Charles Delisi

The kinetics of aggregation phenomena I. Minimal models for patch formation on lymphocyte membranes

Numerous biological processes involve the assembly of one or more monomers into aggregates or networks of interconnected units. In this paper we present the initial aspects of a mathematical theory for network formation on lymphocyte membranes. We assume the fluid mosaic membrane model is valid, that a lymphocyte possesses a homogeneous set of mobile but membrane bound receptors and that these receptors can form bimolecular complexes with antigen. We show that these complexes tend to aggregate and derive expressions for their size distribution as a function of time, antigen valence and concentration, and antigenreceptor affinity. At early times, the mass of the system (receptors plus antigen) is in very small aggregates. However under appropriate conditions, a critical time is reached at which they coalesce in such a way that the mass shifts, becoming concentrated predominantly in large aggregates. We assume that this coalescence (“patch” formation) is a necessary condition for lymphocyte triggering and briefly pursue the consequences. It is shown that the time required for patch formation is a sensitive function of affinity (K), antigen valence and antigen concentration (C), and that if KC is either too high or too low patch formation will not be possible. Moreover within the range of binding constants which can lead to patching, there will be an optimum value which leads to the fastest rate of triggering, and this optimum shifts to higher affinity as the concentration of free antigen surrounding the cell decreases. For optimum KC values we estimate times typically of the order of (10–100) seconds for patch formation. The theory also suggests that if antigen valence is too low, triggering will not be possible within times of interest, without introducing other factors. It thus leads naturally to a requirement for auxiliary cells which would tend to present low valence antigens in such a way that the B lymphocytes see an effective, increased valence. The theory, although primitive, thus meets some minimal requirements in that it distinguishes binding reactions from triggering reactions, makes predictions consistent with observations on affinity maturation and the nonresponsiveness to high doses and low doses of antigen, and suggests the need for helper cells (or their products) in order for low valence antigens to be effective in lymphocyte triggering.

On the mechanism of hemolytic plaque inhibition

Abstract Equations are presented which and can be used to describe the inhibition of γG plaques by hapten. Two types of experimental situations are considered. In the first the Red Blood Cell epitope density is large enough so that intramolecular reaction is favorable, and the reaction with antibody is considered irreversible during the time of the experiment. In the second case the epitopes are considered to be sparsely distributed so that only univalent attachment is possible. Since antibody site epitope interaction occurs rapidly ( ⪆1 sec ) on the time scale of diffusion, local equilibrium is assumed to apply in this case. We show that these models lead to different types of inhibition results. In the first instance a differential plot of the inhibition curve will reflect the affinity distribution as is often assumed. The reason involves the fact that only the interaction between antibody and free hapten is controlled by affinity, the interaction between antibody and Red Blood Cell (RBC) being irreversible. Consequently the interaction with RBC cannot effect the shape of the inhibition curve. In the second model, however, the interaction with RBC bound hapten and free hapten are both controlled by affinity. The model predicts that at a fixed secretion rate inhibition occurs abruptly, In this case, therefore, the breadth of the inhibition curve reflects the spread in the secretion rate distribution.

Antigen binding to receptors on immunocompetent cells. I. Simple models and interpretation of experiments.

Abstract We have developed simple mathematical models for treating the kinetics of binding of multivalent antigen to immune cell receptors when the binding may be in competition with nonspecific binding of antigen to cell surfaces and with the binding of hapten to the receptors. All three kinds of binding are treated as reversible bimolecular reactions. In general, the resulting equations must be solved numerically. When, however, the antigen and hapten concentrations are large compared to receptor concentrations, approximate algebraic solutions are found. It is shown that the most important effect of the hapten-receptor binding and of the nonspecific antigen binding is to slow down the antigen-receptor association; this may be viewed as a decrease in the antigen-receptor association rate constant. We have applied these models to analyze experiments of Davie and Paul on the binding of antigen to receptors on immunocompetent cells. Many difficulties have been found to arise from nonspecific binding. In particular, the association rate constant and equilibrium constant will appear reduced by nonspecific binding and the association rate constant will appear anomalously temperature dependent. We interpret hapten inhibition of antigen binding as a nonequilibrium effect in which hapten reduces the rate of antigen-cell association. In this way the concentration of hapten required to give 50% inhibition of antigen binding is found to decrease, as observed, with time after immunization. If equilibrium were to be achieved we predict that the required concentrations of hapten would be found to increase with time.

A theory of precipitation and agglutination reactions in immunological systems.

Abstract Aggregation phenomena in immunological systems have been profitably treated in the past by applying theories developed to describe polymerization reactions. An interesting result of these theories is the prediction that critical ratios exist for the reactants. These are ratios at which the composition of the system changes abruptly from one characterized primarily by small aggregates to one characterized primarily by large aggregates. In this paper, these ratios are studied as a function of physical and molecular parameters. In particular, the valence of the antibody is allowed to be greater than two and certain types of intramolecular reactions are considered. The effect of antigen and antibody heterogeneity on the results is discussed by developing the theory for a particular heterogeneous system.

Lack of involvement of auto-anti-idiotypic antibody in the regulation of oscillations and tolerance in the antibody response to levan

Abstract Bacterial levan (BL) induces a cyclic (oscillatory) antibody response in both euthymic and athymic BALB/c mice. Significant numbers of plaque-forming cells (PFC) making auto-anti-idiotypic antibody directed against a major cross-reactive idiotype expressed on antibodies specific for BL were detected in euthymic—but not athymic—mice. This suggests that auto-anti-idiotypic antibody, the formation of which requires the participation of mature T cells, does not play a decisive role in generating the cyclic patterns observed. However, such antibody might still influence the proportion of PFC making antibody of complementary idiotype. No relationship between the suppressive property of auto-anti-idiotypic antibody and either the induction or maintenance of immunological tolerance to BL was evident.

