# Membrane budding driven by intra-cellular ESCRT-III filaments

Sk Ashif Akram  Gaurav Kumar  Anirban Sain<EMAIL_ADDRESS>Physics
Department, Indian Institute of Technology-Bombay, Powai, Mumbai, 400076,
India.

###### Abstract

Exocytosis is a common transport mechanism via which cells transport out non-
essential macromolecules (cargo) into the extra cellular space. ESCRT-III
proteins are known to help in this. They polymerize into a conical spring like
structure and help deform the cell membrane locally into a bud which wrapps
the outgoing cargo. we model this process using a continuum energy functional.
It consists of elastic energies of the membrane and the semi-rigid ESCRT-III
filament, favorable adhesion energy between the cargo and the membrane, and
affinity among the ESCRT-III filaments. We take the free energy minimization
route to identify the sequence of composite structures which form during the
process. We show that membrane adhesion of the cargo is the driving force for
this budding process and not the buckling of ESCRT-III filaments from flat
spiral to conical spring shape. However ESCRT-III stabilizes the bud once it
forms. Further we conclude that a nonequilibrium process is needed to pinch
off/separate the stable bud (containing the cargo) from the cell body.

Figure 1:  Schematic diagram showing progression of ESCRT-III mediated
exocytosis in our model. The spherical cargo is partially wrapped by membrane
(in blue) and unwrapped part of the membrane bud is covered by ESCRT-III
filaments (in red). Degree of wrapping of the cargo is parameterised by the
angle $\theta_{0}$ and the area of the bud gradually increases from left to
right ($A_{0}\rightarrow A_{3}$). Unwrapped portion of the membrane is
parameterised by the contour $s$, when $s\in[0,s_{max}]$.

##  I Introduction

Eukaryotic cells use various proteins, which deform the cell membrane, in
order to allow transport of essential macromolecules (cargo) between the cell
and its exterior. ESCRT-III is one such protein in the protein family
Endosomal sorting complex required for transport or ESCRT. It helps in the
formation of a membrane bud and its detachment, carrying non-essential cargo
out of the cell. One of the unique abilities of ESCRT-III filaments compared
to other intracellular filaments is its ability to severe the membrane neck
from inside the membrane bud. Dynamin filaments, for example wraps around the
neck from outside in order to severe it. ESCRT-III are intracellular
filamentous proteins which bind to the cell membrane from inside, around the
cargo in the form of flat spirals. In the absence of any direct visualisation
of the process in vivo, various proposals have emerged about the dynamical
process. It is not clear whether the flat spiral grows after the cargo
attaches to the already deformed membrane or whether the flat ESCRT-III spiral
grows first and the cargo expels the spiral and attaches itself to the
membrane. Irrespective of the initiation of the spiral, it buckles into a
helical shape and push the cargo out of the cell. Subsequently the spiral also
retracts itself back to the cell and disintegrates. During this process the
cargo gradually wraps itself around with the cell membrane in the form of a
membrane bud and the bud eventually pinches off. It is not clear whether the
process is driven by the energy gain due to the shape transformation of the
ESCRT-III coil lenz2009membrane  or due to the adhesion of the cargo molecule
to the cell membrane Lipowsky2016stabilization ; Lipowski2018domes .

The cargo, which is assumed to be a sphere here, adheres to the cell membrane
from the cell-interior and deforms it, see Fig.1 The angular extent of
adhering membrane area is marked by the polar angle $\theta_{0}$. The amount
of membrane area adhering with the cargo is then $A=2\pi
R^{2}(1-\cos\theta_{0})$. The ESCRT-III filament grows around the cargo,
initially as a flat spiral (not shown here). As the cargo wraps itself with
more membrane area (i.e., increasing $\theta_{0}$), the filament lifts off as
a helical spiral, shown in (a and b). As $\theta_{0}$ crosses $\pi/2$ the
spiral is pressed in the middle as a neck develops, shown in (c). As
$\theta_{0}$ approaches $\pi$ the conical spiral begins to retract from the
neck and moves towards a pinch-off configuration. From (a) through (c) the
adhering area grows as $A_{0}\rightarrow A_{1}\rightarrow A_{2}\rightarrow
A_{3}$. The part of the surface not adhering to the sphere is parameterised
PhysRevA.44.1182 ; julicher1994shape ; PhysRevE.94.062404  by the axisymmetric
curve $\vec{r}(s)=[X(s),Z(s)]$ which extends from $s=0$ to $s=s_{max}$. The
location of the point where the sphere looses contact with the membrane is
$X(s=0)=R\sin\theta_{0}$ and $Z(s=0)=Z_{0}+R\cos\theta_{0}$, where
$(X_{0},Z_{0})$ is the center of the sphere. The shape of this surface is
expressed using an angle $\psi(s)$, which is the angle between the local
tangent to the curve and the $X$ axis, see fig(a). Note that
$\psi(s=0)=\theta_{0}$ and we assume that $\psi(s_{max})=0$, i.e., it joins
smoothly with the cell surface which is assumed to be flat. This is justified
because the radius of the cell is much larger than the radius of the bud $R$.

It is also not clear if the attachment of flat spirals precedes attachment of
cargo to the membrane. Question arises if the membrane is already covered with
dense arrays of ESCRT-III flat spirals, then how does the cargo attach onto
the membrane. On the other hand, if the cargo attaches to the membrane first,
the ESCRT-III spirals will be excluded from that patch of the membrane.
However it is plausible that ESCRT-III can still polymerize around the patch
forming bigger circles.

Lenz et al had shown lenz2009membrane  that due to its intrinsic curvature and
its favourable association with the membrane, the ESCRT-III filaments can
undergo a buckling transition from a flat spiral to a helical spiral, all the
while keeping its contact with the membrane. They assumed an attractive
interaction between the turns of the ESCRT-III spral and the membrane of
strength $-\mu$ per unit area. In their model the tightly packed helical
shaped membrane was approximated as a stack of circular layers since the pitch
is much smaller than the radius of the helix. We essentially use a variant of
their model to study the ESCRT-III mediated budding and exocytosis, which in
addition to the filament instability also requires separation of the bud from
the cell. In this context refLipowski2018domes  has highlighted the role of
the adhesion energy between the cargo and the cell membrane in the budding
process. In our model below we include this adhesion energy contribution which
turns out to be the main driver of exocytosis. In Fig.1 we show a schematic
diagram for the process.

Concerning the transition of the flat ESCRT-III spiral into a helical spiral,
a recent study has proposed a conformational change in the binding between the
ESCRT-III molecules and the membrane Saric2019changes ; saric2020 . They
showed that this effect causes successive helical turns in the spiral to have
decreasing radii, thereby inducing the flat spiral to become a conical helix.
However there is no direct experimental evidence for this conformational
transition.

The shape of the membrane bud during its development is not clear either. The
portion of the membrane joining the neck to the basal surface of the cell has
been argued to be cone shaped Lipowski2018domes  rather than dome shaped
kozlov2009 . RefLipowski2018domes , which has focused on the bud scission
rather than development of the bud, has shown that a cone shaped neck requires
lesser adhesion energy, between the filament and membrane, for scission
compared to if it is dome shaped. Further, the process of fission of the neck
has been proposed to be a non-equilibrium active process Saric2019changes
which serves to cross a free-energy barrier.

Ref Lenz2015relaxation , motivated by in vitro experiments on membrane covered
with an array of flat ESCRT-III coils (spirals), have proposed a compressed
coil hypothesis. In this in vitro experiment array the coils seem to be in a
compressed state. This is inferred from the observation that often the outer
layers of the neighboring coils meet at straight lines exhibiting an
approximate hexagonal arrangement of coils. At lower concentration the coils
are circles touching each other. Ref Lenz2015relaxation  argues that beyond a
threshold of compression level the coils buckle out of the plane. However, it
is not evident whether the flat spirals are in a compressed state in the in
vivo situation hansenJBC08 .

