Team:Paris/Production modeling

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<span/ id="bottom">[https://2009.igem.org/ iGEM ] > [[Team:Paris#top | Paris]] > [[Team:Paris/Production_overview#top | Production]] > [[Team:Paris/Production_modeling#bottom | Modeling]]
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<span/ id="bottom">[https://2009.igem.org/ iGEM ] > [[Team:Paris#top | Paris]] > [[Team:Paris/DryLab#bottom | DryLab]] > [[Team:Paris/Production_modeling#bottom | Vesicle production Model (delay model)]]
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== Modeling ==
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== DryLab - Delay model: Improving message quality==
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<a class="menu_sub"href="https://2009.igem.org/Team:Paris/DryLab#bottom"> Main </a>|
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<a class="menu_sub_active"href="https://2009.igem.org/Team:Paris/Production_modeling#bottom"> Delay model</a>|
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<a class="menu_sub"href="https://2009.igem.org/Team:Paris/Production_modeling2#bottom"> Vesicle model</a>|
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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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<a class="menu_sub_active"href="https://2009.igem.org/Team:Paris/Production_modeling#bottom"> Introduction </a>|
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<a class="menu_sub"href="https://2009.igem.org/Team:Paris/Production_modeling#Implementing_the_delay_by_a_transcriptional_cascade"> Transcriptional Cascade</a>|
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<a class="menu_sub"href="https://2009.igem.org/Team:Paris/Production_modeling#Implementing_the_delay_by_feed-forward_motifs"> The feed-forward motif</a>
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==='''A. Genetic Network Regulation'''===
 
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It is crucial to send vesicules only when the concentration of proteins to be sent is at its maximum level in the periplasm. To increase the power of our communication system, we also tried to make this peak of vesicules creation correspond with the peak of protein concentration.
 
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This part aimed at solving the first problem linked to the creation of messengers :
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We had two hypothesis which led us to think over two different genetic regulatory network :
 
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*first the number of vesicules produced is at its maximum when the concentration of TolRII has reached its peak ; this achieve this configuration, we tried to design a delay system based on a transcriptional cascade as described in the part “The delay sytem”.
 
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*then, we also made the hypothesis that the number of vesicules by time units would be more important when the production of TolRII by time units is at its maximum ; this would allow a more powerful signal provided a good concentration of proteins. That is what we tried to put into place thanks to the use of feed-forwards ; see here for more explanations.
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<center>'''To improve the quality of signal sent, how can we get a good synchronization between the maximal vesicles production rate and the maximal concentration of proteins to encapsulate'''</center>
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<span/ id="1"><span/ id="2">
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Obviously, it is crucial to send vesicles only when the concentration of proteins to encapsulate is at its maximum level in the periplasm. To optimize the quality of our message, we tried to make this peak of vesicles creation corresponding to maximum levels of protein concentration. We assumed that the number of vesicles produced is at its maximum when the concentration of TolRII has reached its peak.
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===="The Delay system"====
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In what follows, we consider two different designs for implementing the delay: using a transcriptional cascade and using feed-forward motifs.
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We found that the '''design based on transcriptional cascade is sufficient for obtaining the desired delay'''<sup>[[https://2009.igem.org/Team:Paris/Production_modeling#References 1]]</sup>, whereas the '''design based on feed-forward motifs allows for more flexibility, at the cost of an increased complexity'''<sup>[[https://2009.igem.org/Team:Paris/Production_modeling#References 2]]</sup>.
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These studies are described below.
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The first thing that we wanted to study in modeling was the efficiency of the construction chosen to create a delay. Our first approach was a deterministic analysis of the system using differential equations. The regulation of promoters was described using the Hill functions and the methods described by U. Alon in his book ''An Introduction to Systems Biology''.
 
