Pierre Gabriel

An Efficient Kinetic Model for Assemblies of Amyloid Fibrils and Its Application to Polyglutamine Aggregation

Protein polymerization consists in the aggregation of single monomers into polymers that may fragment. Fibrils assembly is a key process in amyloid diseases. Up to now, protein aggregation was commonly mathematically simulated by a polymer size-structured ordinary differential equations (ODE) system, which is infinite by definition and therefore leads to high computational costs. Moreover, this Ordinary Differential Equation-based modeling approach implies biological assumptions that may be difficult to justify in the general case. For example, whereas several ordinary differential equation models use the assumption that polymerization would occur at a constant rate independently of polymer size, it cannot be applied to certain protein aggregation mechanisms. Here, we propose a novel and efficient analytical method, capable of modelling and simulating amyloid aggregation processes. This alternative approach consists of an integro-Partial Differential Equation (PDE) model of coalescence-fragmentation type that was mathematically derived from the infinite differential system by asymptotic analysis. To illustrate the efficiency of our approach, we applied it to aggregation experiments on polyglutamine polymers that are involved in Huntington’s disease. Our model demonstrates the existence of a monomeric structural intermediate acting as a nucleus and deriving from a non polymerizing monomer (). Furthermore, we compared our model to previously published works carried out in different contexts and proved its accuracy to describe other amyloid aggregation processes.

The contribution of age structure to cell population responses to targeted therapeutics

Cells grown in culture act as a model system for analyzing the effects of anticancer compounds, which may affect cell behavior in a cell cycle position-dependent manner. Cell synchronization techniques have been generally employed to minimize the variation in cell cycle position. However, synchronization techniques are cumbersome and imprecise and the agents used to synchronize the cells potentially have other unknown effects on the cells. An alternative approach is to determine the age structure in the population and account for the cell cycle positional effects post hoc. Here we provide a formalism to use quantifiable lifespans from live cell microscopy experiments to parameterize an age-structured model of cell population response.

Self-similarity in a general aggregation–fragmentation problem. Application to fitness analysis

We consider the linear growth and fragmentation equation with general coefficients. Under suitable conditions, the first eigenvalue represents the asymptotic growth rate of solutions, also called \emph{fitness} or \emph{Malthus coefficient} in population dynamics ; it is of crucial importance to understand the long-time behaviour of the population. We investigate the dependency of the dominant eigenvalue and the corresponding eigenvector on the transport and fragmentation coefficients. We show how it behaves asymptotically as transport dominates fragmentation or \emph{vice versa}. For this purpose we perform suitable blow-up analysis of the eigenvalue problem in the limit of small/large growth coefficient (resp. fragmentation coefficient). We exhibit possible non-monotonic dependency on the parameters, conversely to what would have been conjectured on the basis of some simple cases.

The shape of the polymerization rate in the prion equation

We consider a polymerization (fragmentation) model with size-dependent parameters involved in prion proliferation. Using power laws for the different rates of this model, we recover the shape of the polymerization rate using experimental data. The technique used is inspired from [15], where the fragmentation dependence on prion strains is investigated. Our improvement is to use power laws for the rates, whereas [15] used a constant polymerization coefficient and linear fragmentation.

Non-linear eigenvalue problems arising from growth maximization of positive linear dynamical systems

We study a growth maximization problem for a continuous time positive linear system with switches. This is motivated by a problem of mathematical biology: modeling growth-fragmentation processes and the PMCA protocol (Protein Misfolding Cyclic Amplification). We show that the growth rate is determined by the non-linear eigenvalue of a max-plus analogue of the Ruelle-Perron-Frobenius operator, or equivalently, by the ergodic constant of a Hamilton-Jacobi (HJ) partial differential equation, the solutions or subsolutions of which yield Barabanov and extremal norms, respectively. We exploit contraction properties of order preserving flows, with respect to Hilberts projective metric, to show that the nonlinear eigenvector of the operator, or the “weak KAM” solution of the HJ equation, does exist. Low dimensional examples are presented, showing that the optimal control can lead to a limit cycle.

HIGH-ORDER WENO SCHEME FOR POLYMERIZATION-TYPE EQUATIONS ∗

Polymerization of proteins is a biochimical process involved in different diseases. Mathematically, it is generally modeled by aggregation-fragmentation-type equations. In this paper we consider a general polymerization model and propose a high-order numerical scheme to investigate the behavior of the solution. An important property of the equation is the mass conservation. The fifth-order WENO scheme is built to preserve the total mass of proteins along time.

Identification of the Fragmentation Role in the Amyloid Assembling Processes and Application to their Optimization

The goal is to establish a kinetic model of amyloid formation which will take into account the contribution of fragmentation to the de novo creation of templating interfaces. We propose a new, more comprehensive mathematical model which takes into account previously neglected phenomena potentially occurring during the templating and fragmentation processes. In particular, we try to capture a potential effect of the topology and geometry of prion folding on the elongation and fragmentation properties of a polymer of a given length by separating polymers of the same length into several compartments. Additionally, we apply techniques from geometric control to the new model to design optimal strategies for accelerating the current amplification protocols, such as the Protein Misfolding Cyclic Amplification (PMCA). The objective is to reduce the time needed to diagnose many neurodegenerative diseases. Determining the optimal strategy for accelerated replication in the general problem of fragmentation optimization is still an open question.

Measure solutions to the conservative renewal equation

We prove the existence and uniqueness of measure solutions to the conservative renewal equation and analyze their long time behavior. The solutions are built by using a duality approach. This construction is well suited to apply the Doeblins argument which ensures the exponential relaxation of the solutions to the equilibrium.