The exchange dynamics of biomolecular condensates
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This valuable contribution studies factors that impact molecular exchange between dense and dilute phases of biomolecular condensates through continuum models and coarsegrained simulations. The authors provide convincing evidence that the bouncing of molecules off the interface can lead to interfacial resistance and limit mixing. Results like these can inform how experimental results in the field of biological condensates are interpreted.
https://doi.org/10.7554/eLife.91680.3.sa0Valuable: Findings that have theoretical or practical implications for a subfield
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Abstract
A hallmark of biomolecular condensates formed via liquidliquid phase separation is that they dynamically exchange material with their surroundings, and this process can be crucial to condensate function. Intuitively, the rate of exchange can be limited by the flux from the dilute phase or by the mixing speed in the dense phase. Surprisingly, a recent experiment suggests that exchange can also be limited by the dynamics at the droplet interface, implying the existence of an ‘interface resistance’. Here, we first derive an analytical expression for the timescale of condensate material exchange, which clearly conveys the physical factors controlling exchange dynamics. We then utilize stickerspacer polymer models to show that interface resistance can arise when incident molecules transiently touch the interface without entering the dense phase, i.e., the molecules ‘bounce’ from the interface. Our work provides insight into condensate exchange dynamics, with implications for both natural and synthetic systems.
Introduction
The interior of cells is organized in both space and time by biomolecular condensates, which form and dissolve as needed (Shin and Brangwynne, 2017; Banani et al., 2017). These condensates play key roles in processes ranging from transcription to translation, metabolism, signaling, and more (An et al., 2008; Su et al., 2016; Sabari et al., 2018; Formicola et al., 2019). The complex interactions among their components endow condensates with distinct physical properties, including low surface tension, viscoelasticity, aging, etc. These distinct properties are crucial to the ability of condensates to carry out their biological functions. Here, we focus on one important physical property of condensates – the rate of exchange of material between condensed and dilute phases. This rate of exchange can impact biochemical processes taking place in condensates by limiting the escape of completed products (e.g. ribosomes produced in nucleoli; Yao et al., 2019), or limiting the availability of components or regulatory molecules (e.g. snoRNAs and ribosomal proteins entering nucleoli, or mRNAs entering P bodies or stress granules). The rate of exchange can also control the dynamical response of condensates to a changing environment, and, as exchange between dense and dilute phase is central to coarsening via Ostwald ripening, it can regulate the number, size, and location of condensates within the cell.
The material exchange between a condensate and the surrounding dilute phase can be probed via FRAP experiments, a commonly used approach for measuring condensate fluidity and molecular diffusion coefficients. Exchange dynamics are thus readily measurable and have been reported for a variety of systems (Li et al., 2012; Patel et al., 2015; Burke et al., 2015; Banani et al., 2016; Jain et al., 2016; Aumiller et al., 2016). However, only a very limited number of studies (Taylor et al., 2019; Hubatsch et al., 2021; Bo et al., 2021; Folkmann et al., 2021; Lee, 2021) aimed to understand what controls the timescales of condensate component exchange. Briefly, Taylor et al., 2019 combined FRAP experiments on condensates in vitro and in vivo with different theoretical models to examine the impact of model choice on the physical parameters derived from data fitting (Taylor et al., 2019). Folkmann et al., 2021 and Lee, 2021 proposed that the rate of molecular absorption to the condensate can be ‘conversionlimited’ instead of diffusionlimited and established a mathematical framework for the temporal evolution of droplet sizes in this limit (Folkmann et al., 2021; Lee, 2021). In all these cases, the modeling of interface resistance (Taylor et al., 2019) or conversionlimited material transfer (Folkmann et al., 2021; Lee, 2021) was conducted at the phenomenological level, without aiming to understand the underlying physical mechanism that gives rise to interface resistance. Hubatsch et al., 2021 and Bo et al., 2021 tackled the exchange dynamics problem by developing, respectively, a continuum theory of macroscopic phase separation (Hubatsch et al., 2021) and a stochastic Langevin equation of singlemolecule trajectories (Bo et al., 2021). However, the meanfield approaches in Hubatsch et al., 2021 and Bo et al., 2021 neglect the potentially complex dynamics of molecules at the condensate interface, which can slow down material exchange significantly as suggested by Taylor et al., 2019 and Folkmann et al., 2021.
In the following, we first derive an analytical expression for the timescale of condensate material exchange, which conveys a clear physical picture of what controls this timescale. We then utilize a ‘stickerspacer’ polymer model to investigate the mechanism of interface resistance. We find that a large interface resistance can occur when molecules bounce off the interface rather than being directly absorbed. We finally discuss the characteristic features of the FRAP recovery pattern of droplets when the exchange dynamics is limited by different factors.
Results
Mathematical formulation of exchange dynamics
The exchange of molecules between a condensate and the dilute phase can be investigated through FRAPtype 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 $c(r,t)$ of the molecules initially located in a spherical condensate of radius $R$ (bleached population) can be described by the following continuum diffusion equations (Taylor et al., 2019):
with the initial condition:
and boundary conditions:
where $D}_{\text{den}$ and $D}_{\text{dil}$ are, respectively, the diffusion coefficients of molecules in the dense and dilute phases, and $c}_{\text{den}$ and $c}_{\text{dil}$ 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 ${k}_{}c({R}_{},t){k}_{+}c({R}_{+},t)$, where $k}_{+/$ denotes the entering/exiting rate of molecules at the interface and $c({R}_{+/},t)$ the concentration of bleached molecules immediately outside/inside of the boundary. At thermal equilibrium, this net flux goes to zero, i.e., $k}_{}{c}_{\text{den}}={k}_{+}{c}_{\text{dil}$ so ${k}_{+}={k}_{}({c}_{\text{den}}/{c}_{\text{dil}})$. The net outward flux is therefore ${k}_{}[c({R}_{},t)({c}_{\text{den}}/{c}_{\text{dil}})c({R}_{+},t)]$. The parameter $\kappa \equiv {k}_{}$ 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 $\kappa$ can kinetically limit the flux going through the interface. We therefore term $\kappa$ 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 $t$ is
Clearly, how quickly $f(t)$ recovers from 0 to 1 quantifies the timescale of material exchange between the condensate and the surrounding dilute phase.
