ImmunologyToday, vol. 4, No. '8, 1983
209
The value of theoretical models in immunological research from Judith Rae Lumb With the increase in understanding of immunological phenomena has come an increase in availability of mathematical models which describe them. Here Judith Rae Lumb argues that theoretical models should be more widely appreciated and used, especially since helpful computer programs are now becoming available. In recent years there has been a tremendous increase in the understanding of immunological phenomena at the molecular and cellular levels, and there has been a concomitant increase in models constructed to study these phenomena ~-3. The contribution that a particular model makes to the study of a phenomenon is dependent upon several factors: (1) the extent to which the model represents the phenomenon; (2) the availability of quantitative data to test all aspects of the model; and (3) the extent to which other investigators make use of the model. This report asserts that the third factor is rate-limiting and that theoretical immunology is developing in relative isolation from experimental immunology. A cursory look at the Scientific Citation Index indicates that the work of theoretical immunologists is referenced almost exclusively by other theoretical immunologists. The objectives of this report are: ( 1) to call attention to the field of theoretical immunology by describing several examples of models which might be used by experimental immunologists; (2) to persuade experimental immunologists that theoretical immunology can be of use to them; and (3) to suggest means by which this subdiscipline might be integrated into the mainstream of immunological work. Models have been constructed to represent immunological phenomena at the molecular, cellular and organismal levels. There are models which contribute to the design of experimental methods and interpretation of data, and models which represent aspects of the immune response in a conceptual way. Models may be deterministic and consist of a series of equations which describe the phenomenon mathematically, or they may be stochastic and consist of equations which determine the probabilities of particular events. In either case, a computer may be used to perform the numerical and stochastic analyses required to apply the model to a certain set of conditions. For example, a computer program has been constructed for the interpretation of data from radioimmunoassays (RIA) 4.
Standard data are not linear and thus require more sophisticated curve-fitting than the equation for a straight line. Four alternative methods were evaluated for their ability to accurately interpret RIA data. The suitability of a particular method depended upon the antigen being assayed and other assay characteristics. The progranl is available on request. A computer simulation of the precipitin reaction has been extended to study immune complex disease 5-7. The basic model includes n-valent antigen, bivalent ~mtibody and an association constant which is the same in all interactions. A theoretical precipitin curve is generated which is similar to experimental curves. To explain immune complex disease, antibody was assumed to be produced at an exponential rate, and antigen-antibody complexes were eliminated when they became insoluble. Free antigen, free antibody and the sizes and amounts of antigen-antibody complexes were determined at various time intervals. The resulting zonal centrifugation pattern was generated and compared to that obtained from the serum of patients at different stages of immune complex disease. The model accurately predicts the various stages of the disease. It can be used to investigate different hypotheses of etiology and pathogenesis of the disease, and to evaluate potential diagnostic and therapeutic methods. Two models address the cell-mediated immunity reaction in vivo. One considers tumor immunity 8 and the other the infiltration and consolidation oflymphocytes and macrophages in an infection with Mycobacterium tuberculosis9. In the latter, two models are constructed which vary in the details of the interactions of the monocyte, lymphocyte, tubercleinfected macrophage, antigen, chemotactic factor and migration inhibition factor. The questions were derived but neither model was tested. In the former work a model was constructed which simulated the differentiation of a specific precursor T cell into a proliferating cell and then into a cytotoxic cell. This dif-
ferentiation process is stimulated by the presence of tumor-cell antigen and is dependent upon the amount of antigen. In addition, blocking factors were included in the model. T u m o r elimination was closely approximated in vivo including the escape from elimination by both low and high doses of tumor cells. This model provides a means of conceptuaiizing and testing different hypotheses in regard to the importance of blocking factors in anti-tumor immunity. Cellular kinetic theory has been used to simulate the development of the thymus 1°'11. One such model (THYMS) simulates the development of the thymus in fetal stages to its degeneration in later adult stages. The model involves the proliferation and differentiation of thymus precursor cells, endothelial cells and thymus cells". The