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===Implementing the delay by a transcriptional cascade===
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<span/ id="3"><span/ id="4">
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When the system is activated, there is Arabinose in the medium and the pBad promoters are activated. And the system can be described by this system of differential equations :
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We have decided to construct the following system.
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The addition of arabinose to the medium leads to the activation of the pBad promoters: two proteins are produced: the protein to encapsulate and LacI-LVA<sup>[[https://2009.igem.org/Team:Paris/Production_modeling#References 3]]</sup><sup>[[https://2009.igem.org/Team:Paris/Production_modeling#References 4]]</sup>. The latter inhibits the expression of TetR and thus releases the inhibition of TolRII, responsible for vesicle formation.
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[[Image:Global On.jpg|600px|center]]
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<span/ id="5">
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To study this system, we simply used differential equations. We used the method described by U. Alon in his book ''An Introduction to Systems Biology'', with notably Hill functions to describe the regulation of the promoters<sup>[[https://2009.igem.org/Team:Paris/Production_modeling#References 5]]</sup>.
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The network behavior can be described by a system of differential equations:
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|[[Image:Genetic network equations.png|500px| Modeling Overview]]
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|[[Image:Genetic network equations.png|400px| Modeling Overview]]
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These equations describe the evolution of the concentrations of the messenger protein P, LacI repressor, TetR repressor, and the TolRII protein responsible for vesiculation. The gammas are the degradation rates of the respective compounds, while the sigmoidal functions describe the non-linear production rates. The parameters are estimated according to typical values as discussed below</div>
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</html>
As a first approximation, we assumed that :
As a first approximation, we assumed that :
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*When the pBad promoter is induced, the concentration of arabinose in the medium is very high and constant during he whole study ; as a consequence, we will consider that the creation rate of Protein and LacI* is constant during the experiment :
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*When the pBad promoter is induced, the concentration of arabinose in the medium is very high and constant during he whole study; as a consequence, we will consider that the creation rate of Protein P and LacI* is constant during the experiment:
<center>
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|[[Image:Constant rate copie.png|450px|center| Arabinose constant]]
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|[[Image:Constant rate copie.png|360px|center| Arabinose constant]]
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<span/ id="6">
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*We considered that all the binding constants are identical and of an average of 40nM which correspond to approximately 40 monomers per cell ; we can write that :
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*We considered that all the binding constants are identical<sup>[[https://2009.igem.org/Team:Paris/Production_modeling#References 6]]</sup> and of an average of 40nM which correspond to approximately 40 monomers per cell; we can write that:
<center>
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|[[Image:Constant rate copie.png|450px|center| Binding constants]]
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|[[Image:K equals.jpg|175px|center| Binding constants]]
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*We chose identical intrisinc promoter activities, &beta; , all equal to 4000 proteins/cell cycle :
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*We chose identical intrinsic promoter activities, &beta;, all equal to 4000 proteins/cell cycle:
<center>
<center>
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*All the time units were expressed in units of cell cycle (approximately half an hour) ; as a consequence we chose a dilution rate &gamma; of 1 for protein without special tags. For the Laci protein with a LVA tag, the dilution rate is multiplied by 3:
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*All the time units were expressed in units of cell cycle (approximately half an hour). As a consequence we chose a dilution rate &gamma; of 1 for protein without special tags. For the LacI protein with a degradation tag (LacI-LVA), the dilution rate is 3 times higher<sup>[[https://2009.igem.org/Team:Paris/Production_modeling#References 3]]</sup>:
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|[[Image:Dilution One.png|250px|center| Binding constants]]
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|[[Image:Dilution One.png|200px|center| Binding constants]]
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|[[Image:Dilution LVA.png|100px|center| Binding constants]]
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|[[Image:Dilution LVA.png|80px|center| Binding constants]]
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[[Image:Delay System.jpg|600px|center]]
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For simulation, we used the following initial quantities:
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*4000 units of TetR (steady state level at inactivation)
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*no protein to be encapsulated
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*no LacI
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*no TolRII
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====The Feedforwards====
 
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We got the following dynamics using Matlab software, showing that '''this simple system is capable of generating the desired behavior'''.
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Using the delay system in our genetic regulatory network, we were able to find a way to get a maximum quantity of TolRII when the concentration of exported proteins is at its maximum.
 