Timescale of condensate component exchange
The authors of Taylor et al., 2019 derived an exact solution for $f(t)$ 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 $D}_{\text{dil}}/{D}_{\text{den}$ in the range of 10^{2}–10^{5} (Freeman Rosenzweig et al., 2017; Taylor et al., 2019). We therefore employed the exact solution to derive an approximate solution in the parameter regime $D}_{\text{dil}}\gg {D}_{\text{den}$:
where the timescale of fluorescence recovery is given by
Please refer to Appendix 1 for a detailed derivation. We note that, in practice, $D}_{\text{dil}}>20{D}_{\text{den}$ is sufficient for the validity of the approximation with the approximate $\tau$ 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 $\tau \simeq {R}^{2}/({\pi}^{2}{D}_{\text{den}})$. Second, if instead diffusion in the dilute phase is sufficiently slow, or the concentration in the dilute phase is very low, then $\tau \simeq {c}_{\text{den}}{R}^{2}/(3{c}_{\text{dil}}{D}_{\text{dil}})$, 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 $\kappa$ is very small, the interfacial flux can be rate limiting for exchange, yielding $\tau \simeq R/(3\kappa )$.
Can interface resistance be much larger than predicted by meanfield theory?
What determines the magnitude of the interface conductance $\kappa$? 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 singlemolecule 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 FloryHuggins and CahnHilliardtype meanfield theories. Indeed, if we start with the continuum approach in Hubatsch et al., 2021, where the concentration of bleached components $c(\mathbf{\text{{r}}},t)$ is governed by
with ${c}_{\text{eq}}(\mathbf{\text{{r}}})$ the equilibrium concentration profile and $D[{c}_{\text{eq}}(\mathbf{\text{{r}}})]$ the diffusion coefficient which depends on the local equilibrium concentration, one can obtain an expression for $\kappa$ (see Appendix 1):
where the integral is over the interface region. We would then conclude ${\kappa}^{1}<\delta /{D}_{\text{den}}+\delta {c}_{\text{den}}/({c}_{\text{dil}}{D}_{\text{dil}})$, where $\delta$ 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 LAF1 protein droplets (Taylor et al., 2019) contradicts the above meanfield result. In the experiment, a micronsized LAF1 droplet ($\displaystyle R=1\phantom{\rule{thinmathspace}{0ex}}\text{\xb5}\text{m}$) 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 meanfield approach to be $\tau ={R}^{2}/({\pi}^{2}{D}_{\text{den}})+{c}_{\text{den}}{R}^{2}/(3{c}_{\text{dil}}{D}_{\text{dil}})=64\pm 18\phantom{\rule{thinmathspace}{0ex}}\text{s}$, 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 meanfield theory, and if so, what could be the underlying mechanisms and how does the interface resistance depend on the microscopic features of phaseseparating molecules?
Coarsegrained simulation of ‘stickerspacer’ polymer phase separation
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 ‘stickerspacer’ polymer model (Choi et al., 2020; Semenov and Rubinstein, 1998) to explore one possible mechanism in which molecules can assume nonsticking 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 ‘stickerspacer’ 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 $r$ is the distance between two stickers. Onetoone 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 manytoone binding:
We take $K=0.15\phantom{\rule{thinmathspace}{0ex}}{k}_{\text{B}}T/{\text{nm}}^{2}$, $R}_{0}=10\phantom{\rule{thinmathspace}{0ex}}\text{nm$, ${U}_{0}=14\phantom{\rule{thinmathspace}{0ex}}{k}_{\text{B}}T$, $r}_{0}=1\phantom{\rule{thinmathspace}{0ex}}\text{nm$, $\u03f5=1\phantom{\rule{thinmathspace}{0ex}}{k}_{\text{B}}T$, $\sigma =2\phantom{\rule{thinmathspace}{0ex}}\text{nm}$, and ${r}_{\text{c}}=1.12\sigma$ in all simulations, except in the simulations of Figure 3F where we vary $U}_{0$ systematically from 13.5 to $15\phantom{\rule{thinmathspace}{0ex}}{k}_{\text{B}}T$. For all simulation results we reported below, the standard error of the mean is typically smaller than the symbol size and therefore not shown.
For each of the five sequences shown in Figure 2A, we simulated 1000 polymers in a $500\phantom{\rule{thinmathspace}{0ex}}\text{nm}\times 50\phantom{\rule{thinmathspace}{0ex}}\text{nm}\times 50\phantom{\rule{thinmathspace}{0ex}}\text{nm}$ box with periodic boundary conditions using Langevin dynamics (see Appendix 2 for details). Simulations were performed using LAMMPS moleculardynamics 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 timeaveraged profile of the total polymer concentration. The five different polymer sequences we simulated were chosen to yield a range of dilute and densephase sticker concentrations (Figure 2D) as well as a range of dilute and densephase 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 dilutephase concentrations. This follows because it is entropically unfavorable for these polymers to form multiple selfbonds, which favors the dense phase where these polymers can readily form multiple transbonds. These longblock polymers also have low densephase diffusion coefficients because of their large number of transbonds, which need to be repeatedly broken for the polymers to diffuse.
Interface conductance $\kappa$ from simulations
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 $\kappa$ by applying the 1D, slabgeometry 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 AB 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 $\tau$ is used together with the known dense and dilute phase parameters to infer $\kappa$ from:
where $d$ is the halfwidth of the dilute phase. As shown in Figure 3C, the resulting values of $\kappa$ 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 selfbonds.
