Now at ]Department of Chemistry and Chemical Engineering, University of
California Berkeley, CA 94720

# Local Changes in Protein Filament Properties Drive Large-Scale Membrane
Transformations Involved in Endosome Tethering and Fusion

Ashesh Ghosh1, [  Andrew J. Spkaowitz1,2,3<EMAIL_ADDRESS>1Department
of Chemical Engineering, Stanford University, Stanford, CA 94305
2Biophysics Program, Stanford University, Stanford, CA 94305
3Department of Materials Science & Engineering, Stanford University, Stanford,
CA 94305

###### Abstract

Large-scale cellular transformations are triggered by subtle physical and
structural changes in individual biomacromolecular and membrane components. A
prototypical example of such an event is the orchestrated fusion of membranes
within an endosome that enables transport of cargo and processing of
biochemical moieties. In this work, we demonstrate how protein filaments on
the endosomal membrane surface can leverage a rigid-to-flexible transformation
to elicit a large-scale change in membrane flexibility to enable membrane
fusion. We develop a polymer field-theoretic model that captures molecular
alignment arising from nematic interactions with varying surface density and
fraction of flexible filaments, which are biologically controlled within the
endosomal membrane. We then predict the collective elasticity of the filament
brush in response to changes in the filament alignment, predicting a greater
than 20-fold increase of the effective membrane elasticity over the bare
membrane elasticity that is triggered by filament alignment. These results
show that the endosome can modulate the filament properties to orchestrate
membrane fluidization that facilitates vesicle fusion, providing an example of
how active processes that modulate local molecular properties can result in
large-scale transformations that are essential to cellular survival.

††preprint: APS/123-QED Figure 1: Schematic of an endosomal membrane with EEA1
brush layer. The A and B images show the collective behavior of the brush
layer in the predominantly extended and predominantly collapsed states on the
surface of a patch of elastic membrane. Yellow and Red colored polymers are in
the collapsed flexible and extended rigid conformations, respectively. The
effective elastic rigidity of the membrane and brush is higher in the aligned
state of polymer brush layer (panel A). Molecular-level changes that transform
polymers from extended to flexible conformation (red $\to$ yellow) alter
membrane rigidity and induce large length-scale fluctuations.

Cellular self-organization and large-scale orchestrated transformations are
driven by coordinated biophysical, mechanical, and dynamical processes
involving numerous biomacromolecules and biological membranes. A fundamental
challenge in understanding and predicting cellular self-organization and
dynamics stems from the need to determine how molecular components (_e.g._
RNA, proteins, lipids) leverage subtle physical changes to drive dramatic
cellular transformations across different length and time scales [1, 2, 3, 4].
The complex interplay of microscopic chemical events in membranes, including
lipid raft formation [5], coupled lipid-protein interactions [6], dynamin-
driven membrane fission [7], and active transport of fluid across membranes
[8, 9], determine critical biological fates such as cell shape and function
[10]. Coordinated membrane fusion enables the transport of critical cargo and
biochemicals to and from the cell [11], which is an essential biological
process involving subtle biophysical events.

In this work, we develop a physical model for the physical transformations in
the endosomal membrane that underlie membrane fusion and transport, which
serves as a prototypical example of cellular self-organization and dynamics.
Recently, allosteric changes to the early endosome tether protein EEA1 (Fig.
1) has been shown to play an important role in mechanical pulling of membranes
to close spatial proximity upon binding with vesicle bearing small GTPase Rab5
in its GTP bound form Rab5(GTP). EEA1 is a coiled-coil dimeric molecule with a
contour length of $L\\!\\!\sim\\!\\!222\pm 26$ nm and a persistence length of
$l_{p}^{\text{EEA1}}\\!\\!\sim\\!\\!246\pm 42$ nm. EEA1 predominantly exists
in “extended” or rigid conformation (Fig. 1A) in its free N-terminus unbound
form [12]. Upon binding to Rab5(GTP) the contour length essentially remains
the same. However, the molecule shifts to a more “flexible” conformation (Fig.
1B) with an average end-to-end distance of $\sim\\!\\!122\pm 50$ nm,
coinciding with a persistence length of
$l_{p}^{\text{EEA1(GTP)}}\\!\\!\sim\\!\\!74\pm 3$ nm [12].

Changes in the population density of the structurally rigid (or flexible) EEA1
filaments is thought to play a crucial role in mechanical pulling of membranes
via thermal and active enzymatic fluctuations present in the cellular media.
Moreover, EEA1 is present in high density patches on the membrane surface and
thus forms a switchable polymer brush layer [13]. It is to be noted that such
a structure-driven entropic transition is not unique to EEA1-tether system,
but common in coil-coiled tether proteins, such as GCC185 that binds to
Rab9(GTP) [14]. Hence, these long coil-coiled tethers can act as a mechanical
switch in bringing distant membranes to physical proximity to facilitate
fusion [12, 15].

We provide a theoretical model and explanation of how the collective brush
layer alters the effective elasticity of the membrane and makes it more
susceptible for fusion. More concretely, we construct a general thermodynamic
model of two types of semiflexible polymers in a solvent and specialize to our
case to determine how nematic interactions that originate from hydrophobic
moieties along the chain backbone of coil-coiled proteins dictate chain
alignment and alter the collective membrane rigidity. This work provides
fundamental insight into how the cell can control molecular properties of
specific biomacromolecules to elicit large-scale transformations involved in
major biological events.

We consider an incompressible polymer solution of total volume $V$, with
$n_{s}$ solvent molecules with molecular volume $v_{s}$, $n_{p}^{R}$ rigid
polymer chains with contour length $L_{R}$ and cross-sectional area $A_{R}$
and $n_{p}^{F}$ flexible polymer chains with contour length $L_{F}$ and cross-
sectional area $A_{F}$. The polymer chains are modeled using the wormlike
chain model, which describes the polymer chains as inextensible elastic
threads that are subjected to thermal fluctuations [16, 17]. The polymer
configuration of the $i$th polymer of type $\alpha\in[R,F]$ at arc-length
position $s$ (note, $s\\!=\\!0$ at one end and $s\\!=\\!L_{\alpha}$ at the
opposite end) is defined by the space curve $\vec{r}_{i}^{\alpha}(s)$, and the
local tangent vector
$\vec{u}_{i}^{\alpha}(s)=\partial\vec{r}_{i}^{\alpha}(s)/\partial s$ gives the
tangent orientation of the monomer segment at $s$. Inextensibility is strictly
enforced by ensuring that $|\vec{u}_{i}^{\alpha}(s)|=1$ for all configurations
of the system. The position of the $j$th solvent molecule is given by
$\vec{r}_{j}$. The bending rigidity of the semiflexible polymers are given by
their respective persistence lengths $l_{p}^{\alpha}$, and we also express the
chain length in dimensionless units as the number of Kuhn lengths
$N_{\alpha}=L_{\alpha}/(2l_{p}^{\alpha})$, where the Kuhn length
$b_{\alpha}=2l_{p}^{\alpha}$ gives the statistical segment length of a
polymer.

The system energy includes contributions for polymer bending deformation,
solvent-polymer mixing enthalpy captured through effective Flory-Huggins
$\chi$-parameters [18], and nematic alignment free energy of polymer chains
through an effective Maier-Saupe interaction [19, 20]. The total energy is
given by

