In this talk we present the starting mechanical model of the lamellipodial actin-cytoskeleton meshwork. The model is derived starting from the microscopic description of mechanical properties of filaments and cross-links and also of the life-cycle of cross-linker molecules. We introduce a simplified system of equations that accounts for adhesions created by a single point on which we apply a force. We present the non-dimensionalisation that led to a singular limit motivating our mathematical study. Then we explain the mathematical setting and results already published. In the last part we present the latest developments: we give results for the fully coupled system with unbounded non-linear off-rates. This leads to two possible regimes: under certain hypotheses on the data there is global existence, out of this range we are able to prove blow-up in finite time.

• Thematic program: Models in Biology and Medicine (2016/2017)

In the recent years a significant attention has been directed again to Euler system, which was derived more than 250 years ago by Euler. The system describes the motion of an inviscid fluid. The main attention has been directed to incompressible fluids. Nevertheless, also the system of compressible fluids is an emerging topic, however still very far from a complete understanding. The classical results of Scheffer and Schnirelman pointed out the problem of non-uniqueness of distributional solutions to incompressible Euler system. However the crucial step appeared to be an application of methods arising from differential geometry, namely the celebrated theorem by Nash and Kuiper. This brought Camillo De Lellis and Laszlo Szekelyhidi Jr. in 2010 to the proof of existence of bounded nontrivial compactly supported in space and time solutions of the Euler equations (obviously not conserving physical energy!), basing on the Baire category method, which was highly non-standard kind of proof used in the theory of PDEs. Without a doubt this result is a first step towards the conjecture of Lars Onsager, who in his 1949 paper about the theory of turbulence asserted the existence of such solutions for any Hoelder exponent up to 1/3. As a result very much related to the Onsager conjecture one can find the result of P. Constantin, W. E and E. Titi for incompressible flow proving the energy conservation for any Hoelder exponent above 1/3. Our talk is based on several resent results joint with Eduard Feireisl and Emil Wiedemann and concerns various notions of solutions to compressible Euler equations and some systems of a similar structure.

• Thematic program: Models in Biology and Medicine (2016/2017)

All of biology occurs in ionic solutions that are plasmas in both the physical and biological meanings of the word. The composition of these ionic mixtures has profound effects on almost all biological functions, whether on the length scale of organs like the heart or brain, of the length scale of proteins, like enzymes and ion channels. Ion channels are proteins with a hole down their middle that conduct ions (spherical charges like Na+ , K+ , Ca2+ , and Clƒ{ with diameter ~ 0.2 nm) through a narrow tunnel of fixed charge (¡¥doping¡¦) with diameter ~ 0.6 nm. Ionic channels control the movement of electric charge and current across biological membranes and so play a role in biology as significant as the role of transistors in computers: almost every process in biology is controlled by channels, one way or the other. Ionic channels are manipulated with the powerful techniques of molecular biology in hundreds of laboratories. Atoms (and thus charges) can be substituted a few at a time and the location of every atom can be determined in favorable cases. Ionic channels are one of the few living systems of great importance whose natural biological function can be well described by a tractable set of equations. Ions can be studied as complex fluids in the tradition of physical science although classical treatments as simple fluids have proven inadequate and must be abandoned in my view. Ion channels can be studied by Poisson-Drift diffusion equations familiar in plasma and semiconductor physics ¡X called Poisson Nernst Planck or PNP in biology. Ions have finite size and so the Fermi distribution must be introduced to describe their filling of volume. The PNP-Fermi equations form an adequate model of current voltage relations in many types of channels under many conditions if extended to include correlations, and can even describe ¡¥chemical¡¦ phenomena like selectivity with some success. My collaborators and I have shown how the relevant equations can be derived (almost) from stochastic differential equations, and how they can be solved in inverse, variational, and direct problems using models that describe a wide range of biological situations with only a handful of parameters that do not change even when concentrations change by a factor of 107. Variational methods hold particular promise as a way to solve problems outstanding for more than a century because they describe interactions of ¡¥everything with everything¡¦ else that characterize ions crowded into channels. An opportunity exists to apply the well established methods of computational physics to a central problem of computational biology. The plasmas of biology can be analyzed like the plasmas of physics.

• Thematic program: Models in Biology and Medicine (2016/2017)

We consider a scalar conservation law with a flux containing spatial heterogeneities of bounded variation, where the number of discontinuities may be infinite. We address the question of existence of an adapted entropy solution in the BV framework. A sufficient key condition guaranteeing existence is identified and new BV estimates are given. This provides the most general BV theory available. Moreover, we show with a counter-example that if this hypothesis is violated, the problem may be ill-posed in the BV framework.

• Thematic program: Models in Biology and Medicine (2016/2017)

I will discuss the long time asymptotic of the solutions to the growthfragmentation equation, presenting several results and approaches. I will then focus on the spectral analysis and semigroup approach for which I will give some more details about the proof.

• Thematic program: Models in Biology and Medicine (2016/2017)
• Event: Workshop on: "Coagulation and Fragmentation Equations" (2017)

Growth-fragmentation equations can exhibit various asymptotic behaviours. In this talk we illustrate this diversity by working in suitable weighted L^p spaces which are associated to entropy functionals. We prove that, depending on the choice of the coefficients, the following behaviours can happen: uniform exponential convergence to the equilibrium, non-uniform convergence to the equilibrium, or convergence to periodic solutions. This is a joint work with Etienne Bernard and Marie Doumic.

• Thematic program: Models in Biology and Medicine (2016/2017)
• Event: Workshop on: "Coagulation and Fragmentation Equations" (2017)

The growth-fragmentation equation describes a system of growing and dividing particles, and arises in models of cell division, protein polymerisation and even telecommunications protocols. Several important questions about the equation concern the asymptotic behaviour of solutions at large times: at what rate do they converge to zero or infinity, and what does the asymptotic profile of the solutions look like? Does the rescaled solution converge to its asymptotic profile at an exponential speed? These questions have traditionally been studied using analytic techniques such as entropy methods or splitting of operators. In this work, we present a probabilistic approach to the study of this asymptotic behaviour. We use a Feynman–Kac formula to relate the solution of the growth-fragmentation equation to the semigroup of a Markov process, and characterise the rate of decay or growth in terms of this process. We then identify the spectral radius and the asymptotic profile in terms of a related Markov process, and give a spectral interpretation in terms of the growth-fragmentation operator and its dual. In special cases, we obtain exponential convergence.

