Oscillations in Neuronal Activity: A Neuron-Centered Spatiotemporal Model of the Unfolded Protein Response in Prion Diseases.

Many neurodegenerative diseases (NDs) are characterized by the slow spatial spread of toxic protein species in the brain. The toxic proteins can induce neuronal stress, triggering the Unfolded Protein Response (UPR), which slows or stops protein translation and can indirectly reduce the toxic load. However, the UPR may also trigger processes leading to apoptotic cell death and the UPR is implicated in the progression of several NDs. In this paper, we develop a novel mathematical model to describe the spatiotemporal dynamics of the UPR mechanism for prion diseases. Our model is centered around a single neuron, with representative proteins P (healthy) and S (toxic) interacting with heterodimer dynamics (S interacts with P to form two S's). The model takes the form of a coupled system of nonlinear reaction-diffusion equations with a delayed, nonlinear flux for P (delay from the UPR). Through the delay, we find parameter regimes that exhibit oscillations in the P- and S-protein levels. We find that oscillations are more pronounced when the S-clearance rate and S-diffusivity are small in comparison to the P-clearance rate and P-diffusivity, respectively. The oscillations become more pronounced as delays in initiating the UPR increase. We also consider quasi-realistic clinical parameters to understand how possible drug therapies can alter the course of a prion disease. We find that decreasing the production of P, decreasing the recruitment rate, increasing the diffusivity of S, increasing the UPR S-threshold, and increasing the S clearance rate appear to be the most powerful modifications to reduce the mean UPR intensity and potentially moderate the disease progression.

Seventy Years of Dose-response Models: From the Target Theory to the Use of Big Databases Involving Cell Survival and DNA Repair.

Radiobiological data, whether obtained at the clinical, biological or molecular level has significantly contributed to a better description and prediction of the individual dose-response to ionizing radiation and a better estimation of the radiation-induced risks. Particularly, over the last seventy years, the amount of radiobiological data has considerably increased, and permitted the mathematical formulas describing dose-response to become less empirical. A better understanding of the basic radiobiological mechanisms has also contributed to establish quantitative inter-correlations between clinical, biological and molecular biomarkers, refining again the mathematical models of description. Today, big data approaches and, more recently, artificial intelligence may finally complete and secure this long process of thinking from the multi-scale description of radiation-induced events to their prediction. Here, we reviewed the major dose-response models applied in radiobiology for quantifying molecular and cellular radiosensitivity and aimed to explain their evolution: Specifically, we highlighted the advances concerning the target theory with the cell survival models and the progressive introduction of the DNA repair process in the mathematical models. Furthermore, we described how the technological advances have changed the description of DNA double-strand break (DSB) repair kinetics by introducing the important notion of DSB recognition, independent of that of DSB repair. Initially developed separately, target theory on one hand and, DSB recognition and repair, on the other hand may be now fused into a unified model involving the cascade of phosphorylations mediated by the ATM kinase in response to any genotoxic stress.

Toward an Early Diagnosis for Alzheimer's Disease Based on the Perinuclear Localization of the ATM Protein.

Alzheimer's disease (AD) is the most common neurodegenerative dementia, for which the molecular origins, genetic predisposition and therapeutic approach are still debated. In the 1980s, cells from AD patients were reported to be sensitive to ionizing radiation. In order to examine the molecular basis of this radiosensitivity, the ATM-dependent DNA double-strand breaks (DSB) signaling and repair were investigated by applying an approach based on the radiation-induced ataxia telangiectasia-mutated (ATM) protein nucleoshuttling (RIANS) model. Early after irradiation, all ten AD fibroblast cell lines tested showed impaired DSB recognition and delayed RIANS. AD fibroblasts specifically showed spontaneous perinuclear localization of phosphorylated ATM (pATM) forms. To our knowledge, such observation has never been reported before, and by considering the role of the ATM kinase in the stress response, it may introduce a novel interpretation of accelerated aging. Our data and a mathematical approach through a brand-new model suggest that, in response to a progressive and cumulative stress, cytoplasmic ATM monomers phosphorylate the APOE protein (pAPOE) close to the nuclear membrane and aggregate around the nucleus, preventing their entry in the nucleus and thus the recognition and repair of spontaneous DSB, which contributes to the aging process. Our findings suggest that pATM and/or pAPOE may serve as biomarkers for an early reliable diagnosis of AD on any fibroblast sample.

Influenza transmissibility among patients and health-care professionals in a geriatric short-stay unit using individual contact data.

