# Collective deformation modes promote fibrous self-assembly in protein-like
particles

Hugo Le Roy<EMAIL_ADDRESS>Université Paris-Saclay, CNRS, LPTMS, 91405,
Orsay, France  Institute of Physics, École Polytechnique Fédérale de
Lausanne—EPFL, 1015 Lausanne, Switzerland  M. Mert Terzi  Université Paris-
Saclay, CNRS, LPTMS, 91405, Orsay, France  PMMH, CNRS, ESPCI Paris, PSL
University, Sorbonne Université, Université de Paris, F-75005, Paris, France
Martin Lenz<EMAIL_ADDRESS>Université Paris-Saclay,
CNRS, LPTMS, 91405, Orsay, France  PMMH, CNRS, ESPCI Paris, PSL University,
Sorbonne Université, Université de Paris, F-75005, Paris, France

###### Abstract

The self-assembly of particles into organized structures is a key feature of
living organisms and a major engineering challenge. While it may proceed
through the binding of perfectly matched, puzzle-pieces-like particles, many
other instances involve ill-fitting particles that must deform to fit
together. These include some pathological proteins, which have a known
propensity to form fibrous aggregates. Despite this observation, the general
relationship between the individual characteristics of the particles and the
overall structure of the aggregate is not understood. To elucidate it, we
analytically and numerically study the self-assembly of two-dimensional,
deformable ill-fitting particles. We find that moderately sticky particles
tend to form equilibrium self-limited aggregates whose size is set by an
elastic boundary layer associated with collective deformations that may extend
over many particles. Particles with a soft internal deformation mode thus give
rise to large aggregates. Besides, when the particles are incompressible,
their aggregates tend to be anisotropic and fiber-like. Our results are
preserved in a more complex particle model with randomly chosen elastic
properties. This indicates that generic protein characteristics such as
allostery and incompressibility could favor the formation of fibers in protein
aggregation, and suggests design principles for artificial self-assembling
structures.

Soft matter $|$ Theoretical biophysics $|$ Frustrated self-assembly $|$ Self-
limiting assembly $|$ Protein aggregation

††preprint: APS/123-QED

Functional structures in living cells are often self-assembled from several
copies of a single protein, from microtubules and clathrin cages to viral
capsids in the shape of cylinders or spheres [1, 2, 3]. The radius of such
assemblies is dictated by the curvatures of the individual particles that
precisely fit together to form them. Similarly, artificial self-assembly often
relies on fitting well-adjusted particles together to build structures with a
controlled size [4, 5, 6, 7].

In other instances however, the shapes of the individual particles are ill-
fitting and do not obviously dictate the structure of the aggregate. This is
the case in the pathological aggregation of normally soluble proteins, _i.e._
, of proteins not evolutionarily optimized to self-assemble into a well-
defined structure [8, 9, 10, 11]. Despite the diversity of the shapes and
interactions involved, the aggregation of these ill-fitting proteins produces
fibrous structures with remarkable consistency. These fibers display varied
widths and internal structures [12, 13, 14], and the proteins within are often
significantly deformed in ways that depend on the assembly protocol [15].
Deformations are common in proteins, and many display physiologically relevant
deformation modes that facilitate self-assembly [16, 17], perform a motor
function [18], participate in their biochemical activity [19], or serve to
mechanically transmit a signal, a function known as allostery [20, 21].
Nevertheless, the generic implications of the deformability of proteins on
their ill-fitting aggregation is not understood.

Beyond proteins, particle deformations have long been suggested as a mechanism
to regulate aggregate size in self-assembly [22]. In this picture, ill-fitting
particles are forced to deform as they tightly bind to one another. As more
and more particles are added to the aggregate, the distortions build up until
they become so severe as to prevent any further assembly [23, 24]. The
accumulation of stresses resulting from such distortions may govern the
structure of DNA origami assemblies [25, 26] and prevent the indefinite
bundling of preexisting protein fibers [27, 28, 29]. Beyond merely fixing the
overall size of an aggregate, the accumulation of deformations can moreover
dramatically alter its shape. This has been proposed to drive a transition
from cylindrical to tape-like fiber bundles [30, 31]. Finally, it can also
drive sticky, deformable particles to form anisotropic aggregates that grow
into infinite one-dimensional structures reminiscent of pathological protein
fibers [32]. The underlying mechanism and the nature of the particle
properties that determine the dimensionality of the final aggregate however
remain elusive.

In this paper, we provide a detailed analytical understanding of the emergence
of self-limited and fibrous aggregates in a two-dimensional deformable
particle system. We first introduce a minimal model based on highly
symmetrical particles. It gives rise to an emergent elastic boundary layer
length $\ell$, allowing us to map it onto a continuum description in the limit
of large $\ell$. We use this description to compare the energies of several
candidate structures and establish an aggregation phase diagram, which we then
validate using numerical simulations. Finally, we introduce a much broader
class of elastic particles, and demonstrate that the results derived in the
idealized model still apply there, including in cases where the values of
$\ell$ are moderate.

## Elastic aggregation model

To understand the interplay between particle deformations and aggregate
structure, we first discuss a minimal one-dimensional example where ill-
fitting particles deform upon aggregation. This leads to a deformation
gradient from moderately deformed particles at the edge of the aggregate to
highly deformed ones in its bulk. We then introduce a two-dimensional model
that allows for a much wider diversity of aggregate structures, including
fibers and planes. This model is analytically intractable in its general form,
which leads us to design a continuum limit to enable further analysis.

