The interplay of heterogeneity and product detachment in templated polymer copying

Guntoro, Jeremy E. B.; Qureshi, Benjamin J.; Ouldridge, Thomas E.

doi:10.1063/5.0245687

Author Notes

Templated copolymerization, in which information stored in the sequence of a heteropolymer template is copied into another polymer product, is the mechanism behind all known methods of genetic information transfer. A key aspect of templated copolymerization is the eventual detachment of the product from the template. A second key feature of natural biochemical systems is that the template-binding free energies of both correctly matched and incorrect monomers are heterogeneous. Previous work has considered the thermodynamic consequences of detachment and the consequences of heterogeneity for polymerization speed and accuracy, but the interplay of both separation and heterogeneity remains unexplored. In this work, we investigate a minimal model of templated copying that simultaneously incorporates both detachment from behind the leading edge of the growing copy and heterogeneous interactions. We first extend existing coarse-graining methods for models of polymerization to allow for heterogeneous interactions. We then show that heterogeneous copying systems with explicit detachment do not exhibit the subdiffusive behavior observed in the absence of detachment when near equilibrium. Next, we show that heterogeneity in correct monomer interactions tends to result in slower, less accurate copying, while heterogeneity in incorrect monomer interactions tends to result in faster, more accurate copying, due to an increased roughness in the free energy landscape of either correct or incorrect monomer pairs. Finally, we show that heterogeneity can improve on known thermodynamic efficiencies of homogeneous copying, but these increased thermodynamic efficiencies do not always translate to increased efficiencies of information transfer.

I. INTRODUCTION

Genetic information transfer, through the processes of translation, transcription, and replication, underpins all life.¹ Underlying these processes is the same fundamental motif, wherein a long information-carrying heteropolymer, known as a template, has the information stored in its sequence of monomers copied into another polymer, known as the copy. The copy itself may or may not be composed of the same types of monomers. As demonstrated by the sheer breadth of biological form and function, this templated polymer copying motif is capable of specifically assembling a massive space of structures from a small handful of subunits. To give a brief idea of the scale of the challenge, consider the converse approach of self-assembly without an underlying template. In this counterfactual biology, a cell would need to obtain all its proteins (tens of thousands in humans), in their correct proportions, by mixing together the 20 natural amino acids. This task would be impossible as there is not enough information stored in the interactions between the 20 amino acids to create anything but a mess of partially formed or malformed assemblies.²

Given the central role of templated polymer copying in biology, understanding its fundamental principles will likely play a crucial role in attempts at building artificial life³ or understanding life’s origins.⁴ A key test of our understanding of these principles is our ability to build synthetic copying systems without relying on highly evolved enzymes such as polymerases. Our progress here has been far slower than in template-free self-assembly,^5–10 highlighting the difficulties that come with engineering templated copying systems. The earliest attempts at building synthetic copiers resulted in copies that remained stuck to the template, rendering the template unable to catalyze the production of more copies.¹¹ Later designs tended to utilize time-varying external conditions such as temperature and pressure to induce separation.^12–15

The need to separate copies from templates is, in fact, the crux of the difficulty in engineering synthetic polymer copying systems. Accurately producing a state where the correct copy monomers are bound to their correct positions along the template is relatively straightforward, simply requiring a large free-energy advantage for correct monomer pairings.^5–10 However, the copy and template will then tend to remain bound due to cooperative interactions, preventing the production of further copies. Cells are able to supply the free energy needed to separate chemically, without requiring changes in conditions external to the cell, through the use of ATP-consuming enzymes. This separation occurs after the completion of copying in the case of replication¹⁶ or during copying in the case of transcription¹⁷ and translation.¹⁸ Mimicking this autonomy without enzymes is challenging, but approaches based on DNA nanotechnology^19–21 and organic chemistry^22,23 that use the free energy of backbone formation to drive the product off the template have shown some promise.

Limited progress in the development of artificial copying systems highlights a lack of understanding of the basic theory of polymer copying. To remedy this knowledge gap, high-level models have been devised that emphasize the thermodynamic consequences of separation from the template.^24–27 The fundamental challenge in templated copying is the production of a polymer or ensemble of polymers with low sequence entropy. Although specific bonds form during the production of a copy, they are transient and, thus, cannot supply the free energy to compensate for this drop in entropy relative to random sequences. Hence, extra chemical work is required to produce low entropy polymer ensembles, and an efficiency can be defined by comparing this chemical work to the free energy stored in the low entropy ensemble.^24,25 Furthermore, maintaining such an ensemble in a non-equilibrium, low entropy steady state imposes thermodynamic constraints on the network that maintains the steady state.^26,27

More detailed models have also been made that focus on the polymerization mechanism, typically discrete state Markov models. The simplest ones involve one step per added monomer;^28–34 more complex ones have multiple steps³⁵ and sometimes added cycles.^36–43 Some models explicitly consider microscopic reversibility,^{28–34,43,44} while others incorporate irreversible steps.^{36–38,40–42} Another distinction is whether they assume a symmetry between all correct pairings and, separately, all incorrect pairings (homogeneous)^{28,29,31,33,35–44} or consider explicit differences between them (heterogeneous).^32,34 Finally, some approaches are not preoccupied with copy–template separation.^{28,30,31,34,36–40,42,44} On the other hand, other works consider models in which the copy explicitly separates from the template,^41,45 including some that attempt to understand the thermodynamic consequences of separation.^29,33,35,43

Although there are multiple theoretical frameworks to treat heterogeneity in the context of separating copolymerization systems^32,45–47 (in particular,⁴⁵ explicitly applying their framework to RNA polymerase), existing work has not considered both separation and heterogeneity in models with thermodynamic self-consistency. Thermodynamically self-consistent models have been constructed for homogeneous systems,^29,33,35,43 but it remains unclear if and how the effects of heterogeneity interplay with the thermodynamic consequences of separation. We fill this gap by incorporating heterogeneity into a known thermodynamically consistent model of polymer copying that incorporates separation from behind the growing tip of the product, analogous to transcription and translation.³³ The addition of each monomer in our model consists of multiple steps, following.³⁵ We first introduce a method of coarse-graining the model to a single-step description, allowing existing methods (Refs. 32 and 43) of analysis to be used. Using these methods, we demonstrate qualitative differences between heterogeneous copying with and without separation. We then identify regimes where heterogeneity facilitates or hinders fast, accurate, and efficient copying.

We organize our work as follows: In Secs. II A and II B, we introduce our coarse-grained and fine-grained Markov copying models with separation. We briefly describe Gaspard’s³² and Qureshi et al.’s⁴³ analysis methods in Sec. II C and our extended iteration for state visits in Sec. II D. In Sec. II E, we describe the parameter sets we initially consider. We begin our results in Sec. III A by comparing velocities of heterogeneous copying with and without separation. In Sec. III B, we investigate the effect of heterogeneity on error and completion time for our limiting regimes. Finally, in Sec. III C, we show that in the low driving regime, the thermodynamic efficiency of copying can be improved by heterogeneity. We conclude our paper in Sec. IV.

II. METHODS

A. General coarse-grained model of templated polymer copying with heterogeneity and product separation

We consider the coarse-grained model of polymer copying in Fig. 1(a), well-known in the literature.^30,32,33,44 In this section, we will define the model in a general sense, without enforcing separation or thermodynamic consistency (we shall define a parameterization of this model that fulfills these conditions in Sec. II E). The state of the model (x, y) is defined by the two sequences x = n₁n₂ … n_L of the template polymer and y = m₁m₂ … m_l of the copy polymer (Note that L and l are indices of the last monomer in x and y, respectively). The sequences draw their elements from finite templates and copy alphabets {1, …, N} and {1, 2, …, M}, respectively. We list a few notational conventions before we proceed. First, for both x and y, we use n_i:i′ and m_i:i′ to refer to the subsequences of x and y between indices i and i′, inclusive. Second, we will often use & as a shorthand for m_1:l−2 for convenience, writing y = &m_l−1m_l.

FIG. 1.

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Polymer copying with continuous separation of the product from the template. (a) Illustration of the coarse-grained model for a two-monomer system. Each monomer type is assigned a unique number (1 or 2 in this case) and color (light blue for 1 and red for 2). Square monomers correspond to the template polymer, while round monomers correspond to the copy polymer. A coarse-grained state is represented by the full sequence of the copy and the template polymers. Purple dotted arrows represent the coarse-grained state transitions corresponding to either monomer addition (forward arrow) or monomer removal (backward arrow). Each coarse-grained transition may be further subdivided into fine-grained steps. (b) Illustration of the fine-grained steps that occur between coarse-grained states. In particular, the fine-grained steps corresponding to the leftmost purple dot in subfigure (a) are depicted (only monomers in the final 3 positions of the copy and template are depicted). First, a new monomer binds to the template with both a generic (dashed line) and sequence-specific (solid line) bond. Then, the newly bound monomer is covalently incorporated into the growing polymer, breaking the generic bond in the process. Finally, the previously added monomer unbinds from the template. (c) Labeling of fine-grained states for a given coarse-grained state, with indices f denoted by bold bracketed numbers. Arrows here correspond to the fine-grained state transitions. Transitions to other states of index f = 0 (denoted here by the dotted arrows) change the coarse-grained state (x, y). Note that these fine-grained state indices f are not valid when returning from a different coarse-grained state (via a dashed arrow) as f is dependent on the last completed state visited. For simplicity, general and sequence-specific interactions are not depicted separately in subfigures (a) and (c).

From each state (x, &m_l−1m_l), two types of transitions are allowed. Monomer addition at a propensity of Φ⁺(m_l+1, x, &m_l−1m_l) results in the state (x, &m_l−1m_lm_l+1), while monomer removal at a propensity of Φ⁻(x, &m_l−1m_l) results in the state (x, &m_l−1) (we use the term propensity at the coarse-grained level to distinguish Φ^± from the rates of the fine-grained process introduced in Sec. II B). Hence, changes to the template are forbidden in the model, as are monomer additions and removals away from the tip of the copy. From this generic framework, the only additional constraint imposed is that propensities of monomer addition and removal are only dependent on copy monomers within a small locality of the copy tip, such that $Φ^{+} (m_{l + 1}, x, & m_{l - 1} m_{l}) = Φ_{l}^{+} (x, m_{l} m_{l + 1})$ and $Φ^{-} (x, & m_{l - 1} m_{l}) = Φ_{l}^{-} (x, m_{l - 1} m_{l})$ ⁠.^30,32,33,44 These forms are physically justified by our mechanism in Sec. II E, but for now it is sufficient to note that these forms are an appropriate first-order approximation for processes such as transcription and translation that are composed of reactions that occur in a small neighborhood of the tip of the growing polymer.^30,32 Hence, local transitions and propensities from (x, y) are fully determined by (x, m_l−1m_l, l), known as the tip state.

For finite L, once the copy reaches the length of the template l = L, the copy has a certain propensity of detaching from the template. Detachment terminates copying, and hence a detached copy is an absorbing state of the model. Formally, we need to distinguish between a complete copy still attached to the template and one completely detached; purely as a trick of notation, we append the integer 0 to the end of a detached copy sequence (resulting in l = L + 1).

B. Fine-grained reaction steps

The monomer addition and removal steps in Sec. II A are not elementary chemical reactions, and in a general model will be described through a series of smaller, fine-grained steps. Properties of the fine-grained system can be preserved in the coarse-grained system if an appropriate coarse-graining procedure is followed (we cover this procedure for our case in Secs. II C and II D; refer to Qureshi et al.⁴³ for more detail). Fine-grained steps for our system follow from earlier work³⁵ on a copying system that separates as it is being copied; in brief, monomer binding, polymerization, and unbinding of the copy from the template are each treated as separate steps. Note that although we use the term fine-grained, these states should still be understood as biochemical macrostates.⁴⁸

To define the state space, we assume that no more than two monomers are bound to the template at any given time, and fully incorporating a new monomer onto the copy requires breaking the existing template-copy bond. The exact process of monomer addition, summarized in Fig. 1(b) and Table I, consists of three steps. First, a monomer binds to the next empty site on the template. Then, a covalent bond is formed between the new monomer and the tip of the copy. Finally, the previous tip detaches from the template. The reverse processes occur for monomer removal. Copy–template bonds have a generic component and a sequence-specific component. The generic bond is broken during polymerization; this mechanism was found to reduce product inhibition in Ref. 35, and so we incorporate it here.

TABLE I.

A summary of the fine-grained events and their associated rates.

Event	Forward rate	Backward rate
Binding a monomer to the template	$K_{bind}^{+} (n_{l} n_{l + 1}, m_{l} m_{l + 1})$	$K_{bind}^{-} (n_{l - 1} n_{l}, m_{l - 1} m_{l})$
Polymerization of the tip monomer	$K_{pol}^{+} (n_{l} n_{l + 1}, m_{l} m_{l + 1})$	$K_{pol}^{-} (n_{l - 1} n_{l}, m_{l - 1} m_{l})$
to the newly bound monomer
Detachment of the previous	$K_{tail}^{-} (n_{l} n_{l + 1}, m_{l} m_{l + 1})$	$K_{tail}^{+} (n_{l - 1} n_{l}, m_{l - 1} m_{l})$
tip monomer (tail unbinding)

Event	Forward rate	Backward rate
Binding a monomer to the template	$K_{bind}^{+} (n_{l} n_{l + 1}, m_{l} m_{l + 1})$	$K_{bind}^{-} (n_{l - 1} n_{l}, m_{l - 1} m_{l})$
Polymerization of the tip monomer	$K_{pol}^{+} (n_{l} n_{l + 1}, m_{l} m_{l + 1})$	$K_{pol}^{-} (n_{l - 1} n_{l}, m_{l - 1} m_{l})$
to the newly bound monomer
Detachment of the previous	$K_{tail}^{-} (n_{l} n_{l + 1}, m_{l} m_{l + 1})$	$K_{tail}^{+} (n_{l - 1} n_{l}, m_{l - 1} m_{l})$
tip monomer (tail unbinding)

We use the notation (x, y, f) to specify the fine-grained states, consisting of the coarse-grained state as well as an index f ∈ {0, …, 6} that specifies the fine-grained state given the coarse-grained state [Fig. 1(c)]. We treat the fine-grained process as a series of sub-processes in which the system transitions between “completed states” with f = 0 and “transitory states” with f ≠ 0, as in Ref. 43. The coarse-grained label (x, y) is given by the completed state that was visited most recently, with f then defined relative to that state. Hence, biochemical macrostates with f ≠ 0 have two different fine-grained state descriptors, y and f, depending on whether they were approached “from the back” or “from the front.” For example, starting from (x, &m_l, 0), binding and then polymerization of a type 1 monomer would yield (x, &m_l, 3). On the other hand, the same state would be labeled (x, &m_l1, 1) if (x, &m_l1, 0) was the last completed state visited.

C. Solving for distributions of complete polymers

We will cover here and in Sec. II D, the analysis methods used in this work. Readers interested only in our physical results may skip to Sec. II E. To be able to discuss our methods and results precisely, we must first distinguish between the various stochastic processes that arise from our models and clarify the relationships between them. Let t refer to (continuous) time, and k refer to the discrete number of steps taken (number of state transitions). For a fixed template X = x, the stochastic processes Y(t) and F(t), giving the coarse-grained and fine-grained index of the copy, respectively, together define a Markov process $S_{F}^{x} (t) = (Y (t), F (t) | X = x)$ ⁠. From $S_{F}^{x} (t)$ ⁠, we can extract an embedded discrete-time Markov chain $S_{F}^{x} [k]$ representing the sequence of states visited. If we remove any information about the fine-grained index F(t) and only record Y(t), we obtain the continuous, non-Markovian stochastic process ${\tilde{S}}_{F}^{x} (t)$ and the embedded discrete sequence of states ${\tilde{S}}_{F}^{x} [k]$ ⁠. If we are only interested in the distribution of final copies, it is sufficient to consider ${\tilde{S}}_{F}^{x} [k]$ ⁠, since it records any change in the composition of the copy.

