# Molecular Assembly Lines in Active Droplets

Tyler Harmon1,2* and Frank Jülicher1,3,4 1Max Planck Institute for the Physics
of Complex Systems, Nöthnitzerstr. 38, 01187 Dresden, Germany,
2Max Planck Institute of Molecular Cell Biology and Genetics, Pfotenhauerstr.
108, 01307 Dresden,
3Center for Systems Biology Dresden, 01307 Dresden, Germany.
4Cluster of Excellence Physics of Life, TU Dresden, 01062 Dresden, Germany.

###### Abstract

Large protein complexes are assembled from protein subunits to form a specific
structure. In our theoretic work, we propose that assembly into the correct
structure could be reliably achieved through an assembly line with a specific
sequence of assembly steps. Using droplet interfaces to position compartment
boundaries, we show that an assembly line can be self organized by active
droplets. As a consequence, assembly steps can be arranged spatially so that a
specific order of assembly is achieved and incorrect assembly is strongly
suppressed.

##  I Introduction

Large complexes play important roles in several critical aspects of life.
Examples are bacterial flagellar motors, viral capsids, proteasomes, and
ribosomes. Many of such complexes must be assembled into a specific
arrangement in order to function. An important question is how such complexes
can be created at high efficiency and fast rate, avoiding the formation of
incorrect and incomplete assemblies Gartner et al. (2020); Grant et al.
(2011); Hagan et al. (2011); Chen et al. (2008); Hagan and Elrad (2010);
Lazaro and Hagan (2016); Reinhardt and Frenkel (2014); Jacobs et al. (2015);
Ke et al. (2012); Jacobs and Frenkel (2015); Whitelam (2015); Sear (2007); Wei
et al. (2012); Rotskoff and Geissler (2018). One strategy of self assembly is
complex formation through a thermodynamic process where the assembled state is
a free energy minimum Hagan et al. (2011); Grant et al. (2011). There could
exist several accessible minima such that self assembly would not lead to a
unique structure Gartner et al. (2020); Hagan et al. (2011); Grant et al.
(2011); Whitelam (2015). Furthermore, assembly can often be slowed by trapping
in metastable intermediates. An example of complexes that have been proposed
to employ a thermodynamic assembly strategy are viral capsids Chen (2016).

Alternative strategies based on non-equilibrium assembly could avoid the
problems of thermodynamic self assembly such as kinetic traps and allow for
the assembly of structures that cannot be reached by minimizing the free
energy Gartner et al. (2020); Whitelam (2015); Rotskoff and Geissler (2018).
This can for example be achieved when components assemble in a specific
temporal order. One of these strategies is separation of assembly timescales.
Here the assembly of the components that need to be added early have high
association kinetics while components that need to be added late have slow
association kinetics. This biases the addition of subunits towards a correct
temporal ordering. For this strategy to yield a robust temporal ordering, it
requires kinetics that span many orders of magnitude. Another non-equilibrium
strategy uses a carefully timed sequential synthesis to provide subunits when
specific assembly phases have completed. In this process, incorrect binding of
subunits is prevented by allowing them to encounter the nascent complex when
only the correct binding site is accessible. This strategy requires specific
gene expression programs which limit the rate of complex formation. It is for
example employed in the assembly of the bacterial flagella motor Alon (2006).

These examples of non-equilibrium assembly lead us to the question of whether
complex assembly could emerge as a non-equilibrium pattern in a self-organized
process. More precisely, we ask whether reliable and high-throughput assembly
of a complex with a specific structure can be achieved in a steady state where
components flow in and complexes flow out. We show that this is possible in a
self-organized assembly line. The assembly line organizes assembly steps in
space, thereby providing a specific temporal order during complex assembly.
Such an assembly line could be organized within a biochemical compartment.

