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

Rates of diffusion controlled reactions in one, two and three dimensions

The dimensionality of diffusion may markedly affect the rate and economy of diffusion controlled reactions. Moreover, the degree of dependence of the steady state rate of these reactions on the concentration of each of the two reacting species is also dictated by the dimensionality and it ranges from linear dependence in the three dimensional case to a nearly square dependence in the one dimensional case. These theoretical observations emerge from a direct analysis of the steady state diffusion controlled rates which are derived here using a simple straightforward approach. This approach is based on the conjecture that in the steady state the rate of diffusional encounters between the two reaction partners equals to the sum of the encounter rates of two independent processes which are obtained by alternately immobilizing one of the reaction partners while the other partner diffuses freely. Unlike Smoluchowski's classical approach, the presented point of view permits to obtain in a unified fashion reaction rates for all dimensionalities.

Percolation of clusters with a residence time in the bond definition: Integral equation theory

We consider the clustering and percolation of continuum systems whose particles interact via the Lennard-Jones pair potential. A cluster definition is used according to which two particles are considered directly connected (bonded) at time t if they remain within a distance d, the connectivity distance, during at least a time of duration tau, the residence time. An integral equation for the corresponding pair connectedness function, recently proposed by two of the authors [Phys. Rev. E 61, R6067 (2000)], is solved using the orthogonal polynomial approach developed by another of the authors [Phys. Rev. E 55, 426 (1997)]. We compare our results with those obtained by molecular dynamics simulations.

Evaluation of data in terms of two-dimensional random walk model: Interaction between NADH-cytochrome b5 reductase and cytochrome b5

Normally, bimolecular reactions are analyzed in terms of the Smoluchowski theory. However, when one attempts to generalize this analysis to cases where diffusion proceeds in two other than in three dimensions, one soon encounters severe conceptual difficulties. Although kinetic studies of membrane enzymes are generally difficult because the usual kinetic formalism refers to nonaggregated homogenous solutions, a major goal of our research is to define the molecular mechanism(s) by which alterations in membrane-bound substrate contents affect the enzyme activity in the same membrane. For that purpose, a simplified random-walk model was adopted in the present work. The enzyme reaction in the two-dimensional membrane could be calculated theoretically by applying the classical analysis of heat equation. As a result, the theoretical rate equation well accounting experimental findings was derived on the model of the liver microsomal NADH-cytochrome b5 reductase reaction. Furthermore, it was found that the modification of the simple rigid-sphere collision theory by including a term called the steric factor was not necessary in this derived equation.

The mast cell function-associated antigen and its interactions with the type I Fcepsilon receptor

Rat mucosal-type mast cells of the RBL-2H3 line express a glycoprotein termed the MAst cell Function-associated Antigen (MAFA). When MAFA is clustered by its specific monoclonal antibody G63, secretion normally triggered by aggregating these cells' type I Fcepsilon receptor (FcepsilonRI) is substantially inhibited. The nature of MAFA-FcepsilonRI interactions giving rise to this inhibition remains unclear. Rotational diffusion of a membrane protein is a sensitive probe of its involvement in intermolecular interactions. We have therefore studied by time-resolved phosphorescence anisotropy the rotational behavior of both MAFA and FcepsilonRI as ligated by various reagents involved in FcepsilonRI-induced degranulation and MAFA-mediated inhibition thereof. From 4 to 37 degrees C, the rotational correlation times (mean +/- SD) of FcepsilonRI-bound, erythrosin-conjugated IgE resemble those observed for MAFA-bound, erythrosin-conjugated G63 Fab, 82 +/- 17 and 79 +/- 31 micros at 4 degrees C, respectively. Clustering the FcepsilonRI-IgE complex by antigen or by anti-IgE increases the phosphorescence anisotropy of G63 Fab and slows its rotational relaxation. Lateral diffusion of G63 Fab is also slowed by antigen clustering of the receptor. Taken together, these results indicate that unperturbed MAFA associates with clustered FcepsilonRI. They are also consistent with its interaction with the isolated receptor.

Equilibrium binding of multivalent ligands to cells: effects of cell and receptor density

We study the equilibrium binding properties of multivalent ligands to cell surface receptors. We examine the effects of cell density and number of receptors per cell, that is, receptor concentration, on ligand binding. These parameters can significantly affect the formation of receptor aggregates and cross-links. We then use our general results to show that ligand-induced cell proliferation may be self-limiting, since ligand depletion reduces the signal received by individual cells once the cell population has expanded. We discuss the concept of avidity and show its limitations. As a specific example, we examine the binding of haptenated polymers to B cells and reinterpret experiments related to the immunon theory of B-cell activation.

