The microrheology of sickle hemoglobin gels

Abstract

Sickle cell disease is a rheological disease, yet no quantitative rheological data exist on microscopic samples at physiological concentrations. We have developed a novel method for measuring the microrheology of sickle hemoglobin gels, based on magnetically driven compression of 5- to 8-microm-thick emulsions containing hemoglobin droplets approximately 80 microm in diameter. Using our method, by observing the expansion of the droplet area as the emulsion is compressed, we were able to resolve changes in thickness of a few nanometers with temporal resolution of milliseconds. Gels were formed at various initial concentrations and temperatures and with different internal domain structure. All behaved as Hookean springs with Young's modulus from 300 to 1500 kPa for gels with polymerized hemoglobin concentration from 6 g/dl to 12 g/dl. For uniform, multidomain gels, Young's modulus mainly depended on the terminal concentration of the gel rather than the conditions of formation. A simple model reproduced the quadratic dependence of the Young's modulus on the concentration of polymerized hemoglobin. Partially desaturated samples also displayed quadratic concentration dependence but with a smaller proportionality coefficient, as did samples that were desaturated in steps; such samples were significantly less rigid than gels formed all at once. The magnitude of the Young's modulus provides quantitative support for the dominant models of sickle pathophysiology.

Figure 1

(a) Typical droplet arrangement, viewed at 432 nm. The large droplet central to the picture is the reporter droplet, whose thickness change will be measured by observing the change in area of its image. In the upper left corner is the target droplet, which has been photolyzed and subsequently has polymerized. The dark region shows the enhanced absorbance from deoxyHb at this wavelength. The lighter gray region is the reservoir, an area that was not photolyzed because the laser was masked and did not illuminate that segment. The reservoir contributes monomers to the gel until the monomer concentration in the gel equals that in reservoir. (b) A side schematic of the sample holder (not to scale). A urethane washer is attached to a coverslip, and in its center is placed an emulsion of COHbS in castor oil, suspended in turn in heptane. On top of the emulsion is placed a round coverslip, called the piston, to which a nickel ring has previously been glued. The remainder is filled with heptane, and an upper coverslip is sealed to the top. The thickness of the emulsion is typically ∼5 μm. Magnets placed below the sample holder pull on the nickel ring, compressing the sample.

Figure 2

A typical response of droplets to compression. In each case, forces were chosen at random, and the deflection was measured by observing the area of the reporter droplet. Note the high accuracy and repeatability provided by this method. The upper (black) points were measured without photolysis and thus represent the elastic response of the whole slide. The lower (red) points were measured with laser photolysis and show the diminished compressibility of the sample when a single droplet has been photolyzed. Points were fit by a straight line constrained to pass through the origin. The upper (control) slope was 52.5 ± 0.4 nm/mN, whereas the lower slope was 46.6 ± 0.3 nm/mN. The sample concentration was 29.1 g/dl, and measurements were performed at 25°C.

Figure 3

Young's modulus as a function of the concentration of Hb that is polymerized for samples that were fully photolyzed. Data taken at 20°C are represented by triangles, those at 25°C by circles, and those at 30°C by diamonds (see Table 1). Higher concentrations are indicated by darker symbols. The solid curve is the best fit to the simple theory (Eq. 1) derived in the Appendix.

Figure 4

Young's modulus as a function of the concentration of Hb that is polymerized for partially photolyzed samples. All data were collected at 25°C (see Table 1). Higher concentrations are indicated by darker symbols. The solid curve is the best-fit theory for the full-photolysis experiments of Fig. 3, and it accurately fits some but not all of the data. The lower concentrations were also fit by Eq. 1 with a smaller value of a, which we interpret as the consequence of a different internal gel structure.

Figure 5

Young's modulus as a function of the concentration of Hb that is polymerized for experiments in which an iris masked the initial photolysis area but was subsequently opened to illuminate the entire droplet. Thus, full photolysis was created but with a different underlying gel structure than that seen in the experiments represented in Fig. 3. All data were collected at 25°C (Table 1). The higher the concentration is, the darker the symbols are. The solid curve is the fit to the full-photolysis data of Fig. 3 and lies above the data shown. Dashed curves using different a values are drawn to illustrate the difference that gel formation can generate.

Figure 6

A sketch of a simple cubic fiber gel of mesh size ξ. Cross-links, roughly equivalent to entanglement, are shown as cross-shaped symbols. (Left) Unsheared state. (Right) After shear. Note that the sections of fiber between cross-links are strained into an S-shape by the shear.