###### Abstract

The COVID-19 pandemic caused by the Severe Acute Respiratory Syndrome
Coronavirus 2 (SARS-CoV-2) has spurred unprecedented and concerted worldwide
research to curtail and eradicate this pathogen. SARS-CoV-2 has four
structural proteins: Envelope (E), Membrane (M), Nucleocapsid (N), and Spike
(S), which self-assemble along with its RNA into the infectious virus by
budding from intracellular lipid membranes. In this paper, we develop a model
to explore the mechanisms of RNA condensation by structural proteins, protein
oligomerization and cellular membrane–protein interactions that control the
budding process and the ultimate virus structure. Using molecular dynamics
simulations, we have deciphered how the positively charged N proteins interact
and condense the very long genomic RNA resulting in its packaging by a lipid
envelope decorated with structural proteins inside a host cell. Furthermore,
considering the length of RNA and the size of the virus, we find that the
intrinsic curvature of M proteins is essential for virus budding. While most
current research has focused on the S protein, which is responsible for viral
entry, and it has been motivated by the need to develop efficacious vaccines,
the development of resistance through mutations in this crucial protein makes
it essential to elucidate the details of the viral life cycle to identify
other drug targets for future therapy. Our simulations will provide insight
into the viral life cycle through the assembly of viral particles de novo and
potentially identify therapeutic targets for future drug development.

###### keywords:

SARS-CoV-2; assembly; budding; genome condensation; protein intrinsic
curvature

1 1 0 https://doi.org/ Biophysical Modeling of SARS-CoV-2 Assembly: Genome
Condensation and Budding Siyu Li 1 and Roya Zandi 2

##  1 Introduction

Challenging every aspect of daily lives, the severe acute respiratory syndrome
coronavirus 2 (SARS-CoV-2), the etiologic agent of COVID-19, has driven
scientists all around the world to rapidly develop technology and research
advancing our understanding of this virus and the means to combat it. The
coronavirus family extracts a significant burden on the global population
through the sheer number of viruses from this family causing disease. Before
the pandemic, research had been hampered by the limited scope of severe
infections dying out before becoming wide spread health risks (e.g., the
original SARS-CoV) and by the high level of complexity involved in the
formation and assembly pathways of these viruses. Up to now, nearly all
vaccines target the spike (S) proteins inhibiting viral attachment to the host
cell membrane thus preventing viral entry into the host cell. Focusing on this
single mechanism to disarm the virus can easily be rendered ineffective by
mutations in the S protein.

While the S protein does play a crucial role in the life-cycle of the virus,
it is not integral to the assembly process. Coronaviruses have three other
structural proteins: Envelope (E), Membrane (M), and Nucleocapsid (N), which
are known essential constituents for the assembly and formation of the viral
envelope, see Figure 1. It is known that the N proteins attach to the genomic
RNA and form a compact viral ribonucleoprotein complex (RNP) Chang et al.
(2009); ke Chang et al. (2013, 2014); Klein et al. (2020); Mark Fischetti and
Christiansen (2020). It appears that the electrostatic interaction between the
positively charged N proteins and the negatively charged RNA is crucial for
the genome condensation. Since coronaviruses have the largest genome
($\sim$30,000 nucleotides) among all RNA viruses, the role of N proteins is
specifically very important in condensing and packaging of viral RNA. The
virus then forms as the compact RNP buds at the membrane of the Endoplasmic
Reticulum-Golgi Intermediate Compartment (ERGIC) where other structural
proteins (M, E, S) are located Klein et al. (2020); Martínez-Menárguez et al.
(1999); Klumperman et al. (1994).

Figure 1:  Schematic view of the structure of a coronavirus. The figure shows
the structural proteins: Spike (S) proteins (magenta), Membrane (M) proteins
(green) and Envelope (E) proteins (yellow). The complex of genome and
Nucleocapsid (N) proteins (beige) are enclosed in a lipid membrane called an
envelope (red). The picture is adapted from
https://www.scientificanimations.com/coronavirus-symptoms-and-prevention-
explained-through-medical-animation.

The morphology of the formed RNP and how the N proteins interact with and
condense the genomic RNA resulting in its packaging by a lipid envelope
decorated with structural proteins inside a host cell are still unclear. Two
models have been proposed based on Electron Microscopy (EM) images of RNP
composite structures. Some studies ke Chang et al. (2013, 2014); Macnaughton
et al. (1978); Bárcena et al. (2009) show that N proteins and RNA form a
helical fiber structure (9–16 nm in diameter), which attaches to the interior
of the viral envelope. Other studies Klein et al. (2020); Yao et al. (2020)
suggest that the entire RNP consists of 35–40 copies of small but connected
clusters in which each cluster contains 12 N proteins attached to 800
nucleotides forming a spherical layer with a thickness of 15 nm.

In addition to the uncertainty about the structure within the virus envelope,
the process of packing of RNP and its budding from the organelles inside the
host cell is not well understood. In fact, it is believed that the interaction
between N and M proteins play an important role in the RNA encapsulation by
the lipid envelope Kuo and Masters (2002); Kuo et al. (2016); Hurst et al.
(2005); Huang et al. (2004); Arndt et al. (2010). Using cryo-EM, Neuman et al.
have shown that the M protein forms a matrix which is responsible for the
curvature of the viral envelope Neuman et al. (2011); Neuman and Buchmeier
(2020). Several other studies show that the presence of M proteins is required
for virus budding and envelope formation Mortola and Roy (2004); Huang et al.
(2004); Siu et al. (2008); Plescia et al. (2021). It is worth mentioning that
the M protein is the most abundant structural protein in the viral particle.
However, due to limitations of resolution in electron microscopy imaging, our
knowledge remains rudimentary about the properties of all the structural
proteins including M proteins composing the viral envelope.

In general, a careful study of the assembly experiments reveals that there are
very few known details about the formation and assembly of coronaviruses.
Computational models have often used to shed light on the myriad of
complexities contributing to the assembly of other viruses not easily gleaned
from experiments Li et al. (2018); Panahandeh et al. (2022); Perlmutter et al.
(2013); Chen et al. (2007); Fejer et al. (2010); Rapaport (2004); Patel et al.
(2015); Vernizzi and Olvera de la Cruz (2007); Paquay et al. (2016). All-atom
molecular dynamics simulations provide powerful analytic tools in analyzing
protein–protein interactions. However, because these processes represent
miniscule portions of an immensely larger enterprise and occur over time
scales on the order of nanoseconds, they are nearly impossible to apply to the
entire assembly and budding process Collins et al. (2021); Yu et al. (2021);
Cao et al. (2022). To this end, coarse-grained (CG) modeling has been employed
to explore the kinetics of assembly of spherical icosahedral viruses. While a
considerable number of CG computer simulations and the in vitro self-assembly
experiments of other viruses such as HBV Ruan et al. (2018); Nair et al.
(2018) have had remarkable impact on deciphering the factors that contribute
to the assembly of viruses and the means to combat them Perlmutter et al.
(2014); Hagan (2014); Twarock and Luque (2019); Dykeman et al. (2014); Garmann
et al. (2016, 2019); Chevreuil et al. (2018); Zandi et al. (2020); Comas-
Garcia et al. (2014), similar systematic studies aimed at understanding the
assembly of coronaviruses are missing.

To investigate and understand the assembly pathway of SARS-CoV-2 and gain
insight into behavior over longer time scales, we have designed coarse-grained
models for RNA, phospholipid membrane, and N and M proteins. Using Molecular
Dynamics simulations, we have examined what types of interactions between the
membrane, proteins, and genome are necessary to give rise to the packaging of
RNA and the assembly of the virus. Our simulations reveal the role of each
structural protein in the formation of the SARS-CoV-2, the mechanisms
underlying the assembly process and the physical factors contributing to the
virus morphology, see Video S6 and Video S7.

We find that, while the positively charged N monomers can condense the
negatively charged RNA, oligomerization of N proteins is essential for the
packaging of the very long genome of coronaviruses. Without N oligomerization,
the RNA will not be compact enough to be packaged. As a result of the
oligomerization of N proteins, RNA and N-proteins form several small helical
clusters connected through small segments of RNA, which leads to the
condensation of the long genome and its packaging. The structures of RNP in
our simulations reveal why some experimental results show that RNP has a
helical structure ke Chang et al. (2013, 2014); Macnaughton et al. (1978);
Bárcena et al. (2009), while others find that it is a combination of several
clusters with no particular order and symmetry Klein et al. (2020); Yao et al.
(2020). In fact, we find that it is a combination of the two observed cases.
Our study shows that small helical clusters can join and condense RNA
resulting in its packaging; however, one single long helical structure renders
RNP too stiff to be packaged.

Furthermore, considering the length of RNA and the size of the virus, our
study shows that the intrinsic curvature of M proteins is necessary for viral
budding. Under no circumstances were we able to observe the packaging and
formation of an envelope through budding in the absence of the M protein
intrinsic curvature. Consistent with the experimental data on most
coronaviruses Neuman and Buchmeier (2020); Siu et al. (2008); Plescia et al.
(2021), we find that, while the interaction between the M and N proteins is
not absolutely necessary for the budding process of an empty envelope, it is
necessary for the genome packaging and significantly facilitates the virus
budding through the host cell membrane.

Our work elucidates the fundamental process of viral formation essential to
its reproduction, survival and ultimately its infectivity. Even though our
knowledge about the coronavirus proteins responsible for the virus assembly is
still very basic, our CG model based on the known protein structures has been
able to successfully condense the RNA and shed light on the kinetic pathways
of the viral assembly. Our simulations have deciphered to some extent the
elusive assembly process of SARS-CoV-2, which could be the basis and guide for
the design of future experiments.

Understand that the factors contributing to the formation of coronaviruses can
ultimately prevent viral replication through disruption of the assembly. It is
through inhibition of multiple viral mechanisms that resistance is most
effectively thwarted and viral replication suppressed.

##  2 Materials and Methods

###  2.1 N Protein

We construct our coarse-grained model for the N proteins based on the fact
that they are highly positively charged and mainly form dimers in solution.
Furthermore, NMR, small angle X-ray scattering, and other microscopy
experiments Zeng et al. (2020); Yang et al. (2021); McBride et al. (2014)
reveal that the structure of N proteins (built from 419 amino acids) can be
divided into five regions, namely: the N-terminal disordered region
(N${}_{\text{IDR}}$), the RNA binding domain (RBD), the Linker disordered
region (Linker${}_{\text{IDR}}$), the C-terminal dimerization domain (CTD),
and the C-terminal disordered region (C${}_{\text{IDR}}$), in which
N${}_{\text{IDR}}$, Linker${}_{\text{IDR}}$ and C${}_{\text{IDR}}$ form the
highly flexible intrinsically disorder regions (IDRs) of the protein Cubuk et
al. (2021); Perdikari et al. (2020). The in vitro experiments using
reconstitution approaches and cellular assays have shown that the Linker
region plays a crucial role in phase separation of the N protein with RNA Lu
et al. (2021), whereas the CTD mainly contribute to the N protein
oligomerization ke Chang et al. (2014).