The kinetics of hemolytic plaque formation III. inhibition of plaques by antigen

Abstract We present a mathematical theory of hapten inhibition of hemolytic plaque formation. The treatment is based upon the mathematical model for plaque growth presented by DeLisi & Bell (1974). The lymphocyte under consideration is embedded in an infinite three-dimensional medium, and is secreting antibodies isotropically at a constant rate. As the antibodies diffuse from the source they can bind reversibly to hapten, and in the most general case reversibly to red blood cell (RBC) epitope. The model leads to a non-linear diffusion equation coupled to a set of first order differential equations. The system must, in general, be solved numerically. However, in many cases of experimental interest simplifications arise which permit closed form solutions to be obtained. In this paper we have developed solutions for three special cases. In the first example antibodies can bind only univalently to RBCs, as would be expected if the epitopes are sparsely distributed. In this case reaction between antibody site and RBC epitope is rapid ( ⪆ 1 sec) and reversible and local equilibrium is assumed. This leads to a “pure” diffusion equation in the free antibody concentration, but with a reduced diffusion coefficient. In another example univalent attachment of an antibody site to a RBC epitope is followed by a rapid irreversible intramolecular reaction. This might be expected for example if the epitope density is large. An exact solution to the resulting diffusion equation was also found in this case. In order to assess an intermediate situation, we also solved the equations for a model in which intramolecular reaction is slow and irreversible. The theory predicts that the type of information one can obtain from inhibition experiments depends critically upon the preparation of the RBC. If the cell is sufficiently haptenated so that rapid irreversible multivalent attachment is favorable, a differential plot of the inhibition curve will reflect the affinity distribution of antibody sites for free hapten. If only univalent attachment with RBCs is possible, so that antibody sites bind to RBC hapten in the same way they bind to free hapten, then a differential plot of the inhibition curve will reflect the secretion rate distribution.

The kinetics of hemolytic plaque formation. IV. IgM plaque inhibition.

Abstract An analysis of the inhibition of hemolytic plaques formed against IgM antibodies is presented. The starting point is the equations of DeLisi & Bell (1974) which describe the kinetics of plaque growth, and DeLisi & Goldstein (1975) which describe inhibition of IgG plaques. However, the physical chemical models which were used previously to describe IgG inhibition data are shown to be inadequate for describing the characteristics of IgM inhibition curves. Moreover, it is shown that the experimental results place severe restrictions on the possible choices of physical chemical models for IgM upon which to base the calculations. It is argued that in order to account even qualitatively for all the data, one must assume (1) a very restricted motion of IgMs about the Fab hinge region and (2) a very narrow secretion rate distribution of IgM by antibody secreting cells.

Immunodiffusion in gels containing erythrocyte antigen. I. Theory for diffusion of antiserum from a circular well.

A theory for the diffusion of antibodies in a gel containing erythrocyte antigen is presented. We show that when t ⪢ (k2−1 (K2 is the reverse rate constant for the binding of an antibody to an erythrocyte binding site) the free antibody concentration appears to diffuse with a reduced diffusion coefficient D∗ = D2/(1 + Kϱ0). D2 is the antibody diffusion coefficient in the gel, ϱ0 is the density of binding sites, and K the equilibrium constant for binding of an antibody to an erythrocyte binding site. The theory is developed for experiments in which a circular well is cut from the gel and antiserum placed within the well. Expression for the free and bound antibody concentrations as a function of position and time are derived. We show how these expressions can be used to (1) determine diffusion coefficients and (2) gain information about the distribution of equilibrium constants which characterize the antiserum.

Plaque morphology as an antibody specificity marker: an analysis of the physical chemical foundations of the method.

Abstract We present an analysis of the factors which affect plaque morphology. It is argued that in order for morphology differences to be sensitive to affinity variations, the antibody RBC interaction should be equilibrium controlled during the time of the experiment. Since a typical plaque assay lasts about fifty minutes this constrains application of the technique to antibodies with sufficiently rapid reverse rate constants. Moreover, the mode of attachment (univalent or multivalent) and perhaps the class of antibody may also be important. When local equilibrium conditions are met, morphology changes may reflect affinity variation, but there is no a priori reason to rule out the possibility that secretion rate variation is responsible for a particular morphological difference. It is shown that in general secretion rate will contribute to morphology unless the affinity and secretion rate distributions satisfy certain constraints. It is also shown that the criteria used to score morphology may have an important effect on how morphology variation is to be interpreted. Our analysis indicates that there is a rational basis for using morphology as a specificity marker but the range of applicability of the method may be restricted to rather special conditions and it should therefore by used cautiously.

A systematic and graphical method for generating the kinetic equations governing the growth of aggregates

We present a systematic method for deriving differential equations which govern the kinetics of formation of large aggregates from two types of multifunctional monomers. The problem of formulating the dynamic equations is complicated by the fact that there can be an arbitrarily large number of different species, each of different composition and with different numbers of reactive sites. It is shown that aggregates can easily be represented by graphs. Graphical methods are then used to simplify a number of counting problems and to derive two parameters which totally describe the reactive state of an aggregate. All possible reactions between aggregates are tabulated in terms of these parameters and the kinetic equations describing the aggregation phenomena are systematically derived.