Model Definition.

We consider a copy polymer M=M1,…,Ml, made up of a series of subunits or monomers Mx, growing with respect to a template N=N1,…,NL (l≤L). Inspired by transcription and translation, we consider a copy that detaches from the template as it grows; Fig. 1B shows the simplest model of this type. We consider whole steps in which a single monomer is added or removed, encompassing many individual chemical substeps (21, 24). After each step, there is only a single interpolymer bond at position l, between Ml and Nl. As a new monomer joins the copy at position l+1, the bond position l is broken, contrasting with previous models of TSA (20⇓⇓⇓⇓–25) (Fig. 1A). Importantly, as explained in the next paragraphs of this section, each step therefore depends on both of the two leading monomers, generating extra correlations within the copy sequence.

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Fig. 1.

Free-energy landscapes for simple examples of (A) TSA, in which the monomers remain bound to the template during the copy process and (B) persistent copying, in which the monomers detach from the template after they have been incorporated into the polymer. Both diagrams show the addition of three monomers to a growing polymer, driven by a chemical free energy of backbone polymerization ΔGpol. In each subfigure, two scenarios are considered: the addition of two incorrect monomers, followed by a correct one (Top), and the addition of three correct monomers (Bottom). Local minima in the landscape represent macrostates following complete incorporation of monomers; intermediate configurations, illustrated schematically for the first transition, are part of the effective barriers. In TSA, the chemical free-energy cost of previously incorporated mismatches is retained as the daughter grows (20⇓⇓⇓⇓–25). Thus, in A, each mismatch in the daughter increases the chemical free energy by ΔGD relative to the perfect match. In persistent copying (in B), the chemical free-energy penalty for incorporating wrong monomers is only temporary; it arises when the incorrect monomer is added to the growing polymer, but is lost when that monomer subsequently detaches from the template. As a result, the overall chemical free-energy change of creating an incorrect polymer is the same as that for a correct one. Analyzing the consequences of this constraint, which is a generic feature of copying but does not arise in TSA, is the essence of this work. The figure also shows that, in our specific model, incorporating a wrong monomer after a correct one tends to reduce the chemical free-energy drop to ΔGpol−ΔGTT, and incorporating a correct monomer after an incorrect one tends to increase it to ΔGpol+ΔGTT; however, adding a wrong monomer to a wrong one, and adding a correct monomer to a correct one, does not change the free-energy drop ΔGpol.

Following earlier work, we assume that both polymers are copolymers, and that the two monomer types are symmetric (20⇓⇓⇓⇓–25). Thus, the relevant question is whether monomers Ml and Nl match; we ignore the specific sequence of N and describe Ml simply as right or wrong. Thus, Ml∈r,w, with example chain M=rrwwrrrrrwrr. An excess of r indicates a correlation between template and copy sequences. Breaking this symmetry would favor specific template sequences over others, disfavoring the accurate copying of other templates and compromising the generality of the process.

Given the model’s state space, we now consider state free energies (which must be time-invariant for autonomy). We treat the environment as a bath of monomers at constant chemical potential (20⇓⇓⇓⇓–25). By symmetry, extending the polymer while leaving the copy–template interaction unchanged involves a fixed polymerization free energy. We thus define −ΔGpol as the chemical free-energy change for the transition between any specific sequence m1,…,ml and any specific sequence m1,…,ml+1, ignoring any contribution from interactions with the template. We then define ΔGTT as the effect of the free-energy difference between r and w interactions with template. This bias can be describes as “temporary thermodynamic” (TT), since it only lasts until that contact is broken.

Overall, each forward step makes and breaks one copy–template bond. There are four possibilities: either adding r or w at position l+1 to a template with Ml=r or adding r or w in position l+1 to a template with Ml=w. The first and last of these options make and break the same kind of template bond, so the total free-energy change is −ΔGpol. For the second case, there is a r bond broken and a w bond added, implying a free-energy change of −ΔGpol+ΔGTT. Conversely, for the third case, there is a w bond broken and a r bond added, giving a free-energy change of −ΔGpol−ΔGTT. These constraints are highlighted in Fig. 1B; the contribution of this work is to study the consequences of these constraints. Models of TSA (Fig. 1A) of equivalent complexity can be constructed, but they are not bound by these constraints, and hence the underlying results and biophysical interpretation are distinct.

Having specified model thermodynamics, we now parameterize kinetics. We assume that there are no “futile cycles,” such as appear in kinetic proofreading (17). Reactions are thus tightly coupled: Each step requires a well-defined input of free energy determined by −ΔGpol and ±ΔGTT (29), and no free-energy release occurs without a step.

A full kinetic treatment would be a continuous time Markov process incorporating the intermediate states shown schematically in Fig. 1B. To identify sequence output, however, we need only consider the state space in Fig. 1B and the relative probabilities for transitions between these explicitly modeled states, ignoring the complexity of nonexponential transition waiting times (21). We define propensity ψxy+ as the rate per unit time in which a system in state &x starts the process of becoming &xy and, propensity ψxy− as the equivalent quantity in the reverse direction (& is an unspecified polymer sequence). Our system has eight of these propensities (ψrr±, ψrw±, ψwr±, and ψww±); the simplest TSA models require four (20, 21, 23⇓–25).

