# Surfactancy in a tadpole model of proteins

O. T. Dyer  Department of Physics, University of Warwick, Coventry CV4 7AL,
United Kingdom  R. C. Ball  Department of Physics, University of Warwick,
Coventry CV4 7AL, United Kingdom

###### Abstract

We model the environment of eukaryotic nuclei by representing macromolecules
by only their entropic properties, with globular molecules represented by
spherical colloids and flexible molecules by polymers. We put particular focus
on proteins with both globular and intrinsically disordered regions, which we
represent with ‘tadpole’ constructed by grafting single polymers and colloids
together. In Monte Carlo simulations we find these tadpoles support phase
separation via depletion flocculation, and demonstrate several surfactant
behaviours, including being found preferentially at interfaces and forming
micelles in single phase solution. Furthermore, the model parameters can be
tuned to give a tadpole a preference for either bulk phase. However, we find
entropy too weak to drive these behaviours by itself at likely biological
concentrations.

##  1 Introduction

Liquid-liquid phase separation (LLPS) has been identified as a mechanism for
eukaryotic cells to organise their constituent biomolecules without needing
the lipid bilayer membranes used by conventional organelles [1, 2, 3]. The
resulting droplets are sometimes called ‘membraneless organelles’ [4]. The
best and longest known example of these is the nucleolus, which is involved in
the production of ribosomes [5], but more recent work has identified a variety
of similar - albeit smaller - liquid aggregates inside both the nucleus and
the cell cytoplasm [4]. Their locations, functions and compositions vary, but
they frequently feature proteins with long sections of polypeptide chain that
remain flexible instead of folding into rigid secondary and tertiary
structures [3, 6, 7, 8]. These sections are called intrinsically disordered
regions (IDRs) and can be found anywhere along the polypeptide chain. In this
article we will model IDRs at either the carboxy terminal domain (CTD) or
amino terminal domain, leading to proteins with a flexible chain attached to a
single globular region.

The role of these IDRs in LLPS remains uncertain, with most work to date
focusing on weak attractive interactions between IDRs, especially those
between charge dipoles, aromatic groups and net charges in amino acid residues
[9]. The idea being that having longer IDRs provides more weak interactions
with which to hold the droplet together.

These energetic considerations are not alone in driving phase behaviour,
however, as entropy also plays a role. This has several contributions: entropy
of mixing drives systems to mix, and thus hinders LLPS; conformational entropy
of IDRs multiplies the number of microstates utilising the chemistry above
[10]; and even without chemistry the conformational entropy introduces
crowding effects that give IDRs an effective repulsion from globular
molecules. Crowding effects, which are the focus of this work, are not new to
cell biology. Studies have included the organisation of DNA at different
length scales [11, 12, 13, 14, 15], the influence of crowding on transcription
rates [16] and on LLPS [11, 17, 18, 19, 20]. However, to our knowledge purely
replusive models of proteins with both IDRs and globular regions have not yet
been studied.

Physics can provide insights into such systems by drawing analogy to polymer-
colloid systems, which are characterised by the ratio of polymer radius of
gyration and colloid radius $q=R_{g}/R_{\mathrm{col}}$. Behaviour in the
‘colloid limit’, $q\ll 1$, is very well established, with mixtures of non-
adsorbing polymers and colloids separating via depletion flocculation [21, 22,
23]. The story in the ‘protein limit’, $q>1$, is similar, but with the key
difference that phase separation is less sensitive to $R_{g}$ than the
correlation length of the polymer mesh [24]. Recent work has argued similar
mechanisms can play a role in biology where it can separate nuclear transport
receptors in nuclear pores [25].

Interesting behaviour also arises when the polymers are grafted to proteins.
In the colloid limit, it is common practice to coat the colloids with
polymers, stabilising them against the van der Waals forces driving
flocculation by exploiting the entropic repulsion of the polymers [26]. In the
protein limit, polymer-colloid grafts are often called grafted nanoparticles
(GNPs), and have been studied extensively in the last two decades [27, 28, 29,
30, 31, 32, 33, 34]. This work has focused largely on materials science,
usually aiming to improve the properties of nanocomposites [32] or using
specific interactions of the GNPs in the design of materials [33]. Research
into GNPs with only one grafted polymer - an appropriate analogue to proteins
with a single IDR - has been comparatively sparse [27, 28, 29, 33].

Similarly to the biological LLPS literature, we are unaware of any work that
has sought to quantify the importance of entropy in GNP behaviour. Hence, by
modeling proteins with IDRs as single-chain GNPs with no attractive
interactions, this article simultaneously fills gaps in both fields.

Finally, we recognise solvent entropy can also play a role via hydration
effects and has been shown to be important in protein aggregation, even for
proteins without IDRs [35, 36, 37, 38, 39]. In this work we only consider
explicitly the entropic properties of the macromolecules, with solvent entropy
folded into net local solute interactions which we assume to be fully
repulsive.

