The exchange of molecules between a condensate and the dilute phase can be investigated through FRAP-type experiments in which, e.g., fluorescence is locally bleached and recovery as a function of time recorded (Figure 1A and B). Theoretically, the time evolution of the concentration profile ${\displaystyle c(r,t)}$ of the molecules initially located in a spherical condensate of radius ${\displaystyle R}$ (bleached population) can be described by the following continuum diffusion equations (Taylor et al., 2019):

Figure 1

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with the initial condition:

and boundary conditions:

where ${\displaystyle D_{\text{den}}}$ and ${\displaystyle D_{\text{dil}}}$ are, respectively, the diffusion coefficients of molecules in the dense and dilute phases, and ${\displaystyle c_{\text{den}}}$ and ${\displaystyle c_{\text{dil}}}$ are, respectively, the equilibrium concentrations in the dense and dilute phases. The second boundary condition corresponds to flux balance at the interface of the condensate. Specifically, the flux exiting the dense phase (left) equals the flux entering the dilute phase (middle) and also equals the flux passing through the interface (right).

To understand the physical origin of the last term in the second boundary condition in Equation 3, we note that the net outward flux across the interface can be written as ${\displaystyle k_{-}c(R_{-},t)-k_{+}c(R_{+},t)}$, where ${\displaystyle k_{+/-}}$ denotes the entering/exiting rate of molecules at the interface and ${\displaystyle c(R_{+/-},t)}$ the concentration of bleached molecules immediately outside/inside of the boundary. At thermal equilibrium, this net flux goes to zero, i.e., ${\displaystyle k_{-}{c}_{\text{den}}=k_{+}{c}_{\text{dil}}}$ so ${\displaystyle k_{+}=k_{-}(c_{\text{den}}/c_{\text{dil}})}$. The net outward flux is therefore ${\displaystyle k_{-}[c(R_{-},t)-(c_{\text{den}}/c_{\text{dil}})c(R_{+},t)]}$. The parameter ${\displaystyle \kappa \equiv k_{-}}$ is a transfer coefficient that governs the magnitude of this net flux. When the ratio of the concentrations on the two sides of the interface deviates from the equilibrium ratio, a small ${\displaystyle \kappa }$ can kinetically limit the flux going through the interface. We therefore term ${\displaystyle \kappa }$ the interface conductance, the inverse of interface resistance.

For the model described by Equations 1–3, the fraction of molecules in the condensate which are unbleached at time ${\displaystyle t}$ is

Clearly, how quickly ${\displaystyle f(t)}$ recovers from 0 to 1 quantifies the timescale of material exchange between the condensate and the surrounding dilute phase.

The authors of Taylor et al., 2019 derived an exact solution for ${\displaystyle f(t)}$ in an integral form using Laplace transforms. However, it is not directly apparent from the integral expression what physics governs the timescale of fluorescence recovery. In addition, the lengthy integral form of the expression also presents an impediment to its practical experimental applications. To obtain a more intuitive and concise result, we note that diffusion of biomolecules in the dilute phase is typically much faster than diffusion in the dense phase, with measured ${\displaystyle D_{\text{dil}}/D_{\text{den}}}$ in the range of 10²–10⁵ (Freeman Rosenzweig et al., 2017; Taylor et al., 2019). We therefore employed the exact solution to derive an approximate solution in the parameter regime ${\displaystyle D_{\text{dil}}\gg D_{\text{den}}}$:

where the timescale of fluorescence recovery is given by

Please refer to Appendix 1 for a detailed derivation. We note that, in practice, ${\displaystyle D_{\text{dil}}>20D_{\text{den}}}$ is sufficient for the validity of the approximation with the approximate ${\displaystyle \tau }$ in Equation 6 within 10% of the exact value.

Equation 6 conveys a clear physical picture of what controls the timescale of condensate material exchange. First, for large condensates and slow internal diffusion, exchange is limited by the rate of mixing within the condensate, so that ${\displaystyle \tau \simeq R^2/({\pi }^2{D}_{\text{den}})}$. Second, if instead diffusion in the dilute phase is sufficiently slow, or the concentration in the dilute phase is very low, then ${\displaystyle \tau \simeq c_{\text{den}}{R}^2/(3c_{\text{dil}}{D}_{\text{dil}})}$, which is the time required to replace all molecules in the condensate if molecules incident from the dilute phase are immediately absorbed (see Appendix 1). Finally, if the interface conductance ${\displaystyle \kappa }$ is very small, the interfacial flux can be rate limiting for exchange, yielding ${\displaystyle \tau \simeq R/(3\kappa )}$.

What determines the magnitude of the interface conductance ${\displaystyle \kappa }$? From a theoretical perspective, transitions between dense and dilute phases have been modeled both from the continuum theory approach (Hubatsch et al., 2021) and by considering single-molecule trajectories (Bo et al., 2021). However, for any particular systems, the magnitude of the interface conductance depends on microscopic features of the biomolecules, such as internal states, which may not be captured by Flory-Huggins and Cahn-Hilliard-type mean-field theories. Indeed, if we start with the continuum approach in Hubatsch et al., 2021, where the concentration of bleached components ${\displaystyle c(\mathbf{\text{{r}}},t)}$ is governed by

with ${\displaystyle c_{\text{eq}}(\mathbf{\text{{r}}})}$ the equilibrium concentration profile and ${\displaystyle D[c_{\text{eq}}(\mathbf{\text{{r}}})]}$ the diffusion coefficient which depends on the local equilibrium concentration, one can obtain an expression for ${\displaystyle \kappa }$ (see Appendix 1):

where the integral is over the interface region. We would then conclude ${\displaystyle {\kappa }^{-1}<\delta /D_{\text{den}}+\delta c_{\text{den}}/(c_{\text{dil}}{D}_{\text{dil}})}$, where ${\displaystyle \delta }$ is the width of the interface. As the interface is typically narrow, this inequality would imply that in practice the interfacial term in Equation 6 would always be smaller than the sum of the other two terms, and thus could be neglected.

