The cardiac bidomain model and homogenization

We provide a rather simple proof of a homogenization result for the bidomain model of cardiac electrophysiology. Departing from a microscopic cellular model, we apply the theory of two-scale convergence to derive the bidomain model. To allow for some relevant nonlinear membrane models, we make essential use of the boundary unfolding operator. There are several complications preventing the application of standard homogenization results, including the degenerate temporal structure of the bidomain equations and a nonlinear dynamic boundary condition on an oscillating surface.Comment: To appear in Networks and Heterogeneous Media, Special Issue on Mathematical Methods for Systems Biolog

Multiscale analysis and simulation of a signalling process with surface diffusion

We present and analyze a model for cell signaling processes in biological tissues. The model includes diffusion and nonlinear reactions on the cell surfaces and both inter- and intracellular signaling. Using techniques from the theory of two-scale convergence as well the unfolding method, we show convergence of the solutions to the model to solutions of a two-scale macroscopic problem. We also present a two-scale bulk-surface finite element method for the approximation of the macroscopic model. We report on some benchmarking results as well as numerical simulations in a biologically relevant regime that illustrate the influence of cell-scale heterogeneities on macroscopic concentrations

Homogenization of parabolic problems with dynamical boundary conditions of reactive-diffusive type in perforated media

This paper deals with the homogenization of the reaction-diffusion equations in a domain containing periodically distributed holes of size ε, with a dynamical boundary condition of reactive-diffusive type, i.e., we consider the following nonlinear boundary condition on the surface of the holes ∇uε · ν + ε ∂uε ∂t = ε δ∆Γuε − ε g(uε), where ∆Γ denotes the Laplace-Beltrami operator on the surface of the holes, ν is the outward normal to the boundary, δ > 0 plays the role of a surface diffusion coefficient and g is the nonlinear term. We generalize our previous results (see [3]) established in the case of a dynamical boundary condition of pure-reactive type, i.e., with δ = 0. We prove the convergence of the homogenization process to a nonlinear reaction-diffusion equation whose diffusion matrix takes into account the reactive-diffusive condition on the surface of the holes

Existence, uniqueness and concentration for a system of PDEs involving the Laplace-Beltrami operator

In this paper we derive a model for heat diffusion in a composite medium in which the different components are separated by thermally active interfaces. The previous result is obtained via a concentrated capacity procedure and leads to a non-stantard system of PDEs involving a Laplace-Beltrami operator acting on the interface. For such a system well-posedness is proved using contraction mapping and abstract parabolic problems theory. Finally, the exponential convergence (in time) of the solutions of our system to a steady state is proved

Modelling the molecular mechanisms of biocompatibility of artifical materials

One of the most common reasons for implant failure is immune rejection. Implant rejection leads to additional surgical intervention and, ultimately, increases health cost as well as recovery time. Within a few hours after implantation, the implant surface is covered with host proteins. Adsorption of fibrinogen, a soluble plasma glycoprotein, is responsible in triggering the immune response to a given material and, subsequently, in determining its biocompatibility. The work presented here is focused on modeling the interaction between artificial surfaces and plasma proteins at the microscopic level by taking into account the physico-chemical properties of the surfaces. Carbon-based nanomaterials are chosen as a model system due to their unique bioadhesive and contradictory biocompatible properties as well as the possibility of functionalization for specific applications. Graphene and its derivatives, such as graphene oxide and reduced graphene oxide, demonstrate controversial toxicity properties in vitro as well as in vivo. In this study, by covalently adding chemical groups, the wettability of graphene surfaces and the subsequent changes in its biocompatibility are being examined. An empirical force field potential (AMBER03) molecular dynamic simulation code implemented in the YASARA software package was utilized to model graphene/biomolecule interactions. The accuracy of the force field choice was verified by modeling the adsorption of individual amino acids to graphene surface in a vacuum. The obtained results are in excellent agreement with previously published ab initio findings. In order to mimic the natural protein environment, the interaction of several amino acids with graphene in an explicit solvent was modeled. The results show that the behaviour of amino acids in aqueous conditions is drastically different from that in vacuum. This finding highlights the importance of the host environment when biomaterial-biomolecule interfaces are modeled. The surface of Graphene Oxide (GO) has been shown to exhibit properties that are useful in applications such as biomedical imaging, biological sensors and drug delivery. An assessment of the intrinsic affinity of amino acids to GO by simulating their adsorption onto a GO surface was performed. The emphasis was placed on developing an atomic charge model for GO that was not defined before. Next, the simulation of a fibrinogen fragment (D-domain) at the graphene surface in an explicit solvent with physiological conditions was performed. This D-domain contains the hidden (not expressed to the solvent) motifs (PI 7190-202 and P2 7377-395, and specifically P2-C portion 7383-395) that were experimentally found to be responsible for attracting inflammatory cells through CDllb/CD18 (Mac-1) leukocyte integrin and, consequently, promoting the cascade of immune reactions. It was hypothesized that the hydrophobic nature of graphene would cause critical changes in the fibrinogen D-domain structure, thus exposing the sequences and result in the foreign body reaction. To further study this issue, molecular mechanics was used to stimulate the interactions between fibrinogen and a graphene surface. The atomistic details of the interactions that determine plasma protein affinity modes on surfaces with high hydrophobicity were studied. The results of this work suggest that graphene is potentially pro-inflammatory surface, and cannot be used directly (without alterations) for biomedical purposes. A better understanding of the molecular mechanisms underlying the interaction between synthetic materials and biological systems will further the ultimate goal of understanding the biocompatibility of existing materials as well as design of new materials with improved biocompatibility