Figure 1: Examples of our model molecules. Note they are 3 dimensional, so
apparent overlaps are not real. The tadpole’s ‘head’ (large bead) is
equivalent to a colloid, while the ‘tail’ is equivalent to a polymer. All
beads are hard spheres and interact with potentials in Eqs. (1) and (2).

##  2 Methods

To study the role of IDRs’ entropy in LLPS, we run dynamic Monte Carlo
simulations in continuous, 3-dimensional, periodic boxes. The box dimensions
are $L_{x}\times L_{y}\times L_{z}$, where we always set $L_{x}=L_{y}$ and
take $L_{z}$ as the box’s longest dimension. These dimensions will vary
between sections, where they are stated.

We coarse grain macromolecules into collections of spheres where the $i$th
particle has hard core radius $R_{i}$. Since we focus solely on their entropic
properties, these particles experience no attractions and repel each other
with the hard sphere potential

$U_{\mathrm{HS}}(r_{ij})=\left\\{\begin{array}[]{ll}\infty&r_{ij}<R_{i}+R_{j}\\\
0&\mathrm{otherwise}\end{array}\right.$ (1)

when particles $i$ and $j$ are distance $r_{ij}$ apart.

Spherical particles are joined together into the 3 molecule species depicted
in Fig. 1. Colloids are single large beads with radius $R_{\mathrm{col}}$
representing low-entropy (globular) macromolecules. Polymers are flexible,
linear chains of $N_{\mathrm{mono}}$ small beads, each with radius
$R_{\mathrm{mono}}$, that represent molecules with high conformational
entropy. Finally, ‘tadpoles’ represent IDR-containing proteins with a globular
region attached to a single IDR. These two regions are represented by a large
‘head’ bead with radius $R_{\mathrm{head}}$, and a ‘tail’ chain of
$N_{\mathrm{tail}}$ small beads with radii $R_{\mathrm{tail}}$. To reduce the
number of parameters in this work, we set $R_{\mathrm{head}}=R_{\mathrm{col}}$
and $R_{\mathrm{tail}}=R_{\mathrm{mono}}$, so a tadpole can be viewed as a
colloid grafted to a polymer when $N_{\mathrm{tail}}=N_{\mathrm{mono}}$. This
helps minimise the number of parameters used in this work, but limits our
ability to make quantitative comparisons to experiments.

All intra-molecular bonds are enforced with the hard square well potential

$U_{\mathrm{bond}}(r_{i,i+1})=\left\\{\begin{array}[]{ll}0&r_{i,i+1}<R_{i}+R_{i+1}+3R_{\mathrm{tail}}\\\
\infty&\mathrm{otherwise}\end{array}\right.,$ (2)

which acts alongside $U_{\mathrm{HS}}$.

These molecules are placed in an implicit solvent and do not experience any
hydrodynamic effects beyond their diffusivity being set by the usual Stokes-
Einstein expression $D_{i}=k_{\mathrm{B}}T/(6\pi\eta R_{i})$. Rather than
specify the viscosity $\eta$ explicitly, we define the time unit by the time
taken for small particles to diffuse their own size, as listed in Table 1.
Similarly, we use the thermal energy $k_{\mathrm{B}}T$ as our energy unit.

This diffusivity is achieved by displacing particles with the noise term used
in the Brownian dynamics algorithm [40],

$\textbf{r}_{i}(t+\delta t)=\textbf{r}_{i}(t)+\sqrt{2D_{i}\,\delta
t}\textbf{W},$ (3)

where W is a Gaussian-distributed random vector with zero mean and unit
variance in all 3 directions. The conservative interactions are implemented
with a Metropolis acceptance test, passing with probability

$P_{\mathrm{accept}}=\mathrm{min}\left[1,\exp(-\Delta
U/k_{\mathrm{B}}T)\right].$ (4)

With our hard systems, this means only accepting displacements if the
particle’s final energy is zero. To maintain a high acceptance rate in this
hard environment we set the time increment $\delta t$ such that
$\sqrt{2D_{i}\,\delta t}<R_{\mathrm{tail}}$.

The values of parameters used in this work are listed in Table 1. Our choices
for particle sizes lead to $q\approx 2.3$ when $N_{\mathrm{tail}}=30$, which
approximates the ratio for human RNA polymerase II’s globular region and the
flexible CTD on its largest subunit [41, 7].

Our criterion for equilibration is described in Appendix A. Except where
otherwise stated, our quantitative results average over the last 10 system
configurations in each simulation, each separated in time by at least $3\times
10^{4}\tau$ to allow for local re-equilibration. We then further averaged over
at least 7 independent simulation runs for each system, giving us a minimum of
70 configurations contributing to each data point.