As mentioned earlier, we work with an augmented version of the model by Lenz
et al. lenz2009membrane . We focus on the development of the bud rather than
its scission. We minimize the equilibrium free energy to look for the optimum
state where the bending energy of the deformed membrane, adhesion energy of
the cargo and the bending energy of the filament are jointly minimized.
Description of the model is given below:

##  II Model

As shown in figure 1 a spherical shaped outgoing cargo adheres to the dome
shaped membrane at the top Lipowski2018domes ; fabrikant2009computational ;
julicher1994shape ; Lenz2015relaxation ; Lipowsky2016stabilization ;
PhysRevA.44.1182 ; PhysRevE.94.062404 ; rozycki2012membrane ;
Sabyasachi2013wrapping ; Saric2019changes . Our goal is to find an optimized
surface configuration of the membrane and its evolution during buckling. Dense
spiral filaments (attached to the membrane) are simplified to be concentric
circular rings with uniform surface density throughout the evolution. The
equilibrium shape of the membrane, with its ESCRT filaments and the adhering
cargo attached to it, is found by minimizing the free energy of the composite
system. To describe the composite system consisting of a membrane and polymer,
we use a model similar to that in lenz2009membrane . We study the process
using an equilibrium membrane elasticity based continuum model
lenz2009membrane  (below) which includes the elasticity of the ESCTR-III
filaments, lateral attraction between the filaments, adhesion between the
filament and the membrane and adhesion between the membrane and the cargo.The
bending energy cost of the membrane and the ESCRT filaments are given by the
Helfrich energy and worm-like chain model, respectively. The adhesion energy
of the cargo is an addition to the model.

We take into account the membrane-filament attraction, but we do not go into
the details of the molecular structure of filaments since we are only
interested in their overall coarse-grained contribution to the total energy of
the system. A spherical bud gets created and eventually leads to the scission
from larger membrane at the neck (the minimum radius region). The adhesion
energy between the cargo and the membrane works to overcome the potential
barrier induced by rest of the terms, thus making buckling process
energetically favourable. The expression of the free energy is given by,
$E_{T}=E+E_{c}$. Here $E_{c}$ is proportional to the cap area of the membrane
bud which partially wraps the cargo and it consists of favorable adhesion
energy between the cargo and the membrane, and the elastic bending energy cost
of the membrane. $E$ corresponds to the rest of the bud, below the cargo,
which is covered by ESCRT-III filaments.

$E_{c}=-2\pi R^{2}(\sigma_{0}-\frac{\kappa}{2R^{2}})(1-\cos\theta_{0})$ (1)

and,

$\displaystyle E$ $\displaystyle=$
$\displaystyle\int_{0}^{s_{\max}}ds\Bigg{[}2\pi
X(s)\Bigg{\\{}\frac{k}{2}\left(\frac{1}{X(s)}-\frac{1}{\chi_{0}}\right)^{2}$
(2) $\displaystyle+$
$\displaystyle\frac{\kappa}{2}\left(\frac{\sin\psi(s)}{X(s)}+\psi^{\prime}(s)\right)^{2}+(\sigma_{1}-\mu)\Bigg{\\}}$
$\displaystyle+$
$\displaystyle\lambda(s)\Big{\\{}X^{\prime}(s)-\cos\psi(s)\Big{\\}}+\eta(s)\Big{\\{}Z^{\prime}(s)+\sin\psi(s)\Big{\\}}\,\Bigg{]}$
$\displaystyle+$ $\displaystyle 2\pi\gamma[X(0)+X(s_{max})]$

Here, $k$ is bending stiffness of the filament, $\kappa$ the bending modulus
of the membrane, $-\mu$ the attractive chemical potential between the
filaments and the membrane, $\sigma_{1}$ is the surface tension and
$\sigma_{0}$ is the attractive adhesion coefficient between the membrane and
the spherical shaped cargo.

The first term represents the bending energy of the protein filament ESCRT-
III. The tightly wound spiral is approximated as a stack of circular coils of
finite thickness with variable radii $X(s)$, while $\chi_{0}$ is the intrinsic
radius of curvature of these coils. The radius and the height variables $X(s)$
and $Z(s)$ represent the space curve ($X(s),Z(s)$) parameterized by the
contour length $s$, running between $0$ to $s_{max}$. An azimuthally symmetric
deformed membrane (the partial bud) is the surface of revolution of this space
curve ($X(s),Z(s)$). One important assumption implicit here is that the
filaments are closely packed and dense enough to cover whole of the surface.
We assume that the filament density per unit area of the membrane remains
uniform during the process. Note that this term will favor a helix of radius
$\chi_{0}$ as opposed to a flat coil where all the concentric layers, except
the one with radius $\chi_{0}$ will be strained and therefore have an energy
cost.

The second term denotes the Helfrich free energy of the deformed membrane
represented by the space curve $\vec{r}(s)=[X(s),Z(s)]$. $\psi(s)$ is the
angle between the tangent and the $X$ axis. All derivatives here are with
respect to curve length $s$. The two $\psi$ dependent terms are the local
principal curvatures on the azimuthally symmetric surface Any sharp bending of
the membrane, for example at the neck of the bud, costs high bending energy.

The third term, proportional to the membrane area which is in contact with the
filament, includes effect of surface tension $\sigma_{1}$ of the membrane and
the attractive interaction $-\mu$ between the filament and the membrane, also
known as membrane affinity of the filament. Since $\sigma_{1}-\mu$ is what
matters we define $\sigma_{1}-\mu=-\mu^{\prime}$ and look for variation of
shapes with respect to $\mu^{\prime}$. Note that higher $\mu^{\prime}$ will
favor larger membrane area covered with the filament which requires longer
filament length as well. This will promote the membrane bud to have a wider
base rather than lifting off higher in the $Z$ direction. Lagrange multipliers
$\lambda(s)$ and $\eta(s)$ ensure the relations between $X(s),Z(s)$ and $\psi$
PhysRevA.44.1182 . The next term is crucial to include the attractive
interactions between the filaments (which are closely packed) and
characterized by the line tension $\gamma$.

The second last term involving $\gamma$ captures the lateral attraction
between the filaments lenz2009membrane  which show up as line tensions at the
two open edges of the helical coil. These open edges are essentially 1-D
interface between membrane covered with filaments and the bare membrane. Just
as 2D surface tension at a liquid-air interface emerge from the fact that
liquid molecules have greater attraction among themselves (cohesion) than that
with the air molecules, here the attraction between the ESCRT-III filaments
give rise to line tension which is positive definite like surface tension.
This term can be rewritten as

$2\pi\gamma[X(0)+X(s_{max})]=2\pi\gamma\Big{[}\int_{0}^{s_{\max}}X^{\prime}(s)ds+2X(0)\Big{]}$
(3)

Note that for a given $\theta_{0}$ the term $X(s_{max})$ is dependent on the
shape which is unknown apriori, but $X(0)$ is fixed.

The last term combines the adhesion interaction energy between the surface of
the cargo and the membrane wrapping around it, and bending energy of this part
of the membrane with curvature $1/R$. Both the terms are proportional to the
area of the partial sphere (dependent on $\psi(0)=\theta_{0}$), which is
covered by the membrane. Both $\sigma_{0}$ and $\kappa/R^{2}$ has the unit of
energy per unit area. The adhesion energy is negative (since attractive) while
the bending energy is positive. We define an effective adhesion energy
$\sigma=\sigma_{0}-\frac{\kappa}{2r^{2}}$ and assume $\sigma>0$. Note that
this adheson energy term monotonically decreases as $\theta_{0}$ increases and
hence promotes complete wrapping of the cargo by the membrane.

As Fig.1 shows, the evolving bud can be parameterized by the angle
$\theta_{0}$. Therefore $\theta_{0}$ serves as a proxy for time in our
investigation. For different $\theta_{0}$ values, we solve for the lowest free
energy configuration of the system. The path connecting these different
configurations represent the quasi-static evolution of the system in the
configuration space. We track how the curve length $s_{max}$ and the area of
the unwrapped portion of the bud $A$ in the minimum energy configuration
change as $\theta_{0}$ is varied. Thereby we generate the potential energy
profiles for the system as a function of $\theta_{0}$, for a given set of
parameters.

