supplement.tex

# Minimal mechanism for cyclic templating of length-controlled copolymers
under isothermal conditions

Jordan Juritz  Department of Bioengineering and Centre for Synthetic Biology,
Imperial College London, London SW7 2AZ, United Kingdom  Jenny M. Poulton
Foundation for Fundamental Research on Matter (FOM) Institute for Atomic and
Molecular Physics (AMOLF), 1098 XE Amsterdam, The Netherlands  Thomas E.
Ouldridge<EMAIL_ADDRESS>Department of Bioengineering and Centre
for Synthetic Biology, Imperial College London, London SW7 2AZ, United Kingdom

###### Abstract

The production of sequence-specific copolymers using copolymer templates is
fundamental to the synthesis of complex biological molecules and is a
promising framework for the synthesis of synthetic chemical complexes. Unlike
the superficially similar process of self-assembly, however, the development
of synthetic systems that implement templated copying of copolymers  under
constant environmental conditions has been challenging. The main difficulty
has been overcoming product inhibition, or the tendency of products to adhere
strongly to their templates – an effect that gets exponentially stronger with
template length.  We develop coarse-grained models of copolymerisation on a
finite-length template and analyse them through stochastic simulation. We use
these models first to demonstrate that product inhibition prevents reliable
template copying, and then ask how this problem can be overcome to achieve
cyclic production of polymer copies of the right length and sequence in an
autonomous and chemically-driven context. We find that a simple addition to
the model is sufficient to generate far longer polymer products that initially
form on, and then separate from, the template. In this approach, some of the
free energy of polymerisation is diverted into disrupting copy-template bonds
behind the leading edge of the growing copy copolymer. By additionally
weakening the final copy-template bond at the end of the template, the model
predicts that reliable copying with a high yield of full-length, sequence-
matched products is possible over large ranges of parameter space, opening the
way to the engineering of synthetic copying systems  that operate
autonomously.

††preprint: AIP/123-QED

##  I Introduction

Copolymers – polymers formed from two or more types of monomer unit – are
ubiquitous in biology. DNA, RNA and proteins are biological copolymers for
which the information pertaining to the function and/or the structure of the
molecule is encoded into the sequence of the copolymer[1]. There is a huge
diversity of copolymer sequences to be found in biology; the human proteome,
for instance, is comprised of roughly 25,000 different proteins, each
consisting of linear chains assembled from the 20 amino acid residues [2],
with a median length of 375 amino acids [3].

How are these complex molecules formed? In general, it is impossible to encode
and consistently assemble thousands of distinct and essentially arbitrary
macromolecules, each with approximately $375$ units, through the self-assembly
of just 20 types of building blocks[4]. Instead, these copolymers are produced
by copying copolymer templates, such as DNA and mRNA. In templated copying
processes, complementary interactions between monomers and the template direct
the assembly of the product so that distinct, arbitrary copolymer sequences
can be reliably created from a common set of monomers [5]. In extant
organisms, templated copying is supported by a large amount of the cellular
resources and is aided by ensembles of complex molecular machines, such as DNA
polymerases involved in DNA replication [6, 7], RNA polymerases in
transcription [8, 9, 10], and ribosomes in translation [11, 12, 13, 14, 15,
16]. The physics of templated copolymer copying therefore lies at the heart of
the synthesis of the diverse and complex molecules of biology [1].

While recent decades have seen the engineering of remarkably complex self-
assembling systems made of DNA [17, 18, 19, 20, 21], proteins [22, 23], and
combinations of DNA and proteins [24], advances in the field of synthetic
templating have been comparatively slow. This disparity suggests a poor
understanding of templating relative to self-assembly, and represents a missed
opportunity to harness biology’s most important mechanism of producing
chemical complexity - both to generate synthetic sequence-controlled polymers
[25] and in the context of combinatorial molecular discovery [26, 27]. In this
work, we simulate coarse-grained models to investigate a class of templated
copolymerisation reactions that may enable enzyme-free copying of templates
analogously to the copying processes found in biology.

We look to biological systems to set the requirements and conditions of
operation for the non-enzymatic, synthetic template-copying system we wish to
engineer. First, biological copying systems are capable of the accurate
reproduction of copolymer sequences from arbitrary but specific lengths of
copolymer template, whether by replication [6, 7], transcription [8, 9, 10],
or translation [11, 12, 13, 14, 15, 16]. Secondly, each template can be reused
many times to produce many copies; the copying process net generates complex
molecules that persist separately from their template[28, 5, 29, 30]. This
fact is crucial, since templates themselves are necessarily complex molecules
[28, 5, 29, 30]. If each new copy consumes – or remains bound to – a template,
then a new template must be generated from scratch for each copy formed.
Thirdly, natural copying systems are capable of operating in spatio-temporally
constant environmental conditions, without perturbations, exploitable spatial
gradients or other external interventions; they are driven solely by the
chemical free energy of the dissolved building blocks and "fuel" molecules,
such as ATP [31]. We describe copying that does not rely on such external
factors as autonomous.

The phenomenon of "product inhibition" presents a major challenge to achieving
reliable, repeatable production of specific copolymers by templating under
autonomous conditions[32]. As monomers polymerise on the template, the
product’s length increases and the binding free energy between product and
template typically grows linearly. Therefore, the likelihood that a product
spontaneously detaches from the template is exponentially suppressed with
length, inhibiting subsequent copying of the template. As we explore in
Section III.1, products of varying – but short – lengths result when copying
long templates in such a setting. Due to product inhibition, long copies may
fail to be released from the template, sequestering the template and slowing
the copying rate, while short, fragmented copies may be released rapidly.
Therefore product inhibition poses significant challenges for the reliability
of both length-control and template reuse.

Michaelis et al. [32] emphasised that, in non-enzymatic systems restricted to
isothermal operation, strategies that reduce the affinity between products and
the template are required to achieve high turnovers (defined as the final
ratio of products to templates), even for dimeric templates[33]. Enforcing
separation by reducing the affinity between the copy and template, without
compromising the template’s reuse, has also proved challenging. Osuna Gálvez
and Bode recently reported a templated reaction in which the dimerisation of
the reactants and the disruption of both the reactant-template bonds occurred
simultaneously, drastically reducing the product-template affinity[34].
However, this came at the cost of scarring the template, which was was unable
to promote further reactions [34].

Instead, time-varying, non-autonomous environmental conditions are commonly
used to drive cyclic templating and product release. In polymerase chain
reactions, for instance, cycles of heating dissociate the product from the
template and enable further rounds of amplification[35]. Various other time-
varying strategies have also been used to drive enzyme-free, dimeric [36, 37]
and longer template-copying and replicating systems [38, 39, 40, 41, 42].

Others have employed non-chemical energy or exploited spatial gradients to
engineer systems that spontaneously separate copies from the template, thereby
favouring the cyclic assembly and separation of longer copies [43, 35, 44, 45,
46, 47]. In Schulman’s replicator, shear flow was used to fragment the layers
of information-bearing DNA tile-based crystals, revealing more reusable
templating surfaces, which enabled exponential self-replication of the
crystal[43]. Braun and colleagues have exploited the convective currents
generated by thermal gradients, inspired by oceanic thermal vents [48], with
templating and polymerisation occurring in cool regions and separation in hot
regions[35, 44, 45]. Similar environments with spatially and/or temporally
varying conditions are thought to have played an important role in the
emergence of life [49, 50, 51, 52, 53, 54, 55, 56, 57], though these
conditions are not required for the operation of modern biological copying
enzymes.

Recently, a novel DNA strand displacement motif, handhold-mediated strand
displacement, was used to autonomously drive dimer formation directed by
templates [5]. Here, binding between monomers on a template weakens the
connection of one monomer to the template, limiting product inhibition. In
principle, the mechanism would allow efficient dimerisation without template
scarring. However, it is unclear whether such a mechanisms can scale to
overcome product inhibition and reliably produce length- and sequence-
controlled copies on longer templates.

From a theoretical perspective, many authors have considered the permanent
deposition of a copy on a template [58, 59, 60, 61, 62, 63, 64, 65], though
effects of subsequent separation has not been a focus until recently.
Moreover, while these previous works have considered mechanisms of sequence-
control without copy separation, mechanisms for precise length-control as
observed in nature have been neglected.

More recently, models of templated copolymerisation have been used to study
self-replicating molecules under prebiotically plausible conditions [57, 66].
Tupper and Higgs have argued that a rolling-circle mechanism could have
overcome product inhibition to promote non-enzymatic RNA replication in an
RNA-world [66]. In this high-level model, the chemical details of the
directional polymerisation mechanism were not considered, and nor was precise
length control. In a recent theoretical and experimental work on pre-enzymatic
templated assembly, Rosenberger et al. demonstrated that polymer aggregates
with increased length can be generated from short building blocks under
isothermal conditions, though, these polymers were bound up in complexes, not
separated from their templates [57]. Moreover, observing the gradual
lengthening of copolymers on average was the priority, not precise length
control, as required for perfect information copying.

The thermodynamic constraints that copy-template separation place on sequence-
control were considered in Ref. [28], though non-chemical means were invoked
to separate template and copy. In Refs. [29, 30], the thermodynamic and
kinetic consequences of an isothermal mechanism for generating sequence-
controlled copolymers with product separation were considered, but the actual
mechanistic details were implicit. Nonetheless, a key theoretical result of
these works is that separating an accurate copy from its template necessitates
producing a state that is extremely far from equilibrium. This thermodynamic
argument also applies to length control: in the absence of residual
interactions with a template, an ensemble of copolymers of a specific (but
arbitrary and template-selected) length is extraordinarily far from the
equilibrium of a broad, exponential distribution of lengths. It is this
fundamental physical principle that is practically manifest as the challenge
of product inhibition: how can interactions be tuned to allow templates to act
as a catalyst for the production of a specific, far-from-equilibrium product
state, while avoiding a stable equilibrium of copy-template complexes?

In this paper we argue that decoupling the length of a copy from its affinity
for the template, by the channelling the free energy released in
polymerisation to the disruption of copy-template bonds as achieved by HMSD
[5], would be sufficient to overcome product inhibition even on long templates
in an isothermal environment. Further, we hypothesise that additionally
weakening the copy-template connection at the final template site would result
in the selective release of complete copies rather than shorter fragments. We
explore these arguments through simulations of coarse-grained models.

In Section II.1 we introduce a basic model of isothermal templated
copolymerisation with separation. In Section III.1, we demonstrate that
cooperative binding to the template prohibits the release of long polymers
under a basic growth mechanism, demonstrating that it is necessary to have
more complex interactions between copy and template in order to promote
reliable separation in a constant environment. In Section III.2, we propose
and investigate a mechanism which could alleviate the cooperative effect of
product inhibition resulting in an increase in the mean polymer length.
Extending this mechanism in Section III.3, we bias the production of complete
polymers by selectively weakening the final site on the template. Finally, in
Section III.4, we demonstrate that long and accurate copies of the template
can be generated if the correct and incorrect monomers have varying binding
rates.