A careful review of the entire 419 residues of N proteins show that the
majority of basic residues are located in N${}_{\text{IDR}}$ (+4, residues
1–43), the middle of RBD (+5, residues 89–110), Linker${}_{\text{IDR}}$ (+6,
residues 181–243), and the N-terminal of the CTD (+9, residues 244–278), as
shown in Figure 2. Since our focus is on the electrostatic interaction between
the N proteins and RNA, we assume that the N protein is composed of three
regions based on its charge distribution. A schematic view of our N protein
model is shown in Figure 2. Each N protein consists of three hard particles:
N-terminal (N1, red), Linker (N2, yellow), and C-terminal (N3, black) regions,
respectively. Note that the C-terminal region contains C${}_{\text{IDR}}$ and
CTD, while the N-terminal region contains N${}_{\text{IDR}}$ and RBD. We note
that, since the charges in CTD are located at the N-terminal sites, we combine
them with the Linker${}_{\text{IDR}}$ charges, which form the N2 region with
+15 charges, the yellow particle in Figure 2.

We further perform a set of molecular dynamics simulations to adjust the
strength of N–N interaction based on the experimental data reporting that most
N proteins form dimers in solution Zeng et al. (2020). We find that the
oligomerization of N proteins largely depends on the relative size of N1, N2,
and N3 domains as shown in Figure S1. This is mainly due to the fact that a
smaller N3 effectively increases the electrostatic repulsion between charged
regions (N1 and N2) rendering the higher order oligomerization (trimers,
tetramers, …) less favorable. Based on the analysis of the distributions of
the N protein oligomerization, we set the diameters of N1, N2, and N3
particles equal to $1.2a$, $0.5a$, and $0.8a$, respectively, and set
$\epsilon_{N_{3}N_{3}}=20$ for the N3– N3 interaction, where $a$ is the system
unit length corresponding to 3 nm. With these parameters, the majority of
oligomers form dimers regardless of the N protein concentration, consistent
with the experimental data.

Figure 2: Schematic view of an N protein. The dimerization domain (CTD) of the
N protein and the RNA binding domain (RBD) are obtained from PDB 6ZCO and
6YI3, respectively Zinzula et al. (2021); Dinesh et al. (2020). The intrinsic
disorder regions (C${}_{\text{IDR}}$, Linker${}_{\text{IDR}}$,
N${}_{\text{IDR}}$) are obtained from UniProt P0DTC9. The visualization is
performed through RCSB 3D-view, where the concentrated positive charges are
colored light green; the rest of the regions have a zero net charge. The
shaded circles indicate the coarse-grained model including three spherical
particles: the N-terminal region (N1, red), the linker region (N2, yellow),
and the C-terminal region (N3, black).

###  2.2 Membrane and Membrane Proteins

We model the ERGIC membrane as a spherical vesicle built of a triangular
lattice; see Figure 3A. The energy of the triangular network involves both the
bond stretching and bending energies. The stretching energy can simply be
expressed by a harmonic potential summed over all bonds as follows:

$\displaystyle E_{s}/k_{B}T$ $\displaystyle=$
$\displaystyle\sum_{<bond>}\frac{1}{2}k_{s}(l-l_{0})^{2},$ (1)

where $l_{0}$ is equal to $a$, the system length scale corresponding to 3 nm
as described before, and $k_{s}=20a^{-2}$ is the stretching modulus. The
bending energy is determined by summing over all pairs of triangular subunits
sharing an edge and can be written as

$\displaystyle E_{b}/k_{B}T$ $\displaystyle=$
$\displaystyle\sum_{<bond>}k_{b}(1-\cos(\theta-\theta_{0})),$ (2)

with $\theta_{0}=\pi$ the preferred angle and $k_{b}=20$ the bending modulus.
To make the membrane incompressible, we have placed a spherical hard particle
with diameter $a$ (green sphere in Figure 3A) on each membrane vertex.

As for the M protein, we have modeled it as a rigid body composed of three
particles, M1, M2, and M3, corresponding to N-terminal, transmembrane, and
C-terminal domains, respectively, see Figure 3B. We have then replaced a
fraction $f$ of the membrane vertices with membrane proteins as illustrated in
Figure 3A.

The M2–M2, M3–M3, N3–N3, and N3–M3 (see Figure 3B,C) interactions are modeled
by Lennard–Jones potential,

$\displaystyle LJ_{ij}/k_{B}T$
$\displaystyle=\sum_{i}4\epsilon_{ij}((\frac{\sigma}{r_{ij}})^{12}-(\frac{\sigma}{r_{ij}})^{6}),$
(3)

where $\sigma=2^{-1/6}d$ with $d$ the distance between two interacting
particles when touching each other and $\epsilon_{ij}$ and $r_{ij}$ are the
interaction strength and distance between two particles $i$ and $j$,
respectively. The interactions are subject to a cutoff distance $r_{cut}=3a$.

The electrostatic interactions between N2, N3, and the genome are modeled by
Debye–Hückle interaction potential,

$\displaystyle DH_{ij}/k_{B}T$ $\displaystyle=l_{b}Z_{i}Z_{j}*\frac{e^{\kappa
d}}{1+\kappa d}\frac{e^{-\kappa r_{ij}}}{r_{ij}},$ (4)

where the Bjerrum length $l_{b}=0.24a$ corresponds to the length scale over
which the electrostatic interaction between two elementary charges is
comparable to the thermal energy, $Z_{i}$, and $Z_{j}$ are the charge numbers,
$d$ is the sum of the radius of two particles, and $\kappa^{-1}=0.5a$ is the
Debye–Hückel screening length. In addition, the excluded volume interaction
between all particles is presented with a shifted LJ repulsion potential

$LJ^{rep}_{ij}/k_{B}T=\sum_{<i,j>}4\epsilon((\frac{\sigma}{r_{ij}})^{12}-(\frac{\sigma}{r_{ij}})^{6})+\epsilon,$
(5)

with $\epsilon=1$, $r_{cut}=d$ and $\sigma=2^{-1/6}d$.

Figure 3: (A) A schematic view of ERGIC. The membrane vertices (Mc) are green
and the M protein particles are colored blue-purple-white. The right figure
shows the bond connections in the ERGIC membrane built from a triangular
lattice. (B) a schematic view of an M protein. The coarse-grained M protein
model contains three hard particles: M1 (N-terminal domain, blue); M2
(Transmembrane domain, purple), and M3 (C-terminal or cytosolic domain,
white). The arrows indicate the attractive M2–M2 and M3–M3 interactions. The
diameters of M1 and M2 are equal while that of M3 is smaller than the other
two, which is the source of the intrinsic curvature of M proteins; (C)
illustration of the N–N (left) and N–M (right) interactions. The attractive
interactions between domains are indicated with arrows, where the C-terminal
of the N proteins (N3) attracts each other, and N3 also attracts the
C-terminal of M proteins (M3).

###  2.3 Genome

While the length of SARS-CoV-2 RNA is 30k nucleotides, it is important to note
that 63.8% of the RNA is base-paired Cao et al. (2021). Given that dsRNA forms
an A-form helix which is 20–25% shorter than dsDNA helix Lipferta et al.
(2014); Arias-Gonzalez (2014), we assume each bead is built of 11–12 bp. Thus,
we use 800 beads to present the double-stranded segments of RNA. We note that
the single-stranded segments of RNA are mostly at the branching points or stem
loops Klein et al. (2020). For simplicity, we did not explicitly model the
single-stranded segments but rather implicitly consider their role in RNA
flexibility. To this end, we model RNA as a negatively charged linear chain
composed of L = 800 beads whose diameter is 1.0a and are connected by
unextendable bonds. The chain is the so-called freely-jointed model if we do
not consider the electrostatic repulsion. Because of the base-pairing and the
secondary structures, RNA has often been modeled as a branched polymers
Perlmutter et al. (2013); Li et al. (2017); Nguyen and Bruinsma (2006). Thus,
we also consider the case of branched polymers with the same total length
$L=800$.

Taking into account the condensation of counter ions around the genome Manning
(1978), we set the effective number of charges of each coarse-grained genome
bead equal to $Z_{g}=\frac{b}{zl_{b}}Q_{g}$ with $b$ the distance between the
genome phosphate charges, $l_{b}$ the Bjerrum length as described above, $z$
the valence of the counter ions, and $Q_{g}$ the original charge number of
each bead. In the following sections, we perform a number of simulations for
$Z_{g}\in[-2,-10]$ and employ the Debye–Hückle theory to model the protein–RNA
electrostatic interactions.

###  2.4 Simulations

To study the kinetics of budding, we use the Lagenvin integrator to integrate
the membrane vertices movement, with time step $dt=10^{-3}s$. At every 100
time steps, we perform Monte Carlo (MC) simulations combined with the bond
flipping method to change the membrane connectivity. The details of Bond Flip
moves are shown in Figure 4. Each MC step consists of $N_{b}$ attempts of
removing a randomly chosen bond between two triangles and attaching two
vertices that were not linked before as shown in Figure 4, where $N_{b}$ is
the total number of bonds. Detachment and attachment of the bonds are such
that the total number of vertices and bonds remain constant. The simulation is
performed on NVIDIA GeForce RTX 3090 where a 8000 s run takes approximately 3
h. We have employed HOOMD-blue Anderson et al. (2020) for the Molecular
Dynamics (MD) simulations and built a plug-in for Bond-Flip MC simulations.
All simulations are visualized by OVITO software Stukowski (2010).

Figure 4: A schematic diagram of the Bond-Flip Monte Carlo method in which the
shared edge of two adjacent triangles are clipped and reconnected to the
diagonal vertices to change the local topology of the structure Panahandeh et
al. (2018).

##  3 Results

###  3.1 N Protein and RNA Assemble into RNP with Multiple Protein Clusters

As noted in the Introduction, coronaviruses have the largest genome among all
single-stranded RNA viruses. Several experiments have shown that the presence
of N proteins is necessary for the packaging of such a long genome by the
viral envelope. In fact, it is now widely accepted that the electrostatic
interaction between the positively charged N proteins and negatively charged
RNA is the driving force for genome encapsulation.

In this section, we explore the condensation of RNA by N proteins into a
compact ribonucleocapsid structure (RNP) as a function of the concentration of
N proteins and the strength of genome–N protein interaction. Following the
literature on the structure of the N protein, we have modeled it as described
in the Methods section; see Figure 2. Note that we have chosen the N–N
interaction based on the structure shown in Figure S1, where the dimers are
dominant structures in the solution. It is worth mentioning that the stronger
N–N interaction results in the formation of higher-ordered oligomers, but, as
it is illustrated in Figure S3, they do not modify the final structure of RNP.
Using these subunits, in the majority of cases, we find that N proteins form
several clusters along the length of RNA. Quite interestingly, as illustrated
in Figure 5, RNA wraps around N proteins to form several clusters with helical
structures that are connected by small segments of naked RNA.

To quantify the structure of RNPs and to create a basis for the comparison
with the future experiments, we have analyzed the clusters and assigned a
unique index number and color to each one, see Figure S2. If the distance
between two neighboring N proteins is less than a cutoff distance $1.2a$ with
$a$ the system length scale 3 nm, then we consider that they belong to the
same cluster.

