stable configuration, the effective rate of the initial attachment reaction is
found to be

$r_{\sigma\to\sigma+4}=24\,\bar{\nu}^{\prime}c_{\sigma}m^{4}$ (45)

with

$\displaystyle\bar{\nu}^{\prime}=\nu\left(\frac{\nu}{\delta_{1}}\right)^{3}\,.$
(46)

Consequently, the corresponding effective attachment rate per monomer for
${\sigma\leq s\leq\sigma+3}$ is given by

$g_{s}=\tilde{r}_{s\to s+1}=96\,\bar{\nu}^{\prime}c_{s}m^{4}\,.$ (47)

We modified the effective theory Eq. (4) for the system with triangle-shaped
monomers accordingly to account for the higher order of the initial attachment
reaction.

#### Numerical solution of the kinetic rate equations

To numerically solve the resulting kinetic rate equations given by Eq. (4), we
used Matlab’s ode15s, a semi-implicit solver designed for solving stiff
problems. Typically, semi-implicit ODE solvers allow one to provide the
Jacobian matrix or a sparsity pattern of the Jacobian for the specific
problem. The sparsity pattern indicates only the nonzero elements of the
Jacobian without specifying them. For the given problem, Eq. (4), the sparsity
pattern is easily found by observing that the temporal evolution of $c_{s}$
depends only on itself, on $c_{s-1}$ and the monomer concentration
${m=c_{1}}$. Hence, the sparsity pattern is given by a ${S\times S}$ matrix
with ones only in its diagonal, its subdiagonal, as well as its first column
and first row, while all other entries are zero. By providing the sparsity
pattern for the Jacobian, the performance of the solver could be accelerated
significantly, especially for large structure sizes $S$. In this way, Eq. (4)
can be solved quickly despite the very small absolute and relative error
tolerance of $10^{-12}$ that we used. The speed could be further increased by
providing an analytic Jacobian, which was unnecessary in this case.

#### Growth rate for tube-like structures

In Sec. V.6 we investigate the growth of two-dimensional tubes (cylinders)
with a constant width (circumference) of 6 monomers and a variable height of
$S/6$ monomers. Hence, in contrast to the case of symmetric target structures,
the tubes have one dimension (the width) that is significantly shorter than
the other one (the height). Thus, as soon as the structures close in the short
dimension, they continue growing with a constant growth rate since the
structure’s perimeter remains constant. To adopt the effective model
accordingly, we determine the tubes’ constant growth rate $g_{s}$. To this
end, we note that, in the case of square and hexagonal building blocks, the
number of possible binding sites $p_{s}$ for the first monomer to form a new
seed is $p_{s}=2\cdot 6$ since the monomer can attach either to the upper or
the lower edge of the tube. In contrast, in the case of triangular monomers,
$p_{s}=2\cdot 3$, since a row of width $l_{s}=6$ monomers only has 3 binding
sites. The rate of seed formation is then given by Eq. (40) with the
multiplicity factors $q_{s}$ shown in Table 2. Due to the domino effect
following seed formation, $l_{s}=6$ monomers in total are added as the result
of a single seed-forming event in the case of square or triangular monomers
($l_{s}=1$ for hexagonal monomers due to $\gamma=1$). Hence, with Eq. (42), we
obtain the constant effective growth rate $g_{s}=a\bar{\nu}m^{\gamma}$, with
$a=144/108/12$ for square, triangular and hexagonal monomers, respectively.
For simplicity, we assume that up to size $s=30$, clusters grow with the same
(size dependent) growth rate $g_{s}$ as symmetric structures and for $s>30$
with the constant growth rate derived above. The effective model modified this
way accurately matches the simulation data for tube-like structures, as shown
in Fig. 7.

#### Interaction of oligomers

Equation (4) describes the basic reaction kinetics of reversible self-
assembly. The basic theory can easily be extended by accounting for additional
effects. For example, in App. E, we investigate how reactions between larger
oligomers might affect the assembly time [see Fig. 15(b)]. To estimate this
effect with the help of the effective theory, we additionally accounted for
the reactions of any two clusters of size ${i\geq\sigma}$ and ${j\geq\sigma}$
with ${i+j\leq S}$ to a cluster of size ${i+j}$ at rate $\nu$:

$r_{(i,j)\to i+j}=\nu\,c_{i}\,c_{j}\,.$ (48)

To this end, we augmented the right-hand side of Eq. (4) by the following
terms:

$\partial_{t}c_{s}=...\,+\frac{1}{2}\,\nu\sum\limits_{\begin{subarray}{c}i,j\geq\sigma\\\
i+j=s\end{subarray}}c_{i}c_{j}-\nu
c_{s}\sum\limits_{\begin{subarray}{c}i\geq\sigma\\\ i+s\leq
S\end{subarray}}c_{i}\,,$ (49)

for all ${\sigma\leq s\leq S}$, where the factor $1/2$ in front of the first
sum avoids double counting and serves as a stoichiometric factor in the case
${i=j}$.

#### Hierarchical self-assembly system

Furthermore, we used the effective theory to simulate the hierarchical self-
assembly scenario in Sec. VI [cf. Fig. 9]. To this end, we have formally
distinguished between weak and strong bonds of the monomers. This causes the
monomers to first form either dimeric rhombuses or hexameric hexagons, as
shown in Fig.9.

In the first case, we simulated the irreversible dimerization of two
(triangle-shaped) monomers via strong bonds into rhomboids at rate
${\mu=\nu}$. In the second case, we considered the formation of hexagons by
explicitly simulating the nucleation process of triangle-shaped monomers
depicted in Appendix Fig. 10 with detachment rate
${\delta_{\text{str}}=10^{-4}\cdot\delta_{1}}$. The rhomboids and hexagons,
once formed, are stable and react in the usual way as described by Eq. (4)
mediated by the weak bonds. To simulate the assembly of these higher-order
particles, we used Eq. (4) together with the morphological parameters for
square- or hexagon-shaped particles, respectively, and the corresponding
detachment rate for the weak bonds.

Interactions between (triangle-shaped) monomers and incomplete hexagons with
larger oligomers are entirely neglected, which is justified by the time
interval between the reactions mediated by strong and weak bonds. In the
hexagonal case, the rate of detachment $\delta_{\text{str}}$ of the strong
bonds must be neither too large nor too small relative to $\delta_{1}$: In
both cases, the formation of the hexagons would be slow, leading to an
increase in the total build-up time. We found that a ratio of
${r_{h}:=\delta_{\text{str}}/\delta_{1}\approx 10^{-4}}$ between the
detachment rates of the strong and weak bonds minimizes about $T_{90}$.

##  Appendix C Scaling theory for yield onset

Here we analyze the solutions of the rate equations in the strong binding
limit, Eq. (4), in a parameter regime where the assembly process has a low
yield. The solutions are shown to exhibit scale invariance. Using this scale
invariance, we can derive a scaling law for the parameter condition for yield
onset, i.e., the condition to obtain a nonzero yield, in dependence on the
target structure size. The same scaling law also describes how the optimal
parameter value scales as a function of the target structure size, which we
will show rigorously in section D.

#### Effective dynamics of monomers and clusters

Using the asymptotic form $f_{s}=as^{\omega}$ for the combinatorial factor,
one can rewrite the sum on the right-hand side of the rate equations, Eq.
(4a), governing the evolution of the concentration of monomers, as

$\sum_{s=\sigma}^{S}s^{\omega}c_{s}=K\sum_{s=\sigma}^{S}s^{\omega}\frac{c_{s}}{K}=K\,\langle
s^{\omega}\rangle\,.$ (50)

Here ${K:=\sum\nolimits_{s=\sigma}^{S}c_{s}}$ denotes the total concentration
of all complexes above the nucleation size and $\langle s^{\omega}\rangle$
defines the $\omega$-moment of the distribution of cluster sizes. For
${\omega=1}$, which holds for square- and triangle-shaped monomers, $\langle
s^{\omega}\rangle$ reduces to the average cluster size $\langle s\rangle$. As
long as the yield is zero, the average cluster size equals the total number of
bound monomers divided by the total number of clusters: ${\langle
s\rangle=(C-m)/K}$. For general $\omega$, there is no such exact identity, but
to make progress, we approximate the $\omega$– moment by ${\langle
s^{\omega}\rangle\approx\langle s\rangle^{\omega}}$. Since $\omega$ is usually
close to 1 (e.g., for the hexagon-shaped monomers ${\omega\approx 1/2}$), we
expect the approximation to be generally sufficiently accurate. Hence, as long
as the yield is zero, the dynamics of the monomer concentration are governed
by [cf. Eq. (4a)]

$\partial_{t}m(t)=-a\bar{\nu}\,m^{\gamma}\left(C-m\right)^{\omega}K^{1-\omega}-\sigma\,\bar{\mu}\,m^{\sigma}\,.$
(51)

Moreover, the total concentration of complexes $K$ increases by nucleation
events and thus follows the equation

$\partial_{t}K(t)=\bar{\mu}\,m^{\sigma}\,.$ (52)

Given initial conditions for both the concentrations of monomers $m$ and
complexes $K$, Eq. (51) and Eq. (52) form a closed system of differential
equations from which the number of particles that remain in the monomer pool
and the number of complexes can be calculated.

#### Continuum limit of cluster assembly

However, to determine the condition under which nonzero yield is achieved in
the self-assembly process, we need to know how the distribution of cluster
sizes evolves: Nonzero yield is achieved if the outermost front of the cluster
size distribution eventually reaches the target structure size $S$, i.e.,
${\lim_{t\to\infty}c_{S}(t)\neq 0}$. To determine a condition for the onset of
the yield, we rewrite Eq. (4c) in a continuum limit by considering
${c(s)=c_{s}}$ as a continuous function of the cluster size $s$. Since the
cluster size $s$ is typically large, we approximate ${(s-1)^{\omega}\approx
s^{\omega}}$ and then use a second order Taylor expansion ${c_{s-1}\approx
c(s)-\partial_{s}c(s)+\frac{1}{2}\partial_{s}^{2}c(s)}$ to arrive at

$\partial_{t}c(s,t)=-a\,\bar{\nu}s^{\omega}m^{\gamma}\partial_{s}c(s,t)+\tfrac{1}{2}a\,\bar{\nu}s^{\omega}m^{\gamma}\partial_{s}^{2}c(s,t)\,.$
(53)

This constitutes an advection-diffusion equation for the time-dependent
cluster size distribution $c(s,t)$ with coefficients depending on the cluster
size $s$ and the instantaneous monomer concentration $m(t)$. The drift
velocity reads ${v(s)=a\,\bar{\nu}s^{\omega}m^{\gamma}}$ and the diffusion
coefficient ${D(s)=\tfrac{1}{2}a\,\bar{\nu}s^{\omega}m^{\gamma}}$. A similar
approach of approximating the system of rate equations by a continuous
advection-diffusion equation has been used previously to describe virus capsid
assembly [68, 19, 69].

Equation (53) can be interpreted as a Fokker-Planck equation for $c(s,t)$. The
corresponding jump moments for small time increments $\Delta t$ are given by
(note that $D(s)=\frac{1}{2}v(s)$)

$\displaystyle\begin{split}\langle\Delta s\rangle&=v(s)\,\Delta t\,,\\\
\langle(\Delta s)^{2}\rangle&=v(s)\,\Delta t\,,\end{split}$ (54)

where the average is taken over the cluster size distribution. Hence the Fano
factor (variance-to-mean ratio) ${F=1}$, i.e., fluctuation effects are fairly
large. Here, as we are interested in the asymptotic scaling for large target
sizes $S$, we neglect the diffusive contribution against the advective
contribution. However, as it turns out, and is expected from the value of the
Fano factor, the diffusive contribution is one of the main reasons why the
yield and assembly time for small structure sizes deviate from their
asymptotic scaling. Considering only the advective contribution, i.e., the
average time evolution of the cluster distribution, and assuming that several
complexes nucleate at ${s=\sigma}$ at time ${t=0}$, the leading edge of the
density profile evolves according to

$\partial_{t}s(t)=v(s(t))=a\,\bar{\nu}\,s(t)^{\omega}\,m(t)^{\gamma}\,,$ (55)

with initial condition ${s(0)=\sigma}$. In this advective limit, the onset
condition for the existence of a finite yield reads
${\lim_{t\to\infty}s(t)=S}$, i.e., to obtain a finite yield, the leading edge
of the cluster size distribution must eventually reach the target structure
size $S$.

