoff, The Abstract Characteristics of Living Systems, in Systems

Thinking, ed. by F. E. Emery. Baltimore: Penguin Books, 1969, pp. 147

202. Theorem appears on p. 188.

APPENDIX

I wrote this report in the E-Prime Language proposed by Bourland (General Semantics Bulletin Nos. 32 & 33, 1965/66, pp. 111-116), which consists of the English language with the exclusion of all inflectional forms of ‘to be’. Likewise, I shall translate the set theory sentences into words according to the conventions of E-Prime.

1) The field and the O ( E Cartesian product space consist of the same elements.

2) The section over the element o (at time ti) comprises a set composed of all those elements e (at time ti) for which the couple of o with each of these e’s (at time ti) comprises an element of the field.

3) The first projection (the projection with respect to the first variable _

†瘠牡慩汢_915306938

and assumes Aristotle’s notion of ‘the rational soul’. Impolitely stated, that means this traditional terminology proves equivalent to the assumption that a little demon does the organism’s ‘regulating’ for it. To the best of my knowledge, no one has succeeded in providing rigorous definitions for those terms. And although in the last two decades a literature concerning mathematical models of ‘adaptation’ has accumulated, still, no fundamental revision of this archaic terminology has yet achieved general acceptance.

Sommerhoff2 provides a rigorously defined mathematical construct which (he claims) proves similar in structure to (a "good map" of) the apparently ‘purposive’ activities of living systems. This construct, termed directive correlation, comprises a rigorously

defined form of joint-causation involving a temporally

ordered sequence composed of coenetic variables (‘initial conditions’), and responses of environment and of organism to the coenetic variables, which in turn interact so as to achieve of the couple (o,e)) comprises an element o (at time ti) which constitutes an element of the set the organism.

4) The second projection of the couple (o,e) comprises an element e (at time ti ) which constitutes an element of the environment.

5) If there exists the couple (o,e) (at time ti), then there exists an outcome (at time tj), which comprises an element of the set of Outcomes.

6) Every element of the boundary comprises an element of the O ( E Cartesian product space, such that every element of the set composed of the union of sensory receptors, proprioceptors, motor apparatus, or integument comprises an element of the boundary.

7) There exists a set of physical functions, f , such that the set of the environment gets mapped into the set of stimuli.

7a) If x (at time ti) comprises an element of the environment, then physical functions of x (at time ti) comprise elements of the set of stimuli.

8) If there exists the couple (s,e) (at time t0) with s an element of the set of sensory receptors and e an element of the set of supraliminal stimuli, then (at time ti) there exists an activated sensory element, which comprises an element of the boundary (and every element of the boundary comprises an element of the O ( E Cartesian product space).

9) An activated sensory element (at time ti) comprises an element of the set of activated sensory receptors (at time ti); every element of the set of activated sensory receptors (at time ti) comprises an element of the set of sensory receptors, which itself comprises a subset of the boundary.

With this introduction (and the appended account of the notation), probably readers will have little difficulty with the other simple sentences, e.g. 10-15. Sentences (16) and (17) seem of "intermediate’’ difficulty.

16) The enzyme Z comprises a subset of the boundary (which comprises a subset of the O ( E field) such that if (at time ti) there exists a concentration of the enzyme Z , and a greater-than-zero concentration of the substrate A , and the concentration of the inhibiting modifier Q approximates zero, then (at time tj) the forward velocity of the Z reaction system equals some value given by the mapping theta of the activated

state term (Z*(A)) (at time ti+1); but if the concentration of the inhibiting modifier Q exceeds zero (at time ti), then (at time ti+1) there exists Q-inhibition of the activated

state term (Z*(A)) such that the Q-inhibited forward velocity of the Z reaction systems remains less than that of the uninhibited Z reaction system (at time tj).

17)

18)

19)

20)

21)

22) The enzyme phosphohexose isomerase (Phi) comprises a subset of the organism (a subset of the boundary) such that if, given that there exists an appropriate concentration of the enzyme, a concentration of G6p greater than zero, and a concentration of F6p approximately equal to zero, and the couple of enzyme with the substrate G6p (at time ti) precedes the activated state Phi*(G6p) (at time ti+1), then (at time tj) there exists a concentration of the enzyme, and the concentration of F6p (at time tj) exceeds its previous value (at time ti), and the concentration of G6p (at time tj) has a lower value than that of G6p (at time ti), such that the ratio of concentrations of F6p to G6p (at time tj) approximates the value of the equilibrium constant, and the forward velocity of the Phi reaction system (at any time tq)exceeds some limit v( intrinsic to the organism.