$\displaystyle\beta E$
$\displaystyle=\displaystyle\sum_{\alpha\in\\{R,F\\}}\Bigg{\\{}\sum_{i=1}^{n_{p}^{\alpha}}\frac{l_{p}^{\alpha}}{2}\int_{0}^{L_{\alpha}}ds\left(\frac{\partial\vec{u}_{i}^{\alpha}(s)}{\partial
s}\right)^{2}$ $\displaystyle+\chi_{\alpha}\int
d\vec{r}\,\hat{\phi}_{s}(\vec{r})\hat{\phi}_{p}^{\alpha}(\vec{r})-\frac{a_{\alpha}}{2}\int
d\vec{r}\,\hat{\mathbf{S}}_{\alpha}(\vec{r}):\hat{\mathbf{S}}_{\alpha}(\vec{r})\Bigg{\\}}$
$\displaystyle+\chi_{RF}\int
d\vec{r}\,\hat{\phi}_{p}^{R}(\vec{r})\hat{\phi}_{p}^{F}(\vec{r})-a_{RF}\int
d\vec{r}\,\hat{\mathbf{S}}_{R}(\vec{r}):\hat{\mathbf{S}}_{F}(\vec{r}),$ (1)

where $\hat{\phi}_{s}(\vec{r})$ and $\hat{\phi}_{p}^{\alpha}(\vec{r})$
respectively define the local dimensionless density (volume fraction) of the
solvent and polymer molecules of type $\alpha$ at spatial location $\vec{r}$,
and $\hat{\mathbf{S}}_{\alpha}(\vec{r})$ is the tensorial nematic order
parameter density of state $\alpha$ at $\vec{r}$ [21, 22, 23]. Defining a
generalized volume fraction in terms of real spherical harmonics [24] allows
us to write the thermodynamic energy in a consolidated form as done in Ref.
[21] for a one-component polymer system (details in Supplemental Materials,
Secs. I-IV). Writing the canonical partition function followed by a series of
standard polymer field-theoretic transformations [18] and using the saddle-
point approximation [18], we write the homogeneous mean-field free energy
density of the system as

$\displaystyle\beta f_{0}$
$\displaystyle=\frac{1-\phi_{p}}{v_{s}}\ln{(1-\phi_{p})}+\sum_{\alpha\in\\{R,F\\}}\Bigg{\\{}\frac{\phi_{p}^{\alpha}}{L_{\alpha}A_{\alpha}}\ln{(\phi_{p}^{\alpha})}$
$\displaystyle+\chi_{\alpha}\phi_{p}^{\alpha}(1-\phi_{p})-\frac{a_{\alpha}}{3}({\phi}_{p}^{\alpha})^{2}m_{\alpha}^{2}+\frac{\phi_{p}^{\alpha}}{L_{\alpha}A_{\alpha}}\ln{(q_{\alpha})}\Bigg{\\}}$
$\displaystyle+\chi_{RF}\phi_{p}^{R}\phi_{p}^{F}-\frac{2a_{RF}}{3}{\phi}_{p}^{R}{\phi}_{p}^{F}m_{R}m_{F},$
(2)

where $\phi_{p}^{\alpha}$ is the bulk volume fraction of polymer $\alpha$, and
the total polymer volume fraction is
$\displaystyle\phi_{p}=\phi_{p}^{R}+\phi_{p}^{F}$. The nematic order parameter
$m_{\alpha}$ of polymer $\alpha$ is defined as $\displaystyle
m_{\alpha}=\frac{1}{L_{\alpha}}\int_{0}^{L_{\alpha}}ds\left\langle
P_{2}(u_{z}^{\alpha}(s))\right\rangle_{0_{\alpha}}$ 111$\displaystyle
P_{2}(x)=(3x^{2}-1)/2$ is the Legendre polynomial of order 2.. The average
$\langle\dots\rangle_{0_{\alpha}}$ is taken with respect to the single-chain
mean field energy, defined by

$\displaystyle\beta\mathcal{E}_{0}^{\alpha}=\int_{0}^{L_{\alpha}}ds\left\\{\frac{l_{p}^{\alpha}}{2}\left(\frac{\partial\vec{u}^{\alpha}(s)}{\partial
s}\right)^{\\!2}-\gamma_{\alpha}\left[(u_{z}^{\alpha}(s))^{2}-\frac{1}{3}\right]\right\\},$
(3)

where
$\gamma_{\alpha}=(a_{\alpha}\phi_{p}^{\alpha}m_{\alpha}A_{\alpha}+a_{RF}\phi_{p}^{\beta}m_{\beta}A_{\beta})$
gives the molecular field strength for the $\alpha$-type polymer. The single-
chain partition function $q_{\alpha}$ of polymer $\alpha$ is evaluated from a
Boltzmann-weighted sum over all polymer-chain conformations with respect to
the mean-field energy $\beta\mathcal{E}_{0}^{\alpha}$. We note that in the
absence of any molecular quadrupole field interactions ($a\to 0$ and $q\to 1$
limit), Eq. 2 reduces to a multicomponent Flory-Huggins (FH) mean-field theory
[18]. This FH free energy corresponds to the isotropic state of the
multicomponent system.

In this work, we assume the volume fractions in the isotropic and nematic
states are the same in order to model transitions in the filament brush for a
fixed membrane surface coverage, and we assume $a_{R}=a_{F}=a_{RF}=a$. With
these assumptions, we write the Helmholtz free energy of the nematic phase
relative to the isotropic phase for the alignment strength
$\hat{a}=a\phi_{p}LA$ to be

$\displaystyle\beta\Delta
f_{0}=\frac{\hat{a}}{3}\left(f_{R}m_{R}+f_{F}m_{F}\right)^{2}-f_{R}\ln{q_{R}}-f_{F}\ln{q_{F}},$
(4)

where $f_{\alpha}=\phi_{p}^{\alpha}/\phi_{p}$. The Maier-Saupe parameter
$\hat{a}$ should be understood as altering the surface coverage of filaments
on the membrane. The stability of the nematic phase is determined by finding
the curvature of $\beta\Delta f_{0}$ with respect to the overall variational
parameter, defined through the relation $\langle
m\rangle=f_{R}m_{R}+f_{F}m_{F}$. Finally, we note that the end-to-end distance
can be calculated from the tangent correlation functions, similar to Ref. [25,
21].

Fluctuations around the mean-field state [18] of the polymer brush layer
predict the Frank elastic energies of the system that contribute to the
effective membrane bending rigidity. Frank elastic energies provide energetic
contributions of the polymer layer with respect to the aligned state in terms
of the normal modes of deformation such as bend, twist, and splay [26, 27,
21]. Considering the Canham-Helfrich Hamiltonian [28, 29] using a Monge-Gauge
representation [30], we note the following relation in the presence of a
polymer brush layer (for details see, Supplemental Materials, Sec. V):

$\displaystyle\kappa_{\text{mem}}^{\text{eff}}=\kappa_{\text{mem}}+K_{\text{splay}}\sqrt{R_{\parallel}^{2}}\geq\kappa_{\text{mem}},$
(5)

where only the splay modulus $K_{\text{splay}}$ is seen to contribute to
altering the effective membrane bending modulus, and $R_{\parallel}^{2}$
denotes the mean-squared end-to-end distance of the polymer along the nematic
alignment direction. Previous work [31] determines the membrane elasticity to
be coupled to the lipid-bilayer splay modulus, which in spirit is similar to
what we obtain. However, our single layer membrane elasticity is coupled to
splaying of the outer protein-brush layer, and the nematic state of the brush
layer becomes crucial for determining the effective membrane rigidity. The
stretching modulus of the membrane remains unchanged, since modes of
deformation that are affected by the brush layer are associated with
directions that are predominantly perpendicular to the in-plane stretching
modes. For our case of two different polymers, the effective bending modulus
is determined to be the population average $\sim
K_{\text{splay}}\sqrt{R_{\parallel}^{2}}$. Details of a microscopic theory of
how fluctuations around the mean-field solution can be used to calculate
Frank-elastic constants for semiflexible polymers is given in Ref. [21].
Equation 5 suggests that when $K_{\text{splay}}\\!=\\!0$ in the absence of
nematic alignment, the effective membrane elasticity comes from internal
rigidity of the lipid bilayer. Hence, thermal (and perhaps active)
fluctuations present in the system result in larger height fluctuations of the
membrane [32] (_cf._ Fig. 1B). Under conditions of alignment of the polymer
molecules, $K_{\text{splay}}\\!>\\!0$, and effective membrane elasticity
increases. Physically, this corresponds to suppressed height fluctuations [33]
of the membrane due to the extended configurations within the polymer brush
layer (_cf._ Fig. 1A).

We show results for the two types of polymers with variable dimensionless
chain length $N_{\alpha}=L/2l_{p}^{\alpha}$, where the contour length and
persistence lengths are set by experimental measurements of EEA1 filaments. We
also set all Maier-Saupe parameters $a_{\alpha}$ to be the same and cross-
sectional area is considered to be the same for EEA1 protein in its extended
and flexible states. We note that for a fixed composition (i.e. fractions of
rigid and flexible polymers) transitioning to a liquid-crystalline phase, the
nematic order parameter for the two polymers exhibit a first-order phase
transition.

