Viral infection and immune response Work in collaboration with Anna Marciniak-Czochra. Already submitted.

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The role of Ca2+ in spine twitching and synaptic plasticity

One of the main questions addressed by Holcman and his colaborators in [33] and [34] was to understand the spine twitching and the synaptic plasticity in terms of the binding reactions between the Ca2+ ions and some proteins inside the spine. They attributed the twitching motion of the spine to the contraction of actin-myosin AM proteins in the following way. They considered that once an AM protein has four Ca2+ ions bound there occurs a local contraction of the AM, and that all local contractions at a given time produces a global contraction of the spine, which has two consequences: first, it is responsible for the rapid twitching motion of the spine, and second, it produces a hydrodynamical movement of the cytoplasmic fluid in the direction of the dendrite. This motion is thus responsible of the transport of the ions, not only Ca2+ but also Na+, into the neuron, which constitutes the electric potential we mentioned in Section 1.1.1. Holcman et al [33] also mention that a protein with four Ca2+ ions bound contracts at a fixed rate until one Ca2+ ion unbinds. The contraction due to Ca2+ binding also appears in the works of Farah et al [19], Klee et al [36] and Shiftman et al [59], but in all three cases the proteins suffer a conformational change each time a Ca2+ ion binds, and not only when they have four Ca2+ ions. We will consider this experimental evidence for the new model we will propose.
The synaptic plasticity is defined as changes in the synaptic strength, i.e. in the intensity of the signal transmission between two neurons. These changes can be short-term if they occur in the range of milliseconds or minutes, or long-term if their duration is measured in hours, days, weeks or longer. The long-lasting changes in synapses are related to cognitive processes like learning and memory. These changes are divided in two: Long-Term Potentiation (LTP), if there is an increase in the synaptic strength, or Long-Term Depression (LTD), if there is a decrease in the synaptic strength. The major determinant of whether LTP or LTD appears seems to be the amount of Ca2+ in the post-synaptic cell: small rises in Ca2+ lead to depression, whereas large increases trigger potentiation (see Purves et al [54], Chapter 24, pp. 575-610).

Modeling the twitching motion of the spine

In Section 1.1.2 we mentioned that each time a Ca2+ ion binds to a protein this latter suffers a contraction, and that the addition of all these local contractions have two effects: the twitching of the spine and changes in the cytoplasmic flow field V (x, t). In order to take into account both effects we will assume that the spine movement depends on the cytoplasmic velocity at the spine surface 􀀀, and that this value depends on the total number of Ca2+ ions that are bound to the proteins. More precisely, if we define λ(t) := Z  W(x, t) d , (1.12). which is the total number of occupied binding sites at time t, then our assumption is that the spine surface 􀀀 moves with velocity V · n proportional to λ(t).

On the diffusion coefficient

If we wish to take into account the existence of obstacles inside the spine, like organelles or macromolecules, we can add them in two forms: either as “exterior domains”, i.e. we take out a tiny section from the domain and suppose that the boundary of the section belongs to the boundary of , or either by considering that the diffusion coefficients are no longer constant.
The results we presented here are still valid in both situations provided that the exterior domains have C1 boundaries, D, d ∈ C1(¯ × [0, T]) and 0 < D1 ≤ D(x, t) ≤ D2 , 0 < d1 ≤ d(x, t) ≤ d2 .

On the reactions between calcium and the proteins

As in Holcman and Schuss [34], we assumed in the model that all binding sites have the same affinity for the Ca2+ ions, but this is not the real case. Indeed, calcineurin has one binding site with high affinity and three with low affinity (see Klee et al [36]), AM-type proteins like Troponin have two low affinity sites and two high affinity sites (see Farah et al [19]), and CaM with two Ca2+ ions bound has more affinity to bind calcium than CaM with no Ca2+ ions bound (see Shiftman et al [59]). Nevertheless, such distinctions were not considered here in order to keep things as simple as possible.

