#  Microscopic model for aging of biocondensates

Hugo Le Roy<EMAIL_ADDRESS>Institute of Physics, École Polytechnique
Fédérale de Lausanne—EPFL, 1015 Lausanne, Switzerland  Paolo De Los Rios
Institute of Physics, École Polytechnique Fédérale de Lausanne—EPFL, 1015
Lausanne, Switzerland

###### Abstract

Biomolecular condensates are membraneless compartments in the cell that are
involved in a wide diversity of biological processes. These phase-separated
droplets usually exhibit a viscoelastic mechanical response. A behavior
rationalized by modeling the complex molecules that make up a condensate as
stickers and spacers, which assemble into a network-like structure. The proper
functioning of biocondensates requires precise control over their composition,
size, and mechanical response. For example, several neurodegenerative diseases
are associated with dysfunctional condensates that solidify over a long period
of time (days) until they become solid. A phenomenon usually described as
aging. The emergence of such a long timescale of evolution from microscopic
events, as well as the associated microscopic reorganization leading to aging,
remains mostly an open question. In this article, we explore the connection
between the mechanical properties of the condensates and their microscopic
structure. We propose a minimal model for the dynamic of stickers and spacers,
and show that entropy minimization of spacers leads to an attractive force
between stickers. Our system displays a surprisingly slow relaxation toward
equilibrium, reminiscent of glassy systems and consistent with the liquid-to-
solid transition observed. To explain this behavior, we study the clustering
dynamic of stickers and successfully explain the origin of glassy relaxation.

Soft matter $|$ Theoretical biophysics

††preprint: APS/123-QED

## Introduction

Liquid-liquid phase separation has emerged as a major organizing principle for
the compartmentalization of cellular processes into membraneless organelles
known as biomolecular condensates. Condensates are associated with various
functions of the cell, although their precise role often remains elusive [1],
to the point that it is not even clear if some of them are functional,
incidental or pathological [2]. Similarly, little is known about the complete
composition of condensates; however, one or several scaffolding components
that drive the phase separation can usually be identified, including proteins
, DNA [3, 4], or RNA [5, 6]. Despite their diversity, condensates also display
common features. It is usually accepted that their formation requires weak
multivalent interactions [7], mediated by disordered portions of proteins [8]
or by RNA/DNA-protein interactions [9]. Similarly, micro-rheology experiments
suggest that biocondensates exhibit a general viscoelastic mechanical response
[10, 11]. This picture has been rationalized through the sticker and spacer
model, which describes proteins as generic sequences of sticky (stickers) and
non-sticky (spacers) regions [12]. The complex interactions that make up a
condensate give rise to an emergent long-time transition to a solid-like
state, usually described as aging [13, 14]. This process has been associated
with protein denaturation [15], amyloid formation [16] and, from a
physiological perspective, with neurodegenerative diseases [17]. Current
physical pictures of condensate aging either rely on the modeling of specific
proteins’ interaction – like $\beta$-sheet stacking [18] –, or do not model
the microscopic dynamic of the components [19, 20]. Yet, aging seems to be a
fairly general behavior, essentially insensitive to the details of the
condensate’s composition, which suggests that a universal physical mechanism
may govern this transition. A general microscopic picture explaining how large
timescales (days or more) emerge from a collection of microscopic processes,
that typically occurs on much shorter timescales, ($\mu s$) is missing.

In this work, we use the sticker and spacer framework to model the microscopic
structural rearrangement taking place in condensates, and leading to aging.
This approach essentially considers proteins as a heterogeneous collection of
localized, attractive regions (stickers) and longer, neutral regions
(spacers). Stickers reversibly associate via non-covalent interactions, while
spacers provide flexibility and connection between stickers, resulting in the
formation of a percolating network. The ensuing mechanical behavior of this
network is typical of an associative gel, characterized by a viscoelastic
response to stress. At short timescales, the gel displays solid, elastic-like
behavior due to the entropic elasticity of the spacers, resisting deformation.
Conversely, over longer timescales, the reversible binding and unbinding of
stickers enable the material to flow, displaying liquid-like characteristics.
Homogeneous associative gels display a soft relaxation over a single
characteristic timescale, directly related to the unbinding dynamics of the
stickers [21]. In contrast, heterogeneous gels feature multiple coexisting
timescales, each corresponding to a different sub-region within the system,
and a complex relaxation behavior emerges from their superposition [22].
Strong heterogeneity results in a broad distribution of timescales, sometime
heavy tailed. In these extreme cases, a portion of these timescales are
arbitrarely large and relaxation seemingly never ends, a phenomena typical of
glasses, and known as aging [23, 24]. To investigate how aging can emerge from
the collective rearrangement of the gel components, we model them as sticker
and spacer. the former acting as diffusing particles that can bind to (and
unbind from) the latter, that represent Gaussian polymers, possibly capturing
intrinsically disordered regions of proteins or long RNA strands. Using a
combination of analytical results and simulations, we show that an entropic
Casimir-like force drives the clustering of stickers, in accordance with
several other works [15, 3, 25]. We find that collective rearrangement of the
stickers induce a slow-down of the dynamic of the system. As a consequence,
the relaxation toward equilibrium is surprisingly slow and reminiscent of the
usual glass dynamic [26]. To explain our observations, we introduce a minimal
model for the dynamic of clusters, which provide a scaling law for the
dynamical slowing-down. Despite its simplicity, our model displays a rich
phenomenology leading to glassy dynamic which includes emergence of large
timescales, ergodicity breaking and aging. By focusing on the minimal
ingredients required for the emergence of a soft glassy relaxation, our model
proposes a universal mechanism to aging of complex associative gels.