Figure 1:  The assembly of ill-fitting particles results in collective
deformations, shown here in a minimal 1D model. (a)  _left:_ Individual
particle at rest, _right:_ colored springs schematizing the elasticity of the
particles. In this 1D model, particles are allowed to aggregate only along the
horizontal direction. Their corners then become colocalized (_black arrows_),
which requires deforming at least some of the springs. (b) Schematic of a one-
dimensional particle aggregate showing the state of the springs therein. While
the yellow and red springs are able to assume different lengths in the
vicinity of the edge of the aggregate, they are forced to have the same
lengths in the bulk. The resulting energetic penalty hampers the formation of
space-filling, bulky aggregates. In two or three dimensions, a similar penalty
may result in the formation of fibrous aggregates.

### One-dimensional toy model

We consider a collection of identical isosceles trapezoids [Fig. 1(a)]. Each
such particle can aggregate with its left and right neighbors by fusing its
vertical sides with theirs. In the special case where the trapezoids are well-
adjusted, _i.e._ , if they are rectangles, such binding does not require any
deformation. Conversely, particles whose top and bottom faces have different
lengths are ill-fitting and must deform to bind. We model the energetic cost
of this deformation using four springs: two representing the top and bottom
faces of the particles (yellow and red) with rest lengths $1\pm\epsilon$ and
spring constant $k$, and two connecting springs with spring constants
$k_{c}/2$ and rest lengths $\epsilon$ that tend to center the top and bottom
faces (blue). Summing the contributions of these four springs, we write the
deformation energy of the central particle of Fig. 1(b) as

$\displaystyle e_{d}^{(i)}=$
$\displaystyle\frac{k}{2}[(x_{i+1}^{\uparrow}-x_{i}^{\uparrow})-(1+\epsilon)]^{2}+\frac{k}{2}[(x_{i+1}^{\downarrow}-x_{i}^{\downarrow})-(1-\epsilon)]^{2}$
$\displaystyle+\frac{k_{c}}{4}(x_{i+1}^{\uparrow}-x_{i+1}^{\downarrow}-\epsilon)^{2}+\frac{k_{c}}{4}(x_{i}^{\downarrow}-x_{i}^{\uparrow}-\epsilon)^{2},$
(1)

where the $\\{x_{i}^{\uparrow}\\}$, $\\{x_{i}^{\downarrow}\\}$ denote the
coordinates of particle corners. This model does not involve any explicit
prestresses; including some would not make any difference within the linear
response regimes studied here and in the rest of this work.

Defining the shift between an upper and lower corner as
$\delta_{i}=x_{i}^{\uparrow}-x_{i}^{\downarrow}$, force balance dictates that
inside the aggregate

$k(\delta_{i+1}-2\delta_{i}+\delta_{i-1})=k_{c}\delta_{i}\quad\Rightarrow\quad\delta_{i}\propto\epsilon\sinh(i/\ell),$
(2)

where we define $i=0$ as the center of the aggregate and where

$\displaystyle\ell$
$\displaystyle=1/\ln\left[1+k_{c}/k+\sqrt{2k_{c}/k+(k_{c}/k)^{2}}\right]$
$\displaystyle\underset{k_{c}\ll k}{\sim}\sqrt{{k}/{2k_{c}}}.$ (3)

The full prefactor of the last expression of (2) is fixed through the force
balance condition at the aggregate’s left and right edges. In a large
aggregate, it results in an exponential decay
$\delta_{i}\propto\epsilon\exp(-|i-i_{\text{edge}}|/\ell)$ close to these
edges. The initially trapezoidal particles at the center of the aggregate are
thus forced into a rectangular shape ($\delta_{i}=0$), in contrast with the
particles that reside within an edge-associated elastic boundary layer of size
$\ell$.

In the limit $k_{c}/k\rightarrow 0$, the boundary layer size diverges. This
regime is characterized by very rigid yellow and red springs, implying that
the yellow and red springs close to the edge of the aggregate are almost at
their equilibrium lengths. Going deeper into the aggregate, each blue spring
exerts a small compressive (tensile) force on the yellow (red) chain. These
forces add up over long distances, implying a progressive change of the yellow
and red strain over a length scale much larger than the particle size.

### Two-dimensional particle model

One-dimensional aggregates are very simple geometrically, and are entirely
characterized by the number of particles that they contain. Aggregation in
higher dimension allows for a much broader variety of aggregate structures. To
study the emergence of complex shapes there, we introduce a two-dimensional
model based on hexagonal particles. Well-adjusted particles are represented by
regular hexagons. Ill-fitting hexagons, by contrast, have alternating
protruding (yellow) and withdrawn (red) corners [Fig. 2(a)]. The corners
belonging to each category form a yellow and a red equilateral triangle whose
sides are springs with rest lengths $1\pm\epsilon$ and a spring constant $k$.
The (blue) sides of the hexagon itself play the role of connecting springs
with spring constant $k_{c}$, and are at rest when the yellow and red
equilateral triangles are themselves at rest and centered. These hexagonal
particles are three-fold symmetric, which rules out an intrinsic, particle-
level preference for forming one-dimensional fibers. Any fiber formed from
their aggregation will thus be an emergent symmetry-broken structure.

Figure 2: 2D model of ill-fitting self-assembly. (a) Individual particles have
two kind of vertices, which define three interconnected elastic networks. (b)
As the particles are aggregated by matching the vertices with the same color,
individual particles undergo deformations.

The response of these particles to shear and uniform compression cannot be
independently varied while holding $k_{c}/k$ (and therefore the boundary layer
size) constant. To enable particles that range from fully compressible to
incompressible, we thus additionally endow both yellow and red triangles with
an areal rigidity. We implement it through an energy
$e^{\uparrow/\downarrow}_{\text{area}}=k_{\text{area}}(A^{\uparrow/\downarrow}-A_{0}^{\uparrow/\downarrow})^{2}/2A_{0}^{\uparrow/\downarrow}$,
where $\uparrow$ and $\downarrow$ respectively refer to the yellow and red
triangle and $A$ ($A_{0}$) are the associated triangle areas (rest areas).