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We also have thought of another device designed to have a reaction rate of TolRII production at its higher level when the concentration of exported is at its maximum. To this, end, we decided to combine two feed-forward loops to create a "burst" in the production of each protein.
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[[Image:Delay System.jpg|800px|center]]
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<center>
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Plot showing that vesiculation happens only once the protein encapsulated has reached its maximal concentration.
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</center>
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===Implementing the delay by feed-forward motifs===
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=====The Feed Forward Motif=====
 
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Using the previous design, we were able to find a way to get the maximal vesicle formation (ie maximal TolRII quantity) when the concentration of proteins to encapsulate is at its maximum. This system satisfies our requirements.
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However, it is not very flexible. In particular, the concentration of the protein to encapsulate remains always high after induction. This could be detrimental for the cell if this protein is toxic. Consequently, we '''considered an alternative design allowing for more flexibility''', in which the expression of the protein to encapsulate can be '''transient'''.
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The feed forward motif was studied and described by U. Alon in his book "An Introduction to Synthetic Biology" where he describes the different form of existing feed forward ; they can be either coherent or incoherent.
 
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In each case, there exist a different network organisation to create a feed-forward loop.
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====The Feed Forward Motif====
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<span/ id="7">
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The feed forward motif was studied and described by U. Alon in his book "An Introduction to Synthetic Biology" where he describes the different forms of existing feed forward; they can be either coherent or incoherent<sup>[[https://2009.igem.org/Team:Paris/Production_modeling#References 7]]</sup>.
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In each case, there exist a different network organization to create a feed-forward loop as shown on the picture
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The coherent feed forward loop allow the creation of a delay either on activation or on desactivation depending on the logical function between the two input X and Y.
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When the output logical gate (as shown on scheme) is an AND function, the delay appears on activation, while when the output gate is an OR function, the delay appears on desactivation.
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[[Image:FFd global.jpg|250px|left]]
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An example of caracterised feed-forward loop is the fliA system which was used by the [https://2008.igem.org/Team:Paris| 2008 Paris team] in order to create a oscillator.
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The X and Y inputs can be combined either using a logical AND or OR gate thus giving different dynamics properties in each case.
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The coherent feed forward loop allow the creation of a delay either on activation or on deactivation depending on the logical function between the two input X and Y.
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When the output logical gate is an AND function, the delay appears on activation, while when the output gate is an OR function, the delay appears on deactivation.
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An example of characterized feed-forward loop is the fliA system which was used by the [https://2008.igem.org/Team:Paris| 2008 Paris team] in order to create a oscillator.
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Two elementary incoherent feed forward motifs can be combined in order to create two time delayed bursts as described by U.Alon in its description of the development of ''Bacillus Subtilis'' spore. Anyway, it seems quite difficult to find a molecule which would be both an activator and a repressor ; we decided to try and think about another combination of these two feed forwards so as to get the dynamics we wanted as shown on the following part.
 
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=====Our design proposition=====
 
