Team:Tokyo Tech/Modeling/RPS-game/RPS-game

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	<ul>
		<li><a href="#1.1">1. Introduction</a></li>
		<li><a href="#1.2">2. Adjusting the Model to create a True Randomizer</a></li>
		<li><a href="#1.3">3. How the Three Types of Bacteria Compete for Survival</a></li>
		<li><a href="#1.4">4. The Old Model</a></li>
		<li><a href="#1.5">5. Our New Model</a></li>
		<li><a href="#1.6">6. The Biological Meaning of our Model</a></li>
		<li><a href="#1.7">7. Making it Obvious</a></li>
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	<h1 id="1"> Survival of One Strain</h1>

	<p>
	
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			<img src="https://static.igem.org/mediawiki/2011/2/28/Image042.png" width="500px" />
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		<h2 id="1.1">1. Introduction: Very small differences determine who will survive</h2>
	
		<p>
			In this section we will show a shocking scenario of evolution: 
			the future of each of three different rival strains (whether the strain will die or survive) 
			is marked by very small differences between the initial population densities of the strains(a phenomenon also known as the "butterfly effect"). 
			Furthermore, we will also show that we can apply this very interesting result to create a 
			randomizer that can be used in our Rock-Paper-Scissors game, due to the fact that only one 
			of the rival strains will survive. More specifically, we assign to each of the three 
			rival strains either of Rock, Paper or Scissors, make them compete for survival and take 
			the surviving strain to represent the bacteria’s choice for the RPS game.
		</p>
	               
		<h2 id="1.2">2. Adjusting the Model to create a True Randomizer</h2>
		
		<p>
			The idea for creating this randomizer was born from a paper written in 1996 by Durrett and Levin. 
			In it, the authors described a system of three types of bacteria that competed for 
			survival in dynamic that resembled a Rock-Paper-Scissors (RPS) game. However, 
			the model proposed in this paper is not fully appropriate for our RPS randomizer, 
			since one of the three types of bacteria cannot ultimately survive (although it 
			can dominate the system, i.e. have the highest population density, for definite periods 
			of time). We will discuss more on the limitations we found in this model to be adopted as 
			a randomizer and the modifications we made to create a true randomizer.
		</p>
	                
		<h2 id="1.3">3. How the Three Types of Bacteria Compete for Survival</h2>
		
		<p>
			The three types of bacteria that compete for survival use three tactics to 
			outcompete their rivals: the production of a toxin (a bacteriocin called colicin)
			 that is toxic to other strains, resistance to the toxin produced by other strains, 
			 and a higher birth rate than their rival strains. Namely, the three types 
			 of bacteria are: colicin-producing <span class="name">E. coli</span> (R), 
			 colicin-resistant <span class="name">E. coli</span> (P) and colicin-sensitive 
			 <span class="name">E. coli</span> (S). The colicin-producer outcompetes the 
			 colicin-sensitive by producing the colicin. The colicin-sensitive bacteria 
			 outcompetes the colicin-resistant because its birth rate is higher 
			 than that of the colicin-resistant. The colicin-resistant outcompetes the 
			 colicin producer because its birth rate is higher than that of the colicin producer. 
			 The colicin resistant bacteria are also able to produce colicin, 
			 but at a lower energetic cost, which allows them to have a higher birth rate.
		</p>
	
		<div align="center">
			<img src="https://static.igem.org/mediawiki/2011/b/bf/Image044.png" width="500px" />
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		<p>
			The system was described by the following general differential equations
		</p>
		
		<div align="center">
			<img src="https://static.igem.org/mediawiki/2011/2/28/Colisin1.png" />
		</div>

		<p>
			Where
		</p>

		<div align="center">
			<img src="https://static.igem.org/mediawiki/2011/c/c0/Colisin2.png" />
		</div>


		<h2 id="1.4">4. The Old Model</h2>
		
		<p>
			In the model described by Durrett and Levin’s paper the equations were as follows:
		</p>

		<div align="center">
			Producer<br />
			<img src="https://static.igem.org/mediawiki/2011/1/1d/Colisin3.png" /><br />
			Resistant<br />
			<img src="https://static.igem.org/mediawiki/2011/0/04/Colisin4.png" /><br />
			Sensitive<br />
			<img src="https://static.igem.org/mediawiki/2011/3/36/Colisin5.png" /><br />
		</div>

		<p>
			These equations show that the colicin-resistant bacteria are completely immune to colicin 
			(there is not death factor associated to colicin in the equation for du<sub>2</sub>/dt). However, as will be explained afterwards, this results in a loss of balance that does not allow building a true randomizing system.
		</p>

		<p>
			Now, setting the parameters as follows, the graph below was created by Durrett and Levin.
		</p>

		<div align="center">
			<img src="https://static.igem.org/mediawiki/2011/4/41/Colisin6.png" /><br />
			<img src="https://static.igem.org/mediawiki/2011/e/e3/Colisin7.png" />
			<img src="http://partsregistry.org/wiki/images/b/bd/Modeling1.png" width="354px"/><br />
			<img src="https://static.igem.org/mediawiki/2011/thumb/b/b1/Colisin8.png/800px-Colisin8.png" width="800px"/><br />
			<img src="https://static.igem.org/mediawiki/2011/5/55/Modeling2.png" width="354px"/><br />
		</div>

