Phase separation in soft repulsive polymer mixtures: foundation and implication for chromatin organization
Naoki Iso Yuki Norizoe Takahiro Sakaue



1]Naoki Iso
1]Yuki Norizoe
1]Takahiro Sakaue^∗
[1]Department of Physical Sciences, Aoyama Gakuin University

Phase separation in soft repulsive polymer mixtures: Foundation and implication for chromatin organization
    [
    
==========================================================================================================

§ ABSTRACT 
Given a wide range of length scales, the analysis of polymer systems often requires coarse-graining, for which various levels of description may be possible depending on the phenomenon under consideration. Here, we provide a super-coarse grained description, where polymers are represented as a succession of mesosopic soft beads which are allowed to overlap with others.
We then investigate  the phase separation behaviors in mixture of such homopolymers based on a mean-field theory, and discuss universal aspects of the miscibility phase diagram in comparison with the numerical simulation. We also discuss an extension of our analysis to the mixtures involving random copolymers, which might be interesting in the context of chromatin organization in cellular nucleus.


[0]^* Department of Physical Sciences, Aoyama Gakuin University, 5-10-1 Fuchinobe, Chuo-ku, Sagamihara, Japan. E-mail:sakaue@phys.aoyama.ac.jp




§ INTRODUCTION

Phase separations in polymer solution and blend have a long history of research due to its importance in fundamental science as well as industrial applications <cit.>. Recently, its pivotal role in the field of biophysics has been recognized as a basic mechanism to organize various cellular and nuclear bodies <cit.>.
Here, given the complexity in biological systems, standard approaches such as the Flory-Huggins theory to analyze the phase separation do not always suffice, and various extensions or modifications may be called for depending on the phenomena under consideration.

In this paper, we provide one such example, where we investigate the phase separation behavior of polymer mixtures made of mesoscopic segments.
Our work has been motivated by the recent attempts to simulate chromatin organization in cellular nucleus. 
In ref. <cit.>, Fujishiro and Sasai constructed a polymer model of the whole genome of human cells, where each chromatin is modeled as a succession of soft-core monomers. 
Here, individual monomers (beads) represent ∼ 10^2 kbp of DNA, which is much larger than the conventional monomers defined in standard theory or simulation of polymer systems. They argued that the interaction between such  mesoscopic segments is soft and repulsive, and the imbalance in such repulsion in systems with e.g., eu- and hetero-chromatic monomers would trigger the phase separation. 
Similar modelings of large scale behavior of chromatin with soft-core potential naturally arises after the coarse-graining, hence have been employed in other works as well <cit.>, where the soft potential incorporates the entropic effect relevant to the mesoscipic segments.


How can we describe such a phase separation phenomena in chromatin theoretically? The immediate complication lies in the copolymer nature of the chromatin model <cit.>. However, even if we let aside the sequence effect and restrict our attention to a binary mixture of homopolymers, the application of a Flory-Huggins theory is hampered because of the allowed overlap between monomers due to the soft-core nature of the inter-monomer potentials. 
A key insight would thus be obtained by the phase behavior of the mixture of soft particles. This problem has been extensively studied by groups of Likos, Löwen and Kahl<cit.>.
Very recently, Staňo, Likos and Egorov have extended the framework to the system of chains of soft beads<cit.>. Although their primal target is a mixture of linear polymers and ring polymers (or polycatenanes), we expect that the same physics applies to chromatin system, too.

Our first aim is thus to recapitulate and to numerically validate the theoretical framework for the binary mixture of polymers made of soft monomers, which allows one to analyze the phase behavior.
In Sec.  <ref>, we introduce the mean-field free energy for our system. From the analysis of the free energy, we present in Sec.  <ref>, the miscibility phase diagram and compare it with the result from molecular dynamic (MD) simulations. Sec.  <ref> is devoted to discussions on universal aspects of the phase diagram, comparison with a conventional Flory-Huggins theory, and connection to the Gaussian core model. Building on the framework, we also discuss its extension to a system containing copolymers in Sec. <ref>.




