Quantitative theory for the diffusive dynamics of liquid condensates

  1. Lars Hubatsch
  2. Louise M Jawerth
  3. Celina Love
  4. Jonathan Bauermann
  5. TY Dora Tang
  6. Stefano Bo
  7. Anthony A Hyman
  8. Christoph A Weber  Is a corresponding author
  1. Max Planck Institute for the Physics of Complex Systems, Germany
  2. Max Planck Institute of Molecular Cell Biology and Genetics, Germany
  3. Center for Systems Biology Dresden, Germany

Abstract

Key processes of biological condensates are diffusion and material exchange with their environment. Experimentally, diffusive dynamics are typically probed via fluorescent labels. However, to date, a physics-based, quantitative framework for the dynamics of labeled condensate components is lacking. Here, we derive the corresponding dynamic equations, building on the physics of phase separation, and quantitatively validate the related framework via experiments. We show that by using our framework, we can precisely determine diffusion coefficients inside liquid condensates via a spatio-temporal analysis of fluorescence recovery after photobleaching (FRAP) experiments. We showcase the accuracy and precision of our approach by considering space- and time-resolved data of protein condensates and two different polyelectrolyte-coacervate systems. Interestingly, our theory can also be used to determine a relationship between the diffusion coefficient in the dilute phase and the partition coefficient, without relying on fluorescence measurements in the dilute phase. This enables us to investigate the effect of salt addition on partitioning and bypasses recently described quenching artifacts in the dense phase. Our approach opens new avenues for theoretically describing molecule dynamics in condensates, measuring concentrations based on the dynamics of fluorescence intensities, and quantifying rates of biochemical reactions in liquid condensates.

Introduction

Liquid phase separation has emerged as an organizing principle in biology and is thought to underlie the formation of various membrane-less cellular organelles (Banani et al., 2017). Hallmark properties of such organelles are their rapid formation and dissolution, their fusion, and their wetting to membranes (Hyman et al., 2014). Moreover, phase-separated organelles exchange material with their environment leading to dynamic sequestration of molecules, which affects biochemical processes by spatial redistribution of reactants (Moon et al., 2019; Lyon et al., 2021; Saha et al., 2016; Guillén-Boixet et al., 2020; Sanders et al., 2020; Yang et al., 2020). Probing the dynamics of condensate components is thus imperative for a quantitative understanding of how they affect the cellular biochemistry (Mir et al., 2019).

To probe the dynamics of condensates, biomolecules are typically labeled with fluorescent tags. In general, in systems with tagged molecules, various methods exist to characterize molecular properties such as binding rates and diffusion coefficients, including fluorescence correlation spectroscopy (FCS) (Ries and Schwille, 2012; Rigler and Elson, 2012), single-particle tracking (SPT) (Tinevez et al., 2017; Saxton and Jacobson, 1997), and fluorescence recovery after photobleaching (FRAP) (Diaspro, 2010; Stasevich et al., 2010). However, interpretation of the experimental data acquired from such methods requires a rigorous derivation accounting for the underlying physicochemcial processes. This derivation has been achieved for some biological systems and processes, but is lacking for condensates formed by liquid phase separation. Processes that are well-understood include membrane-cytoplasmic exchange and transport (Sprague et al., 2004; Robin et al., 2014; Goehring et al., 2010) as well as chemical reactions (Elson, 2001) or filament turnover (McCall et al., 2019). For liquid condensates, various phenomenological fit functions have been proposed in the literature (e.g. Patel et al., 2015; Banerjee et al., 2017; Hubstenberger et al., 2013, for a broader summary see Taylor et al., 2019). However, it was recently shown that these fits lead to wildly differing estimates of the diffusion constant inside, Din (Taylor et al., 2019). Taylor et al. showed that these discrepancies were attributed to unrealistic assumptions, for example infinitely large droplets or infinitely fast diffusion outside the bleach area.

Here, we first introduce a quantitative FRAP method to extract the diffusion coefficient inside, Din, purely based on fluorescence measurements inside droplets, without resorting to unrealistic assumptions or requiring knowledge about the partition coefficient, P or diffusion outside, Dout. Using irreversible thermodynamics, we then derive the theory that connects dynamics inside and outside of the droplet via transport across a finite interface. We use the corresponding dynamic equation to derive a relationship between P and Dout, which we use to investigate effects of salt addition on P. We show that this dynamic equation agrees with our experimentally observed dynamics. By numerically solving the underlying equations, we show that in theory all relevant parameters of the system, P, Din, and Dout, can be extracted purely based on knowledge of the dynamics inside the droplet. We find that our measurements are agnostic to breaking radial symmetry, for example by introducing a coverslip or neighboring droplets. Our approach does not suffer from typical limitations of fluorescence-based concentration measurements, such as low fluorescence in the dilute phase and fluorophore quenching in the rich phase. We anticipate that this new understanding will open the door to characterizing dynamical properties such as chemical rates and rheological parameters in multi-component, phase-separated systems.

Results

Determining the diffusion constant inside liquid condensates

First, we discuss a quantitative method to extract diffusion coefficients of biomolecules in a condensate. After photobleaching, bleached molecules diffuse out and unbleached molecules diffuse into the condensate, until the unbleached components reach the spatially homogeneous level prior to bleaching (Figure 1a, left and middle). Inside a spherical condensate of radius R, the concentration of unbleached components, cu(r,t), follows a diffusion equation (for derivation, see subsequent section),

(1a) tcu(r,t)=-𝒋u,
(1b) 𝒋u=-Dincu,
(1c) cu(r=R-,t),r=R,

where cu(r=R-,t) is the time-dependent concentration directly inside the interface at r=R-. Here, r denotes the radial distance to the center of the condensate. The flux 𝒋u is given by Fick’s law (Equation (1b)). It vanishes at the condensate center. Moreover, we have =𝒆rr, with 𝒆r denoting the radial unit vector. During FRAP, the concentration at the interface, cu(r=R-,t), changes with time (Figure 1a, middle) and is determined by the physical properties of the condensate environment. This environment is characterized by the diffusion constant and the concentration of unbleached components outside, the distribution of neighboring condensates as well as system boundaries like the coverslip.

Quantitatively measuring Din by extracting intensity at condensate interface.

(a) (1) Spatial time course of FRAP recovery after full bleach for PLYS/ATP coacervate droplet. Azimuthal average (orange) is highlighted. (2) To obtain the boundary condition c(r=R-,t), fluorescence intensity is extracted approximately 1.4μm away from the interface (see methods). Data closer to the droplet boundary can be subject to optical artefacts giving rise to an artificially broad interface. Lower panel shows azimuthal averages at different time points. (3) Cost function for fitting. For each fitting step, a numerical solution to Equation (1a) is calculated with a trial Din and the experimentally measured boundary condition from step (2). Subsequently simulation and experiment are compared according to the cost function, before choosing the next trial Din. Scale bar, 5μm. (b) Flurorescence recovery inside a PGL-3 condensate, along the radial direction (azimuthal averages, see (a)). Din is extracted by global fitting of Equation (1b) to the experimental profiles, using the experimentally extracted initial and boundary conditions (see panel (a) (1) and (2)). The gray line indicates an offset that comes about due to incomplete bleaching, and a small but visible, fast and uniform recovery with unknown cause (see Materials and methods). (c) Same as (b) but for ATP/PLYS coacervate droplet. (d) Same as (b) but for a CMD/PLYS coacervate droplet. (e) Cost function (see panel (a)) used to extract Din for a PGL-3 droplet. Due to the smoothness of the cost function, the minimum can be determined with high accuracy. (f) Comparison between Din and viscosity η for PGL-3 condensates at different salt concentrations. Viscosity data taken from Jawerth et al., 2018 for untagged PGL-3. Note, GFP-tagged PGL-3 has a higher viscosity than untagged PGL-3, which means the hydrodynamic radius of PGL-3:GFP cannot be directly computed from this panel (Jawerth et al., 2020). Replicates: salt concentration [mM]/#: 50/2, 60/3, 75/3, 90/3, 100/6, 120/4, 150/3, 180/3. (g) Diffusion coefficients for coacervate systems. Each data point represents Din for a single droplet time course. Note the low spread of the measured values. Replicates: CMD/PLYS: 9, PLYS/ATP:16.

To initially bypass this complex dependence on the condensate environment, we propose to extract the fluorescence concentration of unbleached molecules cu(r=R-,t) directly inside of the spherical phase boundary between the condensate and the dilute phase from experimental data after photobleaching (Figure 1a). To achieve spherical symmetry of the recovery inside the droplet, the entire spherical droplet should be bleached. Using the experimentally determined dynamic boundary condition cu(r=R-,t), we can accurately determine the diffusion constant inside a condensate, Din (Figure 1f,g), provided R2/Dinτbound. Here, τbound is the time scale of recovery at the boundary (see Figure 1a, middle), which in general features a complex dependence on bleach geometry in the dilute phase, the coverslip, and neighboring droplets. Following this idea, we fit the solutions of Equation (1a) to spatio-temporal experimental data, with Din as the only fit parameter. We find very good agreement between experimentally measured and fitted concentration profiles (Figure 1b–d and supp. Videos 1, 2 and 3). Specifically, we consider condensates composed of PGL-3, a main protein component of P granules in the C. elegans embryo (Brangwynne et al., 2009; Griffin, 2015), as well as two synthetic polyelectrolye-complex coacervate systems, Polylysine/ATP (PLYS/ATP) and Carboxymethyldextran/Polylysine (CMD/PLYS).

Video 1
FRAP dynamics in a PGL-3 droplet Left: Representative in vitro droplet after full bleach.

Time course starts after a small time lag due to a fast uniform recovery (see methods). Middle: Diffusion Equation (1) fit to azimuthally averaged droplet intensity with a global fit parameter Din. Right: Full model Equation (6) fit to azimuthally averaged droplet intensity.

Video 2
FRAP dynamics in a CMD/PLYS coacervate.

For description see Video 1.

Video 3
FRAP dynamics in a PLYS/ATP coacervate.

For description see Video 1.