The quadratic spring and areal energies introduced above all vanish in the
particle’s resting state. Any deformation away from this state implies an
energetic cost, and such deformations are required to accommodate particle
binding. In our model, two particles can bind along a blue side by merging one
yellow and one red corner each. The merging of corners with different colors
is not allowed. Each pair of bound sides is rewarded by an energy $-g$
regardless of the particles’ state of deformation, which defines a zero-range
interaction between particles. These rules favor the assembly of hexagons into
a triangular “particle lattice” [Fig. 2(b)] where all pairs of neighboring
particles are bound, which we consider throughout. The aggregate topology,
_i.e._ , the specification of which particles bind to which others through
which sides, can thus be entirely described by considering a triangular
lattice and specifying a list of the lattice sites that are occupied by a
particle. In the following we use the symbol $\mathcal{T}$ to refer to this
topology. Since the binding energy is fully determined by the number of bound
particle sides, it only depends on $\mathcal{T}$.

### Continuum formalism

Finding the most favorable aggregate in our 2D particle model requires two
steps: to compute the optimal deformation energy for each fixed topology
$\mathcal{T}$, and then to determine which topology has the lowest optimal
energy. Here we introduce a continuum approximation that renders the first
step analytically tractable in several important cases. This approximation is
formally valid in the limit where the 2D counterpart of the boundary layer
size $\ell$ is much larger than the particle size.

To define our continuum limit, we note that in large aggregates where all
sites of the particle lattice discussed above are occupied (_i.e._ , without
holes), the yellow and red springs arrange into triangular spring lattices. In
the regime $k_{c}\ll k$ where connecting (blue) springs are much softer than
triangle (yellow and red) springs, the strain within the yellow and red
triangular spring lattices varies slowly over space. This is similar to the
behavior of our one-dimensional model. As a result, we can assimilate each of
these triangular spring lattices to a continuum sheet, giving rise to a
continuum elastic energy

$\displaystyle E_{d}=$
$\displaystyle\iint\left[\frac{\lambda}{2}\left(\partial_{\alpha}u_{\alpha}^{\uparrow}-2\epsilon\right)^{2}+\mu\left(\frac{\partial_{\alpha}u_{\beta}^{\uparrow}+\partial_{\beta}u_{\alpha}^{\uparrow}}{2}-\epsilon\delta_{\alpha\beta}\right)^{2}\right]\,\text{d}A$
$\displaystyle+\iint\left[\frac{\lambda}{2}\left(\partial_{\alpha}u_{\alpha}^{\downarrow}+2\epsilon\right)^{2}+\mu\left(\frac{\partial_{\alpha}u_{\beta}^{\downarrow}+\partial_{\beta}u_{\alpha}^{\downarrow}}{2}+\epsilon\delta_{\alpha\beta}\right)^{2}\right]\,\text{d}A$
$\displaystyle+\iint\frac{\kappa_{c}}{2}(u_{\alpha}^{\uparrow}-u_{\alpha}^{\downarrow})^{2}\,\text{d}A,$
(4)

where the superscripts ↑ and ↓ refer to the yellow and red sheets
respectively, and where the summation over repeated indices is implied while
$\delta$ denotes the Kronecker delta. The displacement fields
$\mathbf{u}^{\uparrow/\downarrow}(\mathbf{r})$ of either sheet are computed
with respect to the infinite-aggregate, bulk state where all hexagons are
regular, a state akin to a row of length-one rectangles in the one-dimensional
model. Neither elastic sheet is at rest in this reference state, and
$\mathbf{r}$ is the position vector in this state. The displacement gradient
$\partial_{\alpha}u^{\uparrow/\downarrow}_{\alpha}$ thus plays the same role
as the finite difference
$(x^{\uparrow/\downarrow}_{i+1}-x^{\uparrow/\downarrow}_{i}-1)$ of (1). The
first integral of (4) is a two-dimensional generalization of the first term of
(1), and gives the elastic energy of an isotropic elastic sheet with Lamé
coefficients $\lambda$ and $\mu$ whose resting state is characterized by an
isotropic strain
$\partial_{\alpha}u^{\uparrow}_{\beta}=\epsilon\delta_{\alpha\beta}$. As a
reminder, $\mu$ is the usual shear modulus and the sheet’s Young modulus and
Poisson ratio are $Y=4\mu(\lambda+\mu)/(\lambda+2\mu)$ and
$\nu=\lambda/(\lambda+2\mu)$. The integration element $\text{d}A$ runs over
the reference area $A$. Finally, the last integral captures the energy of the
connecting springs. It penalize any shift between the centers of the yellow
and red triangles of a given particle, and therefore any difference in the
displacements of the two sheets.

The harmonic form of (4) is valid for small deformations, which can be
obtained for any aggregate topology considered here given a small enough
$\epsilon$. Within this limit, the particles’ binding energy can be described
through

$E_{b}=\gamma L,$ (5)

where $\gamma$ is a line tension and $L$ is the total length of aggregate edge
in the reference state, including any internal holes. The parameters of the
continuum model are mapped onto those of the 2D particles in the _Supporting
Information_ , yielding

$\displaystyle\mu$ $\displaystyle=\frac{\sqrt{3}}{4}k$ (6a) $\displaystyle\nu$
$\displaystyle=\frac{\sqrt{3}k+2k_{\rm area}}{3\sqrt{3}k+2k_{\rm area}}$ (6b)
$\displaystyle\kappa_{c}$ $\displaystyle=2\sqrt{3}k_{c}$ (6c)
$\displaystyle\gamma$ $\displaystyle=\frac{4}{\sqrt{3}}g$ (6d)