We note that one can alternatively obtain $\kappa$ by directly measuring the flux of molecules that enter the dense phase. Mathematically, this flux equals $k}_{+}{c}_{\text{dil}}=\kappa {c}_{\text{den}$. We show in Appendix 2 that the values of $\kappa$ found via this method are consistent with results reported in Figure 3C.
‘Bouncing’ of molecules can lead to large interface resistance
What gives rise to the very different values of $\kappa$? To address this question, we first consider the predicted interface conductance $\kappa}_{0$ if polymers incident from the dilute phase simply move through the interface region with a local diffusion coefficient that crosses over from $D}_{\text{dil}$ to $D}_{\text{den}$. Then according to Equation 8 (see Appendix 1)
However, as shown in Figure 3D, the actual values of $\kappa$ in our simulations can be a factor of ∼50 smaller than $\kappa}_{0$. 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 densephase 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 stickerspacer 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 stickersticker binding strength, the dilute and densephase polymer concentrations, and the width of the interface:
where $n$ is the number of monomers in a polymer, $s$ is the global stoichiometry (i.e. $c}_{\text{A}}/{c}_{\text{B}$), $f}_{\text{dilA/dilB}$ and $f}_{\text{denA/denB}$ are the fractions of unbound $\text{A}/\text{B}$ monomers in the dilute and dense phases, respectively. In support of this picture, we find that all our simulation results for $\kappa /{\kappa}_{0}$ collapse as a linear function of a lumped parameter $u$ (Figure 3D):
which expresses the availability of free stickers, where all parameters in $u$ 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 equalstoichiometry sequence A6B6, we find that the extra A stickers substantially increase the interface conductance $\kappa$. 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 $U}_{0$ 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 $\kappa$ of the A6B6 system indeed drops by a factor of 5 as the value of $U}_{0$ increases from 13.5 to $15\phantom{\rule{thinmathspace}{0ex}}{k}_{\text{B}}T$.
Direct simulation of droplet FRAP
Above we simulated phase separation of stickerspacer polymers in a slab geometry, and discussed how the extracted interface conductance $\kappa$ 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 $286\phantom{\rule{thinmathspace}{0ex}}\text{nm}$ 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 dilutephase 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 $t=0$, 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 $A\approx 0.5$ 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.
Fitting the recovery curve by
which takes into account the effect of the finitesize dilute phase (see Appendix 2 for a derivation), yielded a recovery time $\tau =0.071\phantom{\rule{thinmathspace}{0ex}}\text{s}$. We compare the measured FRAP recovery time for the small droplet $R=37\phantom{\rule{thinmathspace}{0ex}}\text{nm}$ (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 $\displaystyle \kappa =0.14\phantom{\rule{thinmathspace}{0ex}}\text{\xb5}\text{m}/\text{s}$ lower than the measured $\displaystyle {\kappa}^{\text{d}}=0.20\phantom{\rule{thinmathspace}{0ex}}\text{\xb5}\text{m}/\text{s}$ of the droplet system, which in turn reflects the difference in the dilutephase concentrations of the two systems, as $\kappa \sim {c}_{\text{dil}}$ from Equation 14.
Signatures of interface resistance
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 FRAPtype experiments, making it very difficult to measure $\kappa$. However, as shown in Figure 3, the bouncing effect can reduce $\kappa$ 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 dilutephase 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 $R$ while the other terms increase as $R}^{2$ (Figure 5A). Therefore, one expects a crossover for the recovery from being interfaceresistance dominated (small $R$) to being either dilutephasediffusion or densephasemixing dominated (large $R$). 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 crossover 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 LAF1 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 crossover for LAF1 droplets to be around $R=71\phantom{\rule{thinmathspace}{0ex}}\mu \text{m}$, which in principle can be tested experimentally.
Discussion
The dynamic exchange of condensate components with the surroundings is a key feature of membraneless organelles, and can significantly impact condensate biological function. In this work, we combined analytical theory and coarsegrained simulations to uncover physical mechanisms that can control this exchange dynamics. Specifically, we first derived an analytical expression for the exchange rate, which conveys the clear physical picture that this rate can be limited by the flux of molecules from the dilute phase, by the speed of mixing inside the dense phase, or by the dynamics of molecules at the droplet interface. Motivated by recent FRAP measurements (Taylor et al., 2019) that the exchange rate of LAF1 droplets can be limited by interface resistance, which contradicts predictions of conventional meanfield theory, we investigated possible physical mechanisms underlying interface resistance using a ‘stickerspacer’ model. Specifically, we demonstrated via simulations a notable example in which incident molecules have formed all possible internal bonds, and thus bounce from the interface, giving rise to a large interface resistance. Finally, we discussed the signatures in FRAP recovery patterns when the exchange dynamics is limited by different factors.
What are potential mechanisms that could lead to the bouncing of molecules from the interface and hence to a substantial interface resistance? The essential requirement is that molecules in the dilute phase and molecules at the interface should not present ‘sticky’ surfaces to each other. Since these same molecules must be capable of sticking to each other in order to phase separate, a natural scenario is that these molecules assume nonsticky conformations due to the shielding of interacting regions, e.g., burial of hydrophobic residues in the core of a protein, or, in the scenario explored in the simulations, the saturation of stickerlike bonds. Examples of systems with strong enough bonds to allow bond saturation include SIMSUMO (Banani et al., 2016) and nucleic acids with strong intramolecular basepairing. Interestingly, a recent coarsegrained simulation of RNA droplets of (CAG)_{47} (Nguyen et al., 2022) illustrated that a (CAG)_{47} molecule in a closed hairpin conformation fails to integrate into a droplet but rather bounces off the droplet interface. Another possible scenario is that charged molecules could arrange themselves to form a charged layer at the interface, resulting in a high energetic barrier from electrostatic repulsion for a dilutephase component to reach and cross the interface (Ray et al., 2023; Dai et al., 2023; Majee et al., 2024). In the case of LAF1, we note that the values of interface conductance $\kappa$ obtained in our simulations are a factor of 10^{3} to 10^{4} higher than the experimentally measured $\kappa$ for the LAF1 droplet. While we do not aim to specifically simulate the LAF1 system in this work and the value of $\kappa$ in simulations can in principle be tuned by adjusting the bond strength $U}_{0$, the large disparity between simulation and experiment renders the mechanism responsible for the inferred large interface resistance in LAF1 droplets unclear. We hope that our study will motivate further experimental investigations into the anomalous exchange dynamics of LAF1 droplets and potentially other condensates, and the mechanisms underlying interface resistance.
