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Growth and dissolution of macromolecular Markov chains

The kinetics and thermodynamics of free living copolymerization are studied for processes with rates depending on k monomeric units of the macromolecular chain behind the unit that is attached or detached. In this case, the sequence of monomeric units in the growing copolymer is a kth-order Markov chain. In the regime of steady growth, the statistical properties of the sequence are determined analytically in terms of the attachment and detachment rates. In this way, the mean growth velocity as well as the thermodynamic entropy production and the sequence disorder can be calculated systematically. These different properties are also investigated in the regime of depolymerization where the macromolecular chain is dissolved by the surrounding solution. In this regime, the entropy production is shown to satisfy Landauer's principle.

The growth of macromolecular chains composed of different species of monomeric units is a kinetic process that is fundamental in polymer science and biology [1][2][3]. Such macromolecules are called copolymers or heteropolymers, as opposed to homopolymers composed of identical monomeric units [4,5]. In general, the sequence of monomeric units is irregular in copolymers and a key issue is to understand their statistical properties. They can be obtained experimentally by sequencing methods or NMR analysis, revealing that the sequences follow statistical laws described as Markov chains of different orders [6][7][8]. In principle, these statistical properties are determined by the conditions prevailing to the synthesis of the copolymers.
The complexity of such kinetic processes holds in the fact that they generate a multitude of possible sequences. These processes are thus defined in the infinite space of all the corresponding concentrations, which all change in time. Typically, the kinetic equations are nonlinear if the reaction network is such that the sequences may attach between each other (and reversely break into any possible subsequences), which is favored by chain transfer reactions. In the case of living copolymerization, the reaction proceeds without chain termination, without chain transfer reaction, and after a rapid chain initiation avoiding nucleation [5]. This is the case if the process evolves only by the attachment and detachment of monomeric units to the chain, for instance with a catalyst located at the tip of the copolymer. Free living copolymerization is most important in polymer science where macromolecules composed of various types of monomeric units can be synthesized with newly advanced organometallic catalyst systems [9][10][11][12][13][14][15][16][17].
Since the forties, the kinetic theory of living copolymerization has been developed for the fully irreversible regime where the chain grows only by the attachment of monomeric units, ignoring the detachment events [18][19][20]. Remarkably, the kinetics can also be solved analytically in the presence of both detachment and attachment events, as recently shown for the growth of Bernoulli and first-order Markov chains [21,22].
The purpose of the present paper is to extend these results to living copolymerization with attachment and detachment rates depending not only on the last monomeric unit incorporated in the chain, but also on k previously incorporated monomeric units. As suggested by previous work [21,22] devoted to the k = 0 and k = 1 cases, k th -order Markov chains are generated if the kinetics depends on k lastly incorporated monomeric units [23]. In the present paper, the analytical method previously developed in Ref. [22] is generalized from first to arbitrary order k. The central idea of this method is that the properties of the growing chain can be deduced by solving a set of self-consistent equations, directly established in terms of the attachment and detachment rates [22]. This theoretical method is much more efficient than numerical Monte Carlo simulations. In this way, the composition of the macromolecular chain can be determined, as well as the mean growth velocity and the corresponding thermodynamic quantities, including the entropy production. A fluctuation relation is also established.
Furthermore, the dissolution of the macromolecular chain is investigated and analytical formulas are obtained for the mean depolymerization velocity and the thermodynamic entropy production, generalizing previous results as well [22,24]. The erasing of information possibly contained in the macromolecular sequence is shown to obey Landauer's principle [25], and also a stronger bound coming from the order k of the reaction network. There is an interesting analogy with the growth and dissolution of crystals, which are the three-dimensional extensions of the one-dimensional macromolecular chains that are here studied.
In the following developments, a more abstract analogy is made between the sequences of monomeric units in the macromolecular chains and the symbolic dynamics of chaotic systems [26], the analogy providing useful techniques for mathematical analysis. The paper is organized as follows. Section II is devoted to the growth regime. In Subsection II A and Appendix A, the kinetic equations are solved in the regime of steady growth. In Subsection II B, the results are used to obtain the thermodynamic entropy production, necessary and sufficient conditions for equilibrium, and a fluctuation relation that is partly deduced in Appendix B. Section III is dealing with the dissolution of the macromolecular chains by depolymerization. The kinetics of this regime is solved in Subsection III A and the thermodynamics is obtained in Subsection III B. Section IV compares the theoretical results with kinetic Monte Carlo simulations for two illustrative examples. Section V contains the conclusion.
II. GROWTH OF k th -ORDER MARKOV CHAINS A. Kinetics

Master equations
The growth proceeds by living copolymerization, i.e., the successive attachments and detachments of different species of monomers m j ∈ {1, 2, ..., M } to macromolecular chains ω = m 1 m 2 · · · m l , elongating without termination. The chains -also called copolymers -and the monomers are in solution where they move and react together. The growth is assumed to take place only at one of both ends of each chain: The rates of attachment and detachment, respectively w ±m l |m l−1 ···m l−k , depend not only on the monomer m l that is on the verge of being attached or detached, but also on k previously incorporated monomeric units m l−k · · · m l−1 .
The number k determines the order of the Markov chain ruling the copolymer sequence, as will be shown in detail here below using the method of Ref. [22]. The number of monomeric units in the macromolecular chain l = |ω| ∈ N defines the length of the copolymer. The chain is growing if attachment is more frequent than detachment, otherwise the chain dissolves by depolymerization. The reactions (1) may be induced by a catalyst located at the growing tip of the copolymer. The process is supposed to be isothermal at the temperature T and to evolve in a dilute solution. If attachment and detachment are the elementary steps of the reaction (1), the rates are determined by the mass action law. In this respect, the attachment rates are proportional to the concentrations c m of the monomers in solution, while the detachment rates are independent of the concentrations. Moreover, if an external force f is exerted on the tip of the copolymer or the catalyst therein located, the rates have an Arrhenius-type dependence on the change of activation energy induced by the external force, which is often exponential [27]. Under such circumstances, the rates can be modeled by attachment rates: detachment rates: where k ±m l |m l−1 ···m l−k are the rate constants, β = T −1 the inverse temperature in units where Boltzmann's constant is equal to unity, d ±m l |m l−1 ···m l−k the transition-state displacements in the direction of the external force f , and c m l the concentration of the monomer species that is attached [27,28]. The solution is large enough so that the concentrations c m of the monomers remain constant during the whole process. If the reactions (1) are not elementary, the attachment and detachment rates are more complicated than in Eqs. (2) and (3), for instance, with Michaelis-Menten dependences on the monomer concentrations. However, the following considerations remain general because they hold without specifying the dependences of the rates w ±m l |m l−1 ···m l−k , except otherwise stated. In the solution, the copolymers are dilute enough so that they do not interact with each other and the process can be considered at the level of a single copolymer. In this case, the concentration of copolymers with the length l and the sequence ω = m 1 m 2 · · · m l is proportional to the probability that a single copolymer has these properties at the time t: where N is the total number of copolymers in the solution of volume V . This probability evolves in time according to the master equations: forming an infinite hierarchy of coupled linear ordinary differential equations of first order in time. Accordingly, the growth is described as a stochastic jump process on the coarse-grained states: which form a tree growing as M l with the length l. Equation (5) holds for l = k + 1, k + 2, . . . . For l = 0, 1, 2, . . . , k, the same expression can still be used with m 0 = ∅, m −1 = ∅, ... , m −k = ∅, ... by defining the following initiation rates w ±m k |m k−1 ···m2m1∅ , w ±m k−1 |m k−2 ···m1∅∅ , w ±m1|∅···∅∅∅ , and by setting w ±∅|∅···∅∅∅ = 0. Consequently, the total probability is conserved, ω P t (ω) = 1, with the sum running over the states (6). In order to describe the process at the growing tip of the copolymer, the hierarchy of master equations (5) can be transformed into another hierarchy for the probabilities: . . .
They are interpreted as follows: p t (l) is the probability that the copolymer has the length l at the time t; p t (m l , l) the probability that it has the length l and m l is the last monomeric unit at its tip; p t (m l−1 m l , l) is the probability that it has the length l, its last monomeric unit is m l , and its penultimate unit is m l−1 ; etc. These probabilities are related to each other according to . . .

Regrouping these probabilities into the infinite vector
the full set of equations can be written in the form: with infinite matrices A, B, and C of constant coefficients given by the rates.

Solutions of the master equations
Since Eq. (14) is linear in the probabilities, its general solution can be decomposed as a linear superposition of the particular solutions: (15) expressed in terms of the wavenumber −π < q ≤ +π and a dispersion relation, which is expected to take the form: because the growth process is similar to a random walk of drift velocity v and diffusivity D to be determined. The drift velocity v is the mean growth velocity or mean elongation rate counted in monomeric units incorporated in the macromolecular chain per unit time. Equation (15) can also be expressed as for the probability to find the subsequence m l−r+1 · · · m l−1 m l at the growing tip of the copolymer with r = 0, 1, 2, 3, .... We notice that the quantities φ q (m l−r+1 · · · m l−1 m l ) are related to each other by Eqs. (10), (11), (12),... in the same way as the corresponding probabilities are. Inserting the particular solution (15) into Eq. (14), the dispersion relation is given by the eigenvalue problem: The right and left eigenvectors are respectively denoted Φ Φ Φ q andΦ Φ Φ q . By expanding these eigenvectors and the eigenvalue s q in powers of the wavenumber q and taking the limit q = 0, Eq. (18) gives the mean growth velocity as where Because of probability conservation, the left eigenvector at q = 0 has identical elements, which can be fixed to the unit value:Φ Φ Φ 0 = {1, 1, 1, ...}. Renormalizing the right eigenvector at q = 0, the following probabilities are defined . . . as the consequence of Eqs. (10), (11), (12), etc. These quantities have the following interpretations: In the regime of steady growth, µ(m l ) is the probability to find the monomeric unit m l at the tip of the copolymer, µ(m l−1 m l ) is the probability that m l is the last monomeric unit and m l−1 is the penultimate unit of the chain, etc.
Supposing that these time-independent probabilities are known, a closed equation can be deduced for the timedependent probability p t (l) that the chain has the length l ≫ k at the time t: where the coefficients are constant and expressed as Solving Eq. (28), the dispersion relation is obtained as with −π < q ≤ +π, whereupon the mean growth velocity is given by and the diffusivity by D = (a + b)/2. As a consequence, the length probability distribution obeys the central limit theorem: meaning that, in the long-time limit, the random variable x ≡ (l − vt)/ √ 2Dt has a Gaussian distribution of zero mean and unit variance [29]. Accordingly, the mean length is growing linearly in time as l t ≃ v t, and the standard deviation as where the notation l n t ≡ l p t (l) l n is used with n = 1, 2. We notice that, beyond the time scale t ≃ k 2 /(2D), the width (34) of the length probability distribution becomes larger than the number k of monomeric units determining the rates, which justifies the steady growth at a constant mean velocity v after a long enough time. In this regime, the probabilities can therefore be supposed to behave as P t (m 1 · · · m l−1 m l ) ≃ p t (l) µ(m 1 · · · m l−1 m l ) (35) in terms of the time-dependent length probability distribution (7) and the time-independent probabilities (22), (23), (24), etc, which is commonly assumed [30,31]. Now, these stationary probabilities are obtained by solving the hierarchy of Eqs. (20), which are explicitly written down in Appendix A for the probabilities µ(m l−r+1 · · · m l−1 m l ) of subsequences with an increasing number r of monomeric units. The key observation is that these equations are similar to each other for r = k + 1, k + 2, ..., which shows that they can be solved if the growing copolymer is assumed to be a k th -order Markov chain, implying the factorization: in terms of the M k+1 conditional probabilities µ(m l−k |m l−k+1 · · · m l ) and the M k tip probabilities µ(m l−k+1 · · · m l ). These probabilities are normalized according to Using the factorization (36) in the second term of the expression (32) for the mean growth velocity, this latter can be written as by defining the M k partial velocities: As shown in Appendix A, these partial velocities are directly determined in terms of the attachment and detachment rates by solving the M k following self-consistent equations: which play a central role in the following developments. In the growth regime, these equations can be solved by numerical iteration in the M k -dimensional space of positive partial velocities, which shows convergence. Once the partial velocities are known, the tip probabilities are obtained by solving the M k following equations: and the M k+1 conditional probabilities are given by which completely determines the k th -order Markov chain. The mean growth velocity is finally obtained with Eq. (39) by averaging the partial velocities over the tip probabilities (42). We notice that Eqs. (42) are recovered by multiplying Eq. (43) with the tip probability µ(m l−k+1 · · · m l ) of the denominator, summing over m l−k , and using the normalization (37) of the conditional probabilities.
The bulk probabilities are defined as the stationary probabilities of the Markov chain according tō which gives the probability to find the subsequence m j · · · m j+k−1 of length k, anywhere in a long enough chain of length l ≫ k. The bulk and tip probabilities are related to each other bȳ Indeed, replacing Eq. (45) into Eq. (44) and using Eq. (43), we go back to the self-consistent equations (41). Moreover, the expression (39) for the mean growth velocity is recovered by summing Eq. (45) over m l−k+1 · · · m l , which completes the proof of Eq. (45).

Correlation functions characterizing the sequences
We notice that the statistical properties of the copolymer can be characterized in terms of correlation functions: between some observables A(ω) and B(ω) defined over the subsequences ω = m i · · · m i+k−1 . In Eq. (46), σ denotes the shift of the subsequence by one monomeric unit σω = m i+1 · · · m i+k . The decay of such correlation functions is determined by the eigenvalues formed by the conditional probabilities (43). We notice that the leading eigenvalue is equal to unity, Λ 1 = 1, due to the conservation of probability (37). These results have already been established for first-order Markov chains in the case k = 1 [22].

The fully irreversible growth regime
In this regime, the detachment of the last monomeric unit becomes negligible with respect to the attachment of new units so that the detachment rates can be assumed to be vanishing: Hence, the partial velocities are simply given by summing the attachment rates according to Eq. (40): The corresponding tip and conditional probabilities are inferred by Eqs. (42)- (43) under the conditions (48). In this regime, which has been much studied since the forties, we recover previously established results, in particular, the famous Mayo-Lewis formula for first-order Markov chain [18][19][20].
B. Thermodynamics

Entropy production
The entropy production is given by which is equivalent to the expression known for dilute solutions in terms of the concentrations (4) [32][33][34][35]. In the long-time limit, the probability distribution factorizes according to Eq. (36) and the length probability distribution becomes broader and broader because of the central limit theorem (33) and Eq. (34). Therefore, we may suppose that p t (l − 1) ≃ p t (l). Hence, the entropy production reaches the following stationary value expressed in terms of the net rates and the affinities The net rates (52) can be written first in terms of the partial velocities by using Eq. (43) and, thereafter, in terms of the bulk probabilities with Eq. (45) to get Now, the affinities (53) are expanded as follows Replacing into Eq. (51), we find that the last two terms cancel each other because of Eq. (44) and the normalization conditions (37). We thus find that the entropy production takes the form in terms of the mean growth velocity v, the free-energy driving force [36] and the Shannon disorder per monomeric unit as established in Ref. [37]. The free-energy driving force is proportional to the mean free-enthalpy g per incorporated monomeric unit. Combining Eqs. (57) and (58) defines the mean affinity: which is the thermodynamic entropy production per incorporated monomeric unit. We recover the expressions of Refs. [21,22] in the cases of the processes with k = 0 and k = 1, respectively generating Bernoulli chains and first-order Markov chains. Since the entropy production (56) is always non-negative according to the second law of thermodynamics, the free-energy driving force is bounded as i.e., in the growth regime. Therefore, the growth of the copolymer can be driven not only by a positive free-energy driving force ǫ > 0, but also by the entropic effect of sequence disorder if −D < ǫ ≤ 0 [37]. The thermodynamic entropy production (56) becomes infinite in the fully irreversible regime, because the freeenergy driving force (57) diverges under the conditions (48), although the mean growth velocity and the Shannon disorder keep finite values.

Equilibrium
The principle of detailed balance holds at thermodynamic equilibrium so that the net rates (52) and the affinities (53) both vanish. Consequently, the conditional probabilities are given by Comparing with Eq. (43), we can conclude that the partial velocities are also vanishing at equilibrium: v m l−k+1 ···m l ,eq = 0. Therefore, the mean growth velocity is vanishing v = 0 together with the entropy production and the affinity. As a consequence, the free-energy driving force is related to the Shannon disorder per monomeric unit by ǫ eq = −D eq . Summing Eq. (61) over m l−k and using the normalization conditions (37), we see that the equilibrium tip probability should satisfy with the coefficients The which is composed of the elements (63) located at the row m l−k · · · m l−1 and the column m l−k+1 · · · m l . The necessary condition to solve the set (62) of homogeneous linear equations is that implying that at least one of the eigenvalues z α of the matrix (64) is equal to unity. Beyond, a necessary and sufficient condition for the system to be at equilibrium is that the spectral radius of the matrix (64) is equal to unity: The copolymer is growing if ρ(Z) > 1 and depolymerizing if ρ(Z) < 1.

Fluctuation relation
A remarkable result is that the incorporation of monomeric units in the chain is ruled by a fluctuation relation, which is established as follows in the framework of large-deviation theory [46,47].
Suppose that the chain has been growing in the past till reaching the length l 0 ≫ 1. From this initial situation, the growth process goes on and we count the M k+1 numbers ∆N = ∆N mj ···m j+k of the different possible subsequences m j · · · m j+k of length k + 1 that are formed at the tip of the chain every time a new monomeric unit is incorporated. Due to the possible detachment events, these numbers can take negative values if the chain length decreases and some multiplets of length k + 1 disappear. Accordingly, the counters may fluctuate in time between positive and negative values ∆N ∈ Z M k+1 , even if they grow on average. As shown in Appendix B, the probability p t (∆N) that these counters take the values ∆N at the time t obeys the fluctuation relation: where A = A mj ···m j+k denotes the M k+1 affinities (53). This result generalizes the relations already obtained in the special cases of Bernoulli and first-order Markov chains to k th -order Markov chains with k > 1 [28,38]. We notice that the fluctuation relation (67) implies the non-negativity of the entropy production in accordance with the second law of thermodynamics. Indeed, the entropy production can be obtained in terms of the Kullback-Leibler divergence between the probability distributions p t (∆N) and p t (−∆N): Replacing the ratio of probabilities in the logarithm by the fluctuation relation (67), we find that the entropy production can be expressed as in terms of the net rates (52) or (54) given by which is equivalent to Eq. (56) in terms of the mean growth velocity (32) and the mean affinity (59). In this regard, the fluctuation relation (67) concerns the currents of multiplets incorporated in the chain.