The binding energy of an aggregate depends only on topology $\mathcal{T}$,
which we denote as $E_{b}(\mathcal{T})$. By contrast, the deformation energy
$E_{d}(\mathcal{T},\\{\mathbf{u^{\uparrow/\downarrow}}(\mathbf{r})\\})$
depends both on the topology and on the displacement fields
$\mathbf{u}^{\uparrow/\downarrow}(\mathbf{r})$. As described in the beginning
of this section, the optimal energy associated with a given aggregate topology
is thus

$E(\mathcal{T})=E_{b}(\mathcal{T})+\underset{\\{\mathbf{u}^{\uparrow/\downarrow}\\}}{\min}\,E_{d}(\mathcal{T},\\{\mathbf{u}^{\uparrow/\downarrow}\\}).$
(7)

Once this minimization is performed, finding the most favorable aggregate
structure requires finding the topology $\mathcal{T}$ that minimizes
$E(\mathcal{T})$.

## Aggregation phase diagram

To establish an aggregation phase diagram, we consider a system with a fixed
but large number of particles and ask which binding topology minimizes the
total energy of the system. We first use our continuum formalism to compute
the energies of an infinite bulk, an elongated fiber and a disk-like
aggregate, thus offering a first comparison of the stability of two-, one- and
zero-dimensional structures. We then numerically compare the energies of a
wider range of putative aggregate structures in our discrete particle model.
Finally, we confirm the converging results of these two approaches using
numerical simulations devoid of _a priori_ constraints on the aggregate
topology.

### Continuum phase diagram

In 2D space-filling, infinite aggregate, all particles are forced into a
regular hexagonal shape. The continuum energy of (4) is then minimal for
$\mathbf{u}^{\uparrow}=\mathbf{u}^{\downarrow}=\mathbf{0}$, yielding an
optimal energy per unit reference surface

$e_{\text{bulk}}=\mu\frac{1+\nu}{1-\nu}4\epsilon^{2}.$ (8)

Denoting the reference area per particle by $a$, the total energy of a set of
$N$ particles thus reads $Nae_{\text{bulk}}$ in the large-$N$ limit. The
binding energy is proportional to the perimeter of the aggregate
($E_{b}\propto\sqrt{N}$), and is thus negligible.

Figure 3:  Large binding energies favor bulk aggregation and incompressible
particles tend to form fibrous aggregates. (a) Analytical phase diagram
derived from Eqs. (8), (11) and (Continuum phase diagram). Fiber widths always
diverge when approaching the transition to the bulk. Conversely, in disks the
radius discontinuously jumps from a finite value to $+\infty$ at the
transition. The point where the three phases meet is $\Gamma=1$, $\nu=1/2$.
(b) Phase diagram based on the numerical comparison of the energies of the
aggregates shown in Fig. 4(a) for $\ell=5$. The fiber region of the phase
diagram is larger than in the continuum model, and the disk radius $R$ again
jumps discontinuously at the transition with the bulk. Smaller values of
$\ell$ lead to a very similar phase diagram, albeit with an extended “bulk
with holes” region (_Supporting Information_). Monte-Carlo simulations for the
conditions indicated by the small squares are shown as small panels, and are
consistent with the phase diagram. The bottom line of snapshots uses $300$
particles and a box of $30\times 30$ sites, while the others use $200$
particles and a box of $60\times 60$ sites.

To determine whether fiber formation is favored over bulk aggregation, we
minimize (4) over $\\{\mathbf{u}^{\uparrow/\downarrow}(\mathbf{r})\\}$ for an
infinite strip of width $W$ and find (_Supporting Information_)

$\mathbf{u}^{\uparrow}(x)=-\mathbf{u}^{\downarrow}(x)=\ell\epsilon\left[(1+\nu)\frac{\sinh(x/\ell)}{\cosh(W/2\ell)}\right]\,\hat{\mathbf{x}},$
(9)

where $x$ is the direction perpendicular to the fiber and where

$\ell=\sqrt{\frac{\lambda+2\mu}{2\kappa_{c}}}.$ (10)

The quantity $\lambda+2\mu$ is known as the P-wave modulus of the sheet, and
charcterizes the cost of compressing it along one axis without allowing it to
deform in the perpendicular direction. Similar to (1) and Fig. 1(b), the
profile of (9) implies bulk-like, highly deformed particles in the center of
the fiber, while close to the edges the red and yellow sheets gradually relax
within a boundary layer of width $\ell$. Defining the dimensionless line
tension $\Gamma=2\gamma/[(1+\nu)\ell e_{\text{bulk}}]$, line tension cost
associated with the fibers’ edges reads $E_{b}=2\Gamma Na/W$ and the mean
energy per unit surface reads

$\frac{e_{\text{fiber}}(W)}{e_{\text{bulk}}}=1-(1+\nu)\frac{\tanh(W/2\ell)}{W/\ell}+(1+\nu)\frac{\Gamma}{W/\ell}.$
(11)