Biochemical compartments in cells either have a membrane or are membraneless.
Membraneless compartments are often liquid like biomolecular condensates that
can be described as droplets that phase separate from their surroundings Li et
al. (2012); Banani et al. (2017); McCall et al. (2018); Weber et al. (2019). A
rich diversity of geometries for coexisting droplets have been observed in
cells. One example is a smaller droplet inside a larger droplet Feric et al.
(2016); Boisvert et al. (2007). It remains an open question of what functions
such an arrangement of compartments can have.

Here we present the theory of non-equilibrium complex formation by a self
organized assembly line. These emerge as patterns in an active reaction-
diffusion system that is confined in droplets. These patterns consist of
separated bands of distinct reactions which correspond to different assembly
steps. Chemical patterns are typically viewed through the lens of patterns of
concentrations. Note that the bands discussed here reflect the localization of
distinct reactions while the molecular species are not localized. The spatial
arrangement of different assembly steps defines the temporal order in which
subunits are added to the complex. This process can occur at steady state with
a constant influx of subunits and constant outflux of finished complexes. This
scenario allows a high rate of assembly and ensures the formation of correctly
assembled complexes.

We first develop these ideas using a simplified one dimensional model where
the position and width of the reaction bands can be calculated analytically.
We consider a 1D geometry of three distinct volumes separated by two
boundaries, see figure 1a. These volumes could correspond to an arrangement of
droplets. Subunits enter the central stage volume from the left and right
volumes. Assembly of complexes occurs in the stage volume, and is considered
irreversible. We show that if the assembly is fast, association happens at a
specific location in the stage volume. This location depends on the physical
properties of the components and their supply at the boundaries.

We then consider a simplified model in 3-dimensions based on two concentric
droplets, see figure 1b. Here the stage volume is defined by a spherical
droplet that coexisting with both an outer liquid phase and an inner droplet.
Components enter the stage via the boundaries from opposite directions. Such a
geometry of droplets inside droplets can be found in cells, for example in the
nucleolus Boisvert et al. (2007); Weber and Brangwynne (2015); Feric et al.
(2016).

Figure 1:  Spatial organization of macromolecular assembly. a) 1D arrangement
of compartments for molecular assembly. Component $C_{1}$ is produced in the
left volume (green) and component $C_{2}$ is produced in the right volume
(blue). Assembly of the complex $C_{12}$ occurs in the stage volume (orange).
Assembly can be localized to a band between the left and right volumes (dashed
line). The complex is exported via the right volume. b) 3D arrangement of
compartments for molecular assembly. Component $C_{1}$ is produced in the
inner volume (green) and component $C_{2}$ is produced in the outer volume
(blue). Assembly of the complex $C_{12}$ occurs in the stage volume (orange).
Assembly can be localized to a spherical band between the inner and outer
volumes (dashed line). The complex is exported via the outer volume.

##  II Localizing assembly steps in one dimension

First we examine a minimal model in one dimension where molecular assembly
steps can be positioned along a line and separated in space. The region,
$[0,L]$, corresponds to the stage where assembly steps occur. The left
boundary connects the stage to the inner volume, and the right boundary
connects the stage to the outer volume which serve as reservoirs that provide
components. We consider two components $C_{1}$ and $C_{2}$, which enter from
the left and right boundaries respectively. These components bind irreversibly
to form a complex,

$C_{1}+C_{2}\rightarrow C_{12}\;.$

After the formation of the complex $C_{12}$ it will leave the stage into the
outer volume via diffusion. All molecular species move by diffusion and
associate following mass action kinetics. The system is thus described by the
association diffusion equations along the position coordinate x given by

$\displaystyle\frac{\partial}{\partial t}n_{1}$
$\displaystyle=D\frac{\partial^{2}}{\partial x^{2}}n_{1}-kn_{1}n_{2}\;,$ (1a)
$\displaystyle\frac{\partial}{\partial t}n_{2}$
$\displaystyle=D\frac{\partial^{2}}{\partial x^{2}}n_{2}-kn_{1}n_{2}\;.$ (1b)