§ FREE ENERGY OF THE SOFT REPULSIVE POLYMER MIXTURE


Following Staňo  <cit.>, we adopt the following free energy density for the mixture of polymers modeled as the succession of soft beads which represent mesoscopic segments

    f/k_BT    =    c_a/N_alnc_a +  c_b/N_blnc_b
          + 1/2χ_aa c_a^2 +  1/2χ_bb  c_b^2+ χ_ab c_a c_b

where k_BT is thermal energy, c_x and N_x are, respectively, the number density of beads and the chain length (number of beads per chain) of component x (=a  or  b). Parameters χ_xy (>0) represent the strength of the repulsive interaction between beads x and y. 

Note that in this representation, the parameters χ_xy have a unit of volume, and we measure them in unit of the volume of individual beads. In other words, we assume, for simplicity, the characteristic size (σ) of beads a and b are equal, which is taken to be the unit of length. Although there is no attraction, the phase separation may be induced by the asymmetry in the repulsion, i.e., χ_aa≠χ_bb. At first sight, Eq. (<ref>) looks as a free energy in the second virial approximation valid for low concentrations. As we shall show below, however, the free energy (<ref>) is capable of describing the phase separation in concentrated regime (c_a + c_b) σ^3 > 1 (see Seq. <ref> for discussion). 

Let us first clarify a mathematical aspect relevant to the phase equilibria condition in the system described by the free energy Eq. (<ref>). If the homogeneous mixture of polymer A and polymer B separates into phase 1 and phase 2,  the demixed state is specified by the concentrations of both components in respective phases, i.e., (c_a^(1),  c_b^(1)) and (c_a^(2),  c_b^(2)). The number of unknowns is thus n_u=4.  
On the other hand, the phase equilibria between two phases indicates the equalities of chemical potentials μ_x^(α)=∂ f/∂ c_x^(α) between two phases (α=1 or 2) for both components (x=a or b), i.e., μ_a^(1)=μ_a^(2) and μ_b^(1)=μ_b^(2) and also the mechanical balance ensured by the equality of pressure  P^(1)=P^(2) , where P(c_a,c_b) = c_a [∂ f (c_a,c_b)/ ∂ c_a] + c_b  [∂ f (c_a,c_b)/ ∂ c_b]  - f(c_a,c_b), leading to the number of condition to determine the phase equilibria n_c=3.
Comparing the number of unknowns and that of conditions, we expect that the dimensionality of the phase boundary, i.e., binodal is d_pb=n_u-n_c=1.




§ PHASE DIAGRAM OF THE SOFT REPULSIVE POLYMER MIXTURE

 
In Fig. <ref> (a), we show an example of the miscibility phase diagram obtained from the free energy Eq. (<ref>) under the fixed interaction parameters.
As we have discussed, the phase diagram is two-dimensional spanned by c_a and c_b, in which the uniform state (bottom left) and the demixed state (upper right) are separated by one-dimensional phase boundary. Tie lines, which connect (c_a^(1),  c_b^(1)) and (c_a^(2),  c_b^(2)) in demixed state, are negatively sloped, indicating the phase separation is a segregative type. 
As expected, the region for demixing widens with the increase in either chain length or the repulsion strength, see Fig.  <ref> (b).
Note that despite the symmetry in the chain length N_a = N_b in the examples shown here, the phase diagram exhibits the asymmetry about the diagonal c_a=c_b. The asymmetry is caused by the difference in physical properties of type a and b beads, which leads to the phase rich in softer beads b being more concentrated than the other. Such a feature can be made more evident by re-plotting the phase diagram in total concentration c = c_a + c_b and the composition ψ = c_a/c plane (Fig. <ref> (c)).

 
 


To check the validity of the free energy prediction, we have performed numerical simulations of the polymer mixture.
Briefly, the system is a mixture of two types of linear homopolymers A and B, where A (B) polymer is made of a succession of N_a (N_b) beads of type a (b) (see Appendix for details of the simulation model). To represent the soft repulsion between monomers, we employ the Gaussian potential, see Eq. (<ref>) in Appendix, where the strength of the repulsion between x-bead and y-bead is ϵ_xy in unit of k_BT.

