Hydrodynamic trapping measures the interaction between membrane-associated molecules

How membrane proteins distribute and behave on the surface of cells depends on the molecules’ chemical potential. However, measuring this potential, and how it varies with protein-to-protein distance, has been challenging. Here, we present a method we call hydrodynamic trapping that can achieve this. Our method uses the focused liquid flow from a micropipette to locally accumulate molecules protruding above a lipid membrane. The chemical potential, as well as information about the dimensions of the studied molecule, are obtained by relating the degree of accumulation to the strength of the trap. We have used this method to study four representative proteins, with different height-to-width ratios and molecular properties; from globular streptavidin, to the rod-like immune cell proteins CD2, CD4 and CD45. The data we obtain illustrates how protein shape, glycosylation and flexibility influence the behaviour of membrane proteins, as well as underlining the general applicability of the method.

with both low and high trapping force as seen in the different maximum ε hydro values. The shape of the interaction curves is essentially independent of the trapping strength.

Surface coverage [%]
A B

Conversion between fluorescence intensity and protein density
A 1 pM solution of labelled proteins was added to a silicone well on a glass cover slide. The

Determining ε hydro and σ hydro from finite element simulations
Different factors such as the geometry of the pipette, the distance between the pipette tip and the SLB and the pressure applied over the pipette affects εhydro. The distance between the tip of the pipette and the underlying SLB can be related to the change in ion current as the pipette approaches the surface using finite element simulations. 3 To avoid having to do a new simulation for each experiment, the following approximate expression, valid for the lowtapered pipettes used in this work, was used: where Rpipette is the electric resistance over the pipette, K is the ion conductivity in the medium,

R0 the inner tip radius of the pipette and k(h/R0) a function of the dimensionless parameter h/R0
with h being the distance between the tip of the pipette and the SLB. The inner tip radius is obtained from D0 in Fig. 5 by multiplying with 0.29, where it is assumed that the ratio between inner and outer diameter of the pipette is constant along the length of the pipette (Rinner/Router = 0.58). The parameter ϕeff is the voltage inner half cone angle of the multi-segmented pipette, and is defined by: where: and it is assumed that the ratio between inner and outer diameter of the pipette is constant along the length of the pipette. It is enough to do a series of finite element simulations at different heights h above a surface for a reference pipette after which Supplementary equation (S2) can be used to determine Rpipette and its dependence of h for an arbitrarily-sized pipette.
The value for εhydro, or σhydro, can also be determined from finite element simulations. 3 The shear force σhydro will, in the creeping flow regime, be given by: where η is the viscosity of the fluid and j is a function of the two parameters r/R0 and h/R0, where r is the radial distance from the centre of the trap at the surface and h is again the distance between the pipette tip and the surface. The parameter Q is the liquid flow rate out of the pipette, which for the small cone angles considered in this work can be shown to have the following approximate dependence on the applied pressure Δp over the pipette: where f is a function that only depends on the parameter h/R0 and θeff is the pressure inner half cone angle which for a pipette with multiple segments is given by the expression: The value for σhydro as a function of r was determined for a reference pipette at a series of different values of h using finite element simulations as previously described. 3 With this tabulated data for the reference pipette and Supplementary equations (S5) and (S6) it is possible to determine σhydro for an arbitrarily-sized pipette at different distances h from the surface.

Converting from A hydro to h c
Equation (5)  (S10)

Trapping of two proteins
For the trapping of CD2 and CD45, it is as a first approximation assumed that the chemical potential of CD45 is given by: and that of CD2 by: where in Supplementary equation (S14) it is assumed that the shielding of the hydrodynamic force on CD2 is dominated by CD45 and varies similarly with concentration as for CD45.
Combining these four equations results in: where μCD45 is defined by Supplementary equation (S11) and we as an approximation have set f(cCD45) = cCD45. The radial concentration profile of CD45 in the double trap was used as input values, cCD2,0 = 610 molecules/μm 2 and cCD45,0 = 140 molecules/μm 2 , and the excess chemical potential for CD45 was taken from the fitted curve in Fig. 2B. The excess chemical potential of CD2 was assumed to obey a hard-disk model with radius 5 nm. The values for Ahydro,CD2 and S10 Ahydro,CD45 were measured from the double-trap interaction curves to be 419 nm 2 and 1015 nm 2 .
Solving Supplementary equation (S15) with these parameters gave the curves in Supplementary   Fig. S7. The system is assumed to be in quasi equilibrium at all times, which, especially at lower times, is a rather crude approximation, but the obtained curves still capture the essential behaviour of the experimental data.