In summary, Eqs. (51, 52, 55) constitute a closed set of equations that allows
us to compute the onset of yield by the condition ${S=s(\infty)}$.

#### Analytic solution

We were able to find a closed analytic solution only for systems with square-
shaped monomers (${\sigma=4}$, ${\omega=1}$, and ${\gamma=2}$) using
elementary calculus. Specifically, the complex concentration $K$ drops out of
Eq. (51) if $\omega=1$, which can therefore be solved for the monomer
concentration $m$. Using $m$ in Eq. (55) then gives the solution

$\displaystyle
s(\infty)=\sigma\,\frac{1+\sqrt{1-4\,\tilde{\eta}}}{1-\sqrt{1-4\,\tilde{\eta}}}$
(56)

with ${\tilde{\eta}:=\sigma\,\bar{\mu}\,C/(a\bar{\nu})}$, which, for large $S$
(and ${\eta\ll 1}$), implies

$\displaystyle\frac{\delta_{1}^{\text{on}}}{C\nu}\sim\frac{\mu}{\nu}\,S\,.$
(57)

#### Scaling analysis

To determine the asymptotic scaling exponents for general cluster
morphologies, however, finding an explicit solution to the above equations is
not necessary. It is sufficient to demonstrate that in the limit of large
target, structure sizes $S$, the equations obey a scale invariance, which can
be exploited to determine the exponents without finding explicit solutions.

To reveal this scale invariance, we first note that in the limit of large
structure sizes, the nucleation term can be neglected in the dynamics of the
monomer concentration Eq. (51) since nucleation events are rare compared to
the attachment of monomers to existing clusters. The rate equations can then
be written in dimensionless form by measuring concentrations in units of $C$
and time in units of the effective time scale
$(a\,\bar{\nu}\,C^{\gamma})^{-1}$. Specifically, with the variable
transformations ${m\to m\,C}$, ${K\to K\,C}$, and ${t\to
t/(a\,\bar{\nu}\,C^{\gamma})}$ one obtains

$\displaystyle\partial_{t}m$
$\displaystyle=-{m}^{\gamma}(1-m)^{\omega}K^{1-\omega}\,,$ (58a)
$\displaystyle\partial_{t}K$ $\displaystyle=\eta\,{m}^{\sigma}\,,$ (58b)
$\displaystyle\partial_{t}s$ $\displaystyle=s^{\omega}m^{\gamma}\,,$ (58c)

where we have introduced the dimensionless parameter

$\displaystyle\eta:=\frac{\bar{\mu}}{a\,\bar{\nu}}\,C^{\sigma-\gamma-1}\,.$
(59)

Using a scaling ansatz

$\displaystyle m$ $\displaystyle=\tilde{m}(\eta^{z}t)\,,$ (60a) $\displaystyle
K$ $\displaystyle=\eta^{x}\tilde{K}(\eta^{z}t)\,$ (60b) $\displaystyle s$
$\displaystyle=\eta^{y}\tilde{s}(\eta^{z}t)\,,$ (60c)

with the dynamical exponent ${z=(1{-}\omega)/(2{-}\omega)}$, and the scaling
exponents ${x=(2{-}\omega)^{-1}}$ and ${y=-(2{-}\omega)^{-1}}$ for the cluster
density and the size $s$ of the largest clusters, one can easily verify that
the resulting set of equations for the scaling functions becomes independent
of the parameter $\eta$:

$\displaystyle\partial_{\tilde{t}}\,\tilde{m}(\tilde{t})$
$\displaystyle=-{\tilde{m}}^{\gamma}(1-\tilde{m})^{\omega}\tilde{K}^{1-\omega}\,,$
(61a) $\displaystyle\partial_{\tilde{t}}\,\tilde{K}(\tilde{t})$
$\displaystyle={\tilde{m}}^{\sigma}\,,$ (61b)
$\displaystyle\partial_{\tilde{t}}\,\tilde{s}(\tilde{t})$
$\displaystyle=\tilde{s}^{\omega}\tilde{m}^{\gamma}\,,$ (61c)

where ${\tilde{t}=\eta^{z}\,t}$. Therefore, the scaling functions $\tilde{m}$,
$\tilde{K}$ and $\tilde{s}$ still depend on the parameters $\sigma$, $\gamma$
and $\omega$ characterizing the morphology, but are independent of $\eta$. The
fact that the dynamics obey scale invariance can now be exploited to write the
onset condition for a finite yield as

$\displaystyle S=s(\infty)=\eta^{y}\,\tilde{s}(\infty)\sim\eta^{y}\,,$ (62)

since $\tilde{s}(\infty)$ is just a number that is independent of $\eta$. We
conclude that there must be a scaling relation

$\eta^{-1}\sim S^{2-\omega}\,.$ (63)

which, by using the definition of the dimensionless parameter $\eta$ [Eq.
(59)] and the definitions for $\bar{\mu}$ and $\bar{\nu}$ from Eqs. (2) and
(3), can be written as a scaling relation for the detachment rate

$\frac{\delta_{1}^{\text{opt}}}{C\nu}\sim\frac{\delta_{1}^{\text{on}}}{C\nu}\sim\left(\frac{\mu}{\nu}\right)^{\frac{1}{\sigma-\gamma-1}}S^{\phi}\,,$
(64)

with the control parameter exponent

$\displaystyle\phi=\frac{2-\omega}{\sigma-\gamma-1}\,.$ (65)

This relation is consistent with the result obtained by the closed analytical
solution above for the system with square-shaped monomers.

Since nucleation is the time-limiting step, the assembly time $T_{90}$ can be
estimated as the time required for $0.9\,C/S$ nucleation events to happen.
Denoting the effective nucleation rate resulting from
${\delta_{1}=\delta_{1}^{\text{opt}}}$ by $\bar{\mu}^{\text{opt}}$, one hence
obtains

$\displaystyle T^{\text{min}}_{90}$
$\displaystyle\sim\frac{C/S}{\bar{\mu}^{\text{opt}}\,C^{\sigma}}\sim(C\nu)^{-1}\left(\frac{\mu}{\nu}\right)^{\frac{\gamma-1}{\sigma-\gamma-1}}S^{\theta}\,,$
(66)

with the asymptotic time complexity exponent given by

$\displaystyle\theta=\frac{(1-\omega)\,\sigma+\gamma+2\,\omega-3}{\sigma-\gamma-1}\,.$
(67)

Alternatively, as described in the main text, the time complexity exponent can
also be read off directly from Eq. (61). Specifically, by reintroducing the
proper dimensions, the time scale of the dynamics described by Eq. (61) is
found to obey

$\tau_{\text{assem}}\sim\left(a\,\bar{\nu}\,C^{\gamma}\,\eta^{z}\right)^{-1}\sim\delta_{1}^{\alpha}\,,$
(68)

with ${\alpha=\gamma-1+z\,(\sigma-\gamma-1)}$. Using Eqs. (64) and (65), one
thus obtains

$\tau_{\text{assem}}\sim S^{\phi\alpha}:=S^{\theta}\,$ (69)

with $\theta$ given by Eq. (67).

##  Appendix D General scaling symmetry

A similar scaling symmetry that we found for Eq. (58) can be shown to hold
approximately even more generally, namely for the entire dynamic system
described by the rate equations, Eq. (4). Note that in Eq. (58), we assumed
that complexes can grow indefinitely, and therefore the derived scaling
symmetry holds only as long as the yield of completed structures is zero. In
contrast, a scaling symmetry for the entire system would have even more far-
reaching consequences. It would prove, among other things, our basic
assumption underlying the analysis in the previous section that the threshold
rate for nonzero yield scales identically as the optimal rate to achieve 90%
yield in the shortest time.

To see how a scaling symmetry also holds for the entire system, Eq. (4), we
again use a continuous (hydrodynamic) approximation and write the dynamic
system as a partial differential equation. To this end, we interpret the
density $c(s)$ as a continuous function of a real variable $s$ over the
interval $[\sigma,S]$. Expanding the first term in Eq. (4)c to first order in
$\partial_{s}$, one obtains the advection equation,

$\displaystyle\partial_{t}c(s,t)$
$\displaystyle=\big{(}s^{\omega}c(s)-\partial_{s}[s^{\omega}c(s)]-s^{\omega}c(s)\big{)}\,a\,\bar{\nu}\,m(t)^{\gamma}$
$\displaystyle=-a\,\bar{\nu}\,m(t)^{\gamma}\partial_{s}[s^{\omega}c(s)]\,,$
(70)

where ${m:=c_{1}}$ denotes the concentration of monomers. This can also be
written as a continuity equation ${\partial_{t}c(s,t)=-\partial_{s}J(s)}$ with
the flux (of cluster sizes) given by
${J(s)=a\,\bar{\nu}\,m^{\gamma}s^{\omega}c(s)}$. Note that this first-order
expansion, where we neglect the diffusion term and higher-order contributions,
becomes exact in the limit ${S\to\infty}$, where advection becomes the
dominant driving force for the transfer of cluster mass. The equation for the
concentration of nuclei, Eq. (4b), translates into a boundary condition at
${s=\sigma}$, where the the influx of complexes
${J_{\text{in}}=\bar{\mu}m^{\sigma}}$ must match the outflux towards larger
complexes ${J(\sigma)=a\,\bar{\nu}\,m^{\gamma}\sigma^{\omega}c(\sigma)}$. At
${s=S}$, we set an absorbing boundary condition, which removes completed
structures from the system. Finally, in the equation for the monomers, Eq.
(4a), the sum is replaced by an integral,
${\sum_{\sigma}^{S-1}s^{\omega}c(s)\to\int_{\sigma}^{S}s^{\omega}c(s)\,ds}$
and we neglect the first term describing the loss of monomers by nucleation
versus the second term describing the loss by attachment of monomers to
clusters. The latter approximation is accurate whenever there is significant
growth of the structures and $S$ is sufficiently large so that, in total,
there are many more attachment events than nucleation events.