There are many alternative ways that $S_{F}^{x} (t)$ can be coarse-grained with a view to avoiding the need to handle non-Markovian processes. Qureshi et al.’s coarse-graining procedure⁴³ can be applied if the full state space of the system can be partitioned into transitory and completed states such that every transitory state is only encountered during a transition between two unique completed states. In such systems, between any two completed states is a “petal” of transitory states. Each petal between completed states can then be analyzed independently.

For our model, transitory states (x, &m_lm_l+1, f) for f ∈ {1, 2} [Fig. 1(c)] are accessible only during transitions from the completed state (x, &m_lm_l+1, 0) to (x, &m_l, 0). All other transitory states are accessible only during transitions from (x, &m_lm_l+1, 0) to (x, &m_lm_l+1m_l+2, 0) for some m_l+2. Hence, the entire set of states and transitions in Fig. 1(b), which connect two completed states together, is a “petal” that can be analyzed independently of others. Qureshi et al.’s procedure⁴³ is based on a graphical interpretation of discrete-state Markov processes, where states in a petal are nodes in a directed weighted graph and transitions between states are directed edges in the graph with weights equal to transition rates. Qureshi et al. then consider rooted spanning trees of this graph for a single petal—subgraphs without cycles where the out-degree of every node is equal to 1 except for the root node, which has an out-degree of zero (we say the spanning tree is “rooted at” the root node). Given a starting state (x, y, 0) and target state (x, y*, 0), the key quantity

Λ_{x}^{+} ({y, y}^{*})

can be calculated by multiplying the rates of all transitions in each spanning tree rooted at (x, y*, 0) and then summing over these spanning trees.

Λ_{x}^{-} ({y, y}^{*})

can be obtained by doing the same over spanning trees rooted at (x, y, 0). Summing over spanning trees rooted at (x, y, 0) for a modified process where all transitions to (x, y*, 0) are redirected back to (x, y, 0) yields a second key quantity, A_x(y, y*). Qureshi et al.’s procedure⁴³ assigns the coarse-grained backward and forward propensities from y to y* as follows:

Φ_{x}^{\pm} (y, y^{*}) = \frac{Λ_{x}^{\pm} ({y, y}^{*})}{A_{x} ({y, y}^{*})} .

(1)

This coarse-graining procedure, yielding both the Markov process

S_{C}^{x} (t) = (Y_{C} (t) | X = x)

and the embedded sequence of states

S_{C}^{x} [k]

⁠, has two advantages. First, the resultant addition and removal rates obey the constraints

Φ^{+} (m_{l + 1}, x, y) = Φ_{l}^{+} (x, m_{l} m_{l + 1})

and

Φ^{-} (x, y) = Φ_{l}^{-} (x, m_{l - 1} m_{l})

⁠, facilitating the use of Gaspard’s iterated function system,³² which we will describe shortly. Second, it preserves the distribution of the sequence of coarse-grained states visited:

S_{C}^{x} [k] = {\tilde{S}}_{F}^{x} [k]

⁠. Hence, if we are interested in the distribution of complete polymers (defined as the distribution of polymers obtained after complete detachment from the template) and derivative quantities such as error rate, then we can work with the Markovian coarse-grained processes

S_{C}^{x} (t)

and

S_{C}^{x} [k]

⁠, while completely ignoring the fine-grained steps other than when calculating

Λ_{x}^{\pm} ({y, y}^{*})

and A_x(y, y*). However, in general,

S_{C}^{x} (t) \neq {\tilde{S}}_{F}^{x} (t)

⁠, which means that polymer completion times and velocities are not preserved by this mapping. In the presence of cycles in the fine-grained steps, free-energy consumption will also not be preserved (this fact is less important for us as our fine-grained system has no cycles). Qureshi et al. describe a method for recovering these properties in the homogeneous case,⁴³ but we need to derive an additional result (Sec. II D) in order to adapt his method to the heterogeneous case.

Having argued that quantities of the fine-grained model can be obtained from solutions of a Markovian coarse-grained model, we now proceed to discuss the method we use to actually solve for quantities of the coarse-grained model. Gaspard and Andrieux showed that the distribution of consecutive copy monomers, conditioned on the entire template, is Markovian if transition rates are only dependent on tip states.^30,32 Moreover, the conditional probability distributions P_x,l(m_l|m_l−1) for complete polymers given a fixed template x and position indices l can be obtained through the solution of an iterated function system. We discuss here a brief intuition for Gaspard’s iterated function system³² in terms of absorbing probabilities of Markov chains and performing a full derivation for our system in Appendix A (refer to Ref. 32 for a full treatment).

Consider a system at a state (x, &m_l−1m_l). We can construct a Markov chain that includes all possible coarse-grained transitions from (x, &m_l−1m_l), including monomer addition to form (x, &m_l−1m_lm_l+1) and monomer removal to form (x, &m_l−1). (x, &m_l−1) is treated as an absorbing state. Trajectories that reach each of the monomer-added states (x, &m_l−1m_lm_l+1) must eventually either get absorbed into a complete polymer (x, &m_l−1m_lm_l+1…m_L0) with a probability of Q or undergo monomer removal to (x, &m_l−1m_l) with a probability of 1 − Q. Due to the local nature of propensities Φ⁺ and Φ⁻, Q cannot depend on &m_l−1 and can be written as a function Q_x,l+1(m_lm_l+1). Refer to Fig. 2 for an example. Assuming that we know all the values of Q_x,l+1(m_lm_l+1) for this Markov chain centered on (x, &m_l−1m_l), we can calculate the absorption probability into the backward absorbing state R_x,l(−|m_l−1m_l) and each of the forward absorbing states R_x,l(m_l+1|m_l−1m_l) through standard methods (details for a generic chain of the form in Fig. 2 are given in Appendix A). Then,

Q_{x, l} (m_{l - 1} m_{l}) = Σ_{m_{l + 1}} R_{x, l} (m_{l + 1} | m_{l - 1} m_{l}) .

(2)

FIG. 2.

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Markov chain absorption probability problem for the calculation of complete polymer distributions with a length l product polymer as an initial state. Single arrows (without corresponding reverse counterparts) represent the transitions to absorbing states (monomer removal to a copy of length l − 1 and copy completion to some complete polymer for some given monomer at position l + 1), and the probability of absorption to each absorbing state is considered. The length of the copy for each state is given at the top of the figure (note the jump from l + 1 to L, indicating absorption into a complete and fully detached polymer). The p_x,l(m_l+1|m_l−1m_l) variables are local transition probabilities that can be obtained by considering coarse-grained propensities $Φ_{l}^{+}$ or $Φ_{l}^{-}$ ⁠. The Q variables are local probabilities of absorption (into some arbitrary complete polymer) occurring before monomer removal. The absorption probabilities R_x,l(m_l+1|m_l−1m_l) follow from values of Q for the next position index Q_x,l+1(m_lm_l+1). The Q values (since these are the probabilities of absorption before removal) for the current iteration step can then be calculated by summing over all the non-backward absorption probabilities; in this case, Q_x,l(21) = R_x,l(1|21) + R_x,l(2|21). With this current Q value, the next step of the backward iteration may proceed.

As each R_x,l(m_l+1|m_l−1m_l) is wholly determined by local propensities

Φ_{l}^{+} (x, m_{l} m_{l + 1})

and

Φ_{l}^{-} (x, m_{l - 1} m_{l})

as well as Q_x,l+1(m_lm_l+1), Eq. implies that Q_x,l(m_l−1m_l) is obtainable from Q_x,l+1(m_lm_l+1) and the local propensities. From this construction, the entire ensemble of probabilities Q_x,l(m_l−1m_l) can be obtained through a backward iteration. These variables can in turn be used to obtain conditional monomer probabilities (in the ensemble of complete polymers) P_x,l+1(m_l+1|m_l) as follows:

P_{x, l + 1} (m_{l + 1} | m_{l}) = \frac{Φ_{l}^{+} (x, m_{l} m_{l + 1}) Q_{x, l + 1} (m_{l} m_{l + 1})}{Σ_{m_{l + 1}^{'}} Φ_{l}^{+} (x, m_{l} m_{l + 1}^{'}) Q_{x, l + 1} (m_{l} m_{l + 1}^{'})} .

(3)

This form for P_x,l+1(m_l+1|m_l) comes from analyzing a modified version of the Markov chain in Fig. 2 that completely omits the probability of moving back to (x, &m_l−1). This procedure works because the rate of production of complete copies …m_lm_l+1…0 is the rate of addition of m_l+1 after m_l such that m_l is never removed again ( Appendix A and Ref. 32 have further details).

For the very first monomer, analysis of these Markov chains results in a row vector of probabilities P_x,1 representing the distribution of monomers in the first index for complete polymers. Conditional distributions for other template positions are defined by matrices

P_{x, l}

⁠. The absolute distributions P_x,l can, therefore, be obtained by serial multiplication of the conditional probability matrices as follows:

P_{x, l} = P_{x, 1} \prod_{j = 2}^{l} P_{x, j} .

(4)

Once we define correct and incorrect monomer pairings, these absolute distributions can then be used to define an error probability. In this work, we will consider systems where the copy (excluding the end of copy character 0) and template monomers draw from the same set {1, 2}, and pairings are considered correct if the copy monomer at a given location equals the template monomer.

D. Recovering further properties of the fine-grained system from the coarse-grained system

While the probabilities P_x,l can be directly obtained from analysis of the coarse-grained system, other properties of interest, such as the average free-energy consumption and time to completion, cannot be directly obtained in this manner, as they are dependent on the details of the fine-grained system. However, if an observable O(x) is extensive in the number of visits to coarse-grained tip states (x, m_l−1m_l, l) (for instance, the time elapsed and the free energy consumed), then we can evaluate its expectation by first performing a count of the local tip states visited. Let V_x,l(m_l−1m_l) be a random variable representing the number of visits to a tip state (x, m_l−1m_l, l), and let O_x,l(m_l−1m_l) be a random variable representing the value of the observable O for a given visit of the tip state (x, m_l−1m_l, l). Throughout this paper, we use angled brackets ⟨⟩ to refer to per-site averages (i.e., averages over l) in the L → ∞ limit for a stationary distribution of templates X (that is, joint probability distributions of monomers in X at some fixed distance from each other do not change with polymer length; in practice we use Bernoulli template distributions throughout, which are always stationary) and assuming a self-averaging variable. On the other hand, we use the expectation operator

E []

(without any subscripts) acting on a variable with dependencies on x or l of the form O_x,l, which refers to expected values associated with a particular position l for some fixed x.

E_{V} []

is an expectation for each tip state visit. Then,⁴³

⟨ O ⟩ = \lim_{L \to \infty} \frac{1}{L} Σ_{l = 1}^{L} Σ_{m_{l - 1} m_{l}} E [V_{x, l} (m_{l - 1} m_{l})] E_{V} [O_{x, l} (m_{l - 1} m_{l})] .

(5)

Since our fine-grained model lacks cycles, we do not need to perform this procedure to calculate the free energy consumed for each visit. In Appendix B, we show how the expected waiting time for each visit of a given coarse-grained copy tip state

E_{V} [θ_{x, l} (m_{l - 1} m_{l})]

can be calculated for our specific case.

E [V_{x, l + 1} (m_{l} m_{l + 1})]

can be obtained through the following iteration:

\begin{align} E [V_{x, l + 1} (m_{l} m_{l + 1})] & = \sum_{m_{l - 1}} E [V_{x, l + 1} (m_{l}, m_{l + 1}) | m_{l - 1} m_{l}] \\ \times E [V_{x, l} (m_{l - 1} m_{l})] . \end{align}

(6)

Here,

E [V_{x, l} (m_{l}, m_{l + 1}) | m_{l - 1}, m_{l}]

is the expected number of visits to tip state (x, m_lm_l+1, l + 1) from each visit of (x, m_l−1m_l, l).

E [V_{x, l} (m_{l}, m_{l + 1}) | m_{l - 1}, m_{l}]

can be obtained from the Markov chain constructed in Fig. 3. This Markov chain is similar to that in Fig. 2 with two important changes. First, instead of considering absorption probabilities from (x, &m_l−1m_lm_l+1), we further extend the chain to (x, &m_l−1m_lm_l+1m_l+2) before allowing absorption (these absorption probabilities are again the Q probabilities from Gaspard’s iteration in Sec. II C), allowing visits to (x, &m_l−1m_lm_l+1) from (x, &m_l−1m_lm_l+1m_l+2) to be counted. Second, monomer removal from (x, &m_l−1m_lm_l+1) results in an artificial absorbing state instead of returning to (x, &m_l−1m_l). This approach prevents the double counting of visits to (x, &m_l−1m_lm_l+1) from distinct visits of (x, &m_l−1m_l). Note that we have not imposed any additional assumptions unique to our system for this iteration to work, and it is applicable to any polymerization system whose coarse-grained form (applying Qureshi et al.’s procedure⁴³) obeys the assumptions in Ref. 32. The derivation of a general form for

E [V_{x, l + 1} (m_{l} m_{l + 1}) | m_{l - 1} m_{l}]

is provided in Appendix D.

FIG. 3.

View large Download slide

Markov chain from which expected visitation counts per visit of the previous tip state $E [V_{x, l} (m_{l}, m_{l + 1}) | m_{l - 1}, m_{l}]$ can be calculated. Solid arrows represent the local transition probabilities that can be calculated from coarse-grained rates Φ⁺ or Φ⁻, while dotted arrows represent the Q variables obtained via Gaspard’s iteration³² (Fig. 2). To avoid over-counting, the tip states whose visitation counts are of interest are not permitted to move back into the initial state; rather, they point to a virtual absorbing state at a rate equal to the rate of monomer removal. Complete polymers are again treated as absorbing states. Starting from the initial state, we can now count, through standard methods, the average number of times either tip state of interest [denoted by “ $T i p {(m_{l}, m_{l} + 1)}_{l}$ ”] is visited before either the initial tip state is revisited or polymerization terminates.

E. Thermodynamically consistent parameterization for heterogeneous separating copiers

Refer back to Fig. 1(b) and Table I for the rates of our fine-grained system. In this section, we will assign parameters to these fine-grained rates such that thermodynamic consistency is maintained. We first assume that monomer concentrations [M] are equal and chemostatted (unchanging over the copy process). We assume that binding of a monomer to a template obeys mass-action, so $K_{bind}^{+} (n_{l - 1} n_{l}, m_{l - 1} m_{l}) = k_{bind} (n_{l - 1} n_{l}, m_{l - 1}) [M]$ ⁠. The rate constants for monomer binding are set to be invariant between the different tip states, as they are chemically difficult to tune, so $K_{bind}^{+} (n_{l - 1} n_{l}, m_{l - 1} m_{l}) = k_{bind} [M]$ ⁠. Tail binding is similarly difficult to tune and obeys a pseudo-mass action rule where rates are proportional to an effective local concentration of the tail monomer around the tip of the growing polymer, so $K_{tail}^{+} (n_{l - 1} n_{l}, m_{l - 1} m_{l}) = k_{tail} {[M]}_{eff}$ ⁠. Monomer polymerization rates, on the other hand, can conceivably be engineered, and here we envision a parameterization $K_{pol}^{+} (n_{l - 1} n_{l}, m_{l - 1} m_{l}) = k_{pol} (n_{l - 1} n_{l}, m_{l - 1} m_{l})$ ⁠.

Since copying terminates on detachment, there is a net flow of molecules producing complete polymers, so the system does not satisfy a global detailed balance and is thus out of equilibrium. However, each fine-grained reaction step constitutes an elementary reaction step, and hence a generalized local detailed balance condition applies as follows:⁴⁹

\frac{k^{+}}{k^{-}} = \exp (\frac{- Δ G + δ}{k_{B} T}) .

(7)

Here, k⁺ is the rate of some elementary forward reaction, k⁻ is the rate of its corresponding backward reaction, ΔG is the free-energy change of the forward reaction, δ is an external driving, k_B is Boltzmann’s constant, and T is the reservoir temperature. This generalized local detailed balance condition, along with the need for the polymer to separate as it copies, enforces constraints on the rates of the fine-grained reactions based on the free-energy changes incurred during transitions. Note that we use negative free energies when defining free-energy parameters (i.e., more positive parameter values correspond to greater stability) and then include the minus signs in the rate expressions. Furthermore, there is a single global heat reservoir, and all free-energy terms are measured in units of k_BT.