Figure 5A shows the snapshots of the simulations of the formation of several
clusters built from N proteins along a linear genome for two cases of weak and
strong RNA–N protein interactions. The top row illustrates the RNA–N protein
(RNP) complex structures. For clarity, the middle row displays RNA only,
whereas the bottom row illustrates only the N protein clusters (NPC) without
showing the genome. The simulations indicate that multiple nucleation sites
simultaneously form along the genome. At the early stages of the assembly, we
observe that RNA wraps around N proteins (see $t$ = 1000 s in Figure 5A) and
several small clusters made up of two or three N proteins are formed. As time
passes, the small clusters either fuse with neighboring clusters or attract
more proteins from the solution to form larger clusters.

We note that the strength of RNA–N protein interaction in the simulation
depends on the effective genome charge density, which depends on the
concentration of salt and counterions in solution Belyi and Muthukumar (2006).
As the effective charge density increases, both the RNA–N protein attractive
interaction and the RNA self-repulsion become stronger. At this regime, larger
N protein clusters form with an average of 10 proteins in each cluster. As
time passes, multiple clusters join and form a large cylindrical configuration
which is pretty rigid, as shown on the right side of Figure 5A.

We recall that the focus of our work is on the regimes in which electrostatics
dominates all the other interactions between RNA and N proteins. If, however,
the strength of RNA–N protein interaction were weak, the presence of packaging
signals or other specific interactions would be necessary to produce a similar
genome condensation process de Bruijn et al. (2022). In that case, each
packaging signal would act as a nucleation site for the N proteins to start
forming clusters along RNA.

The snapshots of simulations of the formation of RNPs using branched polymers
are shown in Figure 5B. In each case, $30$ branch points were randomly
generated. The length of each branch was randomly chosen between one and forty
monomers while the total length of the branch polymer was kept the same as the
linear chain model ($L=800$). Our simulations show that, at the weak RNA–N
protein interaction, several clusters immediately nucleate along the longest
length of the polymer, which later will grow in all directions. However, as we
increase the strength of RNA–N protein interaction (the effective genome
charge density), the N proteins nucleate first along the small branches and
then they grow towards the branch point, see Video S1.

Moreover, while the branchiness of the polymers seems to speed up the
condensation process, we find that the impact of the secondary structure of
RNA becomes more significant when the branch number is larger than $10$. At
the smaller branch numbers ($\leq$10), the condensation of branch polymers
follows more or less a similar pathway as a linear chain (see Video S2). We
note that, while previous experimental and theoretical investigations on small
icosahedral viruses like CCMV and BMV have shown that branchiness increases
packaging efficiency and lowers the free energy of the formation of virus
particles Comas-Garcia et al. (2012); Perlmutter et al. (2013); Li et al.
(2017), the effect of branchiness on the rate and the kinetics of virus
assembly has not yet been studied.

Figure 5: (A) N proteins condense a linear chain with $L=800$ subunits and
form RNP. Both cases of the weak ($Z_{g}=-2$) and strong ($Z_{g}=-5$) RNA–N
protein interactions are presented. The N protein concentration is
25$\mathrm{\SIUnitSymbolMicro}\mathrm{M}$. In each window, the upper snapshot
shows the RNP with genome in blue and N proteins in red and black. The middle
and lower snapshots display the RNA and N protein clusters (NPCs),
respectively. Note that, if the distance between the N-terminals of N proteins
(black) are less than a cut-off distance, they are considered to belong to the
same cluster. (B) N proteins condense RNA. RNA is modeled as a branched
polymer with the branch number $N=30$. Both cases of the weak and strong RNA–N
protein interactions are considered. The branched points and end points are
colored in green and yellow, respectively. At the weak RNA–N protein
interaction, the N proteins aggregate into clusters in the same way as they do
in the case of linear chains, regardless of the branching structure of the
chain. The situation differs for the case of the strong RNA–N protein
interaction, where the N proteins condense along the branches first, which
later aggregate to form a compact RNP.

We have also explored the effect of the protein concentration on the structure
of RNP (see Videos S3 and S4). The results of the simulations indicate that
the protein concentration does not have a significant impact on the final RNP
structures if the genome effective charge density is low. Nevertheless, a
higher protein concentration speeds up the condensation time as expected. For
the case of the genomes with a higher effective charge density, a higher
protein concentration, however, facilitates the adhesion of neighboring
clusters, rigidifies RNP configuration, and increases the number of clusters
and the number of N proteins per cluster, leading to a more compact RNP
configuration.

Finally, we examine the impact of the strength of the attractive hydrophobic
interaction between N proteins on the RNP structure. We find that the N–N
attractive interaction plays a key role in the formation of N protein clusters
along RNA. Even a weak interaction between N proteins can induce the formation
of several small clusters. In fact, for a wide range of the strength of the
N–N interaction, the RNP configurations are almost the same, and the number of
N proteins in clusters remains mainly the same (see Figure S3). Quite
interestingly, we find that these clusters are crucial for making RNP compact
enough to ensure its packaging. In the absence of the N–N interaction, the
positive charges on N proteins condense the negatively charged RNA but not as
much as it is necessary for its complete packaging. The presence of N protein
clusters along the genome is essential for RNP packaging if the genome is very
long as in the case of SARS-CoV-2, see the structures of RNP in Figure S3 for
$\epsilon_{N_{3}N_{3}}=0$ and $\epsilon_{N_{3}N_{3}}=30$.

###  3.2 Spontaneous Radius and Line Tension of M Proteins Are Important for
Budding

As noted in the Introduction, despite the intense experimental and theoretical
studies, the most basic questions about the mechanism and relevant time scales
associated with the budding process of coronaviruses are still lingering.
Using a combination of electrophoresis, fluorescence and electron microscopy,
the earlier experiments on coronaviruses have revealed that M, E, and N
proteins are sufficient for the budding of virus-like particles (VLPs),
whereas the shape of VLPs is mainly determined by M proteins Mortola and Roy
(2004); Huang et al. (2004); DeDiego et al. (2007); Siu et al. (2008); Neuman
et al. (2011); Xu et al. (2020); Plescia et al. (2021). It is important to
note that, according to the previous experiments, the E proteins are required
for budding; it appears that they contribute to the completion of assembly and
the scission of the budding neck. In the absence of enough information about
the role of E proteins, we do not explicitly include their effects here, and
thus investigate the general properties of the proteins embedded in the lipid
membrane resulting in virus budding. More specifically, we investigate the
minimum requirement for the packaging and budding of RNP at the ERGIC membrane
and explore the contribution of the mechanical properties of M proteins to the
formation of viral envelope.

As explained in the Methods section, we model ERGIC as a fluid membrane and
model M proteins based on the literature investigating its structure Cao et
al. (2022); Thomas (2020); Heo and Feig (2020); Dolan et al. (2022). To
explore the impact of M–M interaction on the formation of the envelope, we
first build a two-dimensional phase diagram as a function of the interaction
between the transmembrane domains (M2–M2) and endodomains or cytosolic domains
(M3–M3) of M proteins, see Figure 3. We note that, for all simulations
presented in this section, we have assumed that M proteins have already
aggregated into a circular region and that ERGIC is relaxed with several
pentameric defects located at various positions. It is worth mentioning that
we have explored different scenarios and found that the locations and the
initial numbers of defects do not affect the final budding configuration. We
explain in the later section why the initial distribution of M proteins in the
membrane does not have an impact on the final results.

Figure 6A shows the snapshots of the M proteins budding at ERGIC in the
absence of a genome. The strength of the interaction between the segments of
the proteins embedded in the membrane (M2–M2) increases from top to bottom
while the strength of interaction between cytosolic domains, the segments
outside of the membrane (M3–M3), increases from left to right. Figure 6B shows
the plots of the size of the perimeters of the partially budding vesicles (red
regions in Figure 6A) vs. time. The snapshots and time analysis of the growing
vesicles indicate that the stronger transmembrane-transmemebrane (M2-M2) and
endodomain-endodomain (M3–M3) interactions facilitate the budding process
within the regime explored in Figure 6A. We emphasize that we have set the
size of endodomains smaller than the transmembrane domains, which gives rise
to the M protein’s effective spontaneous curvature, leading to the bending of
the membrane. A stronger attractive interaction between endodomains
facilitates budding and overcoming the energy barrier for the formation of
vesicles.

It is worth mentioning that, since the transmembrane domains are embedded in
the membrane, their interactions with each other and the membrane create a
line tension, promoting the budding process through reducing the perimeter of
the budding region. However, if the strength of the transmembrane-
transmembrane interaction is beyond a certain limit, the M proteins can easily
get trapped in a metastable state where the proteins form a hexagonal lattice
and cannot bud. Thus, we were not able to locate a regime in which the line
tension by itself is sufficient for virus budding, see the first column in
Figure 6A. The spontaneous curvature of M proteins is essential for bending
the membrane to complete the budding and viral formation. It is interesting to
note that, even in the absence of the M2–M2 attractive interaction, if the
strength of M3–M3 attractive interaction is high enough, the budding will take
place (see Figure 6A.c).

We note that recent computational models on SARS-CoV-2 show that M proteins
can induce the membrane curvature through the protein–lipid interaction
Collins et al. (2021), which can be equivalent to the M3–M3 (cytosolic
domains) interaction in our model. However, based on the fact that the
formation of empty envelope is not optimal for a virus, we think that the
effective M3–M3 interaction of SARS-CoV-2 is not sufficient for budding and
that the RNP is required to trigger the budding so that the genomic material
can be packaged more efficiently. In fact, some experiments on coronaviruses
suggest that RNP interact with M proteins through the electrostatic
interaction between the negatively charged portion of the C-terminal domain of
the N protein and the positively charged cytosolic domain of the M protein Kuo
and Masters (2002); Kuo et al. (2016); Verma et al. (2006); Luo et al. (2006).
In the next section, we show how N proteins can induce viral budding even if
M3–M3 interaction is weak.

Figure 6: (A) Snapshots of budding through the ERGIC membrane as a function of
the strength of the interaction between the M protein transmembrane domains
($y$-axis) and between the endodomains ($x$-axis). The interactions become
stronger from top to bottom and from left to right. The lipid–lipid
interaction strength is kept constant $\epsilon_{McMc}=1$ for all simulations;
(B) the perimeters (in the unit of $a$) of the cap built from the M proteins
as a function of time in the budding process. The dotted, dashed, and solid
lines correspond to the endodomain interaction strengths
$\epsilon_{M_{3}M_{3}}$ = $0$ (a, d, g), $1$ (b, e, h) and $2$ (c, f, i),
respectively. As the strength of $\epsilon_{M_{3}M_{3}}$ becomes stronger (the
solid line), the budding process becomes faster.

###  3.3 Two Different Models for RNA Packaging

We now explore the effect of the interaction between N and M proteins on the
packaging of RNP at the ERGIC membrane. We consider two different scenarios:
(1) The genome has not yet been condensed but is surrounded by N proteins and
is located in the vicinity of M proteins embedded in ERGIC (Video S6), and (2)
the compact RNP has been formed and transported to the ERGIC interface before
interacting with the M proteins (Video S7). The snapshots of the simulations
of two scenarios are shown in Figure 7.

