Team:Paris/Production modeling2

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== DryLab ==
 
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<center> [[team:Paris/DryLab#DryLab| Main]] - [[team:Paris/Modeling#bottom|Introduction]] - [[Team:Paris/Production_modeling#bottom | Delay model]] - [[Team:Paris/Production_modeling2#bottom | Vesicle model]] - [[Team:Paris/Transduction_modeling#bottom | Fec simulation]] </center>
 
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<center> '''Vesicle model'''</center>
 
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=='''Modeling Vesicles creation==
 
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===Unequal pressures create small blebbing:===
 
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We modelised the reaction of the membrane under the osmotic pressure. We used a simple 2 dimension model to show the creation of blebbing under pressure increase. We simplified the equation given by Ou-Yang and Helfrich [Phys. Rev. Lett. 59 (1987) 2486], which describes the  membrane conformation under unequal pressure:
 
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[[Image:Diferential System.png|400px|center]]
 
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::Where H is the mean gaussian curvature and K is the Gaussian curvature :
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== DryLab - Vesicles biophysics model==
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[[Image:Final Theorical Curvature.png|150px|center]]
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::We first simplified the membrane equation to a 2 dimensional equation system based on a polar curvature simplification approximation :  
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::the curvature in polar is:
 
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[[Image:Curvature.png|200px|center]]
 
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::With the hypothesis of a cylinder form the mean and gaussian curvatures can be written as:
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::indeed we have in those hypothesis:
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[[Image:Principal curvatures.png|75px|center]]
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<div id="left-side"></div>
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<div id="middle-side"><center>
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<a class="menu_sub"href="https://2009.igem.org/Team:Paris/DryLab#bottom"> Main </a>|
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::Thous we can concider the polar system of equation:
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<a class="menu_sub"href="https://2009.igem.org/Team:Paris/Production_modeling#bottom"> Delay model</a>|
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<a class="menu_sub_active"href="https://2009.igem.org/Team:Paris/Production_modeling2#bottom"> Vesicle model</a>|
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[[Image:Systeme_Membrane.png|400px|center]]
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<a class="menu_sub"href="https://2009.igem.org/Team:Paris/Transduction_modeling#bottom"> Fec simulation</a>
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::to have a unique solution we must use some boundary conditions to compute our model (Cauchy Lipshitz thorem for the mathematician ;)) those conditions are simple without the peptidoglycan attachement lypoproteins (Pal). As the model is dependent of the angle we must impose the fact that r(0)=r(2&pi;). In addition the &Delta;p must be concidered as dependent of the quantity of periplasmic species.
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::Indeed the osmotic pressure can be written in this way:
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[[Image:Osmotic pressure.png|200px|center]]
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::And by concidering that we have low concentrations we can concider a simpliest formula analogue to the perfect gases law:
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[[Image:Osmotic pressure Analogue.png|100px|center]]
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::Finaly we can write:
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[[Image:Delta pressure.png|200px|center]]
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::And Thus the whole parameters are given. In addition, we can translate the role of the Tol/Pal System has boundary condition on the whole system: a cluster of Tol-Pal can be considered as a point with a radius equal to the peptidoglycan ones. Finally, we were able to obtain this type of behavior:
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[[Image:Vesicle wiki.jpg|400px|center]]
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==''Travail  à faire''==
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<font color=red>Reprendre pourquoi cette modélisation et introduire ce quel'on fait ici. Par ex.
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In this section we are going to develop an original biophysical model developed to explain the formation of vesicles. This study aims at a better understanding of the mechanisms leading to the production of vesicles by the cell a process still very poorly described [Références] despite the natural occurrence of this phenomenon in gram-negative bacteria. Knowledge of the underlying processes can prove valuable in the perspective of optimizing inter-cellular communication by vesicle transport.
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</font>
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Our model is based on three different physical phenomena: (i)The lipid surface conformation,  (ii) Tol-Pal proteins diffusion and  (iii) the increase of the osmotic pressure in the periplasm
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===The lipid surface conformation===
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<font color=red>The conformation of the lipid outer membrane is a well...</font>
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The Lipid conformation of the outer membrane is a well known problem: at 35°c the lipid bilayer behaves like a liquid which conformation character is ruled by an energy called the bending energy. This energy represents the fact that the lipid bilayer will search a special conformation depending on the shape and the chemical properties of  its constituents.  
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We first use the bending energy has a rough shape for our model and its understanding:<font color=red> Phrase pas claire. Par ex. simplement</font>
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<font color=red>  In order to explain the way lipids organize together we need an expression for the membrane bending energy. This  is given by</font>
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{|
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|[[Image:Bending energy.png|200px]]
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<font color=red>where E is the energy....</font>
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<div id="left-side2"></div>
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<font color=red>OUT(</font>::This formula give use the abilities to explain the way lipids will organize together<font color=red>)</font>
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<div id="middle-side2"><center>
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<a class="menu_sub_active"href="https://2009.igem.org/Team:Paris/Production_modeling2#bottom">Intro</a>|
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E is the energy of a whole lipid bilayer (or monolayer). Kb and Kg are Bending and Gaussian moduli which can be obtained by experiments. &gamma;0 is the intrinsic curvature of the outer membrane which describes the local form of a lipid bilayer when it is at is lowest state of energy ,the more stable. &gamma;m and Hg <font color=red> ce n'est pas plutôt γg?</font>are the mean curvature and the gaussian curvature<font color=red>Impérativement une référence</font>. <font color=red>dS is a surface  infinitesimal element and the previous formula relates how the local surface energy varies with the local mean curvature. (JUSTE?)</font>
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<a class="menu_sub_active"href="https://2009.igem.org/Team:Paris/Production_modeling2#2">Membrane model</a>|
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<a class="menu_sub_active"href="https://2009.igem.org/Team:Paris/Production_modeling2#3">Tol/Pal model</a>|
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<font color=red>Let us calculate first </font>the energy of two different shapes of membranes, (i) a model of the shape of E. coli before budding, and  (ii) after budding of a vesicle.
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<a class="menu_sub_active"href="https://2009.igem.org/Team:Paris/Production_modeling2#4">Bleb initiation</a>|
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<a class="menu_sub_active"href="https://2009.igem.org/Team:Paris/Production_modeling2#5">Tol/Pal accumulation</a>|
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<a class="menu_sub_active"href="https://2009.igem.org/Team:Paris/Production_modeling2#6">Vesiculation</a>
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<font color=red>SUPPRIMER A. We compute the shape of E. coli before the division and then the vesicles' shape.</font>
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<font color=red>Pourquoi cette configuraiton correspond à la région de division de E.coli?</font>
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::(i) We consider the shape of E.coli as a cylinder of radius r =0.3 μm and of infinite length.
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<font color=red>Pourquoi longueur nifinie?</font>The aim of this first representation is to estimate this energy in the division region of E.coli before division. With this approximation &gamma;m is equal to 2/r and &gamma;g=0. In this case:
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<center>
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{|
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|[[Image:BendingE AeraEcoli.png|150px]]
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<font color=red>Comme tu l'utilises après, explicite Ee ici, il me semble que c'était un élément que tu voulais citer mais qui manquait</font>
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Thus for the  E.coli lipids membrane  that the bending energy is:
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<font color=red>A PLACER Expression pour Ee=</font>
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</center>
</center>
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</div>
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::(ii) For the vesicles we consider their basic shape as a sphere of rayon r’ so the bending energy by lipid area units is:
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<div id="right-side2"></div>
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<center>
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<span/ id="1">
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{|
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This part aims at investigating an important issue regarding the creation of vesicles:
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|- style="background: #0d3e99; text-align: center; color:white;"
 