Figure 2: Critical strength of aligning field as a function of the fraction of
rigid polymer (lower X-axis) is plotted. The order parameter variation is
shown for three states corresponding to $f_{R}=0.04,0.50\text{ and }0.96$
respectively [blue line is $m_{R}$, orange is $m_{F}$ and the black dashed
line follows $\langle m\rangle$].

The main plot of Fig. 2 shows the field strength at the nematic transition
$\hat{a}_{c}$ as a function of fraction of rigid polymers $f_{R}$. The
transition field strength $\hat{a}_{c}$ systematically decreases with
increasing $f_{R}$, indicating the system transitions to a nematic state at
lower effective strength of alignment interactions. Phase diagrams for the
three states [$f_{R}\\!=0.04$ (A), $0.50$ (B), and $0.96$ (C)] showing the
nematic order parameters for the flexible and rigid polymers ($m_{F}$ and
$m_{R}$, respectively) are given in the inset plots of Fig. 2. The two colors
in the inset figures represent $m_{R}$ (rigid, blue) and $m_{F}$ (flexible,
orange). The population-weighted average nematic order parameter is shown in
black dashed line. The limit of stability of the nematic phase $\hat{a}_{s}$
is shown as the dots in the inset phase diagrams. Both the rigid and flexible
polymers exist in the isotropic state with $m_{\alpha}=0$ for $a\\!<\\!a_{s}$.
The polymer brush layer exhibits nematic alignment for $\\!a\\!>\\!a_{s}$,
resulting in non-zero nematic ordering for both rigid and flexible polymers.

Figure 3: Mean-squared end-to-end distance as a function of aligning field
strength for the three states in Fig. 2. Dotted line represents an isotropic
state and solid and dashed lines represent $\langle R_{\parallel}^{2}\rangle$
(along the aligning field) and $\langle R_{\perp}^{2}\rangle$
($\perp^{\text{r}}$ to the aligning field) respectively [blue
$\equiv\text{rigid}$ and orange $\equiv\text{flexible}$].

The isotropic state of the system exhibits orientational symmetry, such that
the mean-squared end-to-end distance has equivalent contributions from all
directions [25, 21]. Upon transitioning to the nematic state, the broken
symmetry of correlation lengths along parallel and perpendicular to the field
directions result in an ‘elongated’ polymer along the nematic-director
direction [21] (i.e. perpendicular to the membrane). We show the end-to-end
distance as a function of aligning field strength for three representative
states in Fig. 3. For $\hat{a}\\!<\\!\hat{a}_{c}$, we obtain a single end-to-
end distance in the two directions, and at $\hat{a}\\!=\\!\hat{a}_{c}$, there
is a discontinuous jump associated with elongation along the parallel
direction and retraction along the perpendicular direction. With increasing
field strength, the polymers become more elongated along the nematic director
direction, and perpendicular fluctuations monotonically decrease. The limit of
high strength of alignment corresponds to a rigid rod-like state, where the
end-to-end distance in the director direction approaches the contour length of
the polymer, _i.e._ $\langle R_{\parallel}^{2}\rangle\to L^{2}$.

Figure 4: The top plot (A) shows the change in free energy with respect to the
isotropic state is shown as a function of average order parameter variation
for the three states of Fig. 2 corresponding to rigid-polymer fractions of
$0.04$, $0.50$ and $0.96$ ($A\to B\to C$) for a fixed
$\hat{a}=LAa\phi_{p}=27$. The bottom plot (B) shows the ratio of effective
membrane rigidity as a function of aligning field strength for the three
states.

Within the mean-field approximation, we now consider free energies of the
system as given by the purely nematic energy density in Eq. 4. The order
parameter of the entire system can be given as $\langle m\rangle$, and
variational optimization of the free-energy density as a function of the order
parameter gives the optimal value of $\langle m\rangle$ for a particular
state. We plot the excess nematic free energy density in Fig. 4A for the three
defined states for a fixed $\hat{a}=27$. As we note from Fig. 2, state $A$ has
$\hat{a}_{c}>27$, meaning the system would be in isotropic state and
variational minimization of free energy predicts the system resides at
$\langle m\rangle_{0}\\!=\\!0$, as noted by the circle. For both state B and
C, Fig. 2 states $\hat{a}$ should produce a nematic state ($\langle
m\rangle_{0}\\!>\\!0$), which is shown in Fig. 4. We note that as the fraction
of rigid polymers increase from $A\to B\to C$, $\langle m\rangle_{0}$
monotonically grows from the isotropic to the nematic state towards $\langle
m\rangle_{0}\\!\to\\!1$ indicating perfectly aligned polymer configurations.
As the fraction of rigid polymers increase, the nematic free energy of the
system decreases indicating stabilizing forces in the system. With the same
surface coverage of EEA1 brush-like polymers on a membrane patch, we
anticipate a decreasing free energy as the brushes exist in the rigid state.

We assign values of the EEA1 contour length $L=222$ nm, cross-sectional area
$A=3.1$ nm2 [13, 34], and EEA1 volume fraction $\phi_{p}=0.01$ [12, 35]. The
bare membrane bending elasticity is set to $\kappa_{\text{mem}}=5k_{B}T$,
which is within the measured physiological range [36]. These parameters permit
the evaluation of the effective membrane rigidity for the three rigid-polymer
fractions shown in Fig. 4A. Figure 4B gives the ratio of the effective
membrane rigidity to the bare-membrane rigidity
$\kappa_{\text{mem}}^{\text{eff}}/\kappa_{\text{mem}}$ as a function of
aligning field strength. For $\hat{a}\\!<\\!\hat{a}_{c}$ in each state, the
effective membrane elasticity
$\kappa_{\text{mem}}^{\text{eff}}\\!=\\!\kappa_{\text{mem}}$, since the splay
modulus $K_{\text{splay}}$ of the filament brush is zero in the isotropic
state. At the nematic transition $\hat{a}\\!=\\!\hat{a}_{c}$, the effective
membrane elasticity undergoes a first-order transition associated with the
sudden increase in the nematic ordering. Further increase in the aligning
field strength leads to a greater than 20-fold increase in the effective
elasticity of the membrane and the EEA1 brush, leading to an overall picture
where filament alignment dramatically modulates the membrane elasticity.

Notably, this dramatic change in the effective membrane elasticity is enabled
by the collective nematic transition. The persistence lengths of the rigid and
flexible states have a ratio of
$l_{p}^{\text{EEA1}}/l_{p}^{\text{EEA1(GTP)}}\\!\approx\\!3.3$, indicating a
limiting ability for a single filament to physically manipulate exterior
cargo. However, the collective elastic behavior of the EEA1 brush would
facilitate large-scale property changes that facilitate large deformations of
the membrane necessary for membrane fusion.

In conclusion, we provide a theoretical model for collective transitions in
the EEA1 brush on an endosomal membrane, and our analyses reveal how large-
scale membrane fluctuations that are essential for membrane fusion are
controlled by subtle physical and structural changes of the molecular
constituents. Our work suggests a possible mechanism of membrane fusion that
relies on membrane fluidity (or reduced mechanical rigidity), which is
modulated by the brush-layer protein filaments. Extended EEA1 molecules that
behave as effective rod-like polymers, increases the effective elastic
rigidity of membrane through their collective alignment within the protein
brush. Conformational changes of EEA1 molecules (from a rigid to a flexible,
floppy state) help physical proximity due to less steric hindrance and reduces
the mechanical rigidity, allowing fusion to take place.

Our theoretical model and subsequent analyses addresses the role the coil-
coiled dimeric structure of EEA1 protein in the mechanics of the rigid-
flexible transition, kinetics of Rab5-GTP binding, ATP$\to$ADP hydrolysis and
Rab5-GDP unbinding, and the molecular mechanisms that are responsible for
rigid-to-flexible conformational change. Our results have broad impact beyond
the membrane-fusion problem. The model demonstrates how biophysical processes
that happen on large, collective length and time scales can originate from
orchestrated chemical and physical events at much finer scales, which is a
general principle that underlies a range of critical life processes.

_Acknowledgements._ A.G. acknowledges funding support from the Human Frontier
Science Program, Grant No. HFSP/REF RGP0019/20. A.J.S. was supported by the
NSF program Condensed Matter and Materials Theory, Grant No. 1855334.

