\section{Introduction}
Every quantitative analysis of the primordial black hole (PBH) number
and mass spectrum \cite{carr75}
requires knowledge of the threshold parameter, $\delta_{\rm c}$, separating
perturbations that form
black holes from those that do not, and the resulting black hole mass,
$M_{\rm bh}$, as a function of distance from the threshold. In order to
determine $\delta_{\rm c}$ and $M_{\rm bh}$ for various initial conditions,
we performed one-dimensional,
general relativistic simulations of the hydrodynamics of PBH
formation in the radiation-dominated phase of the early
universe. Three families of perturbation shapes were
chosen to represent ``generic'' classes of initial data, reflecting the lack of
information about the specific shape of primordial fluctuations. The
numerical technique is sketched
in Section \ref{numerics}. Defined as the excess mass within the
horizon sphere at the onset of the collapse, we find $\delta_{\rm c} \approx 0.7$
for all three perturbation shapes, indicating that the threshold value
may indeed be universal (Section \ref{hydro}). A numerical confirmation of the
previously suggested power-law scaling of $M_{\rm bh}$ with
$\delta - \delta_{\rm c}$ \cite{niejed97}, related to the well-known behavior of
collapsing space--times at the critical point of black hole formation
\cite{chop93}, is presented in Section \ref{scaling}. In this framework,
the PBH mass spectrum is determined by the dimensionless coefficient,
$K$, the scaling exponent, $\gamma$, and the initial horizon mass,
$M_{\rm h}$, such that
\begin{equation}
\label{scale}
M_{\rm bh} = K M_{\rm h} (\delta - \delta_{\rm c})^\gamma\,\,.
\end{equation}
We provide numerical results for $K$ and $\gamma$ for
the three perturbation families. A more detailed description of the
numerical technique and further results will be presented in a
forthcoming publication \cite{niejed98}.

\section{Numerical technique}
\label{numerics}
The dynamics of collapsing density perturbations in the Early Universe
are determined by the general relativistic
equations of motion for a perfect fluid, the field equations, the first
law of thermodynamics, and a radiation-dominated equation of state. The
assumption of spherical symmetry is well justified for large
fluctuations in a Gaussian distribution \cite{barea86}, reducing the
problem to one spatial dimension.

For our simulations, we chose the formulation of the hydrodynamical
equations by Hernandez and Misner \cite{hermis66} as implemented by
Baumgarte et al.~ \cite{baum95}. Based on the original equations by
Misner and Sharp
\cite{missha64}, Hernandez and Misner proposed to exchange the
Schwarzschild time variable, $t$, with the outgoing null coordinate,
$u$. In so doing, the hydrodynamical equations retain the Lagrangian
character of the Misner--Sharp equations but avoid crossing into the
event horizon of a black hole once it has formed. Covering the entire
space--time outside while asymptotically approaching the event horizon,
the Hernandez--Misner equations are perfectly suited to
follow the evolution of a black hole for long times after its
formation without encountering coordinate singularities. This allowed
us, in principle, to study the
accretion of material onto newly formed PBHs for arbitrarily long times (in
contrast with earlier calculations \cite{Nade}).
Since the expanding outer regions of our simulated piece of the
universe are most conveniently tracked in a comoving numerical
reference frame, the Lagrangian form of the Hernandez--Misner equations
is their second major asset. It also provides a simple prescription
for the outer boundary condition, which is defined to match the exact
solution of the Friedmann equations for a radiation
dominated flat universe. Hence, the pressure follows the
analytic solution
\begin{equation}
P = P_0 \left(\frac{\tau}{\tau_0}\right)^{-2}\,\,,
\end{equation}
where $\tau$ is the proper time of the outermost fluid element
(corresponding to the cosmological time $t$ in a
Friedmann--Robertson--Walker (FRW) universe) and
$P_0$ and $\tau_0$ are the initial values for pressure and proper
time.

\section{Threshold for black hole formation}
\label{hydro}
We studied the spherically symmetric evolution of three families of curvature
perturbations. Initial conditions were chosen to be
perturbations of the energy density, $\epsilon = \rho_0 e$, in
unperturbed Hubble flow specified
at horizon crossing. The first family of perturbations is described by
a Gaussian-shaped overdensity that asymptotically approaches the FRW
solution at large radii. The other two families of initial conditions
involve a spherical
Mexican Hat function and a fourth order polynomial. These functions
are characterized by
rarefaction regions outside of the horizon radius, $R_{\rm h}$, that
identically compensate
for the additional mass of the overdensities inside the horizon
volume, so that the mass derived from the total integrated
density profile is equal to that of an unperturbed FRW solution.
In our numerical experiments, the amplitude, $A$, of the perturbations
is used to tune the
initial conditions to sub- or supercriticality with respect to black
hole formation. The critical amplitude, $A_{\rm c}$, is
strongly shape-dependent, varying between $A_{\rm c} = 3.04$ for
Mexican-Hat-shaped perturbations and $A_{\rm c} = 2.05$ for the
Gaussian curve. If, however, we define the control
parameter $\delta$ as the additional mass
inside $R_{\rm h}$ in units of the horizon mass, we find strikingly similar
values --- $\delta_{\rm c} = 0.67$ (Mexican Hat), $\delta_{\rm c} = 0.70$ (Gaussian curve), and
$\delta_{\rm c} = 0.71$ (polynomial) --- for all three families of initial data in our
study. This number is considerably greater than the previously
employed threshold $\delta_{\rm c} = 1/3$
following from analytical estimates \cite{carr75}. Given
the qualitative difference of the functional forms of the different
perturbation families, the result that $\delta_{\rm c}$ of the Gaussian
perturbation lies in between
the critical values of the mass compensated functions is
surprising. Our results suggest that $\delta_{\rm c} \approx 0.7$ (with $\delta$
defined as above) is a universal statement, i.e., true for all
perturbation shapes in the radiation-dominated regime. This remains to
be verified by means of additional experiments.

\section{Scaling of PBH masses with distance to the threshold}
\label{scaling}

For a variety of matter models, it is well-known that the dynamics of
near-critical collapse
exhibit continuous or discrete self-similarity and power-law scaling
of the black hole mass with the offset from the critical point
\{Eq.~(\ref{scale}) \cite{chop93,critrev}\}. In particular, Evans
and Coleman \cite{evans94} found
self-similarity and mass scaling in numerical experiments of a
collapsing radiation fluid. They numerically determined the scaling exponent
$\gamma \approx 0.36$, followed by a linear perturbation analysis of
the critical solution by Koike et al.~ \cite{koike95} that yielded $\gamma \approx
0.3558$. Until recently, it was believed that entering the scaling regime
requires a degree of fine-tuning of the initial data
that is unnatural for any astrophysical application. It was noted
\cite{niejed97} that fine-tuning to criticality
occurs naturally in the case of PBHs forming from a steeply declining
distribution of primordial density fluctuations, as generically predicted by
inflationary scenarios. In the radiation-dominated
cosmological epoch, the only difference with the fluid collapse
studied by Evans and Coleman \cite{evans94} is the expanding,
finite-density-background space--time of a FRW universe. Assuming that
self-similarity and mass scaling are consequences of an intermediate
asymptotic solution that is independent of the asymptotic boundary
conditions, Eq.~(\ref{scale}) is applicable to PBH masses,
allowing the derivation of a universal PBH initial mass function
\cite{niejed97}.
\begin{figure}
\epsfysize=8cm
\epsfbox{scaling.eps}
\vspace{-13mm}
\caption{\label{f7} Black hole masses as a function of $\delta - \delta_{\rm c}$
for three different
perturbation shape families. The best fit parameters to equation
(\ref{scale}) are:
$\gamma = 0.36$, $K = 2.85$, $\delta_{\rm c} = 0.6745$ (Mexican Hat
perturbation, triangles); $\gamma = 0.37$, $K = 2.39$, $\delta_{\rm c} = 0.7122$
(polynomial perturbation, crosses); $\gamma = 0.34$, $K = 11.9$,
$\delta_{\rm c} = 0.7015$ (Gaussian curve perturbation, diamonds).}
\end{figure}
Figure (\ref{f7}) presents numerical evidence for mass scaling
according to Eq.~(\ref{scale}) in black hole collapse in an
asymptotic FRW space--time. All three perturbation families give rise
to scaling solutions with a scaling exponent $\gamma
\approx 0.36$.

\section{Conclusions}
This work discusses numerical collapse simulations of three
generic families of energy density perturbations, one with a finite
total excess mass
with respect to the unperturbed FRW solution and two mass compensated
ones. Among various possible definitions for the collapse control parameter
$\delta$, the total excess
gravitational mass of the perturbed space--time with respect to the
unperturbed FRW background enclosed in the initial horizon volume is
the only one that gives rise to a
similar threshold value for all three shape families, $\delta_{\rm c} \approx
0.7$. Whether this result is an indication for universality of $\delta_{\rm c}$
in this specific definition needs to be verified with the help of
additional simulations using a larger sample of initial perturbation
shapes.

The previously suggested \cite{niejed97} scaling relation between
$M_{\rm bh}$ and $\delta - \delta_{\rm c}$, based on the analogy with critical
phenomena observed in near-critical black hole collapse in
asymptotically non-expanding space--times \cite{chop93}, is confirmed
numerically for an asymptotic FRW background. For the smallest black
holes in this investigation, the scaling exponent is $\gamma \approx
0.36$, which is identical to the non-expanding numerical and
analytical results \cite{evans94,koike95} within our numerical accuracy. The
parameter $K$ of Eq.~(\ref{scale}), needed in addition to
$\gamma$ to evaluate the two-parameter PBH
IMF derived in \cite{niejed97}, ranges from $K \approx 2.4$ to $K
\approx 12$.