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|[[Image:Vesicle Energy Aera.png|200px]]
 
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|}
 
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</center>
 
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<font color=red>ORGINAL avec confusions me semble t'il Thus as the area of a sphere is known and  is independent of the location on the sphere we can write: In the same way we can write for an area of  E.coli lipids that the bending energy is:</font>
 
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::Thus as the area of a sphere is known and  is independent of the location on the sphere we can write:
 
<center>
<center>
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{|
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'''What is the link between Tol/Pal expression and vesicle formation?'''
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|- style="background: #0d3e99; text-align: center; color:white;"
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|[[Image:Bending energy Vesicul.png|200px]]
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|}
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</center>
</center>
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<font color=red>Tu es sûr? Essentiellement on multiplie ici par la surface de la sphère A\pi r'^2  pourquoi le dernier terme à droite est seulement Kg?</font>
 
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The relations can provide a basic vision of the statistical repartition of vesicles in case of absence of integrity control system in the outer membrane.
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Or, more precisely, '''can we explain vesicle formation solely by the diffusion of the doubly anchored Tol/Pal complex in the membrane''' ?
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Understanding this connection is instrumental, since '''our project relies on the hypothesis that an increased rate of vesicle formation can be obtained simply by destabilization of the Tol/Pal complexes'''.
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<font color=red>Quelle "first" energy et quelle "second" one?? Pas clair</font> The first energy is the potential energy of the lipid area in E.coli outer membrane necessary to the construction of a vesicle and the second one the potential energy of the same lipid area but in the conformation of a vesicle shape. So the energy which must be given to the whole system to create a vesicle is:
 
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<font color=red>J'interprète.... vois si on est d'accord: Ee </font> is the potential energy of the lipid area in E.coli outer membrane before construction of a vesicle and <font color=red>Ev </font> is the energy of the same lipid area but in the conformation after budding including a vesicle shape. So the energy which must be given to the whole system to create a vesicle is:
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To answer this question, we developed a '''biophysical model of the cellular membranes and Tol/Pal complexes''', that incorporates '''outer membrane deformation''' and '''Tol/Pal diffusion''' in membranes by Brownian motion.
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<center>
 
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{|
 
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|- style="background: #0d3e99; text-align: center; color:white;"
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Our initial '''simulation results suggested a three-stage process''' for vesicle formation. 
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|[[Image:Whole energy.png|400px]]
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Thus, in what follows, we decomposed the original main question in three parts:
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* Can we explain the formation of small blebs by differences of osmotic pressures between intra- and extra-cellular environments?
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* Can we explain the accumulation of Tol/Pal molecules at the basis of nascent blebs?
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* Can we explain vesiculation by constriction of blebs basis by accumulated Tol/Pal molecules?
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|}
 
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</center>
 
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We can suppose that the most easily created vesicles will be the ones which requires a minimum energy. By derivation we find that the minimum his obtain for:
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Before answering these questions, we start by describing how we modeled '''membrane deformation''' and '''Tol/Pal diffusion'''.
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<center>
 