Detailed information are lacking on influenza transmissibility in hospital although clusters are regularly reported. In this pilot study, our goal was to estimate the transmission rate of H3N2 2012-influenza, among patients and health care professionals in a short-term Acute Care for the Elderly Unit by using a stochastic approach and a simple susceptible-exposed-infectious-removed model. Transmission parameters were derived from documented individual contact data collected by Radio Frequency IDentification technology at the epidemic peak. From our model, nurses appeared to transmit infection to a patient more frequently with a transmission rate of 1.04 per day on average compared to 0.38 from medical doctors. This transmission rate was 0.34 between nurses. These results, even obtained in this specific context, might give a relevant insight of the influenza dynamics in hospitals and will help to improve and to target control measures for preventing nosocomial transmission of influenza. The investigation of nosocomial transmission of SARS-COV-2 might gain from similar approaches.

Theoretical evidence of osteoblast self-inhibition after activation of the genetic regulatory network controlling mineralization.

Bone is a hard-soft biomaterial built through a self-assembly process under genetic regulatory network (GRN) monitoring. This paper aims to capture the behavior of the bone GRN part that controls mineralization by using a mathematical model. Here, we provide an advanced review of empirical evidence about interactions between gene coding (i) transcription factors and (ii) bone proteins. These interactions are modeled with nonlinear differential equations using Michaelis-Menten and Hill functions. Compared to empirical evidence - coming from osteoblasts culture -, the two best systems (among 126=2,985,984 possibilities) use factors of inhibition from the start of the activation of each gene. It reveals negative indirect interactions coming from either negative feedback loops or the recently depicted micro-RNAs. The difference between the two systems also lies in the BSP equation and two ways for activating and reducing its production. Thus, it highlights the critical role of BSP in the bone GRN that acts on bone mineralization. Our study provides the first theoretical evidence of osteoblast self-inhibition after activation of the genetic regulatory network controlling mineralization with this work.

Why Are Periodic Erythrocytic Diseases so Rare in Humans?

Many studies have shown that periodic erythrocytic (red blood cell linked) diseases are extremely rare in humans. To explain this observation, we develop here a simple model of erythropoiesis in mammals and investigate its stability in the parameter space. A bifurcation analysis enables us to sketch stability diagrams in the plane of key parameters. Contrary to some other mammal species such as rabbits, mice or dogs, we show that human-specific parameter values prevent periodic oscillations of red blood cells levels. In other words, human erythropoiesis seems to lie in a region of parameter space where oscillations exclusively concerning red blood cells cannot appear. Further mathematical analysis show that periodic oscillations of red blood cells levels are highly unusual and if exist, might only be due to an abnormally high erythrocytes destruction rate or to an abnormal hematopoietic stem cell commitment into the erythrocytic lineage. We also propose numerical results only for an improved version of our approach in order to give a more realistic but more complex approach of our problem.

A reaction-diffusion model of spatial propagation of A[Formula: see text] oligomers in early stage Alzheimer's disease.

The misconformation and aggregation of the protein Amyloid-Beta (A[Formula: see text]) is a key event in the propagation of Alzheimer's Disease (AD). Different types of assemblies are identified, with long fibrils and plaques deposing during the late stages of AD. In the earlier stages, the disease spread is driven by the formation and the spatial propagation of small amorphous assemblies called oligomers. We propose a model dedicated to studying those early stages, in the vicinity of a few neurons and after a polymer seed has been formed. We build a reaction-diffusion model, with a Becker-Döring-like system that includes fragmentation and size-dependent diffusion. We hereby establish the theoretical framework necessary for the proper use of this model, by proving the existence of solutions using a fixed point method.

A unifying model for the propagation of prion proteins in yeast brings insight into the [PSI+] prion.

The use of yeast systems to study the propagation of prions and amyloids has emerged as a crucial aspect of the global endeavor to understand those mechanisms. Yeast prion systems are intrinsically multi-scale: the molecular chemical processes are indeed coupled to the cellular processes of cell growth and division to influence phenotypical traits, observable at the scale of colonies. We introduce a novel modeling framework to tackle this difficulty using impulsive differential equations. We apply this approach to the [PSI+] yeast prion, which is associated with the misconformation and aggregation of Sup35. We build a model that reproduces and unifies previously conflicting experimental observations on [PSI+] and thus sheds light onto characteristics of the intracellular molecular processes driving aggregate replication. In particular our model uncovers a kinetic barrier for aggregate replication at low densities, meaning the change between prion or prion-free phenotype is a bi-stable transition. This result is based on the study of prion curing experiments, as well as the phenomenon of colony sectoring, a phenotype which is often ignored in experimental assays and has never been modeled. Furthermore, our results provide further insight into the effect of guanidine hydrochloride (GdnHCl) on Sup35 aggregates. To qualitatively reproduce the GdnHCl curing experiment, aggregate replication must not be completely inhibited, which suggests the existence of a mechanism different than Hsp104-mediated fragmentation. Those results are promising for further development of the [PSI+] model, but also for extending the use of this novel framework to other yeast prion or amyloid systems.