The author wishes to thank K. Jedamzik for a fruitful collaboration,
T. Baumgarte for providing the original version of
the hydrodynamical code, and Joan George for valuable stylistic corrections.


\section{Introduction}
$\quad$Cosmic muons are the
important contributors to background processes when
search    for the  conversion of a muon to an  electron\cite{come,mu2e}. Cosmic
ray veto geometry surrounding the detectors and stopping target should
be carefully eliminated this background. Passive and active  shielding
should provide background of $\sim$0.1 events in signal window for 3 year run of
experiments. Scintillator-based active shielding will consist of four layers.
The scintillator strips will read out by multiclad  wavelength
shifting (WLS) fibers connected to photodetectors. Most promising devices
based on new technologies  used as photodetectors are the  silicon photomultipliers
(SiPM)\cite{golo}. Photon  detection efficiency of SiPM is  dependent on wavelength of
the photon, temperature, dark count\cite{yang} etc., and very difficult to be
measured and even more difficult to model\cite{kjha}. For this reason, we are
limited only to the study of the light propagation and collection.
Even for
this purpose, a  Monte Carlo simulation can adequately predict the
experimental results only if the detector parameters are
sufficiently close to their true values. Some of parameter, e.g. surface
boundaries descriptions, can be tuned by using measurements for
particular scintillator strip configurations.
\input{texfig1.tex}

\section{ Some features of the plastic scintillator  strip modeling }
$\quad$The Monte Carlo
simulation  with all possible processes play a crucial
 role in the feasibility study of the proposed detector module and in identifying
detector parameter values. Low-energy optical photons (photons with a
wavelength much greater than the typical atomic spacing) undergo the
following processes: bulk absorption, Rayleigh scattering, reflection
and refraction at medium boundaries, and wavelength shifting.
\input{texfig2.tex}

\quad The boundary processes on all scintillator play an important role in  tracing
 photons in strips. Compared to   them, photon self-absorption  in
scintillator is less significant\cite{knoll}. In Geant4.10.06 \cite{gea4}
 simulation we combined
the polished scintillator surface finishes with the backpainted wrapping
option which represents diffuse (Lambertian) reflection. In this
simulation we use the UNIFIED model for the processes between two
dielectric materials.
\input{texfig31.tex}

\section{Simulation and results}

$\quad$ The peak of emission light of a plastic scintillator
(e.g., Saint Gobain BC400 series) does not matches the peak sensitivity of used
photodetectors. To solve this problem it is necessary to use the WLS fiber to
transfer light to the photodetector. In this simulation the fiber are multiclad
consisting of a scintillating core surrounded by an acrylic inner cladding
\input{texfig3.tex}
and an outer cladding which made of a fluor-acrylic material (similar
to the Kuraray double clad fibers of type Y11(200)\cite{kura}).
It was assumed that in a scintillator strip a mean value of 10000
scintillation photons per MeV of deposited energy  were emitted.
For this scintillator, the maximum emission is at a wavelength of 431\,nm
and refractive index is 1.58.

\quad  For WLS fiber attenuation length of 500\,cm for its own radiation, and for
plastic scintillator attenuation length of 300\,cm are assumed.
The total diameter
of fiber is 1.2\,mm. The total thickness of  cladding structure is 6\%
of the diameter of a fiber. In this simulation the strip contains one or two
 co-extruded grooves with 3\,mm depth and 1.3\,mm width
 for insertion of the WLS fiber. The selected strip and fiber parameters
are close to those used in the test-beam measurements at JINR (Dubna, Russia).
\input{texfig9.tex}

\quad  This simulation was performed using Geant4  for plastic
scintillator with the dimension 4*1*300\,cm${}^3$  and co-extruded TiO$_{2}$
white diffuse reflective (R=98$\%$) coating.
The strip contain one at the center or two grooves at a
distance of  2\,cm from each other along the entire length of scintillator strip.
 We collect photons from a
WLS fiber at one of the strip ends (hereinafter referred to as
photodetector side). On the photodetector side at the fiber end
the photons are fully absorbed.  The opposite ends of the fibers are
blackened.
\input{texfig4.tex}

\quad The $^{90}Sr$   source was  simulated in the Geant4  framework.
 The source provides an electron flux in a wide energy  range
up to 2.3\,MeV (see Fig.1). The radiation source was enclosed in a shell with
a lead collimator. The diameter of the collimator outlet was 1\,mm.
The source was located  at a distance of 2\,mm above or below the scintillator strip.

\quad Cosmic muons were generated according to\cite{volk}  in the range
0.3-5000\,GeV. The zenith   angle and energy distribution for simulated cosmic
muons are displayed in Figure\,2.
\input{texfig5.tex}
\input{texfig6.tex}

\quad In Figure\,3, we show the distribution of the energy deposited in the scintillation
strip when the middle of the  side 4*300\,cm${}^{2}$ is irradiated with a ${}^{90}Sr$ source.

\quad Figure\,4 shows the light intensity distribution in the end  of a fiber
as seen by the photodetector side. This simulation study   show that
the light intensity increases towards the edge of the fiber core. The mean
wavelength of light collected by the photodetector is 535\,nm.

\quad The distribution of the photon number at the photodetector side
when the middle of the strip side 4*300\,cm${}^{2}$ is
irradiated with  a ${}^{90}Sr$ is shown in Figure\,5.

\quad  Figure\,6 show the light yield when strip with one fiber irradiated
with a  $^{90}Sr$  source which located over or  under the strip at Z=0.0\,cm.
The strip is located at X=$\pm$\,2.0\,cm, Y=$\pm$\,0.5\,cm, Z=$\pm$\,150.0\,cm.
The first point is 0.5\,mm away from the side (X=19.5\,mm)  and  each step is
1\,mm (X-coordinate).
Points close to X=0.0\,mm are located at a distance of 0.5\,mm.
In this and the following Figures, 500 events were simulated for each point.

\quad  In Figures\,7  and 8,  the same thing  is shown as in Figure\,6  but for
the case with two fibers in the strip.
Note that, in both cases (strip with one and two
fibers), the behaviors of the light yield when a radiation source is above and
below the strip differ from each other. But this difference is not significant.

\quad  Figure\,9 shows the relation between the mean number of optical
photons detected at the photodetector side and the distance between point
of impact of electrons from the radiation source and photodetector side.
This graph is fitted by a function
\vskip 0.5 cm
$\qquad\qquad\quad  N_{phot}(z) = A*e^{-z/\lambda_{1}} + B*e^{-z/\lambda_{2}}.$
\vskip 0.5 cm

Note that this formula was proposed by Kaiser et al.\cite{kais} for the case
when light is collected from the ends of the scintillator using a photomultipliers.
The first term is the transmission behavior for photons that travel directly
to the photodetector side.
\input{texfig7.tex}
The second term is the transmission behavior for photons that hit the detector
after a series of reflection on a scintillator surface.
The first point in the Figure is 75 mm away from the photodetector side
and each step is 75\,mm. The curve in the figure corresponds to the parameters
 $\lambda_{1}$=43.5\,m and $\lambda_{2}$=1.75\,m  with almost 100\% errors.

$\quad$To study the light attenuation when the strip surface is  irradiated
by cosmic muons  we retreated on each side of the surface by 1\,mm  and
divided it into 40 equal parts.   Each sector has been
uniformly  irradiated by 500 muons with energies, azimuth and
zenith angles modeled accordingly to\cite{volk}. In Figure 10, we demonstrate
the light attenuation for this case. The points in the Figure are located in the
center of each of the 40 sections. For the given points, the results of one
exponential and double exponential fit are the same (blue curve in the Figure),
$\lambda_{1}=\lambda_{2}$=5.88\,m. The green curve in  Figure corresponds
to the fit by the formula
\vskip 0.5 cm
$\qquad\qquad\quad  N_{phot}(z) = A*e^{-z/\lambda} + B,$
\vskip 0.5 cm
where $\lambda=$2.32\,m.
\input{texfig8.tex}

\section{Conclusion}
\qquad In this note, we modelled  the  light output and attenuation in a scintilltion
strip with dimensions of  4*1*300\,cm${}^{3}$.
The  simulated  radiation source ${}^{90}Sr$ and cosmic muons were used as beam particles.
The scintillation
strip was irradiated both from the side of the embedded fibers and from
the opposite side along and across the strip.
Optical photons was collected
from one and two fibers embedded in the strip along the entire length. It
was shown that the attenuation of light depending on the distance to the
photodetector is described by a double exponential function.

$\quad$ We are sincerely grateful to  Z.\,Tsamalaidze and
 Yu.\,Davydov for initiating this work.
}
\small{
\bibliographystyle{plain}


\section{Introduction}
\label{sec:intro}
Membrane remodeling is required for critical cellular processes including endocytosis, formation of multivesicular bodies, retrograde trafficking and exosome formation. Viruses and other pathogens also reshape cellular membranes during different stages of their lifecycles including entry into the host cell, formation of replication complexes, construction of assembly factories, and exit (also called egress or budding). Understanding the mechanisms of viral budding and the forces that drive this process would advance our fundamental understanding of viral lifecycles, and shed light on analogous cellular processes in which membrane remodeling and vesicle formation are essential for function.
In parallel, understanding fundamental determinants of budding and membrane dynamics would facilitate the design of viral nanoparticles.  There is keen interest in reengineering enveloped viral nanoparticles to be used as targeted transport vehicles capable of crossing cell membranes through viral fusion \cite{Lundstrom2009,Cheng2013,Rowan2010,Petry2003,Rohovie2017}.