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{|
 
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|- style="background: #0d3e99; text-align: center; color:white;"
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The main features of our membrane model is that we take into account the presence of '''osmotic pressure differences''' between intra and extra cellular environments and the existence of an outer '''membrane intrinsic preferred curvature'''.
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|[[Image:Final gamma0.png|250px]]
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The main feature of our Tol/Pal diffusion model is that we take into account that '''diffusion''' may happen '''on non-flat surfaces''', and that the '''Tol/Pal complex is anchored''' both in the '''intracellular''' (Tol) and '''extracellular''' (Pal) '''membranes'''.
 +
We assumed that the intracellular membrane position is fixed (pushed against the peptidoglycan layer).  
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<span/ id="2">
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</center>
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===Modeling membrane at steady state===
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::Hence as we know that the range of created vesicles radii is 25 nm to 175nm we can suppose that the r’ is somehow about 100 nm. This leads us to an estimate of the outer membrane intrinsic cuurvature:
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The objective of this part is to discuss the existence of an intrinsic curvature of the outer membrane at steady state, and to obtain an estimate of its characteristic parameter.
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<center>
 
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{|
 
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|- style="background: #0d3e99; text-align: center; color:white;"
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Lipopolysaccharides (LPSs) are important constituents of the ''E. coli'' outer membrane.
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|[[Image:Num value.png|150px]]
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They possess long polycaccharide chains pointing outwards.
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These extracellular sugar extensions create mutual attraction forces that curb the membrane.
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This situation is depicted in the figure below.  
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|}
 
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</center>
 
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[[Image:LPS_CLUSTER_Wiki.png|250px|center]]
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Which is realistic in the order of magnitude.
 
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In fact we know that the E.coli lipid bilayer is built of distinct types of lipids: Lippopolysacharides (LPS) and simple phospholipids. LPS are located in the exterior lipid layer of the outer membrane. The others are located in the interior lipid bilayer. Moreover, those LPS present a sugar extension toward the medium. Those sugars can bind to each other. So we can assume that they are going to create clusters and to curve the membrane toward the exterior of E.coli.
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From a physical point of view, the lipid bilayer behaves like a liquid at 37°c.
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To each membrane conformation, one can associate an energy, called the bending energy [1].
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This is given by :
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<font color=red>AU MOINS UN SCHÉMA SERAIT IMPORTANT POUR ETTAYER CE QUI PRECEDE</font>
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[[Image:Bending energy.png|220px|center]]
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where E is the energy of the membrane, K<sub>b</sub> and K<sub>g</sub> are Bending and Gaussian moduli, H<sub>g</sub> is the Gaussian curvature and c<sub>0</sub> is a parameter describing the intrinsic curvature of the outer membrane at steady state (ie, minimal energy). The shape of the membrane is described locally by the principal curvature variables c<sub>1</sub> and c<sub>2</sub> (see [http://en.wikipedia.org/wiki/Principal_curvature here] for details). dS is an infinitesimal surface element.
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<font color=red>ON PASSE DANS UNE AUTRE PARTIE:  petite conclusion de ce qui précède et introduction de la suite</font>
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Under simple assumptions on the shape of ''E. coli'' (seen as a cylinder) and assuming that on average the size of the vesicles is the one requiring a minimum of energy, we can using the above relation relate the radius of ''E. coli'' r, the mean radius of vesicles r', and the γ<sub>0</sub> parameter:
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==="Tol-Pal proteins diffusion"===
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[[Image:Relations.png|180px|center]]
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Tol and Pal are membrane proteins which are located respectively in the outer and the inner membrane. The diffusion of proteins in those lipid bilayers can be modelled by a probabilistic Brownian movement. This diffusion model gives us the law of probability for the location of Tol and Pal on the membranes. It has been observed that the Tol and Pal proteins interact with each other, which is linked to the membrane stability: indeed the Tol and Pal will bind inner and outer membrane and furthermore stabilize the outer membrane using the peptidoglycan rigidity. <font color=red>FAIRE UN RENVOIT A LA PARTIE DU WIKI QUI PRESENTE LE SYSTEME TOL-PAL OU INCLURE ICI UN SCHEMA</font>
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Using r= 300 nm and r’=100 nm (the vesicle size ranges from 25 nm to 175 nm), we estimate the outer membrane intrinsic curvature as
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[[Image:gamma0.png|150px|center]]
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==="Periplasmic osmotic pressure increase"===
 
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The  osmotic pressure in the periplasm is the same that the medium  pressure in normal time. But during the division period of the bacteria the peptidoglycan is degraded to be recycled in a new cell wall. During this phenomena of turn-over a  part of the peptidoglycan is released in the periplasm which increased its osmotic pressure.
 
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Obtaining an estimation of this parameter is essential to model and simulate the membrane dynamic deformation under osmotic pressure differences as we will see.
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Thus, we can consider that if there is not enough Tol-Pal linked proteins the outer membrane will distort to create a beginning of vesicle. But in this part of the membrane the tol pal proteins will not have the possibility to bind themselves and they will be free to diffuse in other parts of  the membranes. The surface shape will guide the proteins to the border of the vesicle and stabilized the shape of the vesicle.
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===Modeling membrane deformation===
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<font color=red>A FIGURE IS WORTH 1000 WORDS !</font>
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To obtain a reasonably simple model, we assumed that ''E. coli'' cells can be represented as cylinders, as a first approximation.
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This way, we could develop a 2D model of a transversal section of cells, using cylindrical coordinates.
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To describe thevesicles creation we first have to use a more complete approach and need to use an equation  relation the osmotic pressure difference between the inside and outside of the cell and the equilibrium shape (Ou-Yang and Helfrich [Phys. Rev. Lett. 59 (1987) 2486]):
 
