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We investigate the adsorption of a charged macromolecule to a charged surface under the influence of a weak electrostatic attraction. The probability density P(x, N ) to find the end point of a molecule with N monomer units at distance x from the surface is calculated analytically. In the limit N -+ CO this function shows a phase transition at a critical value (e,) of the adsorption energy in units of kBT: for % > Bc P(x, N ) is peaked near the surface and the molecule is effectively adsorbed to the surface; for 0 < Bc the heat motion is strong enough to remove the molecule from the surface and P(x, N ) is effectively constant throughout the whole available volume. We comment briefly upon a possible role of such conformational phase transitions in the regulation of cellular metabolism.

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We develop a theory of ligand diffusion in the presence of a central potential toward or away from receptor patches that are uniformly distributed over a spherical cell. The current onto the receptors is reduced to less than that onto the sphere by a factor that is a nonlinear function of the number of free receptors, their size, and their potential energy. Similarly, under conditions defined by the theory, the dissociation rate from a receptor is reduced by the probability ofrebinding to some other receptor on the same cell. This complicates the kinetic analysis, leading to the possibility of an occupancy-dependent dissociation rate, but has no effect on the interpretation of thermodynamic data.The transmission of biological information from the external environment ofa cell to its interior often involves the interaction of exogenous ligands-e.g., antigens, peptide hormones, and neurotransmitters-with cell surface receptors. The thermodynamics and kinetics of these reactions are being investigated in a wide variety of systems in an effort to gain insight into the mechanisms underlying the early stages in the transduction of signals that ultimately lead to a cellular response (e.g., see ref.1 and references cited therein). Sound interpretation of the data, however, requires a clear understanding ofthe implication of the Brownian movement ofligands in the presence of spherically distributed receptors (2). Berg and Purcell (3) recently made an important contribution to this problem by developing a theory of the diffusive forward rate constant for ligands impinging upon receptors distributed over a spherical cell. In this paper, we develop a generalized theory by considering diffusive ligand currents both toward and away from cell-bound receptors when an external force is present and briefly discuss some of the implications of the results.GENERAL CONSIDERATIONS Bimolecular reaction between ligands and receptors, whether the receptors are cell bound or not, can generally be considered a two-step process: diffusion of the reactive pair to a distance and orientation appropriate for reaction, followed by complexation (4). Thus, if L and R represent the free reactive partners, L-R is the encounter complex in which the units are appropriately positioned but have not yet reacted, and LR is the product, thenbe replaced by a single-step process that has effective forward and reverse rate constants, kf and kr, where kf = k+kl/(k-+ kl)[2] and k_ = kk-1/(k-+ kl). [3] One can show rigorously (5) that Eqs. 2 and 3 represent the forward and reverse rate constants for the zeroth order term in the singular perturbation solution for the equations representing [1]. The equations remain approximately valid in the presence of orientation requirements provided orientational diffusion is fast compared with translational diffusion and provided the ligand is spherically symmetric. The effects ofnonspherical ligand geometry must be calculated separately for each geometry (6) and will not be considered in this paper.For conditions u...

Summary Using recent progress in biological scaling, we explore the way in which the immune system of an animal scales with its mass ( M ). It is shown that the number of cells in a single clone of B cells should scale as M and that the B-cell repertoire scales as ln ( cM ), where c is a constant. The time that a B cell needs to circulate once through the organism is shown to scale as M 1/4 ln ( cM ). It is suggested that the scaling of other cell populations in the immune system could be derived from these scaling relations for B cells.