A new bell-shaped function for idiotypic interactions based on cross-linking

Most recent models of the immune network are based upon a phenomenological log bell-shaped interaction function. This function depends on a single parameter, the "field," which is the sum of all ligand concentrations weighted by their respective affinities. The typical behavior of these models is dominated by percolation, a phenomenon in which a local stimulus spreads globally throughout the network. The usual reason for employing a log bell-shaped interaction function is that B cells are activated by cross-linking of their surface immunoglobulin receptors. Here we formally derive a new phenomenological log bell-shaped function from the chemistry of receptor cross-linking by bivalent ligand. Specifying how this new function depends on the ligand concentrations requires two fields: a binding field and a cross-linking field. When we compare the activation functions for ligand-receptor pairs with different affinities, the one-field and the two-field functions differ markedly. In the case of the one-field activation function, its graph is shifted to increasingly higher concentration as the affinity decreases but keeps its width and height. In the case of the two-field activation function, the graph of a low-affinity interaction is nested within the graphs of all higher-affinity interactions. We show that this difference in the relations among activation functions for different affinities radically changes the network behavior. In models that described B cell proliferation using the one-field activation function, network behavior was dominated by low-affinity interactions. Conversely, in our new model, the high-affinity interactions are the most significant. As a consequence, percolation is no longer the only typical network behavior.

Cross-linking reconsidered: binding and cross-linking fields and the cellular response

We analyze a model for the reversible cross-linking of cell surface receptors by a collection of bivalent ligands with different affinities for the receptor as would be found in a polyclonal anti-receptor serum. We assume that the amount of cross-linking determines, via a monotonic function, the rate at which cells become activated and divide. In addition to the density of receptors on the cell surface, two quantities, the binding field and the cross-linking field, are needed to characterize the cross-linking curve, i.e., the equilibrium concentration of cross-linked receptors plotted as a function of the total ligand site concentration. The binding field is the sum of all ligand site concentrations weighted by their respective binding affinities, and the cross-linking field is the sum of all ligand site concentrations weighted by the product of their respective binding and cross-linking affinity and the total receptor density. Assuming that the cross-linking affinity decreases if the binding affinity decreases, we find that the height of the cross-linking curve decreases, its width narrows, and its center shifts to higher ligand site concentrations as the affinities decrease. Moreover, when we consider cross-linking-induced proliferation, we find that there is a minimum cross-linking affinity that must be surpassed before a clone can expand. We also show that under many circumstances a polyclonal antiserum would be more likely than a monoclonal antibody to lead to cross-linking-induced proliferation.

Monte Carlo simulation of ligand-receptor interactions on a cell surface

In this paper we develop a compartmentalized, discrete simulation model for investigating the spatial distribution and dynamic properties of receptor crosslinking on the surface of a cell. Results generated by the model are compared with some of the major results of existing analytical models, and differences are discussed in relation to differences in the model assumptions. Finally, the model is used to evaluate the dynamic effects of a time-varying non-homogeneous ligand concentration.

The antigen-antibody interaction algebraically interpreted as a relational process

The antigen-antibody interaction occurring previous to the triggering of the immunological response is analyzed as a relational process in terms of lattices. Accordingly, this process is expressed as a lattice belonging to a pseudo-Boolean algebraic variety. The Heyting arrow operation, which appears in this kind of algebra, is used to analyze behaviors between non-comparable biological states expressed by the lattice. The resulting states coming from the arrows are connected with the influence of increasing and decreasing energies involved in the linking process.

On antigen-antibody binding distribution

The probability distributions for the number of bound antigens and antibodies during immune response are obtained in this paper. Biological significance of this work and directions for further application are discussed along with some illustrative numerical results.

