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A free boundary model for transport induced neurite growth

Greta Marino, Jan-Frederik Pietschmann, Max Winkler

TL;DR

A free boundary model consisting of systems of drift-diffusion equations describing the evolution of the density of antero- and retrograde vesicles in each neurite coupled to reservoirs located at the soma and the growth cones of the neurites, respectively is introduced.

Abstract

We introduce a free boundary model to example the effect of vesicle transport onto neurite growth. It consists of systems of drift-diffusion equations describing the evolution of the density of antero- and retrograde vesicles in each neurite coupled to reservoirs located at the soma and the growth cones of the neurites, respectively. The model allows for a change of neurite length depending on the vesicle concentration in the growth cones. After establishing existence and uniqueness for the time-dependent problem, we briefly comment on possible types of stationary solutions. Finally, we provide numerical studies on biologically relevant scales using a finite volume scheme. We illustrate the capability of the model to reproduce cycles of extension and retraction.

A free boundary model for transport induced neurite growth

TL;DR

A free boundary model consisting of systems of drift-diffusion equations describing the evolution of the density of antero- and retrograde vesicles in each neurite coupled to reservoirs located at the soma and the growth cones of the neurites, respectively is introduced.

Abstract

We introduce a free boundary model to example the effect of vesicle transport onto neurite growth. It consists of systems of drift-diffusion equations describing the evolution of the density of antero- and retrograde vesicles in each neurite coupled to reservoirs located at the soma and the growth cones of the neurites, respectively. The model allows for a change of neurite length depending on the vesicle concentration in the growth cones. After establishing existence and uniqueness for the time-dependent problem, we briefly comment on possible types of stationary solutions. Finally, we provide numerical studies on biologically relevant scales using a finite volume scheme. We illustrate the capability of the model to reproduce cycles of extension and retraction.
Paper Structure (18 sections, 8 theorems, 101 equations, 6 figures)

This paper contains 18 sections, 8 theorems, 101 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Relation to existing work
  3. Contribution and outline
  4. Mathematical model
  5. Existence of weak solutions
  6. Preliminaries
  7. Transformation for a fixed reference domain
  8. Notion of weak solution and existence result
  9. Proof of Theorem \ref{['thm:existence']}
  10. Stationary solutions
  11. Constant stationary solutions
  12. Finite volume scheme and scaling
  13. Finite volume scheme
  14. Non-dimensionalisation of the model
  15. Numerical studies
  16. ...and 3 more sections

Key Result

Theorem 3.3

Let the assumptions (H$_0$)--(H$_6$) hold. Then, for every $T>0$ there exists a unique weak solution $(\boldsymbol{f}_1, \boldsymbol{f}_2, \Lambda_{\emph{som}}, \Lambda_1, \Lambda_2, L_1, L_2)$ to eq:discr1--ode_transformed, free-boundary in the sense of Definition def-1.

Figures (6)

  • Figure 1: Sketch of a developing neuron. Here a) represents the cell nucleus/soma where vesicles are produced, b) a neurite and c) a growth cone, i.e., the location where vesicles are inserted/removed into the cell membrane.
  • Figure 1: Sketch of the model neuron: it consists of two neurites modelled by two intervals $(0, L_1(t))$ and $(0, L_2(t))$. The squares correspond to pools where vesicles can be stored. More precisely, the pool in the middle corresponds to the soma while the others stand for the corresponding growth cones. The interaction between neurites and pools is realised via boundary fluxes and the parameters governing their respective strength are displayed along with arrows of the transport direction. For an easy visualisation, $(0, L_1(t))$ is illustrated as a mirrored copy of $(0, L_2(t))$.
  • Figure 1: The vesicle densities $f_{\pm, j}$, $j=1,2$, and pool capacities $\Lambda_k$, $k\in\{\text{som},1,2\}$, for the example from Section \ref{['sec:experiment1']} plotted at different time points.
  • Figure 2: The neurite lengths $L_j$, $j=1,2$, and pool capacities $\Lambda_k$, $k\in\{\text{som},1,2\}$, for the example from Section \ref{['sec:experiment1']} plotted over time.
  • Figure 3: The vesicle densities $f_{\pm,j}$, $j=1,2$, and pool capacities $\Lambda_k$, $k\in\{\text{som},1,2\}$, for the example from Section \ref{['sec:experiment2']} plotted at different time points.
  • ...and 1 more figures

Theorems & Definitions (18)

  • Remark 2.1
  • Definition 3.1
  • Remark 3.2: Interpretation of the assumptions
  • Theorem 3.3
  • Lemma 3.4
  • Proof 1
  • Lemma 3.5
  • Proof 2
  • Lemma 3.6
  • Proof 3
  • ...and 8 more