Physical observables to determine the nature of membrane-less cellular sub-compartments

Version of Record

Accepted for publication after peer review and revision.

Download
Cite
Share
CommentOpen annotations (there are currently 0 annotations on this page).

Version of Record published: November 17, 2021 (This version)
Accepted Manuscript published: October 22, 2021 (Go to version)
Accepted: October 21, 2021
Received: April 7, 2021

1. Of interest
A unified approach to dissecting biphasic responses in cell signalling

Vaidhiswaran Ramesh, J Krishnan

Research Article Dec 6, 2023
Further reading

Abstract
Editor's evaluation
Introduction
Results
Discussion
Materials and methods
Appendix 1
Appendix 2
Data availability
References
Decision letter
Author response
Article and author information
Metrics

Abstract

The spatial organization of complex biochemical reactions is essential for the regulation of cellular processes. Membrane-less structures called foci containing high concentrations of specific proteins have been reported in a variety of contexts, but the mechanism of their formation is not fully understood. Several competing mechanisms exist that are difficult to distinguish empirically, including liquid-liquid phase separation, and the trapping of molecules by multiple binding sites. Here, we propose a theoretical framework and outline observables to differentiate between these scenarios from single molecule tracking experiments. In the binding site model, we derive relations between the distribution of proteins, their diffusion properties, and their radial displacement. We predict that protein search times can be reduced for targets inside a liquid droplet, but not in an aggregate of slowly moving binding sites. We use our results to reject the multiple binding site model for Rad52 foci, and find a picture consistent with a liquid-liquid phase separation. These results are applicable to future experiments and suggest different biological roles for liquid droplet and binding site foci.

Editor's evaluation

There has been a lively debate recently concerning the multiplicity of reported observations of phase-separated compartments inside of cells. Specifically, some claims of phase separation have been challenged, and an alternative model has been put forward that explains clustering of observed particles as resulting from colocalization of binding sites with no phase separation. The current study does an admirable job of proposing and analyzing ways of distinguishing these two scenarios.

https://doi.org/10.7554/eLife.69181.sa0

Introduction

The cell nucleus of eukaryotic cells is not an isotropic and homogeneous environment. In particular, it contains membrane-less sub-compartments, called foci or condensates, where the protein concentration is enhanced for certain proteins. Even though foci in the nucleus have been observed for a long time, the mechanisms of their formation, conservation, and dissolution are still debated (Strom et al., 2017; Altmeyer et al., 2015; Larson et al., 2017; Patel et al., 2015; Boehning et al., 2018; Pessina et al., 2019; McSwiggen et al., 2019b; McSwiggen et al., 2019a; Oshidari et al., 2020; Gitler et al., 2020; Erdel et al., 2020). An important aspect of these sub-compartments is their ability to both form at the correct time and place, and also to dissolve after a certain time. One example of foci are the structures formed at the site of a DNA double strand break (DSB) in order to localize vital proteins for the repair process at the site of a DNA break (Lisby et al., 2001). Condensates have also been reported to be involved in gene regulation (Hnisz et al., 2017; Bing et al., 2020) and in the grouping of telomeres in yeast cells (Meister and Taddei, 2013; Ruault et al., 2021). More generally, a vast number of membrane-less cellular sub-compartments that have been reported in the literature with different names. Here, we consider a focus to be a spherical condensate of size smaller than a few hundreds nanometers.

Different hypotheses have been put forward to explain focus formation in the context of chromatin, among which two main ones (discussed in the particular context of DSB foci in Miné-Hattab and Taddei, 2019): the Polymer Bridging Model (PBM) and the Liquid Phase Model (LPM). The Polymer Bridging Model is based on the idea that specific proteins form bridges between different chromatin loci by creating loops or by stabilizing interactions between distant loci on the DNA (Figure 1A, left). These interactions can be driven by specific or multivalent weak interactions between chromatin binding proteins and chromatin components. In this case, the existence of sub-compartments relies on both the binding and bridging properties of these proteins. By contrast, the LPM posits that membrane-less sub-compartments arise from a liquid-liquid phase separation. In this picture, first proposed for P granules involved in germ cell formation (Brangwynne et al., 2009), proteins self-organize into liquid-like spherical droplets that grow around the chromatin fiber, allowing certain molecules to become concentrated while excluding others (Figure 1A, right).

Figure 1

Download asset Open asset

Schematic setup of the models.

(A) In the middle, the observed signal from a fluorescently tagged Rad52 protein inside the nucleus following a double-stand break. Left: Schematic figure showing the Polymer Bridging Model (PBM). Proteins binding specifically to the chromatin stabilize it, effectively trapping the motion of other molecules. Right: Schematic figure showing the Liquid Phase Model (LPM). Liquid-liquid phase separation results in the formation of a droplet foci with a different potential and different effective diffusion properties than outside the droplet. (B) Details of the PBM model. Particles diffuse freely with diffusivity $D_{n}$ until they hit one of the $N$ spherical binding sites, themselves diffusing with diffusivity $D_{b}$ . The focus is formed due a high concentration of binding sites. The binding sites are only partially absorbing, so that not all collision events result in a binding even. Once bound, the particle stays attached to the binding site, and then unbinds with rate $k_{-}$ .

Although some biochemical and wide field microscopy data support the LPM hypothesis for DSB foci (Altmeyer et al., 2015; Larson et al., 2017; Strom et al., 2017; McSwiggen et al., 2019b), these observations are at the optical resolution limit, and a more direct detection of these structures is still missing. Coarse-grained theoretical models of the LPM exist (Statt et al., 2020; Grmela and Öttinger, 1997), but predictions of microscale behavior that can be combined with a statistical analysis of high-resolution microscopy data to discriminate between the hypotheses has not yet been formulated. Previously, we analyzed in detail single-particle tracking data in the context of yeast DSB foci (Miné-Hattab et al., 2021). We found that the behavior of Rad52 foci was consistent with a liquid droplet, based on several observations, including the diffusion coefficient of proteins inside the focus relative to that of the whole focus, the size of the focus following two double-strand breaks, and its dissolution upon adding aliphatic alcohol hexanediol.

Here, we build a general physical framework for understanding and predicting the behavior of each model under different regimes. The framework is general and applicable to many different types of foci, although we chose to focus on the regime of parameters relevant to yeast DSB foci, for which we can directly related our results to experimental measurements. While the LPM and PBM models have often been presented in the literature as opposing views, here we show under what conditions the PBM may be reduced to an effective description that is mathematically equivalent to the LPM, but with specific constraints linking its properties. We discuss the observables of the LPM and PBM and derive features that can be used to discriminate these two scenarios.

Results

Two models of foci

To describe the situation measured in single particle tracking experiments, we consider the diffusive motion of a single molecule within the nucleus of a cell in the overdamped limit, described by the Langevin equation in three dimensions (using the Itô convention, as we will for the rest of this work):

d r = d t [\nabla D (r) - \frac{D (r)}{k_{B} T} \nabla U (r)] + \sqrt{2 D (r)} d W,

where $W$ is a three-dimensional Wiener process, $U (r)$ is the potential exerted on the particle, and $D (r)$ is a position-dependent diffusion coefficient. The $\nabla U$ term corresponds to a force divided by the drag coefficient $k_{B} T / D (r)$ , which is given in terms of $D$ and temperature according to Einstein’s relation. The $\nabla D$ term comes from working within the Itô convention. The steady state distribution of particles is given by the Boltzmann distribution:

p (r) = \frac{1}{Z} \exp [- \frac{U (r)}{k_{B} T}],

where $Z$ is a normalization constant.

In the LPM, we associate the focus with a liquid droplet characterized by a sudden change in the energy landscape. We model the droplet as a change in the potential $U (r)$ , and a change in the diffusion coefficient $D (r)$ inside the droplet focus compared to the diffusion coefficient in the rest of the nucleus $D_{n}$ . We assume both the diffusion coefficient and the potential are spherically symmetric around the center of the focus, and have sigmoidal forms:

D (r) = D_{0} + \frac{D_{n} - D_{0}}{1 + e^{- b (r - r_{f})}},

U (r) = \frac{A}{1 + e^{- b (r - r_{f})}},

where D₀ is the diffusion coefficient inside the focus, r_f is the radial distance to the center of the focus, and the coefficients are defined in Table 1. Different relations between the diffusion coefficient and the surface potential are possible.

Table 1

Parameters used in this study with their typical values, and the ranges we have considered.

Experimental values are from Miné-Hattab et al., 2021 (see Materials and Methods for details on estimating diffusion coefficients and free energy differences). D₀ and $A$ are model parameters in the LPM, but also effective observables in the PBM. The diffusivity of binding sites is taken to be that of Rfa1 molecules in the focus, which bind to single-stranded DNA in repair foci, and are thus believe to follow the diffusion of the chromatin (Miné-Hattab et al., 2021). The number $N$ of binding site is related to their density $ρ$ inside the focus through $N = (4 / 3) π ρ r_{f}^{3}$ .

Variable	Model	Description	Value	Range	Exp. value	Units
r_f	both	radius of focus	100	50–200		$nm$
r_n	both	radius of nucleus	500	300–1000		$nm$
$D_{n}$	both	Diffusion coefficient in nucleus	1.0	0.5–2.0	1.08	$μ m^{2} / s$
$σ$	both	Experimental noise level	30	30	30	$nm$
D₀	LPM	Diffusion coefficient inside droplet	0.05	0.01–0.5	0.032	$μ m^{2} / s$
$A$	LPM	Surface potential	5.0	0–10	5.5	$k_{B} T$
$b$	LPM	Steepness in potential	1,000	500–10000		$μ m^{- 1}$
$ρ$	PBM	Density of binding sites inside focus	$4.8 \cdot 10^{4}$	$1 \cdot 10^{3}$ - $8.4 \cdot 10^{4}$		$μ m^{- 3}$
$D_{b}$	PBM	Diffusion coefficient of binding sites	0.005	0-0.1	0.005	$μ m^{2} s^{- 1}$
r_b	PBM	Radius of binding sites	10	5-20		nm
$k_{-}$	PBM	Unbinding rate	500	10-10,000		$s^{- 1}$
$κ$	PBM	Absorption parameter	100	0-1000		$μ m / s$

In the PBM, we describe the dynamics of particles using a model where the focus has $N$ binding sites, each of which is a partially reflecting sphere (Bryan, 1891; Duffy, 2015; Carslaw and Jaeger, 1992) with radius r_b (Figure 1B and C). Binding sites can themselves diffuse with diffusion coefficient $D_{b}$ , and are confined within the focus by a potential $U_{b} (r)$ , so that their density is $ρ (r) \propto e^{- U_{b} (r)}$ according to the Boltzmann distribution. While not bound, particles diffuse freely with diffusion constant $D_{n}$ , even when inside the focus. However, the movement of the particle is affected by direct interactions with the binding sites. Binding is modeled as follows. As the particle crosses the spherical boundary of a binding site during an infinitesimal time step $d t$ , it gets absorbed with probability $p_{b} = κ \sqrt{π d t / D_{n}}$ (Figure 1C), where $κ$ is an absorption parameter consistent with the Robin boundary condition at the surface of the spheres, $D n \cdot \nabla p (x) = κ p (x)$ (Erban and Chapman, 2007; Singer et al., 2008), where $x$ is a point on the surface of the sphere, and $n$ is the unit vector normal to it.

While bound, particles follow the motion of their binding site, described by:

d r = - d t \frac{D_{b}}{k_{B} T} \nabla U_{b} (r) + \sqrt{2 D_{b}} d W,

where $W$ is a three-dimensional Wiener process. A bound particle is released with a constant rate $k_{-}$ . Since the potential $U_{b}$ is constant within the bulk and its only function is to keep binding sites within the focus, the PBM can be described by five parameters: $N$ , r_b, $D_{b}$ , $κ$ and $k_{-}$ . Their typical values can be found in Table 1.

Comparison between simulated and experimental traces

In recent experimental work (Miné-Hattab et al., 2021), we used single particle tracking to follow the movement of Rad52 molecules, following a double-strand break in S. cerevisiae yeast cells, which causes the formation of a focus. These experiments show that temporal traces of Rad52 molecules concentrate inside the focus, as shown for a representative cell in Figure 2A.

Figure 2 with 1 supplement see all

Download asset Open asset

Diffusion properties and effective free energy.

(A) Example of experimental tracking of Rad52 molecules visiting a double-strand break (DSB) focus. Different connected traces correspond to distinct Rad52 molecules. (B) Example trajectory of a particle visiting the focus from simulations in the PBM (left) and LPM (right). The simulated trajectories are visually similar to the data in B. (C) Displacement histogram (jump sizes) for the PBM, LPM and experiments, for an interval $δ = 20$ ms. D. Displacement histogram for the PBM for small values of $k_{-} d t$ (top) and high values (bottom). Here we varied the interval from $δ t = 1$ ms (top) to $δ t = 15$ ms (bottom). (E) Hypothesis testing using a two sided KS-test, comparing the displacement histogram of a free diffusion process (black line in D) and the displacement histogram of diffusion inside the focus (green line in D). Parameters are the same as in (D) $δ t$ was varied from 1 to 25 ms. (F) Effective diffusion coefficient as a function of distance to the focus center $r$ , estimated from simulations of the PBM calculating $\tilde{D} = ⟨ δ r^{2} ⟩ / (2 d δ t)$ in each radial segment. (G) Particle density $p (r)$ as a function of $r$ , estimated from simulations of the PBM. Error bars are standard errors on the mean. (H) Relation between the ratio $(D_{0} - D_{b}) / (D_{n} - D_{b})$ versus the ratio of densities inside and outside the focus (From Equation 11), both estimated from simulations of the PBM (green crosses), compared to the identity prediction (Equation 12, black line). Blue cross shows the experimental observation for Rad52 in DSB loci (see text related to Figure 7 and Table S1 in Miné-Hattab et al., 2021). Parameter values as in Table 1 except: $r_{n} = 1 μ m$ for B; $A = 2.5 k_{B} T$ for B-C; $r_{n} = 0.3 μ m$ , $r_{f} = 0.15 μ m$ and $D_{n} = 0.5 μ m^{2} / s$ for D-E, $κ = 300 μ m / s$ for D, $r_{n} = 0.75 μ m$ for F-H. In H we varied $κ = 1$ – $400 μ m / s$ , $k_{-} = 1$ – $1, 500$ s-1, and $ρ = 2.4$ – $4.8 \cdot 10^{4} μ m$ . See Figure 2—source data 1.