However, a recent FRAP experiment on LAF-1 protein droplets (Taylor et al., 2019) contradicts the above mean-field result. In the experiment, a micron-sized LAF-1 droplet (${\displaystyle {\displaystyle R=1\text{µ}\text{m}}}$) was bleached and fluorescence recovery measured as a function of time (Figure 1C). It was observed that recovery of that droplet occurs on a timescale of ∼ 1.3hr. Given the measured parameters of the system, one can estimate the recovery time in the mean-field approach to be ${\displaystyle \tau =R^2/({\pi }^2{D}_{\text{den}})+c_{\text{den}}{R}^2/(3c_{\text{dil}}{D}_{\text{dil}})=64\pm 18\text{s}}$, much shorter than the measured recovery time. A large interface resistance was proposed as a possible explanation for this discrepancy (Taylor et al., 2019). Motivated by this surprising experimental result, we sought to investigate if it is possible for the interface resistance to be much larger than predicted by mean-field theory, and if so, what could be the underlying mechanisms and how does the interface resistance depend on the microscopic features of phase-separating molecules?

As noted above, if all molecules incident from the dilute phase are immediately absorbed into the dense phase, the interfacial flux can’t be rate limiting. The existence of a large interface resistance then necessarily implies a strongly reduced flux of molecules successfully crossing the interface. This can occur either because the molecules incident from the dilute phase fail to incorporate into the interface, or they transiently incorporate but fail to enter the dense phase. In both cases, the molecules effectively ‘bounce’ from the interface leading to a large interface resistance. Mechanistically, bouncing can occur for a variety of reasons, which we discuss in the Discussion section below. Here, we employ a ‘sticker-spacer’ polymer model (Choi et al., 2020; Semenov and Rubinstein, 1998) to explore one possible mechanism in which molecules can assume non-sticking conformations by saturating all their possible binding sites. These molecules incident from the dilute phase typically fail to form bonds with the dense phase, thus ‘bouncing’ off of the condensate.

The ‘sticker-spacer’ model provides a conceptual framework for understanding biomolecular phase separation, wherein the ‘stickers’ represent residues or larger domains that are capable of forming saturable bonds, while the ‘spacers’ connect the stickers to form polymers. Specifically, we simulated polymers consisting of type A and type B stickers connected by implicit spacers in the form of stretchable bonds (Kremer and Grest, 1990; Figure 2A):

where ${\displaystyle r}$ is the distance between two stickers. One-to-one heterotypic bonds between A and B are implemented via an attractive potential:

while stickers of the same type interact through a repulsive potential to prevent many-to-one binding:

We take ${\displaystyle K=0.15k_{\text{B}}T/{\text{nm}}^2}$, ${\displaystyle R_0=10\text{nm}}$, ${\displaystyle U_0=14k_{\text{B}}T}$, ${\displaystyle r_0=1\text{nm}}$, ${\displaystyle ϵ=1k_{\text{B}}T}$, ${\displaystyle \sigma =2\text{nm}}$, and ${\displaystyle r_{\text{c}}=1.12\sigma }$ in all simulations, except in the simulations of Figure 3F where we vary ${\displaystyle U_0}$ systematically from 13.5 to ${\displaystyle 15k_{\text{B}}T}$. For all simulation results we reported below, the standard error of the mean is typically smaller than the symbol size and therefore not shown.

Figure 2

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Figure 3

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For each of the five sequences shown in Figure 2A, we simulated 1000 polymers in a ${\displaystyle 500\text{nm}\times 50\text{nm}\times 50\text{nm}}$ box with periodic boundary conditions using Langevin dynamics (see Appendix 2 for details). Simulations were performed using LAMMPS molecular-dynamics simulator (Plimpton, 1995). Figure 2B shows a snapshot of coexisting dense and dilute phases after equilibration of the A6B6 polymers (6A stickers followed by 6B stickers), while Figure 2C shows the time-averaged profile of the total polymer concentration. The five different polymer sequences we simulated were chosen to yield a range of dilute- and dense-phase sticker concentrations (Figure 2D) as well as a range of dilute- and dense-phase diffusion coefficients (Figure 2E). As found previously (Weiner et al., 2021), polymers like A6B6 with long blocks of stickers of the same type have low dilute-phase concentrations. This follows because it is entropically unfavorable for these polymers to form multiple self-bonds, which favors the dense phase where these polymers can readily form multiple trans-bonds. These long-block polymers also have low dense-phase diffusion coefficients because of their large number of trans-bonds, which need to be repeatedly broken for the polymers to diffuse.

Having determined the concentrations and diffusion coefficients in the dense and dilute phases, we are now in a position to extract the values of interface conductance from simulations. Figure 3 depicts a simple protocol that allows us to infer ${\displaystyle \kappa }$ by applying the 1D, slab-geometry version of Equations 1–6 to simulation results (see Appendices 1 and 2 for details): (i) All polymers in the dilute phase are initially considered ‘labeled’, (ii) any labeled polymer that forms a lasting A-B bond with a polymer in the dense phase becomes permanently unlabeled (Figure 3A), (iii) the remaining fraction of labeled dilute phase polymers is fit to an exponential decay (Figure 3B), and (iv) the resulting decay time constant ${\displaystyle \tau }$ is used together with the known dense and dilute phase parameters to infer ${\displaystyle \kappa }$ from:

where ${\displaystyle d}$ is the half-width of the dilute phase. As shown in Figure 3C, the resulting values of ${\displaystyle \kappa }$ span more than an order of magnitude for our selected polymer sequences, despite the fact that all five polymers can in principle form the same number (6) of self-bonds.

We note that one can alternatively obtain ${\displaystyle \kappa }$ by directly measuring the flux of molecules that enter the dense phase. Mathematically, this flux equals ${\displaystyle k_{+}{c}_{\text{dil}}=\kappa c_{\text{den}}}$. We show in Appendix 2 that the values of ${\displaystyle \kappa }$ found via this method are consistent with results reported in Figure 3C.