##  II Model and methods

###  II.1 Model

####  II.1.1 State space

As shown in Fig. 1, we consider a copolymer template
$T=t_{1}t_{2}t_{3}...t_{L_{T}}$, where $t_{i}$ is an integer indicating the
monomer type that can take values ${0,1,...,\alpha_{T}-1}$. $\alpha_{T}$ is
the number of distinct types of monomeric unit in the template. A single
template copolymer is suspended in a large-volume, well-mixed bath of a second
distinct type of monomers that interact with the template. The identity of
these monomers is labelled by an integer that can take values
${0,1,...,\alpha_{C}-1}$. There are $\alpha_{C}$ distinct types of copy
monomer, the $i^{th}$ of which has a concentration $[\textrm{M}]_{i}$. These
monomers can bind onto the template at any unoccupied site on the template. In
this paper, we consider both homogeneous copy-template systems in which there
is a single monomer type, $\alpha_{T}=\alpha_{C}=1$, and also information-
bearing binary systems in which there are two monomer types for each of the
copy and template monomers $\alpha_{T}=\alpha_{C}=2$. Following Ref. [29], we
assume that copy/template interactions are symmetric with respect to
interchange of 0 and 1, giving identical dynamics for all template sequences.
We shall use the uniform template $T=000...0$ in all simulations. When the
copy-template alphabets are binary, $\alpha_{T}=\alpha_{C}=2$, monomers of
type 0 and 1 in the copy copolymer can then simply be interpreted as “correct"
and “incorrect" matches, respectively. We will refer to all molecules as
polymers unless their nature as copolymers is important.

Figure 1: The state of the system is described by $t_{\textrm{occ}}$,
$\mathbf{C}_{\textrm{seq}}$ and $\mathbf{BB}$. Sites on the template are
indexed with i. The array $T(i)$ stores the template sequence.
$t_{\textrm{occ}}$ is an array containing the polymer label of each copy unit
attached to the template. The label is the row of $\mathbf{C}_{\textrm{seq}}$
in which the polymer’s sequence is stored. $\mathbf{BB}$ stores the backbone
bonds. In this example, three polymers (labelled 0, 1 and 3) are attached by
one, two and one bond/s to the template, respectively.

A copy unit with no neighbouring backbone bonds is a monomer. As shown in Fig.
1, copy polymers are chains of copy units that share a ‘backbone’ linking one
unit to the next. Each copy monomer may form up to two backbone bonds, one
with a copy unit ahead and one with a copy unit behind with the boundary
condition that backbone bonds can only be formed ahead at site $0$, and behind
at site $L_{T}-1$. As can be seen in Fig. 1, copy polymers can occupy
contiguous stretches of template sites and may have ‘tails’, portions of their
length which are not directly bound to the template, but are indirectly
tethered to the template by bound units with which they share a backbone.
These tails may be present at either end of the polymer. Copy polymer
‘bridges’, unbound stretches between bound units within the same copy polymer,
are neglected. It is assumed that such behaviour is strongly suppressed due to
the constrained close proximity enforced by the bound units, just as internal
bubbles are strongly suppressed between complementary nucleic acid duplexes
[67].

We assume that the concentration of the monomer species $[M_{i}]$ in the bath
is buffered to be constant. In the simulations that follow, we will initiate
the system with no polymers in solution, and with the template empty (all
sites unoccupied). Under these conditions, and, again, given that the volume
of the bath is large, we further assume that the concentration of copy
polymers in the bath remains at a negligible level compared to the
concentrations of copy monomer. That is to say, once polymers fully unbind
from the template, they diffuse into the bath and do not bind to the template
again. We therefore neglect possible product inhibition due to very high
concentrations of products, a secondary problem relative to product inhibition
due to products that simply never detach from the template. In the models
presented here, polymers cannot bind to the template from solution and bridges
cannot form loops that cause the contour length of a copy polymer to exceed
the length of the template, $L_{T}$, and, therefore, only copy polymers with
length $L\leq L_{T}$ can be produced.

The state of the system is thus uniquely specified by the sequence of monomers
in all growing copy polymers, including those with only one unit, and by
specifying which site each monomer is attached to on the template (if it is
attached to no template site, the monomer exists in a tail) as outlined in
Fig. 1. We do not consider the conformations of the system within each of
these macrostates.

Figure 2: Simple transition rules between coarse grained states of a templated
polymerisation process. Each site on the template can be occupied by one
monomeric unit. Monomers may bind to and unbind from the template from
solution and from the tails of copolymers. Neighbouring units on the template
may polymerise or depolymerise. All transitions, aside from the final release
of a polymer from the template, are reversible, and can occur at any site on
the template.

As we indicate in Fig. 1, sites in the system are indexed with $i$ running
from $0$ to $L_{T}-1$. The $i^{th}$ site on the template can be unoccupied or
occupied by a copy unit; either a monomer or part of a polymer. Each isolated
monomer or polymer on the template is given a unique label. If two copy units
share a backbone, even indirectly via other monomers, they must be part of the
same polymer, and therefore they share the same label. The template can be
occupied by at most $L_{T}$ distinct monomers or polymers at any moment, hence
polymer labels run from $0$ to $L_{T}-1$. We introduce this label to enable
easy accounting when extending, fragmenting or recombining polymers. The
template occupation labels are stored in a vector $t_{\textrm{occ}}$, where
the $i^{th}$ component, $t_{\textrm{occ}}(i)$, is the label $l$ of the copy
unit bound to the template at site $i$, or $-1$ if the site is empty.

The sequence of all polymers - including units in tails- and monomers in the
system is stored in an $L_{T}\times L_{T}$ matrix,
$\mathbf{C}_{\textrm{seq}}$, in which the $l^{th}$ row contains the sequence
of the copy unit with label $l$, buffered with -1s before and after the
sequence. Therefore, $\mathbf{C}_{\textrm{seq}}(t_{\textrm{occ}}(i),i)$ is the
identity of the copy unit occupying the template at site $i$.

Each copy unit may form up to two backbone bonds, one with a copy unit ahead
and one with a copy unit behind, with the boundary condition that backbone
bonds can only be formed ahead at site $0$, and behind at site $L_{T}-1$. The
$L_{T}\times L_{T}-1$ matrix $\mathbf{BB}$ stores the state of all backbone
bonds in the system. The $i^{th}$ column of the $l^{th}$ row of $\mathbf{BB}$
is $1$ if there is a backbone bond between copy units with label $l$ at sites
$i$ and $i+1$ (not necessarily connected to the template), and $0$ otherwise.

####  II.1.2 Transition rules

Figure 3: Seven types of transition constitute the basic model of templated
polymerisation. Each transition modifies the system at a single site, or, in
the case of polymerisation, between neighbouring sites. The free energy change
associated with the transition constrains the log of the ratio of backwards to
forwards transitions according to the principle of local detailed balance,
which reduces the number of free parameters in our model and assures that the
model is thermodynamically self-consistent. Final release of a polymer is
assumed to be irreversible, and hence is associated with an infinite free
energy change.

The transition rules that define the permitted state changes are listed in
Fig. 3 and illustrated in in Fig. 2. All transitions in this model are
microscopically reversible, apart from the final detachment of copolymers,
since the concentration of products in solution, and hence the rate of polymer
rebinding, is assumed to be zero. The transitions that feature in this basic
model are as follows:

  1. 1.

A copy monomer in solution may diffuse close enough to an unoccupied site $i$
on the template for an attractive interaction to cause it to bind there.

  2. 2.

The reverse reaction may also occur, in which a monomer bound at site $i$ on
the template may unbind and freely diffuse away from the template.

  3. 3.

A unit of a copy polymer that is either bound at the end of a polymer or is
the last bound unit before a tail may unbind from the template. This increases
the length of the tail by $1$.

  4. 4.

In reverse, the first unit in the tail of a polymer hovering over site $i$
(which must have a template bound and backbone-linked neighbour at either site
$i-1$ or site $i+1$) may bind to site $i$ if the site is not already occupied.

  5. 5.

If the last bond between a polymer and the template breaks, the polymer is
released. This liberated polymer mixes with the large volume and never returns
to bind to the template.

  6. 6.

Polymerisation can only occur between available, template-bound neighbours.
Specifically, if a bound copy unit at site $i$ can make a forward backbone
bond to site $i+1$, and a bound copy unit at site $i+1$ is capable of
receiving a backbone bond from site $i$, then a backbone bond may form.

  7. 7.

Depolymerisation, the breaking of backbone bonds, can only occur between
template-bound copy units which share a backbone bond, a consequence of this
being that units in copolymer tails cannot spontaneously depolymerise.

We assume that polymerisation or depolymerisation of molecules not attached to
the template in the baths or on the end of free copolymer tails is negligible.

####  II.1.3 Concentrations and transition energies

The concentration of monomer units in the baths are set to be equal,
$[\textrm{M}]_{i}=[\textrm{M}]$, for all monomer types $i$, defined as a
dimensionless quantity relative to an arbitrary reference concentration. We
are interested in copying systems in which any template can be copied with the
same efficiency as any other template from the same set of building blocks.
Templates with an over-abundance of a certain monomer type could be accurately
copied more easily if the corresponding copy monomer was over-represented in
the pool too [30]. However, a biased monomer pool would, on average, provide
no advantage for copying an arbitrary template sequence.

The monomeric units in the tails of the polymers are constrained to a small
volume in closer proximity to the template and neighbouring monomers than the
free monomers in the baths. We therefore define an effective concentration
$[\textrm{M}_{\textrm{eff}}]\gg[\textrm{M}]$ in order to parameterize the free
energy change of polymerization and the rebinding rate of monomers in the tail
to the template.

Bond formation is parameterized through standard free-energy change of
reactions at the reference concentration ($[\textrm{M}]=1$). In this work we
define all free energies as dimensionless quantities relative to $kT=1$. The
formation of the backbone bond between monomers in isolation is associated
with a standard free-energy change of $\Delta
G_{\textrm{BB}}+\ln[\textrm{M}_{\textrm{eff}}]$; $\Delta G_{\textrm{BB}}$
reflects the chemical bond strength and $\ln[\textrm{M}_{\textrm{eff}}]$ the
loss of entropy associated with binding from solution. More negative values of
$\Delta G_{\textrm{BB}}$ favour polymer growth.

Initially in Section III.1, we consider a simple model of cooperative copy-
template interactions that is depicted in Fig. 3. In this picture, individual
monomers bind to the template with a standard free energy change of $\Delta
G_{\textrm{spec}}$ – called “specific" because it depends on whether the copy
and template monomers are complementary. The standard free-energy change for
binding of a copy polymer of length $l$, forming exactly $l$ bonds with the
template is

$\sum_{i=0}^{l-1}\Delta
G_{\textrm{spec}}^{i}-(l-1)\ln\left([\textrm{M}_{\textrm{eff}}]/[\textrm{M}]\right),$
(1)

where $\Delta G_{\textrm{spec}}^{i}$ is the specific bond free energy of the
$i^{\textrm{t}h}$ copy-template pair. Here the factor
$(l-1)\ln[\textrm{M}_{\textrm{eff}}]$ captures the cooperative nature of the
bond between a long template and a long copy that is responsible for product
inhibition. This formulation is an extremely simple model of cooperativity: a
copolymer of length $l$ gains $l$ favourable bonds of the same strength but,
for $l-1$ of those bonds, the entropic cost is reduced enormously relative to
binding from solution. In Section III.2 we extend the model to allow for less
cooperative behaviour, and in III.3 also allow sites at the end of the
template to bond more weakly with the copy.