All viruses contain a capsid protein, which primarily functions to protect the viral genome during viral transmission. In enveloped viruses, the internal capsid is surrounded by a host-derived lipid bilayer and viral glycoproteins (GPs) embedded in this membrane.  Enveloped viruses can be sub-divided into two groups based on their sequence of virion assembly and budding.  For the first group (e.g. alphaviruses, hepatitis B, herpes) budding requires the assembly of a preformed nucleocapsid core (NC), which may be ordered or disordered depending on the virus. The core then binds to membrane-bound GPs and initiates budding \cite{Sundquist2012,Hurley2010,Welsch2007,Garoff1998}. For the second group, (e.g. influenza, type C retroviruses (HIV)) capsid assembly occurs concomitant with budding \cite{Welsch2007,Sundquist2012,Vennema1996}. The advantage of one assembly mechanism over another is not obvious; particle infectivity, morphology, and stability may all influence the preferred budding process.

The importance of preformed capsids in Alphavirus assembly is of particular interest because the presence of a capsid in the particle is not necessary for production of infectious particles. The traditional view is that alphaviruses follow the preassembled NC budding pathway \cite{Garoff1978,Garoff2004,Strauss1994,Wilkinson2005}, based on the observation of high concentrations of NCs in the cytoplasm \cite{Acheson1967}, and evidence that GP-GP and NC-GP interactions are required for virion formation \cite{Suomalainen1992,Lopez1994}. However, several studies have challenged this conclusion. In particular, Forsell et al.\xspace \cite{Forsell2000} reported successful assembly and budding of alphavirus despite mutations which inhibited NC assembly by impairing interactions between NC proteins, while Ruiz-Guillen et al.\xspace \cite{Ruiz-Guillen2016} observed budding of infectious alphavirus particles from cells which did not express the capsid gene. In both cases infectious particles were assembled and released or budded from the cell.  These observations suggest that GP interactions may be sufficient for alphavirus budding. This begs the question: Why do enveloped viruses have internal nucleocapsid cores? Is there an advantage to having a NC during budding?

Molecular dynamics (MD) simulations can be a useful tool to bridge the gaps between the different steps of assembly that cannot be experimentally characterized. Computational studies have already provided insightful information about virus NC assembly \cite{Perlmutter2015}, as well as the interactions between proteins and lipid membranes \cite{Reynwar2007,Simunovic2013,Bradley2016}. Previous simulations on budding of nanoscale particles led to important insights but did not consider the effect of GPs (\cite{Smith2007,Vacha2011,Ruiz-Herrero2012,Deserno2002,Jiang2015,Li2010a, Li2010,Yang2011}), although budding directed by GP adsorption or capsid assembly have been the subject of continuum theoretical modeling \cite{Lerner1993,Tzlil2004,Zhang2008}. The formation of clathrin cages during vesicle secretion, a process which bears similarities to viral budding, has also been the subject of modeling studies \cite{Foret2008,Cordella2014, Cordella2015,Matthews2013}. Most closely related to our work are previous simulations on the assembly and budding of 12-subunit capsids, which found that membrane adsorption can lower entropic barriers to assembly \cite{Matthews2012,Matthews2013a} and that membrane microdomains can facilitate assembly and budding \cite{Ruiz-Herrero2015}. In contrast to these earlier works, we consider the presence of a nucleocapsid, a larger shell (80 trimer subunits), and a different subunit geometry. We find that these modifications lead to qualitatively different assembly pathways and outcomes in some parameter ranges.

In this article, we perform MD simulations on a coarse-grained model for GPs, the NC, and a lipid bilayer membrane to elucidate the forces driving enveloped virus budding. Our model is motivated by the alphavirus structure and experimental observations on alphavirus budding \cite{Forsell2000,Ruiz-Guillen2016,Garoff2004}, but we consider our results in the broader context of enveloped viruses. To evaluate the relative roles of a preassembled NC compared to the assembly of transmembrane glycoproteins in driving budding, we perform two sets of simulations. The first focuses entirely on glycoprotein-directed budding (Fig. \ref{fig:budding}a) by including only the membrane and model GPs, whose geometry and interactions drive formation of an icosahedral shell with the geometry of the alphavirus envelope. This model directly applies to experiments on budding from cells in which capsid assembly was eliminated  \cite{Forsell2000,Ruiz-Guillen2016}.  The second set of simulations includes model GPs and a preassembled NC, thus allowing for NC-directed budding (Fig. \ref{fig:budding}b).

We present phase diagrams describing how assembly morphologies depend on the strength of GP-GP and NC-GP interactions. The results demonstrate that the competition between the elastic energy of membrane deformations and deviations from preferred protein curvature can lead to polymorphic morphologies, and that templating by the NC can significantly decrease the resulting polydispersity. In the presence of a preassembled NC, there is a threshold strength of NC-GP interactions above which pathways transition from GP-directed to NC-directed budding. Our simulations enable visualization of the intermediates along each of these pathways as well as analysis of their relative timescales. In both pathways, assembly proceeds rapidly until budding is approximately 2/3 complete, after which curvature of the membrane at the neck of the bud imposes a barrier to subunit diffusion that significantly slows subsequent assembly and budding.  We discuss possible implications of this slowdown for enveloped viruses such as HIV that bud with incompletely formed capsids.

\begin{figure}[hbt]
  \begin{center}
  \includegraphics[width=\columnwidth]{figures/scheme_mechanisms2.pdf}
    \caption{ Mechanisms of enveloped virus budding on membranes: glycoprotein(GP)-directed budding and nucleocapsid(NC)-directed budding. }
  \label{fig:budding}
  \end{center}
\end{figure}

\begin{figure*}[hbt]
  \begin{center}
  \includegraphics[width=0.9\textwidth]{figures/scheme_model_capsid3.pdf}
    \caption{{\bf a)} Cryoelectron microscopy (CryoEM) density distribution of Sindbis virus. This central cross-section shows the inner structure of alphaviruses, with the RNA molecule (purple) enclosed by the NC (orange, red) and the lipid membrane (green) with the transmembrane GPs (blue). [{\bf b-g}] Computational model of the alphavirus GPs and NC. {\bf b)} Comparison between the GP trimer as revealed by CryoEM (PDB ID: 3J0C \cite{Zhang2002}) and our coarse-grained model trimer. Trimers are modeled as rigid bodies comprising three cones, with each cone formed by 6 pseudoatoms of increasing diameter. {\bf c)} Each of the 4 inner pseudoatoms of a cone (green) interacts with its counterpart in a neighboring cone through a Morse potential with well-depth $\epsilon_{\text{gg}}$. All pseaudoatoms, including the `excluders' (red), interact through a repulsive Lennard-Jones potential.  {\bf d)} Model glycoproteins are trapped in the membrane by `membrane-excluders' (purple) which interact with membrane pseudoatoms through a repulsive Lennard-Jones potential.  {\bf e)} Complete trimer subunit embedded in the membrane. To aid visibility, in subsequent figures only the membrane excluders are shown. {\bf f)} Snapshot of a typical capsid assembled by model glycoproteins in the absence of a membrane, consisting of 80 trimer subunits. {\bf g)} Snapshot of typical capsid assembled by glycoproteins around the model nucleocapsid (NC, blue) in the absence of a membrane. The NC is modeled as a rigid spherical particle. NC pseaudoatoms interact with the lowermost pseudoatom in each GP cone through a Morse potential with depth $\epsilon_{\text{ng}}$.}
  \label{fig:scheme}
  \end{center}
\end{figure*}

 \begin{figure*}[hbt]
  \begin{center}
  \includegraphics[width=0.8\textwidth]{figures/simulation_trajectoryd2_all.pdf}
    \caption{ {\it (Top)} Typical simulation trajectory of GP-directed budding, at simulation times 600, 1800, 2700 and 4200$\tau_{0}$, from left to right, with GP-GP interaction strength $\epsilon_{\text{gg}}=2.3$.  {\it (Bottom)} Simulation trajectory of NC-directed budding, at 600, 1800, 2700 and 5600 $\tau_{0}$, with $\epsilon_{\text{gg}}=2.5$ and NC-GP interaction strength $\epsilon_{\text{ng}}=3.5$. The second timepoint in each row corresponds to an example of the intermediate with a constricted neck described in the text. Except where noted otherwise, the membrane bending modulus $\kappa_{\text{mem}}\approx 14.5k_{\text{B}}T$ throughout the manuscript.}
  \label{fig:trajectory}
  \end{center}
\end{figure*}

\begin{figure}
\caption{{\bf (A): Simulation Movie 1.} Animation of a typical simulation showing GP-directed budding, for $\epsilon_{\text{gg}}=2.5$ and $\kappa_{\text{mem}}=14.5 k_{\text{B}}T$. Colors are as follows: GPs, magenta; membrane head groups, cyan; membrane tails, yellow. To show the membrane neck geometry more clearly the inactive subunits are rendered invisible in this animation. {\bf (B): Simulation Movie 2.} The same simulation trajectory as in {\bf (A)}, but rendered to show a central cross-section of the budding shell and the membrane. Inactive subunits are rendered brown in this animation. }
\label{fig:MovieGPdirected}
\end{figure}

\begin{figure}
\caption{{\bf (C): Simulation Movie 3.} Animation of a typical simulation showing NC-directed budding, for $\epsilon_{\text{gg}}=2.5$, $\epsilon_{\text{ng}}=3.5$, and $\kappa_{\text{mem}}=14.5 k_{\text{B}}T$. Colors are as in Simulation Movie 1, and the NC is colored blue. To show the membrane neck geometry more clearly we do not show inactive subunits in this animation. {\bf (D): Simulation Movie 4.} Animation showing a central cross-section of the NC-directed budding (same simulation trajectory as {\bf (C)}). Inactive subunits are shown in brown in this animation. }
\label{fig:MovieNCdirected}
\end{figure}