We first compared Din of PGL-3 for different salt concentrations between 50mM and 180mM (see Methods). We find that Din varies by roughly one order of magnitude, between 0.009μm2s1 and 0.070μm2s1. Our trend is in agreement with reported measurements of the viscosity η, determined by active micro-rheology (Jawerth et al., 2018) for untagged PGL-3 (Figure 1f). Using viscosity data for GFP-tagged PGL-3 (Jawerth et al., 2020), we use the Stokes-Sutherland-Einstein relationship Din=kBT/(6πaη) to estimate the hydrodynamic radius of PGL-3:GFP, a=1.5nm (Einstein, 1905; Sutherland, 1905; von Smoluchowski, 1906). This estimate is consistent with the value reported in Liarzi and Epel, 2005. Across all salt concentrations, the average coefficient of variation per condition is found to be cv=0.22. Due to the smooth cost function (Figure 1e) this is unlikely to be a stochastic artefact. It rather seems to reflect variation within the experimental assay.

For the coacervate systems, we find for the diffusion coefficients inside Din=1.68±0.09μm2s1 for PLYS/ATP coacervates and Din=5.65±0.32μm2s1 for CMD/PLYS coacervates; see (Figure 1g). The coefficient of variation of these measurements is low enough such that even a single measurement provides a good estimate of Din. We find cv=0.05 and cv=0.06 for PLYS/ATP and CMD/PLYS coacervates respectively. Interestingly, Din for the coacervate droplets is about ten times smaller than the diffusion constant of the dilute polyelectrolytes, Dout (Arrio-Dupont et al., 1996; Morga et al., 2019).

Theory for the dynamics of labeled molecules in phase-separated systems

To understand the physical origin of the time-dependent concentration of unbleached components at the condensate interface and the phenomenological Equations (1a, b, c) for the dynamics inside a condensate, we need a theory that encompasses diffusion inside, outside and across phase boundaries. Here, we derive such a theory for a system that can be described by a binary, incompressible mixture prior to photobleaching. This binary mixture is composed of condensate material and solvent. The condensate material has a concentration profile, ctot(𝒙,t)=ϕtot(𝒙,t)/ν, which can be expressed in terms of a volume fraction profile ϕtot(𝒙,t) by dividing through the molecular volume of the condensate material, ν. Due to incompressibility, the solvent volume fraction in such a binary mixture is given by (1-ϕtot). The system prior to photobleaching is assumed to be either at equilibrium, that is, a single droplet and tϕtot(x,t)=0, or close to equilibrium, that is, a system composed of many droplets undergoing slow Ostwald ripening and fusion and tϕtot(𝒙,t)0. Thus, the (quasi-) stationary profile ϕtot(𝒙) prescribes a physical constraint for FRAP dynamics.

After photobleaching, the system becomes a ternary incompressible mixture composed of bleached (b) and unbleached (u) components, as well as solvent (Figure 2a). Introducing the volume fraction of the bleached and unbleached components, ϕb and ϕu, the physical constraint for the FRAP dynamics reads

(2) ϕtot(x,t)=ϕu(x,t)+ϕb(x,t),

where the profiles depend on space and time. The above constraint reflects particle number conservation of bleached and unbleached molecules and becomes a local constraint due to incompressibility. In our work, we focus on FRAP dynamics at thermodynamic equilibrium, where the total volume fraction ϕtot(𝒙) depends on space only. Please note that that the derivation below can be generalized to non-equilibrium situations where ϕtot exhibits a flux (Bo et al., 2021).

Ternary mixture accounts for the dynamics of bleached and unbleached molecules.

(a) Before bleaching, a droplet that is composed of fluorescently labeled molecules can be described by a binary mixture, namely unbleached molecules and solvent. After bleaching, the system is composed of three components, bleached molecules, unbleached molecules and solvent. If the system was at equilibrium prior to bleaching, the sum of bleached and unbleached molecules forms a stationary, non-uniform profile ϕtot(r) (see panel b). (b) Snapshot of model dynamics at t=0.22R2/Din. Initial conditions are ϕu(r,t=0)=ϕoutΘ(r-R), corresponding to a fully bleached droplet. Note that at any time we have ϕtot=ϕu+ϕb. (c) Time course of spatial recovery. For long times, when nearly all bleached material has been exchanged, ϕu approaches ϕtot. Panels (c,d) use radial symmetry for illustration purposes, however, the theory is general (see Figure 3).

Immediately after photobleaching, unbleached molecules diffuse into the condensate leading to FRAP dynamics of unbleached molecules inside (Figure 2b,c). At long times, the concentration profile of unbleached molecules approaches the profile prior to photobleaching, ϕtot(𝒙). The dynamics of both concentration profiles, bleached and unbleached molecules, ci=ϕi/νi (i=b,u), with νi denoting the molecular volumes, is described by the following conservation laws (j=u,b),

(3a) tci=-𝒋i,
(3b) 𝒋i=-Γiμi-Γ¯ijμj,

Here, Γi are the Onsager transport coefficients, often referred to as mobilities, and Γ¯ij are the Onsager cross coupling coefficients obeying Γ¯ij=Γ¯ji=:Γ¯. In general, both mobility coefficients depend on the volume fractions. According to irreversible thermodynamics, the flux 𝒋i is driven by gradients in exchange chemical potentials μi and μj. In the following, we consider linear response for simplicity. The exchange chemical potentials, μi=δF/δci, are linked to the free energy, F=d3xf, where f denotes the free energy density. Expressing concentrations in terms of the volume fractions, ϕi=ciνi, we describe our incompressible ternary mixture after photobleaching by a Flory-Huggins free energy density (Flory, 1942; Huggins, 1942; Krüger et al., 2018):

(4) f=kBTνsol[ϕunulnϕu+ϕbnblnϕb+(1ϕuϕb)ln(1ϕuϕb)+χusϕu(1ϕuϕb)+χbsϕb(1ϕuϕb)+χubϕuϕb+κu2(ϕu)2+κb2(ϕb)2+κub2ϕuϕb],

where we write the molecular volumes of bleached and unbleached components in ni multiples of the solvent molecular volume νsol, that is, νi=niνsol. Moreover, χij denote dimensionless Flory-Huggins parameters characterizing the interactions between different components i and j, where subscript s indicates the solvent. The parameters κi and κub characterize the free energy penalties for spatial heterogeneities and are linked to the surface tensions.

If photobleaching does not affect the molecular interactions or molecular volumes, the free energy density above can be simplified significantly (Appendix of Krüger et al., 2018). In this case, the interactions between unbleached and solvent, and bleached and solvent components are equal, χus=χbs=:χ, and cross interactions vanish, χub=0. Moreover, molecular volumes of bleached and unbleached components are equal, nu=nb=:n, and the parameters characterizing free energy penalties for spatial heterogeneities obey κu=κb=:κ and κub=2κ. Thus, the simplified free energy reads

(5) f=kBTνsol[ϕunlnϕu+ϕbnlnϕb+(1-ϕu-ϕb)ln(1-ϕu-ϕb)+χ(1-ϕu-ϕb)(ϕu+ϕb)+κ2((ϕu+ϕb))2].

To ensure a constant diffusion coefficient in the dilute limits of the bleached and unbleached components, we employ the scaling ansatz for a ternary mixture, Γi=Γ0ϕi[(1-ϕtot)+(Γ¯0/Γ0)ϕj] and Γ¯=-Γ¯0ϕuϕb. In general, both mobility functions, Γ0 and Γ¯0, depend on the total volume fraction ϕtot. For the limiting case where bleached and unbleached molecules are identical particles, we can choose Γ0=Γ¯0. Applying the equilibrium FRAP condition (2) and using Equation (3a, b), we find that the concentration of unbleached components is governed by the following diffusion equation

(6) tcu=[D(ϕtot)(cu-cuϕtotϕtot)],

 with a ϕtot(𝒙)-dependent diffusivity, D(ϕtot)=kBTΓ0(ϕtot). As we show in Bo et al., 2021 a similar approach can be used to investigate single-molecule dynamics across phase boundaries.

Similar to Equation (1), the diffusion equation above is linear in cu. However, the dynamics of unbleached components are affected by gradients in ϕtot(𝒙) and components diffuse with different diffusion coefficients inside and outside the condensate, where in each phase ϕtot=0 (Figure 2b). The position-dependence of ϕtot(𝒙) is given by the equilibrium condition of a homogeneous chemical potential of the binary mixture prior to photobleaching, which implies tϕtot=0. For a radially symmetric system with r denoting the radial coordinate, ϕtot(r)=ϕouteq+(ϕineq-ϕouteq)H((r-R)/), where ϕineq and ϕouteq are the equilibrium volume fractions inside and outside, respectively, R is the droplet radius, and denotes the width of the interface. Moreover, H((r-R)/) is a function that decreases from one to zero at r=R on an interface width . For phase separation close to the critical point and large droplet sizes, H(x)=(1+tanh(x))/2 (Bray, 1994; Weber et al., 2019). We numerically solve Equation (6) using a finite element method (Anders et al., 2012) in a finite domain of size L which is much larger than the droplet radius R and fit the solution to experimental data.

In summary, our model has seven parameters. Four of these, namely, the equilibrium volume fractions ϕineq and ϕouteq, the interface width , and the droplet radius R, characterize the equilibrium profile prior to bleaching ϕtot(𝒙). The remaining parameters are the system size L and the diffusion coefficients inside and outside, which are given as

(7a) Din=kBTΓ0(ϕineq),
(7b) Dout=kBTΓ0(ϕouteq).

For the case of a single, spherical droplet with an infinitely thin interface (Weber et al., 2019), we can derive an effective droplet model for the unbleached component from Equation (6), where the dynamics of unbleached components inside and outside are given by diffusion equations that are coupled by boundary conditions (see Appendix 1 for the derivation):

(8a) tcu(r,t)=Din2cu,forr<R,
(8b) tcu(r,t)=Dout2cu,forr>R,
(8c) -Din𝒆rc|R-=-Dout𝒆rc|R+,
(8d) cu(r=R-,t)=Pcu(r=R+,t).

Here, R- and R+ denote the radial position directly inside and outside the droplet interface, respectively. Equation (8c) describes an equality of the fluxes directly inside and outside of the interface, respectively, and thereby expresses particle number conservation at the interface r=R. Equation (8d) describes a jump in concentration of unbleached components, which is determined by the thermodynamic partition coefficient

(9) P=ϕineqϕouteq.

