Force–velocity relation for copolymerization processes

A study is reported of copolymerization processes subjected to an external force in the light of recent advances on the kinetics and thermodynamics of copolymer growth. Different processes are considered: the free copolymerization of Bernoulli and first-order Markov chains and copolymerization with a template. For every process, the dependence on the force is analyzed for the elongation rate, the growth velocity, the disorder in the copolymer sequence, the thermodynamic entropy production, as well as related quantities. It is shown that disorder in the copolymer sequence can generate forces of entropic origin. They are characterized by the value of their stall force in the force–velocity relation of the corresponding process.

These processes can be influenced by external forces exerted on such motors, for instance with optical tweezers. Motion can even be stopped by opposing an external force, called the stalling force. At this value of the force, the detachment rate of monomeric units compensates the attachment rate, so that elongation is stopped. These considerations have been much developed for processes taking place in periodic potentials or with identical monomeric units. However, many processes involve different species of monomers or proceed along a heterogeneous filament, as it is the case for DNA or RNA polymerases [20,21]. For such copolymerization processes, theoretical work has shown that disorder in the sequence of monomeric units remarkably contributes to the thermodynamic entropy production and may even drive the growth of the copolymer [22][23][24][25][26][27][28]. In this regard, we may wonder if this sequence disorder could generate a pushing force of entropic origin during copolymerization.
The purpose of the present paper is to show that this can indeed be established on the basis of recent advances [23][24][25]. According to these advances, the thermodynamic entropy produced per monomeric unit during the growth of copolymers is equal to the sum of the contributions from both chemical free energy and disorder in the growing sequence. If an external force is applied, its mechanical work should also be included. Under the assumption of tight mechano-chemical coupling, the disorder per monomeric unit in the growing copolymer sequence would thus contribute to the value of the stalling force, besides chemical free energy. For certain concentrations of monomers, the copolymer could even generate a pushing force although the pure polymers grown in the absence of comonomers could not.
A stochastic approach is used to describe the random attachments and detachments of monomeric units to the growing copolymer. In this approach, the thermodynamics of these nonequilibrium processes can be established and the entropy production obtained. Fluctuation relations are used in order to characterize the growth processes at the level of the fluctuating numbers of monomeric units incorporated in the copolymer.
The general results of this paper are presented in section 2 for copolymerization processes without or with a template. The cases of growing Bernoulli and first-order Markov chains are developed respectively in sections 3 and 4. The case of copolymerization with a template is given in section 5. Conclusions are drawn in section 6. Furthermore, a multivariate fluctuation relation is proved in the appendix for the numbers of monomeric units or doublets incorporated in a growing copolymer.

