Team:Tokyo Tech/Projects/RPS-game/index.htm

From 2011.igem.org

(Difference between revisions)
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<p>
<p>
-
<h2 id="1.">1. The Human-Bacteria Rock, Paper and Scissors game</h2>
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-
 
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<h2 id="1">1. The Hands</h2>
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<h3 id="1.1">1.1 The Hands</h3>
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<p>
<p>
-
The first step towards making an RPS game that can be played between humans and
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-
bacteria is giving each player a set of signaling molecules through which
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-
they can communicate their choice of rock, paper or scissors.
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-
For that purpose we created two sets of three signaling molecules corresponding
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-
each to rock, paper or scissors. For humans we used IPTG, aTc and salicylate,
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-
respectively. For E. coli we used 3O-C6-HSL, 3O-C12-HSL and AI-2, respectively.
+
</p>
</p>
-
<h3 id="1.2">1.2 The Judge</h3>
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<h2 id="2">2. The Judge</h2>
<p>
<p>
-
We have our set of six signaling molecules that can be used to play the RPS game,
+
 
-
but we still need to find a way to know who wins the game.
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-
To know who the winner of each game is, we designed a Judge E. coli that has a built
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-
in AND-gate promoter and can let us know its decisions by expressing a particular
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-
fluorescent protein. GFP, RFP or CFP to indicate whether humans win,
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-
lose or it is a tie, respectively.
+
</p>
</p>
-
<h4 id="1.2.1">1.2.1 Creating New Parts</h4>
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<h3 id="2.1">2.1 Using AND-Gate promoters to create Judges</h3>
<p>
<p>
-
<p>
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The first step to make the Judge E. coli was to make sure there were AND-gates
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-
promoters that could work properly. We tested the AND-gate promoter designed by
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-
iGEM 2006's team Tokyo Alliance and confirmed it works well and can be used
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as a logic device to know who the winner of the RPS game is.
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Since in our human-bacteria RPS game each of the two players has one set of three
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-
different signaling molecules, we need 9 different AND-gate promoters,
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-
each corresponding to one of the 9 possible two signal combinations.
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-
</p>
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-
<p>
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In the process of constructing enough AND-gates that could suffice the needs of
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-
our RPS game design we discovered two faulty BioBricks and made new parts that can
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-
replace them. Namely, we made a lasI promoter and a lsrR promoter that work well.
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-
These promoters are single input promoters but they are useful as a reference to
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-
construct AND gate promoters. Also note that LsrR is essential for regulating the
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-
AND gate promoter. Therefore, we solved important issues and made significant
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-
advances towards constructing AND-gate promoters.
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-
</p>
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</p>
</p>
-
<h4 id="1.2.2">1.2.2 The Design</h4>
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<h3 id="2.2">2.2 Creating Parts that responded correctly to our set of Signaling Molecules</h3>
<p>
<p>
-
We designed AND-gate promoters that use two operators:
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-
one that uses activators as regulators, and other that uses repressors as regulators.
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-
After an activator binds to an inducer the resulting complex binds to the
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-
corresponding operator and activates it. However, since the AND-gate promoter needs
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-
its both operators to be active, transcription will not start until the operator
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that uses repressors as regulators has been de-repressed by the appropriate signaling
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-
molecule. This mechanism assures that transcription will not start until both players
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have added the corresponding signaling molecules to the place where the Judge
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-
bacterium is, so it fits perfectly in our RPS game design. To see the game results
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-
easily, we added a reporter gene downstream of the AND-gate promoter, so either of gfp,
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-
rfp or cfp is expressed when humans win, lose or tie the game, respectively.
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</p>
</p>
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<h5 id="1.2.1.1">1.2.1.1</h5>
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<h3 id="2.3">2.3 Improving PlsrA</3>
  <p>
  <p>
-
  <img alt="TokyoAliance AndGate" />
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<div class="graph_title">Tokyo Alliance AND-gate promoter</div>
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We used this Plux-lac hybrid promoter which contains a LacI operator,
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a LuxR operator and luxR. We introduced this plasmid into a LacI expressing E. coli
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strain pTrc99A (JM2.300). Because IPTG controls the binding of LacI to two
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LacI-operator parts and 3OC6-HSL controls the binding of LuxR to a LuxR-operator part,
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GFP of the reporter part is dually regulated by IPTG and 3O-C6-HSL.
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We also used promoterless pSB3K3-GFP (BBa_J54103) as a negative control, and Pλ-GFP
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(chloramphenicol-resistance), which constitutively expressed GFP, as a positive control.
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</p>
</p>
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<h5 id="1.2.1.2">1.2.1.2</h5>
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<h3 id="2.4">2.4 Improving PlasI</h3>
<p>
<p>
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<img alt="Signal table" />
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</p>
</p>
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<h5 id="1.2.1.3">1.2.1.3</h5>
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<h2 id="3">3. The Randomizers</h2>
<p>
<p>
-
<p>
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When doing experiments to make more AND-gate promoters, we found that the parts
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</p>
-
corresponding to the plasI promoter (BBa_###) and plsrA promoter (BBa_###) did not
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work as expected.
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</p>
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<p>
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<img alt="PlasI previous" />
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<div class="graph_title">Not working lasI promoter (BBa_J64010)</div>
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<img alt="Plasmid map" />
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<h3 id="3.1">3.1 Single Colony Isolation</h6>
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<img alt="PlsrA prev" />
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<div class="graph_title">Not working lsrA promoter (BBa_K117002)</div>
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</p>
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<p>
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We confirmed that the lsrA promoter BBa_K117002 does not work
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(samples used our experiments are listed in Table 1.)  We measured transcriptional
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activity of lsrA promoter by introducing a gfp gene downstream of this promoter
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(Fig.1).The intensity level of GFP fluorescence of BBa_1K17002 were lower
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even than those of the negative control (Fig.2), which clearly shows that
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lsrA promoter(BBa_K117002) does not work properly.
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</p>
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<p>
<p>
-
Because of these faulty parts, our game design became flawed and the Judge E. coli
 