Stability analysis of a steady state of a model describing Alzheimer's disease and interactions with prion proteins.

Alzheimer's disease (AD) is a neuro-degenerative disease affecting more than 46 million people worldwide in 2015. AD is in part caused by the accumulation of A[Formula: see text] peptides inside the brain. These can aggregate to form insoluble oligomers or fibrils. Oligomers have the capacity to interact with neurons via membrane receptors such as prion proteins ([Formula: see text]). This interaction leads [Formula: see text] to be misfolded in oligomeric prion proteins ([Formula: see text]), transmitting a death signal to neurons. In the present work, we aim to describe the dynamics of A[Formula: see text] assemblies and the accumulation of toxic oligomeric species in the brain, by bringing together the fibrillation pathway of A[Formula: see text] peptides in one hand, and in the other hand A[Formula: see text] oligomerization process and their interaction with cellular prions, which has been reported to be involved in a cell-death signal transduction. The model is based on Becker-Döring equations for the polymerization process, with delayed differential equations accounting for structural rearrangement of the different reactants. We analyse the well-posedness of the model and show existence, uniqueness and non-negativity of solutions. Moreover, we demonstrate that this model admits a non-trivial steady state, which is found to be globally stable thanks to a Lyapunov function. We finally present numerical simulations and discuss the impact of model parameters on the whole dynamics, which could constitute the main targets for pharmaceutical industry.

Generalizing a mathematical model of prion aggregation allows strain coexistence and co-stability by including a novel misfolded species.

Prions are proteins capable of adopting misfolded conformations and transmitting these conformations to other normally folded proteins. Prions are most commonly known for causing fatal neurodegenerative diseases in mammals but are also associated with several harmless phenotypes in yeast. A distinct feature of prion propagation is the existence of different phenotypical variants, called strains. It is widely accepted that these strains correspond to different conformational states of the protein, but the mechanisms driving their interactions remain poorly understood. This study uses mathematical modeling to provide insight into this problem. We show that the classical model of prion dynamics allows at most one conformational strain to stably propagate. In order to conform to biological observations of strain coexistence and co-stability, we develop an extension of the classical model by introducing a novel prion species consistent with biological studies. Qualitative analysis of this model reveals a new variety of behavior. Indeed, it allows for stable coexistence of different strains in a wide parameter range, and it also introduces intricate initial condition dependency. These new behaviors are consistent with experimental observations of prions in both mammals and yeast. As such, our model provides a valuable tool for investigating the underlying mechanisms of prion propagation and the link between prion strains and strain specific phenotypes. The consideration of a novel prion species brings a change in perspective on prion biology and we use our model to generate hypotheses about prion infectivity.

A Novel Multiscale Mathematical Model for Building Bone Substitute Materials for Children.

Bone is an engineering marvel that achieves a unique combination of stiffness and toughness exceeding that of synthesized materials. In orthopedics, we are currently challenged for the child population that needs a less stiff but a tougher bone substitute than adults. Recent evidence suggests that the relationship between inter-molecular connections that involve the two main bone building blocks, TropoCollagen molecules (TC) and carbonated Hydroxyapatite (cAp), and bone macroscopic mechanical properties, stiffness and toughness, are key to building bone substitute materials for children. The goal of our study is to establish how inter-molecular connections that occur during bone mineralization are related to macroscopic mechanical properties in child bones. Our aim is to link the biological alterations of the TC-cAp self assembly process happening during bone mineralization to the bone macroscopic mechanical properties' alterations during aging. To do so, we have developed a multiscale mathematical model that includes collagen cross links (TC⁻TC interface) from experimental studies of bone samples to forecast bone macroscopic mechanical properties. Our results support that the Young's modulus cannot be a linear parameter if we want to solve our system. In relation to bone substitute material with innovative properties for children, our results propose values of several biological parameters, such as the number of crystals and their size, and collagen crosslink maturity for the desired bone mechanical competence. Our novel mathematical model combines mineralization and macroscopic mechanical behavior of bone and is a step forward in building mechanically customized biomimetic bone grafts that would fit children's orthopedic needs.