Free copolymerization
We consider the growth of a copolymer. Its description is coarse grained at the level of the sequence 1 remain fixed during the growth process. Moreover, the solution is also a heat bath so that the growth proceeds at constant temperature T. One end of the copolymer is anchored on a surface while monomers are attached to the other end. At this end, a catalyst or enzyme may lower the activation energies of monomer attachments. The reversed reaction of detachment is also possible. Accordingly, the chain length l undergoes a biased random walk. An external force F is exerted at the growing end of the chain, for instance on the catalyst or enzyme, as shown in figure 1(a). We assume a tight coupling between the mechanical movement of the growing end at the velocity v, and the chemical process of chain elongation at the rate r. Moreover, the dynamics is supposed to be quasi onedimensional. In other circumstances, the growing copolymer can exert a pushing force on a wall if the copolymer is rigid enough to maintain the directionality of the generated force [13][14][15][16].
Since the attachment and detachment events are random, the process is stochastic and described by the master equation for the probability ω P ( ) t to find the sequence ω at the time t. The rates of opposite transitions have ratios given by is the chemical free enthalpy or Gibbs' free energy of the copolymer ω in the surrounding solution, and ω x is the position of the copolymer growing end, on which the external force F is exerted.
At the time t, the total entropy of the system is given by where the first term is the average contribution of the entropy ω S ( )of the copolymer if it has the sequence ω, while the second term is the contribution due to the distribution of the probability ω P ( ) t over the different possible sequences ω { } [29]. The contribution of vibrational, rotational, and translational atomic movements is taken into account in the entropy ω S ( ). Figure 1. Schematic representations of force generation by growing copolymers in the cases: (a) of free copolymerization, (b) of copolymerization with a template. F denotes the force exerted on the copolymer chain end, while v is its growth velocity. Now, the total entropy (3) has a time evolution ruled by the master equation (1). The time derivative of the total entropy can be decomposed as and the entropy production This latter is always non-negative in accordance with the second law of thermodynamics.
Copolymerization is assumed to proceed in a regime of steady growth at a constant mean elongation rate t and mean growth velocity δ being the average size of a monomeric unit. In this regime and after a long enough time t, the probability ω P ( ) t is supposed to factorize as into the time-dependent probability p t (l) that the copolymer contains l monomeric units and the stationary probability distribution μ ω ( ) l that the copolymer of length ω = | | l has the sequence ω [23][24][25]. Using these assumptions and equation (2), the entropy production (6) can be expressed as c l l c is the mean chemical free enthalpy per monomeric unit and is the Shannon disorder per monomeric unit in the sequence. We notice that the entropy production (11) can be written as = rA , as the product of the elongation rate r with the affinity [23][24][25]. The tight mechano-chemical coupling explains that a single rate and a single affinity describe this process.
The expression (11) shows that the elongation of the chain can be powered by three possible mechanisms: the first one is the mechanical force F, which can drive elongation if it is positive and large enough, but which is opposed to elongation if it is negative. The second one is the chemical free energy of monomeric attachment, which is driving elongation if the free-energy landscape goes down as the chain grows, i.e., if < g 0 c . The third one is the disorder D in the sequence, which is always non-negative and can thus also drive elongation if it is large enough to dominate the two other terms [23].
At equilibrium, the elongation rate r and the affinity A are vanishing together with the thermodynamic entropy production (11). Accordingly, the stall force, which stops elongation r = 0, is given by The stall force is negative if it is opposed to elongation, as it should in particular if the free-energy landscape is favorable to the growth when < g 0 c . For the growth of a polymer composed of a single monomeric species, the disorder is vanishing D = 0, in which case the stall force is solely determined by the chemical free energy. However, for the growth of a copolymer, the sequence disorder also contributes to the stall force. We see that, even if the free-energy landscape is not favorable to the growth when > g 0 c , the sequence disorder can nevertheless generates a contribution such that the stall force is negative opposing the growth if β > D g c . In this case, the sequence disorder generates a pushing force of entropic origin, although the copolymer evolves in an adverse free-energy landscape with > g 0 c .

Copolymerization with a template
In the fundamental biological processes of DNA replication or transcription, the copolymerization of DNA or RNA takes place with a template of sequence α = ⋯ ⋯ − + n n n n n l l l 1 2 1 1 , which has its own statistical distribution ν α ( ) l . Now, the transition rates ω ω → ′ W ( )also depend on the template sequence, for instance, if the monomers should first form Watson-Crick pairs with the corresponding monomeric units of the template, before their attachment to the growing DNA or RNA strand by some polymerase, as depicted in figure 1(b).
The previous framework continues to hold. If elongation is again assumed to proceed in a regime of steady growth, the probability ω P ( ) t that the copy has the sequence ω at the time t can also be factorized as into the probability p t (l) that its length is equal to ω = | | l and the probability μ ω α | ( ) l that the copy has the sequence ω given that the template has the sequence α [23]. By a reasoning similar as in the previous section 2.1, the thermodynamic entropy production is obtained as in terms of the mean chemical free enthalpy per monomeric unit c l l l c , and the conditional Shannon disorder per monomeric unit of the copy ω with respect to the template α: Besides this conditional Shannon disorder, we can introduce the overall Shannon disorder of the copy as l l l with the probability distribution of the copy whatever the template sequence can be: l l l We can also introduce the mutual information between the copy and the template as . This mutual information is always lower or equal to the overall disorders in the copy and the template: By a standard formula of information theory [32], the mutual information can be written as which characterizes fidelity in the copying process [23]. The thermodynamic entropy production (16) becomes Consequently, the stall force is here given by The stall force is negative and opposed to elongation if the chemical free-energy landscape is favorable to the growth or if sequence disorder in the copy is large enough. Instead, the mutual information has a non-negative contribution to the stall force. In the following sections, these general results are illustrated in particular cases.
3. The growth of Bernoulli chains 3.1. The kinetic process In the simplest growth processes, the rates only depend on the attaching or detaching monomeric unit: The attachment and detachment rates are supposed to obey mass action law and to have an exponential dependence on the external force F: where C m is the concentration of monomers m in the surrounding solution [33,34]. The ratio of these transition rates is given by We notice that, in general, the rates may have non-exponential dependences on the force, as long as their ratio has the required exponential dependence [35].
The master equation of this process takes the following form [24] ∑ Inserting the factorization (10), the stationary probability distribution of the sequences is found to factorize itself into the product of the probabilities of the monomeric units where r is the mean elongation rate given by the non-negative root of if it exists. Therefore, the growing copolymer forms a Bernoulli chain [24].

