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<h1>Noise analysis-model</h1>
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<h1 style="border:none;">Noise analysis-model</h1>
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<p><strong>Aim:</strong><span></span>to find out whether the system is steady.
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<p><strong>Aim:</strong><span></span>To figure out whether output is robust when presented with intrinsic noise.
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<p><strong>Steps:</strong>
<p><strong>Steps:</strong>
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<br/><span></span>1   Using the Gillespie algorithm to draw the stochastic picture.
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<br/><span></span>1. Use the Gillespie algorithm to perform the stochastic analysis;
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<br/><span></span>2.   drawing the probability distribution figure (pdf) is to observe the result whether it is around the certainly value.
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<br/><span></span>2. Draw probability distribution figures (pdf);
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<br/><span></span>3. computing the noise result
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<br/><span></span>3. Compute the noise statistically.
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<br/><span></span>4. comparing the noise result
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<p><strong>Model preparation：</strong><span></span>After having determined the parameter range, we would make the noise analysis.According to theoretical predictions elementary chemical reactions involved in biochemical processes exhibit substantial stochastic fluctuations when low numbers of reactant molecules are involved within the small volume of a living cell. The existence of significant stochastic fluctuations in biochemical processes has been confirmed by numerous experiments including tracking of individual protein molecules in individual cells in gene expression processes[Zhou L, Gregori G, Blackman J,Robinson J, Wanner B (2005) Stochastic activation of the response regulator PhoB by noncognate histidine kinases.J Integrative Bioinformatics 2: 11.] From that we could conclude the noise plays an important role in affecting the system and it is meaningful for us to have noise analysis. At the same time, we refer to the most classical Gillespie algorithm and lots of improvements have been made by later generation. For example, Sagar Indurkhya et al mentioned that Dynamic Monte Carlo methods are a common means of simulating the time-evolution of chemical systems.The Gillespie Algorithm (SSA) [1] is the standard algorithm for this process, and has inspired a variety of derivative methods that speed up computation, including the Optimized Direct Method (ODM) [2] and the Next Reaction Method (NRM) [3]. These methods, however, are still computationally costly. Speeding up the Gillespie Algorithm and related hybrid methods will likely play an important role in advancing the productivity of computational systems biology. We have not applied the improved method into our model to ensure that the algorithm would match our model to the largest degree, because during the model computation we gain much help from the internet center of our school. So, we still use the classical algorithm.
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<p><strong>Background:</strong><span></span>After having determined the parameter range, we would make the noise analysis.According to theoretical predictions, elementary chemical reactions involved in biochemical processes exhibit substantial stochastic fluctuations when low numbers of reactant molecules are involved within the small volume of a living cell.  
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The existence of significant stochastic fluctuations in biochemical processes has been confirmed by numerous experiments including tracking of individual protein molecules in individual cells in gene expression processes[Zhou L, Gregori G, Blackman J,Robinson J, Wanner B (2005) Stochastic activation of the response regulator PhoB by noncognate histidine kinases.J Integrative Bioinformatics 2: 11.]  
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We refer to the most classical Gillespie algorithm and lots of improvements have been made by later generation. For example, Sagar Indurkhya et al mentioned that Dynamic Monte Carlo methods are a common means of simulating the time-evolution of chemical systems.The Gillespie Algorithm (SSA) [1] is the standard algorithm for this process, and has inspired a variety of derivative methods that speed up computation, including the Optimized Direct Method (ODM) [2] and the Next Reaction Method (NRM) [3]. These methods, however, are still computationally costly.Now,we still use the classical algorithm.:
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<br/><br/>
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<p><strong>Model assumption：</strong><span></span> It is more appropriate for us to use differential equations get a correct description, because there are large amounts of molecules(about 1023) in common chemical reactions. Differential equation, however, is not suitable for the biological and chemical reactions in biological systems. There are two main reasons: Firstly, a biological reaction involves much less molecules and the probability of reactant collision has decreased sharply, so there is a probability for collision. Secondly, the reactions are affected by the thermal fluctuation and enough activation energy is needed to make the reaction to happen even though the reactants collide together. But activation energy is also affected by thermal fluctuation and is stochastic. Therefore, stochastic process is an important way to describe biological and chemical reactions. For stochastic simulation, we use Gillespie algorithm. There are three reasons:
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<p>ηij is used to describe the noise, where：
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<br/><span></span>1.The stochastic simulation algorithm is exact.
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<br/><span></span>2. A  stochastic  simulation  computer  program  will normally  require very  little in  the  way  of  computer memory space.
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<br/><span></span>3.  In terms  of  the  “statistical  ensemble of systems” envisioned  in the  master equation approach, the  stochastic simulation  algorithm essentially  provides  us  with information on  the  time  behavior  of  individual  ensemble members, such as one  would observe in an ideal laboratory experiment.
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<br/>The construction of model: Through the parameter data of parameter sweep, we get the following graphics by using Gillespie algorithm. ηij   is use to describe the noise.
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<br/><a><img style="margin-left:270px;" src="https://static.igem.org/mediawiki/2012/9/9a/Math21.JPG" /></a>
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<p><span></span>We can determine the value of ηi2 by getting the answers of its mean value and variance. Then, we can determine whether the parameter value is drown in noise by comparing the value of noise to the 5 times of the difference m equlibrium value between as1:as2=1:1 and as1:as2=1:2.
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<p><span></span>The value of noise is ηi2 .
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If the mean([m]1:1)-mean([m]1:2) >> the value of noise, we identify the output wouldn’t be drown in noise.
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<br/><span></span>
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<a><img src="https://static.igem.org/mediawiki/2012/8/89/Ouc-parameter_range.png" /></a>
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<h2>Ratio sensor</h2>
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<h2>Result_Ratio sensor</h2>
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<p><span></span>To get how the system behaves in the stochastic regime, we plotted the mRNA and sRNA molecule numbers as function of time(at left). Then we obtain the histogram(at right) The histogram make clear that the mean is around the mean, thus we can conclude that the system is stable.  Since the numbers of mRNA molecules do not jump between two or more distinct values we can assume that the system is not bistable or multistable.
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<p>  
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<a><img src="https://static.igem.org/mediawiki/2012/1/18/Ouc-modeling-math22.jpg"></a>
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<a><img style="margin-left:50px;" src="https://static.igem.org/mediawiki/2012/1/1b/Ouc-R1L.jpg" /></a>
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<a><img style="margin-left:380px; margin-top: -290px;" src="https://static.igem.org/mediawiki/2012/6/6a/Ouc-modeling-math23.jpg"></a>
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<a><img style="margin-left:380px; margin-top:-230px;" src="https://static.igem.org/mediawiki/2012/d/d0/Ratiop0R.jpg" /></a>
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<a><img src="https://static.igem.org/mediawiki/2012/a/a9/Ouc-modeling-math24.jpg"></a>
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<a><img style="margin-left:50px; margin-top:-30px;"src="https://static.igem.org/mediawiki/2012/6/6c/Ouc-R2L.jpg" /></a>
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<a><img style="margin-left:380px; margin-top: -290px;" src="https://static.igem.org/mediawiki/2012/b/bc/Ouc-modeling-math25.jpg"></a>
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<a><img style="margin-left:380px; margin-top:-230px;" src="https://static.igem.org/mediawiki/2012/7/74/Ratiop1R.jpg" /></a>
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<a><img src="https://static.igem.org/mediawiki/2012/8/8d/Ouc-modeling-math26.jpg"></a>
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<a><img style="margin-left:50px; margin-top:-30px;" src="https://static.igem.org/mediawiki/2012/9/98/Ouc-R3L.jpg" /></a>
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<a><img style="margin-left:380px; margin-top: -290px;" src="https://static.igem.org/mediawiki/2012/8/8c/Ouc-modeling-math27.jpg"></a>
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<a><img style="margin-left:380px; margin-top:-230px;" src="https://static.igem.org/mediawiki/2012/b/b0/Ratiop2R.jpg" /></a>
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<p style="text-align:center; font-size:90%;">Figure 1: the parameters which is from the parameter sweep are used in the Gillespie algorithm  the left are the sRNA and mRNA change with time goes by. The right are the mRNA of probability distribution frequency (pdf)
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<p style="text-align:center; font-size:90%;">Fig.1 Pdf means probability distribution frequency  
</p>
</p>
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<p>The respective mean value and variance are shown as follows:
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<p>The statistics is given below:
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<a><img style="margin-left:65px;" src="https://static.igem.org/mediawiki/2012/c/c8/Ouc-modeling-math28.jpg"></a>
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<a><img src="https://static.igem.org/mediawiki/2012/5/52/Ouc-rs_analysis2.png"></a>
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<br/><br/><p style="text-align:center; font-size:90%;">Table 1:using the Gillespie algorithm, we get the mean and variance when as1:as2 equal to 0.5,1,2.</p>
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<p>then we caculate the noise :
 