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<center>
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Because of osmotic pressure differences, we assumed that the intracellular membrane is pushed against the -perfectly circular- peptidoglycan layer. So, only outer membrane deformations have to be modeled.
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{|
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The difference of osmotic pressure between the intra-cellular environment and the periplasm, and between the periplasm and the extra-cellular environment comes notably from the turnover of peptidoglycan molecules <sup>[[https://2009.igem.org/Team:Paris/Production_modeling2#References 1]]</sup>. This hypothesis has been made by  Zhou & Doyle <sup>[[https://2009.igem.org/Team:Paris/Production_modeling2#References 3]]</sup>
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|- style="background: #0d3e99; text-align: center; color:white;"
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To model membrane deformation, we use the equation proposed by Ou-Yang and Helfrich <sup>[[https://2009.igem.org/Team:Paris/Production_modeling2#References 4]]</sup>:
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|[[Image:Formula1.png|500px]]
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[[Image:Diferential System.png|300px|center]]
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</center>
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<font color=red>DECRIRE LES PARAMÈTRES</font> 
 
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<font color=red>J'ai réécrit en partie ici. Relis stp</font> 
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After simplifications, we obtain the following equality.
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'''Simplified approach: without brownian motion of the intermembrane Tol-Pal links'''
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[[Image:Simplified_Zhong_can.png|400px|center]]
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In the following we first consider a mode model without brownian movement of the Tol-Pal links that are considered immobilized. We look also here at a simplified one dimensional membrane. Our aim is to get a visual representation of the equilibrium shape of the outer membrane for a given osmotic pressure difference. In one dimension and in polar coordinates the membrane equation of the vesicle simplifies well.
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In this equation, &gamma; is the variable membrane curvature, &Delta;p is the osmotic pression difference, c<sub>0</sub> is the membrane intrinsic curvature, and other parameters are as described in Ou-Yang and Helfrich <sup>[[https://2009.igem.org/Team:Paris/Production_modeling2#References 4]]</sup>.
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In polar coordinates the curvature is:
 
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<center>
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A formal connection between the membrane curvature &gamma; and the cylindric coordinate variables ''r'' and &theta; can be obtained by the following polar curvature simplification approximation:
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|[[Image:Curvature.png|200px]]
 
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<font color=red>Definir r' et r. Faire apparaître la coordonnée angulaire</font> 
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[[Image:Curvature.png|120px|center]]
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where
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[[Image:derivative.png|60px|center]]
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<font color=red>CHANGE AVEC CE QUI SUIT  ::With the hypothesis of a cylinder form the mean and gaussian curvatures can be written as:</font>
 
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Here also we assume a cylindrical mean shape, and hence the mean and gaussian curvatures can be written as:
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Lastly, by combining the two equations given above, we obtain the following set of '''differential equations describing membrane deformations''':  
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<center>
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[[Image:Systeme_Membrane.png|300px|center]]
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{|
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|- style="background: #0d3e99; text-align: center; color:white;"
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Considering that we have low concentrations we can obtain a simpler formula analogue to the perfect gases law:
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|[[Image:Curvatures.png|75px]]
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|}
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[[Image:Osmotic pressure Analogue.png|80px|center]]
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</center>
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::indeed we have in those hypothesis:<font color=red>???? COMPRENDS PAS</font>
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Finaly we have:  
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<center>
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[[Image:Delta pressure.png|150px|center]]
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{|
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Here we observe that the model is depending of the volume V which will stabilized the equation and totally define  all the parameters of the system.
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|[[Image:Principal curvatures.png|75px]]
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|}
+
In addition, '''we decided to model the role of the Tol/Pal system as boundary condition for the system of differential equations : cluster of Tol-Pal are considered as a point with a radius equal to peptidoglycan's one plus the length of the protein. ''' Furthermore, as the surface is closed, we must impose the fact that r(0)=r(2&pi;) to account for closing the vesicle.
-
</center>
+
 +
<span/ id="3">
 +
===Modeling Tol/Pal diffusion===
-
<font color=red>::Thous  we can concider the polar system of equation:  REMPLACÉ PAR</font>
 
-
With these relations the membrane equation becomes:
 
 +
Tol and Pal are membrane proteins which are located respectively in the outer and the inner membrane.
 +
Tol and Pal proteins interact with each other, forming Tol/Pal complexes.
 +
By acting like press studs binding the inner and outer membranes, thus stabilizing the outer membrane using the peptidoglycan rigidity, the Tol/Pal complexes play a central role to preserve membrane integrity <sup>[[https://2009.igem.org/Team:Paris/Production_modeling2#References 5]]</sup><sup>[[https://2009.igem.org/Team:Paris/Production_modeling2#References 6]]</sup>.
 +
[[Image:TolPal Ancored.png|300px|center]]
-
<center>
 