## Results

### Dynamic of stickers and spacers

We first introduce a physical model for the dynamic of stickers and spacers.
Starting with spacers, which correspond to RNA or DNA strands, or to
intrinsically disordered segments of proteins. Their dynamics are similar to
those of a polymer, and we consider the dilute regime where excluded volume or
topological constraints due to entanglement are negligible. Therefore, we
model spacers as linear Gaussian polymers. In this work, we will focus on the
dynamic of a single polymer strand and treat the others effectively. To model
protein interactions, we consider that polymers interacts with a background
disordered potential, effectively representing the surrounding environment as
represented in Fig. 1. The positions of the minima of this potential
correspond to the positions of the stickers. We propose that the long time
evolution of the condensates results from the slow evolution of the background
potential, that we model through the diffusion of stickers. While the
diffusion dynamics of stickers may differ from a free particle depending on
the exact nature of what they represent, we expect that the details of the
microscopic dynamics will not significantly affect the long-time evolution of
our system. Additionally, considering that stickers correspond to proteins or
sticky regions of a protein, we treat them as effectively always bound to
another polymer strand in the background. As a consequence, whenever the
polymer binds to a sticker, it effectively creates a cross-link that impairs
its diffusion and constraints the conformations of the polymer. Here, we
consider the limiting case where stickers do not diffuse at all when bound to
the polymer, thus creating an anchoring point. This dynamic provides enough
freedom for the system to rearrange while capturing the constraining effect of
binding.

Figure 1: Schematic representation of our sticker and spacer model. The
disordered background potential is represented as a color coding, while
stickers are represented as diffusing particles. We focus on a single polymer
dynamic highlighted in bold, while the other effective polymers are in shaded.
(a) Represents the initial state in which stickers are homogeneously spread in
space. (b) Qualitatively represent a typical final state as predicted by our
model.

To describe the dynamic of the stickers, we now consider the probability of
the $i$-th sticker to be in position $\bm{r}_{i}$ at time $t$ in the bound
state along the lineic position $\ell$ of the polymer denoted by
$p_{\text{b}}(\bm{r}_{i},\ell,t)$. Similarly, we write the probability to be
unbound $p_{\text{ub}}(\bm{r}_{i},t)$. Given the previously introduced
dynamics, the time evolution of these two probability distributions is
governed by the two coupled master equations:

$\displaystyle\partial_{t}p_{\text{b}}(\bm{r}_{i},\ell,t)=k_{\text{b}}(\bm{r}_{i},\ell)p_{\text{ub}}(\bm{r}_{i},t)-k_{ub}p_{\text{b}}(\bm{r}_{i},\ell,t)$
(1)
$\displaystyle\partial_{t}p_{\text{ub}}(\bm{r},t)=k_{ub}p_{\text{b}}(\bm{r}_{i},\ell,t)-k_{\text{b}}(\bm{r}_{i},\ell)p_{\text{ub}}(\bm{r}_{i},t)+D\nabla^{2}p_{ub}(\bm{r}_{i},t),$

where $k_{\text{b}}$ and $k_{\text{ub}}$ are respectively the binding and
unbinding rates of the sticker with the considered polymer. We first write the
unbinding rate using a single energy scale under the Kramers approximation:

$k_{\text{ub}}=\frac{1}{\tau_{0}}e^{-\beta E_{\text{b}}},$ (2)

where $\beta=1/(k_{B}T)$ is the inverse temperature, $E_{b}$ the binding
energy scale, and $\tau_{0}$ a timescale associated with the microscopic
escape process. While the complexity of real proteins’ interaction should lead
to a distribution of binding energies, we show here that this assumption is
not necessary for the system to display a glassy dynamic, and address the
effect of a more complex model in the discussion section.

