Non-specific adhesive forces between filaments and membraneless organelles

Many membraneless organelles are liquid-like domains that form inside the active, viscoelastic environment of living cells through phase separation. To investigate the potential coupling of phase separation with the cytoskeleton, we quantify the structural correlations of membraneless organelles (stress granules) and cytoskeletal filaments (microtubules) in a human-derived epithelial cell line. We find that microtubule networks are substantially denser in the vicinity of stress granules. When microtubules are depolymerized, the sub-units localize near the surface of the stress granules. We interpret these data using a thermodynamic model of partitioning of particles to the surface and bulk of the droplets. In this framework, our data are consistent with a weak (≲kBT) affinity of the microtubule sub-units for stress granule interfaces. As microtubules polymerize, their interfacial affinity increases, providing sufficient adhesion to deform droplets and/or the network. Our work suggests that proteins and other objects in the cell have a non-specific affinity for droplet interfaces that increases with the contact area and becomes most apparent when they have no preference for the interior of a droplet over the rest of the cytoplasm. We validate this basic physical phenomenon in vitro through the interaction of a simple protein–RNA condensate with microtubules.

Note that granules with locations that fall within the cell nucleus or outside the cell at cell i + 1 are discarded. d, Radial curves corresponding to the average images in panel a. e, g(r) corresponding to the distribution maps in panel b. f, g(r) corresponding to the distribution maps in panel c.
Note that we do not show standard error here because it is smaller than the fluctuations seen in g(r) of the shifted images.      corresponding to the distribution maps in panel c.  ⟨·⟩ gives the average intensity of the respective channel.

A. Partition Coefficients
To evaluate under which conditions a tubulin sub-unit will prefer to reside on the granule surface or in bulk cytosol or bulk granule, we identify the chemical potential µ for each state assuming tubulin is dilute: where c 0 is an arbitrary reference concentration. Note that the surface concentration c s is also a volumetric concentration. This is becausewe measure intensity per voxel in the experiment. Moreover, the assumption that tubulin is dilute implies that tubulin sub-units do not feel and interact with each other.
We assume a local equilibrium of the chemical potential of tubulin in and around the granule. We justify this for once, because we consider a small subset of the cell, i.e. inside and outside of the granule are close together (about 1 µm) such that we expect diffusion to relax gradients in chemical potential, and because we assume that no chemical reactions occur as tubulin moves from one phase to the other. Equating the chemical potentials of a sub-unit in the cytosol and inside the granule, we can define the partition coefficient between granule and cytosol with affinity for the bulk of the granule −∆G g = µ c (c 0 , T ) − µ g (c 0 , T ) between a sub-unit in the granule compared to the cytosol. Analogously, we define the partitioning coefficient to the surface relative to the cytosol as with surface affinity −∆G s = µ c (c 0 , T ) − µ s (c 0 , T ).
In order to calculate the free energy differences ∆G g and ∆G s , we present two models, which differ in the definition of the interface between stress granule and cytosol. The tubulin sub-unit is, in both cases modeled as a colloidal particle. FIG. S10. Tubulin sub-unit modelled as a colloidal particle t interacting with a thin interface between stress granule g and cytosol c. a, Reference state of a sub-unit with surface area A 0 in bulk cytosol. b, A sub-unit wetting the interface with contact angle θ. A c is the surface area of the spherical cap of the sub-unit exposed to the cytosol, A g = A 0 − A c respectively the spherical cap exposed to the granule.