Here $n_{1}$ and $n_{2}$ are the concentrations of components $C_{1}$ and
$C_{2}$ respectively and $D$ and $k$ are the diffusion constant and
association rate. For simplicity we have chosen equal diffusivities for both
species. These equations are complemented by boundary conditions that connect
the stage to the inner and outer volumes. We consider the case where there are
no sinks in the inner volume and $C_{1}$ and $C_{2}$ are produced only in the
inner and outer volumes respectively. Accordingly, we impose constant influx
$J$ of component $C_{1}$ and no flux of component $C_{2}$ at $x=0$. At $x=L$
we impose boundary conditions that depend on the concentrations $n_{1}$ and
$n_{2}$ which describe a partitioning of molecules between the stage and the
outer volume. The boundary conditions then read

$\displaystyle D\partial_{x}n_{1}\rvert_{x=0}=$ $\displaystyle-J\;,$ (2a)
$\displaystyle D\partial_{x}n_{2}\rvert_{x=0}=$ $\displaystyle 0\;,$ (2b)
$\displaystyle D\partial_{x}n_{1}\rvert_{x=L}=$
$\displaystyle-\beta_{1}n_{1}(x=L)\;,$ (2c) $\displaystyle
D\partial_{x}n_{2}\rvert_{x=L}=$ $\displaystyle\alpha-\beta_{2}n_{2}(x=L)\;.$
(2d)

Here $\alpha$ corresponds to the source of $C_{2}$ and $\beta_{1}$ and
$\beta_{2}$ correspond to sinks of $C_{1}$ and $C_{2}$ respectively.

Of particular interest to us is the steady state solution of equations (1)
with boundary conditions (2), and the profile of the association flux
$\Phi\equiv kn_{1}n_{2}$ as a function of $x$. In figure 2a we show examples
of stationary profiles of the association flux that reveal the localization of
the association to different positions depending on the magnitude of $\alpha$,
the source of $C_{2}$. Interestingly, by changing the magnitude of of
$\alpha$, the location $x_{M}$ of the peak of the association flux is changed,
but the shape remains essentially unchanged. Figure 2b shows the corresponding
steady state concentration profiles for component $C_{1}$ and $C_{2}$ as
dashed and solid lines, respectively.

Away from the region of high association flux, the concentration profiles are
linear, corresponding to a constant flux. These fluxes for the two components
are equal and opposite because assembly consumes an equal number of both
components. As a result equation (1) has symmetric solutions with respect to
$x_{M}$ if the boundaries are far from the assembly volume. In this limiting
case, assembly is fully efficient within $x=[0,L]$, and we can calculate the
exact position $x_{M}$ and the variance $\sigma^{2}$ of the association flux
profile in a Gaussian approximation, see appendix,

$x_{M}=L-D\frac{\alpha-J}{J\beta_{2}}\;,$ (3)
$\sigma\approx\sqrt{\frac{2}{\pi}}(1+\frac{\pi}{2}+\sqrt{1+\pi})^{1/3}\bigg{(}\frac{D^{2}}{kJ}\bigg{)}^{1/3}\;.$
(4)

The limit where boundaries are far from the assembly volumes corresponds to
$\sigma\ll x_{m}$ and $\sigma\ll L-x_{m}$.

This result shows that one can spatially localize an assembly step in a region
of width $\sigma$ centered at position $x_{M}$. This localization of assembly
steps can be used to generate an assembly line. In this assembly line, the
complex grows as it diffuses towards one boundary which we choose to be the
one on the right side. Complex assembly is initiated by a molecule that enters
the stage from the left. It first encounters a single component from the right
to which it can bind. As the assembling complex diffuses toward the right, it
will encounter sequentially additional components that arrive from the right
boundary. These components can thus be incorporated sequentially to ensure
correct assembly. This is schematically represented in figure 2c. We call the
molecule entering from the left and initiating complex formation the
foundation, denoted $F$, and the components entering from the right the
assembly bricks, denoted $B_{i}$. Assembly regions can be arranged in space
such that they are separated by more than a distance $\sigma$. This implies
that the addition of one type of component is completed before the complex
encounters the next component. The spatial order of assembly regions specifies
the dominant temporal order of assembly steps. This assembly line can operate
at steady state with all assembly steps being conducted in parallel.