Figure 2—source data 1 Compressed ZIP file containing all the data plotted in the panels of Figure 2 as CSV and TXT files.: https://cdn.elifesciences.org/articles/69181/elife-69181-fig2-data1-v2.zip
Download elife-69181-fig2-data1-v2.zip

Using both the PBM and LPM models described above, we can construct traces that look similar to the data (Figure 2B). To mimic the data, we only record and show traces in two dimensions and added detection noise corresponding to the level reported in the experiments (Miné-Hattab et al., 2021). Based on these simulations, we gather the statistics of the particle motion to create a displacement histogram representing the probability distribution of the observed step sizes between two successive measurements. For this choice of parameters (see Figure 2 caption), both of the models and the experimental data look very similar (Figure 2C).

In principle, we could have expected the displacement histogram of particles inside the bulk of the focus (where traces are not close to the boundary) to look markedly different between the PBM and the LPM. While the LPM should follow the prediction from classical diffusion (given by a Gaussian radial distribution, $p (| δ r |) \propto | δ r |^{2} e^{- | δ r |^{2} / (4 D δ t)}$ for a small interval $δ t$ in the bulk), the PBM prediction is expected to be in general non-Gaussian because of intervals during which the particle is bound and almost immobile (as the chromatin or single-stranded DNA carrying the binding sites moves very slowly), creating a peak of very small displacements. Simulations show that departure from Gaussian displacements is most pronounced when the binding and unbinding rates are slow compared to the interval $δ t$ (Figure 2D, top), but is almost undetectable when they are fast (Figure 2D, bottom). With our parameters, the binding rate is $k_{+} ρ \approx 3, 000$ s-1, and $k_{-}$ ranges from $10$ to $10, 000$ s-1, with $δ t = 20$ ms. For comparison, assuming weak binding to DNA, $K_{d} = k_{+} / k_{-} \approx 1 μ M$ would give $k_{-} \sim 40$ s-1, and assuming strong specific binding, $K_{d} \sim 1$ nM, implies $k_{-} \sim 0.04$ s-1. We stress that there is a lot of uncertainty in the values for experimentally measured rate constants, and a recent study (Saotome et al., 2018) found the dissociation constant k_d for Rad52 in yeast to vary between two observed sites from 5.6 nM to 101 nM. Figure 2E shows how the detectability of non-Gaussian displacements gets worse as $k_{-} δ t$ increases, and is further degraded by the presence of measurement noise.

The experimental findings of single Rad52 molecules in yeast repair foci (Miné-Hattab et al., 2021) suggest that the movement inside the focus are consistent with normal diffusion and its Gaussian distribution of displacements (Figure 2C). This observation excludes a wide range of slow binding and unbinding rates in the PBM, as this would lead to non-Gaussian statistics (Figure 2D, top). However, it does not rule out the PBM itself, which is undistinguishable from classical diffusion for fast binding and unbinding rates (Figure 2D, bottom). In addition, separating displacements inside the focus from boundary-crossing ones can be very difficult in practice, and errors in that classification may result in spurious non-Gaussian displacement distributions that would confound this test. Therefore, it is important to find observables that can distinguish the two underlying models.

Effective description of the polymer bridging model

Motivated by experimental observations, we want to find a coarse-grained description of the PBM that can be reduced to a classical diffusion process under an effective potential and with an effective position-dependent diffusivity, and relate its parameters to the properties of the binding sites. To do so, we analyze the PBM in a mean-field approximation, which is valid in the limit where binding and unbinding events are fast relative to the traveling time of the particles. In this regime, a particle rapidly finds binding sites with rate $k_{+} ρ (r)$ (where $ρ (r)$ is the density of binding sites) and unbinds from them with rate $k_{-}$ . While in principle rebinding events complicate this picture, it has been showed that the period where rebindings to the same binding site occurs can be included in the time they are bound, and thus can be renormalized into a lower effective unbinding rate (Kaizu et al., 2014). Assuming that interactions between binding sites do not affect their binding to the particle of interest, the binding rate can be approximated in the presence of partially reflecting binding sites by the Smoluchowski rate (Nadler and Stein, 1996; Berezhkovskii et al., 2019) (Appendix References):

k_{+} = \frac{4 π D_{n} r_{b}}{1 + \frac{D_{n}}{r_{b} κ}} .

If the processes of diffusion, binding, and unbinding are in equilibrium, the steady state distribution of a particle can be derived using the Boltzmann distribution. The equilibrium assumption is justified by the fact that our time of observation is much smaller than the time scales of focus formation, and that the focus is of constant size during our observations. It is possible that active fluctuations are present inside the focus, but the Rad52 molecules that we are observing are not actively involved in the chemical reactions that take place over the measurement timescale. In this sense, Rad52 can be considered a passive agent, and this description is therefore an effective description of its motion inside the focus. This is supported by the fact that the Rad52 diffusion properties look constant across our observation period.

At each position $r$ , the unbound state is assigned weight 1, and the bound state weight $ρ (r) / K_{d}$ , where $K_{d} = k_{-} / k_{+}$ is the dissociation constant. Then the probability distribution of the particle’s position is given by:

p (r) \propto (1 + \frac{ρ (r)}{K_{d}}) = \frac{1}{p_{u} (r)},

where

p_{u} (r) = \frac{k_{-}}{k_{-} + k_{+} ρ (r)}

is the probability of being unbound conditioned on being at position $r$ .

Here, we assume that binding and unbinding is fast compared to variations of $ρ (r)$ experienced by the tracked particles in the measured time intervals. This assumption holds if the density of binding sites is large, which is a fundamental assumption of the PBM. In this limit, the dynamics of particles are governed by an effective diffusion coefficient, which is a weighted average between the free diffusion of tracked molecules, and the diffusion coefficient of the binding sites:

\tilde{D} (r) = p_{u} (r) D_{n} + (1 - p_{u} (r)) D_{b} = \frac{D_{n} k_{-} + D_{b} k_{+} ρ (r)}{k_{-} + k_{+} ρ (r)} .

Likewise, particles are pushed by an effective confinement force: when they are bound to binding sites, they follow their motion which is confined inside of the focus. The resulting drift is given by that of the binding sites, but weighted by the probability of being bound to them:

\begin{array}{ll} ⟨ d r ⟩ & = - d t (1 - p_{u} (r)) \frac{D_{b}}{k_{B} T} \nabla U_{b} (r) \\ = d t [- \frac{\tilde{D} (r)}{k_{B} T} \nabla \tilde{U} (r) + \nabla \tilde{D} (r)], \end{array}

where in the second line, we have rewritten the dynamics in terms of an effective potential $\tilde{U} (r) = k_{B} T \ln (1 + k_{+} ρ (r) / k_{-})$ , using $ρ (r) \propto e^{- U_{b} (r) / k_{B} T}$ . Thus, the effective dynamics may be described by the Langevin equation of the same form as the LPM (1) but with the relation between $\tilde{U} (r)$ and $\tilde{D} (r)$ constrained by their dependence on $ρ (r)$ :

\tilde{U} (r) = k_{B} T \ln [\frac{\tilde{D} (r) - D_{b}}{D_{n} - D_{b}}],

with the convention that $\tilde{U} = 0$ far away from the focus where $ρ = 0$ . As a consistency check, one can verify that the equilibrium distribution $p \propto e^{- \tilde{U} / k_{B} T}$ gives back Equation 7. Equation 11 reveals a fundamental relation about the dynamics of molecules inside the PBM, and is therefore an important fingerprint to test the nature of foci.

Scaling relation between concentration and diffusivity in the PBM

Experiments or simulations give us access to the effective diffusivity through the maximum likelihood estimator $\tilde{D} = ⟨ δ r^{2} ⟩ / (2 d δ t)$ , where $δ t$ is the time between successive measurements, $δ r$ is the measured displacement between two measurements, and $d$ the dimension in which motion is observed. Within the PBM, Equation 11 allows us to establish a general relation between the particle concentration $p (r)$ , which can also be measured, and the effective diffusivity $\tilde{D}$ , through:

p (r) \propto \frac{1}{\tilde{D} (r) - D_{b}} .

Typically in experiments we have $D_{b} ≪ \tilde{D} ≪ D_{n}$ , in which case this relation may be approximated by $p (r) \tilde{D} (r) = c o n s t$ .

We validated Equation 12 in simulations of the PBM. We divided the radial coordinate $r$ into small windows of $10^{- 3} μ m$ and plotted the measured effective diffusion coefficient $\tilde{D} (r)$ , as a function of $r$ (Figure 2F), as well as the density of tracked particles $p (r)$ (Figure 2G). $\tilde{D} (r)$ takes an approximately constant value inside the focus, defined as D₀ by analogy with the LPM, and is equal to $D_{n}$ well outside the focus where diffusion is free. Likewise the density $p (r)$ decreases from p_in inside to p_out outside the focus. Note that D₀ in the PBM is not a free parameter, but rather emerges from the mean-field description and depends on the properties of binding site. We extracted those values numerically from the simulations. Figure 2H shows that Equation 12 predicts well the relationship between these four numbers, for a wide range of parameter choices of the PBM (varying $κ$ from 1 to 400 $μ m / s$ , $k_{-}$ from 5 to 1500 $s^{- 1}$ and $ρ$ from $23873 - 47746 μ m^{- 3}$ , while keeping $D_{b} = 5 \cdot 10^{- 3} μ m^{2} / s$ and the other parameters to values given by Table 1). While this relation was derived in the limit of fast binding and unbinding, it still holds for the slower rates explored in our parameter range (see Figure 2H). However, it breaks down in the limit of strong binding, when we expect to see two populations (bound and unbound), making the effective diffusion coefficient an irrelevant quantity (see Figure 2D).

We can compare this prediction to estimates from the experimental tracking of single Rad52 molecules in yeast repair foci (Miné-Hattab et al., 2021) (see Materials and methods for details), assuming that the diffusivity of the binding sites is well approximated by that of the single-stranded DNA-bound molecule Rfa1, measured to be $D_{b} = 5 \cdot 10^{- 3} μ m^{2} / s$ . This experimental point, shown as a blue cross in Figure 2H, substantially deviates from the PBM prediction: Rad52 particles spend much more time inside the focus than would be predicted from their diffusion coefficient based on the PBM. To agree with the data, the diffusion coefficient of binding sites would have to be increased to $D_{b} = 0.0314 μ m^{2} / s$ , which is almost an order of magnitude larger than what was found in experiments. The existence of multiple binding sites could in principle lead to an enhanced level of molecular crowding. This would in fact decrease the effective diffusion coefficient inside the focus, moving points of the PBM simulations in Figure 2H to the left, further away from the experimental observation. However, we checked numerically that this effect was small, by adding inert spheres of the same size as the binding sites to generate crowding (Figure 2—figure supplement 1).

Diffusion coefficient and concentration predict boundary movement in the PBM

Another observable that is accessible through simulations and experiments is the radial displacement near the focus boundary. In practice, we gather experimental traces around the focus, and estimate the radius of the focus as shown in Figure 3A. Using many traces, we can find the average radial displacement $⟨ δ r ⟩$ during $δ t$ , as a function of the initial radial position of the particle $r$ (Figure 3B). Under the assumption of spherical symmetry, within the PBM this displacement is given by:

\begin{array}{ll} ⟨ δ r ⟩ & ≃ δ t (- (1 - p_{u} (r)) \frac{D_{b}}{k_{B} T} \partial_{r} U_{b} (r) + \frac{\tilde{D} (r)}{r}) \\ = δ t (- \frac{\tilde{D} (r)}{k_{B} T} \partial_{r} \tilde{U} (r) + \partial_{r} \tilde{D} (r) + \frac{\tilde{D} (r)}{r}), \end{array}

Figure 3 with 1 supplement see all

Download asset Open asset

Radial and angular dynamics.

(A) Experimental trace of a single Rad52 in a DSB focus (Miné-Hattab et al., 2021). The inset shows the definition of the radial movement $δ r$ . Here, concentric circles are shown to define the radius relative to the focus center, where a particle moves a specific distance away from the center of the focus, as well as the angle $θ$ between consecutive displacements. (B) Data extracted from experiments (Miné-Hattab et al., 2021) to estimate the average radial displacement of the tracked particle multiplied by the radius. Here and throughout the x-axis represents the radial position at the beginning of the timestep. Error bars are standard errors on the mean. (C) Simulations showing the radial displacements in the PBM with slowly (top) and rapidly (bottom) moving binding sites. Black lines are predictions based on the measurement of $\tilde{D}$ (Equation 12). Error bars are standard deviations on the mean. (D) Radial displacement from simulations of the LPM. Light-green line shows the (wrong) prediction made while assuming the PBM, using the measurement of the effective diffusion coefficient $\tilde{D} (r)$ (Equation 13). We call the discrepancy between data and the PBM prediction the ‘‘Maximal positive difference.’’ Error bars are standard errors on the mean. (E) Heatmap showing the maximal positive difference in LPM simulations as a function of D₀ and $A$ . (F) Distribution of angles (represented radially on the left, and linearly on the right) between the displacements of consecutive steps of length $δ t$ , from experiments and simulations. Multiple curves for the PBM correspond to different parameter choices corresponding to the points of Figure 2H. Parameter values as in Table 1, except $r_{n} = 1 μ$ m for F. For F parameters are varied with $κ = 100$ – $300 μ$ m/s, $k_{-} = 500$ – $1, 500$ s $^{- 1}$ , $r_{f} = 0.1$ – $0.14 μ$ m. See Figure 3—source data 1 and Figure 3—source data 2.

Figure 3—source data 1 Compressed ZIP file containing all the data plotted in the panels of Figure 3 as CSV files.: https://cdn.elifesciences.org/articles/69181/elife-69181-fig3-data1-v2.zip
Download elife-69181-fig3-data1-v2.zip
Figure 3—source data 2 Compressed ZIP file containing all the data plotted in Figure 3—figure supplement 1 as CSV files.: https://cdn.elifesciences.org/articles/69181/elife-69181-fig3-data2-v2.zip
Download elife-69181-fig3-data2-v2.zip

where the term $\tilde{D} / r$ comes from the change to spherical coordinates.