simulated thymus recovers from irradiation damage with a pattern which is similar to experimental data for a n u m b e r of doses of irradiation. The first model of the cellular kinetics of clonal selection (ABIGAIL) assumed that considerations of chemical equilibrium could be used to describe the binding of antigen to surface immunoglobulin. It also assumed that such considerations could be used to show that immunoglobulin maintains the same affinity constant whether it is on the surface of B cells or secreted by the plasma cell, and that after a threshold value of antigen is bound to receptors on one cell, that particular cell will proliferate 12q4. ABIGAIL has been interfaced with T H Y M S to simulate the thymus-indeppendent immune response 15. T H Y M S was used as a source of B cells assuming that B cells are produced with kinetics similar to T cells. Those cells were used as input to ABIGAIL. The effect of irradiation was tested in the simulation: the results compared favorably with experimental data. More recently reported models show further refinements of the concepts used in the above models. A model using a systems approach is similar to ABIGAIL except that the affinity constants are assumed to be continuous within a range so that more than one cell clone can be considered at a time 16. The results of this model are similar to experimental results. Another, more refined model considers the role of T cells in the humoral process 17. At the molecular level, B-cell activation by the binding of polymeric T-independent antigens has been considered 1s'19. The distribution between bound subunits (trains) of the polymeric antigen and the unbound loops is determined under different conditions of
Immunology Today, vol. 4, No. 8, 1983
210 polymeric length and subunit concentration. T h e n u m b e r of trains and loops and their average length were calculated, assuming the subunit was 5 units long and using chain-generating functions. It was concluded that 5-10 subunits are needed to trigger the antibody response. This will decrease as the affinity constant increases. Recently the notion that specific immunoglobulin contains antigenic determinants (idiotypes) in the variable region has led to a network theory of the control of the i m m u n e response 2°-23. T h e theory is that when antibody is produced its idiotype is foreign to the system and will therefore induce anti-idiotype antibody. T h e extrapolation of this phenomenon is a network of idiotype-antiidiotype interactions. It is suggested that these interactions may be important in the control of the i m m u n e response. This subject is quite conducive to study by theoretical immunology. This list of examples of immunological models was not intended to be exhaustive or complete, but merely to whet the appetite of the experimental immunologist. A computer simulation model may be an excellent adjunct to experimental work: it can be used to resolve conflicting hypotheses and can clarify experimental design. For example, a model may suggest when to take samples, which concentrations to use and which parameters to vary. Trivial experiments m a y be discarded and the important ones pointed out. T h e model may help to keep the experiments focused in the direction of understanding basic immunological principles. Indeed, the process of constructing a model requires that the p h e n o m e n a be conceptualized in a much more complete way than does the process of interpreting data. T h e ideal situation would have the
persons doing modelling work in close collaboration with the persons doing experiments. T h e scientist modelling should be involved in the design of experiments which allow the model to be tested and further refined, and the e.xperimental immunologists should be involved in the conceptualization and in the testing of the model. In order to bring this under-used tool into the mainstream of immunological research, the programs of conferences might include theoretical reports in the same sessions as the relevant experimental reports. Investigators planning experimental projects might consider including a theoretical component to the project. Models could be used to consider the quantitative aspects of factorreceptor interactions in all areas of the i m m u n e response, including T-cell growth factors, B-cell growth factors, B-cell differentiation factors, T-cell replacing factors, suppressor factors, cytotoxic factors and other lymphokines. Models could be used to consider the regulation of immunoglobulin gene expression and rearrangement. Cellular kinetics models could be used to study lymphocyte differentiation. Models could be used to study the quantitative aspects o f tolerance and the control of the i m m u n e response. Cellular interactions in vivo could be quantitated and simulated. In general, the study of immunology is now at a stage where the specific events are being characterized individually. These events are very complicated and involve many components. It is possible to conceptualize each one separately, but the integration of all of these events with all their quantitative aspects is impossible for the h u m a n mind. However, the computer can be used as an extension of the m i n d for fast calculation and integration of the concepts.