As in Ref. 35, we assume the binding free energy between monomers n_l and m_l can be divided into a specific monomer-dependent free energy ΔG_TT(n_l, m_l) and a generic monomer-independent free energy ΔG_gen [Fig. 1(b)]. Coupling the breaking of this generic bond to backbone formation was found to reduce product inhibition and allow detachment.³⁵ Monomer binding incurs a free-energy change of −ln[M] − ΔG_TT(n_l, m_l) − ΔG_gen, polymerization incurs a free-energy change of −ΔG_BB + ΔG_gen, and tail unbinding results in a net free-energy change of ΔG_TT(n_l−1, m_l−1) + ln[M]_eff.

Due to the need to separate, copy–template interactions must be transient. Observe that ΔG_gen cancels during the incorporation of a single monomer. Similarly, the −ΔG_TT(n_l, m_l) free-energy change from monomer binding is offset by the ΔG_TT(n_l, m_l) change from tail unbinding during the incorporation of the next monomer. The persistent net polymerization free-energy change per incorporated monomer is then $- Δ G_{pol} = - Δ G_{B B} - \ln \frac{[M]}{{[M]}_{eff}}$ ⁠, free of any ΔG_TT or ΔG_gen terms and, hence, consistent with transient copy–template interactions. The resulting forward and backward rates, with the ratio determined by generalized local detailed balance, are summarized in Table II.

TABLE II.

Parameterized forms of each of the fine-grained reaction steps after local detailed balance is imposed and further simplifications are made.

Forward reaction	Forward rate	Parameterized form	Backward rate	Parameterized form
Binding	$K_{bind}^{+} (n_{l - 1} n_{l}, m_{l - 1} m_{l})$	k_bind[M]	$K_{bind}^{-} (n_{l - 1} n_{l}, m_{l - 1} m_{l})$	$k_{bind} e^{- Δ G_{T T} (n_{l}, m_{l}) - Δ G_{gen}}$
Polymerization	$K_{pol}^{+} (n_{l - 1} n_{l}, m_{l - 1} m_{l})$	k_pol(n_l−1n_l, m_l−1m_l)	$K_{pol}^{-} (n_{l - 1} n_{l}, m_{l - 1} m_{l})$	$k_{pol} (n_{l - 1} n_{l}, m_{l - 1} m_{l}) e^{- Δ G_{B B} + Δ G_{gen}}$
Tail unbinding	$K_{tail}^{-} (n_{l - 1} n_{l}, m_{l - 1} m_{l})$	$k_{tail} e^{- Δ G_{T T} (n_{l - 1}, m_{l - 1})}$	$K_{tail}^{+} (n_{l - 1} n_{l}, m_{l - 1} m_{l})$	k_tail[M]_eff

Forward reaction	Forward rate	Parameterized form	Backward rate	Parameterized form
Binding	$K_{bind}^{+} (n_{l - 1} n_{l}, m_{l - 1} m_{l})$	k_bind[M]	$K_{bind}^{-} (n_{l - 1} n_{l}, m_{l - 1} m_{l})$	$k_{bind} e^{- Δ G_{T T} (n_{l}, m_{l}) - Δ G_{gen}}$
Polymerization	$K_{pol}^{+} (n_{l - 1} n_{l}, m_{l - 1} m_{l})$	k_pol(n_l−1n_l, m_l−1m_l)	$K_{pol}^{-} (n_{l - 1} n_{l}, m_{l - 1} m_{l})$	$k_{pol} (n_{l - 1} n_{l}, m_{l - 1} m_{l}) e^{- Δ G_{B B} + Δ G_{gen}}$
Tail unbinding	$K_{tail}^{-} (n_{l - 1} n_{l}, m_{l - 1} m_{l})$	$k_{tail} e^{- Δ G_{T T} (n_{l - 1}, m_{l - 1})}$	$K_{tail}^{+} (n_{l - 1} n_{l}, m_{l - 1} m_{l})$	k_tail[M]_eff

Now that the fine-grained system has been parameterized, we can proceed to define transition propensities.⁴³ These propensities are derived in full in Appendix B, and the results are given in Eqs. and . Due to the forms taken by our fine-grained steps, we can impose a more restrictive condition on locality than required by Gaspard’s method: that the propensities are also dependent only on local template monomers, as well as copy monomers, Φ⁺(m_l+1, x, &m_l−1m_l) = Φ⁺(n_ln_l+1, m_lm_l+1) and Φ⁻(n_l−1n_l, &m_l−1m_l) = Φ⁻(n_l−1n_l, m_l−1m_l). We find

\begin{align} Φ^{+} (n_{l - 1} n_{l}, m_{l - 1} m_{l}) \\ = \frac{k_{bind} k_{pol} (n_{l - 1} n_{l}, m_{l - 1} m_{l}) k_{tail} e^{- Δ G_{T T} (n_{l}, m_{l})} [M]}{\begin{matrix} e^{- Δ G_{B B} - Δ G_{T T} (n_{l}, m_{l})} k_{bind} k_{pol} (n_{l - 1} n_{l}, m_{l - 1} m_{l}) + e^{- Δ G_{T T} (n_{l - 1}, m_{l - 1}) - Δ G_{T T} (n_{l}, m_{l}) - Δ G_{gen}} k_{bind} k_{tail} \\ + e^{- Δ G_{T T} (n_{l - 1}, m_{l - 1})} k_{pol} (n_{l - 1} n_{l}, m_{l - 1} m_{l}) k_{tail} \end{matrix}}, \end{align}

(8)

\begin{align} Φ^{-} (n_{l - 1} n_{l}, m_{l - 1} m_{l}) \\ = \frac{k_{bind} k_{pol} (n_{l - 1} n_{l}, m_{l - 1} m_{l}) k_{tail} e^{- Δ G_{B B} - Δ G_{T T} (n_{l}, m_{l})} {[M]}_{eff}}{\begin{matrix} e^{- Δ G_{B B} - Δ G_{T T} (n_{l}, m_{l})} k_{bind} k_{pol} (n_{l - 1} n_{l}, m_{l - 1} m_{l}) + e^{- Δ G_{T T} (n_{l - 1}, m_{l - 1}) - Δ G_{T T} (n_{l}, m_{l}) - Δ G_{gen}} k_{bind} k_{tail} \\ + e^{- Δ G_{T T} (n_{l - 1}, m_{l - 1})} k_{pol} (n_{l - 1} n_{l}, m_{l - 1} m_{l}) k_{tail} \end{matrix}} . \end{align}

(9)

The binding rate constants k_bind and k_tail are unlikely to be very different from each other (although binding and unbinding rates can be tuned by the concentration terms), and hence we can set k_bind = k_tail. To start with, we assume a sequence-independent k_pol, although we later relax this assumption. We begin by investigating the following limits:

Slow binding and unbinding of the free monomers, [M] → 0 and ΔG_gen, ΔG_BB → ∞. For simplicity, we let $[M] = e^{- Δ G_{gen}}$ and we normalize rates such that k_bind[M] = 1. Then,
$Φ^{+} (n_{l - 1} n_{l}, m_{l - 1} m_{l}) = 1,$
(10)
$Φ^{-} (n_{l - 1} n_{l}, m_{l - 1} m_{l}) = e^{- Δ G_{pol} + Δ G_{T T} (n_{l - 1}, m_{l - 1}) - Δ G_{T T} (n_{l}, m_{l})} .$
(11)
Slow polymerization, k_pol ≪ k_bind. Here, rates are normalized so that k_pol[M] = 1. Then,
$Φ^{+} (n_{l - 1} n_{l}, m_{l - 1} m_{l}) = e^{Δ G_{T T} (n_{l}, m_{l})},$
(12)
$Φ^{-} (n_{l - 1} n_{l}, m_{l - 1} m_{l}) = e^{- Δ G_{pol} + Δ G_{T T} (n_{l - 1}, m_{l - 1})} .$
(13)

Case 1 corresponds to a discrimination on backward propensities (consistent with the “temporary thermodynamic discrimination” described in Ref. 33). Case 2, on the other hand, corresponds to a thermodynamically driven discrimination on forward propensities (consistent with “combined kinetic and thermodynamic discrimination,” as described in Ref. 33).

III. RESULTS

A. Consequences of separation for the velocity and energy landscapes of heterogeneous copying

We wish to compare the velocity profile of separating templated copolymerization (STC) and non-separating templated copolymerization (NTC; note the phrase “templated self-assembly” has been used in some literature³³) as studied by Gaspard.³² In Fig. 4(a), we illustrate a fine-grained model of NTC, which we will compare against our model of STC. The fine-grained dynamics is similar to that in Sec. II E, with the exception that the tail binding step has been removed. We assume here that the binding free energy of any given monomer pair is independent of any other monomer pair, giving rise to the parameterization in Table III. We consider here the slow binding limit, leading to the following coarse-grained propensities:

Φ^{+} (n_{l - 1} n_{l}, m_{l - 1} m_{l}) = 1,

(14)

Φ^{-} (n_{l - 1} n_{l}, m_{l - 1} m_{l}) = e^{- Δ G_{pol} - Δ G_{P T} (n_{l}, m_{l})} .

(15)

This choice again corresponds to discrimination on backward propensities. We denote the sequence-independent non-equilibrium drive provided by monomer binding and backbone formation by ΔG_pol = ΔG_BB + ln[M] (similar to copying with separation, but with the [M]_eff term absent). However, observe that in the absence of separation, the sequence-dependent free-energy change ΔG_PT incurred on the addition of each monomer persists as the copy is produced (note ΔG_TT was used in Ref. 33 to refer to a “temporary thermodynamic” discrimination factor, which is no longer present after separation. To maintain consistency with the intended meaning in Ref. 33, we have introduced a “permanent thermodynamic” discrimination factor ΔG_PT for NTC).

FIG. 4.

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Fine-grained steps for a model of copying where the copy does not separate from the template. Only monomers in the vicinity of the copy tip are depicted. (a) New monomers bind and polymerize in separate steps, but there is no tail unbinding of the previously added monomer. (b) Labeling of fine-grained states associated with a given coarse-grained state. Indices f are denoted by the bold bracketed numbers. Transitions to other states of index f = 0 (denoted here by the dotted arrows) change the coarse-grained state (x, y).

TABLE III.

Parameterized forms of each of the fine-grained reaction steps for a model of copying where the copy does not separate from the template.

Forward reaction	Forward rate	Backward rate
Binding	$K_{bind}^{+} (n_{l - 1} n_{l}, m_{l - 1} m_{l})$ k_bind[M]	$K_{bind}^{-} (n_{l - 1} n_{l}, m_{l - 1} m_{l})$ $k_{bind} e^{- Δ G_{P T} (n_{l}, m_{l})}$
Polymerization	$K_{pol}^{+} (n_{l - 1} n_{l}, m_{l - 1} m_{l})$ k_pol(n_l−1n_l, m_l−1m_l)	$K_{pol}^{-} (n_{l - 1} n_{l}, m_{l - 1} m_{l})$ $k_{pol} (n_{l - 1} n_{l}, m_{l - 1} m_{l}) e^{- Δ G_{B B}}$

Forward reaction	Forward rate	Backward rate
Binding	$K_{bind}^{+} (n_{l - 1} n_{l}, m_{l - 1} m_{l})$ k_bind[M]	$K_{bind}^{-} (n_{l - 1} n_{l}, m_{l - 1} m_{l})$ $k_{bind} e^{- Δ G_{P T} (n_{l}, m_{l})}$
Polymerization	$K_{pol}^{+} (n_{l - 1} n_{l}, m_{l - 1} m_{l})$ k_pol(n_l−1n_l, m_l−1m_l)	$K_{pol}^{-} (n_{l - 1} n_{l}, m_{l - 1} m_{l})$ $k_{pol} (n_{l - 1} n_{l}, m_{l - 1} m_{l}) e^{- Δ G_{B B}}$

We begin by considering how the velocity profiles for STC change with increasing nonequilibrium drive ΔG_pol. As with NTC, we consider backward propensity discrimination (that is, the slow binding limit). The binding free energies of the incorrect monomer (we remind readers that a correct pair is defined by m_l = n_l) pairs are kept constant, such that ΔG_TT(1, 2) = ΔG_TT(2, 1) = 0. The free energy of a correct monomer 1 pair is held at ΔG_TT(1, 1) = 2, while ΔG_TT(2, 2) is varied from 2 to 10. Gaspard’s iterated function system³² was applied to obtain the absorption probabilities Q for the copying of a polymer sequence of length L = 10 000. Then, the methods in Sec. II D and Ref. 43 were used to find τ_x,l, defined as the average amount of total time spent at position l for a template sequence x before a complete polymer forms and detaches. τ_x,l is calculated by multiplying visits to each state by the waiting time for each visit to said state and then summing over all states of copy length l. The average completion times t_c = Σ_lτ_x,l can then be calculated, and the average copying velocity was calculated as $v = \frac{L}{t_{c}}$ ⁠. As in Ref. 33, the value of ΔG_pol at equilibrium is given by ΔG_pol,eq = −ln 2, and we report copy velocities as a function of the difference in ΔG_pol from this equilibrium value up to ΔG_pol = 2 in Fig. 5(b).

FIG. 5.

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Comparison of velocity profiles and free energy landscapes for STC and NTC. Graphs of normalized average velocity as a function of ΔG_pol − ΔG_pol,eq are shown for NTC in (a) and STC in (b). Equilibrium G_x,l − ⟨G⟩ free-energy landscapes (at ΔG_pol = ΔG_pol,eq) are shown for NTC in (c) and STC (d). A zoom inset is provided in (d). The long tails of zero velocity for copying without separation, caused by a rougher free-energy profile, do not occur for copying with simultaneous separation, where the velocity profile is linear close to ΔG_pol − ΔG_pol,eq = 0.

We now consider velocity profiles for NTC. We must first calculate the equilibrium point ΔG_pol,eq, which follows from the partition function of the product. As the monomer binding free energies are uncorrelated in our NTC system, the (negative of) free energy G_x,l of the ensemble of length l product polymers can be calculated as follows:

z_{1} = e^{Δ G_{P T} (1, 1)} + e^{Δ G_{P T} (2, 1)},

(16)

z_{2} = e^{Δ G_{P T} (2, 2)} + e^{Δ G_{P T} (1, 2)},

(17)

Z_{x, l} = Z_{x, 1} e^{(l_{1} + l_{2}) Δ G_{pol}} z_{1}^{l_{1}} z_{2}^{l_{2}},

(18)

G_{x, l} = \ln [Z_{l}] .

(19)

Here, z_i is the partition function for a template monomer of type i over its copies, Z_x,l is the partition function of length l products for a template x, Z_x,1 is the partition function of the very first copy monomer, l₁ and l₂ are the numbers of monomers of types 1 and 2, respectively, in the template x up to index l excluding the first monomer, and l = l₁ + l₂ + 1. We now introduce a new expectation operator

E_{x}

averaging over all templates x for a fixed template distribution. Then, for equilibrium between polymer growth and shrinking, the system free energy averaged over templates

E_{x} [G_{x, l}]

should not change with length index l. Furthermore, the average over positional indices in the L → ∞ limit ⟨G⟩ should be the same as the per-site expectation

E_{x} [G_{x, l}]

⁠,

E_{x} [G_{x, l}] = E_{x} [G_{x, l}] = ⟨ G ⟩ .

(20)

As the system free energy is self-averaging, the equilibrium drive ΔG_pol,eq can be calculated as follows:

Δ G_{pol,eq} = \lim_{l, L \to \infty} \frac{- \ln z_{1}^{l_{1}} z_{2}^{l_{2}}}{(l_{1} + l_{2})} .

(21)

We plot the normalized copy velocity (obtained by dividing velocity by its maximum value over the range of ΔG_pol considered; in practice this maximum value always occurs at our maximum considered drive ΔG_pol = 0)

\bar{v} = v / (\max_{Δ G_{pol}} v)

as a function of ΔG_pol − ΔG_pol,eq for NTC in Fig. 5(a). Comparing Figs. 5(a) and 5(b), we observe a few common features. First, velocity tends to zero as the equilibrium ΔG_pol is reached. At equilibrium, there should be no net flux, and hence this behavior is expected. Copy velocity increases with increasing ΔG_pol, as the rate of monomer removal decreases.