-
{|
 
-
|- style="background: #0d3e99; text-align: center; color:white;"
+
The diffusion of proteins in these lipid bilayers can be modeled by '''Brownian motion'''. This diffusion model gives the probability law for the location of Tol and Pal in the membranes.
-
|[[Image:Systeme_Membrane.png|400px]]
+
The following three equations model the Brownian motion of Tol, Pal and Tol/Pal complexes, represented as particles.
-
|}
+
[[Image:Diffusion.png|400px|center]]
-
</center>
+
 +
In the above equations, C<sub>X</sub> denotes the concentration of protein X, and D<sub>X</sub> denotes the constant kinetic coefficient associated to X. The '''k'''<sub>TolPal</sub> and '''k'''*<sub>TolPal</sub> denote the reaction constants for Tol-Pal complexation.
 +
In the last equation, &Phi; is an unknown function representing the Tol-Pal complex motility.
 +
However, because the Tol-Pal complex is significantly larger than the other proteins, and is doubly anchored in membranes, we made the assumption that '''the mobility of Tol-Pal complexes is negligable in comparison to the mobility of isolated Tol and Pal molecules'''. Stated differently, we set &Phi; to zero.
 +
Lastly, additional constraints on Tol/Pal diffusion have been added to enforce a constant distance between Tol and Pal proteins.
-
<font color=red>We can now integrate this system and obtain a representation of the equilibrium shape of the outer membrane. To compute the model and to obtain </font>a unique solution we must use some boundary conditions (Cauchy Lipshitz thorem for the mathematician ;)). <font color=red>These are given bythe case  </font> without the peptidoglycan attachement lypoproteins (Pal) <font color=red>QUELQUES DETAILS EN PLUS</font>.  As the model is dependent of the angle we must impose the fact that r(0)=r(2&pi;) <font color=red>to account for closing the vesicle.</font>.
+
In the above equations, '''the laplacians capture the non-homogeneous molecular diffusion on non-flat membranes'''.
 +
Indeed, they link the evolution of protein concentrations with the local shape of the membranes, since we know that on a two dimensional space:
-
<font color=red>Finally, we include also in the model the fact that</font> the &Delta;p must be considered as dependent of the quantity of periplasmic species.
+
[[Image:Laplacian.png|100px|center]]
-
Indeed the osmotic pressure can be written in this way:
+
-
<center>
+
where x and y are the coordinate of the surface and where the different partial derivatives are depending on the cartesian coordinates of the surface we are evolving on.
-
{|
+
-
|- style="background: #0d3e99; text-align: center; color:white;"
+
<span/ id="4">
-
|[[Image:Osmotic pressure.png|200px]]
+
===Unequal osmotic pressures create small blebbing===
-
|}
 
-
</center>
 
-
 
-
And by considering that we have weak concentrations we obtain a simpler formula analogue to the perfect gas law:
 
-
 
-
<center>
 
-
{|
 
-
 
-
|- style="background: #0d3e99; text-align: center; color:white;"
 
-
|[[Image:Osmotic pressure Analogue.png|100px]]
 
-
 
-
|}
 
-
 
-
</center>
 
 +
Using the equations for membrane deformation, we can compute numerically the shape of the outer membrane at steady state under various assumptions.
 +
If we assume that Tol/Pal complexes acts like '''press studs''' locally imposing a fixed distance between the two membranes, and that their initial distribution  in the membrane is '''not totally homogeneous''', then we obtain results of the following type.
-
Giving finally for the osmotic pressure difference:
 
<center>
<center>
-
{|
+
[[Image:Vesicle wiki.jpg|400px|center]]
-
 
+
::Shape at equilibrium of the outer membrane presenting blebs (red) resulting from osmotic pressure differences and with Tol/Pal complexes locally imposing a fixed distance between the inner (green) and outer membranes.
-
|- style="background: #0d3e99; text-align: center; color:white;"
+
-
|[[Image:Delta pressure.png|200px]]
+
-
 
+
-
|}
+
-
 
+
</center>
</center>
 +
<span/ id="5">
 +
===Tol/Pal accumulates at the basis of nascent blebs===
-
<font color=red>MAIS COMMENT TU UTILISES CETTE EXPRESSION? \Delta P,  ou bien 'n' n'est ce pas un paramètre que tu imposes (et éventuellement par la suite feras varier pour faire pousser les vésicules? </font>
 
-
<font color=red>ABSOLUMENT PAS CLAIR...::And Thous the whole parameters are given. In addition we can translate the role of the Tol/Pal System has boundary condition on the whole system: a cluster of tol-Pal can be translate as a point of radius equal to the peptidoglycan ones. Finally we can have this type of results:</font>
+
The value of the Laplacian operators used to model Tol and Pal diffusions in the membranes (see equations above) depends on the membrane curvature: in non-flat membranes, the "efficiency" of the diffusion depends on the direction.
 +
In fact, '''molecules anchored in the outer membrane''', like Pal, '''diffuse less efficiently in regions of negative curvature, and consequently tend to accumulate in these regions'''.
-
<font color=red>The integration of the polar membrane equations gives results of the following type:</font>
+
Using our computer program, we have been able to experimentally demonstrate this accumulation.
 +
The bottom plot presents the accumulation of Pal proteins in the membrane represented on the top plot.
 +
In the bottom plot, the left/right axis represents coordinates along the membrane, whereas the other horizontal axis represent the time evolution of protein concentrations (from time 70 to 100, instead of from 0 to 25 as indicated)
 +
One can show the protein accumulation at coordinates 11 and 43 approximately, corresponding to the regions of negative curvature of the membrane.
-
<center>
 
-
{|
 
-
|- style="background: #0d3e99; text-align: center; color:white;"
+
[[Image:Accumulation2.png|600px|center]]
-
|[[Image:Vesicle wiki.jpg|600px]]
+
-
|}
 
