A Stochastic Framework for Neuronal Morphological Comparison: Application to the Study of <Emphasis Type="Italic">imp</Emphasis> Knockdown Effects in Drosophila Gamma Neurons

A. Razetti; X. Descombes; C. Medioni; F. Besse

doi:10.1007/978-3-319-54717-6_9

International Joint Conference on Biomedical Engineering Systems and Technologies

BIOSTEC 2016: Biomedical Engineering Systems and Technologies pp 145-166 | Cite as

A Stochastic Framework for Neuronal Morphological Comparison: Application to the Study of imp Knockdown Effects in Drosophila Gamma Neurons

Authors
Authors and affiliations

A. Razetti
X. Descombes
C. Medioni
F. Besse

Conference paper

First Online: 04 March 2017

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Part of the Communications in Computer and Information Science book series (CCIS, volume 690)

Abstract

In order to reach their final adult morphology, Gamma neurons in Drosophila brain undergo a process of pruning followed by regrowth of their main axons and branches called remodelling. The mRNA binding protein Imp was identified to play a fundamental role in this process. One of Imp targets, profilin mRNA, encodes for an actin regulator that has been shown to be involved in axon remodelling. In this paper we intend to further understand the role of Imp and the importance of profilin mRNA expression regulation during remodelling. To do so, we propose a stochastic framework to exhaustively compare the adult morphology between wild type (WT), imp knockdown (Imp) and imp knockdown rescued by Profilin (Prof Rescue) neurons. Our framework consists in (i) the selection of the main neuron morphological features, (ii) their stochastic modelling and parameter estimation from data and (iii) a maximum likelihood analysis for each individual neuron to quantitatively assess the similarity or difference between groups. Thanks to this framework we show that imp mutant neurons can be divided in two phenotypical groups with a different aberrancy degree, and that profilin overexpression partially rescues the main axon and branch development thereby it reduces the proportion of neurons with the strongest remodelling phenotype.

Keywords

Gamma neurons Remodelling Stochastic models Maximum likelihood analysis

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Notes

Acknowledgements

This work was supported by the French Government (National Research Agency, ANR) through the « Investments for the Future » LABEX SIGNALIFE: program reference # ANR-11-LABX-0028-01.

All the authors are within Morpheme (a joint team between Inria CRI-SAM, I3S and IBV).

Appendix

From Eq. 6 we can express $ \upmu_{k} \left( {A + 1,p} \right) $ as

$$ \upmu_{k} \left( {A + 1,p} \right) = \sum\nolimits_{k} k \left( {\begin{array}{*{20}c} {k - 1} \\ {A + 1} \\ \end{array} } \right)p^{A + 2} \left( {1 - p} \right)^{k - A - 2} . $$

(14)

Using

$$ \left( {\begin{array}{*{20}c} {k - 1} \\ A \\ \end{array} } \right) + \left( {\begin{array}{*{20}c} {k - 1} \\ {A + 1} \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} k \\ {A + 1} \\ \end{array} } \right), $$

(15)

Equation 14 can be rewritten as

$$ \begin{aligned} &\upmu_{k} \left( {A + 1,p} \right) = \sum\nolimits_{k} k \left( {\begin{array}{*{20}c} k \\ {A + 1} \\ \end{array} } \right)p^{A + 2} \left( {1 - p} \right)^{k - A - 2} \\ &\qquad\quad\quad\quad\quad - \mathop \sum \nolimits_{k}^{{}} k\left( {\begin{array}{*{20}c} {k - 1} \\ A \\ \end{array} } \right)p^{A + 2} \left( {1 - p} \right)^{k - A - 2} . \\ \end{aligned} $$

(16)

Taking out $ \frac{p}{1 - p} $ as a common factor from the second sum in Eq. (16), we obtain

$$ \begin{aligned} & \mathop { \sum }\nolimits_{k}^{{}} k\left( {\begin{array}{*{20}c} {k - 1} \\ A \\ \end{array} } \right)p^{A + 2} \left( {1 - p} \right)^{k - A - 2} \\ & = \frac{p}{1 - p}\mathop \sum \nolimits_{k}^{{}} k\left( {\begin{array}{*{20}c} {k - 1} \\ A \\ \end{array} } \right)p^{A + 1} \left( {1 - p} \right)^{k - A - 1} = \frac{p}{1 - p}\upmu_{k} \left( {A,p} \right) \\ \end{aligned} $$

(17)

Similarly, the first sum in (16) can be worked out to obtain

$$ \upmu_{k} \left( {A + 1,p} \right) = \frac{1}{1 - p}\upmu_{k} \left( {A + 1,p} \right) - \frac{1}{1 - p}. $$

(18)

From (17) and (18) we finally obtain

$$ \upmu_{k} \left( {A,p} \right) = \frac{A}{p}. $$

(19)

From Eqs. (7) and (19) we can express

$$ \upsigma^{2}_{\text{k }} (A,p) = \mathop \sum \nolimits_{k}^{{}} k^{2} \left( {\begin{array}{*{20}c} {k - 1} \\ A \\ \end{array} } \right)p^{A + 1} \left( {1 - p} \right)^{k - A - 1} - \frac{1}{{p^{2} }}A^{2} , $$

(20)

from where

$$ \upsigma^{2}_{{{\text{k}} }} \left( {A + 1,p} \right) = \mathop \sum \nolimits_{k}^{{}} k^{2} \left( {\begin{array}{*{20}c} {k - 1} \\ {A + 1} \\ \end{array} } \right)p^{A + 2} \left( {1 - p} \right)^{k - A - 2} - \frac{1}{{p^{2} }}\left( {A + 1} \right)^{2} . $$

(21)

Equation (21) can be rewritten using Eq. (15) to get $ \upsigma^{2}_{k} \left( {A + 1,p} \right): $

$$ \begin{aligned} &\upsigma^{2}_{{{\text{k }}^{0} }} \left( {A + 1,p} \right) = \mathop \sum \nolimits_{k}^{{}} k^{2} \left( {\begin{array}{*{20}c} k \\ {A + 1} \\ \end{array} } \right)p^{A + 2} \left( {1 - p} \right)^{k - A - 2} \\ & \qquad\qquad\qquad - \mathop \sum \nolimits_{k} k^{2} \left( {\begin{array}{*{20}c} {k - 1} \\ A \\ \end{array} } \right)p^{A + 2} \left( {1 - p} \right)^{k - A - 2} - \frac{1}{{p^{2} }}\left( {A + 1} \right)^{2} , \\ \end{aligned} $$

(22)

Working out Eq. (22) similarly to Eq. (16) we finally get

$$ \upsigma^{2}_{\text{k}} \left( {A,p} \right) = \frac{{\left( {1 - p} \right)A}}{{p^{2} }}. $$

(23)

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Authors and Affiliations

A. Razetti
- 1
Email author
X. Descombes
- 2
C. Medioni
- 3
F. Besse
- 3

1.University of Nice Sophia Antipolis, I3SSophia AntipolisFrance
2.Inria, CRISAMSophia AntipolisFrance
3.Institute of Biology ValroseUniversity of Nice Sophia AntipolisNiceFrance