Moreover, the flux vanishes at the origin r=0, 𝒆rc|r=0=0, and at the system boundary r=L, 𝒆rc|r=L=0.

Above we provide a thermodynamic derivation of Equations (8) in the limit of thin interfaces. These equations were already proposed as a model for FRAP dynamics of protein condensates (Taylor et al., 2019). Interestingly, while we obtained Equations (8) for the dynamics of a phase-separated protein component undergoing fluorescence recovery, similar equations were also used to investigate diffusion of a protein that was added to an already existing two-phase system at thermodynamic equilibrium (Münchow et al., 2008).

In Equations (8) Din, Dout, and P are considered to be independent parameters. Strictly speaking, due to phase separation, the diffusion coefficients Din and Dout are not independent which is evident in Equations (7). For example, in the absence of phase separation or at the critical point, ϕineq=ϕouteq (i.e. P=1), the diffusion coefficients inside and outside must be equal, Din=Dout. For a given condensate with fixed ϕineq and Din, there is a relationship between the diffusivity outside Dout and the partition coefficient P, which can be expressed using Equations (7) as

(10) Dout(P)=DinΓ0(ϕineq)Γ0(ϕineq/P).

However, except for the limit P1, Equation (10) does not impose further constraints for the determination of the parameters since the mobility function Γ0(ϕtot) is unknown. For large P, the missing knowledge of the mobility function renders Dout, Din and P as effectively independent parameters. This provides a theoretical justification for the assumption made by Taylor et al., 2019.

In the following, we use our theory (Equation (6)) to investigate the impact of the condensate environment on the FRAP dynamics. In particular, we consider how a passivated coverslip (no wetting of condensates) and nearby condensates affect the influx of unbleached molecules and thereby the recovery dynamics. Lastly, given the concentration at the droplet boundary, c(r=R,t), we derive a relationship between Dout and P. This can be used to investigate changes of the partition coefficient, for example when changing salt concentrations. Importantly, this method does not rely on absolute fluorescence intensities when measuring concentrations in the dense and dilute phases, which are prone to artefacts (McCall et al., 2020).

Impact of non-wetting coverslip on FRAP dynamics

Here, we investigate the influence of the coverslip surface on the FRAP dynamics of non-wetting spherical droplets. Under typical in vitro conditions, condensates sediment due to gravity, leading to sessile droplets on a coverslip. In many experimental setups, coverslips are passivated, for example pegylated, in order to suppress wetting of condensates on the coverslip surface (Alberti et al., 2018). These experimental conditions lead to almost spherical droplets since capillary effects are typically negligible for micrometer-sized polymer-rich or protein-rich droplets (Park et al., 2013; McCall et al., 2020).

We numerically solved Equation (6) for a spherical condensate on top of a no-flux boundary that mimics the coverslip surface (Figure 3a). We find that the recovery of the average volume fraction inside the condensate can slow down compared to the case without a coverslip (Figure 3b). This slow-down vanishes if droplets have a distance to the coverslip surface larger than a few droplet diameters. Moreover, we find that the slow-down is more pronounced for larger partition coefficients P. This trend can be explained by the size of the region in the dilute phase from where most unbleached molecules are recruited: if P is small, most of the unbleached molecules come from the immediate surroundings of the condensate. Hence the influence of radial asymmetry is minimal and the recovery appears almost unchanged compared to the case without coverslip. However, for large P, the condensate recruits unbleached molecules from distances far away, limiting the recruitment to an effective half-space compared with the case without coverslip. Indeed, for very large P recovery rates slow down maximally by a factor of two.

Impact of droplet environment on recovery dynamics.

(a) Sketch of a typical experimental set-up with a droplet above a passivated coverslip, where the droplet center has a distance h>R to the coverslip. (b) Recovery of average unbleached volume fraction ϕ¯u(t)=d3rϕu(r,t)3/(4πR3) for different heights h above the coverslip at different partition coefficients P. Results were obtained by solving Equation (6) using the finite element method and considering the geometries depicted in (a). For even larger h-values (e.g. no coverslip), results are approximately equal to the blue dashed line. (c) Using the method introduced in Figure 1 on the scenario with the largest influence of the coverslip (droplet sessile on coverslip) h=0.25μm in (a) results in an excellent fit and can reliably extract the input Din. (d) Sketch of neighboring droplets next to a bleached droplet. (e) Total recovery curves for finite element simulations of the geometry depicted in (d), for different distances between neighboring droplet centers, at different partition coefficients P. Note the strong dependence on the distance of neighboring droplets. For even larger d-values (e.g. no neighboring drops), results are approximately equal to the blue dashed line. (f) Same as (c) but for largest influence of neighboring droplets, that is d=0.5μm, where there is no distance between bleached droplet and neighboring droplets.

Interestingly, by extracting the boundary concentration in mid-plane, similar to the procedure in Figure 1, and spatially fitting the solutions of Equation (1a) to the ensuing recovery profiles, we can reliably recover the input diffusion coefficient Din (Figure 3c). The reason for this agreement is that by considering the intensity at the condensate interface, our method is independent of the time scale set by diffusion in the dilute phase.

Impact of neighboring condensates on FRAP dynamics

In this section, we address the impact of neighboring condensates on the recovery dynamics. We solved Equation (6) for a system composed of a bleached condensate with four unbleached neighboring condensates of the same size (Figure 3d) and find that neighboring condensates can significantly speed up the recovery dynamics (Figure 3e). This speed-up is only pronounced for rather close condensates with inter-droplet distance on the order of condensate size. Moreover, similar to the impact of the coverslip, the effects of the recovery dynamics are stronger for larger partition coefficients P. For small P, most unbleached molecules are recruited from the dilute phase leading to almost no effect on the recovery also when condensates are very close to each other (Figure 3e, left). For large partition coefficients P, however, a certain fraction of molecules are recruited from the neighboring condensates causing a significant speed-up of the recovery (Figure 3e, right).

Again, despite this change in total recovery due to close-by neighboring droplets, we can reliably measure Din via our spatial fitting method (Figure 3f). In particular, by extracting the boundary intensity in mid-plane and spatially fitting the solutions of Equation (1a), we find very good agreement with our input Din. This agreement shows that our method is robust for typical experimental systems that deviate from an ideal, isolated condensate.

Interfacial flux relates partition coefficient and outside diffusivity

We have shown that by using the time-dependent fluorescence at the interface of a spherical droplet we can accurately fit our dynamic Equation (1) to our experimental data and thus determine the diffusion constant inside the droplet, Din (Figure 1a). Our theory (see Equation 6) suggests that the fluorescence at the droplet interface is affected by the physical parameters characterizing the droplet environment such as the diffusion coefficient Dout and the partition coefficient P. In particular, the flux through the droplet interface is enlarged for increasing Dout or decreasing P (see Equation (8c) after rescaling the concentration close to the interface). Thus, for a condensate with concentration ϕineq and diffusion coefficient Din, a given flux between both phases through the interface implies a relationship between Dout and P.

Here, we determined the relationship between Dout and P by fitting numerical solutions of Equation (6) to the recovery dynamics inside the droplet (Figure 4b–d and Videos 1, 2 and 3). The diffusion coefficient inside, Din, was independently determined for each experiment via our method introduced in Figure 1a. This leaves P and Dout as independent parameters, which is valid for large P (see discussion after Equation (10)). We thus sampled P (with P1) across three orders of magnitude and obtained the best-fitting Dout for each P (Figure 4e,f). Notably, all the combinations of Dout and P represent relatively good fits, and each experimental condition leads to a unique relationship Dout(P). For large P, we find that the best Dout scales linearly with P (dashed lines in Figure 4e,f); for a discussion on the origin of this scaling, please refer to Appendix 2.

Varying partition coefficient P and diffusivity outside Dout simultaneously can lead to similar recovery kinetics.

(a) Schematics of the fitting approach used to extract the relationship between P and Dout. Although there is one specific pair (P, Dout) with the lowest cost (see Figure 5), there is a line in (P, Dout) space, which produces good fits for a range of values. (b) Spatial recovery of a single PLYS/ATP droplet (blue) with a fit to the full model (black). Note, data close to the droplet boundary cannot be fit, due to optical artefacts giving rise to an artificially broad interface (see Materials and methods and Figure 1a). (c) Same as (b) for a CMD/PLYS coacervate droplet. (d) Same as (b) for a PGL-3 droplet. (e) Given the partition coefficient P, Dout is found by fitting the coacervate data to the model. Note the convergence to a power law, DoutPn with n=1 for large partition coefficients (gray dashed line; for a discussion see Appendix 2). Shaded area around curves: standard deviation. Replicates: same as in Figure 1g. (f) Same as (e) but for PGL-3 with different salt concentrations. Note the order from top to bottom from highest to lowest salt concentration. (g) Based on (f), the change of partition coefficient P(csalt) can be estimated for a given Dout. Confidence intervals not shown for clarity. Similar to panel e. Replicates: same as in Figure 1f.

We would like to draw attention to the values of P, which are higher than expected (A.W. Fritsch and J.M. Iglesias-Artola, personal communication), for a realistic parameter range 50μm2s1<Dout<100μm2s1. One reason for this discrepancy is the assumption of spherical symmetry for the fitting routine leading to absolute values of P that are overestimated up to twofold (see caption of Figure 3 for a discussion). Additionally, our current experiments cannot exclude boundary effects, such as a recently hypothesised interfacial resistance (Taylor et al., 2019). Equation (6) should therefore be interpreted as a minimal model that fits the available data with high accuracy and explains the boundary dynamics, but cannot rule out additional effects.

Thus, the relationship Dout(P) has to be assessed critically. Since P and Dout, in conjunction with Din, set the time scale for recovery at the boundary, any effects that are ignored would change this relationship. Since the boundary time scale is well-described by DoutP, this change would likely manifest itself as a constant prefactor in each curve in Figure 4e,f. We thus only interpret the ratio between curves in Figure 4e,f, instead of their absolute values.