-
<font color=red>Légende:  
+
[[Image:Accumulation3.png|600px|center]]
-
Equilibrium shape of the outer membrane (red) with immobile Tol-Pal links fixed to the inner membrane (green) assumed cylindrical.</font>
 
-
</center>
+
This result is corroborated by recent experimental findings showing that “negative membrane curvatures [act] as a cue for sub-cellular localization of a bacterial protein” <sup>[[https://2009.igem.org/Team:Paris/Production_modeling2#References 7]]</sup>.
-
'''Treatment including  brownian motion of the intermembrane Tol-Pal links'''
 
-
<font color=red>We will now let the Tol-Pal links between the cell outer and inner membranes move freely with a brownian motion.</font> This motion links the local concentration of proteins with the shape of the membrane they are included in. <font color=red>The diffusion of this links is described by:</font>
+
Most notably, we can thus assume that Pal molecules accumulate at the basis of blebs.
 +
This naturally leads to an increase of Tol-Pal complexes formation in these regions.
 +
Lastly, because we additionally assumed that these large complexes diffuse less efficiently than Tol and Pal alone, we finally obtain that '''Tol-Pal complexes accumulate at the basis of the blebs'''.
 +
This is depicted in the following picture.
-
<center>
+
[[Image:Shema_Curvature.png|400px|center]]
-
{|
+
-
|- style="background: #0d3e99; text-align: center; color:white;"
+
<span/ id="6">
-
|[[Image:EquationEinstein.jpg|200px]]
+
-
|}
+
===Rings of accumulated Tol/Pal constricts blebs basis===
-
</center>
+
As explained above, Tol-Pal complexes tend to accumulate at the basis of the blebs, creating a ring of Tol-Pal (if seen from above).
-
where P is the proteins presence density of probability .
+
[[Image:Shema_diffusion.png|400px|center]]
-
<font color=red>PIERRE, JE ME RAPPELLE QUE TOL ET PAL DIFFUSAIENT INDEPENDEMMENT ET POUVAIENT QUAND ILS SE RENCONTRAIENT FORMER DES LIENS. C'EST CE QUI SE PASSE ICI OU BIEN CE SONT DE LIENS TOL-PAL DEJA FORMES QUI DIFFUSENT? ETRE PLUS CLAIR ICI</font>
 
-
The Laplacian of P explains the fact that the concentration of proteins like Pal increased in regions of negative curvature. Such a model can explain some recent observations [Kumaran & Losick] “negative membrane curvatures as a cue for sub-cellular localization of a bacterial protein”.  
+
When Tol-Pal molecules move to the interior of the ring, due to Brownian motion, they narrow the two membranes because they impose to have a constant distance between the two membranes as explained in the picture below: the proteins act like zippers on membranes, as depicted below. This leads to vesicle formation.  
-
According to this we can assume that the Pal will be confined to region with negative curvature. In those regions the probability of linking between Pal and Tol is increased which enhances the local membrane stability and contributes to  enlarge  these regions.
+
-
<font color=red>PAS CLAIR POURQUOI LA VESICULE EST MURE.... Then the vesicle is mature.</font>
+
-
<font color=red>::So Brownian motion is a good explanation for the proteins migration toward the septa during division. PAS CLAIR SI GARDER </font>
+
[[Image:zip.png|400px|center]]
-
::But this Brownian motion can explain the vesiculation too.
+
<html>
-
 
+
</div>
-
<font color=red> This reinforcement process driven by brownian motion can explain the vesiculation.</font>When there are too few proteins we can assume the creation of a vesicle at germinal states (called bleb). At the border of the blebs an amount of Pal protein prevent the whole membrane to swallow and keep the vesicle in form. But at the border between the blebs and the protein rich area curvature is highly negative and so the quantity of proteins in those regions increases enlarging the region toward the centre of the blebs.
+
<div id="paris_content_boxtop">
 +
</div>
 +
<div id="paris_content">
 +
</html>
-
<font color=red>UN SCHEMA IMPERATIF</font>
+
====References====
-
::Then two different options are expectable: the blebs is too small and then the proteins are “zippering” back the blebs. But if the bleb is big enough the zippering became a separation between the vesicle and the bacteria. The good vision of this blebs forming can be understood in this way: the proteins rich region on the border pass from an annular form toward a circular one.
+
<ol class="References">
 +
<li> [[Team:Paris/Production_modeling2#1 | ^]]1991 - Lipowsky - The conformation of membranes, ''Nature'', 349(6309):475-481</li>
 +
<li> [[Team:Paris/Production_modeling2#2 | ^]]2008 - Park & Uehara - How bacteria consume their own exoskeletons, ''Microbiol Mol Biol Rev'', 72(2):211-227</li>
 +
<li> [[Team:Paris/Production_modeling2#3 | ^]]1998 - Zhou ''et al''  - On the origin of membrane vesicles in gram-negative bacteria, ''FEMS microbiology letters'', 163(2):223-228 </li>
 +
<li> [[Team:Paris/Production_modeling2#4 | ^]]1987 - Ou-Yang & Helfrich - Instability and deformation of a spherical vesicle by pressure, ''Phys. Rev. Lett.'', 59:2486-2488</li>
 +
<li> [[Team:Paris/Production_modeling2#5 | ^]]2009 - Deatherage ''et al''  - Biogenesis of bacterial membrane vesicles, ''Mol Microbiol'', 72(6):1395-1407 </li>
 +
<li> [[Team:Paris/Production_modeling2#6 | ^]]2005 - Kuehn & Kesty      - Bacterial outer membrane vesicles and the host pathogen interaction, ''Genes & Dev'', 19:2645-2655 </li>
 +
<li> [[Team:Paris/Production_modeling2#7 | ^]]2009 - Kumaran & Losick  - Negative membrane curvature as a cue for subcellular localization of a bacterial protein. ''PNAS USA'', 106(32):13541-13545
 +
</li>
 +
</ol>

