TOWARDS A UNIFIED INTO THE Ag SYSTEM JAN

HIRSCHFELD,

THEORY

FOR

IMMUNOGENETIC

SYSTEMS.

IV. A COMPLEX-SIMPLE

PROBE

State Institute for Blood Group Serology, Statens riittskemiska laboratorium, S-58185 Linkaping

Sweden.

SUMMARY

The Ag system is a human P-lipoprotein polymorphism of medium complexity discussed in the present article mainly as a model for an analysis of the feedback processes between facts, artifacts and hypotheses. Two Caucasian population materials comprising a total of 530 unrelated individuals are explored at the six-reagent level. Through the new (complexsimple) conceptual framework (language and theory) these 6 x 530 or 3,180 experimental observables or information bits (plus(+) and minus(-) signs) are probed in a most sensitive manner. This probe also permits a perfectly transparent explanation for previously inexplicable-unnoticed pseudo-mysteries with regard to remarkable “clustering” and other relations of observed frequencies for (simple-complex) Ag “genes” and “antigens” within a specified population. The main ambition of this article is to introduce a new model and methodology permitting a more transparent discussion of the complex interrelations between “facts” or “visual sense data” and the (distorting) ability of reason to understand only that which it creates according to its own design. It is concluded that the serologic field (SEF) does indeed have considerable conceptual advantages over other scientific fields with regard to such fundamental epistemological studies concerning the growth of scientific knowledge. The present data illustrate the mutual feedbacks or dualisms between observation and hypotheses where one component has no meaningful existence without the other and where “facts” are not very trustworthy unless produced within a “guiding paradigm”.

Facts

INTRODUCTION and Hypotheses.

Practically all leading scientists agree about the existence between observationalinterrelations complex of experimental facts (EF) and theories or hypotheses acting as “guiding” paradigms in the generation and testing of such facts. Only a few disciplines or “schools” are today primarily obsessed with “facts and nothing but the facts” to the stubborn and sometimes aggressive exclusion of radically new ways of structuralizing old facts according to new methods. In his program for this Journal, Horrobin summarizes the important feedback processes between “facts and fiction” as follows (1): The physical and chemical sciences long ago recognized that observations are not superior to hypotheses in generating scienttjic progress nor are hypotheses superior to observations. Both are necessary.

Margenau attacks the contemporary obsession (according to the Anglo-Saxon tradition) with “facts and facts only” as follows (2): According to this popular view, science is like an enormous picture puzzle; the scientist discovers the pieces and trusts that benevolent nature or providence has shaped and adjusted them so that they will@ together. When enough piecemeal facts are available and put in their proper places, a recognizable pattern results and a problem has been solved . . . A forest of facts unordered by concepts and constructive relations may be cherished for its existential appeal, its vividness, or its nausea; yet it is meaningless and cognitively unavailing unless it be organized by reason.

Newton introduced the epistemological distinction between how and why] questions-you could legitimately study how “facts” can be structuralized or described within an experimental-conceptual framework, but the actual 245

reasons why apples attract each other according to Newton’s rules of the thumb or why light is split by prisms was relegated to the domains of “theorizing’‘-i.e. more a matter for the Creator to answer. Newton’s first sentence in his 1704 Edition of Opticks thus goes as follows (3): My design in this Book is not to explain the Properties of Light by Hypotheses, hut to propose them, and prove them by Reason and Experiment.

And in his 17 13 edition of Principia he sounded the (somewhat obscure) slogan of Hypotheses non fingo-i.e. I “feign” (invent-pretend-simulate-counterfeit-imagine-test) no hypotheses (4). Kant as well as Goethe commented on the old question concerning observation-observer couplings in science whereby the “facts” now started to become somewhat “soiled” by Reason, the Sensory Apparatus as well as by Methodology: Reason can understand only what it creates according to its own design. (Kant (5)). If the eye were not attuned (sonnenhaft) to the sun, the sun would never be seen. (Goethe (6)).

Hence, Kant did not, as the classical Anglo-Saxon empiricists-Bacon, Locke, Hume, Berkely and Mill-insist that the ultimate source of all knowledge lay in observation or the “facts”. Instead, Kant was a firm (and somewhat stubborn) believer in the inner Light of Reason where: Our intellect does not draw its laws from Nature but imposes its laws upon Nature. In this way, Kant gave to mankind the freedom of choice in structuralizing reality according to man’s own ways (of structuralizing reality). Paradoxically enough, Kant was, however, a firm believer in the “ultimate truth” of e.g. Newton’s laws and Euclidean geometry and hence, in my opinion, missed the whole point of the tenets of Gestalt psychology and structuralism where man is legitimately entitled to impose different “a prioristic” structures on the same set of facts. Obviously Gestalt and

(sicher), and in so far that they are certain, they do not relate to reality. ”

Structure falsify or at least clash with the “facts” by suppressing some “facts” while enhancing other “facts” e.g. as discussed by Stent (7): “Structuralism . . . has provided the insight that knowledge about the world enters the mind not as raw data but in already highly abstracted form, namely as structures. In the preconscious process of converting the primary data step by step, information is necessarily lost, because the creation of structures, or the recognition of patterns, is nothing else than the selective destruction of information. . . . data that cannot be transformed into a structure congruent with canonical knowledge are a dead end; in the last analysis they remain meaningless. That is, they remain meaningless until a way has been shown to transform them into a structure that is congruent with the canon.”

Goethe approached the problem of “facts and fiction” from another angle by stressing the distorting effects of the “experimental method”. Method also falsifies and if light is squeezed through a narrow hole or a lens you would of course be expected, according to Goethe, to obtain an extremely distorted image of reality4.g. a photograph! However, fortunately enough, (for Goethe) photography was not discovered until 1839-seven year’s after his death and 31 years after his Farbenlehre which was a vigorous attack on Newton’s theory (!) of light obtained by the Experimental Method and proved by Reason. Similar structuralistic thoughts, where any attempt (method and/or theory) to impose a structure on reality is bound to falsify and generate “artifacts” rather than “facts”, have of course engaged nearly all professional philosophers and outstanding scientists in other fields such as Eddington, Bohr, Heisenberg, Schrodinger and Einstein. In some cases considerable discrepancies have then been revealed between the “true” world of reality or immediate sense data as contrasted to the “false” world of Reason, Ideas and Experimentation-Methodology. Consider, for example, the two writing Tables of Eddington (8): the substantial emotional-sensory “thing” table as contrasted to the professional physicist’s empty intellectualexperimental “force” table consisting of very little (if any) “matter of fact substance” indeed. Consider also the remarkable ability of the Chinese (or the Swedes) to walk (and think) upside down relative to each other; Nature’s despise of straight lines (except in local areas) and the strange behavior of Eddington’s arrow of Time if you are in a hurry or in situations where time really counts. Consider the difficulties in deciding the true direction of Time even for the best Swiss watch makers as this requires a card-shuffler or some other kind of negative entropy randomizer. And consider that not even the best Swiss precision can supply you with a watch which tells you how much or for whom the absolute bell tolls when the dust particles of the Universe are considered. Einstein stressed this discrepancy between the spinning wheels of the Mind as contrasted to the workings of Matter and simultaneously warned against the generally occurring confusion between the (man-made) Reality and Realityin-itself as follows (9): “‘In so far as the theorems (Sritze) of mathematics relate (beziehen) to reality (Wirklichkeit) they are not certain

Hence, Einstein’s somewhat pessimistic view, no matter how “true” your new frameworks, structures or “gaming rules” are, you are bound to discover, sooner or later, that Nature is playing the game with another set of rules. Consequently, any discrepancies that you can discover between your “theorems” or “rules” and those of Nature will be of immense value for the progress of science where knowledge is obtained because “theory falsifies” and hence, in Einstein’s words (10): “There could be no fairer destiny for any theory than that it should point the way to a more comprehensive theory in which it lives on as a limiting case.”