Considering this ratio allows us to investigate how the addition of salt affects the partition coefficient of PGL-3, assuming that boundary effects and Dout are independent of csalt. Figure 4f indicates that for all salt concentrations we have DoutP for Dout>1μm2s1. For proteins and coacervate components we can safely assume Dout>10μm2s1 in solution. Consequently, we chose Dout=10μm2s1 and calculated how the partition coefficient P changes with salt concentration csalt (Figure 4g). Specifically, for salt concentrations in the range from 50 mM to 180 mM, we find that the estimated partition coefficient P of PGL-3 droplets decreases approximately 10-fold. This trend is probably a result of enhanced screening of charged groups for increasing salt concentration.

Proposal how to determine partition coefficient and outside diffusivity

We next asked, whether we can obtain both parameters, P and Dout, at the same time, without measuring fluorescence intensities outside the droplet. Although each combination (Dout, P) along the lines specified in Figure 4e,f leads to a reasonable fit, we will now show that there is a distinct combination that exhibits a global minimum of the cost function for each Dout(P). Here, we assume no resistance at the interface as proposed in Münchow et al., 2008; Hahn et al., 2011; Hahn and Hardt, 2011. Providing experimental evidence of this global minimum can be hampered by effects due to the droplet environment, such as neighboring droplets or the coverslip surface (Figure 3a,d), inhomogeneous bleaching in the dilute phase, imaging artefacts at the phase boundary and effects at the droplet boundary not accounted for in Equation (6). In particular, in our experimental studies, interdroplet distances are sometimes on the order of the droplet size and diffusive exchange is affected by the coverslip. Thus, we decided to use our model to create in silico data and provide evidence for the existence of a distinct combination of Dout and P for a fixed Din. Fixing Din mimics the approach of initially determining Din, as outlined in Figure 1. To determine the relationship Dout(P), we proceed as described for Figure 4. Figure 5a depicts the Dout(P) relationships corresponding to four parameter combinations in a range relevant for protein condensates and coacervate droplets (Drobot et al., 2018; Riback et al., 2020). In particular, we choose two outside diffusivities Dout of 0.1μm2s1 and 1μm2s1 and two partition coefficients, P=5 and P=150. We find indeed that each cost function exhibits a unique minimum for each of the considered parameter combinations (Figure 5b,c). These findings indicate that all three parameters, Din, Dout and P, can in theory be determined by a single FRAP time course of the droplet intensity inside. Thus, in principle, there is no need to measure kinetic properties of the dilute phase to fully characterise the system in terms of its parameters. This possibility represents a new approach to characterize the partition coefficient P, which is particularly important in light of recent data showing that measurements based on fluorescence intensity can lead to drastic underestimation of P (McCall et al., 2020).

Given the recovery dynamics inside a condensate, the key parameters D𝐢𝐧, D𝐨𝐮𝐭 and P can in theory be determined uniquely without measuring the outside dynamics, using Equation (6).

(a) Given the partition coefficient P, Dout is found by fitting the model to synthetically generated example data. As example systems, indicated by open circles, we consider in silico data, obtained by solving Equation (6) with known parameters P and Dout. To mimic the approach of initially determining Din (see Figure 1) we keep Din=0.01μm2s1 for all in silico datasets. For each example system (open circles), the best fitting Dout is found given a range of P (solid lines). Parameters used were as follows: pink circle: Dout=0.1μm2s1, P=5, orange circle: Dout=1μm2s1, P=5, green circle: Dout=0.1μm2s1, P=150, blue circle: Dout=1μm2s1, P=150. (b) Cost function (colorbar, log-scale) as a function of Dout and P. We note that the global minimum coincides with the parameters used to generate the synthetic data (green circle). The valley in parameter space (dashed line) corresponds to the green line in (a). (c) Minimum of cost function for each P, corresponding to curves shown in (a). This minimum corresponds to the valley indicated by the dashed line in (b). Note the minimum at the input parameter set, which indicates uniqueness of the outside dynamics for given values of Dout and P.

Discussion

The dynamic redistribution of fluorescent molecules has been used to characterize liquid phase separation in biology via a variety of techniques, including SPT, FRAP, and FCS (Elbaum-Garfinkle et al., 2015; Taylor et al., 2019; Moon et al., 2019). Here, we have derived a theory that describes the diffusive motion of labeled molecules based on the physics of phase separation. It can be applied to many state-of-the art fluorescent methods such as FCS, SPT, and FRAP and can thus help extend traditional techniques to the realms of phase separation (Ries and Schwille, 2012). Importantly, this theory enables us to avoid commonly applied approximations such as the frequently used single-exponential recovery (Brangwynne et al., 2009; Frottin et al., 2019; Kaur et al., 2019; Fisher and Elbaum-Garfinkle, 2020; Kistler et al., 2018).

Our theory, essentially represented by Equation (6), governs the dynamics of labeled molecules through interfaces of condensates. As we show in our work, it can be applied to spherical condensates. In addition, our theory could also be used for non-spherical condensates and arbitrary bleach geometries, since spherical symmetry is only assumed for ease of data analysis. We were able to quantify the impact of neighboring droplets and the coverslip on the recovery dynamics. We found that neighboring droplets caused an appreciable speed-up in overall recovery, while emulating a coverslip caused a weak slow-down. In order to experimentally verify our theory, we have used three in vitro droplet systems, two composed of charged synthetic polymers and one with a purified protein component. There is remarkable quantitative agreement between our theory and the diffusion dynamics observed inside such droplets. This agreement shows that proteins and charged, synthetic polymers can form droplets that follow simple diffusive dynamics. Crucially, we use the full spatio-temporal data for fitting and can thus distinguish the timescale set by intra-droplet diffusion from the timescales at play in the dilute phase. We extract the intensity directly at the inside of the droplet interface and fit a spatially resolved diffusion equation to the ensuing recovery. We use the boundary intensity as a dynamic boundary condition and the experimentally measured profile as initial condition. Within the statistical fluctuations, the numerical solutions and the experimental data are not distinguishable (see Figure 1b–d). Throughout the time course, we find excellent agreement with the data and have thus found a method with minimal approximations that can precisely measure the inside diffusion coefficient Din.

Building on the analysis inside the droplet, we show that there is a relationship between partition coefficient P and the diffusion coefficient in the dilute phase, Dout. Data obtained from FRAP experiments define a line in (Dout, P) space, along which a range of parameter sets can reliably account for the boundary dynamics. This relationship allowed us to characterize changes in P upon salt addition, opening an alternative avenue for characterizing P without relying on fluorescence intensities. This is particularly important in light of recent results obtained by quantitative phase microscopy (QPM). These results show that measuring partition coefficients based on fluorescence intensity can lead to strong underestimation of P (McCall et al., 2020). While it is tempting to interpret our values for P as actual partition coefficients, we would like to stress that these values were obtained from a physical model, which does not consider additional effects at the condensate interface such as a potential interfacial resistance. This effect has recently been hypothesized to solve a contradiction in time scales between FRAP and FCS experiments (Taylor et al., 2019). However, introducing an interfacial resistance significantly lowered the fit quality in Taylor et al., 2019, an issue that has yet to be resolved. In this context, it will be interesting to reexamine work for three-component systems (Binks and Lumsdon, 2000; Lin et al., 2003). Some evidence for an interfacial resistance across a PEG/Dextran interface has been found for some types of molecules, for example DNA, bovine serum albumin, and bovine γ-globulin (Hahn and Hardt, 2011; Gebhard et al., 2021). Measuring Dout via FCS, similar to Taylor et al., 2019, and partition coefficients via QPM (McCall et al., 2020), will allow further characterization of the existence of interfacial effects.

Our approach can be readily extended to multi-component systems with an arbitrary number of components, which is particularly useful in vivo. This would hardly be possible for techniques that do not use labeled components, such as QPM or other scattering methods. Of particular interest are multi-component systems with chemical reactions away from equilibrium. Our approach can then be used to determine the diffusion coefficients and concentration levels of reactants, and thereby provide insights into reaction kinetics. Interestingly, introducing the bleached molecules via a ternary mixture also enabled us to derive the Langevin equation governing single-molecule motion in phase-separated media, thus providing a link to SPT (Bo et al., 2021). Approaches for single labeled molecules are relevant since high labeling fractions were shown to alter the viscosity and thus kinetics in dense protein phases (compare viscosity for PGL-3 in Jawerth et al., 2018 with viscosity for tagged PGL-3:GFP Jawerth et al., 2020 ). Finally, by accurately measuring Dout, our technique can also be employed to characterize rheological properties of condensates such as the recently reported glass-like dynamics of protein droplets (Jawerth et al., 2020).

Materials and methods

Coacervate assay

General reagents

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Carboxymethyl-dextran sodium salt (CM-Dex, (C6H10O5)n.(COOH), 10–20 kDa, monomer MW = 191.3g/mol), Poly-L-lysine hydrobromide (PLys, (C6H12N2O)n, 4–15 kDa, monomer MW = 208.1g/mol) and adenosine 5’-triphosphate disodium salt hydrate (ATP, C10H14N5Na2O13P3, MW = 551.1g/mol) were purchased from Sigma Aldrich. FITC-PLys ((C6H12N2O)n.(C21H11NO5S), 25 000g/mol) was purchased from Nanocs, NewYork, USA . Milli Q water was used to prepare aqueous stocks of CM-Dex (1000 mM, pH 8), PLys (200 mM, pH 8) and ATP (100 mM, pH 8). All solutions were stored in the freezer at -20 until use and the pH of all stocks was adjusted using a stock solution of 1M NaOH.

Coacervate preparation

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Stock solutions of CM-Dex, PLys and ATP were first diluted to 25 mM and the PLys solution doped with 1% v/v PLys-FITC. Diluted solutions of CM-Dex/PLys or PLys/ATP were then mixed together at a 4:1 vol ratio (16 μl), resulting in the formation of turbid coacervate solutions. Solutions were left to equilibrate for at least 5 min before imaging, up to a maximum of 15 min when larger droplets were desired.

PGL-3 droplets

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PGL-3 was purified and stored as previously described (Saha et al., 2016). To obtain droplets, 300 mM KCl stock protein solution was diluted to the desired concentration, achieving final salt concentrations of 50–180 mM. A small imaging volume was created by using polystyrene beads, resulting in complete droplet sedimentation after less than five minutes. Droplets were imaged immediately to avoid changes in material properties due to ageing (Jawerth et al., 2020).