To simplify calculation, we use dimensionless parameters and variables. We use
membrane bending modulus $\kappa$ and the fixed radius of the spherical cargo
$R$, as the characteristic energy and length scales, respectively, to make
everything (including curve length $s$) dimensionless. In units of $\kappa$
($=20k_{B}T$) the relevant energy density $\mathcal{E}$ is now given by,

$\displaystyle\mathcal{E}$ $\displaystyle=$ $\displaystyle 2\pi
X(s)\Big{[}\frac{k}{2}\Big{(}\frac{1}{X(s)}-\frac{1}{\chi_{0}}\Big{)}^{2}+\frac{1}{2}\Big{(}\frac{\sin\psi(s)}{X(s)}+\psi^{\prime}(s)\Big{)}^{2}$
(4) $\displaystyle-$ $\displaystyle\mu^{\prime}\Big{]}+2\pi\gamma
X^{\prime}(s)+\lambda(s)[X^{\prime}(s)-\cos\psi(s)]$

Here we retained the same notations for the all the non-dimensionalised
quantities.The corresponding Euler-Lagrange equations are:

$\displaystyle-\frac{k}{\chi_{0}^{2}}+2\mu^{\prime}-\psi^{\prime
2}+\frac{\left(k+\sin^{2}\psi\right)}{X^{2}}+\frac{\lambda^{\prime}(s)}{\pi}=0.$
(5) $\displaystyle 2X^{\prime}\psi^{\prime}+2X\psi^{\prime\prime}-\frac{\sin
2\psi}{X}-\frac{\lambda(s)}{\pi}\sin\psi=0.$ (6)
$\displaystyle\cos\psi(s)-\chi^{\prime}(s)=0.$ (7)

The EL-equation that involves $Z$ is the constraint $Z^{\prime}=-\sin\psi$
which is coupled with the above three equations via $\psi(s)$. But $\psi(s)$
can be obtained by solving the above three equations only. The constraint
equation $Z^{\prime}=-\sin\psi$ can be integrated later, with appropriate
boundary conditions, to compute $Z(s)$, once $\psi(s)$ is known.

The boundary conditions on the membrane shape are set by demanding continuity
of slope between the partial sphere part (adhering to the cargo) and the part
below it (attached to the filament), i.e., $\psi(0)=\theta_{0}$. Also the base
of the bud is assumed to smoothly join onto the flat basal membrane, i.e.,
$\psi(s_{max})=0$. The radius at the junction $\chi(0)$, where the membrane
loose contact with the cargo is computed as $\chi(0)=\frac{\sqrt{4\pi
A-A^{2}}}{2\pi}$, where the wrapped area $A=2\pi(1-\cos\theta_{0})$ in units
of $R=1$. Further, we assume $\lambda(0)=0$. After computing $\psi(s)$ we
compute the space curve by integrating the equations
$Z^{\prime}(s)=-\sin\psi(s)$ and $\chi^{\prime}(s)=\cos\psi(s)$, where we use
$Z(s_{max})=0$. Note that for a given $\theta_{0}$ we do not know $s_{max}$ a
priori. Instead of extremizing the energy density with respect to $s_{max}$,
which would increase number of equations to be solved by one we compute the
lowest energy configurations for different value of $s_{max}$ for a given
$\theta_{0}$ and choose the lowest energy configuration among them.

Figure 2:  a) Curve length $s_{max}$, b) filament covered area $A$, and c)
energy $E$, shown as functions of $\theta_{0}$, at different values of
$\gamma$, at $\sigma=0$, for $\sigma_{1}<\mu$. d) Shows the corresponding
total energy $E_{T}$ versus $\theta_{0}$, at $\gamma=0$, and for different
values of $\sigma_{1}$. Membrane binding of the cargo can pull down the second
minimum and make the bud stable. Dimensionless parameters : $k=10$,
$\chi_{0}=0.75$, $\mu^{\prime}=-0.5$. (a) and (b) exhibits a 1st order jump,
at $\gamma<1$. Figure 3:  a) Curve length $s_{max}$, b) filament covered area
$A$ and c) energy $E$, plotted as a function of $\theta_{0}$, at different
values of $\gamma$, at $\sigma=0$. d) Shows the total energy $E_{T}$ versus
$\theta_{0}$, at $\gamma=0$ and for different values of $\sigma_{1}$. But the
second minimum has higher energy in the absence of adhesion energy due to the
cargo and therefore cannot form a stable bud. Membrane binding of the cargo
can pull down the second minimum and make the bud stable. Dimensionless
parameters : $k=2.5$, $\chi_{0}=0.5$, $\mu^{\prime}=0.5$ Figure 4:  Origin of
1st order jump : (a) Energy as a function of $s_{max}$ for different values of
$\theta_{0}$ (in radian), for parameters corresponding to Fig.2, at
$\gamma=\sigma=0$. For a given $\theta_{0}$, the equilibrium value of
$s_{max}$ is the one that corresponds to the global minimum in energy. The
figure shows a first order jump in the location of the global minimum, from
$s_{max}<0.1$ to $s_{max}\geq 0.6$, in a narrow range of $\theta_{0}$ around
$0.3$ rad. b) Shows the contribution of the ESCRT-III filament (solid line) to
the energy $E(\sigma=0)$ (dashed line), as a function of $\theta_{0}$ seen in
Fig.2-c and Fig.3-c. It implies that the double minima structure is due to the
elastic energy of the filament.

###  II.1 Phenomenological parameters

We first estimate the various parameters for which we solve our model
equations. The membrane bending rigidity $\kappa=20\>k_{B}T$ is standard for
bilayer phospholipid membranes. Radius of the cargo is taken as $R=50\>nm$,
for most of the calculations.

Intrinsic radius of curvature $\chi_{0}$ for ESCRT-III filaments have been
estimated to be $25-30\>nm$ by growing Snf7 (main component of ESCRT-III)
filaments in vitro on supported membrane bilayers Lenz2015relaxation . These
filaments nucleate as rings which later grow many concentric flat layers. Both
electron microscopy (EM) and total internal reflection fluorescence (TIRF)
microscopy measurements give similar results. Other studies (see references in
Lenz2015relaxation ) also give similar estimates. In vivo measurements on cell
membranes also show array of flat Snf7 concentric rings, in cells where Snf7
is over expressed.

Ref.Lenz2015relaxation  and references therein estimate ESCRT-III persistence
length $l_{p}$ in the range $250-800\>nm$. The bending modulus $k$ in our
model, which has dimension of energy, can be extracted from $l_{p}$. In
standard worm like chain (WLC) model filament bending energy
$E=\frac{l_{p}.k_{B}T}{2}\int_{0}^{L}ds\>C^{2}(s)$, where the variable $s$ is
the contour length and $C(s)$ is the local curvature along the contour of the
filament. Introducing filament thickness $\Delta$ in the expression for energy
we rewrite $E=\frac{l_{p}.k_{B}T}{2\Delta}\int dA\>C^{2}$, where $dA=\Delta
ds$ is the area element covered by a segment of the filament of length $ds$.
This yields $k=\frac{l_{p}.k_{B}T}{\Delta}$. Using $\Delta\approx 5nm$
hansenJBC08 ; Lenz2015relaxation  for width of the ESCRT-III single strand we
estimate the dimensionless parameter $k/\kappa\approx 2.5-8$. We note that
reflenz2009membrane  used $k/\kappa=2.5$. We explore a range
$k/\kappa=[2.5-10]$.

Adhesion strength of the cargo is not known precisely. We used the values of
adhesion strength between the ESCRT-III filaments and the membrane, estimated
in Ref.Lipowski2018domes . They had estimated for both conical shaped as well
as dome shaped necks. We use the one for cone shaped neck, see Fig. 6B in Ref
Lipowski2018domes , which in their notation $W\approx 0.05\>mN/m$. In our
dimensionless units $\sigma=WR^{2}/\kappa\approx 1.6$ and we explore a small
range $[0-5]$ around this. Note that $W$ in Ref.Lipowski2018domes  is same as
our $\mu$ which we had taken to be of order $\sigma_{1}=10^{-5}N/m$. Since the
effective adhesion energy $\sigma=\sigma_{0}-\frac{\kappa}{2R^{2}}$ is further
reduced by $\frac{\kappa}{2R^{2}}$ which is of the same order, a low value
$0.05\>mN/m$ is justified.