Figure 7: Snapshots of simulations of the budding process for two different
cases: Scenario 1. Condensation of the genome occurs at the same time as the
budding process. In this case, at the beginning of the simulations, the naked
branched genome is located near the M proteins embedded in the ERGIC membrane
surrounded by N proteins (25 $\mathrm{\SIUnitSymbolMicro}\mathrm{M}$), where
the N-terminals of N proteins attract the genome and their C-terminals
interact with M proteins, triggering both RNP condensation and budding
simultaneously. Scenario 2: N proteins (25
$\mathrm{\SIUnitSymbolMicro}\mathrm{M}$) have already condensed RNA before the
transport of RNP to the ERGIC interface. In this scenario, the N proteins
within RNP are arranged such that their C-terminals interact with the
endodomain of M proteins and trigger the budding process. For clarity, only
the N proteins (not RNA) are shown after the completion of the budding process
in both cases. For both scenarios, the genome length is $L=800$ and genome
charge is $Z_{g}=-2$, the interactions between proteins are
$\epsilon_{McMc}=1$, $\epsilon_{M_{2}M_{2}}=1$, $\epsilon_{M_{3}M_{3}}=0$,
$\epsilon_{N_{3}N_{3}}=20$, and $\epsilon_{N_{3}M_{3}}=10$.

A careful review of the snapshots shows that both scenarios are feasible and
could result in the formation of viral particles. The cluster analysis of the
final structures (Figure S5) reveals that, when the genome is not condensed
but is in the vicinity of the membrane surrounded by N proteins, many small
RNP clusters form right at the beginning of the assembly process. This is
mainly due to the fact that the interaction with M proteins make the small
clusters consisting of two to four N proteins survive over a relatively longer
period of time. The packaged RNP through this mechanism contains more N
proteins than if it were formed while away from the membrane and then would be
transferred to the membrane to interact with the M proteins to acquire its
envelope. In the latter case (Scenario 2), the number of clusters remains
constant during the budding as the clusters have already been formed and fully
relaxed. However, the packaging of RNP will take more time as N proteins in
the clusters need to reorient and expose their C-terminal domains to interact
with the M proteins. It is worth mentioning that the protein concentration
does not affect the rates of condensation and the assembly process, in the
case of the second scenario. However, for the first scenario, the condensation
and budding occur simultaneously so that the protein concentration and thus
rate of condensation will have an impact on the budding rate.

Furthermore, we find that the strength of the attractive interaction between
the N proteins and RNA modifies the structure of RNP inside the envelope. The
clusters along RNP are distributed more sparsely inside the viral particle for
a weak N protein–RNA interaction while RNP assumes a more condensed structure
if the N protein–RNA interaction is stronger, as shown in Figure S6. In the
case of the second scenario, such condensed conformation might prevent RNP
from being packaged at the ERGIC membrane since the N protein clusters are too
compact to reorient their C-terminal and interact with M proteins. A thorough
analysis of RNP structures can reveal the condensation pathway of the
coronavirus genomic RNA. However, the current cryo-em images cannot
distinguish the difference between the structures of RNP formed through
different pathways. The comparison of our simulations with future cryo-em
images and experiments with different mutants of N proteins can shed light on
the mechanism through which RNP is condensed and packaged at ERGIC.

Our simulations also show that, if we set the interaction between transdomains
$\epsilon_{M_{2}M_{2}}=1$ but between endodomains $\epsilon_{M_{3}M_{3}}=0$
(see Figure 3), the M–M interaction cannot bend the membrane and no empty
vesicle can form, as shown in Figure 6Ad. However, in the presence of RNP, the
interaction between M and RNP is enough for the virus to bud even though
$\epsilon_{M_{3}M_{3}}=0$. We emphasize that, if the size of M3 is not smaller
than M2, under no circumstances does the virus bud as discussed in the
previous section, indicating that the spontaneous curvature of M proteins is
absolutely necessary for budding (see Figure S4). It appears that M–N
interaction constitutes the main contribution to the RNP budding if there is
no interaction between endodomains. Because of the attraction between the N
proteins, the M–N interaction induces the M3–M3 attractive interaction
indirectly, which then provides the necessary curvature to trigger the budding
process. This can explain why for some coronaviruses, the interaction of M–N
is crucial for the formation of VLPs. Further experiments is necessary to
understand if there is an attractive interaction between endodomains of M
proteins.

###  3.4 Condensation of RNA by N Proteins Is Essential for Its Packaging

It is worth mentioning that the recent experiments on the SARS-CoV-2 M
proteins show that the endodomains of the M proteins are highly positively
charged, which might have an impact on their interactions with N and S
proteins and RNA Dolan et al. (2022). To examine the effect of the M positive
charges on the packaging of the genome, we also consider the case in which the
endodomain (M3) has positive charges (+6, residues 101–222) (see Figure S7).
Following Collins et al. (2021), we assume that the electrostatic repulsion
between the M protein endodomains is screened out by the POPI lipids through
the protein–lipid interactions, and thus we set the M3–M3 electrostatic
repulsion equal to zero. Intuitively, one would expect that the positive
charges on M proteins might be sufficient for the RNA packaging because of the
attractive electrostatic interactions between RNA and the M endodomain.
However, we find that the sole attractive electrostatic interaction between
the M proteins and RNA is not enough for its packaging in the absence of N
proteins, i.e., the condensation of genome by the N proteins is essential for
the virus formation. This is in contrast to the case of many icosahedral
viruses where the sole attractive interaction between the genome and the
capsid proteins is sufficient for the encapsidation of the genome with viral
capsid proteins Li et al. (2017); Siber et al. (2012). Because of the length
of the genome of coronaviruses and the budding process, the presence of N
proteins are essential for the genome condensation and virus formation. We
have examined that, in our system, the interaction between the positive
charges on M and negative charges on RNA will be enough for budding if the
length of the genome is much shorter.

We note that all the budding simulations are performed using branched polymers
with the number of branch points $N=30$. Although the secondary structure of
RNA does not have an impact on the final configurations of condensed RNPs, the
budding of a long linear chain is more challenging because the radius of
gyration of a free linear chain is larger than a free branched polymer; thus,
it takes much more time for the linear one to condense and be completely
packaged. For example for a linear chain with $L\sim 800$, we have been able
to package only half of the RNP during the time of our simulations (see Figure
S8).

###  3.5 Impact of the Initial Distribution of M Proteins on Virus Budding

We recall that, in the budding simulations presented above, we have always
assumed that the M proteins initially form a spherical cap with a circular
edge. However, the initial distribution of M proteins at the ERGIC membrane
could be different. To this end, we have also considered the condition in
which the initial distribution of the M proteins is random and have repeated
the simulations presented in Figure 6A.d with the same parameters. We first
fully relax the ERGIC membrane and then allow the randomly distributed M
proteins to diffuse and form different structures rather than imposing a
circular cap from the beginning Figure 8A. As we increase the strength of the
interactions between the endodomain part of M proteins, the irregular
branched-like patterns start to form several circular domains pointed with
arrows in Figure 8A, which will later bud into empty vesicles. This
disorder–order transition indicates that the spontaneous curvature of M
proteins makes the branched configuration energetically unfavorable; as such,
they diffuse to form separate circular domains.

Quite interestingly, we find that, even in the absence of the interaction
between endodomains of M proteins, RNP can trigger the formation of circular
caps (Video S8). Figure 8B illustrates that, when we place RNP near the
disordered regions where the M proteins embedded in ERGIC have formed a
branched-like pattern, the M–M interaction starts to bend the membrane due to
the endodomain–RNP interaction. As explained in the previous section, even in
the absence of M3–M3 attractive interaction, the interaction of RNP with the
endodomain of M leads to an effective spontaneous curvature between M
proteins. The curved region attracts more M proteins due to the energetically
favorable endodomain–RNP interaction. After a while, the branched-like regions
start to assume a circular configuration, which later rapidly buds through
ERGIC encapsulating RNP to form the viral particle. We note that with a random
distribution of M proteins, the simultaneous packaging of genome and budding
are not plausible through scenario 1 because then many N proteins basically
follow the branched-like pattern of the M proteins and will be distributed
everywhere on ERGIC. In that case, RNA spreads out to reach those N proteins
and thus it cannot condense to be packaged. Thus, a better understanding of
the M–M interaction while embedded in the membrane can shed light on the
pathway of genome condensation and its budding through the ERGIC membrane.

Figure 8: (A) Snapshots of the simulations of the distribution of the M
proteins embedded in the ERGIC membrane as a function of time. Initially, the
M proteins are randomly distributed in the ERGIC membrane. The first snapshot
at the left illustrates the distribution of the M proteins in ERGIC where the
proteins and membrane have been relaxed for 16,000 s with
$\epsilon_{M_{2}M_{2}}=1$ and $\epsilon_{M_{3}M_{3}}=0$. As a result of
diffusion, the M proteins have formed a branched-like pattern. In the absence
of the M3–M3 interaction, the branched-like pattern does not change (top row).
However, with increasing the endodomain–endodomain attractive interaction
$\epsilon_{M_{3}M_{3}}$, the connected patches of M proteins begin to
aggregate into circular domains (bottom row, indicated with red arrows), which
will eventually bud into empty vesicles. Note that the M proteins cover half
of the surface of ERGIC, i.e., the M protein fraction is $f=50\%$ during the
simulations; (B) snapshots of the simulations of the RNP budding from ERGIC.
Initially, the randomly distributed M proteins form a branched-like pattern on
the surface of ERGIC in the absence of the endodomain interaction. However,
the interaction with RNP results in the aggregation of the proteins in the
vicinity of RNP and the formation of the envelope enclosing RNP. The branched-
like distribution is obtained by relaxing ERGIC and the M proteins for 16,000
s. The RNP structure is obtained by relaxing N proteins (25
$\mathrm{\SIUnitSymbolMicro}\mathrm{M}$) and RNA for 8000 s. RNA is modeled as
a branched polymers with the charge density $Z_{g}=-2$ and the number of
branch points $N=30$. The comparison of (A,B) reveals that RNP can mediate the
interaction between the endodomains giving rise to a preferred curvature
between M proteins.

##  4 Discussion

Even though the replication and assembly are conserved across many members of
the family of coronaviruses, the most rudimentary details of these processes
remain elusive. This could be due to the presence of a number of structural
proteins involved in the formation of viral particles and our limited
knowledge regarding their specific roles. Additionally, viral assembly and
budding occur in the milieu of the intracellular host cell organelle membranes
which further complicates isolating mechanisms specific only to the virus.

In this paper, we have investigated the underlying physical principles for the
formation of SARS-CoV-2 focusing on the role of the structural proteins in the
assembly of the virus. While many unknown parameters are involved in the
packaging of the genome, our primary goal has been to obtain an overall
understanding of under what conditions the large genomic RNA of coronaviruses
can be condensed to bud through the host cell intercellular membrane.
Specifically, we have sought to determine what types of effective
protein–protein, genome–protein and membrane–protein interactions are
necessary for the budding and formation of viral particles.