In this work, we focused on the exchange dynamics of in vitro singlecomponent condensates. How is the picture modified for condensates inside cells? It has been shown that Ddx4YFP droplets in the cell nucleus exhibit negligible interface resistance (Taylor et al., 2019), which raises the question whether interface resistance is relevant to natural condensates in vivo. Future quantitative FRAP and singlemolecule tracking experiments on different types of droplets in the cell will address this question. One complication is that condensates in cells are almost always multicomponent, which can increase the complexity of the exchange dynamics. Interestingly, formation of multiple layers or the presence of excess molecules of one species coating the droplet is likely to increase interface resistance. A notable example is the Pickering effect, in which adsorbed particles partially cover the interface, thereby reducing the accessible area and the overall condensate surface tension, slowing down the exchange dynamics (Folkmann et al., 2021). The development of theory and modeling for the exchange dynamics of multicomponent condensates is currently underway.
Biologically, the interface exchange dynamics also influences the coarsening of condensates. The same interface resistance that governs exchange between phases at equilibrium will control the flux of material from the dilute phase to the dense phase during coarsening, so that bouncing will slow down the coarsening process. Indeed, a recent theoretical study (Ranganathan and Shakhnovich, 2020) of coarsening via mergers of small polymer clusters found anomalously slow coarsening dynamics due to exhaustion of binding sites, paralleling the singlepolymer bouncing effect explored here. Other mechanisms that may slow coarsening include the formation of metastable microemulsions (Welsh et al., 2022; Kelley et al., 2021) and the Pickering effect (Folkmann et al., 2021) mentioned above. In the latter study, additional slow coarsening of PGL3 condensates was attributed to a conversionlimited (i.e. interface resistance) rather than a diffusionlimited flux of particles from the dilute phase into the dense phase. Interestingly, a conversionlimited flux has been shown to lead to qualitatively distinct scaling of condensate size with time (Lee, 2021). As many condensates dissolve and reform every cell cycle (or as needed), we anticipate that interfacial exchange will constitute an additional means of regulating condensate dynamics.
Methods
We perform coarsegrained moleculardynamics 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 onetoone binding between A and B stickers. For each of the five selected polymer sequences, we perform 10 simulation replicates with different random seeds in a slab geometry. Consistency of results is checked across replicates and across the first and second halves of the recorded data. The agreement indicates that the system has reached equilibrium. For details see Appendix 2, Simulation procedures and data recording.
To measure the dilute and densephase 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 $r}_{0}=1\phantom{\rule{thinmathspace}{0ex}}\text{nm$. Connected stickers are then grouped into clusters. To find the concentrations of each phase, we identify the center of mass of the largest cluster in each recording, and recenter the simulation box to this center of mass. The resulting polymer concentration profile has high values in the middle corresponding to the densephase concentration, and low values on the two sides corresponding to the dilutephase concentration (Figure 2C). For details, see Appendix 2, Determining the dilute and densephase concentrations.
To measure the dilute and densephase diffusion coefficients, we perform simulations with a pure dilute phase or dense phase, i.e., with polymers at the measured dilute and densephase concentrations. To find the diffusion coefficients, we compute the timeaveraged mean squared displacement (MSD) for each polymer as a function of the lag time $t}_{\text{lag}$, and average over all polymers in a simulation box and over five replicates. The time and ensembleaveraged MSD is then linearly fit to $\text{MSD}=6D{t}_{\text{lag}}$ to extract the diffusion coefficient. For details, see Appendix 2, Determining the dilute and densephase diffusion coefficients.
To measure the interface conductance $\kappa$, we follow the simple protocol depicted in Figure 3A. Specifically, in this protocol we first define a ‘survival’ variable $S$ for each polymer in the dilute phase as a function of time: $S=1$ if the polymer has remained in the dilute phase, and $S=0$ if the polymer has ever entered the densephase cluster. The obtained $S(t)$ is the average survival probability of polymers in the dilute phase that have never entered the dense phase. We fit $S(t)$ to a decaying exponential to extract the decay time $\tau$. The interface conductance $\kappa$ is then calculated using Equation 12 with the measured decay time and dilute and densephase parameters. For details, see Appendix 2, Determining the interface conductance.
Simulations of the A6B6 spherical droplet system largely follow their counterparts in the slab geometry. 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 ensembleaveraged polymer concentration histogram along the radial direction. The dilute and densephase concentrations ($c}_{\text{dil}}^{\text{d}$ and $c}_{\text{den}}^{\text{d}$) of the droplet system are calculated by averaging the concentration profile over the relevant regions. To obtain the concentration profile of the bleached population at time $t$ after photobleaching in Figure 4C, we label all polymers in the droplet at time $t}_{0$ as bleached and track the concentration profile of these polymers at a later time ${t}_{0}+t$. Results are averaged over all possible choices of $t}_{0$. To obtain the theory curve in Figure 4D, we numerically integrate Equation 1 using a finitedifference method. Interface conductance of the droplet system is determined using the flux method. For details, see Appendix 2, Details of simulation and theory of FRAP recovery of an A6B6 droplet.
