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A bi monomeric nonlinear Becker Do ring type system to capture. oscillatory aggregation kinetics in prion dynamics. Marie Doumic Klemens Fellner Mathieu Mezache Human Rezaei. August 29 2018, In this article in order to understand the appearance of oscillations observed in protein. aggregation experiments we propose motivate and analyse mathematically the differential. system describing the kinetics of the following reactions. W Ci Ci 1 1 i n, Ci V Ci 1 2V 2 i n, with n finite or infinite This system may be viewed as a variant of the seminal Becker Do ring. system and is capable of displaying sustained though damped oscillations. Keywords Protein polymerisation Prion modelling Becker Do ring system Lotka Volterra. system Lyapunov functional stability analysis oscillations asymptotic expansion. Mathematical Subject Classification 34E05 34D08 37L15 92B05. 1 Introduction, The aim of this article is to propose and study a new polymerisation depolymerisation model. capable of explaining oscillations which have been observed experimentally in the time course of. Sorbonne Universite s Inria Universite Paris Diderot CNRS Laboratoire Jacques Louis Lions F 75005 Paris. France marie doumic inria fr Wolfgang Pauli Institute c o university of Vienna Austria. University of Graz Austria Institute of Mathematics and Scientific Computing 8010 Graz. klemens fellner uni graz at, Sorbonne Universite s Inria Universite Paris Diderot CNRS Laboratoire Jacques Louis Lions F 75005 Paris. France mathieu mezache inria fr, INRA UR892 Virologie Immunologie Mole culaires 78350 Jouy en Josas France human rezaei inra fr.
prion protein polymerisation experiments Up to our knowledge such oscillations have never been. observed neither theoretically or numerically in the family of growth fragmentation nucleation. equations which are most often used to model protein polymerisation. Biological background and motivation, In its largest acceptance the prion phenomenon prion being derived from proteinaceous infec. tious only particle involves the self propagation of a biological information through structural. information transfer from a protein in a prion state i e misfolded resp infectious to the same. protein in a non prion state Such a concept is key to the regulation of diverse physiological. systems and to the pathogenesis of prion diseases 9 24 26 Recently prion like mechanisms. have been involved in the propagation and gain of toxic functions of proteins or peptides as. sociated with other neurodegenerative disorders such as Alzheimer Parkinson and Huntington. diseases 17 Elucidating the mechanisms driving prion like aggregation is thus of key impor. tance and as explained below still requires new mathematical modelling and analysis. During the evolution of prion pathology the host encoded monomeric prion protein PrPC. is converted into misfolded aggregating conformers PrPSc 6 PrPSc assemblies have the abil. ity to self replicate and self organise in the brain through a still unresolved molecular mechanism. commonly called templating Differences in disease phenotypes distinctive symptomologies in. cubation times and infectious characters of PrPSc are reported within the same host species. These phenotypic differences are assigned to structural differences in PrPSc assemblies intro. ducing the concept of prion strains based on structural diversity heterogeneity of PrPSc as. semblies In the prion literature a plethora of evidences strongly suggest that within a given. prion strain a PrPSc structural heterogeneity exists which suggests that in a given environ. ment structurally different PrPSc subassemblies with different biological and physico chemical. properties coexist 18 even if the mechanism of this diversification remains elusive To date. very few mathematical models have taken into account the coexistence of multiple prion assem. blies or multiple type of fibrils 10 Indeed most of the aggregation models have been built. using the canonical nucleation elongation fragmentation process seminally reported by Bishop. and Ferrone see e g 5 19 23 which is based on the existence of a structurally unique type of. assemblies characterised only by their size distribution The characterisation of multiple types of. PrPSc subassemblies with different rates of aggregation depolymerisation and exchange requires. new mathematical models and analyses to describe the dynamics and relation between different. subspecies, In order to explore the consequence of the coexistence of structurally different PrPSc assem. blies within the same environment the depolymerisation kinetics of recombinant PrP amyloid. fibrils have been explored by Static Light Scattering SLS 20 A detailed study of those ex. periments revealed a surprising transient oscillatory phenomenon as the time evolution of the. SLS measurement see Appendix D for details shows in Figure 1 First note that when denoting. by ci t the concentrations at time t of the polymers containing i monomers we can interpret. the signal of an experimental SLS measurement as in 23 as the time evolution of the second. moment of the polymers i e, M2 t i2 ci t 1, 0 72 0 725. 0 67 0 705, 15 3 15 35 15 4 15 45 15 5, 3 4 1 4 15 4 2 4 25 4 3. Scattered intensity, 0 65 0 705, 4 4 2 4 4 4 6 4 8 5 15 15 2 15 4 15 6 15 8 16.
