Team:BGU Israel/Model2.html

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<span class="title">A Stochastic Birth-Death Model</span>
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<span class="title">Modelling Results</span>
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<p>
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                  <p>It is possible to describe the behavior of a bacterial population by looking at a specific generation k of bacteria and examining its inter-division rate and death rate:</br></br>
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As explained in the theoretical section on the stochastic birth-death model, 25 combinations of protein leakage and mechanism strength were simulated in-silico. For each combination, 30 repeats ("experiments") were performed. </br> </br>
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<img src="https://static.igem.org/mediawiki/2013/f/fb/BGU_eq1.png" style="margin-left:50px;"/> </br></br>
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Leakage rate was set between 0.1-0.5, and mechanism strengths were given numbers between 1-5, representing the 'a' parameter: protein concentrations functions.</br></br>
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Where &beta; is inter-division rate and µ is death rate. </br>
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A deterministic equation to describe such a population will look like this: </br>
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</br>
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<img src="https://static.igem.org/mediawiki/2013/f/f7/BGU_eq2.png" style="margin-left:50px;"/> </br></br>
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<b>Model Results</b></br>
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This deterministic approach assumes that the doubling time equals the inter-division time, meaning </br></br>
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<ol class="bulletlist">
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<img src="https://static.igem.org/mediawiki/2013/c/c3/BGU_eq3.png" style="margin-left:50px;"/> </br>
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        <li class="bulletlist">Die-out timeframe: 300-750 minutes.</li>
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The stochastic approach for the same issue suggests that there are different inter-division and death time distributions that will fit a specific population behavior, and not just an average time. </br> </br>
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        <li class="bulletlist">Working timeframe: 190-600 minutes.</li>
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        <li class="bulletlist">Generation range: 13-32 generations.</li>
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</ol>
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<p>
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What does a simulation of one of these combinations of protein leakage and mechanism strength look like?  In each example below, 5 experiments out of the 30 are marked in grey, an average curve is red, and standard deviation curves are black.</br></br>
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</p>
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</p>
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One can replace the use of the rates &beta;  and &mu;  with the use of inter-division and death time distributions instead <b>[1]</b>.</br></br>
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<div style="margin-left:30px;">
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<img src="https://static.igem.org/mediawiki/2013/d/d9/Bgu_model_1.png" /></br>
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<h8>Strongest mechanism, Weakest leakage rate</h8></br></br>
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</div>
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There is evidence <b>[2][1]</b> and it is fair to assume that the distributions for cell division and death times are similar to the Gamma distribution. Right after an individual cell has divided, in order for a second proliferation to happen, a set of events in the cell cycle need to occur, therefore proliferation probability is low right after the division, is rising as time advances, and is falling again asymmetrically as time goes by after avg. cell cycle time. Same can be assumed for death time distribution because the cell cycle can be considered as a sensitive time with high death possibility. </br></br>
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<p>
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Example 1: Strongest mechanism, weakest leakage rate. All experiments end with the entire population dying out. Cells experience a log phase of around 90 minutes, a maximum population of around 230 is reached, and the variance is small.</br></br>
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As described in the overview, we defined our problematic parameters that we are modeling as the leakage of the system, and the strength of the mechanism influencing the cell. When a bacterium is released and no more induction occurs, concentration will behave like this: </br>
 