What gives rise to the very different values of ${\displaystyle \kappa }$? To address this question, we first consider the predicted interface conductance ${\displaystyle {\kappa }_0}$ if polymers incident from the dilute phase simply move through the interface region with a local diffusion coefficient that crosses over from ${\displaystyle D_{\text{dil}}}$ to ${\displaystyle D_{\text{den}}}$. Then according to Equation 8 (see Appendix 1)

However, as shown in Figure 3D, the actual values of ${\displaystyle \kappa }$ in our simulations can be a factor of ∼50 smaller than ${\displaystyle {\kappa }_0}$. This reduction can be traced to a ‘bouncing’ effect. As shown schematically in the inset to Figure 3D and for an exemplary simulated trajectory in Figure 3E (more trajectories can be found in Appendix 2), molecules incident from the dilute phase may fail to form bonds with the dense phase, effectively ‘bouncing’ off of the condensate. The differing extent of this bouncing effect for the five sequences we studied reflects differences in their numbers of free stickers in both their dilute- and dense-phase conformations. The fewer such available stickers, the fewer ways for a polymer incident from the dilute phase to bond with polymers at the surface of the dense phase, and thus the more likely the incident polymer is to bounce. More generally, we find that the interface conductance of the sticker-spacer polymers is controlled by the encounter rate of a pair of unbound stickers and the availability of these stickers, which in turn depends on the sticker-sticker binding strength, the dilute- and dense-phase polymer concentrations, and the width of the interface:

where ${\displaystyle n}$ is the number of monomers in a polymer, ${\displaystyle s}$ is the global stoichiometry (i.e. ${\displaystyle c_{\text{A}}/c_{\text{B}}}$), ${\displaystyle f_{\text{dilA/dilB}}}$ and ${\displaystyle f_{\text{denA/denB}}}$ are the fractions of unbound ${\displaystyle \text{A}/\text{B}}$ monomers in the dilute and dense phases, respectively. In support of this picture, we find that all our simulation results for ${\displaystyle \kappa /{\kappa }_0}$ collapse as a linear function of a lumped parameter ${\displaystyle u}$ (Figure 3D):

which expresses the availability of free stickers, where all parameters in ${\displaystyle u}$ are determined directly from simulations. See Appendix 1 for derivations of Equations 14 and 15.

Comparing sequences with unequal sticker stoichiometry A8B6 and A10B6 to their most closely related equal-stoichiometry sequence A6B6, we find that the extra A stickers substantially increase the interface conductance ${\displaystyle \kappa }$. Intuitively, the excess As in both dense and dilute phases of A8B6 and A10B6 provide a pool of available stickers for any unbound B to bind to. By contrast, at equal stoichiometry, both free As and free Bs are rare which maximizes the bouncing effect. This reduction in potential binding partners at equal stoichiometry has also been observed experimentally (Brassinne et al., 2017), and theoretically Ronceray et al., 2022, to cause an anomalous slowing of diffusion within condensates at equal stoichiometry in the regime of strong binding.

Finally, we expect the interface resistance to increase approximately exponentially with the increase of binding strength ${\displaystyle U_0}$ between A and B stickers, as the tighter the binding, the fewer available stickers, and hence the more bouncing of molecules at the interface. We demonstrate in Figure 3F that the interface conductance ${\displaystyle \kappa }$ of the A6B6 system indeed drops by a factor of 5 as the value of ${\displaystyle U_0}$ increases from 13.5 to ${\displaystyle 15k_{\text{B}}T}$.

Above we simulated phase separation of sticker-spacer polymers in a slab geometry, and discussed how the extracted interface conductance ${\displaystyle \kappa }$ depends on sequence pattern, sticker stoichiometry, and binding strength between stickers. In principle, with the parameters measured from such simulations, Equation 6 and Equations 1–4 can be used to predict FRAP recovery times for simulated 3D droplets. To check the consistency between theory and simulation, we simulated a small droplet of the A6B6 polymers and measured its FRAP recovery time. Briefly, we simulated 2000 A6B6 polymers in a cubic box of side length ${\displaystyle 286\text{nm}}$ with periodic boundaries using Langevin dynamics. All interaction potentials and parameters are the same as the simulations in Figure 2. Figure 4A shows a snapshot of the droplet coexisting with the surrounding dilute phase after equilibration. Figure 4B shows the mean polymer concentration profile, which is consistent with the dense- and dilute-phase concentrations of the A6B6 system reported in Figure 2D. We note that both concentrations are slightly higher than their counterparts in the slab geometry due to a surface tension effect (Thomson, 1872). At time ${\displaystyle t=0}$, we labeled all the molecules inside the droplet as ‘bleached’ and tracked the time evolution of the concentration profile of the bleached population (Figure 4C) and obtained the FRAP recovery curve (Figure 4D, black circle). We note that the recovery curve plateaus at ${\displaystyle A\approx 0.5}$ instead of 1 due to limited number of polymers in the dilute phase. Using parameters of the droplet system, we numerically integrated Equation 1 with modified initial and boundary conditions to account for the system’s finite size. The resulting numerical curve agrees almost perfectly with the simulation result (Figure 4D), which validates both our theoretical and simulation approaches.

Figure 4

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Fitting the recovery curve by

which takes into account the effect of the finite-size dilute phase (see Appendix 2 for a derivation), yielded a recovery time ${\displaystyle \tau =0.071\text{s}}$. We compare the measured FRAP recovery time for the small droplet ${\displaystyle R=37\text{nm}}$ (green circle) to theoretical predictions from Equation 6 (gray) and Equations 1–4 (black) in Figure 5A. The FRAP recovery of the simulated droplet is clearly limited by the interface resistance. We note that the small deviation between theory and simulation in Figure 5A is due to the utilization of parameters from the slab geometry for the theory prediction, including a ${\displaystyle {\displaystyle \kappa =0.14\text{µ}\text{m}/\text{s}}}$ lower than the measured ${\displaystyle {\displaystyle {\kappa }^{\text{d}}=0.20\text{µ}\text{m}/\text{s}}}$ of the droplet system, which in turn reflects the difference in the dilute-phase concentrations of the two systems, as ${\displaystyle \kappa \sim c_{\text{dil}}}$ from Equation 14.

