This applied to free IgGs using the estimate in Equation (12) yields: relative to the more calm conformation [24,25]. Similarly, the tilts or wagging and wobbling are: (1) of the whole BCR relative to the major axis of the molecule (along the Fc region, normal to the plasma membrane) VAL-083 denoted here Fc tilt; and (2) of a Fab arm relative to its major axis, denoted here Fab tilt; the range of this tilt is much larger than the Fc tilt range (in some isotypes the angle of the Fab tilt cone may be even larger than [26,27,28]). Considering the rotational movements sketched in Determine 3a,b, once a BCR and an Ag are in the state one can map the binding problem to that of random walk of a paratope on a region of a spherical surface of radius ? [23,29] delimited by altitude and azimuthal angles and diffuses within this restricted spherical surface region until it reaches its cognate epitope ? [30], which is equivalent to an angle (Physique 3c). pressure, and propose some testable predictions that can lead to future research. and molecules to a distance that allows binding (denoted by [13]. We summarize these series of says and processes in Equation (1), where the translational and rotational diffusion and constant rates correspond either to those in 3D or in 2D settings. and are, respectively, the effective binding and unbinding rate constants. The corresponding system of coupled regular differential equations for this simplified global process is usually: and and in terms of the free species (and and [13]: and are the intrinsic or molecular and constant rates, which are, therefore, identical for the 3D and the 2D interactions. The effective association affinity constant is simply the product of the affinities (denoted by as a function of as a function of and because it shows how misleading can be the estimation of and by the effective rate constants and the effective rate will increase linearly for increasing values of methods the value will increase more and more slowly, approaching asymptotically the maximum when (observe Physique 2A). The latter situation could be named diffusion/orientation-limited reactions as the limiting process is not binding itself but getting the Ab or BCR and the Ag close enough and well oriented before the actual molecular binding occurs. Open in a separate window VAL-083 Physique 2 Effective kinetic rates measured experimentally can be misleadingly interpreted as approximations of the underlying molecular binding constant rates. Expected dependency of (panel A) and (panel B) as a function of and for any VAL-083 given value of the effective off rate constant is a good estimation of this estimation is not longer valid, and for the value of becomes instead a good estimation of the true dissociation constant, more precisely (see Physique 2B). In the remaining of this paper we refer to the quantity as the is usually greater or smaller than and constant rates are mathematically different in 2D (membrane) and 3D (answer). Specifically, for unbound BCRs of IgM and IgG isotypes the median translational diffusion constant has been estimated to be [16,17,18] [17]. However, for free IgMs and IgGs in diluted saline solutions, as is the VAL-083 case in many SPR experimental setups, has been reported to be [19,20,21] rate constant of BCRs toward an Ag tethered on a cell membrane can be approximated by the expression: (a clear derivation of it can be found in [22], pp. 151C152, Section 4.2.3) where is the translational diffusion constant of BCRs in the membrane (the contribution to total translational diffusion of Ag tethered around the membrane of a cell like FDCs is very small or none at all), is the the mean distance between BCRs in the membrane of a B cell, and is the encounter radius, which is essentially the VAL-083 radius of a BCR (7.5 nm [23]). Following Lauffenburger and Linderman (observe p. 153 in Reference [22]), to convert into the same unit as a typical bimolecular rate, we IL15RB multiply by (Avogadro constant) and, to give an estimated local volume for cell surface components, by a membrane thickness of nm. Considering a B cell with a surface area of 40 (corresponding to a diameter of 7 from Equation (11) one finds: rate constant can be approximated as: is the effective local cell surface area, defined as [22]. In 3D, the translational diffusion and rate constants of proteins can be approximated by and is, as before, the encounter radius [13]. This applied.