Table 1:  Parameter values in this work, with code units specified in the top rows to help ground our results in their biological context. Where multiple values are used, their ranges are specified.  Parameter | Value in simulation
---|---
Length unit, $\ell$ | 12.5nm
Time unit, $\tau$ | $\pi\eta R_{\mathrm{tail}}^{3}/k_{\mathrm{B}}T$
Energy unit, $\varepsilon$ | $k_{\mathrm{B}}T$
$R_{\mathrm{col}}/\ell$, $R_{\mathrm{head}}/\ell$ | 0.44
$R_{\mathrm{mono}}/\ell$, $R_{\mathrm{tail}}/\ell$ | 0.0908
$N_{\mathrm{col}}$ | 0-376
$N_{\mathrm{poly}}$ | 0-188
$N_{\mathrm{mono}}$ | 30
$N_{\mathrm{tad}}$ | 0-188
$N_{\mathrm{tail}}$ | 15-30
$\delta t/\tau$ | 0.4

##  3 Entropic phase separation

Before showing the results of these simulations, it is helpful to explain the
physics driving phase behaviour in our minimal model. By only considering hard
interactions, all allowed states have the same energy and the temperature does
not affect equilibrium states (although it does still play a role in
dynamics). Free energy differences can therefore be written only in terms of
the entropy:

$dF=-TdS=-TdS_{m}-TdS_{c}.$ (5)

Here we have split the entropy of mixing ($S_{m}$) and conformational entropy
($S_{c}$), characterising arrangements of molecule centres of mass and intra-
molecule arrangements respectively.

This split is relevant for crowding effects where flexible chains are
obstructed by nearby rigid bodies, reducing the number of available
conformations of the chain, and in turn leading to a significant reduction in
$S_{c}$. The archetypal system for this is a mixture of colloids and polymers,
in which the two species can phase separate via depletion flocculation, where
the dominance of $S_{c}$ over $S_{m}$ drives the colloids to cluster to
minimise the contact of the two species, thus allowing the polymers to adopt
the maximum number of conformations [21].

The expectation is that this remains significant in our systems with tadpoles
present, and hence that proteins may have evolved IDRs to exploit it.

##  4 Results

###  4.1 Qualitative phase behaviour

We first confirm depletion flocculation occurs in our simulations by
simulating systems with only polymers and colloids present. For this we used
highly stretched periodic boxes, with dimensions $4\times 4\times 27$, that
produce well-defined planar interfaces perpendicular to the $z$-axis, and
whose widths are much smaller than the bulk phases. The transverse dimensions
of the box are large enough to prevent molecules interacting with their own
periodic images, but are quite small to keep the computational cost of
simulations manageable. Such systems are shown in Fig. 2, in which the denser
system in part (a) shows a clear separation into polymer- and colloid-rich
phases, as expected. The interfaces between these phases are also predictably
close to planar and perpendicular to the box’s long axis.

Reducing the system density allows for a greater concentration of the minority
component in each phase, which is particularly noticeable by the number of
colloids in the polymer-rich phase in Fig. 2(b). Reducing the densities
further ultimately reaches fully mixed states, and this will be analysed more
quantitatively in Sec. 4.2.

Figure 2: Snapshots of equilibrated polymer + colloid systems. Both systems
have box dimensions $4\times 4\times 27$. (a): $N_{\mathrm{col}}=374$,
$N_{\mathrm{poly}}=188$. (b): $N_{\mathrm{col}}=328$, $N_{\mathrm{poly}}=126$.

Next, we add tadpoles to our systems, utilising the properties of our model
molecules to make the replacement polymer + colloid $\rightarrow$ tadpole. In
this way the total number of large particles (colloids and tadpole heads) is
kept constant, as is the number of small particles (in polymers and tadpole
tails). This process can be viewed as the addition of $N_{\mathrm{tad}}$ bonds
fusing polymers to colloids. Starting from the systems in Fig. 2, the
concentrations are high enough that nonlinear terms in the osmotic pressure
dominate, so the osmotic pressure is close to constant in each series of
systems thus created, despite the number of molecules being reduced by
$N_{\mathrm{tad}}$ from the initial polymer + colloid system.

Snapshots from such a series of simulations starting from Fig. 2(a) are shown
in Fig. 3. Part (a) demonstrates that adding a small number of tadpoles this
way has minimal impact on the volumes of the two bulk phases, and that of the
two they prefer the polymer-rich phase. Fig. 4(a), which averages volume
fractions in systems that have fused together 47 tadpoles, quantitatively
shows the tadpole density drops to almost zero in the colloid-rich phase.

Furthermore, in Fig. 3(a) there appears to be a higher density of tadpoles at
the interfaces than in the bulk. This would make intuitive sense as their
heads and tails each prefer different phases when not bonded together, and
hence one might expect tadpoles to want to straddle the interface to satisfy
the preference of both their head and tail. This surfactant-like behaviour is
confirmed quantitatively in Fig. 4(a). Here, the tadpole density peaks at the
interfaces, and close inspection finds the tadpoles tend to orient themselves
so their heads and tails reside in the colloid and polymer phases
respectively, as seen by the peaks of the head density being closer to the
colloid phase than the corresponding peaks in tail density.