• Thematic program: Models in Biology and Medicine (2016/2017)
• Event: Workshop on: "Coagulation and Fragmentation Equations" (2017)

We consider fragmentation equations with non-conservative solutions, some mass being lost to a dust of zero-mass particles as a consequence of an intensive splitting. Under assumptions of regular variation on the fragmentation rate, we describe the large time behavior of solutions. Our approach is based on probabilistic tools: the solutions to the fragmentation equations are constructed via non-increasing self-similar Markov processes that continuously reach 0 in finite time. We describe the asymptotic behavior of these processes conditioned on non-extinction and then deduced the asymptotics of solutions to the equation.

• Thematic program: Models in Biology and Medicine (2016/2017)
• Event: Workshop on: "Coagulation and Fragmentation Equations" (2017)

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• Thematic program: Models in Biology and Medicine (2016/2017)
• Event: Workshop on: "Coagulation and Fragmentation Equations" (2017)

In the last years there has appeared several applications of relative entropy method for strong measure-valued uniqueness of solutions in physical models (see: e.g. incompressible Euler equation [1], polyconvex elastodynamics [2], compressible Euler equation [3], compressible Navier-Stokes equation [4]). The topic of the talk will be application of similar techniques to structured population models. Preliminary result in this direction was obtain in [5]. The talk is based on the joint result with Marie Doumic-Jauffret and Emil Wiedemann. [1] Y. Brenier, C. De Lellis, and L. Sz´ekelyhidi, Jr. Weak-strong uniqueness for measure-valued solutions. Comm. Math. Phys., 305(2):351--361, 2011. [2] S. Demoulini, D.M.A. Stuart, and A.E. Tzavaras. Weak-strong uniqueness of dissipative measure-valued solutions for polyconvex elastodynamics. Arch. Ration. Mech. Anal., 205(3):927--961, 2012. [3] P. Gwiazda, A. Œwierczewska-Gwiazda, and E. Wiedemann. Weak-strong uniqueness for measure-valued solutions of some compressible fluid models. Nonlinearity, 28(11):3873--3890, 2015. [4] E. Feireisl, P. Gwiazda, A. Œwierczewska-Gwiazda and E. Wiedemann Dissipative measure-valued solutions to the compressible Navier-Stokes system, Calc. Var. Partial Differential Equations 55 (2016), no. 6, 55--141 [5] P. Gwiazda, E. Wiedemann, Generalized Entropy Method for the Renewal Equation with Measure Data, to appear in Commun. Math. Sci., arXiv:1604.07657

• Thematic program: Models in Biology and Medicine (2016/2017)
• Event: Workshop on: "Coagulation and Fragmentation Equations" (2017)

The question whether the long-time behaviour of solutions to Smoluchowski's coagulation equation is characterized by self-similar solutions has received a lot of interest within the last two decades. While this issue is by now well-understood for the three solvable cases, the theory for non-solvable kernels is much less developed. For kernels with homogeneity smaller than one existence results for self-similar solutions and some partial uniqueness results are available. In this talk I will report on some recent results on the borderline case of kernels with homogeneity of degree one. For so-called class II kernels we can prove the existence of a family of self-similar solutions. For class I, or diagonally dominant, kernels, it is known that self-similar solutions cannot exist. Formal arguments suggest that the long-time behaviour of solutions is, in suitable variables, to leading order the same as for the Burgers equation. However, in contrast to diffusive regularizations, we obtain phenomena such as instability of the constant solution or oscillatory traveling waves. (Joint work with Marco Bonacini, Michael Herrmann and Juan Velazquez)

• Thematic program: Models in Biology and Medicine (2016/2017)
• Event: Workshop on: "Coagulation and Fragmentation Equations" (2017)

When the coagulation kernel and the overall fragmentation rate are homogeneous of degree ë and ã > 0, respectively, there is a critical value ëc := ã + 1 which separates two different behaviours: all solutions are expected to be mass-conserving when ë < ëc while gelation is expected to take place when ë > ëc, provided the mass of the initial condition is large enough. The focus of this talk is the case ë = ëc for which we establish the existence of mass-conserving self-similar solutions. This is partly a joint work with Henry van Roessel (Edmonton).

• Thematic program: Models in Biology and Medicine (2016/2017)
• Event: Workshop on: "Coagulation and Fragmentation Equations" (2017)

In the talk we shall describe how the substochastic semigroup theory can be used to prove analyticity of a class of fragmentation semigroup. This result is applied to discrete fragmentation processes with growth to analyze their long time behaviour and to prove the existence of classical solutions to equations describing such processes combined with coagulation.

• Thematic program: Models in Biology and Medicine (2016/2017)
• Event: Workshop on: "Coagulation and Fragmentation Equations" (2017)

In this talk, we present two types of linked partial differential equation models that describe the evolution of an interacting neural network and where neurons interact with one another through their common statistical distribution. We will show, according to the choice of EDP studied, what information can be obtained in terms of synchronization phenomena, qualitative and asymptotic properties of these solutions and what are the specific difficulties on each of these models.

• Thematic program: Models in Biology and Medicine (2016/2017)
• Event: Workshop on: "Coagulation and Fragmentation Equations" (2017)

We will present some recent results on the long behaviour of the Becker-Döring equations, mainly involving subcritical solutions: speed of convergence to equilibrium (sometimes exponential, sometimes algebraic) and some new uniform bounds on moments. We will also comment on a continuous model that serves as an analogy of the discrete equations, that seems to exhibit a similar long-time behaviour. This talk is based on collaborations with J. Conlon, A. Einav, B. Lods and A. Schlichting.

• Thematic program: Models in Biology and Medicine (2016/2017)
• Event: Workshop on: "Coagulation and Fragmentation Equations" (2017)

We consider results on discrete and continuous coagulation and coagulation-fragmentation models. For discrete models, we shall present some recent regularity results concerning smoothness of moments and absence of gelation. For the continuous Smoluchowski equation with constant rates, we shall prove exponential, resp. superlinear convergence to equlibrium. This are joint works with M. Breden, J.A. Canizo, J.A. Carrillo and L. Desvillettes.