Taken together, this gives the following set of hydrodynamic equations

$\displaystyle\partial_{t}m(t)$
$\displaystyle=-a\,\bar{\nu}\,m(t)^{\gamma}\int_{\sigma}^{S}s^{\omega}c(s,t)\
ds$ (71a) $\displaystyle\partial_{t}c(s,t)$
$\displaystyle=-a\,\bar{\nu}\,m(t)^{\gamma}\partial_{s}(s^{\omega}c(s,t))\,,$
(71b) with boundary conditions $\displaystyle J(\sigma,t)$
$\displaystyle=a\,\bar{\nu}\,m(t)^{\gamma}\sigma^{\omega}c(\sigma,t)=\bar{\mu}\,m(t)^{\sigma}\,$
(71c) $\displaystyle c(S,t)$ $\displaystyle=0\,.$ (71d) and initial conditions
$\displaystyle m(0)$ $\displaystyle=C\,,$ (71e) $\displaystyle c(s,0)$
$\displaystyle=0\,.$ (71f)

Next, we use the variables transformations ${m\to m\,C}$, ${c\to c\,C}$,
${t\to\tau\,/(\nu C)}$, and ${s\to x\,S}$, to convert the equations into
dimensionless form. In these dimensionless units, the effective attachment and
nucleation rate are given by $\bar{\nu}=\tilde{\delta}_{1}^{1-\gamma}$ and
$\bar{\mu}=\tilde{\mu}\,\tilde{\delta}_{1}^{2-\sigma}$, with
$\tilde{\mu}:=\mu/\nu$. Furthermore, we approximate the system by replacing
the position of the lower boundary $\sigma/S$ (in dimensionless units) with
some small and constant $\epsilon$. Hence, instead of a process in which
clusters grow from a size $\sigma$ to $S$, we consider one in which clusters
only grow from size $\epsilon S$ to $S$, thereby neglecting a fraction
${(\epsilon S-\sigma)/S\approx\epsilon}$ of all assembly steps. If $\omega>1$,
as we will discuss in detail in App. Sec. D.2, this approximation can become
inaccurate because the growth of small clusters from size $\sigma$ to
$\epsilon\,S$ defines the predominant time scale in the growth process of a
cluster. In this case, we will therefore use a different approximation; see
App. Sec. D.2 for details.
In dimensionless form, with the lower boundary of the integral approximated by
the constant, the system becomes

$\displaystyle\partial_{\tau}m(\tau)=-a\,\tilde{\delta}_{1}^{1-\gamma}\,m(\tau)^{\gamma}\,S^{1+\omega}\int_{\epsilon}^{1}x^{\omega}c(x,\tau)\,dx\,$
(72a)
$\displaystyle\partial_{\tau}c(x,\tau)=-a\,\tilde{\delta}_{1}^{1-\gamma}\,m(\tau)^{\gamma}\,S^{\omega-1}\partial_{x}(x^{\omega}c(x,\tau))\,,$
(72b) with the boundary conditions $\displaystyle J(\epsilon,\tau)$
$\displaystyle=a\,\tilde{\delta}_{1}^{1-\gamma}\,S^{\omega}\epsilon^{\omega}m(\tau)^{\gamma}c(\epsilon,\tau)=\tilde{\mu}\,\tilde{\delta}_{1}^{2-\sigma}\,m^{\sigma}$
(72c) $\displaystyle c(1,\tau)$ $\displaystyle=0\,,$ (72d) and initial
conditions $\displaystyle m(0)$ $\displaystyle=1\,,$ (72e) $\displaystyle
c(x,0)$ $\displaystyle=0\,.$ (72f)

Next, we perform the following general scaling ansatz, $\tilde{\delta}_{1}\to
S^{\phi}\,\widehat{\delta}_{1}$; $\tau\to S^{\theta}\,\widehat{\tau}$; $m\to
S^{\chi}\,\widehat{m}$ and $c\to S^{\xi}\,\widehat{c}$. To show that the
system exhibits scale invariance, the exponents $\phi,\theta,\phi$ and $\xi$
must be determined in a way so that the system of the new scaling variables
(those with the hat) is independent of $S$. By equating powers of $S$ on both
sides of the equations after the scale transformation, this results in a set
of linear equations for the exponents which must be simultaneously fulfilled.
Since Eqs. (72d) and (72f) equate to zero, they pose no additional constraint
on the exponents and the initial condition Eq. (72e) immediately implies that
$\chi=0$. The remaining three linear equations then become

$\displaystyle-\theta$ $\displaystyle=\phi\,(1-\gamma)+1+\omega+\xi\,,$ (73a)
$\displaystyle\xi-\theta$ $\displaystyle=\phi\,(1-\gamma)+\omega-1+\xi\,,$
(73b) $\displaystyle\phi\,(2-\sigma)$
$\displaystyle=\phi\,(1-\gamma)+\omega+\xi\,.$ (73c)

Subtracting the second from the first equation yields $\xi=-2$ and solving the
third equation then gives

$\displaystyle\phi=\frac{2-\omega}{\sigma-\gamma-1}\,.$ (74)

The first or second equation can then be solved for $\theta$, which gives the
unique solution

$\displaystyle\theta=\frac{(1-\omega)\sigma+\gamma+2\omega-3}{\sigma-\gamma-1}\,,$
(75)

consistent with the exponents derived in the previous section [cf. Eqs. (65)
and (67)]. This scaling symmetry implies that it is sufficient to understand
the behavior of the scale-free system (for the variables with the hat),
whereupon the functions $m(t)$, $c(s,t)$, etc., for arbitrary (but
sufficiently large) structure size $S$ are obtained by appropriate rescaling.
Specifically, the monomer concentration and complex size distribution obey the
scaling forms

$\displaystyle m(t,\tilde{\delta}_{1},S)$
$\displaystyle=C\,\widehat{m}(S^{-\theta}\,\tau,\,S^{-\phi}\,\tilde{\delta}_{1})\,,$
(76a) $\displaystyle c(s,t,\tilde{\delta}_{1},S)$ $\displaystyle=CS^{-2}\
\widehat{c}(x,\,S^{-\theta}\,\tau,\,S^{-\phi}\,\tilde{\delta}_{1})\,.$ (76b)

To relate the exponent $\theta$ with the time complexity exponent, however, it
is necessary that the condition to realize a yield $Y$, translates into a
scaleless condition depending only on the variables $\widehat{m}$ and
$\widehat{c}$. This can be seen by noting that the yield being $Y$ is
equivalent to the amount of resources that remain in the system (i.e., that
have not been absorbed by the absorbing boundary) being equal to ${1-Y}$:
${m+\int_{\sigma}^{S}s\,c(s)\,ds=1-Y}$. Hence, performing the same variable-
and scale transformations, this condition translates into the scale-free form
${\widehat{m}+\int_{\epsilon}^{1}x\,\widehat{c}\,dx=1-Y}$. Due to the scale
invariance of the governing equation [Eq. (72)], as well as the yield
condition, the exponents $\phi$ and $\theta$ can be identified with the
parameter- and time complexity exponents measured in the simulations.

The general scaling symmetry implies that the same scaling behavior that was
found to hold for the optimal value of the control parameter and the minimal
assembly time, in fact, applies identically for the entire functions
$T_{Y}(\tilde{\delta}_{1})$, denoting the time required to achieve a yield
${Y\in(0,1)}$ as a function of $\tilde{\delta}_{1}$, whenever this has a
finite value:

$T_{Y}(\tilde{\delta}_{1})=S^{\theta}\,\widehat{T}_{Y}(S^{-\phi}\,\tilde{\delta}_{1})\,,$
(77)

In particular, this shows that for the scaling behavior of $T_{Y}$, it is
irrelevant which value we choose for the yield threshold $Y$ (in the main
text, we exclusively used ${Y=0.9}$). Finally, the scaling symmetry also
explains the collapse of the yield curves in Fig. 3 and, in particular, it
shows that the threshold parameter values $\tilde{\delta}_{1}^{\text{on,Y}}$
scale identically as the optimal parameter values
$\tilde{\delta}_{1}^{\text{opt,Z}}$, independently of the demanded values $Y$
and $Z$ for the yield.

###  D.1 Scaling theory for the dimerization barrier $\tilde{\mu}$

With the scaling ansatz in the last section, we have transformed the system
into one that is independent of $S$, giving us the scaling forms Eqs. (76a)
and (76a). However, through the boundary condition Eq. (72c), the solution
still depends on the ratio $\tilde{\mu}=\mu/\nu$ between the dimerization and
the attachment rate. For the simulations in the main text, we set $\mu=\nu$,
corresponding to $\tilde{\mu}=1$. A ratio $\tilde{\mu}<1$ corresponds to the
case where the dimerization rate (or any other subnucleation binding reaction
rate) is smaller than the typical attachment rate $\nu$, e.g., as a result of
cooperative binding effects. With a similar scaling ansatz as before, we can
also eliminate the parameter $\tilde{\mu}$ from the system. Specifically,
transforming the control parameter as
$\tilde{\delta}_{1}\to{\tilde{\mu}}^{\phi^{\prime}}\,\tilde{\delta}_{1}$ and
time as $\tau\to{\tilde{\mu}}^{\theta^{\prime}}\,\tau$, the system can be
shown to become independent of $\tilde{\mu}$ if

$\phi^{\prime}=\frac{1}{\sigma-\gamma-1}\qquad\text{and}\qquad\theta^{\prime}=\frac{\gamma-1}{\sigma-\gamma-1}\,.$
(78)

We call $\phi^{\prime}$ and $\theta^{\prime}$ the complementary control
parameter- and time complexity exponent, respectively. The corresponding
general scaling forms for $m$ and $c$ thereby read:

$\displaystyle m(t,\tilde{\delta}_{1},S,\tilde{\mu})$
$\displaystyle=C\,\widehat{m}(S^{-\theta}{\tilde{\mu}}^{-\theta^{\prime}}\tau,\,S^{-\phi}{\tilde{\mu}}^{-\phi^{\prime}}\tilde{\delta}_{1})\,,$
(79a) $\displaystyle c(s,t,\tilde{\delta}_{1},S,\tilde{\mu})$
$\displaystyle=CS^{-2}\
\widehat{c}(x,\,S^{-\theta}{\tilde{\mu}}^{-\theta^{\prime}}\tau,\,S^{-\phi}{\tilde{\mu}}^{-\phi^{\prime}}\tilde{\delta}_{1})\,.$
(79b)

and for the assembly time, we obtain

$T_{Y}(\tilde{\delta}_{1})=S^{\theta}{\tilde{\mu}}^{\theta^{\prime}}\,\widehat{T}_{Y}(S^{-\phi}{\tilde{\mu}}^{-\phi^{\prime}}\,\tilde{\delta}_{1})\,.$
(80)

This implies that whether or not the minimal assembly time (as a function of
$\tilde{\delta}_{1}$) can be reduced by decreasing $\tilde{\mu}$ depends on
the morphology of the building blocks: For hexagonal building blocks, we have
$\theta^{\prime}=0$, so $T_{90}^{\text{min}}$ can no further be reduced. In
contrast, for square- and triangle-shaped monomers, we have
$\theta^{\prime}=1$, and so the minimal assembly time decreases proportionally
with $\tilde{\mu}$ within a certain range of $\tilde{\mu}$ for which our
effective theory gives an accurate description.
Note that, if the complementary exponents are nonzero, the efficiency of the
simulation can be increased significantly by reducing $\tilde{\mu}$: First,
reducing $\tilde{\mu}$ (for $\theta^{\prime}>0$) diminishes the assembly time,
and thus the simulation time. In addition, if $\phi^{\prime}>0$, the optimal
parameter value decreases as well. Hence, fewer simulation steps are needed to
simulate one time unit if $\delta_{1}$ is close to the respective optimal
value. This offers the possibility to speed up simulations that would normally
be extremely inefficient (if the goal is to determine the minimal assembly
time or its scaling properties). For example, we use this trick for systems in
Sec. J that would otherwise take extremely long to simulate. With a reduced
value of $\tilde{\mu}$, however, we are able to determine their scaling
properties with reasonable simulation effort. Plotting the assembly time in
units of $(C\nu)^{-1}\tilde{\mu}^{\theta^{\prime}}$ and the control parameter
in units of $(C\nu)\tilde{\mu}^{\phi^{\prime}}$ allows one to compare
simulation data obtained with different values of $\tilde{\mu}$.