Spatial effects of viral infection and immunity response

When a virion, i.e., an individual viral particle, enters a healthy cell, it modifies the genetic structure of its host. After infection, the altered biochemical machinery of the host starts to create new virions. The virions are then released from the host cell and may infect other cells. However, the infected cell activates intrinsic host defenses, which include, among others, activation of the innate immunity system and release of biomolecules called interferons (IFN), which communicate with the other cells and induce them to deploy protective defenses. The dynamics of such complex virus-host system results from the intra- and extracellular interactions between invading virus particles and cells producing substances, which confer resistance to virus. These key processes have been addressed by the mathematical model proposed by Getto, Kimmel and Marciniak-Czochra in [24]. The model was motivated by the experiments involving vesicular stomatitis virus (VSV) [16, 42] and respiratory syncytial virus (RSV). The work of P.
Getto et al [24] is focused on the study of the role of heterogeneity of intracellular processes, reflected by a structure variable, in the dynamics of the system and stability of stationary states. It is shown that indeed the heterogeneity of the dynamics of cells in respect to the age of the individual cell infection may lead to the significant changes in the behavior of the model solutions, exhibiting either stabilizing or destabilizing effects.
Another interesting aspect of the dynamics of the spread of viral infection and development of resistance is related to the spatial structure of the system and the effects of spatial processes, such as random dispersal of virions and interferon particles. In a series of experiments on the vesicular stomatitis virus infection [16, 42], it was observed that the spatial structure of the system may influence the dynamics of the whole cell population. The role of spatial dimension and diffusion transport of virions and interferon molecules were experimentally studied using two type of experiments: a one-step growth experiment in which all cells were infected simultaneously, and a focal-infection experiment in which cell population was infected by a point source of virus. A spreading cicular wave of infection followed by a wave of dead cells was observed. The experiments were performed on two different cell cultures: DBT (murine delayed brain tumor) cells, which respond to IFN and can be activated to resist the replication of viruses, and BHK (baby hamster kidney) cells, which are not known to produce or respond to IFN. In case of focal infection  both in DBT and BHK populations spread of infection (rings) was observed. The size of the rings was dependent on the type of the virus (N1, N2, N3, N4 -gene ectopic strains as well as M51R mutant and XK3.1). However, for all virus types, it was observed that in DBT cells the speed of the infection propagation was decreasing with time, while in case of BHK the radius of the infected area was growing linearly in time. Results of the experiments showed that  the rate of infectious progeny production in one-step growth experiments was a key determinant of the rate of focal spread under the absence of IFN production. Interestingly, the correlation between one-step growth and focal growth did not apply for VSV strains XK3.1 and M51R in the cells producing IFN. Focal infection in DBT cells led only to the limited infection, the spread of which stopped after a while.

Table of contents :