Microscopy and FRAP

Confocal imaging

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Droplets were imaged at midplane by visually defining the focal position with the largest droplet area of the droplet of interest. Images were acquired on an Andor spinning disk confocal microscope equipped with an Andor IX 81 inverted stand, a FRAPPA unit, an Andor iXON 897 EMCCD camera, and a 488 nm laser, using a 60x/1.2 U Plan SApo OLYMPUS water objective. Imaging conditions were optimized for minimal bleaching at the required frame rate. Frame rates were optimised for each system: PGL-3, 0.1s<Δt<5s, CMD/PLYS, Δt=0.03s, PLYS/ATP, Δt=0.07s.

Frap

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Droplets were bleached in their entirety by using the minimal FRAP ROI that encompasses the entire droplet. FRAP was performed in three focal planes, equally spaced across the droplet in z-direction, to reduce non-uniform bleaching of the droplet. FRAP rates and dwell times were chosen such that left-over fluorescence intensity above background was smaller than 1% for PGL-3 and smaller than 15% for coacervate droplets to maximize bleaching within the droplet while keeping bleaching impact on the droplet environment minimal.

Data analysis

Azimuthal averaging and normalization

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Time-lapse images were cropped with the droplet of interest in the center. An azimuthal average was performed around the center of the droplet to obtain a 1D profile along the radial coordinate with minimal loss of data, using the radialavg function provided by David J. Fischer on Matlab File Exchange (Fischer, 2016). Camera background was subtracted uniformly from the resulting 1D profiles. The radial intensity profile at the prebleach stage was used for normalization and to correct for optical artefacts that lead to increased fluorescence at the droplet center compared to the droplet-bulk interface. Data close to the droplet interface cannot be used for fitting, since the droplet has an artificially broad boundary due to the point-spread function and likely due to curvature effects. Therefore, on average, the intensity of the ten pixels closest to the boundary were not used for analysis. The droplet boundary was defined as the inflection point of the azimuthally averaged profile in the pre-bleach frame.

Immediately after bleaching, a uniform recovery across the entire droplet can be seen, which cannot be spatially resolved even at frame rates <30ms. This recovery is fast compared to the recovery by diffusion from the outside for all systems under investigation. We thus chose to not account for this uniform recovery in our model and instead start the fitting after a time lag that depends on the system and droplet size. This offset typically consists of less than 5% of the total pre-bleach intensity. Additionally, bleaching is not complete, resulting in an additional offset above the camera background even immediately after bleaching (see gray lines in Figure 1).

Photo-bleaching due to continuous imaging was minimal in all droplet types. We thus chose to not account for imaging-induced photo-bleaching, in order to not introduce additional noise due to necessarily occurring fluctuations within the bleach correction.

Extracting experimental boundary conditions

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cu(r=R-,t) was extracted from the intensity profiles as the value at the outermost pixel. In order to speed up fitting and avoid jumps in cu(r=R-,t), the extracted intensity values were sorted to eliminate small fluctuations.

Fitting of Din by using experimentally measured boundary conditions (Figure 1)

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The resulting spatio-temporal profiles were fit as described in the main text, using Din as a single global fit parameter and using cu(r=R-,t) as described above as the system’s time-dependent boundary condition. Fits were performed in MATLAB (Mathworks), using pdepe to solve the PDE and fminsearch for minimizing the squared distance between data and model. Code is available at https://gitlab.pks.mpg.de/mesoscopic-physics-of-life/DropletFRAP (Hubatsch, 2021a; copy archived at swh:1:rev:7e5b59fff3c634cfce5d0f99a86c807635a090fd).

Numerical solution of Equation (6)

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Equation 6 was solved using either pdepe (MATLAB (Mathworks), Figures 4 and 5, for spherically symmetric systems) or by using the finite element method via the FENICS environment (Anders et al., 2012) for arbitrary 3D geometries (Figure 3). All fits in Figures 4 and 5 were performed using fminsearch based on a squared-difference metric. Code is available at https://gitlab.pks.mpg.de/mesoscopic-physics-of-life/frap_theory (Hubatsch, 2021b; copy archived at swh:1:rev:2c4a972a380df7f9e86ddbbf0ae921443ce0800f).

Appendix 1

Limit of narrow interfaces

Here, we derive the effective droplet model for our dynamic equation

(11) tcu=[D(ϕtot)(cu-cuϕtotϕtot)]
(12) =-𝒋u

by considering the limit of narrow interfaces. In the equation above, 𝒋u denotes the flux of unbleached molecules. Conservation of molecules at the interface implies

(13) limϵ𝒋u(R-)=limϵ𝒋u(R+),

where is the characteristic size of the interface. Moreover, ϵ>0 is a small but non-zero parameter by which we define the position directly left and right of the interface. In this limit, ϕtot|R±ϵ=0 directly left and right of the interface. Thus, we obtain Equation (8c) for narrow interfaces. However, the contribution D(ϕtot)cuϕtot/ϕtot in Equation (11) that determines the dynamics through the interface implies a Dirichlet boundary condition for the concentration of unbleached molecules at the interface r=R. Since ϕtot/ϕtot, we demand

(14) lim0(cu|R-ϕtotϕtot|R-)=lim0(cu|R+ϕtotϕtot|R+)

 in the limit of decreasing interface width . This condition ensures that the dynamics at the interface remain unchanged for decreasing . Parameterizing the interface for example by ϕtot(r)=ϕouteq[1+(P-1)(1+tanh(r/))/2], we find that Equation (14) leads to Equation (8d).

Appendix 2

Solution of effective droplet model for FRAP

In this appendix, we derive the solution for the two diffusion equations coupled at the interface for a spherically symmetric case; see Equations (8). We consider an initial condition with a constant concentration c(r,0)=c0 outside a droplet with radius R and bleaching gives rise to c(r,0)=0 inside the droplet. Performing a Laplace transformation, c^(r,s)=0𝑑te-stc(r,t), of Equations (8), we find

(15a) sc^u(r,s)=Din2c^u(r,s),forr<R,
(15b) sc^u(r,s)=Dout2c^u(r,s)+c0,forrR.

where s is the rate parameter of the Laplace transform. The corresponding solutions read

(16) c^u(r,s)={arsinh(ξinr), for r<R,brexp(-ξoutr)+c0s, for rR,

where ξin/out=s/Din/out. We have selected the solutions with no radial flux at the origin and a finite concentration at infinity. The remaining unknown constants a and b are determined by the conditions at the interface stated in Equations (8). Inside the droplet r<R, the solution reads

(17) c^u(r,s)=c0PR(Dout+RDouts)rs(Dout-PDin+RDouts+PRDinscoth(RsDin))sinh(rsDin)sinh(RsDin).

Even though it is non-trivial to perform the inverse Laplace transformation of this expression we can obtain some analytical understanding of the behavior at short and long timescales by considering the asymptotics of large and small s. Expanding Equation (17) for small s, we obtain:

(18) c^u(r,s)=c0P[1s-(2PDinDout+1)R2-r26Din]+O(s1/2).

This relationship shows that for small s, the rescaled concentration c^u(r,s)/P is affected by Dout only via the PDin/Dout combination. Conversely, for large s, the leading contribution features the term

(19) c^u(r,s)c0PRrs(1+PDinDout)-1e(r-R)sDin

showing that for short times the system is influenced by Dout via the ratio PDin/Dout. Note that for these short times the solution is well approximated by that of a one-dimensional system. These long and short time behaviors are confirmed by the numerical solutions plotted in Appendix 2—figure 1. C, which show that the evolution of the rescaled concentrations match at short times, if PDin/Dout is kept constant but start to deviate for longer times. Conversely, at longer times the dynamics are invariant if PDin/Dout is kept constant except for a shift due to a short timescale transient.

Appendix 2—figure 1
Solution to dynamics inside droplets is unique, but early and late time behavior can be approximated by different scaling regimes.

(Top) Early time behavior of numerical solutions. If a parameter set is chosen (’Reference’), at early times this can be approximately recovered by scaling DoutP2 (orange). A linear scaling does not recover the same inside dynamics at early times (green). (Bottom) Late time behavior of numerical solutions. At late times, the reference dynamics can be approximately recovered by scaling DoutP (green). Meanwhile, DoutP2 does not recover the reference inside dynamics (orange).

Data availability

Code for modelling and data analysis is available at https://gitlab.pks.mpg.de/mesoscopic-physics-of-life/frap_theory and https://gitlab.pks.mpg.de/mesoscopic-physics-of-life/DropletFRAP (copies archived at https://archive.softwareheritage.org/swh:1:rev:2c4a972a380df7f9e86ddbbf0ae921443ce0800f and https://archive.softwareheritage.org/swh:1:rev:7e5b59fff3c634cfce5d0f99a86c807635a090fd, respectively).

References

    1. Ries J
    2. Schwille P
    (2012) Fluorescence correlation spectroscopy
    BioEssays : News and Reviews in Molecular, Cellular and Developmental Biology 34:361–368.
    https://doi.org/10.1002/bies.201100111

Decision letter

  1. Rohit V Pappu
    Reviewing Editor; Washington University in St Louis, United States
  2. José D Faraldo-Gómez
    Senior Editor; National Heart, Lung and Blood Institute, National Institutes of Health, United States
  3. Rohit V Pappu
    Reviewer; Washington University in St Louis, United States

Our editorial process produces two outputs: (i) public reviews designed to be posted alongside the preprint for the benefit of readers; (ii) feedback on the manuscript for the authors, including requests for revisions, shown below. We also include an acceptance summary that explains what the editors found interesting or important about the work.

Acceptance summary:

This work is timely and relevant as the field grapples with the issue of diffusive dynamics across phase boundaries. The numerical formalism in this work will be of broad interest to the condensate field.

Decision letter after peer review:

Thank you for submitting your article "Quantitative Theory for the Diffusive Dynamics of Liquid Condensates" for consideration by eLife. Your article has been reviewed by 2 peer reviewers, including Rohit Pappu as the Reviewing Editor and Reviewer #1, and the evaluation has been overseen by José Faraldo-Gómez as the Senior Editor.

The reviewers have discussed their reviews with one another, and the Reviewing Editor has drafted this to help you prepare a revised submission.