Latest revision as of 03:55, 22 October 2009

iGEM > Paris > DryLab > Vesicle biophysics Model (vesicle model)





Contents

DryLab - Vesicles biophysics model

This part aims at investigating an important issue regarding the creation of vesicles:


What is the link between Tol/Pal expression and vesicle formation?


Or, more precisely, can we explain vesicle formation solely by the diffusion of the doubly anchored Tol/Pal complex in the membrane ? Understanding this connection is instrumental, since our project relies on the hypothesis that an increased rate of vesicle formation can be obtained simply by destabilization of the Tol/Pal complexes.


To answer this question, we developed a biophysical model of the cellular membranes and Tol/Pal complexes, that incorporates outer membrane deformation and Tol/Pal diffusion in membranes by Brownian motion.


Our initial simulation results suggested a three-stage process for vesicle formation. Thus, in what follows, we decomposed the original main question in three parts:

  • Can we explain the formation of small blebs by differences of osmotic pressures between intra- and extra-cellular environments?
  • Can we explain the accumulation of Tol/Pal molecules at the basis of nascent blebs?
  • Can we explain vesiculation by constriction of blebs basis by accumulated Tol/Pal molecules?


Before answering these questions, we start by describing how we modeled membrane deformation and Tol/Pal diffusion.


The main features of our membrane model is that we take into account the presence of osmotic pressure differences between intra and extra cellular environments and the existence of an outer membrane intrinsic preferred curvature. The main feature of our Tol/Pal diffusion model is that we take into account that diffusion may happen on non-flat surfaces, and that the Tol/Pal complex is anchored both in the intracellular (Tol) and extracellular (Pal) membranes. We assumed that the intracellular membrane position is fixed (pushed against the peptidoglycan layer).

Modeling membrane at steady state

The objective of this part is to discuss the existence of an intrinsic curvature of the outer membrane at steady state, and to obtain an estimate of its characteristic parameter.


Lipopolysaccharides (LPSs) are important constituents of the E. coli outer membrane. They possess long polycaccharide chains pointing outwards. These extracellular sugar extensions create mutual attraction forces that curb the membrane. This situation is depicted in the figure below.


LPS CLUSTER Wiki.png


From a physical point of view, the lipid bilayer behaves like a liquid at 37°c. To each membrane conformation, one can associate an energy, called the bending energy [1]. This is given by :

Bending energy.png

where E is the energy of the membrane, Kb and Kg are Bending and Gaussian moduli, Hg is the Gaussian curvature and c0 is a parameter describing the intrinsic curvature of the outer membrane at steady state (ie, minimal energy). The shape of the membrane is described locally by the principal curvature variables c1 and c2 (see [http://en.wikipedia.org/wiki/Principal_curvature here] for details). dS is an infinitesimal surface element.


Under simple assumptions on the shape of E. coli (seen as a cylinder) and assuming that on average the size of the vesicles is the one requiring a minimum of energy, we can using the above relation relate the radius of E. coli r, the mean radius of vesicles r', and the γ0 parameter:

Relations.png


Using r= 300 nm and r’=100 nm (the vesicle size ranges from 25 nm to 175 nm), we estimate the outer membrane intrinsic curvature as


Gamma0.png


Obtaining an estimation of this parameter is essential to model and simulate the membrane dynamic deformation under osmotic pressure differences as we will see.

Modeling membrane deformation

To obtain a reasonably simple model, we assumed that E. coli cells can be represented as cylinders, as a first approximation. This way, we could develop a 2D model of a transversal section of cells, using cylindrical coordinates.


Because of osmotic pressure differences, we assumed that the intracellular membrane is pushed against the -perfectly circular- peptidoglycan layer. So, only outer membrane deformations have to be modeled. The difference of osmotic pressure between the intra-cellular environment and the periplasm, and between the periplasm and the extra-cellular environment comes notably from the turnover of peptidoglycan molecules [1]. This hypothesis has been made by Zhou & Doyle [3]


To model membrane deformation, we use the equation proposed by Ou-Yang and Helfrich [4]:

Diferential System.png


After simplifications, we obtain the following equality.

Simplified Zhong can.png


In this equation, γ is the variable membrane curvature, Δp is the osmotic pression difference, c0 is the membrane intrinsic curvature, and other parameters are as described in Ou-Yang and Helfrich [4].


A formal connection between the membrane curvature γ and the cylindric coordinate variables r and θ can be obtained by the following polar curvature simplification approximation:


Curvature.png

where

Derivative.png


Lastly, by combining the two equations given above, we obtain the following set of differential equations describing membrane deformations:

Systeme Membrane.png

Considering that we have low concentrations we can obtain a simpler formula analogue to the perfect gases law:

Osmotic pressure Analogue.png

Finaly we have:

Delta pressure.png

Here we observe that the model is depending of the volume V which will stabilized the equation and totally define all the parameters of the system.