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The two feed-forward loops used are incoherent and agenced in a way that each molecule is either an activator or a repressor (and not both). The following scheme gives the functionning of the system.
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[[Image:Double IFFD.jpg|450px|right]]
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Two elementary incoherent feed forward motifs can be combined in order to create two time delayed bursts as described by U.Alon in its description of the development of ''Bacillus subtilis'' spore. Anyway, it seems quite difficult to find a molecule which would be both an activator and a repressor; we decided to try and think about another combination of these two feed forwards so as to get the dynamics we wanted as shown on the following part.
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====Our design proposition====
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The two feed-forward loops used are incoherent and organized in a way that each molecule is either an activator or a repressor (and not both). The following scheme gives the functioning of the system.
[[Image:Feed forward.png|300px|center]]
[[Image:Feed forward.png|300px|center]]
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|4
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|Finally, Z3 repressesthe promoter allowing the creation of Z5 and every molecules reaches its steady level concentration.
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|Finally, Z3 represses the promoter allowing the creation of Z5 and every molecules reaches its steady level concentration.
|Z1~   
|Z1~   
Z2~   
Z2~   
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It is possibleto obtain a important scale of values according to the choice of an optimized &#x03b2; for the production of molecule Z3 whose steady concentration and production rates impact on the steady state concentrations of the protein exported as well as the TolRII expressed. The &#x03b2; is chosen initially equal to 40 and 400 molecules by cell cycle and we have plotted the different response for multiples of this initial value.
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To study the dynamics of our model, we decided to use a classical approach using Michaelis-Menten equations to model the activation and repression of the various promoters, including a cooperativity of order 2 for the repressor which often the case for 'classical' repressors such as LacI. We obtained the following system of equations:
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<left>&#x03b2;=40</left>
 
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[[Image:Simu ff.png|450px|left]]
 
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<right>&#x03b2;=40</right>
 
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[[Image:Feed forward bis.png|450px|right]]
 
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==='''B. Modeling Vesicles creation.'''===
 
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Our model is based on three different physical phenomenon:
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[[Image:Equation feed.png|300px|center]]
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*The lipid surface conformation, Tol-Pal proteins diffusion and the increase of the osmotic pressure in the periplasm
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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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::The conformation of the lippopolysaccharides and the normal phospholipids can be seen has a lipid bilayer which conformed itself as a liquid crystal at the growth culture. The conformation of lipid bilayer has been well studied and a lot of theoretical results have been shown. Some of them are of great interest for our project. This conformation is something which can enable us to understand the way vesicle can be produced and how they will be received. In our project we first notice that from a basic calculus we can obtain very interesting results on the way outer membrane vesicles can be produced.
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::We first use the bending energy has a rough shape for our model and its understanding:
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|[[Image:Bending energy.png|200px]]
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</center>
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::This formula give use the abilities to explain the way lipids will organize together. E is the energy of a whole lipid bilayer (or monolayer).Kb and Kg are Bending and Gaussian modulii 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 are the mean curvature and the gaussian curvature. Our first task was to calculate the energy of two different shapes of membranes.
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::We compute the E.coli Shape before the division and then the vesicles' shape.
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::We consider E.coli's shape as a cylinder of rayon r =0.3 μm and of infinite length. The aim of this first representation was 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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{|
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|[[Image:BendingE AeraEcoli.png|150px]]
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</center>
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::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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<center>
 
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{|
 
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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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It is possible to obtain a broad spectrum of different dynamical behaviors by playing with the value of the maximum production rate of Z3, called here &#x03b2; . This maximal production rate impacts deeply both the evolution of the exported protein concentration and the evolution of TolRII concentration. This rate is initially chosen equal to 40 and 400 molecules by cell cycle and we have plotted the different responses for multiples of these initial values.
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</center>
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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: In the same way we can write for an area of E.coli lipids that the bending energy is:
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In each case, the initial conditions are the same :
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*no proteins encapsulated
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*no TolRII
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*no Z1 molecules
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*no Z2 molecules
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*no Z3 molecules
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*a weak amount of Z4 molecules (around a hundred)
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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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<center>Weak maximal Z3 production rate</center>
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|[[Image:Bending energy Vesicul.png|200px]]
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<center>''An example of dynamics for a range of weak &#x03b2; values; for each plot, the maximal Z3 production rate is given by: &#x03b2;=k*40''</center>
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[[Image:Simu ff.png|650px|center]]
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|}
 
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</center>
 
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::The considerations aforementioned 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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::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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{|
 