Introduction
I Reaction-diffusion systems and modelling 
1 Calcium ions in dendritic spines Work in collaboration with Kamel Hamdache. Published in Nonlinear Analysis: Real World Applications, Volume 10, Issue 4, August 2009. 
1.1 Introduction
1.1.1 Dendritic spines
1.1.2 The role of Ca2+ in spine twitching and synaptic plasticity
1.1.3 The original model
1.2 The modified model and main results
1.2.1 New variables
1.2.2 Modeling the twitching motion of the spine
1.2.3 The modified model
1.2.4 Main results
1.3 Proof of Theorem 1.1
1.3.1 A priori estimates
1.3.2 The Fixed Point operator
1.3.3 Conclusion of the proof
1.4 Proof of Theorem 1.2
1.4.1 A priori estimates
1.4.2 The Fixed Point operator
1.4.3 Conclusion of the proof
1.5 Proof of Theorems 1.3 and 1.4
1.5.1 Proof of Theorem 1.3
1.5.2 Proof of Theorem 1.4
1.6 Final remarks
1.6.1 On the cytoplasmic flux
1.6.2 On the diffusion coefficient
1.6.3 On the reactions between calcium and the proteins
1.7 Discussion
2 Viral infection and immune response Work in collaboration with Anna Marciniak-Czochra. Already submitted. 
2.1 Introduction
2.1.1 Spatial effects of viral infection and immunity response
2.1.2 The original model
2.1.3 Biological hypotheses of the new model
2.1.4 The reaction-diffusion (RD) model
2.1.5 The hybrid model
2.2 Main results
2.2.1 Existence and uniqueness results
2.2.2 Asymptotic results for the RD system
2.2.3 Asymptotic results for the hybrid system
2.3 Numerical simulations
2.4 The fixed point operator and a priori estimates
2.4.1 Construction of the fixed point operator R
2.4.2 Positivity of solutions
2.4.3 A priori estimates
2.4.4 Continuity of the operator R
2.5 Proof of the theorems
2.5.1 Proof of Theorem 2.1
2.5.2 Proof of Theorem 2.2
2.5.3 Proof of Theorem 2.3
2.5.4 Proof of Theorem 2.4
2.5.5 Proof of Theorem 2.5
2.5.6 Proof of Theorem 2.6
2.6 Discussion
II Reaction-diffusion equations and systems on manifolds
3 The effect of growth on pattern formation 
3.1 Introduction
3.2 Main results
3.2.1 Reaction-diffusion systems on growing manifolds
3.2.2 Properties of solutions: existence and uniqueness
3.2.3 The anti-blow-up effect of growth
3.2.4 The stabilising effect of growth
3.3 Proof of Theorem 3.1
3.3.1 Parametrisation and Riemannian metric
3.3.2 The general model with growth and curvature
3.3.3 The isotropic growth model
3.4 Proof of Theorem 3.2
3.5 Proof of Theorem 3.3
3.6 Proof of Theorem 3.4
3.7 Proof of Theorem 3.5
3.8 Discussion
4 Generalised travelling waves on manifolds 
4.1 Definition of general travelling waves on manifolds
4.1.1 Complete Riemannian manifolds
4.1.2 Reaction-diffusion equations on manifolds
4.1.3 Fronts, waves and invasions
4.2 Properties of fronts on manifolds
4.3 Proofs
4.3.1 Proof of Theorem 4.1
4.3.2 Proof of Theorem 4.2
4.3.3 Proof of Theorem 4.3
5 Travelling waves on the real line 
5.1 Introduction
5.1.1 Calcium waves and fertilized eggs
5.1.2 The reaction-diffusion model on the sphere
5.1.3 Murray’s approach
5.1.4 Modified equation
5.1.5 On intuition and the real dynamics of the travelling waves
5.2 Main results
5.3 Proofs
5.3.1 Supersolutions and subsolutions
5.3.2 Global solution for N
5.3.3 Global solution for S
5.3.4 Steady-state solutions and blocking of the waves
5.4 Discussion
6 Travelling waves on the sphere Work in collaboration with Henri Berestycki and Fran¸cois Hamel. To be submitted. 
6.1 Elliptic equation on the truncated sphere
6.1.1 Trivial solutions
6.1.2 Non-trivial solution: variational approach
6.1.3 Pair of non-trivial solutions: topological approach
6.2 Reaction-diffusion equations on the truncated sphere
6.3 Elliptic equation on the N-sphere
6.3.1 Trivial solutions
6.3.2 Stability of solutions
6.3.3 Non-trivial solutions
6.4 Reaction-diffusion equation on the N-sphere
6.4.1 Bistable nonlinearity
6.4.2 Monostable nonlinearity
6.5 Discussion
III Elliptic equations and nonlinear eigenvalues on the sphere 
7 Bifurcation and multiple periodic solutions on the sphere Work in collaboration with Henri Berestycki and Fran¸cois Hamel. To be submitted. 
7.1 Bifurcation on S1
7.1.1 Properties of solutions
7.1.2 Proof of Conjecture 6.12 for S1
7.1.3 Bifurcation analysis
7.2 Bifurcation on SN
7.2.1 Eigenvalues and eigenvectors
7.2.2 Groups, actions and equivariance
7.2.3 Symmetries and reduction to an ODE
7.2.4 Existence and uniqueness of axis-symmetric solutions
7.2.5 Bifurcation analysis of axis-symmetric solutions
7.3 Discussion
IV Perspectives 
8 Conclusions 
8.1 Reaction-difusion systems and modelling
8.1.1 Calcium dynamics in neurons
8.1.2 Virus infection and immune response
8.2 Reaction-diffusion equations and systems on manifolds
8.2.1 The effect of growth on pattern formation
8.2.2 Generalised travelling waves on manifolds
8.2.3 Travelling waves on the real line
8.2.4 Travelling waves on the sphere
8.3 Elliptic equations and nonlinear eigenvalues on the sphere
8.3.1 Bifurcation and multiple periodic solutions on the sphere
References

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