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| - | <h3>The Oscillator</h3>
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| - | In this part we describe the design of an oscillator that could be useful in biological networks. We designed one ourselves since we have very specific demands and look forward to the challenge. We even tried to create a system that creates synchronized oscillations without depending heavily on the components used. So our proposal oscillates inherently, and only slightly depends on the parameters of the components used. In this text, we start with an explanation of how this oscillating model fits within the framework of our project. Second, we explain several necessities to obtain a synchronized oscillator, and how we managed to incorporate those within our network. For the thorough study of the network and to see what has been achieved in the lab, we refer to the modeling page and the wetlab page respectively.</p>
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| - | <!-- Modeling -->
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| - | <div class="span2 bg-yellow visible-desktop"> | + | <div class="visible-desktop span3 bg-grey1"> |
| - | <a href = "https://2013.igem.org/Team:KU_Leuven/Project/Oscillator/modeling"> | + | <a href="https://2013.igem.org/Team:KU_Leuven/Project/ Aphid_Background"> |
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| + | <h3>Aphid Background</h3> |
| - | <div class="span10 icon-text "> | + | </a> |
| - | <h3>Modeling</h3> </a> | + | <p>Crashcourse in aphid biology</p> |
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| - | <p align="justify"> On this page we will talk about some more modeling stuff.</p > | + | |
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| - | <!-- Wetlab -->
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| - | <i class="livicon activeicon" data-name="lab" data-onparent="true" data-color="white"></i> | + | <a href="https://2013.igem.org/Team:KU_Leuven/Project/HoneydewSystem"> |
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| - | <div class="span10 icon-text "> | + | <h3>Honeydew System</h3> </a> |
| - | <h3>Wetlab</h3> | + | <p>Our BanAphids react to honeydew</p> |
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| - | <p align="justify"> The C1 FFL coming to life.</p > | + | |
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| - | <h3>Requirements</h3>
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| - | <!-- Why an oscillator? -->
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| - | <h3>Why would we use an oscillator?</h3> | + | <a href="https://2013.igem.org/Team:KU_Leuven/Project/StickerSystem"> |
| - | <p align="justify">An autonomous production system for β-farnesene might be a good solution in order to avoid having to put our systems directly on plants. However, a constitutive production of the pheromone, might rapidly render the aphids insensitive to it (Kunert, Reinhold and Gershenzon, 2010). Consequently we need a solution in which the production of β-farnesene is alternatingly on and off. In order to elaborate on the possibility of such a periodical production we investigated biological oscillating networks. A transcriptional network that exhibits oscillating behavior is the repressilator of Elowitz and Leibler (2000). This has been a cornerstone for synthetic biology since they were among the first to successfully introduce a synthetic model in a living organism. However, their paper mentions the lack of colony-wide synchronization. This is a necessity to achieve a periodical production, otherwise the variation will even out, resulting in a de facto constitutive expression. This means the repressilator does not suffice for a bacterial production unit with a periodical output.</p>
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| - | <div class="span4 greytext"> | + | <h3>Sticker System</h3> |
| - | <h3><br/></h3> | + | </a> |
| - | <img src="https://static.igem.org/mediawiki/2013/e/eb/Repressilator.png" alt="The repressilator"/> | + | |
| - | <p> Figure 1ǀ The repressilator by Elowitz and Leibler</p> | + | |
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| - | <!-- Synchronized oscillator -->
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| - | <h3><br/></h3>
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| - | <img src="https://static.igem.org/mediawiki/2011/e/e1/System3.png" alt="Wageningen Team 2011 project"/>
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| - | <p>Figure 2ǀ The oscillator of the Wageninen 2011 iGEM team.</p>
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| - | <h3>A synchronized oscillator</h3>
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| - | <p align="justify">The <a href="https://2011.igem.org/Team:Wageningen_UR"> iGEM team from Wageningen</a> tried to attain colony wide oscillations in 2011, by using the model proposed by Danino et al. (2010). This model provides a next step in the engineering of genetic circuits and is thoroughly described by the Wageningen 2011 iGEM team. As they mention on their wiki, this model heavily depends on the parameter values. Because we want to use an oscillator as a pace regulator, the eventual system will have multiple other inserted genes. The introduction of another set of genes besides an oscillator means an extra load on the current genetic circuit and this can influence the parameters of the oscillator-network (Shiue and Prather, 2012). To have a synchronized oscillator module that functions ‘independently’ on the presence of other modules, we need a system that gives oscillations for a broad range of parameters. This way it can preserve its oscillating behavior independently of possible other loads on the cell’s metabolism.</p>