The codes for generating simulated data following the abovementioned methods are uploaded as Source code 1. Source code 1 contains MATLAB codes used to generate input files for the LAMMPS Molecular Dynamics Simulator, as well as the generated LAMMPS input files. All data in the manuscript can be reproduced using these files. LAMMPS input files are contained in the folders: FullSystem, DensePhase, and DilutePhase. Codes in FullSystem/In are for simulations in slab geometry at a fixed interaction strength ${U}_{0}=14\phantom{\rule{thinmathspace}{0ex}}{k}_{\text{B}}T$, which generate the data shown in Figures 2 and 3. Codes in FullSystem/In_A are for simulations in slab geometry at varying interaction strengths, which generate the data shown in Figure 2F. Codes in FullSystem/In_Droplet are for simulations of a 3D droplet, which generate the data shown in Figure 4. Codes in DensePhase and DilutePhase are used to measure the diffusion coefficients of molecules in dense and dilute phases, respectively, which generate the data shown in Figure 2E.
Appendix 1
Derivation of the FRAP recovery curve in Equations 5 and 6
The exact solution for $f(t)$ in Equation 4, the fraction of molecules in a spherical condensate of radius $R$ which are unbleached at time $t$, is derived by Taylor et al., 2019, in an integral form using Laplace transforms (Taylor et al., 2019),
where
and $\lambda ={D}_{\text{dil}}/{D}_{\text{den}}$, $\alpha ={c}_{\text{den}}/{c}_{\text{dil}}$, $t}^{\ast}=t{D}_{\text{den}}/{R}^{2$, and $k=R\kappa /{D}_{\text{den}}$, where $\kappa$ is the interface conductance.
To obtain a more intuitive result, we first rearrange $g(u,{t}^{\ast})$ as
We note that the diffusion of biomolecules in the dilute phase is typically much faster than diffusion in the dense phase, i.e., $\lambda ={D}_{\text{dil}}/{D}_{\text{den}}\gg 1$. In this parameter regime, $g(u,{t}^{\ast})$ is sharply peaked at the values of $u$ where
i.e., when the second term in the denominator of $g(u,{t}^{\ast})$ becomes 0. Representative $g(u,{t}^{\ast})$ curves are shown in Appendix 1—figure 1. We can therefore approximate $g(u,{t}^{\ast})$ as
where $u}_{n$ is the $n$ th solution of Equation 20 and $a}_{n}\sim 1/\sqrt{\lambda$ is the inverse of effective width of $n$ th peak of $g(u,{t}^{\ast})$. Clearly, the prefactor of the delta function $\delta (u{u}_{n})$ in Equation 21 drops rapidly with increasing values of $u}_{n$. Consequently, the integral in Equation 17 is always dominated by the contribution from the first mode, and therefore
where
with $u}_{1$ the first root of Equation 20.
At any given values of $k$, $\alpha$, and $\lambda$, 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 $K=k/(1+k\alpha /\lambda )$ versus the first root $u}_{1$ in Appendix 1—figure 2. For small $K$, Taylor series expansion around $u=0$ for the left side of Equation 24 yields
which yields $u}_{1}=\sqrt{3K$. For large $K$, $u}_{1$ plateaus at $\pi$. We therefore approximate $u}_{1$ as
This approximate solution is compared with the exact numerical solution of $u}_{1$ in Appendix 1—figure 2. The maximum error of about 5% occurs at an intermediate value of $K$ (Appendix 1—figure 2, inset). The relaxation time $\tau$ corresponding to the approximate solution in Equation 26 is
Time required to replace all molecules in a spherical droplet in the absorbing boundary limit
The time evolution of the spherically symmetric concentration profile $c(r,t)$ of molecules in the dilute phase around a droplet of radius $R$ is given by
If molecules incident from the dilute phase are immediately and irreversibly absorbed into the dense phase, the boundary condition is then $c(R,t)=0$. The steadystate solution of Equation 28 in this absorbingboundary limit is
where $c}_{\text{dil}$ is the concentration at $r\to \mathrm{\infty}$, which yields a total steadystate flux into the droplet of
It then takes a time
to replace all $4\pi {R}^{3}{c}_{\text{den}}/3$ molecules in the droplet.
Derivation of the interface conductance in the continuum limit, yielding Equations 8 and 13
To derive the interface conductance in the continuum limit (which neglects the bouncing effect), we start with the meanfield formulation developed in Hubatsch et al., 2021, where the concentration of bleached components $c(\mathbf{\text{{r}}},t)$ is governed by
with the flux
where ${c}_{\text{eq}}(\mathbf{\text{{r}}})$ is the equilibrium concentration profile, and $D[{c}_{\text{eq}}(\mathbf{\text{{r}}})]$ the diffusion coefficient which depends on the local equilibrium concentration.
For a spherical condensate of radius $R$, 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 $\hat{r}$ 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 $\delta$ from $R}_{$ to $R}_{+$ with ${R}_{+/}=R\pm \delta /2$, then $c}_{\text{eq}}(R\delta /2)={c}_{\text{den}$ and $c}_{\text{eq}}(R+\delta /2)={c}_{\text{dil}$. Substituting Equation 36 into the second boundary condition in Equation 3, we have
We then obtain an expression for the interface conductance $\kappa$:
The equilibrium concentration ${c}_{\text{eq}}(r)$ transitions from $c}_{\text{den}$ to $c}_{\text{dil}$ between $R\delta /2$ and $R+\delta /2$, and the corresponding diffusion coefficient $D(r)$ transitions from $D}_{\text{den}$ to $D}_{\text{dil}$. Assuming a monotonic sigmoidal transition, along with $D}_{\text{den}}\ll {D}_{\text{dil}$ and $c}_{\text{den}}\gg {c}_{\text{dil}$, 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
Derivation of the 1D, slabgeometry versions of Equations 1–6
For the case of a quasi1D slab geometry, we consider the condensate to sit in the middle in the region $l<x<l$ with the simulation box extending along the $x$axis from $L$ to $L$. The time evolution of the concentration profile $c(x,t)$ 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 $\tau =({D}_{\text{den}}{p}_{n}^{2}{)}^{1}=({D}_{\text{dil}}{q}_{n}^{2}{)}^{1}$ is the relaxation time of the $n$ th mode of the system. For given parameters $c}_{\text{dil}$, $c}_{\text{den}$, $D}_{\text{dil}$, $D}_{\text{den}$, $l$, $L$, and $\kappa$, $\tau$ 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 $\tau =R/(3\kappa )$.