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Supplemental Information
to
Local Changes in Protein Filament Properties Drive Large-Scale Membrane
Transformations Involved in Endosome Tethering and Fusion

###### Contents

  1. I Detailed Derivation of the Polymer Field Theory of a three component polymer nematic solution
    1. I.1 System Definition
    2. I.2 Particle to Field Transformation
  2. II Mean Field Approximation
    1. II.1 Mean Field Equations
    2. II.2 Homogeneous Mean-field Approximation and System Energy
  3. III Fluctuations around mean field solution
  4. IV Chemical potential and system Phase Behavior
  5. V Effective Membrane Rigidity
    1. V.1 Frank Elastic Energy of Polymer Brush Layer
    2. V.2 Helfrich Free Energy of Membrane and Renormalized Bending Rigidity
  6. VI Supplementary Figures

##  I Detailed Derivation of the Polymer Field Theory of a three component
polymer nematic solution

###  I.1 System Definition

We consider an incompressible polymer solution of total volume $V$, with
$n_{s}$ solvent molecule with volume $v_{s}$, $n_{p}^{A}$ polymer chains with
contour length $L_{A}$ and cross-sectional area $A_{A}$ and $n_{p}^{B}$
polymer chains with contour length $L_{B}$ and cross-sectional area $A_{B}$.
The polymer chains are modeled using the wormlike chain model, which describes
the polymer chains as inextensible elastic threads that are subjected to
thermal fluctuations. The polymer configuration of the $i_{\alpha}$th polymer
of type $\alpha\in(A,B)$ at arclength position $s$ (note, $s=0$ at one end and
$s=L_{\alpha}$ at the opposite end) is defined by the space curve
$\vec{r}_{i}^{\alpha}(s)$, and the local tangent vector
$\vec{u}_{i}^{\alpha}(s)=\partial\vec{r}_{i}^{\alpha}(s)/\partial s$ gives the
orientation of the monomer segment at $s$. Inextensibility is strictly
enforced by ensuring that $|\vec{u}_{i}^{\alpha}(s)|=1$ for all configurations
of the system. The position of the $j$th solvent molecule is given by
$\vec{r}_{j}$. The bending rigidity of the semiflexible polymers are given by
their respective persistence lengths $l_{p}^{\alpha}$, and we also express the
chain length in dimensionless units as the number of Kuhn lengths
$N_{\alpha}=L_{\alpha}/(2l_{p}^{\alpha})$, where the Kuhn length
$b_{\alpha}=2l_{p}^{\alpha}$ gives the statistical segment length of a
polymer.

The system energy includes contributions for polymer deformation, solvent-
polymer mixing, and nematic alignment of polymer chains and is given by the
total energy,

$\displaystyle\beta
E=\sum_{\alpha\in\\{A,B\\}}\left\\{\sum_{i=1}^{n_{p}^{\alpha}}\frac{l_{p}^{\alpha}}{2}\int_{0}^{L_{\alpha}}ds\left(\frac{\partial\vec{u}_{i}^{\alpha}(s)}{\partial
s}\right)^{2}+\chi_{\alpha}\int
d\vec{r}\,\hat{\phi}_{s}(\vec{r})\hat{\phi}_{p}^{\alpha}(\vec{r})-\frac{a_{\alpha}}{2}\int
d\vec{r}\,\hat{\mathbf{S}}_{\alpha}(\vec{r}):\hat{\mathbf{S}}_{\alpha}(\vec{r})\right\\}$
$\displaystyle+\chi_{AB}\int
d\vec{r}\,\hat{\phi}_{p}^{A}(\vec{r})\hat{\phi}_{p}^{B}(\vec{r})-a_{AB}\int
d\vec{r}\,\hat{\mathbf{S}}_{A}(\vec{r}):\hat{\mathbf{S}}_{B}(\vec{r})$ (S1)

where, $\hat{\phi}_{s}$ and $\hat{\phi}_{p}^{\alpha}$ are respectively the
local dimensionless density (volume fraction) of the solvent and polymer
molecules of type $\alpha$, and $\hat{\mathbf{S}}_{\alpha}$ is the tensorial
nematic order parameter density of state $\alpha$. These local order
parameters are given by,

$\displaystyle\hat{\phi}_{s}(\vec{r})=v_{s}\sum_{j=1}^{n_{s}}\delta(\vec{r}-\vec{r}_{j})$
(S2)
$\displaystyle\hat{\phi}_{p}^{\alpha}(\vec{r})=A_{\alpha}\sum_{i=1}^{n_{p}^{\alpha}}\int_{0}^{L_{\alpha}}ds\delta(\vec{r}-\vec{r}_{i}^{\alpha}(s))$
(S3)
$\displaystyle\hat{S}_{\alpha}(\vec{r})=A_{\alpha}\sum_{i=1}^{n_{p}^{\alpha}}\int_{0}^{L_{\alpha}}ds\delta(\vec{r}-\vec{r}_{i}^{\alpha}(s))\left(\vec{u}_{i}^{\alpha}(s)\vec{u}_{i}^{\alpha}(s)-\frac{\mathbf{I}}{3}\right)$
(S4)

For the subsequent analyses, we define a generalized volume fraction (tensor)
that incorporates local orientational order, defined as [21],

$\hat{\phi}_{l_{\alpha}}^{m_{\alpha}}(\vec{r}\,)=A_{\alpha}\sqrt{\frac{4\pi}{2l_{\alpha}+1}}\sum_{i=1}^{n_{p}^{\alpha}}\int_{0}^{L_{\alpha}}\\!\\!\\!ds\mathcal{Y}_{l_{\alpha}}^{m_{\alpha}}(\vec{u}_{i}^{\alpha}(s))\delta(\vec{r}-\vec{r}_{i}^{\alpha}(s)).$
(S5)

$\mathcal{Y}_{l_{\alpha}}^{m_{\alpha}}$ is the real-valued spherical harmonic
(i.e. the tesseral spherical harmonic) for state $\alpha$, given by,

$\mathcal{Y}_{l_{\alpha}}^{m_{\alpha}}=\begin{cases}\sqrt{2}(-1)^{m_{\alpha}}\mathrm{Im}\left[Y_{l_{\alpha}}^{|m_{\alpha}|}\right]&\mathrm{for}\,\,\,m_{\alpha}<0\\\
Y_{l_{\alpha}}^{0}&\mathrm{for}\,\,\,m_{\alpha}=0\\\
\sqrt{2}(-1)^{m_{\alpha}}\mathrm{Re}\left[Y_{l_{\alpha}}^{m_{\alpha}}\right]&\mathrm{for}\,\,\,m_{\alpha}>0\end{cases}$
(S6)

where $Y_{l_{\alpha}}^{m_{\alpha}}$ is the standard spherical harmonic
(complex valued). We note that the polymer volume fraction
$\hat{\phi}_{p}^{\alpha}=\hat{\phi}_{0}^{0_{\alpha}}$, and the nematic order
parameter $\hat{\mathbf{S}}_{\alpha}$ can be written in terms of
$\hat{\phi}_{2}^{m_{\alpha}}$. With this definition, we rewrite the system
energy as

$\displaystyle\beta E$ $\displaystyle=$
$\displaystyle\sum_{\alpha\in(A,B)}\left\\{\sum_{i=1}^{n_{p}^{\alpha}}\frac{l_{p}^{\alpha}}{2}\int_{0}^{L_{\alpha}}\left(\frac{\partial\vec{u}_{i}^{\alpha}}{\partial
s}\right)^{\\!\\!2}ds+\chi_{\alpha}\int
d\vec{r}\,\hat{\phi}_{s}(\vec{r}\,)\hat{\phi}_{0}^{0_{\alpha}}(\vec{r}\,)-\frac{a_{\alpha}}{3}\int
d\vec{r}\,\sum_{m_{\alpha}=-2}^{2}\left[\hat{\phi}_{2}^{m_{\alpha}}(\vec{r}\,)\right]^{2}\right\\}$
(S7) $\displaystyle+\chi_{AB}\int
d\vec{r}\,\hat{\phi}_{0}^{0_{A}}(\vec{r}\,)\hat{\phi}_{0}^{0_{B}}(\vec{r}\,)-\frac{2a_{AB}}{3}\int
d\vec{r}\,\sum_{m_{A}=-2}^{2}\sum_{m_{B}=-2}^{2}\hat{\phi}_{2}^{m_{A}}(\vec{r}\,)\hat{\phi}_{2}^{m_{B}}(\vec{r}\,)\delta_{m_{A}m_{B}}$

where the integrand in the last term in Eq. S7 signifies summation over terms
such as,
$\hat{\phi}_{2}^{-2_{A}}(\vec{r}\,)\hat{\phi}_{2}^{-2_{B}}(\vec{r}\,),\,\hat{\phi}_{2}^{-1_{A}}(\vec{r}\,)\hat{\phi}_{2}^{-1_{B}}(\vec{r}\,),\dots$
and so on.