Figure 3: Snapshots of systems in the series replacing colloids and polymers
with tadpoles (orange), starting from the system shown in Fig. 2(a). The
number of tadpoles added is (a) 14, (b) 94, (c) 188 (all polymers replaced).
The opacity of polymers in part (a) is reduced to improve visibility of
tadpoles in the polymer phase and at the interfaces. As in Fig. 2, the box
dimensions are $4\times 4\times 27$.

Figure 4: Plots of species volume fractions as a function of position along
the simulation box, scaled by their global values. Both plots are taken from
systems with $N_{\mathrm{col}}=329$, $N_{\mathrm{poly}}=141$ and
$N_{\mathrm{tad}}=47$, with $N_{\mathrm{tail}}$ as indicated, and are averages
over 80 simulation snapshots, each translated to align their interfaces.

When half of the polymers have been fused into tadpoles, shown in Fig. 3(b),
the two phases remain separated but the polymer-rich phase expands to
accommodate the additional volume of the tadpole heads. The interfaces have
become more jagged, suggesting a reduction in interface tension (a point we
will return to in Sec. 4.4), which is expected if the tadpoles are acting as
surfactants. The tadpoles do not perfectly line the interfaces, however, which
is indicative of the limited strength of the entropic ‘forces’ holding them
there.

Adding yet more tadpoles, ultimately to the complete removal of polymers,
yields the microphase separation seen Fig. 3(c). In this case we find
bicontinuous phases of tadpole tails and tadpole heads + colloids. By changing
species concentrations, we can coax the tadpoles into forming micelles.
Snapshots of these in shorter simulation boxes, $4\times 4\times 8$, are shown
in Fig. 5. In part (a), colloid density has been increased from the
bicontinous phase, producing a cylindrical micelle with tadpole tails inwards,
and whose axis is along the $x$-axis (front-to-back). For part (b), replacing
colloids with polymers inverts the micelle so that the tadpole tails now point
outwards, leaving a rod-like cluster of heads at its centre. These microphases
are commonly produced by surfactant molecules [42], adding further evidence
that tadpoles act as entropic surfactants.

Figure 5: Example systems in which tadpoles form micelles, (a), and inverted
micelles, (b). (a): tadpole + colloid system with
$N_{\mathrm{tad}}=56,N_{\mathrm{col}}=112$. (b): tadpole + polymer system with
$N_{\mathrm{tad}}=42,N_{\mathrm{poly}}=126$. The opacity of colloids and
polymers has been reduced to help see the tadpoles. Both systems have box
dimensions $4\times 4\times 8$.

###  4.2 Phase diagrams

We have constructed a phase diagram for our model in Fig. 6, using species
volume fractions $\phi$ as axes. For the volume of each species, we sum the
volumes of individual particles within them, multiplied by the number of those
molecules present. Since the polymer pervaded volume is more important than
its bare volume fraction for phase behaviour, its volume fraction is shown on
different scale from the colloid to improve the clarity of results; similar
applies for the tadpoles. Note this shears the contours in our ternary plot,
which are shown in Fig. 6(a). Also, because our systems are compressible with
varying total volume occupied by our particles, the implicit solvent needs
including as its own component, hence including the solvent volume fraction
$\phi_{\mathrm{s}}=1-\phi_{\mathrm{col}}-\phi_{\mathrm{tad}}-\phi_{\mathrm{poly}}$.

Our primary motivation is to determine where these systems undergo macrophase
separation, as is believed to happen inside the cell nucleus and cytoplasm. To
that end, we have run several series of simulations as per the previous
section, from which we map the binodal volume from the tie lines of
macrophase-separating states. The end points of these tie lines were found by
measuring the volume fractions in each bulk phase, and averaging this over our
independent simulations. We can then compare this to the range of biologically
relevant concentrations [43], shown as the shaded volume calculated using the
Statistical Associating Fluid Theory (SAFT) formalism detailed in Appendix B
[44, 45, 46, 47]. We use this formalism as a way to estimate the contribution
of bonded chains in the pressure, which is difficult to measure in our hard
systems, making it challenging to compare pressures directly. However, SAFT
differs from our model in that it fixes bond lengths to exactly $R_{i}+R_{j}$,
and we found it over-predicts the number of our polymers required for a given
pressure, as evidenced by the slightly different gradients of our tie lines
and the shaded region’s lower edge in Fig. 6(a). However, this will not alter
our conclusions.