Concerning the binding rate, we assume that the disordered potential
represented by the stickers is partially-quenched. Meaning that it evolves on
a much larger timescale than the polymers. As a consequence, polymers have
time to equilibrate between two binding/unbinding events. Assuming that all
conformations of the polymer are equiprobable, we show in Supplementary
Material that the probability for a given conformation of the polymer to meet
a sticker located at $\bm{r}$ in a volume $a^{3}$ (where $a$ is the monomer
size) at a lineic position $\ell$ is given by:

$p_{\text{meet}}(\bm{r},\ell)=e^{S_{\text{b}}(\bm{r},\ell)-S_{\text{ub}}}.$
(3)

where $S_{\text{b}}$ and $S_{\text{ub}}$ correspond to the entropy of the
polymer strand when bound and unbound to the sticker, respectively. Our
approximation of fast polymer conformational exploration means that the
limiting timescale is the binding time. Thus, we write the binding rate of the
sticker to the polymer as:

$k_{\text{b}}(\bm{r},\ell)=1/\tau_{0}\sum_{\alpha}p^{\alpha}_{\text{meet}}(\bm{r},\ell)=1/\tau_{0}\sum_{\alpha}e^{S^{\alpha}_{\text{b}}(\bm{r},\ell)-S^{\alpha}_{\text{ub}}},$
(4)

Where the sum over $\alpha$ represents the participation of all the polymer
strands in the vicinity of the given sticker. Together with (2), this
expression respects detailed balance, thus guaranteeing that the evolution of
the system is consistent with thermodynamic equilibrium. Notice that
$S^{\alpha}_{\text{b}}(\bm{r},\ell)$ introduces a coupling between the dynamic
of the different stickers through $\ell$ that ultimately depends on where all
the other stickers are bound (see Supplementary Material for the full
expression). As a result, we cannot directly solve this system of equation
without further approximation. In the next section, we use a Gillespie
simulation to study the time evolution of our system.

### Early time behavior

Figure 2: (a) Early time evolution of the Percentage of bound stickers in the
system.Here, we use $L=2.10^{4}$ and $N=100$. (b) Plateau value of the
percentage of bound stickers as a function of the binding energy, rescaled by
the lineic density of stickers. The scattered points correspond to the
measurements obtained from the panel (b).

We simulate a polymer of length $L$, which defines the system size, surrounded
by $N$ initially unbound stickers spread homogeneously in space. First,
looking at the early time evolution of the system, we observe in Fig. 2 (a) a
rapid increase in the number of stickers bound to the polymer until a plateau
is reached. We show in Fig. 2 (b) how the height of the plateau depends on the
binding energy, observing a sharp increase from $0\%$ of the bound stickers to
$100\%$ at a critical value of the binding energy. This phenomenon is
reminiscent of the phase transition observed in the Poland-Scheraga model of
DNA denaturation [27]. In their model, two DNA strands can bind to one another
through sticky base pairs that are regularly spaced along the strands.
Similarly to our model, the energetic binding energy must overcome the loss of
entropy associated with the binding. As the linear density of stickers
increases, it becomes increasingly favorable to bind them as the entropic cost
for successive binding decreases. Their theory predicts a phase transition
with a critical energy scaling, $E_{c}\propto\log(L/N)$. We observe that our
model consistently displays a similar scaling based on the collapsed curves of
Fig.2 As a consequence, for our system, low values of the binding energy (or
high temperature) mean that stickers are mostly unbound, and the system
mechanically responds like a liquid. If the value of the binding energy is
higher than its critical value (or the temperature lower than its critical
value), stickers remain bound most of the time, leading to a viscoelastic
mechanical response similar to that of an associative gel. This type of gel
displays a solid-like mechanical response at timescales lower than the
unbinding time and liquid-like at timescales much larger than the unbinding
time. In this work, we focus on the viscoelastic aging regime; for this
reason, we always consider the binding energy above its critical value. In
this case, unbinding becomes the time-limiting process for the dynamics of the
system. In the following, we do not discuss the influence of the binding
energy, which can be ruled out by rescaling times with respect to
$1/k_{\text{ub}}$. In the high energy regime, the phenomena discussed are
insensitive to the value of the binding energy.

