Noise analysis-model

Aim:To analysis the noise in the ternary system .

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

Model assumption： 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:
1.The stochastic simulation algorithm is exact.
2. A stochastic simulation computer program will normally require very little in the way of computer memory space.
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.
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.

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

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.

Ratio sensor

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.

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)

The respective mean value and variance are shown as follows:

Table 1:using the Gillespie algorithm, we get the mean and variance when as1:as2 equal to 0.5,1,2.

then we caculate the noise :

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.

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.

Comparator

We have done the same thing in the comparator. The differences are the km and ks , which values are from the parameter sweep.

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)

then we compute the noise :

Table 3:Using the Gillespie algorithm, we get the mean and variance values when as1:as2 equal to 0.5,1

Table 4: From the table, We can conclude that the noise is also very small .

Then we calculate the noise value 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.

Conclusion

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.