The numerically determined phase boundary shown in Fig. <ref> (a) is obtained with the interaction strength ϵ_aa=2,  ϵ_bb =1,  ϵ_ab=1.5 and the chain length N_a = N_b=20, where the overall concentrations are varied from (c_a^(o), c_b^ (o)) = (0.1, 0.1) to (c_a^(o), c_b^ (o)) = (0.5, 0.5).
We find that the mixture is homogeneous at (c_a^(o), c_b^ (o)) = (0.1, 0.1), but develops large concentration fluctuation at (c_a^(o), c_b^ (o)) = (0.25, 0.25), and further increase in concentration leads to a well-defined phase separated structure (Fig. <ref> (b)). 
Remarkably, the numerically determined phase diagram resembles that predicted by the free energy analysis (Fig. <ref>). More specifically, we find that the numerical and analytical phase diagrams almost overlap under the correspondence χ_xy/ ϵ_xy≃ 2.5 between interaction parameters in free energy and interaction strengths in simulation.   

 




§ DISCUSSIONS




 §.§ General aspects of phase diagram


In Sec. <ref>, we have shown one example how the phase boundary alters  with the change in chain length or the interaction strength (Fig.  <ref> (b)). To clarify the dependence of the shape of phase diagram on system parameters in a more systematic way, it is desirable to find out universal aspects inherent to the model described by the free energy (<ref>).

To this end, we introduce the rescaled concentrations c̃_a = c_a N_a χ_aa and c̃_b = c_b N_b χ_bb, which enables us to rewrite Eq. (<ref>) as

    f/k_BT =  1/N_a^2 χ_aa[c̃_alnc̃_a +  k_1 c̃_blnc̃_b + 1/2c̃_a^2 +  1/2 k_1 c̃_b^2+ k_2 c̃_ac̃_b]

where irrelevant linear terms in concentrations are dropped, and coefficients are

    k_1 = N_a^2/N_b^2χ_aa/χ_bb,  k_2 = N_a/N_bχ_ab/χ_bb = √(k_1)χ_ab/√(χ_aa χ_bb)

The above free energy density is invariant under the parameter changes which keep k_1 and k_3 = χ_ab/√(χ_aa χ_bb) constant. These conditions are satisfied by the following transformations

    (χ_aa, χ_bb, χ_ab)    ⇒    p (χ_aa, k χ_bb, √(k)χ_ab) 
    
    (N_a, N_b)    ⇒    q (N_a, N_b/√(k))

where p, q, k are positive real numbers.
Therefore, with the change in parameters according to Eqs (<ref>) and (<ref>), the phase diagram drawn in terms of rescaled concentrations remains the same.
A similar analysis has been done by Staňo et. al <cit.>.
Indeed, from the stability analysis of the uniform state, one finds the spinodal curve

    (1+c̃_a)(1+c̃_b)/c̃_ac̃_b = k_3^2

and the critical point is determined by Eq. (<ref>) together with

    c̃_a(1+c̃_a)^3/c̃_b(1+c̃_b)^3 = k_1

Eq. (<ref>) indicates that a necessary condition for the phase separation k_3>1  ⇔χ_ab > √(χ_aaχ_bb), and Eq. (<ref>) determines the location of critical point on the spinodal curve  <cit.>.
In Fig. <ref>, we demonstrate a collapse of the phase diagram upon rescaling. 





 

 §.§ Comparison with Flory-Huggins theory

 It is instructive to compare the present theory with the standard Flory-Huggins theory for polymer blends.
The Flory-Huggins free energy (per lattice site) for a blend of polymer A and B is written as <cit.> <cit.>

    f_ FH/k_BT = ϕ_a/N_alnϕ_a +  ϕ_b/N_blnϕ_b  + χϕ_a ϕ_b

where ϕ_x is the volume fraction of component x, and a non-dimensional parameter χ measures the nature and the strength of interaction. The incompressiblity condition enforces ϕ_a + ϕ_b =1.
Since χ is usually positive, corresponding to attraction among the like-species, such an interaction acts as a driving force to the phase separation. The free energy (<ref>) reduces to that of polymer solution in the limit N_b=1 where the component b represents a solvent.


