Deterministic and Stochastic Becker–Döring Equations: Past and Recent Mathematical Developments

Hingant, Erwan; Yvinec, Romain

doi:10.1007/978-3-319-62627-7_9

Cited by 17 publications

(15 citation statements)

References 95 publications

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“…Treating the uncertainties inherent in experimental data will be the topic of our future studies. The kinetics of cluster formation are described by Becker-Döring equations (Ball et al, 1986;Hingant and Yvinec, 2017), which model cluster birth and death which arise from collisions of the smaller clusters into larger ones and evaporations from the bigger clusters into smaller ones. Precisely, labelling the clusters by i ∈ {1, 2, .…”

Section: Discussionmentioning

confidence: 99%

Identification of molecular cluster evaporation rates, cluster formation enthalpies and entropies by Monte Carlo method

et al. 2020

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Abstract. We address the problem of identifying the evaporation rates for neutral molecular clusters from synthetic (computer-simulated) cluster concentrations. We applied Bayesian parameter estimation using a Markov chain Monte Carlo (MCMC) algorithm to determine cluster evaporation/fragmentation rates from synthetic cluster distributions generated by the Atmospheric Cluster Dynamics Code (ACDC) and based on gas kinetic collision rate coefficients and evaporation rates obtained using quantum chemical calculations and detailed balances. The studied system consisted of electrically neutral sulfuric acid and ammonia clusters with up to five of each type of molecules. We then treated the concentrations generated by ACDC as synthetic experimental data. With the assumption that the collision rates are known, we tested two approaches for estimating the evaporation rates from these data. First, we studied a scenario where time-dependent cluster distributions are measured at a single temperature before the system reaches a steady state. In the second scenario, only steady-state cluster distributions are measured but at several temperatures. Additionally, in the latter case, the evaporation rates were represented in terms of cluster formation enthalpies and entropies. This reparameterization reduced the number of unknown parameters, since several evaporation rates depend on the same cluster formation enthalpy and entropy values. We also estimated the evaporation rates using previously published synthetic steady-state cluster concentration data at one temperature and compared our two cases to this setting. Both the time-dependent and the two-temperature steady-state concentration data allowed us to estimate the evaporation rates with less variance than in the steady-state single-temperature case. We show that temperature-dependent steady-state data outperform single-temperature time-dependent data for parameter estimation, even if only two temperatures are used. We can thus conclude that for experimentally determining evaporation rates, cluster distribution measurements at several temperatures are recommended over time-dependent measurements at one temperature.

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Section: Discussionmentioning

confidence: 99%

Identification of molecular cluster evaporation rates, cluster formation enthalpies and entropies by Monte Carlo method

et al. 2020

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“…The particular system that we study is a modified system of Becker-Döring equations. (For a nice historical review of mathematical developments concerning the Becker-Döring equations, see [16].) As usual for Becker-Döring equations, we suppose that the coagulation of clusters of size with monomers proceeds at the rate a n n 1 , and clusters of size + 1 lose monomers at the rate b +1 n +1 .…”

Section: Introductionmentioning

confidence: 99%

Temporal oscillations in Becker–Döring equations with atomization

Pego

Velázquez

2020

Nonlinearity

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We prove that time-periodic solutions arise via Hopf bifurcation in a finite closed system of coagulation-fragmentation equations. The system we treat is a variant of the Becker-Döring equations, in which clusters grow or shrink by addition or deletion of monomers. To this is added a linear atomization reaction for clusters of maximum size. The structure of the system is motivated by models of gas evolution oscillators in physical chemistry, which exhibit temporal oscillations under certain input/output conditions. All solutions of the system (2)-(4) conserve mass, since ∂ t M =1 n = 0. R. L. Pego and J. J. L. Velázquez 3 1. If β 0 < κ < β k , then any unstable eigenvalue of B is non-real and simple, and satisfies |λ| ≤Ĉ k M −3/2 .

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“…The "Clustering model" may be considered as an extended Becker-Döring (BD) model. Indeed, it shares some common features, such as the fully clustered state achieved at some parameter values [68]. However, this state in the BD model is observed only for a particular set of parameters, and the number of parameters in these sets is unreasonably large, and it is complicated to account for all possible solutions.…”

Section: Discussionmentioning

confidence: 98%

Development of a Simple Kinetic Mathematical Model of Aggregation of Particles or Clustering of Receptors

Dasgupta

Martyanov

Filkova

et al. 2020

Life

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The process of clustering of plasma membrane receptors in response to their agonist is the first step in signal transduction. The rate of the clustering process and the size of the clusters determine further cell responses. Here we aim to demonstrate that a simple 2-differential equation mathematical model is capable of quantitative description of the kinetics of 2D or 3D cluster formation in various processes. Three mathematical models based on mass action kinetics were considered and compared with each other by their ability to describe experimental data on GPVI or CR3 receptor clustering (2D) and albumin or platelet aggregation (3D) in response to activation. The models were able to successfully describe experimental data without losing accuracy after switching between complex and simple models. However, additional restrictions on parameter values are required to match a single set of parameters for the given experimental data. The extended clustering model captured several properties of the kinetics of cluster formation, such as the existence of only three typical steady states for this system: unclustered receptors, receptor dimers, and clusters. Therefore, a simple kinetic mass-action-law-based model could be utilized to adequately describe clustering in response to activation both in 2D and in 3D.

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Deterministic and Stochastic Becker–Döring Equations: Past and Recent Mathematical Developments

Cited by 17 publications

References 95 publications

Identification of molecular cluster evaporation rates, cluster formation enthalpies and entropies by Monte Carlo method

Identification of molecular cluster evaporation rates, cluster formation enthalpies and entropies by Monte Carlo method

Temporal oscillations in Becker–Döring equations with atomization

Development of a Simple Kinetic Mathematical Model of Aggregation of Particles or Clustering of Receptors

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