0 2 4 6 8 10 12 14 16, Time hours, Figure 1 Human PrP amyloid fibrils Hu fibrils depolymerisation monitored by Static Light. Scattering see Appendix D for details A The overall view of the 0 35 M Hu fibrils depolymeri. sation at 550 C B E correspond to a zoom in on different time segments of the depolymerisation. curve A As shown in B from time 4 to time 5h oscillations have been observed when for time. segment corresponding to time 15 3 to 15 5h only noise has been detected D. Hence at the beginning of the experiments after a short lag phase quick depolymerisation. is observed This is followed by a transient phase ranging from approximately 1h to 11 hours. when slow variations were superimposed by fast periodic oscillations with a frequency around. 0 01 to 0 02 Hz see Figures 1B and 1C Both the variations and the oscillations progressively. disappear and a constant signal with noise is observed at the end of the experiments Figure 1 D. and E This specific phenomenon may be used to gain new insight into the underlying biological. A first key question of our study is thus the following What kind of core elements should a. model feature in order to explain the appearance of such oscillations. The most natural departure point in the formulation of a suitable mathematical model is the. Becker Do ring model of polymerisation and depolymerisation 4 The Becker Do ring model is. coherent with other biological measurements 20 and it is viewed in the protein polymerisation. literature as the primary pathway model 5 23, Becker Do ring considers two reverse reactions polymerisation through monomer addition. and depolymerisation due to monomer loss Accordingly the model is characterised by the. following system of reactions where Ci denotes polymers containing i monomers so that C1 are. the monomers and ai bi are the polymerisation resp depolymerisation reaction rate coefficients. C1 Ci Ci 1 i 1, Ci 1 C1 i 2, The Becker Do ring system however satisfies the detailed balance condition 3 which implies. the existence of a Lyapunov functional and no sustained oscillations are possible Also damped. oscillations up to the best of our knowledge have never been observed numerically or evidenced. analytically We thus needed a variant of the Becker Do ring model to explain the experimentally. observed oscillations displayed in Figure 1, In 16 it was recently shown that PrPSc assemblies are in equilibrium with an oligomeric. conformer suPrP encoding the entire strain information and constituting an elementary build. ing block of PrPSc assemblies The fact that such an oligomeric building block appears separately. from the monomeric PrP points towards models with two different quasi monomeric species i e. one monomer and one oligomeric conformer in contrast to the polymer species Ci each of which. playing a role in a different reaction A suitable mathematical model should also to take into ac. count the constraint that large polymers cannot interact directly for reasons of size and order of. magnitude of their concentrations Hence we assume that polymers can only interact indirectly. through the exchange of monomers or small oligomeric conformers. A third crucial modelling aspect concerns the details of the depolymerisation reaction rates. which are linear in the original Becker Do ring system However numerical studies see below. for a more detailed discussion and numerical illustrations as well as the content of this paper. strongly suggests that sustained or damped oscillations require a nonlinear more precisely a. monomer induced depolymerisation process which we detail in the following Section. 2 Introduction of the proposed model system, We propose the following model system Let V and W denote the two monomeric species.