		<h2 id="1.5">5. Our New Model</h2>
		<p>
			As mentioned before, the model proposed by Durrett and Levin has critical limitations as a randomizer for the RPS game. 
			To be able to create a true randomizer, we modified the differential equations of the model taking care to give it a 
			biological meaning. With our new differential equations, any of the three types of bacteria can ultimately survive by 
			outcompeting the other two strains, which will die. 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.  
			Since which strain will be the one that survives is determined by very small differences in the initial concentrations of 
			the three different populations of bacteria, in practice this systems becomes a randomizer because of the imprecisions in 
			the measurements that result, for example, when using micropipettes. This randomizer describes a new competition dynamic 
			that could not be reproduced in the previous model proposed by Durrett and Levin due to the instability along the
			<img src="https://static.igem.org/mediawiki/2011/e/e0/Image088.png" alt="u1">axis.
		</p>
		
		<div align="center">
			<img src="https://static.igem.org/mediawiki/2011/c/c1/Colimodel1.png" alt="our new model" />
		</div>
		<p>If we set the parameters as follows</p>
		<div align="center">
			<img src="https://static.igem.org/mediawiki/2011/0/00/Colimodel2.png" alt="new model's coefficient" />
		</div>

		<p>
			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 (Figure 1, Up).
		</p>
		
		<img src="https://static.igem.org/mediawiki/2011/b/b8/Model2.png" width="815px" style="float:center;"/><br />
		<img src="https://static.igem.org/mediawiki/2011/0/05/Image106.png" width="815px" style="float:center;"/>
		
		<div align="center">
			Figure 1. Up: Our New model. Down: The Old Model
		</div>

		<p>
			These stable points (u<sub>1</sub>,0,0), (0,u<sub>2</sub>,0) and (0,0,u<sub>3</sub>) indicate that for the equations we have set 
			all of the three strains may ultimately survive for infinite peiriods of time. The differences between 
			our model and the model of Durrett and Levin can be seen graphically in Figure 1. These graphs were plotted using Matlab. 
		</p>

		<p>
			Note that the parameters we have set for our equations satisfy the initail 
			conditions of the model proposed by Durrett and Levin (indicated in black font)<br />
			<img src="https://static.igem.org/mediawiki/2011/a/a4/Colimodel4.png" alt="new terms" width="800px"/>
		</p>

		<h2 id="1.6">6. The Biological Meaning of our Model</h2>

		<p>
			From a biological perspective, our model describes the existence of two strains of 
			bacteria that produce two different types of bacteriocins. One of these strains is not 
			completely resistant to its own bacteriocin nor to the bacteriocin produced by its
			 rival strain. This can be justified as the consequence of insufficient/ineffective 
			resistance protein production by the &ldquo;resistant&rdquo; strain. This limitation in 
			the production of resistance protein could be thought of as a consequence of the 
			&ldquo;resistant&rdquo; strain being a mutant of a colicin-sensitive strain.
		</p>

		<h2 id="1.7">7. Making it Obvious</h2>

		<p>
			From the graph of our new model (Figure 1, left) it can be deduced that there are 
			paths that converge at stable points (u<sub>1</sub>,0,0), (0,u<sub>2</sub>,0) and (0,0,u<sub>3</sub>), and that this 
			paths all have an approximately common origin. In this section we would like to show 
			that the origin of these paths is practically the same, and that in that sense we have 
			designed a true randomizer (since, as mentioned before, the imprecisions that result 
			in the experimental measurements will make it impossible to make the initial
			 population density of the three strains the same). 
		</p>

		<p>
			In the following set of graphs we will make it obvious that each of the three different 
			strains of <span class="name">E. coli</span> to survive in a random fashion by very small
			differences on the initial concentrations of each strain.
		</p>

		<p>
			We modeled our results using Matlab. 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.
		</p>
		
		<table>
			<tr>
				<td>
					<a href="https://static.igem.org/mediawiki/2011/thumb/1/19/Image130.png/800px-Image130.png">
						<img src="https://static.igem.org/mediawiki/2011/thumb/1/19/Image130.png/800px-Image130.png" alt="output1" Width="400px" />
					</a>
				</td>
				<td>
					<a href="https://static.igem.org/mediawiki/2011/thumb/a/a1/Image132.png/800px-Image132.png">
						<img src="https://static.igem.org/mediawiki/2011/thumb/a/a1/Image132.png/800px-Image132.png" alt="output2" Width="400px" />
					</a>
				</td>
			</tr>
			<tr>
				<td>
						<img src="https://static.igem.org/mediawiki/2011/1/11/Tt-modeling1.png" alt="coefficient of output1" Width="400px"/>
				</td>
				<td>
						<img src="https://static.igem.org/mediawiki/2011/9/99/Tt-modeling2.png" alt="coefficient of output2" Width="400px"/>
				</td>
			</tr>
			<tr>
				<td>
					<a href="https://static.igem.org/mediawiki/2011/thumb/a/a2/Image134.png/800px-Image134.png">
						<img src="https://static.igem.org/mediawiki/2011/thumb/a/a2/Image134.png/800px-Image134.png" alt="output3" Width="400px" />
					</a>
				</td>
				<td rowspan="2">
					With these graphs it becomes clear that the imprecisions in experimental measurements 
					(i.e. pipetting) are enough to 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).
				</td>
			</tr>
			<tr>
				<td><img src="https://static.igem.org/mediawiki/2011/b/b7/Tt-modeling3.png" alt="coefficient of output3" Width="400px"/></td>
			</tr>
		</table>

		
		

	</p>
<br/>
<br/>
<h2>Reference </h2>
<p>Durrett, R., Levin, S. Allelopathy in Spatially Distributed Populations (1997). Journal of Theoretical Biology, 185, 165-171.</p>

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