Both reviewers have arrived at similar conclusions. FRAP is a method that is routinely used to study internal dynamics of molecules within biomolecular condensates. Of course, FRAP was introduced in a completely different context, and as has been shown before, its adoption needs appropriate adaptation to the context of interest. How should FRAP data be analyzed in the context of studying biomolecular condensates that form via phase separation. The authors build on the work of Hardt and coworkers, and demonstrate the incorporation of a Flory-Huggins free energy alongside diffusion equations to describe the dynamics of unbleached molecules, using features of mass balance. They show that numerical solutions of the derived equations – see Equation (6) – can be used to fit FRAP data for different systems. The authors also suggest that inferences from FRAP data can go beyond extraction of individual parameters. In other words, FRAP data seem to be more information-rich than originally thought. The current version, although very interesting, features opacities that should be remediable by following the recommendations made by both reviewers.

Essential revisions:

1) Two distinct flowcharts summarizing how Equations (1) and (6) are used in the fitting of FRAP data will be essential.

2) Accounting for interfacial tension and / or interfacial resistance (see Taylor et al.) requires discussion.

3) Please delete references to non-equilibrium situations since the model imposes detailed balance throughout. In fact, this point should be made clear.

4) There is considerable confusion regarding the claims regarding being able to extract Din, Dout, and P from single sets of FRAP data and the actual demonstration of this versatility. This is accentuated by considerable confusion caused, for both reviewers, by the introduction of the cost function, which was opaque, and the sweep of parameters for Dout and P that clearly give satisfactory fits to the FRAP data. At this juncture, the claim of being able to extract more insights from FRAP data than one is accustomed to seeing has not been unequivocally demonstrated.

5) Several scholarly issues, specifically pertaining to the work of Hardt and colleagues, and the semantics of what constitutes a phenomenological vs. physical theory description were raised. These should be addressed and there is a strong desire to see a toning down of what were perceived as over-claims.

Reviewer #1 (Recommendations for the authors):

As it currently stands, the average practitioner of FRAP is likely to find the narrative to be rather opaque. Two flowcharts, that summarize (a) how the dynamic boundary condition and its application lead to the internal diffusion coefficients and (b) the use of equation (6) for experimentalists in their analysis of FRAP data need to be added. In doing so, it is really important to explain when and how the analysis can be used, and when and where it cannot be used.

Other recommendations:

1. phi is a conserved order parameter, because the underlying theory imposes a closed system. Therefore, the relations between phitot and phiu etc. come not from incompressibility, but from mass balance in a closed system.

2. The work of Steffen Hardt, mentioned in passing, and labeled as being phenomenological needs rectification. Elaboration of their work and a clear, scholarly contrast between the current effort, the published work of Hardt that goes beyond the 2008 Langmuir paper, and the differences between the work of Taylor et al., would be helpful for the reader who is interested in understanding what is new, and what is different. In this context, please also see Lin et al., (2003) Science 299: 226, Binks and Lumsdon (2000) Langmuir 16: 8622.

3. The authors classify previous efforts as being phenomenological and the current effort as a physical theory. Respectfully, I disagree with this characterization. Inasmuch as previous efforts largely use the same equations as being used by the authors, with the key difference being the use of the FH functional for extracting the chemical potentials, the extant literature and the current manuscript are both phenomenological or both physical theories. It would be best to drop these prefaces and simply state what has been done. Specifically, the work of Hardt and coworkers does account for phase boundaries. They don't use the FH free energy functional, and this is really the only main difference. And many would rightly argue that the FH functional is phenomenological. So, please do not use terms like "first principles" for this work, and phenomenological to inadvertently dismiss other efforts.

4. Throughout the manuscript, there is a lot of self-congratulation about the excellent fits, the striking agreements etc. This is jarring. It is impossible to know if one should be impressed with the agreements or if such agreements are readily achieved. An unvarnished statement of the facts is sufficient.

5. The description of the cost function analysis needs a lot of work. In effect, the implication is that Dout depends on P. Whether this is an implicit function or not is unclear. The cost function has not been defined in the main text, and why there should be a minimum is unclear. There is also a curious leap that happens: at one point, we see that Dout and P that describe the data can span a wide range. This would lead to inference that the FRAP data cannot pin down these parameters absent independent measurements. However, the conclusions paint a more rosy picture, suggesting that knowledge of Din is sufficient, and that fitting the FRAP data will lead to reliable estimates of Dout and P. The section that describes this leap needs a lot of work. The flowcharts recommended in the public review will help.

6) There are important caveats that apply as well. The slow or non-recovery of fluorescence, becomes a scenario wherein Din will be very small or immeasurable. Alternatively, if the material state has changed, then the usage of the dynamic boundary condition might still yield reliable measurements of Din, due to an anisotropic stress tensor, but this would be an erroneous estimate of Din on the inside. Therefore, it is important to lay out a set of requirements for the applicability of the proposed approach, i.e., under what circumstances can the method be used, and under what circumstances can it not be used.

7) As a logical follow up to the preceding point, how might one falsify the theory? Fitting FRAP data is inadequate for such an effort. This will be very useful for experimentalists and computational scientists alike.

8) Finally, do the authors know that the salt concentrations across the phase boundary are equal to one another or is this implicitly assumed? This is particularly relevant for the synthetic coacervates. Sing and coworkers have proposed that there are differences in salt concentrations (even ion concentrations) across phase boundaries. This is relevant because it requires the addition of a Donnan potential to the equations that describe molecular transport. A discussion of this issue would be helpful.

Reviewer #2 (Recommendations for the authors):

Concerns on figures and writing:

1. Recurrent lack of clarity and/or consistency in how figures are presented

When dissecting the data shown in figures, we often found details in presentation that detracted from our understanding of the study. Notable, but not exhaustive, examples are listed:

a. The retrieval of physical parameters in this study frequently alternates between the use of Equation 1 to track FRAP data, and Equation 6 to model droplet dynamics. The authors do not clearly distinguish the purpose of these equations in the theoretical sections, leaving the reader to understand by context.

b. The data in Figure 3 would benefit from control data sets that entirely lack the coverslip and/or neighbouring droplets for comparison. In particular, showing the retrieval of identical Din across all simulations (including no coverslip nor neighboring droplets condition) will convey the major point of the paper very clearly.

c. In the caption of Figure 5 it is unclear why the authors have denoted the four indicated values of Dout and P as "reference systems' as opposed to example parameter sets.

We recommend the authors reassess their figures for clarity of the information that is meant to be communicated.

2. The theory sections lack needed elaboration in some areas.

a. The authors conclude starting on line 153 that P, Din, and Dout can be treated as independent of each other for sufficiently large P. Their justification is that the unknown mobility functions do not impose constraint on Dout(P) other than shown in Equation 10. While the data demonstrates that this is a valid assumption, we find this justification to be opaque and would like to see further elaboration on how independent P, Din, and Dout follows from the mobility functions being unknown.

Suggestions to the authors for bolstering the overall strength of this study below:

1. When interpreting the Results section, we often found that the nature of the experiment was unclear. This is particularly true for Figure 4 and 5. We recommend specifying the procedure followed to acquire the data more clearly and explicitly.

2. In Figure 1c, the label "Dynamic BC" made this figure confusing to interpret, especially because there is a same-color arrow depicting time progression. The authors should consider some other way of noting that dynamic BC is applied at the max r (=R-). Also, for the line going across the earliest data points, "initial condition" rather than the "fit" label may be more appropriate.

3. If available, showing corrected viscosity data from Jawerth et al., 2020 rather than Jawerth et al., 2018 in Figure 1d would bolster this figure via internal consistency with the text in line 98. We would like to be able to back-calculate consistent viscosities using your diffusion data and the Stokes-Sutherland-Einstein relationship.

4. Since the time progression of FRAP recovery is illustrated in Figure 2c, we believe the time point label in Figure 2 to be unnecessary and possibly confusing.

5. We find that the citation of privately communicated and unseen data in line 234 does not add to the preceding statement. Optimum salt concentration is a very believable observation.

6. There appears to be a typo in the subscripts on line 138. Both read "in".

7. Regarding the concluding statement on line 306: We do not find that Jawerth et al., 2018,2020 contain discussions of altering dense phase kinetics by high labeling fraction. Rather, McCall et al., 2020 describes effects closer to this.

8. Regarding Figure 5a, labelling the "ratio" of Dout and P can be confusing. We suggest noting the specific (Dout, P) pairs to specify the points. Also, a clear distinction between "simulation generating parameters (points)" and the "Dout and P dependence (lines) obtained from simulation result and equation 6" is needed.

9. Please consider labelling the point in Figure 5b as "simulation input" or "simulation parameter" rather than "reference simulation".

10. We suggest the following change in line 230 for clarity: "Specifically, for salt concentrations in the range from 50 mM to 180 mM, we find that the estimated partition coefficient P of PGL-3 droplets decreases more than 10-fold."

11. The significance of the shading in Figure 4d is unclear and inconsistent with how the analogous data in Figure 4e is presented.

12. The text accompanying Figure 5 beginning on line 249 describes the range of Dout and P used as "relevant for protein condensates and coacervate droplets' without citation. We recommend backing up the validity of the range of Dout and P used here with evidence from literature.

13. In Figure 1b, individual FRAP recovery curves cut off at seemingly arbitrary points and are difficult to distinguish based on provided color coding. We also note that the curve of 100mM salt concentration does not seem to fit the same trend as the other data sets, but is not discussed by the authors.

14. Figure 1f shows the precision of Din determination for the two coacervate systems, but not the protein condensate system, and a salt concentration for comparison with Figure 1d is not provided.

https://doi.org/10.7554/eLife.68620.sa1

Author response

Essential revisions:

1) Two distinct flowcharts summarizing how Equations (1) and (6) are used in the fitting of FRAP data will be essential.

We are thankful for this suggestion and have added the proposed flow charts; see Figure 1 and Figure 4 in revised manuscript.

2) Accounting for interfacial tension and / or interfacial resistance (see Taylor et al.) requires discussion.

We now discuss our results in the light of a potential interfacial resistance at the phase boundary. We have also now made explicitly clear in several key parts of the paper that we assume no interfacial resistance. For a more detailed discussion, see also replies below. Interfacial tension is related to the interfacial width which is accounted for in our model; see \ell in the paragraph below Equation (6).