So, in a sense it looks as if I have acted as the Devil’s advocate and we are back where we started-the “pure” (Goethe) “facts” or sensory data are the most important things for science while “theory”, photography, watches and methodology only falsifies. So, why make theories or hypotheses when, at best, they are doomed to become limiting cases in new theories with no Chinese boxes’ end in sight? And why try to squeeze out the last drop of Light and Time when the immense sea of light-in-itself and absolute time is already there for everybody to see and feel in its truly undistorted Gestalt? Why introduce “falsifying” chess-rules, when the obvious facts are already on one of the (Eddington) chess tables as a specified (or unspecified) arrangement of empty pieces. Why make watches when nearly everybody can hear his own (inner) time Table? The clue to all these questions lies of course in the distinction between “seeing” and “hearing” as contrasted to “perceiving” and “listening”. Everybody can “see” a forest of chess pieces or “hear” the seconds pass by. But very few (if any) can perceive the chess-problem without knowledge of the other empty (Eddington) chess table-i.e. the chessrules which assign a meaning, structure or Gestalt to, otherwise unstructuralized configurations of chess-pieces on empty (“meaningless”) chess tables. Furthermore, if we know the “rules of the game” any attempts at “cheating” by introducing for example two kings or placing some pieces between the squares will immediately be discovered. Similarly, through a watch, previously unstructured time will now be structuralized. And, hence, if somebody is supposed to be late for a meeting, the watch will locally and relativistically decide who came too early. And perhaps with the aid of another watch it will be found that all were present “in time” despite the fact that some actually left home after the meeting had started. Suppose now that, at this meeting, it is decided to impose a new set of (meta-) chess rules on the same chess-piecetable facts. Obviously a new problem, structure or Gestalt might then be perceived and, under certain conditions, this new “problem position” would be equally true as the old problem and hence require new solution(s) within the new meta-chess framework. Furthermore, previously undetected attempts of “manipulating” the facts and hence falsify Reality might now be immediately revealed through our new “watch”. Some of these “falsifications” may have been purely accidental, the pieces were admittedly perfectly aligned within the squares of the old framework but in the new framework, certain transmission errors may be noticed 246

as the “squares of the table” might now be defined with a greater precision-i.e. a new grid may be superimposed on reality. In particular, most interesting and radical Gestalt switches would be expected to occur if the new set of framework rules could be designed in such a way that they would be as complementary or dualistic and equally logically true as the “old rules” in ordinary chess. If so, the two “problems”. structures and Gestalts created out of the same set of “facts” or theory-invariant data would presumably be completely contradictory to each other but equally true despite the fact that they are generated from

exacti)) the same set of visual sense data or information bits! Facts and artifacts. One interesting aspect of imposing one or more different structures on the same set of visual sense data-e.g. a chess problem or a set of experimental observables-is obviously to analyze whether the “chess pieces” or experimental facts (EF k are “correctly arranged and displayed”. Transmission errors may always be expected to occur when the pieces are put OII the Table (in chess) or in the Table (in science). Without a guiding paradigm we do not know whether a piece or information bit arranged between the “squares” of the Table is “meaningful” information about the problem or a “meaningless” artefact or “noise” introduced e.g. by a or, simply, through “carelessspecified “methodology” ness”: A possible logical or transmission error in this sentence

ma>)be difJicult to ident$v for the non-native non-English speaker lacking a Bertrand Russell or Noam Chomsky sense for the “language”-i.e. the “rules of the game”. Some information bits in the above sentence admittedly look “puzzhng”, but in absence of clear rules for the linguistic game it may not be known with certainty whether I am expressing a “meaningful profoundity” or if I have just made a meaningless transmission or typing error when arranging my thoughts in “empty symbols”. Or, in Wittgenstein’s words: “Whereof one cannot speak thereof one must be silent”. To which Karl Popper’s friend replied: “But it is only here that speaking becomes worth while” (11). Hence, “facts” collected in absence of a “guiding paradigm” may not be trusted and may in fact contain a large number of artifacts or “typing errors”. As examples, Kuhn (12) mentions the chemical law of fixed proportions which before Dalton was an occasional experimental finding of very dubious generality. He also mentions that even after the introduction of Dalton’s law, it took almost another generation of scientists to “beat nature into line” because it is so very hard to make nature fit a paradigm. When finally the new paradigm and methodology are accurately aligned, even the ‘&facts”themselves have changed! Consequently, Kuhn, as well as Hanson discuss at length the radical Gestalt switches which occur after scientific revolutions (12) or in the pattern of discovery (13) whereby even the most fundamental methodand observerdependent facts are completely changed-after such revolutions a sitting rabbit becomes a walking duck in Hanson’s pictorial Gestalt, acting as a metaphor for this “radical meaning variance” school of thinking with close 247

connections to structuralism and classical Gestalt psychology. The whole contains more than its parts-a thesis which I do not think needs any excuses. Obviously, structures imposed on an unstructuralized reality will require the addition of extra information bits not originally present in that which was structuralized or encoded. The Ag System. To exemplify some possible feedback effects between observation and theory, experimental facts emanating from the Ag system-a human low density (LDL) lipoprotein polymorphism-will be briefly discussed. A more exhal)stive discussion of the Ag system will be given elsewhere. For simplification, I will thus only analyze the behavior of the Ag system at its six-reagent level relative to a 362 individual Swiss (14) and a 168 individual English (15) population. Both these materials can be regarded as truly “pre-paradigm shift facts”-i.e. they were obtained within the currently reigning or ruling (simple-complex) paradigm. This simple-complex framework (language and theory) specifies that antibody molecules are ultimately monospecific and hence that for every antibody there is R corresponding one-to-one related “antigen” and a similarly one-to-one related “gene”. In such a simple-complex framework, the Ag system is regarded as emanating from a closely linked multiple locus system where each locus has two and only two contrasting or antithetical alleles showing codominant inheritance (16. 25). At the following six-reagent level of discussion, only a simple-complex three-locus model is needed in order to assign a meaning or structure to the experimental observables: ---_+----_t-__. --_+_-__

Ag” Ag’

Ag”’ Agd

Ag’ Agg

Consequently, this theory does not permit any “doubly negative” samples of types Ag(x-y-); Ag(al-d-) or Ag(c-g-) but does otherwise not impose any restrictions on the experimental observables. For example, all other 33 or 27 different phenotypes are permitted as produced by 36 pairwise combinations of the 23 or eight different chromosomes, “linkage groups” or “haplotypes” as follows:

1. Ag”“l’ 2. Ag”lg 3. Ag-rdc

4. Agxdg

5. AgW 6. AgW 7. AgYdc 8. AgVdc

A rivalling conceptual framework (language and theory) was also proposed at an early stage of the game (17-20, 24,26). In this complementary or dualistic (complex-simple) framework, the antibody reagents are regarded as complex or “cross-reacting” while the genetically non-segregating antigens or “haplotypes’-i.e. the products of one and only one chromosome are permitted one and only one “empty symbol” or specificity. Accordingly, in such a complexsimple framework, the Ag system is regarded as emanating from a system with one and only one chromosomal locus with simple (one-symbol) alleles e.g. as follows:

----v-

_--

following, could have been of very great value for the present study. For simplification, easier comparison and increased logical transparency, I will only discuss the typing results for the Ag(x, y, a,, d, c, g) factors. A more comprehensive discussion of these materials as well as other probes for revealing inconsistencies in the data will be given elsewhere.