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<center>Strong maximal Z3 production rate</center>
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|[[Image:Whole energy.png|400px]]
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<center>''An example of dynamics for a range of strong &#x03b2; values; for each plot, the maximal Z3 production rate is given by: &#x03b2;=k*400''</center>
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[[Image:Feed forward bis.png|650px|center]]
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|}
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====A more flexible design====
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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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<center>
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Regardless of the steady state concentrations of the molecules, the use of two incoherent feed-forward loops allows for '''more flexibility in the behaviors of our system'''. With a simple delay system, we just get one possible dynamic behavior whereas using a variable range of production rates, we can obtain two behaviors:
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|[[Image:Final gamma0.png|250px]]
 
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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 and so:
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*When &#x03b2; is weak (&#x03b2; = 40), the number of vesicles (ie of TolRII) produced slowly increase before returning to zero, whereas the protein to be exported inside the vesicles is created very quickly and remains at a high concentration compared to the TolRII protein. We can therefore send vesicles containing an important amount of interesting proteins and, after a while, stop expressing dangerous proteins for the cell. All of this can be performed with a single input signal.
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<center>
 
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{|
 
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*When &#x03b2; is stronger, the behavior is different and we notice a change of concavity of the TolRII curve when the amount of protein exported is at its maximum. During such a change of concavity, the derivative reaches its maximum - in other words, the amount of TolRII produced by time units is at its highest level. Considering the hypothesis according to which the production rate of vesicles is proportional to TolRII production rate, we get a peak in vesicle production when the concentration of the proteins to exported is at its maximum.
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|[[Image:Num value.png|150px]]
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</center>
 
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::We can consider the orders of magnitude as really realistic. In fact we know that the E.coli lipid bilayer is built of distinct types of lipids: Lippopolysacharides 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.
 
-
*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.
 
 +
Unfortunately, we did not have the time to try to construct this system in the lab and only the first design was implemented. Nevertheless, we thought that '''this analysis was very useful to get a good insight of the diversity of the possible behaviors of the sender cells'''.
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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.
 
 +
{{Template:Paris2009_guided|Production_overview|Production_modeling2}}
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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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::To describe the time variation of the vesicles creation we first have to use a more complicated approach: we need to use an equation which is obtained by derivation of the bending energy:
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====References====
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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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<ol class="References">
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|[[Image:Formula1.png|500px]]
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<li> [[Team:Paris/Production_modeling#1 | ^]]Ultrasensitivity and noise propagation in a synthetic transcriptional cascade S.Hooshangi & R.Weiss
-
 
+
2005  - [http://www.ncbi.nlm.nih.gov/pubmed/15738412 15738412]</li>
-
|}
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<li> [[Team:Paris/Production_modeling#1 | ^]]Network motifs : theory and experimental approaches U.Alon
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</center>
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2007  - [http://www.ncbi.nlm.nih.gov/pubmed/17510665 117510665]</li>
-
 
+
<li> [[Team:Paris/Production_modeling#3 | ^]]New Unstable Variants of Green Fluorescent Protein for Studies of Transient Gene Expression in Bacteria Jens Bo Andersen & Søren Molin 1998  - [http://www.ncbi.nlm.nih.gov:80/pmc/articles/PMC106306/ 106306]</li>
-
Ou-Yang and Helfrich [Phys. Rev. Lett. 59 (1987) 2486]
+
<li> [[Team:Paris/Production_modeling#4 | ^]]Spatiotemporal control of gene expression with pulse-generating networks S.Basu & R.Weiss
-
 
+
2003  - [http://www.ncbi.nlm.nih.gov/pubmed/15096621 15096621]</li>
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:*Our first model which didn't use the brownian movement :
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<li> [[Team:Paris/Production_modeling#5 | ^]]Negative Autoregulation Speeds The Response Times of Transcription Network N.Rosenfold & U.Alon
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::We first conciders that the membrane equation can be first simplified to a one dimentional model :in a first approach we use a one dimensional model based on a polar curvature simplification approximation :
+
2002  - [http://www.ncbi.nlm.nih.gov/pubmed/12417193 2417193]</li>
-
 