Figure 2:  Positioning of assembly steps in narrow regions. (a) Normalized
assembly flux $\Phi$ as a function of normalized position x in the stage
region for different values of the normalized influx $\alpha$ of component
$C_{2}$ ($\alpha=70$ (blue), $50$ (orange), $30$ (green), $10$ (yellow)). (b)
Steady state concentrations of components $C_{1}$(dashed) and $C_{2}$(solid)
for the same values of $\alpha$ as in (a). Shown are numerical solutions of
the minimal model given by equations (2) and (3). (c) Cartoon schematic of how
an assembly line could be organized. The foundation molecule tends to diffuse
to the right progressing through the different association bands. Progressing
to each sequential association bands adds another available brick that can
bind to the foundation. Parameter values are $J=0.31$, $D=0.039$, $k=5.9$
$\beta_{1}=5$, $\beta_{2}=10$, and $L=1$.

##  III Assembly lines in spherical geometry

We now discuss an example of how such an assembly line could be realized in
three dimensions using concentric spherical droplets where a small droplet is
located in the center of a larger droplet, see figure 1b. The foundational
component, denoted $F_{0}$ is produced inside the small central droplet which
defines the inner volume. In our example we consider four assembly bricks,
$B_{i}$, $i=1...4$, which are provided in the outer volume that surrounds both
droplets. The stage volume is the space in the larger droplet between the
inner volume and outer volume.

The foundation and bricks enter the stage from opposite sides via diffusion.
For simplicity we consider a simple model where the assembly bricks must bind
to the foundation in a specific sequential order for correct assembly. We call
the intermediate assemblies assembled in the correct sequential order the on-
pathway intermediates, denoted by $F_{i}$ where $i$ is the index of the last
brick that was incorporated into the complex. In this logic, $F_{4}$ is the
correctly assembled final complex. If binding occurs in a different order then
complex formation is unsuccessful and we call these configurations off-
pathway. Figure 3a shows a schematic of all constructs and associations, both
on and off-pathway. We define our model using concentration fields for all
components and assembly intermediates. Inside the stage these concentrations
are described by association-diffusion equations that capture the diffusion of
components and their association steps,

$\displaystyle\frac{\partial}{\partial t}$ $\displaystyle
n_{F,i}=D_{F,i}\nabla^{2}n_{F,i}-kn_{F,i}n_{B,i+1}-kn_{F,i}\Theta(2-i)\sum_{k=i+2}^{4}n_{B,k}+kn_{F,i-1}n_{B,i}\;,$
(5) $\displaystyle\frac{\partial}{\partial t}$ $\displaystyle
n_{B,i}=D_{B,i}\nabla^{2}n_{B,i}-kn_{F,i-1}n_{B,i}-kn_{B,i}\Theta(i-2)\sum_{k=0}^{i-2}n_{F,k}\;.$

Here $n_{B,i}$ is the concentration of brick $i$, $n_{F,i}$ is the
concentration of the free foundation ($i=0$), the on-pathway intermediates
($i=1,2,3$), and product ($i=4$). Furthermore $D_{F0}$, $D_{Fi}$ and $D_{Bi}$
are the diffusion constants for foundation, complexes and bricks respectively,
and $k$ is the rate constant for on and off pathway associations. The Laplace
operator is denoted $\nabla^{2}$ and the Heaviside function is denoted
$\Theta$ with $\Theta(i)=1$ if $i\geq 0$ and $\Theta(i)=0$ otherwise. The
first term of the right hand sides in equations 5 describe diffusion, the
second terms describe the component loss due to on-pathway assembly, the third
terms describe loss via off-pathway associations that does not lead to correct
assembly in our model, and the fourth term is a source due to on-pathway
assembly. These equations are supplemented by boundary conditions at radii
$r_{1}$ and $r_{2}$,