Figure 5

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Under what circumstances is interface resistance experimentally measurable? If there were no bouncing effect, i.e., if all molecules incident from the dilute phase that touch the interface get immediately absorbed into the condensate, then interface resistance would never dominate the recovery time in FRAP-type experiments, making it very difficult to measure ${\displaystyle \kappa }$. However, as shown in Figure 3, the bouncing effect can reduce ${\displaystyle \kappa }$ substantially. For such systems, the interface conductance can be inferred quantitatively from Equation 6 or by fitting FRAP recovery curves as in Taylor et al., 2019, using the experimentally measured dense- and dilute-phase concentrations and diffusion coefficients.

Even without knowing all parameters, one may still be able to infer the presence of a large interface resistance by observing the pattern of fluorescence recovery in droplets of different sizes. According to Equation 6 the recovery time associated with interface resistance increases linearly with radius ${\displaystyle R}$ while the other terms increase as ${\displaystyle R^2}$ (Figure 5A). Therefore, one expects a cross-over for the recovery from being interface-resistance dominated (small ${\displaystyle R}$) to being either dilute-phase-diffusion or dense-phase-mixing dominated (large ${\displaystyle R}$). In the latter case, the fluorescence profile during recovery will be notably different in large versus small droplets as shown in Figure 5B – for large droplets progressive diffusion of fluorescence into the droplet will be apparent, whereas small droplets will recover uniformly as internal mixing will be fast compared to exchange with the surroundings. Thus observation of such a cross-over of the recovery pattern as a function of droplet size provides evidence for the presence of a large interface resistance, which can be followed up by more quantitative studies. For example, the uniform recovery of the LAF-1 droplet in Figure 1C and the simulated droplet in Figure 4C are indicative of a large interface resistance, as the diffusion in the dilute phase is too fast to be rate limiting. We also predict the cross-over for LAF-1 droplets to be around ${\displaystyle R=71\mu \text{m}}$, which in principle can be tested experimentally.

The exact solution for ${\displaystyle f(t)}$ in Equation 4, the fraction of molecules in a spherical condensate of radius ${\displaystyle R}$ which are unbleached at time ${\displaystyle t}$, is derived by Taylor et al., 2019, in an integral form using Laplace transforms (Taylor et al., 2019),

where

and ${\displaystyle \lambda =D_{\text{dil}}/D_{\text{den}}}$, ${\displaystyle \alpha =c_{\text{den}}/c_{\text{dil}}}$, ${\displaystyle t^{\ast }=t{D}_{\text{den}}/R^2}$, and ${\displaystyle k=R\kappa /D_{\text{den}}}$, where ${\displaystyle \kappa }$ is the interface conductance.

To obtain a more intuitive result, we first rearrange ${\displaystyle g(u,t^{\ast })}$ as

We note that the diffusion of biomolecules in the dilute phase is typically much faster than diffusion in the dense phase, i.e., ${\displaystyle \lambda =D_{\text{dil}}/D_{\text{den}}\gg 1}$. In this parameter regime, ${\displaystyle g(u,t^{\ast })}$ is sharply peaked at the values of ${\displaystyle u}$ where

i.e., when the second term in the denominator of ${\displaystyle g(u,t^{\ast })}$ becomes 0. Representative ${\displaystyle g(u,t^{\ast })}$ curves are shown in Appendix 1—figure 1. We can therefore approximate ${\displaystyle g(u,t^{\ast })}$ as

where ${\displaystyle u_n}$ is the ${\displaystyle n}$ th solution of Equation 20 and ${\displaystyle a_n\sim 1/\sqrt{\lambda }}$ is the inverse of effective width of ${\displaystyle n}$ th peak of ${\displaystyle g(u,t^{\ast })}$. Clearly, the prefactor of the delta function ${\displaystyle \delta (u-u_n)}$ in Equation 21 drops rapidly with increasing values of ${\displaystyle u_n}$. Consequently, the integral in Equation 17 is always dominated by the contribution from the first mode, and therefore

where

with ${\displaystyle u_1}$ the first root of Equation 20.

Appendix 1—figure 1

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At any given values of ${\displaystyle k}$, ${\displaystyle \alpha }$, and ${\displaystyle \lambda }$, the solutions of Equation 20 can be obtained numerically. Alternatively, we can obtain an approximate analytical solution by first rewriting Equation 20 as

We plot the combined parameter ${\displaystyle K=k/(1+k\alpha /\lambda )}$ versus the first root ${\displaystyle u_1}$ in Appendix 1—figure 2. For small ${\displaystyle K}$, Taylor series expansion around ${\displaystyle u=0}$ for the left side of Equation 24 yields

which yields ${\displaystyle u_1=\sqrt{3K}}$. For large ${\displaystyle K}$, ${\displaystyle u_1}$ plateaus at ${\displaystyle \pi }$. We therefore approximate ${\displaystyle u_1}$ as

This approximate solution is compared with the exact numerical solution of ${\displaystyle u_1}$ in Appendix 1—figure 2. The maximum error of about 5% occurs at an intermediate value of ${\displaystyle K}$ (Appendix 1—figure 2, inset). The relaxation time ${\displaystyle \tau }$ corresponding to the approximate solution in Equation 26 is

Appendix 1—figure 2

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The time evolution of the spherically symmetric concentration profile ${\displaystyle c(r,t)}$ of molecules in the dilute phase around a droplet of radius ${\displaystyle R}$ is given by

If molecules incident from the dilute phase are immediately and irreversibly absorbed into the dense phase, the boundary condition is then ${\displaystyle c(R,t)=0}$. The steady-state solution of Equation 28 in this absorbing-boundary limit is

where ${\displaystyle c_{\text{dil}}}$ is the concentration at ${\displaystyle r\to \mathrm{\infty }}$, which yields a total steady-state flux into the droplet of

It then takes a time

to replace all ${\displaystyle 4\pi R^3{c}_{\text{den}}/3}$ molecules in the droplet.