In the majority of this work we consider a uniform template where there is one
type of copy and template monomer, and hence the specific bond strength,
$\Delta G_{\textrm{spec}}$ takes one value. In the final section of this
paper, Section III.4, we consider a model with a binary copy monomer alphabet
with $\alpha_{C}=2$.

####  II.1.4 Parameterisation of transition rates

We assume that each transition described in Section II.1.2 and depicted in
Fig. 3 is well described as an instantaneous process with an average rate.
Since the thermodynamics of copy production is an important feature underlying
its physics [28], we ensure that the thermodynamics of the system is self-
consistent by applying the principle of local detailed balance to set the
transition rates [68]. The principle of local detailed balance states that the
ratio of forwards to backwards rates between any pair of states is constrained
by the chemical free-energy change associated with the forwards transition
$\Delta G$,

$\frac{R_{\textrm{forward}}}{R_{\textrm{reverse}}}=e^{-\Delta G}.$ (2)

We use this equation to parameterise the transition rates between states in
our system, as shown in Fig. 3.

Monomer binding and unbinding is associated with a free-energy change of
$\Delta G=\Delta G_{\textrm{spec}}-\ln[\textrm{M}]$, since one specific bond
is formed in the process of monomer binding. The ratio of the monomer binding
rate $R_{\textrm{bind}}$ to the monomer unbinding rate $R_{\textrm{unbind}}$
is given by

$\frac{R_{\textrm{bind}}}{R_{\textrm{unbind}}}=[\textrm{M}]e^{-\Delta
G_{\textrm{spec}}}.$ (3)

We begin by assuming that monomers bind to the template sites with mass action
kinetics, giving $R_{\textrm{bind}}\propto[\textrm{M}]$, and that the monomers
unbind from the template with a rate that is exponentially dependent on the
strength of the specific bond $\Delta G_{\textrm{spec}}$, and hence we arrive
at

$R_{\textrm{bind}}=k_{0}\,[\textrm{M}]\text{ and
}R_{\textrm{unbind}}=k_{0}\,e^{\Delta G_{\textrm{spec}}}.$ (4)

In Section III.4, we consider the competition between two types of monomer,
and compare the effects of two different ways of parameterising the unbinding
and binding rates, in which either unbinding rate or the binding rate is
exponentially dependent on $\Delta G_{\textrm{spec}}$. Prior to Section III.4,
we maintain the parameterisation given above for simplicity.

The free-energy change associated with copy units in the tails at either end
of polymers binding to a template site is $\Delta G=\Delta
G_{\textrm{spec}}-\ln[\textrm{M}_{\textrm{eff}}]$. Therefore, the rates of
tail binding and unbinding are constrained by

$\frac{R_{\textrm{tail bind}}}{R_{\textrm{tail
unbind}}}=[\textrm{M}_{\textrm{eff}}]e^{-\Delta G_{\textrm{spec}}}.$ (5)

Following a similar argument to that for monomers binding from solution, we
set the rate that a unit in a copolymer tail rebinds to an available site on
the template to be

$R_{\textrm{tail bind}}=k_{0}\,[\textrm{M}_{\textrm{eff}}].$ (6)

The rate at which polymer units unbind from a template site to become part of
a tail is then identical to the rate at which a monomer in the same position
would detach,

$R_{\textrm{tail unbind}}=k_{0}\,e^{\Delta G_{\textrm{spec}}}.$ (7)

In this model, the final detachment and release of a copolymer is
irreversible. Release occurs when the last copy-template bond is broken, and
therefore we set the rate of release to equal the rate at which monomers
unbind from the template, giving

$R_{\textrm{release}}=k_{0}\,e^{\Delta G_{\textrm{spec}}}.$ (8)

Polymerisation between units attached to the template at site $i$ and $i+1$ is
associated with a free energy change of $\Delta G_{\textrm{BB}}$. Therefore

$\frac{R_{\textrm{polym}}}{R_{\textrm{depolym}}}=e^{-\Delta G_{\textrm{BB}}}.$
(9)

We choose $R_{\textrm{polym}}=k$, and therefore
$R_{\textrm{depolym}}=ke^{\Delta G_{\textrm{BB}}}$. Under this
parameterisation, increasing the backbone bond strength increases the average
lifetime of a backbone.

With these assignations for the free-energy change of each reaction, the
overall free-energy change of incorporating a single monomer into a polymer in
solution is

$\Delta G_{\textrm{pol}}=\Delta
G_{\textrm{BB}}-\ln\frac{[\textrm{M}]}{[\textrm{M}_{\textrm{eff}}]},$ (10)

as required.

Figure 4: For the basic model of Fig. 3, a wide search over multiple axes of
the parameter space shows that the mean polymer length is always low. Each
graph shows an average of the mean polymer length produced by the system,
$\langle L\rangle$, with SEM error bars, on a template of length $L_{T}=30$,
against $\Delta G_{\textrm{BB}}$, with $-\Delta G_{\textrm{BB}}$ being the
driving force behind backbone bond formation. The monomer concentration
$[\textrm{M}]$ is varied between rows, and the specific bond strength, $\Delta
G_{\textrm{spec}}$, between the columns, while the polymerisation rate, $k$,
is varied and represented with different colours and marker shapes. Error-bars
are smaller than line width where not visible. A black line gives the value of
$\Delta G_{\textrm{BB}}$ at which $\Delta G_{\textrm{pol}}=0$. In all cases,
$\langle L\rangle\ll L_{T}$, although a moderate peak in $\langle L\rangle$ is
observed at moderate values of $\Delta G_{\textrm{BB}}$.

###  II.2 Methods

####  II.2.1 Gillespie simulation

We simulate the dynamics of this system using Gillespie’s kinetic Monte Carlo
simulation method [69] with two levels of stochastic sampling. The first level
of sampling selects the template site to update, with a probability
proportional to the sum of transition rates at that site, and the second
selects the transition rule to apply at that site with a probability
proportional to its individual transition rate. This two-stage sampling method
is advantageous because the valid transitions of the system only change the
rates for subsequent transitions in a localised region of the system. For
instance, a monomer binding the template at site 3 will not change the
conformation of the system at site 10 or 100, and hence doesn’t change the
valid transitions around site 10 or 100. Given the locality of the model we
have developed, we need only need to recalculate the valid transitions and sum
over the valid rates at the few sites either side of the site which was last
updated at each step.

####  II.2.2 Parameters, initial conditions and stopping criteria

We use dimensionless units by setting $k_{B}T=1$ and $k_{0}=1$. We also use
$[\textrm{M}_{\textrm{eff}}]=100$ throughout. We vary the free parameters $k$,
$[\textrm{M}]$, $\Delta G_{\textrm{spec}}$, $\Delta G_{\textrm{BB}}$, and
$\Delta G_{\textrm{gen}}$ and $\Delta G_{\textrm{end}}$ that will be
introduced later. Unless stated otherwise, the initial condition of the system
in each simulation is an empty template. We simulate systems with templates of
length $L_{T}=10,30$ and $100$. We calculate statistics of the copying
process, such as the mean polymer length and, later when considering systems
with two distinct monomer pools, the error rate (defined as the number of
mistakes contained within a polymer divided by its length), over the first
fixed number of polymers (polymers with length $\geq 2$) that form upon and
are released from the template. We take averages of quantities such as the
mean polymer length by running multiple simulations with the same parameters,
but different random number seeds. Where the number of polymers produced by
the system (the sample size) generated within the allotted compute-time for
each simulation is less than 10, the data are excluded as they are
unrepresentative of steady state. In each case we run between $O(10)$
independent repeats for each set of parameter values, with each independent
simulation stopping after producing $O(1000)$ polymers, though this varied
from case to case. The code and input files required to reproduce the data
presented in this work are freely accessible through the link provided under
Data Availability at the end of this paper.

##  III Results

###  III.1 A simple model of copy-template binding and copy-copy
polymerization cannot reliably produce long polymers

Figure 5: Typical configurations help to explain the observed mean length
$\langle L\rangle$ of copy products in the simple model of templating. We plot
$\langle L\rangle$ against $\Delta G_{\textrm{BB}}$ for a template of length
30, $\Delta G_{\textrm{spec}}=-4$, $[\textrm{M}]=1$ and $k=1$, showing a
moderate peak surrounded by regions of $\langle L\rangle=2$. We extract
typical configurations at the labelled values of $\Delta G_{\textrm{BB}}$: (I)
Weak backbone, many small fragments; (II) Moderate backbone strength, a small
number of longer fragments; (III) Strong backbone, a single long polymer
covers the whole template.

We begin by asking whether long copy polymers can consistently be produced
from a system of monomers governed by the simple model introduced in Section
II.1.4, with transitions parameterised as in Fig. 3. In Fig. 4, we analyse the
polymers produced by the system on a template with length $L_{T}=30$. We show
the mean polymer length $\langle L\rangle$ against the free energy driving
backbone formation on the template, $\Delta G_{\textrm{BB}}$ as we vary the
polymerisation rate $k=[10^{-4},10^{-2},1,10^{2},10^{4}]$, the monomer
concentration $[\textrm{M}]=[0.01,0.1,1]$, the specific copy-template bond
strength $\Delta G_{\textrm{spec}}=[-10,-6,-2,2,6]$, and backbone strength
$\Delta G_{\textrm{BB}}$ ranging from -4.6 to -20. Five independent
simulations were run for each parameter value, with a target of 2000 products.
When parameter values where extreme, long embedded Markov processes were
observed that were time consuming to simulate. Refer to Section S1 of the SI
for a discussion of these parameter values.

As shown in Fig. 4, the mean length of polymers produced by the system remains
low, $2\lesssim\langle L\rangle\lesssim 4$, across the parameter space. We now
explore particular regions of parameter space to explain why. To help with
this analysis, we consider a simple cut through the parameter space, and also
probe the actual state of the template during simulations. To do so, we
consider the specific case with $\Delta G_{\textrm{spec}}=-4$,
$[\textrm{M}]=1$ and $k=1$ keeping other parameters the same as in Fig. 4, and
average over 5 independent repeats of 1000 polymers at each data point as we
vary the backbone bond strength $\Delta G_{\textrm{BB}}$. The results are
reported in Fig. 5, alongside characteristic snapshots from simulations.

Figure 6: Production of dimers in the limit of strong backbone bonds. Long
polymers form on the template when the backbone bond $\Delta G_{\textrm{BB}}$
is sufficiently strong. Cooperative binding slows the complete release of the
long polymer, while dimers are formed and released under its fraying ends.

Parameter values where a template-bound polymer is unstable with respect to
either the dissociation into monomers free in solution ($\Delta
G_{\textrm{BB}}+\Delta G_{\textrm{spec}}-\ln[\textrm{M}]>0$), or unconnected
template-bound monomers ($\Delta G_{\textrm{BB}}>0$) don’t feature heavily in
Fig. 4 and Fig. 5, but are consistent with $\langle L\rangle\rightarrow 2$ as
$\Delta G_{\textrm{BB}}$ gets more positive as the only “polymers" produced
are transiently bound dimers.