The thermodynamic entropy production
The thermodynamic entropy production is here given by equation (11) with the mean monomeric size m M 1 as it should for a Bernoulli chain [24].

The stall force
The stall force has the expression (14), which can be compared with the stall forces of pure polymers grown in the absence of comonomers. If all the concentrations would be vanishing except the one of the species m, the disorder would also be vanishing D = 0 and the stall force would take the value As a corollary, the Shannon disorder per monomeric unit can be obtained as Since the disorder is always non-negative, the stall force of the copolymer is always bounded from above by the weighted sum of the stall forces of the pure polymers grown with equal corresponding concentrations of monomers:

Illustrative example
In order to illustrate the predictions, the growth of Bernoulli chains composed of M = 2 monomeric species can be simulated using Gillespie's algorithm [36,37] with the following rate constants and concentrations , which shifts the stall force towards a negative value according to equation (37). This is an example where the stall force is exerted to oppose a growth driven by sequence disorder. Since the velocity is given by the elongation rate r multiplied by the mean monomeric size (33), δ = v r , the force-velocity relation is shown in figure 2 as the curve with the triangles. Figure 3 shows the different possible regimes in the plane of the control parameters given by the concentration C 1 of monomers 1 and the external force F, otherwise in the conditions (40). If the concentration of monomers 1 is too low or the external force too opposing, the copolymer undergoes depolymerization and shrinks. Above the stall force where A = 0, the copolymer is growing, first driven by the entropic effect of its sequence disorder and, next, by the free energy of binding the monomeric units. The two growth regimes are separated by the condition of vanishing free-energy driving power ε β δ Figure 4 compares the growths of the copolymer and pure polymers composed of monomeric units m = 1 or m = 2. For pure polymers, there is no disorder, D = 0, so that the affinity coincides with the free-energy driving power ϵ = A . The affinity of pure polymers is thus given by their free-energy driving power in figure 4.
Moreover, their stall force varies with the concentration of monomers as shown in equation (36). Accordingly, different comparisons can be made depending on the concentrations chosen for the pure polymers.
In figure 4(a), the concentrations are chosen to illustrate the inequality (39) between the stall forces of the copolymer and the polymers. With this aim, the concentrations should take equal values, because equations (37) and (38) are deduced for the same values of the concentrations in the growths of the copolymer and the corresponding pure polymers. For the concentrations C 1 = 0.01 and C 2 = 0.0005, the copolymer is composed of monomeric units 1 and 2 with the fractions μ = (1) 2 3 and μ = (2) 1 3 and its stall force is thus equal to . According to equation (36), the stall force of polymer 1 is vanishing at the concentration C 1 = 0.01, β δ = . Consequently, the inequality (39) is satisfied, as observed in figure 4(a). Furthermore, the weighted difference (38) between these stall forces gives the value of the Shannon disorder per monomeric unit at the copolymer stall force: = − ≃ D ln 3 (2 3)ln 2 0.6365. In contrast, the concentrations can be chosen for the pure polymers to develop equal stall forces as the copolymer, as shown in figure 4(b). Using equation (36), the conditions  and C 2 = 0.0015 for the growth of polymer 2. As seen in figure 4(b), all the elongation rates and the affinities are vanishing at the same value of the stall force for these concentrations. To get this effect, the concentrations needed for the growths of the pure polymers have to be increased with respect to the concentrations (40) used for the growth of the corresponding copolymer. Different comparisons are thus possible by tuning the concentrations.

