# Enzymatic Aggregation

P. L. Krapivsky  Department of Physics, Boston University, Boston, MA 02215,
USA  Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA  S.
Redner  Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA

###### Abstract

We introduce an autocatalytic aggregation model in which the rate at which two
clusters merge to form a cluster is controlled by the presence of a third
“enzymatic” cluster whose mass must equal to the mass of one of the reaction
partners. The enzyme is unaffected by the joining event and is available to
either participate in or catalyze subsequent reactions. This model is meant to
mimic the self-replicating reactions that occur in models for the origin of
life. We solve the kinetics of enzymatic aggregation in the model with mass-
independent rates and show that the total cluster density decays as
$t^{-1/3}$, while the density of clusters of any fixed mass decays as
$t^{-2/3}$. These behaviors contrast with the corresponding $t^{-1}$ and
$t^{-2}$ scalings for classic aggregation. We extend our model to mass-
dependent reaction rates, to situations where only “magic” mass clusters can
catalyze reactions, and to include steady monomer input.

##  I Introduction and Model

One of the profound mysteries of the natural world is the origin of life.
Self-replication has been invoked as a starting point to understand how the
complex reactions that underlie living system might arise, see, e.g., Eigen
(1971); Kauffman (1971); Gilbert (1986); Nowak and Ohtsuki (2008); Ohtsuki and
Nowak (2009); Fellermann  _et al._ (2017); Tkachenko and Maslov (2018);
Hordijk (2019); Hilhorst (2020) and references therein. In such processes, the
products of a given reaction serve to catalyze the rate of new products,
which, in turn, can catalyze further reactions, leading to potentially to
complex chemistries.

Various types of random catalytic reaction networks have been proposed and
investigated to predict the emergence of autocatalytic cycles in populations
of diverse reactants with general types of catalytic activity Stadler  _et
al._ (1993); Hanel  _et al._ (2005); Filisetti  _et al._ (2012). The outcome
of from studies such as these is that catalytic activity among a set of
molecules is sufficient to promote the appearance of groups of molecules that
are able to replicate themselves through catalytic reactions.

While the behavior of many of these autocatalytic reactions is extremely rich,
it is often not possible to discern which aspects of the complex chemical
reaction networks that have been studied is truly necessary for the emergence
of self replication. A missing element in these model is analytical
tractability—most of the model that have been considered thus far typically
contain many species and many reaction pathways. These complications make an
analytical solution of such models out of reach. Motivated by this disconnect
between complexity and analytical tractability, we formulate a simple
realization of catalytic kinetics in the framework of irreversible
aggregation. While our model is idealized and incomplete, it might provide a
starting point for analytically determining the kinetics of autocatalytic
reactions.

Figure 1: Schematic of the elemental events in: (a) aggregation and (b)
enzymatic aggregation. In our enzymatic aggregation model, the presence of a
catalyst (red) whose mass matches one of the reactants is required. This
catalyst is unaffected by the reaction itself.

In our _enzymatic_ aggregation model, some fraction of the reactions are
catalytic; namely, these enzymes are unaffected by the joining of two other
reactants and can subsequently participate in or catalyze further reactions.
That is, the rate at which a cluster of mass $i$, an $i$-mer, and a $j$-mer
join requires the presence of either another $i$-mer or another $j$-mer to
catalyze the reaction. This constraint that the catalyst mass is equal to one
of the masses of the two clusters that are to be joined is the feature that
makes our model analytically tractable. We may represent this reaction as

$\\{i\\}\oplus[i]\oplus[j]\overset{K_{ij}}{\longrightarrow}\\{i\\}+[i+j]\,.$

Here, reactants inside square brackets undergo aggregation, while the reactant
within the braces is unaffected by the reaction. In the mean-field or perfect-
mixing limit where all reactant concentrations are spatially uniform, we will
show that this reaction has slower kinetics compared to that in conventional
aggregation. We will extend our theory to treat the case where the enzymatic
efficiency of the catalyst is a function of its mass. We will also treat the
situations where only clusters of certain “magic” masses can catalyze
reactions and where the reaction is augmented by a steady monomer input.