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could only sense the Player E. coli's signaling molecule when the latter chose
 
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that signaling molecule corresponding to Rock.
 
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<img alt="e.coli in sad" />
 
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</p>
 
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<p>
 
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To solve this problem we made new parts that can replace these faulty parts.
 
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Below you can see that our plasI (BBa_###) and plasrA (BBa_###) work perfectly!
 
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This allows E. coli to also chose the signaling molecules corresponding to
 
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Paper and Scissors, so we have again a working RPS game design.
 
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<img alt="human helped e.coli" />
 
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<img alt="Assay data"/>
 
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<img alt="plasmid map" />
 
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</p>
 
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<p>
 
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We made new lasI promoter and confirmed it works as expected. In our assay,
 
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we used the same LasR regulator part used in the assay of BBa_ J64010. Clearly,
 
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for our part the fluorescence intensity of 3OC12-HSL+ was higher than that of
 
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3O-C12-HSL-. To see details about our assay method click here.
 
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<img alt="assay data" />
 
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</p>
 
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<p>
 
-
We succeeded in the construction of the first working iGEM lsrA promoter
 
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((BBa_K649100) and made sure it works properly.  This lsrA promoter is repressed
 
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by our new LsrR part. These working parts allow us to use AI-2 as a signaling molecule,
 
-
which is a very powerful tool to build complex Synthetic Biology systems,
 
-
since AI-2's mechanism prevents it from cross-talking with other signaling molecules
 
-
such as AHL.
 
-
</p>
 
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<p>
 
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In our experiment, we compared the transcriptional activity of the previous lsrA
 
-
promoter (BBa_K117002) and our new lsrA promoter (BBa_K649100). We measured the
 
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transcriptional activity of our lsrA promoter by introducing a gfp gene downstream of
 
-
this promoter (Fig.3).  The intensity levels of GFP fluorescence of BBa_K649100 were
 
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much higher than BBa_117002 (Fig.4), which shows that our new lsrA promoter
 
-
(BBa_K649100) works.
 