Figure 6:  Phase diagrams for our model, with axes showing volume fractions
for components - including the implicit solvent, $\phi_{\mathrm{s}}$ \- such
that they sum to 1 everywhere inside each diagram. To better see the binodal
region, the low-$\phi_{\mathrm{s}}$ part of the phase diagrams has been
omitted, and polymer and tadpole axes are shown on different scales, leading
to the non-equal volume fractions at the corners. The end points of the dashed
tie lines within data series have also been connected with lines of that
series’ colour. The key by marker shape indicates the stability of different
points, and errors at the ends of tie lines are shown as diamonds whose
lengths in each direction are twice the standard error on the mean. The key by
colour identifies different series of data, and both keys apply to all parts
of the figure. The grey volume indicates the range of realistic osmotic
pressures in Ref. [43], mapped into model concentrations as discussed in
Appendix B. Panel (a) shows the ternary phase diagram for systems with
$N_{\mathrm{tad}}=0$, where contours are labeled to help read the $\phi$
values of our data. Panel (b) shows the 3D phase diagram including tadpoles
with $N_{\mathrm{tail}}=30$, with the 2D diagram in (a) on its back-right
face. Panel (c) shows a different view of (b) showing how the binodals close
as tadpoles are added. Panel (d) is the equivalent of (c) for systems with
$N_{\mathrm{tail}}=15$.

It is again helpful to focus first on the colloid + polymer systems with no
tadpoles present. This face of the phase diagram is shown in Fig. 6(a), in
which the blue and red data respectively encompass the two systems shown in
Fig. 2. This demonstrates quantitatively that as the density decreases
(towards $\phi_{\mathrm{s}}=1$), the number of colloids found in the polymer-
rich phase increases, although the density of polymers found in the colloid-
rich phase remains small until the systems are a low enough density to mix.
Since such systems are nothing new [23], our main interest is in its overlap
of its binodal and the shaded biological concentrations. While there is indeed
some overlap, we caution that it is confined to the high concentration (low
$\phi_{\mathrm{s}}$) edge, and note that the author of Ref. [43] believed the
lower osmotic pressures to be the most accurate. We are therefore mindful that
the overlap we see may only exist due to inaccurate measurements extending the
shaded region further than it should.

Whether that is the case or not, we infer from the limited overlap that
entropy is unlikely to be strong enough by itself to drive biological phase
separation. Nevertheless, it is strong enough that we believe it should not be
neglected as a contributing factor, especially as it is independent of
chemistry and thus part of the interaction in all systems.

Fig. 6(b) shows the phase diagram produced when we add tadpoles as per Sec.
4.1. The higher density series of systems (in blue) includes those shown in
Fig. 3 and extends out from the blue data in Fig. 6(a). A second, lower
density series likewise extends out from the red data. This viewpoint is
useful for visualising how these series ‘grow’ the binodal out from the
colloid + polymer diagram, ultimately closing when enough tadpoles are added
in this way. Further viewpoints are provided in the Supplementary Material to
help picture this phase diagram in 3D space.

The best viewpoint for seeing how quickly the binodal closes upon addition of
tadpoles is shown in Fig. 6(c), where the tadpole volume fraction increases
rightwards away from the colloid-polymer plane (now found at the left edge).
Here we see both series of simulations map out planes in the binodal that are
viewed nearly edge-on and that run close to parallel to the lower boundary of
the shaded volume. Interpolating the binodal between our series shows us that
adding tadpoles decreases the overlap with the shaded volume. Hence, the
addition of tadpoles does not help entropy drive phase separation, as compared
to the polymer + colloid case. It therefore requires some inter-molecular
attraction to account for biological phase separation.

Finally, we note that the micelle-producing systems in Fig. 5 were
significantly more dense than biological systems (see the green triangles near
the base of Fig. 6(c)). So while this is important behaviour for
characterising the general properties of our tadpoles, we do not expect it to
have much biological relevance.

###  4.3 Differences when $N_{\mathrm{tail}}=15$

We now investigate what changes when reduce the length of our tadpole tails
from 30 to 15 beads, which would biologically represent shorter IDRs, e.g. the
shorter CTDs of RNA Polymerase II in yeast as compared to the human variant
[7]. We keep the length of polymers unchanged, to represent an otherwise
unchanged environment. Consequently, the replacement of colloids and polymers
for an equal number of tadpoles no longer keeps the total number of beads
fixed, and nor does the osmotic pressure remain approximately constant.

The phase diagram for these shorter tails is shown in Fig. 6(d). In this we
show data series starting from the same two colloid + polymer systems as
before, and again more viewpoints are provided in the Supplementary Material.
We kept the same axes as in the $N_{\mathrm{tail}}=30$ phase diagram to aid in
their comparison, but due to the shorter tails, more tadpoles are required to
create the same pressure. This is why the shaded volume now extends down to a
larger value on the $\phi_{\mathrm{tad}}$ axis. Our procedure for adding
tadpoles now also _increases_ $\phi_{\mathrm{s}}$, leading to the data series
rising as tadpoles are added. So, where the tie lines and the base of the
shaded volume were nearly co-planar with $N_{\mathrm{tail}}=30$, they now pass
through each other. On its own, this would help bring the binodal into the
shaded volume.