##  II Classical Aggregation

To set the stage for enzymatic aggregation, we review some essential features
of classical aggregation. Aggregation is a ubiquitous kinetic process in which
a population of clusters continuously merge to form clusters of ever-
increasing mass Leyvraz (2003); Krapivsky  _et al._ (2010). This process
underlies many physical phenomena, such as blood clotting, gravitational
accretion of gas clouds into stars and planets, and gelation. In aggregation,
two clusters of mass $i$ and $j$ join irreversibly at rate $K_{ij}$ to form a
cluster of mass $i+j$ according to

$[i]\oplus[j]\overset{K_{ij}}{\longrightarrow}[i+j]\,.$

The basic observables are the densities of clusters of mass $k$ at time $t$.
These $k$-mer densities depend in an essential way on the reaction rates
$K_{ij}$. Much effort has been devoted to determining these cluster densities
in the perfectly-mixed or mean-field limit, where the shape and spatial
location of the clusters are ignored and the only degree of freedom for each
cluster is its mass Ziff (1980); van Dongen and Ernst (1985a, b, 1988).

Let $c_{k}(t)$ denote the density of $k$-mers at time $t$. In the simplest
aggregation process with mass-independent reaction rates, the Smoluchowski
equations Smoluchowski (1916, 1917); Chandrasekhar (1943) describing the
evolution of densities in the mean-field limit are particularly simple:

$\frac{dc_{k}}{dt}=\sum_{i+j=k}c_{i}c_{j}-2c_{k}c$ (1)

where

$c(t)\equiv\sum_{k\geq 1}c_{k}(t)$ (2)

is the total cluster density. Summing Eqs. (1) over all $k$, one finds that
the total cluster density satisfies $\frac{dc}{dt}=-c^{2}$, with solution
$c(t)=(1+t)^{-1}$.

For the monodisperse initial condition $c_{k}(t\\!=\\!0)=\delta_{k,1}$, the
solution to (1) is

$c_{k}(t)=\frac{t^{k-1}}{(1+t)^{k+1}}\,.$ (3)

In the scaling limit of $t\to\infty$ and $k\to\infty$, with the scaled mass
$kc(t)$ kept finite, the mass distribution (3) has the scaling form

$c_{k}(t)\simeq c^{2}F(ck)\,,$ (4)

with scaled mass distribution $F(x)=e^{-x}$. We will compare these classic
results with the corresponding behavior of the enzymatic aggregation in the
following section.

##  III Enzymatic Aggregation

Suppose now that merging of two clusters can occur only in the presence of a
catalyst according to the reaction
$\\{i\\}\oplus[i]\oplus[j]\longrightarrow\\{i\\}+[i+j]$. We initially assume
that the rate of each of these reactions is independent of the reactant masses
and we set all reaction rates to 1. The time evolution of the cluster
densities now obey

$\displaystyle\frac{dc_{k}}{dt}$
$\displaystyle=\frac{1}{2}\sum_{i+j=k}c_{i}c_{j}(c_{i}+c_{j})-c_{k}\sum_{i\geq
1}c_{i}(c_{i}+c_{k})$
$\displaystyle=\sum_{i+j=k}c_{i}^{2}c_{j}-c_{k}^{2}\,c-c_{k}Q\,,$ (5)

involving, in addition to the total cluster density (2), the quadratic moment
of the mass distribution

$Q(t)\equiv\sum_{k\geq 1}c_{k}(t)^{2}$ (6)

The presence of this quadratic moment renders the governing equations (III)
intractable. To understand why, we recall that one can solve the rate
equations (1) for classical aggregation recursively in terms of the known
cluster density. In enzymatic aggregation, the governing equations (III) are
also recurrent, but they require knowledge of both $c(t)$ _and_ $Q(t)$. Using
Eqs. (III), these quantities obey

$\displaystyle\frac{dc}{dt}$ $\displaystyle=-cQ,$ (7a)
$\displaystyle\frac{dQ}{dt}$ $\displaystyle=2\sum_{i\geq 1}\sum_{j\geq
1}c_{i}^{2}c_{j}c_{i+j}-2c\sum_{k\geq 1}c_{k}^{3}-2Q^{2}\,.$ (7b)

Equation (7b) involves moments higher than quadratic, so Eqs. (7a)–(7b) do not
form a closed system and hence are not solvable.