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</br>
 
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<img src="https://static.igem.org/mediawiki/2013/f/f5/BGU_eq4.png" style="margin-left:50px;"/> </br>
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</p>
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Where k is generation (k=1, [protein]=100%).</br>
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<div style="margin-left:30px;">
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<img src="https://static.igem.org/mediawiki/2013/b/ba/Bgu_model_2.png" /></br>
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<h8>Intermediate combination. Mechanism strength=4, Leakage rate=0.4</h8></br></br>
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</div>
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In this model we chose distributions for inter-division and death times such that they will describe the behavior. Those distributions were initially configured to represent a regular genetically modified E.coli, the initial setting corresponded to a P.A.S.E 1 with 100% CI concentration, or a P.A.S.E 2 with 100% tyrosine synthetase concentration (the state of the organism just after its release when it is fully induced). The algorithm, which is based on a Gillespie algorithm, advanced each individual in the population with time according to those distributions (a decision is being made each iteration for each cell if it will die or proliferate). With each update, individual protein concentration was updated as well. </br>
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<p>
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</br>
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Example 2: Intermediate parameters. Here, the stochasticity of the system takes an important role. Some of the experiments are trending down at the end of the timeframe, while others are trending up. Variance is much bigger. Log phase is about 200 minutes. A maximum population of around 410 is reached.</br></br>
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In order to compare all of the experiments, 3D surfaces projecting the data of all combinations were made. In each one of figures 4-6, a specific parameter was chosen to be compared between the combinations. The coefficient of variation (CV) of the 30 repeats was computed and is represented by the surface’s color:</br></br>
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</p>
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<div style="margin-left:30px;">
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<img src="https://static.igem.org/mediawiki/2013/b/b4/Bgu_model_3.png" height="700" width="1100"/></br>
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<h8>Figure 3: Z axis shows the time it took for 95% of the population to die out. X and Y are the values for the tested parameters, the bacteria symbol represents a non- reliable combination where less than 90% of the experiments died out within the model time frame, and the rest lived on. </h8></br></br>
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</div>
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So how did we introduce protein concentration differentiation and mechanism strength to the system? </br>
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<p>
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</br>
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The graph omits combinations in which the weakest mechanism strength 1 was used, because those combinations never died out. This parameter is robust and has a clear limit of around 2 as defined in the model.</br></br>
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The leakage is a more modular parameter, giving delicate tuning for high mechanism strengths, thus strengthening our hypothesis that leakage can provide nodularity for the system.</br></br>
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A time frame of around 300-750 minutes is seen, as the z-differential between the lowest combination and the highest one.</br></br>
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</p>
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The Gamma(a,b) distribution is defined by its parameters- &rsquo;a&rsquo; is shape (dimensionless) , &rsquo;b&rsquo; is scale (minutes in our case). The Mean of the distribution equals a*b and represents the average inter division time, it was initially set for a*b=35 min, similar to a genetically modified E.coli. Control of the distribution means control over the shape parameter &rsquo;a&rsquo; , therefore we chose the death time shape parameter to be linearly depended on the concentration of protein as follows: </br></br>
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<div style="margin-left:30px;">
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<img src="https://static.igem.org/mediawiki/2013/4/4a/Bgu_model_4.png" height="700" width="1100"/></br>
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<h8>Figure 4: Z axis is the number of generations reached after 1000 minutes of simulation, in combinations that died out - a high number wasn’t reached</h8></br></br></br>
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<img src="https://static.igem.org/mediawiki/2013/e/ee/BGU_eq5.png" style="margin-left:50px;"/> </br> </br>
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<p>A generation frame of 13-32 generations within the effective combinations is seen in this model. This effects the chance of the system to transfer genes horizontally, and the chance for a mutation in the mechanism to occur.</br></br>
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</p>
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Mechanism strength was simply introduced by changing the &rsquo;slope&rsquo; and &rsquo;n&rsquo; parameters to represent a stronger influence of [protein] on &rsquo;a&rsquo;. </br>
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<img src="https://static.igem.org/mediawiki/2013/1/16/Bgu_model_5.png" height="700" width="1100"/></br>
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</br>
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<h8>Figure 5: Z axis is the time it took until population reached a 50% mark from initial conditions.</h8></br></br></br>
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Protein leakage was changed by changing the function that updates the concentration each iteration. </br>
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<p>
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</br>
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This parameter, the measurement of the half-life, can give an indication for the possible working time for the system. A time frame of 190-600 minutes has been calculated.</br></br>
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While inducer concentration usually doesn’t provide adequate control of the system because of the promoter’s bimodality[1] (all or nothing behavior when induced), some control can be achieved from different inducer combinations [2]. Therefore it is possible to induce P.A.S.E-containing bacteria to express only a partial percentage of the maximal protein expression capability, allowing for control of the overall population lifetime.</br></br>
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</p>
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25 combinations of leakages and mechanism strengths were simulated with repeats, and means and deviation of different parameters were analyzed. </br>
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View our <a href="https://static.igem.org/mediawiki/2013/b/b2/Birth_death_iGEM_BGU.m" target="_blank">code.</a>
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</br>
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</div>
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<p>
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</br></br><h6>  Continue the journey: read about our <a href="/Team:BGU_Israel/HPOverview">Human Practice</a>.</h6></br></br>
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</br>
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<hr/>
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<b>References</b></br></br>
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</p>
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<p>
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<b>[1]</b> [1] J Biotechnol. 2007 Feb 1;128(2):362-75. Epub 2006 Oct 17. Cell population heterogeneity in expression of a gene-switching network with fluorescent markers of different half-lives. Portle S, Causey TB, Wolf K, Bennett GN, San KY, Mantzaris N.
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<a href="https://static.igem.org/mediawiki/2013/e/ef/BGU_1-s2.0-S0168165606008613-main_-1-.pdf" target="_blank">View Source</a></br>
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<b>[2]</b> R. Lutz, H. Bujard. Independent and tight regulation of transcriptional units in Escherichia coli via the LacR/O, the TetR/O and AraC/I1-I2 regulatory elements. Nucleic Acids Research 25(6), 1203–1210 (1997). <a href="https://static.igem.org/mediawiki/2013/2/23/BGU_1203.full.pdf" target="_blank">View Source</a>
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</p></br></br>
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   </div>
   </div>