### Glassy relaxation

Figure 3: (a) sticker distance probability distribution at different time-
point. The color coding corresponds to the time, and the analytical solution
is provided as dotted line. We simulate systems with fixed lineic density:
$N/L=5.10^{-2}$. Small systems of length $L=100$ successfully converge toward
the equilibrium distribution. (b) Large systems, of length $L=2000$, evolves
toward the analytic solution without reaching it. (c) We follow the
equilibration process by looking at the decay of the free energy per unit
length over time, the initial binding regime as been removed for improved
visibility. Using a fixed lineic density of stickers, we expect all curves to
converge to the same relative value. Large systems are stuck at a higher value
of the free energy, stressing its inability to reach equilibrium. (d)
Schematic representation of the partial equilibration of the system into sub-
clusters.

Once all stickers are bound, the system’s energy becomes essentially fixed.
However, the entropy of the polymer also depends on the positions of the
stickers. We study the equilibration of the system through the diffusion of
stickers. By setting the time derivative of the master equation to zero and
independently equilibrating the binding/unbinding dynamics and the diffusion
dynamics, we demonstrate in the Supplementary Material that the equilibrium
probability distribution of finding two stickers interacting with a polymer of
length $L$ at a distance $|r|$ from each other is given by:

$p(\bm{r})=\frac{3\text{erfc}\left(\sqrt{\frac{3}{2aL}}|\bm{r}|\right)}{2\pi
aL|\bm{r}|}$ (5)

We derive an approximated formula in Supplementary Material for a system of
$N$ particles, considering polymer strands that mediate interactions as a mean
field. The simulated time evolution of the probability distribution toward the
expected equilibrium analytic solution for smaller systems is depicted in Fig.
3 (a). Contrastingly, Fig. 3 (b) shows that larger systems, while converging
similarly, do not reach the predicted equilibrium solution To further
characterize the relaxation of our system toward equilibrium, we plot, in Fig.
3 (c), the free energy per unit length difference between our system and the
theoretical free energy of an unconstrained polymer. Calculated as
$f_{\text{free}}=F_{\text{free}}/L=NE_{b}/L-\log(4\pi)$ (where $4\pi$ is the
solid angle between two monomers). We use a fixed lineic density of stickers
($N/L=5.10^{-2}$) and different system sizes, characterized by $L$. We observe
a slow logarithmic decay of the free energy, followed by a plateau, where the
total relaxation time depends on the system size. This type of logarithmic
relaxation is usually associated with glassy systems [26], and we propose a
model in the next section to better understand this unique behavior. Looking
at the height of the plateau, we notice that it first increases with the
system size before becoming independent of it. This is an unexpected behavior
as the system is self-similar along the polymer, and the equilibrium free
energy per unit length is an intensive quantity with respect to $L$.
Additionally, we show in Supplementary Material that the height of the plateau
also depends on the diffusion constant, $k_{\text{diff}}$. These results
suggest that systems above a certain size are unable to fully relax to
equilibrium. Instead, they form substructures, each equilibrated, that we call
clusters. We propose in Fig. 3 (d) a 1D schematic of this picture. Starting
from a homogeneous distribution of stickers, they aggregate into equilibrated
clusters. If the total system size is similar to the typical cluster size, the
plateau free energy is the equilibrium one. For a system much larger than the
cluster size, the plateau free energy is higher in average and intensive
again.

To further examine the formation of equilibrated subsystems, we employ a
hierarchical clustering algorithm to group stickers into clusters. This
algorithm uses a single maximum length to define connectivity, here, we use
the characteristic lengthscale of Eq. (5) to match the definition of clusters
as equilibrated subsystems. Fig. 4 (a) illustrates the time evolution of the
relative cluster size, showing that smaller systems tend to converge into a
single cluster, while larger systems form clusters of fixed size, effectively
showing a decay in the relative size.

### Origin of the glassy relaxation

We now investigate the origin of the slow logarithmic relaxation dynamics of
the free energy. Logarithmic decay typically implies a continuous slowing down
of the dynamics during relaxation. To confirm this, we examine the
intermediate scattering function (ISF), defined as:

$I(\bm{k},t)=\sum_{j=0}^{N}\left<e^{i\bm{k}(\bm{r}_{i}(t)-\bm{r}_{j}(0)}\right>.$
(6)