• Thematic program: Models in Biology and Medicine (2016/2017)
• Event: Workshop on: "Coagulation and Fragmentation Equations" (2017)

In this talk we present some PDE models which describe the activity of neural networks by means of the membrane potential. We focus on models based on nonlinear PDEs of Fokker-Planck type. We study the wide range of phenomena that appear in this kind of models: blow-up, asynchronous/synchronous solutions, instability/stability of the steady states ...

• Thematic program: Models in Biology and Medicine (2016/2017)
• Event: Workshop on "Cross diffusion systems and kinetic equations for biology" (2017)

Fractional kinetics is derived from Gaussian processes when the medium where the diffusion takes place is characterized by a population of length-scales [1]. This approach is analogous to the generalized grey Brownian motion [2], and it can be used for modeling anomalous diffusion in complex media. In particular, the resulting stochastic process can show sub-diffusion with a behavior in qualitative agreement with single-particle tracking experiments in living cells, such as the ergodicity breaking, p variation, and aging. Moreover, for a proper distribution of the length-scales, a single parameter controls the ergodic-to-nonergodic transition and, remarkably, also drives the transition of the diffusion equation of the process from nonfractional to fractional, thus demonstrating that fractional kinetics emerges from ergodicity breaking [3]. References: [1] Pagnini G. and Paradisi P., A stochastic solution with Gaussian stationary increments of the symmetric space-time fractional diffusion equation. Fract. Cacl. Appl. Anal. 19, 408–440 (2016) [2] Mura A. and Pagnini G., Characterizations and simulations of a class of stochastic processes to model anomalous diffusion. J. Phys. A: Math. Theor. 41, 285003 (2008) [3] Molina–García D., Pham T. Minh, Paradisi P., Manzo C. and Pagnini G., Fractional kinetics emerging from ergodicity breaking in random media. Phys. Rev. E. 94, 052147 (2016)

• Thematic program: Models in Biology and Medicine (2016/2017)
• Event: Workshop on "Cross diffusion systems and kinetic equations for biology" (2017)

A class of parabolic cross-diffusion systems modeling the interaction of an arbitrary number of population species is analyzed in a bounded domain with no-flux boundary conditions. The equations are formally derived from a random-walk lattice model in the diffusion limit. Compared to previous results in the literature, the novelty is the combination of general degenerate diffusion and volume-filling effects. Conditions on the nonlinear diffusion coefficients are identified, which yield a formal gradient-flow or entropy structure. This structure allows for the proof of global-in-time existence of bounded weak solutions and the exponential convergence of the solutions to the constant steady state. The existence proof is based on an approximation argument, the entropy inequality, and new nonlinear Aubin-Lions compactness lemmas. The proof of the large-time behavior employs the entropy estimate and convex Sobolev inequalities. Moreover, under simplifying assumptions on the nonlinearities, the uniqueness of weak solutions is shown by using the H^{-1} method, the E-monotonicity technique of Gajewski, and the subadditivity of the Fisher information.

• Thematic program: Models in Biology and Medicine (2016/2017)
• Event: Workshop on "Cross diffusion systems and kinetic equations for biology" (2017)

In this talk, we will describe how to mix entropy structure and duality estimates in order to build global weak solutions to a class of cross-diffusion systems.

• Thematic program: Models in Biology and Medicine (2016/2017)
• Event: Workshop on "Cross diffusion systems and kinetic equations for biology" (2017)

We investigate the rigorous fast-reaction limit from a reaction-cross-diffusion system with known entropy to a new class of cross-diffusion systems using entropy and duality estimates. Performing the fast-reaction limit leads to a limiting entropy of the limiting cross-diffusion system. In this way, we are able to obtain new entropies for new classes of cross-diffusion systems. This is a joint work with L. Desvillettes and A. Juengel.

• Thematic program: Models in Biology and Medicine (2016/2017)
• Event: Workshop on "Cross diffusion systems and kinetic equations for biology" (2017)

We show global-in-time existence of bounded weak solutions to systems of cross-diffusion equations in a one dimensional moving domain. These equations stem from the modelization of the evolution of the concentration of chemical species composing a crystalline solid during a physical vapor deposition process. To this aim, we use the so called boundedness-by-entropy technique developed in [1], [2] and [3] based on the formal gradient flow structure of the system. Moreover, we are interested in controlling the fluxes of the different atomic species during the process in order to reach a certain desired final profile of concentrations. This problem is formulated as an optimal control problem to which the existence of a solution is proven. In addition, an investigation of the long time behavior is presented in the case of constant positive external fluxes. Finally, some numerical results and comparison with actual experiments are presented. The material of this talk is a joint work with Virginie Ehrlacher. References [1] M.Burger, M.Di Francesco, J-F. Pietschmann and B. Schalke. Non linear cross diffusion with size exclusion. SIAM J. Math Anal 42 (2010). [2] A. Jüngel and Nicola Zamponi boundedness of weak solutions to cross-diffusion systems from population dynamics. arxiv:1404.6054v1 (2014). [3] A. Jüngel. The boundedness-by-entropy method for cross-diffusion systems. To appear in Nonlinearity, http://www.asc.tuwien.ac.at/ juengel/ (2015).