Latest revision as of 00:52, 29 October 2013

BGU_Israel

Modelling Results

As explained in the theoretical section on the stochastic birth-death model, 25 combinations of protein leakage and mechanism strength were simulated in-silico. For each combination, 30 repeats ("experiments") were performed.

Leakage rate was set between 0.1-0.5, and mechanism strengths were given numbers between 1-5, representing the 'a' parameter: protein concentrations functions.

Model Results

  1. Die-out timeframe: 300-750 minutes.
  2. Working timeframe: 190-600 minutes.
  3. Generation range: 13-32 generations.

What does a simulation of one of these combinations of protein leakage and mechanism strength look like? In each example below, 5 experiments out of the 30 are marked in grey, an average curve is red, and standard deviation curves are black.


Strongest mechanism, Weakest leakage rate

Example 1: Strongest mechanism, weakest leakage rate. All experiments end with the entire population dying out. Cells experience a log phase of around 90 minutes, a maximum population of around 230 is reached, and the variance is small.


Intermediate combination. Mechanism strength=4, Leakage rate=0.4

Example 2: Intermediate parameters. Here, the stochasticity of the system takes an important role. Some of the experiments are trending down at the end of the timeframe, while others are trending up. Variance is much bigger. Log phase is about 200 minutes. A maximum population of around 410 is reached.

In order to compare all of the experiments, 3D surfaces projecting the data of all combinations were made. In each one of figures 4-6, a specific parameter was chosen to be compared between the combinations. The coefficient of variation (CV) of the 30 repeats was computed and is represented by the surface’s color:


Figure 3: Z axis shows the time it took for 95% of the population to die out. X and Y are the values for the tested parameters, the bacteria symbol represents a non- reliable combination where less than 90% of the experiments died out within the model time frame, and the rest lived on.

The graph omits combinations in which the weakest mechanism strength 1 was used, because those combinations never died out. This parameter is robust and has a clear limit of around 2 as defined in the model.

The leakage is a more modular parameter, giving delicate tuning for high mechanism strengths, thus strengthening our hypothesis that leakage can provide nodularity for the system.

A time frame of around 300-750 minutes is seen, as the z-differential between the lowest combination and the highest one.


Figure 4: Z axis is the number of generations reached after 1000 minutes of simulation, in combinations that died out - a high number wasn’t reached


A generation frame of 13-32 generations within the effective combinations is seen in this model. This effects the chance of the system to transfer genes horizontally, and the chance for a mutation in the mechanism to occur.


Figure 5: Z axis is the time it took until population reached a 50% mark from initial conditions.


This parameter, the measurement of the half-life, can give an indication for the possible working time for the system. A time frame of 190-600 minutes has been calculated.

While inducer concentration usually doesn’t provide adequate control of the system because of the promoter’s bimodality[1] (all or nothing behavior when induced), some control can be achieved from different inducer combinations [2]. Therefore it is possible to induce P.A.S.E-containing bacteria to express only a partial percentage of the maximal protein expression capability, allowing for control of the overall population lifetime.

View our code.



Continue the journey: read about our Human Practice.


References

[1] [1] J Biotechnol. 2007 Feb 1;128(2):362-75. Epub 2006 Oct 17. Cell population heterogeneity in expression of a gene-switching network with fluorescent markers of different half-lives. Portle S, Causey TB, Wolf K, Bennett GN, San KY, Mantzaris N. View Source
[2] R. Lutz, H. Bujard. Independent and tight regulation of transcriptional units in Escherichia coli via the LacR/O, the TetR/O and AraC/I1-I2 regulatory elements. Nucleic Acids Research 25(6), 1203–1210 (1997). View Source



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