We use membrane surface tension $\sigma_{1}=10^{-5}N/m$ from
Ref.lenz2009membrane . If we assume the membrane affinity of ESCRT-III to be
of similar order, then $-\mu^{\prime}=\sigma_{1}-\mu$ will also be of order
$\sim 10^{-5}N/m$. Thus in dimensionless units
$\mu^{\prime}R^{2}/\kappa\approx 0.3$. Ref.lenz2009membrane  has estimated
$\sigma_{1}-\mu$ in a different way. They define a dimensionless effective
surface tension
$\Sigma=(\sigma_{1}-\mu)\frac{\chi_{0}^{2}}{\kappa}+\frac{k}{2\kappa}$ and
considered different values of $\Sigma$. Taking $\Sigma=1$ we get
$\mu^{\prime}=1$. We explore a range $\mu^{\prime}=[-0.5,0.5]$. In the regime
where surface tension $\sigma_{1}$ is stronger $\mu^{\prime}$ is negative and
where membrane affinity of ESCRT-III $\mu$ is stronger $\mu^{\prime}$ is
positive.

From Ref. lenz2009membrane , we estimate $\gamma$ to be 0.6. We consider
variation of $\gamma\in[0,1]$, to study its effect on the buckling process.
Specifically, we use $\gamma=0,0.5$ and $1$.

##  III Results

We present below the solutions of the Euler-Lagrange’s equations obtained from
this energy functional. We use experimentally relevant values for the
parameters $\kappa,k,\chi_{0},\gamma$ and $R$, described above, and compute
angular dependence of contour length $S_{max}$, filament covered area $A$ and
the Energies $E$ and $E_{T}$ (see Fig2,3). We considered two broadly different
scenarios : $\sigma_{1}<\mu$ (Fig.2) and $\sigma_{1}>\mu$ (Fig.3). Note that,
$\sigma_{1}<\mu$ can promote polymerization of ESCRT-III on the membrane and
is presumably the expermentally relevant case, but direct measurement of $\mu$
is not available. Therefore we examine the $\sigma_{1}>\mu$ case also. It may
appear that, $\sigma_{1}<\mu$ would promote uncontrolled polymerization at the
base of the bud, however the consequent filament turns with progressively
higher radius would cost higher energy due to the intrinsic curvature term and
that would limit the outer radius of the coil. In Fig2-d and 3-d the total
energy ($E_{T}$) of the system is shown for different values of adhesion
strength $\sigma$.

The common feature evident in the energy landscapes for both the cases, shown
in Fig2 and 3, is the existence of two minima. The minima at small and large
$\theta_{0}$s’ correspond to a nearly flat pre-bud and a fully formed bud,
respectively. Accordingly, the shape of the ESCRT-III spiral is nearly flat at
the first minima and conical at the second minima. Plots of $E$ in Fig2,3-c
reveals that without the contribution from the favorable adhesion energy of
the cargo ($E_{c}$) the bud (the second minima) is metastable since it has
higher energy compared to the 1st minima corresponding to the pre-bud state.
Also since the 1st minima occurs at a finite $\theta_{0}$, it is attained via
the buckling of the flat membrane at $\theta_{0}=0$. But the energy of the
filament and the membrane associated with it cannot drive this buckling,
except for $\gamma\simeq 0$ in the $\sigma_{1}<\mu$ case (see Fig.2-c).
Therefore favorable contribution from adhesion energy is crucial even for this
initial buckling transition. Furthermore, comparison between (c) and (d) in
both Fig2,3 shows that there is a barrier between the nominally buckled pre-
bud state and the metastable bud, which can be overcome only by a strong
adhesion energy.

Formation of the pre-bud at small finite $\theta_{0}$ shows qualitative
difference between the $\sigma_{1}<\mu$ and $\sigma_{1}>\mu$ cases. For the
$\sigma_{1}<\mu$, the pre-bud is a well defined minima of $E_{T}$ while for
$\sigma_{1}>\mu$ it is not. Although in both cases there is no jump in the
energy $E$ or $E_{T}$ around the pre-bud state, for $\sigma_{1}<\mu$ there is
a discontinuous jump in both $s_{max}$ and $A$, at $\gamma<1$. But physically,
this region of small $\theta_{0}$ corresponds to the nucleation of the ESCRT-
III spiral, which could a more complicated activated process, not captured by
our model. Nevertheless, we analysed the origin of this jump in $s_{max}$ in
Fig.4-a. The lowest minima at a given $\theta_{0}$ correspond to the
equlibrium value of $s_{max}$ (shown in Fig.2). In Fig.4-a, we see that, as
$\theta_{0}$ is increased the boundary minima at $\theta_{0}$ loose out to the
developing minima at finite $\theta_{0}$ in the range $[0.3-0.35]$, causing
the system to jump to the new global minima, as it occurs for a 1st order
transitition. For example, a mean field theory with a cubic order parameter
term gives abrupt jump in magnetization, from zero to a nonzero value,
although the free energy changes smoothly at the transition.

The double minima structure in the elastic energy of the filament (Fig.4-b) is
due to its intrinsic curvature. It can be understood by considering
contribution coming from the top most part of the filament which has radius
$X(0)=R\sin\theta_{0}$. With $\chi_{0}=25nm$ and $R=50nm$ the elstic energy
contribution is proportional $(\frac{1}{\sin\theta}-2)^{2}$ which has a double
well structure with a maxima at $\theta=\pi/2$. This is because as
$\theta_{0}$ increases from $0$ to $\pi$, initailly $X(0)$ is smaller than
$\chi_{0}$, then it increases and goes past $\chi_{0}$ and then again it
decreases and falls below $\chi_{0}$. The energy is high whenever
$|X(0)-\chi_{0}|$ is high. The rest of the conical shaped filament also behave
similarly.

In Fig.4-b note that $E-E_{c}$, the elastic energy contribution of the
membrane, increases slowly as its curvature increases with $\theta_{0}$ (as
the bud develops). Also note that the area associated with the filament first
increases and then later decreases (see subfigure (a) in both Fig.2,3). This
indicates that polymerization in the beginning and depolymerization near the
end as we did not constrain the length of the filament in our model. We now
discus how the different features of the energy landscapes shown in Fig.2,3
changes as we change the various parameters of the model.

  1. 1.

Line tension $\gamma$ : Fig.2-a,b, show that, for the $\sigma_{1}<\mu$ case,
the jump associated with the contour length ($s_{max}$) and the filament-
covered membrane area ($A$) gradually disappear as $\gamma$ is increased.
However, for the $\sigma_{1}>\mu$ case, $s_{max}$ and $A$ do not change much,
but the energy $E$ changes significantly with $\gamma$. In particular the
barrier between the pre-bud and bud increases with $\gamma$ and therefore
stronger adhesion ($\sigma$) is needed to cross this barrier. Althugh the
position of the maxima (in $\theta_{0}$) does not change much with $\gamma$,
the position of the second minima shifts right with increase in $\gamma$.. As
mentioned before $\gamma$ does not explicitly enter the Euler-Largarne
equations (Eq.7) but it influences the energy $E$ (see Eq.2) and thereby
influences the equilibrium value of $s_{max}$ for a given $\theta_{0}$.

  2. 2.

Filament stiffness $k$ : Comparison between Fig.2 (with $k=10$) and Fig.3
(with $k=2.5$) shows that, as $k$ decreases, the second minima shifts to
larger $\theta_{0}$ i.e., closer to scission ($\theta_{0}=180^{o}$) (figure
not shown). At the same time the energy barrier for scission of the bud also
goes down. But note that our model does not address scisson which could be due
to a nonequilibrium process.

  3. 3.

Intrinsic filament curvature $\chi_{0}$ : Decrease in $\chi_{0}$ reduces the
jump associated with curve length and area, for the $\sigma_{1}<\mu$ case
(figure not shown). Also the second minima shifts towards higher $\theta_{0}$
helping the scission process.

  4. 4.