In addition, we decided to model the role of the Tol/Pal system as boundary condition for the system of differential equations : cluster of Tol-Pal are considered as a point with a radius equal to peptidoglycan's one plus the length of the protein. Furthermore, as the surface is closed, we must impose the fact that r(0)=r(2π) to account for closing the vesicle.

Modeling Tol/Pal diffusion

Tol and Pal are membrane proteins which are located respectively in the outer and the inner membrane. Tol and Pal proteins interact with each other, forming Tol/Pal complexes. By acting like press studs binding the inner and outer membranes, thus stabilizing the outer membrane using the peptidoglycan rigidity, the Tol/Pal complexes play a central role to preserve membrane integrity [5][6].

TolPal Ancored.png


The diffusion of proteins in these lipid bilayers can be modeled by Brownian motion. This diffusion model gives the probability law for the location of Tol and Pal in the membranes. The following three equations model the Brownian motion of Tol, Pal and Tol/Pal complexes, represented as particles.

Diffusion.png

In the above equations, CX denotes the concentration of protein X, and DX denotes the constant kinetic coefficient associated to X. The kTolPal and k*TolPal denote the reaction constants for Tol-Pal complexation. In the last equation, Φ is an unknown function representing the Tol-Pal complex motility. However, because the Tol-Pal complex is significantly larger than the other proteins, and is doubly anchored in membranes, we made the assumption that the mobility of Tol-Pal complexes is negligable in comparison to the mobility of isolated Tol and Pal molecules. Stated differently, we set Φ to zero. Lastly, additional constraints on Tol/Pal diffusion have been added to enforce a constant distance between Tol and Pal proteins.


In the above equations, the laplacians capture the non-homogeneous molecular diffusion on non-flat membranes. Indeed, they link the evolution of protein concentrations with the local shape of the membranes, since we know that on a two dimensional space:

Laplacian.png

where x and y are the coordinate of the surface and where the different partial derivatives are depending on the cartesian coordinates of the surface we are evolving on.

Unequal osmotic pressures create small blebbing

Using the equations for membrane deformation, we can compute numerically the shape of the outer membrane at steady state under various assumptions. If we assume that Tol/Pal complexes acts like press studs locally imposing a fixed distance between the two membranes, and that their initial distribution in the membrane is not totally homogeneous, then we obtain results of the following type.


Vesicle wiki.jpg
Shape at equilibrium of the outer membrane presenting blebs (red) resulting from osmotic pressure differences and with Tol/Pal complexes locally imposing a fixed distance between the inner (green) and outer membranes.

Tol/Pal accumulates at the basis of nascent blebs

The value of the Laplacian operators used to model Tol and Pal diffusions in the membranes (see equations above) depends on the membrane curvature: in non-flat membranes, the "efficiency" of the diffusion depends on the direction. In fact, molecules anchored in the outer membrane, like Pal, diffuse less efficiently in regions of negative curvature, and consequently tend to accumulate in these regions.

Using our computer program, we have been able to experimentally demonstrate this accumulation. The bottom plot presents the accumulation of Pal proteins in the membrane represented on the top plot. In the bottom plot, the left/right axis represents coordinates along the membrane, whereas the other horizontal axis represent the time evolution of protein concentrations (from time 70 to 100, instead of from 0 to 25 as indicated) One can show the protein accumulation at coordinates 11 and 43 approximately, corresponding to the regions of negative curvature of the membrane.


Accumulation2.png


Accumulation3.png


This result is corroborated by recent experimental findings showing that “negative membrane curvatures [act] as a cue for sub-cellular localization of a bacterial protein” [7].


Most notably, we can thus assume that Pal molecules accumulate at the basis of blebs. This naturally leads to an increase of Tol-Pal complexes formation in these regions. Lastly, because we additionally assumed that these large complexes diffuse less efficiently than Tol and Pal alone, we finally obtain that Tol-Pal complexes accumulate at the basis of the blebs. This is depicted in the following picture.


Shema Curvature.png

Rings of accumulated Tol/Pal constricts blebs basis

As explained above, Tol-Pal complexes tend to accumulate at the basis of the blebs, creating a ring of Tol-Pal (if seen from above).

Shema diffusion.png


When Tol-Pal molecules move to the interior of the ring, due to Brownian motion, they narrow the two membranes because they impose to have a constant distance between the two membranes as explained in the picture below: the proteins act like zippers on membranes, as depicted below. This leads to vesicle formation.


Zip.png

References

  1. ^1991 - Lipowsky - The conformation of membranes, Nature, 349(6309):475-481
  2. ^2008 - Park & Uehara - How bacteria consume their own exoskeletons, Microbiol Mol Biol Rev, 72(2):211-227
  3. ^1998 - Zhou et al - On the origin of membrane vesicles in gram-negative bacteria, FEMS microbiology letters, 163(2):223-228
  4. ^1987 - Ou-Yang & Helfrich - Instability and deformation of a spherical vesicle by pressure, Phys. Rev. Lett., 59:2486-2488
  5. ^2009 - Deatherage et al - Biogenesis of bacterial membrane vesicles, Mol Microbiol, 72(6):1395-1407
  6. ^2005 - Kuehn & Kesty - Bacterial outer membrane vesicles and the host pathogen interaction, Genes & Dev, 19:2645-2655
  7. ^2009 - Kumaran & Losick - Negative membrane curvature as a cue for subcellular localization of a bacterial protein. PNAS USA, 106(32):13541-13545