As an alternative, we specialize to the long-time limit, where the cluster
mass distribution should have the scaling behavior (4). We will see that Eqs.
(7a)–(7b) can be solved in this scaling limit. For consistency with (2) and
with mass conservation, $\sum_{k\geq 1}k\,c_{k}=1$, the scaling function
$F(x)$ must satisfy the conditions

$\int_{0}^{\infty}dx\,F(x)=1\quad\text{and}\quad\int_{0}^{\infty}dx\,x\,F(x)=1\,.$
(8)

By substituting the scaling form $c_{k}(t)\simeq c^{2}F(ck)$ into (6) we
obtain

$Q=Ac^{3},\qquad A=\int_{0}^{\infty}dx\,F^{2}(x)\,.$ (9)

Finally, we substitute (9) into (7a) and integrate to obtain the cluster
density in the long time limit:

$c=(3At)^{-{1}/{3}}\,.$ (10)

Now that we have found the cluster density, let us determine the monomer
density. Its governing equation is

$\frac{dc_{1}}{dt}=-c_{1}^{2}c-c_{1}Q=-c_{1}^{2}c-Ac_{1}c^{3}\,.$ (11)

Dividing (11) by $\frac{dc}{dt}=-cQ=-Ac^{4}$ yields

$\frac{dc_{1}}{dc}=\frac{c_{1}^{2}+Ac_{1}c^{2}}{Ac^{3}}$ (12)

The behavior in classical aggregation, $c_{1}\simeq c^{2}$, suggests a similar
algebraic scaling, $c_{1}\simeq Bc^{\beta}$, in enzyme aggregation.
Substituting this asymptotic into (12) gives

$(\beta-1)Bc^{\beta-1}\simeq\frac{B^{2}}{A}\,c^{2\beta-3}$

Two possibilities emerge: $\beta=1$ when the left-hand side dominates, and
$\beta=2$ when both terms are of the same order, and we further deduce $B=A$.
A more accurate analysis based on plugging $c_{1}\simeq Bc$ into (11) leads to
inconsistent results, and we thus conclude that

$c_{1}=Ac^{2}\,.$ (13)

This equation for $c_{1}$ is consistent with the scaling form (4) only if

$F(0)=A=\int_{0}^{\infty}dx\,F^{2}(x)\,,$ (14)

Collecting (9), (10) and (13) we arrive at

$c\simeq\frac{1}{(3At)^{1/3}}\,\quad
c_{1}\simeq\frac{A^{1/3}}{(3t)^{2/3}}\,,\qquad Q\simeq\frac{1}{3t}\,.$ (15)

Therefore we have determined the asymptotic behavior of the quadratic moment,
while the densities of monomers and clusters are expressed in terms of the
unknown amplitude $A$.

One can, in principle, continue this analysis to determine the $k$-mer
densities one by one. However, it is more expedient to invoke scaling. Thus we
substitute the scaling form (4) into the rate equations (III), from which we
can directly obtain the entire scaled mass distribution. After some
straightforward algebra, the rate equations transform to the integro-
differential equation

$F^{2}-A\left[x\,\frac{dF}{dx}+F\right]=\int_{0}^{x}dy\,F^{2}(y)F(x-y)\,.$
(16a) Notice that for $x=0$, the condition (14) that $F(0)=A$, is
automatically satisfied.

The transformation $\xi=Ax$ and $F(x)=A\Phi(\xi)$ recasts (16a) into

$\Phi^{2}-\Phi-\xi\,\Phi^{\prime}=\int_{0}^{\xi}d\eta\,\Phi^{2}(\eta)\Phi(\xi-\eta)$
(16b)

where the prime denotes differentiation with respect to $\xi$. In these new
variables, Eq. (14) becomes

$1=\Phi(0)=\int_{0}^{\infty}d\xi\,\Phi^{2}(\xi)\,.$ (17)

The scaled mass distribution approaches to $\Phi(0)=1$ in the small-mass limit
of $\xi\to 0$. To find the next correction, we write $\Phi=1-\epsilon$ with
$\epsilon\ll 1$, and substitute this ansatz into (16b) to find

$\xi\epsilon^{\prime}-\epsilon=\xi$ (18)

to leading order. The solution is $\epsilon=\xi(\ln\xi+a)$, with $a$ some
constant. Thus we conclude that the scaled mass distribution has the small-
mass tail

$\Phi=1-\xi(\ln\xi+a)+\ldots$ (19)

as $\xi\to 0$. This small-mass behavior suggests that the scaled mass
distribution is more complicated than the scaled distribution $\Phi=e^{-\xi}$
in classical aggregation. While we have found the small-mass tail of the
scaled mass distribution, we have been unable to determine the large-mass
tail.

We can extend the above approach to treat enzymatic aggregation in which the
reaction rate depends on the mass of the enzyme. Such a generalization
accounts for the possibility that the efficacy of the enzyme depends on
geometrical constraints; for example, if the enzyme serves as a physical
scaffold upon which the reaction takes place and a larger-area scaffold is
more efficient.