\section{Results}
\label{sec:results}
 Although intact viruses can be simulated at atomistic or near-atomistic resolution \cite{Freddolino2006,Zhao2013,Reddy2015,Perilla2015,Huber2016}, the time scales for alphavirus  assembly (ms-minutes) are prohibitive at such resolution. We thus consider a coarse-grained description for the viral GPs and the membrane, which enables tractable simulation of a large membrane over biologically relevant timescales while retaining the essential physical features of membranes and virus capsid and transmembrane proteins (see Fig. \ref{fig:scheme} and section \ref{sec:methods}). The membrane is represented by a solvent-free model which can be tuned to match properties of biological membranes while allowing simulation of large systems \cite{Cooke2005}.  Our model GPs are designed to roughly match the triangular shape, dimensions and aspect ratio of Sindbis virus GP trimers ~\cite{Mukhopadhyay2006,Zhang2002}. They experience lateral interactions, which in the absence of a membrane drive assembly into capsids containing 80 subunits, consistent with the 80 trimers in the alphavirus glycoprotein shell. In our simulations the GPs are embedded within the membrane, where they freely tilt and diffuse but cannot escape on simulation timescales.
  Motivated by the recent observation that alphavirus nucleocapsids do not require icosahedral symmetry \cite{Wang2015} to be infectious, we model the nucleocapsid as a rigid isotropic sphere. To account for experimental observations of capsid and GP conformational dynamics, the model GPs interconvert between assembly-inactive and assembly-active conformations which are respectively compatible or incompatible with assembly (see section~\ref{sec:methods}).

To compare GP-directed and NC-directed budding, we performed two sets of simulations that respectively included or did not include a NC. To understand how these pathways depend on parameters which can be controlled in experiments or varied under evolutionary pressures, we simulated assembly as a function of parameters controlling the GP-GP interaction strength $\epsilon_{\text{gg}}$, the NC-GP interaction strength $\epsilon_{\text{ng}}$ (when a NC is present), and the membrane bending modulus $\kappa_{\text{mem}}$.  All energies are reported in units of the thermal energy, $k_{\text{B}}T$. For notational convenience, we refer to particles assembled from GPs only as GP-particles, and GPs assembled around the NC as GPNC-particles.

\subsection*{Budding includes an intermediate with a constricted neck}
 We show snapshots from typical trajectories for simulations in the presence and absence of a NC in Fig. \ref{fig:trajectory}. In both cases, assembly and membrane deformation proceed rapidly until the GP shell is approximately 2/3 complete (the second timepoint in each row in Fig. \ref{fig:trajectory}), when the budding region is connected to the rest of the membrane by a narrow neck. Subunits within the neck experience restricted configurations due to the high membrane curvature. Thus, the neck acts as an entropic barrier that impedes subunit diffusion to the growing shell, causing the assembly rate to slow dramatically as the shell nears completion (Fig. \ref{fig:timescales}). The neck continues to narrow as additional GPs assemble until it becomes a tether connecting the bud and membrane.  In this article we do not consider ESCRT or related scission-inducing proteins, and thus the bud separates from the membrane only when a large thermal fluctuation induces membrane fission leading to scission of the tether.

\begin{figure}[hbt]
  \begin{center}
  \includegraphics[width=\columnwidth]{figures/phasediagramGP.pdf}
    \caption{ Predominant end-products for assembly without a NC as a function of GP-GP interaction strength, along with simulation snapshots that exemplify each class of end-product.The distribution of end-products for several representative values of $\epsilon_{\text{gg}}$ is shown in Fig.  \ref{fig:distributionoutcomes}.}
  \label{fig:phasediagramGP}
  \end{center}
\end{figure}

\subsection*{Glycoprotein-directed budding leads to complete but polydisperse particles}
\label{sec:gpdriven}
We first consider assembly in the absence of a NC, so that budding is necessarily GP-directed. 	Fig. \ref{fig:phasediagramGP} shows the most frequent end-product obtained as a function of the GP-GP interaction strength. For weak interactions ($\epsilon_{\text{gg}}<1.4$) assembly is unfavorable.  In contrast, bulk simulations (model GPs in the absence of a membrane) exhibit shell assembly for $\epsilon_{\text{gg}}>0.97$ at trimer concentration $\phi_{3D}=3.4 \cdot 10^{-5}\sigma^{-3}$ (see Fig. ~\ref{fig:capsidsbulk}a in the SI), well below the effective concentration of trimers on the membrane $\phi_{2D}^{eff}=0.001\sigma^{-3}$ \footnote{To compare the subunit concentration in bulk $\phi_{3D}=N_{s}/L^{3}$, where $N_{s}$ is the number of subunits and $L$ the box size, with the subunit concentration on the membrane, we measure the mean deviation of the height fluctuations of the subunits on the membrane, $\lambda\approx 4.5$, and the effective concentration on the membrane is then given by  $\phi_{2D}^{eff}=N_{s} / L^2 \lambda$.} . This result demonstrates that the membrane rigidity can introduce a substantial barrier to assembly~\cite{Ruiz-Herrero2012}.

Within a narrow range of interaction strengths  $1.4< \epsilon_{\text{gg}}<1.7$, assembly and budding stalls at the constricted neck intermediate described above. For these parameters, the intermediate remains upon extending the simulation length to 10,500$\tau_{0}$, suggesting that it corresponds to a true steady state or a very long-lived kinetic trap.  This configuration resembles partially assembled states that were predicted theoretically \cite{Zhang2008,Foret2014}, but arises due to different physics. We find that the range of $\epsilon_{\text{gg}}$ over which the state arises depends on the subunit geometry, but the state exists for any geometry we considered (see section \ref{sec:subunit} in the SI). A similar configuration was observed during simulations of assembly and budding of a 12-subunit capsid on a membrane \cite{Ruiz-Herrero2015}, suggesting it is a generic feature of assembly and budding. However, in that work assembly never proceeded past the partially assembled state for any parameter set on a homogeneous membrane, possibly due to the to the small size of the simulated capsid.

For stronger interactions, we observe complete budding. However, the morphologies of the resulting GP-particles depend on the interaction strength in two ways. First, overly strong interactions ($\epsilon_{\text{gg}}>9.0$) drive rapid assembly which can proceed simultaneously along multiple fronts within a shell, leading to the formation of holey GP-particles (Fig. \ref{fig:phasediagramGP}). This result is consistent with holey capsids that assemble under strong interactions in bulk simulations \cite{Hagan2006,Rapaport2012}. Second, for moderate interactions ($1.6< \epsilon_{\text{gg}}<9.0$) the shells are complete, but their size  depends on the interaction strength, with  typical sizes ranging over 95--140 subunits. The origin of this polymorphism is  discussed later in this section.

\subsection*{Nucleocapsid-directed budding leads to more monodisperse particles}
\label{sec:ncdriven}

The predominant end-products of assembly in the presence of a NC are shown in Fig.~\ref{fig:phasediagramNC} as a function of the two interaction parameters: GP-GP ($\epsilon_{\text{gg}}$) and NC-GP ($\epsilon_{\text{ng}}$).
We observe complete assembly  and budding  for $\epsilon_{\text{ng}} > 0.9$ and $1\le \epsilon_{\text{gg}} \le 6$ (Fig.~\ref{fig:phasediagramNC}, blue region), low to moderate GP-GP interactions. Compared to the GP-directed pathway, the presence of a NC allows assembly to occur at a lower $\epsilon_{\text{gg}}$ as evidenced by obtaining complete shells even for $\epsilon_{\text{gg}}$ as low as 0.9.
Outside of this range, several other end-products arise. For $\epsilon_{\text{ng}}<0.4$ (Fig.~\ref{fig:phasediagramNC}, pink region), the NC interactions are sufficiently weak that budding is entirely GP-directed (\textit{i.e.}\xspace GP shells assemble and bud, but not around the NC). In the range  $0.4< \epsilon_{\text{ng}}<0.9$ (Fig.~\ref{fig:phasediagramNC}, brown region) we observe an intermediate regime in which the NC promotes nucleation but fails to act as a perfect template. The GP shell initially starts assembling on the NC surface, but eventually separates from the surface to form a larger shell due to the effect of membrane rigidity (see Section \ref{sec:polymorphism}). The result is an asymmetric shell that is partially attached to the NC; with a typical size of 95 subunits it is smaller than a GP-particle but considerably larger than the intrinsic preferred shell size. These two assembly outcomes (GP shells partially attached or unattached to the NC) demonstrate that the presence of a NC does not necessarily imply NC-directed budding; there is a minimum NC-GP interaction strength required for the NC to direct the assembly and budding pathway.

 \begin{figure*}[hbt]
  \begin{center}
  \includegraphics[width=0.8\textwidth]{figures/phasediagram_NC.pdf}
    \caption{ Predominant end-products of the NC-directed budding, as a funtion of the NC-subunit interaction $\epsilon_{\text{ng}}$ and the subunit-subunit interaction $\epsilon_{\text{gg}}$, and snapshots showing representative examples of each outcome. }
  \label{fig:phasediagramNC}
  \end{center}
\end{figure*}

Strong GP-GP interactions ($\epsilon_{\text{gg}}>6$, Fig.~\ref{fig:phasediagramNC}, upper grey region) lead to holey particles. This result can be explained as in the case of holey GP-particles described above; however, notice that the threshold value of $\epsilon_{\text{gg}}$ is smaller than in the absence of the NC ($\epsilon_{\text{gg}}=9$).  Interestingly, we also observe holey GPNC-particles when strong NC-GP interactions are combined with weak GP-GP interactions ($\epsilon_{\text{gg}}<1.0$, Fig.~\ref{fig:phasediagramNC}, lower grey region). In this regime, NC uptake proceeds rapidly, but GP subunits do not associate quickly enough form a complete shell as budding proceeds.