23) The organism (at time t0) exists such that G6p inhibition of the activated state (Hxk*(Glu, ATP)) occurs (at time t0), such that if the concentration of G6p equals 0.0447 (M/ml packed cells (at time t0), then the (forward) velocity of the Hxk system equals the sum of the velocity of elaboration of carbon dioxide plus the (forward) velocity of the Pfk system (all at time t0), which equals 1.132 (M/hr/ml packed cells; and if (at time ti) the difference term obtained by subtracting the G6p concentration from 0.0447 exceeds the threshold of the Hxk system, then (at time ti+1) the velocity of the Hxk system will exceed the sum of the velocity of production of carbon dioxide plus the velocity of the Pfk system; and if this condition holds, then (at time ti+2) the concentration of G6p will exceed its previous value (at time ti)’ and (at time ti+3) the G6p inhibition of (Hxk*(Glu, ATP)) will change, such that (at time ti+4) the velocity of the Hxk system will have a lower value than previously (at time ti), and other steps of this form may occur, until (at time tj) the organism exists such that the difference term obtained by subtracting the G6p concentration from 0.0447 does not exceed the threshold of the Hxk system, and the velocity of the Hxk system (at time tj) equals the sum of the velocity of elaboration of carbon dioxide plus the velocity of the Pfk system (at time tj), which equals the previous value of this sum (at time t0); whereas if (at time ti) the converse difference term (obtained by subtracting 0.0447 from the G6p concentration value) exceeds the threshold of the Hxk system, then an equivalent set of operations (but with the direction of the inequality signs reversed) will occur, until (at time tj) the organism exists such that the difference between the G6p concentration value and 0.0447 does not exceed the threshold of the Hxk system, and the velocity of the Hxk system (at time tj) equals the sum of the velocity of elaboration of carbon dioxide plus the velocity of the Pfk system (at time tj), which equals the previous value of this sum (at time t0).

Sentences (24) through (31) present a concerted "story":

24) The organism (at time t0) exists such that its coenetic variables (at time t0) include the situation that the environmental phosphate concentration equals some constant j , and the organism’s G6p concentration equals some constant l , and its total rate of glucose utilization equals m , and its ATP:ADP concentration ratio equals n , and the organism’s focal conditions (at time t0) include the situation that (at any time t0+1) the velocity of the Hxk system will equal the sum of the velocity of elaboration of carbon dioxide plus the velocity of the Pfk system, which will equal the total rate of glucose utilization, which in turn will equal the constant m , such that (at time t0+1) the ATP:ADP concentration ratio will equal n .

25) At time tl , the organism exists such that if G6p inhibition of the activated state (Hxk*(Glu, ATP)) occurs such that the G6p concentration equals l (at time t1), then (at time t2) the velocity of the Hxk system equals the sum of the velocity of elaboration of carbon dioxide plus the velocity of the Pfk system (at time t2), which equals the constant m ; also (at time t1) the activated state (Phi*(G6p,F6p)) exists such that (at time t2) the velocity of the Phi system greatly exceeds the velocity of the Hxk system (which itself equals the sum of the velocity of elaboration of carbon dioxide plus the velocity of the Pfk system, at time t2), and the concentration ratio of F6p:G6p (at time t2) approximates the equilibrium constant of the Phi system; and if (at time t1) ATP inhibition of (Pfk*(F6p, ATP)) occurs such that the ATP:ADP concentration ratio (at time t1) equals n and the concentration ratio of F6p:G6p (at time t1) approximates the value of the equilibrium constant of the Phi system, then (at time t2) the velocity of the Pfk system equals the difference term obtained by subtracting the velocity of elaboration of carbon dioxide from the velocity of the Hxk system (at time t2); and this situation comprises a subset of the organism’s focal conditions of time t0 .

26) At time t3 , the organism exists such that its coenetic variables include an increase in the environmental concentration of phosphate to a value of j + (j , and its focal conditions (at time t3) include a total rate of glucose utilization (at time ti) greater than the previous value at time t0 , and a concentration of G6p at time ti equal to its previous value at time t0.

27) At time t4 , the organism exists such that phosphate activation of the G6p inhibited activated state (Hxk*(Glu, ATP)) occurs (at time t4), such that (at time t5) the velocity of the Hxk system equals the sum of its velocity at time t0 plus an increment vHxk (at time t5); and (at time t4) phosphate activation of the ATP inhibited activated state (Pfk(F6p, ATP)) occurs such that (at time t5) the velocity of the Pfk system equals the gum of the velocity at time to plus an increment vPfk (at time t5).

28) If the velocity of the Hxk system (at time t5) exceeds its previous value (at time t0), and the velocity of the Pfk system (at time t5) exceeds its previous value (at time t0), then the ATP:ADP concentration ratio at time t6 will have a lower value than it did at time t0 ; and if that condition holds, then (at time t7) a change in the ATP inhibition of the activated state (Pfk*(Glu, ATP)) will occur, such that the value of the velocity of the Pfk system at time t8 will exceed its value at time t5 , which in turn exceeded its value at time t0 .

29) If the value of the velocity of the Pfk system at time t8 greatly exceeds that of the Hxk system at time t5 , then at time t9 the concentration of G6p will have a lower value than it did at time t0 ; and if that condition holds, then at time t10 the velocity of the Hxk system will exceed its previous value at time t5 .

30) If the velocity of the Pfk system at time t8 exceeds its (or not achieve) the organism’s focal conditions (‘goals’, outcomes which prove somehow ‘favorable’ from the point of view of the organism).