The different distribution of enzymatic collagen cross-links found in adult and children bone result in different mechanical behavior of collagen.

Enzymatic collagen cross-linking has been shown to play an important role in the macroscopic elastic and plastic deformation of bone across ages. However, its direct contribution to collagen fibril deformation is unknown. The aim of this study is to determine how covalent intermolecular connections from enzymatic collagen cross-links contribute to collagen fibril elastic and plastic deformation of adults and children's bone matrix. We used ex vivo data previously obtained from biochemical analysis of children and adults bone samples (n = 14; n = 8, respectively) to create 22 sample-specific computational models of cross-linked collagen fibrils. By simulating a tensile test for each fibril, we computed the modulus of elasticity (E), ultimate tensile and yield stress (σu and σy), and elastic, plastic and total work (We, Wp and Wtot) for each collagen fibril. We present a novel difference between children and adult bone in the deformation of the collagen phase and suggest a link between collagen fibril scale and macroscale for elastic behavior in children bone under the influence of immature enzymatic cross-links. We show a parametric linear correlation between We and immature enzymatic collagen cross-links at the collagen fibril scale in the children population that is similar to the one we found at the macroscale in our previous study. Finally, we suggest the key role of covalent intermolecular connections to stiffness parameters (e.g. elastic modulus and We) in children's collagen fibril and to toughness parameters in adult's collagen fibril, respectively.

Estimates and impact of lymphocyte division parameters from CFSE data using mathematical modelling.

Carboxyfluorescein diacetate succinimidyl ester (CFSE) labelling has been widely used to track and study cell proliferation. Here we use mathematical modelling to describe the kinetics of immune cell proliferation after an in vitro polyclonal stimulation tracked with CFSE. This approach allows us to estimate a set of key parameters, including ones related to cell death and proliferation. We develop a three-phase model that distinguishes a latency phase, accounting for non-divided cell behaviour, a resting phase and the active phase of the division process. Parameter estimates are derived from model results, and numerical simulations are then compared to the dynamics of in vitro experiments, with different biological assumptions tested. Our model allows us to compare the dynamics of CD4+ and CD8+ cells, and to highlight their kinetic differences. Finally we perform a sensitivity analysis to quantify the impact of each parameter on proliferation kinetics. Interestingly, we find that parameter sensitivity varies with time and with cell generation. Our approach can help biologists to understand cell proliferation mechanisms and to identify potential pathological division processes.

Mathematical models of radiation action on living cells: From the target theory to the modern approaches. A historical and critical review.

Cell survival is conventionally defined as the capability of irradiated cells to produce colonies. It is quantified by the clonogenic assays that consist in determining the number of colonies resulting from a known number of irradiated cells. Several mathematical models were proposed to describe the survival curves, notably from the target theory. The Linear-Quadratic (LQ) model, which is to date the most frequently used model in radiobiology and radiotherapy, dominates all the other models by its robustness and simplicity. Its usefulness is particularly important because the ratio of the values of the adjustable parameters, α and ß, on which it is based, predicts the occurrence of post-irradiation tissue reactions. However, the biological interpretation of these parameters is still unknown. Throughout this review, we revisit and discuss historically, mathematically and biologically, the different models of the radiation action by providing clues for resolving the enigma of the LQ model.

First passage times in homogeneous nucleation: Dependence on the total number of particles.

Motivated by nucleation and molecular aggregation in physical, chemical, and biological settings, we present an extension to a thorough analysis of the stochastic self-assembly of a fixed number of identical particles in a finite volume. We study the statistics of times required for maximal clusters to be completed, starting from a pure-monomeric particle configuration. For finite volumes, we extend previous analytical approaches to the case of arbitrary size-dependent aggregation and fragmentation kinetic rates. For larger volumes, we develop a scaling framework to study the first assembly time behavior as a function of the total quantity of particles. We find that the mean time to first completion of a maximum-sized cluster may have a surprisingly weak dependence on the total number of particles. We highlight how higher statistics (variance, distribution) of the first passage time may nevertheless help to infer key parameters, such as the size of the maximum cluster. Finally, we present a framework to quantify formation of macroscopic sized clusters, which are (asymptotically) very unlikely and occur as a large deviation phenomenon from the mean-field limit. We argue that this framework is suitable to describe phase transition phenomena, as inherent infrequent stochastic processes, in contrast to classical nucleation theory.