However, we also find that the binodal now closes at a lower tadpole density
for both data series. This shrinking of the binodal is the larger effect of
the two, resulting in an overall reduction in the capacity for tadpoles with
shorter tails to support phase separation, as would be expected from their
smaller conformational entropy. We acknowledge, however, that longer tails
also provide more interactions sites when chemistry is accounted for, such
that analogous differences in experiments cannot easily be attributed solely
to differences in entropy.

Another major change with shorter tails is their shifting preferences for bulk
phases. Where the $N_{\mathrm{tail}}=30$ tadpoles had a strong preference for
the polymer-rich phase, Fig. 4(b) shows the concentration of tadpoles with
$N_{\mathrm{tail}}=15$ is approximately equal in both bulk phases. Tail length
can therefore be utilised to tune a macromolecule’s preference for the
separating phases. It also has the advantage of being biologically accessible
through mutations that repeat polypeptide sequences, as is thought to have
happened with the CTD of RNA Polymerase II, without changing chemical
interactions in the process.

###  4.4 Interfacial tensions

We now seek to quantitatively confirm that the interfacial tension $\gamma$ in
our systems decreases with the addition of tadpoles. $\gamma$ is difficult to
access with hard potentials due to having infinitely poor statistics of the
infinite forces present, so for this section we soften our system by using the
Weeks-Chandler-Anderson (WCA) potential for particle repulsion,

$U_{\textrm{W}}(s)=\left\\{\begin{array}[]{ll}4u_{\textrm{W}}\left(s^{-12}-s^{-6}+\frac{1}{4}\right)&\mathrm{for}~{}s<2^{1/6}\\\
0&\mathrm{otherwise}\end{array}\right.$ (6)

where $s=r/R_{\textrm{W}}$ scales the particle separation, $r$, by the length
scale of the WCA potential, $R_{\textrm{W}}$. Bond potentials are also
softened by using finitely extensible non-linear elastic (FENE) bonds with
energy

$U_{\textrm{F}}(r)=\left\\{\begin{array}[]{ll}-\frac{1}{2}k_{\textrm{F}}R_{\textrm{F}}^{2}\ln\\!\left[1-\left(\dfrac{r}{R_{\textrm{F}}}\right)^{2}\right]&\mathrm{for}~{}r<R_{\textrm{F}}\\\
\infty&\mathrm{otherwise}\end{array}\right..$ (7)

The new parameters were set to $u_{\textrm{W}}=k_{\mathrm{B}}T=\varepsilon$,
$R_{\textrm{W}}=R_{i}+R_{j}$, $R_{\textrm{F}}=R_{i}+R_{j}+4R_{\mathrm{tail}}$
between particles of species $i$ and $j$. To match mean bond lengths to the
hard systems, we set $k_{\textrm{F}}=7.822\varepsilon/\ell^{2}$ between pairs
of tail/polymer beads, and $k_{\textrm{F}}=2.558\varepsilon/\ell^{2}$ between
tadpole heads and tails.

With these potentials we have well-defined forces that can be calculated from
system configurations. This in turn lets us calculate the stress tensor
$\bm{\sigma}$ and hence $\gamma$ via the Kirkwood-Buff expression [48]

$\gamma=-\frac{1}{2}\int\limits_{0}^{L_{z}}dz\left(\sigma_{zz}-\frac{1}{2}(\sigma_{xx}+\sigma_{yy})\right),$
(8)

where having 2 interfaces in the system contributes the front factor of 1/2.

Finally, we change the box dimensions to $6\times 6\times 12$, which preserves
the volume and hence particle numbers, but increases the ratio of interface
area to bulk volume, improving our statistics. In these shorted boxes we ran 3
sets of simulations equivalent to those in the higher density series in Sec.
4.1 fusing colloids and polymers into tadpoles with $N_{\mathrm{tail}}=30$. To
improve our statistics we ran these for longer to generate 200 independent
snapshots, and in which we measured the interfacial tensions in Table 2. These
show a clear reduction in $\gamma$ with the addition of tadpoles, as expected
if they act as surfactants. These absolute values are in line with the
commonly used estimate $\gamma\approx k_{\mathrm{B}}T/\delta^{2}$, with
$\delta$ being the interface width. We can also compare to the experimental
values of 1 to $5\mu\mathrm{Nm}^{-1}$ measured by Jawerth et al. [49], which
overlaps with our range when we convert our units to SI, as shown in the
fourth column of Table 2.