• Thematic program: Models in Biology and Medicine (2016/2017)
• Event: Workshop on "Cross diffusion systems and kinetic equations for biology" (2017)

We consider the one-dimensional multi-species Boltzmann system of equations [2] in the diffusive scaling. Suppose that the Mach and the Knudsen numbers are of the same order of magnitude epsilon > 0 small enough. For each species i of the mixture, we define the macroscopic quantity of matter and flux as the moments 0 and 1 in velocity of the distribution functions f_i, solutions of the Boltzmann system associated to the scaling parameter epsilon. Using the moment method [4], we introduce a proper ansatz for each distribution function f_i in order to recover a Maxwell-Stefan diffusion limit-type as in [1]. In this way we build a suitable numerical scheme for the evolution of these macroscopic quantities in different regimes of the parameter epsilon. We prove some a priori estimates (mass conservation and nonnegativity) and well-posedness of the discrete problem. We also present numerical examples where we observe that the scheme shows an asymptotic preserving property similar to the one presented in [3]. This is a joint work with L. Boudin and B. Grec. References [1] L. Boudin, B. Grec and V. Pavan, The Maxwell-Stefan diffusion limit for a kinetic model of mixtures with general cross sections, Nonlinear Analysis: Theory, Methods and Applications, 2017. [2] L. Desvillettes, R. Monaco and F. Salvarani, A kinetic model allowing to obtain the energy law of polytropic gases in the presence of chemical reactions, Eur. J. Mech. B Fluids, 24(2005), 219-236. [3] S. Jin and Q. Li, A BGK-penalization-based asymptotic-preserving scheme for the multispecies Boltzmann equation, Numer. Methods Partial Differential Equations, 29(3), pp. 1056-1080, 2013. [4] C. D. Levermore, Moment closure hierarchies for kinetic theories, J. Statist. Phys., 83(5-6):1021-1065, 1996

• Thematic program: Models in Biology and Medicine (2016/2017)
• Event: Workshop on "Cross diffusion systems and kinetic equations for biology" (2017)

In this talk, we consider some specific reaction-diffusion equations in order to understand the equivalence between asymptotic Turing instability of a steady state and backwardness of some parabolic equations or cross-diffusion equations in the formal limit of fat reaction terms. We will see that the structure of the studied equations involves some Lyapunov functions which leads to a priori estimates allowing to pass rigorously for the fast reaction terms in the case without Turing instabilities.

• Thematic program: Models in Biology and Medicine (2016/2017)
• Event: Workshop on "Cross diffusion systems and kinetic equations for biology" (2017)

The main feature of the two-dimensional Keller-Segel model is the blow-up behaviour of solutions for supercritical masses. We introduce a regularisation of the fully parabolic system by adding a cross-diffusion term to the equation for the chemical substance. This regularisation provides another helpful entropy dissipation term allowing to prove global existence of weak solutions for any initial mass. For the proof we first analyse an approximate problem obtained from a semi-discretisation and a carefully chosen regularisation by adding higher order derivatives. Compactness arguments are used to carry out the limit to the original system. A similar approach can be used to analyse a pedestrian dynamics model for two groups moving in opposite direction. The evolutionary equations are driven by cohesion and aversion and are formally derived from a 2d lattice based approach. Also numerical simulations illustrating lane formation will be presented. These methods are extended to a crossing pedestrian model, where we additionally analyse the stability of stationary states in the corresponding 1d model.

• Thematic program: Models in Biology and Medicine (2016/2017)
• Event: Workshop on "Cross diffusion systems and kinetic equations for biology" (2017)

We study interacting particles behaving according to a reaction-diffusion equation with non-linear diffusion and non-local attractive interaction. This class of equations has a very nice gradient flow structure that allows us to make links to homogeneous functionals and variations of well-known functional inequalities (Hardy-Littlewood-Sobolev inequality, logarithmic Sobolev inequality). Depending on the non-linearity of the diffusion, the choice of interaction potential and the dimensionality, we obtain different regimes. Our goal is to understand better the asymptotic behaviour of solutions in each of these regimes, starting with the fair-competition regime where attractive and repulsive forces are in balance. This is joint work with José A. Carrillo and Vincent Calvez.

• Thematic program: Models in Biology and Medicine (2016/2017)
• Event: Workshop on "Cross diffusion systems and kinetic equations for biology" (2017)

Amyloid-â peptide (Aâ) isoforms of different lengths and aggregation propensities coexist in vivo. These different isoforms are able to nucleate or frustrate the assembly of each other. Nterminal truncated Aâ(11-40) and Aâ(11-42) make up one fifth of plaque load yet nothing is known about their interaction with full-length Aâ(1-40/42). Here we show that in contrast to C-terminal truncated isoforms which do not co-fibrillise, deletions of ten residues from the N-terminus of Aâ have little impact on its ability to co-fibrillise with the full-length counterpart. As a consequence N-terminal truncated Aâ will accelerate fibre formation and co-assemble into short rod-shaped fibres with its full-length Aâ counterpart. Furthermore we show Cu2+ forms a very tight tetragonal complex with truncated Aâ(11-40) with a femtomolar affinity. These observations have implications for the assembly kinetics, morphology and toxicity of all Aâ isoforms. The process by which amyloid-â (Aâ) disrupts synaptic activity, and causes neuronal cell death in Alzheimer’s disease remains poorly understood. A potential mechanism of toxicity is in the ability of Aâ to form, membrane-spanning ion channels. However, there has been a mismatch between the channel forming properties of Aâ isoforms, 40 and 42 amino acids long, and their known relative pathogenicity. We observe ion channel formation by oligomeric Aâ42, but also show Aâ40 does not form ion channels in cellular membranes. This makes a strong link between ion channel formation and the pathology of Aâ isoforms. Molecules that block these ion channels may represent therapeutic targets. [1] Ion Channel Formation by Amyloid-â42 Oligomers but not Amyloid-â40 in Cellular Membranes DC Bode, MD Baker, JH Viles* (2017) J of Biol Chem 292, 1404-1413 [2] Truncated Amyloid-â (11-40/42) from Alzheimer's Disease Binds Copper2+ with a Femtomolar Affinity and Influences Fibre Assembly J D Barritt, J H. Viles* (2015) J of Biol Chem, 290, 27791-27802 [3] The Rapid Exchange of Zinc2+ Enables Trace Levels to Profoundly Influence Amyloid-â Misfolding and Dominates Assembly Outcomes in Cu2+/Zn2+ Mixtures C J Matheou, N D Younan, J H Viles* (2016) J Mol Biol 428, 2832-2846

• Thematic program: Models in Biology and Medicine (2016/2017)
• Event: Workshop on "Protein Aggregation: Biophysics and Mathematics" (2017)

The protein Lgl (Lethal giant larvae) is part of a conserved protein complex, which is responsible for the asymmetric localisation of cell-fate determinants, for instance, in Drosophila SOP precursor cells. We formulate continuum models, which consider the phosphorylated and the unphosphorylated conformations of Lgl within the cell cytoplasm and on the cell cortex. After presenting illustrative numerical simulations, we prove first the equilibration of the underlying complex-balance volumesurface reaction-diffusion system and perform further a rigorous quasi-steady-state-approximation in a fast-reaction limit.

• Thematic program: Models in Biology and Medicine (2016/2017)
• Event: Workshop on "Protein Aggregation: Biophysics and Mathematics" (2017)

Angélique Igel-Egalon1¶, Mohammed Moudjou1¶, Florent Laferrière1¶, Tina Knäpple1, Laetitia Herzog1, Fabienne Reine1, Hubert Laude1, Human Rezaei1*, Vincent Béringue1* 1VIM, INRA, Université Paris-Saclay, 78350 Jouy-en-Josas, France ¶Equal contributors, *Senior authorship Mammalian prions are proteinaceous pathogens responsible for fatal, neurodegenerative disorders in human and animals. They are formed of misfolded assemblies (PrPSc) of the host-encoded cellular prion protein (PrPC). In the infected species, prions replicate by seeding the conversion and polymerization of host PrPC. Distinct prion strains are recognized within the same host-species, exhibiting defined PrPSc biochemical properties and stereotyped biological traits. While strain information is encoded within the conformation of PrPSc assemblies, the storage of the structural information and the molecular requirements for self-perpetuation remain uncertain. In particular, the polymerization steps and its dynamic nature remains mostly hypothetical. It is widely believed that monomeric PrPC is constantly recruited within the forming aggregates allowing PrPSc fibril growth. Fibril fragmentation is supposed to provide further converting seeds, favouring prion exponential replication. Whether this proposed mechanism is versatile or strain-dependent remains to be determined, as is the real contribution of fragmentation. We have investigated this issue by analysing the dynamic of PrPSc assembling during cell-free prion amplification by protein misfolding cyclic amplification (PMCA). We show that: i) prion amplification occurs through preferential amplification of small oligomeric forms of PrPSc that can further assemble into larger aggregates; ii) disassembling rather than fragmentation sustains the self-perpetuation of the process, iii) different prion strains exhibit similar amplification dynamic. Thus, prion replication may proceed through an assembly/disassembly process.

• Thematic program: Models in Biology and Medicine (2016/2017)
• Event: Workshop on "Protein Aggregation: Biophysics and Mathematics" (2017)

This work is motivated by protein aggregation phenomena in neurodegenerative diseases. A key observation of in-vitro spontaneous polymerization experiments of prion protein is the large variability of the so-called 'nucleation time', which is experimentally defined as the lag time before the polymerization of proteins truly starts (typically several hours in a 10-20 hours experiment). In this context, we study a stochastic version of a well-known nucleation model in physics, namely the Becker-Döring model [1]. In this model, aggregates may increase or decrease their size one-by-one, by capturing or shedding a single monomer particle. We will present numerical and analytical investigation of the nucleation time defined as a first passage time problem [2, 3]. Finally, we will present limit theorem techniques to study the link from the discrete size Becker-Döring model to a continuous size version (the Lifshitz-Slyozov model), which may be of importance to study large size aggregates formation. For general coefficients and initial data, we introduce a scaling parameter and show that the empirical measure associated to the Becker-Döring system converges in some sense to the Lifshitz-Slyozov equation when the scaling parameter goes to 0. When the aggregation is favorable, we derive a mean-field transport PDE limit together with an entrant boundary condition, leading to an effective reduced dynamical model [4]. When the aggregation is initially unfavorable, we shed light on metastable behavior and phase transition phenomena. [1] E. Hingant, R. Y., arXiv:1609.00697 (2016). [2] R. Y., M. R. D'Orsogna, and T. Chou. J. Chem. Phys., 137:244107, (2012). [3] R. Y., S. Bernard, E. Hingant, L. Pujo-Menjouet, J. Chem. Phys., 144(3):034106, (2016). [4] Julien Deschamps, Erwan Hingant, R.Y., arXiv:1605.08984 (2016).

• Thematic program: Models in Biology and Medicine (2016/2017)
• Event: Workshop on "Protein Aggregation: Biophysics and Mathematics" (2017)

Cassandra Terrya Adam Wenborna Nathalie Grosa Jessica Sellsa Susan Joinera Laszlo L.P. Hosszua M. Howard Tattuma Silvia Panicob Daniel K. Clareb, John Collingea, Helen R. Saibilb and Jonathan D.F. Wadswortha* a, MRC Prion Unit and Department of Neurodegenerative Disease, UCL Institute of Neurology, Queen Square, London WC1N 3BG, UK b, Institute of Structural and Molecular Biology, Department of Biological Sciences, Birkbeck College, University of London, Malet Street, London WC1E 7HX, UK Prions cause lethal neurodegenerative diseases in mammals, including scrapie in sheep and goats, bovine spongiform encephalopathy (BSE) in cattle and Creutzfeldt–Jakob disease (CJD) in humans. Mammalian prions are hypothesised to be fibrillar or amyloid forms of prion protein (PrP) which self-propagate by means of seeded protein polymerisation but structures observed had not been definitively correlated with infectivity and the three-dimensional structure of prions remained unknown. We developed new methods to obtain pure preparations of intact prions from mouse brain1 and showed that pathogenic PrP is assembled into rod-like assemblies (PrP rods) that faithfully transmit prion strain-specific phenotypes when inoculated into mice. We have utilised the precision of cell culture prion infectivity assays to define the physical relationship between PrP rods and prion infectivity and used electron tomography to define their architecture. Our 3D analysis2 demonstrates that ex vivo infectious PrP rods from different strains observed have a common hierarchical assembly comprising twisted pairs of short fibres with repeating substructure which are markedly different to non-infectious PrP fibrils generated in vitro. References 1. A. Wenborn, C. Terry, N. Gros, S. Joiner, L. D’Castro, S. Panico, J. Sells, S. Cronier, J. Linehan, S. Brandner, H.R. Saibil, J. Collinge, J.D.F Wadsworth, Sci. Rep. A novel and rapid method for obtaining high titre intact prion strains from mammalian brain, 2015, 5, 10062. C. Terry, A. Wenborn, N. Gros, J. Sells, S. Joiner, L.L.P Hosszu, M.H. Tattum, S. Panico, D.K. Clare, J. Collinge, H.R. Saibil, J.D.F Wadsworth. Open Biology. Ex vivo mammalian prions are formed of paired double helical prion protein fibrils, 2016, 6, 160035.

• Thematic program: Models in Biology and Medicine (2016/2017)
• Event: Workshop on "Protein Aggregation: Biophysics and Mathematics" (2017)

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 then to propose an original hypothesis able to reconcile the in vitro data with the predictions of a mathematical model describing the problem. The model presented here, has been 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. Experimental data have been compared to the prediction of our model leading to a new hypothesis for the formation of infectious prion amyloids. In the last part, we will introduce a new model describing another dangerous liaison: the interaction between prion proteins and Abeta peptides that may lead to Alzheimer’s disease.

• Thematic program: Models in Biology and Medicine (2016/2017)
• Event: Workshop on "Protein Aggregation: Biophysics and Mathematics" (2017)

Autophagosomes are double membrane-bound organelles that are formed de novo during a process called autophagy. Autophagosomes mediate the bulk degradation of cytoplasmic material such as aggregated proteins, dysfunctional or surplus mitochondria and intracellular pathogens. Autophagy is conserved from yeast to human and has been shown to protect the organism from conditions such as starvation, neurodegeneration and infectious diseases. During autophagosome formation initially small membrane structures termed isolation membranes are formed. These isolation membranes expand and thereby gradually enclose cytoplasmic cargo. Finally, isolation membranes close to give rise to mature autophagosomes. After their formation autophagosomes fuse with lysosomes within which their inner membranes and the contents are degraded. Autophagy has the ability to selectively capture and subsequently degrade aggregated and ubiquitinated proteins. This is mediated by the p62 cargo receptor, which is required for the aggregation of these proteins into larger structures. These structures then serve as templates for autophagosome formation. I will present our results from a fully reconstituted system, which enabled us to dissect the interplay between p62 and ubiquitin positive proteins during protein aggregation in selective autophagy.

• Thematic program: Models in Biology and Medicine (2016/2017)
• Event: Workshop on "Protein Aggregation: Biophysics and Mathematics" (2017)

Some recent contributions to the modeling of the polymerization and depolymerization of actin filaments will be reviewed. Some results of the embedding of these models into the Filament Based Lamellipodium Model will be presented.

• Thematic program: Models in Biology and Medicine (2016/2017)
• Event: Workshop on "Protein Aggregation: Biophysics and Mathematics" (2017)

Mathematical modelling of protein polymerisation is a challenging topic, with wide applications, from actin filaments in myocytes (muscle tissues) to the so-called amyloid diseases (e.g. Alzheimer's, Parkinson's or Creuzfeldt-Jakob's diseases). In this talk, we will give an overview of recent results for both deterministic - where statistical mechanical fluctuations arising from intrinsic noise are negligible - and stochastic approaches, envisaged as giving complementary insights on the still largely mysterious intrinsic mechanisms of polymerisation. A data assimilation approach is developed in parallel of more specific methods for fragmentation estimation. The results we will present are partly joint work with A. Armiento, J. Calvo, S. Eugène, M. Escobedo, P. Moireau, B. Perthame, H. Rezaei, P. Robert, M. Tournus and W.F. Xue.

• Thematic program: Models in Biology and Medicine (2016/2017)
• Event: Workshop on "Protein Aggregation: Biophysics and Mathematics" (2017)

Nicola Vettore and Alexander K. Buell, Institute of Physical Biology, University of Düsseldorf Amyloid fibrils are thermodynamically very stable [1], but the origin of their enhanced stability with respect to the native state has not yet been elucidated in molecular detail. The high stabilities of amyloid fibrils render the study of their equilibrium behaviour challenging. One way to approach this issue, in direct analogy to the study of protein folding equilibria is denaturation with commonly used denaturants, such as GdmCl or Urea. A theoretical framework to extract from such measurements the free energy difference between the fibril state and the soluble state, based on Oosawa's linear polymerisation model, was proposed in [2]. Here we present experimental results of amyloid fibril equilibrium denaturation measured via capillary fluorescence over a wide range of temperatures. The data highlight how the influence of temperature seems of primary importance not only for the kinetics of fibril formation, but also for the thermodynamic stability of the fibrillar structures. We will also present our attempts to describe the temperature-dependence of fibril stability within a general thermodynamic framework. [1] A. J. Baldwin, T. P. J. Knowles, G. G. Tartaglia, A. W. Fitzpatrick, G. L. Devlin, S. L. Shammas, C. A. Waudby, M. F. Mossuto, S. Meehan, S. L. Gras, J. Christodoulou, S. J. Anthony-Cahill, P. D. Barker, M. Vendruscolo, and C. M. Dobson, J. Am. Chem. Soc. 133, 14160 (2011). [2] T. Narimoto, K. Sakurai, A. Okamoto, E. Chatani, M. Hoshino, K. Hasegawa, H. Naiki, and Y. Goto, FEBS Lett. 576, 313 (2004).

• Thematic program: Models in Biology and Medicine (2016/2017)
• Event: Workshop on "Protein Aggregation: Biophysics and Mathematics" (2017)

We investigate the oscillatory behaviour of the PrP protein during the polymerization/depolymerization process. In order to modelize this oscillatory process, we study a simplified Belousov-Zhabotinsky reaction from a kinetic point of view. This simplified oscillatory system of chemical reactions allows us to introduce a modified Becker-Döring system where the trajectories oscillate. A key to have a closed oscillatory polymerization/depolymerization system is to consider different specices of polymers and monomers. We finally present several system where the numerical simulations show a more or less sustained oscillatory behaviour.