The first line of Equation 13 shows that the average change in radial position of single particles $⟨ δ r ⟩$ cannot be negative in the PBM for steady binding sites ( $D_{b} = 0$ ). This result does not hold for moving binding sites ( $D_{b} > 0$ ), as we will see below. This is reproduced in simulations, for different absorption probabilities, as shown in Figure 3C.

By constrast, in the LPM there is no constraint on the sign of the displacement $⟨ δ r ⟩$ since the relation between the diffusion coefficient and the surface potential is not constrained like in the PBM. Even when binding sites can move, this prediction can be used to falsify the PBM. Equation 13 makes a prediction for the average radial displacement of the tracked molecule in the PBM, solely as a function of the diffusivity and concentration profiles $\tilde{D} (r)$ and $p (r)$ , using $\tilde{U} (r) = k_{B} T \ln p (r)$ . Accordingly, this prediction agrees well with simulations of the PBM (Figure 3C).

Using Equation 13 that is derived for the PBM, along with the definition of $\tilde{U}$ as a function of $\tilde{D}$ in Equation 11, to analyze a simulation of the LPM leads to large disagreement between the inferred and true parameters. This PBM-based analysis underestimates the depth of the potential (Figure 3D, green lines compared to the red). It predicts a negative displacement $⟨ d r ⟩$ when $D_{b}$ is inferred using the PBM formula $p_{in} / p_{out} = (D_{n} - D_{b}) / (D_{0} - D_{b})$ , although its magnitude is underestimated. But when taking the experimental value of $D_{b} = 0.005 μ m^{2} / s$ , $⟨ d r ⟩$ is always positive even at the boundary. This spurious entropic ‘reflection’ is an artifact of using the wrong model, since the distinct relation between the observed diffusion coefficient and the equilibrium distribution for the PBM leads to a specific shape around the boundary which is not the same for the LPM. The inference using the PBM of such a positive displacement at the surface of the focus can therefore be used to reject the PBM. Figure 3E represents the magnitude of that discrepancy as a function of two LPM parameters — diffusivity inside the droplet and surface potential — showing that the PBM is easier to reject when diffusivity inside the focus is high.

In summary, the average radial diffusion coefficient can predict the radial displacement of tracked molecules within the PBM, and deviations from that prediction can be used as a means to reject the PBM using single-particle tracking experiments.

Distribution of angles between consecutive time steps

To go beyond the average radial displacement, we considered a commonly used observable to study diffusive motion in complex environment: the distribution of angles between two consecutive displacements in two dimensions. While this distribution is uniform for a homogenous environment (Liao et al., 2012), it is expected to be asymmetric in presence of confinement and obstacles (Izeddin et al., 2014).

We computed this distribution from simulations of the PBM and LPM, and compared them to experiments in yeast repair foci (Figure 3F), calculating the angle between the vector relating the first two points and the vector relating the last two points. These distributions are all asymmetric, with an enrichment of motion reversals (180 degree angles). Since the LPM assumes standard diffusion within a potential, the asymmetry in that model can be entirely explained by the effect of confinement, which tends to push back particles at the focus boundary. With the parameters of Table 1, the LPM agrees best with the data, while the PBM shows a more moderate asymmetry across a wide range of parameters. Therefore, both the LPM and the PBM are expected to show asymmetric diffusion around the boundary of the focus, but one could expect that the PBM (and not the LPM) revealed an additional asymmetry inside the bulk of the focus, due to the interactions of the tracked molecules with the binding sites, which causes reflections and hinders motion. To isolate this effect from boundary effects, we simulated the PBM in an infinite focus with a constant density of binding sites (Figure 3—figure supplement 1) and found that this expectation is confirmed. However, this asymmetry is seen only when the measurement time step is small or comparable to the binding time. For finite foci, it must also be corrected for boundary effects. These difficulties make the asymmetry criterion unfit to discriminate between the two models in the context of yeast repair foci.

Foci accelerate the time to find a target, but only moderately in the PBM

Foci keep a higher concentration of molecules of interest within them through an effective potential. We wondered if this enhanced concentration of molecules could act as a ‘funnel’ allowing molecules to find their target (promoter for a transcription factor, repair site, etc) faster.

To address this question, we consider an idealized setting with spherical symmetry, in which the target is a small sphere of radius r₀ located at the center of the focus, of radius r_f (Figure 4A). We further assume that the nucleus is a larger sphere of radius r_n, centered at the same position. We start from a general Langevin equation of the form in Equation 1, and assume that the target is perfectly absorbing, creating a probability flux $J = τ_{a}^{- 1}$ , equal to the rate of finding the target for a single particle. The corresponding Fokker-Planck equation can be solved at steady state, giving (Appendix A):

τ_{a} = \int_{r_{0}}^{r_{n}} d r r^{2} e^{- U (r) / k_{B} T} \int_{r_{0}}^{r} \frac{d r^{'}}{D (r^{'}) r^{' 2}} e^{U (r^{'}) / k_{B} T} .

Figure 4

Download asset Open asset

First passage times to a target site inside the focus.

A. Schematic figure showing the setup of the tracked molecule and the effective target. B. Time to reach the specific target in simulations for the LPM. Black curve shows the predicted result from the analytical derivation (Equation 15). Here we use parameters: $A = - 5.5 k_{B} T$ , $D_{0} = 0.05 μ^{2} / s$ , $r_{n} = 1.0 μ m$ , $D_{n} = 0.8 μ^{2} / s$ . C. Same as B, but for the PBM. Same parameters as in B, but with $D_{0} = D_{n} \cdot e^{U (r) / k_{B} T}$ . D. Heatmap showing the expected search time as a function of the droplet size (x-axis) and the focus diffusion coefficient (y-axis) (Equation 15). Green point corresponds to experimental observations. E. Heatmap showing the expected search time as a function of the droplet size (x-axis) and the height of the surface potential (y-axis) (Equation 15). Green point corresponds to experimental observations. Parameter values as in Table 1, except $b = 2, 000 μ m^{- 1}$ , $D_{0} = 0.04 μ m^{2} / s$ , $A = 5.5 k_{B} T$ for B, E, and F, $κ = 50 μ m^{2} / s$ , $k_{-} = 100$ s $^{- 1}$ , $ρ = 2.4 \cdot 10^{4} μ m^{- 3}$ for C-D. See Figure 4—source data 1.

Figure 4—source data 1 Compressed ZIP file containing all the data plotted in the panels of Figure 4 as CSV files.: https://cdn.elifesciences.org/articles/69181/elife-69181-fig4-data1-v2.zip
Download elife-69181-fig4-data1-v2.zip

Taking the particular form of Equations 3; 4, with a sharp boundary $b r_{f} ≫ 1$ , the integral can be computed explicitly:

\begin{array}{ll} τ_{a} = & \frac{r_{f}^{3} - r_{0}^{3}}{3 D_{0} r_{0}} + \frac{r_{0}^{2} - r_{f}^{2}}{2 D_{0}} + e^{- \frac{A}{k_{B} T}} (\frac{r_{n}^{3} - r_{f}^{3}}{3 D_{0} r_{0}} + \frac{r_{f}^{3} - r_{n}^{3}}{3 D_{0} r_{f}}) \\ + \frac{r_{n}^{3} - r_{f}^{3}}{3 D_{n} r_{f}} + \frac{r_{f}^{2} - r_{n}^{2}}{2 D_{n}} . \end{array}

In the limit $r_{0} ≪ r_{f} ≪ r_{n}$ and of a strong potential $A ≫ k_{B} T$ , Equation 15 simplifies to:

τ_{a} \approx \frac{r_{f}^{3}}{3 D_{0} r_{0}} + \frac{r_{n}^{3}}{3 D_{n} r_{f}},

which is exactly the sum of the time it takes to find the focus from the edge of the nucleus, and the time it takes to find the target from the focus boundary.

Expression (15) can be related to the celebrated Berg and Purcell bound (Berg and Purcell, 1977), which sets the limit on the accuracy of sensing small ligand concentration by a small target, due to the limited number of binding events during some time $t$ . This bound puts a physical constraint on the accuracy of biochemical signaling, and has been shown to be relevant in the context of gene regulation (Gregor et al., 2007). With a mean concentration of ligands $c$ in the cell nucleus, there are $m = (4 π / 3) r_{n}^{3} c$ such ligands, and their rate of arrival at the target is $m / τ_{a} = 4 π c r_{n}^{3} / (3 τ_{a})$ , so that the number of binding events during $t$ is equal to $n \sim 4 π c r_{n}^{3} t / (3 τ_{a})$ on average. Random Poisson fluctuations of $n$ result in an irreducible error in the estimate of the concentration $c$ :

\frac{δ c^{2}}{c^{2}} \sim \frac{δ n^{2}}{n^{2}} \sim \frac{1}{n} \sim \frac{3 τ_{a}}{4 π c r_{n}^{3} t} .

Replacing $τ_{a}$ in Equation 17 with the expression in Equation 16, we obtain in the limit of large nuclei ( $r_{n} \to \infty$ ):

\frac{δ c^{2}}{c^{2}} \sim \frac{1}{4 π c t} [\frac{1}{D_{n} r_{f}} + \frac{e^{- A / k_{B} T}}{D_{0}} (\frac{1}{r_{0}} - \frac{1}{r_{f}})] .

One can further check that in the limit of a strong potential, or when there is no focus, $r_{0} = r_{f}$ , we recover the usual Berg and Purcell limit for a perfectly absorbing spherical measurement device, $δ c / c \sim 1 / \sqrt{4 π D_{n} c r_{f} t}$ .

Equation 15 agrees well with simulations in the general case (Figure 4B), where we used parameters obtained for Rad52 in a repair focus (Miné-Hattab et al., 2021). Equation 15 typically admits a minimum as a function of r_f, meaning that there exists an optimal focus size that minimizes the search time. Using the measured parameters for Rad52, we find an optimal focus size of $r_{f}^{*} \approx 120$ nm, which matches the estimated droplet size $r_{f} = 124$ nm in these experiments (Miné-Hattab et al., 2021) (dashed line in Figure 4B). In these experiments, the estimated experimental noise level was $\approx 30$ nm, but r_f could be extracted accurately by fitting the confinement radius as well as gathering statistics for the radial steady state distribution. We observe that the theoretical curve in Figure 4B is rather flat around its minimum, suggesting an optimal range of droplet sizes rather than a single one. In the limit where $r_{n} ≫ r_{0}$ , the optimal size takes the explicit form:

{(r_{f}^{*})}^{4} = r_{0} r_{n}^{3} \frac{\frac{D_{0}}{D_{n}} - e^{- \frac{A}{k_{B} T}}}{3 (1 - e^{- \frac{A}{k_{B} T}})} = r_{0} r_{n}^{3} \frac{\frac{D_{0}}{D_{n}} - \frac{p_{out}}{p_{in}}}{3 (1 - \frac{p_{out}}{p_{in}})} .

This optimum only exists for $D_{0} e^{A / k_{B} T} > D_{n}$ or $D_{0} p_{i n} > D_{n} p_{o u t}$ , that is, when the benefit of spending more time in the focus compensates the decreased diffusion coefficient. Incidentally, in that case the Berg and Purcell bound on sensing accuracy generalizes to:

\frac{δ c^{2}}{c^{2}} \sim \frac{1}{π c t} (\frac{p_{out}}{4 p_{in} D_{0} r_{0}} + \frac{1}{3 D_{n} r_{f}}) .

The previous formulas for the search time and sensing accuracy are valid for the general Langevin Equation 1, which describes both the LPM and the PBM in the mean-field regime. Figure 4C and D show the search time as a function of the focus size for the specific case of the PBM, where diffusion and potential are further linked. The relation between $\tilde{U}$ and $\tilde{D}$ , given by Equation 11, imposes $D_{0} e^{A / k_{B} T} = D_{n} + D_{b} (e^{A / k_{B} T} - 1) > D_{n}$ , giving the optimal focus size:

{(r_{f}^{*})}^{4} = r_{0} r_{n}^{3} \frac{D_{b}}{3 D_{n}} .

For the physiologically relevant regime of very slow binding sites, $D_{b} ≪ D_{n}$ , this optimal focus size shrinks to 0, meaning that the focus offers no benefit in terms of search time, because binding sites ‘sequester’ or ‘titre out’ the molecule, preventing it from reaching its true target.

These results suggest to use the search time, or equivalently the rate for binding to a specific target, as another measure to discriminate between the LPM and the PBM. In the case of slowly diffusing binding sites, the search time in the PBM does not have a clear local minimum (see Figure 4D), and depends less sharply on the focus size than in the LPM. Therefore, identifying an optimal focus size would suggest to rule out the PBM. Conversely, a monotonic relation between the search time and the focus size would be consistent with the PBM (without excluding the LPM). Testing for the existence of such a minimum would require experiments where the focus size may vary, and where reaching the target can be related to a measurable quantity, such as gene expression onset in the context of gene regulation.

Discussion

The PBM and the LPM are the two leading physical models for describing the nature of nuclear foci or sub-compartments. In this work, we analyzed how the traces of single particle tracking experiments should behave in both models. Using statistical mechanics, we derived a mean field description of the PBM that shares the general functional form of the LPM (Equation 1), but with an additional constraint linking concentration and diffusion inside the focus: the denser the focus, the higher the viscosity. This constraint does not appear to be satisfied by the experimental data on Rad52 in repair foci, favoring the liquid droplet hypothesis. We use our formulation of the PBM to predict the behaviour of the mean radial movement around the focus boundary, which may differ markedly from observation of traces inside a liquid droplet (described by the LPM). We find the range of LPM parameters where this difference would be so significant that it would lead to ruling out the PBM. This work provides a framework for distinguishing the LPM and PBM, and should be combined with modern inference techniques to accurately account for experimental noise and limited data availability (for instance accounting for molecules going out of the optimal focus). Future improvements in single-particle tracking experiments will allow for longer and more accurate traces necessary to deploy the full potential of these methods.