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| - | <!-- Our very own oscillator. -->
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| - | <h3>Our very own oscillator</h3>
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| - | <p align="justify">The most important feature of our model is the sustained colony wide oscillations for a broad range of parameters. Meanwhile, for both the amplitude and frequency we do not pose stringent requirements. This is because the production of β-farnesene should definitely not be constitutive, but the height of the peaks in production and the time in between different peaks can vary as long as they remain within reasonable ranges. Since we know very well what we exactly desire from the oscillator we decided to design one ourselves. Another, equally important, reason for this decision is the fact that this design would offer a great journey on its own and give us the opportunity to learn a tremendous amount of new things, which is one of the things iGEM is about. </p>
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| - | <!-- The components -->
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| - | <h3><br/></h3> | + | <a href="https://2013.igem.org/Team:KU_Leuven/Project/E.coligy"> |
| - | <img src="https://static.igem.org/mediawiki/2013/0/09/KU_Leuven_team2.png" alt="Aphid milking"/>
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| - | <p>Figure xǀ Tekst.</p>
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| - | <div class="span8"> | + | <h3><i>E. coligy</i></h3> |
| - | <h3>The components</h3> | + | </a> |
| - | <p align="justify">We will design our model starting from an engineering perspective, inspired by the introduction to systems biology from Alon (2006). As the previously discussed models from the literature, we will also use a transcription factor network that creates oscillations. However, we also try to stretch the features of our model further by having the system not depend too much on what exactly are the used components. With this we mean that the several promoters and transcription factors we use, should be replaceable, without changing the fact that oscillations are produced. This is a very nice feature and is possible because our system will oscillate for a broad range of parameters. Previously we discussed the fact that a broad range of oscillating parameter is a necessity in order to preserve the oscillating behavior despite of other loads on the cell’s metabolism. However allowing components to be changed without altering the qualitative behavior of the system greatly increases the need for a broad range of oscillating parameters.</p> | + | <p> Validating the BanAphids in vivo</p> |
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| - | <h3>How we attain synchronization</h3>
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| - | <!-- Quorum Sensing. -->
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| - | <h3>Quorum sensing molecules</h3>
| + | <a href="https://2013.igem.org/Team:KU_Leuven/Parts"> |
| - | <p align="justify">As for a means of synchronization we will use molecules that oscillate colony-wide, for which the quorum sensing molecules used by Danino et al. (2010) offer a good example. Instead of individual intracellular oscillations that are synchronized by a mechanism that influences the entire colony, we aim for a colony-wide oscillation, which is intrinsically synchronized.<br/>
| + | <i class="livicon activeicon" data-name="gears" data-color="white"></i></div> |
| - | We will use two distinct colony-wide molecules, instead of only one, since that way we can have a less parameter sensitive oscillator. This happens when the two colony-wide molecules (indirectly) repress each other’s production, which will of course be the case in our model. When this is the case then when for instance there is an excess production of the colony-wide molecules, these excesses will partially repress each other. This implicates that a broader spectrum of colony densities should oscillate. As to make sure the evolution towards a stable steady state is avoided for reasonable parameters, we also require delaying steps to ascertain a separation in time of the successive peaks in production. By studying the possibilities for colony-wide transcription factors, we found quorum-sensing molecules as good candidates, since the ones we encountered serve as transcription factors (combined with intracellular receptors) and can diffuse out of the cells (Fugua, Winans and Greenberg, 1994). Because the quorum-sensing molecules are produced by enzymes, instead of being directly transcribed and translated, this extra step in which the enzymes are produced already induces an extra delay. On top of that the processes leading from the activation of a gene towards an active protein also take a finite amount of time, which also adds an extra delay. </p>
| + | <div class="span7 icon-text"> |
| | + | <h3>Parts</h3> </a> |
| | + | <p>BioBrick 'm all!</p> |
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| - | <h3><br/></h3>
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| - | <img src="https://static.igem.org/mediawiki/2013/6/67/Quorum_sensing.png" alt="Quorum sensing molecules"/>
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| - | <p>Figure xǀ Possible quorum sensing molecules.</p>
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| - | <img src="https://static.igem.org/mediawiki/2013/a/ac/Bacterial_Quorum_Sensing_by_CarolineDahl.png" alt="Quorum sensing model"/>
