Modeling
This is our Modeling DesignIntroduction
Microtubule is made up of 13 protofilaments. Now there is an widely accepted feature about the microtubule that microtubule has highly complicated dynamic instability. Under the fixed vitro cultures conditions, on the one hand, subunits will polymerize automatically forming the required structure when the condition is above the critical concentration; on the other hand, the microtubule will depolymerize into subunits when the condition is under the critical concentration. Apart from that, the single microtubule will always in the stage of polymerization and depolymerization.
Tax(Taxol), the efficient anti-cancer medicine, can promote the polymerization of the subunit and restrain the depolymerization of the microtubule, which can make the microtubule be in a stable condition and restrain the mitosis. Therefore, it’s important to study the tax’s mechanism of action during the microtubule’s dynamic assembling process. In order to research tax’s influence on this dynamic procedure from the microcosmic level, we analyze the dynamic procedure and build our mathematical model by four steps.
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Expound the theory of the microtubule’s depolymerization
Visual Simulation -
Verify tax’s influence degree about microtubule
Analysis of variance> -
Tax’s influence on the length of the microtubule
The probability distribution statistic of the Microtubule’s length -
Simulate tax’s mechanism of action to the microtubule
Differential equation modeling
One-way analysis of variance
1.0 - Theory of the one-way analysis of variance
By constructing the F-test statistics, we can use the one-way analysis of variance to study whether classification of the independent variable’s different levels can make significant influence on the variation of the continuous variable. If the levels have a significant influence, we can further give the 95% confidence interval of the dependent variable means under the different levels of the independent variable, and then we can analyze the degree of the different levels. But the precondition is that the data should satisfy the homogeneity of variance, in other words, the variance of the data should be the independent identically distributed. In the next part of the modeling, we will use the one-way analysis of variance to analyze the data, and then deal with the data.
2.0 - The homogeneity test of variance
We use the SPSS to do the homogeneity test of variance with the data we got, the outcome is shown in the figure below:
From the figure, we can see the data’s variance is XXX, nearly zero. Therefore, we can think the data meets the requirement about the homogeneity of variance and we can use the one-way analysis of variance to deal with the data.
3.0 - Construct the F-test statistics
The independent variable is a classified variable which values 0 and 1 to describe whether the tax is added into the test tube. The dependent variable is the change of the micrutubule’ length, our modeling is shown below:
$$ y = u_i + \varepsilon_{ij} $$
y is the dependent variable, the change of the microtubule’s length. is the j observed value of the independent variable under the i level. is the mean of dependent variable under the I level. stands for the residual between dependent variable’s value and it’s mean value, also obey the normal distribution \(N(0, \sigma_i ^2)\)
Then we construct the F test statistics. First, we define the quadratic sum of the residual:
$$ SSE = \sum_{i=1}^k \sum_{j=1}^{n_i} (y_{ij}-\overline y_1)^2 $$
And the quadratic sum of the elements:
$$ SSA = \sum_{i=1}^k n_i (\overline y_{1}-\overline y)^2 $$
SSA reflects the variance between different levels and the difference is made by the different elements; SSE reflects the variance in a certain level and this random difference is due to the selected sample’s random. For example, the measured length of the microtubule will be different when we add the TAX into the test tube.
On the basis of the theory, our F test statistics is:
$$ F = \frac{SSA/(n-k)}{SSE/(k-1)} \sim F(n-k, k-1) $$
The numerator of the equation is a part of the dependent variable which can be explained by the change of the independent variable, while the denominator of the equation can be explained by other random elements except the change of the independent variable. The proportion of the change of independent variable in all change of the dependent variable becomes bigger, in other words, F has a higher value, independent variable influence dependent variable more.
4.0 - The F-test on the data
The numerator of the equation is a part of the dependent variable which can be explained by the change of the independent variable, while the denominator of the equation can be explained by other random elements except the change of the independent variable. The proportion of the change of independent variable in all change of the dependent variable becomes bigger, in other words, F has a higher value, independent variable influence dependent variable more.
stands for that different values of the independent variable( whether the TAX is added into the tube or not) make no difference to the mean value of the dependent variable(microtubule’s length), in other words, the independent is not important to the dependent variable. Then we use \(R\) software to conduct F-test, the outcome is shown below:
Visual Simulation
We applied to programing visualization in this complex process based on certain laws of Microtubule dynamic instability.
Tubulin is made up of two tubulin monomers which are nearly the same as each other. These two tubulin monomers are named α tubulin monomer and β tubulin monomer. Microtubule is made up of 13 protofilaments polymerized by tubulin dimers end to end. And microtubule can be the hollow tube with 13 protofilaments coiled into helix with each other, water in hollow part. The tube wall is 4～5nm thick.
Tubulin dimers are incorporated into the growing lattice in the GTP-bound form and stochastically hydrolyze to GDP-tubulin, thus forming a GTP-cap. It is thought that the switching from growth to shrinkage occurs due to the loss of the GTP-cap.
Caplow M[1] research shows that when the cap structure of microtubule plus end subunit containing GDP- beta tubulin instead of GTP- beta tubulin, microtubule becomes unstable and will quickly depolymerize.
As is shown in the figure, there are two kinds of Dimer, called GDP and GTP, with blue and red two connected to the circular. These dimers have close relationship with each other, and there are three important modes of their action:
- GTP-tubulin dimer in endpoint can aggregate new GTP to make the single protofilament grow, and microtubules extend.
- At the same time, the endpoint GTP may also be made off, thereby protofilaments shorter.
- Any place of GTP (in addition to the right endpoints of the GTP) made made random hydrolyzed to GDP have a chance.
We built a simple GUI interface to simulate the Microtubule dynamic instability. As for a tubulin, we can adjust the parameters of K, R, h, GDP and GTP to display number and length of tubulin in real time. Among them, K, h, R is the number obeying certain distributions.
According to above principles, we built the simulation process of the visual program in MATLAB@, Fig 2 is the schematic diagram of the principle of GTP hydrolysis. Among them means GTP, D means GDP, R means the probability of endpoint GTP polymerizing with new GTP, K means the probability of endpoint falling off, h means the probability of GTP hydrolysis into GDP. It should be noted that once GTP is transferred to GDP, it will not have polymerization, fall off, or hydrolysis, and will become stable state.