The LPM and PBM have often been presented as opposing models (Miné-Hattab and Taddei, 2019), driven by attempts to compare the macroscopic properties of different membraneless sub-compartments to the original example of liquid-like P granules (Brangwynne et al., 2009). The LPM is a macroscopic description of a liquid droplet in the cytoplasm (Hyman et al., 2014), which concentrates some molecules inside the droplet, and alters their different diffusion properties. The droplet is formed by a phase transition, which means it will be recreated if destroyed, and will go back to its spherical shape if sheared or merged. Conversely, the PBM describes the motion and effective diffusion coefficient inside the focus as a result of fundamental interactions, which provides an explicit binding mechanism by which a focus is formed. Here, we clarified the link between the two from the point of view of single molecules. We confirmed mathematically the intuition that, in the limit of very fast binding and unbinding, the PBM is a particular case of the LPM model. Going further, we show that the PBM imposes a strong constraint between the effective diffusion of molecules in the sub-compartment, $D (r)$ , and the effective potential, $\tilde{U} (r)$ (Equation 11). The LPM is compatible with this choice, but does not impose it in general, although alternative mechanistic implementations of the LPM may impose similar constraints with different functional forms. The correspondence between the two models breaks down when binding and unbinding are slow. However, for this regime to be relevant, experimental observations need to be fast enough to capture individual binding or unbinding events, which is expected to be hard in general, and was not observed in the case of repair foci in yeast.

We found another way in which the two models behave very differently: in the LPM, the focus may act as a funnel accelerating the search for a target inside the focus, and we calculated the optimal focus size that minimizes the search time. In the PBM, such an improvement is negligible unless binding sites themselves have a fast diffusive motion. This difference between the two models could potentially be tested in experiments where the focus size varies. It is not clear whether this optimality argument is relevant for DSB: the merger of two foci leads to larger condensates, suggesting that the focus size is not tightly controlled. But the argument may be relevant for gene expression foci, especially in the context of development where transcription factors need to reach their regulatory target fast in order to ensure rapid cell-fate decision making (Bialek et al., 2019). On the contrary, if a focus is created in order to decrease the probability of specific binding, such as in silencing foci (Brown et al., 1997), a PBM implementation may be more advantageous. Binding sites, which act as decoys (Burger et al., 2010), sequester proteins involved in gene activation, thus increasing the time it takes to reach their target and suppressing gene expression. In that picture, genes would be regulated by the mobility and condensation of these decoy binding sites. Therefore, while this difference between the two models may be hard to investigate experimentally, it provides be a very important distinction in terms of function.

More generally, foci or membraneless sub-compartments are formed in the cells for very different reasons and remain stable for different timescales. For example, repair foci are formed for short periods of time (hours) to repair double strand breaks, and then dissolve. In this case the speeds of both focus formation and target finding are important for rapid repair, but long-term stability of foci is not needed. Gene expression foci (Hnisz et al., 2017; Bing et al., 2020) can be long lived, and their formation may be viewed as a way to ‘prime’ genes for faster activation. However, given the high concentrations of certain activators, not all genes may require very fast search times of the transcription factors to the promoter. While molecularly the same basic elements are available for foci formation – binding and diffusion – different parameter regimes exploited in the LPM and PBM may lead to different behaviour covering a vast range of distinct biological requirements.

Materials and methods

Simulation of PBM

Request a detailed protocol

In order to simulate the bridging model we generated $N$ binding sites of radius r_b. We simulate a diffusing molecule through the free overdamped Langevin equation in three dimensions, and at each time-step, we find the closest binding site to the particle. If the distance of the particle ( $Δ r$ ) is smaller than r_b we bind the molecule with probability $p_{b} = κ \sqrt{π δ t / D_{n}}$ . If the particle does not bind, it is reflected so the new distance to the center of the particular binding site is $2 r_{b} - Δ r$ . At this new position we evaluate the position of all other binding sites they all diffuse with diffusion coefficient $D_{b}$ , and if the molecule is within the radius of another binding site (happens extremely rarely), it is again accepted to bind with the same probability p_b. If a particle binds, it stays at the position of the intersection with the binding site, and at each time step it can be released with probability $k_{-} δ t$ . We choose $δ t$ small so that $p_{b} ≪ 1$ and $\sqrt{2 D_{n} δ t} ≪ r_{b}$ , which for the considered parameter ranges in Table 1 is typically obtained for values of $δ t = 10^{- 6} s$ .

Simulation of LPM

Request a detailed protocol

To simulate the LPM, we use the Milstein algorithm to calculate the motion of a particle. As in the PBM, the particle is reflected at the nucleus boundary, and can otherwise move freely in the nucleus. We typically choose the same value of $δ t$ as the PBM, since the surface potential typically has a very steep gradient, given by $b \approx 1000$ as shown in Table 1.

Experimental measurements

Request a detailed protocol

Experimental details about single-particle tracking are given in Miné-Hattab et al., 2021. Briefly, the x- and y-values of single particles were sampled at 50 Hz for molecules inside the visible z-frame ( $\approx \pm 150 n m$ thick). Therefore one cannot separate whether molecules are inside the focus or above/below it, but since the radius of the focus is $\approx 125 n m$ , this effect is very small, and statistically it is possible to take this effect into account when calculating the radial concentration of molecules.

The diffusion coefficient inside the focus was calculated as follows. The distributions of displacements was fitted by a mixture of two Gaussians corresponding to a slow (inside focus) and a fast (outside focus) population. Diffusion inside the focus was extracted from the mean-squared displacement of the slow population, taking the confinement and experimental uncertainty into account (see text related to Figure 2G in Miné-Hattab et al., 2021). Free energy differences were estimated based on the size of the focus and the concentration of particles inside the focus compared to outside (see text related to Figure 7 and Table S1, ibid.). These estimates are not sensitive to radial effects, such as the definition and size of the focus, or to the issue of some particles being above or below the focus.

Appendix 1

Binding rate by a partially absorbing sphere

We consider a particle with diffusivity $D$ , which can be partially absorbed by a spherical binding site of radius r_b and absorption parameter $κ$ . Its Fokker-Planck equation takes the following form, in spherical coordinates projected onto the distance to the center of the binding site, $r$ :

\partial_{t} p = \frac{D}{r^{2}} \partial_{r} r^{2} \partial_{r} p .

The boundary conditions are $p (r = \infty) = 1 / V$ , where $V$ is the total volume, assumed to be much larger than that of the binding site, and the Robin condition:

D \partial_{r} p (r_{b}) = κ p (r_{b}) .

The solution of Equation 22 at steady state with these bondary conditions reads:

p (r) = \frac{1}{V} (1 - \frac{κ r_{b} / r}{κ + D / r_{b}}),

the total diffusive flux is then given by

J = 4 π D r_{b}^{2} \partial_{r} p (r_{b}) = \frac{1}{V} \frac{4 π D r_{b}}{1 + \frac{D}{r_{b} κ}} .

Normalizing by the volume factor gives the association rate for binding, $k_{+} = V J = 4 π D r_{b} / (1 + D / κ r_{b})$ .

Appendix 2

Searching for a target in a funneling potential

We consider a problem similar to that of the previous appendix. Now the spherical object is a target, which is perfectly absorbing. It is at the center of a liquid droplet, which we model by a spherically symmetric potential $U (r)$ .

The probability distribution of a molecule is denoted by $p (r) = p (r)$ . The probability density of being at distance $r$ from the center, $q (r)$ , is related to $p (r)$ through $q (r) = 4 π r^{2} p (r)$ , accounting for the volume of the sphere. The evolution of $r$ is described by the stochastic differential equation:

d r = \frac{2 D}{r} + \partial_{r} D - \frac{D}{k_{B} T} \partial_{r} U + \sqrt{2 D} d W,

where $W$ is a 1-dimensional Wiener process. The corresponding Fokker-Planck equation reads:

\partial_{t} q = - \partial_{r} [(\frac{2 D}{r} + \partial_{r} D - \frac{D}{k_{B} T} \partial_{r} U) q] + \partial_{r}^{2} (D q) ≐ - \partial_{r} J .

At steady state with a non-vanishing flux $J = const$ , we have:

(\frac{2 D}{r} - \frac{D}{k_{B} T} \partial_{r} U) q = D \partial_{r} q - J,

or equivalently:

q \partial_{r} ϕ + \partial_{r} q = \frac{J}{D}

with $ϕ ≐ - 2 \ln (r) + U / k_{B} T$ . Multiplying both sides of the equation by $e^{ϕ}$ , we obtain:

\partial_{r} (e^{ϕ} q) = \frac{J}{D} e^{ϕ} .

The general solution to that equation is:

q (r) = C e^{- ϕ (r)} + J e^{- ϕ (r)} \int_{r_{0}}^{r} \frac{e^{ϕ (r^{'})}}{D (r^{'})} d r^{'} .

We have $C = 0$ because of the absorbing boundary condition $q (r_{0}) = 0$ . The constant $J$ is determined by the normalization $\int_{r_{0}}^{r_{n}} d r q (r) = 1$ , yielding:

J^{- 1} ≐ τ_{a} = \int_{r_{0}}^{r_{n}} d r e^{- ϕ (r)} \int_{r_{0}}^{r} d r^{'} \frac{e^{ϕ (r^{'})}}{D (r^{'})},

This in turns gives the result of the main text after replacing $ϕ (r)$ by its definition.

Data availability

All data generated or analysed during this study are included in the manuscript and supporting files.

The following previously published data sets were used

1. Miné-Hattab J
2. Heltberg M
3. Villemeur M
4. Guedj C
5. Mora T
6. Walczak AM
7. Dahan M
8. Taddei A
(2021) Zenodo
ID 4495116. Single molecule microscopy reveals key physical features of repair foci in living cells.

https://zenodo.org/record/4495116

References

1. Altmeyer M
2. Neelsen KJ
3. Teloni F
4. Pozdnyakova I
5. Pellegrino S
6. Grøfte M
7. Rask M-BD
8. Streicher W
9. Jungmichel S
10. Nielsen ML
11. Lukas J
(2015) Liquid demixing of intrinsically disordered proteins is seeded by poly(ADP-ribose)
Nature Communications 6:1–12.
https://doi.org/10.1038/ncomms9088
- Google Scholar
(2019) Trapping of diffusing particles by small absorbers localized in a spherical region
The Journal of Chemical Physics 150:064107.
https://doi.org/10.1063/1.5083808
- Google Scholar
1. Berg HC
2. Purcell EM
(1977) Physics of chemoreception
Biophysical Journal 20:193–219.
https://doi.org/10.1016/S0006-3495(77)85544-6
- PubMed
- Google Scholar
Preprint
(2019) Action at a Distance in Transcriptional Regulation
arXiv.

https://arxiv.org/abs/1912.08579
- Google Scholar
1. Bing XY
2. Batut PJ
3. Levo M
4. Levine M
5. Raimundo J
(2020) SnapShot: The Regulatory Genome
Cell 182:1674.
https://doi.org/10.1016/j.cell.2020.07.041
- Google Scholar
1. Boehning M
2. Dugast-Darzacq C
3. Rankovic M
4. Hansen AS
5. Yu T
6. Marie-Nelly H
7. McSwiggen DT
8. Kokic G
9. Dailey GM
10. Cramer P
11. Darzacq X
12. Zweckstetter M
(2018) RNA polymerase II clustering through carboxy-terminal domain phase separation
Nature Structural & Molecular Biology 25:833–840.
https://doi.org/10.1038/s41594-018-0112-y
- Google Scholar
(2009) Germline P granules are liquid droplets that localize by controlled dissolution/condensation
Science 324:1729–1732.
https://doi.org/10.1126/science.1172046
- PubMed
- Google Scholar
1. Brown KE
2. Guest SS
3. Smale ST
4. Hahm K
5. Merkenschlager M
6. Fisher AG
(1997) Association of transcriptionally silent genes with Ikaros complexes at centromeric heterochromatin
Cell 91:845–854.
https://doi.org/10.1016/S0092-8674(00)80472-9
- PubMed
- Google Scholar
1. Bryan G
(1891)
Note on a problem in the linear conduction of heat

Proc Camb Phil Soc 7:246–248.
- Google Scholar
(2010) Abduction and asylum in the lives of transcription factors
PNAS 107:4016–4021.
https://doi.org/10.1073/pnas.0915138107
- PubMed
- Google Scholar
Book
1. Carslaw HS
2. Jaeger JC
(1992)
Conduction of Heat in Solids