-
</p>
 
</p>
</p>
-
<h6 id="1.2.1.3.1">1.2.1.3.1 Our New LsrR part</h6>
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<h3 id="3.2">3.2 Conditional Knockout by Recombination</h6>
<p>
<p>
-
We confirmed that our new LsrR part represses the lsrA promoter.
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-
We measured LsrR repression activity by introducing a gfp gene downstream of PlsrA
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-
(Fig.5).  The intensity of GFP fluorescence of sample3 was three times as large
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-
as that of sample 4 (Fig.6). This result shows that LsrR successfully repressed PlsrA.
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-
<img alt="assaydata" />
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<table>
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<tr>
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<th>name</th>
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<th>strain</th>
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<th>plasmid</th>
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-
</tr>
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<tr>
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<td>sample1</td>
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<td>JM2.300</td>
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<td>pSB6A1 Ptet GFP RBS1-12</td>
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-
</tr>
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-
<tr>
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<td>sample2</td>
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<td>JM2.300</td>
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<td>pSB6A1 &Delta;P GFP</td>
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-
</tr>
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<tr>
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<td>sample3</td>
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<td>MG1655</td>
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<td>pSB3K3 PlsrA GFP</td>
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-
</tr>
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-
<tr>
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<td>sample4</td>
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<td>MG1655</td>
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<td>pSB3K3 PlsrA GFP PlsrR lsrR</td>
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-
</tr>
+
-
</table>
+
</p>
</p>
-
<h6 id="1.2.1.3.2">1.2.1.3.2 The AI-2 Mechanism</h6>
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<h3 id="3.3">3.3 The Randomizers</h3>
<p>
<p>
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<p>
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AI2 is synthesized by LuxS and accumulates extracellularly. Then AI2 is imported by
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-
LsrACDB transporter and AI2 is phosphorylated by the LsrK kinase. Phospho-AI2
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relieves LsrR inhibition and LsrR is the repressor of the lsr operon and itself.
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-
<img alt="AI2 signaling" />
+
</p>
</p>
 +
 +
<h3 id="3.4">3.4 Survival of one strain</h3>
<p>
<p>
-
First AI-2 is synthesized by LuxS and accumulates outside the cell. Then AI-2 is
 
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imported into the cell by the LsrACDB transporter. Inside the cell, AI-2 is
 
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phosphorylated by the LsrK kinase. Phospho-AI2 relieves LsrR inhibition from the
 
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lsr operon, therefore activating PlsrA.
 
-
</p>
 
-
</p>
 
-
 
-
<h3 id="1.3">1.3 The Randomizers</h3>
 
-
<p>
 
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To complete our RPS game design, we need to make sure E. coli follows the rules of
 
-
the game by synthetizing only one signaling molecule every time it plays. Additionally,
 
-
we want it to be able to choose its signal randomly. In view of these needs,
 
-
we designed three randomizers that satisfy the conditions for the game Single Colony
 
-
Isolation, Survival of the Fittest, and Conditional Knockout.
 
-
</p>
 
-
<h4 id="1.3.1">1.3.1 Three Randomizers</h4>
 
-
<p>
 
-
<p>
 
-
Our set of six signaling molecules allows us to play RPS with E. coli, but in order
 
-
to be able to play properly we must make sure E. coli can chose either of its three
 
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signaling molecules with the same probability. To do so, we designed three types of
 
-
randomizers: two of them are three-bacterium randomizers and the other is a
 
-
single-bacterium randomizer.
 
-
</p>
 
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<p>
 
-
Namely, the three-bacterium randomizers are Single Colony Isolation and Survival of
 
-
the Fittest. The single-bacterium randomizer is Conditional Knockout by Recombination.
 
-
</p>
 
</p>
</p>
-
<h5 id="1.3.1.1">1.3.1.1 Single Colony Isolation</h5>
 
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<p>
 
-
This is our simplest randomizer design. To make sure E. coli choses any of its signaling
 
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molecules with equal probability, we put the constructs for each molecule inside a
 
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different bacterium, so we create three types of bacteria: one synthetizing the
 
-
corresponding signaling molecule for rock, other synthetizing the corresponding
 
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signaling molecule for paper, and lastly one synthetizing the corresponding signaling
 
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molecule for scissors. By randomly isolating a single colony out of these three
 
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colonies we get a random output as E. coli's choice for the RPS game.
 