To further elucidate the interplay between the two interactions, Fig.~\ref{fig:energy} (SI) compares the total energetic contributions from GP-GP and NC-GP interactions for budded GPNC-particles.  We see that GP-GP interactions account for the majority of the attractive energy stabilizing the shell, with the NC-GP providing as little as 10-20\% of the total energy. These results highlight the delicate balance between GP-GP and NC-GP interactions required to obtain a well-formed GPNC-particle.

\subsection*{Membrane-induced polymorphism}
\label{sec:polymorphism}

 Although both GP-only budding and NC-directed budding lead to the formation of complete particles, the morphology of budded shells significantly differs between both mechanisms. Fig. \ref{fig:bending}a shows the mean shell size as a function of interaction strength (averaged over all closed particles). For GP-directed budding, we see a strong dependence of particle size on subunit interactions: weak interactions lead to ovoid particles containing up to 140 trimers. As $\epsilon_{\text{gg}}$ increases, particles become smaller and more spherical, more closely resembling the shells that assemble in bulk simulations. We show snapshots of typical GP-shells assembled at weak and strong interactions. On the contrary, the size of GPNC-particles is nearly constant with $\epsilon_{\text{gg}}$ and only slightly larger (81-83 subunits) than the preferred size in bulk simulations.

{\textbf{A theoretical model for membrane-induced polymorphism.} Although shell assembly is necessarily out-of-equilibrium in finite-length simulations, we can understand the dependence of size on interaction strength from a simple equilibrium model that uses the Helfrich model \cite{Helfrich1973} to account for the elastic energy associated with membrane deformation (which has no spontaneous curvature and thus favors flat configurations) and deviation of the GP shell from its preferred curvature. The calculation is detailed in the Appendix. Minimizing the total free energy for a system with fixed number of GP subunits obtains that the most probable number of subunits in a shell $n$ corresponds to the value which minimizes the elastic energy per subunit, given by
\begin{align}
n = n_{0} \left( 1 + \frac{\kappa_{\text{mem}}}{ \kappa_{\text{shell}}} \right)^{2},
\label{eq:nsubs}
\end{align}
where $n_{0}$ is the number of subunits in the equilibrium configuration in the absence of a membrane ($n_{0}=80$ in our model), and  $\kappa_{\text{mem}}$ and $\kappa_{\text{shell}}$ as the membrane and shell bending moduli. Thus the preferred GP-particle size is determined by the ratio $\kappa_{\text{mem}} / \kappa_{\text{shell}}$, which quantifies the competition between the membrane and shell deformation energies.
 Only in the limit where the shell rigidity dominates, $\kappa_{\text{mem}} / \kappa_{\text{shell}} \to 0$, will GP-particles exhibit the size observed in bulk simulations.

To compare the theoretical estimate to the shell sizes observed in simulations, we estimated the relationship between the GP interaction strength and the shell bending modulus as $\kappa_{\text{shell}} \approx 25.66 \epsilon_{\text{gg}}$ (we show the complete derivation in section \ref{sec:bendingestimation} in the SI), leading to a range of shell bending rigidities of $\approx 40-250k_{\text{B}}T$.  This range coincides with bending rigidity values measured in AFM experiments on virus capsids (see the Discussion) \cite{Michel2006,Roos2007,May2011}. The prediction of Eq.~\eqref{eq:nsubs} using this estimate and the estimated membrane rigidity $\kappa_{\text{mem}}=14.5k_{\text{B}}T$ is shown in Fig \ref{fig:bending}a.  The prediction is also compared against simulated particle sizes as a function of the parameter $\kappa_{\text{mem}} / \kappa_{\text{shell}}$ for different membrane bending rigidities in Fig.~\ref{fig:bending}b.  For moderate values of $\kappa_{\text{mem}} / \kappa_{\text{shell}}$  we observe good agreement between the theory and simulation results, especially considering that there is no fit parameter. The agreement breaks down for $\kappa_{\text{mem}} / \kappa_{\text{shell}}\gtrsim0.3$, likely for several reasons. Firstly, our theory assumes a closed GP shell, whereas the size of the incomplete region of the GP shell increases in size with $\kappa_{\text{mem}} / \kappa_{\text{shell}}$, as illustrated by the snapshots in Fig.~\ref{fig:bending}b. Secondly, subunits within the largest GP particles are far from their preferred interaction angle, and thus their elastic response could be nonlinear. Finally, finite-size effects could become non-negligible for the largest buds.

In contrast to GP-directed assembly, Figs.~\ref{fig:phasediagramGP} and \ref{fig:phasediagramNC} demonstrate that a NC can dramatically change the morphology of a GP shell by acting as a template over a broad range of interaction strengths. The observed monodispersity in GPNC-particles can be understood from Eq.~\eqref{eq:nsubs} by noting that the NC is modeled as a perfectly rigid sphere in our simulations, and thus corresponds to the limit $\kappa_{\text{shell}}\rightarrow \infty$ if it acts as a perfect template for the GP shell. The relevance of this approximation to enveloped viruses is considered in the Discussion.

\begin{figure}[hbt]
  \begin{center}
  \includegraphics[width=\columnwidth]{figures/bending2.pdf}
    \caption{ {\bf a)} Average number of subunits in budded particles as a function of GP-GP affinity, for  GP-directed budding (\textcolor{blue}{$\blacksquare$} symbols) and NC-directed budding with $\epsilon_{\text{ng}}=4.5k_{\text{B}}T$ (\textcolor{red}{$\CIRCLE$} symbols). The solid green line gives the theoretical prediction (Eq.~\eqref{eq:nsubs})  for the estimated capsid rigidity $\kappa_{\text{shell}}=25.66 \epsilon_{\text{gg}}$ (see section \ref{sec:bendingestimation} (SI)) and $\kappa_{\text{mem}}=14.5k_{\text{B}}T$. {\bf b)} Average number of subunits in GP-shells as a function of the ratio between membrane and shell bending modulii, $\kappa_{\text{mem}} / \kappa_{\text{shell}}$. The data includes different sets of simulations in which either $\kappa_{\text{mem}} $ or $\kappa_{\text{shell}}$ is maintained constant, and we sweep over the other parameter: $\kappa_{\text{mem}}=14.5$ (\textcolor{blue}{$\blacksquare$}), $\kappa_{\text{mem}}=21.5$ (\textcolor{red}{$\blacktriangledown$}), $\kappa_{\text{shell}}=51.3$ (\textcolor{green}{$\CIRCLE$}), and $\kappa_{\text{shell}}=154.0$ (\textcolor{dpurple}{$\blacktriangle$}). The theoretical prediction (Eq.~\eqref{eq:nsubs}) is shown as a black solid line.}
  \label{fig:bending}
  \end{center}
\end{figure}

\subsection*{The nucleocapsid influences timescales for late stage budding}
\label{sec:timescales}
As noted above assembly can be divided into two stages. The shell grows rapidly until about 2/3 completion, after which neck curvature significantly slows subunit association (Fig. \ref{fig:timescales}). The timescale for the second stage depends on the interaction parameters and whether a NC is present --- for the small GPNC-particles assembly is completed quickly $\sim 450\tau_{0}$, whilst in the large GP-particles with the broadest necks it may require up to $3,000\tau_{0}$. In contrast, the timescale for the first stage is almost independent of interaction strengths and the NC, and depends only weakly on membrane bending modulus.  Furthermore, as shown in section \ref{sec:conformational} (SI), conformational switching is not rate limiting, implying that assembly rates during the first stage are limited by subunit diffusion.

This observation parallels models for clathrin-independent receptor-mediated endocytosis, in which the endocytosis timescale is estimated from the time required for membrane receptors to diffuse to the enveloped particle \cite{Gao2005,Bao2005}.  Applying the same analysis to our simulations, the timescale for GPs to diffuse to the budding site is given by $\tau\sim l^{2}/2D$, with $D_{\text{sub}}=\sigma^{2}/\tau_{0}$ the GP diffusion constant in our simulations, and $l\approx 45\sigma$ as the radius of the region around the budding site initially containing 80 trimers, enough to envelop the particle. This estimate yields $\sim1200\tau_{0}$, which is reasonably close to the typical timescale for stage 1 observed in the simulations, $\sim1000\tau_{0}$. Note that this model does not describe the timescale of the latter stage of assembly, since the curved neck region imposes a barrier to subunit diffusion which increases as the particle nears completion.