However, the models approach zero velocity in different ways. As noted by Gaspard,³² the heterogeneous model of NTC displays a long tail of near-zero velocity as it approaches equilibrium. Such behavior is not observed in our model of heterogeneous copying [Fig. 5(b)]. This difference can be explained by considering sample free-energy landscapes in the heterogeneous NTC model and the STC model. We plot free-energy deviation G_x,l − ⟨G⟩ for a fixed sample template at various copy lengths l for both heterogeneous NTC [Fig. 5(c)] and heterogeneous STC [Fig. 5(d)].

For heterogeneous STC, the free-energy deviations switch stochastically between two levels, determined by the template monomer at position l, and G_x,l − ⟨G⟩ is, therefore, constrained to be close to zero. By contrast, the free-energy deviations for heterogeneous NTC are large. On scales of length l_d we expect to see barriers of height $\sim \sqrt{l_{d}}$ ⁵⁰ that trap the system, resulting in slow sub-diffusive motion close to equilibrium driving.^32,50 Hence, we observe a region of zero velocity in Fig. 5(a). These tall barriers do not occur for separating copiers, as the length of the product interacting with the template at any given time is bounded. In our system, only two monomers may be bound at a given time, but this intuition generalizes to other parameter sets and even completely different models that include separation as the copy grows (even realistic models of transcription and translation). If only a finite number of copy monomers bind to the template simultaneously, the roughness of the free-energy landscape associated with sequence heterogeneity is inherently limited.

B. Effects of separation and heterogeneity on error probability and average time spent per site

We now investigate the effect of heterogeneity on two key indicators of a copy system: the error probability, ϵ_x,l, and the total time spent at a site l before the monomer is permanently incorporated, τ_x,l (contrast with θ_x,l, the waiting time for each visit of a tip state). We also consider visitations to a given position V_x,l for reasons that will become apparent. As indicated by the subscripts, these three quantities depend on the positional index l underlying template x. We assume these properties are self-averaging if the underlying template distribution is stationary,³² such that in the limit L → ∞, $⟨ ϵ ⟩ = \frac{Σ_{l = 1}^{L} ϵ_{x, l}}{L}$ ⁠, $⟨ V ⟩ = \frac{Σ_{l = 1}^{L} V_{x, l}}{L}$ ⁠, and $⟨ τ ⟩ = \frac{Σ_{l = 1}^{L} τ_{x, l}}{L}$ are well-defined. Throughout this section, we will calculate these averages for a single long template x of length L = 10⁵ whose monomers are drawn from a Bernoulli distribution B(L, p_t) with p_t representing the average proportion of monomers of type 2, similar to Ref. 32.

The quantities ⟨ϵ⟩, ⟨V⟩, and ⟨τ⟩ are then dependent on p_t, the proportion of monomers of type 2. Consider now random variables ϵ₁ and ϵ₂. ϵ₁ is the error rate (in the large L limit) for copying a homogeneous template with monomers of type 1, while ϵ₂ is the error rate for copying a homogeneous template with monomers of type 2. Let ϵ_I,x,l be a random variable that takes on a value ϵ₁ if the monomer at position l for template x is of type 1, and ϵ₂ if the monomer at position l for template x is of type 1. In the long L limit, for a given probability p_t of monomer 2, then ⟨ϵ_I⟩ = (1 − p_t)ϵ₁ + p_tϵ₂. Similarly, we can apply analogous definitions to τ and V such that ⟨V_I⟩ = (1 − p_t)V₁ + p_tV₂ and ⟨τ_I⟩ = (1 − p_t)τ₁ + p_tτ₂. If the copying of subsequent monomers were independent of each other, we would expect ⟨ϵ⟩ = ⟨ϵ_I⟩, ⟨V⟩ = ⟨V_I⟩, and ⟨τ⟩ = ⟨τ_I⟩. However, these equalities generally do not hold due to inter-monomer correlations in the product.^30,32,33 To infer whether heterogeneity tends to improve or worsen copying performance for a particular parameter set, we now consider, for specific parameter regimes, the sign of log-ratio-averages $\ln \frac{⟨ ϵ ⟩}{⟨ ϵ_{I} ⟩}$ ⁠, $\ln \frac{⟨ V ⟩}{⟨ V_{I} ⟩}$ ⁠, and $\ln \frac{⟨ τ ⟩}{⟨ τ_{I} ⟩}$ ⁠. As an example, $\ln \frac{⟨ ϵ ⟩}{⟨ ϵ_{I} ⟩} > 0$ would imply that for a given parameter set, heterogeneity in monomer interactions tends to increase average errors.

We will separately consider both limits mentioned in Sec. II E, that is, the slow binding limit with backward propensity discrimination and the slow k_pol limit with forward propensity discrimination. Where parameters tend to ∞, a multiplier of e¹² is used for the numerics. For the backward propensity discrimination case, [M] = e⁻¹² and k_bind = e¹². In Sec. III B 1, we consider the case where the interactions between correct monomer pairs (1, 1) and (2, 2) are made heterogeneous, while in Sec. III B 2, we consider the case where the interactions between incorrect pairs (2, 1) and (1, 2) are made heterogeneous. For each section, we will be plotting $\ln \frac{⟨ ϵ ⟩}{⟨ ϵ_{I} ⟩}, \ln \frac{⟨ τ ⟩}{⟨ τ_{I} ⟩}$ and $\ln \frac{⟨ V ⟩}{⟨ V_{I} ⟩}$ as a function of p_t, and we provide a framing for our findings in Sec. III B 3.

1. Heterogeneity in correct monomer interactions

Consider the coarse-grained dynamics given in Eqs. (10)–(13). To investigate the effects of heterogeneous correct monomer interactions, we fix ΔG_TT(2, 2) = 2 and ΔG_TT(1, 2) = ΔG_TT(2, 1) = 0, while varying ΔG_TT(1, 1) from 2 to 10. Keep in mind that with backward propensity discrimination, ΔG_TT modifies the rates of monomer unbinding, while in the case of forward propensity discrimination, ΔG_TT modifies the rate of monomer binding (Sec. II E). Throughout, ΔG_pol is arbitrarily held at 0, which corresponds to relatively weak but non-zero driving.

a. Backward propensity discrimination.

Plots are shown in Figs. 6(a)–6(c). We see that $\ln \frac{⟨ ϵ ⟩}{⟨ ϵ_{I} ⟩} > 0$ ⁠, implying that in this regime errors tend to be increased by heterogeneity. The trend in $\ln \frac{⟨ τ ⟩}{⟨ τ_{I} ⟩}$ is more complex, as there appears to be a small dip below 0 for high p_t. Note that the time spent per site is obtained by modulating the total number of state visits with the average waiting time at each tip state (Sec. II D). If we instead turn our attention to the visitation counts, we once again obtain $\ln \frac{⟨ V ⟩}{⟨ V_{I} ⟩} > 0$ ⁠, implying that heterogeneity in this regime tends to increase the number of tip states visited before completion.

FIG. 6.

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Deviations in ϵ, τ, and V due to heterogeneity in correct monomer interactions, as a function of the monomer 2 content p_t, relative to homogeneous copying. Log-ratio-means $\ln \frac{⟨ ϵ ⟩}{⟨ ϵ_{I} ⟩}$ ⁠, $\ln \frac{⟨ τ ⟩}{⟨ τ_{I} ⟩}$ ⁠, and $\ln \frac{⟨ V ⟩}{⟨ V_{I} ⟩}$ are plotted for backward propensity discrimination in (a)–(c) and forward propensity discrimination in (d)–(f).

b. Forward propensity discrimination.

Plots are shown in Figs. 6(d)–6(f). Again, $\ln \frac{⟨ ϵ ⟩}{⟨ ϵ_{I} ⟩} > 0$ ⁠, so errors tend to increase in this regime due to heterogeneity. Here, both time spent per site τ and the number of state visits tend to increase as a result of heterogeneity.

2. Heterogeneity in incorrect monomer interactions

We now consider the case where the binding strengths of incorrect monomers are heterogeneous, ΔG_TT(1, 2) ≠ ΔG_TT(2, 1) while ΔG_TT(1, 1) = ΔG_TT(2, 2) = 6. ΔG_pol is again arbitrarily held at 0. ΔG_TT(2, 1) = 2 is kept constant and ΔG_TT(1, 2) is varied from 0 to 6.

a. Backward propensity discrimination.

Plots are shown in Figs. 7(a)–7(c). We see that $\ln \frac{⟨ ϵ ⟩}{⟨ ϵ_{I} ⟩} < 0$ ⁠, implying that in this regime errors tend to be decreased by heterogeneity. Unlike in the case where heterogeneity is applied to the binding free energies of correct pairs, both $\ln \frac{⟨ τ ⟩}{⟨ τ_{I} ⟩} < 0$ and $\ln \frac{⟨ V ⟩}{⟨ V_{I} ⟩} < 0$ ⁠. Hence, heterogeneity here tends to decrease visitation counts. The graphs have similar shapes, and hence waiting time modulation does not result in qualitative differences in the total time spent per site.

FIG. 7.

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Deviations in ϵ, τ, and V due to heterogeneity in incorrect monomer interactions, as a function of the monomer 2 content p_t, relative to homogeneous copying. Log-ratio-means $\ln \frac{⟨ ϵ ⟩}{⟨ ϵ_{I} ⟩}$ ⁠, $\ln \frac{⟨ τ ⟩}{⟨ τ_{I} ⟩}$ ⁠, and $\ln \frac{⟨ V ⟩}{⟨ V_{I} ⟩}$ are plotted for backward propensity discrimination in (a)–(c) and forward propensity discrimination in (d)–(f).

b. Forwards propensity discrimination.

Plots are shown in Figs. 7(d)–7(f). We see that there is no clear trend in $\ln \frac{⟨ ϵ ⟩}{⟨ ϵ_{I} ⟩}$ ⁠. Let us turn to a different measure, $⟨ \ln \frac{ϵ}{ϵ_{I}} ⟩$ ⁠. This average-log-ratio of errors is a measure of changes in relative error at each site that occur due to heterogeneity in monomer interactions. For example, $⟨ \ln \frac{ϵ}{ϵ_{I}} ⟩ > 0$ if the average factor of error increase for one monomer is greater than the average factor of error decrease for the other monomer. We will argue the significance of this error measure in Subsection III B 3. For now, observe that $⟨ \ln \frac{ϵ}{ϵ_{I}} ⟩ < 0$ in Fig. 8, implying that relative errors on each site are, on average, reduced. We continue to observe $\ln \frac{⟨ τ ⟩}{⟨ τ_{I} ⟩} < 0$ and $\ln \frac{⟨ V ⟩}{⟨ V_{I} ⟩} < 0$ ⁠.

FIG. 8.

Mean-log-ratios of error ⟨lnϵϵI⟩ for forward propensity discrimination with heterogeneity on incorrect monomers. ⟨lnϵϵI⟩ is negative for all values of pt and ΔGTT,12 with heterogeneous interactions.

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Mean-log-ratios of error $⟨ \ln \frac{ϵ}{ϵ_{I}} ⟩$ for forward propensity discrimination with heterogeneity on incorrect monomers. $⟨ \ln \frac{ϵ}{ϵ_{I}} ⟩$ is negative for all values of p_t and ΔG_TT,12 with heterogeneous interactions.

3. Discussion

Heterogeneity on the correct monomers tends to make copying both slower (up to modifications due to waiting time) and more error-prone, while heterogeneity on incorrect monomers tends to make copying faster and more accurate (in some cases, only the average relative error is made better). We now attempt to explain these results. We can divide the overall coarse-grained state-space into two: one in which the correct monomer is bound at the tip and one in which the incorrect monomer is bound at the tip. Due to the detachment of the copy behind the tip, this division is enough to specify the chemical free energy at each step (the same does not apply to models of NTC). Copying can then be thought of as a special example of 1D diffusion with a choice of the free-energy landscape at each step (Fig. 9). The addition of a correct monomer corresponds to moving through a more favorable landscape (the “correct landscape”), while the addition of an incorrect monomer involves moving through a less favorable landscape (the “incorrect landscape”). To understand the effects of heterogeneity, we only need to consider transitions with free-energy changes modified by heterogeneity (purple lines in Fig. 9), as all other transitions would be found in equivalent forms in corresponding homogeneous copying systems. Observe that the introduction of heterogeneity in the interaction of correct monomers results in an increased roughness of the correct landscape, while the introduction of heterogeneity in the interaction of incorrect monomers results in an increased roughness of the incorrect landscape. This relative roughness is particularly evident as the overall variability in ΔG_x,l is constrained within finite bounds [Fig. 5(d)].

FIG. 9.

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Conceptual diagram of the “choice of landscape” model. Three cases are considered: (a) Heterogeneous correct monomer interactions ΔG_TT = (2 0, 0 4). (b) Heterogeneous incorrect monomer interactions ΔG_TT = (4 0, 2 4). (c) Homogeneous copying ΔG_TT = (2 0, 0 2). The dashed red line represents the landscape that must be traversed to form incorrect pairings and the dashed blue line represents the landscape that must be traversed to form correct pairings. The dashed green line is a sample path including both correct and incorrect monomers. This sample path is colored purple instead when transitions occur within a landscape with free-energy changes altered as a result of heterogeneity. The dotted lines in graphs (a) and (b) represent the landscapes of homogeneous analogs (i.e., free energy landscapes of the two homogeneous systems that involve a template with only monomer 1 and only monomer 2). A template x = 1112212112 is used throughout, and the sample path taken by the dashed green line corresponds to y = 1212122121.

Upward slopes tend to slow down motion more than downward slopes tend to speed up motion, and this asymmetry drives the well-documented observation that rough landscapes tend to be harder to traverse than smoother ones.⁵⁰ A “choice of landscape” model, therefore, explains why heterogeneity in the interaction of correct monomers tends to increase errors—traversing the incorrect landscape becomes more favorable relative to the correct one (compared to a copying system with homogeneous interactions). Similarly, incorrect monomer interaction heterogeneity tends to reduce errors, as the incorrect landscape becomes harder to traverse. This effect is most robust when viewed in terms of averaging over relative errors at each site rather than average absolute errors. A sweep over copying systems with various heterogeneous discrimination factors for both backward and forward discrimination at ΔG_pol = 0 (we expect the effect to be stronger with weaker driving) uniformly shows $⟨ \ln \frac{ϵ}{ϵ_{I}} ⟩ > 0$ when correct monomer interactions are heterogeneous and $⟨ \ln \frac{ϵ}{ϵ_{I}} ⟩ < 0$ when incorrect monomer interactions are heterogeneous (figures are presented in Appendix E).

The sign of $⟨ \ln \frac{ϵ}{ϵ_{I}} ⟩$ influences the sign of $\ln \frac{⟨ ϵ ⟩}{⟨ ϵ_{I} ⟩}$ ⁠, but the latter may experience a sign reversal, for instance, if monomer 2 is the monomer that experiences an error reduction and ϵ₂ ≪ ϵ₁ already. This argument suggests that, generally, the potential benefits of heterogeneity on copying error rates are more limited than the potential drawbacks. Heterogeneity will not tend to increase the probability of finding a correct monomer after another correct monomer but instead can reduce the probability of finding an incorrect monomer placed after another incorrect monomer by making the landscape of incorrect monomers more difficult to traverse. However, finding an incorrect monomer after another incorrect monomer is usually a rare event for a good copying system, and reducing the occurrences of such events would have a smaller impact on the average error rates of a copying system. Conversely, deleterious effects would be expected to be larger as correct monomer pairs would be dominant in a good copying system, and there is more room to reduce the probability of consecutive correct monomers. This reasoning is consistent with the fact that the deterioration factors in Fig. 6 tend to be larger than the improvement factors in Fig. 7.

The observed trends in $\ln \frac{⟨ V ⟩}{⟨ V_{I} ⟩}$ also follow from this “choice of landscape” model after a few additional considerations. When correct monomer interactions are made heterogeneous, moving through the correct landscape becomes more difficult, and hence more state visits are required. This trend is again very robust ( Appendix E). On the other hand, when incorrect monomer interactions are made heterogeneous, barriers to movement in the incorrect landscape are increased. Here, we need to consider two separate effects. First, traversal of the incorrect landscape gets harder, increasing the number of visits to states on average. Second, the system preferentially traverses the correct landscape. The expected number of steps of the coarse-grained model along the correct landscape will be less than that along the incorrect landscape since the correct tip will be harder to remove on average, thus decreasing state visits. The second effect is usually dominant ( Appendix E), but exceptions occur when the correct and incorrect landscapes have a small free-energy gap (implying a small benefit to visitation counts when traversing the correct landscape). The sign of $\ln \frac{⟨ V ⟩}{⟨ V_{I} ⟩}$ feeds forward into the sign of $\ln \frac{⟨ τ ⟩}{⟨ τ_{I} ⟩}$ ⁠, but the latter may experience a sign reversal depending on the waiting times at each template monomer.

We perform a sweep over parameters to identify regimes where heterogeneity results in the greatest improvement in error rate (results in Appendix F). Interestingly, our parameter sweep revealed parameter sets where a heterogeneous system has a lower mean error than either of its constituent monomers used in isolation. Consider, for instance, the plot in Fig. 10. There is a clear minimum in ⟨ϵ⟩ at p_t = 0.4, implying that some combination of the two considered monomers performs better than either one on its own. Thus, it is generally untrue that the error performance of a heterogeneous copying system is bounded by the error performance of its constituent monomers.

FIG. 10.

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Graph of ⟨ϵ⟩ against p_t showing ⟨ϵ⟩ dipping below the values of ⟨ϵ⟩ at both p_t = 0 and p_t = 1. Parameters are ΔG_pol = −0.3, ΔG_TT,11 = ΔG_TT,22 = 6.0, ΔG_TT,12 = 5.0, and ΔG_TT,21 = 0.0.

C. Entropy, mutual information, and efficiency in the low ΔG_pol regime

Recall that Y is the random variable representing the product sequence, and X is the random variable representing the template sequence. As the sequence of copy polymers Y conditioned on a template x is Markov, the template-conditioned entropy rate of complete polymers can be calculated as follows:^30,32,33

\begin{align} h (Y | X = x) & = \lim_{L \to \infty} \frac{1}{L} Σ_{l = 1}^{L} H (Y_{l} | Y_{1 : l - 1}, X = x) \\ = \lim_{L \to \infty} \frac{1}{L} Σ_{l = 1}^{L} H (Y_{l} | Y_{l - 1}, X = x) \\ = - \lim_{L \to \infty} \frac{1}{L} Σ_{l = 1}^{L} Σ_{i j} P_{x, l - 1} (m_{l - 1}) \\ \times P_{x, l} (m_{l} | m_{l - 1}) \ln P_{x, l} (m_{l} | m_{l - 1}) . \end{align}

(22)

Assuming this entropy rate is self-averaging for stationary distributions p(x) of X,³² h(Y|X = x) is the same for any typical⁵¹ instance x of X. Then,

h (Y | X) = Σ_{x} p (x) h (Y | X = x) = h (Y | X = x) .

(23)

For our investigations here, we average entropy over the middle 80% of a long template to mitigate edge effects. Similarly to the error rate, the entropy rate captures uncertainty in the identity of successive monomers in the copy. It is, however, the more thermodynamically relevant parameter, as it is directly bound by the drive ΔG_pol. For ΔG_pol > 0, copying can, in principle, be arbitrarily accurate. However, if ΔG_pol < 0 (as $Δ G_{pol} = Δ G_{B B} + \ln \frac{[M]}{{[M]}_{eff}}$ and generally [M] < [M]_eff, this regime corresponds to one where the chemical free-energy drop due to backbone formation cannot compensate for the entropic drop as a result of attaching a monomer to the polymer tail), then there is a fundamental limit on how accurate a copying system can be when operating in the limit of long polymer length.³³

At equilibrium, copying has no accuracy, as every possible polymer product has the same chemical free energy given by (L − 1)ΔG_pol. For our model of copying, this equilibrium point occurs when ΔG_pol = −ln 2, as this is the point where the forward drive provided by the sequence entropy of the copied polymer is exactly canceled by the polymerization drive ΔG_pol, resulting in no net free-energy change with increasing length of the copy polymer.³³ Furthermore, if −ln 2 < ΔG_pol < 0, then the entropy rate decreases relative to the equilibrium entropy rate of ln 2 per monomer, ln 2 − h(Y|X), is bounded by ΔG_pol + ln 2, leading to a thermodynamic measure of efficiency,

η_{Therm} = \frac{\ln 2 - h (Y | X)}{Δ G_{pol} + \ln 2} .

(24)

η_Therm is the efficiency with which free energy is converted into the low entropy of the product state. Poulton observed in Ref. 33 that the entropy drop during (homogeneous) copying tends to be quite far from the fundamental bound at most values of ΔG_pol, and hence it would be instructive to see if we can get closer to this fundamental bound by permitting heterogeneity. One important caveat in this discussion is that this bound is only valid in the infinite length limit.²⁵ At the polymer length we consider, L = 10⁴, the behavior of most monomers far from polymer boundaries approaches the behavior for the infinite length limit, and hence, while methodological limitations force us to work with finite length polymers, we are endeavoring to make statements on the efficiency of copying of infinite length polymers.

Setting L = 10⁴, and using the coarse-grained model defined by the propensities in Eqs. (10)–(13) with ΔG_TT(1, 1) = ΔG_TT(2, 2) = 6, ΔG_TT(2, 1) = 0, and varying ΔG_TT(1, 2) from 0 (the homogeneous case) to 6, we plot $\max_{p_{t}} η_{Therm}$ in the low ΔG_pol region as a function of ΔG_pol for copying with backward propensity and forward propensity discrimination in Figs. 11(a) and 11(c), respectively. We see quite significant increases in efficiency relative to the homogeneous case. The source of this increased efficiency is the mechanism identified in Sec. III B: roughness in the free-energy landscape of incorrect monomers makes correct monomers more favorable. Interestingly, these increases in efficiency occur despite combining a monomer with another monomer having worse discrimination than itself; hence, (congruent with our observations of error performance in Sec. III B), the thermodynamic performance of a heterogeneous copying system is not bounded by the performance of its constituent monomers. Note that because of finite size effects and sampling, h(Y|X) does not quite achieve ln 2 at ΔG_pol = −ln 2, so the leftmost data points are calculated at ΔG_pol = −ln 2 + 0.0001 and L = 10⁵ to prevent numerical instabilities.

FIG. 11.

Thermodynamic (ηTherm) and (estimated) information (η̃Inf) efficiencies as a function of ΔGpol when incorrect monomer interactions are made heterogeneous by varying ΔGTT(1, 2) from 0. Plots for backward propensity discrimination are shown in (a) and (b), and plots for forward propensity discrimination are shown in (c) and (d). For both types of discrimination, the entropy rate is reduced by heterogeneity, and hence ηTherm is increased in (a) and (c). η̃Inf for backward propensity discrimination is increased by heterogeneity up to about ΔGpol = −0.2 to −0.3, after which it starts decreasing. For forward propensity discrimination, heterogeneity tends to reduce ηInf except for very low values of ΔGpol < −0.55.

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Thermodynamic (η_Therm) and (estimated) information $({\tilde{η}}_{Inf})$ efficiencies as a function of ΔG_pol when incorrect monomer interactions are made heterogeneous by varying ΔG_TT(1, 2) from 0. Plots for backward propensity discrimination are shown in (a) and (b), and plots for forward propensity discrimination are shown in (c) and (d). For both types of discrimination, the entropy rate is reduced by heterogeneity, and hence η_Therm is increased in (a) and (c). ${\tilde{η}}_{Inf}$ for backward propensity discrimination is increased by heterogeneity up to about ΔG_pol = −0.2 to −0.3, after which it starts decreasing. For forward propensity discrimination, heterogeneity tends to reduce η_Inf except for very low values of ΔG_pol < −0.55.

Instead of the entropy rate drop ln 2 − h(Y|X), we may instead wish to consider the mutual information rate

\dot{I} (X, Y) = h (Y) - h (Y | X)

⁠, where h(Y) is the product entropy rate unconditioned on the template,

h (Y) = \lim_{L \to \infty} \frac{1}{L} Σ_{l = 1}^{L} H (Y_{l} | Y_{1 : l - 1}) .

(25)

Per Shannon’s coding theorem,⁵¹ the mutual information quantifies our ability to uniquely distinguish a message encoded in our original template from our obtained copy or, equivalently, tell which template a copy came from. The mutual information maximized over input distributions defines a channel capacity

C = \max_{p (X)} \dot{I} (X, Y)

⁠. We shall now explore this channel capacity, although for practical purposes we will consider optimizing over Bernoulli input distributions. In the case of homogeneous copying, the channel capacity is exactly equal to the entropy rate drop ln 2 − h(Y|X), as the entropy rate h(Y|X) is independent of the template distribution, and due to symmetry, h(Y) = h(X) = ln 2 when the template is drawn from a Bernoulli distribution with

p_{t} = \frac{1}{2}

⁠. To understand how C differs from the entropy drop ln 2 − h(Y|X) in the case of heterogeneous copying, consider a hypothetical copying system that always maps template monomers to a copy monomer of type 1. This copying system would have a maximum entropy rate drop of ln 2 but a mutual information rate of 0. For a given template distribution,

\dot{I} (X, Y) = h (Y) - h (Y | X) \leq \ln 2 - h (Y | X)

and ln 2 − h(Y) can be regarded as the proportion of the entropy drop from the equilibrium distribution that does not contribute toward information transfer. Maximizing over p(X) should maintain our inequality such that C = max_p(X)h(Y) − h(Y|X) ≤ max_p(X) ln 2 − h(Y|X). We may thus define an information efficiency η_Inf ≤ max_p(X)η_Therm as follows:

η_{Inf} = \frac{C}{Δ G_{pol} + \ln 2} .

(26)

The entropy rate h(Y) is difficult to calculate exactly for heterogeneous polymer copying. It can be expressed as follows:

\begin{align} h (Y) & = \lim_{L \to \infty} \frac{1}{L} Σ_{l = 1}^{L} H (Y_{l} | Y_{1 : l - 1}) \\ = - \lim_{L \to \infty} \frac{1}{L} Σ_{l = 1}^{L} Σ_{m_{1 : l}} p (m_{1}, m_{2}, \dots, m_{l}) \\ \times \ln p (m_{l} | m_{1}, m_{2}, . ., m_{l - 1}), \end{align}

(27)

These probabilities are marginalized over the template X. The joint and conditional probabilities can be expanded as follows:

\begin{align} p (m_{l} | m_{1}, m_{2}, \dots m_{l - 1}) \\ = Σ_{x} p (m_{l} | m_{1}, m_{2}, . ., m_{l - 1}, x) p (x | m_{1}, m_{2}, \dots m_{l - 1}) \\ = Σ_{x} p (m_{l} | m_{l - 1}, x) p (x | m_{1}, m_{2}, \dots m_{l - 1}), \end{align}

(28)

\begin{align} p (m_{1}, m_{2}, \dots m_{l - 1}, m_{l}) \\ = Σ_{x} p (m_{1}, m_{2}, . ., m_{l - 1}, m_{l}, x) \\ = p (m_{1}, m_{2}, \dots m_{l - 1}) Σ_{x} p (m_{l}, x | m_{1}, m_{2}, \dots m_{l - 1}) \\ = p (m_{1}, m_{2}, \dots m_{l - 1}) p (m_{l} | m_{1}, m_{2}, \dots m_{l - 1}) . \end{align}

(29)

In Eq. (28), the p(x, m₁, m₂, …, m_l−1) term means that we cannot assume p(m_l|m₁, m₂, …, m_l−1) = p(m_l|m_l−1). In order to proceed, we first make the assumption that Y, having marginalized over X, has finite-length, templated-mediated correlations and can be approximated as an ith-order Markov process ( Appendix G). Remarkably, numerical evaluations show that the change in estimated entropy going from i = 1 to i = 8 is not significant for the parameters we consider when sampling the middle 80% of a length L = 10⁴ template. Furthermore, by sampling length L = 10⁵ and L = 10⁶ templates at select parameter values, we found that a significant proportion of this error (at least for some regimes) is likely attributable to sampling issues instead of genuine long-range correlations ( Appendix G), and hence we are justified in estimating entropies by treating Y as a first-order Markov chain. Formally, this assumption means that the information efficiency we calculate ${\tilde{η}}_{Inf}$ is an upper bound of some true information efficiency η_Inf for Bernoulli templates. However, Appendix G suggests that ${\tilde{η}}_{Inf}$ is a very tight upper bound. On the other hand, η_Inf may be higher when considering non-Bernoulli templates.

In Figs. 11(b) and 11(d), we plot our estimated ${\tilde{η}}_{Inf}$ as a function of ΔG_pol for the backward propensity and forward propensity discrimination regimes, respectively. As expected, we observe ${\tilde{η}}_{Inf} \leq \max_{p_{t}} η_{Therm}$ ⁠. Increases in efficiency relative to the homogeneous case ΔG_TT(1, 2) = 0 are better preserved with backward propensity discrimination compared to forward propensity discrimination. P(Y) is, in essence, more skewed in the forward propensity discrimination case, which is detrimental to mutual information.

IV. CONCLUSION

Using minimal models with fine-grained steps,³⁵ we have investigated how heterogeneity interacts with separation to affect copying error, velocity, and thermodynamic efficiency in STC. We have thus far investigated heterogeneity in monomer binding energies and kinetics; the effects of heterogeneous monomer concentrations, which would naturally manifest as heterogeneity in binding rates, will be an interesting topic for future research. Our first contribution is an approach for extending Qureshi et al.’s coarse-graining method⁴³ to generalize to heterogeneous systems. We have thus far used this method to find polymer completion times for our coarse-grained model. A natural extension would be to use the method to investigate a heterogeneous version of kinetic proofreading,^36,43 where it can be used to find the average expected free-energy consumption per polymer copy.

Using this method, we were able to characterize the velocity profiles of a heterogeneous separating templated copolymerization system. In contrast to non-separating templated copolymerization systems,³² we do not observe a long tail of zero velocity as equilibrium is approached. This absence makes sense in light of the template-dependent free-energy profiles of partial copies, which have significantly higher barriers (scaling as $\sqrt{l_{D}}$ for a length scale l_D) in the case of NTC. Put another way, each monomer type in heterogeneous NTC has a different pseudo-equilibrium point, and copying with a drive lower than a monomer’s pseudo-equilibrium point makes traversing a long stretch of said monomer more difficult. This effect is totally absent in the case of STC since the sequence-specific interactions with the template are transient. We expect that this observation would generalize to more realistic models of transcription or translation, as long as the copy continuously separates from the template and the number of copy monomers interacting with the template at a given time is effectively bounded.

Our results on the effect of heterogeneity on error rates are more surprising. It was not initially obvious that discriminating on correct monomer interactions, arguably the more natural form of heterogeneous discrimination, would tend to increase errors. There is evidence that in protein translation, the ribosome grips on tRNA have identical strengths.⁵² This grip is analogous to our transient copy–template bond, as it is not persistent, and so it is plausible that the homogeneity of this grip was selected due to similar mechanisms that increase heterogeneous error rates (consider that in our case, error increase factors of up to e^1.25 = 3.5-fold were observed). In the context of artificial systems, our results would suggest that it would be wise to minimize interaction heterogeneity on correct pairs of monomers.