3) Please delete references to non-equilibrium situations since the model imposes detailed balance throughout. In fact, this point should be made clear.

We agree that in our manuscript we focus on the case where the total volume fraction (composed of bleached and unbleached molecules) is at thermodynamic equilibrium leading to partialt phitot = 0. Please note that our theory can be applied to non-equilibrium situations, i.e., in the presence of fluxes. For example, see e.g. Equation (6) in Bo et al., (reference in manuscript), where we use this coarse-grained theory to derive a single molecule description also away from thermodynamic equilibrium.

Based on the reviewer’s remark, we have revised the paragraph around Equation (2) and now stress more clearly that we focus on the cases where detailed balance holds.

4) There is considerable confusion regarding the claims regarding being able to extract Din, Dout, and P from single sets of FRAP data and the actual demonstration of this versatility. This is accentuated by considerable confusion caused, for both reviewers, by the introduction of the cost function, which was opaque, and the sweep of parameters for Dout and P that clearly give satisfactory fits to the FRAP data. At this juncture, the claim of being able to extract more insights from FRAP data than one is accustomed to seeing has not been unequivocally demonstrated.

We have now made more explicit that this refers to a theoretical possibility of extracting all three parameters, by introducing a new section heading and adding descriptive text. Also, the flow charts should help, in particular regarding the cost function. We have reevaluated our conclusion regarding being able to extract P if Dout is known and vice versa. While we still think this is a promising avenue, we agree that the current data are better interpreted more carefully. We now conclude that we find a unique relationship between P and Dout for every condition, which allows us to extract relative changes in P for different salt conditions. This was not possible previously.

5) Several scholarly issues, specifically pertaining to the work of Hardt and colleagues, and the semantics of what constitutes a phenomenological vs. physical theory description were raised. These should be addressed and there is a strong desire to see a toning down of what were perceived as over-claims.

We have carefully revised and extended the discussion of the previous literature, such as the works of the Hardt group and Taylor et al. Moreover, we have removed statements such as “phenomenological” and similar formulations.

Reviewer #1 (Recommendations for the authors):

As it currently stands, the average practitioner of FRAP is likely to find the narrative to be rather opaque. Two flowcharts, that summarize (a) how the dynamic boundary condition and its application lead to the internal diffusion coefficients and (b) the use of equation (6) for experimentalists in their analysis of FRAP data need to be added. In doing so, it is really important to explain when and how the analysis can be used, and when and where it cannot be used.

We have added the flow charts according to the reviewer’s suggestion. Moreover, we say explicitly when our analysis can be used and when not (see text below Equation 1).

Other recommendations:

1. phi is a conserved order parameter, because the underlying theory imposes a closed system. Therefore, the relations between phitot and phiu etc. come not from incompressibility, but from mass balance in a closed system.

Our system conserves particle number of each component after photobleaching. Since we consider an incompressible system, the volume fractions of bleached and unbleached molecules add up to the total volume fraction at each position (Equation (2) in manuscript). Please note that the latter is also true for open systems that are incompressible.

2. The work of Steffen Hardt, mentioned in passing, and labeled as being phenomenological needs rectification. Elaboration of their work and a clear, scholarly contrast between the current effort, the published work of Hardt that goes beyond the 2008 Langmuir paper, and the differences between the work of Taylor et al., would be helpful for the reader who is interested in understanding what is new, and what is different. In this context, please also see Lin et al., (2003) Science 299: 226, Binks and Lumsdon (2000) Langmuir 16: 8622.

We are grateful for the additional references and now discuss the work of Hardt and colleagues more thoroughly, covering additional references. Note, both the works by Hardt and colleagues as well as Lin/Science and Binks/Langmuir, assume a third species that adsorbs to the interface. However, we consider a binary mixture of protein and solvent and currently have no evidence that the protein component exists in several states. While we also find the possibility of an interfacial resistance compelling, we didn’t find any direct evidence of it.

We cite Taylor et al., on several occasions in the manuscript, where we felt it was appropriate. They provide a very nice overview of currently used heuristic fit functions and their (severe) limitations. They also give an effective droplet model (referenced), without derivation. However, as discussed in their work, their fitting yields unrealistic results, and even after introducing an additional fit parameter (interfacial resistance), the fits get worse, instead of better. This is curious, given that the interfacial tension model is a superclass of the model without interfacial tension and should thus fit at least as well, if not better than the original model. In our method to determine Din (Figure 1 and Equation (1)), we completely circumvent any boundary effects, can measure Din reliably by fitting the spatial profiles, and thus provide a significant advance. Figures 3-5 have no overlap with any of the references.

3. The authors classify previous efforts as being phenomenological and the current effort as a physical theory. Respectfully, I disagree with this characterization. Inasmuch as previous efforts largely use the same equations as being used by the authors, with the key difference being the use of the FH functional for extracting the chemical potentials, the extant literature and the current manuscript are both phenomenological or both physical theories. It would be best to drop these prefaces and simply state what has been done. Specifically, the work of Hardt and coworkers does account for phase boundaries. They don't use the FH free energy functional, and this is really the only main difference. And many would rightly argue that the FH functional is phenomenological. So, please do not use terms like "first principles" for this work, and phenomenological to inadvertently dismiss other efforts.

We have now dropped all occurrences of ‘phenomenological’ when referring to the work of others.

4. Throughout the manuscript, there is a lot of self-congratulation about the excellent fits, the striking agreements etc. This is jarring. It is impossible to know if one should be impressed with the agreements or if such agreements are readily achieved. An unvarnished statement of the facts is sufficient.

We have toned down our language. The fits however, are “excellent” in our eyes, since we get almost complete agreement with the data within the statistical fluctuations due to radial averaging. From the fits in Figures 1 b-d, it is clear that there are barely any systematic deviations between model and data. Almost all deviations stem from statistical fluctuations of neighbouring pixels. These get stronger towards r=0, since fewer pixels can be used for radial averaging. In this context, please also note that even integrated intensity curves with more parameters are less well described by existing models (cf. Taylor et al.,).

5. The description of the cost function analysis needs a lot of work. In effect, the implication is that Dout depends on P. Whether this is an implicit function or not is unclear. The cost function has not been defined in the main text, and why there should be a minimum is unclear. There is also a curious leap that happens: at one point, we see that Dout and P that describe the data can span a wide range. This would lead to inference that the FRAP data cannot pin down these parameters absent independent measurements. However, the conclusions paint a more rosy picture, suggesting that knowledge of Din is sufficient, and that fitting the FRAP data will lead to reliable estimates of Dout and P. The section that describes this leap needs a lot of work. The flowcharts recommended in the public review will help.

We have now added the requested flow charts and believe they will be very helpful. The distinction between what is currently feasible based on experiments (namely only getting an approximate relationship between Dout and P) and what is theoretically possible with additional assumptions and perfect data has now been made clearer by introducing an extra section heading and additional descriptive text.

6) There are important caveats that apply as well. The slow or non-recovery of fluorescence, becomes a scenario wherein Din will be very small or immeasurable. Alternatively, if the material state has changed, then the usage of the dynamic boundary condition might still yield reliable measurements of Din, due to an anisotropic stress tensor, but this would be an erroneous estimate of Din on the inside. Therefore, it is important to lay out a set of requirements for the applicability of the proposed approach, i.e., under what circumstances can the method be used, and under what circumstances can it not be used.

7) As a logical follow up to the preceding point, how might one falsify the theory? Fitting FRAP data is inadequate for such an effort. This will be very useful for experimentalists and computational scientists alike.

We agree that our method has restricted applicability. As indicated by the reviewer, the underlying kinetic equation does no more apply when the condensed phase is for example a complex liquid with more than one time-scale on the time-scale of the recovery kinetics. However, we think that in this case shapes of spatial profiles should differ between such a complex liquid condensate and a simple liquid. We have added a paragraph to the Results section, pointing out the restricted applicability (below Equation (1)).

8) Finally, do the authors know that the salt concentrations across the phase boundary are equal to one another or is this implicitly assumed? This is particularly relevant for the synthetic coacervates. Sing and coworkers have proposed that there are differences in salt concentrations (even ion concentrations) across phase boundaries. This is relevant because it requires the addition of a Donnan potential to the equations that describe molecular transport. A discussion of this issue would be helpful.

The referee raises an interesting point. Indeed, theoretical studies suggest that salt concentrations can be different, though weakly, inside and outside of condensates (as shown for example in Sing 2020 JCP). Due to the large differences in molecular/particle volumes between salt ions and coacervate components / proteins, however, only weak differences of salt concentrations between phases are expected. Thus, we feel confident that the assumption of equal salt concentrations in both phases represents a reasonable approximation, at least for our systems. In addition, our results are consistent with unpublished data by Fritsch and colleagues.

Reviewer #2 (Recommendations for the authors):

Concerns on figures and writing:

1. Recurrent lack of clarity and/or consistency in how figures are presented

When dissecting the data shown in figures, we often found details in presentation that detracted from our understanding of the study. Notable, but not exhaustive, examples are listed:

a. The retrieval of physical parameters in this study frequently alternates between the use of Equation 1 to track FRAP data, and Equation 6 to model droplet dynamics. The authors do not clearly distinguish the purpose of these equations in the theoretical sections, leaving the reader to understand by context.

We think that this should now be clear from the flow charts and figure legends.

b. The data in Figure 3 would benefit from control data sets that entirely lack the coverslip and/or neighbouring droplets for comparison. In particular, showing the retrieval of identical Din across all simulations (including no coverslip nor neighboring droplets condition) will convey the major point of the paper very clearly.

Panels c and f serve this purpose. In fact, box size is 8*8*8 um, such that the case of h=4um (panel b) gives almost the same results as the case of no coverslip and no neighbors. However, even in the scenario with largest influence of coverslip and neighbours (droplet very close to coverslip, neighbours very close to droplet) the fit can barely be distinguished from the simulated data set (c and f). This is now made clear in the legend.

c. In the caption of Figure 5 it is unclear why the authors have denoted the four indicated values of Dout and P as "reference systems' as opposed to example parameter sets.