 
Figure <ref> shows  the phase diagram calculated from Eq. (<ref>).
When we fix the interaction parameter at the value χ > χ_c, where χ_c=(√(1/Na) + √(1/Nb))^2/2 is the critical value for the phase separation, the phase diagram as a function of ϕ_a is one-dimensional line with two points ϕ_a^(1) and ϕ_a^(2) representing the phase boundaries. If the overall concentration falls in between these two points, the uniform state is metastable (outside spinodal region) or unstable (inside spinodal region) and the system phase separates into the A-poor (dilute) and A-rich (concentrated) phases with the volume fractions ϕ_a^(1) and ϕ_a^(2), respectively. Note that the dimensionality of the phase boundary at fixed χ is d_ pb=0, i.e., points, which is a consequence of the equality of number of conditions (n_c=2, i.e.,  μ_a^(1)=μ_a^(2) and μ_b^(1)=μ_b^(2)) and the number of unknowns (n_u=2, i.e., ϕ_a^(1) and ϕ_a^(2)).
We note that a set of conditions μ_a^(1)=μ_a^(2) and μ_b^(1)=μ_b^(2) is equivalent to μ_a^(1)=μ_a^(2) and Π^(1)=Π^(2), where Π(ϕ_a) = ϕ_a  [df_ FH(ϕ_a)/dϕ_a] - f_ FH(ϕ_a) is the osmotic pressure, the use of which may be more common in polymer solution, where the component b is regarded solvent.

These two methods are equivalent due to the relation 
 
    -Π(ϕ_a) v_0 = μ_b(ϕ_a)

 which follows from the incompressible condition <cit.>, where v_0 is the volume of the monomers and solvents.
 In contrast, in our description of polymer mixture with soft potential, the solvent degrees of freedom is already integrated out, and c_a and c_b are independent variables without constraint, i.e., free from the incompressible condition. One can conceive that our system under consideration is a three component system (two solute A and B plus solvent), and the free energy density (<ref>) represents a mesoscopic description after coarse-graining. 
 
 It is also known that the interaction part in the Flory-Huggins free energy initially takes the form (χ_aa/2) ϕ_a^2 + (χ_bb/2) ϕ_b^2 + χ_abϕ_a ϕ_b. Rewriting it into the form of Eq. (<ref>) with a single parameter χ = χ_aa/2 + χ_bb/2 - χ_ab is made using the incompressibility condition. Again, it does not apply to our soft polymer description. Since our system originally possesses three components, we naturally need three interaction parameters to characterize the system.
 We also note that the critical χ parameter in blends of long polymers N_A, N_B ≫ 1 is χ_c → 0 in Flory-Huggins theory. A necessary condition for the phase separation in this limit is thus χ > 0 ⇔ (χ_aa+χ_bb)/2 > χ_ab. In contrast, as discussed in Sec. <ref>, the corresponding condition in our soft polymer mixtures is χ_ab > √(χ_aaχ_bb) independent of the chain length<cit.>.
 


 §.§ Relation with Gaussian core model


In our model, polymers are described as N successive soft beads, where these beads are already mesoscopic entities with their internal degrees of freedom integrated out. We note here that, starting from a microscopic model, there is a freedom to choose N, i.e., the degree of the coarse-graining.
Although the extreme limit of the choice is N=1, in which individual polymers are described as single soft particles, the validity of such a description may break down at high concentration<cit.>.