We write this mass-dependent enzymatic reaction as

$\\{i\\}\oplus[i]\oplus[j]\xrightarrow{\text{rate}~{}E_{i}}\\{i\\}\oplus[i+j]\,,$
(20)

with mass-dependent reaction rates $E_{i}$. A natural situation is when the
reaction rate is algebraic in the mass: $E_{i}=i^{\nu}$. On the physical
grounds, the reaction rate cannot grow faster than linearly in the mass, i.e.,
the exponent should satisfy $\nu\leq 1$. The $\nu>1$ range is not merely
questionable physically, but the emergent behavior is mathematically
pathological. In the infinite system the process completes in zero time. This
phenomenon is known as instantaneous gelation and its has been studied in the
context of classical aggregation van Dongen (1987); Malyshkin and Goodman
(2001); Ball  _et al._ (2011). This instantaneous gelation also occurs in
enzyme aggregation when $\nu>1$. When $\nu<1$, enzymatic aggregation admits a
scaling treatment parallel to that given in this section for the constant-rate
model ($\nu=0$). Details are given in Appendix A.

##  IV Enzymes with magic masses

In many catalytic reactions, only a small subset of the reactants are
catalytic. Since the cluster mass is the only parameter in our modeling, the
spectrum of masses for the catalytic reactants should be sparse, so that
catalysts are rare. Here we treat an extreme model where only monomers are
catalytic. In Appendix B we briefly consider the model where clusters with
‘magic’ masses $2^{n}$ are catalytic.

###  IV.1 Only monomers are catalytic

If only monomers are catalytic, the enzymatic aggregation reaction is
$\\{1\\}\oplus[1]\oplus[j]\xrightarrow{\text{rate}~{}1}\\{1\\}\oplus[1+j]$.
The class of models (20) with algebraic reaction rates $E_{i}=i^{\nu}$ reduces
to the model where only monomers are catalytic in the $\nu\to-\infty$ limit.

The cluster densities now evolve according to

$\displaystyle\frac{dc_{1}}{dt}$ $\displaystyle=-c_{1}^{2}(c+c_{1})\,,$ (21a)
$\displaystyle\frac{dc_{k}}{dt}$ $\displaystyle=c_{1}^{2}(c_{k-1}-c_{k}),\quad
k\geq 2\,.$ (21b)

Essentially the same equations describe the phenomenon of submonolayer
islanding Brilliantov and Krapivsky (1991); Krapivsky  _et al._ (2010). In the
islanding reaction, monomers adsorb and diffuse freely on a surface. When two
monomers meet or a monomer meets a cluster of mass $k\geq 2$ merging takes
place and all clusters of mass $k\geq 2$ are immobile Brilliantov and
Krapivsky (1991); Krapivsky  _et al._ (2010). The only difference between
submonolayer islanding and enzymatic aggregation with catalytic monomers is
the factor $c_{1}^{2}$ instead of $c_{1}$ on the right-hand sides of Eqs.
(21).

By introducing the modified time variable

$\tau=\int_{0}^{t}dt^{\prime}\,c_{1}^{2}(t^{\prime})$ (22)

we linearize (21) and obtain

$\displaystyle\begin{split}\frac{dc_{1}}{d\tau}&=-c-c_{1}\,,\\\
\frac{dc_{k}}{d\tau}&=c_{k-1}-c_{k},\quad k\geq 2\,,\\\
\frac{dc}{d\tau}&=-c\,.\end{split}$ (23)

The last equation is not independent, as it is obtained by summing the rate
equations for all the $c_{k}$. Solving this last equation gives

$c(\tau)=e^{-\tau}\,.$ (24a) Then we solve the equation for $c_{1}$ and find
$c_{1}(\tau)=(1-\tau)e^{-\tau}\,.$ (24b) Finally, we solve the rate equations
for $c_{k}$ for $k\geq 2$ recursively and find
$c_{k}(\tau)=\left(\frac{\tau^{k-1}}{(k-1)!}-\frac{\tau^{k}}{k!}\right)e^{-\tau}\,,$
(24c)

for the monodisperse initial condition.