Reducing $\tilde{\mu}$ decreases the nucleation rate relative to the growth
rate of clusters. Thus, it has the same effect on the assembly dynamics as
increasing the detachment rate $\tilde{\delta}_{1}$. Hence, we can also ask
how the assembly time scales with the structure size if instead of
$\tilde{\delta}_{1}$ we change only $\tilde{\mu}$ to control the assembly
process. In terms of our scaling approach, this corresponds to switching the
roles of $\tilde{\delta}_{1}$ and $\tilde{\mu}$. Specifically, we now use the
scaling ansatz $\tilde{\mu}\to
S^{\phi_{\mu}}{\tilde{\delta}_{1}}^{\phi_{\mu}^{\prime}}\,\widehat{\mu}$;
$\tau\to
S^{\theta_{\mu}}{\tilde{\delta}_{1}}^{\theta_{\mu}^{\prime}}\,\widehat{\tau}$;
$c\to S^{\xi_{\mu}}\,\widehat{c}$ on the system Eq. (72). This yields two sets
of linear equations for the exponents of $S$ and $\tilde{\delta}_{1}$ on both
sides of the equations. Solving the linear systems analogously, as shown
above, we obtain the exponents

$\phi_{\mu}=\omega-2\,,\qquad\theta_{\mu}=1-\omega\,,\qquad\xi_{\mu}=-2\,,$
(81)

and complementary exponents

$\phi_{\mu}^{\prime}=\sigma-\gamma-1\,,\qquad\quad\theta_{\mu}^{\prime}=\gamma-1\,.$
(82)

For the scaling of the assembly time, this implies

$T_{Y}(\tilde{\mu})=S^{\theta_{\mu}}{\tilde{\delta}_{1}}^{\theta_{\mu}^{\prime}}\,\widehat{T}_{Y}(S^{-\phi_{\mu}}\,\tilde{\delta}_{1}^{-\phi_{\mu}^{\prime}}\,\tilde{\mu})\,.$
(83)

By tuning the dimerization barrier, we find that one achieves the same minimal
time complexity exponent $1-\omega$ as by controlling $\delta_{1}$ for the
case $\gamma=1$. Indeed, both these scenarios are essentially the same since
either of the two scenarios (decreasing $\tilde{\mu}$ or increasing
$\tilde{\delta}_{1}$ in the case $\gamma=1$) reduces the speed of nucleation
while leaving the speed of cluster growth unaffected. Note, however, that the
time complexity exponent in the $\tilde{\delta}_{1}$-scenario exists only if
$\sigma>\gamma+1$, whereas for the $\tilde{\mu}$-scenario, $\theta_{\mu}$ is
independent of $\sigma$ and $\gamma$. This implies that controlling
$\tilde{\mu}$ allows to achieve high resource and time efficiency in the case
where $\tilde{\delta}_{1}$ is very small so that single bonds can already be
considered stable on the relevant time scales. The corresponding nucleation
size in this case is $\sigma=2$ and the attachment order $\gamma=1$. Assuming
a growth exponent $\omega=1/2$ for two-dimensional structures would thus yield
a time complexity exponent $\theta_{\mu}=1/2$. However, controlling the
parameter $\tilde{\mu}$ in an experiment is presumably tricky since it relies
on allosteric or cooperative binding effects. In contrast, the parameter
$\tilde{\delta}_{1}$ can be controlled more effectively, e.g., by changing the
temperature or the monomer concentration, among other possibilities (cf. Sec.
II).

###  D.2 Large growth exponents: 3D structures

Figure 14: Assembly time and optimal detachment parameter for three-
dimensional structures. (a) The assembly time $T_{90}$ for the self-assembly
of cube-shaped monomers into three-dimensional cubic structures is plotted
against the detachment rate $\delta_{1}$. Markers show the results of
stochastic simulations while dash-dotted lines show the prediction of the
effective model for three different sizes of the target structure. Stochastic
simulations were performed with particle numbers $N=500\,S$ and averaged over
10 independent runs. The effecitve model was integrated with parameters
$\sigma=4,\,\gamma=2,\,\omega=4/3,\,a=15$ and $\mu=2\,\nu$; see App. Sec. D.2.
(b) The minimal assembly time $T_{90}^{\text{min}}$ and the optimal detachment
rate $\tilde{\delta}_{1}^{\text{opt}}$ inferred from the stochastic
simulations (markers and solid lines) and effective theory (dashed-dotted
lines) are plotted against the size $S$ of the target structure. The tables
show their respective scaling exponents inferred from the last three data
points of the stochastic simulation in comparison with their theoretical
values according to Eq. (86).

This section investigates the self-assembly of structures characterized by
growth exponents $\omega>1$. In particular, three-dimensional structures can
have growth exponents $\omega>1$. To see this, we consider as an example the
self-assembly of cube-shaped monomers into a three-dimensional cube-shaped
structure of edge length $L=S^{1/3}$. This example corresponds to the three-
dimensional generalization of the square-shaped monomer morphology discussed
in the main text: As in the two-dimensional case, $\sigma=4$ cube-shaped
monomers form a stable nucleus. Subsequently, clusters grow by adding one face
at a time in a domino effect that starts after a leading order attachment
event involving $\gamma=2$ monomers has occurred. The face of a cube-shaped
cluster of size $s$ comprises $l_{s}=s^{2/3}$ monomers and the total perimeter
is thus given by $p_{s}=6\,s^{2/3}$. Therefore, Eq. (42) ff. implies that the
growth exponent is $\omega=4/3$ in this case.
For $\omega>1$, the approximation of the lower boundary of the integral in Eq.
(72a), which we used to derive the scale invariance, is no longer fully
accurate, as we will elaborate in the following. Consequently, our previous
results for the time complexity and control parameter exponent [Eq. (75) and
(74)] no longer accurately describe the scaling behavior for growth exponents
larger than one. We recall that, to derive the scale invariance, we
approximated the lower boundary $\sigma$ of the integral in the equation for
the monomer concentration, Eq. (72a), by a small target-size dependent value
$\epsilon S$. In this way, a small fraction $\epsilon$ of all assembly steps
is neglected. This approximation is accurate as long as $\omega\leq 1$ but
becomes problematic when $\omega>1$. The reason is that, despite neglecting
only a small fraction of the assembly steps, the time scale for the growth of
a cluster from size $\sigma$ to $\epsilon S$,

$T_{\sigma\to\epsilon S}=\int_{\sigma}^{\epsilon
S}\frac{1}{m^{\gamma}\bar{\nu}S^{\omega}}\,ds\sim s^{1-\omega}|_{\epsilon
S}^{\sigma}\approx\sigma^{1-\omega}$

still exceeds the time scale for the growth from size $\epsilon S$ to the
final size $S$, $T_{\epsilon S\to S}\sim(\epsilon S)^{1-\omega}$. Hence,
neglecting the slow initial assembly steps significantly underestimates the
overall time scale of the assembly process.
A better approximation in the case where $\omega>1$ is obtained by
approximating the upper boundary of the integral instead. Specifically, since
large clusters grow very fast on average, we truncate the growth process at a
fixed cluster size $K$ and assume that all subsequent growth steps are
infinitely fast. On average, therefore, each assembly step from size $\sigma$
to $K$ consumes $(S-\sigma)/(K-\sigma)\approx S/K$ monomers. This
stoichiometric factor needs to be included as a prefactor to the integral to
guarantee that the total number of monomers is conserved. The equation for the
monomer concentration replacing Eq. (71a) is thus given by

$\partial_{t}m(t)=-a\,\bar{\nu}\,m(t)^{\gamma}\tfrac{S}{K}\int_{\sigma}^{K}s^{\omega}c(s,t)\
ds\\\ \,.$ (84)

Using the same scaling ansatz as in Sec. D, results in the linear system

$\displaystyle-\theta$ $\displaystyle=\phi\,(1-\gamma)+1+\xi\,,$ (85a)
$\displaystyle\xi-\theta$ $\displaystyle=\phi\,(1-\gamma)+\xi\,,$ (85b)
$\displaystyle\phi\,(2-\sigma)$ $\displaystyle=\phi\,(1-\gamma)+\xi\,,$ (85c)

which is readily solved to give

$\phi=\frac{1}{\sigma-\gamma-1}\,,\qquad\theta=\frac{\gamma-1}{\sigma-\gamma-1}\,.$
(86)

This suggests that if $\omega$ exceeds 1, the time complexity and control
parameter exponent remain constant as functions of $\omega$. Note that, for
$\omega=1$, Eqs. (74) and (75) yield the same values for the exponents as Eq.
(86). Comparing with the results from stochastic simulations [cf. Fig. 14], we
find that the control parameter exponent accurately matches the theoretical
value $\phi_{\text{th}}=1$ predicted by Eq. (86), while the time complexity
exponent is slightly overestimated by the theoretical value
$\theta_{\text{th}}=1$. The estimates are, however, significantly better than
those obtained with Eqs. (74) and (75), which would yield
$\phi_{\text{th}}=2/3$ and $\theta_{\text{th}}=1/3$. The overestimation of the
time complexity exponent results from the fact that we average the rate of
monomer consumption over the entire growth process of a cluster (accounting
for the constant prefactor $S/K$ in Eq. (84)). This approximation particularly
affects the kinetics in the regime of the U-shape, where clusters grow
simultaneously and, thus, the precise temporal dynamics of the monomer
concentration are important. In contrast, in the regime characterized by the
power-law scaling, clusters tend to nucleate and grow consecutively, and the
approximation is therefore quite accurate. Indeed, we find that the assembly
time simulated with the effective model at a constant, large value of
$\tilde{\delta}_{1}$ (e.g. $\tilde{\delta}_{1}=10^{4}$) scales with large
target structure sizes as $\sim S^{-1.00}$ – fully in accordance with the
theoretical exponents $\phi_{\text{th}}=\theta_{\text{th}}=1$ and the power-
law scaling $T_{90}\sim S^{\sigma-2}$. We note from Fig. 14(a) that the form
of the U-shape broadens when $S$ increases. Therefore, there can only be an
approximate scaling relation but no exact one for the entire $T_{90}$-curve as
in the case where $\omega\leq 1$.

Three-dimensional structures with higher coordination numbers that self-
assemble with attachment order $\gamma=1$ most likely have a growth exponent
$\omega\approx 2/3$ (since there cannot be a domino effect if $\gamma=1$). In
this case, Eq. (75) suggests a time complexity exponent of
$\theta_{\text{th}}=1/3$. This value is smaller than the value obtained for
$\gamma\geq 2$ according to Eq. (86) for any realistic nucleation size
$\sigma$. Hence, also in the case of three-dimensional structures, self-
assembly is fastest if the attachment order is small. Structures with higher
coordination numbers, therefore, tend to assemble faster. This is consistent
with the results in reference [52], showing that high coordination structures
typically have steeper free energy gradients.

Effective model for 3D structures. The prefactors for the effective model used
for the plots in Fig. 14 are derived as follows. Once the first monomer has
attached to the face of a three-dimensional structure, the second monomer
forming the seed can attach in $q_{s}=4$ equivalent ways ($q_{s}$ binding
sites) in the case of a large cluster and $q_{s}=2$ ways in the case of a
small cluster. Hence, with the total perimeter surface given by
$p_{s}=6\,s^{2/3}$, Eq. (42) predicts a value for the multiplicity factor $a$
between 12 and 24. Indeed, a good fit with the stochastic simulation is
obtained with $a=15$ in Fig. 14. Furthermore, we note that in the nucleation
process, the multiplicity for a dimer to grow into a trimer is 8, while the
detachment rate of both the trimer and the dimer is $2\,\delta_{1}$. Hence, we
obtain an overall prefactor of the effective nucleation rate of 2 (for 2D
structures, the prefactors for the nucleation processes canceled to 1 for all
three monomer morphologies, cf. Fig. 10). We account for the additional
prefactor in the nucleation process by setting $\mu=2\,\nu$ [cf. Eq. (2)]. To
further improve the fit, we explicitly set the growth rates for the first four
post-nucleation assembly steps to $g_{s}=64\,\bar{\nu}\,m^{\gamma}$. This
improves the fit since the nucleus does not yet have a cube shape, as assumed
for the growth of larger clusters.

##  Appendix E The role of cluster-cluster interactions

In the analysis in the main text, we have worked under the simplifying
assumption of an ideal assembly process, meaning that interactions among
clusters are negligible. This assumption is frequently used in the literature
on self-assembly. It is motivated by the observation from experiments and
molecular dynamics simulations that intermediate assembly products are
typically only present in very low concentrations [84, 15, 16]. Here, we would
like to investigate and quantify to which extent cluster-cluster interactions
can influence the time efficiency of self-assembly and whether the morphology
of the building blocks is relevant in this regard. To this end, we first
introduce a simple measure that informs whether cluster interactions can
potentially influence the assembly process. Afterward, we extend our effective
model to include cluster interactions and estimate the effect that those might
have on the assembly time.