We note that, in principle, Equation 49 can be used to infer the value of $\kappa$ for a system using the relaxation time $\tau$ from simulation. However, due to the relatively small simulation sizes, the interface regime can constitute a significant fraction of the densephase condensate. This can result in uncertainties in the determination of the densephase width $l$ and diffusion coefficient $D}_{\text{den}$, 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 ratelimiting 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 densephase parameters. To relate $\kappa$ to the decay time of the dilutephase molecules, we note that the time evolution of the concentration profile $c(x,t)$ of the dilutephase molecules which have never entered the dense phase is given by
with the initial condition $c(x,0)={c}_{\text{dil}}$ for $l<x<L$ and boundary conditions:
Going through a similar procedure as for Equations 46–49, we obtain
The interface conductance $\kappa$ 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 $\tau$ (Figure 3C, top), of molecules that remain in the dilute phase without ever entering the dense phase.
Derivation of the unboundsticker parameter $u$, Equations 14 and 15
In the interfaceresistancedominated regime, the decay time $\tau$ of the number of dilutephase 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 $j$ entering the dense phase
where $V$ is the volume of the dilute phase, and $A$ the crosssectional 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 $j$ 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 $j$ 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 diffusionlimited monomermonomer binding, but with some modifications: First, we can write the concentration of unbound monomers of type A in the dilute phase as
where $n}_{\text{A}$ is the number of type A monomers per polymer and $f}_{\text{dilA}$ is the fraction of these monomers that are unbound. This concentration implies a diffusionlimited binding flux onto each unbound densephase type B monomer in the interface region
where $r}_{0$ 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 $\rho}_{\text{denB, unbound}$ of available unbound Btype monomers in the densephase interface region, since each one will contribute the above flux (Equation 59). We can write
where $\delta$ is the width of the interface region, $n}_{\text{B}$ is the number of type B monomers per polymer, and $f}_{\text{denB}$ is the fraction of unbound B monomers on densephase polymers in this region. Finally, we can combine the above equations, and include the binding of dilutephase Btype monomers to densephase Atype monomers, to obtain
The interface conductance $\kappa$ is then
Using this expression, we can then estimate the ratio between the true $\kappa$ and its continuum limit ${\kappa}_{0}={c}_{\text{dil}}{D}_{\text{dil}}/(\delta {c}_{\text{den}})$ to be
where $n={n}_{\text{A}}+{n}_{\text{B}}$ is the length of a polymer and $s={c}_{\text{A}}/{c}_{\text{B}}$ is the global stoichiometry. Note that we have used $n}_{\text{A}}{n}_{\text{B}}={n}^{2}s/(1+s{)}^{2$ 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 righthand side of Equation 63 as a lumped parameter $u$.
Appendix 2
Simulation procedures and data recording
We perform coarsegrained moleculardynamics 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 onetoone 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 $90\phantom{\rule{thinmathspace}{0ex}}\text{nm}<x<90\phantom{\rule{thinmathspace}{0ex}}\text{nm}$ 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 ${U}_{0}=0$ to 14 over 2.5×10^{7} 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 ${U}_{0}=14$ for another $2.5\times {10}^{7}$ steps and then the confinement is removed. The system is equilibrated for $2\times {10}^{8}$ 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 $2.5\times {10}^{5}$ 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 $\overrightarrow{r}}_{i$ is the coordinate of particle $i$, $m$ is its mass, $\gamma$ is the friction coefficient, $\overrightarrow{f}$ is random thermal noise, and the potential energy $U({\overrightarrow{r}}_{1},...,{\overrightarrow{r}}_{N})$ contains all interactions between particles, including bonds and stickersticker interactions (Equations 9–11). We take temperature $T=300\phantom{\rule{thinmathspace}{0ex}}\text{K}$, damping factor $\tau =m/\gamma =10\phantom{\rule{thinmathspace}{0ex}}\text{ns}$, step size $dt=0.1\phantom{\rule{thinmathspace}{0ex}}\text{ns}$, and mass of particle $m=188.5\phantom{\rule{thinmathspace}{0ex}}\text{ag}$. These parameters give each sticker the correct diffusion coefficient $D={k}_{\text{B}}T/(3\pi \eta d)$, where $\eta$ is the water viscosity $0.001\phantom{\rule{thinmathspace}{0ex}}\text{kg}/\text{m}/\text{s}$ and $d=2\phantom{\rule{thinmathspace}{0ex}}\text{nm}$ 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 dilutephase concentrations derived from the first and second halves of the recorded data. The agreement indicates that the system has reached equilibrium.
Determining the dilute and densephase concentrations
To measure the dilute and densephase 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 $r}_{0}=1\phantom{\rule{thinmathspace}{0ex}}\text{nm$. 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 $x$ 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 densephase concentration, and low values on the two sides corresponding to the dilutephase concentration (Figure 2C). The dilute and densephase concentrations in Figure 2D are calculated by averaging the concentration profile over the regions $({\displaystyle x\le 150\phantom{\rule{thinmathspace}{0ex}}\text{nm}\phantom{\rule{thinmathspace}{0ex}}\mathrm{o}\mathrm{r}\phantom{\rule{thinmathspace}{0ex}}x\ge 150\phantom{\rule{thinmathspace}{0ex}}\mathrm{n}\mathrm{m})}$ and $({\displaystyle 10\phantom{\rule{thinmathspace}{0ex}}\text{nm}\le x\le 10\phantom{\rule{thinmathspace}{0ex}}\text{nm})}$, respectively.