The time evolution ends at $\tau_{\text{max}}=1$, which corresponds to
$t=\infty$. At this moment the density of monomers vanishes and the reaction
freezes. The $k$-mer densities at this final time are

$c_{k}(t\\!=\\!\infty)=\frac{k-1}{k!}\,e^{-1},\quad c(t\\!=\\!\infty)=e^{-1}$
(25)

While the dependence of the densities in Eqs. (24) in terms of the modified
time $\tau$ is the same as in submonolayer islanding Brilliantov and Krapivsky
(1991); Krapivsky  _et al._ (2010), the dependence on the physical time is
different. To determine the dependence on physical time, we exploit the fact
that the monomer density vanishes, $c_{1}(t)\to 0$, as $t\to\infty$. Using
this fact, together with $c(\infty)=e^{-1}$, we simplify (21a) to

$\displaystyle\frac{dc_{1}}{dt}\simeq-\frac{c_{1}^{2}}{e}\,,$

from which

$c_{1}(t)\simeq\frac{e}{t}\,.$ (26)

In contrast for submonolayer islanding, the density of monomers decays
exponentially with time, $c_{1}\sim e^{-t/e}$. The asymptotic approach of all
the other $k$-mer densities to their final values is also algebraic:

$c_{k}(t)-c_{k}(\infty)\simeq-\frac{k^{2}-3k+1}{k!}\,\frac{e}{t}\,.$ (27)

While enzymatic aggregation with an initial population of catalytic monomers
is solvable, it has the obvious limitation that all reactions terminate in a
finite time. For this reaction with only monomers being catalytic to continue
ad infinitum, it is necessary to postulate the existence of a source of
monomers. This is the subject of the next section.

###  IV.2 Input of monomers

We now extend the model (21) and postulate that monomeric catalysts are
injected at a constant rate. Indeed, in mimicking the origin of life it is
natural to consider open systems. Clusters can spontaneously arise via
external processes which we do not describe; instead, we merely account for
them as a steady input of catalysts. One may anticipate that the balance
between input and the increase of mass due to aggregation manifests itself by
driving the system to a steady state. This steady-state behavior often arises
in classical aggregation (see e.g., Field and Saslaw (1965); Hayakawa (1987);
Krapivsky and Connaughton (2012)). However, the outcome in the present case is
continuous evolution, as we now demonstrate.

With monomer input, we add the source to Eq. (21a):

$\displaystyle\frac{dc_{1}}{dt}=-c_{1}^{2}c-c_{1}^{3}+J$ (28)

where $J$ is the source strength. The $k$-mer densities with $k\geq 2$ again
satisfy Eqs. (21b). The time dependence of the density of non-enzymatic
clusters,

$N=\sum_{k\geq 2}c_{k}\,,$ (29)

can be found by summing Eqs. (21b) for $k\geq 2$ and gives

$\displaystyle\frac{dN}{dt}=c_{1}^{3}$ (30)

It is also useful to rewrite (28) as

$\displaystyle\frac{dc_{1}}{dt}=-c_{1}^{2}N-2c_{1}^{3}+J\,.$ (31)

Equations (30)–(31) do not admit an exact solution. However, we can determine
the asymptotic behavior. The analysis is parallel to that given for the case
of submonolayer island growth Brilliantov and Krapivsky (1991); Blackman and
Wielding (1991); Krapivsky  _et al._ (2010). It turns out (which can be
justified a posteriori) that $c_{1}\to 0$ and $N\to\infty$. Hence from (31) we
obtain $c_{1}^{2}N\simeq J$, so that (30) has the asymptotic form
$\frac{dN}{dt}=\left(\frac{J}{N}\right)^{3/2}$ leading to

$\displaystyle c\simeq N\simeq J^{3/5}\left(\frac{5t}{2}\right)^{2/5},\qquad
c_{1}\simeq J^{1/5}\left(\frac{5t}{2}\right)^{-1/5}\,.$ (32)

By substituting $c_{1}\simeq(2J/5t)^{1/5}$ into (22), we express the modified
time in terms of the physical time and thereby express the density in terms of
the modified time variable:

$\displaystyle c_{1}\simeq\left(\frac{2J}{3\tau}\right)^{1/3}\,.$ (33)

Next we solve $\frac{\partial
c_{k}}{\partial\tau}=c_{k-1}-c_{k}\simeq-\frac{\partial c_{k}}{\partial k}$.
The solution to this wave equation is simply $c_{k}\simeq c_{1}(\tau-k)$.
Therefore

$\displaystyle c_{k}\simeq\left(\frac{2J}{3(\tau-k)}\right)^{1/3}\,,$ (34)

which is valid when $\tau-k\gg 1$.

