The expressions derived in the previous paper for the respective normal, F3, and shear forces, Fshear, acting along and perpendicular to the axis of a doublet of rigid spheres, were used to determine the hydrodynamic forces required to separate two red cell spheres of antigenic type B crosslinked by the corresponding antibody. Cells were sphered and swollen in isotonic buffered glycerol containing 8 X 10(-5) M sodium dodecyl sulfate, fixed in 0.085% glutaraldehyde, and suspended in aqueous glycerol (viscosity: 15-34 mPa s), containing 0.15 M NaCl and anti-B antibody from human hyperimmune antiserum at concentrations from 0.73 to 3.56 vol%. After incubating and mixing for 12 h, doublets were observed through a microscope flowing in a 178-micron tube by gravity feed between two reservoirs. Using a traveling microtube apparatus, the doublets were tracked in a constantly accelerating flow and the translational and rotational motions were recorded on videotape until breakup occurred. From a frame by frame replay of the tape, the radial position, velocity and orientation of the doublet were obtained and the normal and shear forces of separation at breakup computed. Both forces increased significantly with increasing antiserum concentration, the mean values of F3 increasing from 0.060 to 0.197 nN, and Fshear from 0.023 to 0.072 nN. There was no significant effect of glycerol viscosity on the forces of separation. It was not possible to determine whether the shear or normal force was responsible for doublet separation. Measurements of the mean dimensionless period of rotation, TG, of doublets in suspensions containing 0.73 and 2.40% antiserum undergoing steady flow were also made to test whether the spheres were rigidly linked or capable of some independent rotation. A fairly narrow distribution in TG about the value 15.64, predicted for rigidly-linked doublets, was obtained at both antiserum concentrations.

A general method of calculating forces, torques, and translational and rotational velocities of rigid, neutrally buoyant spheres suspended in viscous liquids undergoing a uniform shear flow has been given by Arp and Mason (1977). The method is based on the matrix formulation of hydrodynamic resistances in creeping flow by Brenner and O'Neill (1972). We describe the solution of the Brenner-O'Neill force-torque vector equation in terms of the particle and external flow field coordinates and derive expressions for the normal force acting along, and the shear force acting perpendicular to, the axis of the doublet of spheres, the latter explicitly given for the first time. The equations consist of a term comprising force and torque coefficients obtained from the matrices of the hydrodynamic resistances (functions of the distance h between sphere surfaces which have been computed), and terms comprising the orientation of the doublet axis relative to the coordinates of the external flow field and the shear stress (which can be experimentally determined). We have applied the theory to a system of doublets of sphered, hardened human red cells of group A or B antigenic type cross-linked by the corresponding antibody at a fixed interparticle distance. Working from studies of the breakup of doublets of red cells in an accelerating Poiseuille flow, given in the succeeding paper, we are able to compute the hydrodynamic force required to separate the two spheres. Previous work has shown that the theory can be applied to doublets in a variable shear, Poiseuille flow, provided the ratio of particle to tube diameter is small. In calculating the force-torque coefficients it was assumed that the cells are crosslinked by antibody with h = 20 nm.

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In the flow studies described in two previous papers (Tha, S. P., and H. L. Goldsmith, 1986, Biophys. J. 50:1109-1116; Tha, S. P., J. Shuster, and H. L. Goldsmith, 1986, Biophys. J. 50:1117-1126), hydrodynamic forces of the order of 10(-11) N (mu dyn) were applied to measure the force of separation of doublets of hardened, sphered human red blood cells cross-linked by anti-B antibody. The same cell preparation and hyperimmune antiserum has here been used to carry out experiments with micropipet aspiration techniques. One cell of a doublet was aspirated onto a holding pipet, and a second aspiration pipet was brought into proximity of the other cell so that the two pipets and the doublet were colinear. Suction was then raised until the two cells separated. Some doublets were assembled by aspiration of a singlet, bringing a second singlet into apposition with the first, and releasing it from the pipet which was then withdrawn. Cells could be repeatedly assembled and separated. At 3.56% vol/vol antiserum, the mean normal force of separation was 0.45 +/- 0.11 nN in phosphate-buffered saline suspensions containing 2.5 x 10(4) cells/microliter; at 1.22% vol/vol antiserum, the value was 0.22 +/- 0.11 nN. The above values of the force were approximately 2.5 x greater than those from the flow studies. The data could be fitted to a Poisson distribution with 0.05 nN as the force needed to break a single cross-bridge (c.f. 0.024 nN from the previous hydrodynamic data). The forces of separation of randomly assembled doublets were lower than those of preexisting doublets. Repeated assembly and separation of doublets showed that the cell surfaces are nonuniform in adhesion strength both over the local scale less than 0.25 micron2 and the cell population.