Table 2:  Interfacial tensions measured using Eq. 8 in our softened systems. Values in the third column have been scaled by the estimated interfacial width $\delta=2R_{\mathrm{col}}(1+q)$. In the fourth column we have converted to SI units with $T$=300K, and $\delta=36.3$nm. $N_{\mathrm{tad}}$ | $N_{\mathrm{tad}}/N_{\mathrm{poly}}$ | $\gamma\;\delta^{2}/k_{\mathrm{B}}T$ | $\gamma\>\>(\mu\mathrm{Nm}^{-1})$
---|---|---|---
0 | 0 | $9.2\pm 1.0$ | $28.9\pm 3$
47 | 1/3 | $4.1\pm 1.1$ | $12.9\pm 3.5$
94 | 1 | $0.4\pm 1.0$ | $1.3\pm 3.1$

##  5 Conclusions

We have used minimal, equilibrium Monte Carlo simulations of hard sphere
systems to investigate the role of entropy in biological liquid-liquid phase
separation. By constructing phase diagrams for solutions of polymers, colloids
and tadpole molecules representing biomolecules with different entropic
properties, we were able to compare the concentrations at which such systems
are unstable to macrophase separation with concentrations found in real cell
nuclei. In doing so, we found entropy to be able to drive phase separation
only at the upper estimates of biological concentrations. We conclude that the
entropy from macromolecules is a significant contribution, but insufficient
without some further attractions.

An important agenda for further work would be to assess the source, strength
and impact of attractive macromolecular interactions, and whether that leaves
them weak enough for IDRs to remain flexible. Within the implicit solvent
paradigm, this requires intermolecular interactions net of solvent effects,
and the balance of exposed hydrophobic vs hydrophilic residues is a possible
starting point. Beyond this there are explicit solvent effects to be
considered, including solvent entropy. The latter has been argued to be a
significant driver of some LLPS, though its relevance where the solvent is not
excluded from one phase is unclear [50, 51, 6].

Our simulations with tadpoles identified these molecules as entropic
surfactants, exhibiting several typical surfactant behaviours: microphase
separations such as micelles and bicontinuous phases; a preference for
interfaces; and a reduction in interfacial tensions as tadpoles are added. If
such behaviour is present in vivo, it would be a major step forwards in
understanding the biological importance of IDRs and their influence on liquid
drop formation and structure. Indeed, if they exhibit strong segregation to
the liquid drop interface, then the structure of ‘membraneless organelles’
might be closer to that of regular organelles than was previously realised.

Another avenue for further work could be to model IDR-containing proteins with
different morphologies, such as a “pearl necklace” model for IDRs connected to
globular regions at both ends. We speculate that pearl necklace molecules
would make less effective surfactants since both ends have low conformational
entropy and would prefer the colloid-rich phase. However, this will be
sensitive to the length of the IDR, which could tune the preference for the
two phases and the interface like the length our tadpoles’ tails does.

Finally, in this article we simplified our model by matching particle sizes.
There remains the task of exploring parameter space where particle sizes are
not matched, which will be an important step towards obtaining quantitative
results for comparison with experiment.

## Authors’ contributions

O.T.D.: data curation, formal analysis, investigation, methodology, software,
visualisation, writing - original draft, and writing - review & editing.
R.C.B.: conceptualisation, formal analysis, funding acquisition, supervision,
and writing - review & editing.

## Funding

This work was funded by EPSRC (Grant number: EP/T002794/1).

## Acknowledgements

We thank Dan Hebenstreit and Juntai Liu, and all the research team associated
with that grant, for many discussions throughout the project. We acknowledge
the Warwick Scientific Computing RTP for providing the computational resources
used.

## Data accessibility

The simulation and analysis codes can be found along with all data in the
repository: http://wrap.warwick.ac.uk/163395/

##  Appendix A Verifying equilibration

To determine when systems have equilibrated, we first calculate the radial
distribution functions between all species $i$ and $j$, at each time $t$:
$g_{ij}(r,t)$. When doing this, we exclude pairs of particles within the same
molecule.

We then integrate these like so

$I_{ij}(t)=\int\limits_{0}^{2(R_{i}+R_{j})}dr\,r^{2}g_{ij}(r,t),$ (9)

where the upper limit roughly corresponds to the radius of the first minimum
in a typical liquid RDF. $I_{ij}(t)$ then quantifies the relaxation of the
RDF. Fig. 7 shows examples of this for a variety of systems evolving between
different mixed and separated states. In all plots we show
$I_{\mathrm{cc}},~{}I_{\mathrm{pp}}$ and $I_{\mathrm{cp}}$ to show the
relaxation of the two bulk phases. Where tadpoles are present, we also show
$I_{\mathrm{ct}}$ and $I_{\mathrm{tp}}$ to characterise their relaxation with
respect to both phases.

We used Fig. 7(a) to confirm our systems relax to separated states, and
generated pre-separated states for systems containing tadpoles. In Figs. 7(b)
and (c), which correspond to systems shown in Fig. 4, the tadpoles started in
the polymer phase where they mostly stayed in (b) though they do prefer the
interface leading to a notable increase in $I_{ct}$ over time. In part (c),
many tadpoles leave the polymer phase, leading to a larger increase in
$I_{\mathrm{ct}}$ and decrease in $I_{\mathrm{tp}}$. The high densities made
this a slow process, but exponential fits to our relaxation curves show we
simulated out to about 6 times the relaxation time, making changes small
enough at the end of the time series to treat as equilibrated.