As $\Delta G_{\textrm{BB}}$ becomes less positive/more negative, a significant
number of longer polymers appear on the template. An example configuration is
shown in Fig. 5 I. However, $\langle L\rangle\approx 2$ remains true, because,
although longer copies are present on the template in significant numbers,
they tend to detach more slowly than dimers due to the cooperative bond with
the template. Therefore dimers dominate the distribution of products.

At extremely negative values of $\Delta G_{\textrm{BB}}$, configurations such
as Fig. 5 III are obtained. In this limit, a single long polymer covers the
whole template, This configuration is extremely stable due to the difficulty
in breaking the backbone bond, and the cooperativity with which the long copy
binds to the template. However, this cooperativity does not prevent the ends
of the long polymer from “fraying" (undergoing transitory detachment from the
template) which allows dimers to form under the raised tails; these dimers are
subsequently released into solution must faster than the longer polymer, and
are thus registered as the only product (see Fig. 6).

For some sets of parameters, a moderate peak in $\langle L\rangle$ is observed
between these limits. In this case, although $\Delta G_{\textrm{BB}}$ is
negative, backbone bonds within the copies are broken at an appreciable rate,
resulting in relatively long fragments on the template. In some cases, via a
mechanism of fraying and polymerization at the junction between fragments, one
fragment can displace the other from the template in a step-by-step fashion.
An intermediate state in this process is shown in Fig. 5 II. We observe that
this peak occurs at slightly negative values of $\Delta G_{\textrm{pol}}$,
which can be rationalised by noting that adding a single monomer to the tail
of template-attached polymer is associated with a free-energy change of
$\Delta G_{\textrm{pol}}$, and so states with long tails are thermodynamically
unfavourable for $\Delta G_{\textrm{pol}}>0$.

Although this mechanism can provide some increase in $\langle L\rangle$, a
longer polymer is more likely to push a shorter one off the template, and no
complete polymers are produced. Importantly, therefore, even the systems that
give $\langle L\rangle\neq 2$ still produce outputs that are much smaller than
the template, and do not have a well-defined length. Either template-bound
polymer formation is too unfavourable, or cooperative template binding
prevents the release of long stable polymers (product inhibition). Putative
displacement mechanisms can be initiated at any location and produce
fragmented polymers with a bias towards shorter products. These results
suggest that, for systems that are well-described by the basic model of
Section II.1.4, the basic templating mechanism is incapable of generating and,
crucially, releasing long polymers reliably. Indeed, in Ref. [57], in a
bottom-up, coarse-grained model of a replicating system that obeyed similar
dynamics to that described by the model in Fig. 3, the authors observed the
formation of only long polymers that remained bound in complexes under
isothermal conditions.

In the next section, we extend the model to allow for a more complex
polymerisation mechanism. This scheme can be viewed simultaneously as a way to
avoid product inhibition, and a way to favour effective displacement
mechanisms, by disrupting the cooperativity of copy-template interactions in a
directional manner.

###  III.2 A polymerisation mechanism that disrupts copy-template binding
allows for long copies

Figure 7: The generic bond model is an extension of the basic model presented
in Fig. 3. Here there are two kinds of copy-template bond, the specific bond
(dashed red line) that is potentially sensitive to the match between copy and
template units, and the generic bond (solid black line), that is not sensitive
to the match. The generic bond is formed during monomer binding and breaks
upon polymerisation, and vice versa for the reverse reactions. Units at the
leading edge of copy polymers  (at the right-hand edge) bind to the template
more strongly than other monomers in copies due to the modified polymerisation
mechanism. The generic bond energy, $\Delta G_{\textrm{gen}}$, like the
specific bond energy, $\Delta G_{\textrm{spec}}$, doesn’t contribute to the
overall free energy change of polymer extension $\Delta G_{\textrm{pol}}$.

In the context of simple catalysts, Ref. [70] argued that product inhibition
can be reduced if part of the free energy of product formation is diverted
into destabilizing the interaction of catalyst and product. With that idea in
mind, we develop a model in which backbone bonds in the copy can only be
formed at the expense of breaking a bond between copy and template. The
scheme, illustrated in Fig. 7, could describe a range of chemical systems with
competitive bond formation, but in particular resembles the handhold-mediated
strand displacement mechanism introduced by Cabello-Garcia et al. [5].

Formally, we implement this mechanism by splitting the copy-template bond into
two parts, one of which must be broken for a backbone bond to form. The free-
energy change of monomers binding to the template from solution is now given
by $\Delta G=\Delta G_{\textrm{spec}}+\Delta
G_{\textrm{gen}}-\ln[\textrm{M}]$, where the generic bond $\Delta
G_{\textrm{gen}}$ is not dependent on the match between copy and template. We
assume that the polymers have a directional asymmetry (as is typical in
macromolecular polymers like DNA and RNA [31]), and represent this in diagrams
such as Fig. 7 by drawing the specific bond (dashed) on the left and the
generic bond (solid) on the right.  This asymmetry will be retained throughout
the manuscript.

As shown in reactions 6 and 7 of Fig. 7, the generic bond in the leftmost
monomer breaks when polymerisation occurs between neighbouring monomers,
leaving the leftmost monomer less tightly bound to the template.  In visual
representations, polymerisation will always disrupt the bond of the left or
‘lagging’ monomer; the generic bond of the right or ‘leading’ monomer is
unaffected. The total free-energy change of the template-attached
polymerisation step is then $\Delta G=\Delta G_{\textrm{BB}}-\Delta
G_{\textrm{gen}}$. $\Delta G_{\textrm{gen}}$, like $\Delta G_{\textrm{spec}}$,
does not contribute to the the overall free energy change of extending a
polymer tail (or a polymer in solution), $\Delta G_{\textrm{pol}}=\Delta
G_{\textrm{BB}}-\ln[\textrm{M}]/[\textrm{M}_{\textrm{eff}}]$.  We assume that
the disruption of the generic bond during polymerisation happens at the same
time as the polymerisation reaction; i.e., the bond is transferred from
template to polymer via an “attack" or an “invasion" mechanism, as occurs in
Ref. [5].

Having modified the thermodynamics of the model, we modify the kinetics as
follows. As shown in Fig. 7, generic bonds are only formed during the binding
of monomers from solution, when the leading monomer in a copy polymer rebinds
to the template from a tail state, or when the backbone between two template-
attached monomers is disrupted. For simplicity, we assume that all unbinding
reactions that break a generic bond have rates that scale as
$R_{\textrm{unbind}}\propto e^{\Delta G_{\textrm{gen}}}$ (so $\Delta
G_{\textrm{gen}}$ does not appear in the binding rates), and that
$R_{\textrm{depolym}}=ke^{\Delta G_{\textrm{BB}}-\Delta G_{\textrm{gen}}}$,
$R_{\textrm{polym}}=k$.

Figure 8: Coupling polymerisation to disrupting copy-template bonds can
generate copies with average length $\langle L\rangle\sim L_{T}/2$. Above, we
show surface plots of the mean polymer length, $\langle L\rangle$, against the
backbone bond strength $\Delta G_{\textrm{BB}}$ and the generic bond strength
$\Delta G_{\textrm{gen}}$ for monomer concentrations $[\textrm{M}]={1,10}$ and
$\Delta G_{\textrm{spec}}={-4,2,8}$ on a template of length $L_{T}=30$. Below,
we provide snapshots of the system for the indicated points within the
parameter space as the simulation time passed $t=200000$. White dashed line
constraint: $\Delta
G_{\textrm{BB}}=\ln([\textrm{M}]/[\textrm{M}_{\textrm{eff}}])$. Above the
white line, the template is either occupied by monomers and dimers (I and I’)
or is empty 0 because the backbone bond is too weak. The system is only
capable of producing dimers $\langle L\rangle\approx 2$. Dashed cyan line
constraint: $\Delta G_{\textrm{gen}}<-\Delta
G_{\textrm{spec}}+\ln\frac{2k_{0}}{L_{T}\langle\tau_{\textrm{disp}}\rangle}$.
To the left of this line, the generic bond is strong enough that a lagging
polymer tends to step forward faster than it detaches. To the right of the
cyan constraint, we see either an empty template at 0 (since everything
detaches rapidly) or, when the specific bonds are stronger, a template with a
fully bound copy, since invading copies cannot force it off the template
(configuration II). Here the system still produces dimers $\langle
L\rangle\approx 2$. Dashed red line constraint: $\Delta
G_{\textrm{gen}}<-\Delta
G_{\textrm{spec}}+\ln\frac{2k_{0}}{L_{T}^{2}\langle\tau_{\textrm{disp}}\rangle}$.
To the left of this constraint, lagging polymers step forwards fast enough to
create a dense, orderly polymer brush, as in III and III’. A uniform product
length distribution is observed with a mean length $\langle L\rangle=L_{T}/2$.
In between the red and cyan constraints, at IV, slower advancement of the
lagging polymers leads either to a less dense brush with some tails occupying
the template (IV), or the spontaneous appearance of gaps that encourage the
initiation of shorter polymers within the brush (V). In configurations such as
IV, $\langle L\rangle\gtrsim L_{T}/2$ is observed, whereas for parameters that
lead to configurations like V, the mean length is slightly lower $\langle
L\rangle<L_{T}/2$. Figure 9: Model to estimate the time,
$\langle\tau_{\textrm{disp}}\rangle$, taken for a lagging polymer to extend by
one unit, displacing the tail of the polymer ahead.

In Fig. 8, we present multiple contour plots of the averaged mean length of
polymers produced via a template of length $L_{T}=30$ as the backbone bond
strength $\Delta G_{\textrm{BB}}$ and the generic bond strength $\Delta
G_{\textrm{gen}}$ take values in the range $[0,-1,...,-17]$. For each point
sampled, we averaged the mean length of 1000 polymers each produced by 5
independent simulations. Data are excluded for simulations that didn’t produce
this 1000 polymers within the allowed window. We set $k=1$, and consider
$[\textrm{M}]=[1,10]$, $\Delta G_{\textrm{spec}}=[-4,2,8]$. Additionally, we
provide snapshots showing the wide variety of conformations reached by the
system at a selected points in the parameter space when the simulation time
passed $200,000$ units.

In Fig. 8, we observe a high mean length of around $L_{T}/2=15$ over large
regions of parameter space. This behaviour corresponds to a fully-occupied
template, with a “brush" of polymers attached to it (regions exemplified by
III, III’, IV and V). In all of these regimes, the coupling of polymerisation
to the disruption of generic bonds with the template has two important
effects. Firstly, provided the disrupted bond is strong enough, the mechanism
can overcome the cooperative effect that leads to product inhibition by a
single long polymer. The leading monomer of a copy binds strongly, but the
lagging monomers bind much more weakly. Undesirable confirmations with a
single long polymer bound to the template are recovered (see configuration II
in 8) when $\Delta G_{\textrm{gen}}$ is not large and negative, but the
specific bond is strong. Secondly, the asymmetry introduced by the generic
bond solves the problems associated with the unbiased strand displacement
mechanisms mention in Section III.1. The leading edge of a lagging polymer
out-competes the tail of the polymer ahead by binding to the template with
greater strength at their junction, allowing the system to act as a ratchet in
which biased strand displacement causes shorter, lagging polymers to push
longer polymers ahead off the template.  The emergent direction of
displacement and extension is indicated in Fig. 8 by a red arrow. As shown in
configuration III of 8, the result is a brush of polymers connected to the
template by only one strong bond across the length of the template. The random
detachment of polymers from this brush produces a distribution with an average
polymer length $\langle L\rangle\sim L_{T}/2=\sim 15$.