-
</p>
 
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<h5 id="1.3.1.2">1.3.1.2 Survival of the Fittest</h5>
 
-
<p>
 
-
<p>
 
In 1996 Durret and Levin described a system of three types of bacteria that competed  
In 1996 Durret and Levin described a system of three types of bacteria that competed  
for survival in dynamic that resembled a Rock-Paper-Scissors (RPS) game. The bacteria  
for survival in dynamic that resembled a Rock-Paper-Scissors (RPS) game. The bacteria  
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The colicin resistant bacteria were also able to produce colicin, but at a lower  
The colicin resistant bacteria were also able to produce colicin, but at a lower  
energetic cost, which allowed them to have a higher birth rate.
energetic cost, which allowed them to have a higher birth rate.
-
</p>
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-
<p>
+
The system was described by the following general differential equations
The system was described by the following general differential equations
-
</p>
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-
<p>
+
∑_(i=1)^n?u_i  &lt; 1<br />
∑_(i=1)^n?u_i  &lt; 1<br />
(du_i)/dt=β_i u_i u_0- u_i ( δ_i+∑_(j=1)^(n-1)??γ_j u_j ?)<br />
(du_i)/dt=β_i u_i u_0- u_i ( δ_i+∑_(j=1)^(n-1)??γ_j u_j ?)<br />
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Note that these parameters satisfy the following relations:
Note that these parameters satisfy the following relations:
β_3>β_2>β_1, γ_1> γ_2, and δ_1=δ_2= δ_3
β_3>β_2>β_1, γ_1> γ_2, and δ_1=δ_2= δ_3
-
</p>
+
-
<p>
+
<img alt="Durret and Levin" />
<img alt="Durret and Levin" />
Durret &amp; Levin (1996)<br />
Durret &amp; Levin (1996)<br />
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either to u_3 or u_2. We highlighted this fact by circling the point in the graph below.
either to u_3 or u_2. We highlighted this fact by circling the point in the graph below.
<img alt="Durret and Levin remark" />
<img alt="Durret and Levin remark" />
-
</p>
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-
<p>
+
The existence of this instability along the u_1 axis does not allow us to construct a  
The existence of this instability along the u_1 axis does not allow us to construct a  
randomizer that can function based on minimal differences in the initial concentrations  
randomizer that can function based on minimal differences in the initial concentrations  
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and additionally the resistant strain would also be vulnerable to the colicin produced  
and additionally the resistant strain would also be vulnerable to the colicin produced  
by the colicin producer.
by the colicin producer.
-
</p>
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-
<p>
+
Producer
Producer
(du_1)/dt=β_1 u_1 u_0- u_1 δ_1
(du_1)/dt=β_1 u_1 u_0- u_1 δ_1
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γ_1=2,γ_2=0.5,? ? γ?_1'=0.1 ,γ?_2'=0.002
γ_1=2,γ_2=0.5,? ? γ?_1'=0.1 ,γ?_2'=0.002
δ_1=δ_2= δ_3=1
δ_1=δ_2= δ_3=1
-
</p>
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-
<p>
+
and we graph this equations using a Matlab program, we get a graph which clearly  
and we graph this equations using a Matlab program, we get a graph which clearly  
shows there are stable points on each of the three axes.
shows there are stable points on each of the three axes.
<img alt="new 3d" />
<img alt="new 3d" />
-
  </p>
+
 