The modulus of the ISF characterizes the spatial and temporal dependence of
the relaxation dynamics, It is commonly used to study glassy relaxations [28].
Here, we focus on the wave vector associated with the inter-sticker
distance,$k=1/<r>$, where $<r>$ is the average sticker distance computed from
Eq. (5). In Fig. 4 (b), we plot the decay of the ISF over time, quantifying
the decorrelation of the stickers’ positions after a lag time denoted
$t_{\text{lag}}$ along the relaxation. Unlike an exponential decay consistent
with a normal random walk, we observe a stretched exponential relaxation
defined by $\exp((-t/\tau)^{\alpha})$, with $\alpha\approx 0.7$. This
stretched exponential decay is characteristic of the $\alpha$ relaxation in
glassy liquids [29, 30] and indicates a relaxation slower than exponential,
usually associated with subdiffusive microscopic processes [31]. By fitting
the measured ISF with a stretched exponential, we extract the characteristic
relaxation time of the system. The inset of Fig. 4 (b) shows the increase of
the characteristic relaxation time $\tau$ as a function of the simulation
time, confirming our intuition of a dynamical slow-down.

We now propose a model for the origin of this dynamical slow-down. In the
previous section, we demonstrated that system equilibration involves the
growth of clusters. Individual exchanges of stickers between clusters do not
modify the average size; to increase the average cluster size, it is necessary
to dissolve another cluster. To compute the dissolution time of a cluster, we
consider a system with $N$ stickers spread across clusters. The average number
of stickers in a cluster is denoted $\bar{n}$. We model the exchange of
stickers as a two-step process: unbinding from one cluster and then rebinding
in a random cluster, selected randomly and with equal probability from all
available clusters. Each exchange occurs according to a Poisson process with a
constant rate $1/\tau_{\text{exch}}$, which effectively encapsulates a complex
process of diffusion coupled to the binding/unbinding dynamic, that we do not
investigate here. For a given cluster, the probability $P(n,t)$ to contain $n$
stickers at time $t$ evolves as:

$\displaystyle\frac{\text{d}P(n,t)}{\text{d}t}=$ $\displaystyle
1/\tau_{\text{exch}}\left[P(n+1,t)(n+1)+P(n-1,t)\bar{n}\right.$ (7)
$\displaystyle\left.-P(n,t)(n+\bar{n})\right],$

derived from the master equation, where a cluster with $n$ stickers loses one
at a rate $1/\tau_{\text{exch}}n$ and gains one from any of the $N/\bar{n}$
clusters at a rate
$N/(N/\bar{n})/\tau_{\text{exch}}=\bar{n}/\tau_{\text{exch}}$. Assuming that
the smallest cluster is the one dissolving first, and $N\rightarrow\infty$, we
compute in the Supplementary Material the average dissolution time:

$\tau(\bar{n})\underset{N\rightarrow\infty}{\sim}\tau_{\text{exch}}\frac{e^{\bar{n}}}{\bar{n}^{2}},$
(8)

We now investigate the long-time collective growth of clusters resulting from
the dissolution of other clusters. Assuming that clusters dissolve at a rate
$1/\tau(\bar{n})$, and its sticker is redistributed among the $N/\bar{n}-1$
other clusters. We describe the dissolution events as continuous in time, and
write a differential equation for the time evolution of the average cluster
size:

$\frac{\text{d}\bar{n}}{\text{d}t}=1/\tau_{\text{exch}}e^{-\bar{n}}\frac{\bar{n}^{2}}{N/\bar{n}-1},$
(9)

This non-linear differential equation cannot be solved analytically; however,
we show in the Supplementary Material that for a large systems ($N$ and
$\bar{n}$ large) $\bar{n}\propto\log(t)$. This logarithmic scaling can be used
together with the dominant factor of Eq. (8): $\tau(\bar{n})\propto
e^{\bar{n}}$. Which gives $\tau(t)\propto t^{\gamma}$, where $\gamma$ is a
constant that depends on $\tau_{\text{exch}}$. This power-law qualitatively
agrees with the inset of Fig. 4 (b), which confirms the relation between the
relaxation time of the ISF, and the dissolution time of growing clusters.
Here, we focused on the dominant mathematical factors, and neglected many
processes that could provide a more quantitative agreement with the
simulation. This choice is motivated by the initial strong simplification of
our sticker and spacer model. And we believe that only qualitative results
will be robust to the addition of real microscopic details.

Figure 4:  (a) average cluster size evolution, clusters are built using a
hierarchical algorithm that use a minimum distance linkage criteria. The
distance used is the average sticker distance computed using the pair
correlation function of Eq. (5), in our case $\bar{d}=3.6$. (b) Modulus, of
the intermediate scattering function for $|\bm{k}|=1/\bar{r}=0.37$, where
$\bar{r}$ is the average distance between stickers. The stretched exponential
fits are in good agreement, and the corresponding $\tau$ values obtained are
plotted in the inset. The value of $\alpha$ does not evolve significantly over
time, and remains $\approx 0.7$.

