Testing for high frequency features in a noisy signal

Given nonstationary data, one generally wants to extract the trend from the noise by smoothing or filtering. However, it is often important to delineate a third intermediate category, that we call high frequency (HF) features: this is the case in our motivating example, which consists in experimental measurements of the time-dynamics of depolymerising protein fibrils average size. One may intuitively visualise HF features as the presence of fast, possibly nonstationary and transient oscillations, distinct from a slowly-varying trend envelope. The aim of this article is to propose an empirical definition of HF features and construct estimators and statistical tests for their presence accordingly, when the data consists of a noisy nonstationary 1-dimensional signal. We propose a parametric characterization in the Fourier domain of the HF features by defining a maximal amplitude and distance to low frequencies of significant energy. We introduce a data-driven procedure to estimate these parameters, and compute a p-value proxy based on a statistical test for the presence of HF features. The test is first conducted on simulated signals where the ratio amplitude of the HF features to the level of the noise is controlled. The test detects HF features even when the level of noise is five times larger than the amplitude of the oscillations. In a second part, the test is conducted on experimental data from Prion disease experiments and it confirms the presence of HF features in these signals with significant confidence

Strategic Workforce Planning and sales force : a demographic approach to productivity

Sales force Return on Investment (ROI) valuation with marketing mix frameworks is nowadays common. Sizing discussions then generally follow based on market data and business assumptions. Yet, according to our knowledge, little has been done to embed sales force demographic data (age/experience, tenure, gender etc...) as well as dynamics (especially turnover) in order to investigate the impact of the salesforce characteristics on sales. This paper illustrates such an attempt. It shows that sales force ROI valuation can benefit from a correction on turnover and that optimizinga sales rep hiring policy can unleash additional ROI points. The results are yet heavily dependent in the data structure of the study and their generalization would have to be investigated

Processus oscillatoires lors de l'agrégation et la fragmentation de fibres amyloïdes

This thesis focuses on the study of the process of protein aggregation and fragmentation. In particular, oscillatory kinetic phenomena are identified in experiments on prion diseases, a subcategory of amyloid diseases. First, we notice that attenuated and localized oscillations at specific locations on the experimental signals are observable. These oscillations highlight the presence of complex, underlying kinetic phenomena during protein kinetic processes. We define a parametric characterization of oscillations in the frequency domain. Then, we construct a statistical hypothesis test to confirm the presence of oscillations in the experimental signals. In a second step, we introduce and mathematically analyze a kinetic model capable of generating oscillations. The model considers two species of monomers: a pathological monomer that polymerizes and a healthy monomer that depolymerizes. The model combines a Lotka-Volterra system for monomers with a growth/fragmentation system: Becker-Döring in the discrete case in size, Lifshitz-Slyozov in the continuous case. The mathematical study of these models leads to new and interesting problems and improves the understanding of the underlying physical phenomena.Cette thèse se focalise sur l'étude du processus d'agrégation et de fragmentation des protéines. Plus particulièrement, des phénomènes cinétiques oscillatoires sont identifiés lors d’expériences sur les maladies à prions, une sous-catégorie des maladies amyloïdes. Dans un premier temps, nous remarquons que des oscillations atténuées et localisées à des endroits spécifiques sur les signaux expérimentaux sont observables. Ces oscillations mettent en avant la présence de phénomènes cinétiques complexes, sous-jacents, lors des processus cinétiques de protéines. Nous définissons une caractérisation paramétrique des oscillations dans le domaine fréquentiel. Puis, nous construisons un test statistique d'hypothèses permettant de confirmer la présence d'oscillations dans les signaux expérimentaux. Dans un second temps, nous introduisons et analysons mathématiquement un modèle cinétique capables d'engendrer des oscillations. Le modèle est considère deux espèces de monomères: un monomère pathologique qui polymérise et un monomère sain qui dépolymérise. Le modèle couple un système Lotka-Volterra pour les monomères à un système de croissance/fragmentation: Becker-Döring dans le cas discret en taille, Lifshitz-Slyozov dans le cas continu. L'étude mathématique de ces modèles conduit à de nouveaux problèmes intéressants et améliore la compréhension des phénomènes physiques sous-jacents

A bi-monomeric, nonlinear Becker-Döring-type system to capture oscillatory aggregation kinetics in prion dynamics

In this article, in order to understand the appearance of oscillations observed in protein aggregation experiments, we propose, motivate and analyse mathematically the differential system describing the kinetics of the following reactions:$$\left \{\!\!\begin{array}{lclr}{\cal V}+{\cal W}&\xrightarrow{k}&2{\cal W},\\{\cal W}+{\cal C}_i&\xrightarrow{a_i} &{\cal C}_{i+1}, \quad & 1\leq i\leq n,\\{\cal C}_i+{\cal V} &\xrightarrow{b_i} &{\cal C}_{i-1}+2{\cal V}, \quad & 2\leq i\leq n,\\\end{array}\right.$$with n finite or infinite. This system may be viewed as a variant of the seminal Becker-Döring system, and is capable of displaying sustained though damped oscillations

A bi-monomeric, nonlinear Becker-Döring-type system to capture oscillatory aggregation kinetics in prion dynamics