-
<p>
+
These stable points (u_1,0,0), (0,u_2,0) and (0,0,u_3) indicate that for the  
These stable points (u_1,0,0), (0,u_2,0) and (0,0,u_3) indicate that for the  
equations we have set all of the three strains may ultimately survive for infinite  
equations we have set all of the three strains may ultimately survive for infinite  
peiriods of time even if the initial density of the other two strains is positive.  
peiriods of time even if the initial density of the other two strains is positive.  
-
</p>
+
-
<p>
+
To see the difference between our model and Durret and Levin's model more clearly,  
To see the difference between our model and Durret and Levin's model more clearly,  
we also plotted Durret and Levin's model using Matlab.
we also plotted Durret and Levin's model using Matlab.
<img alt="prev. 3D" />
<img alt="prev. 3D" />
-
</p>
+
-
<p>
+
Note that the parameters we have set for our equations satisfy the initail conditions  
Note that the parameters we have set for our equations satisfy the initail conditions  
of the model proposed by Durret and Levin (indicated in black font) in the sense  
of the model proposed by Durret and Levin (indicated in black font) in the sense  
that β_3&gt;β_2&gt;β_1, γ_1&gt;γ_2  &gt;γ_1'&gt;γ?_2', and δ_1=δ_2= δ_3.  
that β_3&gt;β_2&gt;β_1, γ_1&gt;γ_2  &gt;γ_1'&gt;γ?_2', and δ_1=δ_2= δ_3.  
The two new terms we added are indicated in red.  
The two new terms we added are indicated in red.  
-
</p>
+
-
<p>
+
From a biological perspective, this model describes the existence of two strains of  
From a biological perspective, this model describes the existence of two strains of  
bacteria that produce two different types of bacteriocin. One of these strains is not  
bacteria that produce two different types of bacteriocin. One of these strains is not  
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in its bacteriocin resistance does not produce enough/effective resistance protein,  
in its bacteriocin resistance does not produce enough/effective resistance protein,  
which could be a consequence of it being a mutant of a colicin-sensitive strain.
which could be a consequence of it being a mutant of a colicin-sensitive strain.
-
</p>
+
-
<p>
+
The next step is to find a set of parameters that satisfy the conditions set by the  
The next step is to find a set of parameters that satisfy the conditions set by the  
model of Durret and Levin, the above condition of allowing stable points on each of  
model of Durret and Levin, the above condition of allowing stable points on each of  
Line 734: Line 549:
<img alt="sensitive" />
<img alt="sensitive" />
-
</p>
+
-
<p>
+
We found, by writing another program in Matlab, that the following set of parameters  
We found, by writing another program in Matlab, that the following set of parameters  
satisfy all of the above mentioned conditions.  
satisfy all of the above mentioned conditions.  
-
</p>
+
-
<p>
+
β_3=3, β_2=3.2,? β?_1=5.1
β_3=3, β_2=3.2,? β?_1=5.1
γ_1=2,γ_2=1,? ? γ?_1'=0.2 ,γ?_2'=0.01
γ_1=2,γ_2=1,? ? γ?_1'=0.2 ,γ?_2'=0.01
δ_1=δ_2= δ_3=1
δ_1=δ_2= δ_3=1
-
</p>
+
-
<p>
+
As can be seen in the graphs below, each of the strains can survive if their initial  
As can be seen in the graphs below, each of the strains can survive if their initial  
density in only tree hundredths (a.u.) greater than the other two strains' initial  
density in only tree hundredths (a.u.) greater than the other two strains' initial  
concentrations.  
concentrations.  
-
</p>
+
-
<p>
+
The producer strain survives if u_1= 0.32, u_2= 0.29, u_3= 0.29 (a.u.).<br />
The producer strain survives if u_1= 0.32, u_2= 0.29, u_3= 0.29 (a.u.).<br />
The limited-resistance strain survives if u_1= 0.29, u_2= 0.32, u_3= 0.29 (a.u.).<br />  
The limited-resistance strain survives if u_1= 0.29, u_2= 0.32, u_3= 0.29 (a.u.).<br />  
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that this randomizer is not only feasible but also practical and effective  
that this randomizer is not only feasible but also practical and effective  
(let alone interesting).
(let alone interesting).
-
</p>
 
-
</p>
 
-
 
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<h5 id="1.3.1.3">1.3.1.3 Conditional Knockout by Recombination</h5>
 
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<p>
 
-
<p>
 
-
Our third randomizer differs from our other two randomizers in that it is a
 
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single-bacterium randomizer.  It is based on the recombination mechanism of the enzyme
 
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Cre and the lox sequences. We designed a Cre-Lox system which allows E. coli to
 
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express one of its three signaling molecules by means of conditional knockout.
 