The canonical partition function, $\mathcal{Z}$ is written as,

$\displaystyle\mathcal{Z}=\frac{1}{n_{s}!n_{p}^{A}!n_{p}^{B}!}\frac{1}{v_{s}^{n_{s}}}\frac{1}{(L_{A}A_{A})^{n_{p}^{A}}(L_{B}A_{B})^{n_{p}^{B}}}\int\prod_{j=1}^{n_{s}}d\vec{r}_{j}\int\prod_{i_{A}=1}^{n_{p}^{A}}\mathcal{D}[\vec{r}_{i_{A}}]\int\prod_{i_{B}=1}^{n_{p}^{B}}\mathcal{D}[\vec{r}_{i_{B}}]\prod_{\vec{r}}\delta(\hat{\phi}_{s}+\hat{\phi}_{0}^{0_{A}}+\hat{\phi}_{0}^{0_{B}}-1)$
$\displaystyle\times\prod_{i_{A}=1}^{n_{p}^{A}}\prod_{i_{B}=1}^{n_{p}^{B}}\prod_{s_{A},s_{B}}\delta(|\partial_{s_{A}}\vec{r}_{i_{A}}|-1)\delta(|\partial_{s_{B}}\vec{r}_{i_{B}}|-1)\exp{(-\beta
E)}$ (S8)

where
$\displaystyle\prod_{\vec{r}}\delta(\hat{\phi}_{s}+\hat{\phi}_{0}^{0_{A}}+\hat{\phi}_{0}^{0_{B}}-1)$
accounts for the incompressibility constraint
$\hat{\phi}_{s}+\hat{\phi}_{0}^{0_{A}}+\hat{\phi}_{0}^{0_{B}}=1$ at all
locations within the system. The inextensibility of each polymer chain is
enforced by the constraint. The integration over
$\mathcal{D}[\vec{r}_{i\alpha}]$ implies path integration over all
conformations of the polymer $\alpha$. We use the solvent volume $v_{s}$ and
polymer volume $L_{\alpha}A_{\alpha}$ as volume scales instead of the de-
Broglie wavelengths cubed, which merely shifts the chemical potential by a
constant quantity and does not affect the thermodynamic behavior of our
system.

###  I.2 Particle to Field Transformation

Next, we perform a series of field manipulations that enable systematic
approximation of the thermodynamic behavior. For each instantaneous volume
fraction $\hat{\phi}$, we introduce a volume-fraction field variable by noting
the property[spakowitz2004exact, 21]

$\displaystyle f[\hat{\phi}]$ $\displaystyle=$
$\displaystyle\int\mathcal{D}\phi\prod_{\vec{r}}\delta\\!\left[\phi-\hat{\phi}\right]f[\phi]$
(S9) $\displaystyle=$
$\displaystyle\int\mathcal{D}W\mathcal{D}\phi\exp\left\\{i\int
d\vec{r}\,W(\vec{r}\,)\left[\hat{\phi}(\vec{r}\,)-\phi(\vec{r}\,)\right]\right\\}f[\phi]$

where $W$ emerges from a Fourier representation of the spatial delta function.
This field manipulation is applied to each volume-fraction ($\hat{\phi}_{s}$,
$\hat{\phi}_{0}^{0_{\alpha}}$, and $\hat{\phi}_{2}^{m_{\alpha}}$), resulting
in the introduction of the conjugate fields $W_{s}$, $W_{0}^{0_{\alpha}}$, and
$W_{2}^{m_{\alpha}}$. After these field manipulations, we can write the
canonical partition function as,

$\displaystyle\mathcal{Z}=\int\mathcal{D}W_{s}\prod_{\alpha\in(A,B)}\left\\{\mathcal{D}\phi_{0}^{0_{\alpha}}\mathcal{D}W_{0}^{0_{\alpha}}\prod_{m_{\alpha}=-2}^{2}\mathcal{D}W_{2}^{m_{\alpha}}\mathcal{D}\phi_{2}^{m_{\alpha}}\right\\}\exp\Bigg{\\{}i\int
d\vec{r}\,W_{s}(1-\phi_{0}^{0_{A}}-\phi_{0}^{0_{B}})+i\int
d\vec{r}\sum_{\alpha\in(A,B)}W_{0}^{0_{\alpha}}\phi_{0}^{0_{\alpha}}$
$\displaystyle+i\int
d\vec{r}\sum_{\alpha\in(A,B)}\sum_{m_{\alpha}=-2}^{2}W_{2}^{m_{\alpha}}\phi_{2}^{m_{\alpha}}-\sum_{\alpha\in(A,B)}\chi_{\alpha}\int
d\vec{r}\,\phi_{0}^{0_{\alpha}}(1-\phi_{0}^{0_{A}}-\phi_{0}^{0_{B}})+\sum_{\alpha\in(A,B)}\frac{a_{\alpha}}{3}\int
d\vec{r}\,\sum_{m_{\alpha}=-2}^{2}\left(\phi_{2}^{m_{\alpha}}\right)^{2}$
$\displaystyle-\chi_{AB}\int
d\vec{r}\,\phi_{0}^{0_{A}}\phi_{0}^{0_{B}}+\frac{2a_{AB}}{3}\int
d\vec{r}\,\sum_{m=-2}^{2}\phi_{2}^{m_{A}}\phi_{2}^{m_{B}}+{n_{s}}\ln{\left(\frac{z_{s}[W_{s}]}{n_{s}v_{s}}\right)}+\sum_{\alpha\in(A,B)}n_{p}^{\alpha}\ln{\left(\frac{z_{p}^{\alpha}[W_{0}^{\alpha},W_{2}^{m_{\alpha}}]}{n_{p}^{\alpha}L_{\alpha}A_{\alpha}}\right)}\Bigg{\\}}$
(S10)
$\displaystyle=\int\mathcal{D}W_{s}\prod_{\alpha\in(A,B)}\left\\{\mathcal{D}\phi_{0}^{0_{\alpha}}\mathcal{D}W_{0}^{0_{\alpha}}\prod_{m_{\alpha}=-2}^{2}\mathcal{D}W_{2}^{m_{\alpha}}\mathcal{D}\phi_{2}^{m_{\alpha}}\right\\}\exp\left[-\beta\mathcal{F}\right]$

where, we define, the single solvent partition function as,

$z_{s}[W_{s}]=\int d\vec{r}\,\exp\left[-iv_{s}W_{s}(\vec{r}\,)\right]$ (S11)

and the single polymer partition functions are written as,

$\displaystyle
z_{p}^{\alpha}[W_{0}^{0_{\alpha}},W_{2}^{m_{\alpha}}]=\int\mathcal{D}[\vec{r}\,(s)]\exp\left\\{-\frac{l_{p}^{\alpha}}{2}\int_{0}^{L_{\alpha}}\left(\frac{\partial\vec{u}_{\alpha}}{\partial
s}\right)^{\\!\\!2}ds-
iA_{\alpha}\int_{0}^{L_{\alpha}}\\!\\!\\!dsW_{0}^{0_{\alpha}}(\vec{r}\,(s))\right.$
$\displaystyle\hskip
144.54pt\left.-iA_{\alpha}\int_{0}^{L_{\alpha}}\\!\\!\\!ds\sum_{m_{\alpha}=-2}^{2}W_{2}^{m_{\alpha}}(\vec{r}\,(s))\sqrt{\frac{4\pi}{5}}\mathcal{Y}_{l_{\alpha}}^{m_{\alpha}}(\vec{u_{\alpha}}(s))\right\\}$
(S12)

The partition function $\mathcal{Z}$ has not been subjected to any
approximations so far and is not exactly solvable as it is, although it is
amenable to systematic approximations. Below we formulate the general saddle-
point approximation, which determines the thermodynamic state within a mean-
field (generally inhomogeneous) and specialize to the homogeneous mean-field
approximation.

