Supplementary Materials: Viscosity of Protein Solutions Exposed to an Oscillatory Stress

Fig. S1 depicts a unit of a substance placed between two surfaces. The bottom surface remains fixed, while the top is free to oscillate. A no-slip condition is assumed in which the layer of substance in direct contact with the oscillating surface moves in unison with it while the layer contacting the stationary surface remains fixed to it. An initial force or stress is applied to start lateral flow (step A). Subsequent oscillation at an applied angular frequency (ω) exposes the substance to a periodic shear stress (σ, units of force per area of the surface) which opposes the direction of flow and eventually reverses it (steps B and C). The substance responds to the stress by deforming. This deformation, or shear strain (γ, unitless), is the ratio D:H where D is the lateral displacement of the oscillating surface relative to the layer at half the distance separating the two surfaces (H). It should be noted that we have defined strain differently from most textbook examples where D is the lateral displacement between both surfaces and H is the entire surface separation distance. This is because the textbook examples do not apply to an oscillating stress, but simply to a single application. The change in stress and strain with time (t) are determined from the following complex equations ([1]):

σ = σ0 cos(ωt) + i σ0 cos(ωt) (S1)

γ = γ0 cos(ωt - δ) + i γ0 cos(ωt - δ) (S2)

The symbol “i" is , σ0 is the amplitude of stress, γ0 is the amplitude of strain, and δ is the phase lag. Purely viscous substances have a 90° phase lag while that for a purely elastic substance is 0° (Fig. S2). Using Euler’s equation, cos(x) + i sin(x) = eix, the complex modulus (G*) which is the ratio of the stress to the strain can be determined:

(S3)

This can also be expressed as

G* = G’ + iG’’ where and (S4, S5, S6)

Also, notice that

(S7)

Furthermore, the magnitude of the complex viscosity ( |η*| ) is calculated as

(S8)

G’ is the storage modulus and is a measure of the energy stored in the substance as it is exposed to the oscillating stress. G’’ is the loss modulus and is the energy dissipated as heat or lost through friction during oscillation. Since cos(δ) = 1 and sin(δ) = 0 when δ = 0°, G’ is the sole component of the complex viscosity for a purely elastic substance. Similarly, for δ = 90° a purely viscous substance is composed only of G’’. Typically, solids or more elastic materials have much smaller values of γ0 than viscous substances.

G’ and G’’ are related to the average relaxation time (τ) of the substance by the following equations:

(S9, S10)

Fig. S3 shows this relationship when ω equals 2π × 10 MHz. G' increases with increasing τ while G'' initially increases but decreases when τ > ω-1. The figure also depicts the relationship between δ and τ which can be determined from Eq. S7. At low τ when G’ ~ 0, δ ~ 90°. Conversely, at high τ, G’ is maximum and δ ~ 0°. From these relationships, it is apparent that changes in solution viscosity affect τ. Solutions with low viscosity have low τ values while elastic substances have higher τ values. From previous high frequency rheology studies, we have determined that the ratios of G’’ to G’ are greater than 1 for protein solutions examined at an angular frequency of 2π × 10 MHz ([2],[3]). This means that τ of those solutions must be less than (2π)-1 × 10-7 s which corresponds to previously reported values of 10-7-10-9 s for protein relaxation (refer to the circled area in Fig. S3) ([4],[5]). Since our studies examined extremely large proteins, i.e. antibodies, at very high concentrations (up to 120 mg/ml), it is possible that τ due to protein relaxation is higher than those reported values but the presence of bound and free water molecules with τ values of 10-10-10-11 and 10-12 s, respectively, shift the solution τ lower ([6],[7]). In addition, as the solutions are diluted, the protein contribution to solution τ decreases. Below 40 mg/ml, the protein contribution to viscosity becomes negligible and τ of solution is simply the τ of water.

Our group has previously determined G’ to be an ideal parameter for quantitating the PPI of highly concentrated protein solutions (3,4). G’ was found to be sensitive across a high concentration range, e.g. 80-250 mg/ml, but was below the level of resolution for more dilute solutions (0-40 mg/ml). Also, as shown in Fig. S3, G’ is unique over all values of τ whereas G’’ is not because its profile is symmetric on either side of ω-1. However, this does not mean that G’’, and therefore |η*|, cannot be used to quantitate PPI. According to Eqs. S9 and S10, both moduli depend solely on τ when ω is constant. Since G’ is a measure of stored energy during oscillation, its relationship to PPI is apparent. Greater interaction among protein molecules means a higher potential energy of the solution, and hence a higher G’. In contrast, the relationship between G’’ and PPI is less intuitive. For a viscous solution (τ < ω-1), a portion of the energy applied during oscillation deforms the solution and the remaining is lost as heat or friction. G’’ quantitates this lost energy. A greater G’’ correlates with greater |η*| via Eq. S8 and lower strain γ0 via Eq. S6. More energy lost to friction means less is available to deform the solution, thus resulting in lower strain and higher viscosity. Friction can be increased if components of a solution are forced to flow as a unit rather than individually. Protein molecules that are strongly attracted to each other will tend to flow together for a greater period of time on average than proteins that are weakly interacting. A solution of the former will therefore have a higher viscosity and G’’. In this way G’’ is a measure of PPI when τ < ω-1.

Figure S1. A substance (clear box) is placed between two black surfaces. The bottom surface remains fixed, while the top is free to oscillate. The layer of substance contacting the oscillating surface moves in unison with it, i.e. no-slip condition. An initial force (step A) starts lateral flow. Periodic shear stresses (via F) oppose and reverse flow direction (steps B and C). Because the stress is oscillating, shear strain is defined D:H where D is the lateral displacement of the oscillating surface relative to the layer at half the surfaces’ separation distance (H).

Figure S2. Deformation of a block of a viscous versus elastic substance exposed to an oscillating stress for one time cycle shown. The blocks are divided into infinitesimally small horizontal layers. Only the top (white), middle (hatched), and bottom (black) layers are shown. The bottom layer is in contact with an oscillating surface and moves in unison with it. The top layer is in contact with a stationary surface. Both have the no-slip condition. The solid vertical line is the center frame of reference. Because the stress is oscillating, strain is derived from the bottom layer’s center relative to the middle layer’s center. Stress is minimum when the black layer is centered and maximum at the highest displacement from the center line.

Figure S3. G’, G’’, and |η*| versus average relaxation time for a hypothetical substance changing from viscous to viscoelastic and subjected to an oscillating stress at 2π × 10 MHz. The three rheological properties and δ were determined with Eqs. S7, S8, S9, and S10. The rheological properties of all protein solutions studied in our laboratory fall within the circled area.


Supplementary Materials References

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