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<li> [[Team:Paris/Production_modeling#6 | ^]]A Synthetic oscillatory network of transcriptionnal regulators  M.Ellowitz & S.Leibler 1999  - [http://www.ncbi.nlm.nih.gov/pubmed/10659856 10659856]</li>
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::the curvature in polar is:
+
<li> [[Team:Paris/Production_modeling#7 | ^]]Structure and function ot the feed-forward Loop Network Motif S.Mangan & U.Alon
-
 
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2003  - [http://www.ncbi.nlm.nih.gov/pubmed/14530388 14530388]</li>
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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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|[[Image:Curvature.png|200px]]
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::With the hypothesis of a cylinder form the mean and gaussian curvatures can be written as:
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{|
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|[[Image:Curvatures.png|75px]]
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</center>
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::indeed we have in those hypothesis:
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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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|[[Image:Principal curvatures.png|75px]]
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</center>
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::Thous we can concider the polar system of equation:
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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.
+
-
::Indeed the osmotic pressure can be written in this way:
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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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|[[Image:Osmotic pressure.png|200px]]
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|}
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</center>
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::And by concidering that we have weak concentrations we can concider a simpliest formula analogue to the perfect gases law:
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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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|[[Image:Osmotic pressure Analogue.png|100px]]
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::Finaly we can write:
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{|
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|- style="background: #0d3e99; text-align: center; color:white;"
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|[[Image:Delta pressure.png|200px]]
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::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:
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Latest revision as of 23:54, 21 October 2009

iGEM > Paris > DryLab > Vesicle production Model (delay model)




Contents

DryLab - Delay model: Improving message quality


This part aimed at solving the first problem linked to the creation of messengers :


To improve the quality of signal sent, how can we get a good synchronization between the maximal vesicles production rate and the maximal concentration of proteins to encapsulate


Obviously, it is crucial to send vesicles only when the concentration of proteins to encapsulate is at its maximum level in the periplasm. To optimize the quality of our message, we tried to make this peak of vesicles creation corresponding to maximum levels of protein concentration. We assumed that the number of vesicles produced is at its maximum when the concentration of TolRII has reached its peak.


In what follows, we consider two different designs for implementing the delay: using a transcriptional cascade and using feed-forward motifs. We found that the design based on transcriptional cascade is sufficient for obtaining the desired delay[1], whereas the design based on feed-forward motifs allows for more flexibility, at the cost of an increased complexity[2]. These studies are described below.


Implementing the delay by a transcriptional cascade

We have decided to construct the following system. The addition of arabinose to the medium leads to the activation of the pBad promoters: two proteins are produced: the protein to encapsulate and LacI-LVA[3][4]. The latter inhibits the expression of TetR and thus releases the inhibition of TolRII, responsible for vesicle formation.


Global On.jpg

To study this system, we simply used differential equations. We used the method described by U. Alon in his book An Introduction to Systems Biology, with notably Hill functions to describe the regulation of the promoters[5].


The network behavior can be described by a system of differential equations:

Differential System
Modeling Overview

These equations describe the evolution of the concentrations of the messenger protein P, LacI repressor, TetR repressor, and the TolRII protein responsible for vesiculation. The gammas are the degradation rates of the respective compounds, while the sigmoidal functions describe the non-linear production rates. The parameters are estimated according to typical values as discussed below


As a first approximation, we assumed that :

  • When the pBad promoter is induced, the concentration of arabinose in the medium is very high and constant during he whole study; as a consequence, we will consider that the creation rate of Protein P and LacI* is constant during the experiment:
Arabinose constant

  • We considered that all the binding constants are identical[6] and of an average of 40nM which correspond to approximately 40 monomers per cell; we can write that:
Binding constants


  • We chose identical intrinsic promoter activities, β, all equal to 4000 proteins/cell cycle:
Binding constants



  • All the time units were expressed in units of cell cycle (approximately half an hour). As a consequence we chose a dilution rate γ of 1 for protein without special tags. For the LacI protein with a degradation tag (LacI-LVA), the dilution rate is 3 times higher[3]:
Binding constants
Binding constants


For simulation, we used the following initial quantities:

  • 4000 units of TetR (steady state level at inactivation)
  • no protein to be encapsulated
  • no LacI
  • no TolRII


We got the following dynamics using Matlab software, showing that this simple system is capable of generating the desired behavior.