To derive the interface conductance in the continuum limit (which neglects the bouncing effect), we start with the mean-field formulation developed in Hubatsch et al., 2021, where the concentration of bleached components ${\displaystyle c(\mathbf{\text{{r}}},t)}$ is governed by

with the flux

where ${\displaystyle c_{\text{eq}}(\mathbf{\text{{r}}})}$ is the equilibrium concentration profile, and ${\displaystyle D[c_{\text{eq}}(\mathbf{\text{{r}}})]}$ the diffusion coefficient which depends on the local equilibrium concentration.

For a spherical condensate of radius ${\displaystyle R}$, if the interface width is narrow, we can assume that the flux going through the interface is uniform in space along the radial direction, i.e.,

where ${\displaystyle \hat{r}}$ denotes the unit vector in the radial direction. Therefore,

The solution to the above equation is

We assume that the interface spans a width of ${\displaystyle \delta }$ from ${\displaystyle R_{-}}$ to ${\displaystyle R_{+}}$ with ${\displaystyle R_{+/-}=R\pm \delta /2}$, then ${\displaystyle c_{\text{eq}}(R-\delta /2)=c_{\text{den}}}$ and ${\displaystyle c_{\text{eq}}(R+\delta /2)=c_{\text{dil}}}$. Substituting Equation 36 into the second boundary condition in Equation 3, we have

We then obtain an expression for the interface conductance ${\displaystyle \kappa }$:

The equilibrium concentration ${\displaystyle c_{\text{eq}}(r)}$ transitions from ${\displaystyle c_{\text{den}}}$ to ${\displaystyle c_{\text{dil}}}$ between ${\displaystyle R-\delta /2}$ and ${\displaystyle R+\delta /2}$, and the corresponding diffusion coefficient ${\displaystyle D(r)}$ transitions from ${\displaystyle D_{\text{den}}}$ to ${\displaystyle D_{\text{dil}}}$. Assuming a monotonic sigmoidal transition, along with ${\displaystyle D_{\text{den}}\ll D_{\text{dil}}}$ and ${\displaystyle c_{\text{den}}\gg c_{\text{dil}}}$, we obtain

which leads to an interface conductance in the continuum limit

In the simulations in Figure 3, we are only interested in molecules that remain in the dilute phase without entering the dense phase, and the corresponding interface conductance in the continuum limit is then

For the case of a quasi-1D slab geometry, we consider the condensate to sit in the middle in the region ${\displaystyle -l<x<l}$ with the simulation box extending along the ${\displaystyle x}$-axis from ${\displaystyle -L}$ to ${\displaystyle L}$. The time evolution of the concentration profile ${\displaystyle c(x,t)}$ of molecules initially located in the dense phase (bleached population) is then given by the 1D diffusion equations:

with the initial condition:

and boundary conditions:

The general solution for the diffusion Equation 42 with the boundary condition in Equation 44 is:

Applying the boundary condition in Equation 45 to this general solution yields:

which leads to

where ${\displaystyle \tau =(D_{\text{den}}{p}_n^2{)}^{-1}=(D_{\text{dil}}{q}_n^2{)}^{-1}}$ is the relaxation time of the ${\displaystyle n}$ th mode of the system. For given parameters ${\displaystyle c_{\text{dil}}}$, ${\displaystyle c_{\text{den}}}$, ${\displaystyle D_{\text{dil}}}$, ${\displaystyle D_{\text{den}}}$, ${\displaystyle l}$, ${\displaystyle L}$, and ${\displaystyle \kappa }$, ${\displaystyle \tau }$ can be obtained numerically using the above equation. In the regime where the interface conductance is small, we derive an analytical expression for the relaxation time

which resembles the corresponding relaxation time for a spherical droplet when interface conductance is small ${\displaystyle \tau =R/(3\kappa )}$.

We note that, in principle, Equation 49 can be used to infer the value of ${\displaystyle \kappa }$ for a system using the relaxation time ${\displaystyle \tau }$ from simulation. However, due to the relatively small simulation sizes, the interface regime can constitute a significant fraction of the dense-phase condensate. This can result in uncertainties in the determination of the dense-phase width ${\displaystyle l}$ and diffusion coefficient ${\displaystyle D_{\text{den}}}$, etc., leading to errors in the determination of the interface conductance using Equation 49. Such errors can be significant when slow diffusion in the dense phase becomes rate-limiting for overall system relaxation. Therefore, instead of sticking to the ‘FRAP protocol’, we find it more convenient to track the molecules that remain in the dilute phase without ever entering the dense phase (Figure 3), as this minimizes the errors caused by any inaccuracies in dense-phase parameters. To relate ${\displaystyle \kappa }$ to the decay time of the dilute-phase molecules, we note that the time evolution of the concentration profile ${\displaystyle c(x,t)}$ of the dilute-phase molecules which have never entered the dense phase is given by

with the initial condition ${\displaystyle c(x,0)=c_{\text{dil}}}$ for ${\displaystyle l<|x|<L}$ and boundary conditions:

Going through a similar procedure as for Equations 46–49, we obtain

The interface conductance ${\displaystyle \kappa }$ for simulated systems in Figure 3C (bottom) is obtained from the relationship in Equation 54, which is Equation 12 in the main text, using the measured relaxation time ${\displaystyle \tau }$ (Figure 3C, top), of molecules that remain in the dilute phase without ever entering the dense phase.