Multivariate fluctuation relation
In order to follow the stochastic evolution of the copolymerization process, we may consider the random numbers . The multivariate fluctuation relation can be used in order to determine the monomer-specific affinities by setting to zero all the numbers except one. In the case where M = 2, we get both relations:  The multivariate fluctuation relation is applied to copolymerization in the conditions (40) without external force in figure 5 and at the stall force in figure 6. In the absence of external force, the copolymer is growing so that the probability distribution p N N ( , )

The growth of first-order Markov chains
Similar results hold for growing Markov chains.  As before, the rates obey mass action law and are supposed to depend exponentially on the external force F: as proved in [25].
In the limit = ± | ± w w n m n , where the rates no longer depend on the previously incorporated monomeric unit, the conditional, tip, and bulk probabilities coincide, so that the partial elongation rates are equal to the mean one, = r r m for all = … m M 1, 2, , , and we recover Bernoulli chains.

The thermodynamic entropy production
The thermodynamic entropy production (11) is here given in terms of the mean monomeric size

The stall force
The stall force is thus given by the expression (14), which here reads Provided that the present process is at equilibrium if the external force takes its stall value, an equivalent way to obtain the stall force is to require that detailed balancing is satisfied and the partial elongation rates are vanishing r m = 0 for all = … m M 1, 2, , . This leads to the condition

Illustrative example
The growth of Markov chains composed of M = 2 monomeric species is simulated using Gillespie's algorithm [36,37] Figure 8 shows how the affinity A and the elongation rate r, i.e., the growth velocity δ = v r , depend on the rescaled external force β δ F for the copolymer, as well as for the pure polymers grown with different concentrations of monomers, in order to illustrate the effects of sequence disorder in the copolymer. As aforementioned, the affinity of pure polymers coincides with their free-energy driving power, ϵ = A , because they have a vanishing sequence disorder D = 0.
In figure 8(a), the concentrations chosen for the growth of pure polymers take the same value as for the growth of the copolymer, but in the absence of their respective comonomers. In this case, the copolymer has a negative stall force, which is opposed to the growth driven by disorder, although the pure polymers have positive stall forces.
In figure 8( for the growth of polymer 2. Under such conditions, the pure polymers develop the same stall forces as the copolymer, as seen in figure 8(b). We notice that the affinity coincides with the free-energy driving power ε = A for the pure polymers, while they are different for the copolymer because of disorder in its sequence.    This is illustrated for the growth of first-order Markov chains in the conditions (61) without external force in figure 9(a), and at the stall force in figure 9(b). In the absence of external force, the four affinities (63) are positive, although they vanish at the stall force, which corresponds to equilibrium conditions. In figure 9, we see the agreement between numerical simulations with Gillespie's algorithm and the theoretical expectation (63), which supports the multivariate fluctuation relation (62).

The growth of copolymers with a template
In this section, an example is considered illustrating copolymerization processes with a template.

The kinetic process
The kinetics proceeds by random attachments and detachments of monomeric units to a growing chain in contact with a template so that the rates depend not only on the involved monomer, but also on the corresponding monomeric unit of the template: The attachment and detachment rates are here also supposed to obey mass action law and to have an exponential dependence on the external force F: where C m is the concentration of monomers m in the surrounding solution so that their ratios are given by 1, 2, , . We notice that, for processes with DNA or RNA polymerases, the kinetics is of Michaelis-Menten type, which requires a specific study beyond the aims of the present paper.
The process defined by equations (64)-(66) can be simulated with Gillespie's algorithm [36,37] to obtain the different quantities of interest given by equations (17)-(23). We consider an example with M = 2 monomeric species. The template is taken as a Bernoulli chain of probabilities ν ν = = (1) (2) 1 2, in which case the copy is itself a Bernoulli chain and the stationary probability distribution of the sequences introduced with equation (15)

Illustrative example
The following rate constants and concentrations are chosen for simulations: . In contrast, the free-energy driving power ε β δ = − F g ( ) c vanishes at the positive value β δ ≃ F 0.1 of the rescaled external force. Again, the growth is driven by sequence disorder, which is dominant between these two values of the external force. This disorder is characterized by the conditional Shannon disorder (18) and the overall disorder (19), which are depicted in figure 10. The difference between the overall and conditional disorders gives the mutual information (22) between the copy and the template. As seen in figure 10, the mutual information is significantly lower than the disorders, which means that the copying process is affected by a lot of errors in the present example. , ω D ( )the overall Shannon disorder of the copy (19), and ω α I ( , )the mutual information (22) between the copy and the template. The template is a Bernoulli chain of probabilities ν ν = = (1) (2) 1 2. The symbols connected by dashed lines are the results of numerical simulations with Gillespie's algorithm.