## Discussion

We have introduced a minimal model to describe the dynamics of protein or
RNA/DNA-protein condensates. By utilizing the general sticker and spacer
framework, we examined the long-term structural evolution of these systems.
Our investigation primarily focused on two general effects: the entropic cost
associated with binding and the increasing difficulty of the system to
rearrange upon equilibration. Our model reveals two distinct mechanisms: the
emergence of a Casimir attractive force between stickers mediated by spacers
and the clustering of stickers leading to glassy relaxation. The first
mechanism is reminiscent of bridging-induced phase separation, where RNA or
DNA-protein binding is driven by a similar entropic force, triggering
condensation [3, 9]. In our framework, this entropic attractive force appears
for high binding energy, as the simultaneous unbinding of multiple stickers
would disrupt the attractive interaction This approximation might not be
universally valid for all biological systems. in fact, we anticipate that in
many cases, the Casimir force will not be the primary driver of structural
reorganization. Nonetheless, in our model, this attractive force is only
useful to drive sticker clustering, which is ultimately responsible for glassy
relaxation. This clustering can also be driven by system-specific mechanisms,
such as three-body stabilizing interactions or multiple $\beta$-sheet
stacking, and has been observed in experimental systems [25, 15, 32]. In fact,
such clustering is also a general behavior of random walkers in disordered
media, commonly known as localization in this context [33]. For these reasons,
we are confident that the clustering of stickers described here is a fairly
universal feature, from which we have only highlighted the entropic
minimization effect.

Our model shares similarities with previously studied models of random walker
with trap [34, 35], which reproduce key features, such as glass transition,
subdiffusion, or localization, reminiscent of the phenomenology observed here.
However, these models use a prior distribution of binding energies (e.g.,
exponential), aging occurs through the exploration of deeper and deeper traps
by the random walker. On the one hand, real proteins exhibit a distribution of
interaction energy, which is sufficient to induce complex relaxation of a
single protein [36]. On the other hand, proteins’ interactions –electrostatic,
hydrophobic, or hydrogen bonding– are all of the same order of magnitude
($\approx 10k_{B}T$), which should lead to a fairly narrow distribution of
interaction energy. For this reason, intrinsic protein complexity does not
justify a priori the existence of ever-deepening traps needed for long-time
aging. Our model offers a rationale for considering a broad distribution of
energetic interactions, as proposed in previous studies [19, 37]. By focusing
on the exponential part of Eq. (8), we find that the dissolution time of
clusters is similar to a Kramer escape time for a trap of depth $\bar{n}$ time
larger than an isolated sticker. Such mechanism is known for enhancing small
heterogeneities of stickers, leading to complex viscoelastic relaxation [38].

###### Acknowledgements.

We thank Lara Koehler for her critical reading of the manuscript. This work
was supported by the Swiss National Science Foundation under grant CRSII5
193740.