B. Thin interface
First, let us consider the colloidal model tubulin particle of radius R interacting with an interface of width 2w that is much thinner than the size of the particle R >> w. (Note that the smallest possible tubulin particle is a sub-unit but we cannot ensure that all tubulin particles are sub-units in nocodazole-treated cells.) This corresponds to the classical picture of a particle at an interface as shown in Fig. S10. −∆G g = µ c (c 0 , T )−µ g (c 0 , T ) is the affinity of a sub-unit towards the bulk phase, i.e. the energy difference a sub-unit experiences when it is moved from bulk cytosol to inside the granule, where a positive value means that the inside of the granule is energetically favorable. −∆G g is then given as the surface area of the sub-unit A 0 times the difference in the surface tension of the sub-unit towards the cytosol γ tc and towards the bulk of the granule γ tg In order to calculate the surface affinity −∆G s = µ c (c 0 , T ) − µ s (c 0 , T ) between a sub-unit in bulk cytosol and adhered to the surface of the granule, let us first consider the energy of a sub-unit at the interface. This energy is given as the balance of the area of the interface taken up by the sub-unit A x times the surface tension of the granule towards the cytosol γ cg and the surface areas A c and A 0 − A c of the granule exposed to the cytosol and the granule multiplied with the respective surface tension: The surface affinity is then Using Young's law with contact angle θ, we find To define the geometric quantities, we now consider a spherical sub-unit, as illustraded in Fig. S10. The sub-unit with radius R then has a surface area A 0 = 4πR 2 , surface area exposed to the cytosol (in the shape of a spherical cap) of A c = 2πR 2 (1 − cos(θ)) and an area A x = πR 2 sin 2 (θ) that the sub-unit takes up on the surface of the granule, with contact angle θ as defined in Fig. S10 b. Thus we finally arrive at −∆G s = 4πR 2 − 2πR 2 (1 − cos(θ) γ cg cos(θ) + πR 2 γ cg sin 2 (θ) = πR 2 γ cg 2 cos 2 (θ) + sin 2 (θ) + 2 cos(θ) = πR 2 γ cg (1 + cos(θ)) 2 . (10) The energy difference between a particle in bulk cytosol compared to bulk granule is then analogously −∆G g = 4πR 2 (γ tc − γ tg ) = 4πR 2 γ cg cos(θ).
(11) For comparison to our data, we express ∆G s in terms of ∆G g Note that this expression has removed the explicit dependence on contact angle. −∆G g is a function of both γ cg and θ. The histogram of observed −∆G g (Fig. 5 (d)), however, shows both positive and negative values with a mean close to zero. If θ was constant, this would call for negative surface tension γ cg , which is not physical. We thus assume that variations in −∆G g are dominated by variations in contact angle θ and hold γ cg constant.

C. Thick interface
Tubulin sub-units, as well as the constituents of stress granules, are proteins, i.e. the assumption that the particle interacting with a droplet interface is much larger than the interface width is not given. To set up a theory for a thick interface (R ≪ w), we follow the approach presented by Cahn and Hillard [1]. We consider a two-phase system characterized by an intensive scalar quantity ϕ (other than temperature or pressure) that transitions smoothly from one state (inside the bulk of the stress granule) to another state (bulk cytosol).
Here we express ϕ as the local mesoscale composition where ψ g gives the local volume fraction of stress granule components and ⟨·⟩ meso gives the local average over a volume sufficiently large such that ϕ is smooth [2]. The spatial coordiante x is normal to a flat interface between granule and cytosol, with the midpoint of the interface at x = 0, without loss of generality.
We assume a Helmholtz free energy per unit volume of the system given as with parameter a = a(T ) that is negative under conditions in which the system separates and positive constants b and κ [1-3]. The volume terms capture the phase behavior of the system and the gradient term describes interfacial energies. For negative a, i.e. below the critical temperature, f has two minima at ±ϕ b = −a/b [2]. Assuming a constant chemical potential µ everywhere in the system, it can be shown that the local composition takes the form [2,3] with width of the interface [3] w = −2κ a .
Integrating the free energy over the interface, one finds the surface tension between both phases [2] γ cg = 2a 2 w 3b .
To evaluate the energy per unit volume around the interface, we expand ϕ(x) around Inserting into equation 14 we find Far from the interface, we find for either bulk phase Moving a particle with surface area A 0 and volume V 0 from bulk cytosol (x → ∞) to position x we then change the free energy by

IV. CALCULATION OF THE BENDOCAPILLARY LENGTH
To estimate the stress granule radius at which we can expect microtubules to bend around the granule we perform a gedankenexperiment where a microtubule wraps a stress granule once. The elastic energy to bend a length l of filament to a radius of curvature matching the radius of the stress granule R SG is then with bending rigidity of the microtubule K M T and line integral dl. We can approximate K M T from the persistence length l p of microtubules, which we take to be about one millimeter [4]: