Final Report Summary - MODELLING DIFFUSION (Mathematical models for diffusion controlled systems: diffusion on cell membrane, cluster formation and maintenance)
In the first part of the project "Mathematical models for diffusion controlled systems: diffusion on cell membrane, cluster formation and maintenance", we developed a mathematical model for describing the early stages in neuronal polarity and a software that was based on our algorithm to distinguish between random Brownian motion and confined movement. During the return phase, we extended the work done in the first part considering the effect of fluctuations in neuronal polarity and applying our algorithm/software to analyze AMPA diffusion. We also begin to study the role of MARCKS during aging.
-Fluctuations and neuronal polarity: results and conclusions
Neurons are among the cell types with the most prominent asymmetry. The correct establishment of polarized domains in neurons enables their directional migration and polarized axon-dendrite formation and is thus one of the most critical steps in brain development. Neuronal polarization starts with the selection of the site from which the first neurite will grow before morphological changes are evident. During the first part of the project we proposed a mathematical model that was based in those proposed by Altschuler et al. (Nature 2008) and Turing (Philos. T. R. Soc. B 1952). Our model assumes an activator-inhibitor dynamics and diffusion-driven instabilities. Extensive analyses on the equilibrium points and phase and bifurcation diagrams were done, allowing us to study spontaneous symmetry breaking. This work has already been published in PLoS ONE (Menchón SA, Gärtner A, Román P, Dotti CG, 2011 Neuronal (Bi)Polarity as a Self-Organized Process Enhanced by Growing Membrane. PLoS ONE 6(9): e24190. doi:10.1371/journal.pone.0024190).
Although the model was novel and useful to address the question we were dealing with during the first part, it did not consider stochastic fluctuations. In particular, intrinsic noise can induce spatio-temporal ordered patterns such as Turing patterns, i.e. with the same quantitative characteristics of self-organization. In order to study these Stochastic Turing patterns, we developed a stochastic version of our deterministic model.
We consider a system with two kinds of molecules, the activator, A, and the inhibitor, B, we assume that they inhabit patches which are situated in the sites of a d-dimensional lattice (with periodic boundary conditions). The maximum number of molecules allowed at each patch is fixed, and we call it M. We also consider the “empty spaces”, E, at each patch, i.e. for the i-th patch we have Ai+Bi+Ei=M. These components can interact locally according to few rules we have defined and they can also migrate between neighbors patches with a given rate. With these rules we can write the transition probabilities between different states and arrive to a master equation that describes the system evolution. The master equation contains information of both, the mean field dynamics and the finite corrections. Performing a van Kampen expansion and a perturbative calculation, it is possible to obtain a system of partial differential equations for the macroscopic quantities and a Fokker-Planck equation to analyze the fluctuations. Although the theoretical approach is done in an arbitrary dimension, d, in order to compare with our deterministic previous work we did most of the calculation with d=1.
The first difference to be noticed is the appearance of the cross-diffusion terms in the mean-field equations. The cross-diffusion terms appear as a consequence of the requirement that there must be an empty space available in the patch where the molecule is moving to. When compared to the Turing analysis performed before, (without the cross-diffusion terms), now the region of self-organization becomes bigger. On the other hand, stochastic fluctuations can give rise to stochastic Turing patterns due to an inherent resonant mechanism. In order to analyze this, instead of using the Fokker-Planck equation, we study the equivalent Langevin equation and calculate the power spectrum. In order to validate the theoretical results, we also perform simulations. We implement the Gillespie algorithm, which is a variety of a dynamic Monte Carlo method and gives us the system temporal evolution.
This work is in the final stage. The manuscript is being written to be submitted to Physical Review E. A pdf file is attached to this form.