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| - | <p>Figure xǀ Typical model of quorum sensing.</p>
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| - | <!-- The logical circuit. -->
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| - | <div class="row-fluid"> | + | <a href="https://2013.igem.org/Team:KU_Leuven/Project/DataPage"> |
| - | <div class="span4 greytext"> | + | <i class="livicon activeicon" data-name="info" data-color="white"></i> |
| - | <h3><br/></h3> | + | </div> |
| - | <img src="https://static.igem.org/mediawiki/2013/0/09/KU_Leuven_team2.png" alt="Aphid milking"/>
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| - | <p>Figure xǀ Tekst.</p>
| + | <h3>Data Page</h3> </a> |
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| + | <p> All our achievements on one small page!</p> |
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| - | <h3>The logical circuit</h3> | + | |
| - | <p align="justify"> Figure X displays a logical circuit with our proposal for an oscillating model. It consists of two analogous halves that are meant to exhibit sequential peaks. In this system A and B represent the colony-wide molecules. They are produced by enzymes, which is indicated by the dotted lines, and of which the expression is controlled by the logical AND gate. We incorporated this enzyme step already because we will propose a practical implementation that uses quorum-sensing molecules later on . We will first explain how this system produces oscillations and afterwards, with a more thorough explanation of the subsystems and components, we will show how this inherently creates a synchronized oscillation.</p> | + | |
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| - | <!-- Basic mechanism. -->
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| - | <h3>Basic mechanism</h3>
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| - | <p align="justify">When there is a high level of A, this induces the production of both X and C, of which the latter will repress A and induce B. The production of C will not stop, even after A has disappeared below its threshold, since there is still X present that induces C. This effect creates an extended repression of A, in order to make sure the level of A drops sufficiently, so there is no production of X and C while the other halve of the system is active. This extended period in which C is present, also induces the production of B and the same story can be told for this halve. This means there are sequential peaks in the different components, which means this is an oscillating system.<br/>
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| - | However, we aren’t satisfied yet with only this. Our ultimate goal is to provide synchronized oscillations throughout the colony with a low parameter dependence. In order to describe this system we will first explain the importance of the prolonged expression of C (or D). Second, we will describe how this system attains synchronization and rapid resynchronization and lastly we will discuss the other parameter dependencies. </p>
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| - | <h3><br/></h3>
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| - | <img src="https://static.igem.org/mediawiki/2013/0/09/KU_Leuven_team2.png" alt="Aphid colors"/>
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| - | <p>Figure xǀ Text</p>
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| - | <!-- Prolonged production. -->
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| - | <h3><br/></h3>
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| - | <img src="https://static.igem.org/mediawiki/2013/0/09/KU_Leuven_team2.png" alt="Aphid milking"/>
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| - | <p>Figure xǀ Tekst.</p>
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| - | <h3>Prolonged production</h3>
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| - | <p align="justify">As mentioned above, there is a prolonged production of C (D) after the disappearance of A (B). However, when there is A (B) present, the production of C (D) starts immediately. This means there’s an asymmetric time delay for the production of the repressing molecules. The network motif that exhibits this behavior is called a ‘coherent type 1 feed forward loop’ (C1-FFL) with an OR logical gate (Mangan and Alon, 2003), of which one is highlighted in 3. This extended period in which one of the colony-wide molecules is repressed and the other produced, helps to reduce the effect of cell-to-cell variability in the different parameters, which is a necessity in order to have a synchronized oscillation. Even if not all cells produce a sufficient amount of repressing molecules, the prolonged repression by the others still results in an effective reduction in the amount of that colony-wide molecule, and analogously also a rise in the concentration of the other.</p>
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| - | <!-- Rapid resynchronization. -->
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| - | <h3>Rapid resynchronization</h3>