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<a><img style="margin-left:65px;" src="https://static.igem.org/mediawiki/2012/9/9a/Ouc-modeling-math29.jpg"></a>
 
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<br/><br/><p style="text-align:center; font-size:90%;">Table 2: From the table, We can conclude that the noise is very small and it’s nearly the same when as1:as2 equal 0.5 and as1:as2 equal 2 .it’s fit the ode equations.<br/>
 
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<p><span></span>From the table2 , we can conclude that the noise almost the same. And the mean of mRNA>>difference value between as1:as2=1 .At the same time, the mean of mRNA>>difference value between as1:as2=1 and 2. So when as1:as2 is equal to 1:1,the equilibrium concentration isn’t drown the noise.
 
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<h2>Comparator</h2>
<h2>Comparator</h2>
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<p><span></span>We have done the same thing in the comparator. The differences are the km and ks , which values are from the parameter sweep.
 
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<a><img style="margin-left:50px;" src="https://static.igem.org/mediawiki/2012/d/de/Comparator0.jpg" /></a>
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<a><img style="margin-left:380px; margin-top:-230px;" src="https://static.igem.org/mediawiki/2012/5/51/Ouc-R1R.jpg" /></a>
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<a><img style="margin-left:50px; margin-top:-30px;"src="https://static.igem.org/mediawiki/2012/2/27/Comparator1.jpg" /></a>
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<a><img style="margin-left:380px; margin-top:-230px;" src="https://static.igem.org/mediawiki/2012/0/02/Ouc-R2R.jpg" /></a>
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<a><img style="margin-left:50px; margin-top:-30px;" src="https://static.igem.org/mediawiki/2012/c/c3/Comparator2.jpg" /></a>
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<a><img style="margin-left:380px; margin-top:-230px;" src="https://static.igem.org/mediawiki/2012/b/b8/Ouc-R3R.jpg" /></a>
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<br/><p style="text-align:center; font-size:90%;">Fig.2 Pdf stands for probability distribution frequency.</p>
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<a><img src="https://static.igem.org/mediawiki/2012/1/18/Ouc-modeling-math22.jpg"></a>
 
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<a><img style="margin-left:380px; margin-top: -290px;" src="https://static.igem.org/mediawiki/2012/b/b2/Ouc-modeling-math31.jpg"></a>
 
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<a><img src="https://static.igem.org/mediawiki/2012/1/15/Ouc-modeling-math32.jpg"></a>
 
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<a><img style="margin-left:380px; margin-top: -290px;" src="https://static.igem.org/mediawiki/2012/b/bc/Ouc-modeling-math25.jpg"></a>
 
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<a><img src="https://static.igem.org/mediawiki/2012/e/e2/Ouc-modeling-math34.jpg"></a>
 
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<a><img style="margin-left:380px; margin-top: -290px;" src="https://static.igem.org/mediawiki/2012/9/9b/Ouc-modeling-math35.jpg"></a>
 