AgA AgB &f AgD Obviously, this new structure for the Ag system does, when used as a “guiding paradigm”, impose a large number of additional and very severe restrictions on the experimental observables which can be permitted in such a sixreagent and four-allele framework. For example, only (4 x 5)/z or 10 different phenotypes and genotypes will now be permitted due to pairwise combinations of the four simple Ag-antigens Ag(A-D) as produced by the coresponding genes for Ag (Ag”-AgD). Principles

Arrangement

of approach.

Through the new complex-simple framework, important Gestalt switches are obviously accomplished-i.e. the experimental observables are structuralized in entirely new (and previously unknown!) ways. Briefly, what has, up to the present, been regarded as complex and essentially inexplicable relations between genes and antigens in the contemporary (simple-complex) conceptual framework is radically shifted, in a complex-simple framework, to perfectly obvious set relations between the reaction ranges for antibodies and reagents used in revealing these (simplecomplex) “genes” and “antigens”. Consequently, the following strategy will be used: Identify the hitherto unknown reaction range (RR) values for the six anti-Ag reagents relative to certain sample categories in the Swiss and English materials. Arrange the six reagents in series with subset-related or, in set terms, comparable and not disjoint (CNDrelation) RR-values---e.g. the CND-series or “inclusion group” anti-ABC > anti-AC > anti-A. Identify, through this CND-probe, those samples and sample categories which are non-aligned by giving reactions with (one or more) of the subset-related or “short” reagents(s) without also reacting with the “broad” reagent which, in its RR-value (e.g. anti-ABC), also includes that of the “short” or more “specific reagents (e.g. anti-AC; anti-A). For example, no samples would be expected to react with the “specific” anti-A reagent without also reacting with the broader anti-AC and anti-ABC reagents which also include the anti-A “specificity” or “empty symbol”. Discuss the findings both in terms of the contemporary (simple-complex) and rivalling (complex-simple) framework structures or “rules of the game”.

Population

MATERIALS materials

AND

of matrix facts and sample categories.

In Table 1, the observed results of the Ag-typings are given as matrix facts for the Swiss and English materials. The data are arranged in decreasing order of sample frequencies relative to the total material. When arranged in this new fashion, it is immediately seen that, at least the Swiss material, can be divided in two distinct groups or classes of sample categories-a “common” Group I category (S 1-S 10) with lo-55 members in each category and a *‘rare” Group II category (Sl l-S 19) with only 1-2 members belonging to each sample category or phenotype. No less than 350/362 or 96.7 per cent of the Swiss samples do thus belong to the “common” Group I phenotypes while only 149/168 or 88.7 per cent of the English material are members of the Group I sample categories (S 1-S 10): Obviously, the actual existence of the rare Group II sample categories is less well experimentally verified than the common Group I phenotypes-at best Sl l-S19 represent rare phenotypes; at worst nothing but technical errors or artefacts. Nevertheless, in the currently reigning simple-complex) framework all 19 sample categories (S 1-S 19) are equally legitimate and equally well aligned to this, very permissive, framework. TABLE 1. In Table 1, the reagents (Rl-R6) are “traditionally arranged” relative to the corresponding @; Agnljd and AgC’g chromosomal loci and the simple-complex labels (x; y; at; d; c; g) for these reagents are given within brackets. A plus (+) sign at the intersection between a reagent column and a sample row indicates that the corresponding sample is assigned the same symbol as that of the reagent. And hence, in a simple-complex framework, Sl giving the reaction pattern ~ + + + + + is labeled Ag(x- y+ ai+ d+ c+ g+) corresponding to the genotypes Agydc/AgJ’al g or AgyddAgY“J ! Even with this traditional arrangement of anti-Ag reagents, it is immediately seen that, also in theory-invariant terms, some reagentse.g. R2(y) and R6(g )--appear “broader” than other reagents relative to the Group I sample categories and samples. R2(y) and R6(g) thus “cross-react” with no less than nine out of the 10 Group I sample categories or phenotypes which are responsible for respectively 97.8 and 91.0 per cent of the 499 Group I samples. In contrast, RI(x) and R5(c) only react with four out of the 10 Group I sample categories which are only responsible for 36.3 and 52.5 per cent of the Group I samples. An interesting problem is obviously why this theory-invariant distinction between “broad” and narrow” reagents does not apply for the Group II samples. For example, RI(x) reacts with no less than 23/31 or 74.2 per cent of the Group II samples but only with 181/499 or 36.3 per cent of the Group I samples. Actually, all anti-Ag reagents react with 6-7 of the Group II sample categories. Table 1 also shows that the sample categories or phenotypes can be arranged in “broad” and “short” families or classes relative to the six anti-Ag reagents. For example, four narrow sample categories (S7-SlO) can be identified which only react with three out of the six anti-Ag reagents while the remaining Group I and Group II sample categories (Sl-S6; S l l-S19) all react with four to six anti-Ag reagents and hence appear less “specific”. Precisely these narrow sample categories (S7SlO) are also, in the reigning simple-complex framework, regarded as being “homozygous” at the A~YY; Agardand Agclgloci. In a complex-simple framework, permitting only one-symbol chromosomes or “haplotypes”, these phenotypes are assigned a unique oneletter symbol (Ag(A)-Ag(D)) and the corresponding (complex-simple) genotypes are simply given as AgA/AgA-AgVAgD under the assumption of four simple alleles (AgA-AgO) at one single chromosomal locus. Instead, the antibody reagents (Rl-R6) are regarded as “cross-reacting” or “polyfactorial” with different reaction ranges.

METHODS

The Ag typings and a detailed description of the methodology for the 362 mainly Swiss (14) and 168 English (15) samples are given in the original publications (14,15). In both studies a passive haemagglutination test was used and all sera were stated to have been “typed in duplicate” (14) or “at least twice” (15). Unfortunately, no data were, however, presented showing the actual typing accuracy or empirical reproducibility rates of these “duplicate” or “at least twice” studies. Furthermore, one of the research groups (Walton and Valente) has refused to communicate their empirical reproducibility rates which, as will be shown in the

Assignment Obviously.

248

of complex-simple it would be of considerable

reaction ranges (RR) theoretic

and pragmatic

interest

with the A-D “antigens”. For example, R2(y) interacts with the three sample categories S7-S9 which are labeled A, B and C. Accordingly, R2 is assigned the RR-value of anti-ABC. Consequently, all samples which react with R2(y) or anti-ABC must, in a complex-

of their reactions

to be able to describe

the reaction range (RR) values for the various anti-Ag reagents in discontinuous, discrete or “atomistic” terms. Through the identification of the “pure” or “homozygous” sample categories S7S 10 in Table I. discrete RR-values for the various antiLAg reagents can now be precisely identified by simply labeling the reagents with the symbols (A-D) corresponding to the “pure” samples with which they react. The detailed rules for this meta-serologic game with empty symbols has been given elsewhere (24.26-29). In Table 2. the six antiLAg reagents are labeled relative to the four homozygous or pure sample categories or phenotypes (S7-SIO) in terms

Table

I.

A set of matrix facts (MF) obtained

reagents (R lR6).

Simple-complex

SlO) and “rare” SAMPLE

Group

notations

II (Sl I-S19) sample

530 unrelated

Swiss and English individuals

are given in brackets.

against

six different

The material is divided into “common”

categories.

REAGENT

No.