Despite all the experiments, the quantitative values of the interactions
critical to modeling remain experimentally unknown. To this end, we have made
several phase diagrams exploring how the relative strength of interactions
between the different virus constituent parts influence the virus assembly and
budding. Our simulations show that the essential requirement for the virus
budding is the spontaneous curvature of the membrane proteins. Considering the
length of SARS-CoV-2 RNA, under no circumstances were we able to observe virus
budding in the absence of the M protein spontaneous curvature. We note that
the attractive interaction between the cytosolic domain of M proteins or the
presence of external scaffolding such as N proteins or RNP interacting with
the cytosolic domain of M proteins can give rise to the spontaneous curvature
of M proteins, see Figure 8.

The attractive interaction between the transmembrane domain of M proteins
could also lower the budding energy barrier by introducing the line tension. A
very strong interaction between the transmembrane domains might create a high
line tension, which could lead to virus budding. However, if the M–M
interaction is very strong, we observe that the M proteins form immobile large
islands, trapped in metastable configurations.

We have also investigated the mechanism of packaging of the long genomic RNA
during the virus budding. Although several experiments show that the N
proteins and RNA phase separate Cubuk et al. (2021); Savastano et al. (2020);
Perdikari et al. (2020); Carlson et al. (2020); Wang et al. (2021); Wu et al.
(2021), the exact location and when this process occurs remain unclear. Our
simulations offer two distinct plausible models: Either the RNP condensation
and budding occur simultaneously at the ERGIC membrane or RNP is formed
remotely and then transported to the ERGIC for budding. In both models, the
RNP forms a spherical layer under M proteins with a hollow center, consistent
with previous observations Klein et al. (2020). In the first scenario,
however, the order of arrival of N proteins and RNA to the ERGIC regions seems
vital since the N proteins may initiate the budding process in the absence of
genome and form empty particles (see Figure S9). If the entire RNP is to be
encapsulated, it will largely depend on the protein concentration, the rate of
RNP condensation, and the initial M protein distribution.

In the alternative model that we offer in this paper, the preformed RNP is
transported to the ERGIC region and triggers the budding process by the M–N
interaction. There will be no extra N proteins in the immediate vicinity of
ERGIC in this model; thus, the environmental factors have less impact on the
budding process. However, if the RNP configurations are specifically compact,
in the case of the strong N proteins–RNA interactions, the packaging of the
RNP will be difficult as the N proteins in the clusters need to reorient and
expose their C-terminal domains to interact with M proteins. Quite
interestingly, a recent paper on SARS-CoV-2 shows that N proteins aggregate
around ER membranes that connect to viral RNA replication foci, whereas M, E,
and S proteins accumulate at the ERGIC membrane where the budding and
maturation happen Scherer et al. (2022). Since N proteins are located near
RNA, it is possible that RNP is pre-formed before meeting M, E, and S proteins
in ERGIC. Furthermore, the study shows that N proteins and genome are observed
at 5 h post infection (hpi), whereas M, E, and S proteins are not observed
until 7.5 hpi Scherer et al. (2022). The fact that N proteins are expressed
earlier than the other structural proteins indicates the possibility that RNA
is condensed before being transported to the location of ERGIC. Considering
these temporospatial aspects, we think that scenario 2 is completely
plausible. However, more experiments are still needed to decipher where and
when the genome is condensed. The comparison of the our simulations with the
future in vivo experiments focused on the kinetic pathways of RNP condensation
can shed light on virus formation at the ERGIC membrane.

We note that, while some experiments regarding the assembly of both SARS-CoV
and SARS-CoV-2 demonstrate that the minimal requirements for the virus budding
are M and N proteins Huang et al. (2004); Plescia et al. (2021), other
experiments show that M and E proteins are essential for the VLP formation
Mortola and Roy (2004); Xu et al. (2020). Interestingly, another set of
experiments on SARS-CoV suggests that E proteins might not be required for the
VLP assembly but rather enhance the production level Siu et al. (2008);
DeDiego et al. (2007). Even with the unprecedented deluge of research the
pandemic has wrought, the role the structural proteins (E, M and N) play in
the SARS-CoV-2 life cycle continue to remain speculative and pose an area of
intense debate.

Our work can explain to some extent why there are inconsistencies in the
experiments regarding the budding mechanism of various coronaviruses.
Different requirements for budding could be due to the various strength of
interactions between the structural proteins from one type of coronavirus to
another, or the composition of lipids in different host cells. In one case,
the M–M interaction might be weak and the M–N interaction must be essential
for the budding to occur, while in another case, the presence of the E
proteins confers a higher spontaneous curvature to the membrane Ruch and
Machamer (2012); Schoeman and Fielding (2019); and as such, when E proteins
are incorporated, a significant enhancement of the viral particle production
is observed Siu et al. (2008).

Our aim is to shed light on the elusive assembly process of coronaviruses and
particularly that of SARS-CoV-2, which could be the basis and guide for the
design of future experiments too. A systematic comparison of our simulations
and the experiments with various mutations on M and N proteins could
significantly advance our knowledge on this virus and the factors contributing
to its formation. Understanding the complex life cycle of this virus
inherently involves how the virus reproduces itself in such an extraordinarily
successful way. Since there is always the danger of reduced effectiveness or
even ineffectiveness of vaccines as a result of novel mutations, a better
understanding of the structural properties of SARS-CoV-2 could lead to
alternative treatments for COVID-19 infections beyond the limited means we now
have.

###### Acknowledgements.

R.Z. acknowledges support from NSF DMR-2131963, NSF DMR-2034794 and the
University of California Multicampus Research Programs and Initiatives (Grant
No. M21PR3267). S.L. acknowledges support from NSFC No.12204335\. We thank
Paul van der Schoot for many helpful suggestions and Thomas Kuhlman for
critically reading several sections of the paper. We thank Yuanzhong Zhang,
Ajay Gopinathan and Umar Mohideen for many helpful discussions.  All
simulation data are present in the manuscript. HOOMD-blue plug-in is available
at https://github.com/virusassembly/hoomd_bondflip_plugin. References