It is known that the effective pair potential between two isolated polymer coils in dilute solution can be well approximated by a simple Gaussian potential

    U(r)/k_BT = ϵexp( -r^2/R^2)

where ϵ≃ 2 and the width R is of order of the gyration radius the coil  <cit.>. The fact that the energy scale of the interaction is of order of thermal energy indicates entropic nature of the interaction. It has been shown that the above potential also provides a reasonable description for the effective interaction in semidilute solution, where polymers are overlapped.
Thermodynamic properties of fluid composed of soft particles interacting through Eq. (<ref>), i.e., the Gaussian core model, has been analyzed in detail by Louis et. al. <cit.>. 
Phase separation in binary mixture of such fluids has been also extensively studied <cit.>.
As discussed in Sec. <ref>, our free energy (<ref>) can formally be mapped to that case by (N_a, N_b, c_a, c_b) → (1,1,c_a/N_a, c_b/N_b). 
The analysis in Sec. <ref> may then indicate that the miscibility phase diagram is intact if we simultaneously transform the interaction parameters as (χ_aa, χ_bb, χ_ab) → (χ_aaN_a^2, χ_bbN_b^2, χ_abN_a N_b). 
One may then conclude that the introduction of the “polymerization index" N_a and N_b might be auxiliary for the description of homopolymer mixtures. 
However, there are, at least, two reasons we need the polymeric description.
First, N=1 description is known to suffer from a significant concentration dependence of the effective repulsive interactions once polymer coils start to overlap deep in semidilute concentration regime. This motivates the multisegment description with N>1  <cit.>, where a suitable choice for N would be guided by the overlapping condition for mesoscopic segments.
Second, once there arises a characteristic length scale in the problem, we need the polymeric description as a succession of beads with appropriate degree of coarse-graining. In Sec. <ref>, we provide one such example, where we analyze the effect of modulation in local physical properties along polymers, (i.e., due to post translational modification) on the phase separation.

Another point deserving comment is the relation between the strength of the interaction potential ϵ_xy and the interaction parameter χ_xy in our free energy (<ref>). We have shown in Sec.<ref> that the simulation results quantitatively match with the free energy prediction under the relation χ_xy/ ϵ_xy≃ 2.5.
Since the free energy (<ref>) takes apparently the same form as the virial expansion up to second order, one may expect that the ϵ_xy - χ_xy relation would be obtained from χ_xy= - ∫ (e^-U_xy(r)/k_BT-1) dr⃗.
As emphasized in ref. <cit.>, however, the free energy (<ref>) is based on the random phase approximation (RPA)<cit.>. As such, it becomes more and more accurate in higher concentration regimes in contrast to the second virial approximation<cit.>. In fact, unlike the virial expansion, the quadratic form of the free energy in concentrations is a consequence of the RPA closure, where the direct correlation functions, which appears in the Ornstein-Zernike relation, is independent of the concentrations. The analysis of equation of state with RPA leads to the identification χ_xy= (1/k_BT) ∫ U_xy(r) dr⃗ = π^3/2σ_xy^3 ϵ_xy.
Given the resultant ratio  χ_xy/ ϵ_xy = π^3/2 is considered to be an upper bound compared to a more accurate estimate e.g., obtained from hypernetted chain closure  <cit.>, we find our result χ_xy/ ϵ_xy≃ 2.5 reasonable and providing an overall consistency of the soft core model description of the phase separation based on the free energy Eq.(<ref>).




§ MIXTURES WITH RANDOM COPOLYMERS


So far discussed is a foundation for the coarse-grained description of polymer mixture, where polymers are represented as succession of soft mesoscopic beads. In particular, we have focused on the phase behavior of mixture of homopolymers.
As stated in Introduction,  however, one of the main motivations to necessitate such a description is its relevance to describe the large scale behavior of chromatin in cellular nucleus. In this section, we would like to discuss a simple extension of our theory, which may be linked to a certain aspect of chromatin organization in living cells.