\subsection*{GP conformational changes avoid kinetic traps}
Finally, we note that when subunit conformation changes are not accounted for (\textit{i.e.}\xspace all subunits are in the active state), we observe complete assembly and budding only under a narrow range of interaction strengths for NC-directed budding and not at all for GP-directed budding (section \ref{sec:conformational}, SI).  For most interaction strengths the simulated densities of GPs led to multiple small aggregates which failed to drive significant membrane deformation. This behavior is indicative of kinetic trapping, known to occur in assembly reactions at high concentrations or binding affinities \cite{Hagan2014,Whitelam2015}. The ability of an inactive conformation to avoid this trap is consistent with simulations of bulk assembly \cite{Lazaro2016,Grime2016}, and the ability of budding to proceed in the presence of high subunit concentrations (when conformational changes are accounted for) is consistent with the observation of high densities of GPs in the membranes of cells infected with Sindbis virus \cite{Bonsdorff1978}.

\begin{figure}[hbt]
  \begin{center}
  \includegraphics[width=\columnwidth]{figures/timescales4_outcomes.pdf}
    \caption{ Number of GP trimers in budding shells as a function of time for trajectories at different parameter values: GP-directed budding with $\epsilon_{\text{gg}}=1.4$ (black) or $\epsilon_{\text{gg}}=1.75$ (blue) and NC-directed budding with  $\epsilon_{\text{ng}}=4.5$ and   $\epsilon_{\text{gg}}=1.75$ (red). The snapshots to the right show the assembly products. For each parameter set the thick line shows an average over three trajectories, and the two thin lines show two individual trajectories to give a sense of the size of fluctuations. The lines end when budding occurs, except for the GP-directed case with $\epsilon_{\text{gg}}=1.4$ which ended in the stalled partially budded state.  }
  \label{fig:timescales}
  \end{center}
\end{figure}

\section{Discussion and conclusions }
\label{sec:discussion}
\subsection*{Advantages of NC-driven budding}
We have described dynamical simulations of the assembly and budding of GPs in the presence and absence of a preassembled NC, and presented phase diagrams showing how assembly pathways and products depend on the relevant interaction parameters.
The key difference between NC-directed and GP-directed budding identified by our results  is variability of the budded particle size and morphology.
The presence of the core directs the morphology of the particle, which  may have direct consequences on particle stability during transmission and the conformational changes that occur during particle entry into a new host cell.
The number of GPs in GP-particles (containing no NC)  varies by $>$50\% over the range of interaction parameters in which we observe successful budding, in comparison to a variation of less than 5\% for GPNC-particles (containing a NC). A simple equilibrium model accounting for the competition between the energy costs associated with membrane deformation and deviation of the GP shell from its preferred curvature was qualitatively consistent with the simulation results. The membrane bending energy favors formation of larger particles, while a higher bending rigidity of the GP shell favors smaller particles.  This prediction applies to any form of assembly on a fluid membrane, and thus is relevant to capsid-directed budding as well as the GP-directed budding studied here.

It is worth considering this observation in the context of recent experimental observations on budding in the absence of a NC, as well as experimental measurements on viral particle elastic properties.
Several experimental studies have reported GP-directed budding from cells in which the NC proteins are impaired \cite{Forsell2000,Ruiz-Guillen2016}. In particular, Ruiz-Guillen et al.\xspace \cite{Ruiz-Guillen2016} recently showed that cells expressing the genome and GPs, but not the capsid protein, for Sindbis and Semliki Forest virus generate infectious viral particles that can propagate in mammalian cells.  For the wild-type virus, in which the GPs assemble around the NC, they estimated a virion diameter of 60-70nm, whereas the GP-only particles were typically 100-150nm. This increase in particle size is consistent with our simulations and theoretical model, suggesting that the relatively low rigidity of the GP shell leads to the formation of large particles.  However there are two important caveats to this interpretation.  First, the bending rigidity of the alphavirus GP shell has not been measured, so we cannot directly predict the increase in particle size.  Second, our model assumes that the preferred curvature of the GP shell is commensurate with its size in a wild type virion, for which there is no direct evidence. A recent study of herpes simplex virus nuclear egress complex (NEC), which consists of two viral envelope proteins, found that NEC particles budded in the absence of capsid are smaller than native viral particles \cite{Hagen2015}. In our model this observation would require that the intrinsic spontaneous curvature radius of the NEC complex is smaller than that of the capsid, as suggested by the authors \cite{Hagen2015}.

Despite different tendencies for polymorphism, our results show that GP-directed and NC-directed budding share many similarities, presenting a challenge for distinguishing between NC-directed and assembly-directed budding.
For example, assembly timescales depend only weakly on the presence of a NC and interaction parameters, instead being limited by diffusion of GP subunits to the budding site. We anticipate that this result would be unchanged by adding additional ingredients to the model, since the conditions we simulated (high density of GPs in the surrounding membrane) correspond to the lower bound of GP diffusion timescales.  The same principle would apply if GPs are directly targeted to the budding site (rather than diffusing on the membrane) as has been suggested for alphavirus budding \cite{Martinez2014}, except that the diffusive flux would be replaced by a targeting flux.

\subsection*{Requirement for scission machinery}
A second commonality between the two scenarios investigated is that assembly slows down considerably after the GP shell reaches approximately 2/3 completion, because the high curvature of the neck region imposes a barrier to subunit diffusion. For weak interactions, this leads to a long-lived partially budded state. For stronger interactions budding eventually completes, but completion of the shell can be preempted by scission.
 Since spontaneous scission is a rare event, most viruses actively drive scission by either recruiting host cell machinery, such as the ESCRT protein complex in the case of HIV \cite{Baumgartel2011}, or encoding their own macinery, such as the M2 protein in Influenza \cite{Rossman2010}. In alphaviruses specifically the scission machinery has not been identified, though it is known that alphavirus budding is independent of ESCRT proteins \cite{Chen2008}. Although in the present article we do not consider the action of these scission-inducing mechanisms, our observation of a slowdown of assembly rates due to neck curvature could be relevant to HIV budding. If the association rate becomes sufficiently slow, ESCRT-directed scission will occur before assembly completes, leaving up to 40\% of the shell incomplete, as is observed for immature HIV virions \cite{Briggs2009}. In support of this possibility, we note that although scission is a rare event in our simulations because there is no ESCRT, it usually occurs before the final 1-3 subunits assemble causing the budded particle to have small hole at the scission site.

\subsection*{Virus elasticity and GP spontaneous curvature affect budding morphology}
Since Fig. \ref{fig:phasediagramGP} suggests that the elastic properties of the different viral components are key determinants of the assembly product,
it is worth considering the validity of the model parameters.
The mechanical properties of viruses have been extensively studied using atomic force microscopy (AFM) indentation \cite{Michel2006,Roos2007} and fluctuation spectrum analysis \cite{May2011}. Typical estimates lie within the range $\kappa=30-400k_{\text{B}}T$, with considerable variation depending on the specific virus and experimental technique. A recent work by \citet{Schaap2012} explicitly compared the stiffness of the capsid protein coat of influenza virus with that of a lipid membrane, by AFM indentation of similarly sized particles, with the goal of identifying the contribution of the matrix proteins to the virus stiffness. They found that matrix coats are approximately ten times stiffer than bare membranes,  $\kappa_{\text{mem}} / \kappa_{\text{shell}} \sim 0.1$. This value lies within the range explored in our simulations.
Similarly,  Kol et al.\xspace \cite{Kol2007} investigated the effect of HIV maturation on its mechanical properties.
The immature HIV particle consists of a gag polyprotein capsid surrounded by a lipid bilayer containing viral envelope proteins. During maturation, the NC and capsid portions of gag are cleaved, leaving only a thin matrix layer and the envelope proteins in contact with the bilayer. Kol et al.\xspace found that this cleavage softens the particles by an order of magnitude, suggesting that inter-protein contacts of the underlying capsid layer are necessary for the high rigidity of immature virions.

Finally, we note that the membrane elasticity properties also play a role in determining particle morphology. Depending on the virus family and host cell type, enveloped viral particles bud through different cellular membranes (the plasma membrane, the ER, the ERGIC, or the nuclear membrane), all of which have different lipid compositions and thus different bending properties. Moreover, many viruses create and/or exploit membrane microdomains with different compositions (such as lipid rafts) as preferential locations for budding \cite{Waheed2010,Welsch2007,Rossman2011}. The effect of inhomogeneous membrane elastic properties on particle morphology thus deserves further exploration. While these ingredients can be incorporated into the model, the results described here demonstrate that the interplay between the elastic properties of membranes and viral proteins and the presence of an interior core can shape the morphology of a budding particle.

\section{Methods}
\label{sec:methods}

\subsection{Glycoproteins and capsid}
\label{sec:capsid}
Our coarse-grained GP model is motivated by the geometry of Sindbis virions as revealed by cryoelectron microscopy ~\cite{Mukhopadhyay2006,Zhang2002}. The outer layer of Sindbis is comprised from heterodimers of the E1 and E2 GPs.  Three such heterodimers form a tightly interwoven trimer-of-heterodimers, and 80 of these trimers are organized into a T=4 lattice. On the capsid surface each trimer forms a roughly equilateral triangle with edge-length $\sim 8$nm. In the radial direction, each E1-E2 heterodimer spans the entire lipid membrane and the ectodomain spike, totaling $\sim12$nm  in length. In our model, we consider the GP trimer as the basic assembly subunit, assuming that the formation of trimers is fast relative to the timescale for assembly of trimers into a complete capsid.