Delay System.jpg

Plot showing that vesiculation happens only once the protein encapsulated has reached its maximal concentration.


Implementing the delay by feed-forward motifs

Using the previous design, we were able to find a way to get the maximal vesicle formation (ie maximal TolRII quantity) when the concentration of proteins to encapsulate is at its maximum. This system satisfies our requirements. However, it is not very flexible. In particular, the concentration of the protein to encapsulate remains always high after induction. This could be detrimental for the cell if this protein is toxic. Consequently, we considered an alternative design allowing for more flexibility, in which the expression of the protein to encapsulate can be transient.


The Feed Forward Motif

The feed forward motif was studied and described by U. Alon in his book "An Introduction to Synthetic Biology" where he describes the different forms of existing feed forward; they can be either coherent or incoherent[7].

In each case, there exist a different network organization to create a feed-forward loop as shown on the picture


FFd global.jpg



The X and Y inputs can be combined either using a logical AND or OR gate thus giving different dynamics properties in each case.


The coherent feed forward loop allow the creation of a delay either on activation or on deactivation depending on the logical function between the two input X and Y. When the output logical gate is an AND function, the delay appears on activation, while when the output gate is an OR function, the delay appears on deactivation. An example of characterized feed-forward loop is the fliA system which was used by the 2008 Paris team in order to create a oscillator.


In an incoherent feed forward loop, the two inputs of the last gate function have two different logical levels; one molecule (molecule X for example) is an activator while the other molecule is a repressor of the promoter controlling the expression of the output. This way, when the input signal is on, the expression of molecule X starts activating both the promoters controlling the expression of Y and Z ; transcription starts and the amount of Y and Z molecules increases. Nevertheless, when the concentration of Y has reached its repression threshold, repression starts and the concentration of Z molecule decrease to its steady state.



Double IFFD.jpg



Two elementary incoherent feed forward motifs can be combined in order to create two time delayed bursts as described by U.Alon in its description of the development of Bacillus subtilis spore. Anyway, it seems quite difficult to find a molecule which would be both an activator and a repressor; we decided to try and think about another combination of these two feed forwards so as to get the dynamics we wanted as shown on the following part.

Our design proposition

The two feed-forward loops used are incoherent and organized in a way that each molecule is either an activator or a repressor (and not both). The following scheme gives the functioning of the system.

Feed forward.png


Steps Description Changes in concentration
1 When the input signal is ON, molecule1 is created and activates the promoter upstream the coding sequence for molecules Z2 and Z3 as well as Z4 since there is no molecule Z3 in the medium (see the AND gate function) at the beginning  ; there is an initial amount of Z5 molecule but no Z2 so the no Z6 molecule is produced. Z1 +++

Z2+ Z3+ Z4++ Z5~ Z6~

2 After a while, there is enough Z3 to start the repression of Z4 promoter and there is also enough Z2 to start producing Z6 signficantly. Z1+++

Z2+++ Z3+++ Z4- Z5~ Z6+

3 Then, because of the repression of Z3, the Z4 molecule reaches its steady level ; in the same time, Z3 starts repressing the production of Z5 and the amount of Z6 starts decreasing. Z1+

Z2+ Z3+ Z4~ Z5- Z6+

4 Finally, Z3 represses the promoter allowing the creation of Z5 and every molecules reaches its steady level concentration. Z1~

