A numerical scheme for the early steps of nucleation-aggregation models

Abstract

In the formation of large clusters out of small particles, the initializing step is called the nucleation, and consists in the spontaneous reaction of agents which aggregate into small and stable polymers called nuclei. After this early step, the polymers are involved in a number of reactions such as polymerization, fragmentation and coalescence. Since there may be several orders of magnitude between the size of a particle and the size of an aggregate, building efficient numerical schemes to capture accurately the kinetics of the reaction is a delicate step of key importance. In this article, we propose a conservative scheme, based on finite volume methods on an adaptive grid, which is capable of simulating well the early steps of the reaction as well as the later chain reactions.

Information content in data sets: A review of methods for interrogation and model comparison

“…In [52] and subsequent efforts in [12,13], the authors sought to investigate several questions including (i) understanding the key polymerization mechanisms, (ii) numerical approximation of the models, and (iii) selection of parameters and calibration of the model. Here we focus on illustrating the methodology used in (iii).…”

“…Following [12,53], we define a small parameter ε = 1 i M , and let x i = iε with i M 1 being the average polymer size defined by…”

“…A formal derivation for a full model, also including nucleation, is carried out in [52]. For a fixed integer N 0 we obtain after some arguments [12,13,53] the system…”

Information content in data sets for a nucleated-polymerization model

“…We developed methodology for forward solutions in [7]. In considering these forward solutions we first observed that the desired spatial computational domain is very large as determined by the maximum size of observed polymers, with range up to 10 6 and the peak in the distribution is at the left side of the domain of interest; for larger polymer sizes, the distribution is almost linearly decreasing.…”

Estimation from moments measurements for amyloid depolymerisation

“…Note that there are many other possible applications of this model, such as phase transition, which was the original application for which it had been designed [14]. The problem now stands: Measuring z sls (t) or z tht (t), or more generally the timedependence of a n-th moment defined by 0 x n u(x, t)dx, with u(x, t) solution of System (1) (2), what may be possibly estimated among the unknown quantities, i.e. the initial state u 0 (x) and the parameter functions a(x) and b(x)?…”