-
The design is depicted in Figure 1.
 
-
</p>
 
-
<p>
 
-
(a)
 
-
<img alt="cre" />
 
-
 
-
(b)
 
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<img alt="plasmid" />
 
-
Figure 1 - (a) Cre recombinase construction.
 
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(b) Lox cassettes distribution for the randomizer design
 
-
</p>
 
-
<p>
 
-
Basically, each pair of lox sites (indicated by the same color) mark the points which
 
-
the enzyme Cre will excise (they will be cut off the backbone along with the
 
-
sequence between them). For a design that allows choosing randomly one of E. coli's
 
-
three signaling molecules, at least two cassettes of lox sites are needed.
 
-
When these two cassettes of lox sites and protein coding sequences are arranged as in
 
-
Figure 1(b), only one signaling molecule is produced.
 
-
</p>
 
-
<p>
 
-
For example, when the blue cassette of Lox sites is excised, the signaling molecule
 
-
coded by LasI (3OC6-HSL) will be produced. Likewise, when the purple Lox cassette is
 
-
excised, the signaling molecule coded by LuxS (AI-2) will be produced.
 
-
On the other hand, a third possible outcome is that recombination does not take place.
 
-
In this case, the signaling molecule coded by LuxI (3OC6-HSL) will be produced.
 
-
Also note that excision of one kind of lox cassette removes the remaining cassette,
 
-
thereby preventing further recombination.
 
-
</p>
 
-
<p>
 
-
To avoid excision of lox sites from different cassetes (for example one blue lox site
 
-
and one purple lix site), we used a lox71-lox66 cassette and a lox2272-lox2272 cassette.
 
-
It is well known that lox71/lox2272 and lox66/lox2272 are incompatible
 
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(Zorana Carter and Daniela Delneri et al, Yeast 2010).
 
-
</p>
 
-
<p>
 
-
Also, to be able to control the LuxI gene's expression, we used an inducible promoter
 
-
instead of a constitutive promoter. A constitutive promoter will cause LuxI gene to be
 
-
expressed beforehand and could lead to an E.coli producing two signaling molecules
 
-
at the same time (the equivalent of showing two hands in the RPS game). In contrast,
 
-
the inducible promoter prevents a particular gene from expressing preferentially.
 
-
</p>
 
-
<p>
 
-
Now that we have designed a recombination mechanism that allows the expression of
 
-
three signaling molecules one at a time, the next step towards making a randomizer
 
-
is to make sure these three signals are expressed with the same probability,
 
-
so that E. coli's choice in the RPS game cannot be predictable.
 
-
So, how can we make the probability of expressing each of these signals the same?
 
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</p>
 
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</p>
 
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<h6 id="1.1.1.3.1">1.1.1.3.1</h6>
 
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<p>
 
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<p>
 
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How to make each of the Outcomes (R, P, and S) Equally Probable<br />
 
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The first step towards making the probability of each of the three genes' expressions
 
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to be the same, is to examine the recombination frequency of each lox cassette.
 
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</p>
 
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<p>
 
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For that purpose, three kinds of BioBricks were constructed. Two of them are expected
 
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to express GFP when the lox sites are excised and RFP when they are not: <br />
 
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<ul>
 
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<li>PlacIQ-lox2272-rfp-lox2272-gfp (BBa_640201)</li>
 
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<li>PlacIQ-lox71-rfp-lox66-gfp (BBa_640202),</li>
 
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</ul>
 
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By comparing levels of GFP fluorescence and RFP fluorescence we could determine the
 
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relative recombination frequency of BBa_640201 and BBa_640202 (the recombination
 
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frequency levels when compared to one another).
 