## References

  * Chang et al. (2009) Chang, C.K.; Hsu, Y.L.; Chang, Y.H.; Chao, F.A.; Wu, M.C.; Huang, Y.S.; Hu, C.K.; Huang, T.H.  Multiple Nucleic Acid Binding Sites and Intrinsic Disorder of Severe Acute Respiratory Syndrome Coronavirus Nucleocapsid Protein: Implications for Ribonucleocapsid Protein Packaging.  J. Virol. 2009, 83, 2255–2264.  https://doi.org/10.1128/jvi.02001-08.
  * ke Chang et al. (2013) Chang, C.K.; Chen, C.M.; Chiang, M.H.; Hsu, Y.L.; Huang, T.H.  Transient Oligomerization of the SARS-CoV N Protein—Implication for Virus Ribonucleoprotein Packaging.  PLoS ONE 2013, 8, e65045.  https://doi.org/10.1371/journal.pone.0065045.
  * ke Chang et al. (2014) ke Chang, C.; Hou, M.H.; Chang, C.F.; Hsiao, C.D.; huang Huang, T.  The SARS coronavirus nucleocapsid protein – Forms and functions.  Antivir. Res. 2014, 103, 39–50.  https://doi.org/https://doi.org/10.1016/j.antiviral.2013.12.009.
  * Klein et al. (2020) Klein, S.; Cortese, M.; Winter, S.L.; Wachsmuth-Melm, M.; Neufeldt, C.J.; Cerikan, B.; Stanifer, M.L.; Boulant, S.; Bartenschlager, R.; Chlanda, P.  SARS-CoV-2 structure and replication characterized by in situ cryo-electron tomography.  Nat. Commun. 2020, 11, 5885.  https://doi.org/10.1038/s41467-020-19619-7.
  * Mark Fischetti and Christiansen (2020) Mark Fischetti, Veronica Falconieri Hays, B.G.; Christiansen, J.  A Visual Guide to the SARS-CoV-2 Coronavirus.  Sci. Am. 2020, 323, 32.  https://doi.org/10.1038/scientificamerican0720-32.
  * Martínez-Menárguez et al. (1999) Martínez-Menárguez, J.A.; Geuze, H.J.; Slot, J.W.; Klumperman, J.  Vesicular tubular clusters between the ER and Golgi mediate concentration of soluble secretory proteins by exclusion from COPI-coated vesicles.  Cell 1999, 98, 81–90.  https://doi.org/10.1016/S0092-8674(00)80608-X.
  * Klumperman et al. (1994) Klumperman, J.; Locker, J.K.; Meijer, A.; Horzinek, M.C.; Geuze, H.J.; Rottier, P.J.  Coronavirus M proteins accumulate in the Golgi complex beyond the site of virion budding.  J. Virol. 1994, 68, 6523–6534.  https://doi.org/10.1128/jvi.68.10.6523-6534.1994.
  * Macnaughton et al. (1978) Macnaughton, M.R.; Davies, H.A.; Nermut, M.V.  Ribonucleoprotein-like structures from coronavirus particles.  J. Gen. Virol. 1978, 39, 545–549.  https://doi.org/10.1099/0022-1317-39-3-545.
  * Bárcena et al. (2009) Bárcena, M.; Oostergetel, G.T.; Bartelink, W.; Faas, F.G.; Verkleij, A.; Rottier, P.J.; Koster, A.J.; Bosch, B.J.  Cryo-electron tomography of mouse hepatitis virus: Insights into the structure of the coronavirion.  Proc. Natl. Acad. Sci. USA 2009, 106, 582–587.  https://doi.org/10.1073/pnas.0805270106.
  * Yao et al. (2020) Yao, H.; Song, Y.; Chen, Y.; Wu, N.; Xu, J.; Sun, C.; Zhang, J.; Weng, T.; Zhang, Z.; Wu, Z.; et al.  Molecular Architecture of the SARS-CoV-2 Virus.  Cell 2020, 183, 730–738.e13.  https://doi.org/10.1016/j.cell.2020.09.018.
  * Kuo and Masters (2002) Kuo, L.; Masters, P.S.  Genetic Evidence for a Structural Interaction between the Carboxy Termini of the Membrane and Nucleocapsid Proteins of Mouse Hepatitis Virus.  J. Virol. 2002, 76, 4987–4999.  https://doi.org/10.1128/jvi.76.10.4987-4999.2002.
  * Kuo et al. (2016) Kuo, L.; Hurst-Hess, K.R.; Koetzner, C.A.; Masters, P.S.  Analyses of Coronavirus Assembly Interactions with Interspecies Membrane and Nucleocapsid Protein Chimeras.  J. Virol. 2016, 90, 4357–4368.  https://doi.org/10.1128/jvi.03212-15.
  * Hurst et al. (2005) Hurst, K.R.; Kuo, L.; Koetzner, C.A.; Ye, R.; Hsue, B.; Masters, P.S.  A Major Determinant for Membrane Protein Interaction Localizes to the Carboxy-Terminal Domain of the Mouse Coronavirus Nucleocapsid Protein.  J. Virol. 2005, 79, 13285–13297.  https://doi.org/10.1128/jvi.79.21.13285-13297.2005.
  * Huang et al. (2004) Huang, Y.; Yang, Z.y.; Kong, W.p.; Nabel, G.J.  Generation of Synthetic Severe Acute Respiratory Syndrome Coronavirus Pseudoparticles: Implications for Assembly and Vaccine Production.  J. Virol. 2004, 78, 12557–12565.  https://doi.org/10.1128/jvi.78.22.12557-12565.2004.
  * Arndt et al. (2010) Arndt, A.L.; Larson, B.J.; Hogue, B.G.  A Conserved Domain in the Coronavirus Membrane Protein Tail Is Important for Virus Assembly.  J. Virol. 2010, 84, 11418–11428.  https://doi.org/10.1128/jvi.01131-10.
  * Neuman et al. (2011) Neuman, B.W.; Kiss, G.; Kunding, A.H.; Bhella, D.; Baksh, M.F.; Connelly, S.; Droese, B.; Klaus, J.P.; Makino, S.; Sawicki, S.G.; et al.  A structural analysis of M protein in coronavirus assembly and morphology.  J. Struct. Biol. 2011, 174, 11–22.  https://doi.org/10.1016/j.jsb.2010.11.021.A.
  * Neuman and Buchmeier (2020) Neuman, B.; Buchmeier, M.  Supramolecular Architecture of the Coronavirus Particle.  Adv. Virus Res. 2020, 105, 93–116.  https://doi.org/10.1016/bs.aivir.2016.08.005.
  * Mortola and Roy (2004) Mortola, E.; Roy, P.  Efficient assembly and release of SARS coronavirus-like particles by a heterologous expression system.  FEBS Lett. 2004, 576, 174–178.  https://doi.org/10.1016/j.febslet.2004.09.009.
  * Siu et al. (2008) Siu, Y.L.; Teoh, K.T.; Lo, J.; Chan, C.M.; Kien, F.; Escriou, N.; Tsao, S.W.; Nicholls, J.M.; Altmeyer, R.; Peiris, J.S.M.; et al.  The M, E, and N Structural Proteins of the Severe Acute Respiratory Syndrome Coronavirus Are Required for Efficient Assembly, Trafficking, and Release of Virus-Like Particles.  J. Virol. 2008, 82, 11318–11330.  https://doi.org/10.1128/jvi.01052-08.
  * Plescia et al. (2021) Plescia, C.B.; David, E.A.; Patra, D.; Sengupta, R.; Amiar, S.; Su, Y.; Stahelin, R.V.  SARS-CoV-2 viral budding and entry can be modeled using BSL-2 level virus-like particles.  J. Biol. Chem. 2021, 296, 100103.  https://doi.org/10.1074/jbc.RA120.016148.
  * Li et al. (2018) Li, S.; Roy, P.; Travesset, A.; Zandi, R.  Why large icosahedral viruses need scaffolding proteins.  Proc. Natl. Acad. Sci. USA 2018, 115, 10971–10976.  https://doi.org/10.1073/pnas.1807706115.
  * Panahandeh et al. (2022) Panahandeh, S.; Li, S.; Dragnea, B.; Zandi, R.  Virus Assembly Pathways Inside a Host Cell.  ACS Nano 2022, 16, 317–327.  https://doi.org/10.1021/acsnano.1c06335.
  * Perlmutter et al. (2013) Perlmutter, J.D.; Qiao, C.; Hagan, M.F.  Viral genome structures are optimal for capsid assembly.  eLife 2013, 2013, 1–21.  https://doi.org/10.7554/eLife.00632.
  * Chen et al. (2007) Chen, T.; Zhang, Z.; Glotzer, S.C.  A Precise Packing Sequence for Self-Assembled Convex Structures.  Proc. Natl. Acad. Sci. USA 2007, 104, 717.  https://doi.org/10.1073/pnas.060423910.
  * Fejer et al. (2010) Fejer, S.; Chakrabarti, D.; Wales, D.  Emergent Complexity from Simple Anisotropic Building Blocks: Shells, Tubes, and Spirals.  Nano Lett. 2010, 4, 219.  https://doi.org/10.1021/nn9013565.
  * Rapaport (2004) Rapaport, D.C.  Self-assembly of polyhedral shells: A molecular dynamics study.  Phys. Rev. E 2004, 70, 051905.  https://doi.org/10.1103/PhysRevE.70.051905.
  * Patel et al. (2015) Patel, N.; Dykeman, E.C.; Coutts, R.H.A.; Lomonossoff, G.P.; Rowlands, D.J.; Phillips, S.E.V.; Ranson, N.; Twarock, R.; Tuma, R.; Stockley, P.G.  Revealing the density of encoded functions in a viral RNA.  Proc. Natl. Acad. Sci. USA 2015, 112, 2227–2232.  https://doi.org/10.1073/pnas.1420812112.
  * Vernizzi and Olvera de la Cruz (2007) Vernizzi, G.; Olvera de la Cruz, M.  Faceting ionic shells into icosahedra via electrostatics.  Proc. Natl. Acad. Sci. USA 2007, 104, 18382–18386.  https://doi.org/10.1073/pnas.0703431104.
  * Paquay et al. (2016) Paquay, S.; Kusumaatmaja, H.; Wales, D.J.; Zandi, R.; Van Der Schoot, P.  Energetically favoured defects in dense packings of particles on spherical surfaces.  Soft Matter 2016, 12, 5708–5717.  https://doi.org/10.1039/C6SM00489J.
  * Collins et al. (2021) Collins, L.T.; Elkholy, T.; Mubin, S.; Hill, D.; Williams, R.; Ezike, K.; Singhal, A.  Elucidation of SARS-Cov-2 Budding Mechanisms through Molecular Dynamics Simulations of M and E Protein Complexes.  J. Phys. Chem. Lett. 2021, 12, 12249–12255.  https://doi.org/10.1021/acs.jpclett.1c02955.
  * Yu et al. (2021) Yu, A.; Pak, A.J.; He, P.; Monje-Galvan, V.; Casalino, L.; Gaieb, Z.; Dommer, A.C.; Amaro, R.E.; Voth, G.A.  A multiscale coarse-grained model of the SARS-CoV-2 virion.  Biophys. J. 2021, 120, 1097–1104.  https://doi.org/10.1016/j.bpj.2020.10.048.
  * Cao et al. (2022) Cao, Y.; Yang, R.; Wang, W.; Jiang, S.; Yang, C.; Liu, N.; Dai, H.; Lee, I.; Meng, X.; Yuan, Z.  Probing the formation, structure and free energy relationships of M protein dimers of SARS-CoV-2.  Comput. Struct. Biotechnol. J. 2022, 20, 573–582.  https://doi.org/10.1016/j.csbj.2022.01.007.
  * Ruan et al. (2018) Ruan, L.; Hadden, J.A.; Zlotnick, A.; Ou, J.H.J.  Assembly Properties of Hepatitis B Virus Core Protein Mutants Correlate with Their Resistance to Assembly-Directed Antivirals.  J. Virol. 2018, 92, e01082-18.  https://doi.org/10.1128/JVI.01082-18.
  * Nair et al. (2018) Nair, S.; Li, L.; Francis, S.; Turner, W.W.; VanNieuwenhze, M.; Zlotnick, A.  Use of a Fluorescent Analogue of a HBV Core Protein-Directed Drug To Interrogate an Antiviral Mechanism.  J. Am. Chem. Soc. 2018, 140, 15261–15269.  https://doi.org/10.1021/jacs.8b07988.
  * Perlmutter et al. (2014) Perlmutter, J.D.; Perkett, M.R.; Hagan, M.F.  Pathways for Virus Assembly Around Nucleic Acids.  J. Mol. Biol. 2014, 426, 3148–3165.  https://doi.org/https://doi.org/10.1016/j.jmb.2014.07.004.
  * Hagan (2014) Hagan, M.F.  Modeling Viral Capsid Assembly.  Adv. Chem. Phys. 2014, 155, 1.  https://doi.org/10.1002/9781118755815.ch01.
  * Twarock and Luque (2019) Twarock, R.; Luque, A.  Structural Puzzles in Virology Solved with an Overarching Icosahedral Design Principle.  Nat. Commun. 2019, 10, 4414.  https://doi.org/https://doi.org/10.1038/s41467-019-12367-3.
  * Dykeman et al. (2014) Dykeman, E.C.; Stockley, P.G.; Twarock, R.  Solving a Levinthal’s Paradox for Virus Assembly Identifies a Unique Antiviral Strategy.  Proc. Natl. Acad. Sci. USA 2014, 111, 5361–5366.  https://doi.org/10.1073/pnas.1319479111.
  * Garmann et al. (2016) Garmann, R.F.; Comas-Garcia, M.; Knobler, C.M.; Gelbart, W.M.  Physical Principles in the Self-Assembly of a Simple Spherical Virus.  Acc. Chem. Res. 2016, 49, 48–55.  https://doi.org/10.1021/acs.accounts.5b00350.
  * Garmann et al. (2019) Garmann, R.F.; Goldfain, A.M.; Manoharan, V.N.  Measurements of the Self-Assembly Kinetics of Individual Viral Capsids Around Their RNA Genome.  Proc. Natl. Acad. Sci. USA 2019, 116, 22485–22490.  https://doi.org/10.1073/pnas.1909223116.