Our subunit model aims to capture the triangular shape, aspect ratio, and preferred curvature of the GP trimers while minimizing computational detail. To this end, we employ the conical particles studied by Chen et al.\xspace \cite{Chen2007}, modified in two ways.
First, our GP trimer subunit comprises three cones, which are fused together and simulated as a rigid body. Second, the cones are truncated, so that they form a shell with an empty interior, as shown in Fig. \ref{fig:scheme}. The cone length and trimer organization within the capsid are consistent with the Sindbis structure (see section \ref{sec:subunit} (SI) for full details). Note that while the domains primarily responsible for curvature of alphavirus GPs are located to the exterior of the envelope, the conical regions which drive curvature of the model subunit oligomers are located within and below the plane of the membrane. We found that this arrangement facilitated completion of assembly (see section \ref{sec:subunit} in the SI).

Each cone consists of a linear array of six beads of increasing diameter.
Two nearby cones experience repulsive  interactions,
 mediated by a repulsive Lennard-Jones potential between all pairs of beads, with size parameter $\sigma$ equal to the bead diameter.  In addition, each of the four inner beads experiences an attractive interaction with its counterpart (the bead with the same diameter) in the neighboring subunit, modeled by a Morse potential. The Morse potential depth $\epsilon_{\text{gg}}$ determines the subunit-subunit interaction strength, which is related to the GP-GP binding affinity. The equilibrium distance of the Morse potential $r_{\text{e}}$, and the Lennard-Jones diameter $\sigma$ for each interacting pair is chosen to drive binding towards a preferred trimer-trimer angle.  We set the
preferred angle so that in bulk simulations (in the absence of membrane) the subunits predominantly assemble into aggregates with the target size, 80 subunits. However, there is a small amount of polydispersity, with some capsids having sizes between 79 and 82 subunits (Fig.~\ref{fig:capsidsbulk}). Although the subunit geometry locally favors hexagonal packing, formation of a closed capsid requires 12 five-fold defects \cite{Grason2016}. We find that the spatial distribution of these defects is typically not fully consistent with icosahedral symmetry for dynamically formed capsids. It is unclear whether this is a kinetic effect or indicates that icosahedral symmetry is not the free energy minimum at this particle size. However, the relatively high monodispersity observed suggests that the 80-subunits capsid is a free energy minimum and assembly is robust at these conditions.

\subsection{Lipid membrane}
\label{sec:membrane}
The lipid membrane is represented by the implicit solvent model from Cooke and Deserno \cite{Cooke2005}. This model enables on computationally accessible timescales the formation and reshaping of bilayers with physical properties such as rigidity, fluidity, and diffusivity that can be tuned across the range of biologically relevant values. Each lipid is modeled by a linear polymer of three beads connected by FENE bonds; one bead accounts for the lipid head and two beads for the lipid tail.  An attractive potential between the tail beads represents the hydrophobic forces that drive lipid self-assembly. In section \ref{sec:bendingestimation} in the SI we estimate the bending rigidity of the membrane in our simulations by analyzing their fluctuations spectra. Unless otherwise specified, our simulations used $\kappa_{\text{mem}}\approx 14.5 k_{\text{B}}T$ as a typical rigidity of plasma membranes.

\subsection{Glycoprotein-membrane interactions}
The effect of individual GPs on the behavior of the surrounding membrane has not been well characterized. Moreover, to facilitate interpretation of our simulation results, we require a model in which we could independently vary subunit-subunit interactions and subunit-membrane interactions. Therefore, we use the following minimal model for the GP-membrane interaction.
We add six membrane excluder beads to our subunit, three at the top and three at the bottom of the subunit, with top and bottom beads separated by 7nm (Fig.~\ref{fig:scheme}c,d).  These excluder beads interact through a repulsive Lennard-Jones potential with all membrane beads, whereas all the other cone beads do not interact with the membrane pseudoatoms. In a simulation, the subunits are initialized with membrane located between the top and bottom layer of excluders.  The excluded volume interactions thus trap the subunits in the membrane throughout the length of the simulation, but allow them to tilt and diffuse laterally. Separating the subunit pseudoatoms that interact with the membrane from those which control the subunit-subunit potential allows us to independently vary subunit-subunit and subunit-membrane interactions.  The position of the subunit-subunit interaction beads (cones) relative to the membrane excluders has little effect on the initial stages of assembly and budding, but strongly affects its completion (described in detail in section  \ref{sec:subunit} in the SI).

We note that the model does not account for local distortions within the lipid hydrophobic tails in the vicinity of the GPs. Such interactions could drive local membrane curvature and membrane-mediated subunit interactions which could either enhance or inhibit assembly and budding. Understanding these interactions is an active area (e.g. Refs. \cite{Weikl1998,Semrau2009,Reynwar2011,Goulian1993,Deserno2009}) but beyond the scope of the present study.

\subsection{Nucleocapsid}
\label{sec:NC}
The NC is represented in our model by a rigid spherical particle. This minimal representation is based on two experimental observations.  We model it as spherically symmetric because asymmetric reconstructions by Wang et al.\xspace \cite{Wang2015} showed that the alphavirus NC does not exhibit icosahedral symmetry in  virions (assembled in host cells) or viruslike particles (assembled in vitro). Second, within the NC-directed hypothesis the NC assembles completely in the endoplasmic reticulum and is then transported by the secretory pathway to the budding site at the plasma membrane. The complete NC has been shown to have a significantly higher rigidity than lipid membrane or GP-coated vesicles \cite{Schaap2012,Kol2007}; thus, we model it as infinitely rigid.

Our model NC as constructed from 623 beads distributed on a spherical surface with radius $r_{\text{NC}}=19.0\sigma$, and subjected to a rigid body constraint. To represent the hydrophobic interactions between GPs cytoplasmic tails and the capsid proteins, NC beads and the third bead of the GP subunits (counting outwards) experience an attractive Morse potential, with well-depth $\epsilon_{\text{ng}}$. The radius of the NC sphere was tuned using bulk simulations to be commensurate with a capsid comprising 80 GPs. To minimize the number of parameters, we do not consider an attractive interaction between the NC and membrane, but the NC beads experience a repulsive Lennard-Jones potential with all membrane beads.

\subsection{Conformational change and subunit concentration}
\label{sec:conformationalmodel}
Experiments on several viral families suggest that viral proteins interconvert between `assembly-active' and `assembly-inactive' conformations, which are respectively compatible or incompatible with assembly into the virion \cite{Packianathan2010,Deshmukh2013,Zlotnick2011}. Computational modeling suggests that such conformational dynamics can suppress kinetic traps \cite{Lazaro2016,Grime2016}. Conformational changes of the alphavirus GPs E1 and E2 are required for dimerization in the cytoplasm, and it has been proposed that the GPs interconvert between assembly-inactive and assembly-active conformations \cite{Zlotnick2011}, possibly triggered by interaction with NC proteins \cite{Forsell2000}. Based on these considerations, our GP model includes interconversion between assembly-active and assembly-inactive conformations. The two conformations have identical geometries, but only assembly-active conformations experience attractive interactions to neighboring subunits. We adopt the `Induced-Fit' model of Ref.~\cite{Lazaro2016}, meaning that interaction with an assembling GP shell or the NC favors the assembly-active conformation. For simplicity, we consider the limit of infinite activation energy. In particular, with a periodicity of $\tau_\text{c}$ all the inactive subunits found within a distance 1.0$\sigma$ of the capsid are switched to the active conformation, while any active subunits further than this distance from an assembling shell convert to the inactive conformation.  Results were unchanged when we performed simulations at finite activation energies larger than $4k_{\text{B}}T$.

In simulations performed at a constant total number of GPs  the assembly rate progressively slows over the course of the simulation due to the depletion of unassembled subunits. This is an unphysical result arising from the necessarily finite size of our simulations. Moreover, during an infection additional GPs would be targeted to and inserted into the membrane via non-equilibrium process (powered by ATP).  Therefore, our simulations are performed at constant subunit concentration within the membrane (outside of the region where an assembling shell is located). To achieve this, we include a third subunit type called `reservoir subunits', which effectively acts as a reservoir of inactive subunits. These subunits interact with membrane beads but experience no interactions with the other two types of GP subunits. With a periodicity of $\tau_\text{c}$, reservoir subunits located in a local region free of active or inactive subunits (corresponding to a circumference of radius 1.5 times the radius of the largest subunit bead) are switched to the assembly-inactive state.

\subsection{Simulations}
\label{sec:simulations}
We performed simulations in HOOMD-blue\cite{Nguyen2011}, version 1.3.1, which uses of GPUs to accelerate molecular simulations. Both the subunits and the NC were simulated using the Brownian dynamics algorithm for rigid bodies. The membrane dynamics was integrated using the NPT algorithm, a modified implementation of the Martina-Tobias-Klein thermostat-barostat. The box size changes in the membrane plane, to allow membrane relaxation and maintain a constant lateral pressure. The out-of-plane dimension was fixed at $200\sigma$.