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Besides these two BioBricks we made another one with the construction
 
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PlacIQ-lox2272-gfp-lox2272(BBa_640200).  For this last BioBrick, in vivo and in vitro
 
-
assays were made.
 
-
</p>
 
-
<p>
 
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The in vitro assay of BBa_640200 allowed us to confirm that the Cre-mediated
 
-
recombination on lox2272 cassette works as designed.  In the assay, the part was
 
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made linear using restriction enzymes. Cre recombinase (xconcentration, xvolume)
 
-
was added to the linear DNA (x concentration, x volume) and incubated for 0.5, 2,
 
-
and 4 hours.  Images of the experiments have been added below.
 
-
</p>
 
-
IMAGES
 
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<p>
 
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These images of the electrophoresis experiments show that there are several bands in
 
-
samples to which Cre was added, which indicates that excision of the lox sites
 
-
successfully occurred.
 
-
</p>
 
-
<p>
 
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For the in vivo assay, we prepared a competent cell JM2.300 into which
 
-
Pbad/araC-Cre(pSB1A2, BBa_ ####) had been constructed. Subsequently, our BioBrick
 
-
was constructed into the cell's genome. The strain was grown in a 3ml liquid culture,
 
-
and 75μl of 2M arabinose was added to induce Cre expression. For the control
 
-
experiments we used the same strain without arabinose induction and a JM2.300
 
-
strain which had only our BioBrick.  All the strains were cultured each for periods of
 
-
0.5, 1, 2, and 4 hours, and in each case the florescence levels were measured by
 
-
flow cytometer and fluorescence microscopy.
 
-
</p>
 
-
</p>
 
-
</p>
 
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Revision as of 01:53, 4 October 2011

