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Asymptotic Analysis of a bi-monomeric nonlinear Becker-Döring system

Résumé

To provide a mechanistic explanation of sustained {then} damped oscillations observed in a depolymerisation experiment, a bi-monomeric variant of the seminal Becker-D\"oring system has been proposed in~(Doumic, Fellner, Mezache, Rezaei, J. of Theor. Biol., 2019). When all reaction rates are constant, the equations are the following: \begin{align*} \frac{dv}{dt} & =-vw+v\sum_{j=2}^{\infty}c_{j}, \qquad \frac{dw}{dt} =vw-w\sum_{j=1}^{\infty}c_{j}, \\ \frac{dc_{j}}{dt} & =J_{j-1}-J_{j}\ \ ,\ \ j\geq1\ \ ,\ \ \ J_{j}=wc_{j}-vc_{j+1}\ \ ,\ \ j\geq1\ \ ,\ J_{0}=0, \end{align*} where $v$ and $w$ are two distinct unit species, and $c_i$ represents the concentration of clusters containing $i$ units. We study in detail the mechanisms leading to such oscillations and characterise the different phases of the dynamics, from the initial high-amplitude oscillations to the progressive damping leading to the convergence towards the unique positive stationary solution. We give quantitative approximations for the main quantities of interest: period of the oscillations, size of the damping (corresponding to a loss of energy), number of oscillations characterising each phase. We illustrate these results by numerical simulation, in line with the theoretical results, and provide numerical methods to solve the system.
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Dates et versions

hal-04635659 , version 1 (04-07-2024)

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  • HAL Id : hal-04635659 , version 1

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Marie Doumic, Klemens Fellner, Mathieu Mezache, Juan J L Velázquez. Asymptotic Analysis of a bi-monomeric nonlinear Becker-Döring system. 2024. ⟨hal-04635659⟩
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