III. DISSOLUTION OF k th -ORDER MARKOV CHAINS
The dissolution of macromolecular chains is the process that is reverse to the growth. This dissolution proceeds by the depolymerization of the copolymer due to the dominance of monomeric detachment over attachment. In this section, we generalize the results of Refs. [22,24] to depolymerization processes where the attachment and detachment rates w ±m l |m l−1 ···m l−k depend on the last k + 1 monomeric units m l−k · · · m l−1 m l at the tip of the copolymer.
Contrary to the growth where the copolymer is generated by a kinetic process with properties essentially independent of the initial conditions, the depolymerization does depend on the initial copolymer. Indeed, this latter is introduced in the solution at the beginning of dissolution and it has thus been previously synthesized under conditions different from those of the solution where depolymerization takes place. Consequently, the rate of depolymerization and the other properties depend on the initial sequence of the copolymer. This latter is supposed to be long enough so that a regime of steady depolymerization can be reached before the complete dissolution of the initial copolymer.

A. Kinetics
The mean depolymerization velocity −v > 0 can be obtained by a first-passage problem [39,40]. For non-vanishing attachment rates, the depolymerization is a succession of detachment and attachment events. A transient growth may happen between two detachment events. The mean time before the monomeric unit m l is detached is given by summing over all the possible transient growths during which r = 0, 1, 2, 3, ... monomeric units are attached. If r = 0, the removal of m l lasts the time (w −m l |m l−1 ···m l−k ) −1 . If r = 1, the transient growth is caused by the transitions m l → m l m l+1 → m l , so that the removal lasts the time (w −m l |m l−1 ···m l−k ) −1 m l+1 z m l+1 |mi···m l−k+1 , which is expressed in terms of the ratios (63) of attachment to detachment rates. If r > 1, the transient growth continues up to the elongation of the copolymer by the subsequence m l m l+1 · · · m l+r , so that the corresponding time is given by a similar expression with a product of ratios (63) and a sum over the monomeric units m l+1 · · · m l+r . Adding all these contributions, the mean time is obtained as During depolymerization, such transient growths may occur for all the successive monomeric units of the initial copolymer. Ifμ k+1 (m 0 m 1 · · · m k ) denotes the probability to find the multiplet m 0 m 1 · · · m k in the sequence of the initial copolymer, the mean depolymerization velocity is thus given by averaging as (72) Using the matrix notation (64), we find that with m j , n j ∈ {1, 2, ..., M }. The mean velocity v is negative because ρ(Z) < 1 in the depolymerization regime.

B. Thermodynamics
Since the initial copolymer has a specific sequence, there is no sequence disorder that is generated during depolymerization contrary to what happens during growth. In the regime of steady depolymerization, the thermodynamic entropy production thus takes the form where the free-energy driving force is here given bȳ in terms of the probabilityμ k+1 (m 0 m 1 · · · m k ) of the corresponding multiplet in the initial copolymer. The free-energy driving force (75) is bounded by Landauer's principle [25], according to which erasing possible information contained in the initial copolymer during its dissolution has a thermodynamic cost. This bound can be established by considering the Kullback-Leibler divergence (per monomeric unit) between the sequence probability distribution of the initial copolymer and the one of the copolymer that would be grown at thermodynamic equilibrium: According to the principle of detailed balance, the equilibrium probability distribution is determined by Eq. (61) in terms of the ratios of transition rates and it is related to the free enthalpy G(ω) of the copolymer in the solution by µ (eq) l (ω) = l−k j=1 µ eq (m j |m j+1 · · · m j+k ) µ eq (m l−k+1 · · · m l ) = µ eq (m 1 · · · m k ) The Kullback-Leibler divergence (76) thus becomes whereḡ = −β −1ǭ is the free enthalpy per monomeric unit averaged over the sequence of the initial copolymer and given by (75), whileĪ is the Shannon disorder per monomeric unit of the initial copolymer. This quantity also characterizes the information possibly coded in the initial copolymer and that is erased during depolymerization. Therefore, the general bound implied by Landauer's principle is that the entropy production given by Eq. (74) should remain larger than the Shannon information, both evaluated per monomeric unit: However, the depolymerization process only depends on no more than k + 1 monomeric units at the copolymer tip. We can thus also consider the Kullback-Leibler divergence between the sequence probability distribution of the initial copolymer truncated to a k th -order Markov chain and the equilibrium one to get with the Shannon information contained in the multiplets of length k + 1: The thermodynamic entropy production per monomeric unit is thus bounded by 1 |v| which holds if the rates depend on no more than k + 1 monomeric units behind the tip of the copolymer. In general,Ī k+1 ≥Ī ∞ so that the bound (83) is stronger than (80) and Landauer's principle is always obeyed.

IV. ILLUSTRATIVE EXAMPLES
In this section, several illustrative examples are considered for the growth and depolymerization of 2 nd -order Markov chains and composed of two monomeric species m ∈ {1, 2}, so that k = 2 and M = 2. The previous theoretical results are compared with numerical simulations using Gillespie's kinetic Monte Carlo algorithm [41,42]. In each figure, the dots depict data from numerical simulations and the lines the corresponding theoretical results. In the illustrative examples, the external force is set equal to zero f = 0 and the mass action law applies. The attachment rates are thus proportional to the concentrations of the monomers that are attached, while the detachment rates do not depend on the concentrations according to Eqs. (2)-(3).

A. Growth of period-two copolymers and dissolution
In the first example, the rate constants are taken as  The concentration c 1 of monomers 1 is varying, while the concentration of monomers 2 is fixed to the value: c 2 = 0.005. The attachment and detachment rates are thus given respectively by w +m3|m2m1 = k +m3|m2m1 c m3 and w −m3|m2m1 = k −m3|m2m1 with m 1 , m 2 , m 3 ∈ {1, 2}.

Growth
The example (84) illustrates the growth of an alternating copolymer, which would be of period two (12) ∞ if all the rates were equal to zero except k ±1|21 and k ±2|12 . Since the other rates are non-vanishing, the growing chains are not perfectly periodic and some disorder manifests itself.
In the case where k = 2 and M = 2, the self-consistent equations (41) for the partial velocities are given by These equations are solved by numerical iterations starting from positive initial values. Away from equilibrium, convergence up to ten digits is achieved in a few dozens of iterations. At equilibrium where Eq. (66) is satisfied, the convergence is slower and goes as the inverse of the number of iterations. An accuracy of ten digits is obtained over the whole concentration range with a thousand iterations.
Once the partial velocities are obtained, the tip and conditional probabilities are successively calculated with Eqs. (42) and (43). The mean growth velocity is given by Eq. (39), the free energy driving force by Eq. (57), the Shannon disorder per monomeric unit by Eq. (58), the mean affinity by the sum A = ǫ + D, and the thermodynamic entropy production by Eq. (56). These different quantities are shown in Fig. 1 as a function   In Fig. 1, we see that the mean growth velocity, the mean affinity, and the entropy production are vanishing at the critical concentration c 1,eq ≃ 0.00064027 corresponding to thermodynamic equilibrium. This critical concentration is calculated by solving Eq. (65) and selecting the value where the necessary and sufficient condition (66) is satisfied. At equilibrium, the Shannon disorder per monomeric unit takes the value D eq = −ǫ eq ≃ 0.09276. However, the free-energy driving force is vanishing ǫ = 0 at the larger concentration c 1,ǫ=0 ≃ 0.00077711. In the interval c 1,eq < c 1 ≤ c 1,ǫ=0 , the growth is driven by the entropic effect of sequence disorder [37]. The free-energy driving force becomes positive and plays its role for larger concentrations: c 1,ǫ=0 < c 1 . This is confirmed by the correlation functions computed with N sequences of total length L. They are shown in Fig. 3 for the concentrations c 1 = 0.01, 0.1 and 1. The alternating character manifests itself by the fact that the correlation function changes sign, which is due to the presence of a negative leading eigenvalue for the matrix (47) formed by the conditional probabilities µ(m 1 |m 2 m 3 ). The calculation of these eigenvalues again confirms this conclusion: (87) We note that the leading eigenvalue Λ 2 is significantly smaller in absolute value if the concentration is c 1 = 1, which explains that the corresponding correlation function decays faster in Fig. 3 and the alternating character is much less marked in this case where the monomeric unit 1 is most frequent in the chain sinceμ 3 (111) ≃ 0.745 when c 1 = 1.