Clarendon press.
- Google Scholar
Book
1. Duffy DG
(2015) Green’s Functions with Applications
cRc press.
https://doi.org/10.1201/9781315371412
- Google Scholar
1. Erban R
2. Chapman SJ
(2007) Reactive boundary conditions for stochastic simulations of reaction-diffusion processes
Physical Biology 4:16–28.
https://doi.org/10.1088/1478-3975/4/1/003
- PubMed
- Google Scholar
1. Erdel F
2. Rademacher A
3. Vlijm R
4. Tünnermann J
5. Frank L
6. Weinmann R
7. Schweigert E
8. Yserentant K
9. Hummert J
10. Bauer C
11. Schumacher S
12. Al Alwash A
13. Normand C
14. Herten DP
15. Engelhardt J
16. Rippe K
(2020) Mouse Heterochromatin Adopts Digital Compaction States without Showing Hallmarks of HP1-Driven Liquid-Liquid Phase Separation
Molecular Cell 78:236–249.
https://doi.org/10.1016/j.molcel.2020.02.005
- PubMed
- Google Scholar
1. Gitler AD
2. Shorter J
3. Ha T
4. Myong S
(2020) Just Took a DNA Test, Turns Out 100% Not That Phase
Molecular Cell 78:193–194.
https://doi.org/10.1016/j.molcel.2020.03.029
- PubMed
- Google Scholar
(2007) Probing the limits to positional information
Cell 130:153–164.
https://doi.org/10.1016/j.cell.2007.05.025
- PubMed
- Google Scholar
1. Grmela M
2. Öttinger HC
(1997) Dynamics and thermodynamics of complex fluids. I. Development of a general formalism
Physical Review E 56:6620–6632.
https://doi.org/10.1103/PhysRevE.56.6620
- Google Scholar
(2017) A Phase Separation Model for Transcriptional Control
Cell 169:13–23.
https://doi.org/10.1016/j.cell.2017.02.007
- PubMed
- Google Scholar
(2014) Liquid-Liquid Phase Separation in Biology
Annual Review of Cell and Developmental Biology 30:39–58.
https://doi.org/10.1146/annurev-cellbio-100913-013325
- PubMed
- Google Scholar
1. Izeddin I
2. Récamier V
3. Bosanac L
4. Cissé II
5. Boudarene L
6. Dugast-Darzacq C
7. Proux F
8. Bénichou O
9. Voituriez R
10. Bensaude O
11. Dahan M
12. Darzacq X
(2014) Single-molecule tracking in live cells reveals distinct target-search strategies of transcription factors in the nucleus
eLife 3:e02230.
https://doi.org/10.7554/eLife.02230
- PubMed
- Google Scholar
(2014) The berg-purcell limit revisited
Biophysical Journal 106:976–985.
https://doi.org/10.1016/j.bpj.2013.12.030
- PubMed
- Google Scholar
1. Larson AG
2. Elnatan D
3. Keenen MM
4. Trnka MJ
5. Johnston JB
6. Burlingame AL
7. Agard DA
8. Redding S
9. Narlikar GJ
(2017) Liquid droplet formation by HP1α suggests a role for phase separation in heterochromatin
Nature 547:236–240.
https://doi.org/10.1038/nature22822
- PubMed
- Google Scholar
1. Liao Y
2. Yang SK
3. Koh K
4. Matzger AJ
5. Biteen JS
(2012) Heterogeneous single-molecule diffusion in one-, two-, and three-dimensional microporous coordination polymers: Directional, trapped, and immobile guests
Nano Letters 12:3080–3085.
https://doi.org/10.1021/nl300971t
- PubMed
- Google Scholar
(2001) Rad52 forms DNA repair and recombination centers during S phase
PNAS 98:8276–8282.
https://doi.org/10.1073/pnas.121006298
- PubMed
- Google Scholar
1. McSwiggen DT
2. Hansen AS
3. Teves SS
4. Marie-Nelly H
5. Hao Y
6. Heckert AB
7. Umemoto KK
8. Dugast-Darzacq C
9. Tjian R
10. Darzacq X
(2019a) Evidence for DNA-mediated nuclear compartmentalization distinct from phase separation
eLife 8:e47098.
https://doi.org/10.7554/eLife.47098
- Google Scholar
1. McSwiggen DT
2. Mir M
3. Darzacq X
4. Tjian R
(2019b) Evaluating phase separation in live cells: diagnosis, caveats, and functional consequences
Genes & Development 33:1619–1634.
https://doi.org/10.1101/gad.331520.119
- PubMed
- Google Scholar
1. Meister P
2. Taddei A
(2013) Building silent compartments at the nuclear periphery: A recurrent theme
Current Opinion in Genetics & Development 23:96–103.
https://doi.org/10.1016/j.gde.2012.12.001
- PubMed
- Google Scholar
1. Miné-Hattab J
2. Taddei A
(2019) Physical principles and functional consequences of nuclear compartmentalization in budding yeast
Current Opinion in Cell Biology 58:105–113.
https://doi.org/10.1016/j.ceb.2019.02.005
- Google Scholar
1. Miné-Hattab J
2. Heltberg M
3. Villemeur M
4. Guedj C
5. Mora T
6. Walczak AM
7. Dahan M
8. Taddei A
(2021) Single molecule microscopy reveals key physical features of repair foci in living cells
eLife 10:e60577.
https://doi.org/10.7554/eLife.60577
- PubMed
- Google Scholar
1. Nadler W
2. Stein DL
(1996) Reaction–diffusion description of biological transport processes in general dimension
The Journal of Chemical Physics 104:1918–1936.
https://doi.org/10.1063/1.471810
- Google Scholar
1. Oshidari R
2. Huang R
3. Medghalchi M
4. Tse EYW
5. Ashgriz N
6. Lee HO
7. Wyatt H
8. Mekhail K
(2020) DNA repair by Rad52 liquid droplets
Nature Communications 11:1–8.
https://doi.org/10.1038/s41467-020-14546-z
- PubMed
- Google Scholar
1. Patel A
2. Lee HO
3. Jawerth L
4. Maharana S
5. Jahnel M
6. Hein MY
7. Stoynov S
8. Mahamid J
9. Saha S
10. Franzmann TM
11. Pozniakovski A
12. Poser I
13. Maghelli N
14. Royer LA
15. Weigert M
16. Myers EW
17. Grill S
18. Drechsel D
19. Hyman AA
20. Alberti S
(2015) A liquid-to-solid phase transition of the ALS protein FUS accelerated by disease mutation
Cell 162:1066–1077.
https://doi.org/10.1016/j.cell.2015.07.047
- PubMed
- Google Scholar
1. Pessina F
2. Giavazzi F
3. Yin Y
4. Gioia U
5. Vitelli V
6. Galbiati A
7. Barozzi S
8. Garre M
9. Oldani A
10. Flaus A
11. Cerbino R
12. Parazzoli D
13. Rothenberg E
14. d’Adda di Fagagna F
(2019) Functional transcription promoters at DNA double-strand breaks mediate RNA-driven phase separation of damage-response factors
Nature Cell Biology 21:1286–1299.
https://doi.org/10.1038/s41556-019-0392-4
- PubMed
- Google Scholar
(2021) Sir3 mediates long-range chromosome interactions in budding yeast
Genome Research 31:411–425.
https://doi.org/10.1101/gr.267872.120
- PubMed
- Google Scholar
1. Saotome M
2. Saito K
3. Yasuda T
4. Ohtomo H
5. Sugiyama S
6. Nishimura Y
7. Kurumizaka H
8. Kagawa W
(2018) Structural basis of homology-directed DNA repair mediated by RAD52
Iscience 3:50–62.
https://doi.org/10.1016/j.isci.2018.04.005
- PubMed
- Google Scholar
1. Singer A
2. Schuss Z
3. Osipov A
4. Holcman D
(2008) Partially reflected diffusion
SIAM Journal on Applied Mathematics 68:844–868.
https://doi.org/10.1137/060663258
- Google Scholar
(2020) Model for disordered proteins with strongly sequence-dependent liquid phase behavior
The Journal of Chemical Physics 152:075101.
https://doi.org/10.1063/1.5141095
- PubMed
- Google Scholar
1. Strom AR
2. Emelyanov AV
3. Mir M
4. Fyodorov DV
5. Darzacq X
6. Karpen GH
(2017) Phase separation drives heterochromatin domain formation
Nature 547:241–245.
https://doi.org/10.1038/nature22989
- PubMed
- Google Scholar

Decision letter

Agnese Seminara

Reviewing Editor; University of Genoa, Italy
José D Faraldo-Gómez

Senior Editor; National Heart, Lung and Blood Institute, National Institutes of Health, United States
Pierre Ronceray

Reviewer

In the interests of transparency, eLife publishes the most substantive revision requests and the accompanying author responses.

Decision letter after peer review:

Thank you for submitting your article "Physical observables to determine the nature of membrane-less cellular sub-compartments" for consideration by eLife. Your article has been reviewed by 3 peer reviewers; this evaluation has been overseen by Agnese Seminara as Reviewing Editor and José Faraldo-Gómez as the Senior Editor. The following individuals involved in review of your submission have agreed to reveal their identity: Pierre Ronceray (Reviewer #3).

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

Essential revisions:

The reviewers believe the work is suitable for eLife but also believe the manuscript could be improved upon – as noted below. Please consider their concerns and recommendations and implement the suggested changes whenever possible.

Reviewer #1:

The authors propose several ways of leveraging single-particle tracking experiments to distinguish between intracellular phase separation and an alternative model of clustered binding sites. The first proposed scheme is particularly intuitively appealing: in the binding site scenario, the local density of binding sites both increases particle density and slows effective particle diffusion, leading to a definite relationship between these two quantities, while the phase separation scenario would not necessarily couple these two quantities. The additional schemes based on particle movement near a cluster boundary, angles between consecutive steps, and search times add to the arsenal of potential analysis tools. Overall, the work is timely, rigorous, and generally clearly presented and given the growing list of reported observations of phase separation, will appeal to a broad audience.

1. The authors don't explicitly address the effects of crowding that might occur inside a cluster of binding sites. Crowding can change both the free density and the free diffusion coefficient of the particles in the cluster. How would such crowding affect the relation between observed particle density and observed diffusion, particularly if crowding scales with density of binding sites?

2. I found the discussion of the angles between consecutive steps hard to follow at points. In particular, what do the authors have in mind by the statement that binding sites can "reflect" the motion of tracked molecules (line 219)? I also wasn't sure what the final sentence of that section was meant to convey – some more guidance on the conditions under which this approach is useful would help.

3. The paper is well written but could use some additional proofreading for spelling, e.g. "dropblet", "membranelss", "rapide", "displacememnt", "mimick".

Reviewer #2:

Heltberg et al., investigate two possible mechanisms for the formation of nuclear foci and how these mechanisms can be distinguished experimentally, based on single-particle tracking of molecules that are up-concentrated in the focus. First, liquid-liquid phase separation (here: Liquid Phase Model, LPM) is treated as one of the major mechanisms currently hypothesized. Second, as an alternative mechanism, a polymer-bridging model (PBM) is investigated, in which the focus is held together by polymer bridges and contains binding sites, which can lead to local enrichment, appearing as a focus.

The theory is presented in a clean way, and while the Langevin equation for single molecules in a phase-separated liquid comes without derivation, it is plausible, and in fact backed up by our own calculations. A similar Langevin equation is found for the PBM and it is subsequently shown that both models can lead to very similar displacement distributions, thus showing that this simple observable cannot always distinguish between PBM and LPM.

Subsequently, the authors derive an effective description of the PBM, based on the experimental observation that potential binding sites on the DNA (proxied by Rfa1, a DNA-binding protein) diffuse much more slowly than a typical repair factor (represented by Rad52). Thus there is a separation of time scales between the two relevant diffusion processes, which is used to constrain the possible parameter combinations for the PBM. Based on these constraints, the authors shown that PBM is incompatible with their previous experimental results.

The remainder of the paper deals with a number of interesting observables, such as the angular distribution of displacements and search time to find a repair target, which can also be used to distinguish PBM and LPM with an ideal setup.

Strengths:

Heltberg et al., present a clean way to distinguish LPM on the one hand, and a realization of PBM on the other hand, based on theory. This is validated by comparison to data they obtained in previous work. The theory is rigorous and the data analysis is well carried out, save for minor ambiguities, which can likely be eliminated during revision. The paper draws its main strength from its interdisciplinarity.

Conclusions and Discussion:

The authors have achieved their goal of distinguishing LPM and PBM. The corresponding theory will be of great use for everyone in the field aiming to make this distinction based on single molecule tracking, a strategy that has been attempted numerous times, but eventually always failed due to the lack of an appropriate theoretical framework. Heltberg et al., have gone on to show a striking difference between experimentally constrained PBM realizations and the experimental measurements themselves, rendering the PBM much less likely than the LPM.

The manuscript could use some additional proofreading for grammar/typos.

Regarding the references to Miné-Hattab et al., 2021 we were uncertain with regard to two points:

1. Figure 4b shows a striking agreement between the focus size and the minimum in the time-to-find-a-target function. Measuring the focus radius of 120 nm with such a high accuracy requires exquisite microscope resolution, while we believe that this is in principle possible with PALM, we couldn't find any supporting data for this resolution in the original paper, would it be possible to point us either to the right place in the manuscript or to general references that make clear that this is a standard resolution for live-PALM the way it was used in Miné-Hattab et al.?

2. Please point us to the measurements of inverse partitioning (p_out/p_in). We weren't able to immediately find them in the previous paper.

Abstract/Intro:

Foci as a word works, however, in particular in the intro condensate and foci need more of distinction. A nucleolus (as a very prominent example of a nuclear condensate) wouldn't be called focus in the literature.

The abstract seems to be missing a key point of the paper, namely that based on the available observables/data a call can be made that yeast repair foci are more likely LPM, rather than PBM.

L48: The references seem to be a mix of reviews and primary research, we believe this strong statement calls for only primary references.

L54: What did Miné-Hattab et al., find? How did you make the call in your previous paper?

Results:

L73: maybe 'by the Boltzmann factor' is a bit easier to google for the uninitiated reader, rather than 'Boltzmann's law', which isn't really a thing.

L76: 'While this description is general' is unclear, but could simply be left out.

L80ff: Motivate why this potential is a good way to model binding sites? In the PMB model, the diffusion model for the binding sites ends up being very similar to the one used for LPM. Is it then fair to claim that this model is more "microscopic"? It seems that PBM is more microscopic only in words and not really in the modeling. One might say that the binding sites are just themselves following diffusion in an LPM. Then the difference between the two models is more about the fact that for LPM the tracer molecule diffuses in the LPM and in the other case the tracers do not feel directly the LPM but just bind to molecules (or binding sites) that are diffusing in the LPM. In this respect, the distinction between the two models does not contain much information on whether the foci are formed by phase separation or not. In view of the importance of this distinction and the ongoing controversies, the authors should clearly discuss this issue. To have a more microscopic model it might be useful to adopt a polymer model description, rather than a simple potential.

L92: Why exclude potential? Relevant for confinement, shape of droplet boundary (which might be a significant part of the droplet, for such small droplets). Also, six isn't much worse than five parameters.

L105: Maybe say that parameter values are mentioned in Figure legend.

L109: Why just the simple Gaussian of free diffusion? Shouldn't there be confinement effects for small droplets?

L116: Would be good to have references for k- in yeast. At least ballpark. Otherwise also look at other k-.

L120: Does Rad52 ever appear in groups, e.g. dimers or multiple times within the same repair site?

L127ff: This entire section is a bit less clear than the rest. Maybe a slightly longer, more systematic exposition could help?

L131: Rebinding description is a bit vague, maybe elaborate a bit? Can this be connected to appendix 1?

Equation 7: Shouldn't the second ∝ be an equal sign?

141: How realistic is this limit? Doesn't it imply also fast diffusion? In other words, it seems like equation 6 gives many constraints on what your rate can be. Please comment. How well does the time scale separation hold? If it doesn't, many of the conclusions would have to be modified.

L152: Recall the definition of δ r. Is average from equation 10 removed?

L161: Not sure about D₀, according to table this is only part of LPM? Also not mentioned in L93.

L164: Typo in unit for rho.