To estimate the extent to which interactions among oligomers can potentially
influence the dynamics, we define the assembly ideality index that is
calculated under the assumption of an ideal assembly process,

$I:=\frac{S}{C}\,\int\limits_{0}^{T_{90}}\nu K^{2}(t)\,\text{d}t\,,$ (87)

where, as above, ${K(t):=\sum\nolimits_{s=\sigma}^{S-1}c_{s}(t)}$ denotes the
total concentration of incomplete complexes in the system at time $t$, and we
integrate over the time window $[0,T_{90}]$ to estimate the total number of
cluster-cluster interactions until time $T_{90}$. The index, therefore,
estimates the expected number of cluster interactions per completed structure
until time $T_{90}$ (note that the concentration of completed structures at
90% yield is ${0.9\,C/S\approx C/S}$), assuming the same reaction rate $\nu$
between two clusters as between a cluster and a monomer. Small values ${I\ll
1}$ indicate that cluster interactions occur only with a small probability $I$
during the growth of a structure and can thus be neglected. Hence, the index
provides a simple consistency check for the ideality assumption of a
simulation. Note that assuming the same reaction rate $\nu$ between clusters
as between clusters and monomers most likely overestimates the reaction rate
between clusters since larger clusters typically diffuse and therefore react
more slowly than monomers, and only a fraction of their interactions would
lead to stable configurations. Hence, the ideality index must be interpreted
as an upper limit for the expected number of cluster interactions. Small
values ${I\ll 1}$ are considered a sufficient condition for the ideality
assumption. In contrast, large values ${I>1}$ indicate that cluster
interactions might potentially influence the dynamics, depending on the
specifics of the system.

Figure 15: Beyond ideal self-assembly: the role of cluster-cluster
interactions. (a) The ideality index $I$, which estimates the number of
interactions of an oligomer with other oligomers until time $T_{90}$ [see
definition Eq. (87)], is plotted against the detachment rate
$\tilde{\delta}_{1}$ for the assembly processes with three different particle
morphologies (${S=100}$). The index has been calculated numerically by
integrating the effective model, Eq. (4). Values of $I$ larger than 1 (dashed
line) indicate that cluster interactions may be important for the assembly
dynamics. Thus, a strong dependence of the potential role of cluster
interactions on particle morphology is revealed. (b) Assembly time $T_{90}$
with (drawn line) and without (dotted line) cluster interactions as a function
of the detachment rate $\tilde{\delta}_{1}$. Assembly times were obtained by
integrating the effective theory that was extended by additional terms
accounting for all possible cluster reactions
${c_{s}+c_{p}\overset{\nu}{\to}c_{s+p}}$ subject to the constraint ${s+p\leq
S}$ (see App. B for details). Since, in an actual system, the rate of
reactions between clusters would likely be smaller, this must be considered an
upper estimate of the effect expected by cluster interactions. The minimal
assembly time is only slightly reduced as a consequence of cluster
interactions and – as in the ideal case – the assembly time increases like
${T_{90}\sim\tilde{\delta}_{1}^{\sigma-2}}$ for large $\tilde{\delta}_{1}$.

Figure 15(a) shows the numerically determined ideality index as a function of
the detachment rate $\delta_{1}$ for the assembly processes with the three
different particle morphologies. The index reveals a strong dependence of the
potential role of cluster-cluster interactions on the morphology of the
constituents: While cluster-cluster interactions can be safely neglected for
hexagonal-shaped monomers (${I\ll 1}$), the magnitude of the index observed
for the triangle-shaped monomers suggests that these interactions play a
significant role in the assembly process. Furthermore, the index behaves
differently as a function of the detachment rate $\tilde{\delta}_{1}$: While
in the system with hexagonal monomers, cluster-cluster interactions become
less critical when $\tilde{\delta}_{1}$ is increased, their potential
influence further increases in the system with triangle-shaped monomers as
$\tilde{\delta}_{1}$ grows. Hence, the ideality index seemingly correlates
with the time efficiency of a self-assembly system: Particles with a
morphology that assemble time efficiently tend to have a smaller ideality
index in general than particles with a morphology that assemble less
efficiently.

In order to estimate the extent to which cluster interactions affect the
assembly time, we extend the effective model by additional terms accounting
for reactions of any two clusters of sizes ${i,j\geq\sigma}$ with $i+j\leq S$
to a cluster of size ${s=i+j}$ at rate $\nu$:

$\partial_{t}c_{s}=...\,+\tfrac{1}{2}\,\nu\sum\limits_{\begin{subarray}{c}i,j\geq\sigma\\\
i+j=s\end{subarray}}c_{i}c_{j}-\nu\,c_{s}\sum\limits_{\begin{subarray}{c}i\geq\sigma\\\
i+s\leq S\end{subarray}}c_{i}\,,$ (88)

for all ${\sigma\leq s\leq S}$. The factor of $\frac{1}{2}$ in front of the
first sum avoids double counting and serves as a stoichiometric factor in the
case ${i=j}$. As before, assuming that clusters react at the same rate $\nu$
as monomers will strongly overestimate the effect that cluster interactions
have on the assembly time, and thus, the model describes an upper limit for
the impact that is to be expected by these additional interactions.

Figure 15(b) compares the assembly time of the ideal system with that obtained
from integrating the extended model accounting for cluster interactions. The
comparison suggests that cluster-cluster interactions can indeed reduce the
assembly time in systems with triangles—and to a much lesser extent also in
systems with square-shaped monomers. However, the reduction of the assembly
time is relatively small in general. The reason is that, as the size of a
cluster increases, the number of its possible binding partners decreases:
according to Eq. (88), a cluster of size $i$ can only react with clusters of
size ${j\leq S-i}$. Therefore, cluster interactions are not effective in
avoiding kinetic traps.

If we also take into account that the model probably overestimates the
reaction rate between clusters, the reduction in assembly time due to cluster-
cluster interactions will presumably be even smaller in any actual system that
uses the respective types of particles. On the other hand, cluster-cluster
interactions might enhance the occurrence of assembly errors and defects. For
example, occasionally, clusters might react that do not fit together
perfectly, thus leaving defects and holes or distorting the structure.
Operating at a small ideality index might thus be preferable for general self-
assembly systems as it might enhance the robustness of the process against
errors. In conclusion, the morphology of the constituents could be an
essential determinant not only of the time efficiency but also of the
reliability of the assembly process.

##  Appendix F Invariance of the scaling results to the yield threshold

Figure 16:  Assembly time $T_{50}$ (solid) in comparison with assembly time
$T_{90}$ (faded). (a) The assembly times $T_{50}$ (solid) and $T_{90}$(faded)
required to achieve 50% or 90% yield, respectively, in the strong binding
limit are plotted against the detachment rate $\delta_{1}$ (in units of
$C\nu$) for systems with triangle- (red), square- (blue), and hexagon-shaped
(green) building blocks for a fixed target structure size $S=98$ (triangle-
shaped monomers) or $S=100$ (square and hexagonal monomers). Markers represent
averages of 10 independent runs of the stochastic simulation (performed with
$N=500\,S$ and $N=300\,S$ (triangle-shaped monomers)). The dash-dotted colored
lines represent the prediction of the effective theory obtained by numerically
integrating Eq. (4). Black dashed lines show the analytic result for the
assembly times given by Eqs. (31) and (32) in the slow nucleation regime
(ranges for $\tilde{\delta}_{1}$ are chosen manually). (b) The minimal
assembly time $T_{50}^{\text{min}}$ (solid) and $T_{90}^{\text{min}}$ (faded)
inferred from the stochastic simulation (markers and solid lines) and
effective theory (dashed-dotted lines) are plotted against the size S of the
target structure. Although the assembly times $T_{90}$ and $T_{50}$ may differ
significantly, they exhibit approximately the same scaling in dependence on
the structure size $S$. The scaling exponents $\theta_{\text{sim}}$ inferred
from the last three data points of the stochastic simulation for $T_{50}$ (and
their theoretical asymptotic values $\theta_{\text{th}}$, cf. Eq. (17)) are
summarized in the table (compare with the analogous exponents
$\theta_{\text{sim}}$ for $T_{90}$ exhibited in the table in Fig. 3(b)).

In the main text and the appendix so far, we have studied the behavior of the
assembly time $T_{90}$ required to achieve 90% yield. Alternatively, one could
also choose a different threshold for the yield, e.g., 50%, and analyze the
behavior of the corresponding assembly time $T_{50}$. For our scaling results
to be informative and valuable, it is important that these results are
invariant to the yield threshold and that the qualitative behavior of the
assembly time $T_{Y}$ is the same, independent of the yield threshold $Y$. We
will verify this in the following by repeating the same plots shown in the
main text for $T_{90}$ now with $T_{50}$.

Figure 16(a) shows the assembly time $T_{50}$ as a function of the detachment
rate $\delta_{1}$ in comparison with $T_{90}$ for the different particle
morphologies in the strong binding limit. We find that $T_{50}$ and $T_{90}$
behave qualitatively similar and, in particular, both exhibit the same scaling
$\sim\tilde{\delta}_{1}^{\sigma-2}$ for large enough $\delta_{1}$. However,
the values of the assembly times $T_{50}$ and $T_{90}$ can differ quite
dramatically depending on the particle morphology. In particular, differences
in the minimal assembly times between the different particle morphologies tend
to become less pronounced for a lower yield threshold. This is consistent with
our analysis of the regime in which nucleation is very slow [cf. section V.4].
In particular, Eq. (31) and (32) (visualized as black dashed lines in Fig.
16(a)) predict that the assembly time in this regime depends on the yield
threshold $Y$ as $T_{Y}\sim(1-Y)^{1-\sigma}$. Thus, the assembly time in this
regime depends more strongly on the demanded resource efficiency if the
nucleation size $\sigma$ is larger. However, the scaling of the minimal
assembly time with the target structure size (i.e., the time complexity
exponent $\theta$) remains independent of the yield threshold as we verify in
Fig. 16(b): Comparing the measured time-complexity exponents for $T_{50}$ with
those measured for $T_{90}$ (see table in Fig. 3(b)) shows that they are
almost identical. Thus, despite the assembly time as such might depend
strongly on the yield threshold, all scaling results are invariant to the
yield threshold. Differences in the assembly times between the different
particle morphologies studied in this work tend to become less pronounced if
the requirement for resource efficiency is relaxed.

##  Appendix G Seeding

Figure 17: Yield and assembly time in seeded systems (according to the
effective model). (a) The maximal yield of target structures of size $S=100$
in the effective model in the limit of large $\tilde{\delta}_{1}$ (here
$\tilde{\delta}_{1}=10^{12}$) is plotted against the initial concentration
$\Sigma$ of nuclei (seeds). Parameters of the effective model were chosen to
simulate the self-assembly of square-shaped monomers ($\sigma=4$, $\gamma=2$,
$\omega=1$ and $a=5.3$), and the initial monomer concentration was
$m(0)=C-\sigma\Sigma$. (b) Assembly time $T_{90}$ as a function of
$\tilde{\delta}_{1}$ for seeded systems with initial seed concentration
$\Sigma=C/S$ (drawn lines) and corresponding unseeded systems (dash-dotted
lines, faded). (c) Minimal assembly time $T_{90}^{\text{min}}$ as a function
of $S$ for seeded systems with initial seed concentration $\Sigma=C/S$ (drawn
lines) and corresponding unseeded systems (dash-dotted lines, faded). Black
dashed line shows the assembly time for a putative system of triangular
monomers in which the first assembly step after nucleation has attachment
order $\gamma^{\prime}=3$ instead of $\gamma^{\prime}=4$. Table shows the time
complexity exponents $theta_{\text{sim}}$ measured in the range of target
structure sizes $400\leq S\leq 1000$ and their corresponding theoretical
values $theta_{\text{th}}$.