Dissolution
For lower concentrations c 1 < c 1,eq , the chain undergoes depolymerization. The initial chain should thus have a long enough length l 0 , which decreases during the process simulated with the same algorithm as the growth. In order to show the important dependence on the sequence of the initial copolymer, depolymerization is compared between three different initial sequences: a Bernoulli chain of probabilitiesμ 1 (1) =μ 1 (2) = 0.5 with the Shannon informations I ∞ =Ī 3 = ln 2; the periodic chain (11121222) ∞ withĪ ∞ = 0 but exactly the same singlet, doublet, and triplet bulk probabilities as the Bernoulli chain so thatĪ 3 = ln 2; and the periodic chain (12) ∞ withĪ ∞ =Ī 3 = 0.  In Fig. 4, we see the remarkable result that the depolymerization velocity as well as the free-enthalpy per monomeric unit take precisely the same values for Bernoulli chains as for the periodic chain (11121222) ∞ because they have the same triplet bulk probabilitiesμ 3 (m 1 m 2 m 3 ) = 1/8. This is indeed expected from Eqs. (73) and (75) with k = 2. We also observe that the free enthalpy per monomeric unit undergoes an important jump at the critical equilibrium concentration between the values βg ≤ βg eq = D eq ≃ 0.093 in the growth regime and those βḡ ≥ βḡ 0 ≃ 2.1 for the depolymerization of a Bernoulli chain or the periodic chain (11121222) ∞ , where the notation βḡ 0 = lim v→0 βḡ is used.
However, the jump is much smaller in Fig. 5 for the depolymerization of the periodic chain (12) ∞ , in which case βḡ ≥ βḡ 0 ≃ 0.02. The reason is that the periodic chain (12) ∞ has statistical propertiesμ (k) closer to the equilibrium probability distribution µ (eq) , which is determined by the rates according to Eq. (77), than a Bernoulli chain or the periodic chain (11121222) ∞ .
In order to further test the bound (81), Figure 6 shows the minimum free enthalpy per monomeric unit βḡ 0 at vanishing depolymerization velocity v = 0 together with the triplet Shannon informationĪ 3 for the depolymerization of the periodic sequences [(12) n 2] ∞ as a function of 2 ≤ n ≤ 100. At the critical equilibrium concentration c 1,eq ≃ 0.00064027 where the velocity is vanishing v = 0, the minimum free enthalpy of depolymerization βḡ 0 can be calculated with Eq. (75) for the periodic sequences [(12) n 2] ∞ to be equal to although the triplet Shannon information can be evaluated with Eq. (82) for k = 3 as For the sequence (122) ∞ at n = 1, these quantities take the values βḡ 0 ≃ 1.1472 andĪ 3 = 0. For n → ∞, we recover the aforementioned values βḡ 0 ≃ 0.02 andĪ 3 = 0 of the periodic chain (12) ∞ . The difference βḡ 0 −Ī 3 is also depicted in Fig. 6, which is the smallest for n = 31, but always positive. Accordingly, the bound βḡ ≥Ī 3 determined by the order k of the kinetic scheme is always satisfied in agreement with Eq. (81).
Here, the concentration of monomers 2 is fixed to the value: c 2 = 0.1, while the concentration c 1 is varying.

Growth
The example (90) illustrates the growth of a copolymer of period three (122) ∞ in the limit where the only nonvanishing rates would be k ±1|22 , k ±2|12 and k ±2|21 . Such sequences can be synthesized by experimental techniques [43].
Here also, the partial velocities are efficiently obtained by iterating Eqs. (85). They are used to get the tip and conditional probabilities of the Markov chain, allowing us to determine the mean growth velocity (39), the free energy driving force (57), the Shannon disorder (58), the mean affinity A = ǫ + D, and the thermodynamic entropy production (56), which are depicted in Fig. 7 as a function of the concentration c 1 . Here, the critical equilibrium concentration where the mean velocity vanishes together with the mean affinity and the entropy production takes the value c 1,eq ≃ 3.2643 × 10 −6 where the Shannon disorder is given by D eq = −ǫ eq ≃ 0.035125. In this example, the free-energy driving force vanishes at the concentration c 1,ǫ=0 ≃ 3.6306 × 10 −6 . The growth is driven by the entropic effect of sequence disorder in the interval c 1,eq < c 1 ≤ c 1,ǫ=0 and by the free-energy driving force if c 1,ǫ=0 < c 1 , in accordance with the predictions of Ref. [37].
To confirm this conclusion, the correlation functions (86) are depicted in Fig. 9 for the concentrations c 1 = 0.01, 0.1 and 1. Here, we observe damped oscillations of period three. The eigenvalues of the matrix (47) formed by the which explains the behavior seen in Fig. 9.

Dissolution
Depolymerization happens for concentrations below equilibrium: c 1 < c 1,eq with c 1,eq ≃ 3.2643 × 10 −6 .  Figure 10 shows the depolymerization velocity (73) as a function of the rescaled free enthalpy (75) for three different initial copolymers: a Bernoulli chain of probabilitiesμ 1 (1) =μ 1 (2) = 0.5 withĪ ∞ =Ī 3 = ln 2; the periodic chain (11121222) ∞ withĪ ∞ = 0 but exactly the same Shannon informationĪ 3 = ln 2 as the Bernoulli chain; and the periodic chain (122) ∞ withĪ 3 =Ī ∞ = 0. Since the Bernoulli chain and the periodic chain (11121222) ∞ have the same triplet bulk probabilitiesμ 3 (m 1 m 2 m 3 ) = 1/8, their depolymerization velocity and free enthalpy per monomeric unit coincide, as seen in Fig. 10 and as predicted by Eqs. (73) and (75). Moreover, the bound (81) is satisfied for the Bernoulli chain and (11121222) ∞ because where βḡ 0 is the minimum value of the rescaled free enthalpy. However, the gap between βḡ 0 andĪ 3 is relatively large since the Bernoulli chain and (11121222) ∞ have triplet probabilities that are significantly different from those generated by the kinetics (90), which is instead favoring the formation of the period-three chain (122) ∞ . Accordingly, the free enthalpy can take much smaller values for the depolymerization of this periodic chain, as seen in Fig. 10. Nevertheless, the bound (81) is again satisfied, becauseĪ 3 is vanishing for (122) ∞ : Therefore, the results confirm that the bound βḡ ≥ βḡ 0 ≥Ī 3 predicted by Eq. (81) is satisfied.