OBSERVED

RI

R2

R3

R4

R5

R6

(x1

(y)

(al)

(d)

(c)

(g)

+ + _

+ + + _

+ + + -

i + + t +

_--

No. OF SAMPLES

SWISS

ENGLISH

$2

+ + + _

+

s5 Sh $7

+ f + -

+ _

S8 s9

+

SlO

-

55

35

43

21

90 64

48

I4

63

45

8

53

37

16

53

34

19

53

32

13

c

29

II

45 41

t

17

9

26

+

10

1

11

350

149

499

(96.7%)

(88.7%)

(94.2%)

+ _

A

Total

GROUP

II

_

t +

Sll s12

+

s14 s15

t + + +

S16 s17 S18 s19

+ + +

+

+ + + +

s13

+ + + + + _ _

+ +

+

_ + + +

2

+

2

-

2

+

2

+

2

c

0

+ +

0 1 1

Total

-19

12 (3.3%)

sample

TOTAL

I

ii4

Table

anti-Ag

Group I (Sl-

CATEGORY

--

CiROLP -S! ‘il

by testing

for the reagents

simple framework (language and theory) be assigned at least one of the three “empty symbols” or “antigens” A. B or C. Conversely, all samples which do not react with anti-ABC must be regarded as lacking all the “symbols-antigens” A, B and C if the rules of the complex-simple game should be adhered to.

2.

Complex-simple

categories

reaction

(S7SlO)

range (RR) values for the six anti-Ag reagents (R lR6)

in Table I. The reagents

REAGENT:

R2(y)

R6(g)

R3(ai)

RR-VALUE:

ABC

BCD

BD

+ i-

-

+ + +

+

(11.3%)

SAMPLE CATEGORY

GENOTYPE

R5(c)

AC

D

A

_

-t

s7

Ag A/Ag .(

-

_ _ _~

S8

AgR/AgH

s9 SlO

Agq.4gC Ag?Ag”

f 249

or “pure”

relative to S7-SlO.

R l(x)

+ _

(5.8%)

relative to the four homozygous

of “broadness”

R4(d)

+

+ _

in three classes

“NARROW” REAGENTS

“MEDIUM” REAGENTS

“BROAD” REAGENTS

c

are arranged

31

COMPLEXSIMPLE

SIMPLECOMPLEX

Ag’dc/Ag.@ Ag?‘“l5’/&W’“1P Ag?dcfAg?‘dn Ag‘:,$/Ag-rar ‘:

From this labeling of the anti-Ag reagents (Rl-R6) relative to S7SlO it is now found that they can be arranged in three classes of “broadness”: the broad-range reagents R2 and R6 reacting with three out of the four antigens; the medium-range reagents R3 and R4 reacting with two out of the four antigens and the narrow-range or even “monospe&c” reagents R 1 and RS reacting exclusively with one and only one of the four antigens, sample categories or “homozygous phenotypes”. Notice, however, that these RR-values are only relative to the four sample categories or “antigens” used in this labeling process. Also notice that only four sample categories (S7-S 10) have been “used up” in this complex-simple labeling operation. And hence, no less than 15 completely free, unsoiled or virgin sample categories (Sl-S6; Sl LS19) totalling 407 individual samples and 6 x 407 or 2,442 experimental “facts” or informationbits remainin Table 1 for the actual probing.

anti-A and anti-D and hence must be assigned the label AD with no additional problems. And, in the same way, the reader should carefully go over the labeling of the remaining samples in terms of this new meta-serologic calculus. It will then be found that all the labels in Table 3 are logically true and perfectly unambiguous-there is no way of “cheating”.

RESULTS Complex-simple labeling of sample categories An obvious but exciting test for the general alignment

between the remaining experimental facts in Table 1 and the complex-simple framework structure in Table 2 is to explore which of the 15 remaining sample categories (Sl-S6; Sl lS19) can be labeled by the use of four and only four symbols. Obviously, only 4;5 or 10 different sample categories or “phenotypes” can be labeled by an alphabet with four symbols and the complex-simple grammatical restriction that, in a diploid species, a sample category can be assigned a maximum of two and only two empty symbols if “heterozygous” and one and only one symbol if “homozygous”. In Table 3, it is shown that all the “common” Group I sample categories (S 1-S10) can unambiguously be labeled in this new complex-simple framework-i.e. by assuming only four alleles (&A-Ago) at one single chromosomal locus. This is very encouraging as the common Group I sample categories could scarcely be expected to represent “technical errors” and hence, they should be aligned to the new framework. In contrast, not a single sample category belonging to the “rare” Group II samples can be labeled. These 3 1 samples and nine phenotypes (Sl l-S19) simply can not exist within the present six-reagent and four allele framework. Another rather remarkable (and satisfactory) finding is that all the 10 Group I samples can be unambiguously labeled with one and only one set of symbols. And consequently, a previodsly unknown obligatory one-to-one relation is introduced between a phenotype and its corresponding genotype for the Ag system at this six-reagent and four-allele level of analysis. For example, Sl does not react with anti-D(R1) and hence can have, at most, two of the three symbols A, B and C. However, Sl reacts with anti-A (R5) and hence must be assigned the symbol “A”. Sl also reacts with anti-BD(R3) and, because “D” has already been excluded, S 1 must also be assigned the symbol “B”. Accordingly, Sl can be unambiguously labeled as AB (corresponding to the heterozygous genotype AgA/AgB> and the positive reactions of Sl also with R2(y), R6(g) and R4(d) only serve as redundant or independent verification showing that Sl does indeed have the “antigens” A and B-as would be expected or “predicted” Sl also reacts with anti-ABC(R2), anti-BCD(R6) and anti-AC(R4). Similarly, S2 reacts with the “monospecific” reagents 250

In contrast, the currently ruling simple-complex framework for the Ag system permits no less than 23 or 8 different Ag-chromosomes or “linkage groups” corresponding to *;9 or 36 different genotypes! Accordingly, for example the S2 sample category or phenotype would (in a simplecomplex framework represent one or more of no less than four d@rent genotypes as follows: 1. Agxal qAg@g 2. Ag”l YAg”d’ 3. Agxdc/Agyalg 4. AgxdglAgyal’ Similarly, also Sl and S5 would, in a traditional simplecomplex framework, correspond to two different and “hidden” genotypes Calculation

of complex-simple

gene frequencies

Another interesting complex-simple probe of the “facts” in Table 1 is therefore to investigate the alignment between actually observed and theoretically calculated or “expected” samples in the various Group I sample categories. For example, in the established (simple-complex) conceptual framework, the S2 sample category could contain no less than four different “hidden” genotypes representing various combinations of no less than eight different anti-Ag chromosomes. In contrast, the new complex-simple framework does, at its four-allele level, explicitly specify that e.g. S2 does represent one and only one genotypic constellation: Ag;JIAgD which, in the simple-complex framework, would correspond to only the genotype Agxalg/Ag+ Accordingly, the denial of the existence of the remaining three Ag genotypes might be expected to show up as a considerable excess of S2 samples relative to their calculated values. Similarly, it would be expected that Sl and S5 with the two “hidden” simple-complex genotypes would also be poorly aligned to the new framework which only permits one single genotype in each sample category. In Table 4, it is, however, found that at least the Swiss (P=O.9-0.8) and total (P= 0.5-0.3) materials show a very good agreement between observed and expected values for the samples belonging to the 10 common Group I sample categories despite the fact that we have “shaved off’ no less than four Ag-chromosomes and/or (simple-complex) linkage groups in the new (complex-simple) framework! Furthermore, due to the strict one-to-one correspondence between a phenotype and a genotype in this new framework, no elaborate gene calculations are required to obtain the expected frequencies. The different Ag genes are simply counted! For example, the Swiss material comprising 350 individuals or 700 chromosomes has 2 10 AgAchromosomes and accordingly the AgA gene frequency is 210/700 or 0.3000. Obviously, this gene-counting technique does have a considerable “self-correcting” or redundant character but still

formula xwhere p is the relative frequency of actually observed homozygous samples. When the gene frequencies for AgA-AgDare calculated in this way, it is found that they only add up to 97.96 per cent for the Swiss and, even worse, only 90.69 per cent for the English material. Hence, in order to give the English a kind of (perhaps needed) “handicap”, the actual gene frequencies used in calculation of expected values are adjusted in order to add up to 100.00 per cent-e.g. the adjusted value for the Swiss AgA gene becomes$&$ or 0.3087.

it is obvious that the rather high XL-value of 13,768O (P= 0.2-0.1) could be due to a certain technical error rate for the English material. In order to subject the data to a much more severe probing as regards the alignment between observed and expected values, Table 5 gives another way of obtaining gene frequencies by using only the homozygous sample categories S7-SlO. From the observed frequencies of these four homozygotes (AgAIAgA-AgD/AgD the gene frequencies for the alleles AgA-AgO are simpiy obtained from the Table 3.