Some simple physical reasoning can be used to explain the regions of parameter
space in which various behaviours are observed. For long polymer copies to
form easily, incorporating monomers into copy polymers must be
thermodynamically favourable [30]. Thus $\Delta G_{\textrm{pol}}=\Delta
G_{\textrm{BB}}-\ln([\textrm{M}]/[\textrm{M}_{\textrm{eff}}])<0$, and we
require a sufficiently strong backbone:

$\Delta G_{\textrm{BB}}\ll\ln([\textrm{M}]/[\textrm{M}_{\textrm{eff}}]).$ (11)

The line representing this constraint is shown in white in Fig. 8. Above this
line, the system exhibits either an empty template as in configuration 0 if
the copy-template interaction if weak ($\Delta G_{\textrm{spec}}+\Delta
G_{\textrm{gen}}>\ln[\textrm{M}_{\textrm{eff}}]$); or a template largely
covered in monomers, as in I and I’, if the copy-template interaction is
strong ($\Delta G_{\textrm{spec}}+\Delta
G_{\textrm{gen}}<\ln[\textrm{M}_{\textrm{eff}}]$). When the constraint of Eq.
11 is not satisfied, the system is only capable of producing short polymers,
much for the same reasons we presented in Section III.1 and Fig. 5, as long
polymers are unstable structures.

Given a sufficiently stable backbone, long polymers can form. However, the
strength of the generic bond $\Delta G_{\textrm{gen}}$ can drastically change
the conformations that these long polymers settle into, and thereby change the
typical length of polymers that are produced by the system. When the generic
bond is weak, as in II where $\Delta G_{\textrm{gen}}=-2$, we tend to observe
single, long polymers stuck on template, and dimers are created and released
under the fraying end of the polymer as in III.1. Note that when $\Delta
G_{\textrm{gen}}=0$, the system reverts back to the model presented in Section
III.1. When the generic bond is strong and takes very negative values, as in
III and III’, we observe conformations where the polymer tails form a dense
brush, and the mean length is roughly half the template length $\langle
L\rangle=L_{T}/2$. We also observe regions with configurations like IV and V,
in which the copy polymers form a less dense, less regular brush.

The key criterion for whether a brush of long polymers forms relates to two
timescales. The first is the time taken for a polymer to polymerise forward
into a space that is occupied by the lagging edge of the polymer in front. We
call this time $\langle\tau_{\textrm{disp}}\rangle$, since the lagging tail of
the polymer in front is effectively displaced (although in many cases its
binding is weak to begin with). The second is the timescale on which the
leading edge of a polymer detaches, $\langle\tau_{\textrm{unbind}}\rangle$. We
can estimate $\langle\tau_{\textrm{disp}}\rangle$ with the simple discrete-
state, continuous-time model in Fig. 9. We depict the four states involved in
a single forward step alongside their transition rates. Treating this
subprocess in isolation, we obtain a rate matrix K with entries:

$\textbf{K}=\begin{pmatrix}-k_{0}e^{\Delta
G_{\textrm{spec}}}&k_{0}[\textrm{M}_{\textrm{eff}}]&0&0\\\ k_{0}e^{\Delta
G_{\textrm{spec}}}&-k_{0}[\textrm{M}_{\textrm{eff}}]-k_{0}[\textrm{M}]&k_{0}e^{\Delta
G_{\textrm{gen}}+\Delta G_{\textrm{spec}}}&0\\\
0&k_{0}[\textrm{M}]&-k_{0}e^{\Delta G_{\textrm{gen}}+\Delta
G_{\textrm{spec}}}-k&0\\\ 0&0&k&0\end{pmatrix}$

The mean first-passage time $\langle\tau_{\textrm{disp}}\rangle$ from state 0
to state 3 is given by the $(3,0)$ element of the Drazin inverse of K, which
takes a value

$\displaystyle\langle\tau_{\textrm{disp}}\rangle=$
$\displaystyle(k(1+([\textrm{M}]+[\textrm{M}_{\textrm{eff}}])e^{-\Delta
G_{\textrm{spec}}})$ (12) $\displaystyle+$ $\displaystyle k_{0}(e^{\Delta
G_{\textrm{gen}}+\Delta G_{\textrm{spec}}}+[\textrm{M}]+e^{\Delta
G_{\textrm{gen}}}[\textrm{M}_{\textrm{eff}}]))/(kk_{0}[\textrm{M}]),$

$\langle\tau_{\textrm{unbind}}\rangle$ is simply given by the inverse of the
monomer unbinding rate,
$\langle\tau_{\textrm{unbind}}\rangle=(R_{\textrm{unbind}})^{-1}=\frac{1}{k_{0}}e^{-\Delta
G_{\textrm{gen}}-\Delta G_{\textrm{spec}}}$. To produce copies of length $\sim
L_{T}/2$, it is necessary that $\langle\tau_{\textrm{disp}}\rangle
L_{T}/2<\langle\tau_{\textrm{unbind}}\rangle$, otherwise the growing polymer
will detach too rapidly. By rearranging this inequality, we arrive at

$\Delta G_{\textrm{gen}}<-\Delta
G_{\textrm{spec}}+\ln\frac{2k_{0}}{L_{T}\langle\tau_{\textrm{disp}}\rangle}$
(13)

which can be implicitly solved for $\Delta G_{\textrm{gen}}$. This constraint
is represented by the cyan dashed line in Fig. 8. When this constraint isn’t
satisfied, newly-initiated polymers tend to detach before they extend. When
the total copy-template affinity $\Delta G_{\textrm{spec}}+\Delta
G_{\textrm{gen}}$ is strong, as at II, we observe a single, long polymers
stuck to the template, and when when the affinity is weak, at 0, we observe
empty templates. Polymer brush configurations III, III’, IV and V, which all
produce a relatively long distributions, all lie to the right of this
constraint.

For a dense, ordered brush conformation as in III and III’ to be stable, $\sim
L_{T}/2$ sequential extension and displacement steps must occur on a timescale
shorter than that at which any of the $L_{T}$ polymers fall off, enabling the
system to heal the brush after any detachment events. Therefore, a dense,
brush-like conformation will be reached when $\langle\tau_{\textrm{\rm
disp}}\rangle L_{T}/2<\langle\tau_{\textrm{\rm unbind}}\rangle/L_{T}$.
Rearranging, we obtain

$\Delta G_{\textrm{gen}}<-\Delta
G_{\textrm{spec}}+\ln\frac{2k_{0}}{L_{T}^{2}\langle\tau_{\textrm{disp}}\rangle},$
(14)

represented by the red dashed line in Fig. 8. In a maximally-displaced, brush-
like conformation, all polymers are only attached by their leading edge. Hence
detachment can occur equally from any point and the product length
distribution is uniform, with a mean length very close to $\langle
L\rangle=L_{T}/2$.

When $\Delta G_{\textrm{gen}}$ lies between the constraints Eqs. 13 and 14,
the generic bond is strong enough to enable long polymers to form but
insufficient to reach a maximally-displaced, brush-like conformation. If the
specific bond strength is strong enough that lagging tails tend to bind back
to the template, and are only removed via displacement, we see conformations
similar to IV. Here, we see gaps between the leading edges of polymers, with
weakly-bound tails occupying the intervening sites. Some shorter polymers
occupy large swathes of the template, while some longer polymers have been
pushed to the end of the template where they can advance no further. Here we
see a slight inversion of the cooperative effect; longer polymers which bunch
at the end of the template tend to have fewer template connections than
shorter polymers, which skews the product length distribution slightly toward
long polymers giving $\langle L\rangle>L_{T}/2$.

Figure 10: The dynamics of the system at the weakened end (green square) of
the template. We extend the generic bond model presented in Fig. 7 and modify
the copy-template interaction strength at the end of the template with an
energetic term $\Delta G_{\textrm{end}}$. When $\Delta G_{\textrm{end}}>0$,
the bond made between copy units and the final template site is weakened.  As
before, the right hand edge of the copy polymer is the leading edge.

This effect is not observed when the specific bond is weak enough that polymer
tails can spontaneously detach from the template without the need for
displacement, as seen at V where $\Delta
G_{\textrm{spec}}=8>\ln{[\textrm{M}_{\textrm{eff}}]}$. Here gaps appear
quicker than they can be filled by extension of polymers already on the
template, and monomers that bind within these gaps can sometimes form dimers.
The tails of the dimers may spontaneously detach from the template, preventing
the incorporation of the dimer into a the polymer behind, and initiating a new
copy from the centre of the template. As a consequence, the ordered polymer
brush conformation of III or III’ is disrupted by shorter polymers that have
incorrectly initiated in the middle of the template. Consequently, the system
produces polymers with a mean length $\langle L\rangle<L_{T}/2$.

We have not observed any parameter values in which the template sites are
typically unoccupied but in which long copies are frequently produced. To get
long polymers, monomer binding in isolation must be stable ($R_{\rm
bind}/R_{\rm unbind}>1$) since we have deliberately disrupted cooperative
binding. Therefore the only way that the template sites can typically be
available is if both (a) lagging tails tend to unbind, and (b) if stepping
forward is too fast for monomers to fill in behind. But a monomer filling in
happens at least as quickly as a polymer can step forwards, by definition, and
much faster if the polymer runs into a traffic jam of other polymers. So empty
sites tend to be a rare commodity unless binding to the template is just
pathologically unstable.

We have found that applying the two constraints of Eqs. 11 and 13 is generally
sufficient to identify the region in which long copies are produced (Fig. 8).
We note that in the upper left triangle of Fig. 8, where $G_{BB}\gg G_{gen}$,
the system failed to produce 1000 polymers during the allotted run-time and
the data were excluded. When $G_{BB}\gg G_{gen}$, the depolymerisation step in
the model becomes very fast causing repeated polymerisation and
depolymerisation events. In this region, the embedded process of polymer
growth becomes very long and requires ever longer run-times to simulate. In
the under-sampled regions, we have no evidence to suggest that the polymer
length distributions produced by the system deviate from the simple arguments
presented above.

By introducing a mechanism that channels the free energy released during
polymerisation into destabilising the interactions of polymers tails and the
template, we have shown that far longer polymers can be produced in a simple
model of templating. However, for true copying, the polymers produced by the
system must have identical lengths to the template. In the following section,
we bias the production complete polymers simply by weakening the copy-template
interaction strength at end of the template.

###  III.3 Weakening the final site on the template biases the production of
complete polymers

Figure 11:  A. The free energy of a polymer, $\Delta G_{\textrm{}}$, decreases
with length, $L$ when $\Delta G_{\textrm{pol}}<0$ in all models. B. The
component of the free energy due to the copy-template bonds is given by
$\Delta G_{\textrm{}}-L\,\Delta G_{\textrm{pol}}$. The copy-template free
energy decreases with length in the basic model (red line and conformations
iii and iv) which suppresses the release of long polymers. In the maximally
displaced conformations (i and ii) generated under the generic bond model
(blue line), all polymers are only connected to the template at their leading
edge, and hence the free energy landscape is flat. Weakening the final site on
the template by an amount $\Delta G_{\textrm{end}}$ destabilises polymers with
length $L=L_{T}$ (ii), leading to their selective release.