It is known that interphase chromatin in early embryo is quite homogeneous inside nucleus, which is, in certain sense, reminiscent to a uniform solution of homopolymers <cit.>. With the progress of the development stage, however, several characteristic structures, such as heterochromatin foci and transcriptional factories, start to appear <cit.>. Responsible for such structure formations would be a phase separation, which is driven by local alternation of chromatin monomers caused by e.g., post-translational modification. The change in the chemical state in chromatin monomers likely induces the modulation of physical properties along chromatin polymer, which could be represented by a copolymer model. 
Since the variation in repulsive forces primarily reflects the difference in density of core-bearing monomers within the coarse-grained segments<cit.>, the segment “a" represents regions where chromatin exists in a relaxed, less condensed state, reminiscent of euchromatin, while the segment “b" corresponds to more condensed regions akin to heterochromatin. The structure formation under consideration could thus be treated by the appearance of copolymers in the matrix of homopolymers. With this in mind, let us consider a mixture of homopolymers H (with length N_h), which composes of type a beads only, and copolymer C (with length N_c) , which composes of types a and b beads.
The monomer concentration of H and C polymers are c_h and c_c, respectively. For the analytical tractability in a simple mean-field description, we assume the latter to be a random copolymer, which is characterized by the fraction α of b beads, i.e.,  the number of b beads in a copolymer C is α N_c.

The free energy of the mixture is written as

    f/k_BT    =    c_h/N_hlnc_h +  c_c/N_clnc_c
          + 1/2χ_hh c_h^2 +  1/2χ_cc  c_c^2+ χ_hc c_h c_c

which takes the same form as Eq. (<ref>) except for the appearance of new interaction parameters. 
While χ_hh=χ_aa trivially from the definition of the homopolymer H, the others χ_cc and χ_hc are nontrivial, which appear instead of χ_bb and χ_ab , respectively.
Given the randomness in the sequence of the copolymer, we can evaluate these interaction parameters as mean values of the inter-beads interactions  χ_aa,  χ_bb,  χ_ab;

    χ_cc   =   (1-α)^2 χ_aa + α^2 χ_bb + 2 α (1-α) χ_ab
    χ_hc   =   (1-α) χ_aa + αχ_ab









In Fig. <ref>, we show phase diagrams obtained from the free energy (<ref>) with Eqs. (<ref>) and (<ref>) for a fixed α. Note that α=0 reduces to a homopolymer solution (with only type a bead), and α=1 corresponds to a blend of homopolymers A and B analyzed in earlier sections. Here we show the cases for α=0.3, 0.5. As expected, the region for phase separation enlarges with the fraction α. In addition, the results depend on a relative stiffness between beads a and b.
As shown, the system is more prone to phase separation when the matrix polymer is softer χ_aa < χ_bb reflecting the asymmetry in the phase diagram of homopolymer mixtures (Sec. <ref>).

To check the validity of the free energy prediction, we again performed numerical simulations using Gaussian potentials to represent bead-bead soft repulsions. In Figs. <ref> and <ref>, we compare the theoretical phase diagram in Fig. <ref> with simulation results, where the repulsion strengths for Gaussian potentials are set to be ϵ_xy = χ_xy/2.5 between beads x and y as determined from the result of homopolymer mixtures. As shown, the agreement is rather satisfactory, demonstrating that the overall trend of phase separation is well captured by the proposed free energy. In Figs. <ref> and <ref>, we also show the spatial profiles of the monomer concentration c_h of homopolymer together with the corresponding typical snapshots. 











§ DISCUSSIONS AND SUMMARY

Numerical simulations of large scale chromatin organization in cellular nucleus often adopt highly coarse-grained models, in which chromatin polymer is represented as a succession of soft beads <cit.>. Unlike a model which employs nucleosomes as monomers of chromatin polymer, each of beads here represents a substantial amount of nucleosomes, thus regarded as a mesoscopic entity, allowing mutual overlaps with entropic penalty. It has been shown that the effective interaction between such mesoscipic segments is soft and repulsive, qualitative feature of which is well approximated by the Gaussian potential <cit.>.

We have considered binary mixtures of such soft repulsive polymers and investigated how the imbalance in repulsive interactions between different species leads to the phase separation. 


After summarizing universal aspects of the phase diagram based on invariant property of the free energy upon changes in parameter values, we have  extended the theory to mixtures including random copolymers, which may have some implications to chromatin phase separation. 