-The role of MARCKS during aging: results and conclusions
Cognitive and motor performances decline during aging. Although it is clear that such signs reflect synaptic compromise, the underlying mechanisms are not yet defined. This dysfunction in the aged is paralleled by a large number of structural and biochemical alterations. It is kwon that membrane cholesterol and sphingomyelin content vary significantly with age. However, it is not clear how these changes contribute to functional decay during aging. Recently, performed investigations in Dotti’s Lab (Nature Neuroscience, 16, 449-455, 2013) have shown than the levels and activity of PIP2 are significantly reduced during aging and this seems to be due to reduced levels of MARCKS on the membrane. We are working in a mathematical model to analyze and study the PIP2, PIP3 and MARCKS membrane concentrations during aging. Our goal is to determine if there is an optimal amount of MARCKS to be introduced, in order to revert the reduced levels of PIP2. Our approach consists in writing the master equation of the system to analyze its temporal evolution. Although it is still an ongoing project we have already presented some preliminary data in a National Conference (TREFEMAC 2013 – La Plata - Argentina).
-AMPA receptors movement: results and conclusions
We used the software developed during the first part of the project, (Menchón SA, Martín, MG, Dotti CG, “APM GUI: analyzing particle movement on the cell membrane and determining confinement”, BMC Biophysics 2012, 5(4) doi:10.1186/2046-1682-5-4) to compare diffusion regimes of AMPA receptors between young and old neurons, showing that the lateral diffusion of AMPA receptors does not increase upon activation with exogenous glutamate in hippocampal neurons with decreased cholesterol levels. This work has been submitted to Nature Neuroscience and is under review process, (“Constitutive hippocampal cholesterol loss underlies poor cognition in old rodents”, MG Martín, T Ahmed, A Korovaichuk, C Venero, SA Menchón, I Salas Hernández, S Munck, O Herreras, D Balschun and Carlos G. Dotti, submitted to Nature Neuroscience).
-Socio-economic impact of the project
The work carried out in this project has been presented in the meeting of young researchers in neuroscience. The goal of this meeting is to encourage young students to follow their academic career doing a PhD in Neuroscience. The National University of Córdoba has created this PhD in 2010, its main aim is to promote the interdisciplinary study of neuroscience in Córdoba.
The work carried out in this project has also been presented in the National University of Río Cuarto. This University has created the career of Physics few years ago and they promote activities to contact their undergraduate students with researchers in other Universities.
Seminars in the National University of Córdoba as well as in National Conferences were given.
I also gave a talk and discussed my results about fluctuation and Turing patterns during my visit to Dotti’s Lab (VIB – Leuven - Belgium) and Kappen’s group (SNN Adaptive Intelligence – Nijmegen – The Nederlands). Prof. Horacio Wio (Universidad de Cantabria - Santander - Spain) helps me to formulate the mathematical model to consider stochastic Turing patterns.
-Fluctuations and neuronal polarity: results and conclusions
Neurons are among the cell types with the most prominent asymmetry. The correct establishment of polarized domains in neurons enables their directional migration and polarized axon-dendrite formation and is thus one of the most critical steps in brain development. Neuronal polarization starts with the selection of the site from which the first neurite will grow before morphological changes are evident. During the first part of the project we proposed a mathematical model that was based in those proposed by Altschuler et al. (Nature 2008) and Turing (Philos. T. R. Soc. B 1952). Our model assumes an activator-inhibitor dynamics and diffusion-driven instabilities. Extensive analyses on the equilibrium points and phase and bifurcation diagrams were done, allowing us to study spontaneous symmetry breaking. This work has already been published in PLoS ONE (Menchón SA, Gärtner A, Román P, Dotti CG, 2011 Neuronal (Bi)Polarity as a Self-Organized Process Enhanced by Growing Membrane. PLoS ONE 6(9): e24190. doi:10.1371/journal.pone.0024190).
Although the model was novel and useful to address the question we were dealing with during the first part, it did not consider stochastic fluctuations. In particular, intrinsic noise can induce spatio-temporal ordered patterns such as Turing patterns, i.e. with the same quantitative characteristics of self-organization. In order to study these Stochastic Turing patterns, we developed a stochastic version of our deterministic model.