We now propose a simple adjustment to the model presented in Section III.2
that allows for the systematic release of complete polymers from the template
when the system is in a brush-like conformation. Here, the copy-template bond
at the final template site is weakened with an free-energetic factor $\Delta
G_{\textrm{end}}>0$. This change is somewhat analogous to a stop codon in
translation [11]. Crucially, since the end of the template is a unique site,
this change can be made in a way that doesn’t interfere with the copy-template
interactions at other sites.

The effect of this adjustment is illustrated in the free energy profiles shown
in Fig. 11. In the previous section we saw that disrupting the copy-template
bond upon polymerisation could allow the lagging tail of polymers to detach
from the template, meaning that, although polymerisation was thermodynamically
"down-hill", the binding strength of the copy to the template did not increase
as the copy extends; the template binding free energy landscape was flat.
Here, by weakening the copy-template interaction strength at the end of the
template, $\Delta G_{\textrm{end}}$, we expected to destabilise the binding of
completed polymers and promote their release over incomplete polymers.

The thermodynamics and kinetics of the modified system are identical to the
model presented in Section III.2 and Fig. 7, apart from at the very end of the
template. In Fig. 10, we outline the thermodynamic and kinetic changes to the
model. The free-energy change of binding at the end of the template is now
given by $\Delta G=\Delta G_{\textrm{spec}}+\Delta G_{\textrm{gen}}+\Delta
G_{\textrm{end}}-\ln[\textrm{M}]$, and the free-energy change of polymer tail
binding at the end of the template is now given by $\Delta G=\Delta
G_{\textrm{spec}}+\Delta G_{\textrm{gen}}+\Delta
G_{\textrm{end}}-\ln[\textrm{M}_{\textrm{eff}}]$. The rates of monomer
unbinding, leading-edge polymer unbinding and polymer termination at the end
of the template all take the value $R=k_{0}e^{\Delta G_{\textrm{spec}}+\Delta
G_{\textrm{gen}}+\Delta G_{\textrm{end}}}$; binding rates are unchanged
relative to Section III.2.

Figure 12: Destabilising the last site on the template with $\Delta
G_{\textrm{end}}$ increases the probability of producing a complete polymer
$p_{\textrm{complete}}$ when the configuration of copies forms a dense brush
as seen in conformation III in Fig. 8. We plot the averaged probability of
producing a complete polymer, $p_{\rm complete}$, against the destabilising
energy penalty, $\Delta G_{\textrm{end}}$, for a range of system parameters.
We compare the data to a simple analytical model that accounts for the flux of
complete polymers and uniformly distributed polymers from the template, and
also indicate the points at which
$\langle\tau_{\textrm{disp}}\rangle=\langle\tau_{\textrm{end}}\rangle$ (red
line) and $\Delta G_{\textrm{end}}=-\Delta G_{\textrm{gen}}-\Delta
G_{\textrm{spec}}+\ln[\textrm{M}_{\textrm{eff}}]$ (green line) to guide
interpretation, as explained in the text.  Figure 13:  The full length
distributions of the data shown in Fig. 12 for $[\textrm{M}]=1$ and $\Delta
G_{\textrm{spec}}=-4$ and a template of length $L_{T}=10$, averaged over 5
independent simulations at each value of $\Delta G_{\textrm{end}}$. At $\Delta
G_{\textrm{end}}=0$, we see a approximately uniform product distribution. At
low $\Delta G_{\textrm{end}}$ we see a sharp increase in the probability of
producing complete polymers with $\Delta G_{\textrm{end}}$ before a plateau is
reached. At higher $\Delta G_{\textrm{end}}$, a peak in the production of
dimers is observed. As $\Delta G_{\textrm{end}}$ is increased further, there
is a final peak in the production of full length products, after which the
system produces a uniform distribution of products excluding complete
polymers.  Figure 14:  Weakening the end of the template with $\Delta
G_{\textrm{end}}$ affects the typical conformations of the system on the
templates. We illustrate typical configurations from the simulations reported
in Fig. 13. A At $\Delta G_{\textrm{end}}=0$, the system reaches a dense
brush. B At $\Delta G_{\textrm{end}}=10$, completed polymers are released
sooner than displacement can compensate, and so the brush of copies becomes
less dense. C At $\Delta G_{\textrm{end}}=22$, the final template site has low
occupancy. Here dimers form under the fraying ends of completed polymers. D At
$\Delta G_{\textrm{end}}=35$, the occupancy of the final site is so low that
incomplete polymers detach sooner than they can complete by polymerising at
the end of the template. E Niche formation. The leading edge of completed
polymers may fray, creating a protective niche that promotes the cyclical
turnover of dimers. If the completed polymer falls off faster than a dimer can
form, then the dimer can be incorporated into the preceding polymer.

In Fig. 12, we plot the probability of producing a full-length copy polymer,
$p_{\textrm{complete}}$ against the destabilising free-energy penalty of the
last site, $\Delta G_{\textrm{end}}$, at a few values of the generic bond
strength $\Delta G_{\textrm{gen}}$, the monomer concentration $[\textrm{M}]$
and the template length $L_{T}$, as a function of $\Delta G_{\textrm{end}}$.
We set $k=1$, $\Delta G_{\textrm{spec}}=-4$ , and $\Delta
G_{\textrm{BB}}=-20$, which puts the system into a dense brush steady state
conformation when $\Delta G_{\textrm{end}}=0$ as seen in Section III.2. For
each parameter point sampled we ran 5 independent repeated simulations that
would produce up to 2000 polymers or truncated after 4hrs run-time. In Fig.
13, we show the full product length distribution for $[\textrm{M}]=1$, $\Delta
G_{\textrm{spec}}=-4$, $L_{T}=10$.

Fig. 12 shows that weakening the end of the template can increase the
proportion of complete products, $p_{\textrm{complete}}$, and that
$p_{\textrm{complete}}\approx 1$ is possible under certain conditions. When
$\Delta G_{\textrm{end}}=0$, the polymer brushes encountered in Section III.2
tend towards a jammed state, as depicted in Fig. 14 A, with detachment events
occurring at any point on the template. Weakening the final site on the
template creates a release point that allows this polymer ‘traffic’ to flow
through, with unbinding predominantly occurring at the final site where
complete polymers are attached by a weakened final copy-template bond,  as
seen in the binding free energy landscape in Fig. 11.

To probe this behaviour in more detail, we construct a simple model to explain
the proportion of complete polymers in the product pool,
$p_{\textrm{complete}}$, produced by the system when a dense conformation has
been reached. This model considers only the rate of formation and release of
complete polymers and the rate of a competing unwanted process of release of a
uniform distribution of polymers from the body of the template. In the
densely-packed regime, as depicted in Fig. 14 A, all polymers have one bond
with the template. The timescale on which one of $L_{T}-1$ incomplete polymers
are released from a template of length $L_{T}$ is approximately,

$\langle\tau_{\textrm{uniform}}\rangle=(L_{T}-1)^{-1}R_{\rm
lead\,\,release}^{-1}\approx(L_{T}-1)^{-1}\,k_{0}\,e^{-\Delta
G_{\textrm{gen}}-\Delta G_{\textrm{spec}}}.$ (15)

Next we estimate the time it takes the system to complete a growing polymer
and release it from the template.To extend the polymer attached to the
penultimate site, a free monomer must occupy the final site. The fraction of
time that this final site is occupied (as opposed to empty), $g$, can be
approximated by the binding equilibrium of the monomer

$g\approx\frac{k_{0}\,[\textrm{M}]}{k_{0}\,[\textrm{M}]+k_{0}\,e^{\Delta
G_{\textrm{gen}}+\Delta G_{\textrm{spec}}+\Delta G_{\textrm{end}}}}$ (16)

Given a polymerisation rate $k$, the average timescale on which a polymer with
length $L_{T}-1$ is completed by polymerising with a monomer at the end site
on the template is

$\langle\tau_{\textrm{form}}\rangle=1/gk\approx\frac{[\textrm{M}]+e^{\Delta
G_{\textrm{gen}}+\Delta G_{\textrm{spec}}+\Delta
G_{\textrm{end}}}}{[\textrm{M}]\,k}$ (17)

In this simple model we assume that the time it takes this newly completed
polymer to fall off the template is rate limited by either the timescale on
which the bond with the end of the template is broken,

$\langle\tau_{\textrm{end}}\rangle=1/k_{0}\,e^{\Delta G_{\textrm{gen}}+\Delta
G_{\textrm{spec}}+\Delta G_{\textrm{end}}},$ (18)

or the time it takes for the polymer unit at the penultimate template site to
be displaced by a polymer behind, which is given by
$\langle\tau_{\textrm{disp}}\rangle$ as calculated in Eq. 12 in Section III.2.

Taken together, the timescale on which complete polymers fall off the template
is approximately

$\langle\tau_{\textrm{fall}}\rangle\approx\langle\tau_{\textrm{end}}\rangle+\langle\tau_{\textrm{disp}}\rangle,$
(19)

and the total timescale of producing and releasing a complete polymer is

$\langle\tau_{\textrm{complete}}\rangle\approx\langle\tau_{\textrm{form}}\rangle+\langle\tau_{\textrm{fall}}\rangle.$
(20)

We can then approximate the steady state value of $p_{\textrm{complete}}$ as

$p_{\textrm{complete}}\approx\frac{1/\langle\tau_{\textrm{complete}}\rangle}{1/\langle\tau_{\textrm{complete}}\rangle+1/\langle\tau_{\textrm{uniform}}\rangle}.$
(21)

With no free fitting parameters, this simple model – which accounts only for
the production rate of the end and uniform components of the product
distribution – captures remarkably well the range of $\Delta G_{\textrm{end}}$
over which $p_{\textrm{complete}}$ is high and the maximum value of
$p_{\textrm{complete}}$ over the whole range of $\Delta G_{\textrm{end}}$, as
seen in Fig. 12 and Fig. S5. This fact suggests that the physics incorporated
into this simple model explains the majority of the behaviour.

Specifically, in Fig. 12, at $\Delta G_{\textrm{end}}=0$, there is no bias for
completed polymers to fall off the template. Products are dominated by
dissociation from the body of the template, and we see a roughly uniform
product length distribution as encountered in Section III.2 and
$p_{\textrm{complete}}\approx 1/L_{T}$. At small values of $\Delta
G_{\textrm{end}}$, breaking the copy-template bond at the end of the template
remains the rate limiting step in producing a full-length copy and
$\langle\tau_{\textrm{complete}}\rangle\approx\langle\tau_{\textrm{end}}\rangle$.
As $\Delta G_{\textrm{end}}$ increases, the time to break the end copy-
template bond, $\langle\tau_{\textrm{end}}\rangle$, decreases exponentially
and therefore $p_{\textrm{complete}}$ increases exponentially. Eventually,
however, $p_{\textrm{complete}}$ saturates when
$\langle\tau_{\textrm{end}}\rangle<\langle\tau_{\textrm{disp}}\rangle$; at
this point,
$\langle\tau_{\textrm{complete}}\rangle\approx\langle\tau_{\textrm{disp}}\rangle$
and breaking the copy-template bond at the end is no longer rate-limiting.
Here the model suggests that no further increase in $p_{\textrm{complete}}$
can be achieved by increasing $\Delta G_{\textrm{end}}$; the value of the
saturating level is determined by the relative timescales of displacement
$\langle\tau_{\textrm{disp}}\rangle$ and spontaneous detachment from the body
of the template $\langle\tau_{\textrm{uniform}}\rangle$.