As discussed in Sec. <ref>, random copolymer model is inspired by epigenetic modification of chromatin. This modification is performed and maintained by enzymes, thus includes energy consuming nonequilibrium process. In this sense, our description based on equilibrium framework should be considered a useful effective description to elucidate the impact of phase separation on chromatin organization. The same remark applies to many of current biophysical modelings of chromatin, not only for its structural organization but also for its dynamics. Yet, there are several other works, which emphasize possible impacts of various nonequilibrium effects on chromatin<cit.>. 
Perhaps, some of these effects associated with nonequilibrium activities could be described by effective equilibrium models. Such a strategy may well work to understand some aspect of the problem, but may fail to capture other aspects.  In our opinion, it remains to be seen how and when nonequilibrium factors are critically important in chromatin biophysics.
The same comment would apply to topological constraints, another  factor presumably important in chromatin, but not explicitly included in our description <cit.>.  
In this regard, it is interesting to note that as discussed in <cit.>,  the similar physics as described in present paper may be important in blend of non-concatenated ring polymers, where the soft repulsion arises from the so-called topological volume due to topological constraints <cit.>.

As possible extensions of our work, we first note that our theory deals with the macro-phase separation, hence does not capture the possible appearance of mesophases. However, the occurrence of  the “micro-phase separation" is naturally expected in copolymer systems, the elucidation of which should provide further insight into the problem of chromatin organization in nuclei.
Secondly, although we have only analyzed bulk property based on mean-field theory,  we expect that the effect of correlations and interfacial properties at phase boundary and near a confining wall could be analyzed by following an approach outlined in ref. <cit.>. It would be interesting to see how such an analysis can be compared to the chromatin spatial profile, e.g. near the nuclear membrane.

Finally, we point out that there are several studies on compressibility effects in polymer solutions, which become evident, for instance, in pressure-induced phase separation  <cit.>. Here, interesting phenomena such as the acousto-spinodal decomposition have been predicted  <cit.>. Although comparison with these studies may be interesting, we note that the loss of incompressible condition in our description results after coarse-graining, i.e., integrating out solvent degrees of freedom. Therefore, to address the kinetic effect, we need to take properly solvent effects into account.
 





§ APPENDIX



 §.§ Simulation model

The system is a mixture of two types of linear homopolymers A and B, where A (B) polymer is made of a succession of N_a (N_b) beads of type a (b). 
The potential energy in the system has two contributions. The first is the intrachain bonding potential

    U^ (b)(r)/k_BT = 1/2 k_b (r-r_0)^2

which acts the bonding pairs to maintain the linear connectivity of the chain, where r and r_0 denotes the separation between bead centers and the natural bond length, respectively. We set the spring constant k_b = 70.0/ σ^2 to keep the bond length nearly constant r_0 =   σ, where σ is the unit of the length (see below). The thermal energy, k_BT, is chosen as the unit energy in the simulation system. The second is the non-bonded interaction potential, which represents the soft repulsion between monomers. We employ the Gaussian core potential; the pair potential between one bead with type X and another bead with type Y reads

    U^( int)_xy(r)/k_BT =  ϵ_xyexp(-r^2/σ_xy^2)

where ϵ_xy and σ_xy, respectively, measure the strength and the range of repulsive interaction between x and y beads. For simplicity, we set the range of repulsive interaction for all types of pair equal, i.e., σ_aa = σ_bb = σ_ab = σ, and adopt this range (denoted as σ) as the unit length. 
Note that this interaction is acting on all the bead pairs except for the nearest neighbors along the chain (bonded pairs).