In this article, in order to understand the appearance of oscillations observed in protein aggregation experiments, we propose, motivate and analyse mathematically the differential system describing the kinetics of the following reactions:$$\left \{\!\!\begin{array}{lclr}{\cal V}+{\cal W}&\xrightarrow{k}&2{\cal W},\\{\cal W}+{\cal C}_i&\xrightarrow{a_i} &{\cal C}_{i+1}, \quad & 1\leq i\leq n,\\{\cal C}_i+{\cal V} &\xrightarrow{b_i} &{\cal C}_{i-1}+2{\cal V}, \quad & 2\leq i\leq n,\\\end{array}\right.$$with n finite or infinite. This system may be viewed as a variant of the seminal Becker-Döring system, and is capable of displaying sustained though damped oscillations

Testing for high frequency features in a noisy signal

Given nonstationary data, one generally wants to extract the trend from the noise by smoothing or filtering. However, it is often important to delineate a third intermediate category, that we call high frequency (HF) features: this is the case in our motivating example, which consists in experimental measurements of the time-dynamics of depolymerising protein fibrils average size. One may intuitively visualise HF features as the presence of fast, possibly nonstationary and transient oscillations, distinct from a slowly-varying trend envelope. The aim of this article is to propose an empirical definition of HF features and construct estimators and statistical tests for their presence accordingly, when the data consists of a noisy nonstationary 1-dimensional signal. We propose a parametric characterization in the Fourier domain of the HF features by defining a maximal amplitude and distance to low frequencies of significant energy. We introduce a data-driven procedure to estimate these parameters, and compute a p-value proxy based on a statistical test for the presence of HF features. The test is first conducted on simulated signals where the ratio amplitude of the HF features to the level of the noise is controlled. The test detects HF features even when the level of noise is five times larger than the amplitude of the oscillations. In a second part, the test is conducted on experimental data from Prion disease experiments and it confirms the presence of HF features in these signals with significant confidence

Tumorigenesis and axons regulation for the pancreatic cancer: a mathematical approach

The nervous system is today recognized to play an important role in the development of cancer. Indeed, neurons extend long processes (axons) that grow and infiltrate tumors in order to regulate the progression of the disease in a positive or negative way, depending on the type of neuron considered. Mathematical modelling of this biological process allows to formalize the nerve-tumor interactions and to test hypotheses in silico to better understand this phenomenon. In this work, we introduce a system of differential equations modelling the progression of pancreatic ductal adenocarcinoma (PDAC) coupled with associated changes in axonal innervation. The study of the asymptotic behavior of the model confirms the experimental observations that PDAC development is correlated with the type and densities of axons in the tissue. In addition, we study the identifiability of the model parameters. This informs on the adequacy between the parameters of the model and the experimental data. It leads to significant insights such that the transdifferentiation phenomenon accelerates during the development process of PDAC cells. Finally, we give an example of a simulation of the effects of partial or complete denervation that sheds lights on complex correlation between the cell populations and axons with opposite functions

Asymptotic Analysis of a bi-monomeric nonlinear Becker-D\"oring system

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

A continuous approach of modeling tumorigenesis and axons regulation for the pancreatic cancer

The pancreatic innervation undergoes dynamic remodeling during the development of pancreatic ductal adenocarcinoma (PDAC). Denervation experiments have shown that different types of axons can exert either pro- or anti-tumor effects, but conflicting results exist in the literature, leaving the overall influence of the nervous system on PDAC incompletely understood. To address this gap, we propose a continuous mathematical model of nerve-tumor interactions that allows in silico simulation of denervation at different phases of tumor development. This model takes into account the pro- or anti-tumor properties of different types of axons (sympathetic or sensory) and their distinct remodeling dynamics during PDAC development. We observe a “shift effect” where an initial pro-tumor effect of sympathetic axon denervation is later outweighed by the anti-tumor effect of sensory axon denervation, leading to a transition from an overall protective to a deleterious role of the nervous system on PDAC tumorigenesis. Our model also highlights the importance of the impact of sympathetic axon remodeling dynamics on tumor progression. These findings may guide strategies targeting the nervous system to improve PDAC treatment

Early stage prion assembly involves two subpopulations with different quaternary structures and a secondary templating pathway

International audienceThe dynamics of aggregation and structural diversification of misfolded, host-encoded proteins in neurodegenerative diseases are poorly understood. In many of these disorders, including Alzheimer’s, Parkinson’s and prion diseases, the misfolded proteins are self-organized into conformationally distinct assemblies or strains. The existence of intrastrain structural heterogeneity is increasingly recognized. However, the underlying processes of emergence and coevolution of structurally distinct assemblies are not mechanistically understood. Here, we show that early prion replication generates two subsets of structurally different assemblies by two sequential processes of formation, regardless of the strain considered. The first process corresponds to a quaternary structural convergence, by reducing the parental strain polydispersity to generate small oligomers. The second process transforms these oligomers into larger ones, by a secondary autocatalytic templating pathway requiring the prion protein. This pathway provides mechanistic insights into prion structural diversification, a key determinant for prion adaptation and toxicity