References 1 Bell, G. I., Perelson, A. S. and Pimbely, G. (eds) (1978) Theoretical Immunology, Marcel Dekker, Inc., New York 2 Bruni, C., Doria, G., Koch, G. and Strom, R. (eds) (1979) Systems Theory in Immunology, Springer-Verlag, Heidelberg 3 Delisi, C. (1981) Fed. Proc. Fed. Am. Soc. Exp. Biol. 40, 1471 4 Schoneshofer, M. (1977) Clin. Chim. Acta 77, 101 5 Steensgaard, J., Johansen, H. K. W. and Moiler, N. P. H. (1975) Immunology 29, 571 6 Moiler, H. P. H. (1978) Scand. J. Rheumatol. 7, 151 7 Steensgaard, J. and Frich, J. R. (1979) Immunology 36, 279 8 Grossman, A. and Berke, G. (1980)J. Theor. Biol. 83, 267 9 Aris, R. (1978)in TheoreticalImmunology (Bell, G. I., Perelson, A. S. and Pimbely, G., eds), p. 519, Marcel Dekker, Inc., New York 10 Lumb,J. R. and MacFarland, B. L. (1972)J. ReticuloendothelialSoc. 12, 80 11 Lumb, J. R. (1975) Comput. Biorned. Res. 8, 379 12 Bell, G. I. (1970)J. Theor. Biol. 29, 191 13 Bell, G. I. (1971)J. Theor. Biol. 30, 339 14 Bell, G. I. (1974)J. Theor. Biol. 33, 379 15 Lumb, J. R. (1981) Comput. Biomed. Res. 14, 220 16 Bruni, C., Giovenco, M. A., Koch, G. and Strom, R. (1978) in Theoretical Immunology (Bell, G. I., Perelson, A. S. and Pimbely, G., eds), p. 379, Marcel Dekker, Inc., New York 17 Mohler, R. R., Barton, C. F. and Hsu, C. S. (1978) in TheoreticalImmunology (Bell, G. I., Perelson, A. S. and Pimbely, G., eds), p. 415, Marcel Dekker, Inc., New York 18 Perelson, A. S. and Wiegel, F. W. (1981)Fed. Proc. Fed. Am. Soc. Exp. Biol. 40, 1479 19 Wiegel, F. W. and Perelson, A. S. (1981)J. Theor. BioL 88, 533 20 Jerne, N. K. (1974) Ann. Immunol. (Paris) 125C, 373 21 Richter, P. H. (1975) Eur. J. Immunol. 5,350 22 Richter, P. H. (1980) Eur. J. Immunol. 5, 539 23 Heirnaux, J. (1981) Fed. Proc. Fed. Am. Soc. Exp. Biol. 40, 1484
Judith Roe Lumb is in the Biology Department, Atlanta University, Atlanta, GA 30314, USA.
209
The value of theoretical models in immunological research from Judith Rae Lumb With the increase in understanding of immunological phenomena has come an increase in availability of mathematical models which describe them. Here Judith Rae Lumb argues that theoretical models should be more widely appreciated and used, especially since helpful computer programs are now becoming available. In recent years there has been a tremendous increase in the understanding of immunological phenomena at the molecular and cellular levels, and there has been a concomitant increase in models constructed to study these phenomena ~-3. The contribution that a particular model makes to the study of a phenomenon is dependent upon several factors: (1) the extent to which the model represents the phenomenon; (2) the availability of quantitative data to test all aspects of the model; and (3) the extent to which other investigators make use of the model. This report asserts that the third factor is rate-limiting and that theoretical immunology is developing in relative isolation from experimental immunology. A cursory look at the Scientific Citation Index indicates that the work of theoretical immunologists is referenced almost exclusively by other theoretical immunologists. The objectives of this report are: ( 1) to call attention to the field of theoretical immunology by describing several examples of models which might be used by experimental immunologists; (2) to persuade experimental immunologists that theoretical immunology can be of use to them; and (3) to suggest means by which this subdiscipline might be integrated into the mainstream of immunological work. Models have been constructed to represent immunological phenomena at the molecular, cellular and organismal levels. There are models which contribute to the design of experimental methods and interpretation of data, and models which represent aspects of the immune response in a conceptual way. Models may be deterministic and consist of a series of equations which describe the phenomenon mathematically, or they may be stochastic and consist of equations which determine the probabilities of particular events. In either case, a computer may be used to perform the numerical and stochastic analyses required to apply the model to a certain set of conditions. For example, a computer program has been constructed for the interpretation of data from radioimmunoassays (RIA) 4.
