Abstract
This paper analyzes the biochemical equilibria between bivalent receptors, homo-bifunctional ligands, monovalent inhibitors, and their complexes. Such reaction schemes arise in the immune response, where immunoglobulins (bivalent receptors) bind to pathogens or allergens. The equilibria may be described by an infinite system of algebraic equations, which accounts for complexes of arbitrary size n (n being the number of receptors present in the complex). The system can be reduced to just 3 algebraic equations for the concentrations of free (unbound) receptor, free ligand and free inhibitor. Concentrations of all other complexes can be written explicitly in terms of these variables. We analyze how concentrations of key (experimentally-measurable) quantities vary with system parameters. Such measured quantities can furnish important information about dissociation constants in the system, which are difficult to obtain by other means. We provide analytical expressions and suggest specific experiments that could be used to determine the dissociation constants.
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Notes
Mathematically, L tot increasing through the value R tot+1/2 corresponds to the “crossing over” of the two positive roots mentioned above: the smaller root increases through the value 1/2 and then ceases to be relevant, while the larger root decreases through 1/2 and becomes the relevant one. Moreover, at this value of L tot, examination of (37) gives Z 1=−1/2 (an unphysical result) except in the special case L=1/2. Thus for L tot=R tot+1/2 we always require L=1/2.
In case (ii), considered next, the ligand is long enough to form cyclic singletons, and this is the dominant complex in the equilibrium mixture; Fig. 5 suggests that in this limit \(\tilde{f}_{1} \to 0\) uniformly in K intra and that therefore this limit may be difficult to use predictively.
References
Barisas, B. G. (2003). Aggregation and gelation of divalent cell surface receptors by rigid polyvalent ligands: examination by theoretical, kinetic and thermodynamic techniques. Thermochim. Acta, 400, 1–20.
Colvin, J., Monine, M. I., Faeder, J. R., Hlavacek, W. S., Von Hoff, D. D., & Posner, R. G. (2009). Simulation of large-scale rule-based models. Bioinformatics, 25, 910–917.
Connors, K. A. (1987). Binding constants: the measurement of molecular complex stability. New York: Wiley.
Dembo, M., & Goldstein, B. (1978). Theory of equilibrium binding of symmetric bivalent haptens to cell surface immunoglobulin: application to histamine release from basophils. J. Immunol., 121, 345–353.
Faeder, J. R., Blinov, M. L., & Hlavacek, W. S. (2005). Graphical rule-based representation of signal transduction networks. Proc. ACM Symp. Appl. Comput., 1, 133–140.
Goldstein, B. (1988). Desensitization, histamine release and the aggregation of IgE on human basophils. In A. S. Perlson (Ed.), Theoretical immunology, part one, SFI studies in the sciences of complexity (Vol. II, pp. 3–41). Redwood City: Addison-Wesley.
Goldstein, B., & Wofsy, C. (1980). Theory of equilibrium binding of a bivalent ligand to cell surface antibody: the effect of antibody heterogeneity on cross-linking. J. Math. Biol., 10, 347–366.
Hendrickson, O. D., Zherdev, A. V., Kaplun, A. P., & Dzantiev, B. B. (2002). Experimental study and mathematical modeling of the interaction between antibodies and antigens on the surface of liposomes. Mol. Immunol., 39, 413–422.
Hlavacek, W. S., Perelson, A. S., Sulzer, B., Bold, J., Paar, J., Gorman, W., & Posner, R. G. (1999). Quantifying aggregation of IgE-FcεRI by multivalent antigen. Biophys. J., 76, 2421–2431.
Hlavacek, W. S., Faeder, J. R., Blinov, M. L., Posner, R. G., Hucka, M., & Fontana, W. (2006). Rules for modeling signal transduction systems. Sci. STKE, 344, re6.
Mack, E. T., Perez-Castillejos, R., Suo, Z., & Whitesides, G.M. (2008). Exact analysis of ligand-induced dimerization of monomeric receptors. Anal. Chem., 80, 5550–5555.
Mack, E. T., Cummings, L. J., & Perez-Castillejos, R. (2011). Mathematical model for determining the binding constants between immunoglobulins, bivalent ligands and monovalent ligands. Anal. Bioanal. Chem., 399, 1641–1652.
Murphy, K., Travers, P., & Walport, M. (2008). Janeway’s immunobiology (7th ed.). New York: Garland Science.
Posner, R. G., Wofsky, C., & Goldstein, B. (1995a). The kinetics of bivalent ligand—bivalent receptor aggregation: ring formation and the breakdown of the equivalent site approximation. Math. Biosci., 126, 171–190.
Posner, R. G., Subramanian, K., Goldstein, B., Thomas, J., Feder, T., Holowka, D., & Baird, B. (1995b). Simultaneous cross-linking by two non-triggering bivalent ligands causes synergistic signaling of IgE-FcεRI complexes. J. Immunol., 155, 3601–3609.
Posner, R. G., Savage, P. B., Peters, A. S., Macias, A., DelGado, J., Zwartz, G., Sklar, L. A., & Hlavacek, W. S. (2002). A quantitative approach for studying IgE-FcεRI aggregation. Mol. Immunol., 38, 1221–1228.
Sklar, L. A., Edwards, B. S., Graves, S. W., Nolan, J. P., & Prossnitz, E. R. (2002). Flow cytometric analysis of ligand-receptor interactions and molecular assemblies. Annu. Rev. Biophys. Biomol. Struct., 31, 97–119.
Wofsy, C. (1980). Analysis of a molecular signal for cell function in allergic reactions. Math. Biosci., 49, 69–86.
Wofsy, C., & Goldstein, B. (1987). The effect of co-operativity on the equilibrium binding of symmetric bivalent ligands to antibodies: theoretical results with application to histamine release from basophils. Mol. Immunol., 24, 151–161.
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Cummings, L.J., Perez-Castillejos, R. & Mack, E.T. Analysis of Biochemical Equilibria Relevant to the Immune Response: Finding the Dissociation Constants. Bull Math Biol 74, 1171–1206 (2012). https://doi.org/10.1007/s11538-012-9716-2
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DOI: https://doi.org/10.1007/s11538-012-9716-2