##  II Mean Field Approximation

###  II.1 Mean Field Equations

The lowest-order approximation to the canonical partition function is given by
the saddle point, which approximates the functional integrals in Eq. S10 by
the maximum term. By setting the first variation of the argument of the
exponential within Eq. S10 to zero (i.e.
$\displaystyle\frac{\delta(\beta\mathcal{F})}{\delta f}=0$ for any arbitrary
field $f$), we arrive at the saddle-point equations. Saddle point equations
are given as (we use $\bar{\phi}_{0}^{0_{\alpha}}=\phi_{p}^{\alpha}$, and
$iW=w$),

$(1-\phi_{p}^{A}-\phi_{p}^{B})=\frac{v_{s}n_{s}}{\bar{z}_{s}}\exp{(-v_{s}\bar{w}_{s})},$
(S13)
$-\chi_{A}(1-2\phi_{p}^{A}-\phi_{p}^{B})+\chi_{B}\phi_{p}^{B}+\bar{w}_{0}^{0_{A}}-\bar{w}_{s}-\chi_{AB}\phi_{p}^{B}=0,$
(S14)
$-\chi_{B}(1-\phi_{p}^{A}-2\phi_{p}^{B})+\chi_{A}\phi_{p}^{A}+\bar{w}_{0}^{0_{B}}-\bar{w}_{s}-\chi_{AB}\phi_{p}^{A}=0,$
(S15)
$\frac{2a_{A}}{3}\bar{\phi}_{2}^{m_{A}}+\frac{2a_{AB}}{3}\bar{\phi}_{2}^{0_{B}}+\bar{w}_{2}^{m_{A}}=0$
(S16)
$\frac{2a_{B}}{3}\bar{\phi}_{2}^{m_{B}}+\frac{2a_{AB}}{3}\bar{\phi}_{2}^{m_{A}}+\bar{w}_{2}^{m_{B}}=0$
(S17)
$\bar{\phi}_{p}^{A}+\frac{n_{p}^{A}}{\bar{z}_{p}^{A}}\frac{\delta\bar{z}_{p}^{A}}{\delta\bar{w}_{0}^{0_{A}}}=0$
(S18)
$\bar{\phi}_{p}^{B}+\frac{n_{p}^{B}}{\bar{z}_{p}^{B}}\frac{\delta\bar{z}_{p}^{B}}{\delta\bar{w}_{0}^{0_{B}}}=0$
(S19)
$\bar{\phi}_{2}^{m_{A}}+\frac{n_{p}^{A}}{\bar{z}_{p}^{A}}\frac{\delta\bar{z}_{p}^{A}}{\delta\bar{w}_{2}^{m_{A}}}=0$
(S20)
$\bar{\phi}_{2}^{m_{B}}+\frac{n_{p}^{B}}{\bar{z}_{p}^{B}}\frac{\delta\bar{z}_{p}^{B}}{\delta\bar{w}_{2}^{m_{B}}}=0$
(S21)

Next, we specialize our treatment of the saddle point to the homogeneous
uniaxial nematic state with nematic alignment along $\hat{\delta}_{z}$. This
gives $\bar{\phi}_{2}^{m_{\alpha}}=0$ for $m_{\alpha}\neq 0$ and
$\alpha\in(A,B)$.

This leads to the self-consistent mean-field equation, (for
$\alpha\in\\{A,B\\}$)

$\bar{\phi}_{2}^{0_{\alpha}}=\frac{\phi_{p}^{\alpha}}{L_{\alpha}}\int_{0}^{L_{\alpha}}ds\left\langle\left(\frac{3}{2}[u_{z}^{\alpha}(s)]^{2}-\frac{1}{2}\right)\right\rangle_{0_{\alpha}}=\phi_{p}^{\alpha}m_{N}^{\alpha}$
(S22)

where the average $\langle\dots\rangle_{0_{\alpha}}$ is taken with respect to
the single chain mean-field energy,

$\beta\mathcal{E}_{0}^{\alpha}=\frac{l_{p}^{\alpha}}{2}\int_{0}^{L_{\alpha}}ds\left(\frac{\partial\vec{u}^{\alpha}(s)}{\partial
s}\right)^{2}-(a_{\alpha}\phi_{p}^{\alpha}m_{N}^{\alpha}A_{\alpha}+a_{AB}\phi_{p}^{\beta}m_{N}^{\beta}A_{\beta})\int_{0}^{L_{\alpha}}ds\left\\{[u_{z}^{\alpha}(s)]^{2}-\frac{1}{3}\right\\}$
(S23)

It is understood that $0\leq m_{N}^{\alpha}\leq 1,\forall\alpha\in\\{A,B\\}$.
The overall nematic order of the system can be thought of as an weighted
average as, $\displaystyle
m_{\text{total}}=\frac{1}{\phi_{p}}\sum_{\alpha}\phi_{p}^{\alpha}m_{N}^{\alpha}$.
For a perfectly aligned state ($m_{N}^{\alpha}=1,\forall\alpha$) we get,
$\displaystyle\bar{\phi}_{2}^{0}=\sum_{\alpha}\bar{\phi}_{2}^{0_{\alpha}}=\sum_{\alpha}\phi_{p}^{\alpha}=\phi_{p}$.

###  II.2 Homogeneous Mean-field Approximation and System Energy

Under homogeneous mean-field approximation the zeroth order energy becomes
(per unit volume $V$, so this is technically an energy density),

$\displaystyle\beta
f_{0}^{\text{iso.nem.}}=\frac{1-\phi_{p}}{v_{s}}\ln{(1-\phi_{p})}+\sum_{\alpha}\frac{\phi_{p}^{\alpha}}{L_{\alpha}A_{\alpha}}\ln{(\phi_{p}^{\alpha})}+\sum_{\alpha}\chi_{\alpha}\phi_{p}^{\alpha}(1-\phi_{p})-\sum_{\alpha}\frac{a_{\alpha}}{3}(\bar{\phi}_{2}^{0_{\alpha}})^{2}+\sum_{\alpha}\frac{\phi_{p}^{\alpha}}{L_{\alpha}A_{\alpha}}\ln{(q_{\alpha})}$
$\displaystyle+\chi_{AB}\phi_{p}^{A}\phi_{p}^{B}-\frac{2a_{AB}}{3}\bar{\phi}_{2}^{0_{A}}\bar{\phi}_{2}^{0_{B}}$
(S24)

where $f_{0}=\mathcal{F}_{0}/V$ and an intensive property of the system. Here
$q_{\alpha}$ is the partition function given as,

$\begin{split}q_{\alpha}=\int d\vec{u}\int
d\vec{u}_{0}\int_{\vec{u}_{0}}^{\vec{u}}\mathcal{D}[\vec{u}(s)]\exp{\left(-\beta\mathcal{E}_{0}^{\alpha}[\vec{u}(s)]\right)}\end{split}$
(S25)

To note,

$\displaystyle\beta
f_{0}^{\text{iso.}}=\frac{1-\phi_{p}}{v_{s}}\ln{(1-\phi_{p})}+\sum_{\alpha}\frac{\phi_{p}^{\alpha}}{L_{\alpha}A_{\alpha}}\ln{(\phi_{p}^{\alpha})}+\sum_{\alpha}\chi_{\alpha}\phi_{p}^{\alpha}(1-\phi_{p})+\chi_{AB}\phi_{p}^{A}\phi_{p}^{B}$
(S26)

Here, it should be noted that in absence of any aligning field (_i.e._
$\gamma_{\alpha}=(A_{\alpha}a_{\alpha}m_{N}^{\alpha}\phi_{p}^{\alpha}+A_{\beta}a_{AB}m_{N}^{\beta}\phi_{p}^{\beta})$),
the orientational partition is unity. Below we simplify to the notation
$m_{N}^{\alpha}\equiv m_{\alpha}$.