Complex-simple labeling of Group I sample categories (Sl-SIO) in Table 1.

REAGENT:

R2(y) R6(g) R3(a,)

R4(d) Rl(x)

RR-VALUE:

ABC

BCD

BD

AC

D

A

+ t+ + + + + + c _

+ + + + + + -

+ +

+ + + -

+ _

+ + +

+ +

-

+ + +

+ _

+ + +

+ t + -

A: A; A; B; B; B; A B C D

D B; A; A A; B; A; A; A;

-

+

NOT EXCLUDED SYMBOLS

EXCLUDED SYMBOLS

+

+ -

+

R5(c)

D C D C; C; B; B;

D D D C

SAMPLE CATEGORY

LABEL

B; C B; C; D C D C; D C

I

AB AD AC BD CD BC A B c D

Sl s2 s3 s4 s5 S6 S7 S8 s9 SlO

Observed and expected genotype frequencies for the Swiss and English Group I sample categories. Expected values are obtained through ordinary gene-counting for, respectively, the Swiss, English and pooled materials. Simple-complex “linkage groups” corresponding to Ag 4-AgD alleles are given in brackets.

Table 4.

FREQUENCIES SIMPLE-COMPLEX PHENOTYPE

SAMPLE CATEGORY

GENOTYPE

SWISS OBS. EXP.

Sl

a,-d+ a,+d+ a,-d+ a,+d+ a,+da,+d+ a,+dM~+Y+ Ad= y+ a,-d+ Adx+ Y+ a,+d+ Adx+ Y- a,+d-

Ag(x- y+ W-y+ k&-y+ Mx+y+ Ag(x- Y+ &(x-y+

Sl s3 s2 S8 S6 s4 s9 S5 SlO

c+g-) c+g+) c+g+ ) c+g+) c-g+ ) c-g+) c-g+) c-g+? c-g+ ) c-g+)

32 55 48 43 29 34 45 17 37 10

31.5 57.6 45.9 43.5 26.3 42.0 39.8 16.7 31.7 15.0

350

350.0

A/A AIB A/C AID B/B B/C BID C/C CID DID

TOTAL P (9 d.f.) Ag”

(AgJd’)

Ag’ Ag ’

(AgJ’V) (Ag.“dd

210/700 1921700 1531700 1451100

= = = =

the

calculations

of

these

expected

values

TOTAL X2

OBS. EXP.

X2 -

0.0079 0.1174 0.096 1 0.005 7 0.2112 1.5238 0.6194 0.0054 0.886 1 1.6667

13 35 15 21 12 19 8 9 16 1

15.8 28.0 22.1 15.3 12.4 19.6 13.6 7.8 10.7 3.7

0.4962 1.7500 2.2810 2.1235 0.0129 0.0184 2.3059 0.1846 2.6252 1.9703

45 90 63 64 41 53 53 26 53 11

47.2 85.5 68.0 59.1 38.7 61.6 53.5 24.5 42.5 18.5

0.1025 0.2368 0.3676 0.4063 0.1367 1.2006 0.0047 0.0918 2.5941 3.0405

5.2657

149

149.0

13.7680

499

499.1

8.1816

0.2-O. 1

0.3000 0.2143 0.2186 0.2071

In Table 5, the observed and expected values for the “unsoiled” heterozygous sample categories S 1-S6 are given where consequently only the observed frequencies for the homozygous sample categories (S7SlO) have been “used in

OBS. EXP

0.9-0.8

AgD (Ag.ralg)

Up”

ENGLISH X2

911298 86/298 681298 411298

= = = =

0.5-0.3 0.3255 0.2886 0.2282 0.1577

3071998 2181998 2211998 1921998

= = = =

0.3016 0.2786 0.2214 0.1924

heterozygotes. The high X*-values for the English S2 and S5 sample categories due in both cases to a considerable excess of observed samples relative to the expected values renders

for

Walton 251

and Valente

an exclamation

mark (!) but does not

Table 5. Observed and expected frequencies for the Swiss and English Group I sample categories. Expected values now obtained through the formula fi where p is the observed relative frequency of a homozygous sample category. In the calculation, “adjusted”

gene-frequencies (adding up to 1.0000) have been used. CATEGORY GENOTYPE

FREQUENCIES SWISS

Sl

A/B AID A/C

s2 s3 s4 S5 S6

BID C/D B/C

TOTAL P

ENGLISH

OBS.

EXP.

X2

OBS.

EXP.

55

43 48 45 37 34

63.5 37.3 48.6 35.5 27.2 46.3

1.1378 0.8710 0.0074 2.5423 3.5309 3.2676

35 21 15 8 16 19

30.4 8.8 26.3 8.4 7.3 25.3

262

258.4

11.3570

114

106.5

(6 d.f.)

0.10-0.05

TOTAL X2

OBS.

EXP.

X2

0.696 1 16.9136(!) 4.855 1 0.0190 10.3685(!) 1.5688

90 64 63 53 53 53

92.5 47.9 73.7 45.7 36.4 70.3

0.0676 5.4115 1.5535 1.1661 7.5703 4.2573

376

366.5

20.0263

34.42 11

d>c) and the Ag (g>a,>x) series. 252

;;;;; . 1.0000

Obviously, in our meta-serologic calculus, no samples could exist which react with a subset-related reagent-e.g. R5(c) with a RR-value of anti-A- without also reacting with the broader reagent (sti.e. R4(d) or anti-AC and/or R2(y) or anti-ABC-which “include” the reaction range (A) of the subset-related reagent. We now immediately realize that one formal reason for the unambiguous labeling of all Group I samples in Table 3 was, precisely due to these subset-relations for the various anti-Ag reagents revealed through the new complex-simple framework. This is immediately obvious from Table 7, where the anti-Ag reagents (Rl-R6) are now arranged according to these subset-series or “inclusion groups” in Table 6. It is now immediately seen that not a single individual of the 499 samples in the common Group I category will refute this CND-probe. In contrast, all (!) 31 Group II samples corresponding to no less than nine sample categories or phenotypes (S 11-S 19) are refuted by this “logically true” CND-probe! For example, the seven Sl 1 samples lack the antigens A, B and C through negative reactions with anti-ABC or R2 But, positive reactions with anti-AC or R4 make them have at least one of the “antigens” A or C. Similarly, in the S 12 phenotype with five members, the antigen “D” is obviously excluded through negative reactions with anti-BD or R3. But this finding is completely contradicted through the positive reactions with anti-D or R 1! Similarly, all the “rare” sample

Table 6. Arrangement of the six anti-Ag reagents in two “inclusion” groups where the reagents have subset-related or CND-related reaction ranges (RR). The RR-values are obtained from Table 2. INCLUSION

Reagent

GROUP

SIMPLE-COMPLEX

NOTATION

COMPLEX-SIMPLE

I.