$\displaystyle\frac{dn_{F,0}}{dr}\biggr{\rvert}_{r_{1}}=$
$\displaystyle-\frac{J}{D_{F}4\pi r_{1}^{2}}\;,$ (6a)
$\displaystyle\frac{dn_{F,i}}{dr}\biggr{\rvert}_{r_{2}}=$
$\displaystyle-\frac{\beta_{F}n_{F,i}(r_{2})}{D_{F}4\pi r_{2}^{2}}\;,$ (6b)
$\displaystyle\frac{dn_{B,i}}{dr}\biggr{\rvert}_{r_{2}}=$
$\displaystyle\frac{\alpha_{B,i}-\beta_{B,i}n_{B,i}(r_{2})}{D_{B,i}4\pi
r_{2}^{2}}\;,$ (6c)

where $J$ is the number of foundation molecules produced per unit time in the
inner volume, $\alpha_{Bi}$ is the influx of brick $B_{i}$ from the outer
volume, $\beta_{F}$ and $\beta_{B,i}$ describes the concentration dependence
of the outflux of the components into the outer volume due to partitioning.
Furthermore, we have imposed no flux boundary conditions at $r_{1}$ for
intermediates $F_{i}$ where $i\geq 1$ and the bricks $B_{i}$. These
expressions can be derived for a droplet in a steady state environment.

In spherical coordinates we define the total association flux as

$\Phi_{i}=4\pi r^{2}kn_{F,i-1}n_{B,i}\;.$ (7)

In steady state the total association flux exhibits a maximum at position

$r_{M,i}\approx r_{2}-\frac{4\pi
r_{2}^{2}D_{B,i}}{\beta_{B,i}}\bigg{(}\frac{\alpha_{B,i}-J}{J}\bigg{)}\;,$ (8)

and a width given by

$\sigma_{i}\approx 1.33\bigg{(}\frac{4\pi
r_{M,i}^{2}D_{F}D_{B,i}}{kJ}\bigg{)}^{1/3}\;,$ (9)

see appendix. Figure 3b shows the fraction of complete, incomplete, and
incorrect assemblies leaving the stage at steady state as a function of
association rate. The fraction of complete assemblies defines the efficiency
of the system. Figure 3c and d show the radial profiles of the total
association fluxes for two different association rates. The association rate
primarily changes the width of the peaks of the total association flux but not
the position, see equations (8) and (9). For more sharply peaked total
association fluxes, the efficiency of assembly is higher because different
assembly steps are better separated in space. Accordingly, assembly efficiency
increases as the association rate increases and off-pathway associations are
highly suppressed, see figure 3b. The general requirement for assembly lines
to improve assembly fidelity is that the spacing between association bands is
larger than the width of the bands. Note that in spherical geometry the
assembly steps occur in concentric shells around the inner volume.