In the interface-resistance-dominated regime, the decay time ${\displaystyle \tau }$ of the number of dilute-phase molecules that have not entered the dense phase can be obtained from Equation 54:

In the slab geometry, this decay time is controlled by the flux per unit area ${\displaystyle j}$ entering the dense phase

where ${\displaystyle V}$ is the volume of the dilute phase, and ${\displaystyle A}$ the cross-sectional area of the interface between the dilute and dense phases (the factor of 2 accounts for the two interfaces). Combining these two equations, we have

For our simulations of polymers with A and B type stickers, we can approximate ${\displaystyle j}$ by assuming that a polymer incident from the dilute phase will join the dense phase if and only if an unbound monomer on the polymer binds to an unbound monomer in the dense phase somewhere in the interface region. To find an approximate formula for ${\displaystyle j}$ we therefore need to estimate the rate of such binding events per unit area of the interface. To this end, we can use the formula for diffusion-limited monomer-monomer binding, but with some modifications: First, we can write the concentration of unbound monomers of type A in the dilute phase as

where ${\displaystyle n_{\text{A}}}$ is the number of type A monomers per polymer and ${\displaystyle f_{\text{dilA}}}$ is the fraction of these monomers that are unbound. This concentration implies a diffusion-limited binding flux onto each unbound dense-phase type B monomer in the interface region

where ${\displaystyle r_0}$ is the sticker radius, and we have assumed that the diffusion rate is set by the whole polymer. Now we need an estimate for the areal density ${\displaystyle {\rho }_{\text{denB, unbound}}}$ of available unbound B-type monomers in the dense-phase interface region, since each one will contribute the above flux (Equation 59). We can write

where ${\displaystyle \delta }$ is the width of the interface region, ${\displaystyle n_{\text{B}}}$ is the number of type B monomers per polymer, and ${\displaystyle f_{\text{denB}}}$ is the fraction of unbound B monomers on dense-phase polymers in this region. Finally, we can combine the above equations, and include the binding of dilute-phase B-type monomers to dense-phase A-type monomers, to obtain

The interface conductance ${\displaystyle \kappa }$ is then

Using this expression, we can then estimate the ratio between the true ${\displaystyle \kappa }$ and its continuum limit ${\displaystyle {\kappa }_0=c_{\text{dil}}{D}_{\text{dil}}/(\delta c_{\text{den}})}$ to be

where ${\displaystyle n=n_{\text{A}}+n_{\text{B}}}$ is the length of a polymer and ${\displaystyle s=c_{\text{A}}/c_{\text{B}}}$ is the global stoichiometry. Note that we have used ${\displaystyle n_{\text{A}}{n}_{\text{B}}=n^{2}s/(1+s{)}^2}$ to derive the above expression. In practice, we find this expression to be quite accurate up to a constant prefactor (Figure 3D), and we define the right-hand side of Equation 63 as a lumped parameter ${\displaystyle u}$.

We perform coarse-grained molecular-dynamics simulations using LAMMPS (Plimpton, 1995) to simulate phase separation of ‘sticker and spacer’ polymers. Individual polymers are modeled as linear chains of spherical stickers of types A and B connected by implicit spacers (Figure 2A) with the interaction potentials in Equations 9–11, which ensure one-to-one binding between A and B stickers.

For each of the five selected sequences (Figure 2A), we simulate 1000 polymers in a 500 nm × 50 nm × 50 nm box with periodic boundary conditions. Following the simulation procedures in Zhang et al., 2021, we first initialize the simulation by confining polymers in the region ${\displaystyle -90\text{nm}<x<90\text{nm}}$ to promote phase separation and ensure that only a single dense condensate is formed. The attractive interaction between A and B stickers (Equation 10) is gradually switched on from ${\displaystyle U_0=0}$ to 14 over 2.5×10⁷ time steps. This annealing procedure leads to the formation of a dense phase close to its equilibrated concentration. The dense condensate is equilibrated at fixed ${\displaystyle U_0=14}$ for another ${\displaystyle 2.5\times {10}^7}$ steps and then the confinement is removed. The system is equilibrated for ${\displaystyle 2\times {10}^8}$ more time steps to allow for the formation of a dilute phase and further relaxation of the dense phase. We then record the positions of all particles every ${\displaystyle 2.5\times {10}^5}$ steps for 800 recordings.

Through the entire simulation, we equilibrate the system using a Langevin thermostat implemented with LAMMPS commands fix nve and fix langevin, i.e., the system evolves according to Langevin, 1908

where ${\displaystyle {\overrightarrow{r}}_i}$ is the coordinate of particle ${\displaystyle i}$, ${\displaystyle m}$ is its mass, ${\displaystyle \gamma }$ is the friction coefficient, ${\displaystyle \overrightarrow{f}}$ is random thermal noise, and the potential energy ${\displaystyle U({\overrightarrow{r}}_1,...,{\overrightarrow{r}}_N)}$ contains all interactions between particles, including bonds and sticker-sticker interactions (Equations 9–11). We take temperature ${\displaystyle T=300\text{K}}$, damping factor ${\displaystyle \tau =m/\gamma =10\text{ns}}$, step size ${\displaystyle dt=0.1\text{ns}}$, and mass of particle ${\displaystyle m=188.5\text{ag}}$. These parameters give each sticker the correct diffusion coefficient ${\displaystyle D=k_{\text{B}}T/(3\pi \eta d)}$, where ${\displaystyle \eta }$ is the water viscosity ${\displaystyle 0.001\text{kg}/\text{m}/\text{s}}$ and ${\displaystyle d=2\text{nm}}$ is the sticker diameter.

We perform 10 simulation replicates with different random seeds for each of the five selected polymer sequences. Consistency of results is checked across replicates. To test if the system has reached equilibrium, we compare the dense- and dilute-phase concentrations derived from the first and second halves of the recorded data. The agreement indicates that the system has reached equilibrium.

