Figure 1

Buffering via non-cooperative ligand binding: "Langmuir buffering". The prototype of Langmuir buffering is the buffering of H⁺ ions in a solution of a weak acid. A, Relation between the three variables of a "Langmuir"-type buffer. Concentrations of free ligand (red), bound ligand (blue), and total ligand in a solution of a weak acid. The relations between the three variables are computed from the equation , where K_d stands for the dissociation constant of the buffer-ligand complex, and [buffer] for total buffer concentration. [buffer] and K_d are assumed to be constant. Plotted for [buffer] = 5 and K_d = 1. B, Describing "Langmuir buffering" using the four buffering measures t, b, T, and B. Titration of a "Langmuir buffer" with increasing concentrations of ligand; constant parameters are: [buffer] = 100, K_d = 10, arbitrary concentration units. Characteristic system states shown are the "half-saturation point" of buffer (asterisk), the "equipartitioning point" where half of the added ligand remains free, and the other half is bound by the buffer (open circle), and the "break even point" where the ligand inside the system is half bound, half free (closed circle). Top panel, left: Transfer function τ, i.e., free ligand concentration (ordinate) as a function of total ligand (ordinate). Top panel, right: buffering function β, i.e., bound ligand concentration as a function of total ligand. Middle panel, left: Transfer coefficient t, i.e., the (differential) fraction of added ligand that enters the pool of free ligand. Middle panel, right: Buffering coefficient b, i.e., the (differential) fraction of added ligand that becomes bound to buffer. Bottom panel, left: Transfer ratio T = d(free)/d(bound), i.e., the differential ratio of additional free ligand over additional bound ligand. Bottom panel, right: Buffering ratio B = d(bound)/d(free), i.e., the differential ratio of additional bound ligand over additional free ligand. The parameters b and B provide two complementary measures of buffering strength. C, Buffering strength of a Langmuir buffer as a function of both total ligand concentration and affinity. Wireframe surface: The buffering ratio B is shown on the vertical axis; affinity expressed as 1/K_d; concentration of ligand, [ligand], and K_d in arbitrary concentration units. Contours on bottom: Lines connect states of identical buffering strength. For a buffer with a given K_d, buffering strength decreases monotonically with increasing ligand concentration. However, at a fixed ligand concentration, buffering strength as a function of affinity runs through a maximum. D, Visualizing Langmuir buffering by two-dimensional plots (same data as in Figure C). Left hand, linear plot; white lines, states of identical buffering strength; black lines, states of identical fractional buffer saturation. Right, double-logarithmical plot. black lines, states of identical buffering strength; red lines, states of identical fractional buffer saturation. E, Using the "buffering angle" to visualize Langmuir buffering: cylinder plot. As shown in Buffering I, the specific angle α for which [α = arccos(T) and α = arctan(B)] can unambiguously represent the buffering parameters t(x), b(x), T(x) and B(x) at a given point on the x axis. Consequently, a curve on the surface of a unit cylinder can represent the buffering behavior for an entire range of x values, yielding a "state portrait". State portraits of several Langmuir buffers are shown. Curves with Roman numerals (I-IV) of different color: effect of decreasing ligand affinity at fixed total concentration. Curves with Arabic numerals (1–4) of different size: effect of increasing total buffer concentration. Less intuitively, yet more practically, the cylinder surface may be "flattened" out and represented in two dimensions (not shown). Blue segment: buffering angle α for curve 4. F, Using the "buffering angle" to visualize Langmuir buffering: polar graph. Alternative form of a buffering state portrait: each point on the curve is characterized by a "buffering angle α " with the vertical axis (clockwise) and a radius (here plotted logarithmically), which correspond to buffering angle α and total ligand concentration, respectively. Open circles, equipartitioning points, i.e., where t = b and α = 45°.