  * Chevreuil et al. (2018) Chevreuil, M.; Law-Hine, D.; Chen, J.; Bressanelli, S.; Combet, S.; Constantin, D.; Degrouard, J.; Möller, J.; Zeghal, M.; Tresset, G.  Nonequilibrium Self-Assembly Dynamics of Icosahedral Viral Capsids Packaging Genome or Polyelectrolyte.  Nat. Commun. 2018, 9, 3071.  https://doi.org/10.1038/s41467-018-05426-8.
  * Zandi et al. (2020) Zandi, R.; Dragnea, B.; Travesset, A.; Podgornik, R.  On virus growth and form.  Phys. Rep. 2020, 847, 1–102.  https://doi.org/10.1016/j.physrep.2019.12.005.
  * Comas-Garcia et al. (2014) Comas-Garcia, M.; Garmann, R.F.; Singaram, S.W.; Ben-Shaul, A.; Knobler, C.M.; Gelbart, W.M.  Characterization of viral capsid protein self-assembly around short single-stranded RNA.  J. Phys. Chem. B 2014, 118, 7510–7519.  https://doi.org/10.1021/jp503050z.
  * Zeng et al. (2020) Zeng, W.; Liu, G.; Ma, H.; Zhao, D.; Yang, Y.; Liu, M.; Mohammed, A.; Zhao, C.; Yang, Y.; Xie, J.; et al.  Biochemical characterization of SARS-CoV-2 nucleocapsid protein.  Biochem. Biophys. Res. Commun. 2020, 527, 618–623.  https://doi.org/10.1016/j.bbrc.2020.04.136.
  * Yang et al. (2021) Yang, M.; He, S.; Chen, X.; Huang, Z.; Zhou, Z.; Zhou, Z.; Chen, Q.; Chen, S.; Kang, S.  Structural Insight Into the SARS-CoV-2 Nucleocapsid Protein C-Terminal Domain Reveals a Novel Recognition Mechanism for Viral Transcriptional Regulatory Sequences.  Front. Chem. 2021, 8, 1–12.  https://doi.org/10.3389/fchem.2020.624765.
  * McBride et al. (2014) McBride, R.; van Zyl, M.; Fielding, B.C.  The coronavirus nucleocapsid is a multifunctional protein.  Viruses 2014, 6, 2991–3018.  https://doi.org/10.3390/v6082991.
  * Cubuk et al. (2021) Cubuk, J.; Alston, J.J.; Incicco, J.J.; Singh, S.; Stuchell-Brereton, M.D.; Ward, M.D.; Zimmerman, M.I.; Vithani, N.; Griffith, D.; Wagoner, J.A.; et al.  The SARS-CoV-2 nucleocapsid protein is dynamic, disordered, and phase separates with RNA.  Nat. Commun. 2021, 12, 1936.  https://doi.org/10.1038/s41467-021-21953-3.
  * Perdikari et al. (2020) Perdikari, T.M.; Murthy, A.C.; Ryan, V.H.; Watters, S.; Naik, M.T.; Fawzi, N.L.  SARS-CoV-2 nucleocapsid protein phase-separates with RNA and with human hnRNPs.  EMBO J. 2020, 39, e106478.  https://doi.org/10.15252/embj.2020106478.
  * Lu et al. (2021) Lu, S.; Ye, Q.; Singh, D.; Cao, Y.; Diedrich, J.K.; Yates, J.R.; Villa, E.; Cleveland, D.W.; Corbett, K.D.  The SARS-CoV-2 nucleocapsid phosphoprotein forms mutually exclusive condensates with RNA and the membrane-associated M protein.  Nat. Commun. 2021, 12, 502.  https://doi.org/10.1038/s41467-020-20768-y.
  * Zinzula et al. (2021) Zinzula, L.; Basquin, J.; Bohn, S.; Beck, F.; Klumpe, S.; Pfeifer, G.; Nagy, I.; Bracher, A.; Hartl, F.U.; Baumeister, W.  High-resolution structure and biophysical characterization of the nucleocapsid phosphoprotein dimerization domain from the Covid-19 severe acute respiratory syndrome coronavirus 2.  Biochem. Biophys. Res. Commun. 2021, 538, 54–62.  https://doi.org/10.1016/j.bbrc.2020.09.131.
  * Dinesh et al. (2020) Dinesh, D.C.; Chalupska, D.; Silhan, J.; Koutna, E.; Nencka, R.; Veverka, V.; Boura, E.  Structural basis of RNA recognition by the SARS-CoV-2 nucleocapsid phosphoprotein.  PLoS Pathog. 2020, 16, e1009100.  https://doi.org/10.1371/journal.ppat.1009100.
  * Cao et al. (2021) Cao, C.; Cai, Z.; Xiao, X.; Rao, J.; Chen, J.; Hu, N.; Yang, M.; Xing, X.; Wang, Y.; Li, M.; et al.  The architecture of the SARS-CoV-2 RNA genome inside virion.  Nat. Commun. 2021, 12, 3917.  https://doi.org/10.1038/s41467-021-22785-x.
  * Lipferta et al. (2014) Lipferta, J.; Skinnera, G.M.; Keegstraa, J.M.; Hensgensa, T.; Jagera, T.; Dulina, D.; Köber, M.; Yu, Z.; Donkers, S.P.; Chou, F.C.; et al.  Double-stranded RNA under force and torque: Similarities to and striking differences from double-stranded DNA.  Proc. Natl. Acad. Sci. USA 2014, 111, 15408–15413.  https://doi.org/10.1073/pnas.1407197111.
  * Arias-Gonzalez (2014) Arias-Gonzalez, J.R.  Single-molecule portrait of DNA and RNA double helices.  Integr. Biol. 2014, 6, 904–925.  https://doi.org/10.1039/c4ib00163j.
  * Li et al. (2017) Li, S.; Erdemci-Tandogan, G.; Wagner, J.; Van Der Schoot, P.; Zandi, R.  Impact of a nonuniform charge distribution on virus assembly.  Phys. Rev. E 2017, 96, 022401.  https://doi.org/10.1103/PhysRevE.96.022401.
  * Nguyen and Bruinsma (2006) Nguyen, T.T.; Bruinsma, R.F.  RNA condensation and the wetting transition.  Phys. Rev. Lett. 2006, 97, 108102.  https://doi.org/10.1103/PhysRevLett.97.108102.
  * Manning (1978) Manning, G.S.  The molecular theory of polyelectrolyte solutions with applications to the electrostatic properties of polynucleotides.  Q. Rev. Biophys. 1978, 11, 179–246.  https://doi.org/10.1017/S0033583500002031.
  * Anderson et al. (2020) Anderson, J.A.; Glaser, J.; Glotzer, S.C.  HOOMD-blue: A Python package for high-performance molecular dynamics and hard particle Monte Carlo simulations.  Comput. Mater. Sci. 2020, 173, 109363.  https://doi.org/https://doi.org/10.1016/j.commatsci.2019.109363.
  * Stukowski (2010) Stukowski, A.  Visualization and analysis of atomistic simulation data with OVITO-the Open Visualization Tool.  Modelling Simul. Mater. Sci. Eng. 2010, 18, 015012.  https://doi.org/10.1088/0965-0393/18/1/015012.
  * Panahandeh et al. (2018) Panahandeh, S.; Li, S.; Zandi, R.  The Equilibrium Structure of Self-Assembled Protein Nano-Cages.  Nanoscale 2018, 10, 22802–22809.
  * Belyi and Muthukumar (2006) Belyi, V.A.; Muthukumar, M.  Electrostatic origin of the genome packing in viruses.  Proc. Natl. Acad. Sci. USA 2006, 103, 17174–17178.  https://doi.org/10.1073/pnas.0608311103.
  * de Bruijn et al. (2022) de Bruijn, R.; Wielstra, P.; Calcines-Cruz, C.; van Waveren, T.; Hernandez-Garcia, A.; van der Schoot, P.  A nucleation-and-growth model for the packaging of genome in linear virus-like particles: Impact of multiple packaging signals.  bioRxiv 2022.  https://doi.org/10.1101/2022.02.24.481677.
  * Comas-Garcia et al. (2012) Comas-Garcia, M.; Cadena-Nava, R.D.; Rao, A.L.N.; Knobler, C.M.; Gelbart, W.M.  In Vitro Quantification of the Relative Packaging Efficiencies of Single-Stranded RNA Molecules by Viral Capsid Protein .  J. Virol. 2012, 86, 12271–12282.  https://doi.org/10.1128/jvi.01695-12.
  * DeDiego et al. (2007) DeDiego, M.L.; Álvarez, E.; Almazán, F.; Rejas, M.T.; Lamirande, E.; Roberts, A.; Shieh, W.J.; Zaki, S.R.; Subbarao, K.; Enjuanes, L.  A Severe Acute Respiratory Syndrome Coronavirus That Lacks the E Gene Is Attenuated In Vitro and In Vivo.  J. Virol. 2007, 81, 1701–1713.  https://doi.org/10.1128/jvi.01467-06.
  * Xu et al. (2020) Xu, R.; Shi, M.; Li, J.; Song, P.; Li, N.  Construction of SARS-CoV-2 Virus-Like Particles by Mammalian Expression System.  Front. Bioeng. Biotechnol. 2020, 8, 1–6.  https://doi.org/10.3389/fbioe.2020.00862.
  * Thomas (2020) Thomas, S.  The structure of the membrane protein of sars-cov-2 resembles the sugar transporter semisweet.  Pathog. Immun. 2020, 5, 342–363.  https://doi.org/10.20411/pai.v5i1.377.
  * Heo and Feig (2020) Heo, L.; Feig, M.  Modeling of Severe Acute Respiratory Syndrome Coronavirus 2 (SARS-CoV-2) Proteins by Machine Learning and Physics-Based Refinement.  bioRxiv 2020.  https://doi.org/10.1101/2020.03.25.008904.
  * Dolan et al. (2022) Dolan, K.A.; Dutta, M.; Kern, D.M.; Kotecha, A.; Voth, G.A.; Brohawn, S.G.  Structure of SARS-CoV-2 M protein in lipid nanodiscs.  bioRxiv 2022.  https://doi.org/10.1101/2022.06.12.495841.
  * Kuo et al. (2016) Kuo, L.; Koetzner, C.A.; Masters, P.S.  A key role for the carboxy-terminal tail of the murine coronavirus nucleocapsid protein in coordination of genome packaging.  Virology 2016, 494, 100–107.  https://doi.org/10.1016/j.virol.2016.04.009.
  * Verma et al. (2006) Verma, S.; Bednar, V.; Blount, A.; Hogue, B.G.  Identification of Functionally Important Negatively Charged Residues in the Carboxy End of Mouse Hepatitis Coronavirus A59 Nucleocapsid Protein.  J. Virol. 2006, 80, 4344–4355.  https://doi.org/10.1128/jvi.80.9.4344-4355.2006.
  * Luo et al. (2006) Luo, H.; Wu, D.; Shen, C.; Chen, K.; Shen, X.; Jiang, H.  Severe acute respiratory syndrome coronavirus membrane protein interacts with nucleocapsid protein mostly through their carboxyl termini by electrostatic attraction.  Int. J. Biochem. Cell Biol. 2006, 38, 589–599.  https://doi.org/10.1016/j.biocel.2005.10.022.
  * Siber et al. (2012) Siber, A.; Bozic, A.L.; Podgornik, R.  Energies and pressures in viruses: Contribution of nonspecific electrostatic interactions.  Phys. Chem. Chem. Phys. 2012, 14, 3746.  https://doi.org/10.1039/c1cp22756d.
  * Savastano et al. (2020) Savastano, A.; Ibáñez de Opakua, A.; Rankovic, M.; Zweckstetter, M.  Nucleocapsid protein of SARS-CoV-2 phase separates into RNA-rich polymerase-containing condensates.  Nat. Commun. 2020, 11, 6041.  https://doi.org/10.1038/s41467-020-19843-1.
  * Carlson et al. (2020) Carlson, C.R.; Asfaha, J.B.; Ghent, C.M.; Howard, C.J.; Hartooni, N.; Safari, M.; Frankel, A.D.; Morgan, D.O.  Phosphoregulation of Phase Separation by the SARS-CoV-2 N Protein Suggests a Biophysical Basis for its Dual Functions.  Mol. Cell 2020, 80, 1092–1103.e4.  https://doi.org/10.1016/j.molcel.2020.11.025.
  * Wang et al. (2021) Wang, S.; Dai, T.; Qin, Z.; Pan, T.; Chu, F.; Lou, L.; Zhang, L.; Yang, B.; Huang, H.; Lu, H.; et al.  Targeting liquid–liquid phase separation of SARS-CoV-2 nucleocapsid protein promotes innate antiviral immunity by elevating MAVS activity.  Nat. Cell Biol. 2021, 23, 718–732.  https://doi.org/10.1038/s41556-021-00710-0.
  * Wu et al. (2021) Wu, Y.; Ma, L.; Cai, S.; Zhuang, Z.; Zhao, Z.; Jin, S.; Xie, W.; Zhou, L.; Zhang, L.; Zhao, J.; et al.  RNA-induced liquid phase separation of SARS-CoV-2 nucleocapsid protein facilitates NF-$\kappa$B hyper-activation and inflammation.  Signal Transduct. Target. Ther. 2021, 6.  https://doi.org/10.1038/s41392-021-00575-7.
  * Scherer et al. (2022) Scherer, K.M.; Mascheroni, L.; Carnell, G.W.; Wunderlich, L.C.; Makarchuk, S.; Brockhoff, M.; Mela, I.; Fernandez-Villegas, A.; Barysevich, M.; Stewart, H.; et al.  SARS-CoV-2 nucleocapsid protein adheres to replication organelles before viral assembly at the Golgi/ERGIC and lysosome-mediated egress.  Sci. Adv. 2022, 8, eabl4895.  https://doi.org/10.1126/sciadv.abl4895.
  * Ruch and Machamer (2012) Ruch, T.R.; Machamer, C.E.  The coronavirus E protein: Assembly and beyond.  Viruses 2012, 4, 363–382.  https://doi.org/10.3390/v4030363.
  * Schoeman and Fielding (2019) Schoeman, D.; Fielding, B.C.  Coronavirus envelope protein: Current knowledge.  Virol. J. 2019, 16, 69.  https://doi.org/10.3390/ijerph17082932.