Standard data are not linear and thus require more sophisticated curve-fitting than the equation for a straight line. Four alternative methods were evaluated for their ability to accurately interpret RIA data. The suitability of a particular method depended upon the antigen being assayed and other assay characteristics. The progranl is available on request. A computer simulation of the precipitin reaction has been extended to study immune complex disease 5-7. The basic model includes n-valent antigen, bivalent ~mtibody and an association constant which is the same in all interactions. A theoretical precipitin curve is generated which is similar to experimental curves. To explain immune complex disease, antibody was assumed to be produced at an exponential rate, and antigen-antibody complexes were eliminated when they became insoluble. Free antigen, free antibody and the sizes and amounts of antigen-antibody complexes were determined at various time intervals. The resulting zonal centrifugation pattern was generated and compared to that obtained from the serum of patients at different stages of immune complex disease. The model accurately predicts the various stages of the disease. It can be used to investigate different hypotheses of etiology and pathogenesis of the disease, and to evaluate potential diagnostic and therapeutic methods. Two models address the cell-mediated immunity reaction in vivo. One considers tumor immunity 8 and the other the infiltration and consolidation oflymphocytes and macrophages in an infection with Mycobacterium tuberculosis9. In the latter, two models are constructed which vary in the details of the interactions of the monocyte, lymphocyte, tubercleinfected macrophage, antigen, chemotactic factor and migration inhibition factor. The questions were derived but neither model was tested. In the former work a model was constructed which simulated the differentiation of a specific precursor T cell into a proliferating cell and then into a cytotoxic cell. This dif-
ferentiation process is stimulated by the presence of tumor-cell antigen and is dependent upon the amount of antigen. In addition, blocking factors were included in the model. T u m o r elimination was closely approximated in vivo including the escape from elimination by both low and high doses of tumor cells. This model provides a means of conceptuaiizing and testing different hypotheses in regard to the importance of blocking factors in anti-tumor immunity. Cellular kinetic theory has been used to simulate the development of the thymus 1°'11. One such model (THYMS) simulates the development of the thymus in fetal stages to its degeneration in later adult stages. The model involves the proliferation and differentiation of thymus precursor cells, endothelial cells and thymus cells". The simulated thymus recovers from irradiation damage with a pattern which is similar to experimental data for a n u m b e r of doses of irradiation. The first model of the cellular kinetics of clonal selection (ABIGAIL) assumed that considerations of chemical equilibrium could be used to describe the binding of antigen to surface immunoglobulin. It also assumed that such considerations could be used to show that immunoglobulin maintains the same affinity constant whether it is on the surface of B cells or secreted by the plasma cell, and that after a threshold value of antigen is bound to receptors on one cell, that particular cell will proliferate 12q4. ABIGAIL has been interfaced with T H Y M S to simulate the thymus-indeppendent immune response 15. T H Y M S was used as a source of B cells assuming that B cells are produced with kinetics similar to T cells. Those cells were used as input to ABIGAIL. The effect of irradiation was tested in the simulation: the results compared favorably with experimental data. More recently reported models show further refinements of the concepts used in the above models. A model using a systems approach is similar to ABIGAIL except that the affinity constants are assumed to be continuous within a range so that more than one cell clone can be considered at a time 16. The results of this model are similar to experimental results. Another, more refined model considers the role of T cells in the humoral process 17. At the molecular level, B-cell activation by the binding of polymeric T-independent antigens has been considered 1s'19. The distribution between bound subunits (trains) of the polymeric antigen and the unbound loops is determined under different conditions of
Immunology Today, vol. 4, No. 8, 1983