## References

  * Banani  _et al._ [2017] S. F. Banani, H. O. Lee, A. A. Hyman, and M. K. Rosen, Biomolecular condensates: Organizers of cellular biochemistry, Nat Rev Mol Cell Biol 18, 285 (2017).
  * Shin and Brangwynne [2017] Y. Shin and C. P. Brangwynne, Liquid phase condensation in cell physiology and disease, Science 357, eaaf4382 (2017).
  * Ryu  _et al._ [2021] J.-K. Ryu, C. Bouchoux, H. W. Liu, E. Kim, M. Minamino, R. de Groot, A. J. Katan, A. Bonato, D. Marenduzzo, D. Michieletto, F. Uhlmann, and C. Dekker, Bridging-induced phase separation induced by cohesin SMC protein complexes, Sci Adv 7, eabe5905 (2021).
  * Rippe [2022] K. Rippe, Liquid–Liquid Phase Separation in Chromatin, Cold Spring Harb Perspect Biol 14, a040683 (2022).
  * Hondele  _et al._ [2019] M. Hondele, R. Sachdev, S. Heinrich, J. Wang, P. Vallotton, B. M. A. Fontoura, and K. Weis, DEAD-box ATPases are global regulators of phase-separated organelles, Nature 573, 144 (2019).
  * Aulas  _et al._ [2017] A. Aulas, M. M. Fay, S. M. Lyons, C. A. Achorn, N. Kedersha, P. Anderson, and P. Ivanov, Stress-specific differences in assembly and composition of stress granules and related foci, Journal of Cell Science 130, 927 (2017).
  * Lee  _et al._ [2020] C.-Y. S. Lee, A. Putnam, T. Lu, S. He, J. P. T. Ouyang, and G. Seydoux, Recruitment of mRNAs to P granules by condensation with intrinsically-disordered proteins, eLife 9, e52896 (2020).
  * Boeynaems  _et al._ [2018] S. Boeynaems, S. Alberti, N. L. Fawzi, T. Mittag, M. Polymenidou, F. Rousseau, J. Schymkowitz, J. Shorter, B. Wolozin, L. V. D. Bosch, P. Tompa, and M. Fuxreiter, Protein Phase Separation: A New Phase in Cell Biology, Trends in Cell Biology 28, 420 (2018).
  * Brackley  _et al._ [2013] C. A. Brackley, S. Taylor, A. Papantonis, P. R. Cook, and D. Marenduzzo, Nonspecific bridging-induced attraction drives clustering of DNA-binding proteins and genome organization, Proceedings of the National Academy of Sciences 110, E3605 (2013).
  * Jawerth  _et al._ [2018] L. M. Jawerth, M. Ijavi, M. Ruer, S. Saha, M. Jahnel, A. A. Hyman, F. Jülicher, and E. Fischer-Friedrich, Salt-Dependent Rheology and Surface Tension of Protein Condensates Using Optical Traps, Phys. Rev. Lett. 121, 258101 (2018).
  * Alshareedah  _et al._ [2021] I. Alshareedah, M. M. Moosa, M. Pham, D. A. Potoyan, and P. R. Banerjee, Programmable viscoelasticity in protein-RNA condensates with disordered sticker-spacer polypeptides, Nat Commun 12, 6620 (2021).
  * Choi  _et al._ [2020] J.-M. Choi, A. S. Holehouse, and R. V. Pappu, Physical Principles Underlying the Complex Biology of Intracellular Phase Transitions, Annual Review of Biophysics 49, 107 (2020).
  * Jawerth  _et al._ [2020] L. Jawerth, E. Fischer-Friedrich, S. Saha, J. Wang, T. Franzmann, X. Zhang, J. Sachweh, M. Ruer, M. Ijavi, S. Saha, J. Mahamid, A. A. Hyman, and F. Jülicher, Protein condensates as aging Maxwell fluids, Science 370, 1317 (2020).
  * Linsenmeier  _et al._ [2022] M. Linsenmeier, M. Hondele, F. Grigolato, E. Secchi, K. Weis, and P. Arosio, Dynamic arrest and aging of biomolecular condensates are modulated by low-complexity domains, RNA and biochemical activity, Nat Commun 13, 3030 (2022).
  * Farag  _et al._ [2022] M. Farag, S. R. Cohen, W. M. Borcherds, A. Bremer, T. Mittag, and R. V. Pappu, Condensates formed by prion-like low-complexity domains have small-world network structures and interfaces defined by expanded conformations, Nat Commun 13, 7722 (2022).
  * Ray  _et al._ [2020] S. Ray, N. Singh, R. Kumar, K. Patel, S. Pandey, D. Datta, J. Mahato, R. Panigrahi, A. Navalkar, S. Mehra, L. Gadhe, D. Chatterjee, A. S. Sawner, S. Maiti, S. Bhatia, J. A. Gerez, A. Chowdhury, A. Kumar, R. Padinhateeri, R. Riek, G. Krishnamoorthy, and S. K. Maji, $\alpha$-Synuclein aggregation nucleates through liquid–liquid phase separation, Nat. Chem. 12, 705 (2020).