This has been addressed now, please see below.

We recommend the authors reassess their figures for clarity of the information that is meant to be communicated.

2. The theory sections lack needed elaboration in some areas.

a. The authors conclude starting on line 153 that P, Din, and Dout can be treated as independent of each other for sufficiently large P. Their justification is that the unknown mobility functions do not impose constraint on Dout(P) other than shown in Equation 10. While the data demonstrates that this is a valid assumption, we find this justification to be opaque and would like to see further elaboration on how independent P, Din, and Dout follows from the mobility functions being unknown.

We are sorry for not being clear on this. As stressed in the manuscript and as shown for example by Equation (10), P, Din and Dout are not independent. If we knew the mobility as a function of phitot, we could use e.g. Equation (10) to calculate from two of the three parameters the third one. However, mobility as a function of phitot is not known. We have now revised the paragraphs around Equation (10).

Suggestions to the authors for bolstering the overall strength of this study below:

1. When interpreting the Results section, we often found that the nature of the experiment was unclear. This is particularly true for Figure 4 and 5. We recommend specifying the procedure followed to acquire the data more clearly and explicitly.

We have added a flow chart describing the procedure in figure 4 and hope that the experimental procedure is now clearer.

2. In Figure 1c, the label "Dynamic BC" made this figure confusing to interpret, especially because there is a same-color arrow depicting time progression. The authors should consider some other way of noting that dynamic BC is applied at the max r (=R-). Also, for the line going across the earliest data points, "initial condition" rather than the "fit" label may be more appropriate.

Following the reviewer’s suggestions, Figure 1 has been redone, including the addition of a flowchart.

3. If available, showing corrected viscosity data from Jawerth et al., 2020 rather than Jawerth et al., 2018 in Figure 1d would bolster this figure via internal consistency with the text in line 98. We would like to be able to back-calculate consistent viscosities using your diffusion data and the Stokes-Sutherland-Einstein relationship.

Unfortunately, Jawerth et al., 2020 only investigate csalt=75mM, we can thus not correct the entire graph, but can only note an order of magnitude given the available data point.

4. Since the time progression of FRAP recovery is illustrated in Figure 2c, we believe the time point label in Figure 2 to be unnecessary and possibly confusing.

We have shifted this information to the legend and removed the label.

5. We find that the citation of privately communicated and unseen data in line 234 does not add to the preceding statement. Optimum salt concentration is a very believable observation.

We have deleted the superfluous statement.

6. There appears to be a typo in the subscripts on line 138. Both read "in".

Corrected.

7. Regarding the concluding statement on line 306: We do not find that Jawerth et al., 2018,2020 contain discussions of altering dense phase kinetics by high labeling fraction. Rather, McCall et al., 2020 describes effects closer to this.

Here, we are referring to the change in droplet viscosity, when labelling PGL-3 with GFP vs unlabelled PGL-3 (Figure S5G in Jawerth et al., 2020 vs Figure 3G in Jawerth et al., 2018). This has been clarified.

8. Regarding Figure 5a, labelling the "ratio" of Dout and P can be confusing. We suggest noting the specific (Dout, P) pairs to specify the points. Also, a clear distinction between "simulation generating parameters (points)" and the "Dout and P dependence (lines) obtained from simulation result and equation 6" is needed.

We have deleted the legend and noted the parameter values in the figure caption. A sentence was added to the caption to distinguish open circles and solid lines.

9. Please consider labelling the point in Figure 5b as "simulation input" or "simulation parameter" rather than "reference simulation".

We went for ‘Example system’, trying to stay coherent with panel a.

10. We suggest the following change in line 230 for clarity: "Specifically, for salt concentrations in the range from 50 mM to 180 mM, we find that the estimated partition coefficient P of PGL-3 droplets decreases more than 10-fold."

This has been incorporated.

11. The significance of the shading in Figure 4d is unclear and inconsistent with how the analogous data in Figure 4e is presented.

We have removed the shading in 4d.

12. The text accompanying Figure 5 beginning on line 249 describes the range of Dout and P used as "relevant for protein condensates and coacervate droplets' without citation. We recommend backing up the validity of the range of Dout and P used here with evidence from literature.

We have added two references for P. Dout can be seen for example in Figure 1.

13. In Figure 1b, individual FRAP recovery curves cut off at seemingly arbitrary points and are difficult to distinguish based on provided color coding. We also note that the curve of 100mM salt concentration does not seem to fit the same trend as the other data sets, but is not discussed by the authors.

Figure 1b was previously shown to illustrate the dynamic boundary condition and to indicate a trend in recovery speed vs. salt concentration, which mostly holds, except for the noted outlier. However, to be able to quantitatively compare droplet recovery speed based on integrated fluorescence, droplets need to be of the same size, which is not entirely fulfilled in this graph. Since it seems to also have caused confusion with the overall fitting procedure, we chose to remove this panel and instead opt for a cleaner presentation of a single curve in Figure 1a.

14. Figure 1f shows the precision of Din determination for the two coacervate systems, but not the protein condensate system, and a salt concentration for comparison with Figure 1d is not provided.

Since we did not have a large enough sample size for any individual condition for PGL-3, we initially chose to not show a graph that corresponds to Figure 1f (now 1g). Here in Author response image 1 we include the requested graph for the reviewer’s convenience. We agree that it is useful to give a measure of measurement precision also for the protein system and now discuss this in the text. The average CoV for PGL-3 across all measurements is approximately four times larger than for the coacervates, likely due to variations in the experimental assay (see text).

Author response image 1
https://doi.org/10.7554/eLife.68620.sa2

Article and author information

Author details

  1. Lars Hubatsch

    1. Max Planck Institute for the Physics of Complex Systems, Dresden, Germany
    2. Max Planck Institute of Molecular Cell Biology and Genetics, Dresden, Germany
    3. Center for Systems Biology Dresden, Dresden, Germany
    Contribution
    Conceptualization, Data curation, Software, Formal analysis, Validation, Investigation, Visualization, Methodology, Writing - original draft, Writing - review and editing
    Competing interests
    No competing interests declared
    ORCID icon "This ORCID iD identifies the author of this article:" 0000-0003-1934-7437
  2. Louise M Jawerth

    1. Max Planck Institute for the Physics of Complex Systems, Dresden, Germany
    2. Max Planck Institute of Molecular Cell Biology and Genetics, Dresden, Germany
    Contribution
    Conceptualization, Resources, Validation, Writing - review and editing
    Competing interests
    No competing interests declared
    ORCID icon "This ORCID iD identifies the author of this article:" 0000-0002-7221-939X
  3. Celina Love

    Max Planck Institute of Molecular Cell Biology and Genetics, Dresden, Germany
    Contribution
    Resources, Investigation
    Competing interests
    No competing interests declared
  4. Jonathan Bauermann

    Max Planck Institute for the Physics of Complex Systems, Dresden, Germany
    Contribution
    Formal analysis, Validation, Methodology
    Competing interests
    No competing interests declared
    ORCID icon "This ORCID iD identifies the author of this article:" 0000-0002-0301-7655
  5. TY Dora Tang

    Max Planck Institute of Molecular Cell Biology and Genetics, Dresden, Germany
    Contribution
    Resources, Funding acquisition, Writing - review and editing
    Competing interests
    No competing interests declared
  6. Stefano Bo

    Max Planck Institute for the Physics of Complex Systems, Dresden, Germany
    Contribution
    Formal analysis, Supervision, Methodology
    Competing interests
    No competing interests declared
    ORCID icon "This ORCID iD identifies the author of this article:" 0000-0002-2738-867X
  7. Anthony A Hyman

    1. Max Planck Institute of Molecular Cell Biology and Genetics, Dresden, Germany
    2. Center for Systems Biology Dresden, Dresden, Germany
    Contribution
    Conceptualization, Funding acquisition, Writing - review and editing
    Competing interests
    No competing interests declared
  8. Christoph A Weber

    1. Max Planck Institute for the Physics of Complex Systems, Dresden, Germany
    2. Max Planck Institute of Molecular Cell Biology and Genetics, Dresden, Germany
    3. Center for Systems Biology Dresden, Dresden, Germany
    Contribution
    Conceptualization, Supervision, Funding acquisition, Validation, Investigation, Methodology, Writing - original draft, Project administration, Writing - review and editing
    For correspondence
    christoph.weber@physik.uni-augsburg.de
    Competing interests
    No competing interests declared
    ORCID icon "This ORCID iD identifies the author of this article:" 0000-0001-6279-0405

Funding

Deutsche Forschungsgemeinschaft (SPP 2191)

  • Lars Hubatsch
  • Anthony A Hyman
  • Christoph A Weber

The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.

Acknowledgements

We are grateful to P McCall, A Fritsch, JM Iglesias-Artola, G Bartolucci, T Wiegand, M Karnat, E Filippidi, T Harmon, F Jülicher and members of the Weber and Hyman groups for stimulating discussions, and P McCall and J Pfanzelter for very valuable and insightful comments on the manuscript. L Hubatsch, AA Hyman and C Weber acknowledge the SPP 2191 ‘‘Molecular Mechanisms of Functional Phase Separation’’ of the German Science Foundation for financial support.

Senior Editor

  1. José D Faraldo-Gómez, National Heart, Lung and Blood Institute, National Institutes of Health, United States

Reviewing Editor

  1. Rohit V Pappu, Washington University in St Louis, United States

Reviewer

  1. Rohit V Pappu, Washington University in St Louis, United States

Publication history

  1. Preprint posted: March 8, 2021 (view preprint)
  2. Received: March 21, 2021
  3. Accepted: October 11, 2021
  4. Accepted Manuscript published: October 12, 2021 (version 1)
  5. Version of Record published: November 10, 2021 (version 2)

Copyright

© 2021, Hubatsch et al.

This article is distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use and redistribution provided that the original author and source are credited.

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  1. Lars Hubatsch
  2. Louise M Jawerth
  3. Celina Love
  4. Jonathan Bauermann
  5. TY Dora Tang
  6. Stefano Bo
  7. Anthony A Hyman
  8. Christoph A Weber
(2021)
Quantitative theory for the diffusive dynamics of liquid condensates
eLife 10:e68620.
https://doi.org/10.7554/eLife.68620

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