In the $\epsilon\rightarrow 0$, small-particle-mismatch limit, all deformation
energies scale as $\epsilon^{2}$. Thus the parameter $\Gamma$ encloses both
the $\gamma$ and the $\epsilon$ dependence of all self-assembly outcomes
studied in this paper. When $\Gamma<1$, $e_{\text{fiber}}(W)$ displays a
minimum at a finite fiber width $W^{*}$. This optimal width diverges in the
limit $\Gamma\rightarrow 1$, and the corresponding fiber is always more stable
than the bulk [Fig. 3(a)]. To understand this stability, consider a semi-
infinite aggregate that fills half of the plane. While its energy per unit
surface far from its edge is equal to $e_{\text{bulk}}$, the presence of the
edge brings about two energetic contributions. The first is a bare line
tension cost $\gamma$ per unit edge length. The second is the deformation
energy gain in the boundary layer, which is of the order of $e_{\text{bulk}}$
per unit area. Since the width of the layer is $\ell$, the resulting energy
gain per unit edge length is of the order of $\ell e_{\text{bulk}}$. For
$\gamma\lesssim\ell e_{\text{bulk}}$, forming a new edge thus results in a net
energy gain. At the scaling analysis level, this is equivalent to $\Gamma<1$.
This argument implies that infinite bulks can lower their energy by breaking
up into fibers in this regime. However, if these fibers are made so narrow
that their widths become of order $\ell$ or smaller, the boundary layers
associated with their two edges start to overlap. Such narrow fibers can only
claim a fraction of the deformation energy reduction described above. As a
result, very narrow fibers are penalized. This implies the existence of an
optimal width $W^{*}$ that is of order $\ell$ when $\Gamma$ is of order one
but smaller than one.

In an aggregate whose resting shape is a disk of radius $R$, the displacement
field is given by (_Supporting Information_)

${\mathbf{u}^{\uparrow}(r)}=\ell\epsilon\frac{(1+\nu)I_{1}(r/\ell)}{I_{0}(R/\ell)+I_{2}(R/\ell)+\nu[I_{0}(R/\ell)-I_{2}(R/\ell)]}\,\hat{\mathbf{r}}$
(12)

and ${\mathbf{u}^{\downarrow}(r)}=-{\mathbf{u}^{\uparrow}(r)}$. Here $r$
denotes the radial coordinate and the $I_{\alpha}$s are the modified Bessel
functions of the first kind. Just like Eqs. (2) and (9), (12) predicts an
exponential decay of the boundary displacement at the edge of large
aggregates. Again, the deformation cost is smaller at the aggregate edge than
in its center, yielding an average energy per unit area

$\displaystyle\frac{e_{\text{disk}}(R)}{e_{\text{bulk}}}=1$
$\displaystyle-\frac{I_{0}(R/\ell)-I_{2}(R/\ell)}{I_{0}(R/\ell)+I_{2}(R/\ell)+\nu[I_{0}(R/\ell)-I_{2}(R/\ell)]}$
$\displaystyle+(1+\nu)\frac{\Gamma}{R/\ell}.$ (13)

This expression displays an optimal finite aggregate size $R^{*}$ at low
values of $\Gamma$. As in the fiber case, this optimal disk is more stable
than the bulk up to values of $\Gamma$ of order one, although the exact
criterion differs due to the curved geometry of the interface [Fig. 3(a)].

Our phase diagram indicates that fibers are more stable than disks for large
Poisson ratios, _i.e._ , they are favored in the aggregation of incompressible
particles (characterized by $\nu=1$ in 2D). To understand this, we compare a
vertical fiber and a disk at $\nu=1$. Symmetry forbids vertical (orthoradial)
displacements in the fiber (disk), allowing only horizontal (radial)
displacements. However, no such displacement is possible without violating the
incompressibility condition of the yellow or red sheet. As a result, the
sheets must remain in their resting states, implying displacements
$u_{x}^{\uparrow}=-u_{x}^{\downarrow}=2\epsilon x$
($u_{r}^{\uparrow}=-u_{r}^{\downarrow}=2\epsilon r$) to lowest order in
$\epsilon$ with respect to the fictitious bulk reference state. All the
deformation cost thus comes from the connecting springs. The resulting
connecting spring energy per unit area reads $8\kappa\epsilon^{2}x^{2}$
($8\kappa\epsilon^{2}r^{2}$), proportional to the square of the distance from
the center of the aggregate. This is where the difference between fibers and
disks manifests itself. The edge of a fiber is just as long as its centerline,
while the center of a disk is much smaller than its perimeter. As a result, a
smaller proportion of connecting springs are highly extended in the fiber than
in the disk, making the former energetically cheaper. Conversely, in the limit
of low Poisson ratio $\nu\rightarrow 0$, the gain per unit length of the
straight, fiber-like and of the curved, disk-like boundary layers become
identical. Forming such boundary layers is favorable overall for $\Gamma<1$.
Since the disk tends to have more boundary layer per unit area than the fiber,
it is more favorable in this limit.

### Comparison of pre-made discrete aggregate structures

The phase diagram of Fig. 3(a) focuses on continuum sheets, leaving open the
question of whether the formation of holes within the aggregate or regimes
where the particle size is comparable to the boundary layer thickness could
result in different aggregation behaviors. To assess its robustness to these
effects, we numerically implement our discrete particle model in a computer.
We consider several aggregates with predetermined topologies including
periodic bulks with and without holes, fibers of various widths as well as
hexagonal aggregates approximating disks of different radii [Fig. 4(a)].
Isolated particles are taken into account as disks with one particle. We use a
conjugate gradient algorithm to minimize the energy of each aggregate over the
position of the all particle corners, analogous to the minimization over the
deformation field in (7). We build three different fiber structures by cutting
the bulk along distinct directions. One of the fibers is left-right asymmetric
and spontaneously curves, although that curvature vanishes in the large-$W$
limit. In practice we obtain the energy of infinitely long fibers by
extrapolating from long ones with increasing lengths.