A micellar on-pathway intermediate step explains the kinetics of prion amyloid formation.

In a previous work by Alvarez-Martinez et al. (2011), the authors pointed out some fallacies in the mainstream interpretation of the prion amyloid formation. It appeared necessary to propose an original hypothesis able to reconcile the in vitro data with the predictions of a mathematical model describing the problem. Here, a model is developed accordingly with the hypothesis that an intermediate on-pathway leads to the conformation of the prion protein into an amyloid competent isoform thanks to a structure, called micelles, formed from hydrodynamic interaction. The authors also compare data to the prediction of their model and propose a new hypothesis for the formation of infectious prion amyloids.

Alzheimer's disease: analysis of a mathematical model incorporating the role of prions.

We introduce a mathematical model of the in vivo progression of Alzheimer's disease with focus on the role of prions in memory impairment. Our model consists of differential equations that describe the dynamic formation of ß-amyloid plaques based on the concentrations of Aß oligomers, PrP(C) proteins, and the Aß-x-Aß-PrP(C)complex, which are hypothesized to be responsible for synaptic toxicity. We prove the well-posedness of the model and provided stability results for its unique equilibrium, when the polymerization rate of Aß-amyloid is constant and also when it is described by a power law.

Dynamics of polymerization shed light on the mechanisms that lead to multiple amyloid structures of the prion protein.

It is generally accepted that spongiform encephalopathies result from the aggregation into amyloid of a ubiquitous protein, the so-called prion protein. As a consequence, the dynamics of amyloid formation should explain the characteristics of the prion diseases: infectivity as well as sporadic and genetic occurrence, long incubation time, species barriers and strain specificities. The success of this amyloid hypothesis is due to the good qualitative agreement of this hypothesis with the observations. However, a number of difficulties appeared when comparing quantitatively the in vitro experimental results with the theoretical models, suggesting that some differences should hide important discrepancies. We used well defined quantitative models to analyze the experimental results obtained by in vitro polymerization of the recombinant hamster prion protein. Although the dynamics of polymerization resembles a simple nucleus-dependent fibrillogenesis, neither the initial concentration dependence nor off-pathway hypothesis fit with experimental results. Furthermore, seeded polymerization starts after a long time delay suggesting the existence of a specific mechanism that takes place before nucleus formation. On the other hand, polymerization dynamics reveals a highly stochastic mechanism, the origin of which appears to be caused by nucleation heterogeneity. Moreover, the specific structures generated during nucleation are maintained during successive seeding although a clear improvement of the dynamics parameters (polymerization rate and lag time) is observed. We propose that an additional on-pathway reaction takes place before nucleation and it is responsible for the heterogeneity of structures produced during prion protein polymerization in vitro. These amyloid structures behave like prion strains. A model is proposed to explain the genesis of heterogeneity among prion amyloid.

Adding self-renewal in committed erythroid progenitors improves the biological relevance of a mathematical model of erythropoiesis.

We propose a new mathematical model of erythropoiesis that takes a positive feedback of erythrocytes on progenitor apoptosis into account, and incorporates a negative feedback of erythrocytes on progenitor self-renewal. The resulting model is a system of age-structured equations that reduces to a system of delay differential equations where the delays account for progenitor compartment duration and cell cycle length. We compare this model with experimental data on an induced-anemia in mice that exhibit damped oscillations of the hematocrit before it returns to equilibrium. When we assume no self-renewal of progenitors, we obtain an inaccurate fitting of the model with experimental data. Adding self-renewal in the progenitor compartment gives better approximations, with the main features of experimental data correctly fitted. Our results indicate the importance of progenitor self-renewal in the modelling of erythropoiesis. Moreover, the model makes testable predictions on the lifespan of erythrocytes confronted to a severe anemia, and on the progenitors behavior.

A mathematical analysis of the dynamics of prion proliferation.

How do the normal prion protein (PrP(C)) and infectious prion protein (PrP(Sc)) populations interact in an infected host? To answer this question, we analyse the behavior of the two populations by studying a system of differential equations. The system is constructed under the assumption that PrP(Sc) proliferates using the mechanism of nucleated polymerization. We prove that with parameter input consistent with experimentally determined values, we obtain the persistence of PrP(Sc). We also prove local stability results for the disease steady state, and a global stability result for the disease free steady state. Finally, we give numerical simulations, which are confirmed by experimental data.