The relative error reduction observed when discriminating on incorrect monomers is equally surprising. Applying heterogeneity to incorrect monomer interactions could be a useful design motif for the design of accurate copiers, and we have made some attempts toward this goal by scanning over parameter space. We find that having backward and forward propensity discrimination on separate monomers, with roughly equal effect sizes, tends to produce the best results. However, we emphasize that the usefulness of this design motif depends on the space of accessible parameter sets. Under some chemical restrictions (in particular, if we had to operate in the low ΔG_pol regime, or if our copying mechanism only permits backward propensity discrimination), it may make sense to apply heterogeneity to incorrect monomer interactions for the roughly 10%–20% improvement in error rates. Note that operating under low ΔG_pol may be necessary for synthetic systems. Reassuringly, there does not appear to be a trade-off between using heterogeneity to reduce error or using it to speed up copying, as regimes that tend to reduce error tend to reduce copying time as well. Both performance measures prefer heterogeneity on incorrect monomer interactions as opposed to correct monomer interactions, as heterogeneity on incorrect pairs makes it harder to move through the landscape of incorrect monomers.

Our final result relates to the entropy drop and mutual information in heterogeneous copying systems. In homogeneous copying, the entropy drop from equilibrium is exactly the mutual information when template monomers are equally distributed. In contrast, we showed that there is a meaningful difference between this entropy drop and mutual information in the case of heterogeneous copying due to the skewing of the copy polymer distributions. For a given template distribution, we can evaluate a channel capacity, the mutual information maximized over input distributions. Channel capacity can in turn be used to define an information efficiency measure. We showed that both thermodynamic and information efficiencies can be improved by heterogeneity in the incorrect monomer interactions at low ΔG_pol, but that information efficiency is always less than or equal to thermodynamic efficiency. Per Shannon’s channel coding theory, channel capacity represents the minimal bitrate above which arbitrarily accurate decoding is possible, and hence some of the thermodynamic entropy drop in heterogeneous copying systems does not contribute to reducing this bitrate. On the other hand, for homogeneous copying systems, there is always a template distribution $(p_{t} = \frac{1}{2})$ where the thermodynamic entropy drop in the copy (relative to equilibrium) fully contributes to the reduction of this bitrate.

ACKNOWLEDGMENTS

J.E.B.G. was supported by an Imperial College President’s Ph.D. Scholarship, B.J.Q. by the European Research Council under the European Union’s Horizon 2020 Research and Innovation Program (Grant Agreement No. 851910), and T.E.O. by a Royal Society University Research Fellowship.

AUTHOR DECLARATIONS

Conflict of Interest

All authors certify that they have no affiliations with or involvement in any organization or entity with any financial interest or non-financial interest in the subject matter or materials discussed in this manuscript.

Author Contributions

J.E.B.G., B.J.Q., and T.E.O. planned the research. J.E.B.G. performed the research. J.E.B.G., B.J.Q., and T.E.O. wrote the manuscript.

Jeremy E. B. Guntoro: Conceptualization (equal); Investigation (lead); Methodology (lead); Software (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (equal). Benjamin J. Qureshi: Conceptualization (equal); Methodology (supporting); Supervision (equal); Writing – review & editing (equal). Thomas E. Ouldridge: Conceptualization (equal); Methodology (supporting); Supervision (equal); Writing – review & editing (equal).

DATA AVAILABILITY

Code and data are available at doi.org/10.5281/zenodo.14003309.

APPENDIX A: GASPARD’S METHOD VIA ABSORBING PROBABILITIES OF MARKOV CHAINS

Consider again Fig. 2 in the main text. We wish to calculate the probability R_x,l(m_l+1|m_l−1m_l) of being absorbed into a complete polymer with m_l+1 after m_l starting from the initial state (x, &m_l−1m_l), without going back a step. To aid in our calculations, we introduce the absorption probabilities R_x,l(m_l+1|m_l−1m_lm_r), the probability of completing polymerization with m_l+1 after m_l starting from a system state (x, &m_l−1m_lm_r) in the context of the Markov chain given in Fig. 2 (m_l+1 may be the same or different from m_r). To clarify, this probability includes the probability that we move backward from (x, &m_l−1m_lm_r) to (x, &m_l−1m_l) and then, after an arbitrary non-absorbing set of moves, eventually absorb into (x, &m_l−1m_lm_l+1 … 0), where 0 indicates a detached, complete polymer. Note: moving back two consecutive times from m_r always results in the backward absorbing state. In addition, if m_l+1 = m_r, then absorption without moving back is included in the absorption probability R_x,l(m_l+1|m_l−1m_lm_r).

We can now use the fact that for an arbitrary Markov chain, the absorbing probability to an absorbing state A from a transient state x can be calculated as p_abs(A|x) = Σ_ip(x_i|x)p_abs(A|x_i) + p(A|x), where x_i are all states with transitions from x [setting p(A|x) = 0 if x cannot directly transition into A]. Reminding readers that p_x,l(m_l+1|m_l−1m_l) are the local transition probabilities for the addition of monomer m_l+1 in a single coarse-grained step, the system of Eqs. and is obtained,

R_{x, l} (m_{l + 1} | m_{l - 1} m_{l}) = Σ_{m_{r}} p_{x, l} (m_{r} | m_{l - 1} m_{l}) R_{x, l} (m_{l + 1} | m_{l - 1} m_{l} m_{r}),

(A1)

R_{x, l} (m_{l + 1} | m_{l - 1} m_{l} m_{r}) = \{\begin{cases} (1 - Q_{x, l + 1} (m_{l} m_{r})) R_{x, l} (m_{l + 1} | m_{l - 1} m_{l}), & m_{l + 1} \neq m_{r}, \\ (1 - Q_{x, l + 1} (m_{l} m_{r})) R_{x, l} (m_{l + 1} | m_{l - 1} m_{l}) + Q_{x, l + 1} (m_{l} m_{l + 1}), & m_{l + 1} = m_{r} . \end{cases}

(A2)

Substitution of into yields

\begin{align} R_{x, l} (m_{l + 1} | m_{l - 1} m_{l}) = p_{x, l} (m_{l + 1} | m_{l - 1} m_{l}) Q_{x, l + 1} (m_{l} m_{l + 1}) \\ + Σ_{m_{r}} p_{x, l} (m_{r} | m_{l - 1} m_{l}) (1 - Q_{x, l + 1} (m_{l} m_{r})) R_{x, l} (m_{l + 1} | m_{l - 1} m_{l}) . \end{align}

(A3)

Equation can be rearranged with all R_x,l(m_l+1|m_l−1m_l) terms to the left, yielding

R_{x, l} (m_{l + 1} | m_{l - 1} m_{l}) = \frac{p_{x, l} (m_{l + 1} | m_{l - 1} m_{l}) Q_{x, l + 1} (m_{l} m_{l + 1})}{1 - Σ_{m_{r}} p_{x, l} (m_{r} | m_{l - 1} m_{l}) + Σ_{m_{r}} p_{x, l} (m_{r} | m_{l - 1} m_{l}) Q_{x, l + 1} (m_{l} m_{r})} .

(A4)

We can perform the substitutions

p_{x, l} (m_{l + 1} | m_{l - 1} m_{l}) = \frac{Φ_{l}^{+} (x, m_{l} m_{l + 1})}{Φ_{l}^{-} (x, m_{l - 1} m_{l}) + Σ_{m_{r}} Φ_{l}^{+} (x, m_{l} m_{r})}

to obtain Eq. , and then perform the summation in Eq. of the main text to obtain Eq. ,

R_{x, l} (m_{l + 1} | m_{l - 1} m_{l}) = \frac{Φ_{l}^{+} (x, m_{l} m_{l + 1}) Q_{x, l + 1} (m_{l} m_{l + 1})}{Φ_{l}^{-} (x, m_{l - 1} m_{l}) + Σ_{m_{r}} Φ_{l}^{+} (x, m_{l} m_{r}) Q_{x, l + 1} (m_{l} m_{r})},

(A5)

Q_{x, l} (m_{l - 1} m_{l}) = \frac{Σ_{m_{r}} Φ_{l}^{+} (x, m_{l} m_{r}) Q_{x, l + 1} (m_{l} m_{r})}{Φ_{l}^{-} (x, m_{l - 1} m_{l}) + Σ_{m_{r}} Φ_{l}^{+} (x, m_{l} m_{r}) Q_{x, l + 1} (m_{l} m_{r})} .

(A6)

A slight difference from the derivation in Ref. 32 is that we have opted to express our iteration in terms of Q instead of local velocities v_x,l, as our derivation of expected visits in Appendix D uses Q_x,l rather than v_x,l. However, we emphasize that the underlying mathematics is the same, and the local velocities v_x,l in Ref. 32 can be obtained using

v_{x, l} = Σ_{m_{l + 1}} Φ_{l}^{+} (x, m_{l} m_{l + 1}) Q_{x, l + 1} (m_{l} m_{l + 1})

⁠.

The monomeric conditional probabilities P_x,l+1(m_l+1|m_l) can be calculated by considering the Markov chain in Fig. 2 and then omitting the possibility of removal of monomer m_l. To understand why, note that for complete polymers, the distribution of m_l+1 is only affected by the copying trajectory after m_l is added for the final time and then never removed. P_x,l+1(m_l+1|m_l) is then simply the absorption probability into (x, &m_l−1m_lm_l+1 … 0) for this modified Markov chain. The calculation of absorbing probabilities is identical to our previous derivation up to Eq. . Our final substitution now must omit the backward rate such that

p_{x, l} (m_{l + 1} | m_{l - 1} m_{l}) = \frac{Φ_{l}^{+} (x, m_{l} m_{l + 1})}{Σ_{m_{r}} Φ_{l}^{+} (x, m_{l} m_{r})}

⁠, resulting in

P_{x, l + 1} (m_{l + 1} | m_{l}) = \frac{Φ_{l}^{+} (x, m_{l} m_{l + 1}) Q_{x, l + 1} (m_{l} m_{l + 1})}{Σ_{m_{r}} Φ_{l}^{+} (x, m_{l} m_{r}) Q_{x, l + 1} (m_{l} m_{r})} .

(A7)

APPENDIX B: DERIVATION OF COARSE-GRAINED PROPENSITIES

We use the method in Ref. 43 to derive our coarse-grained propensities for the model with detachment. Refer to Eq. and assume a starting completed state (as a reminder, completed states are states with f = 0) (x, &m_l−1, 0) and a second completed state (x, &m_l−1m_l, 0). In Fig. 1(b), there is only one spanning tree rooted at either of these completed states (the leftmost and rightmost states), and a straightforward multiplication of forward rates results in

Λ_{x}^{+} (n_{l - 1} n_{l}, m_{l - 1} m_{l})

[Eq. ], while a multiplication of backward rates results in

Λ_{x}^{-} (n_{l - 1} n_{l}, m_{l - 1} m_{l})

[Eq. ],

\begin{align} Λ_{x}^{+} (n_{l - 1} n_{l}, m_{l - 1} m_{l}) & = K_{bind}^{+} (n_{l - 1} n_{l}, m_{l - 1} m_{l}) K_{pol}^{+} (n_{l - 1} n_{l}, m_{l - 1} m_{l}) K_{tail}^{-} (n_{l - 1} n_{l}, m_{l - 1} m_{l}) \\ = k_{bind} k_{pol} (n_{l - 1} n_{l}, m_{l - 1} m_{l}) k_{tail} e^{- Δ G_{T T} (n_{l}, m_{l})} [M], \end{align}

(B1)

\begin{align} Λ_{x}^{-} (n_{l - 1} n_{l}, m_{l - 1} m_{l}) & = K_{bind}^{-} (n_{l - 1} n_{l}, m_{l - 1} m_{l}) K_{pol}^{-} (n_{l - 1} n_{l}, m_{l - 1} m_{l}) K_{tail}^{+} (n_{l - 1} n_{l}, m_{l - 1} m_{l}) \\ = k_{bind} k_{pol} (n_{l - 1} n_{l}, m_{l - 1} m_{l}) k_{tail} e^{- Δ G_{B B} - Δ G_{T T} (n_{l}, m_{l})} {[M]}_{eff} . \end{align}

(B2)

The modified process used to obtain A_x(n_l−1n_l, m_l−1m_l) [with transitions to (x, &m_l−1m_l) redirected back to (x, &m_l−1)] and its spanning trees rooted at (x, &m_l−1) are depicted in Figs. 12(a) and 12(b), respectively. The resulting A_x(n_l−1n_l, m_l−1m_l) is given in the following equation:

\begin{align} A_{x} (n_{l - 1} n_{l}, m_{l - 1} m_{l}) & = K_{bind}^{-} (n_{l - 1} n_{l}, m_{l - 1} m_{l}) K_{pol}^{-} (n_{l - 1} n_{l}, m_{l - 1} m_{l}) \\ + K_{bind}^{-} (n_{l - 1} n_{l}, m_{l - 1} m_{l}) K_{tail}^{-} (n_{l - 1} n_{l}, m_{l - 1} m_{l}) \\ + K_{pol}^{+} (n_{l - 1} n_{l}, m_{l - 1} m_{l}) K_{tail}^{-} (n_{l - 1} n_{l}, m_{l - 1} m_{l}) \\ = e^{- Δ G_{B B} - Δ G_{T T} (n_{l}, m_{l})} k_{bind} k_{pol} (n_{l - 1} n_{l}, m_{l - 1} m_{l}) \\ + e^{- Δ G_{T T} (n_{l - 1}, m_{l - 1}) - Δ G_{T T} (n_{l}, m_{l}) - Δ G_{gen}} k_{bind} k_{tail} \\ + e^{- Δ G_{T T} (n_{l - 1}, m_{l - 1})} k_{pol} (n_{l - 1} n_{l}, m_{l - 1} m_{l}) k_{tail} . \end{align}

(B3)

Equations – are consistent with the coarse-grained propensities in Eqs. and . Now refer to Fig. 4(a) for the case of non-separating templated copolymerization.

Λ_{NTC, x}^{+} (n_{l - 1} n_{l}, m_{l - 1} m_{l})

[Eq. ] and

Λ_{NTC, x}^{-} (n_{l - 1} n_{l}, m_{l - 1} m_{l})

[Eq. ] can again be obtained by a straightforward multiplication of forward and backward rates,

\begin{align} Λ_{NTC, x}^{+} (n_{l - 1} n_{l}, m_{l - 1} m_{l}) & = K_{bind}^{+} (n_{l - 1} n_{l}, m_{l - 1} m_{l}) K_{pol}^{+} (n_{l - 1} n_{l}, m_{l - 1} m_{l}) \\ = k_{bind} k_{pol} (n_{l - 1} n_{l}, m_{l - 1} m_{l}) [M], \end{align}

(B4)

\begin{align} Λ_{NTC, x}^{-} (n_{l - 1} n_{l}, m_{l - 1} m_{l}) & = K_{bind}^{-} (n_{l - 1} n_{l}, m_{l - 1} m_{l}) K_{pol}^{-} (n_{l - 1} n_{l}, m_{l - 1} m_{l}) \\ = k_{bind} k_{pol} (n_{l - 1} n_{l}, m_{l - 1} m_{l}) e^{- Δ G_{B B} - Δ G_{T T} (n_{l}, m_{l})} . \end{align}

(B5)

For A_NTC,x(n_l−1n_l, m_l−1m_l), note that the only two spanning trees for a modified process with polymerization redirected back toward the initial state are the single-step processes of unbinding and polymerization, leading to the form in the following equation:

\begin{align} A_{NTC, x} (n_{l - 1} n_{l}, m_{l - 1} m_{l}) & = K_{bind}^{-} (n_{l - 1} n_{l}, m_{l - 1} m_{l}) + K_{pol}^{+} (n_{l - 1} n_{l}, m_{l - 1} m_{l}) \\ = e^{- Δ G_{T T} (n_{l}, m_{l})} k_{bind} + k_{pol} (n_{l - 1} n_{l}, m_{l - 1} m_{l}) . \end{align}

(B6)

FIG. 12.

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Enumeration of spanning trees for the calculation of coarse-grained propensities. (a) The modified network for the calculation of A. (b) Spanning trees of the network rooted at the initial coarse-grained state.