Prior literature on TSA (23) has differentiated between purely “kinetic” discrimination, in which r and w have an equal template-binding free energy but different binding rates, and purely thermodynamic discrimination in which r and w bind at the same rate, but r is stabilized in equilibrium by stronger binding interactions. Eventually, all discrimination is kinetic for persistent copying, since there is no lasting equilibrium bias (Fig. 1B). However, by analogy with TSA, we do consider two distinct mechanisms for discrimination: a kinetic one, in which r is added faster than w to the growing tip, and one based on the TT bias toward correct matches at the tip of the growing polymer due to short-lived favorable interactions with the template, quantified by ΔGTT>0 (Fig. 1B). The kinetic mechanism should not be conflated with fuel-consuming “kinetic proofreading” cycles that are not considered.

We parameterize the propensities as follows. Assuming, for simplicity, that the propensity for adding r or w is independent of the previous monomer, we have ψrr+=ψwr+ and ψrw+=ψww+=1, also defining the overall timescale. Kinetic discrimination is then quantified by ψxr+/ψxw+=exp(ΔGK/kBT). Forward propensities are thus differentiated solely by ΔGK; backward propensities are set by fixing the ratios ψxy+/ψxy− according to the free-energy change of the reaction, which follows from ΔGpol and ΔGTT (Fig. 1B). Thus, setting kBT=1,ψrr+=eΔGK,ψrr−=e−ΔGpoleΔGK,[1]ψrw+=1,ψrw−=e−ΔGpoleΔGTT,[2]ψwr+=eΔGK,ψwr−=e−ΔGpoleΔGKe−ΔGTT,[3]ψww+=1,ψww−=e−ΔGpol.[4]For a given ΔGK, ΔGpol, and ΔGTT, Eqs. 1–4 describe a set of models with distinct intermediate states that yield the same copy sequence distribution. We can thus analyze the simplest model in each set, which is Markovian at the level of the explicitly modeled states, and has ψxy± as rate constants.

Model Analysis.

We use Gaspard’s method to solve the system (28); we note that the underlying kinetic equations can also be mapped to models of distinct physical systems that have different constraints on the parameters (26). In this approach, the tip monomer identity probabilities μ(ml), the joint tip and penultimate monomer identity probabilities μ(ml−1,ml), and the conditional probabilities μ(ml−1∣ml) become stationary for a long polymer and can be calculated. One must first calculate the partial velocities, vr and vw. The quantity vxμ(x) is the net rate at which monomers are added after an x,vx=ψxr+−μ(x∣r)μ(r)μ(x)ψxr−+ψxw+−μ(x∣w)μ(w)μ(x)ψxw−.[5]Following ref. 28, these velocities can be solved in terms of the propensities. In turn, the velocities and propensities determine tip and conditional probabilities μ(ml) and μ(ml−1∣ml) (28). Further details are provided in SI Appendix.

Gaspard’s method describes the chain while it is still growing through the stochastic variables Ml and Ml−1, with the index l being the current length of the polymer. We, however, are interested in the identity of the monomer at position n when l≫n. We label this “final” state of the monomer at position n as Mn∞. As discussed in SI Appendix, Mn∞ is distinct from Mn near the tip. Mn∞ is described by the error probability ϵ and the conditional error probabilities ϵr and ϵw, defined as the probability that Mn+1∞=w given that Mn∞=r or Mn∞=w, respectively. In SI Appendix, we show that ϵ, ϵr and ϵw are sufficient to describe the final state, by proving that the M∞ is a Markov chain of r and w monomers (this requirement is distinct from the Markovian growth dynamics). We note that ϵ ≠ ϵr ≠ ϵw as a direct result of the dependence of the transition propensities on the current and previous tip monomers, which in turn arises from detachment.

To calculate ϵ, ϵr, and ϵw, we define currents Jxy that are related to ψxy±, and to ϵ, ϵr, and ϵw, separately. The current Jxy is the net rate per unit time at which transitions &x→&xy occur: Jxy=ψxy+μ(x)−ψxy−μ(x,y). By considering the transitions in our system as a tree, as in Fig. 2, we can relate the current through a branch to the overall rate at which errors are permanently incorporated into a polymer growing at total velocity v=vrμ(r)+vwμ(w),Jrr=(1−ϵ)(1−ϵr)v=μ(r)ψrr+−μ(r,r)ψrr−,[6]Jrw=(1−ϵ)ϵrv=μ(r)ψrw+−μ(r,w)ψrw−,[7]Jwr=ϵ(1−ϵw)v=μ(w)ψwr+−μ(w,r)ψwr−,[8]Jww=ϵϵwv=μ(w)ψww+−μ(w,w)ψww−.[9]Eliminating ϵ from the simultaneous Eqs. 6–9 yields ϵr and ϵw in terms of known quantities. To find ϵ, note that the final sequence itself is a Markov chain with a transition matrix parameterized by ϵr and ϵw, with the overall error ϵ given by its dominant eigenvector. As detailed in SI Appendix, we obtain ϵ=ϵr/(1+ϵr−ϵw). From ϵ, ϵr, and ϵw, we calculate copy properties in terms of ψxy± and thus the free energies. We corroborate the results with simulation (see SI Appendix).

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Fig. 2.

Transitions of an arbitrary polymer &. To relate the final chain to the growing chain, it is useful to consider fluxes through interfaces in this transition diagram. Using the tip and combined probabilities, along with relative propensities, it is possible to describe the fluxes through interfaces 4 to 7 in terms of properties of the growing chain. Equally, by considering errors and conditional errors and taking fractions of the overall growth velocity, it is possible to find the fluxes through interfaces 4 to 7 in terms of properties of the final chain and growth velocity.