Cargo size $R$ : We checked that at larger cargo size (eg., $R=100nm$) the
jumps in $s_{max}$ and $A$ disappear, and the second minima shifts rightward
(figure not shown). This is because the process gets completely dominated by
the adhesion energy of the cargo.

##  IV Discussion

The common theme which emerges is that, a) the transition of ESCRT-III from
flat spiral to conical spiral is not driven by the intrinsic cirvature of
ESCRT-III filaments (Fig.4-b). b) The evolution of mildly deformed pre-bud
like structure to a bud structure requires drive from adhesion energy. c) This
pre-bud structure and the final complete bud structure constitute two energy
minima separated by a barrier whose magnitude depends on the adhesion energy
and at strong adhesion the barrier can even vanish clearing way for a smooth
passage to complete bud like structure. However even in the presence of a
finite barrier energy, non-equilibrium processes Saric2019changes  can help in
the barrier crossing. We did not use any conformational transition in the
orientation of the molecular bonds in the filaments, which can generate tilt
in the turns of the filament promoting a helix, as proposed in Ref
Saric2019changes .

Figure 5: Evolution of surface with increasing initial surface angle for
parameter set: $k=10$, $L=0.75$ and $\mu^{\prime}=-0.5$

Note that the adhesion of the cargo to the cell membrane breaks the up-down
symmetry of the membrane deformation and makes the filament buckle in the
direction away from the cell body (convex) instead of towards the interior of
the cell body (concave). However, without the adhesion, in our model the
membrane is absolutely symmetric with respect to budding out or budding in. In
case of inward budding the ESCRT-III filament would wrap on the neck of the
bud on its outer surface which may not be possible in reality as ESCRT-III is
always known to cut membrane necks from inside. But in our model we do not
distinguish between filament wrapping the membrane from outside or inside.

In our model the length of the filament is not fixed. The filament has an
affinity for the membrane and it is found to polymerize first and then
depolymerize near the completion of the process which is consistent with
observations.

A relevant question has been what promotes the filament to become a conical
spiral instead of a helical spiral. Our model shows that the shape is neither
a uniform cone nor a cylinder. It is worth noting that, i) the filament starts
out as a flat spiral hansenJBC08 , ii) a helical spiral requires sharp
membrane bending at the base, and iii) as adhesion of the cargo increases the
radius of the neck reduces below $\chi_{0}$ promoting a conical structure.

It is not evident from experiments whether the filament grows as a flat spiral
first on the membrane and then it transits to a helical shape as the bud
starts to grow or does it polymerize only when the budding starts, driven by
cargo-membrane adhesion. Question arises, in the first case if the membrane
surface is already covered by a flat spiral then how does the cargo adhere to
the membrane. Our model assumes the second possibility that the polymerization
starts with the budding.

Note that our model does not answer how membrane scission takes place at the
end of budding. After the second minima the total energy $E$ starts to rise
sharply (see Fig.2) due to the bending energy costs of both the membrane and
the filament, making the minima stable at an angle $\theta_{0}<\pi$. This asks
for a separate mechanism for scission of the bud.

Also, we have implicitly assumed that the filament-spiral ends where the
membrane touches the flat surface at $s_{max}$, which means the filament is
not preferred on flat surface. This is certainly true for $\sigma_{1}<\mu$,
which is experimentally relevant.

## References

  * [1] Martin Lenz, Daniel JG Crow, and Jean-François Joanny.  Membrane buckling induced by curved filaments.  Physical review letters, 103(3):038101, 2009.
  * [2] Jaime Agudo-Canalejo and Reinhard Lipowsky.  Stabilization of membrane necks by adhesive particles, substrate surfaces, and constriction forces.  Soft Matter, 12(39):8155–8166, 2016.
  * [3] Jaime Agudo-Canalejo and Reinhard Lipowsky.  Domes and cones: Adhesion-induced fission of membranes by escrt proteins.  PLoS computational biology, 14(8):e1006422, 2018.
  * [4] Udo Seifert, Karin Berndl, and Reinhard Lipowsky.  Shape transformations of vesicles: Phase diagram for spontaneous- curvature and bilayer-coupling models.  Phys. Rev. A, 44:1182–1202, Jul 1991.
  * [5] Frank Jülicher and Udo Seifert.  Shape equations for axisymmetric vesicles: a clarification.  Physical Review E, 49(5):4728, 1994.
  * [6] Gaurav Kumar and Anirban Sain.  Shape transitions during clathrin-induced endocytosis.  Phys. Rev. E, 94:062404, Dec 2016.
  * [7] Lena Harker-Kirschneck, Buzz Baum, and Andela Šarić.  Changes in escrt-iii filament geometry drive membrane remodelling and fission in silico.  BMC biology, 17(1):1–8, 2019.
  * [8] Anna-Katharina Pfitzner, Vincent Mercier, Xiuyun Jiang, Joachim Moser von Filseck, Buzz Baum, Andela Šarić, and Aurélien Roux.  An escrt-iii polymerization sequence drives membrane deformation and fission.  Cell, 182(5):1140–1155, 2020.
  * [9] Gur Fabrikant, Suman Lata, James D Riches, John AG Briggs, Winfried Weissenhorn, and Michael M Kozlov.  Computational model of membrane fission catalyzed by escrt-iii.  PLoS Comput Biol, 5(11):e1000575, 2009.
  * [10] Nicolas Chiaruttini, Lorena Redondo-Morata, Adai Colom, Frédéric Humbert, Martin Lenz, Simon Scheuring, and Aurélien Roux.  Relaxation of loaded escrt-iii spiral springs drives membrane deformation.  Cell, 163(4):866–879, 2015.
  * [11] Phyllis I Hanson, Robyn Roth, Yuan Lin, and John E Heuser.  Plasma membrane deformation by circular arrays of escrt-iii protein filaments.  The Journal of cell biology, 180(2):389–402, 2008.
  * [12] Gur Fabrikant, Suman Lata, James D Riches, John AG Briggs, Winfried Weissenhorn, and Michael M Kozlov.  Computational model of membrane fission catalyzed by escrt-iii.  PLoS Comput Biol, 5(11):e1000575, 2009.
  * [13] Bartosz Różycki, Evzen Boura, James H Hurley, and Gerhard Hummer.  Membrane-elasticity model of coatless vesicle budding induced by escrt complexes.  PLoS Comput Biol, 8(10):e1002736, 2012.
  * [14] Sabyasachi Dasgupta, Thorsten Auth, and Gerhard Gompper.  Wrapping of ellipsoidal nano-particles by fluid membranes.  Soft Matter, 9(22):5473–5482, 2013.


# Capacity of the Gaussian Relay Channel with Correlated Noises to Within a
Constant Gap

Lei Zhou, Student Member, IEEE and Wei Yu, Senior Member, IEEE

###### Abstract

This paper studies the relaying strategies and the approximate capacity of the
classic three-node Gaussian relay channel, but where the noises at the relay
and at the destination are correlated. It is shown that the capacity of such a
relay channel can be achieved to within a constant gap of
$\frac{1}{2}\log_{2}3=0.7925$ bits using a modified version of the noisy
network coding strategy, where the quantization level at the relay is set in a
correlation dependent way. As a corollary, this result establishes that the
conventional compress-and-forward scheme also achieves to within a constant
gap to the capacity. In contrast, the decode-and-forward and the single-tap
amplify-and-forward relaying strategies can have an infinite gap to capacity
in the regime where the noises at the relay and at the destination are highly
correlated, and the gain of the relay-to-destination link goes to infinity.

###### Index Terms:

Relay channel, approximate capacity, noise correlation, noisy network coding.

##  I Introduction

The relay channel models a communication scenario where an intermediate relay
is deployed to assist the direct communication between a source and the
destination. Although the capacity of the relay channel is still not known
exactly even for the Gaussian case, much progress has been made recently in
the characterization of its approximate capacity [1, 2, 3].

In the classic Gaussian relay channel, the noises at the relay and at the
destination are independent. In many practical systems, however, the noises at
the relay and at the destination may be correlated. This may arise, for
example, due to the presence of a common interference, which in a practical
system is often treated as a part of the background noise, but nevertheless
contributes to the correlation between the noises.

