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Analysis of Biochemical Equilibria Relevant to the Immune Response: Finding the Dissociation Constants

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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

  1. 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.

  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.

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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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