R2 R4 R5

anti-&(y) anti-Ag(d) anti-Ag(c)

anti-ABC anti-AC anti-A

II.

R6 R3 Rl

anti-Ag(g) anti-Ag(a,) anti-Ag(x)

anti-BCD anti-BD anti-D

RR-VALUE

Table 7. The 3,180 information bit matrix facts in Table I after arranging the six anti-Ag reagents in two subset-related series or “inclusion groups”. RR-values for the reagents (Rl-R6) and “adjusted” sample categories for the “non-aligned” Group II samples are given in brackets. SAMPLE CATEGORY REACTIONS

WITH REAGENTS

PHENOTYPE

OBSERVED No. OF SAMPLES SWISS

(A:C) GROUP

pA:

(B:;)

(ii)

+ + + + + +

+ + + + + -

+ + +

+ +

+ + + + + + + + + _

TOTAL

+ + + + + + _

+ + +

+

+

+ + +

+

AB AD AC BD CD BC A B C D

43 48 45 31 34 32 29 17 10

35 21 15 8 16 19 13 12 9 1

90 64 63 53 53 53 45 41 26 11

350

149

499

2 2 2 2 2 0 0 1 1

5 3 3 3

7 5 5 5 3 2 2 1 1

55

II

SI l(S10) S12(S9) S13(S7) s 14@4) SlS(S8) S16(SlO) S 17(S2) S18(S3) S19(S7)

+ +

+ +

-

-I-

+

+

+

+

+ -

+

+

+

+

+

+

+

+

+

+ + + + + + +

+ + + + + + -

+ + + + + + +

-

Typing errors? A challenging question is of course if all the “rare” Group II sample categories and samples represent (trivial or non-trivial) typing errors. If so, the Swiss have an overall error rate of 121362 or 3.3 per cent while the English produce a more impressive figure of no less than 19/168 or 11.3 per cent-i.e. at least one sample in nine was by the Walton-Valente

19

31

Let us test this “typing error” paradigm under the assumption that only one single typing error was committed for each sample and furthermore that this error generated a single false positive result. Under these assumptions, it is found that most of the “rare” Group II sample categories (S 11-S 19) can now be unambiguously shifted (back) into one and only one corresponding common and “legal” Group I sample category @l-S 10). For example, correction of a false positive typing result for Sl 1 with R4 shifts S 1 1 into SlO. Similarly, S12 is shifted into S9 if we assume false positive reactions for R 1; S 13 is shifted into S 7; S 14 into S4 etc. Only for S 17 do we have to assume a false negative

(S 11 -S 19) do (logically) contradict themselves relative to this subset-probe while the common (Sl-S 10) sample categories do (redundantly) confirm themselves.

categories

typed

1 2 2 0 0 -

12

wrongly

ENGLISH

;)

I

Sl s2 s3 s4 s5 S6 Sl S8 s9 SlO

GROUP

&

team. 253

material where the new x2-value is marginally increased to 5.5764. The English go down from 13.7680 to 9.9723 and the pooled materials from 8.18 16 to 5.0678. But, what will happen when the more severe test of calculating expected values as shown in Table 5 is used? Table 8 shows the results and it is now found that the total 3,180 bit material is actually much better aligned after these “error adjustments” than before.

result to avoid a “double positive” error for R4 and RS. We can now test this “error paradigm” by investigating what happens with our observed and expected frequencies when we have shifted all the illegal S 11-S 19 samples back to the Sl-S 10 samples where they might actually belong. Will the data in Tables 4 and 5 improve by this process? With the gene-counting method (not shown) it is found that the $-values are indeed improved except for the Swiss

Observed and expected frequencies for the Swiss, English and total materials in Table 1 (Group I and Group II samples) after shifting the Group II samples into Group I samples through an “error paradigm”. Compare also with Table 5.

Table 8.

CATEGORY GENOTYPE

FREQUENCIES SWISS OBS. EXP.

Sl s2 s3 s4 s5 S6

A/B

AID AIC ND CID B/C

TOTAL

Xz

ENGLISH OBS. EXP.

c/c DID

TOTAL

TOTAL EXP.

X2

64.0 39.8 50.1 37.5. 29.3 47.1

1.2656 0.2573 0.0242 2.4067 2.0235 3.6435

35 23 15 11 16 19

25.1 19.7 24.1 17.7 17.0 21.7

3.9048 0.5528 3.436 1 2.5362 0.0588 0.3359

90 66 64 58 53 53

88.5 59.7 74.3 55.4 46.5 69.0

0.0254 0.6648 1.4279 0.1220 0.9086 3.7101

265

267.8

9.6208

119

125.3

10.8246

384

393.4

6.8588

0.2-O.1 A/A BIB

OBS.

55 43 49 47 37 34

P (6 d.f.) s7 S8 s9 SIO

X2

0.1-0.05

0.5-0.3

35 31 19 12

34.0 30.1 18.5 11.7

0.0294 0.0269 0.0135 0.0077

16 13 12 8

13.9 11.3 10.4 7.0

0.3173 0.2558 0.2462 0.1429

51 44 31 20

47.7 41.1 29.0 18.7

0.2283 0.2046 0.1379 0.0904

97

94.3

0.0775

49

42.6

0.9622

146

136.5

0.6612

GENE FREQUENCIES

0.1821

adjusted 0.3064 0.2884 0.2258 0.1795

0.3086 0.2782 0.2673 0.2182

adjusted 0.2878 0.2594 0.2493 0.2035

0.3 102 0.2881 0.2418 0.1943

adjusted 0.2999 0.2785 0.2338 0.1878

1.0147

1.0001

1.0723

1.0000

1.0344

1.0000

AsA @gyd’)

0.3 109

AgB (AgJ’“l9 AgC (AgYd@ AgD (Ag”l g)

0.2926 0.229 1

TOTAL

It is therefore quite interesting to notice that the new framework does, despite its extreme frugality, have a considerable predictive area. Obviously, the four “empty symbols” or “specificities” (A-D) do thus permit the prediction of no less than 24-2 or 14 “specific” anti-Ag reagents with various reaction range values. Hence, in a simple-complex framework also 14 corresponding (simplecomplex) “genes” and “antigens” become predictable of which only six have been discussed. Furthermore, the actual frequencies of these eight “predicted” new Ag “genes” and “antigens” or “factors” can be exactly specified by knowing the RR-value for corresponding anti-Ag reagent and the gene frequencies for the (complex-simple) AgA-AgD genes! In Table 9, these predictable (simple-complex) genes are given together with their expected gene frequencies for the Group I samples as calculated from the Ag%4gD frequencies in Table 4. For example, the gene frequency for the predicted Agk gene is the sum of the gene frequencies for AgA + AgB + AgC. Furthermore, all these (simple-complex) genes can be

Thrcugh this error adjustment, the total English material now obtains a x2-value of only 10.8245 while, the previous value was a staggering 34.4211 despite the fact that all the 19 “impossible” Group II samples had been eliminated from Table 5. Predictability

of the new framework.

Obviously, the new framework has not only made it possible to identify what with a high degree of probability amounts to nothing but (trivial or non-trivial) “typing errors” but also to shave off no less than nine Ag phenotypes (S l,l-S19) which would have required at least four additional Ag chromosomes or, in a simple-complex framework, four new “linkage groups”. Furthermore, this Occam razor principle has reduced the number of “empty symbols”, “genes”, “antigens’” and “factors” from six to only four! The number of chromosomal loci has been reduced from three to only one! And, by this process, the alignment between actually observed and theoretically calculated samples has been considerably improved. 254

Table 9.