The residual Helmholtz free energy (density) of the nematic phase relative to
the isotropic phase is given by,

$\beta f_{\text{residual}}=\beta f_{0}^{\text{iso.}}-\beta
f_{0}^{\text{iso.nem.}}=\sum_{\alpha}\left(\frac{a_{\alpha}}{3}(\phi_{p}^{\alpha}m_{\alpha})^{2}-\frac{\phi_{p}^{\alpha}}{L_{\alpha}A_{\alpha}}\ln{\left\\{q_{\alpha}(m_{\alpha})\right\\}}\right)+\frac{2a_{\alpha\beta}}{3}\phi_{p}^{\alpha}m_{\alpha}\phi_{p}^{\beta}m_{\beta}$
(S27)

For a constant $a_{\alpha}$, constant $L_{\alpha}=L$ and $A_{\alpha}=A$, we
get the free energy (by multiplying with the relevant volume $LA/\phi_{p}$) as
written in the main manuscript,

$\beta\Delta
f_{0}=\frac{\hat{a}}{3}\left(f_{R}m_{R}+f_{F}m_{F}\right)^{2}-f_{R}\ln{q_{R}}-f_{F}\ln{q_{F}}$
(S28)

where, $\hat{a}=a\phi_{p}LA$ and $f_{\alpha}=\phi_{p}^{\alpha}/\phi_{p}$. In
the limit of $f_{R}\to 1$ (hence, $f_{F}\to 0$) the above equation reduces to
(superscript ‘1’ represents single type of polymer component),

$\beta\Delta f_{0}^{(1)}=\frac{\hat{a}}{3}m^{2}-\ln{q}$ (S29)

that matches with ref. [spakowitz2004exact].

##  III Fluctuations around mean field solution

Now we specialize to fluctuation effects upto quartic order with respect to
the homogeneous mean field basis. The vector of auxiliary variables are
defined as,
$\mathcal{W}\equiv[W_{s},\\{\phi_{0}^{0_{\alpha}}\\},\\{W_{0}^{0_{\alpha}}\\},\\{\phi_{2}^{m_{\alpha}}\\},\\{W_{2}^{m_{\alpha}}\\}]$
(total of $25$ terms) and the vector containing all the fluctuations are
defined using, $\Delta=[\\{\delta\phi_{0}^{0_{\alpha}}\\},\\{\delta
W_{0}^{0_{\alpha}}\\},\\{\delta\phi_{2}^{m_{\alpha}}\\},\\{\delta
W_{2}^{m_{\alpha}}\\}]$ (total of $24$ terms due to the incompressibility
constraint). Performing Gaussian integral over the fluctuating conjugate
fields (not volume fraction fluctuation fields), we arrive at,

$-\beta\mathcal{F}=-\beta\mathcal{F}_{0}-\frac{1}{2}\tilde{\Gamma}_{\tilde{1}\tilde{2}}^{(2,\phi)}\Delta_{\tilde{1}}\Delta_{\tilde{2}}+\sum_{n=3}^{\infty}\frac{1}{n!}\tilde{\Gamma}_{\tilde{1}\dots\tilde{n}}^{(n,\phi)}\Delta_{\tilde{1}}\dots\Delta_{\tilde{n}}$
(S30)

where the above expansion is formally exact and written using modified
Einstein notation to sum over fluctuating volume fraction fields of all
constituents and integration over the respective Fourier variables. The
quadratic order term can be explicitly written as,

$\tilde{\Gamma}_{\tilde{1}\tilde{2}}^{(2,\phi)}\Delta_{\tilde{1}}\Delta_{\tilde{2}}=\frac{1}{(2\pi)^{6}}\sum_{\gamma_{1},\gamma_{2}}\int
d\vec{k}_{1}d\vec{k}_{2}\tilde{\Gamma}_{\gamma_{1}\gamma_{2}}^{(2,\phi)}(\vec{k}_{1},\vec{k}_{2})\Delta_{\gamma_{1}}(\vec{k}_{1})\Delta_{\gamma_{2}}(\vec{k}_{2})$
(S31)

where, $\gamma$ runs over the three species present in the system, namely
solvent, polymer $A$ and polymer $B$. In order to write the full matrix
$\mathbf{\Gamma}^{(2,\phi)}$ in terms of its elements, we write the structure
factors for polymer species $\alpha$ as,

$\displaystyle S_{(l_{1},m_{1}),(l_{2},m_{2})}^{(\alpha)}(\vec{k})$
$\displaystyle=$
$\displaystyle\frac{1}{L_{\alpha}^{2}}\left(\sqrt{\frac{4\pi}{(2l_{1}+1)(2l_{2}+1)}}\right)^{m_{1}+m_{2}}$
(S32)
$\displaystyle\times\int_{0}^{L_{\alpha}}ds_{1}\int_{0}^{L_{\alpha}}ds_{2}\left\langle\mathcal{Y}_{l_{1}}^{m_{1}}(\vec{u}(s_{1}))\exp{\left[i\vec{k}\cdot(\vec{r}(s_{1})-\vec{r}(s_{2})\right]}\mathcal{Y}_{l_{2}}^{m_{2}}(\vec{u}(s_{2}))\right\rangle$

The entire $\mathbf{\Gamma}^{(2,\phi)}$ matrix is a $12\times 12$ matrix
containing all the volume fraction fluctuations. The matrix is sparse
containing only certain non-zero elements ($\beta\neq\alpha$ in equations
below and both indicates $A$ and $B$):

$\Gamma_{(0,0),(0,0)}^{(\alpha)}(\vec{k})=\frac{1}{v_{s}(1-\phi_{p})}-2\chi_{\alpha}+\frac{1}{A_{\alpha}L_{\alpha}\phi_{p}^{\alpha}}[S_{(0,0),(0,0)}^{(\alpha)}]^{-1}(\vec{k})$
(S33)
$\Gamma_{(0,0),(2,m)}^{(\alpha)}(\vec{k})=\frac{1}{A_{\alpha}L_{\alpha}\phi_{p}^{\alpha}}[S_{(0,0),(2,m)}^{(\alpha)}]^{-1}(\vec{k})$
(S34)
$\Gamma_{(2,m_{1}),(2,m_{2})}^{(\alpha)}(\vec{k})=-\frac{2a_{\alpha}}{3}\delta_{m_{1}m_{2}}+\frac{1}{A_{\alpha}L_{\alpha}\phi_{p}^{\alpha}}[S_{(2,m_{1}),(2,m_{2})}^{(\alpha)}]^{-1}(\vec{k})$
(S35) $\Gamma_{(0,0),(0,0)}^{(\alpha\beta)}(\vec{k})=-2\chi_{\alpha\beta}$
(S36)
$\Gamma_{(2,m_{1}),(2,m_{2})}^{(\alpha\beta)}(\vec{k})=-\frac{2a_{\alpha\beta}}{3}\delta_{m_{1}m_{2}}$
(S37)

To note, there is cross correlation of structure since $A$ and $B$ polymers
are separate from one another (_i.e_ not a copolymer like structure). The
formulation above gives the exact free energy up-to quadratic order.

Next steps involve, (1) writing the quadratic order free energy within the RPA
approximation in terms of the eigenvalues of $\mathbf{\Gamma}^{(2,\phi)}$
matrix, (2) defining a non-interacting reference free energy, (3) subtract
this reference free energy and (4) using the property,
$\displaystyle\prod_{\eta\in\\{\text{fields}\\}}\gamma_{\eta}=\text{det}[\mathbf{\Gamma}]$
(where, $\gamma$ are the eigenvalues of the determinant and the number of
eigenvalues are same as the number of fluctuating volume fraction fields) to
arrive at,

$\beta\mathcal{F}=\beta\mathcal{F}_{0}+\frac{V}{4\pi^{2}}\int_{0}^{\Lambda}dk\
k^{2}\ln{\left(\frac{\text{det}[\mathbf{\Gamma}^{(2,\phi)}]}{\text{det}[\mathbf{\Gamma}_{0}^{(2,\phi)}]}\right)}$
(S38)

In the above equation, we have introduced a $\text{high-}k$ cutoff,
$\displaystyle\Lambda=\min_{\\{l_{p}^{\alpha}\\}}\frac{2\pi}{l_{p}^{\alpha}}$
to account for the ultraviolet divergence in field theory.