Figure 4:  Comparison of the energies of a collection of aggregates with pre-
determined topologies. (a) List of the aggregate topologies included in the
trial: bulk with holes, three types of fibers obtained by piling the particles
in three different ways, hexagonal aggregates (including single particles) and
bulk. The white lines and shading in the bulk show the three different types
of cuts used to produce the fibers shown on the left. In practice, the
isolated particles and uncurved fibers are never the most stable structures.
We compare a broad range of fiber widths and hexagonal aggregates, despite
representing only a few here. (b)  _left:_ When measured in units of number of
particles, the optimal fiber width $W^{*}$ (computed here for $\nu=0.95$) and
disk radius $R^{*}$ ($\nu=0.2$), strongly depend on the particles’ elastic
properties through $\ell$. _Right:_ Rescaling both lengths by $\ell$ however
leads to an excellent collapse with the analytical prediction (dashed red
line), even for small $\ell$.

To compute the aggregation phase diagram, we minimize the deformation energy
of aggregates with width (radii) ranging from 1 to 35 (25) for $\ell=5$ and
for values of $\nu$ ranging from 0 to 1. For each value of $\Gamma$ in Fig.
3(b) we select the aggregate with the lowest total energy. Bulks with holes,
which in our model have zero deformation energy, dominate the assembly only at
very small surface tensions. The rest of the phase diagram is essentially
identical to the continuum one, except for an expansion of the fiber region
against both bulks and disks. This may be due to the increased stability of
curved fibers, which are always more stable than their straight counterparts.

To assess the consistency of the morphology of finite-$\ell$ aggregates with
the continuum ($\ell\rightarrow\infty$) expectation of Fig. 3(a), we
numerically determine the most favorable fiber width and disk radius for a
range of $\ell$ and $\Gamma$ in Fig. 4(b). While the two approaches are
guaranteed to agree only in the large-$\ell$ limit, in practice the continuum
approximation yields very accurate predictions all the way down to values of
$\ell$ equal to the particle size. This indicates that the boundary layer
physics revealed by our continuum model remains an excellent qualitative and
quantitative description of the aggregation process even when the stiffness of
the connecting springs $k_{c}$ is comparable to that of the others
($k_{c}\lesssim k\Leftrightarrow\ell\gtrsim 1$). The length $\ell$ thus
provides a robust tool to predict the typical number of particles in the
cross-section of a fiber or a disk.

### Monte-Carlo validation of the phase diagram

As a final validation of our phase diagram, we remove any restriction on the
aggregate’s structure and use a Monte-Carlo algorithm to evolve its topology.
We simulate a triangular lattice where each site can be empty or occupied by a
particle. We start with randomly placed particles, and attempt Monte-Carlo
moves where a randomly chosen particle is moved to a randomly chosen empty
site. We compute the optimal energy of the resulting new topology using our
conjugate gradient method. The move is accepted according to a Metropolis
criterion with temperature $T$. Since the deformation energy is optimized
before the application of the Metropolis criterion, this temperature applies
only to the system’s topological degrees of freedom. To look for an
approximation of the system’s topological ground state, we perform a simulated
annealing procedure whereby $T$ is slowly lowered from a large value to zero
over the course of the simulation.

The computational cost and limited particle number in our simulation make it
difficult to construct a full Monte-Carlo phase diagram. Instead, we simulate
select parameter regimes to validate our main findings. Specifically, we
perform three line scans at $\ell=5$ whose final aggregation states are shown
in the small panels of Fig. 3(b). First, a range of increasing $\Gamma$ at
$\nu=0.9$ shows the dominance of fibers at large Poisson ratios, and the
overall tendency of the fiber widths to increase with increasing line tension.
A second horizontal scan at $\nu=0.4$ shows disk-like aggregates whose radii
increase with increasing $\Gamma$. Finally, a vertical scan at fixed
$\Gamma=0.5$ shows a transition between disks and elongated aggregates at the
predicted value of $\nu$. The shapes of the aggregates resulting from the
simulations are somewhat variable, as can be assessed from the $\Gamma=0.5$,
$\nu=0.9$ condition which belongs to two different line scans and for which
the outcomes of two independent simulations is shown in Fig. 3(b). Our results
show that despite this limitation and the relatively small number of particles
in our system, our phase diagram accurately predicts the overall outcome of an
unconstrained assembly.

## Extension to random particles

The discrete model of Fig. 2 has been specifically designed with the continuum
approximation in mind, making it unclear whether our results apply to more
generic particle types. To test whether this is the case, we define a much
broader class of hexagonal particles with the same shape at rest and the same
binding rules but with a considerably more generic deformation energy. We
parametrize the shape of each particle by nine distances $d_{1}$,
$d_{2},\ldots,d_{9}$ defined in Fig. 5(a). The vector $\mathbf{d}$ of these
lengths completely characterizes a particle’s shape, and we denote its value
at rest by $\mathbf{d}_{0}$. The deformation energy of one particle is an
arbitrary quadratic form of the deviation from that state:

$e_{d}=\frac{1}{2}(\mathbf{d}-\mathbf{d}_{0})^{T}\cdot\mathbf{M}\cdot(\mathbf{d}-\mathbf{d}_{0}).$
(14)

In the following we draw the matrix $\mathbf{M}$ from a random distribution
that ensures that it is positive semi-definite and three-fold symmetric
(_Supporting Information_). Such particles allow not only for differences in
the elastic constants of the colored springs of Fig. 2(a), but also for new
couplings between them. For instance, one of these new couplings dictates that
compressing a yellow spring makes the red spring to its right shrink or
extend. It also opens regimes where the coupling springs are not much softer
than the others.

Figure 5: The aggregation behavior of particles with random elasticity is
accurately captured by our continuum theory. (a) The shape of a particle is
fully characterized by a tuple of nine distances $(d_{1},\ldots,d_{9})$. (b)
To numerically compute the boundary layer thickness $\ell$, we simulate a
half-plane full of particles (colored here by absolute deformation energy). We
then perform an exponential fit of the excess elastic energy relative to the
bulk as a function of distance from the edge, shown here as a line on a lin-
log plot. Finally we define $\ell$ as the associated decay length. (c) The
random particle boundary layer thickness $\ell$ measured using this protocol
is well predicted by the value $\ell^{\text{pred}}$ inferred from an
extrapolation of (10) (_red line_). (d) Radius and width of the best disk and
fibers obtained for random particles. The continuum predictions are shown as
red lines (the position of the line does not depend on $\nu$ for fibers).