  * Shen  _et al._ [2023] Y. Shen, A. Chen, W. Wang, Y. Shen, F. S. Ruggeri, S. Aime, Z. Wang, S. Qamar, J. R. Espinosa, A. Garaizar, P. St George-Hyslop, R. Collepardo-Guevara, D. A. Weitz, D. Vigolo, and T. P. J. Knowles, The liquid-to-solid transition of FUS is promoted by the condensate surface, Proc Natl Acad Sci U S A 120, e2301366120 (2023).
  * Ranganathan and Shakhnovich [2022] S. Ranganathan and E. Shakhnovich, The physics of liquid-to-solid transitions in multi-domain protein condensates, Biophysical Journal 121, 2751 (2022).
  * Takaki  _et al._ [2023] R. Takaki, L. Jawerth, M. Popović, and F. Jülicher, Theory of Rheology and Aging of Protein Condensates, PRX Life 1, 013006 (2023).
  * Garaizar  _et al._ [2022] A. Garaizar, J. R. Espinosa, J. A. Joseph, G. Krainer, Y. Shen, T. P. Knowles, and R. Collepardo-Guevara, Aging can transform single-component protein condensates into multiphase architectures, Proceedings of the National Academy of Sciences 119, e2119800119 (2022).
  * Alberto Parada and Zhao [2018] G. Alberto Parada and X. Zhao, Ideal reversible polymer networks, Soft Matter 14, 5186 (2018).
  * Song  _et al._ [2023] J. Song, N. Holten-Andersen, and G. H. McKinley, Non-Maxwellian viscoelastic stress relaxations in soft matter, Soft Matter 19, 7885 (2023).
  * Lieleg  _et al._ [2011] O. Lieleg, J. Kayser, G. Brambilla, L. Cipelletti, and A. R. Bausch, Slow dynamics and internal stress relaxation in bundled cytoskeletal networks, Nature Mater 10, 236 (2011).
  * Berthier [2011] L. Berthier, Dynamic Heterogeneity in Amorphous Materials, Physics 4, 42 (2011).
  * Dar  _et al._ [2024] F. Dar, S. R. Cohen, D. M. Mitrea, A. H. Phillips, G. Nagy, W. C. Leite, C. B. Stanley, J.-M. Choi, R. W. Kriwacki, and R. V. Pappu, Biomolecular condensates form spatially inhomogeneous network fluids, Nat Commun 15, 3413 (2024).
  * Götze and Sperl [2002] W. Götze and M. Sperl, Logarithmic relaxation in glass-forming systems, Phys Rev E Stat Nonlin Soft Matter Phys 66, 011405 (2002).
  * Poland and Scheraga [1966] D. Poland and H. A. Scheraga, Phase Transitions in One Dimension and the Helix—Coil Transition in Polyamino Acids, The Journal of Chemical Physics 45, 1456 (1966).
  * Janssen [2018] L. M. C. Janssen, Mode-Coupling Theory of the Glass Transition: A Primer, Frontiers in Physics 6 (2018).
  * Wu  _et al._ [2018] Z. W. Wu, W. Kob, W.-H. Wang, and L. Xu, Stretched and compressed exponentials in the relaxation dynamics of a metallic glass-forming melt, Nat Commun 9, 5334 (2018).
  * Kob and Andersen [1995] W. Kob and H. C. Andersen, Testing mode-coupling theory for a supercooled binary Lennard-Jones mixture. II. Intermediate scattering function and dynamic susceptibility, Phys. Rev. E 52, 4134 (1995).
  * Bouchaud [2008] J.-P. Bouchaud, Anomalous Relaxation in Complex Systems: From Stretched to Compressed Exponentials, in  _Anomalous Transport_ (John Wiley & Sons, Ltd, 2008) Chap. 11, pp. 327–345.
  * Wu  _et al._ [2023] T. Wu, M. R. King, M. Farag, R. V. Pappu, and M. D. Lew, Single fluorogen imaging reveals spatial inhomogeneities within biomolecular condensates (2023).
  * Compte and Bouchaud [1998] A. Compte and J.-P. Bouchaud, Localization in one-dimensional random random walks, J. Phys. A: Math. Gen. 31, 6113 (1998).
  * Bouchaud [1992] J. P. Bouchaud, Weak ergodicity breaking and aging in disordered systems, J. Phys. I France 2, 1705 (1992).
  * Bertin and Bouchaud [2003] E. M. Bertin and J.-P. Bouchaud, Subdiffusion and localization in the one-dimensional trap model, Phys. Rev. E 67, 026128 (2003).
  * Morgan  _et al._ [2020] I. L. Morgan, R. Avinery, G. Rahamim, R. Beck, and O. A. Saleh, Glassy Dynamics and Memory Effects in an Intrinsically Disordered Protein Construct, Phys. Rev. Lett. 125, 058001 (2020).
  * Sollich  _et al._ [1997] P. Sollich, F. Lequeux, P. Hébraud, and M. E. Cates, Rheology of Soft Glassy Materials, Phys. Rev. Lett. 78, 2020 (1997).
  * Le Roy  _et al._ [2024] H. Le Roy, J. Song, D. Lundberg, A. V. Zhukhovitskiy, J. A. Johnson, G. H. McKinley, N. Holten-Andersen, and M. Lenz, Valence can control the nonexponential viscoelastic relaxation of multivalent reversible gels, Science Advances 10, eadl5056 (2024).