Figure 3:  Self-organized molecular assembly lines. (a) Scheme of the on
pathway assembly steps (blue, red, green, and yellow arrows) and off-pathway
dead ends (pink arrows). Here $F_{0}$ represents the foundation and $B_{1}$,
$B_{2}$, $B_{3}$, $B_{4}$ represent the bricks. The on-pathway intermediates
are denoted $F_{1}$, $F_{2}$, $F_{3}$, and $F_{4}$ is the correctly assembled
product. (b) Fraction of completed (green), incomplete (yellow), and incorrect
(red) assemblies that leave the stage in steady state as a function of the
association rate $k$. (c) Radial profiles of total assembly fluxes $\Phi=4\pi
r^{2}kn_{F,i}n_{B,j}$ at steady state for the four on-pathway associations
(blue, red, green, yellow) and the off-pathway associations (pink) for
association rate $k=0.038\;\rm{nM}^{-1}\rm{s}^{-1}$. Here $n_{F,i}$ and
$n_{B,j}$ are concentrations of associating components. (d) same as (c) but
for $k=0.27\;\rm{nM}^{-1}\rm{s}^{-1}$. The curves shown in (c) and (d)
correspond to the assembly steps shown in (a) in the same color. Shown in
(b-d) are numerical solutions for spherical geometry and parameter values
$J=10\;\rm{\mu m}^{3}\rm{nMs}^{-1}$, $\beta_{F}=1.26\;\rm{\mu
m}^{3}\rm{s}^{-1}$, $\beta_{B}=1.26\;\rm{\mu m}^{3}\rm{s}^{-1}$,
$r_{1}=0.25\;\rm{\mu m}$, $r_{2}=1.0\;\rm{\mu m}$, $D_{F}=0.01\;\rm{\mu
m}^{2}\rm{s}^{-1}$, $D_{B,i}=0.01\;\rm{\mu m}^{2}\rm{s}^{-1}$, and
$\alpha_{B,i}=(276,163,100,57)\;\rm{\mu m}^{3}\rm{nMs}^{-1}$ for
$i=(1,2,3,4)$.

##  IV Discussion

Cells need to assemble molecular complexes in a correct and efficient manner
and avoid off-pathway dead ends. We have shown that molecular assembly lines
can be self-organized in cellular compartments such as phase separated
condensates. Such an assembly line requires a spatial organization of
association steps which distinguishes it from well mixed association
scenarios. This spatial organization can lead to a high-throughput assembly by
operating at steady state. Additionally, high-fidelity of assembly can be
achieved by controlling the temporal order in which subunits are added.
Control over temporal order is possible by the spatial separation of the
assembly bands. The spatial arrangement of these bands could be controlled by
concentrations of components at their sources, which we note can be a distance
from the assembly line. The scenario proposed here has four hallmarks that
could be seen in experiments: i) association but not concentration is confined
to distinct spatial bands, ii) unbound components can be found close to their
source but not past the position where they are added to the complex, iii) the
average complex size increases towards the outer boundary, and iv) changing
the concentration ratio of components yields incomplete or incorrectly
assembled complexes because assembly bands can become rearranged.

Self organized molecular assembly lines can be compared to assembly under well
mixed conditions where concentrations are constant in space. Using the
parameter values given in the caption for figure 3b, but considering assembly
in well mixed conditions, we find an export fraction of correct assemblies
$E=\prod_{i}^{4}kn_{\infty,i}/\sum_{j=i}^{4}kn_{\infty,j}\simeq 15\%$. This is
much smaller than values approaching $100\%$ which can be obtained in the
self-organized assembly line for the same parameter values, see figure 3b.
This reduction in export fraction of correct assemblies occurs because
intermediates are exposed to all the binding bricks and incorrect binding
steps therefore become likely. This contrasts with an efficient assembly line
where binding bricks are added sequentially in the correct order. In well
mixed conditions, high fidelity can be achieved if time scales are separated
such that different assembly steps occur at different rates. In particular,
components that need to bind early should have the fastest association
kinetics. However, in order to achieve an export fraction $E$ of correctly
assembled complexes, this strategy requires association rates that span at
least a factor of $(E^{-1/(N-1)}-1)^{N-1}$ where $N$ is the number of assembly
bricks in the complex, see appendix. For $80\%$ efficiency of assembly with
four bricks, matching our assembly line example, the association rates would
need to span a factor of 2200. Such a large range of rates suggests that some
steps must be slow and thus rate limiting for assembly. This is in contrast to
the assembly line proposed here where the example in figure 3 uses parameters
that span a factor of 5. Indeed, a large span of assembly rates manifests as a
equally large range of concentration levels at steady state. Therefore, the
well mixed scenario requires much larger amounts of unfinished complexes to
achieve the same export rate. This consumes more resources and materials and
is therefore less efficient.