##  V Summary

We introduced an aggregation in which the reaction requires the presence of
catalysts. These catalysts can both facilitate the aggregation process and
they can also directly participate in the aggregation reaction. While we
vaguely have in mind the self-replicating reactions that occur in models for
the origin of life, our modeling is more naive in character and is focused on
devising a set of reactions that is analytically tractable. By imposing the
constraint that the mass of the catalyst equals that of either of the
participant in the aggregation reaction, we have devised a process that is
analytically tractable. For the case where the the reaction starts with a
population of monomers, we solved the the $k$-mer density in the scaling limit
and found that $c_{k}$ asymptotically decays as $t^{-2/3}$, while the total
cluster density decays as $t^{-1/3}$.

We also extended our approach to deal with open systems. This feature of input
of elemental reactants is a necessary ingredient to devise models of
artificial life. In our modeling, we treated the situation were only monomers
are catalytic and they are injected into the system in order to sustain a
continuously evolving set of reactions. For this process, we can again solve
for the kinetics of the reaction by using classical tools of non-equilibrium
statistical physics. Here we found that the cluster density continuously
evolves, rather than reaching a steady state, with $c(t)$ growing with time as
$t^{2/5}$, while the $k$-mer densities for fixed $k$ all decay with time as
$t^{-1/5}$.

We thank Steen Rasmussen and John Straub for helpful conversations. This work
has been partially supported by the National Science Foundation under Grant
No. DMR-1910736 and by the Santa Fe Institute.

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##  Appendix A Algebraic merging rates

For the enzymatic reaction

$\\{i\\}\oplus[i]\oplus[j]\xrightarrow{\text{rate}~{}E_{i}}\\{i\\}\oplus[i+j]\,,$
(35)

with the reaction rate $E_{i}=i^{\nu}$, the $k$-mer densities obey

$\frac{dc_{k}}{dt}=\sum_{i+j=k}i^{\nu}c_{i}^{2}c_{j}-kc_{k}^{2}\,c-c_{k}Q_{\nu}\,,$
(36)

with

$Q_{\nu}=\sum_{k\geq 1}k^{\nu}c_{k}^{2}\,.$ (37)

Similarly, the cluster density evolves according to

$\displaystyle\frac{dc}{dt}=-c\,Q_{\nu}\,.$ (38)

Combining (37) with the scaling form (4) we obtain the analog of Eq. (9):

$Q_{\nu}=Ac^{3-\nu},\quad A_{\nu}=\int_{0}^{\infty}dx\,x^{\nu}\,F^{2}(x)\,.$
(39)

Substituting (39) into (38) and integrating, we obtain, for the density of
clusters,

$c=[(3-\nu)A_{\nu}t]^{-1/(3-\nu)}$ (40)

in the long-time limit. Substituting (40) into (39) we then find the
asymptotic behavior

$Q_{\nu}=\frac{1}{(3-\nu)t}\,.$ (41)

The monomer density satisfies

$\frac{dc_{1}}{dt}=-c_{1}^{2}\,c-c_{1}\,Q_{\nu}\,,$ (42)

from which

$c_{1}=(1-\nu)A_{\nu}c^{2-\nu}\,,$ (43)

with $c$ given by Eq. (40). Equation (43) is compatible with the scaling
prediction $c_{1}=c^{2}F(x)$ if

$F(x)=\frac{(1-\nu)A_{\nu}}{x^{\nu}}\qquad\text{as}\quad x\to 0\,.$ (44)

We can now obtain the governing equation for the scaled mass density $F(x)$ by
substituting the scaling form $c_{k}(t)\simeq c^{2}F(ck)$ into (36) to give
the analog of Eq. (16a):

$x^{\nu}F^{2}-A\left[x\,\frac{dF}{dx}+F\right]=\int_{0}^{x}dy\,y^{\nu}F^{2}(y)F(x-y)\,.$
(45)

As in the case of Eq. (16a), the full equation is not tractable, but it is
possible to extract partial information about the scaling functions in the
limits of small- and large-$x$.

##  Appendix B Only clusters of mass $2^{n}$ are catalytic

Here we study the process in which only reactants with ‘magic’ masses $2^{n}$
are catalytic. This leads to the following generalization of the reaction
(III)

$\\{2^{i}\\}\oplus[2^{i}]\oplus[j]\xrightarrow{\text{rate}~{}1}\\{2^{i}\\}\oplus[2^{i}+j]\,.$

The time evolution of cluster densities with non-magic masses, $k\neq 2^{n}$,
obey

$\frac{dc_{k}}{dt}=\sum_{2^{i}+j=k}b_{i}^{2}c_{j}-c_{k}Q\,,$ (46a) where
$b_{i}\equiv c_{2^{i}}$ are the densities of clusters with magic masses, while
the density of magic-mass clusters obey
$\frac{db_{i}}{dt}=\sum_{2^{\ell}+m=2^{i}}b_{\ell}^{2}c_{m}-b_{i}^{2}c-b_{i}Q\,.$
(46b)