In another context1,3,4,5 the notion of directive correlation has undergone and survived extensive scrutiny: In the process of performing a logical analysis of a new theory of the structure of human psycho

dynamics6,7 we incorporated this notion into the logical superstructure of an axiomatic system, stated in a set

theory notation, which starts from first principles and treats human behavior

and-experience as a special case of the apparently ‘purposive’ activities of living systems.

Recently it seemed worthwhile to try to obtain an estimate of the degree of adequacy of the notion of directive correlation itself, in some context more or less independent of human behavior. To do this, I have utilized this construct to account for the structure of and the inter

relations between the mechanisms of ‘control’ of carbohydrate metabolism in the "minimal" living system presented by mature mammalian red blood cells. This report presents some of the findings of that study.

THE BOUNDARY

‘Living systems’ display an intrinsic ‘relatedness’. The notion of ‘organism’ (or ‘living system’) implies an ‘environment’ for this ‘organism, and vice versa. And both terms imply a dynamic ‘boundary’ between ‘organism’ and ‘environment’ which in a curious way "belongs" to both, in that it serves as the ‘organ of interaction’ between them.8 In this context, a holistic viewpoint appears indisÜƒ

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*METABOLIC ‘CONTROL' IN MATURE ERYTHROCYTES

C. A. Hilgartner

C. A. Hilgartner

METABOLIC ‘CONTROL’ IN MATURE ERYTHROCYTES

C. A. Hilgartner

Institute of General Semantics

Lakeville, Connecticut 06039, USA

INTRODUCTION

By about 1950, the main outlines of the major metabolic pathways of plant and animal organisms seemed well known. Since then, those mechanisms which determine the steady-state concentrations of metabolic intermediates or the overall rates of the various metabolic pathways have received increasing attention. But although our understanding of the minute details which make up living systems has steadily increased, our comprehension of what constitutes a ‘living system’ has not.

As I have pointed out elsewhere, the usual terminology of ‘adaptation’, ‘coordination’, ‘integration’, ‘control’, ‘regulation’, etc., implies and assumes Aristotle’s notion of ‘the rational soul’. Impolitely stated, that means this traditional terminology proves equivalent to the assumption that a little demon does the organism’s ‘regulating’ for it. To the best of my knowledge, no one has succeeded in providing rigorous definitions for those terms. And although in the last two decades a literature concerning mathematical models of ‘adaptation’ has accumulated, still, no fundamental revision of this archaic terminology has yet achieved general acceptance.

Sommerhoff2 provides a rigorously defined mathematical construct which (he claims) proves similar in structure to (a "good map" of) the apparently ‘purposive’ activities of living systems. This construct, termed directive correlation, comprises a rigorously

defined form of joint-causation involving a temporally

ordered sequence composed of coenetic variables (‘initial conditions’), and responses of environment and of organism to the coenetic variables, which in turn interact so as to achieve (or not achieve) the organism’s focal conditions (‘goals’, outcomes which prove somehow ‘favorable’ from the point of view of the organism).

In another context1,3,4,5 the notion of directive correlation has undergone and survived extensive scrutiny: In the process of performing a logical analysis of a new theory of the structure of human psycho

dynamics6,7 we incorporated this notion into the logical superstructure of an axiomatic system, stated in a set

theory notation, which starts from first principles and treats human behavior

and-experience as a special case of the apparently ‘purposive’ activities of living systems.

Recently it seemed worthwhile to try to obtain an estimate of the degree of adequacy of the notion of directive correlation itself, in some context more or less independent of human behavior. To do this, I have utilized this construct to account for the structure of and the inter

relations between the mechanisms of ‘control’ of carbohydrate metabolism in the "minimal" living system presented by mature mammalian red blood cells. This report presents some of the findings of that study.

THE BOUNDARY

‘Living systems’ display an intrinsic ‘relatedness’. The notion of ‘organism’ (or ‘living system’) implies an ‘environment’ for this ‘organism, and vice versa. And both terms imply a dynamic ‘boundary’ between ‘organism’ and ‘environment’ which in a curious way "belongs" to both, in that it serves as the ‘organ of interaction’ between them.8 In this context, a holistic viewpoint appears indisÜƒ

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9previous value at time t8 , then at time t9 , the rate of production of ATP will exceed its previous value at time t0 ; and if that condition holds, then the value of the ATP:ADP concentration ratio at time t10 will exceed its initial value at time t0 , which in turn exceeds its value at time t6 , and if these conditions hold, then the rate of utilization of ATP at time t11 will exceed its initial value at time t0 , and (at time t11) the ATP inhibition of the activated state (Pfk*(Glu, ATP)) will change in such a way that (at time t12) the velocity of the Pfk system will have a lower value than it did at time t5 ; and further steps of these types may occur.

31) At time ti , the organism exists such that the environmental phosphate concentration remains at a value of j + (j : , and the G6p concentration at time ti equals its initial value of and the velocity of 1, and the Hxk system (at time ti) equals of the velocity of elaboration of carbon dioxide plus the velocity of the Pfk system (at time ti), which equals the total rate of glucose utilization, which in turn equals its initial value an increment, m + (m , and (at time ti) the ATP:ADP concentration ratio has increased to a value of n + (n ; and these conditions taken together comprise a subset of the organism’s focal conditions of time t3 .

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