where the second conformer species is taken monomeric for the sake of simplicity but a slight. modification of the model would allow to consider it as oligomeric Let Ci be the polymers. containing i monomers where polymerisation signifies the amendment of a monomer W while. depolymerisation only occurs when induced via the monomeric species V More precisely we. W Ci Ci 1 1 i n 2, Ci V Ci 1 2V 2 i n, with a reaction rate constant k for the monomer conformer dynamics and polymerisa. tion depolymerisation coefficients ai and bi Note that large values for k compared to ai bi. introduce a slow fast behaviour into 2 and yields a mechanism of oscillations which is detailed. in a fully rigorous way for a two polymer system i e n 2 in Section 3. We emphasise the two main differences of 2 compared to the classical Becker Do ring system. First instead of one monomeric species C1 we now consider two interacting species of monomers. or conformers V and W Secondly depolymerisation is modelled as a monomer induced. nonlinear process which requires the catalytic action of V Note that this process is reminiscent. of the cyclical behaviour of the three species system. which is known to produce sustained periodic oscillations see 28 where it is called the Ivanova. system or 27 where it is referred to as a simplification of the Belousov Zhabotinsky system. To reiterate and further illustrate the reasons which guided us towards model 2 let us. isolate those two main ingredients Firstly let us modify the Becker Do ring system by taking. two monomeric species 16 but with a standard linear depolymerisation reaction i e we consider. the following system, W Ci Ci 1 1 i n 4, Ci Ci 1 V 2 i n. Figure 2 compares the behaviours of the bi monomeric Becker Do ring system 4 to model 2. under conditions when both feature oscillations which is systematic in the nonlinear depoly. merisation model 2 yet only occurs for some parameters in the bi monomeric Becker Do ring. system 4 Nevertheless even if the bi monomeric Becker Do ring system 4 shows oscillatory. behaviour those oscillations are far less sustained and cannot serve as an explanation of the. experimental observations, 0 5 10 15 20 0 20 40 60 80 100. Figure 2 Left images Comparison of the oscillatory behaviour of the monomer concentration v. of the proposed model 2 blue with the bi monomeric Becker Do ring system 4 with linear. depolymerisation red subject to the same initial distribution Right image. Interestingly nonlinear depolymerisation leads not only to much more sustained oscillations. but also yields faster convergence to its size distribution equilibrium data not shown while. the linear bi monomeric Becker Do ring system 4 exhibits similar metastability as observed for. the Becker Do ring system 21, Secondly when considering a monomeric Becker Do ring system with second order depoly. merisation reaction, V Ci Ci 1 1 i n Ci V, Ci 1 2V 2 i n 5.
numerical simulations do not display any kind of oscillations see second row in Figure 3. Figure 3 Numerical results corresponding to SLS measurement i e the quantity M2 defined. by 1 Left Column and the evolution of the size distribution of polymers Right Column. First Row The here proposed model 2 with parameters n 50 k 9 5 ai 4 8 bi 8. Second Row The model 5 with c1 multiplied by 10 in order to ignite the reactions in the. system Third row The model 4 with parameters n 50 k 0 95 ai 0 48 bi 0 8. Let us also remark that in model 2 the first polymer species C1 could also denote a smallest. polymer of size n0 1 i e it represents the smallest active polymer size This means that no. nucleation as modelled by C1 C1 C2 in the Becker Do ring system is considered This is in. line with the time scale of the considered experiment where nucleation is negligible compared. with other reactions, Finally the original Becker Do ring system for n allows to model phase transitions. where polymers of infinite size are created in finite time depending on the polymerisation co. efficients a phenomenon called gelation or also Ostwald ripening 3 In this paper we shall. consider both finite or infinite systems and discuss similarities and differences However in view. of our application background of understanding amyloid fibrils we are never interested in the. appearance of gelation or Ostwald ripening and only consider polymerisation coefficients where. the average size of polymers though possibly large remains finite. A bi monomeric nonlinear Becker Doring type system to capture oscillatory aggregation kinetics in prion dynamics Marie Doumic Klemens Fellnery Mathieu Mezachez Human Rezaeix August 29 2018 Abstract In this article in order to understand the appearance of oscillations observed in protein

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