We first determine whether the length $\ell$ in a random particle is
determined by the same ratio of uniaxial-compression to triangle-shifting
moduli as in (10). To access a wide range of $\ell$ values, we randomly draw a
large number ($3\times 10^{7}$) of instances of matrix $\mathbf{M}$. For each
instance, we operationally define $\ell$ as the thickness of the elastic
boundary layer in a simulation of a semi-infinite aggregate [Fig. 5(b)]. We
then relate this measured value to the properties of a single particle, by
numerically measuring random-particle proxies of the two moduli $\lambda+2\mu$
and $\kappa_{c}$ and using (10) to compute a predicted boundary layer
thickness $\ell^{\text{pred}}$ (_Supporting Information_). As shown in Fig.
5(c), we find a good correlation between $\ell$ and $\ell^{\text{pred}}$ for a
randomly selected subsample of $10^{5}$ matrices. This indicates that
frustration at the edge of an aggregate relaxes in similar ways in simple and
random particles, with the two subtriangles of the particles progressively
shifting relative to each other. Similar to the one-dimensional model of Fig.
1, this shift causes a restoring force (denoted $k_{c}\delta_{i}$ in the 1D
model) which is balanced by stiffness of the particles’ other degrees of
freedom (associated with the stiffness $k$ in the 1D model). Equation (10),
which we use to compute $\ell^{\text{pred}}$, uses the P-wave modulus
$\lambda+2\mu$ as a proxy for this stiffness. While this value is as an upper
bound for the cost of deforming the particles, in practice our random
particles may deform in more complex, less costly ways. Such deformations
would result in a lower effective stiffness, and could explain why
$\ell^{\text{pred}}$ tends to overestimate the actual value of $\ell$.

To determine whether the boundary layer thickness $\ell$ controls the
aggregate size in the same way as in our simple model, we study a randomly
selected subsample of $10^{3}$ matrices $\mathbf{M}$ from our total sample. We
use the same discrete aggregate procedures as in Fig. 4(b) to determine which
fiber/disk width is the most favorable for $\Gamma$ ranging from 0 to
$\Gamma_{\text{max}}$, where we define $\Gamma_{\text{max}}$ as the critical
line tension where the bulk becomes the most favorable structure. Both fibers
and disks tend to become larger with increasing $\Gamma$ despite a large
dispersion of the data similar to that observed in Fig. 4(b) (_Supporting
Information_). We however show in Fig. 5(d) that just as in the case of simple
particles, this dispersion is largely abolished by rescaling the aggregate
size by the boundary layer size $\ell$. This demonstrates that the boundary
layer still controls the physics of the assembly in the random particle case.
Moreover, the resulting radii and width distribution are clearly centered
around the values of $W^{*}$ and $R^{*}$ predicted by the continuum limit.

We finally assess the applicability of the phase diagrams of Fig. 3 to random
particles by correlating the most favorable aggregate type with a suitable
measure of the “individual sheet Poisson ratio” of the analytical theory. We
define this measure by decoupling the “yellow” and “red” subtriangles of our
new particles from each other (_i.e._ , set $M_{ij}=0$ for all
$(i,j)\notin[1,3]^{2}\cup[4,6]^{2}$), composing a lattice out of each and
numerically computing their separate Poisson ratios. We use the average of
these two values as our $\nu$. We then segregate our $3\times 10^{7}$
instances of the matrix $M$ into three groups with $\ell<1$, $1<\ell<1.5$ and
$1.5<\ell$. We randomly select a few thousand particles within each group to
obtain three near-uniform distribution of Poisson ratios. Fig. 6 shows the
type and width of the aggregates obtained in each case. Fibers form at large
$\nu$ and $\Gamma$ in all three groups, thus demonstrating the robust
influence of these parameters. Disks also form in the expected parameter
regimes, although they tend to be replaced by bulks with holes for the
smallest values of $\ell$. Finally, fibers are even more predominant here than
in our initial model, illustrating the broad relevance of our fiber formation
mechanism upon ill-fitting self-assembly.

Figure 6: Random particle aggregation diagram showing good agreement with the
analytical results of Fig. 3. Color coding is as in Fig. 3(b). Each horizontal
line in the diagrams corresponds to an instance of the elasticity matrix
$\mathbf{M}$. Although the boundaries between the different regions of the
diagram fluctuate due to the random origin of $\mathbf{M}$, together they
outline very consistent regions where bulk with holes, disks, fibers and bulks
dominate. The diagrams with larger values of $\ell$ cover a more restricted
range of $\nu$ due to the relative scarcity of particles with both large
$\ell$ and small $\nu$ in matrices $\mathbf{M}$ produced by our random
generation procedure. See _Supporting Information_ for details.

## Discussion

Protein aggregation in disease involves the self-assembly of deformable, ill-
fitting sticky particles. Compact aggregates of such particles display a
deformation gradient between a relatively unconstrained edge and a strongly
deformed, frustrated core. This deformed core is energetically costly, while
the aggregate’s surface tension implies that its edge is also costly. As a
result, the cheapest part of the aggregate lies in between the core and the
edge, in the shallow bulk region. The minimization of the aggregate’s energy
thus requires its structure to comprise as much of this shallow bulk as
possible while keeping both core and surface small. Here we show that for a
rather general model of deformable particles, this results in aggregate size
limitation and emergent anisotropy. This geometrically nontrivial optimization
is _a priori_ strongly dependent on the elastic properties of the particles.
We nevertheless identify two surprisingly simple particle-level predictors of
its outcome, namely an elastic screening length $\ell$ and particle
incompressibility.