L167: Is this shown? Is this Figure 2d?

L181 and equation 13: Inconsistent use of δ vs d.

Equation 13: In measuring this in 3C, is the average r as a function of r computed from displacements starting at r or arriving at r?

L164: Is it obvious that dr cannot be negative? What are the consequences of this? Is it true at the focus boundary?

L195: Why underestimate and how is this seen? Underestimate with respect to what ground truth? Overall the discussion of 3C, D should be extended and made clearer. It is one of the main signatures used to tell PBM from LPM so it should be very clearly discussed.

L197: Didn't you say above it was never negative? Now only for this value?

L199: Can you expand this a bit for a clearer picture?

L210: Without confinement in shouldn't it be uniform in any dimension?

L250: Can you comment more on the Burger Purcell limit? How relevant is the discussion in the present context? If it is relevant, it should be expanded.

L263: Missing 'of'.

L269ff: This seems quite weak as a test, compared to the previous results (in particular 2H). The PBM clearly also has a minimum, how different are these for different parameter ranges? It also seems very hard to control focus size in vivo.

L297: Using 'condensates' as a verb required rereading this sentence for clarification, maybe rephrase.

Figures:

General: It would be helpful to have bold one-sentence figure caption summaries. Please make sure to always mention the equations that each panel refers to in the figure caption.

1: Panel B: might be good to also have shading for the background of the nucleus. Otherwise the boundary could be mistaken for a nuclear membrane happened to us at first glance (also compare Figure 4b).

2: Panel D: caption: are (bottom) and (top) reversed?

Panel F: Caption: please clarify 'using displacement histograms'.

Panel G: Displacement is misleading for the x-axis label, how about 'r' or something like 'dist. from nucleus'.

Panel H: reorder legend, otherwise data point 'experimental observation' looks like legend.

X axis has D₀, shouldn't this be D^~? Same in caption for H.

To us this panel has the strongest evidence that the model of choice should be LPM,

is this true? If yes, please clarify in text.

Caption for H: Could you refer to the figures/sections in Miné-Hattab where the

partitioning is calculated? After a quick skim we weren't able to find it.

3: Panel B: What do the error bars refer to?

Panel C: Why the dip in the bottom panel? Equation 13 is said to never be negative? Is only D_b varied between the bottom and top panels? If D^~ is kept constant, shouldn't the plateaus be the same?

Panel D: Caption: black instead of blue line?

Are these plots purely based on equation (13)? Same as in C, how can this be negative?

Reviewer #3:

Membraneless condensates have recently become a central focus of the molecular and cellular biophysics communities. While the dominant paradigm for their formation, liquid-liquid phase separation (LLPS), has been well established in a number of cases for large, optically resolved droplets, there are significant concerns regarding the generality of this mechanism for smaller foci or puncta, and other mechanisms have been proposed to explain their formation. The problem is that it is very difficult to distinguish experimentally between these mechanisms for sub-optical resolution condensates. In this article, Heltberg et al., propose a novel method, based on the analysis of single molecule tracks, that allows discriminating between the liquid phase model (LPM) and one of the challenger mechanisms, the "polymer bridging model" (PBM). This method relies on the statistics of individual displacements – diffusion, radial displacements, angular changes – which are showed theoretically to exhibit different signatures for the two models. With realistic data this is sufficient to discriminate between the models: for instance in the case of double strand break foci (DSB), building on a recent work by some of the same authors, this article convincingly rules out the PBM in favor of the LPM. The author also investigate the influence on these two models on the search time to reach a specific small target – a commonly invoked role of condensates – and show that only the LPM substantially accelerates this, which could provide additional means to experimentally discriminate between the mechanisms, on top of the intrinsic interest of this finding.

This article is a welcome addition to the literature in this field, as it will help clarify the nature of these condensates, in particular below the optical resolution. It is well-written, interesting and the conclusions are justified. I particularly appreciate the effort to employ simulated data that are realistic for actual experiments, which strengthens the claims of applicability. Some aspects of the data analysis and of the modeling, however, are insufficiently discussed and would need to be precised / expanded.

1) The modeling is made under the assumption of thermal equilibrium, without further discussion. The authors should comment on why this is reasonable, in particular in view of the presence of active fluctuations and of chemical reactions in these condensates.

2) How is the diffusivity measured? Are these measures corrected for experimental error (e.g. using three-point estimators)?

3) The conditioning of the averages should be discussed, e.g. in Equation 13: I assume that it is in the Ito convention? Similarly for the angle changes.

https://doi.org/10.7554/eLife.69181.sa1

Author response

Essential revisions:

The reviewers believe the work is suitable for eLife but also believe the manuscript could be improved upon – as noted below. Please consider their concerns and recommendations and implement the suggested changes whenever possible.

Reviewer #1:

The authors propose several ways of leveraging single-particle tracking experiments to distinguish between intracellular phase separation and an alternative model of clustered binding sites. The first proposed scheme is particularly intuitively appealing: in the binding site scenario, the local density of binding sites both increases particle density and slows effective particle diffusion, leading to a definite relationship between these two quantities, while the phase separation scenario would not necessarily couple these two quantities. The additional schemes based on particle movement near a cluster boundary, angles between consecutive steps, and search times add to the arsenal of potential analysis tools. Overall, the work is timely, rigorous, and generally clearly presented and given the growing list of reported observations of phase separation, will appeal to a broad audience.

We thank the referee for the positive attitude towards our work and are happy for the insightful comments that have increased the value of this paper. We have taken all comments into account in the point-to-point response as can be seen below.

1. The authors don't explicitly address the effects of crowding that might occur inside a cluster of binding sites. Crowding can change both the free density and the free diffusion coefficient of the particles in the cluster. How would such crowding affect the relation between observed particle density and observed diffusion, particularly if crowding scales with density of binding sites?

We agree that crowding could be an important effect of the cellular environment that should be addressed more explicitly. In the typical view of crowding, the presence of multiple large proteins affects the diffusion coefficient in the way:

\frac{Dpure}{Dcrowd} = \frac{ncrowd}{npure}

where D is the relative diffusion coefficients and η is the viscosity. Therefore crowding should affect the diffusion coefficient by making it lower, without affecting the free energy, since the interactions are assumed to be non-binding with molecules leading to crowding. This means that the equilibrium distribution of density of Rad52 particles, p_in and p_out, should not be affected. Taken together if crowding should have a significant effect, this would lead to points to the upper left of the line of Figure 2H. Thereby one could still exclude the two models for a large range of parameters and this would not affect the main conclusions of this paper.

We tested this effect numerically, by replacing a fraction of the binding sites by inert reflecting spheres. We tried in the range of 20%-80%, with parameters otherwise similar to the ones used in Figure 2H. This mimics crowding since the observed particle would have more reflection events but fewer binding events. However, the effect of these reflections are minimal compared to the effect of bindings. As can be seen in figure 1 of this response, the observed points still fall on the line, which is explained by the fact that binding events are much more dominant for the observed dynamics compared to the reflection events.

Furthermore, for the results in Figure 4, the presence of large molecular crowing would increase the search time even more for the PBM, making it even worse at enhancing the on-rates of specific reactions.

We have added a few sentences in the section following the introduction of Figure 2H, in the final part of the section: ”Scaling relation…”, as well as a new supplemental figure to Figure 2.

2. I found the discussion of the angles between consecutive steps hard to follow at points. In particular, what do the authors have in mind by the statement that binding sites can "reflect" the motion of tracked molecules (line 219)? I also wasn't sure what the final sentence of that section was meant to convey – some more guidance on the conditions under which this approach is useful would help.

We have tried to add some explanatory sentences. When we use the word “reflect,” we mean that the density of binding sites is high, and since they are partially reflecting, the reflections result in an enrichment of the change of angle around π. This effect should only be seen if the time scale of measurement is small or comparable to that of binding and unbinding. On the other hand, since the focus is relatively small, a significant part of the particles hit the focus boundary, affecting the statistics even at relatively long timescales. Therefore, both the reflections from the individual binding sites (only present in the PBM) and the effective potential around the boundary of the focus (present in both the PBM and LPM) can in general lead to an asymmetry in the distributions of angles.

In the experiments, the measurement time scale is too large to see the effect of reflections against the binding sites from the PBM. However, asymetry is observed, which can be attributed to the influence of the focus boundary.

To explore the effect of reflections against binding sites, we recourse to simulations in an infinite medium (where boundaries do not play a role) with shorter observational time scales, as reported in Figure 3—figure supplement 1.

We have rewritten parts of this section in order to make this point clearer in the revised version of the manuscript.

3. The paper is well written but could use some additional proofreading for spelling, e.g. "dropblet", "membranelss", "rapide", "displacememnt", "mimick".

We thank the referee for this comment and apologize for the inconvenience. A serious proofreading has been performed for the revised version.

Reviewer #2:

General:

The manuscript could use some additional proofreading for grammar/typos.

We thank the referee for this comment and apologize for the inconvenience. A serious proofreading has been performed for the revised version.

Regarding the references to Miné-Hattab et al., 2021 we were uncertain with regard to two points:

1. Figure 4b shows a striking agreement between the focus size and the minimum in the time-to-find-a-target function. Measuring the focus radius of 120 nm with such a high accuracy requires exquisite microscope resolution, while we believe that this is in principle possible with PALM, we couldn't find any supporting data for this resolution in the original paper, would it be possible to point us either to the right place in the manuscript or to general references that make clear that this is a standard resolution for live-PALM the way it was used in Miné-Hattab et al.?

The resolution in PALM (in fixed cells) depends on the signal to noise ratio. For our types of experiments, we obtain a standard resolution of 30 nm. For live PALM, the signal to noise ratio is smaller than in fixed cells (the fixation altering a little bit the fluorescence). We measured a pointing accuracy of 20 nm in live PALM. However, the limitation in the resolution comes from the motion of the structure during the time of the acquisition. Therefore, to optimize the acquisition, we used the shortest possible exposure time (20 ms) and performed videos of 1000 frames (20 sec). Since a focus can move in 20 seconds, we also discarded foci that moved too much in x and y during the acquisition. Finally, we also measured the mobility of the whole foci formed by 1 versus 2 DSBs, and observed similar mobilities (Figure 6B from Min´e-Hattab et al., 2021).

To support the conclusions obtained with PALM, we also derived the radius of the focus based on the MSD curves, where we separated the traces belonging to the slow diffusion coefficient. In this way we could do a fit to the MSD for confined diffusion, and extracted the best value and the uncertainty on the parameters. Here we found the same values for the radius of the foci, which confirmed that both methods measured the radius of the focus to be ≈ 120nm.

To clarify this point in the manuscript, we have added some sentences related to Figure 4B. Here we explain how the number of 124 nm was extracted based on the fit to the confinement radius. Furthermore we relax the claim that this correspondence between the number is surprising, by stressing that the theoretical curve is flat around the minimum, so there really is a range of optimal droplet radii.

2. Please point us to the measurements of inverse partitioning (p_out/p_in). We weren't able to immediately find them in the previous paper.

We derived the free energy of this, from which the inverse partitioning can be directly obtained. This was done in the final part of the results of Min´e-Hattab et al., 2021, related to Figure 7.

We have added a subsection of Methods to explain how various measurements were obtained in Min´e-Hattab et al., 2021.

Abstract/Intro:

Foci as a word works, however, in particular in the intro condensate and foci need more of distinction. A nucleolus (as a very prominent example of a nuclear condensate) wouldn't be called focus in the literature.

We agree that the nucleolus would not be considered a typical focus, but at the current stage of the literature, a lot of terms are used not always in a clear way. This is not surprising since the field is still quite young and new results affect how we think of these structures. As argued in the paper by Banani, Lee, Hyman and Rosen from 2017 the ”non-Membrane bound compartments … have been referred to by a variety of names, including cellular bodies, nuclear bodies, membraneless organelles, granules, speckles, aggregates, assemblages, membrane puncta” and they propose the name Biomolecular condensates. Overall, we feel that ”Condensate” is more general and does not refer to a small spherical object, which foci does. Here, we consider a focus to be a spherical condensate of size smaller than a few hundreds nanometers.

We have added text to the first paragraph of the introduction to better discuss this definition.

The abstract seems to be missing a key point of the paper, namely that based on the available observables/data a call can be made that yeast repair foci are more likely LPM, rather than PBM.

We are thankful for this comment, and we agree that this is very important to mention in a clear way.

We have added a sentence to the abstract.

L48: The references seem to be a mix of reviews and primary research, we believe this strong statement calls for only primary references.

We thank the referee for this comment, and we agree on this statement. We have removed one reference, so it is now only primary references.

L54: What did Miné-Hattab et al., find? How did you make the call in your previous paper?

In the paper of “Single molecule microscopy reveals key physical features of repair foci in living cells” of Min´e-Hattab et al., 2021, we suggested (but did not conclude) that Rad52 moved according to the LPM model, which is not the case for the single binding protein Rfa1, also contained inside repair foci. This suggestion was based on 6 different observations that we will briefly list here:

– Rad52 molecules inside the focus were significantly more mobile than the focus itself and than double-strand break flanking region.

– We observed a sharp change in the diffusion coefficient around the boundary of the focus.

– We observed a sharp potential around the boundary of the focus.

– When we induced two DSB’s we observed that there would exist one spherical focus, with a volume exactly two times larger than the focus for one DSB. This suggest that the two individual foci had fused together, which would be a fingerprint of two merging droplets.

– We found that by adding aliphatic alcohol hexanediol (a component proposed as a tool to differentiate liquid-like from solid-like assemblies), the Rad52 foci partially dissolved.

– Inside foci, Rad52 molecules exhibit confined motion, with a confinement radius matching with the size of Rad52 foci measured independently by PALM.

We have added details in the second-to-last paragraph of the introduction about the conclusions found in Min´e-Hattab et al., 2021.

Results:

L73: maybe 'by the Boltzmann factor' is a bit easier to google for the uninitiated reader, rather than 'Boltzmann's law', which isn't really a thing.

We agree that this was a wrong formulation that could be confusing. We have corrected this to the “Boltzmann distribution”.

L76: 'While this description is general' is unclear, but could simply be left out.

We agree that this was an unnecessary formulation.