Prediction of additional (simple-complex) Ag-genes (AGk-Ag’) and their expected gene frequencies in “legitimate” Group I

samples. The gene frequencies are obtained from e.g. Table 4 by knowing the particular combination which represents ALREADY

the “new” (simple-complex)

“OBSERVED”

of the genes Ag&dgD(in

brackets)

genes. GENE

GENES

FREQUENCIES

SWISS

IN GROUP

ENGLISH

I SAMPLES TOTAL

Ag’

(Ag” + AgB + Ag?

0.7929

0.8423

0.8076

AgR

(AgR + Ag= + AgD,

0.7000

0.6745

0.6924

AgF

(Ag’ + AgDJ

0.48 14

0.4463

0.47 10

Agd

(Ag”’ + AgC)

0.5186

0.5537

0.5290

Ag’

(Ag?

0.207 1

0.1577

0.1924

AgC

(Ag“‘1

0.3000

0.3255

0.3076

PREDICTED GENES

Ai &’

Agm 4” 4’ AgP 4’ Agr GENE

+ AgD)

0.7814

0.7718

(AgA + AgC + AgD)

0.7257

0.7114

(AgA + AgBJ (AgA + Ag?

0.5743

0.6141

0.5071

0.4832

(AgB + Agel (Ag” + AgD)

0.4929

0.5168

0.4257

(Ag’J (Ag?

0.2743

0.3859 0.2886

0.2186

0.2282

0.7786 0.7214 0.5862 0.5000 0.5000 0.4138 0.2786 0.2214

(Ag‘4 + A?

FREQUENCIES

USED

FOR

CALCULATIONS

(From

Table 4)

AgA

0.3000

0.3255

0.3076

AgB

0.2143

0.2886

0.2786

AgC

0.2186

0.2282

0.2214

AgD

0.207 1

0.1577

0.1924

or in repulsion. From Table 10 we can immediately identify a large number of such potential candidates for the cisl

assigned various chromosomal loci and Table 10 shows the simple-complex gene map for the Ag system when all these anti-Ag reagents have been found and identified.

truns pseudo-mystery. For example, whenever the “genes” Agrand Aggoccur on the same chromosome, the cis antigen Ag(o) is produced! Why? From Table 9 it is easily seen that whenever two (or more reagents) have NCND-related RR-values then a third reagent with the same RR-value as the intersection of the RR-values for the two NCND-related reagents could, in a simple-complex framework, be described as revealing and reacting with a cis antigen. In the foregoing example, R2(y) and R6(g) thus have the NCND-related RR-values anti-ABC and anti-BCD. They intersect at anti-BC. And, anti-Ag(o) is precisely the reagent which has the RR-value of anti-BC-i.e. the same RRvalue as the intersection of the two NCND-related reagents anti-ABC and anti-BCD. The reader is encouraged to identify formally from Table 9 or informally from Table 10 the remaining no less than 29 additional cis antigens produced by pairwise combinations of simple-complex genes. It is found that it is much more easier to identify these cis antigens from the “predictions” in Table 9 than from the “empirical” chromosome map in Table 10. As further practice, the reader is also encouraged to identify the cis antigens occurring when three or more simple-complex genes occur in cis position. For example, the cis antigen Ag(c) is obviously only produced when the “genes” Ag’ and Agd and Agk and Agl and Ag” and Ag” occur in “cis position”! A quick check in Table 9 reveals

Some simple-complex pseudo-mysteries. A pseudo-mystery can briefly be defined as a (preferably) law-like phenomenon which occurs in one framework but which is totally explained-i.e eliminated after a conceptual framework shift. For example, why do the 14 (simple-complex) genes in e.g. Table 10 occur in antithetical pairs showing codominant inheritance? From Table 9 it is found that this pseudo-mystery or “law” is easily explainable in the new framework. Whenever two reagents have non-overlapping or, in set terms not comparable or disjoint (NCD-relation), reaction ranges and furthermore when these RR-values together encompass (EC-relation) all gene products in a system-then these reagents will, in a simplecomplex framework appear to reveal two “antithetical” or “contrasting” genes at a single locus! Example of such NCD-EC related reagent pairs are anti-Ag(y) and anti-Ag(x) with the RR-values of anti-ABC and anti-D; anti-Ag(k) and anti-Ag(r) with the RR-values of anti-ABD and anti-C etc. Another interesting pseudo-mystery is the occurrence of so-called cis antigens-i.e antigens which are produced only when two (or more) (simple-complex) genes occur on the same chromosome-i.e. in cis position-but not when occurring on different chromosomes-i.e. in truns position 255

The orthodox simple-complex genetic theory for Ag at its predicted 14-reagent level (from Table 9). Seven loci, each with two antithetical alleles showing codominant inheritance would be created out of only four alleles (AgA-Age)at one single chromosomal locus. Some of these (simple-complex) “genes” and “loci” may already have been identilied.

Table 10.

I

I

I

4f

Ag”1

AgY

Agd

I

I

Age

Ag’

Agg

Ag’

I

I

Ag’

Agm

Ag”

Agq

Ag*

Ago

SIMPLE-COMPLEX LINKAGE GROUPS: Ag

ydcklmn

AgYal&?kW’ AgYd&VIPo 4

*a,

gklpn

that all these reagents are NCND-related and “intersect” at anti-A. Not unexpectedly, anti-Ag(c) has precisely the RR-value of anti-A and hence this new law obviously also applies in the extremely generalized case. Finally, another interesting (simple-complex) pseudomystery (which presumably has not even been noticed before) is a remarkable “clustering” of (simple-complex) Ag-gene frequencies. From Table 9, we thus immediately recognize that e.g. the genes Agy and Agg have very similar gene frequencies ranging from 0.67-0.84. Another “cluster” are the Ago* and Agd genes ranging between 0.44-0.55. And finally, a third cluster is formed by the Ag” and Age genes ranging between 0.16-0.33. Why? Table 9 immediately gives the answer-broad-range reagents reveal high frequency genes, narrow-range reagents reveal low frequency genes. For example, Table 9 shows how also the (simplecomplex) gene frequencies can be nicely arranged in subsetrelated series corresponding precisely to the subset-related reaction ranges of the corresponding reagents+.g. the Ag(y>d>c) and Ag (g> a,>x) inclusion groups. The corresponding (simple-complex) “genes” do thus have “subsetfor the total material: related” “ gene-frequencies”-e.g. 0.81>0.53>0.31 and 0.69>0.47>0.19. This is, of course, strong evidence for the correctness of the new complexsimple framework structure.

Metascrology

symbols (A, B, C, D). Furthermore, these new gaming rules precisely identify which of the remaining plus and minus pieces may represent transmission errors or faulty alignment on the board. Through this new set of rules it is found and “logically proved” that all the 6 x 9 or 54 “rare” Group II squares in Table 1 can be eliminated. Metaserology

and information

theory

In an information-theoretical approach, Table 1 represents 6 x 530 or 3,180 information bits which, according to the simple-complex code, can be translated into a meaningful “message” through the use of six symbols (x, y, a,, d, c, g). In a complex-simple code only four symbols (A, B, C, D) are required to encode the data to a message which is at least equally meaningful to the human mind as that of the simple-complex code. Furthermore, the complexsimple code identifies certain transmission errors or “noise”. It is thus found that in the Swiss material at least 12/2,172 or 0.55 per cent of the information bits in Table 1 are incorrectly aligned relative to the complex-simple code-i.e. they can not be meaningfully translated through the use of only four symbols and the “grammar” of this code. In the English material, no less than at least 19/1,008 or 1.88 per cent of the information bits (+ and - signs) are incorrectly aligned relative to the new code-i.e. an error rate of one in 53 bits. Accordingly, if we only permit an overall error rate of 3 l/3,180 bits or 0.97 per cent-i.e. one bit in 103 bits-we can meaningfully structuralize, solve or encode the complete 3,180 bit matrix problem in Table 1 through the use of four and only four symbols! Furthermore, we can accurately predict and meaningfully encode no less than 8 x 530 or 4,440 additional information bits which would be generated if the same material was tested with the eight additional anti-Ag reagents (anti-Ag(k)-anti-Ag(r)) as specified in Table 9. Consequently, the complex-simple code does, despite its extreme “Occam razor” parsimoniousness, have a considerable predictive area.