• Thematic program: Models in Biology and Medicine (2016/2017)
• Event: Workshop on "Protein Aggregation: Biophysics and Mathematics" (2017)

Reine1, Charles-Adrien Richard1, Tina Knäpple1 Vincent Béringue1* and Human Rezaei1* 1: INRA, UR892, Virologie Immunologie Moléculaires, Jouy-en-Josas 78350, France *: Corresponding authors The prion phenomenon is based on autonomous structural information propagation towards single or multiple protein conformation changes. During this last decade the prion concept referring the transmission of structural information has been extended to several regulation systems and pathologies including Alzheimer and Parkinson’s diseases. Despite intensive investigation, the molecular basis of structural information transmission remains obscure. Templating (i.e. secondary nucleation as vector of structural information) has been proposed as origin of autocatalytic structural information perpetuation. However, the templating process does not consider the spreading process which consists in an exponential amplification of structural information. Active fibril fragmentation (AFF) constitutes a solution for exponential spreading and amplification of the structural information as strongly suggested in fungi prions (Shorter and Lindquist, Mol Cell, 2006). In the present work, we demonstrate that mammalian Prion assemblies (PrPSc) are constituted from an oligomeric elementary brick called suPrP. We show that in physiological conditions Prion assemblies are in equilibrium with suPrP. The existence of such equilibrium as simple depolymerization/condensation process is sufficient to spread the replicative unit through the release of suPrP, followed by its Brownian diffusion and condensation into PrPSc and discards the requirement of fragmentation for prion spreading.

• Thematic program: Models in Biology and Medicine (2016/2017)
• Event: Workshop on "Protein Aggregation: Biophysics and Mathematics" (2017)

A number of devastating human disorders, for example Alzheimer's disease (AD), Hungtington's diseases, type 2 diabetes and transmissible spongiform encephalopathies (TSEs), are associated with the abnormal folding and assembly of proteins. The net result of this misfolding is the formation of large insoluble protein deposits and small toxic and transmissible protein particles in a state called amyloid. What are the molecular mechanisms that govern the amyloid fibrils’ potential to seed the formation of new aggregates, to propagate the amyloid state as prion particles, and to damage cells in amyloid-associated diseases? We have developed AFM imaging approaches that are capable of resolving the fibril particle concentrations, their length distributions, as well as their toxic and infective potential to cells. With these approaches, we have shown that the disease-associated properties of amyloid can be linked to small nano-sized amyloid particles created through the breakage of amyloid fibrils. The approaches we have developed offer new opportunities to determine, quantify, and predict the course and the consequences in amyloid assembly of cytotoxic, infectious as well as functional amyloid systems.

• Thematic program: Models in Biology and Medicine (2016/2017)
• Event: Workshop on "Protein Aggregation: Biophysics and Mathematics" (2017)

We consider the pure fragmentation fragmentation equation and address the question of estimating the fragmentation parameters (division rate and fragmentation kernel) from measurements of the size distribution at various times. Under the assumption of a polynomial division rate and a self-similar fragmentation kernel, we use the well-known asymptotic behaviour of the solution to guarantee the well-posedness of our inverse problem and provide a representation formula for the fragmentation kernel. The tools used are the Mellin transform and the Wiener-Hopf method. Motivations for studying this problem and applications to amyloid fibril breakage will be described in the talk of W.F. Xue.

• Thematic program: Models in Biology and Medicine (2016/2017)
• Event: Workshop on "Protein Aggregation: Biophysics and Mathematics" (2017)

Guillaume van der Rest, Human, Rezaei, Frédéric Halgand, Université Paris Sud, Laboratoire de Chimie Physique Prion protein is involved in deadly neurodegenerative diseases. Its pathogenicity is linked to its structural conversion (a-helix to b-strand transition). However, recent studies suggest that prion protein can follow a plurality of conversion pathways which hints towards different conformers that might coexist in solution. We therefore decided to screen the ovine and human PrP monomers using ion mobility coupled to mass spectrometry following electrospray ionization. After a short presentation of ion mobility for studying ionized proteins in the gas phase, we will briefly discuss issues with the collision cross section calibration procedure that we have encountered when using travelling wave ion mobility. We will also discuss the development of an automated data extraction pipeline for which we developed a Python/Qt script base interface. Infusion of monomeric PrP solutions have shown that at least three PrP conformers are observed in the gas phase. PrP monomers are known to lead to the formation of oligomeric species in specific conditions (temperature, pH and buffer), which are not compatible with mass spectrometry. We have therefore developed a size-exclusion chromatography IMS-MS setup with the aim to study the oligomers produced in these conditions. The development of this SEC-IMS-MS methodology will be presented as well as its application for calibration with standard protein complexes. Although we did not achieve resolution of the large (O1 ~36-mer) oligomeric species, optimization of the experimental parameters led to the observation of the small (O3) oligomeric species. One key observation in this process was that the abundance of the gas phase monomeric conformers changed upon the oligomerization process. First results allow us to interpret this as an effect of monomer concentration on the ratio of conformers present in solution, which is observed only in specific buffer conditions.

• Thematic program: Models in Biology and Medicine (2016/2017)
• Event: Workshop on "Protein Aggregation: Biophysics and Mathematics" (2017)

Yi Yin*, 1, Stéphanie Prigent1, Joan Torrent, Dirk Drasdo1, Human Rezaei, and Marie Doumic1 1. INRIA Paris, and Sorbonne Universités UPMC Univ. Paris 6, Laboratoire Jacques-Louis Lions, Paris, France, * yi.yin@inria.fr Protein aggregation into fibrils is a key process in amyloid diseases and also in other biological processes. The quantification of fibrils’ morphology and molecular structures is urgently needed in understanding of the key mechanisms and properties of fibrils. In this study, we propose an automated image analysis procedure to extract and quantify fibril morphological features from transmission electron microscopy (TEM) images. Fibrils are segmented by a ‘maximum entropy’ thresholding method and then the ‘fast marching’ skeletonization is applied to detect the fibril centerlines. The individual information of each fibril is gathered based on the fibril segmentation and extracted centerline, including the length (following the curvature of the fibrils, which are rarely straight lines), the varying width along the length, the curvature, as well as the number, position and length of branches. The intricate overlapping and branching structures are identified based on the angles between fibril segments. The proposed method was tested on experiments on the prion protein (PrP), which also allows us to explain in detail the parameters needed for the image analysis. Our method has high estimation accuracy (e.g. width estimation as shown in the figure). The results from different mutants of the PrP protein fibrils showed the potential of the method in fibrils classification through a statistical analysis. Romain

• Thematic program: Models in Biology and Medicine (2016/2017)
• Event: Workshop on "Protein Aggregation: Biophysics and Mathematics" (2017)