Here, we still denote the cluster density by $c$, while $Q$ is now the
quadratic moment of the mass distribution of magic-mass clusters

$Q=\sum_{n\geq 0}b_{n}^{2}\,.$ (47)

With this definition of catalytic clusters, the first of Eqs. (7a) still
remains valid

$\displaystyle\frac{dc}{dt}=-cQ\,.$ (48)

The validity of scaling is questionable. To appreciate this assertion,
consider the evolution of the densities of small-mass clusters. For $k=1,2,3$
we obtain

$\displaystyle\begin{split}\frac{dc_{1}}{dt}&=-c_{1}^{2}c-c_{1}Q\,,\\\
\frac{dc_{2}}{dt}&=c_{1}^{3}-c_{2}^{2}c-c_{2}Q\,,\\\
\frac{dc_{3}}{dt}&=c_{1}^{2}c_{2}+c_{2}^{2}c_{1}-c_{3}Q\,.\end{split}$ (49)

The asymptotic behavior seeminglyis

$\displaystyle\begin{split}\frac{dc_{1}}{dt}&\simeq-c_{1}^{2}c\,,\\\
\frac{dc_{2}}{dt}&\simeq-c_{2}^{2}c\,,\\\ \frac{dc_{3}}{dt}&\simeq-
c_{3}Q\,.\end{split}$ (50)

The decay of the densities of magic clusters $c_{1}$ and $c_{2}$ is apparently
qualitatively faster than the decay of $c_{3}$. This apparently different
temporal behaviors for small-mass clusters indicates that there no longer is a
scaling description for the cluster-mass distribution.

##  Appendix C Catalytic monomers with mass-dependent reaction rates

The reaction scheme for this process is

$\\{1\\}\oplus[1]\oplus[j]\xrightarrow{\text{rate}~{}j}\\{1\\}\oplus[1+j]\,.$

In contrast to models we considered previously when the rate could depend on
the mass of the catalyst, we now assume that the rate depends on the mass of
the reactant. the enzyme is impossible since all enzymes are monomers.) The
cluster densities now obey

$\displaystyle\frac{dc_{1}}{dt}$ $\displaystyle=-c_{1}^{2}(1+c_{1})\,,$ (51a)
$\displaystyle\frac{dc_{k}}{dt}$
$\displaystyle=c_{1}^{2}[(k-1)c_{k-1}-kc_{k}],\quad k\geq 2\,.$ (51b)

In terms of the modified time (22) we rewrite (51) as

$\displaystyle\frac{dc_{1}}{d\tau}$ $\displaystyle=-1-c_{1}\,,$ (52a)
$\displaystyle\frac{dc_{k}}{d\tau}$ $\displaystyle=(k-1)c_{k-1}-kc_{k},\quad
k\geq 2\,.$ (52b)

Solving these equations recursively subject to the monodisperse initial
condition yields Brilliantov and Krapivsky (1991)

$c_{k}(\tau)=e^{-\tau}(1-e^{-\tau})^{k-1}-k^{-1}(1-e^{-\tau})^{k}\,.$ (53)

The density of monomers is

$c_{1}(\tau)=2e^{-\tau}-1\,,$ (54)

and it vanishes at $\tau_{\text{max}}=\ln 2$ corresponding to $t=\infty$. At
this moment the process freezes. The final densities are

$c_{k}(t\\!=\\!\infty)=\frac{k-1}{k}\,2^{-k},\quad c(t\\!=\\!\infty)=1-\ln
2\,.$ (55)

The latter formula follows from $c(\tau)=1-\tau$ that follows by summing all
the $k$-mer densities (53). In terms of the physical time, the asymptotic
approach of the $k$-mer densities to their final values is algebraic. The
leading behavior of these corrections is inversely proportional to time

$c_{k}(t)-c_{k}(\infty)\simeq-\frac{k-3}{2^{k}}\,t^{-1}\,.$ (56)

The only exception is the density of $3-$mers:

$c_{3}(t)-c_{3}(\infty)\simeq-\frac{1}{4t^{2}}\,.$ (57)

We now inject catalysts with rate $J$ to counterbalance freezing. In the
system of equations (51) only Eq. (51a) is affected. In the case of initially
empty system we find

$\displaystyle\frac{dc_{1}}{dt}=-c_{1}^{2}(Jt+c_{1})+J$ (58)

Thus $c_{1}\simeq t^{-1/2}$ as $t\gg 1$.