Modeling the influence of DNA lesion on the regulation of gene expression

Nucleic acids are organic macromolecules that result from the polymerization of nucleotides. These molecules are generally considered as the support of the genetic information. Two families of nucleic acids are currently known: DNA and RNA. From a structural point of view, the most popular form is the double helix of DNA. However, other forms exist and among them are the G-quadruplex. This is a folding of the DNA, or RNA, in an area rich in guanines. These form quadruplex of guanines, which are stacked on top of each other and are stabilized by a central cation. G-quadruplex structures are increasingly studied. This is not surprising since their biological role involves the regulation of genetic mechanisms. They are notably involved in the regulation of the cell cycle, but they also play a role in cancer, certain neurological or viral diseases. The aim of this PhD thesis is to study G-quadruplex using theoretical chemistry tools. The three years of work raise very important points for the research on G-quadruplex. First, the modeling of a theoretical G-quadruplex structure can be achieved by sequence homology and validated by calculations of a theoretical circular dichroism spectrum. Consequently, it is possible to use these tools to propose and use a G-quadruplex structure if it is not yet experimentally solved. Then, the work done shows that G-quadruplex form a very stable folding since they are globally conserved even when 8-oxo-guanine or strand breaks lesions are introduced at the quartets. Then, the paper focuses on the interaction between G-quadruplex and proteins. It highlights the important role of G-quadruplex RNA in the infection of the viral pathogen SARS-CoV-2. This RNA promotes the dimerization of the SUD protein of the virus, which in turn is responsible for the disruption of the immune system. Finally, this thesis provides a structural explanation for the specific interaction between the DARPin 2E4 protein and the G-quadruplex of the c-Myc promoter

BIOMOLECULE LOCALIZATION AND SURFACE ENGINEERING WITHIN SIZE TUNABLE NANOPOROUS SILICA PARTICLES

Mesoporous silica materials are versatile platforms for biological catalysis, isolation of small molecules for detection and separation applications. The design of mesoporous silica supports for tailored protein and biomolecule interactions has been limited by the techniques to demonstrate biomolecule location and functionality as a function of pore size. This work examines the interaction of proteins and lipid bilayers with engineered porous silica surfaces using spherical silica particles with tunable pore diameters (3 – 12 nm) in the range relevant to biomolecule uptake in the pores, and large particle sizes (5 - 15 µm) amenable to microscopy imaging The differentiation of protein location between the external surface and within the pore, important to applications requiring protein protection or catalytic activity in pores, is demonstrated. A protease / fluorescent protein system is used to investigate protein location and protection as a function of pore size, indicating a narrow pore size range capable of protein protection, slightly larger than the protein of interest and approaching the protease dimensions. Selective functionalization, in this case exterior-only surface functionalization of mesoporous particles with amines, is extended to larger pore silica materials. A reaction time dependent functionalization approach is demonstrated as the first visually confirmed, selective amine functionalization method in protein accessible supports. Mesoporous silica nanoparticles are effective supports for lipid bilayer membranes and membrane associated proteins for separations and therapeutic delivery, although the role of support porosity on membrane fluidity is unknown. Transport properties of bilayers in lipid filled nanoparticles as a function of pore diameter and location in the particle are measured for the first time. Bilayer diffusivity increases with increasing pore size and is independent of bilayer location within the core, mid or cap of the particle, suggesting uniform long range bilayer mobility in lipid filled pores. Application of lipid bilayers on mesoporous silica was examined for membrane associated proteins A unique method to adhere functional proteins in lipid bilayers on mesoporous silica particles is established using vesicles derived from cell plasma membranes and their associated proteins. This method of membrane protein investigation retains proteins within native lipid membranes, stabilizing proteins for investigation on supports