Seeding refers to the experimental method of initially putting a certain
number of stable nuclei (seeds) into the system. In systems that enable the
formation of different structures, like, e.g., algorithmic self-assembly,
seeding is an integral part since the composition of the seed determines the
structure or the pattern that grows (i.e., the algorithm that is performed via
algorithmic self-assembly) [35]. From a kinetic point of view, we expect that
seeding should reduce the assembly time by eliminating the requirement of the
rate-limiting nucleation reactions as the seeds can start growing right from
the beginning. First, in order to determine the optimal number (concentration)
of seeds, we integrate the effective model with an initial concentration
$\Sigma$ of stable nuclei ($c_{\sigma}(0)=\Sigma$) and corresponding monomer
concentration $m(0)=C-\sigma\Sigma$ (to guarantee a fixed total number of
resources). We find that the maximum yield (obtained in the limit of large
$\tilde{\delta}_{1}$) is $1$ if the seed concentration is smaller or equal
$C/S$ and decreases rapidly if this concentration is exceeded [Fig. 17(a)].
This is intuitive since an initial concentration $\Sigma=C/S$ of nuclei
exactly equals the concentration of completed structures when the yield is
100%.

Next, we study the effect of seeding on the assembly time by integrating the
effective model with initial concentration $\Sigma=C/S$ of nuclei
(corresponding to the number $N/S$ of seeds required for 100% yield) and
corresponding monomer concentration $m(0)=C-\sigma\Sigma$ [cf. Fig.
17(a),(c)]. Surprisingly, contrary to our expectation, we find that the
assembly time increases due to seeding if $\gamma>1$ (square- and triangular
monomers). This is because if nuclei are already present in the system, the
control parameter $\tilde{\delta}_{1}$ must be chosen even larger to suppress
formation of additional nuclei, which would lead to an excess of clusters.
Increasing $\tilde{\delta}_{1}$ simultaneously decreases the growth rate if
$\gamma>1$, making the overall process less time efficient. In contrast, for
hexagon-shaped monomers ($\gamma=1$), the growth rate is not affected by
$\tilde{\delta}_{1}$ and the minimal assembly time remains the same. With
structure growth (rather than nucleation) being rate limiting in seeded
systems, increasing $\tilde{\delta}_{1}$ beyond the optimal value makes the
assembly time increase only proportional to $\tilde{\delta}_{1}^{\gamma-1}$
(rather than $\sim\tilde{\delta}_{1}^{\sigma-2}$ in the case of unseeded
systems [compare Fig. 3a]). In particular, in the case of hexagonal building
blocks ($\gamma=1$), the assembly time is thus constant as a function of
$\tilde{\delta}_{1}\geq\tilde{\delta}_{1}^{\text{opt}}$.

In the framework of our scaling analysis, the initial condition for seeded
systems, $c_{\sigma}(t\\!=\\!0)\sim S^{-1}$, is consistent with the scale
invariance described by the scaling forms for the total concentration of
clusters, $K(t,S)\sim S^{-1}\widehat{K}(S^{-\theta}\tilde{t})$ [cf. App. Sec.
C], and for the cluster size density, $c(s,t,S)\sim
S^{-2}\widehat{c}(S^{-1}s,S^{-\theta}\tilde{t})$ [cf. Eq. 24] (note that by
rescaling the cluster size $s\to S^{-1}s$, we get an additional factor of
$S^{-1}$ for the cluster size density). Furthermore, the initial condition
$m(0)=C-C/S$ is approximately consistent with the scaling form for the monomer
concentration, $m(t,S)\sim\widehat{m}(S^{-\theta}\tilde{t})$ [cf. Eq. 23] in
the limit of large $S$ where $C/S$ is negligible and $m(0)\approx C$.
Therefore, the scaling analysis in App. Sec. D can be performed identically
for seeded systems with the modified initial conditions [cf. Eqs. (72e) and
(72f)]. Consequently, the scaling behavior should be the same for seeded
systems with initial concentration of seeds $\Sigma\sim S^{-1}$. This is
confirmed by Fig. 17(c) for square and hexagonal monomers. In the case of
triangular monomers, we observe a slightly smaller time complexity exponent
compared to the unseeded system (but still close to its theoretical value).
This anomalous behavior is likely caused by the higher effective order
$\gamma^{\prime}=4$ of the first assembly step following nucleation [cf. App.
Sec. B] (which we did not explicitly account for in the scaling analysis). To
see this, we simulated a putative system of triangular monomers in which the
attachment order of the first step likewise equals $\gamma=3$ (black dashed
line). The assembly time in this case is significantly larger than for the
true system and the scaling exponent is very close to that of the unseeded
system, confirming that the post-nucleation assembly step is responsible for
the drop in the exponent.
Hence, in general, this shows that our scaling results are very robust and
apply likewise to seeded systems. In line with our general conclusions, using
hexagonal building blocks (or generally monomers with $\gamma=1$) appears to
be highly favorable as the minimal assembly time does not increase as a
consequence of seeding and the assembly time even becomes a constant function
of $\tilde{\delta}_{1}$.

##  Appendix H Annealing protocols

The assembly dynamics are controlled by the ratio between the frequency of
detachment events $\delta_{1}$ and the frequency of attachment events $\nu m$.
Initially, the monomer concentration is $m(0)=C$, but as more and more
monomers get consumed during the assembly process, their concentration
gradually decreases. Consequently, the ratio between the detachment and
attachment rate increases during the assembly process. In order to counteract
this effect, a frequently used experimental approach consists in ‘annealing’
the system by decreasing the temperature [35]. Typically, one starts at a high
temperature and gradually cools the system to room temperature. Since the
detachment rate decreases with decreasing temperature, $\delta_{1}\sim
e^{-E_{B}/(k_{B}T)}$, if applied optimally, annealing allows to keep the ratio
between the detachment rate and attachment rate constant during the assembly
process. Here, we show that our scaling analysis also applies to the case of
an annealing protocol in which the temperature adapts instantaneously to the
momentary concentration of monomers such that the ratio between detachment and
attachment rate remains constant. To describe such an annealing protocol with
the effective theory presented in Sec. V.2 and App. Sec. D, we only need to
replace the control parameter $\tilde{\delta}_{1}=\delta_{1}/(C\nu)$ with a
time-dependent control parameter
$\tilde{\delta}_{1}(t)=\tilde{\delta}_{1}(t=0)\frac{m(t)}{C}$ that gradually
decreases proportional to the monomer concentration. This replacement only
introduces additional factors of $m$ in the dynamic equations, but the monomer
concentration does not scale with $S$ since
$m(t)=\widehat{m}(S^{-\theta}\tilde{t})$ [cf. Eq. (23)]. Hence, the scaling
analysis can be performed in exactly the same way. This implies that all our
scaling results apply identically under annealing. Note that we could also use
any other annealing protocol of the form
$\tilde{\delta}_{1}(t)=\tilde{\delta}_{1}(0)f(S^{-\theta}\tilde{t})$ to
maintain the scale invariance.

Figure 20 shows the average minimal assembly time simulated stochastically
with the above annealing protocol plotted against the target structure size
(for square-shaped monomers). The minimal assembly time was determined by
tuning the parameter $\tilde{\delta}_{1}(t=0)$. We find that the self-assembly
efficiency is indeed significantly increased as a result of annealing.
However, the plot confirms that the scaling of the assembly time remains
invariant under annealing.

##  Appendix I Systems with unlimited cluster growth

Figure 18: Final cluster size distribution. The final cluster size
distributions for a) hexagon-shaped ($\omega\approx 1/2>1$), b) square-shaped
($\omega=1$), and c) 3D cube-shaped ($\omega=4/3$) monomers are shown for the
respective values of the detachment rate $\delta_{1}$ indicated in each plot.
The target structure size in the simulation was chosen very large ($S=3025$
(a,b) and $S=8000$ (c)) so that the yield was zero in each case, thereby
effectively simulating unlimited cluster growth. The initial number of
monomers was $N=3\cdot 10^{5}$, and the distribution was averaged over 40
independent simulation runs (blue circles). Subsequently, moving averages were
taken over windows of $k=10$ (red circles) and $k=50$ (green circles) cluster
sizes. The averaged cluster size distribution decreases exponentially for
large cluster sizes for square- and cube-shaped monomers (i.e., for
$\omega\geq 1$). Black lines show the final cluster size distribution obtained
by integrating the effective model.

So far, we have considered systems with self-limiting cluster growth by
assuming periodic boundary conditions of the target structures. However, our
scaling analysis can straightforwardly be generalized to systems with
unlimited growth of the structures. The only requirement is that the growth
rate of large clusters consistently scales with the cluster size as $\sim
s^{\omega}$, with some arbitrary growth exponent $\omega$.

To describe unlimited cluster growth with the effective theory presented in
Sec. V.2 and App. Sec. D, we only have to drop the absorbing boundary at
$s=S$, which removed finished structures from the system, and replace the
upper bound in the integral in the equation for the monomer concentration, Eq.
(22), resp. Eq. (72a), by infinity, to integrate clusters of arbitrary size.
Clearly, these changes do not affect the scaling analysis. The natural
generalization of the yield $Y_{S}$ represents the fraction of resources that
are bound in clusters of size larger or equal to $S$. The yield can thereby be
written in the same form as Eq. (25) by identifying $1-Y_{S}$ with the number
of resources that are either available as monomers or bound in clusters of
size smaller than $S$. In the form of Eq. (25), the scale invariance of the
yield [cf. Eq. (26)] and of the corresponding assembly time $T_{90}$ (likewise
defined as the time required to achieve 90% yield) [cf. Eq. (27)], becomes
evident. Hence, with the yield and assembly time defined in this way, all
scaling results derived for the case of self-limiting cluster growth can
directly be transferred to the case of unlimited cluster growth.

In principle, we could also define the yield and assembly time in a different
way as long as it is consistently scale invariant. For example, the assembly
time could alternatively be defined as the time when the average size of a
cluster exceeds $S$. The scale invariance of this condition becomes evident by
using the scaling form for the complex concentration, Eq. (24) resp. Eq. (76b)

$S<\frac{\int_{\sigma}^{\infty}sc(s,t)\,ds}{\int_{\sigma}^{\infty}c(s,t)\,ds}=\frac{\int_{\epsilon}^{\infty}S\,x\,S^{-2}\,\widehat{c}(x,S^{-\theta}\tilde{t})\,S\,dx}{\int_{\epsilon}^{\infty}S^{-2}\,\widehat{c}(x,S^{-\theta}\tilde{t})\,S\,dx}\,,$
(89)

since all factors of $S$ on the right and on the left-hand side cancel. Note
that it is not possible to include the monomers as clusters of size 1 in the
definition of the mean cluster size since the contribution of the monomers and
the complexes in the denominator,
$\widehat{m}+S^{-1}\int_{\epsilon}^{\infty}\widehat{c}(x)\,dx$, would scale
differently with $S$ and would thus violate the scale invariance. However, one
can additionally demand that the monomer concentration has dropped below a
certain value $X$, which gives the additional scale-free condition
$\widehat{m}(S^{-\theta}\tilde{t})\leq X$ for the assembly time. This shows
that our scaling results are very general and likewise apply to systems with
unlimited cluster growth.

Cluster size distribution. It is interesting to characterize the final cluster
size distribution in systems with unlimited cluster growth. To this end, we
choose a target structure size in the simulation that is very large and
ensures that the final yield is zero. Therefore, the size limit does not
affect the system’s evolution, effectively mimicking a system with unlimited
cluster growth. Figure 18 shows the final cluster size distributions for
hexagon-, square-, and 3D cube-shaped monomers, which are characterized by
different values of the growth exponent (${\omega<1,\omega=1}$ and
${\omega>1}$, respectively). Since the growth rates of leading and subleading
attachment reactions differ strongly, the distributions are scattered over
broad ranges (in number of clusters, respectively cluster sizes) for square-
and cube-shaped monomers (also compare with Fig. 2). The distributions can be
narrowed down by calculating moving averages in the number of clusters over
increasingly larger windows of $k$ subsequent cluster sizes (Fig. 18 shows the
distributions calculated for $k=1$, $k=10$ and $k=50$).