We consider a system with two kinds of molecules, the activator, A, and the inhibitor, B, we assume that they inhabit patches which are situated in the sites of a d-dimensional lattice (with periodic boundary conditions). The maximum number of molecules allowed at each patch is fixed, and we call it M. We also consider the “empty spaces”, E, at each patch, i.e. for the i-th patch we have Ai+Bi+Ei=M. These components can interact locally according to few rules we have defined and they can also migrate between neighbors patches with a given rate. With these rules we can write the transition probabilities between different states and arrive to a master equation that describes the system evolution. The master equation contains information of both, the mean field dynamics and the finite corrections. Performing a van Kampen expansion and a perturbative calculation, it is possible to obtain a system of partial differential equations for the macroscopic quantities and a Fokker-Planck equation to analyze the fluctuations. Although the theoretical approach is done in an arbitrary dimension, d, in order to compare with our deterministic previous work we did most of the calculation with d=1.
The first difference to be noticed is the appearance of the cross-diffusion terms in the mean-field equations. The cross-diffusion terms appear as a consequence of the requirement that there must be an empty space available in the patch where the molecule is moving to. When compared to the Turing analysis performed before, (without the cross-diffusion terms), now the region of self-organization becomes bigger. On the other hand, stochastic fluctuations can give rise to stochastic Turing patterns due to an inherent resonant mechanism. In order to analyze this, instead of using the Fokker-Planck equation, we study the equivalent Langevin equation and calculate the power spectrum. In order to validate the theoretical results, we also perform simulations. We implement the Gillespie algorithm, which is a variety of a dynamic Monte Carlo method and gives us the system temporal evolution.
This work is in the final stage. The manuscript is being written to be submitted to Physical Review E. A pdf file is attached to this form.
-The role of MARCKS during aging: results and conclusions
Cognitive and motor performances decline during aging. Although it is clear that such signs reflect synaptic compromise, the underlying mechanisms are not yet defined. This dysfunction in the aged is paralleled by a large number of structural and biochemical alterations. It is kwon that membrane cholesterol and sphingomyelin content vary significantly with age. However, it is not clear how these changes contribute to functional decay during aging. Recently, performed investigations in Dotti’s Lab (Nature Neuroscience, 16, 449-455, 2013) have shown than the levels and activity of PIP2 are significantly reduced during aging and this seems to be due to reduced levels of MARCKS on the membrane. We are working in a mathematical model to analyze and study the PIP2, PIP3 and MARCKS membrane concentrations during aging. Our goal is to determine if there is an optimal amount of MARCKS to be introduced, in order to revert the reduced levels of PIP2. Our approach consists in writing the master equation of the system to analyze its temporal evolution. Although it is still an ongoing project we have already presented some preliminary data in a National Conference (TREFEMAC 2013 – La Plata - Argentina).
-AMPA receptors movement: results and conclusions
We used the software developed during the first part of the project, (Menchón SA, Martín, MG, Dotti CG, “APM GUI: analyzing particle movement on the cell membrane and determining confinement”, BMC Biophysics 2012, 5(4) doi:10.1186/2046-1682-5-4) to compare diffusion regimes of AMPA receptors between young and old neurons, showing that the lateral diffusion of AMPA receptors does not increase upon activation with exogenous glutamate in hippocampal neurons with decreased cholesterol levels. This work has been submitted to Nature Neuroscience and is under review process, (“Constitutive hippocampal cholesterol loss underlies poor cognition in old rodents”, MG Martín, T Ahmed, A Korovaichuk, C Venero, SA Menchón, I Salas Hernández, S Munck, O Herreras, D Balschun and Carlos G. Dotti, submitted to Nature Neuroscience).
-Socio-economic impact of the project
The work carried out in this project has been presented in the meeting of young researchers in neuroscience. The goal of this meeting is to encourage young students to follow their academic career doing a PhD in Neuroscience. The National University of Córdoba has created this PhD in 2010, its main aim is to promote the interdisciplinary study of neuroscience in Córdoba.
The work carried out in this project has also been presented in the National University of Río Cuarto. This University has created the career of Physics few years ago and they promote activities to contact their undergraduate students with researchers in other Universities.
Seminars in the National University of Córdoba as well as in National Conferences were given.
I also gave a talk and discussed my results about fluctuation and Turing patterns during my visit to Dotti’s Lab (VIB – Leuven - Belgium) and Kappen’s group (SNN Adaptive Intelligence – Nijmegen – The Nederlands). Prof. Horacio Wio (Universidad de Cantabria - Santander - Spain) helps me to formulate the mathematical model to consider stochastic Turing patterns.