At very high values of $\Delta G_{\textrm{end}}$, the simple model accurately
predicts the drop in $p_{\textrm{complete}}$ to zero shown in Fig. 12. Here,
the end bond is so unstable that the monomer occupancy of the final site on
the template, $g$, tends to zero. Consequently,
$\langle\tau_{\textrm{form}}\rangle$ diverges and no full length copies form.
Instead the system effectively operates as a uniform template of length
$L_{T}-1$, and generates a uniform distribution of polymers with lengths
$L\leq L_{T}-1$, but no polymers with length $L=L_{T}$. This behaviour is
particularly clear in the full length distribution shown in Fig. 13, in which
at high $\Delta G_{\textrm{end}}$ no polymers are produced with $L=L_{T}=10$.

The simple model presented here assumes that the system operates in the dense,
ordered brush regime. However, in Fig. 12, where $[\textrm{M}]=1$ and $\Delta
G_{\textrm{gen}}=-8$, the constraint given in Eq. 14 which predicts whether
the dense conformation is reached, $\langle\tau_{\textrm{\rm disp}}\rangle
L_{T}/2<\langle\tau_{\textrm{\rm unbind}}\rangle/L_{T}$, is not satisfied.
Here the model’s estimates of the level of the plateau in $p_{\rm complete}$
are less accurate. The biggest failure of the simple model, however, is not
predicting the dips in $p_{\rm complete}$ that occur at intermediate values of
$\Delta G_{\textrm{end}}$. In Fig. 12, a single dip in $p_{\textrm{complete}}$
is observed for $[\textrm{M}]=1$, and a dip and an intermediate plateau are
observed for $[\textrm{M}]=10$. From looking at the full polymer length
distributions shown in Fig. 13 and Figs. S2, S3 and S4, we can conclude that
these dips correspond to a large spike in dimer production. Moreover, from
snapshots of simulations we observe that this dimer formation occurs at the
final two sites on the template underneath the fraying end of completed- but
not yet detached- polymers (Fig. 14 E).

Despite significant effort, a simple, quantitative model that predicts these
‘dimer dips’ could not be constructed, however a qualitative understanding of
the factors which cause dimer production has been reached. The production of
dimers occurs at fairly high values of $\Delta G_{\textrm{end}}$, which are
certainly not necessary for this the copying to be effective (it would be
sufficient to just weaken the connection by a few units of thermal energy) and
which may be experimentally inaccessible. Nonetheless, we offer an explanation
of how this unexpected phenomenon arises in the model.

When $\Delta G_{\textrm{end}}$ is weakened sufficiently that
$\langle\tau_{\textrm{end}}\rangle<\langle\tau_{\textrm{disp}}\rangle$ (shown
as a red line in Fig. 12), the leading edge of completed polymers are released
faster than the tails are fully displaced, leading to less dense conformations
such as Fig. 14 B and C. Completed polymers tend to be displaced from the
template from their lagging edge, and can spontaneously fray from their
leading edge at the end of the template. When $\Delta G_{\textrm{end}}>-\Delta
G_{\textrm{gen}}-\Delta G_{\textrm{spec}}+\ln[\textrm{M}_{\textrm{eff}}]$
(equality at green line in Fig. 12), frayed configurations at the leading
edge, such as Fig. 14 c, are more likely than un-frayed alternatives such as
Fig. 14 B. This constraint on $\Delta G_{\textrm{end}}$ locates the dimer dip.
If a completed polymer frays far enough to expose the penultimate site on the
template, a monomer may occasionally bind there with a strong combined generic
and specific bond. This penultimate monomer is unable to polymerise into the
completed chain, and instead dimerises with monomers which occasionally bind
to the final template site. Completed polymers can be understood to create a
’protective niche’ for the formation of dimers. Upon dimerising, the dimer is
rapidly released. This cycle is shown in Fig. 14 E.

If the timescale of formation and release of dimers in the protective niche is
less than the average timescale on which the completed polymers are released,
then multiple dimers may be released for each completed polymer. When $\Delta
G_{\textrm{end}}$ is increased further, the monomer occupancy of the final
site reduces and the time to form a dimer increases. If the timescale of
formation and release of dimers in the protective niche is greater than the
timescale on which the completed polymers are released, then the completed
polymers fall off, removing the protective niche. As in Fig. 14 E, without the
protective niche, then the dimer is able to polymerise into the preceding
polymer, which dramatically reduces the number of dimers that are released
into the product pool. Hence, after the dimer dip, we observe a sharp increase
in $p_{\textrm{complete}}$ again, close to the saturating value that the
simple model predicts, before the eventual drop off to $p_{\rm
complete}\approx 0$ at the highest values of $\Delta G_{\textrm{end}}$. At
high monomer concentrations, such as $[\textrm{M}]=10$ in Fig. 12, we observe
an intermediate plateau in $p_{\textrm{complete}}$ as well as the dimer dip,
which occurs when
$\langle\tau_{\textrm{end}}\rangle<\langle\tau_{\textrm{disp}}\rangle$ shortly
after the model predicts saturation. Due to the increased monomer
concentration, fraying events at the end of the template result in more
monomer invasions at the penultimate site causing more frequent formation of
dimers.

###  III.4 On-rate discrimination enables the reliable production of copies
with the correct length and accurate sequences

Figure 15: Models with on-rate discrimination can yield complete polymers with
high probability, $p_{\textrm{complete}}\approx 1$,and low error rates per
monomer, whereas models with off-rate discrimination cannot reliably generate
complete polymers with low error rates per monomer. We present A the mean of
$p_{\textrm{complete}}$ and B the mean error rate per monomer over 5 repeats
at each value of the discrimination free energy, $\Delta
G_{\textrm{disc}}=\Delta G_{\textrm{w}}-\Delta G_{\textrm{r}}$, for
simulations using the model in Fig. S6 of the Supplementary Material with an
alphabet size of two. Simulations are performed for $L_{T}=$ 10, 30 and 100
using $k=1$, $\Delta G_{\textrm{BB}}=-20$, $\Delta G_{\textrm{gen}}=-12$,
$\Delta G_{\textrm{end}}=7$, and $[M_{0}]=[M_{1}]=1$ for models with on- and
off-rate discrimination, shown with dotted and solid lines, respectively.
Standard Error of the Mean (SEM) is shown as errorbars, which are generally
smaller than line width. In B the error rates per monomer for different
template lengths overlap one another and lie very close to the limiting line
$(1+\exp(\Delta G_{\textrm{disc}}))^{-1}$.

In Section III.3, we identified a mechanism for producing polymers of a single
monomer type with a specific template-determined length in a reliable fashion.
Here we consider whether the mechanisms that grant control over the length
distribution are compatible with mechanisms which generate accurate copies
from a pool of two types of monomer. We extend the models presented in
previous sections and consider the growth of binary copies. The
parameterisation is given in Fig. S6 in the Supplementary Material. It is
identical to that in Fig. 10, except that: (i) $\Delta G_{\textrm{spec}}$ now
depends on the match between template and copy monomer; (ii) we allow the
specific binding free energy to appear either in the binding or the unbinding
rates.

In general we now have four specific bond strengths, $\Delta G_{\textrm{00}}$,
$\Delta G_{\textrm{01}}$, $\Delta G_{\textrm{10}}$, and $\Delta
G_{\textrm{11}}$, where the first and second indices give the copy unit and
template unit respectively. We make the simplifying assumption that all copy
monomer concentrations are equal
$[\textrm{M}_{0}]=[\textrm{M}_{1}]=[\textrm{M}]$ and that all interactions are
symmetric (all mismatches are equal, as are all matches) following [29]. If
the correct pairings, $00$ and $11$, have the same free energy $\Delta
G_{\textrm{00}}=\Delta G_{\textrm{11}}=\Delta G_{r}$, and if the incorrect
pairings are also equal, $\Delta G_{\textrm{01}}=\Delta G_{\textrm{10}}=\Delta
G_{w}$, we can describe the specific bond with two energy parameters, $\Delta
G_{\textrm{spec,r}}=\Delta G_{r}$ or $\Delta G_{\textrm{spec,w}}=\Delta
G_{w}$, the standard free energy of forming a right match or wrong match copy
template bond. Given these simplifying assumptions, we can describe the
process of copy sequence simply in terms of whether the monomers are matched
(right, r) or unmatched (wrong, w) with respect to the template, allowing us
to solve the problem in a way that is independent of the specific template
sequence in question.

Again, we use the principle of local detailed balance constrain the forwards
and backwards transition rates between any pair of states. The ratio of the
monomer binding rates $R_{\textrm{bind,r/w}}$ to the monomer unbinding rates
$R_{\textrm{unbind,r/w}}$ for right or wrong template matches is proportional
to the exponential of the specific bond strength (amongst other factors in
Fig. 10):

$\frac{R_{\textrm{bind,r/w}}}{R_{\textrm{unbind,r/w}}}\propto e^{-\Delta
G_{\textrm{r/w}}}.$ (22)

We compare the effects of two contrasting parameterisations of the model,
detailed in Fig. S6 of the Supplementary Material. If the unbinding rates are
dependent on the match between copy and template,
$R_{\textrm{unbind,r/w}}\propto e^{\Delta G_{\textrm{r/w}}}$, as was the case
in the models presented in previous sections, then the system has "off-rate"
discrimination, and the monomer binding rates are constant. If the monomer
binding rates are dependent on the match between copy and template,
$R_{\textrm{bind,r/w}}\propto e^{-\Delta G_{\textrm{r/w}}}$, then the system
is has "on-rate" discrimination, and the unbinding rates are constant. As
shown in Fig. S6 of the Supplementary Material, we apply the $\Delta
G_{\textrm{spec}}$ sensitivity consistently across all binding and unbinding
transitions for on and off-rate discrimination. As in the Section III.3, all
unbinding rates for units at the end of the template are increased by a factor
$e^{\Delta G_{\textrm{end}}}$.

We take parameter values that produced complete polymers in Section III.3. To
generate the data shown in Fig. 15, we set $k=1$, $\Delta
G_{\textrm{BB}}=-20$, $\Delta G_{\textrm{gen}}=-12$, $\Delta
G_{\textrm{end}}=7$, and $[M_{0}]=[M_{1}]=1$. We set $\Delta
G_{\textrm{spec,w}}=\Delta G_{w}=-1$ for wrong-match monomers, and $\Delta
G_{\textrm{spec,r}}=\Delta G_{r}=\\{-1,-2,\ldots,-8\\}$ for right-match
monomers. As described in the methods section, to generate each data point in
Fig. 15 we ran 5 independent repeats of simulations that each generate up to
5000 copolymers. In addition to calculating the average of
$p_{\textrm{complete}}$, we also calculate the average error rate per monomer
(the number of wrong units in the copolymer divided by the polymer length) for
complete polymers.