##  IV Chemical potential and system Phase Behavior

We only care about the phase behavior at the mean-field level. Due to enforced
incompressibility of the system the Gibbs and Helmholtz free energy of the
system are equivalent to one other. Here, we define the chemical potential of
$\eta^{\text{th}}$ species as,

$\beta\mu_{\eta}=\frac{\partial(\beta\mathcal{F})}{\partial n_{\eta}}=\beta
v_{\eta}f+V\frac{\partial(\beta f)}{\partial n_{\eta}}$ (S39)

where the second equality follows from defining the free energy density, as
$f=\mathcal{F}/V$. The expression for the zeroth order chemical potential of
species $\theta$ follows from the expression of mean field free energy of the
system. We write the mean field energy in a consolidated notation as,

$\displaystyle\beta
f_{0}=\sum_{k}\frac{\phi_{k}}{v_{k}}\ln{\left(\frac{\phi_{k}}{q_{k}}\right)}+\frac{1}{2}\sum_{j,k}\chi_{jk}\phi_{j}\phi_{k}-\frac{1}{3}\sum_{j,k}a_{jk}m_{j}m_{k}\phi_{j}\phi_{k}$
(S40)
$\displaystyle=\sum_{k}\frac{\phi_{k}}{v_{k}}\ln{\left(\frac{\phi_{k}}{q_{k}}\right)}+\frac{1}{2}\sum_{j,k}\Xi_{jk}\phi_{j}\phi_{k}$
(S41)

where $\displaystyle\Xi_{jk}=\chi_{jk}-\frac{2}{3}a_{jk}m_{j}m_{k}$. This
leads to the expression for mean field chemical potential for species $i$ as,
(where, $i\in\\{\text{solvent},\text{polymer }A,\text{polymer }B\\}$)

$\beta\mu_{0}^{(i)}=1+\ln{\left(\frac{\phi_{i}}{q_{i}}\right)}-v_{i}\sum_{j}\frac{1}{v_{j}}\phi_{j}+v_{i}\sum_{j}\Xi_{ij}\phi_{j}-\frac{v_{i}}{2}\sum_{jk}\Xi_{jk}\phi_{j}\phi_{k}$
(S42)

where all the $\phi_{i}$ denote the mean-field volume fraction of species $i$.

##  V Effective Membrane Rigidity

###  V.1 Frank Elastic Energy of Polymer Brush Layer

Figure S1: A membrane patch is shown to schematically show normal at any point
on the membrane that holds a polymer. The polymer structural axis is shown in
yellow solid line and the polymer end-to-end distance along the preferred
direction of alignment in its extended state is shown in cyan double-headed
arrow. Construction of unit normal $\hat{n}_{P}$ at point
$P\equiv(x,y,h(x,y))$ is demonstrated as the normalized cross product of
$\overrightarrow{PP}_{x}$ and $\overrightarrow{PP}_{y}$

Here, we briefly describe how existence of polymer brush layer in its extended
state is responsible for renormalizing the membrane bending rigidity. Noting
that one end of the polymer is tethered to the membrane, we approximate the
Frank elastic energy of the polymer brush layer as follows [21],

$\displaystyle\beta F_{\mathrm{elas}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int
d\vec{r}\,\left\\{K_{\mathrm{bend}}\left(\vec{n}\times\vec{\nabla}\times\vec{n}\right)^{2}+K_{\mathrm{twist}}\left(\vec{n}\cdot\vec{\nabla}\times\vec{n}\right)^{2}+K_{\mathrm{splay}}\left(\vec{\nabla}\cdot\vec{n}\right)^{2}\right\\}$
$\displaystyle\approx\frac{1}{2}\left\\{K_{\mathrm{bend}}\left(\vec{n}_{P}\times\vec{\nabla}\times\vec{n}_{P}\right)^{2}+K_{\mathrm{twist}}\left(\vec{n}_{P}\cdot\vec{\nabla}\times\vec{n}_{P}\right)^{2}+K_{\mathrm{splay}}\left(\vec{\nabla}\cdot\vec{n}_{P}\right)^{2}\right\\}\sqrt{R_{\parallel}^{2}}$

where $\vec{n}_{P}$ is the nematic director direction at polymer leg on the
membrane (point $P$) and couples directly to membrane height fluctuation
field. Hence, $\vec{n}_{P}$ at point $P$ on the membrane can be identified as
a normal to the membrane at $P$. This can be calculated by parametrizing the
membrane as $P\equiv(x,y,h(x,y))$, where $h(x,y)$ represents the height of the
membrane at $2d$ grid point on the membrane $(x,y)$. Hence, the normal at
point $P$ is given as,

$\displaystyle\vec{n}_{P}=\frac{\overrightarrow{PP}_{x}\times\overrightarrow{PP_{y}}}{||\overrightarrow{PP_{x}}\times\overrightarrow{PP_{y}}||}=\frac{-h_{x}\hat{\delta}_{x}-h_{y}\hat{\delta}_{y}+\hat{\delta}_{z}}{\sqrt{1+h_{x}^{2}+h_{y}^{2}}}\approx-
h_{x}\hat{\delta}_{x}-h_{y}\hat{\delta}_{y}+\hat{\delta}_{z}$ (S44)

where, $\displaystyle h_{x}=\frac{\partial h(x,y)}{\partial x}$ and the
approximate identity gives us variations with respect to minimal order changes
in membrane height fluctuations. Looking at the bend and twist terms of Frank
elastic energy we note that,
$\displaystyle(\vec{\nabla}\times\vec{n}_{P})=h_{zy}\hat{\delta}_{x}-h_{zx}\hat{\delta}_{y}$,
where $\displaystyle h_{zy}=\frac{\partial^{2}h(x,y)}{\partial z\partial y}$
and thus are all higher order terms. This implies only splay rigidity of the
polymers contribute to the Frank elastic energy to minimal order and is given
as,

$\displaystyle\beta
F_{\text{elas}}\approx\frac{1}{2}K_{\text{splay}}\sqrt{R_{\parallel}^{2}}\Big{(}h_{xx}+h_{yy}\Big{)}^{2}$
(S45)

###  V.2 Helfrich Free Energy of Membrane and Renormalized Bending Rigidity

For our case, we consider a membrane with the symmetric distribution of all
membrane components (such as lipids and proteins) and the same environment on
both sides of the membrane that may ensure symmetric membrane fluctuations
and, thereby zero spontaneous curvature. The free energy for such a symmetric
membrane is generally expressed as the Helfrich Hamiltonian [28], given as,

$\displaystyle\beta\mathcal{F}[h(\vec{x})]=\int
d^{2}{\vec{x}}\left\\{\frac{\nu_{\text{mem}}}{2}(\vec{\nabla}h(\vec{x}))^{2}+\frac{\kappa_{\text{mem}}}{2}(\nabla^{2}h(\vec{x}))^{2}\right\\}$
(S46)

where, $h(\vec{x})$ describes the height of the membrane at location
$\vec{x}\equiv(x,y)$ with respect to a reference state. In the above equation
$\nu_{\text{mem}}$ and $\kappa_{\text{mem}}$ represent the membrane tension
and (bending) rigidity respectively. The membrane bending rigidity at any
point per above Helfrich Hamiltonian can be written as,
$\displaystyle\frac{\kappa_{\text{mem}}}{2}(h_{xx}+h_{yy})^{2}$. Hence the
membrane rigidity is renormalized due to Frank elastic energies of polymers
and is given by,

$\displaystyle\kappa_{\text{mem}}^{\text{eff}}=\kappa_{\text{mem}}+K_{\text{splay}}\sqrt{R_{\parallel}^{2}}\geq\kappa_{\text{mem}}$
(S47)

Qualitatively, we can infer in the isotropic state of polymer brush layer, the
splay rigidity do not contribute to the overall membrane bending resistance.
However, in the nematic extended state of the polymer brush layer, we have
finite $K_{\text{splay}}$ and effective membrane rigidity increases to larger
stiffness prone to lesser fluctuations due to thermal and active forces in the
medium. In essence the ratio,
$\Theta_{p/m}=K_{\text{splay}}\sqrt{R_{\parallel}^{2}}/\kappa_{\text{mem}}$
controls membrane fluctuations to zeroth order.

##  VI Supplementary Figures

Figure S2: Phase diagram showing the boundary (thick solid line) of isotropic
(gray) and nematic states as a function of rigid polymer fraction, $f_{R}$.
Figures ((a)-(d)) show changes in free energy (cf. Fig.4 in main article) for
state A ($f_{R}=0.04$), state B ($f_{R}=0.50$) and state C ($f_{R}=0.96$) for
$\hat{a}=17,27,37,47$ (panel (a) through (d) respectively). The states that
are isotropic are designated as dashed squares in the left phase diagram while
the states that shows nematic ordering are shown in solid squares in the phase
diagram.