APPENDIX C: DERIVATION OF AVERAGE TIP WAITING TIMES

We apply the method in Ref. 43 to calculate the expected waiting times $E_{V} [θ_{x, l}]$ at a given tip state of the coarse-grained model. Note the waiting times will be dependent on the template as well as copy monomers near the tip; hence, in this section, a tip state will refer to (n_l−1n_ln_l+1, m_l−1m_l). For each coarse-grained tip state, we wish to consider the network of reactions into and out of the completed state corresponding to f = 0 [Fig. 1(c)]. Refer to Fig. 13 for the first passage network from a completed state, with all transitions to other completed states redirected back to the initial completed state. As our propensities are dependent only on local template and copy monomers, $E_{V} [θ_{x, l}] = E_{V} [θ (n_{l - 1} n_{l} n_{l + 1}, m_{l - 1} m_{l})]$ for our case. As per,⁴³ $E_{V} [θ (n_{l - 1} n_{l} n_{l + 1}, m_{l - 1} m_{l})] = 1 / J (n_{l - 1} n_{l} n_{l + 1}, m_{l - 1} m_{l})$ where J(n_l−1n_ln_l+1, m_l−1m_l) is the flux into completed states corresponding to distinct coarse-grained states. To find this flux, we will need to calculate the stationary probability distribution p_ss(x, &m_l−1m_l, f) of fine-grained states in the network in Fig. 13.

FIG. 13.

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Fine-grained network for calculating first passage between coarse-grained states. Transitions leading to other coarse-grained tip states are redirected back to the initial coarse-grained state. Rates for the bottom arm are omitted but are analogous to those for the top arm with (12, 12) in place of (12, 11).

Our discussion for the remainder of this section will take place in the context of a single network of the type in Fig. 13 [with state indices given by Fig. 1(c)]. Hence, x and &m_l−1m_l are fixed, and we can omit them from variables for simplicity. Rates of the form K^±(n_ln_l+1, m_lm_l+1) (with dependence on m_l+1) are abbreviated to K^±(m_l+1), rates of the form K^±(n_l−1n_l, m_l−1m_l) (without dependence on m_l+1) are abbreviated to K^±(−), J(n_l−1n_ln_l+1, m_l−1m_l) is abbreviated to J, and p_ss(x, &m_l−1m_l, f) is abbreviated to p_ss(f). Finally, we introduce a function χ(m_l+1) = 2 + 2 × m_l+1 that maps a monomer m_l+1 to the transitory state index f_exit such that (x, &m_l−1m_l, f_exit) has a transition into (x, &m_l−1m_lm_l+1, 0). That is, χ returns the last fine-grained state encountered before the complete addition of a monomer m_l+1. Then, the flux J can be calculated as follows:⁴³

J = K_{bind}^{-} (-) p_{s s} (2) + Σ_{m_{l + 1}} K_{tail}^{-} (m_{l + 1}) p_{s s} (χ (m_{l + 1})) .

(C1)

Here, the first term is the backward flux, and the second term is the sum of all forward fluxes. The steady state probabilities of states with transitions to other coarse-grained tip states can be found through standard methods, and they are expressed in Eqs. and [and a normalization variable

N

expressed in Eq. ],

p_{s s} (χ (m_{l + 1})) = \frac{K_{bind}^{+} (m_{l + 1}) K_{pol}^{+} (m_{l + 1})}{N (K_{bind}^{-} (m_{l + 1}) (K_{pol}^{-} (m_{l + 1}) + K_{tail}^{-} (m_{l + 1})) + K_{tail}^{-} (m_{l + 1}) K_{pol}^{+} (m_{l + 1}))},

(C2)

p_{s s} (2) = \frac{K_{tail}^{+} (-) K_{pol}^{-} (-)}{N (K_{tail}^{-} (-) (K_{pol}^{+} (-) + K_{bind}^{-} (-)) + K_{bind}^{-} (-) K_{pol}^{-} (-))},

(C3)

\begin{align} N & = 1 + \frac{K_{tail}^{+} (-) K_{pol}^{-} (-)}{K_{tail}^{-} (-) (K_{pol}^{+} (-) + K_{bind}^{-} (-)) + K_{bind}^{-} (-) K_{pol}^{-} (-)} (1 + \frac{K_{pol}^{+} (-) + K_{bind}^{-} (-)}{K_{pol}^{-} (-)}) \\ + Σ_{m_{l + 1}} \frac{K_{bind}^{+} (m_{l + 1}) K_{pol}^{+} (m_{l + 1})}{K_{bind}^{-} (m_{l + 1}) (K_{pol}^{-} (m_{l + 1}) + K_{tail}^{-} (m_{l + 1})) + K_{tail}^{-} (m_{l + 1}) K_{pol}^{+} (m_{l + 1})} (1 + \frac{K_{pol}^{-} (m_{l + 1}) + K_{tail}^{-} (m_{l + 1})}{K_{pol}^{+} (m_{l + 1})}) . \end{align}

(C4)

Analogous definitions and abbreviations can be applied to NTC. The function χ_NTC(m_l+1) = 1 + m_l+1 now maps each m_l+1 to f_exit, its final state before transition. The flux J_NTC is then calculated as follows [refer to Fig. 4(b) for state indices]:

J_{NTC} = K_{bind}^{-} (-) p_{s s, N T C} (1) + Σ_{m_{l + 1}} K_{pol}^{+} (m_{l + 1}) p_{s s} (χ_{NTC} (m_{l + 1})) .

(C5)

Steady state probabilities and normalization variables for NTC without detachment, calculated through standard methods, are given in the following equations:

p_{s s, N T C} (χ_{NTC} (m_{l + 1})) = \frac{K_{bind}^{+} (m_{l + 1})}{N_{NTC} (K_{bind}^{-} (m_{l + 1}) + K_{pol}^{+} (m_{l + 1}))},

(C6)

p_{s s, N T C} (1) = \frac{K_{pol}^{-} (-)}{N_{NTC} (K_{pol}^{+} (-) + K_{bind}^{-} (-))},

(C7)

N_{NTC} = 1 + \frac{K_{pol}^{-} (-)}{K_{pol}^{+} (-) + K_{bind}^{-} (-)} + Σ_{m_{l + 1}} \frac{K_{bind}^{+} (m_{l + 1})}{K_{bind}^{-} (m_{l + 1}) + K_{pol}^{+} (m_{l + 1})} .

(C8)

APPENDIX D: ITERATION FOR THE EXPECTED NUMBER OF VISITS TO A TIP STATE

Let a tip state ${(m_{l}, m_{l + 1})}_{x, l + 1}$ refer to a class of states with template x and a copy of length l + 1 ending in m_l, m_l+1. This definition extends to longer tails; for example, ${(m_{l - 1}, m_{l}, m_{l + 1})}_{x, l + 1}$ would refer to states ending in m_l−1, m_l, m_l+1. In this section, we aim to calculate $E [V_{x, l} (m_{l}, m_{l + 1}) | m_{l - 1}, m_{l}]$ ⁠, the expected number of visits to ${(m_{l}, m_{l + 1})}_{x, l + 1}$ for each visit to ${(m_{l - 1}, m_{l})}_{x, l}$ ⁠. We argue that this quantity can be calculated by considering visits to transient tip states in the Markov chain in Fig. 3. We will use abbreviations of the form $V_{l + 1} (m_{l + 1}) = E [V_{x, l} (m_{l}, m_{l + 1}) | m_{l - 1}, m_{l}]$ for convenience, noting that we always refer to the Markov process in Fig. 3, and hence the conditioned tip m_l−1, m_l, and template x are defined.

Each tip state

{(m_{l - 1}, m_{l}, m_{l + 1})}_{x, l + 1}

will be visited an expected p_x,l(m_l+1|m_l−1, m_l) number of times from each transition out of

{(m_{l - 1}, m_{l})}_{l}

(we call this visit, resulting directly from monomer addition, a “forward” visit). We can count “backward” visits to

{(m_{l - 1}, m_{l}, m_{l + 1})}_{x, l + 1}

from

{(m_{l - 1}, m_{l}, m_{l + 1}, m_{l + 2})}_{x, l + 2}

by noting that each visit of

{(m_{l - 1}, m_{l}, m_{l + 1}, m_{l + 2})}_{x, l + 2}

has a probability of 1 − Q_x,l+2(m_l+1m_l+2) of returning back to

{(m_{l - 1}, m_{l}, m_{l + 1})}_{x, l + 1}

⁠. Hence, denoting forward visits to

{(m_{l - 1}, m_{l}, m_{l + 1}, m_{l + 2})}_{x, l + 2}

in the Markov chain given in Fig. 3 by

V_{l + 2}^{f} (m_{l + 1} m_{l + 2})

⁠, we obtain an expression for V_l+1(m_l+1) in the following equation:

\begin{align} V_{l + 1} (m_{l + 1}) & = p_{x, l} (m_{l + 1} | m_{l - 1} m_{l}) + Σ_{m_{l + 2}} \\ \times (1 - Q_{x, l + 2} (m_{l + 1} m_{l + 2})) V_{l + 2}^{f} (m_{l + 1}, m_{l + 2}) . \end{align}

(D1)

We further know that visits to

{(m_{l - 1}, m_{l}, m_{l + 1}, m_{l + 2})}_{x, l + 2}

are just the visits to

{(m_{l - 1}, m_{l}, m_{l + 1})}_{x, l + 1}

weighted by the probability of adding m_l+2 so that

V_{l + 2}^{f} (m_{l + 1}, m_{l + 2}) = p_{x, l + 1} (m_{l + 2} | m_{l} m_{l + 1}) V_{l + 1} (m_{l + 1})

⁠, leading to the following equation:

\begin{align} V_{l + 1} (m_{l + 1}) & = p_{x, l} (m_{l + 1} | m_{l - 1} m_{l}) + Σ_{m_{l + 2}} (1 - Q_{x, l + 2} (m_{l + 1} m_{l + 2})) \\ \times p_{x, l + 1} (m_{l + 2} | m_{l} m_{l + 1}) V_{l + 1} (m_{l + 1}) . \end{align}

(D2)

Rearranging for V_l+1(m_l+1), we obtain the following equation:

\begin{align} E [V_{x, l} (m_{l}, m_{l + 1}) | m_{l - 1}, m_{l}] = V_{l + 1} (m_{l + 1}) \\ = \frac{p_{x, l} (m_{l + 1} | m_{l - 1} m_{l})}{1 - Σ_{m_{l + 2}} (1 - Q_{x, l + 2} (m_{l + 1} m_{l + 2})) p_{x, l + 1} (m_{l + 2} | m_{l} m_{l + 1})} . \end{align}

(D3)

E [V_{x, l} (m_{l}, m_{l + 1}) | m_{l - 1}, m_{l}]

is then calculable from previously calculated Q variables (and local rates). Then, the absolute visitations to any tip state

{(m_{l - 1} m_{l})}_{x, l}

can be calculated using Eq. of the main text.

APPENDIX E: PARAMETER SWEEP OF RELATIVE ERROR AND TOTAL VISITATIONS

To test the robustness of our qualitative predictions on the effect of heterogeneity on relative error and state visits, we perform a sweep of parameter space. Letting ΔG_pol = 0, L = 10⁴, and considering a high discrimination level ΔG_TT,H and a low discrimination level ΔG_TT,L with ΔG_TT,H ≥ ΔG_TT,L, we consider forward/backward discrimination matrices of the following form for heterogeneity on correct monomers:

Δ G_{T T} = (\begin{matrix} Δ G_{T T, H} & 0 \\ 0 & Δ G_{T T, L} \end{matrix}) .

(E1)

On the other hand, for heterogeneity on incorrect monomers, we consider matrices of the following form:

Δ G_{T T} = (\begin{matrix} Δ G_{T T, H} & Δ G_{T T, L} \\ 0 & Δ G_{T T, H} \end{matrix}) .

(E2)

We then sweep over these parameters for different p_t. In the case of heterogeneity on correct monomers, we want to show that relative error and state visits are always increased, and we thus expect

\min_{p_{t}} ⟨ \ln \frac{ϵ}{ϵ_{I}} ⟩ \geq 0

[Figs. 14(a) and 14(b)] and

\min_{p_{t}} \ln \frac{⟨ V ⟩}{⟨ V_{I} ⟩} \geq 0

[Figs. 15(a) and 15(b)]. On the other hand, for heterogeneity on incorrect monomers,

\max_{p_{t}} ⟨ \ln \frac{ϵ}{ϵ_{I}} ⟩ \leq 0

[Figs. 14(c) and 14(d)] and

\max_{p_{t}} \ln \frac{⟨ V ⟩}{⟨ V_{I} ⟩} \leq 0

[Figs. 15(c) and 15(d)] are expected. In this L = 10⁴ regime, the noise due to edge effects becomes significant, and so we extract values from the central 80% of the polymer. Only the upper right triangle of all of the heat maps is plotted. Despite allowing the system to optimize over p_t, the maximum and the minimum in each case stay very close to 0, implying that error and visits never improve when correct monomer interactions are heterogeneous and never degrade when incorrect monomer interactions are heterogeneous. The exception to this observation is the deviation in

\ln \frac{⟨ V ⟩}{⟨ V_{I} ⟩}

observed in the small discrimination regime for heterogeneity on incorrect monomer interactions. For heterogeneity on incorrect monomer interactions, there is tension between increased visitation in the incorrect landscape and the higher probability of traversing the correct landscape. Hence, when the correct and incorrect landscapes have a small gap, to begin with (true for small discrimination), the effect of increased visitations through the incorrect landscape becomes more significant.

FIG. 14.

Heat map illustrating the resulting relative error change ⟨lnϵϵI⟩ from parameter sweeps for heterogeneity, having extremized over pt. We consider heterogeneity on correct [backward propensity discrimination: (a) and forward discrimination propensity discrimination: (b)] and incorrect [backward propensity discrimination: (c) and forward propensity discrimination: (d)] monomers. In the case of heterogeneity on correct monomer interactions, this relative error change is minimized over pt; for heterogeneity on incorrect monomer interactions, it is maximized over pt. Only the upper triangle ΔGTT,H ≥ ΔGTT,L is plotted. For heterogeneity on correct monomers, minpt⟨lnϵϵI⟩=0, so relative error decreases do not occur, and for heterogeneity on incorrect monomers, maxpt⟨lnϵϵI⟩=0, so relative error increases do not occur.

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Heat map illustrating the resulting relative error change $⟨ \ln \frac{ϵ}{ϵ_{I}} ⟩$ from parameter sweeps for heterogeneity, having extremized over p_t. We consider heterogeneity on correct [backward propensity discrimination: (a) and forward discrimination propensity discrimination: (b)] and incorrect [backward propensity discrimination: (c) and forward propensity discrimination: (d)] monomers. In the case of heterogeneity on correct monomer interactions, this relative error change is minimized over p_t; for heterogeneity on incorrect monomer interactions, it is maximized over p_t. Only the upper triangle ΔG_TT,H ≥ ΔG_TT,L is plotted. For heterogeneity on correct monomers, $\min_{p_{t}} ⟨ \ln \frac{ϵ}{ϵ_{I}} ⟩ = 0$ ⁠, so relative error decreases do not occur, and for heterogeneity on incorrect monomers, $\max_{p_{t}} ⟨ \ln \frac{ϵ}{ϵ_{I}} ⟩ = 0$ ⁠, so relative error increases do not occur.

FIG. 15.

Heat map illustrating the resulting log heterogeneous change in state visits ln⟨V⟩⟨VI⟩ from parameter sweeps for heterogeneity, having extremized over pt. We consider heterogeneity on correct [backward propensity discrimination: (a) and forward discrimination propensity discrimination: (b)] and incorrect [backward propensity discrimination: (c) and forward propensity discrimination: (d)] monomers. In the case of heterogeneity on correct monomer interactions, ln⟨V⟩⟨VI⟩ is minimized over pt; for heterogeneity on incorrect monomer interactions, it is maximized over pt. Only the upper triangle ΔGTT,H ≥ ΔGTT,L is plotted. For heterogeneity on correct monomers, minpt⁡ln⟨V⟩⟨VI⟩=0, so decreases in state visits do not occur. For heterogeneity on incorrect monomers, maxpt⁡ln⟨V⟩⟨VI⟩=0 for the vast majority of parameters, so increases in state visits do not occur. However, deviations maxpt⁡ln⟨V⟩⟨VI⟩>0 are observed in the small discrimination regime for heterogeneity on incorrect monomer interactions.