Tokyo Tech 2011

Tokyo Tech 2011

Rock-Paper-Scissors game

1. The Hands

2. The Judge

2.1 Using AND-Gate promoters to create Judges

2.2 Creating Parts that responded correctly to our set of Signaling Molecules

2.3 Improving PlsrA

2.4 Improving PlasI

3. The Randomizers

3.1 Single Colony Isolation

3.2 Conditional Knockout by Recombination

3.3 The Randomizers

3.4 Survival of one strain

In 1996 Durret and Levin described a system of three types of bacteria that competed for survival in dynamic that resembled a Rock-Paper-Scissors (RPS) game. The bacteria used two main evolutionary stable strategies (ESS) to outcompete their rivals: the production of a toxin (a bacteriocin called colicin) that was toxic to other strains and a higher birth rate than their rival strains. The three types of bacteria described in the model by Durret and Levin were colicin-producing E. coli (R), colicin-resistant E. coli (P) and colicin-sensitive E. coli (S). The colicin producer outcompeted the colicin sensitive by producing the colicin, the colicin sensitive bacteria outcompeted the colicin resistant because it's birth rate was higher than that of the colicin resistant, and the colicin resistant outcompeted the colicin producer because it' birth rate was higher than that of the colicin producer. The colicin resistant bacteria were also able to produce colicin, but at a lower energetic cost, which allowed them to have a higher birth rate. The system was described by the following general differential equations ∑_(i=1)^n?u_i < 1
(du_i)/dt=β_i u_i u_0- u_i ( δ_i+∑_(j=1)^(n-1)??γ_j u_j ?)
u_0=1-∑_(i=1)^n?u_i
Where
u_i i's concentration in arbitrary units (a.u.) β_i i's birth rate δ_i i's death rate γ_j i's death rate due to j's bacteriocin u_0 carrying capacity In the model described by Durret and Levin's paper the equations were as follows: Producer (du_1)/dt=β_1 u_1 u_0- u_1 δ_1 Resistant (du_2)/dt=β_2 u_2 u_0- u_2 δ_2 Sensitive (du_3)/dt=β_3 u_3 u_0- u_i ( δ_3+γ_1 u_1+ γ_2 u_2 ) Setting the parameters as follows, the following graph was created by Durret and Levin. β_1=3, β_2=3.2, β_3=4 γ_1 = 3, γ_2 = 0.5 δ_1 = δ_2 = δ_3 =1 Note that these parameters satisfy the following relations: β_3>β_2>β_1, γ_1> γ_2, and δ_1=δ_2= δ_3 Durret and Levin Durret & Levin (1996)
In this model, however, there is no case where the colicin-producer can survive for infinitively long periods of time if the colicin-resistant's initial concentration is greater than zero. This can be seen in the graph drawn by Durret and Levin, where the lines along the axis for u_1 (in the rightmost lower corner) always converge either to u_3 or u_2. We highlighted this fact by circling the point in the graph below. Durret and Levin remark The existence of this instability along the u_1 axis does not allow us to construct a randomizer that can function based on minimal differences in the initial concentrations of the three different populations of bacteria. For that reason, we modified the differential equations of the model so that any of the three types of bacteria could win for infinitely long periods of time (could win definitely). More specifically, we limited the resistance of the colicin-resistant bacteria in the sense that it would produce a type of bacteriocin that is only toxic to itself and to the sensitive strain, and additionally the resistant strain would also be vulnerable to the colicin produced by the colicin producer. Producer (du_1)/dt=β_1 u_1 u_0- u_1 δ_1 Limited Resistance (du_2)/dt=β_2 u_2 u_0- u_2 (δ_2+γ_1' u_1+γ_2' u_2) Sensitive (du_3)/dt=β_3 u_3 u_0- u_3 ( δ_3+γ_1 u_1+ γ_2 u_2) If we set the parameters as follows β_3=3.1, β_2=3.2,? β?_1=4 γ_1=2,γ_2=0.5,? ? γ?_1'=0.1 ,γ?_2'=0.002 δ_1=δ_2= δ_3=1 and we graph this equations using a Matlab program, we get a graph which clearly shows there are stable points on each of the three axes. new 3d These stable points (u_1,0,0), (0,u_2,0) and (0,0,u_3) indicate that for the equations we have set all of the three strains may ultimately survive for infinite peiriods of time even if the initial density of the other two strains is positive. To see the difference between our model and Durret and Levin's model more clearly, we also plotted Durret and Levin's model using Matlab. prev. 3D Note that the parameters we have set for our equations satisfy the initail conditions of the model proposed by Durret and Levin (indicated in black font) in the sense that β_3>β_2>β_1, γ_1>γ_2 >γ_1'>γ?_2', and δ_1=δ_2= δ_3. The two new terms we added are indicated in red. From a biological perspective, this model describes the existence of two strains of bacteria that produce two different types of bacteriocin. One of these strains is not completely resistant to its own bacteriocin nor to the bacteriocin produced by its rival strain. This can be explained by thinking that the strain with this limitation in its bacteriocin resistance does not produce enough/effective resistance protein, which could be a consequence of it being a mutant of a colicin-sensitive strain. The next step is to find a set of parameters that satisfy the conditions set by the model of Durret and Levin, the above condition of allowing stable points on each of the three axes and also allow each of the three different strains of E. coli to survive in a random fashion by minimal differences on the initial concentrations of each strain. producer limit-registance sensitive We found, by writing another program in Matlab, that the following set of parameters satisfy all of the above mentioned conditions. β_3=3, β_2=3.2,? β?_1=5.1 γ_1=2,γ_2=1,? ? γ?_1'=0.2 ,γ?_2'=0.01 δ_1=δ_2= δ_3=1 As can be seen in the graphs below, each of the strains can survive if their initial density in only tree hundredths (a.u.) greater than the other two strains' initial concentrations. The producer strain survives if u_1= 0.32, u_2= 0.29, u_3= 0.29 (a.u.).
The limited-resistance strain survives if u_1= 0.29, u_2= 0.32, u_3= 0.29 (a.u.).
The sensitive strain survives if u_1= 0.29, u_2= 0.29, u_3= 0.32 (a.u.).
In practice, this minimal variation of concentrations will cause the outcome of Rock, Paper or Scissors signaling molecule to be random. Consequently, we can conclude that this randomizer is not only feasible but also practical and effective (let alone interesting).

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