Throughout the manuscript we report dimensions of length, mass, and energy in units of $\sigma$, $m_{0}$, and the thermal energy $k_{\text{B}}T$.  We fixed temperature at $k_{\text{B}}T/ \epsilon_{0}=1.1$. Physical sizes and timescales can be estimated as follows. We set the diameter of the lipid head as $d_{\text{head}}= \sigma$, so that considering a 5nm-thick bilayer leads to $\sigma \approx 0.9$nm.  The characteristic timescale of the simulation is determined by the subunit diffusion, which in our simulations  is dominated by the interaction with the membrane lipids rather than with the bath. We define our unit of time $\tau_{0}$ as the characteristic time of a subunit to diffuse a distance $\sigma$ on the membrane. Comparing with a typical transmembrane protein diffusion constant $\sim 4\mu$m$^{2}$/s \cite{Ramudarai2009,Goose2013}, we obtain $\tau_{0}=250$ns.

Our simulated equations of motion do not account for hydrodynamic coupling between the membrane and the implicit solvent, which can accelerate the propagation of bilayer perturbations. To assess the significance of this effect, we performed an additional series of simulations which did account for hydrodynamic coupling, by evolving membrane dynamics according to the NPH algorithm in combination with a dissipative particle dynamics (DPD) thermostat.
As expected from Matthews and Likos \cite{Matthews2012}, we found that hydrodynamic interactions did enhance the rate of membrane deformations; however, budding proceeded only 1.1-1.2 times faster than with the NPT scheme. Moreover, the end-product distribution was the same with and without hydrodynamic interactions. Therefore, to avoid the increased computational cost associated with the DPD algorithm, we performed all subsequent simulations with the NPT method. The very limited effect of hydrodynamics can be understood from the fact that assembly timescales in our simulations are more strongly governed by subunit diffusion than by membrane dynamics (Fig. \ref{fig:timescales}).

Our system size was constrained by the capsid dimensions and the need to access long timescales. Taking the Sindbis virion as a reference structure, the bilayer neutral surface radius in the virion is $\approx 24 $nm \cite{Zhang2002}, so the surface area of the membrane envelope is $A_{0}\sim 7200$nm$^2$. We thus needed to simulate membrane patches that were significantly larger than $A_{0}$ to ensure that the membrane tension remained close to zero and that finite-size effects were negligible. Throughout this manuscript we report results from simulations on a membrane patch with size $170\times170$nm$^2$ ($A \sim 28,900$nm$^2$), which contains $51,842$ lipids. We compared membrane deformations, capsid size and organization from these simulations against a set of simulations on a larger membrane ($210\times210$nm$^2$, $A\sim44,100$nm$^2$) and observed no significant differences, suggesting that finite size effects were minimal.
Simulations were initialized with 160 subunits uniformly distributed on the membrane, including 4 active-binding subunits (located at the center of the membrane) with the remainder in the assembly-inactive conformation. In addition, there were 156 subunits in the reservoir conformation uniformly distributed. The membrane was then equilibrated to relax any unphysical effects from subunit placement by integrating the dynamics for 1,500 $\tau_{0}$ without attractive interactions between GPs.  Simulations were then performed for 4,200 $\tau_{0}$ with all interactions turned on. The timestep was set to $\Delta t=0.0015$, and the thermostat and barostat coupling constants were $\tau_{T}=0.4$ and $\tau_{P}=0.5$, respectively. Since the tension within the cell membrane during alphavirus budding is unknown, we set the reference pressure to $P_{0}=0$ to simulate a tensionless membrane. The conformational switching timescale was set to $\tau_\text{c}=3\tau_0$, sufficiently frequent that the dynamics are insensitive to changes in this parameter.
Unless otherwise specified, for each parameter set we perform 8 independent simulations.

\section{Appendix}

\subsection*{Equilibrium model for the dependence of GP shell size on system parameters}
\label{sec:EquilModel}
In this section we give a detailed derivation of Eq.~\ref{eq:nsubs} of the main text. This expression explains the simulation results for GP shell size as a function of control parameters (Fig.~\ref{fig:bending}), and is obtained  from a simple equilibrium model based on the thermodynamics of assembly \cite{Safran1994,Hagan2014} that accounts for the elasticity of the shell and the membrane.

The total free energy for the system of free subunits on the membrane, shell intermediates of size $n$, and complete shells of size $N$ can be expressed as
\begin{align}
F / k_{\text{B}}T = \sum_{n=1}^{N} \rho_{n} [\log{\rho_{n}v_{0}}-1] + \rho_{n} {G_n^{\text{shell}}} / k_{\text{B}}T  ,
\label{eq:totalfreenergy}
\end{align}
where $\rho_{n}$ and  ${G_n^{\text{shell}}}$ are respectively the concentration and interaction free energy for an intermediate with size $n$,  and $v_{0}$ is a standard state volume. Minimization of Eq.~\eqref{eq:totalfreenergy} subject to the constraint of constant subunit concentration yields the well-known law of mass action for the equilibrium distribution of intermediate concentrations,
\begin{align}
\rho_{n}v_{0}=\exp\left[-\left({G_n^{\text{shell}}}-n \mu_1\right)/ k_{\text{B}}T \right] .
\label{eq:LMA}
\end{align}
with \begin{align}
\mu_{1} = \log{\rho_{1}v_{0}}
\label{eq:chempotential1}
\end{align}
the chemical potential of free subunits. Similarly we can compute the chemical potential for intermediates $\mu=\partial F / \partial \rho_{n}$ as
\begin{align}
\mu_{n} = \log{\rho_{n}v_{0}} + {G_n^{\text{shell}}} / k_{\text{B}}T .
\label{eq:chempotentialn}
\end{align}
For large shells, the first term in \eqref{eq:chempotentialn} is neglegible compared to the  free energy of the shell, and the chemical pontential can be approximated as $\mu_{n}\approx {G_n^{\text{shell}}}/ k_{\text{B}}T$. In equilibrium, the chemical potential of free subunits must be equal to that of subunits in shells and intermediates, leading to $\mu_{1}=\mu_{n} / n \approx {G_n^{\text{shell}}} / nk_{\text{B}}T$.

The intermediate size with maximal concentration is determined by the condition
\begin{align}
\frac{d\rho_{n}}{dn}=\rho_{n}\frac{d}{dn}[-{G_n^{\text{shell}}} k_{\text{B}}T + n \mu_{1}] =0,
\label{eq:rhomax}
\end{align}
which using \eqref{eq:chempotential1}  and \eqref{eq:chempotentialn} can be rewriten as
\begin{align}
\frac{d}{dn}[-{G_n^{\text{shell}}}/ k_{\text{B}}T + n \mu_{1}] \approx \left[- \frac{d {G_n^{\text{shell}}}}{dn} + {G_n^{\text{shell}}} / n \right]/ k_{\text{B}}T.
\label{eq:rhomax2}
\end{align}
Thus, the optimal size at equilibrium is that which minimizes the interaction free energy per subunit, ${G_n^{\text{shell}}}/n$.

The interaction free energy includes subunit-subunit interactions and the elastic energy of the shell and the membrane. Assuming that the shell can be described as a continuous, two-dimensional spherical shell, its elastic energy is given by the Helfrich bending energy, with bending modulus $\kappa_{\text{shell}}$ and spontaneous curvature $c_{0}=2/R_{0}$, where $R_{0}$ is the equilibrium radius of the shell. The membrane underneath is a symmetric bilayer with rigidity $\kappa_{\text{mem}}$. The free energy ${G_n^{\text{shell}}}$ thus reads
\begin{align}
{G_n^{\text{shell}}} = n \Delta g_{\text{g}} + \frac{\kappa_{\text{mem}}}{2} \int_{S} c^{2} dS +  \frac{\kappa_{\text{shell}}}{2} \int_{S} (c-c_0)^{2} dS,
\label{eq:helfrich}
\end{align}
where $\Delta g_{\text{g}}$ is the free energy per subunit added to the shell,  $c$ is the total curvature, and $S$ denotes the surface area. Assuming spherical symmetry and accounting for the fact that the subunits are rigid, the shell surface area is $S \approx n S_{0}$, with $S_{0}$ as the area per subunit. We can then express the total curvature as a function of the number of subunits in the particle, $c= 2/R \approx (16\pi / n A_{0})^{1/2}$.
Here we are assuming that the Gaussian modulus of the membrane is unchanged by the presence of the GPs, so that the integrated Gaussian curvature is constant for fixed topology by the Gauss-Bonnet theorem \cite{Deserno2015}. Under the (reasonable) assumption that the shell size is determined before scission, the Gaussian curvature energy then contributes a constant to the free energy and can be neglected. We have also neglected the energy from the 12 disclinations in the shell. Accounting for this could shift the theory curve in Fig.~\ref{fig:bending}b  but would not change the slope.

Finally, recalling that the equilibrium configuration minimizes the interaction free energy per subunit, ${G_n^{\text{shell}}}/n$ results in \eqref{eq:nsubs} of the main text,
\begin{align}
n = n_{0} \left( 1 + \frac{\kappa_{\text{mem}}}{ \kappa_{\text{shell}}} \right)^{2},
\label{eq:nsubs}
\end{align}
where $n_{0}$ is the number of subunits in the equilibrium configuration in the absence of membrane, corresponding to $n_{0}=80$ in our model.

\subsection*{Acknowledgements}
This work was supported by the NIH, Award Number R01GM108021 from the National Institute Of General Medical Sciences (GRL and MFH), the Brandeis Center for Bioinspired Soft Materials, an NSF MRSEC,  DMR-1420382 (GRL), and the NSF, award number MCB1157716 (SM). Computational resources were provided by the NSF through XSEDE computing resources (MCB090163) and the Brandeis HPCC which is partially supported by the Brandeis MRSEC.