We have removed this, and changed it to “Here different…”

L80ff: Motivate why this potential is a good way to model binding sites? In the PMB model, the diffusion model for the binding sites ends up being very similar to the one used for LPM. Is it then fair to claim that this model is more "microscopic"? It seems that PBM is more microscopic only in words and not really in the modeling. One might say that the binding sites are just themselves following diffusion in an LPM. Then the difference between the two models is more about the fact that for LPM the tracer molecule diffuses in the LPM and in the other case the tracers do not feel directly the LPM but just bind to molecules (or binding sites) that are diffusing in the LPM. In this respect, the distinction between the two models does not contain much information on whether the foci are formed by phase separation or not. In view of the importance of this distinction and the ongoing controversies, the authors should clearly discuss this issue. To have a more microscopic model it might be useful to adopt a polymer model description, rather than a simple potential.

We start by emphasising the importance of the PBM with immobile binding sites (i.e. D_b = 0). It is typically assumed that the diffusion coefficient of the binding sites are significantly smaller than the diffusion coefficient of the free proteins, and this is also something we confirm in the paper Min´e-Hattab et al., 2021 (D ≈ 1.0µm²/s vs D_b ≈ 0.006µm²/s). Therefore one can in most situations approximate the binding sites as being static, and in this case the potential of the focus is not relevant, it is merely the position of the binding sites. Here we can assume that there is a region with high density of binding sites (i.e. inside the focus) and a region with no binding sites. We can then apply the techniques used in Figure 2 to derive the relation between the effective diffusion coefficient and the free energy. However, since the diffusion coefficient of the binding sites is non-zero we need to make sure that the theory did not break down by including this, and as soon as the binding sites are diffusing, one needs to introduce a potential in order to keep them inside the focus, since otherwise they would just spread over the entire nucleus. The potential we are introducing here is therefore merely an effective way to to maintain moving binding sites within the focus. In all realistic parameter ranges (i.e. D > D_b) the binding/unbinding interactions with the binding sites, lead to the effective diffusion coefficient in a description that is also valid for diffusing binding sites. This model is therefore more microscopic since we use fundamental interactions to find the effective diffusion coefficient of the particles, whereas in the LPM it is a free parameter in itself that is not directly related to simple and fundamental interactions. We have added the following explanatory part to the discussion:

“To clarify this, we have removed the word “microscopic” in the section where we introduce the PBM, and we have added a sentence describing how the diffusion coefficient in this model is not a free parameter but an observable that we can measure based on the interactions with the binding sites. Furthermore, we have also removed the word “microscopic” from the Discussion section, and added a sentence stressing that the motion inside the focus is a result of the interactions with the binding sites.”

L92: Why exclude potential? Relevant for confinement, shape of droplet boundary (which might be a significant part of the droplet, for such small droplets). Also, six isn't much worse than five parameters.

We agree that this formulation leads to confusion, since we do not exclude the potential, we just don’t consider it as a parameter since the biding sites are kept inside the focus. Therefore we feel that this formulation should be changed.

To clarify this confusion we have changed the description following the equation 5 in the revised manuscript to emphasize that the only role of U_b is to keep binding sites within the focus, and is thus not fitted.

L105: Maybe say that parameter values are mentioned in Figure legend.

We agree that his is helpful to the reader and thank the referee for pointing this out. In the revised manuscript we have added: “(see Figure 2 caption)”

L109: Why just the simple Gaussian of free diffusion? Shouldn't there be confinement effects for small droplets?

For sufficiently small time steps, molecules inside the focus (that do not interact with the surface potential) experience a free diffusion. For longer time-steps more traces will have interacted with the surface potential and this leads to the confined diffusion.

To clarify these points we have specified that we consider traces within the bulk of the focus which are not interacting with the boundary for small time intervals.

L116: Would be good to have references for k- in yeast. At least ballpark. Otherwise also look at other k-.

There are not many results indicating the actual off rate of Rad52 binding sites. In one recent study they found the dissociation constant k_d for Rad52 to vary between two observed sites from 5.6 nM to 101 nM (Saotome et. al, 2018 iScience), however we do not know the number of molecules present in the study and cannot derive the actual unbinding rates for these results. However for many other receptors the actual unbinding rates can differ by orders of magnitude. We also argue that in the first part of the Results section, we show how we expect the dynamics to be if the value of k_ is much smaller. In this case the diffusion coefficient inside the focus should not follow a normal diffusion translocation histogram but be the sum of two separate events. If we had found these different diffusion coefficients in the data, we could have made a more simple conclusion. Since we do not find this in the data, we use this as an argument to say that if the focus was really a PBM, the off rate has to be fast.

To make these numbers visible as well as the large range of values for the dissociation constant, we have added a sentence highlighting this and citing the paper by Saotome et al., in the section related to Table 1, where we discuss the parameter values.

L120: Does Rad52 ever appear in groups, e.g. dimers or multiple times within the same repair site?

Monomeric and multimeric forms of Rad52, have previously been observed in vitro (see for instance: Saotome et al., 2018, Structural basis of Homology-Directed DNA repair mediated by RAD52). In our previous paper Mine-Hattab et al., 2021, we suggested that both forms were found in the nucleus, but since we found one diffusion coefficient inside the focus, we do not think this alters our results here. If it was the case that more diffusion coefficients were present, we would in principle still be able to calculate the average diffusion coefficient, inside and outside the focus, and thus recover the results of Figure 2H to separate the two models.

L127ff: This entire section is a bit less clear than the rest. Maybe a slightly longer, more systematic exposition could help?

We agree that this argument could be explained better, and since it is closely related to Figure 2D, both the top and the bottom panel should be cited in this section.

In the revised version we have added some sentences and cited the findings in Figure 2D in order to guide the argument. Furthermore, we have added an introductory sentence in the section below, in order to make this transition more clear.

L131: Rebinding description is a bit vague, maybe elaborate a bit? Can this be connected to appendix 1?

Right after a particle is released, it will be close to the binding site from which it was just released. If the binding site is perfectly absorbing and the particle is released exactly at the boundary, it will be immediately recaptured. However since the binding sites are partially absorbing, there is a fixed probability that the particles will escape the binding site from which it was just released. In the paper we cite (Kaizu et al., 2014), they show that by including the rebindings, the effective off rate is significantly lowered but one can still assume it is a constant rate and can thus be treated as an effective k_off in the derivations. This simplifies the mathematics of the section, and let us derive the result that we validate in the simulations, which confirms that this assumption is valid for the system.

In the revised version of the manuscript, we have expanded the discussion of how rebinding events can still be modelled as a rate. This is included in the description just before the citation of Kaizu et al., 2014, which is just before the definition of equation 6.

Equation 7: Shouldn't the second ∝ be an equal sign?

That is correct and we thank the referee for noticing this It has been changed in the revised version.

141: How realistic is this limit? Doesn't it imply also fast diffusion? In other words, it seems like equation 6 gives many constraints on what your rate can be. Please comment. How well does the time scale separation hold? If it doesn't, many of the conclusions would have to be modified.

We thank the referee for pointing this out, since this could create a scepticism for many readers. This limit is realistic for many systems. Equation 6 gives a constraint in the value K₊, which arise as κ 7→ ∞. This is the limit where we call the reaction “diffusion limited”. This is contrary to “rate-limited” reaction where κ is very small. However, reguardless of that, large binding rates for the tracked particle (given by K₊ρ(r)), can be achieved by a large density of binding sites ρ(r). This is one of the fundamental assumptions in the PBM, consistent with our choice of parameters.

In order to discuss this issue, we have added two sentences in the beginning of this part, just before the introduction of equation 9, highlighting that the rates should be large and that the defining property is the large density of binding sites.

L152: Recall the definition of δ r. Is average from equation 10 removed?

δr is the experimental displacement observed during δt, which dr is an infinitesimal step. The average in Equation 10 is correct (otherwise there would be a noise term).

We have added additional descriptions of expression in line 181, in order to define the meaning of δr and clarify that hδr2i/(2dδt)is the maximum likelihood estimator of the diffusion coefficient in a small region of space.

L161: Not sure about D₀, according to table this is only part of LPM? Also not mentioned in

L93.

We are sorry about this confusion. D₀ is the effective diffusion coefficient inside the focus. This can be measured in experiments or simulations for the PBM, but it is a function of the other parameters. Therefore it should not be inserted in the table for the PBM, since the other parameters are more fundamental and they lead to a resulting diffusion behaviour inside the focus.

We have added a sentence explaining that D₀ is a parameter that can be extracted based on the simulations in the PBM but not an input itself. This explanation is added to the text just prior to the result of Figure 2H.

L164: Typo in unit for rho.

We thank for this observation.

It has been corrected in the revised version.

L167: Is this shown? Is this Figure 2d?

Yes this is indeed what we show in Figure 2D.

In the revised version we have added this reference for this sentence.

L181 and equation 13: Inconsistent use of δ vs d.

We are sorry about the confusion in this regard. δr is the measured displacement between two successive measurements. Due to the limits of optical resolution, in experiments we have a time between measurements that is short however not infinitesimal. Therefore we refer to δr, when extracting data from experiments (for instance to measure D˜(r) ). However it should be correct that this estimation we make in equation 13, is an approximation of this displacement and therefore we should change it to δr. To clarify this we have added the following sentence to the reviewed version:

“In order to correct for this and avoid confusion, we have changed the definition in equation 13, so d is replaced by a δ and the = is replaced by ~.

Equation 13: In measuring this in 3C, is the average r as a function of r computed from displacements starting at r or arriving at r?

This is an important question that should be specified clearly in the text. This is the starting position of r. However we note that we have tried to calculate this as a function of the midposition as well, but this does not make significant deviations to the main results.

We have specified this in the caption.

L164: Is it obvious that dr cannot be negative? What are the consequences of this? Is it true at the focus boundary?

This is a consequence of equation 13, where it is seen that if D_b = 0 then the expression reduces to:

< d r >= d t \frac{D (r)}{r}

It is not obvious when one thinks about the problem, but comes directly out the mathematical formulation. The consequence is that for immobile binding sites, the dynamics of the PBM looks like the dynamics in a region with no defined potential but with a radial diffusion coefficient, calculated by using the Ito interpretation, without including the spurious term. It should be noted that this is true everywhere – also around the boundary. We have tried to make this statement clearer by adding two sentences after this statement, clarifying the difference between the situation when D_b = 0 and D_b > 0.

L195: Why underestimate and how is this seen? Underestimate with respect to what ground truth? Overall the discussion of 3C, D should be extended and made clearer. It is one of the main signatures used to tell PBM from LPM so it should be very clearly discussed.

This underestimation is seen since the red curve in Figure 3D is significantly more negative than the two others, predicted only by the diffusion and the equlibrium distribution. The reason for this is that there is a defined connection between the diffusion coefficient and the equilibrium distribution in the PBM, but this is not necessarily the case for the LPM. In particular if we “know” the diffusion coefficient of the possible binding sites through other experimental investigations, one can directly see if the observed diffusion coefficient leads to the predicted behaviour around the boundary.

We have tried to highlight this more clearly. When referring to Figure 3D we have added a sentence about how the underestimation is seen. Also we have added a sentence after the sentence trying to explain why the PBM prediction results in a mismatch when comparing to the LPM. This sentence is inserted after the phrase ending on ”the wrong model”.

L197: Didn't you say above it was never negative? Now only for this value?

It can be negative if D_b > 0, see point 21.

This is now clarified after Equation (13).

L199: Can you expand this a bit for a clearer picture?

It is seen for macroscopic objects (for instance a water strider) that the surface pressure can lead to reflection. This is a direct result of the Laplace pressure which is given by ∆p = −γ∇·~n. Microscopically, this occurs due to the energy between interacting molecules in the liquid forms a surface, that counteract deformations in a droplet, for instance created by a macroscopic molecules that hit the surface. However for microscopic molecules, this does not lead to a deformation in the surface and therefore this Laplace pressure does not affect this system. We wanted to include this part, since many readers tend to think of surface tension in this way, and this is something completely different. However, we feel that it adds more confusion and does not clarify the text and we have therefore decided not to include this sentence.

This sentence has been removed in the revised version of the manuscript since we felt it added more confusion and did not add important information to the questions raised in this paper.

L210: Without confinement in shouldn't it be uniform in any dimension?

Yes this is true, but in two dimensions the angle is easy to define and understand. As we increase the number of dimensions, more angles appear which leads to much more tedious derivations.

We agree that this formulation was misleading and we have corrected this by moving the ”2 dimensions” to the ending of the previous sentence.

L250: Can you comment more on the Burger Purcell limit? How relevant is the discussion in the present context? If it is relevant, it should be expanded.

We believe this is important since the work of Berg and Purcell is a classic result that some readers will be familiar with. Therefore it is important to us to comment that our result is in accordance with this fundamental work in the relevant limits. It does not give any new insight to the result, but it might help a big part of the readers to understand and accept the result. Therefore we do not deem it necessary to expand on this since it does not affect our results directly.

To motivate this comparison more, we have added a sentence explaining that this bound has previously been shown to be relevant to gene regulation, citing the work of Gregor et al., 2007.

L263: Missing 'of'.

We thank the referee for noticing this typo.

This has been corrected in the revised version.

L269ff: This seems quite weak as a test, compared to the previous results (in particular 2H). The PBM clearly also has a minimum, how different are these for different parameter ranges? It also seems very hard to control focus size in vivo.

We agree that as an experimental test, this part is not as strong as the previous sections, mainly since it might be experimentally more difficult to carry out in reality. However, we still feel that it is an important part of the paper, since our analysis shows that if the binding sites are immobile (i.e. D_b = 0) then according the equation 21 the optimal focus size is r_f = 0. This means that it does not have a minimum, and even for relatively fast moving binding sites (remember that Figure 4C with D_b = 0.01µ²/s is twice the experimental value), the advantage of having the focus is really minimal.

This result shows another important element that is unrelated to the physical observables: If a focus is a PBM with approximately immobile binding sites, it is not possible to obtain an advantage in terms of enhancing the on-rates of specific molecules. On the contrary, it increases the search time significantly. Therefore this result suggests that if a focus should increase the on-rates (which would be the case for Rad52 and DNA repair) then a focus with immobile binding sites would never be the optimal structure and a liquid droplet would be much more effective. However, if the focus had the purpose of silencing genes, then it would make sense to increase the search time significantly, and here a PBM is a strong suggestion.