In Fig. 7(d) the tadpoles started with uniform density and the minority
polymer component mixes with the colloid phase over time. This is seen by
$I_{\mathrm{pp}}$ relaxing from a large value as polymers lose sight of each
other.

Figure 7: Plots of $I_{ij}(t)$, normalised by their value at the final time,
showing the relaxation of cross-species radial distribution functions. The
legend in part (a) applies to all plots, and the subscripts correspond to
c=colloid, p=polymer and t=tadpole species. The initial system in (a) was
mixed, while the other systems were initialised as separated states. The
numbers of molecules in each panel were: (a):
$N_{\mathrm{col}}=374,~{}N_{\mathrm{poly}}=188,~{}N_{\mathrm{tad}}=0$; (b) and
(c): $N_{\mathrm{col}}=329,~{}N_{\mathrm{poly}}=141,~{}N_{\mathrm{tad}}=47$;
(d): $N_{\mathrm{col}}=235,~{}N_{\mathrm{poly}}=47,~{}N_{\mathrm{tad}}=141$.
$N_{\mathrm{tail}}=30$ in (a) and (b), and $N_{\mathrm{tail}}=15$ in (c) and
(d).

##  Appendix B The SAFT formalism

Statistical associating fluid theory (SAFT) provides a formalism for
estimating the free energy in systems of associating particles, including
polymers. Here we summarise the SAFT expressions in Ref. [47], using our hard
sphere potentials, starting by breaking the Helmholtz free energy is into 4
terms

$F_{\mathrm{SAFT}}=F_{\mathrm{ideal}}+F_{\mathrm{part}}+F_{\mathrm{chain}}+F_{\mathrm{assoc}}.$
(10)

These respectively represent the ideal gas term, the contribution from
individual particles in the absence of bonds, the contribution from intra-
chain interactions, and the contribution from inter-chain interactions.

For our hard systems $F_{\mathrm{assoc}}=0$, while the ideal gas term is the
usual sum over molecule species

$F_{\mathrm{ideal}}=k_{\mathrm{B}}T\sum\limits_{i}N_{i}\left(\ln\left(\frac{N_{i}}{V\lambda_{\mathrm{th},i}^{3}}\right)-1\right),$
(11)

with the system volume $V$ and the thermal wavelength $\lambda_{\mathrm{th}}$.

The particle term uses an average contribution per particle calculated as

$\displaystyle f_{\mathrm{part}}=\frac{1}{\zeta_{0}}$
$\displaystyle\left(\left(\frac{\zeta_{2}^{3}}{\zeta_{3}^{2}}-\zeta_{0}\right)\ln(1-\zeta_{3})\right.$
$\displaystyle\left.+\frac{3\zeta_{1}\zeta_{2}}{1-\zeta_{3}}+\frac{\zeta_{2}^{3}}{\zeta_{3}(1-\zeta_{3})^{2}}\right),$
(12)

where

$\zeta_{\alpha}=\frac{\pi}{6V}\sum\limits_{i}N_{i}\sigma_{i}^{\alpha}.$ (13)

and $\sigma_{i}$ is the diameter of particles of species $i$. Applying this to
all particles gives

$F_{\mathrm{part}}=k_{\mathrm{B}}T(N_{\mathrm{col}}+N_{\mathrm{tad}}(1+N_{\mathrm{tail}})+N_{\mathrm{poly}}N_{\mathrm{mono}})f_{\mathrm{part}}.$
(14)

Finally, the chain term accounts for covalent bonds holding connected bead
centres a distance $(\sigma_{i}+\sigma_{j})/2$ apart. I.e., the hard spheres
are in contact. This is accounted for through the value of the radial
distribution function at contact

$g_{ij}=\frac{1}{1-\zeta_{3}}+\frac{3B_{ij}}{(1-\zeta_{3})^{2}}+\frac{2B_{ij}^{2}}{(1-\zeta_{3})^{3}},$
(15)

in which

$B_{ij}=\frac{\sigma_{i}\sigma_{j}}{\sigma_{i}+\sigma_{j}}\zeta_{2}.$ (16)

Then

$\displaystyle F_{\mathrm{chain}}=$ $\displaystyle-
k_{\mathrm{B}}TN_{\mathrm{tad}}(\ln g_{\mathrm{ht}}+(N_{\mathrm{tail}}-1)\ln
g_{\mathrm{tt}})$
$\displaystyle\;\;\;-k_{\mathrm{B}}TN_{\mathrm{poly}}(N_{\mathrm{mono}}-1)\ln
g_{\mathrm{pp}},$ (17)

counting all the bonds between tadpole heads, h, tadpole tail beads, t, and
polymer beads, p.

From the free energy expression, we get the (osmotic) pressure from
$p_{\mathrm{SAFT}}=-\left(\partial F_{\mathrm{SAFT}}/\partial V\right)_{T,N}$.
In this work, we solved this numerically to identify combinations of molecule
densities with pressure equal to the upper and lower bounds found
experimentally [43]. This range is then shown as the shaded volume in Fig. 6.

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