To measure the dilute- and dense-phase concentrations, we first group polymers into connected clusters in each recording. Two stickers are considered connected if they are part of the same polymer, or if they are within the attraction distance ${\displaystyle r_0=1\text{nm}}$. Connected stickers are then grouped into clusters. In all simulations, we observe one large cluster which contains most of the polymers, and tens to hundreds of very small clusters (Figure 2B). We consider the large cluster to constitute the dense phase, and the smaller clusters to be constituents of the dilute phase. To find the concentrations of each phase, we identify the center of mass of the dense cluster in each recording, and recenter the simulation box to this center of mass. We then compute the polymer concentration histogram along the ${\displaystyle x}$ axis with a bin size 1/50 of box length. The histogram of numbers of stickers per bin is averaged over all recordings and simulation replicates. The polymer concentration profile is derived as the sticker concentration profile divided by the number of stickers per polymer. The resulting polymer concentration profile has high values in the middle corresponding to the dense-phase concentration, and low values on the two sides corresponding to the dilute-phase concentration (Figure 2C). The dilute- and dense-phase concentrations in Figure 2D are calculated by averaging the concentration profile over the regions ${\displaystyle ({\displaystyle x\le -150\text{nm}\mathrm{o}\mathrm{r}x\ge 150\mathrm{n}\mathrm{m})}}$ and ${\displaystyle ({\displaystyle -10\text{nm}\le x\le 10\text{nm})}}$, respectively.

To measure the dilute- and dense-phase diffusion coefficients, we perform simulations with a pure dilute phase or dense phase, i.e., with polymers at the measured dilute- and dense-phase concentrations. Specifically, for the dilute-phase case, we simulate 750 (A2B2)₃, 285 (A3B3)₂, 101 A6B6, 104 A8B6, and 180 A10B6 polymers, each in a ${\displaystyle 150\text{nm}\times 150\text{nm}\times 150\text{nm}}$ box with periodic boundary conditions. For the dense-phase case, we simulate 1000 polymers for all selected sequences in a ${\displaystyle W\text{nm}\times 50\text{nm}\times 50\text{nm}}$ box with periodic boundary conditions, where ${\displaystyle W=116.1}$ for (A2B2)₃, 96.5 for (A3B3)₂, 85.8 for A6B6, 136.5 for A8B6, and 234.2 for A10B6. To equilibrate the system, the attractive interaction between A and B stickers (Equation 10) is gradually switched on from ${\displaystyle U_0=0}$ to ${\displaystyle 14k_{\text{B}}T}$ over ${\displaystyle 2.5\times {10}^7}$ time steps and equilibrated at fixed ${\displaystyle U_0=14k_{\text{B}}T}$ for ${\displaystyle 2\times {10}^8}$ more time steps. We then record the displacement of all particles every ${\displaystyle 2.5\times {10}^5}$ steps for 400 recordings. Five simulation replicates with different random seeds are performed for each selected sequence.

To find the diffusion coefficients, we compute the time-averaged MSD for each polymer as a function of the lag time ${\displaystyle t_{\text{lag}}}$, and average over all polymers in a simulation box and over five replicates. The time- and ensemble-averaged MSD is then linearly fit to ${\displaystyle \text{MSD}=6D{t}_{\text{lag}}}$ to extract the diffusion coefficient.

To measure the interface conductance ${\displaystyle \kappa }$, we follow the simple protocol depicted in Figure 3A. This scheme, based on the rate that particles in the dilute phase join the dense phase, is both computationally efficient and allows us to infer the interface conductance even when slow diffusion in the dense phase is rate-limiting for overall system relaxation. Specifically, in this protocol we first define a ‘survival’ variable ${\displaystyle S}$ for each polymer as a function of time: ${\displaystyle S=1}$ if the polymer belongs to any dilute-phase cluster (including a solo cluster), and ${\displaystyle S=0}$ if the polymer is in the dense-phase cluster. Next, for all polymers starting with ${\displaystyle S=1}$ (i.e. in the dilute phase), we check if there is a period of time (chosen here to be 10 times the average bond lifetime of an isolated A-B pair) for which its ${\displaystyle S}$ value is always 0 (i.e. the polymer has joined the dense-phase cluster). If yes, we set ${\displaystyle S=1}$ at the time points before the joining event and ${\displaystyle S=0}$ at all times afterward. If not, we set ${\displaystyle S=1}$ for this polymer for all time points. We then average ${\displaystyle S(t)}$ over all polymers starting with ${\displaystyle S=1}$ and over the 10 simulation replicates. The obtained ${\displaystyle S(t)}$ is the average survival probability of polymers in the dilute phase that have never entered the dense phase. We fit ${\displaystyle S(t)}$ to a decaying exponential to extract the decay time ${\displaystyle \tau }$. The interface conductance ${\displaystyle \kappa }$ is then calculated using Equation 54 with the measured decay time and dilute- and dense-phase parameters.

In Figure 3, we set the criterion for a polymer to have entered the dense phase as being continuously connected to the dense-phase cluster for a duration longer than ${\displaystyle 10\tau }$, where ${\displaystyle \tau }$ is the average bond lifetime of an isolated A-B sticker pair. Briefly, the bond lifetime of an isolated pair is obtained by simulating a bound pair of A-B stickers in a box and recording the time when they first separate by the cutoff distance of the attractive interaction ${\displaystyle r_0=1\text{nm}}$. The mean bond lifetime ${\displaystyle \tau }$ is found by averaging results of 1000 replicates with different random seeds. In Appendix 1—figure 1, we compare the results for the interface conductance ${\displaystyle \kappa }$ using alternative durations, ${\displaystyle >\tau }$ and ${\displaystyle >20\tau }$, as criteria for joining the dense phase. The value of ${\displaystyle \kappa }$ changes very little between the ${\displaystyle 10\tau }$ and ${\displaystyle 20\tau }$ criteria, suggesting that the results in Figure 3 are robust to the definition of ‘joining’ the dense phase, provided very short-lived bonds are neglected.

As an alternative approach, we calculated ${\displaystyle \kappa }$ by directly measuring the flux ${\displaystyle j}$ of molecules that enter the dense phase and then using ${\displaystyle \kappa =j/c_{\text{den}}}$ (Equation 57). To find this flux, we first define an entering event as occurring when a molecule starting from the dilute phase joins the dense-phase cluster and stays for a duration longer than 10 times the average bond lifetime for an isolated A-B sticker pair. We count the number of total entering events ${\displaystyle N}$ in a simulation (note that some molecule can enter the dense phase multiple times), and the flux is then ${\displaystyle N/(2AT)}$, where ${\displaystyle A}$ is the cross-sectional area of the interface and ${\displaystyle T}$ is the duration of the simulation. We show in Appendix 1—figure 2 that the values of ${\displaystyle \kappa }$ obtained via this method are consistent with the results reported in Figure 3C.