Regarding the focus size in vivo, we showed in the paper Min´e-Hattab et al., 2021 that in the presence of 2 DSB, one focus is formed with a volume approximately double in size. Therefore it might be difficult to construct a continuum of focus sizes but it should be possible to test this at a few different volumes.

We modified this argument in the revised paper. In the final part of the section, following Equation 21, we have changed the description so we do not use the phrase ”rule out the PBM” but on the contrary we write ”…identifying an optimal focus size would be an indication that the model is not a PBM”. Furthermore, we have added a sentence to the discussion, where we argue that this part is not the most optimal test to setup experimentally, but it might be important in the understanding of why foci could be one structure or the other.

L297: Using 'condensates' as a verb required rereading this sentence for clarification, maybe rephrase.

We have replaced that word by “concentrates”.

Figures:

General: It would be helpful to have bold one-sentence figure caption summaries. Please make sure to always mention the equations that each panel refers to in the figure caption.

We thank the referee for this point, that we agree will help the reader.

In the revised version, we have included one sentence for all four figures in the main text, and we have added references to the relevant equations in the figure captions.

1: Panel B: might be good to also have shading for the background of the nucleus. Otherwise the boundary could be mistaken for a nuclear membrane happened to us at first glance (also compare Figure 4b).

We thank the referee for pointing this out. We also realized that it can be confusing in particular when comparing to Figure 4.

In the revised version of the paper, we have updated Figure 1 based on these suggestions.

2: Panel D: caption: are (bottom) and (top) reversed?

This is correct and we thank the referee for pointing this out. They have been corrected in the revised version of the paper.

Panel F: Caption: please clarify 'using displacement histograms'.

We acknowledge that this formulation was not clear. What we mean is that within a small radial segment we calculate all displacement, and use the maximum likelihood estimator D˜ = ‹hδr²›/(2dδt) to calculate the diffusion coefficient inside this radial segment.

In the revised manuscript, we have added a part to this caption, specifying how we obtain this the diffusion coefficient.

Panel G: Displacement is misleading for the x-axis label, how about 'r' or something like 'dist. from nucleus'.

This is true and we thank the referee for helping us clear this up.

This has been changed to “Radial coordinate” for the revised version of the paper.

Panel H: reorder legend, otherwise data point 'experimental observation' looks like legend.

X axis has D₀, shouldn't this be D^~? Same in caption for H.

To us this panel has the strongest evidence that the model of choice should be LPM,

is this true? If yes, please clarify in text.

We thank the referee for pointing this out and we agree on this point. We have tried to move the ‘experimental observation’ label so it fits better. We also thank the referee for raising the question D˜ or D₀. We will try to make this clearer in the main text, since we define D˜(r) as the spatially dependent diffusion coefficient, whereas D₀ is the diffusion coefficient inside the focus. This we will make clearer in the revised version.

We furthermore agree that this panel is the strongest, which is also why it occupies extra space to highlight it. However we believe that the main result is the equation 11 that is tightly related to Figure 2H, where the relation is shown to hold.

In the revised figure, we have inserted the “experimental observation” as a legend, so the data point is still clear to observe. We have added the distinction between D˜(r) and D₀ in the section regarding Figure 2H. Finally, we have added one sentence emphasizing the importance of this figure and the related equation 11, in the main text after introducing equation 11.

Caption for H: Could you refer to the figures/sections in Miné-Hattab where the

partitioning is calculated? After a quick skim we weren't able to find it.

This is calculated in the bottom part of the Results section, related to Figure 7, and use the input as we show in the table of all experimental values (Table S1).

To clarify this, we have added this information in the Materials and methods section, guiding the reader where to look in the previous paper. This has been added to the caption of Figure 2H and to the main text related to Figure 2H.

3: Panel B: What do the error bars refer to?

The error refers the standard deviation of the mean. We have added that information in the caption.

Panel C: Why the dip in the bottom panel? Equation 13 is said to never be negative? Is only D_b varied between the bottom and top panels? If D^~ is kept constant, shouldn't the plateaus be the same?

From Equation 13 we see that as Db grows, the more negative the value for dr is (given that ∂_rU(r) is positive). Therefore the dip stems from the fact that D_b has a large value. Here it is 20 times larger than the experimental value, and for values that we find realistic it will not be negative. However it is possible for fast diffusing binding sites, and this is what we wanted to show in this figure. This also explains the difference in the plateaus, since the value of D˜ is not the same between the figures, since D_b (but only D_b) has been varied.

In the revised version we have tried to better explain this issue of when it can be negative depending on the value of D_b, (text related to the description of Figure 3C).

Panel D: Caption: black instead of blue line?

Are these plots purely based on equation (13)? Same as in C, how can this be negative?

We thank the referee for pointing this out. The explanation that the blue line can be negative is again, that if the PBM has strongly diffusing binding sites, then the value of dr around the boundary can be negative. However it still fails to predict how negative it is, since the LPM has more freedom to diffuse fast inside the focus.

In the revised version of the paper, this has been corrected. Furthermore, we have shifted the blue, and the yellow color to shades of green, since these are related to the PBM model

Reviewer #3:

Membraneless condensates have recently become a central focus of the molecular and cellular biophysics communities. While the dominant paradigm for their formation, liquid-liquid phase separation (LLPS), has been well established in a number of cases for large, optically resolved droplets, there are significant concerns regarding the generality of this mechanism for smaller foci or puncta, and other mechanisms have been proposed to explain their formation. The problem is that it is very difficult to distinguish experimentally between these mechanisms for sub-optical resolution condensates. In this article, Heltberg et al., propose a novel method, based on the analysis of single molecule tracks, that allows discriminating between the liquid phase model (LPM) and one of the challenger mechanisms, the "polymer bridging model" (PBM). This method relies on the statistics of individual displacements – diffusion, radial displacements, angular changes – which are showed theoretically to exhibit different signatures for the two models. With realistic data this is sufficient to discriminate between the models: for instance in the case of double strand break foci (DSB), building on a recent work by some of the same authors, this article convincingly rules out the PBM in favor of the LPM. The author also investigate the influence on these two models on the search time to reach a specific small target – a commonly invoked role of condensates – and show that only the LPM substantially accelerates this, which could provide additional means to experimentally discriminate between the mechanisms, on top of the intrinsic interest of this finding.

This article is a welcome addition to the literature in this field, as it will help clarify the nature of these condensates, in particular below the optical resolution. It is well-written, interesting and the conclusions are justified. I particularly appreciate the effort to employ simulated data that are realistic for actual experiments, which strengthens the claims of applicability. Some aspects of the data analysis and of the modeling, however, are insufficiently discussed and would need to be precised / expanded.

1) The modeling is made under the assumption of thermal equilibrium, without further discussion. The authors should comment on why this is reasonable, in particular in view of the presence of active fluctuations and of chemical reactions in these condensates.

First of all, the experimental measurements are carried out after the formation of the foci, and the time of observation (tens of seconds) is small compared the lifetime of foci (tens of minutes). Therefore we can assume that these measurements are not affected by the effects of formation and disruption of foci. Secondly, the data extracted to compute the results of Figure 2 (in particular for Figure 2H) are not very sensitive to the active fluctuations, since we derive an average diffusion coefficient inside and outside of the focus as well as a free energy difference between an inside and outside level. It is indeed very likely that the soup of proteins that forms the focus is active, however Rad52 is not involved in chemical reactions at the timescales we are looking at may be considered passive. This is supported by our investigations of the experimental results, where we have not seen any statistical differences as a function of the time of measurements, and we have no reason to believe that active fluctuations affect the diffusivity of Rad52 on the observed timescales. Regarding binding sites, they may also diffuse actively along with the genome and chromatin, but we describe this by an effective description of the motion of Rad52 on short time scales, so that active effects are folded into an effective diffusivity (left as a free parameter).

We want to highlight this issue as well as present our arguments of why this description is valid for the experiments considered in this work. We have added text between Equation 6 and Equation 7 summarizing the arguments outlined above.

2) How is the diffusivity measured? Are these measures corrected for experimental error (e.g. using three-point estimators)?

Estimates of the diffusion coefficients in Min´e-Hattab et al., 2021 were obtained in different ways. Our main method is to generate the displacement histogram, and then estimate the number of different diffusion coefficients in the population based on likelihood fitting and KStesting. Then we take for all the traces, and find the ones that we are certain belong to the slowest diffusion coefficient. These traces are the ones in the focus, but by doing it this way, we are not vulnerable to the position of the boundary and to determine which are in the focus based on their position. Then we compute the MSD curves for this distribution of slowly diffusing molecules, and fit the diffusion coefficient based on a confined fit (which has a good p-value). This method is strong since we are fitting a slow diffusion population and typically can reject traces belonging to the fast diffusion coefficient. We also include the possibility of separating traces if the molecule goes from inside the focus to outside or the other way round. The alternative way we calculated the diffusion coefficient, was based on the microscopy data, where we “cropped” all the traces that could be visibly identified as being inside the focus. This method had the strength that we could visibly follow all traces, but the drawback that we could mistakenly identify molecules as being inside the focus, then they could be under or above the focus, as discussed in the section above. However both method yielded similar results. It is also based on these methods that we extract the size of the focus.

In order to clarify this important point, we have added two sentences in the caption to Table I, describing how the diffusion was measured in the experimental paper, and added a new paragraph about experimental measurements in Materials and methods. In addition, we have clarified in the caption of Figure 2F that we extract the maximum likelihood value of D˜(r) in each radial segment.

3) The conditioning of the averages should be discussed, e.g. in Equation 13: I assume that it is in the Ito convention? Similarly for the angle changes.

We assume that the density of the binding sites, follow a radial distribution, with no significant angular dependency. Thus the average displacement ‹dr› is computed as a function of the initial position of the particle and averaged over all initial displacements with similar radial positions. It is indeed formulated in the Ito convention, which is why the “spurious” term appear in the first term of the second line of equation 13.

To clarify that we are using Ito convention, we have stated that we are using Ito convention for this paper, just before the introduction of equation 1. We have furthermore clarified in the section related to equation 13 and the section related to the distribution of the angles that we use the initial position when calculating the difference between the two connected points.

https://doi.org/10.7554/eLife.69181.sa2

Article and author information

Author details

Mathias L Heltberg
1. Laboratoire de physique de l’École normale supérieure, CNRS, PSL University, Sorbonne Université, and Université de Paris, Paris, France
2. Institut Curie, CNRS, PSL University, Sorbonne Université, Paris, France
Contribution
Conceptualization, Investigation, Methodology, Software, Visualization, Writing - original draft, Writing - review and editing

Competing interests
No competing interests declared

"This ORCID iD identifies the author of this article:" 0000-0002-9699-4075
Judith Miné-Hattab

Institut Curie, CNRS, PSL University, Sorbonne Université, Paris, France

Contribution
Conceptualization, Funding acquisition, Resources, Writing - review and editing

Competing interests
No competing interests declared

"This ORCID iD identifies the author of this article:" 0000-0001-9986-4092
Angela Taddei

Institut Curie, CNRS, PSL University, Sorbonne Université, Paris, France

Contribution
Conceptualization, Funding acquisition, Resources, Writing - review and editing

Competing interests
No competing interests declared

"This ORCID iD identifies the author of this article:" 0000-0002-3217-0739
Aleksandra M Walczak

Laboratoire de physique de l’École normale supérieure, CNRS, PSL University, Sorbonne Université, and Université de Paris, Paris, France

Contribution
Conceptualization, Funding acquisition, Investigation, Methodology, Supervision, Writing - original draft, Writing - review and editing

For correspondence
aleksandra.walczak@phys.ens.fr

Competing interests
eLife senior editor

"This ORCID iD identifies the author of this article:" 0000-0002-2686-5702
Thierry Mora

Laboratoire de physique de l’École normale supérieure, CNRS, PSL University, Sorbonne Université, and Université de Paris, Paris, France

Contribution
Conceptualization, Funding acquisition, Investigation, Methodology, Supervision, Writing - original draft, Writing - review and editing

For correspondence
thierry.mora@gmail.com

Competing interests
No competing interests declared

"This ORCID iD identifies the author of this article:" 0000-0002-5456-9361

Funding

Agence Nationale de la Recherche (Q-life 356 ANR-17-CONV-0005)

Mathias L Heltberg
Judith Miné-Hattab
Angela Taddei
Aleksandra M Walczak
Thierry Mora

Centre National de la Recherche Scientifique (80' MITI project PhONeS)

Judith Miné-Hattab
Angela Taddei

H2020 European Research Council (COG 724208)

Mathias L Heltberg
Aleksandra M Walczak
Thierry Mora

Agence Nationale de la Recherche (ANR-15-CE12-0007)

Judith Miné-Hattab
Angela Taddei

Agence Nationale de la Recherche (ANR-12-PDOC- 0035?01)

Judith Miné-Hattab
Angela Taddei

Fondation pour la Recherche Médicale (DEP20151234398)

Judith Miné-Hattab
Angela Taddei

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

Acknowledgements

The authors are grateful to Jean-Baptiste Masson, Alexander Serov, Namiko Mitarai and Ned Wingreen for valuable discussions. The study was supported by the Agence Nationale de la Recherche (Q-life ANR-17-CONV-0005), Centre National de la Recherche Scientifique (80’ MITI project PhONeS), the European Research Council COG 724208, the Labex DEEP (ANR-11-LABEX-0044 DEEP and ANR-10-IDEX-0001?02 PSL), the ANR DNA-Life (ANR-15-CE12-0007), the Fondation pour la Recherche Médicale (DEP20151234398), and the ANR-12-PDOC- 0035?01. The authors greatly acknowledge the PICT-IBiSA@Pasteur Imaging Facility of the Institut Curie, member of the France Bioimaging National Infrastructure (ANR-10-INBS-04). MLH acknowledges the Carlsberg Foundation grant CF20-0621.

Senior Editor

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

Reviewing Editor

Agnese Seminara, University of Genoa, Italy

Reviewer

Pierre Ronceray

Version history

Received: April 7, 2021
Accepted: October 21, 2021
Accepted Manuscript published: October 22, 2021 (version 1)
Version of Record published: November 17, 2021 (version 2)

Copyright

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