We find that these averaged distributions decrease exponentially with large
cluster sizes for square- and cube-shaped monomers. In contrast, for hexagonal
monomers, due to the smaller growth exponent, the distribution is more compact
and has a significantly shorter tail. In the cases of hexagon- and square-
shaped monomers, the averaged final distributions coincide well with the
distribution obtained by integrating the effective model, Eq. (4) (black
curves). However, in the case of cube-shaped monomers, the effective model
predicts a distribution that decreases as a power law rather than
exponentially, thus implying a significantly longer tail than the
stochastically simulated distribution. The reason for this discrepancy is that
if single clusters quickly grow very large in the three-dimensional system,
the time scale of the subleading processes (‘domino effect’, which scales as
${\sim s^{2/3}}$) can no longer be neglected against that of the leading order
processes [cf. Eq. (41)]. Therefore, the effective growth rate of large
clusters is actually smaller than the growth rate predicted by Eq. (42). To
accurately describe the distribution for three-dimensional structures, the
time scale of the domino effect would have to be considered explicitly in Eq.
(41). On the other hand, self-limitation of cluster growth, as assumed in our
original model, prevents single clusters from growing extremely large. Thus,
the effective model accurately describes the assembly time for limited cluster
growth, as seen in Fig. 14.

##  Appendix J Heterogeneous systems

Figure 19:  Heterogeneous systems. Simulating the self-assembly of
heterogeneous structures (information-rich structures), we assume that each
species (labeled by an index $1...S$) is unique and binds only with the
respective neighboring species, as shown. Thus, each species appears only once
in a finished structure and at a specified position. The underlying kinetics
for self-assembly are the same as in the homogeneous case, but the different
species can have distinct binding and detachment rates or varying
concentrations. In App. Sec. J, we investigate how self-assembly behaves for
various heterogeneous systems.

Next, we study how our results generalize to heterogeneous systems with
different species of monomers, each species binding specifically only with
certain other species. Specifically, we consider a fully heterogeneous system
of square-shaped monomers such that each species (labeled $1...S$) can bind
only with four other species ($i+1$, $i-1$, $i-L$, $i+L$), and each site
within a cluster can only be occupied by a specified species; see Fig. 19. We
investigate five different relevant cases: i) all species have the same
reaction rates $\nu$ and $\delta_{1}$ and are available in the same copy
numbers $N$. ii) The reaction rates $\nu_{i}$ for the different species are
chosen randomly. iii) The detachment rates of four specified species that are
able to form a nucleus are set to zero. iv) Four specified species that are
able to form a nucleus are provided in much larger concentrations. v)
Investigating the role of the boundary conditions, we assume that species at
the boundary limit the growth of the clusters such that structures with
explicit boundaries self-assemble.

###  J.1 Identical reaction rates

First, we consider the case in which all species have the same attachment and
detachment rates and are available in identical concentrations. In this ideal
case, one can show mathematically that the system behaves equivalent to a
homogeneous system in the limit of large particle numbers. To establish this
equivalence, the key insight is that, assuming periodic boundary conditions of
the structures, the concentrations of all species are identical since all
species are subject to the same dynamics. We show this rigorously for the
evolution of the monomer concentration. To this end, we first have to
introduce some notation: We denote the species $i$ of a complex as the species
with the lowest label of the constituent monomers of the complex. A complex in
the heterogeneous system is then uniquely characterized by its species index
and its morphology, which specifies the size of a cluster and its shape. We
denote the morphology by another index $\zeta$ that takes values in some
countable set of numbers characterizing the morphology via some mapping, which
we do not further specify. By $c_{j,\zeta}(t)$, we denote the concentration of
complexes of species $j$ and morphology $\zeta$ at time $t$. We use $nn(i)$ as
a shorthand notation of the set of indices $\\{i+1,i-1,i+L,i-L\\}$ with which
species $i$ is able to bind. Furthermore, $nn(i,j,\zeta)$ denotes the set of
indices of species that are direct neighbors of $i$ and constituents of the
complex $(j,\zeta)$. The temporal change of the monomer concentration of
species $i$ is then given by

$\displaystyle\begin{split}\partial_{t}m_{i}=-\mu_{\text{het}}\,m_{i}\sum_{j\in
nn(i)}m_{j}&-\sum_{\begin{subarray}{c}(j,\zeta)\\\ i\notin(j,\zeta)\\\
|nn(i,j,\zeta)|\geq 1\end{subarray}}\nu\,m_{i}\,c_{j,\zeta}\\\
&+\sum_{\begin{subarray}{c}(j,\zeta)\\\ i\in(j,\zeta)\\\
|nn(i,j,\zeta)|=1\end{subarray}}\delta_{1}\,c_{j,\zeta}\,.\end{split}$ (90)

The three terms on the right account for the loss and gain of monomers due to
nucleation (with rate $\mu_{\text{het}}$), attachment to complexes, and
detachment from complexes. For the attachment reactions with monomers of
species $i$, we consider all complexes that contain at least one neighbor of
species $i$ but not species $i$ itself. Similarly, the complexes from which a
monomer of species $i$ can detach are those that contain species $i$ and
exactly one neighbor of $i$. Next, we use that the average concentrations of
different monomer and complex species must be the same ($m_{i}=m_{j}$ and
$c_{i,\zeta}=c_{j,\zeta}$) since they are subject to equivalent dynamics.
Hence, the dimerization term simply becomes $-4\mu_{\text{het}}m_{i}^{2}$. We
furthermore note that, for a given cluster morphology $\zeta$, there is
exactly one species index for each binding site of the cluster fulfilling the
conditions in the sum of the attachment term. Thus, the second term can be
written as

$-\sum_{\zeta}m_{i}\,c_{i,\zeta}\,\tilde{b}(\zeta)\,\nu\,,$ (91)

where $\tilde{b}(\zeta)$ denotes the number of binding sites of the cluster
with morphology $\zeta$, and the sum goes over all possible cluster
morphologies. Note that the concentration of a specified cluster species $i$
equals the total concentration of complexes with the same morphology divided
by $S$: $c_{i,\zeta}=c_{\zeta}/S$. Finally, for the sum in the last term,
there is exactly one species index fulfilling the conditions for each site in
the cluster that has exactly one neighbor. Denoting the number of sites with
one neighbor for a given cluster morphology by $b_{1}(\zeta)$, the last term
thus becomes

$\sum_{\zeta}c_{i,\zeta}\,b_{1}(\zeta)\,\delta_{1}\,.$ (92)

Hence, by setting $\mu=4\,\mu_{\text{het}}$, this gives the same dynamic
equation for the monomer concentration as one would have for a homogeneous
system

$\partial_{t}m=\mu\,m^{2}-\sum_{\zeta}m\,c_{\zeta}\,\tilde{b}(\zeta)\,\nu+\sum_{\zeta}c_{\zeta}\,b_{1}(\zeta)\,\delta_{1}\,.$
(93)

Setting $\mu=4\,\mu_{\text{het}}$ makes sense intuitively since, in the
homogeneous case, two monomers can dimerize in four possible constellations
(four possible binding sites). In contrast, in the heterogeneous case, they
can only dimerize in one constellation.
Similarly, the evolution of the cluster concentrations can be written
independently of the species indices in a form equivalent to that of a
homogeneous system. Thereby, we use the notation $\zeta=\zeta^{\prime}+1$ to
indicate that the morphology $\zeta$ equals $\zeta^{\prime}$ once we add a
single particle to the boundary of $\zeta^{\prime}$. Similarly,
$\zeta^{\prime}+_{1}1$ indicates that we add one particle to $\zeta^{\prime}$
so that it has exactly one neighbor. The evolution of cluster concentrations
then reads

$\displaystyle\begin{split}\partial_{t}c(\zeta)=&\sum_{\begin{subarray}{c}\zeta^{\prime}\\\
\zeta=\zeta^{\prime}+1\end{subarray}}\nu\,mc(\zeta^{\prime})-\nu\,m\,\tilde{b}(\zeta)\,c(\zeta)\\\
+&\sum_{\begin{subarray}{c}\zeta^{\prime}\\\
\zeta^{\prime}=\zeta+_{1}1\end{subarray}}c(\zeta^{\prime})\,\delta_{1}-\delta_{1}\,b_{1}(\zeta)\,c(\zeta)\,.\end{split}$
(94)

Together, this shows that the heterogeneous process decouples into $S$
homogeneous processes, one for each species. In other words, this means that a
heterogeneous process with total particle number $N\cdot S$ behaves
identically on average as a homogeneous process with particle number $N$. Due
to this mathematical equivalence, all scaling laws derived for homogeneous
systems directly transfer to heterogeneous systems in the ideal case. Next, we
check how robust these scaling laws are if we deviate from the ideal case,
e.g., by varying the rate constants or the particle numbers of the different
species or by assembling structures with explicit boundaries rather than
periodic boundary conditions.

###  J.2 Random heterogeneous binding rates

Figure 20: Robustness of the scaling behavior. The minimal time required to
achieve 90% yield ($T_{90}^{\text{min}}$) is plotted in dependence of the
target structure size $S$ for different variants of the model. Diamond and
triangle markers show $T_{90}^{\text{min}}$ for heterogeneous systems with
random heterogeneous binding rates drawn from a (truncated) Gaussian with
coefficient of variation 50% [cf. App. Sec. J.2] or structures with explicit
boundaries [cf. App. Sec. J.3], respectively. Squares and circles show
$T_{90}^{\text{min}}$ for the heterogeneous system with identical rates and
periodic boundaries (equivalent to the homogeneous system, cf. Fig. 3(b)) as
well as for a homogeneous system under annealing (i.e. with a time-dependent
detachment rate $\delta_{1}(t)\sim m(t)$) [cf. App. Sec. H]. Data points were
obtained by averaging 10-20 independent runs of the stochastic simulation
performed with $N=500$ particles per species (heterogeneous case) or $N=500S$
particles (homogeneous case). While the assembly time varies for the different
model variants, the measured time complexity exponents are largely invariant.

First, we consider a system with random heterogeneous binding rates. Clearly,
for distinct species, the assumption of identical binding rates for all
species is an idealization. More realistically, the rates will vary to some
extent. We, therefore, simulated a system with random heterogeneous nucleation
and attachment rates for the different species, drawn independently from a
(truncated) normal distribution with a coefficient of variation of $50\%$. The
normal distribution was truncated for values below $20\%$ of the mean to
ensure that individual rates do not become negative or extremely small. We
measure the assembly time accordingly in units of $(C\nu_{\text{avg}})^{-1}$,
where $\nu_{\text{avg}}$ is the average attachment rate of all species. For
each run, the binding rates were chosen independently, and the assembly times
were averaged over 10-20 independent runs. The measured time complexity
exponent is identical to the one of the system with homogeneous rates,
demonstrating the robustness of the scaling behavior. However, the time
efficiency, in general, is reduced as a consequence of heterogeneous binding
rates. We attribute this to the fact that the ratio between the average
effective nucleation and effective growth rate increases with the coefficient
of variation of the distribution of the rate constants (since the average
nucleation and growth rate correspond to moments of different order of the
distribution). Thus, increasing the coefficient of variation of the rate
constants is equivalent to increasing the dimerization barrier
$\tilde{\mu}=\mu/\nu$, which, according to Eq. 80, increases
$T_{90}^{\text{min}}$.

###  J.3 Structures with explicit boundaries

Heterogeneous systems furthermore offer an opportunity to test the impact of
the boundary conditions of the clusters. In the homogeneous case, we
considered the self-assembly of closed structures, implemented by periodic
boundary conditions of the clusters. The geometry of the clusters thereby
limits their further growth. In a heterogeneous system, structure growth is
automatically limited if the species required for further growth are missing.
Therefore, we think that heterogeneous systems realized, e.g., with DNA bricks
as used in references [40, 50] provide a valuable testing ground for our
scaling results. However, the emerging structures differ in their boundary
conditions from the systems we considered in the main text (periodic vs.
explicit boundaries)121212Note also that explicit boundaries violate the
equivalence between homogeneous and heterogeneous systems [cf. App. Sec. J.1]
due to the distinct properties of the boundary species.. It is, therefore,
important to understand whether and how the boundary conditions influence the
assembly time. Simulating clusters with explicit boundaries, we find that the
assembly time slightly increases, but its scaling behavior as a function of
the target structure size remains invariant [cf. Fig. 20]. Statistically,
i.e., when averaging over many simultaneously growing clusters, explicit
boundaries merely reduce the effective average growth rate of clusters since
clusters stop growing once they reach the boundary. Hence, changing the
boundary conditions is effectively equivalent to changing the binding rate
$\nu$, which affects the assembly time but not its scaling behavior. This
suggests that our scaling results are generally robust with respect to the
boundary conditions limiting structure growth.