In Fig. 15 A, it is evident that on-rate discrimination maintains a high
proportion of complete polymers even at long lengths and large values of
$\Delta G_{\textrm{disc}}=\Delta G_{w}-\Delta G_{r}$. Off-rate discrimination,
however fails to produce long polymers as the discrimination free-energy
difference, $\Delta G_{\textrm{disc}}=\Delta G_{\textrm{0}}-\Delta
G_{\textrm{1}}$, grows larger. The generic bond and weak end mechanisms
presented in the previous sections, which generate and selectively release
long polymers, require the leading edge of polymers to remain bound to the
template until the end of the template is reached. However, if the system
discriminates on the off-rates, then the incorporation of incorrect monomers
into a polymer can cause it to be released prematurely, thereby reducing
$p_{\textrm{complete}}$.

In Fig. 15 B, on-rate discrimination delivers low error rates per monomer,
close to the equilibrium ratio implied by the discrimination free energy
$1/(1+\exp(\Delta G_{\textrm{disc}})$), even though the products are far from
equilibrium after separation from the template. Off-rate discrimination is
inaccurate for moderate discrimination energies as a strong $\Delta
G_{\textrm{gen}}$ allows incorrect monomers to be incorporated into
copolymers. For off-rate discrimination, there is a trade-off between accuracy
and $p_{\textrm{complete}}$ for the mechanisms we have considered here. Hence,
on rate discrimination is needed to gain control of the length and accuracy in
these minimal models of autonomous templated copying.

##  IV Conclusion

Unlike the template-copying machinery found in biological systems that can
cyclically copy templates under constant environmental conditions [6, 7, 8, 9,
10, 11, 12, 13, 14, 15, 16], state-of-the-art synthetic molecular copying
systems require external intervention or non-chemical driving [36, 37, 38, 43,
35, 44, 45, 46] to operate cyclically. Here we have developed coarse-grained
models of isothermal, enzyme-free, templated copolymerisation processes in
which the copy copolymers can spontaneously detach from the template in a
constant environment, driven by chemical free energy alone.

The central challenge of producing polymer copies is overcoming product
inhibition, which is exacerbated by the natural cooperativity of the
interactions between two polymers. We have shown that a surprisingly simple
mechanism – using the free-energy of polymerisation to weaken the interaction
between the monomers and the template behind the leading edge of the copy – is
sufficient to generate long polymers reliably. Combining this mechanism with a
weakened copy-template bond at final site on the template, as in Section
III.3, is sufficient to reliably generate copies of the full template length
under a wide range of conditions.

These proposed mechanisms require: 1) strong template-monomer bonds (so that
the leading edge of growing polymers remain bound to the template for long
enough to extend along its full length); 2) strong backbone bonds between
monomeric units within polymers (so that formed polymers don’t spontaneously
fragment) that, upon formation 3) cause major disruption to the copy-template
bond of the ’back’ monomer (so that the long cooperatively binding tails of
polymers can be displaced from template by other polymers); and 4) a moderate
destabilisation of the copy-template bond at the end of the template (to
enable the selective release of fully-formed polymers). To copy specific
sequences in addition to producing polymers of a fixed length, it is necessary
to also have “on-rate" discrimination, in which matching sequences bind to the
template faster.

The conditions stated above can be met by DNA-based chemical reaction systems
that exploit handhold-mediated DNA strand displacement (HMSD) [5]. In HMSD,
polymerisation between monomers on the template results in the weakening of
the copy-template connection in the ‘lagging’ monomer through the transfer of
bonds via strand displacement. Moreover, templating via HMSD has the potential
to achieve effective on-rate discrimination during monomer binding, using a
mechanism in which a transient match-specific toehold forms first, with the
strong generic bond only forming after an additional (toehold-mediated) strand
displacement reaction[71, 72].

The models we have introduced have a surprisingly rich range of behaviours and
their dynamics did not match our initial expectations. We expected that
optimal copying would involve isolated copies on a template, as is frequently
observed in transcription [8, 9, 10] and translation [11, 12, 13, 14, 15, 16],
and as reflected in previous models of templated copolymerisation [29]. In
such a setting it would be necessary to prohibit copies from starting at
random locations on the template, and thereby generating truncated products.
However, the natural tendency of brush-like configurations observed here to
absorb isolated monomers into growing copies at all points on the template
solves that challenge, which otherwise might require complex activation and
deactivation of template sites ahead of and behind the growing copy. Moreover,
the observed displacement behaviour means that copying is possible even when
the tails of polymers have moderate affinity with the template; indeed, this
regime was more effective in producing longer polymers than regimes in which
tail affinity was so low that they would detach without displacement, as
described in Section S 3 and Fig. S5 of the Supplementary Material.

Extant biological template copying processes, such as transcription [8, 9, 10]
and translation [11, 12, 13, 14, 15, 16], can operate in sparse regimes in
which the copies do not fully occupy the template, although reasonably dense
regimes are also observed[73, 74]. The sparse regime of operation is possible
because the initiation of polymer growth only occurs at specific locations on
the template, at ribosome binding sites in translation for instance [11], in
tightly controlled ways that involve complex enzymatic machinery. Similarly,
the enzymatic machinery of transcription and translation provides a larger,
moving window in which copies are bound to their template [11, 12, 13, 14, 15,
16]. This factor enables off-rate discrimination between the incoming monomers
without causing partially formed polymers to fall off prematurely. It is
challenging see how to engineer this in a simple setting without the interplay
of enzymes that can define a scale for this template-attached region, but such
a scheme may facilitate additional mechanisms such as kinetic proofreading to
increase the accuracy above limit set by the free energetic discrimination
between monomer types.

Mechanisms that rely on displacement of copies by the subsequent copies have
been considered as solutions to product inhibition in an RNA world [66, 75].
In particular, Tupper and Higgs have argued that a rolling-circle copying of a
circular template can alleviate product inhibition via strand displacement
[66]. This rolling circle approach eliminates the tendency of short, newly-
initialised copies to be out-competed by longer polymers on a linear template,
as we observed in Section III.1. The mechanism assumes directional
polymerisation, but polymerisation is still potentially hampered by a
displacement-based competition between polymer ends for binding to the
template, with no intrinsic bias favouring the desired pre-polymerisation
configuration.  The asymmetric destabilisation in our model induces a
directionally-biased strand displacement between copy polymers, which can
overcome product inhibition even on linear templates. Our mechanism may also
work well in conjunction with the rolling circle mechanism of Tupper and
Higgs, providing a directional bias to resolve the competition for the
template. However, the chemistry and bonding stability of individual
nucleotides as explored by Tupper and Higgs [66] – as opposed to larger
oligonucleotide monomers as used in processes like HMSD [5] – may not be well-
suited to the mechanisms we have investigated in this work. Outside of DNA-
based reactions, Osuna Gálvez and Bode’s reaction – in which polymerisation is
directly coupled to the disruption of product-template connections – shares
the most resemblance to our mechanisms, though the templates cannot be reused.
It is an open question whether small organic molecules with the appropriate
chemistry can be identified and used to build synthetic copying systems with
the properties we’ve stipulated.

We have attempted to keep the models as simple and general as possible, so
that they can be used to guide the design of synthetic copying systems in a
wide array of contexts. We have also attempted to make all chemical steps
explicit, rather than invoking a chemical deus ex machina that resolves the
central challenges of the copy processes, just as highly-evolved enzymatic
machinery do in vivo. A key assumption of our model, however, is that
(de)polymerisation only occurs while monomers are attached to the template -
the template acts as a catalyst for bond formation and breakage. A passive
template that simply brings reactants into close proximity could certainly
allow polymerisation reactions that are prohibitively slow in solution to
proceed on the template. However, such a mechanism would not accelerate
depolymerisation as well, and so unless the template has some direct chemical
coupling to the bond formation mechanism, as is typical in enzymes [6, 7, 8,
9, 10, 11, 12, 13, 14, 15, 16], it is hard to see how to justify this
catalytic assumption. Indeed, as we saw in Section III.1, simple templates are
not effective catalysts in our model – they accelerate the polymerisation
reaction by forming a stable complex with the product, preventing cyclic
copying. However, in our modified mechanism presented in Section III.2, the
competition between the polymerisation bond and the generic bond with the
template provides a direct mechanism for catalysis, justifying the assumption
that (de)polymerisation only occurs while monomers are attached to the
template.

We have also assumed that polymers that detach from the template and are
released into the bath do not rebind to the template. One could imagine, if
the concentration of polymers in the bath grew to significant levels, that
rebinding could occur. Indeed, the rebinding and subsequent elongation of
products was presented as a plausible pathway to the generation of longer RNA
oligomers, bridging a portion of the gap between short RNA building blocks and
the long RNA molecules that make up ribozymes in a pair of papers on templated
ligation in an RNA world [57, 56]. However, in our mechanism, polymerisation
is coupled to the disruption of the copy-template bonds, reducing the
interaction strength that most units in the polymer chain could rebind by.
Only the leading edge of a grown polymer could potentially stick tightly to
the template, possibly causing unwanted over-extension of polymers if this
reattachment happened at an earlier site on the template that happened to be
complementary. If one were to design a copying machine based upon the biased
displacement mechanism introduced here, a special monomeric unit could be
reserved to bind effectively only to the ends of templates, and thus cap the
end of completed polymers. This approach would prevent further unwanted
extension and reduce the copy-template affinity for completed products,
alleviating the aforementioned issues.

Another key assumption of this work is that there is no interaction between
polymer tails. Our models suggest a dense brush of polymers is beneficial for
length control, however this may introduce interactions between polymer tails,
especially on long templates, in in vitro settings. The importance of the
interaction between polymer tails and the formation of bulges on the template
should be considered on a case-by-case basis in experimental settings.

Finally, in our results we have not focused on the speed with which copies are
produced, only the relative yield of longer or full-length copies. In this
setting we observed that polymer tails that have some affinity to the template
are advantageous in maximising the likelihood of producing full length copies.
However, such this behaviour likely comes at the cost of somewhat slowed copy
production, since template sites are more likely to be blocked. The optimal
balance will depend on the system in question, and again will require further
investigation on a case-by-case basis.

##  V Data Availability

The code, input files, and analysis scripts required to reproduce the data
presented in this work are available for download from the Zenodo file
repository through the following link: DOI 10.5281/zenodo.5643001.

##  VI Supplementary Material

The Supplementary Material contains a description of parameter values excluded
from Fig. 4 which generated few samples, the full polymer length distributions
accompanying Fig. 12, a demonstration that the asymmetric destabilisation
mechanism is more effective when polymer tails have higher affinity for the
template, and a depiction of the model used in Section III.4, for which there
are two types of monomer.

##  VII Author Contributions

All authors conceived the project. JJ wrote the simulations and performed the
analysis. All authors interpreted the results and co-wrote the paper.

##  VIII Conflict of Interest

The authors declare no conflicts of interest.

###### Acknowledgements.

This work is part of a project supported by the European Research Council
(ERC) under the European Union’s Horizon 2020 research and innovation
programme (Grant agreement No. 851910). JJ is supported by a Royal Society
studentship. TEO is supported by a Royal Society University Research
Fellowship.

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