Z2~ Z3~ Z4~ Z5- Z6~


To study the dynamics of our model, we decided to use a classical approach using Michaelis-Menten equations to model the activation and repression of the various promoters, including a cooperativity of order 2 for the repressor which often the case for 'classical' repressors such as LacI. We obtained the following system of equations:



Equation feed.png



It is possible to obtain a broad spectrum of different dynamical behaviors by playing with the value of the maximum production rate of Z3, called here β . This maximal production rate impacts deeply both the evolution of the exported protein concentration and the evolution of TolRII concentration. This rate is initially chosen equal to 40 and 400 molecules by cell cycle and we have plotted the different responses for multiples of these initial values.

In each case, the initial conditions are the same :

  • no proteins encapsulated
  • no TolRII
  • no Z1 molecules
  • no Z2 molecules
  • no Z3 molecules
  • a weak amount of Z4 molecules (around a hundred)


Weak maximal Z3 production rate
An example of dynamics for a range of weak β values; for each plot, the maximal Z3 production rate is given by: β=k*40
Simu ff.png



Strong maximal Z3 production rate
An example of dynamics for a range of strong β values; for each plot, the maximal Z3 production rate is given by: β=k*400
Feed forward bis.png

A more flexible design

Regardless of the steady state concentrations of the molecules, the use of two incoherent feed-forward loops allows for more flexibility in the behaviors of our system. With a simple delay system, we just get one possible dynamic behavior whereas using a variable range of production rates, we can obtain two behaviors:


  • When β is weak (β = 40), the number of vesicles (ie of TolRII) produced slowly increase before returning to zero, whereas the protein to be exported inside the vesicles is created very quickly and remains at a high concentration compared to the TolRII protein. We can therefore send vesicles containing an important amount of interesting proteins and, after a while, stop expressing dangerous proteins for the cell. All of this can be performed with a single input signal.


  • When β is stronger, the behavior is different and we notice a change of concavity of the TolRII curve when the amount of protein exported is at its maximum. During such a change of concavity, the derivative reaches its maximum - in other words, the amount of TolRII produced by time units is at its highest level. Considering the hypothesis according to which the production rate of vesicles is proportional to TolRII production rate, we get a peak in vesicle production when the concentration of the proteins to exported is at its maximum.



Unfortunately, we did not have the time to try to construct this system in the lab and only the first design was implemented. Nevertheless, we thought that this analysis was very useful to get a good insight of the diversity of the possible behaviors of the sender cells.


Open book.gif

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References

  1. ^Ultrasensitivity and noise propagation in a synthetic transcriptional cascade S.Hooshangi & R.Weiss 2005 - [http://www.ncbi.nlm.nih.gov/pubmed/15738412 15738412]
  2. ^Network motifs : theory and experimental approaches U.Alon 2007 - [http://www.ncbi.nlm.nih.gov/pubmed/17510665 117510665]
  3. ^New Unstable Variants of Green Fluorescent Protein for Studies of Transient Gene Expression in Bacteria Jens Bo Andersen & Søren Molin 1998 - [http://www.ncbi.nlm.nih.gov:80/pmc/articles/PMC106306/ 106306]
  4. ^Spatiotemporal control of gene expression with pulse-generating networks S.Basu & R.Weiss 2003 - [http://www.ncbi.nlm.nih.gov/pubmed/15096621 15096621]
  5. ^Negative Autoregulation Speeds The Response Times of Transcription Network N.Rosenfold & U.Alon 2002 - [http://www.ncbi.nlm.nih.gov/pubmed/12417193 2417193]
  6. ^A Synthetic oscillatory network of transcriptionnal regulators M.Ellowitz & S.Leibler 1999 - [http://www.ncbi.nlm.nih.gov/pubmed/10659856 10659856]
  7. ^Structure and function ot the feed-forward Loop Network Motif S.Mangan & U.Alon 2003 - [http://www.ncbi.nlm.nih.gov/pubmed/14530388 14530388]

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