Shear relaxation governs fusion dynamics of biomolecular condensates

Phase-separated biomolecular condensates must respond agilely to biochemical and environmental cues in performing their wide-ranging cellular functions, but our understanding of condensate dynamics is lagging. Ample evidence now indicates biomolecular condensates as viscoelastic fluids, where shear stress relaxes at a finite rate, not instantaneously as in viscous liquids. Yet the fusion dynamics of condensate droplets has only been modeled based on viscous liquids, with fusion time given by the viscocapillary ratio (viscosity over interfacial tension). Here we used optically trapped polystyrene beads to measure the viscous and elastic moduli and the interfacial tensions of four types of droplets. Our results challenge the viscocapillary model, and reveal that the relaxation of shear stress governs fusion dynamics. These findings likely have implications for other dynamic processes such as multiphase organization, assembly and disassembly, and aging.


Supplementary Text Viscous liquids vs. elastic solids: distinction in shear relaxation
In a viscous liquid, a bead experiences a frictional force that is proportional to its velocity. In contrast, in an elastic solid, the bead experiences a resistance that is proportional to its displacement. A corresponding contrast between the viscous liquid and elastic solid exists when these materials are deformed by shearing. The resulting stress, τ, is proportional to the shear rate ε̇= ε in the liquid but is proportional to the shear strain ε itself in the solid.
Viscous liquids and elastic solids are opposite extremes of viscoelastic fluids. The latter materials, including biomolecular condensates, generally behave as partly liquid and partly solid. There the stress is determined by the entire history of the shear rate: [S1] The function ( ) is called the shear relaxation modulus. This expression for the shear stress is similar in form and substance to the frictional force on a generalized Langevin particle; the counterpart of ( − ′ ) is the memory kernel. Note that ( − ′ ) must be 0 when ′ > (such that future shear rate does not affect present stress); hence the upper limit of the integral can extend to +∞. In a purely viscous liquid (also known as a Newtonian fluid), the stress is affected by the shear rate only at the present time, not any earlier time. That is, where is the viscosity. Here the shear relaxation modulus has no memory, as represented by a delta function and hence shear relaxation is instantaneous. In contrast, the shear relaxation modulus of an elastic solid is a constant (denoted as 0 ), meaning that the stress never relaxes. The result is the expected strain-stress relation Let us further illustrate with a unit-step strain introduced at time = 0 (Supplementary Fig.  1a): Noting the derivative of the Heaviside step function Θ( ) is a delta function, we find Substituting into Eq [S1], we have The shear relaxation modulus thus represents the stress in response to a unit step strain introduced at time = 0. In the viscous liquid, the stress disappears after = 0 ( Supplementary  Fig. 1b, top); i.e., shear relaxation is instantaneously as stated already. In the elastic solid, the stress, once generated at = 0, stays forever ( Supplementary Fig. 1c, top). Viscoelastic fluids fall in between, with shear relaxation occurring over a time period that is between 0 and infinity. An example is a Maxwell fluid, with ( ) given by an exponential function of time ( Supplementary  Fig. 1d The corresponding shear rate is and the stress is The last identity generalizes Eq [S3] and formally defines the complex shear modulus, * ( ). The latter is essentially the Fourier transform of the shear relaxation modulus, For a viscous liquid (Eq [S2b]), we have which has only an imaginary part ( Supplementary Fig. 1b, bottom). On the other hand, for an elastic solid, by comparing Eqs [S3] and [S8], we find which has only a real part ( Supplementary Fig. 1c, bottom). In general, viscoelastic fluids have both real and imaginary parts, The real part is called the elastic (or storage) modulus, whereas the imaginary part is called the viscous (or loss) modulus. The elastic and viscous moduli of a Maxwell fluid are shown in Supplementary Fig. 1d, bottom.

Comparison of condensate viscosities by OT and by FRAP
In a previous study 1 , we fit fluorescence recovery after photobleaching (FRAP) data to an exponential function Here we use the resulting time constant ( FR ) to deduce the viscosity inside condensates. According to Soumpasis 2 , the half-time, 1/2 = (ln 2) FR , and the radius, B , of the bleached region, can be used to find the diffusion constant of the fluorescently labeled species as In the FRAP experiments, B was kept at 1.39 m; FR was found to be 2.1 ± 0.2, 10.6 ± 0.4, 26.8 ± 1.6, and 105.1 ± 2.3 s, respectively for pK:H, P:H, S:P, and S:L condensates. The fluorescently labeled species was H in the first two cases and S in the last cases. The samples were otherwise the same as in the present study, with the following exceptions. The pK and H concentrations in the pK:H FRAP samples were 50 M instead of the 100 M of the present work; the L concentration in the S:L FRAP samples were 2000 M instead of the 300 M of the present work. The much higher L concentration makes the condensates denser and hence more viscous. Thus the FRAP-derived viscosity for the S:L condensates should be somewhat higher than the corresponding OT-derived value.
To find the viscosity in condensates, we compare calculated from FR using Eq [S14] to the diffusion constant, 0 , obtained for H or S determined in water. The viscosity of the condensates is [S15] where w = 8.9  10 -4 Pa s is the viscosity of water at 25 C. For H, 0 for a polymer fraction with molecular weight around 18 kD (as in our samples) was approximately 60 m 2 /s at 20 C 3 . Using the Stokes-Einstein relation and the viscosities of water, 0 for H at 25 C can be found to be 69 m 2 /s. For S in water, we use a scaling relation between 0 (in m 2 /s at 20 C) and molecular weight ( , in Dalton), 10 4 / 0 = 4.0 1/3 − 6.8, derived for globular proteins 4 , along with = 42849 Dalton to find 0 = 75 m 2 /s. The latter translates into 0 = 87 m 2 /s at 25 C.

Shear thickening and thinning in droplet shape recovery
In a recent theoretical study 5  Supplementary Fig. 2 Images of trapped beads at the center or poles of a droplet before measurements. a Set up for measuring viscoelasticity. This experiment was repeated on 3 to 4 different droplets; for each droplet, measurements were carried at 9 to 10 oscillation frequencies.
b Set up for measuring interfacial tension. This experiment was repeated on 10 to 13 different droplets.