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| - | <p align="justify">The synchronized oscillation is an inherent property of this system. Because of the small space-scale, the diffusion distributes the molecules almost evenly throughout the colony, as long as most partake in the production. This colony-wide concentration of the diffusible molecules ascertains the rapid resynchronization of subpopulations that do not follow the others. If for instance, the unsynchronized subpopulation produces the other colony-wide molecule, that concentration does not reach any significant level because of the rapid diffusion at that scale. This can be explained by using an approximation of the diffusion coefficient of a typical quorum sensing molecule; DN-(3-Oxododecanoyl)-L-homoserine lactone equals 4.9 102 µm2/s (Stewart, 2003). <b>The equation</b> (Einstein, 1905) gives the average displacement of molecules because of diffusion in 3 dimensions after time t. In this example the average displacement after 1 second would be 54 µm, which is about equal to 50 times the size of one cell. This affirms the statement that the production of one cell is rapidly dispersed throughout its environment. Next, this also implies all cells have an approximately equal concentration, since the distance between cells in moderate density cultures is approximately 16 µm (= 2*10^8 cells/ml, Park et al., 2003). This distance corresponds with one third of the diffusion length after one second, if we assume the production is evenly distributed among the population. This colony-wide concentration holds the key to a rapid resynchronization as explained below. </p>
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| - | <img src="https://static.igem.org/mediawiki/2013/0/09/KU_Leuven_team2.png" alt="Aphid colors"/>
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| - | <p>Figure xǀ Text</p>
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| - | <h3>How we retain synchronization</h3> | + | <h3 class="bg-green">The Sticker System</h3> |
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| - | <!-- Noise. --> | + | <!-- INTRO --> |
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| - | <img src="https://static.igem.org/mediawiki/2013/0/09/KU_Leuven_team2.png" alt="Aphid milking"/>
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| - | <p>Figure xǀ Tekst.</p>
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| - | <h3>Coping with noise</h3>
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| - | <p align="justify">A principal problem that goes hand in hand with colony synchronization, is the constant presence of noise. This noise is partitioned into two sources: intrinsic noise, which originates within the regarded subsystem and extrinsic noise, which originates outside of it. We will describe this specifically for gene expression since we’ll build a genetic circuit with oscillating behavior. The intrinsic noise is due to the stochastic effects involved in the reaction events leading to transcription and translation. This leads to fluctuations in timing and order of those events, even in cells that are in an equal state. The extrinsic noise is for example due to a difference in the number of RNA polymerases and ribosomes and because they are gene products themselves are thus subject to their own intrinsic noise. Other factors that lead to extrinsic noise include the stage in the cell cycle, the quantity of the protein, the mRNA degradation machinery and the cell environment (Swain, Elowitz and Siggia, 2002). This substantial presence of noise has to be taken into account when developing a genetic circuit, either by suppressing it, or by building a network that is robust to it (Elowitz et al., 2002).</p>
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| - | <!-- Getting back in sync -->
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| + | Our initial system for the production of methyl salicylate and β-farnesene relied on direct interaction between the bacteria and the aphids, our BanAphids would be sprayed on the plant. This is ethically and practically challenging since our <b>BanAphids</b> could end up in the environment. <b>We therefore designed an alternative pheromone production system, where the bacteria are kept in semi-permeable pouches</b>. These allow the pheromones to disperse in the air yet the bacteria themselves remain in the bags. Consequently, the presence of aphids secreting honeydew cannot be used as a trigger for the production of both pheromones. <b>Since constitutive production of β-farnesene rapidly renders aphids insensitive, concentrations should fluctuate to prevent habituation. This is typically achieved in an oscillator.</b> The oscillator is extensively elaborated in the <b><a href="https://2013.igem.org/Team:KU_Leuven/Project/Modelling/Colony_Level">oscillator model section</a></b>. Here we will mainly focus on the design of a β-farnesene oscillator, a part of it will be actually made in the lab. The production of methyl salicylate can be regulated similarly.</p> |
| - | <h3>Getting back in sync</h3>
| + | |
| - | <p align="justify"> The most important impact of noise on the occurrence of synchronized oscillations is the fact that at any time, a proportion of the cells may oscillate in a different phase. It is consequently imperative that our model has rapid resynchronization as a feature. Otherwise this would lead to a netto constitutive expression from the colony perspective.< br/> | + | |
| - | If, by random occurrence, a certain amount of cells have an excess amount of non-diffusible molecules, as for instance one of the repressors, they will produce the colony-wide molecules at a different rate. The next switch in production, however, occurs for all of the cells approximately at the same time and serves as a resynchronization event. This can be explained because it is the colony-wide molecule that is being produced by the majority that first reaches the threshold needed to switch to the production of the other colony-wide molecule in all of the cells. Since the cells that were out of sync had not yet stopped producing that other molecule, they are resynchronized with the majority. This has to be seen in relation with the asymmetric time delay, which prolong the production of molecules that execute the switching events, ascertaining that it does happen.</p>