In this talk I will discuss various aspects of the kinetics and thermodynamics of the self assembly of peptides into amyloid fibrils and crystals. I will present a theoretical framework that allows to determine free energy barriers and entropies from kinetic data of amyloid fibril growth [1,2]. I will contrast the kinetic behaviour of longer, amyloid forming sequences with that of aromatic dipeptides that form crystals, rather than amyloid fibrils [3,4]. Furthermore, I will present the phenomenon of autocatalytic secondary nucleation, whereby new amyloid fibrils nucleate on the surface of existing ones [5,6]. In particular, I will show how this phenomenon manifests itself in kinetic measurements of protein aggregation, and how biosensing can be used to explore its molecular origin [6,7]. [1] A. K. Buell, J. R. Blundell, C. M. Dobson, M. E. Welland, E. M. Terentjev, and T. P. Knowles, Phys. Rev. Lett. 104, 228101 (2010). [2] A. K. Buell, A. Dhulesia, D. A. White, T. P. J. Knowles, C. M. Dobson, and M. E. Welland, Angew. Chem. Int. Ed Engl. 51, 5247 (2012). [3] T. O. Mason, T. C. T. Michaels, A. Levin, E. Gazit, C. M. Dobson, A. K. Buell, and T. P. J. Knowles, J. Am. Chem. Soc. 138, 9589 (2016). [4] T. O. Mason, A. Levin, C. M. Dobson, E. Gazit, T. P.J. Knowles and A. K. Buell, JACS under revision, (n.d.). [5] A. K. Buell, C. Galvagnion, R. Gaspar, E. Sparr, M. Vendruscolo, T. P. J. Knowles, S. Linse, and C. M. Dobson, Proc. Natl. Acad. Sci. 111, 7671 (2014). [6] R. Gaspar, G. Meisl, A. K. Buell, L. Young, C. F. Kaminski, T. P. J. Knowles, E. Sparr, and S. Linse, Q. Rev. Biophys. 50, (2017). [7] A. Šariæ, A. K. Buell, G. Meisl, T. C. T. Michaels, C. M. Dobson, S. Linse, T. P. J. Knowles, and D. Frenkel, Nat. Phys. 12, 874 (2016).

• Thematic program: Models in Biology and Medicine (2016/2017)
• Event: Workshop on "Protein Aggregation: Biophysics and Mathematics" (2017)

We present a new model for multi-agent dynamics where each agent is described by its position and body attitude: agents travel at a constant speed in a given direction and their body can rotate around it adopting different configurations. Agents try to coordinate their body attitudes with the ones of their neighbours. This model is inspired by the Vicsek model. The goal of this talk will be to present this new flocking model, its relevance and the derivation of the macroscopic equations from the particle dynamics. In collaboration with Pierre Degond (Imperial College London) and Amic Frouvelle (Université Paris Dauphine).

• Thematic program: Models in Biology and Medicine (2016/2017)
• Event: Workshop on "Protein Aggregation: Biophysics and Mathematics" (2017)

Davy Martin1, Joan Torrent i Mas1, Stéphanie Prigent1, Mathieu Mezache2, Marie Doumic-Jauffret2, Vincent Béringue1 and Human Rezaei1* 1. National Institute for Agricultural Research (INRA), Pathological Macro-assemblies and Prion Pathology group (MAP2), UR892, Virologie Immunologie Moléculaires, Jouy-en-Josas, 78350-F, France 2. Sorbonne Universités, Inria, UPMC Univ Paris 06, Lab. J.L. Lions UMR CNRS 7598, Paris, France The prion phenomenon is based on autonomous structural information propagation towards single or multiple protein conformational changes. Since this last decade the prion concept referring to the transmission of structural information has been extended to several regulation systems and pathologies including Alzheimer and Parkinson’s diseases. The unified theory in Prion replication implies structural information transference (SIT) from the prion to a non-prion conformer through a mechanism also called improperly, with regards to biophysical considerations “seeding” phenomenon. Therefore considering prion replication as a structural information transduction from a donor (i.e. template) to an acceptor (i.e. substrate) through a transduction interface a new questioning arises: what are molecular mechanisms of the auto-perpetuation of the Prion structural information and its faithfulness? Considering the Prion propagation as more or less faithful perpetuation of structural information, in the present work, we explored the concept of prion quasi-species (i.e. existence of prion heterogeneous assemblies) and highlighted the existence of prion network, which has an autopoietic behaviour (autoreplicative). Our observations strongly suggest that specific criteria in term of: protein structure, delayprocess and thermo-kinetics should be collated before a system become dissipative and autopoietic.

• Thematic program: Models in Biology and Medicine (2016/2017)
• Event: Workshop on "Protein Aggregation: Biophysics and Mathematics" (2017)

We study travelling wave solutions of a Korteweg-de Vries-Burgers equation with a non-local diffusion term. This model equation arises in the analysis of a shallow water flow by performing formal asymptotic expansions associated to the triple-deck regularisation (which is an extension of classical boundary layer theory). The resulting non-local operator is of fractional differential type with order between 1 and 2. Travelling wave solutions are typically analysed in relation to shock formation in the full shallow water problem. We show rigorously the existence of these waves in the case of a quadratic nonlinearity. The travelling wave problem for the classical KdV-Burgers equation is usually analysed via a phase-plane analysis, which is not applicable here due to the presence of the non-local diffusion operator. Instead, we apply fractional calculus results available in the literature and a Lyapunov functional. In addition we discuss the monotonicity of the waves in terms of a control parameter and prove their dynamic stability in case they are monotone. We also discuss some partial results concerning the existence of travelling waves in the case of a cubic nonlinearity. This existence problem and the monotonicity of the waves in the quadratic case for a small dispersion term in relation with the diffusive one are still open problems, for this reason we have also developed numerical schemes in order to support our conjectures. We will discuss in a second part of the talk, a pseudo-spectral method that approximates the initial value problem. The basic idea is, using an algebraic map, to transform the whole real line into a bounded interval where we can apply a Fourier expansion. Special attention is given to the correct computation of the fractional derivative in this setting. Interestingly, there is a connection of the mapping method to fractional calculus, that we will also mention.

• Thematic program: Models in Biology and Medicine (2016/2017)