Andrew Wohlgemuth

Abstract immunogenetic systems

The problem of defining genes, antigens and antibodies for some immunogenetic system is investigated. A formal mathematical object, an abstract immunogenetic system, is introduced to obtain the required precision and descriptive generality. This system is defined so as to be consistent with a given matrix A (of + and - signs) which can be taken as data from a real-world immunogenetic system. The abstract systems that involve the reaction matrix A are those that may provide legitimate definitions for the genes etc. of the real system. The ideas involved in the abstract systems are applied to two real human blood group systems (Rh and Ag) for illustration. Also the abstract model is related to the system models of Hirschfeld.

Modeling immunogenetic specificities

Abstract The mathematical definitions of specificity due to Hirschfeld, to Nau, Markowsky, Woodbury and Amos, and to Wohlgemuth are shown to be essentially equivalent. A natural correspondence is given whereby it is possible to obtain results on specificity in each of these models from corresponding results in others.

Uncovering antibody incidence structures

Abstract In a first order abstract immunogenetic system solutions to the problem of finding basic recognition factors such as antibodies or cytotoxic T-lymphocytes and basic recognized factors such as genes or gene products is given by a factorization M = ϕ 1 × G × ϕ 2 , where M is a reaction relation (matrix) and ϕ 1 labels individuals with genes, G relates genes to antibodies, and ϕ 2 labels reagents with antibodies. For a given M the problem of determining in general whether for a given n such a factorization exists with n antibodies or genes in G is NP-complete if no information other than M is available. In this paper we consider the problem of obtaining the factorization of M into ϕ 1 × G by ϕ 2 when information on effector cell combinations or reagents is given. When enough information is given it is shown that ϕ 1 × G and ϕ 2 are essentially uniquely determined, and an algorithm to obtain them in polynomial time is given. We also relate the computations necessary to uncover an antibody to its behavior in reaction tests. Based on this theory, a best possible solution is also given in cases where not enough information is available to obtain the unique solution.

Mathematical immunogenetics I mathematics as language

This paper summarizes approaches to developing mathematics that can act as a language for immunogenetics. The need for this has been documented by showing inadequacies of the standard symbolism. Apparent distinctions in symbolizing and conceptualizing factors involved in immunogenetics are seen to disappear in the mathematical models presented here. One model, a three-fold Boolean matrix factorization, subsumes all approaches to the idea of specificity and yet is general enough to incorporate data beyond that found only in a reaction matrix.

Mathematical immunogenetics II. Antibody incidence structure.

Mathematical Immunogenetics I argued for the development of mathematics as a language for immunogenetics. A three-fold factorization of a reaction matrix was seen to be the important form of a model of a first order immunogenetic system. In the present paper, results of the authors on determining this factorization are reworked from a physical perspective and presented in an algorithmic form that can be used to compute a labeling matrix from data. Computer programs to perform these computations are in preparation.

The effect of cross reactivity on immunological reaction matrices

Abstract Antigen and antibody compounds playing a role in immunological reactions for a given experiment are modeled by a bipartite graph. Reduced graphs are defined so as to be the smallest graphs needed to account for a given reaction matrix for all phenotypes immunized in certain maximal ways consistent with cross reactivity. A procedure is given for determining a reduced graph from a given reaction matrix. A one to one correspondence between isomorphism classes of reduced graphs and isomorphism classes of finite lattices is given.

The impact of symbolism on immunogenetics: an application to HLA

The genes coding for the class I human lymphocyte antigens (HLA) are located on chromosome 6. These antigens are involved with the immunological interaction between cells. In some immunogenetic systems, such as HLA in humans, genes are defined by antibody/antigen reaction and are denoted by single symbolic identifiers. This symbolization assumes a one-to-one correspondence between antibodies, antigens and genes. Recent molecular studies, however, suggest that HLA antibody/antigen reaction is complex and most HLA class I specific antibodies may not uniquely identify a single allelic product. Where cross-reactivity is present in an immunogenetic system it is important to label each reagent with symbols corresponding to all genes coding for antigens with which the reagent will react. The problems of cross-reactive groups and unexplained linkage relations may be elucidated by the redefinition and clarification of certain HLA antigens. A computer program can suggest such labelling schemes using input given by phenotype reaction patterns with a panel of reagents. When this program was applied to data on the class I HLA antigens a genetic model was suggested that differs somewhat from the currently accepted model. The new model is able to predict what would appear as linkage relations in the accepted model. Our methodology can provide alternate models to guide in typing cloned genes in terms of the HLA locus and alleles.

An interactive program for determining tentative gene assignments from immunological data

This paper announces an interactive program that could be useful for workers investigating new immunogenetic systems and theoreticians seeking to resolve persistent puzzling questions about systems already developed. The program seeks symbolic representations of data and its interpretations. By its ability to work with symbolizations in their most general form it is able to reveal symbolic patterns that may correspond directly to otherwise unexpected genetic models.

Intersection-union systems

Abstract In two recent papers the authors introduced some new approaches to the problem of modeling the phenomena underlying immunological reaction tests. These approaches allow one to easily construct the best possible models for a given amount of immunological test data. In the first paper the authors used a purely Boolean approach, i.e., it was assumed that the experimenter signified whether or not a reaction occurred in a given test. In the second paper the authors assumed that the relative strengths of the reactions were available as data for the modeling process. They showed that in this case strictly better models could be constructed. This paper generalizes the approaches taken in the first two papers and provides a unified approach to this whole subject. Many of the results, e.g., the ability to construct the best model, of the first two papers hold in this more general setting. Moreover, this generalization allows one to assess the tradeoffs involved in using data on the relative strengths of reactions. In particular, we see that using relative strengths is equivalent to using an additional intersection factor in a strictly Boolean approach. This intersection factor it turns out, can be obtained experimentally by using elution in addition to the absorption involved in the first two papers. Finally, the duality between fragments and cofragments becomes apparent using this approach.

Defining specificities, genes, antigens, and antibodies- A matrix approach

We study the consequences of assigning single letter symbols to operationally defined entities such as genes, antigens, specificities, and antibodies. If this is to be done and if reagents are not specific in recognizing the products of single genes or single antigens, then these entities must be defined by a ‘definition matrix’ to avoid mislabeling a matrix of data. A method is given whereby for a given matrix of data all possible definition matrices consistent with this data can be obtained. In particular, all the ways of labeling by the complex-complex code of Hirschfeld can be so obtained.