The Gaussian relay channel with correlated noises has been studied in [4],
where relaying strategies such as the decode-and-forward and the compress-and-
forward schemes are studied in full-duplex or half-duplex modes. Likewise, the
effect of noise correlation for the single-tap amplify-and-forward scheme has
been studied for the diamond network and the two-hop parallel relay network in
[5]. In both papers, noise correlation has been found to be beneficial.
Neither [4] nor [5], however, addresses the question of whether the classic
relaying strategies are able to achieve to within constant bits of the
capacity for the relay channel with correlated noises.

Inspired by the recent work [1] and [3], where the quantize-map-and-forward
and the noisy network coding strategies with fixed quantization level at the
relays are shown to achieve the capacity of arbitrary Gaussian relay networks
with uncorrelated noises to within a constant gap, this paper shows that such
strategies are also capable of approximating the capacity of the three-node
Gaussian relay channel with correlated noises. However, unlike the existing
schemes of [1] and [3], this paper shows that the relay quantization level
needs to be modified to be noise-correlation dependent in the correlated-noise
case. As a corollary, this paper also establishes that the conventional
compress-and-forward scheme [6] achieves to within constant bits of the
capacity for the Gaussian relay channel in the correlated-noise case as well.
Finally, in contrast to the case with uncorrelated noises, the decode-and-
forward and the single-tap amplify-and-forward strategies can have an infinite
gap to capacity, when the noise correlation goes to $\pm 1$ and the gain of
the relay-to-destination link goes to infinity.

##  II Channel Model

Figure 1: Three-node Gaussian relay channel with correlated noises

This paper considers a real-valued discrete-time three-node Gaussian relay
channel as depicted in Fig. 1, which consists of a source $X$, a destination
$Y$, and a relay. The relay observes a noise-corrupted version of the source
signal, denoted by $Y_{R}$, and transmits $X_{R}$ to the destination. The
source-to-destination channel is denoted $h_{SD}$, the relay-to-destination
channel $h_{RD}$, and the source-to-relay channel $h_{SR}$. The additive
Gaussian noises at the relay and at the destination are denoted as $Z_{R}$ and
$Z$ respectively. Mathematically, the channel model is:

$\displaystyle Y_{R}$ $\displaystyle=$ $\displaystyle h_{SR}X+Z_{R},$ (1)
$\displaystyle Y$ $\displaystyle=$ $\displaystyle h_{SD}X+h_{RD}X_{R}+Z.$ (2)

Without loss of generality, the power constraints at the source and at the
relay can both be normalized to one, i.e., $\mathbb{E}[X^{2}]\leq 1$ and
$\mathbb{E}[X_{R}^{2}]\leq 1$, and so can the noise variances, i.e.,
$Z_{R}\sim\mathcal{N}(0,1)$ and $Z\sim\mathcal{N}(0,1)$. Different from most
of the literature that assumes independence between $Z_{R}$ and $Z$, this
paper introduces a correlation between the two noises

$\rho_{z}\triangleq\frac{\mathbb{E}\left[Z_{R}Z\right]}{\sqrt{\mathbb{E}[|Z_{R}|^{2}]\mathbb{E}[|Z|^{2}]}}.$
(3)

Note that $Z$ and $Z_{R}$ are both i.i.d. in time. Further, the relay
operation is causal.

##  III Within Constant Bits of the Capacity

To approach capacity, the relaying strategy must take advantage of the noise
correlation. Consider the limiting scenario of $\rho_{z}\rightarrow\pm 1$. The
relay’s observation becomes more and more useful to the destination in this
case, thus an increasingly fine quantization resolution at the relay is
required — the fixed quantization strategy of [1] and [3] would result in
significant inefficiency. The main contribution of this paper is to introduce
a correlation-aware quantization strategy at the relay, which better exploits
the noise correlation and achieves to within $\frac{1}{2}\log_{2}3$ bits of
the capacity of the Gaussian relay channel with correlated noises.

######  Theorem 1.

The capacity of the three-node Gaussian relay channel with correlated noises,
as shown in Fig. 1, can be achieved to within $\frac{1}{2}\log_{2}3$ bits to
capacity using a noisy network coding strategy with independent Gaussian
inputs $X\sim\mathcal{N}(0,1)$, $X_{R}\sim\mathcal{N}(0,1)$ and Gaussian
quantization at the relay with quantization variance
$\mathsf{q}^{*}=2(1-\rho_{z}^{2})$.

###### Proof:

First, the capacity of the relay channel is upper bounded by the cut-set
bound, i.e.,

$\displaystyle\overline{C}$ $\displaystyle=$
$\displaystyle\max_{p(x,x_{R})}\min\\{I(X,X_{R};Y),I(X;Y,Y_{R}|X_{R})\\}$ (4)
$\displaystyle=$
$\displaystyle\max_{\rho_{x}}\min\left\\{\frac{1}{2}\log(1+h_{SD}^{2}+h_{RD}^{2}+2\rho_{x}h_{SD}h_{RD}),\right.$
$\displaystyle\left.\frac{1}{2}\log\left(1+\frac{(1-\rho_{x}^{2})(h_{SD}^{2}+h_{SR}^{2}-2\rho_{z}h_{SD}h_{SR})}{1-\rho_{z}^{2}}\right)\right\\}$
$\displaystyle\leq$
$\displaystyle\min\left\\{\frac{1}{2}\log(1+h_{SD}^{2}+h_{RD}^{2}+2h_{SD}h_{RD}),\right.$
$\displaystyle\quad\quad\;\;\left.\frac{1}{2}\log\left(1+\frac{h_{SD}^{2}+h_{SR}^{2}-2\rho_{z}h_{SD}h_{SR}}{1-\rho_{z}^{2}}\right)\right\\}$
$\displaystyle=$ $\displaystyle\min\\{R_{UB1},R_{UB2}\\},$

where $\rho_{x}$ is the correlation between $X$ and $X_{R}$.

The achievable rate by noisy network coding or compress-and-forward with joint
decoding can be readily obtained from [7, Proposition 2] and [3, Theorem 1]:

$\displaystyle R$ $\displaystyle=$
$\displaystyle\min\\{I(X,X_{R};Y)-I(Y_{R};\hat{Y}_{R}|X,X_{R},Y),$ (5)
$\displaystyle\quad\quad\;I(X;Y,\hat{Y}_{R}|X_{R})\\}$ $\displaystyle=$
$\displaystyle\min\\{R_{1},R_{2}\\}$

for any distribution

$p(x,x_{R},y_{R},\hat{y}_{R})=p(x)p(x_{R})p(y_{R}|x,x_{R})p(\hat{y}_{R}|x_{R},y_{R}).$

Substitute independent Gaussian distributions $X\sim\mathcal{N}(0,1)$ and
$X_{R}\sim\mathcal{N}(0,1)$ into (5), and set $\hat{Y}_{R}=Y_{R}+e$, where the
quantization noise $e\sim\mathcal{N}(0,\mathsf{q})$ is independent with
everything else, we have

$\displaystyle R_{1}=I(X,X_{R};Y)-I(Y_{R};\hat{Y}_{R}|X,X_{R},Y)$ (6)
$\displaystyle=$
$\displaystyle\frac{1}{2}\log(1+h_{SD}^{2}+h_{RD}^{2})-\frac{1}{2}\log\left(1+\frac{1-\rho_{z}^{2}}{\mathsf{q}}\right),$

and

$\displaystyle R_{2}$ $\displaystyle=$ $\displaystyle
I(X;Y,\hat{Y}_{R}|X_{R})$ (7) $\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}$
$\displaystyle\frac{1}{2}\log(1+h_{SD}^{2})$
$\displaystyle+\frac{1}{2}\log\left(\frac{\mathsf{q}+\sigma^{2}_{h_{SR}X+Z_{R}|h_{SD}X+Z}}{\mathsf{q}+1-\rho_{z}^{2}}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{2}\log\left(1+\frac{\mathsf{q}+(\mathsf{q}+1)h_{SD}^{2}+h_{SR}^{2}-2\rho_{z}h_{SD}h_{SR}}{1-\rho_{z}^{2}}\right)$
$\displaystyle-\frac{1}{2}\log\left(1+\frac{\mathsf{q}}{1-\rho_{z}^{2}}\right),$

where in $(a)$ the conditional variance of $h_{SR}X+Z_{R}$ given $h_{SD}X+Z$
is calculated as