Supplementary Materials: Biophysical modeling of SARS-CoV-2 assembly: genome
condensation and budding

##  1 Supplementary Figure

Figure S1:  Dimerization of the N proteins for various domain sizes and the
strengths of interaction between the C-terminals. The upper row shows N
proteins with an N-terminal size of $1.2a$ and a C-terminal size of $0.8a$,
where $a$ is the system length corresponding to
$3\text{\,}\mathrm{n}\mathrm{m}$. The size of N-terminal and C-terminal are
$1.1a$ and $0.9a$, respectively, in the lower row. The size of the linker
region N2 is kept constant and is $0.5a$. The N-terminals of N proteins are
colored gray and the C-terminals of N proteins have different bright colors
based on their cluster index (see SI Fig. S2 for the definition of the cluster
index). The histograms in the figure show the count of the N protein
oligomers. For the weak C-terminal interactions, we see that nearly all N
proteins are monomers. As the interaction strength increases, more dimers and
trimers form in solution. The interaction strengths $\epsilon_{N_{3}N_{3}}$
are $10$, $15$, $20$ and $50$ from left to right. We find that dimers are the
dominant structure at $\epsilon_{N_{3}N_{3}}=20$.

Figure S2: Cluster analysis of NPCs along RNP for weak (left) and strong
(right) RNA-N protein interactions. We recall that when the distance between
the N-terminals of the N proteins are less than a cut-off distance, they are
defined as a cluster assigned with a specific color and a unique cluster
index. The plots display the number of N proteins in each cluster, where the
clusters are sorted by their sizes. RNA is wrapped around the N proteins and
is not shown in the figure. Figure S3: RNP condensation for various strengths
of interactions between N proteins. The left figure shows the cluster analysis
for various N3-N3 interactions while the right figures display the structures
of RNP, NPCs and RNA from left to right. The strength of interaction between
the C-terminal of N proteins ($\epsilon_{N_{3}N_{3}}$) increases from top to
bottom. As the strength of the N-N interaction increases, the number of N
proteins in each cluster increases, but the changes are small. Also, the size
of RNP remains more or less the same. We note again that the N proteins in
each cluster have the same color and a unique cluster index, and that the
clusters are sorted by their sizes. The other parameters used are $Z_{g}=-2$,
and N protein concentration is $25$\mathrm{\SIUnitSymbolMicro}\mathrm{M}$$.
Figure S4: Illustration of the role of the spontaneous curvature in the
budding process. The M protein has a cylindrical structure (no spontaneous
curvature). All three spherical beads making the M protein have the same
diameter. In particular, the transmembrane particle (purple) has the same size
as endodomain particle (white). The transmembrane interaction strength is
$\epsilon_{M_{2}M_{2}}=1$ and the endodomain (white) interaction strength
$\epsilon_{M_{3}M_{3}}$ is $0$, $1$ and $2$ from top to bottom. There is no
budding in the absence of spontaneous curvature. Figure S5: Cluster analysis
of the N proteins along RNP during the budding. The left figures show the
number of N proteins in different clusters along RNP during the budding
process vs. the cluster index for scenario 1 (top) and scenario 2 (bottom).
The insets show the configurations for which the clusters along RNPs were
studied. Recall that the N proteins in the same cluster have the same color
and index number, and that they are sorted by the size of the clusters. The
right figures show the total number of N proteins attached to RNA as a
function of time during the budding process. Since in the scenario 1, RNP
condensation and budding take place at the same time, the number of N proteins
increases as a function of time. However, in scenario 2, RNP has already
formed and then has been transported near ERGIC before budding. Figure S6:
RNP budding with various N protein-RNA interactions for scenarios 1 (top) and
2 (bottom) at $t=16,000s$. There are three pairs of figures in each row. The
left figure in each pair shows RNP and the right one shows the N proteins
only. In scenario 1, ERGIC is initially located in the vicinity of RNA and the
N proteins, and the RNA-N protein condensation and budding happen at the same
time. In scenario 2, pre-condensed RNA-N proteins are placed near ERGIC, which
later bud into a vesicle. See the assembly and budding movies SI Video S6 and
Video S7. The figure also shows the simulations at various genome effective
charge densities with $Z_{g}$ = $-2$, $-5$, $-10$ from left to right. We find
that when the genome effective charge density is small, both scenarios (1 and
2) are plausible resulting in the formation of viral particles. However, as
the genome effective charge density increases, more N proteins get absorbed to
RNA, which makes it difficult for them to reorient and expose their C-terminal
domains to interact with M proteins.  Figure S7: Snapshots of the simulations
of the RNP budding when the endodomains of M proteins are positively charged.
The figure illustrates that the attractive electrostatic interactions between
RNA and the M proteins is not sufficient for the RNA budding (left figure),
and the presence of N proteins are essential for RNA packaging and budding
(right figure). Note that a stronger N3-M3 interaction
($\epsilon_{N_{3}M_{3}}=20$) is required to trigger the budding compared to
the neutral M protein case where a much weaker $\epsilon_{N_{3}M_{3}}=10$
results in the budding and assembly of the virus (Fig. 7). Both simulations
are performed for $8000s$ and other parameters used are $L=800$,
$\epsilon_{McMc}=1$, $\epsilon_{M_{2}M_{2}}=1$, $\epsilon_{M_{3}M_{3}}=0$,
$Z_{g}=-2$ and $\epsilon_{N_{3}N_{3}}=20$. We recall that the electrostatic
repulsion between M3 and M3 is assumed to be screened and thus is zero. Note
that for shorter chains, it’s possible to package RNA in the absence of N
proteins, but not relevant to the case of coronaviruses. Figure S8: Snapshots
of simulations of the RNP budding using a linear RNA. Note, only shorter
chains with $L=400$ (upper row) can be completely encapsidated. The M proteins
are initially randomly distributed, and ERGIC has been fully relaxed. The
$25$\mathrm{\SIUnitSymbolMicro}\mathrm{M}$$ protein concentration was used to
condense RNA. The resulting RNP is then located near ERGIC. As illustrated in
the figure, when $L=800$ (the bottom row), the chain cannot be fully packaged
and a part of the chain will remain outside of the envelope. For the last two
figures in each row, we show only the configuration of the N proteins. RNA has
wrapped around the N proteins but is not shown in the figure. Note that RNP
could be packaged if RNA were modeled as a branched polymer with the same
length ($L=800$). The other parameters used are $\epsilon_{McMc}=1$,
$\epsilon_{M_{2}M_{2}}=1$, $\epsilon_{M_{3}M_{3}}=0$, $Z_{g}=-2$,
$\epsilon_{N_{3}N_{3}}=20$, and $\epsilon_{N_{3}M_{3}}=10$.  Figure S9:
Formation of the virus like particles through the budding of N proteins in the
absence of RNA. The parameters used are $\epsilon_{M_{2}M_{2}}=1$,
$\epsilon_{M_{3}M_{3}}=0$, $\epsilon_{N_{3}N_{3}}=20$, and
$\epsilon_{N_{3}M_{3}}=10$. Note that in the absence of the N proteins, the M
proteins cannot bud with the same M-M interaction strength parameters.

##  2 Supplementary Video

###  Video S1 RNA condensation by the N proteins

The N proteins of SARS-CoV-2 condense RNA. The concentration of proteins is
$25$\mathrm{\SIUnitSymbolMicro}\mathrm{M}$$. RNA is modeled as a branched
polymer with the number of branch points $N$ = $30$ and the genome effective
charge density $Z_{g}$ = $-5$.

###  Video S2 The impact of the secondary structure of RNA on its condensation

N proteins condense RNA with different number of branch points. The movies
show RNP with the genome in blue and N proteins in red and black. The branch
points are in green and the end points are in yellow. The total running time
is $5000s$. The number of branch points from left to right is $N$ = $0$, $10$,
$20$, and $30$.

###  Video S3 N proteins condense a linear genome for various N protein
concentrations and effective genome charge densities

N proteins condense RNA modeled as a linear polymer for different protein
concentrations and effective genome charge densities. The movies show RNP with
genome in blue and N proteins in red and black. The total running time is
$5000s$. The genome effective charge density increases from left to right with
$Z_{g}$ = $-2$, $-5$, and $-10$.

###  Video S4 N proteins condense RNA at various N protein concentrations and
effective genome charge densities

N proteins condense RNA modeled as a branched polymer at various protein
concentrations and effective genome charge densities. The movies show RNP with
RNA in blue and N proteins in red and black. The number of branch points is
$N$ = $30$. The branched points are colored green and the end points are
colored yellow. The total running time is $5000s$. The genome effective charge
density increases from left to right with $Z_{g}$ = $-2$, $-5$, and $-10$.

###  Video S5 The role of the M protein interactions in budding at the ERGIC
membrane

Budding through the ERGIC membrane for various interaction strengths between
the M protein transmembrane domains on the one hand and between the
endodomains on the other hand. The lipid-lipid interaction strength is kept
constant $\epsilon_{McMc}=1$ for all simulations. The membrane stretching
modulus is $k_{s}=20a^{-2}$ and the bending modulus is $k_{b}=20$. The total
running time is $3000s$.

###  Video S6 RNP budding through the ERGIC membrane for scenario 1

RNP budding through the ERGIC membrane for scenario 1. The N proteins interact
with the RNA and form many clusters while simultaneously interacting with the
M proteins embedded in the lipid membrane. Thus, the RNP condensation and
budding occur simultaneously at the ERGIC membrane.

###  Video S7 RNP budding through the ERGIC membrane for scenario 2

RNP budding through the ERGIC membrane for scenario 2. First, the N proteins
interact with the RNA to form the condensed RNP complex, which later interacts
with the M proteins that are embedded in the lipid membrane. Budding through
the intercellular membrane then completes virus formation.

###  Video S8 RNP budding through the ERGIC membrane for the randomly
distributed M proteins

RNP budding through the ERGIC membrane movie for the randomly distributed M
proteins. See SI Video S7 for the budding through a circular cap made of M
proteins.