DISCUSSION and metachess.

Table 1 with six reagents and 19 sample categories can be compared to e.g. a chess or checkers position on a 114 square board with plus (+) and minus (-) pieces. In absence of any rules of the game, actually 26 or 64 different sample categories or “rows” could exist generating a 6 x 64 or 384 “square board”. The contemporary (simple-complex) gaming rules as discussed in this article permit the existence of 6 x 27 or 162 “squares” filled up with plus and minus signs at the cost of only six empty symbols (x, y, al, c, g). The new complex-simple gaming rules as discussed in this article permit the game to be played on a six reagent and 10 phenotype or 60 square board at the cost of only four

The experimental

customer is always right?

Brillouin claims that “the experimental 256

customer

is

always right” as evidenced e.g. by the following statement (32): Perfect logic and faultless deduction make a pleasant theoretical structure, but it may be right or wrong; the experimenter is the only one to decide, and he is always right. Superficially, this opinion clashes with Kuhn’s opinion concerning the validity of pre-paradigm shift experimental facts (I 2) where, for example, in the case of Dalton’s law of fixed proportions, “theory finally besieged the facts”. However, this “hen and egg” or “Bacon and Egg-head” differences of opinion are superficial and circular. Obviously, theory can only be refuted by experiments of sufficient accuracy. But, sufficient accuracy is, in Kuhn’s opinion, only possible in presence of a “guiding paradigm” -i.e. a pre-existing theory. So, Horrobin’s statement (1) that “observations are not superior to hypotheses . . . nor are hypotheses superior to observations” is, in a sense, logica& and empirically verified in this article. Obviously, it is very difficult to accurately align e.g. the 3,180 information bits or “facts” in Table 1 when you do not know the “new” rules of the game much in the same way as the non-native (proof) reader may have considerable difficulties in discovering logical transmission or typing errors in this article which, partially, is written in a new and hence strange language. As further verification of the Brillouin-Kuhn-Horrobin theses it may be noticed that the Biitler data do at least superficially seem to be perfectly aligned with the ruling (simple-complex) paradigm despite the fact that a technical error rate of about 5 per cent was empirically demonstrated by Hirschfeld with regard to Ag-typings (20). Even if Butler and Walton-Valente would claim that #theynever make any errors it seems nearly impossible to believe that their very sensitive haemagglutination technique combined with considerable lability of the human P-lipoproteins on which the Ag antigens are localized, would permit them to obtain such a perfect alignment with the simple-complex paradigm if this paradigm had not been known. After all, 3,180 experimental observables or “facts” are an impressive collection of data and even an error rate of one error in 100 determinations could generate 32 erroneously typed samples appearing as “new” phenotypes or linkage groups hidden among the 530 samples responsible for the 3,180 experimental data. If further evidence is required as to the importance of a “ruling paradigm” in making your facts “accurate”; Butler’s “New observations of anti-Ag sera” from 1965 might be mentioned (30). In this early study, no less than 3-4 out of 55 individuals-i.e. at least 5.4 per cent-were claimed to refute the Hirschfeld theory for the dominant inheritance for Ag”. Needless to say, subsequent studies have shown that these Butler exceptions must have been technical errors-typing for Ag(x) is used all over the world in e.g. paternity testing and very large family materials have confirmed the dominant inheritance of Ag” (20-25). Similarly, at an early stage of the game, Contu (31) did actually (without knowing it) describe no less than 3/42 or 7.1 per cent Ag(x-y-) individuals-i.e. what today is clearly regarded as nothing but a technical error rate of at 257

least 7.1 per cent. And even in my own early Ag material no less than 6/462 Ag(z-t-) samples were published in 1968(20)-i.e. what today is regarded as a technical error rate of 1.3 per cent! All these “pre-paradigm” technical errors come very close to the average empirical error rate of 5 per cent for the Ag system which was found and published when over 100 samples were tested in at least duplicate tests against over 30 (absorbed and unabsorbed) anti-Ag reagents (20). The complementarity

principle.

Can two complementary and dualistic world views be equally true? In my opinion it is perfectly obvious that they can. The old simple-complex framework (language and theory) distorts and falsifies reality in one direction while the new complex-simple framework presumably falsifies equally much but in the opposite direction. Antibody molecules are presumably not monospecific (e.g. anti-x; anti-y . . . anti-g) and haplotypes are presumably not mono-factorial (e.g. A, B, C, D). Antigens may not be quite as complex as e.g. the simple-complex Ag type (x+ y+ ai+ d+ c+ g+) but also not as simple as the corresponding (complex-simple) type Ag (A+ D+). The new framework shifts a previous complexity of “genes” and “antigens” into a complexity of antibodies and reagents. Suppose that Nature decided to let both genes-antigens and antibodies reagents be complex. If so, neither the simple-complex nor the complex-simple frameworks (languages and theories) can pretend to be more than “halftruths”. But together they seem to complement each other. If so both these “Hirschfeld” frameworks or “half-lies” for the Ag system (16,24) should be able to look forward to their “fair destiny” or “happy retirement” as limiting cases in a more comprehensive (complex-complex) conceptual framework (24,28,29). It is thus obvious that there are frequently two complementary retiring positions as limiting cases in a more comprehensive or unified theory. What is now the meaning of such logically true, complementary and (perhaps tautological?) games where “empty facts” are transformed into equally “empty symbols”. The answer should be obvious at least to the chessplayer. Through the new gaming rules, new meanings, structures, patterns or Gestalts are imposed on the worldbe it a chess problem or a serology matrix table. In this way, Reason gets the much-needed “understanding” or “insight” which obviously can only come when it creates its laws according to its own design. Hence, the facts must be soiled and distorted by falsifying theory in order to be understandable. Presumably, the scientist is mainly living in a world of “logically true” rules of the game (invented by the scientist himself) while the artist lives in another world of nontautological and, frequently, a-logical rules. Perhaps, Goethe did after all have a point in his claim to “understand” light better than the great Newton? Also the Newtonian distinction between the how and the why questions might be more easily understood from this article. The “how” answers are essentially explanations and proofs of phenomena within a conceptual framework (language and theory). How can you solve a chess or

serology problem? The genuine “why” answers seem more to be based on inter-framework correspondence rules (CR) whereby essentially inexplicable laws in one framework or paradigm are naturally explained or actually eliminated in another framework or paradigm. These “laws” may also be regarded as distortions” introduced through the “rules of the game”. Why do you find “clustering” and CND-relations of (simple-complex) Ag genes? Because the (complex-simple) Ag antibodies have CND-related RR-values. Why do you find CND-related (complex-simple) antibodies? Because the (simple-complex) Ag genes occur in certain “linkage groups”. Why can you solve-i.e. explain a chess-problem? Because the (tautological) rules of the game or Hypotheses for chess specify how the problem should be solved or proved by “Reason and Experiment.”

::: 12. 13. 14. 15. 16.

17. 18. 19. 20.

Acknowledgments

21.

I dedicate this article to one of our great leaders in immunogenetics, Alexander Solomon Wiener (1907- 1976). A grant from the Hesselman Foundation, Stockholm as well as leave of absence for research granted by the Swedish Government is gratefully acknowledged.

22. 23. 24. 25.

1. 2.

:: 5. !: 8. 9.

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