$\displaystyle\sigma^{2}_{h_{SR}X+Z_{R}|h_{SD}X+Z}$ (8) $\displaystyle=$
$\displaystyle\mathbb{E}[|h_{SR}X+Z_{R}|^{2}]-\frac{|\mathbb{E}[(h_{SR}X+Z_{R})(h_{SD}X+Z)]|^{2}}{\mathbb{E}[|h_{SD}X+Z|^{2}]}$
$\displaystyle=$
$\displaystyle\frac{1-\rho_{z}^{2}+h_{SR}^{2}+h_{SD}^{2}-2\rho_{z}h_{SR}h_{SD}}{1+h_{SD}^{2}}.$

Comparing $R_{1}$ and the upper bound $R_{UB1}$, we have

$\displaystyle R_{UB1}-R_{1}$ (9) $\displaystyle=$
$\displaystyle\frac{1}{2}\log(1+h_{SD}^{2}+h_{RD}^{2}+2h_{SD}h_{RD})$
$\displaystyle-\frac{1}{2}\log(1+h_{SD}^{2}+h_{RD}^{2})+\frac{1}{2}\log\left(1+\frac{1-\rho_{z}^{2}}{\mathsf{q}}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{2}\log\left(\frac{1+h_{SD}^{2}+h_{RD}^{2}+2h_{SD}h_{RD}}{2+2h_{SD}^{2}+2h_{RD}^{2}}\right)$
$\displaystyle+\frac{1}{2}\log\left(1+\frac{1-\rho_{z}^{2}}{\mathsf{q}}\right)+\frac{1}{2}$
$\displaystyle<$
$\displaystyle\frac{1}{2}\log\left(1+\frac{1-\rho_{z}^{2}}{\mathsf{q}}\right)+\frac{1}{2}.$

Comparing $R_{2}$ and the upper bound $R_{UB2}$, we have

$\displaystyle R_{UB2}-R_{2}$ (10) $\displaystyle=$
$\displaystyle\frac{1}{2}\log\left(1+\frac{h_{SD}^{2}+h_{SR}^{2}-2\rho_{z}h_{SD}h_{SR}}{1-\rho_{z}^{2}}\right)$
$\displaystyle-\frac{1}{2}\log\left(1+\frac{\mathsf{q}+(\mathsf{q}+1)h_{SD}^{2}+h_{SR}^{2}-2\rho_{z}h_{SD}h_{SR}}{1-\rho_{z}^{2}}\right)$
$\displaystyle+\frac{1}{2}\log\left(1+\frac{\mathsf{q}}{1-\rho_{z}^{2}}\right)$
$\displaystyle<$
$\displaystyle\frac{1}{2}\log\left(1+\frac{\mathsf{q}}{1-\rho_{z}^{2}}\right).$

The gap between the cut-set bound $\overline{C}$ and the achievable rate $R$
is then upper bounded by the maximum of (9) and (10), i.e.

$\displaystyle\overline{C}-R$ $\displaystyle\leq$
$\displaystyle\min\\{R_{UB1},R_{UB2}\\}-\min\\{R_{1},R_{2}\\}$ (11)
$\displaystyle\leq$ $\displaystyle\max\\{R_{UB1}-R_{1},R_{UB2}-R_{2}\\}$
$\displaystyle<$
$\displaystyle\max\left\\{\frac{1}{2}\log\left(1+\frac{1-\rho_{z}^{2}}{\mathsf{q}}\right)+\frac{1}{2},\right.$
$\displaystyle\quad\quad\;\;\left.\frac{1}{2}\log\left(1+\frac{\mathsf{q}}{1-\rho_{z}^{2}}\right)\right\\}.$

The first term above monotonically decreases with $\mathsf{q}$, while the
second term monotonically increases with $\mathsf{q}$. To minimize the maximum
of the two terms, we set

$\displaystyle\frac{1}{2}\log\left(1+\frac{1-\rho_{z}^{2}}{\mathsf{q}^{*}}\right)+\frac{1}{2}=\frac{1}{2}\log\left(1+\frac{\mathsf{q}^{*}}{1-\rho_{z}^{2}}\right),$
(12)

which results in $\mathsf{q}^{*}=2(1-\rho_{z}^{2})$. Substituting
$\mathsf{q}^{*}$ into (11), we have
$\overline{C}-R<\frac{1}{2}\log_{2}3=0.7925$. ∎

In addition, it can be shown that the conventional compress-and-forward rate
is also within the same constant gap to capacity. To prove this directly would
have been quite involved (see [2] for the computation of the gap for the case
of $\rho_{z}=0$). Instead, we obtain the result as a direct consequence of
Theorem 1.

######  Corollary 1.

The following rate, which is achieved by the classic compress-and-forward
strategy on the three-node Gaussian relay channel with correlated noises shown
in Fig. 1:

$R_{CF}=\frac{1}{2}\log\left(1+h_{SD}^{2}+\frac{(h_{SR}-\rho_{z}h_{SD})^{2}}{1-\rho_{z}^{2}+\mathsf{q}_{c}}\right),$
(13)

where

$\mathsf{q}_{c}=\frac{(1-\rho_{z}^{2})(1+h_{SD}^{2})+(h_{SR}-\rho_{z}h_{SD})^{2}}{h_{RD}^{2}}$
(14)

is within $\frac{1}{2}\log_{2}3$ bits to the capacity.

###### Proof:

The rate expression $R_{CF}$ for the correlated-noise Gaussian relay channel
has been obtained in [4, Proposition 5]. The derivation is based on the
classic compress-and-forward rate for the relay channel by Cover and El Gamal
[6, Theorem 6], which is $R_{CF}=I(X;\hat{Y}_{R},Y|X_{R})$ subject to
$I(X_{R};Y)\geq I(Y_{R};\hat{Y}_{R}|X_{R},Y)$ for some joint distribution
$p(x)p(x_{R})p(y_{R}|x,x_{R})p(\hat{y}_{R}|x_{R},y_{R})$. Using the same
signaling scheme as in Theorem 1, i.e., $X\sim\mathcal{N}(0,1)$ and
$X_{R}\sim\mathcal{N}(0,1)$ are independent, and $\hat{Y}_{R}=Y_{R}+e$, where
$e\sim\mathcal{N}(0,\mathsf{q}_{c})$ is chosen to satisfy the relay-
destination rate constraint, we obtain (13).

In the following, we prove the constant gap result for the compress-and-
forward rate by showing that $R_{CF}$ in (13) is greater than the noisy
network coding rate, i.e., $R_{CF}\geq\min(R_{1},R_{2})$, where $R_{1}$ and
$R_{2}$ are as in (6) and (7) respectively. Substituting $\mathsf{q}_{c}$ in
(14) as $\mathsf{q}$ in $R_{1}$ and $R_{2}$, it is easy to verify that
$R_{1}(\mathsf{q}_{c})=R_{2}(\mathsf{q}_{c})=R_{CF}$. Since $R_{1}$ increases
with $\mathsf{q}$ and $R_{2}$ decreases with $\mathsf{q}$, we have
$R_{CF}=\min\\{R_{1}(\mathsf{q}_{c}),R_{2}(\mathsf{q}_{c})\\}=\max_{\mathsf{q}}\min\\{R_{1}(\mathsf{q}),R_{2}(\mathsf{q})\\}\geq\min\\{R_{1}(\mathsf{q}^{*}),R_{2}(\mathsf{q}^{*})\\}$
for any $\mathsf{q}*$ and in particular for
$\mathsf{q}^{*}=2(1-\rho_{z}^{2})$. Since it has been show in Theorem 1 that
$\min\\{R_{1}(\mathsf{q}^{*}),R_{2}(\mathsf{q}^{*})\\}$ is within
$\frac{1}{2}\log 3$ bits of the cut-set upper bound, so is $R_{CF}$. ∎

##  IV Suboptimality of Decode-and-Forward and Single-Tap Amplify-and-Forward

The decode-and-forward and the single-tap amplify-and-forward strategies have
been shown to achieve to within a constant gap to the capacity of the Gaussian
relay channel with uncorrelated noises [1, 2]. In this section, we show that
this is no longer the case when noises are correlated.