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| - | <h3><br/></h3>
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| - | <img src="https://static.igem.org/mediawiki/2013/0/09/KU_Leuven_team2.png" alt="Aphid colors"/>
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| - | <p>Figure xǀ Text</p>
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| - | <!-- An example --> | + | <!-- Aim --> |
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| - | <div class="span4 greytext"> | + | <div class="span2 bg-farnesene visible-desktop"> |
| - | <h3><br/></h3>
| + | <a href = "https://2013.igem.org/Team:KU_Leuven/Project/StickerSystem/Design"> |
| - | <img src="https://static.igem.org/mediawiki/2013/0/09/KU_Leuven_team2.png" alt="Aphid milking"/>
| + | <i class="livicon activeicon" data-name="pencil" data-onparent="true" data-color="white"></i> |
| - | <p>Figure xǀ Tekst.</p> | + | |
| | </div> | | </div> |
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| - | <div class="span8"> | + | <div class="span10 icon-text "> |
| - | <h3>An example</h3> | + | <h3>Design</h3> </a> |
| - | <p align="justify">2 shows an example of this principle. These images have been created by solving a partial differential equation, in which 40% of the cells have initially a quadruple amount of one of the repressors and a quarter of the others, in respect to the other 60% of the cells. 2 a and b show the spatial-temporal oscillations of respectively a colony-wide molecule, like A, (2 a) and a non-permeable molecule, like C (2 b). 2 a displays the above mentioned nearly uniform concentration of diffusible molecules. Even though close to half of the population behaves differently, they show a rapid resynchronization. 2 c and d display the production rates of the colony-wide molecules and of the molecules that each repress one and induce the other colony-wide molecule in the two different kind of cells. 2 c displays this for a cell belonging to the majority and 2 d for the other 40%.<br/> | + | <div class="row-fluid"> |
| - | They show only small differences between the production rates of the non-permeable molecules (red and yellow) and the switches in production happen at the same time. This is because those are controlled by the concentration of colony-wide molecules, which are approximately uniformly distributed. The production of the colony-wide molecules (blue and cyan) show a big difference during the first oscillations, but those differences rapidly disappear, until after the second oscillation there is no significant difference anymore and resynchronization is a fact.</p>
| + | <div class="span12"> |
| | + | <p align="justify"> In this part we explain the necessity of the sticker system, as well as its practical execution, such as the properties of the sticker and the implementation of an oscillator. </ p> |
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| - | <h3>Recapitulation</h3>
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| - | <!-- Why an oscillator? -->
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| - | <h3>Low parameter dependence</h3>
| + | <a href="https://2013.igem.org/Team:KU_Leuven/Project/Oscillator/wetlab"> |
| - | <p align="justify">In the above we already mentioned all of the reasons why a low parameter dependence is a necessity. Here we will first run through those once more. First of all this is a necessity in order to accomplish our goal of having a synchronized oscillating system that does not depend heavily on the components (promoters and transcription factors). This definitely demands the lowest amount of parameter dependence, but it is also the least crucial part of our model. Secondly this is important in the respect of the principle of synthetic biology. Building new life requires well defined modules which can be used together in order to create a new function. This means that in a cell that has our oscillator inside of it, other modules will be used at the same time. In our case, for example, we would couple it to a β-farnesene producing module. However, since we are working with a highly complex internal metabolism of the cell, those modules will interact and influence each other’s behavior. This means our model should still produce oscillations despite of those interactions, and thus not change its behavior when the parameters are slightly altered. Lastly a low parameter dependence also means a low dependence on the cell densities, which has been a problem encountered by Danino et al. (2010).</p> | + | <i class="livicon activeicon" data-name="lab" data-onparent="true" data-color="white"></i> |
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| - | <h3><br/></h3> | + | <h3>Integrating the oscillator into the wetlab</h3> </a> |
| - | <img src="https://static.igem.org/mediawiki/2013/0/09/KU_Leuven_team2.png" alt="Aphid colors"/> | + | <div class="row-fluid"> |
| - | | + | <div class="span12"> |
| | + | <p align="justify"> A part of the oscillator was implemented in the wetlab. You can find the genes we chose for our oscillator here, as well as the results of the wetlab experiments. </p > |
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| - | <!-- Staying synchronized -->
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| - | <img src="https://static.igem.org/mediawiki/2013/0/09/KU_Leuven_team2.png" alt="Aphid milking"/>