Determining the dilute and densephase diffusion coefficients
To measure the dilute and densephase diffusion coefficients, we perform simulations with a pure dilute phase or dense phase, i.e., with polymers at the measured dilute and densephase concentrations. Specifically, for the dilutephase case, we simulate 750 (A2B2)_{3}, 285 (A3B3)_{2}, 101 A6B6, 104 A8B6, and 180 A10B6 polymers, each in a $150\phantom{\rule{thinmathspace}{0ex}}\text{nm}\times 150\phantom{\rule{thinmathspace}{0ex}}\text{nm}\times 150\phantom{\rule{thinmathspace}{0ex}}\text{nm}$ box with periodic boundary conditions. For the densephase case, we simulate 1000 polymers for all selected sequences in a $W\phantom{\rule{thinmathspace}{0ex}}\text{nm}\times 50\phantom{\rule{thinmathspace}{0ex}}\text{nm}\times 50\phantom{\rule{thinmathspace}{0ex}}\text{nm}$ box with periodic boundary conditions, where $W=116.1$ 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 ${U}_{0}=0$ to $14\phantom{\rule{thinmathspace}{0ex}}{k}_{\text{B}}T$ over $2.5\times {10}^{7}$ time steps and equilibrated at fixed ${U}_{0}=14\phantom{\rule{thinmathspace}{0ex}}{k}_{\text{B}}T$ for $2\times {10}^{8}$ more time steps. We then record the displacement of all particles every $2.5\times {10}^{5}$ 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 timeaveraged MSD for each polymer as a function of the lag time $t}_{\text{lag}$, and average over all polymers in a simulation box and over five replicates. The time and ensembleaveraged MSD is then linearly fit to $\text{MSD}=6D{t}_{\text{lag}}$ to extract the diffusion coefficient.
Determining the interface conductance
To measure the interface conductance $\kappa$, 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 ratelimiting for overall system relaxation. Specifically, in this protocol we first define a ‘survival’ variable $S$ for each polymer as a function of time: $S=1$ if the polymer belongs to any dilutephase cluster (including a solo cluster), and $S=0$ if the polymer is in the densephase cluster. Next, for all polymers starting with $S=1$ (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 AB pair) for which its $S$ value is always 0 (i.e. the polymer has joined the densephase cluster). If yes, we set $S=1$ at the time points before the joining event and $S=0$ at all times afterward. If not, we set $S=1$ for this polymer for all time points. We then average $S(t)$ over all polymers starting with $S=1$ and over the 10 simulation replicates. The obtained $S(t)$ is the average survival probability of polymers in the dilute phase that have never entered the dense phase. We fit $S(t)$ to a decaying exponential to extract the decay time $\tau$. The interface conductance $\kappa$ is then calculated using Equation 54 with the measured decay time and dilute and densephase parameters.
In Figure 3, we set the criterion for a polymer to have entered the dense phase as being continuously connected to the densephase cluster for a duration longer than $10\tau$, where $\tau$ is the average bond lifetime of an isolated AB sticker pair. Briefly, the bond lifetime of an isolated pair is obtained by simulating a bound pair of AB stickers in a box and recording the time when they first separate by the cutoff distance of the attractive interaction $r}_{0}=1\phantom{\rule{thinmathspace}{0ex}}\text{nm$. The mean bond lifetime $\tau$ 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 $\kappa$ using alternative durations, $>\tau$ and $>20\tau$, as criteria for joining the dense phase. The value of $\kappa$ changes very little between the $10\tau$ and $20\tau$ criteria, suggesting that the results in Figure 3 are robust to the definition of ‘joining’ the dense phase, provided very shortlived bonds are neglected.
As an alternative approach, we calculated $\kappa$ by directly measuring the flux $j$ of molecules that enter the dense phase and then using $\kappa =j/{c}_{\text{den}}$ (Equation 57). To find this flux, we first define an entering event as occurring when a molecule starting from the dilute phase joins the densephase cluster and stays for a duration longer than 10 times the average bond lifetime for an isolated AB sticker pair. We count the number of total entering events $N$ in a simulation (note that some molecule can enter the dense phase multiple times), and the flux is then $N/(2AT)$, where $A$ is the crosssectional area of the interface and $T$ is the duration of the simulation. We show in Appendix 1—figure 2 that the values of $\kappa$ 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.
Details of simulation and theory of FRAP recovery of an A6B6 droplet
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 ${U}_{0}=0$ to 14 over $4\times {10}^{7}$ time steps. The system is equilibrated at fixed ${U}_{0}=14$ for another $1\times {10}^{7}$ 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 dilutephase 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 $5\times {10}^{7}$ more time steps to allow equilibration between the dilute and dense phases. We then record the positions of all particles every $1\times {10}^{6}$ steps for 600 recordings. 10 simulation replicates were performed.
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 ensembleaveraged polymer concentration histogram along the radial direction with a bin size 5.5 nm. The dilute and densephase concentrations ($c}_{\text{dil}}^{\text{d}$ and $c}_{\text{den}}^{\text{d}$) of the droplet system are calculated by averaging the concentration profile over the regions $r\ge 55\phantom{\rule{thinmathspace}{0ex}}\text{nm}$ and $r\le 12.5\phantom{\rule{thinmathspace}{0ex}}\text{nm}$, respectively. To obtain the concentration profile of the bleached population at time $t$ after photobleaching in Figure 4C, we label all polymers in the droplet at time $t}_{0$ as bleached and track the concentration profile of these polymers at a later time ${t}_{0}+t$. Results are averaged over all possible choices of $t}_{0$ 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 ($6\times {10}^{6}$ steps). $\kappa}^{\text{d}$ is then $N/(AT)$, where $A$ is the surface area of the droplet and $T$ is the duration of the simulation.