Conclusions
In the present paper, we have studied how copolymerization processes may depend on an external force exerted on the end of a growing chain. Different processes are considered: the free copolymerization of Bernoulli or firstorder Markov chains and the copolymerization with a template, which concerns the biological processes of DNA replication, transcription of DNA into mRNA, and translation of mRNA into proteins.
A stochastic approach is used to describe the random process of attachment and detachment of monomeric units to a copolymer growing on average in a surrounding solution where the monomers are supplied at fixed concentrations. If an external force is exerted on the process, its transition rates have some dependence on this force. Since the rates of opposite transitions have ratios related to the free energy of the states, these ratios generally depend exponentially on the external force. The master equation of the stochastic process rules the time evolution of every quantity of interest and, in particular, the thermodynamic entropy. In this way, the entropy production can be obtained for copolymerization processes. Surprisingly, the entropy production depends not only on the external force and the chemical free energy of monomer binding to the growing chain, but also on the Shannon disorder per monomeric unit in the copolymer sequence. Consequently, the stall force, for which the mean elongation rate vanishes, is influenced by the sequence disorder. This feature is specific to copolymers and does not arise for pure polymers, in which case the stall force is only determined by the chemical free energy. Accordingly, the growth of a copolymer can proceed in two possible regimes driven either by a favorable free-energy landscape, or by a large enough disorder in the growing sequence although the free-energy landscape is adverse. In this latter regime, the growth is driven by the entropic effect of sequence disorder and the force exerted by the growing copolymer would thus be of entropic origin. This concerns free copolymerization, as well as copolymerization with a template, as illustrated with several examples.
For growing Bernoulli chains, the stall force of copolymers is bounded from above by the average value of the stall forces of pure polymers grown in corresponding conditions, the difference being due to the Shannon disorder per monomeric units, as shown by equations (38) and (39). An illustrative example is presented where the relation between the external force and the growth velocity is calculated theoretically and with numerical simulations by Gillespie's algorithm, together with other quantities of interest including the thermodynamic entropy production. For this example, the stall force is compared between the copolymer and pure polymers grown with different concentrations, demonstrating the influence of sequence disorder in the copolymer. Moreover, the time evolution of copolymerization can be investigated thanks to a multivariate fluctuation relation for the random numbers of monomeric units incorporated in the chain. This fluctuation relation is proved in the appendix.
Similar results are obtained for growing first-order Markov chains. Here also, the kinetic equations can be solved analytically in the long-time limit and compared with numerical simulations by Gillespie's algorithm in order to obtain the dependences of the elongation rate and entropy production on the external force. Here also, an example is presented where the stall force of the copolymer is compared with the stall forces of pure polymers grown in different conditions. The growth process can be analyzed in great detail thanks to the multivariate fluctuation relation, which allows us to determine the individual affinities characterizing the incorporation of monomeric doublets in the Markov chain. At the stall force, these affinities are vanishing as in equilibrium conditions.
For copolymerization with a template, simulations with Gillespie's algorithm shows again that sequence disorder may generate forces of entropic origin in an adverse free-energy landscape.
These results could be tested experimentally with optical trapping techniques to exert forces on growing copolymers or on polymerases catalyzing the chain elongation. Knowing the sequence of growing copolymer or the template, the effect of its disorder could be studied as a function of the exerted force.

Appendix. The multivariate fluctuation relation for copolymerization
In this appendix, the multivariate fluctuation relation is proved for the growth of first-order Markov chains thanks to methods based on previously work [30,31].  The eigenfunctions of this operator take the form ϕ = e k N i · . In the limit → k 0, the leading eigenvalue gives the cumulant generating function: in terms of the affinities (63). By using large-deviation theory, we finally infer the multivariate fluctuation relation (62). In the limit = ± | ± w w n m n , where the transition rates no longer depend on the previously incorporated monomer, the affinities (63) reduce to those of Bernoulli chains given by equation (42)