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| - | <p>Figure xǀ An ant 'milks' an aphid for his honeydew.</p>
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| - | <h3>Staying synchronized</h3>
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| - | <p align="justify">In the above we already showed why our model oscillates inherently and that shows that the system should not depend too heavily on differences in parameters. This is something that has been tested on the modeling page. We will also further elaborate on another aspect of parameter dependencies, namely the effect of cell-to-cell variability which means each of the cells will have slightly different parameters, even though they are genetically identical. This arises due to stochastic effects and our system makes the entire colony oscillate in sync despite this fact. We have already discussed that our system can withstand this thanks to the asymmetric time-delays <b>(HIER LINKEN NAAR DE DERDE PARAGRAAF ONDER DE EERSTE FIGUUR)</b>. We will now also highlight several other features that counter this effect. The cell-to-cell variability in for instance the thresholds is partly compensated because of the fact that, even though some cells have already reached the threshold for repression, the production of colony-wide molecules does not instantaneously drop. This inherent delay in transcription networks helps to cancel out fluctuations in the height of the threshold. Fluctuations in the production of the colony-wide molecules are canceled out because of the rapid dispersion. Next, the production of the non-diffusible molecules is also subject to fluctuations. However since they are produced over an extended period of time, because of the asymmetric time delay, fluctuations are at least partially canceled out over time. </p>
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| - | <h3>Further elaboration</h3>
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| - | <p align="justify">This theoretical work was a lot of fun to develop and is further extended on the <b>modelling page about the subject</b>, were we study whether each of the above mentioned features. On top of that we also begun the work of bringing it in to cells. Bringing the entire model into a living cell poses some problems however, of which the most important one is the required time for the many cloning steps. We do foresee that in the future this will become easier, when the field of synthetic biology gains importance. We chose to focus on a crucial part of the oscillator in the wetlab, since it serves as a useful network motif by itself and it can help other teams in further developing this oscillator. The transcriptional network motif we are making a biobrick of is a coherent type 1 feed forward loop which uses a quorum sensing molecule as the first transcription factor. The oscillator envelops this network motif twice, of which one is highlighted in green in 3. The reason we have chosen this natural occurring network motif is because of its central role in our proposed oscillator model, as well as in many other natural systems (Mangan and Alon, 2003). For the implementation, we refer you to the <b>wetlab page</b>.</p>
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| - | <h3>References</h3>
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| - | <p align="justify">Alon, U. (2006). An introduction to systems biology: design principles of biological circuits. CRC Press, 301.<br/>
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| - | Danino, T. et al. (2010). A synchronized quorum of genetic clocks. Nature, 463:326-330.<br/>
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| - | Einstein, A. (1905). On the movement of small particles suspended in stationary liquids required by the molecular-kinetic theory of heat. Annalen der Physik 17:549-560.<br/>
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| - | Elowitz, M. B., and Leibler, S. (2000). A synthetic oscillatory network of transcriptional regulators. Nature, 403(6767):335-338.<br/>
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| - | Elowitz, M. B. et al. (2002). Stochastic gene expression in a single cell. Science, 297(5584):1183-1186.<br/>
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| - | Fuqua, W. C., Winans, S. C., and Greenberg, E. P. (1994). Quorum sensing in Bacteria: the luxR-luxI family of cell density-responsive transcriptional regulators. J. Bacteriol, 176:269-275.<br/>
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| - | Kunert, G., Reinhold, C. and Gershenzon, J. (2010). Constitutive emission of the aphid alarm pheromone, (E)-β-farnesene, from plants does not serve as a direct defense against aphids. BMC Ecology, 10:23.<br/>
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| - | Mangan, S., and Alon, U. (2003). Structure and function of the feed-forward loop network motif. PNAS, 100:11980-11985.
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| - | Park, S. et al. (2003). Motion to form a quorum. Science, 301:188.<br/>
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| - | Shiue, E., and Prather, K. L. J. (2012). Synthetic biology devices as tools for metabolic engineering. Biochemical Engineering Journal, 65:82-89.<br/>
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| - | Stewart, P. S. (2003). Diffusion in biofilms. Journal of Bacteriology, 185:1485-1491.<br/>
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| - | Swain, P. S., Elowitz, M. B., and Siggia, E. D. (2002). Intrinsic and extrinsic contributions to stochasticity in gene expression. PNAS, 99:12795-12800.</p>
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