Abstract
We present a theoretical investigation of a polymerization process catalyzed by an enzyme. A structural model of enzyme, sliding along the polymer chain as a Brownian particle, is proposed, and a stochastic approach is employed to describe the kinetics of the whole process. The key point of this work is the coupling mechanics/chemistry obtained by assuming that (1) some rates of chemical reaction depend on the position of the enzyme with respect to the polymer chain and (2) the potential energy and the friction coefficient in the Langevin equation depend on the chemical state of the polymerizing complex. We describe an algorithm for computing our stochastic model and a methodology to solve the Langevin equation numerically. We predict in particular: (1) the sudden arrest of the polymerization, (2) the decrease in the relative polydispersity with the increase in the length of the polymer chain, (3) the occurrence of four regimes, (4) the manifestation of the coupling mechanics/chemistry for one regime and (5) the possibility to evaluate the mechanical variables through classical chemical analysis. Although essentially devoted to the elongation phase, this work also briefly addresses the problem of phase termination and we propose a new device aimed at reducing the polydispersity of technical origin in actual polymerization processes.
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Notes
The release of the monomer from \( \textsf{E}_{10}^\nu \) and \( \textsf{E}_{11}^\nu \) is already described by the backward reactions described by Eqs. (7a) and (7c).
For \( \textsf{A} = \textsf{B} \), the expression for the reaction rate transforms into \( K(t) = \kappa \left[\kern-0.15em\left[ \textsf{A} \right]\kern-0.15em\right]\left( {\left[\kern-0.15em\left[ \textsf{A} \right]\kern-0.15em\right] - 1} \right) \), in agreement with the fact that the reaction is impossible when \( \left[\kern-0.15em\left[ \textsf{A} \right]\kern-0.15em\right] = 1 \) (Gillespie 1992).
This remark also applies to the reactions described by Eqs. (7a)–(7g).
\( {\mathcal{H}}(q)\) (q) = 0 for q < 0 and \( {\mathcal{H}}(q) \) = 1 for q ≥ 0.
Here, the classical rates are assumed to be independent of the length of the polymer chain; for this reason, it is unnecessary to attach an index n to the symbol \( \chi_{i}^{ \pm } \).
We have \( p_{i}^{ \pm } = 1 \) since \( {\mathbb D}_{in}^\pm = {\mathbb R} \).
Note that if one would assume that the polymer and the fluid are fixed, while the enzyme is mobile with respect to the laboratory, then the friction coefficient would be associated with the enzyme and consequently independent of n.
The superscript sim in \( \tau_{\text{I}}^{\text{sim}} \) and \( \tau_{\text{II}}^{\text{sim}} \) indicates that these values result from the simulations of our model, while the subscripts I and II refer to the time periods \( {{\mathbb T}_{\text{I}}} \) and \( {{\mathbb T}_{\text{II}}} \), respectively.
That is confirmed by Fig. 13 that evidences a rate of polymerization of about 100 monomers per ms and per enzyme, that is, in average one monomer per enzyme during \( {{\mathbb T}_{\text{I}}} \cup {{\mathbb T}_{\text{II}}} \).
Note that this feature arises because the enzyme is assumed to be immobile with respect to the laboratory frame (see Sect. 3.3).
Note that \( {{k_{B} T} \mathord{\left/ {\vphantom {{k_{B} T} \alpha }} \right. \kern-0pt} \alpha } \) is the diffusion coefficient of the Brownian particle (Kubo et al. 1991).
A more rigourous approach should take into account the deterministic force, but that would make the analysis much more difficult.
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This work was partly supported by ‘Fundacão para a Ciência e a Tecnologia’ (Portugal) through Research Contract No. C2007-443-CENIMAT-1 to A. Véron.
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Véron, A.R., Martins, A.F. Stochastic Mechanochemical Description of a Bioinspired Polymerization Process. Bull Math Biol 81, 155–192 (2019). https://doi.org/10.1007/s11538-018-0522-3
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DOI: https://doi.org/10.1007/s11538-018-0522-3