 Molecular Dynamics (MD) simulations with fixed volume and the constant temperature are performed using the LAMMPS package <cit.>. To integrate the equations of motion, we adopt the velocity Verlet algorithm, in which all beads are coupled to a Langevin thermostat with the damping constant γ = τ_0^-1 with τ_0 = σ(m/k_BT)^1/2, where 
 
 m is the bead mass (assumed to be the same for type a and b). This τ_0 is chosen as the unit time. The integration time step is set at 0.01 τ_0. ( L_x, L_y, L_z ) represents the size of the rectangular parallelepiped system box with periodic boundary conditions. This system box is placed in - L_ω / 2 < ω < L_ω / 2, where ω represents the Cartesian axes x, y, and z. ( L_x = 48 σ, L_y = 16 σ, L_z = 16 σ) is fixed unless ( L_x = 120 σ, L_y = 16 σ, L_z = 16 σ) is utlized for the simulation of the A-B homopolymer mixtures at ( c_a, c_b ) = ( 0.3, 0.3 ).


To prepare the initial state, we start from dilute solution, where the same numbers of the homopolymers A and B are distributed in a large cubic box with the size ( 200 σ, 200 σ, 200 σ ). We run the simulation at ϵ_aa = ϵ_bb = ϵ_ab = 2.0 (mixture of homopolymers) or 2.5 (mixture with random copolymers) for 2 × 10^7 steps with slowly compressing the system box into the final system size ( L_x, L_y, L_z ). In this way, we obtain the desired concentration of polymer mixture, where both the A and B- polymers are homogeneously mixed. For the mixture with copolymers, (1-α)  N_c beads in each B-homopolymer in this initial configuration are randomly chosen, and turned into type-a beads. We then set the interaction strengths to appropriate values in the sebsequent production run (see below). 
Then, we reassign the monomer label to adjust the initial spatial concentration profile to prepare the phase separated initial state.




To perform various statistical analysis, we sampled microstates of the system every 1000 steps after the system reaches equilibrium, i.e., at 2 × 10^6 steps (simulation runs in Sec. 4) and 3 × 10^6 steps (Sec. 5) after setting the interaction strengths to appropriate values.
However, for the case of simulation runs given in Sec. 4 at ( c_a^(0), c_b^(0) ) = ( 0.3, 0.3 ), the sampling starts at 9 × 10^6 steps, and for the case of Sec. 5 at α = 0.3, (ϵ_aa, ϵ_bb, ϵ_ab) = (2.5, 0.5, 1.5), (c_h^(0), c_c^(0)) = ( 0.5, 0.5 ), the sampling starts at 7 × 10^6 steps.

In all simulations, we collect 1001 independent samples of particle configurations in equilibrium. For the simulation at ( c_a^(0), c_b^(0) ) = ( 0.3, 0.3 ), as only one exception, production runs end at 18 × 10^6 steps, and particle coordinates are sampled every 1000 steps from 9 × 10^6 to 18 × 10^6 steps, during which 9001 independent samples of particle configurations in equilibrium are collected for statistical accuracy improvement in the vicinity of the critical point. We have confirmed that the physical properties of the simulation system are not significantly changed when we start the simulation from the homogeneously mixed initial states.












0








 §.§ Tables

Tables typeset in RSC house style do not include vertical lines. Table footnote symbols are lower-case italic letters and are typeset at the bottom of the table. Table captions do not end in a full point.<cit.>




Adding notes to tables can be complicated.  Perhaps the easiest method is to generate these manually.[Footnotes should appear here. These might include comments relevant to but not central to the matter under discussion, limited experimental and spectral data, and crystallographic data.]






§ EQUATIONS


Equations can be typeset inline e.g. y = mx + c or displayed with and without numbers:

 
    A = π r^2



    γ/ϵ x r^2 = 2r




For footnotes in the main text of the article please number the footnotes to avoid duplicate symbols. e.g. . The corresponding author ∗ counts as footnote 1, ESI as footnote 2, e.g. if there is no ESI, please start at [num]=[2], if ESI is cited in the title please start at [num]=[3] etc. Please also cite the ESI within the main body of the text using †. For the reference section, the style file  can be used to generate the correct reference style.









§ CONFLICTS OF INTEREST

`There are no conflicts to declare.



§ ACKNOWLEDGEMENTS

We thank M. Sasai and S. Fujishiro for discussions. This work is supported by JSPS KAKENHI (Grants No. JP23H00369, JP23H04290 and JP24K00602).











 
rsc 