View large Download slide

Heat map illustrating the resulting log heterogeneous change in state visits $\ln \frac{⟨ V ⟩}{⟨ V_{I} ⟩}$ from parameter sweeps for heterogeneity, having extremized over p_t. We consider heterogeneity on correct [backward propensity discrimination: (a) and forward discrimination propensity discrimination: (b)] and incorrect [backward propensity discrimination: (c) and forward propensity discrimination: (d)] monomers. In the case of heterogeneity on correct monomer interactions, $\ln \frac{⟨ V ⟩}{⟨ V_{I} ⟩}$ is minimized over p_t; for heterogeneity on incorrect monomer interactions, it is maximized over p_t. Only the upper triangle ΔG_TT,H ≥ ΔG_TT,L is plotted. For heterogeneity on correct monomers, $\min_{p_{t}} \ln \frac{⟨ V ⟩}{⟨ V_{I} ⟩} = 0$ ⁠, so decreases in state visits do not occur. For heterogeneity on incorrect monomers, $\max_{p_{t}} \ln \frac{⟨ V ⟩}{⟨ V_{I} ⟩} = 0$ for the vast majority of parameters, so increases in state visits do not occur. However, deviations $\max_{p_{t}} \ln \frac{⟨ V ⟩}{⟨ V_{I} ⟩} > 0$ are observed in the small discrimination regime for heterogeneity on incorrect monomer interactions.

APPENDIX F: INVESTIGATIONS ON ERROR REDUCTION THROUGH HETEROGENEITY FOR SEPARATING COPIERS

Section III B revealed an interesting phenomenon that heterogeneity on correct monomers tends to increase error rates, while heterogeneity on incorrect monomers tends to decrease error rates. We believe it would be illustrative to attempt to find levels of forward and backward heterogeneity that are (in a heuristic sense) optimal for extracting benefits from heterogeneity.

In our fine-grained copying model, we can obtain varying forward and backward discrimination factors in the slow polymerization limit, k_pol ≪ k_bind. Instead of forcing k_pol to be sequence-independent, we allow it to be a function of the added monomer pair k_pol(n_l, m_l) and set

k_{pol} (n_{l}, m_{l}) e^{Δ G_{T T} (n_{l}, m_{l})} [M] = e^{Δ G_{K} (n_{l}, m_{l})}

to obtain the following form:

Φ^{+} (n_{l - 1} n_{l}, m_{l - 1} m_{l}) = e^{Δ G_{K} (n_{l}, m_{l})},

(F1)

\begin{align} Φ^{-} (n_{l - 1} n_{l}, m_{l - 1} m_{l}) \\ = e^{Δ G_{K} (n_{l}, m_{l}) - Δ G_{pol} + Δ G_{T T} (n_{l - 1}, m_{l - 1}) - Δ G_{T T} (n_{l}, m_{l})} . \end{align}

(F2)

Copying is thus parameterized by a scalar parameter, ΔG_pol, and two matrix parameters, ΔG_K and ΔG_TT.

Based on Sec. III B, we anticipate that error reduction is maximized when correct monomer pairs are kept homogeneous while the heterogeneity in the incorrect pairs is maximized, and so we make that assumption. We first consider the limit where one monomer is purely backward propensity discriminated, while the other is purely forward propensity discriminated. Then, we gradually shift the discrimination until one monomer has half the forward propensity discrimination (and the other has half the backward propensity discrimination) of the other.

We list our constrained parameter sets in Table IV. For each constrained set, we allow for baseline amounts of forward and backward propensity discrimination, $Δ G_{K}^{*}$ and $Δ G_{T T}^{*}$ ⁠, to vary independently. Two values of ΔG_pol are considered, 0 and −0.3 (note that −0.3 is weaker driving). For each parameter set, we consider the average error ratio $E_{p_{t}} [\frac{⟨ ϵ ⟩}{⟨ ϵ_{I} ⟩}]$ (here, the expectation is taken over a uniform distribution of the parameter p_t), a measure of how much heterogeneity helps decrease errors averaged over the content of monomer 2 for Bernoulli templates. Sweeping over different baseline values $G_{K}^{*}$ and $G_{T T}^{*}$ ⁠, we obtain heat maps of $E_{p_{t}} [\frac{⟨ ϵ ⟩}{⟨ ϵ_{I} ⟩}]$ ⁠, plotted in Fig. 16.

TABLE IV.

Parameter sets investigated in Appendix F.

Index	ΔG_TT	ΔG_K
i	$(\begin{matrix} G_{T T}^{} & G_{T T}^{} \\ 0 & G_{T T}^{*} \end{matrix})$	$(\begin{matrix} G_{K}^{} & 0 \\ G_{K}^{} & G_{K}^{*} \end{matrix})$
ii	$(\begin{matrix} G_{T T}^{} & \frac{3 G_{T T}^{}}{4} \\ 0 & G_{T T}^{*} \end{matrix})$	$(\begin{matrix} G_{K}^{} & 0 \\ \frac{3 G_{K}^{}}{4} & G_{K}^{*} \end{matrix})$
iii	$(\begin{matrix} G_{T T}^{} & \frac{G_{T T}^{}}{2} \\ 0 & G_{T T}^{*} \end{matrix})$	$(\begin{matrix} G_{K}^{} & 0 \\ \frac{G_{K}^{}}{2} & G_{K}^{*} \end{matrix})$

FIG. 16.

Identifying regimes where heterogeneity has large impacts on error. Ept[⟨ϵ⟩⟨ϵI⟩] is plotted for parameter sets indexed (i), (ii), and (iii) in Table IV, for ΔGpol = 0 (a)–(c) and ΔGpol = −0.3 (d)–(f). When ΔGpol = 0, the greatest decreases in error tend to occur when ΔGK*≈ΔGTT* for parameter set (i), and ΔGK*=0 for parameter sets (ii) and (iii). When ΔGpol = −0.3, the greatest decreases in error tend to occur when ΔGK*=0 or ΔGTT*=0.

View large Download slide

Identifying regimes where heterogeneity has large impacts on error. $E_{p_{t}} [\frac{⟨ ϵ ⟩}{⟨ ϵ_{I} ⟩}]$ is plotted for parameter sets indexed (i), (ii), and (iii) in Table IV, for ΔG_pol = 0 (a)–(c) and ΔG_pol = −0.3 (d)–(f). When ΔG_pol = 0, the greatest decreases in error tend to occur when $Δ G_{K}^{*} \approx Δ G_{T T}^{*}$ for parameter set (i), and $Δ G_{K}^{*} = 0$ for parameter sets (ii) and (iii). When ΔG_pol = −0.3, the greatest decreases in error tend to occur when $Δ G_{K}^{*} = 0$ or $Δ G_{T T}^{*} = 0$ ⁠.

Consider first ΔG_pol = 0. We observe that in the limit of pure opposite discrimination (i.e., one monomer is purely forward propensity discriminated, and the other purely backward propensity discriminated), the largest decreases in error tend to be obtained when backward and forward propensity discrimination is present at roughly similar amounts. However, as discrimination is shifted in parameter sets (ii) and (iii), $E_{p_{t}} [\frac{⟨ ϵ ⟩}{⟨ ϵ_{I} ⟩}]$ tends to increase in this region [Figs. 16(a)–16(c)]. This response is consistent with what we saw in Sec. III B, since shifting the discrimination results in a smoothing of the incorrect monomer potential. In the best cases, heterogeneity tends to decrease errors by about 10–20 percent (averaged over all p_t) relative to the naïve, uncorrelated estimate of the error probability ϵ_I. On the other hand, for the lower value of ΔG_pol = −0.3, beneficial effects tend to peak when one monomer is purely backward or forward propensity-discriminated, while the other experiences no discrimination at all [Figs. 16(d)–16(f)].

APPENDIX G: MARKOV APPROXIMATIONS FOR ENTROPY AND INFORMATION

Applying an ith-order l-independent Markov assumption on Y, having marginalized over X, we get the following joint probability distribution:

\begin{align} p (m_{l - i}, \dots m_{l - 1}, m_{l}) \\ = Σ_{x} p (m_{l - i}, \dots m_{l - 1}, m_{l} | x) p (x) \\ = \lim_{L \to \infty} \frac{1}{L} Σ_{l = 1}^{L} p (m_{l - i}, \dots m_{l - 1}, m_{l} | x) \\ = \lim_{L \to \infty} \frac{1}{L} Σ_{l = 1}^{L} p (m_{l - i} | x) Π_{j = l - i}^{l - 1} p (m_{j + 1} | m_{j}, x) . \end{align}

(G1)

We have applied a self-averaging assumption here, which follows from our ith-order Markov assumption. Similarly,

\begin{align} p (m_{l} | m_{l - i}, \dots m_{l - 1}) \\ = Σ_{x} p (m_{l} | m_{l - 1}, x) p (x | m_{l - i}, \dots m_{l - 1}) \\ = Σ_{x} p (m_{l} | m_{l - 1}, x) \frac{p (m_{l - i}, \dots m_{l - 1} | x) p (x)}{Σ_{x} p (m_{l - i}, \dots m_{l - 1} | x) p (x)} \\ = \frac{\lim_{L \to \infty} Σ_{l = 1}^{L} p (m_{l} | m_{l - 1}, x) p (m_{l - i}, \dots m_{l - 1}, m_{l} | x)}{\lim_{L \to \infty} Σ_{l = 1}^{L} p (m_{l - i}, \dots m_{l - 1}, m_{l} | x)} \\ = \frac{\lim_{L \to \infty} Σ_{l = 1}^{L} p (m_{l} | m_{l - 1}, x) p (m_{l - i}, \dots m_{l - 1}, m_{l} | x)}{Σ_{m} \lim_{L \to \infty} Σ_{l = 1}^{L} p (m | m_{l - 1}, x) p (m_{l - i}, \dots m_{l - 1}, m_{l} | x)} . \end{align}

(G2)

The final step of Eq. is based on the trivial identity Σ_mp(m|m_l−1, x) = 1. It is included simply to avoid small deviations in the sums of estimated probabilities away from 1.

Differences in estimates of h(Y) assuming a first order Markov chain, h₁(Y), and an eighth order Markov chain, h₈(Y), for ΔG_TT,21 = 0 and ΔG_TT,11 = ΔG_TT,22 = 6, maximized over p_t for each point, are presented in Fig. 17. The differences are not significant, suggesting that h₁(Y) is a tight estimate of h(Y). Strangely, errors seem to be largest in the homogeneous case ΔG_pol = 0 and ΔG_TT,12 = 0 at p_t = 0.5, where Y should be Markov and h(Y) should be exactly ln 2 due to symmetry. However, for this parameter set, it appears that sampling errors are the primary contributor to the difference h₁(Y) − h₈(Y). To better illustrate the effect of sampling, we plot the estimation error h(Y) − h_i(Y) for both L = 10⁴ and L = 10⁶ in Fig. 18, for a homogeneous backward propensity-discriminated copying system with p_t = 0.5 [note h(Y) is exactly calculable: h(Y) = ln 2 in this regime]. Here, h_i(Y) is the entropy estimated by assuming an ith order Markov process. Strangely, the estimation error actually increases with increasing i. However, as the overwhelming majority of the difference vanishes for L = 10⁶, this estimation error is likely due to sampling (genuine errors due to correlations in Y should persist in the L → ∞ limit).

FIG. 17.

View large Download slide

Heat maps illustrating the difference in estimated entropy h(Y) assuming a first vs eighth order Markov Y for (a) backward propensity and (b) forward propensity discrimination, showing insignificant differences throughout.

FIG. 18.

View large Download slide

Entropy estimation error for h(Y) − h_i(Y) for a homogeneous backward propensity-discriminated copying system. Estimation error is plotted for (a) L = 10⁴ and (b) L = 10⁶. Parameters are ΔG_TT,11 = ΔG_TT,22 = 6, and p_t = 0.5. We observe an error increase due to taking higher order i. However, the error decreases massively going from L = 10⁴ to L = 10⁶, implying that sampling is the source of this error.

Note that we estimate entropy by considering the probability of follow-up monomers conditioned on previous length i strings [Eq. (G2)]. As i increases for a fixed L, we expect the estimate of this conditional probability to become worse as there are fewer samples of each length i string, leading to undersampling. This homogeneous case is unique in that we can attribute all errors to sampling; for most parameters, the error in entropy is likely some combination of sampling errors (these errors likely get worse with increasing i) and genuine correlations in Y (these errors likely get better with increasing i).

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Published open access through an agreement with JISC Collections

Index	ΔG_TT	ΔG_K
i	$(\begin{matrix} G_{T T}^{} & G_{T T}^{} \\ 0 & G_{T T}^{*} \end{matrix})$	$(\begin{matrix} G_{K}^{} & 0 \\ G_{K}^{} & G_{K}^{*} \end{matrix})$
ii	$(\begin{matrix} G_{T T}^{} & \frac{3 G_{T T}^{}}{4} \\ 0 & G_{T T}^{*} \end{matrix})$	$(\begin{matrix} G_{K}^{} & 0 \\ \frac{3 G_{K}^{}}{4} & G_{K}^{*} \end{matrix})$
iii	$(\begin{matrix} G_{T T}^{} & \frac{G_{T T}^{}}{2} \\ 0 & G_{T T}^{*} \end{matrix})$	$(\begin{matrix} G_{K}^{} & 0 \\ \frac{G_{K}^{}}{2} & G_{K}^{*} \end{matrix})$

Index	ΔG_TT	ΔG_K
i	$(\begin{matrix} G_{T T}^{} & G_{T T}^{} \\ 0 & G_{T T}^{*} \end{matrix})$	$(\begin{matrix} G_{K}^{} & 0 \\ G_{K}^{} & G_{K}^{*} \end{matrix})$
ii	$(\begin{matrix} G_{T T}^{} & \frac{3 G_{T T}^{}}{4} \\ 0 & G_{T T}^{*} \end{matrix})$	$(\begin{matrix} G_{K}^{} & 0 \\ \frac{3 G_{K}^{}}{4} & G_{K}^{*} \end{matrix})$
iii	$(\begin{matrix} G_{T T}^{} & \frac{G_{T T}^{}}{2} \\ 0 & G_{T T}^{*} \end{matrix})$	$(\begin{matrix} G_{K}^{} & 0 \\ \frac{G_{K}^{}}{2} & G_{K}^{*} \end{matrix})$

The interplay of heterogeneity and product detachment in templated polymer copying

Author Notes

I. INTRODUCTION

II. METHODS

A. General coarse-grained model of templated polymer copying with heterogeneity and product separation

B. Fine-grained reaction steps

C. Solving for distributions of complete polymers

D. Recovering further properties of the fine-grained system from the coarse-grained system

E. Thermodynamically consistent parameterization for heterogeneous separating copiers

III. RESULTS

A. Consequences of separation for the velocity and energy landscapes of heterogeneous copying

B. Effects of separation and heterogeneity on error probability and average time spent per site

1. Heterogeneity in correct monomer interactions

a. Backward propensity discrimination.

b. Forward propensity discrimination.

2. Heterogeneity in incorrect monomer interactions

a. Backward propensity discrimination.

b. Forwards propensity discrimination.

3. Discussion

C. Entropy, mutual information, and efficiency in the low ΔGpol regime

IV. CONCLUSION

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

APPENDIX A: GASPARD’S METHOD VIA ABSORBING PROBABILITIES OF MARKOV CHAINS

APPENDIX B: DERIVATION OF COARSE-GRAINED PROPENSITIES

APPENDIX C: DERIVATION OF AVERAGE TIP WAITING TIMES

APPENDIX D: ITERATION FOR THE EXPECTED NUMBER OF VISITS TO A TIP STATE

APPENDIX E: PARAMETER SWEEP OF RELATIVE ERROR AND TOTAL VISITATIONS

APPENDIX F: INVESTIGATIONS ON ERROR REDUCTION THROUGH HETEROGENEITY FOR SEPARATING COPIERS

APPENDIX G: MARKOV APPROXIMATIONS FOR ENTROPY AND INFORMATION

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C. Entropy, mutual information, and efficiency in the low ΔG_pol regime