In the long time limit $c_{k}(t)$ approaches the scaling form

$c_{k}(t)\simeq t^{-1}F(k/t)$ (59a) More precisely, this happens in the
scaling limit $t\to\infty,\quad k\to\infty,\quad\frac{k}{t}=\text{finite}$
(59b)

The scaled mass distribution can be extracted from the exact formula in Ref.
Brilliantov and Krapivsky (1991) (which is valid for all $k\geq 2$):

$c_{k}(\tau)=(k-1)\int_{0}^{\tau}du\,c_{1}(\tau-u)e^{-2u}\,\big{[}1-e^{-u}\big{]}^{k-2}$
(60)

Suppose $k=O(1)$. By substituting $c_{1}\simeq t^{-1/2}$ into (22) and
dropping terms that vanish as $t\to\infty$ we deduce that

$\tau=\ln t+\ln C\,.$ (61)

(Fixing the constant $C$ requires an exact solution of (58) which looks
intractable.) The asymptotic $c_{1}\simeq t^{-1/2}$ becomes
$c_{1}(\tau)\simeq\sqrt{C}\,e^{-\tau/2}$ in the modified time variable $\tau$
when $\tau\gg 1$. Substituting this latter form into (60) and using $U\equiv
e^{-u}$ we deduce

$\displaystyle c_{k}(\tau)$ $\displaystyle\simeq$ $\displaystyle
c_{1}(\tau)(k-1)\int_{0}^{1}dU\,\sqrt{U}\,\big{[}1-U\big{]}^{k-2}$
$\displaystyle=$ $\displaystyle
c_{1}(\tau)\,\frac{\Gamma\big{(}\frac{3}{2}\big{)}\,\Gamma(k)}{\Gamma\big{(}k+\frac{1}{2}\big{)}}\,,$

in the long time limit. When $1\ll k\ll t$, we get

$c_{k}(t)\simeq\sqrt{\frac{\pi}{4kt}}\,,$ (62)

which is consistent with the scaling form (59) and gives the small-mass
asymptotic of the scaled mass distribution

$F(x)\simeq\sqrt{\frac{\pi x}{4}}\quad\text{when}\quad x\to 0\,.$ (63)

To extract the asymptotic behavior of the scaling function for $x\gg 1$, we
simplify the last factor in the integrand in (60). Namely, we write $v=\tau-u$
and obtain

$c_{k}(\tau)\simeq(k-1)e^{-2\tau}\int_{0}^{\tau}dv\,c_{1}(v)\,\exp[2v-ke^{v}e^{-\tau}]\,.$
(64)

Using (61) we find $ke^{-\tau}=x/C$. Hence in the exponent we have
$2v-ke^{v}e^{-\tau}=2v-e^{v}x/C$. Since $x\gg 1$, we only need the small-$v$
behavior. We obtain

$c_{k}(\tau)\simeq
k\,e^{-2\tau-x/C}\int_{0}^{\infty}dv\,c_{1}(v)\,e^{-xv/C}\,.$ (65)

To compute the integral, we need to know the asymptotic behavior of $c_{1}(v)$
when $v\ll 1$, as the integrand vanishes exponentially quickly when $v>1/x$.
From (58) we find $c_{1}\simeq Jt$ when $t\ll 1$. Substituting $c_{1}(t)\simeq
Jt$ into (22) we obtain $\tau\simeq J^{2}t^{3}/3$, and hence
$c_{1}(\tau)\simeq(3J\tau)^{1/3}$, which recasts (65) into

$\displaystyle c_{k}(t)$ $\displaystyle\simeq$
$\displaystyle\frac{k}{(Ct)^{2}}\,e^{-x/C}\int_{0}^{\infty}dv\,(3Jv)^{1/3}\,e^{-xv/C}$
(66) $\displaystyle=$ $\displaystyle
t^{-1}\,\left(\frac{3J}{C^{2}}\right)^{1/3}\,\Gamma\big{(}\tfrac{4}{3}\big{)}\,x^{-1/3}\,e^{-x/C}\,,$

which is compatible with the scaling form (59) and gives the large mass
asymptotic of the scaled mass distribution

$F(x)\simeq\left(\frac{3J}{C^{2}}\right)^{1/3}\,\Gamma\big{(}\tfrac{4}{3}\big{)}\,x^{-1/3}\,e^{-x/C}\quad\text{as}\quad
x\to\infty\,.$ (67)