For simplicity we considered simplified geometries in one and three
dimensions, using a spherical geometry motivated by liquid-like condensates.
However, the strategy discussed here is more general and can also work in
other geometries. As illustrated in the one dimensional case, the assembly
steps are positioned at specific distances from the outer surface. Therefore,
our work suggests that in non-spherical geometries, assembly steps will be
positioned to non-intersecting manifolds inside the outer surface. We expect
this mechanism to robustly self organize assembly lines in three dimensions in
a broader collection of geometries, including multiple spherical inner
volumes.

The assembly line described here can control the temporal order of assembly
steps. In principle, complexes can also form without a fixed temporal order.
For example, if many identical components, such as a viral capsid, then the
time order is irrelevant. On the other extreme, if all components are
different then temporal order can be helpful. In this case it is likely that
several local minima exist which correspond to different assembled structures.
In principle a unique structure emerges as the lowest energy configuration but
this may take a long time to reach. However, if the temporal order is
controlled, a specific local minima could be consistently reached. Therefore
time ordered assembly provides a strategy to reliably assemble many components
into a specific structure. For simplicity we have considered irreversible
assembly steps. In practice molecular binding events are not fully
irreversible and the limit of irreversible steps corresponds to a situation
where the time scale for an unbinding event is slower than the time between
subsequent assembly steps. In this case unbinding events become rare enough to
become negligible.

An important example of molecular assembly are RNA-protein complexes. The
components of such complexes can fit well into our simplified assembly scheme,
shown in figure 3a. Here bricks (proteins) are added to a foundational element
(RNA) and the bricks and foundation are supplied from different regions
(translation in the cytoplasm and transcription in the nucleus respectively).

The most prominent example of an RNA-protein complex is the ribosome which
mainly consists of large RNA and many different small proteins Tschochner and
Hurt (2003); Davis and Williamson (2017); Klinge and Woolford (2019).
Ribosomes are assembled in the nucleolus, a liquid-like droplet-inside-droplet
compartment in the cell nucleus. Ribosomal RNA is produced in the inner
compartment of the nucleolus called the Dense Fibrillar Component,
corresponding to the inner volume. Ribosomal proteins are provided in the
nucleoplasm, corresponding to the outer volume. The assembly occurs in the
Granular Component of the nucleolus corresponding to the stage volume. Given
this arrangement, ribosome assembly is an ideal candidate for the molecular
assembly line proposed here. It will be interesting to explore whether
signatures of an assembly line can be found in ribosome assembly. An
interesting question is whether assembly bands could be directly observed
experimentally. This poses a challenge because labeling components alone may
not reveal the spatial organization of an assembly line. This is because
florescent labels would be attached to a component as well as to the assembled
complex. Even if the assembly is organized into distinct bands, the florescent
signal would not show this structure. Techniques such as FRET could
differentiate between bound and unbound components and thus are promising to
reveal spatial patterns of assembly processes.

In addition to the nucleolus, other liquid like compartments could play a role
to organize the assembly of macromolecular complexes. An example is p-granules
in C. elegans which can be located on the nuclear membrane and cover nuclear
pores Updike et al. (2011). This setting has the important feature that
p-granules can communicate with two different compartments at the same time,
the cytoplasm and nucleoplasm, allowing opposing fluxes from both
compartments. This suggests that p-granules are also naturally suited for
reliable molecular processing using assembly lines of the type proposed here.
More generally, we expect that liquid like compartments are used to self
organize biochemical processes in space. Our work therefor provides new
insights into the possible role of liquid like condensates for biological
processes.

##  V acknowledgments

We thank Rabea Seyboldt, Anthony A. Hyman, Richard W. Kriwacki, and Diana
Mitrea for useful discussions.

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