Immune network theory

Theoretical ideas have played a profound role in the development of idiotypic network theory. Mathematical models can help in the precise translation of speculative ideas into quantitative predictions. They can also help establish general principles and frameworks for thinking. Using the idea of shape space, criteria were introduced for evaluating the completeness and overlap in the antibody repertoire. Thinking about the distribution of clones in shape space naturally leads to considerations of stability and controllability. An immune system which is too stable will be sluggish and unresponsive to antigenic challenge; one which is unstable will be driven into immense activity by internal fluctuations. This led us to postulate that the immune system should be stable but not too stable. In many biological contexts the development of pattern requires both activation and inhibition but on different spatial scales. Similar ideas can be applied to shape space. The principle of short-range activation and long-range inhibition translates into specific activation and less specific inhibition. Application of this principle in model immune systems can lead to the stable maintenance of non-uniform distributions of clones in shape space. Thus clones which are useful and recognize antigen or internal images of antigen can be maintained at high population levels whereas less useful clones can be maintained at lower population levels. Pattern in shape space is a minimal requirement for a model. Learning and memory correspond to the development and maintenance of particular patterns in shape space. Representing antibodies by binary strings allows one to develop models in which the binary string acts as a tag for a specific molecule or clone. Thus models with huge numbers of cells and molecules can be developed and analyzed using computers. Using parallel computers or finite state models it should soon be feasible to study model immune systems with 10(5) or more elements. Although idiotypic networks were the focus of this paper, these modeling strategies are general and apply equally well to non-idiotypic models. Using bit string or geometric models of antibody combining sites, the affinity of interaction between any two molecules, and hence the connections in a model idiotypic network, can be determined. This approach leads to the prediction of a phase transition in the structure of idiotypic networks. On one side of the transition networks are small localized structures much as might be predicted by clonal selection and circuit ideas.(ABSTRACT TRUNCATED AT 400 WORDS)

Use of the average antibody-antigen bond concept and probability theory to simplify modeling of linear and circular antibody-antigen complex formation

We illustrate use of a simple approach to describe the equilibrium interactions of mixtures of monoclonal antibodies and antigens. This procedure is based on elementary concepts in probability theory and is readily suited to describing interactions of antibodies and antigens which form circular as well as linear complexes. The method is also suited to describing the inhibitory effects of antibodies which compete for overlapping epitopes and an example is provided to show how the procedure can be used to describe the interactions of antibodies which inhibit circular complex formation. We also outline simple strategies for preparing computer programs to simulate binding of antigens to defined antibody mixtures. The methods described should facilitate design of immunoassay procedures based on the use of defined mixtures of monoclonal antibodies.

Statistical mechanics of lipid membranes. Protein correlation functions and lipid ordering

An expression is derived for the lipid-mediated intermolecular interaction between protein molecules embedded in a lipid bilayer. It is assumed that protein particles are accommodated by the bilayer, but they distort the lipids in some manner from their equilibrium protein-free configuration. We treat this situation by expanding the free energy density in the plane of the membrane as a Taylor series in some arbitrary parameter and its gradient. Minimization of the total membrane energy for a given particle configuration yields the interparticle interaction energy for that configuration. A test of the model is provided by measurement of the protein-protein pair distribution function from freeze-fracture micrographs of partially aggregated membranes. The measured functions can be simulated by adjustment of two parameters (a) a lipid correlation length that characterizes the distance over which a distortion of the bilayers is transmitted laterally through the bilayer, and (b) a term quantifying the energy of the protein-lipid interaction at the protein-lipid boundary. Correlation lengths obtained by fitting the calculated particle distribution functions to the data are found to be several nanometers. Protein-lipid interaction energies are of the order of a few kT.

Kinetics of rouleau formation. II. Reversible reactions

Red blood cells aggregate face-to-face to form long, cylindrical, straight chains and sometimes branched structures called rouleaux. Here we extend a kinetic model developed by R. W. Samsel and A. S. Perelson (1982, Biophys. J. 37:493-514) to include both the formation and dissociation of rouleaux. We examine thermodynamic constraints on the rate constants of the model imposed by the principle of detailed balance. Incorporation of reverse reactions allows us to compute mean sizes of rouleaux and straight chain segments within rouleaux, as functions of time and at equilibrium. Using the Flory - Stockmayer method from polymer chemistry, we obtain a closed-form solution for the size distribution of straight chain segments within rouleaux at any point in the evolution of the reaction. The predictions of our theory compare favorably with data collected by D. Kernick , A.W.L. Jay , S. Rowlands , and L. Skibo (1973, Can. J. Physiol. Pharmacol. 51:690-699) on the kinetics of rouleau formation. When rouleaux grow large, they may contain rings or loops and take on the appearance of a network. We demonstrate the importance of including the kinetics of ring closure in the development of realistic models of rouleaux formation.

Aggregation of cell surface receptors by multivalent ligands

Using a combination of branching processes and kinetic equations a solution is provided to the problem of describing the size of aggregates formed on cell surfaces when multivalent ligands bind and cross-link multivalent receptors. A criterion is given for the onset of gelation in infinite 2-dimensional systems, which may be relevant to the phenomenon of ligand-induced receptor patching.