To obtain the theory curve in Figure 4D, we numerically integrate Equation 1 using a finitedifference method. We used the modified initial and boundary conditions:
and
to account for the system’s finite size with parameters: $c}_{\text{den}}^{\text{d}}=8.1\phantom{\rule{thinmathspace}{0ex}}\text{mM$, $c}_{\text{dil}}^{\text{d}}=0.073\phantom{\rule{thinmathspace}{0ex}}\text{mM$, $\displaystyle {D}_{\text{den}}=0.013\phantom{\rule{thinmathspace}{0ex}}\text{\xb5}{\text{m}}^{2}/\text{s}$, $\displaystyle {D}_{\text{dil}}=17\phantom{\rule{thinmathspace}{0ex}}\text{\xb5}{\text{m}}^{2}/\text{s}$, $\displaystyle R=0.037\phantom{\rule{thinmathspace}{0ex}}\text{\xb5}\text{m}$, and $\displaystyle {\kappa}^{\text{d}}=0.20\phantom{\rule{thinmathspace}{0ex}}\text{\xb5}\text{m}/\text{s}$ directly extracted from simulations, and a spherical confinement of radius $\displaystyle {R}_{\text{box}}=0.177\phantom{\rule{thinmathspace}{0ex}}\text{\xb5}\text{m}$, 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 $0.5\phantom{\rule{thinmathspace}{0ex}}\text{nm}$ and a time step of $5\phantom{\rule{thinmathspace}{0ex}}\text{ns}$ 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 interfaceresistancedominant regime diffusion in the dilute and dense phases are both fast, therefore the bleached molecules can be assumed to have uniform concentration profiles, ${c}_{\text{in}}(t)$ and ${c}_{\text{out}}(t)$, inside and outside of the droplet, respectively. The net outward flux across the interface can then be written as $j(t)=\kappa [{c}_{\text{in}}(t)({c}_{\text{den}}/{c}_{\text{dil}}){c}_{\text{out}}(t)]$, 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 $c}_{\text{in}}(0)={c}_{\text{den}$ can be solved to yield an analytical solution for ${c}_{\text{in}}(t)$:
where
corresponds to the maximum recovery intensity and $\tau =R/(3\kappa )$ corresponds to the recovery time for a system of infinite size in the interfaceresistancedominant regime. The FRAP recovery curve is then
We note that in the limit where the system is infinite in size ($R}_{\text{box}}\to \mathrm{\infty$), we recover the familiar fluorescence recovery curve in Equation 5.
Data availability
The current manuscript is a computational study. The codes for generating simulated data are uploaded as Source code 1.
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Article and author information
Author details
Funding
National Science Foundation (PHY1734030)
 Yaojun Zhang
 Andrew GT Pyo
 Ned S Wingreen
National Institutes of Health (R01 GM140032)
 Yaojun Zhang
 Ned S Wingreen
Howard Hughes Medical Institute
 Clifford P Brangwynne
Air Force Office of Scientific Research (FA95502010241)
 Clifford P Brangwynne
Princeton Center for Complex Materials (DMR1420541)
 Clifford P Brangwynne
 Howard A Stone
The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.
Acknowledgements
This work was supported in part by the National Science Foundation, through the Center for the Physics of Biological Function (PHY1734030), NIH Grants R01 GM140032, the Howard Hughes Medical Institute, and the Air Force Office of Scientific Research (FA95502010241 to CPB). YZ and RK were partially supported by a startup fund at Johns Hopkins University. CPB and HAS were partially supported by Princeton University’s Materials Research Science and Engineering Center DMR1420541. We also thank the Princeton Biomolecular Condensate Program for funding support. The authors acknowledge that the work reported in this paper was performed using the Princeton Research Computing resources at Princeton University and the Advanced Research Computing at Hopkins core facility (rockfish.jhu.edu, supported by NSF grant number OAC1920103).
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Further reading

 Physics of Living Systems
We propose the Self Returning Excluded Volume (SREV) model for the structure of chromatin based on stochastic rules and physical interactions. The SREV rules of return generate conformationally defined domains observed by singlecell imaging techniques. From nucleosome to chromosome scales, the model captures the overall chromatin organization as a corrugated system, with dense and dilute regions alternating in a manner that resembles the mixing of two disordered bicontinuous phases. This particular organizational topology is a consequence of the multiplicity of interactions and processes occurring in the nuclei, and mimicked by the proposed return rules. Single configuration properties and ensemble averages show a robust agreement between theoretical and experimental results including chromatin volume concentration, contact probability, packing domain identification and size characterization, and packing scaling behavior. Model and experimental results suggest that there is an inherent chromatin organization regardless of the cell character and resistant to an external forcing such as RAD21 degradation.

 Developmental Biology
 Physics of Living Systems
Cell division is fundamental to all healthy tissue growth, as well as being ratelimiting in the tissue repair response to wounding and during cancer progression. However, the role that cell divisions play in tissue growth is a collective one, requiring the integration of many individual cell division events. It is particularly difficult to accurately detect and quantify multiple features of large numbers of cell divisions (including their spatiotemporal synchronicity and orientation) over extended periods of time. It would thus be advantageous to perform such analyses in an automated fashion, which can naturally be enabled using deep learning. Hence, we develop a pipeline of deep learning models that accurately identify dividing cells in timelapse movies of epithelial tissues in vivo. Our pipeline also determines their axis of division orientation, as well as their shape changes before and after division. This strategy enables us to analyse the dynamic profile of cell divisions within the Drosophila pupal wing epithelium, both as it undergoes developmental morphogenesis and as it repairs following laser wounding. We show that the division axis is biased according to lines of tissue tension and that wounding triggers a synchronised (but not oriented) burst of cell divisions back from the leading edge.