In our model, the shallow bulk manifests as an elastic boundary layer with
size $\ell$. This is reminiscent of the size limitation mechanisms at work in
ribbons self-assembled out of planar materials with an intrinsic negative
Gaussian curvature [33] or specially designed warped jigsaw puzzle particles
[24]. Particles with a frustrated continuum spin-like degree of freedom and
coupled incommensurate lattices in the absence of phase slips also give rise
to a boundary layer [34, 35, 36]. In contrast with these specific particle
geometries however, here this behavior emerges in a wide variety of particles
with randomly chosen elastic properties. This suggests that the mechanism may
also apply in packings of complex ill-fitting proteins. Our observation of a
connection between soft deformation modes at the particle level, large values
of $\ell$ and consequently large self-limited aggregates thus suggests an
analogy with allosteric proteins, which mechanically transmit a signal by
undergoing a concerted conformational change along a soft deformation mode
[37, 38, 39].

While extended boundary layers appear to require some particle deformation
modes to be much softer than others [Fig. 5(c)], not all particle-level soft
modes result in one (Fig. S5). Indeed, once embedded in an aggregate, an
initially soft deformation mode may couple to and be stiffened by the presence
of neighboring particles. While our six-vertices, two-dimensional particles
comprise only relatively simple soft modes compatible with collective
deformations [namely the triangle-shifts illustrated in Fig. S4(a-b)], more
complex objects are likely to allow many more. For instance, generalizations
of our model in three dimensions, where the qualitative physics highlighted
here still applies, could additionally involve frustration and soft modes
associated with chiral particle twisting. Further investigations are required
to identify the geometrical requirements for a soft mode to be compatible with
collective deformations, which in turn allows the build-up of a thick boundary
layer. These requirements will shed light on the specific elastic parameters
that most influence the particles’ aggregation behavior, which could include
the formation of clusters, fibers or sheets in three dimensions. In the
language of proteins, this would allow us to assess the likelihood for a given
allosteric mode to dictate the size of a frustrated protein aggregate. This
would constitute a very general tool for predicting the structure of an
aggregate from the individual properties of its proteins.

The three-fold symmetry of our model particles implies that their propensity
to form fibers is an emergent property as opposed to an intrinsic preference
for uniaxial aggregation. This breaking of symmetry is reminiscent of the
strain-induced, elongated structures formed during frustrated epitaxial growth
[40, 41, 42]. Both our most simple model and our generic, random-elasticity
particles indicate that fiber formation is most advantageous in incompressible
particles. This behavior is not specific to hexagonal particles, and we show
in the _Supporting Information_ that an alternative model based on triangular
objects displays an essentially identical phase diagram. These results
suggests that our mechanism is more relevant for dense globular proteins,
which tend to be largely incompressible, than for proteins with fluffy
intrinsically disordered domains. Globular proteins are indeed known to form
fibers in, _e.g._ , sickle cell anemia and amyotrophic lateral sclerosis [8,
11, 10]. The binding free energies involved in these assemblies are typically
at least one order of magnitude larger than the thermal energy $k_{B}T$,
justifying the analogy with our zero-temperature model. The particle
deformations there moreover remain modest, consistent with the analytically
tractable, small-frustration ($\epsilon\ll 1$) regime studied here. In more
asymmetric particles than those studied here, the mechanism outlined in our
work could be complemented by fiber-formation mechanisms based, _e.g._ , on
the presence of two specific binding sites on either side of the particle.
Kinetic effects such as diffusion-limited aggregation, which hampers the
formation of bulky aggregates, may also favor fibers.

Beyond proteins, the principles outlined here could be harnessed to control
the assembly of artificial nano-objects. In DNA origami, soft deformation
modes can be engineered to control aggregate size in a simple one-dimensional
chain [25]. Two- or three-dimensional extensions of such designs should be
prone to frustration-induced fibrillation. Fibrous morphologies also emerge in
DNA origami systems into which this feature is not intentionally designed
[43]. While systems involving rigid nanoparticles are less straightforwardly
mapped onto an elastic continuum than DNA origami, fibrous morphologies have
been observed in packings of tetrahedral particles [44] and successfully
rationalized with an elastic model [45] based on a elastic frustration
originating from a metric incompatibility. Our study does not rely on this
very strong, somewhat specialized type of frustration and thus demonstrates
that it is not required for frustration-induced fiber formation. Rigid
colloids with short-range attractive and long-range repulsive interactions
also display frustration-induced fibrous structures [46]. In this case, the
importance of the distance dependence of the particle interaction profile
falls outside of the scope of our small-frustration ($\epsilon\ll 1$)
formalism. More generally, frustration build-up in the presence of nonlinear
elasticity and strain-induced particle unbinding can lead to the emergence of
new aggregate patterns that have only begun to be explored [47] and could play
a crucial role in the physical implementation of the principles described
here.

###### Acknowledgements.

ML was supported by Marie Curie Integration Grant PCIG12-GA-2012-334053,
“Investissements d’Avenir” LabEx PALM (ANR-10-LABX-0039-PALM), ANR grants
ANR-15-CE13-0004-03, ANR-21-CE11-0004-02, ANR-22-ERCC-0004-01 and
ANR-22-CE30-0024-01, as well as ERC Starting Grant 677532. ML’s group belongs
to the CNRS consortium AQV.

M. Mert Terzi and Hugo Le Roy contributed equally to this work. MMT performed
the analytical calculations and HLR conducted the numerical simulations. All
authors participated in designing the project and writing the paper.

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