Antigen-specific mouse lymphocyte stimulation by DNP-conjugated T-independent antigens studied by photobleaching recovery

Fluorescence photobleaching recovery techniques have allowed us to measure the lateral mobility of T-independent antigens bound to antigen-specific mouse B cells. The in vitro immunogenicity or tolerogenicity of antigens we have examined, DNP-polymerized flagellin (DNP-POL), and DNP-linear dextran (DNP-DEX), depend upon the antigen dose and epitope density. These factors also determine the mobility of antigen bound to B cell surfaces. For DNP-POL bound to DNP-specific cells, the observed diffusion constants D decrease monotonically with increasing antigen dose and epitope density. Values of D range from 10.4 X 10(-11) cm2 sec-1 for DNP0.4-POL at 0.15 microgram/ml to 0.8 X 10(-11) cm2 sec-1 for DNP3.5-POL at 30 microgram/ml. For receptor bound DNP-DEX, D depends strongly on antigen epitope density but not observably on antigen concentration. For epitope densities of 1.2 or less, D is close to the value of 21 X10(-11) cm2 sec-1 observed for single sIg receptors. By an epitope density of 4.8, D has fallen to 2.1 X 10(-11) cm2 sec-1. Peak immunogenicities for DNP-POL and DNP-DEX are observed when antigen-receptor aggregates have mobilities 14-fold and 3-fold lower, respectively, than a single sIg molecule.

Theory of equilibrium binding of a bivalent ligand to cell surface antibody: the effect of antibody heterogeneity on cross-linking

We investigate the equilibrium binding of symmetric bivalent ligands to a heterogeneous population of symmetric bivalent cell surface receptors. The receptors are heterogeneous in their binding affinities (equilibrium binding constants) for the ligand. For any distribution of receptor binding affinities we show how to calculate the total concentration of receptors that are cross-linked by the ligand, i.e., the concentration of cell surface aggregates composed of two or more receptors, as well as the concentration of any given aggregate. We show that certain qualitative properties of cross-linking which hold for homogeneous antibody populations fail to hold in the heterogeneous case. We use our results to interpret certain in vitro experiments in which synthetic bivalent haptens are used to trigger histamine release from basophils which have on their surface antibody specific for the hapten.

The biophysics of ligand-receptor interactions

For the cells of an organism to act in the coordinated fashion necessary for complex functioning, they must be able to receive and transmit information. Information transfer is mediated by molecules released by the cells and may be local, as in the case of neurotransmitters, or long range, as in the case of hormones. It is apparent, however, that irrespective of the range of interaction, a cell must be able to distinguish, with a high degree of precision, the signals relevant to it from an enormous flow of background noise.

Molecular recognition at the cell surface is mediated by receptors: cell surface glycoproteins that usually form an integral part of the plasma membrane (see, for example, Cuatrecasas & Greaves, 1978). Typically, receptors bind the ligands they are designed to recognize with affinities of the order of 108 M-1, and they translate that interaction into a sequence of signals that ultimately lead to biological activity.

Models for the specific adhesion of cells to cells

A theoretical framework is proposed for the analysis of adhesion between cells or of cells to surfaces when the adhesion is mediated by reversible bonds between specific molecules such as antigen and antibody, lectin and carbohydrate, or enzyme and substrate. From a knowledge of the reaction rates for reactants in solution and of their diffusion constants both in solution and on membranes, it is possible to estimate reaction rates for membrane-bound reactants. Two models are developed for predicting the rate of bond formation between cells and are compared with experiments. The force required to separate two cells is shown to be greater than the expected electrical forces between cells, and of the same order of magnitude as the forces required to pull gangliosides and perhaps some integral membrane proteins out of the cell membrane.

Optimal strategies in immunology. I. B-cell differentiation and proliferation

The optimal strategy available to the immune system for responding to a non-replicating thymus-independent antigen is examined. By applying Pontryagin's maximum principle to a set of mathematical models of lymphocyte populations and their antibody production, it is found that the optimal strategy of bang-bang control appears robust. In a variety of structurely related biological models, similar behaviour is observed. The models that we consider assume that antigen triggers a population of B-lymphocytes. These triggered lymphocytes can either proliferate and secrete modest amounts of antibody or differentiate into nondividing plasma cells which secrete large amounts of antibody. For biologically reasonable parameter values it is found that for low doses of antigen, immediate differentiation into plasma cells is optimal, while for high antigen doses a proliferative state followed by differentiation is the best strategy.