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<br/><p style="text-align:center; font-size:90%;">Figure 2: the same principle is used the comparator. the left are the sRNA and mRNA change with time goes by. The right are the mRNA of probability distribution frequency (pdf)</p>
 
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<p>The statistics is given below:
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<p>then we compute the noise :
 
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<a><img src="https://static.igem.org/mediawiki/2012/c/c4/Ouc-c_analysis2.png"></a>
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<a><img style="margin-left:65px;" src="https://static.igem.org/mediawiki/2012/5/58/Ouc-modeling-math36.jpg"></a>
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<br/><br/><p style="text-align:center; font-size:90%;">Table 3:Using the Gillespie algorithm, we get the mean and variance values when as1:as2 equal to 0.5,1</p><br/>
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<a><img style="margin-left:65px;" src="https://static.igem.org/mediawiki/2012/7/7f/Ouc-modeling-math37.jpg"></a>
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<br/><br/><p style="text-align:center; font-size:90%;">Table 4: From the table, We can conclude that the noise is also very small .<br/>
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<br/><br/><p><span></span>Then we calculate the noise value
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From the table 4 ,we can observe that the mean of mRNA>>difference value between as1:as2=1 and 0.5.At the same time, the mean of mRNA>>difference value between as1:as2=1 and 2. So when as1:as2 is equal to 1:1,the equilibrium concentration isn’t drown the noise.
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<h2>Conclusion</h2>
<h2>Conclusion</h2>
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<p><span></span> By using the noise analysis, we can conclude that the mRNA isn’t drawn the noise,. that’s  by simulating the real organism, our model also meets the requirements to design the experiment.
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<p><span></span>Both ratio sensor and comparator, the change of mRNA concentration is robust enough when presented with noise(mean([m]1:1)-mean([m]1:2) >> noise).It is rational to design our device by this parameter sets.
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Latest revision as of 04:04, 27 October 2012

Noise analysis-model

Aim:To figure out whether output is robust when presented with intrinsic noise.

Steps:
1. Use the Gillespie algorithm to perform the stochastic analysis;
2. Draw probability distribution figures (pdf);
3. Compute the noise statistically.

Background:After having determined the parameter range, we would make the noise analysis.According to theoretical predictions, elementary chemical reactions involved in biochemical processes exhibit substantial stochastic fluctuations when low numbers of reactant molecules are involved within the small volume of a living cell. The existence of significant stochastic fluctuations in biochemical processes has been confirmed by numerous experiments including tracking of individual protein molecules in individual cells in gene expression processes[Zhou L, Gregori G, Blackman J,Robinson J, Wanner B (2005) Stochastic activation of the response regulator PhoB by noncognate histidine kinases.J Integrative Bioinformatics 2: 11.] We refer to the most classical Gillespie algorithm and lots of improvements have been made by later generation. For example, Sagar Indurkhya et al mentioned that Dynamic Monte Carlo methods are a common means of simulating the time-evolution of chemical systems.The Gillespie Algorithm (SSA) [1] is the standard algorithm for this process, and has inspired a variety of derivative methods that speed up computation, including the Optimized Direct Method (ODM) [2] and the Next Reaction Method (NRM) [3]. These methods, however, are still computationally costly.Now,we still use the classical algorithm.:

ηij is used to describe the noise, where：

σ ij stands for standard deviation and represents the mean value. If i=j then



The value of noise is ηi2 . If the mean([m]1:1)-mean([m]1:2) >> the value of noise, we identify the output wouldn’t be drown in noise.

Result_Ratio sensor




Fig.1 Pdf means probability distribution frequency



The statistics is given below:





Comparator




Fig.2 Pdf stands for probability distribution frequency.



The statistics is given below:



Conclusion


Both ratio sensor and comparator, the change of mRNA concentration is robust enough when presented with noise(mean([m]1:1)-mean([m]1:2) >> noise).It is rational to design our device by this parameter sets.

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