We show in Appendix 2—figure 3, a few representative trajectories of A6B6 (top) and A10B6 (bottom) polymers ‘bouncing’ off the interface between dilute and dense phases. More bouncing events per unit time are observed in the A6B6 system compared to A10B6 system, consistent with the presence of a larger interface resistance in the A6B6 system.

Appendix 2—figure 1

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Appendix 2—figure 2

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Appendix 2—figure 3

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Simulations of the A6B6 droplet system largely follow their counterparts in the slab geometry. 2000 polymers were placed inside a cubic box with periodic boundary conditions. Half of the polymers are initially confined in a sphere of radius 55 nm and the other half kept outside. The attraction between A and B stickers is gradually turned on from ${\displaystyle U_0=0}$ to 14 over ${\displaystyle 4\times {10}^7}$ time steps. The system is equilibrated at fixed ${\displaystyle U_0=14}$ for another ${\displaystyle 1\times {10}^7}$ steps and then the spherical confinement is removed. The above procedures ensure the formation of a single droplet and a uniform dilute phase at a desired concentration. We started with simulation boxes of side lengths 300 nm and 275 nm, which correspond to initial dilute-phase concentrations of 0.06 mM and 0.08 mM. The droplets shrank and grew accordingly over time in these simulations, which allowed us to extrapolate to the correct box size of side length 286 nm for a stable droplet (Appendix 2—figure 4). The system is equilibrated for ${\displaystyle 5\times {10}^7}$ more time steps to allow equilibration between the dilute and dense phases. We then record the positions of all particles every ${\displaystyle 1\times {10}^6}$ steps for 600 recordings. 10 simulation replicates were performed.

Appendix 2—figure 4

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To obtain the concentration profile in Figure 4B, we identify the center of mass of the droplet and recenter the simulation box to this center of mass in each recording. We then compute the time- and ensemble-averaged polymer concentration histogram along the radial direction with a bin size 5.5 nm. The dilute- and dense-phase concentrations (${\displaystyle c_{\text{dil}}^{\text{d}}}$ and ${\displaystyle c_{\text{den}}^{\text{d}}}$) of the droplet system are calculated by averaging the concentration profile over the regions ${\displaystyle r\ge 55\text{nm}}$ and ${\displaystyle r\le 12.5\text{nm}}$, respectively. To obtain the concentration profile of the bleached population at time ${\displaystyle t}$ after photobleaching in Figure 4C, we label all polymers in the droplet at time ${\displaystyle t_0}$ as bleached and track the concentration profile of these polymers at a later time ${\displaystyle t_0+t}$. Results are averaged over all possible choices of ${\displaystyle t_0}$ and over 10 simulation replicates. Interface conductance of the droplet system is determined using the flux method. We count the total number of entering events, N, in which a polymer starts from the dilute phase and joins the droplet for a duration longer than 10τ (${\displaystyle 6\times {10}^6}$ steps). ${\displaystyle {\kappa }^{\text{d}}}$ is then ${\displaystyle N/(AT)}$, where ${\displaystyle A}$ is the surface area of the droplet and ${\displaystyle T}$ is the duration of the simulation.

To obtain the theory curve in Figure 4D, we numerically integrate Equation 1 using a finite-difference method. We used the modified initial and boundary conditions:

and

to account for the system’s finite size with parameters: ${\displaystyle c_{\text{den}}^{\text{d}}=8.1\text{mM}}$, ${\displaystyle c_{\text{dil}}^{\text{d}}=0.073\text{mM}}$, ${\displaystyle {\displaystyle D_{\text{den}}=0.013\text{µ}{\text{m}}^2/\text{s}}}$, ${\displaystyle {\displaystyle D_{\text{dil}}=17\text{µ}{\text{m}}^2/\text{s}}}$, ${\displaystyle {\displaystyle R=0.037\text{µ}\text{m}}}$, and ${\displaystyle {\displaystyle {\kappa }^{\text{d}}=0.20\text{µ}\text{m}/\text{s}}}$ directly extracted from simulations, and a spherical confinement of radius ${\displaystyle {\displaystyle R_{\text{box}}=0.177\text{µ}\text{m}}}$, which corresponds to the same volume as the cubic simulation box. Equation 1 was integrated over time with a forward Euler scheme with a radial step of ${\displaystyle 0.5\text{nm}}$ and a time step of ${\displaystyle 5\text{ns}}$ to ensure numerical stability and accuracy. The resulting bleached concentration profile is then used in conjunction with Equation 4 to obtain the FRAP recovery curve.

To derive Equation 16 in the main text, we note that in the interface-resistance-dominant regime diffusion in the dilute and dense phases are both fast, therefore the bleached molecules can be assumed to have uniform concentration profiles, ${\displaystyle c_{\text{in}}(t)}$ and ${\displaystyle c_{\text{out}}(t)}$, inside and outside of the droplet, respectively. The net outward flux across the interface can then be written as ${\displaystyle j(t)=\kappa [c_{\text{in}}(t)-(c_{\text{den}}/c_{\text{dil}})c_{\text{out}}(t)]}$, which reduces the total number of bleached molecules inside the droplet according to:

The above equation together with the total number of bleached molecules

and the initial condition ${\displaystyle c_{\text{in}}(0)=c_{\text{den}}}$ can be solved to yield an analytical solution for ${\displaystyle c_{\text{in}}(t)}$:

where

corresponds to the maximum recovery intensity and ${\displaystyle \tau =R/(3\kappa )}$ corresponds to the recovery time for a system of infinite size in the interface-resistance-dominant regime. The FRAP recovery curve is then

We note that in the limit where the system is infinite in size (${\displaystyle R_{\text{box}}\to \mathrm{\infty }}$), we recover the familiar fluorescence recovery curve in Equation 5.