210 polymeric length and subunit concentration. T h e n u m b e r of trains and loops and their average length were calculated, assuming the subunit was 5 units long and using chain-generating functions. It was concluded that 5-10 subunits are needed to trigger the antibody response. This will decrease as the affinity constant increases. Recently the notion that specific immunoglobulin contains antigenic determinants (idiotypes) in the variable region has led to a network theory of the control of the i m m u n e response 2°-23. T h e theory is that when antibody is produced its idiotype is foreign to the system and will therefore induce anti-idiotype antibody. T h e extrapolation of this phenomenon is a network of idiotype-antiidiotype interactions. It is suggested that these interactions may be important in the control of the i m m u n e response. This subject is quite conducive to study by theoretical immunology. This list of examples of immunological models was not intended to be exhaustive or complete, but merely to whet the appetite of the experimental immunologist. A computer simulation model may be an excellent adjunct to experimental work: it can be used to resolve conflicting hypotheses and can clarify experimental design. For example, a model may suggest when to take samples, which concentrations to use and which parameters to vary. Trivial experiments m a y be discarded and the important ones pointed out. T h e model may help to keep the experiments focused in the direction of understanding basic immunological principles. Indeed, the process of constructing a model requires that the p h e n o m e n a be conceptualized in a much more complete way than does the process of interpreting data. T h e ideal situation would have the
persons doing modelling work in close collaboration with the persons doing experiments. T h e scientist modelling should be involved in the design of experiments which allow the model to be tested and further refined, and the e.xperimental immunologists should be involved in the conceptualization and in the testing of the model. In order to bring this under-used tool into the mainstream of immunological research, the programs of conferences might include theoretical reports in the same sessions as the relevant experimental reports. Investigators planning experimental projects might consider including a theoretical component to the project. Models could be used to consider the quantitative aspects of factorreceptor interactions in all areas of the i m m u n e response, including T-cell growth factors, B-cell growth factors, B-cell differentiation factors, T-cell replacing factors, suppressor factors, cytotoxic factors and other lymphokines. Models could be used to consider the regulation of immunoglobulin gene expression and rearrangement. Cellular kinetics models could be used to study lymphocyte differentiation. Models could be used to study the quantitative aspects o f tolerance and the control of the i m m u n e response. Cellular interactions in vivo could be quantitated and simulated. In general, the study of immunology is now at a stage where the specific events are being characterized individually. These events are very complicated and involve many components. It is possible to conceptualize each one separately, but the integration of all of these events with all their quantitative aspects is impossible for the h u m a n mind. However, the computer can be used as an extension of the m i n d for fast calculation and integration of the concepts.
References 1 Bell, G. I., Perelson, A. S. and Pimbely, G. (eds) (1978) Theoretical Immunology, Marcel Dekker, Inc., New York 2 Bruni, C., Doria, G., Koch, G. and Strom, R. (eds) (1979) Systems Theory in Immunology, Springer-Verlag, Heidelberg 3 Delisi, C. (1981) Fed. Proc. Fed. Am. Soc. Exp. Biol. 40, 1471 4 Schoneshofer, M. (1977) Clin. Chim. Acta 77, 101 5 Steensgaard, J., Johansen, H. K. W. and Moiler, N. P. H. (1975) Immunology 29, 571 6 Moiler, H. P. H. (1978) Scand. J. Rheumatol. 7, 151 7 Steensgaard, J. and Frich, J. R. (1979) Immunology 36, 279 8 Grossman, A. and Berke, G. (1980)J. Theor. Biol. 83, 267 9 Aris, R. (1978)in TheoreticalImmunology (Bell, G. I., Perelson, A. S. and Pimbely, G., eds), p. 519, Marcel Dekker, Inc., New York 10 Lumb,J. R. and MacFarland, B. L. (1972)J. ReticuloendothelialSoc. 12, 80 11 Lumb, J. R. (1975) Comput. Biorned. Res. 8, 379 12 Bell, G. I. (1970)J. Theor. Biol. 29, 191 13 Bell, G. I. (1971)J. Theor. Biol. 30, 339 14 Bell, G. I. (1974)J. Theor. Biol. 33, 379 15 Lumb, J. R. (1981) Comput. Biomed. Res. 14, 220 16 Bruni, C., Giovenco, M. A., Koch, G. and Strom, R. (1978) in Theoretical Immunology (Bell, G. I., Perelson, A. S. and Pimbely, G., eds), p. 379, Marcel Dekker, Inc., New York 17 Mohler, R. R., Barton, C. F. and Hsu, C. S. (1978) in TheoreticalImmunology (Bell, G. I., Perelson, A. S. and Pimbely, G., eds), p. 415, Marcel Dekker, Inc., New York 18 Perelson, A. S. and Wiegel, F. W. (1981)Fed. Proc. Fed. Am. Soc. Exp. Biol. 40, 1479 19 Wiegel, F. W. and Perelson, A. S. (1981)J. Theor. BioL 88, 533 20 Jerne, N. K. (1974) Ann. Immunol. (Paris) 125C, 373 21 Richter, P. H. (1975) Eur. J. Immunol. 5,350 22 Richter, P. H. (1980) Eur. J. Immunol. 5, 539 23 Heirnaux, J. (1981) Fed. Proc. Fed. Am. Soc. Exp. Biol. 40, 1484
Judith Roe Lumb is in the Biology Department, Atlanta University, Atlanta, GA 30314, USA.