###  J.4 Heterogeneous detachment rates

The question arises whether, by choosing the reaction rates for the
heterogeneous system in a concerted way, the assembly time can be reduced.
Since nucleation is the rate-limiting process, a simple idea to achieve this
is to increase the effective rate of seed formation by making the bonds
between four specified neighboring species very strong. Specifically, to test
this scenario, we set the detachment rates for the four bonds between species
$1$, $2$, $L+1$, and $L+2$ to zero in the simulation. The intention is that
the specified species will quickly form the required seeds that can
subsequently grow into complete structures. However, the simulation reveals
that the minimal assembly time increases significantly as a result of this
modification of the reaction rates. In fact, to achieve a yield of 90%, the
uniform detachment rate $\delta_{1}$ of the remaining bonds had to be chosen
significantly larger than in the homogeneous case, making the simulation
extremely inefficient. We therefore had to simulate the scenario with a
reduced nucleation rate $\mu/\nu=0.1$ (see Sec. D.1) and correspondingly
plotted the assembly time in units of $(C\nu)^{-1}\,\mu/\nu$ (assuming a
complementary time complexity exponent of 1 as in the homogeneous case [cf.
App. Sec. D.1]). Figure 21 indeed shows a strong increase of the minimal
assembly time as compared to the system with identical rates (purple line in
Fig. 21). The reason for this increase is the same as in the scenario of
seeding discussed in App. Sec. G: By enhancing the nucleation rate of four
selected species, $\delta_{1}$ has to be chosen much larger for all remaining
species to avoid spurious nucleation. This drastically reduces the effective
growth rate and hence the overall time efficiency. In fact, if the time scale
of nucleation of the four species becomes negligible against the overall
assembly time, there is effectively no difference from putting a corresponding
number of seeds into the system. Therefore, in the limit of large $S$, the
assembly time is approximately the same as for the homogeneous seeded system;
compare Fig. 17(c) and dash-dotted line in Fig. 21. Note that the minimal
assembly time will not increase if hexagonal instead of square-shaped monomers
are used because the growth rate of hexagonal monomers is independent of
$\tilde{\delta}_{1}$.

###  J.5 Heterogeneous concentrations

Figure 21: Heterogeneous concentrations and heterogeneous detachment rates.
Dark blue line with crosses shows the minimal assembly time of a heterogeneous
system in which species $1,2,L+1$ and $L+2$ [cf. Fig. 19] are provided in
sixfold higher concentration compared to the other species. Bright blue line
with diamond markers shows the minimal assembly time when the detachment rates
for bonds between species $1,2,L+1$ and $L+2$ are set to zero. In both cases,
data points were obtained by averaging 10 independent runs of the stochastic
simulation performed with $N=500$ particles per species and a dimerization
barrier $\tilde{\mu}:=\mu/\nu=0.1$ to increase the efficiency of the
simulations [cf. App. Sec. D.1] (note that the y-axis is scaled in units of
$\tilde{\mu}$ accordingly). For comparison, the blue dash-dotted line shows
$T_{90}^{\text{min}}$ for the homogeneous system under seeding determined with
the effective model (cf. Fig. G), and the purple line with square markers
shows $T_{90}^{\text{min}}$ for the heterogeneous system with identical rates
and concentrations (equivalent to the homogeneous system, cf. Fig. 3(b)).

In ref. [25], the authors describe how the yield in heterogeneous systems can
be increased by providing a few neighboring species in significantly larger
concentrations compared to the other species. In this way, the authors argued
that specific assembly pathways are favored, leading to higher yields as
resources are used more effectively. Here, we want to analyze the time
efficiency of this strategy. To this end, we set the initial particle number
of species $1$, $2$, $L+1$, and $L+2$ to $6N$ while all other species are
provided in $N$ copies per species. In order not to restrict the achievable
yield, we do not count the additional $20N$ monomers when calculating the
yield. Nevertheless, we find that to achieve an elevated yield of 90%, the
detachment rate must be chosen very large, causing the simulation to become
extremely inefficient. We therefore again simulate this scenario with a
reduced dimerization rate $\mu/\nu=0.1$ (and plot the assembly time in units
of $(C\nu)^{-1}\mu/\nu$) as in the last paragraph. As for heterogeneous
detachment rates, providing four species in higher concentrations increases
the effective nucleation rate. Consequently, the assembly time increases
drastically since $\tilde{\delta}_{1}$ must be chosen large to suppress
spurious nucleation [cf. Fig. 21]. Specifically, the rate of nucleation of
species $1$, $2$, $L+1$, $L+2$ is $\bar{\mu}\,(6N)^{4}$ while the nucleation
rate of each of the $S-4$ remaining nucleus species is $\bar{\mu}N^{4}$.
Hence, providing species in heterogeneous concentrations has a similar effect
as scaling the (homogeneous) dimerization rate parameter
$\tilde{\mu}:=\mu/\nu$ as $\tilde{\mu}\to\tilde{\mu}\,(1+6^{4}/S)$. With Eq.
(80), this suggests that the assembly time scales roughly as
$T_{90}^{\text{min}}\sim 6^{4}+S$. Hence, as long as $S$ is small compared to
$6^{4}$, we expect the minimal assembly time to increase only very slowly with
$S$. Indeed, Fig. 21 shows that the assembly time transiently becomes almost
constant as a function of $S$. In the limit of large $S$, we expect that the
effect of heterogeneous concentrations declines more and more (provided the
concentrations remain constant) and the assembly time ultimately behaves
similarly as in the homogeneous case (purple line in Fig. 21). Hence, we find
that neither the yield nor the time efficiency can be enhanced through
heterogeneous concentrations since heterogeneous concentrations increase the
effective nucleation rate, which reduces the efficiency according to Eq. (80).
In their work [25], Murugan et al. considered a system of one-dimensional
rings that self-assemble irreversibly. Furthermore, they work in a regime
where the yield is very low. We expect that their finding of a yield increase
due to heterogeneous concentrations is associated with these system-
specificities and has a lot to do with avoiding stochastic effects that arise
in irreversible systems [53].

###  J.6 Hierarchical heterogeneous self-assembly

We realize that any measures leading to an increase in the effective
nucleation rate are counterproductive in enhancing self-assembly efficiency.
The key to increasing time efficiency instead lies in enhancing the speed of
cluster growth by reducing the attachment order $\gamma$. Hierarchical self-
assembly, as discussed in Sec. VI, offers a great opportunity in this regard,
particularly for heterogeneous systems. For example, we showed that
rectangular building blocks with 6 binding sites that self-assemble ‘on gap’
[cf. Fig. 8 and Sec. VI] are topologically equivalent to hexagons and thus
self-assemble with the same efficiency as hexagonal monomers. In our
heterogeneous system with square-shaped building blocks, this could be
realized by making the bonds between every second pair of species (i.e.,
$1-2$; $3-4$;… and then ($L+2$)-($L+3$); ($L+4$)-($L+5$); etc.) significantly
stronger than the other bonds. This will induce the monomers to form
rectangle-shaped dimers with 6 binding sites, which then assemble into the
final structures with a minimal time complexity exponent $\theta\approx 1/2$.
We simulated this scenario with the effective model [cf. App. Sec. B]
analogous to the hierarchical assembly scenarios in Fig. 9 in the main text.
Thereby, we simulated the dimerization of monomers with rate $\mu=\nu$ and
detachment rate $\delta_{1}^{\text{strong}}=0$ and the the subsequent self-
assembly of the dimers with the same parameters $\sigma=3,\gamma=1$ and
$\omega=1/2$ as in the self-assembly of hexagonal monomers. By tuning the
detachment rate $\tilde{\delta}_{1}^{\text{weak}}$ for the bonds between the
dimers, we obtain the minimal assembly time plotted in Fig. 22 against the
target structure size. Comparing with Fig. 3(b) (or the green dash-dotted line
in Fig. 22), we find that the minimal assembly time is roughly a factor 2
larger than in the case of hexagonal monomers (since the dimers first have to
form in the first assembly step).

Hence, the ability to control each bond individually in a heterogeneous system
offers plentiful additional possibilities to reduce the attachment order via
hierarchical self-assembly and increase the time efficiency relative to a
homogeneous system.

Figure 22: Improving heterogeneous self-assembly: hierarchical self-assembly
and heterogeneous dimerization rates. Blue dash-dotted line shows the minimal
assembly time obtained with the effective model for the hierarchical self-
assembly scenario in which square-shaped monomers first form rectangle-shaped
dimers ($\delta_{1}^{\text{strong}}=0$), which then assemble ‘on gap’ as shown
in Fig. 8. Notice that this hierarchical scenario can only be realized with
heterogeneous building blocks. Dark blue line with square markers shows the
minimal assembly time of a heterogeneous system in which only species 1 and 2
can dimerize with rate $\mu_{12}=\nu$ and the dimerization rates of all other
pairs of species as well as the detachment rates are zero. Simulations were
performed with $N=1000$ particles per species and averaged over 10 independent
runs. Purple line with square markers shows the assembly time in a
heterogeneous system with identical rates for comparison. Furthermore, green
line with star markers and green dash-dotted line show the assembly time of a
homogeneous system with hexagonal monomers (stochastic simulation and
effective model, respectively) copied from Fig. 3(b) for comparison. With
hierarchical self-assembly or heterogeneous detachment rates the required
assembly time can be reduced drastically.

###  J.7 Heterogeneous dimerization rates

Another possibility to increase self-assembly efficiency is by reducing the
nucleation rate. In principle, this can be achieved with heterogeneous
dimerization rates. Specifically, let us assume that only species 1 and 2 can
form a dimer (with rate $\mu_{12}=\nu$), whereas all other pairs of species
are unable to form dimers ($\mu_{ij}=0$ for $i,j\neq 1,2$), but the monomers
of those species can still attach to existing clusters with rate $\nu$. In
nature, such scenarios are quite common. For example, flagellin proteins, the
building blocks of flagellae in E. Coli do not react freely in the cytosol
[1]. Instead, the molecules only attach irreversibly to the end of an existing
flagellum when they are passed through the flagellum [86].

The fact that only species 1 and 2 can dimerize ensures that just the right
number of nuclei will form and thus the detachment rates of all species can be
chosen arbitrarily small. Thereby self-assembly becomes very efficient.
Indeed, simulating this scenario with square monomers and detachment rate
$\delta_{1}=0$ [cf. Fig. 22], we find that roughly the same time efficiency is
achieved as with hexagonal monomers where the detachment rate is chosen
optimally; compare with Fig. 3(b). Roughly the same minimal assembly time is
also achieved by optimizing the dimerization rate in a homogeneous system [cf.
App. Sec. D.1].

Controlling the dimerization rate is very effective and, in the heterogeneous
case, it does not even require fine-tuning of the rate constants. However,
implementing such a scenario with artificial components would be very
challenging since it requires a particular molecular design that causes strong
allosteric (cooperative) binding effects or the use of specialized enzymes. In
contrast, controlling the ratio $\tilde{\delta}_{1}$ between detachment and
attachment rates is simpler as it can be tuned by several experimentally
accessible parameters [cf. discussion in Sec. II].

In conclusion, due to the exact equivalence between homogeneous and
heterogeneous systems in the ideal case, both types of systems behave very
similarly and can be typically related to each other. We found that the
scaling properties of the assembly time in heterogeneous systems are robust to
noise (e.g., random variations of the rate parameters) and to modifications
regarding the boundary conditions. However, the (transient) scaling behavior
is affected by concerted modifications, e.g., of the detachment rates or by
using heterogeneous concentrations. The resulting effects on the assembly time
can qualitatively be well understood on the basis of the results on
homogeneous systems.

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