###  IV-A Decode-and-Forward

Consider a decode-and-forward strategy as described in [1, Appendix A], in
which when the source-to-relay link is weaker than the source-to-destination
link, i.e., $h_{SR}\leq h_{SD}$, the relay is simply ignored, otherwise the
relay decodes and forwards a bin index to the destination as in the original
scheme of [6]. The following rate is achievable:

$R_{DF}=\max\left\\{\frac{1}{2}\log(1+h_{SD}^{2}),\right.\\\
\left.\min\left\\{\frac{1}{2}\log(1+h_{SR}^{2}),\frac{1}{2}\log(1+h_{SD}^{2}+h_{RD}^{2})\right\\}\right\\}$
(15)

In the extreme scenario where $\rho_{z}=1$ and

$h_{RD}^{2}\gg h_{SR}^{2}\gg h_{SD}^{2}\gg 1,$ (16)

the above decode-and-forward rate (15) becomes

$\displaystyle R_{DF}=\frac{1}{2}\log(1+h_{SR}^{2}).$ (17)

Meanwhile, when $\rho_{z}=1$, the cut-set bound (4) becomes

$\overline{C}=\frac{1}{2}\log(1+h_{SD}^{2}+h_{RD}^{2}+2h_{SD}h_{RD}).$ (18)

Comparing (17) with (18), we observe that

$\displaystyle\overline{C}-R_{DF}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\log\left(\frac{1+h_{SD}^{2}+h_{RD}^{2}+2h_{SD}h_{RD}}{1+h_{SR}^{2}}\right)$
(19) $\displaystyle\rightarrow$
$\displaystyle\frac{1}{2}\log\left(\frac{h_{RD}^{2}}{h_{SR}^{2}}\right),$

which is unbounded in the asymptotic regime (16). This is not unexpected,
because the decoding at the relay eliminates the noise. Therefore, noise
correlation is not exploited.

###  IV-B Single-Tap Amplify-and-Forward

In the single-tap amplify-and-forward, the relay scales the current
observation and forwards to the destination in the next time instance, i.e.,

$X_{R}[i]=\alpha(h_{SR}X[i-1]+Z_{R}[i-1]),$ (20)

where $\alpha\leq\frac{1}{\sqrt{1+h_{SR}^{2}}}$ is chosen to satisfy the power
constraint at the relay. Since $Y[i]=h_{SD}X[i]+h_{RD}X_{R}[i]+Z[i]$, the
relay channel is now converted into a single-tap inter-symbol-interference
(ISI) channel:

$Y[i]=h_{SD}X[i]+\alpha h_{RD}h_{SR}X[i-1]+Z[i]+\alpha h_{RD}Z_{R}[i-1].$ (21)

The capacity of the Gaussian ISI channel is given by

$R_{AF}=\max_{S(\omega)}\frac{1}{2\pi}\int_{0}^{2\pi}\frac{1}{2}\log\left(1+S(\omega)\frac{|H(\omega)|^{2}}{N(\omega)}\right)d\omega,$
(22)

subject to

$\frac{1}{2\pi}\int_{0}^{2\pi}S(\omega)d\omega\leq
1,\;\;\textrm{and}\;\;S(\omega)\geq 0,\quad 0\leq\omega\leq 2\pi,$ (23)

where $N(\omega)=1+\alpha^{2}h_{RD}^{2}+2\rho_{z}\alpha h_{RD}\cos(\omega)$ is
the power spectrum density of the noise, and $H(\omega)=h_{SD}+\alpha
h_{RD}h_{SR}e^{j\omega}$ is the Fourier transform of the channel coefficients,
and $S(\omega)=\left(\lambda-\frac{N(\omega)}{|H(\omega)|^{2}}\right)^{+}$ is
the water-filling power allocation over the frequencies.

Consider again the case of $\rho_{z}=1$ and the asymptotic regime of (16),
i.e. $h_{RD}^{2}\gg h_{SR}^{2}\gg h_{SD}^{2}\gg 1$. In this high signal-to-
noise ratio regime, it is easy to verify that the water-filling power spectrum
converges to an equal power allocation, i.e., $S(\omega)=1$, $0\leq\omega\leq
2\pi$. Substituting $N(\omega)$, $H(\omega)$ and $S(\omega)=1$ into (22) and
using table of integrals, after some algebra, it is possible to show that

$\displaystyle R_{AF}\leq\frac{1}{2}\log(2+h_{SR}^{2}+h_{SD}^{2}).$

Comparing the above with the cut-set bound, we see that

$\displaystyle\overline{C}-R_{AF}$ $\displaystyle\geq$
$\displaystyle\frac{1}{2}\log\left(\frac{1+h_{SD}^{2}+h_{RD}^{2}+2h_{SD}h_{RD}}{2+h_{SR}^{2}+h_{SD}^{2}}\right)$
(24) $\displaystyle\rightarrow$
$\displaystyle\frac{1}{2}\log\left(\frac{h_{RD}^{2}}{h_{SR}^{2}}\right)$

in the asymptotic regime of (16), which is unbounded.

##  V Numerical Simulation

This section numerically compares the cut-set upper bound and the achievable
rates of different relaying schemes. Here, the noisy network coding rate is
computed with $\mathsf{q}^{*}=2(1-\rho_{z}^{2})$. We consider two examples:
Fig. 2 shows the case for $h_{SD}^{2}=20$dB, $h_{SR}^{2}=40$dB and
$h_{RD}^{2}=60$dB, corresponding to an extreme scenario of $h_{RD}^{2}\gg
h_{SR}^{2}\gg h_{SD}^{2}\gg 1$. Fig. 3 shows the case for $h_{SD}^{2}=5$dB,
$h_{SR}^{2}=10$dB, and $h_{RD}^{2}=10$dB. It is clear that in both cases,
compress-and-forward is always better than the noisy network coding scheme
with the specific $\mathsf{q}^{*}$, and both are within a constant gap to the
cut-set upper bound for all values of $\rho_{z}$.

The decode-and-forward rate is always independent of $\rho_{z}$. In the
asymptotic regime as shown in Fig. 2, the single-tap amplify-and-forward rate
is almost independent of $\rho_{z}$ as well, and it coincides with the decode-
and-forward rate. Both can have an unbounded gap to the cut-set bound as
$h^{2}_{RD}\rightarrow\infty$ and $\rho_{z}\rightarrow\pm 1$. Compress-and-
forward, on the other hand, closely tracks the cut-set bound. (Note that the
above observations are not true in the non-asymptotic SNR regime as shown in
Fig. 3.) The noisy-network-coding scheme, although not as good as compress-
and-forward, nevertheless is always within a constant gap to the cut-set
bound.

Figure 2: Comparison of different relaying schemes for $h_{SD}^{2}=20$dB,
$h_{SR}^{2}=40$dB and $h_{RD}^{2}=60$dB Figure 3: Comparison of different
relaying schemes for $h_{SD}^{2}=5$dB, $h_{SR}^{2}=10$dB, and
$h_{RD}^{2}=10$dB

It is interesting to see that the noisy-network-coding rate resembles the
shape of the cut-set upper bound as shown in both Fig. 2 and Fig. 3. It is
also interesting to note that the decode-and-forward curve touches the cut-set
bound at a particular value of $\rho_{z}$. This is because at this value of
$\rho_{z}$, the relay channel becomes degraded [4, Theorem 1].

##  VI Conclusion

This paper investigates different relaying strategies for the three-node
Gaussian relay channel with correlated noises. It is shown that both the
proposed correlation-aware noisy network coding scheme and the conventional
compress-and-forward relaying scheme can achieve to within a constant gap to
the capacity, while the decode-and-forward scheme and the single-tap amplify-
and-forward scheme cannot.

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