V. CONCLUSION
In the present paper, the theory has been developed for living copolymerization processes with attachment and detachment rates depending not only on the monomer that is newly attached to or lastly detached from the molecular chain, but also on k successive monomeric units behind the tip of the copolymer. Such processes generate the growth of k th -order Markov chains.
In the regime of steady growth, the theory allows us to determine analytically the statistical properties of the growing chains in terms of partial velocities obeying self-consistent equations. These equations only depend on the attachment and detachment rates and they can be solved by numerical iterations. Once the partial velocities are known, the conditional probabilities of the Markov chain as well as its bulk and tip probabilities can be deduced. The mean growth velocity is given by averaging the partial velocities over the tip probabilities. The diffusivity of the length during growth can also be obtained. These results are provided by extending the method already developed for the growth of Bernoulli and first-order Markov chains in Refs. [21,22].
With the probability distribution of the monomeric sequences in the growing macromolecular chain, the nonequilibrium thermodynamics of the copolymerization process can be established. The entropy production is explicitly shown to be given in terms of the mean growth velocity, the free energy driving force, and the Shannon disorder per monomeric unit of the k th -order Markov chain by Eq. (56). Accordingly, the growth can be driven either by the free-energy driving force away from equilibrium or by the sequence disorder as equilibrium is approached [37]. A fluctuation relation is also deduced for the probability distribution of the random numbers of multiplets incorporated in the chain.
Necessary and sufficient conditions are formulated to find the equilibrium state where the entropy production vanishes together with the mean growth velocity. This latter can become negative in the regime of depolymerization. Interestingly, the properties of depolymerization are determined by the sequence of the initial macromolecular chain, which is typically different from the sequences self-generated during growth. Indeed, the initial chain is synthesized under conditions different from those prevailing in the solution where depolymerization happens. In this regime, analytical formulas are also obtained for the mean velocity of depolymerization and the thermodynamic entropy production, which both depend only on the statistical distribution of the multiplets composed of k + 1 monomeric units and no more. Consequently, the entropy production of depolymerization is always larger than the Shannon information per monomeric units in these multiplets and, thus, always larger than the overall Shannon information per monomeric unit. Therefore, Landauer's principle is satisfied, according to which the thermodynamic entropy production of a process erasing some molecular structure cannot be smaller than the information it may contain [25].
Theory is compared with numerical simulations by Gillespie's kinetic Monte Carlo algorithm [41,42] for two illustrative examples of 2 nd -order kinetics, leading to the growth of period-two and period-three copolymers composed of two monomeric species or the corresponding depolymerization. In both examples, theory is in nice agreement with the numerical simulations, justifying the assumptions of stationary growth or depolymerization at the basis of the analytical results. In the examples, the different quantities of interest are studied as a function of monomeric concentrations in the surrounding solution for the regime of steady growth and depolymerization. In between, there exists a critical value of concentration corresponding to the state of thermodynamic equilibrium. Although the mean velocity remains continuous across the transition between growth and depolymerization, the entropy production per monomeric unit does not, its discontinuity depending on the initial copolymer sequence that is depolymerized. This discontinuity can be reduced if the initial sequence has statistical properties close to the sequence that would naturally be grown by the kinetic process. The study of these illustrative examples confirms that the mean depolymerization velocity as well as the free enthalpy per monomeric unit and, thus, the entropy production only depend on the statistics of triplets in the initial sequence, the length k + 1 of these multiplets corresponding to the order k of the kinetic scheme (and of the Markov chains in the growth regime).
To conclude, this work completes the analysis of free living copolymerization processes of arbitrary order started in Refs. [21,22]. The analytical formulas are spectacularly more efficient than Monte Carlo simulations so that they can be used to quickly predict the composition of copolymers once the reaction rates of their synthesis by living copolymerization are known.
Beyond, the present theory can also be applied to study the autonomous processes of living copolymerization with a template [37], or the cyclic processes used for the synthesis of sequence-controlled copolymers with encoded information [44,45].
Since the sum extends over all the multiplets of length k + 1, the indices can be arbitrarily changed as m l−k · · · m l → m j · · · m j+k for any integer j. The net rates (52) or (54) are obtained by differentiating the cumulant generating function (B10) with respect to the counting parameters: The remarkable property is that the cumulant generating function (B10) obeys the symmetry relation Q(λ λ λ) = Q(A − λ λ λ) , with respect to the affinities (53). Using large-deviation theory [46,47], the fluctuation relation (67) is thus established. Q.E.D.