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| | </header> | | </header> |
| | <h2>Introduction</h2> | | <h2>Introduction</h2> |
| − | <p>Dynamic instability of microtubule during its lifetime is very interesting and worth researching. When a cancer patient is cured by taxol treatment fortunately, it should be owed to taxol's interaction with microtubule in dynamic growth process. So it's significant to explore principles of this process. We established a series of math works to clarify mechanisms of microtubule dilution system. Then we got data from wet parts of our project. Our model predicts observed data and phenomenon. In conclusion, we have presented a model that utilizes basic kinetics and it has been proved by experimental observations. We can use it to predict results in other conditions for guidance.</p> | + | <p> It’s essential for us to get accurate growth condition of Chlamydomonas reinhardtii in the natural environment to keep the concentration of toxin at a lethal level. But in fact, it is almost impossible to test concentration anywhere due to the lack of equipment and skills. Therefore, building the growth model can help determine the amount of Chlamydomonas reinhardtii they should use and when they need to add more. To build an accurate growth model, BNU-China team members who have much experience in the mathematics helped us to achieve it.</p> |
| − | <p>What we have done to analyze the dynamic process can be divided into three steps.</p>
| + | <p>Contacting with data provided by wet laboratory, we can draw the diagram of variation trend of algae population. Then we can get the key point where rate of algae population increment meets the maximal value so that the results can guide to culture of algae in their wet part. To control quantity of aquatic larva of mosquito by applying expression of specific protein in algae. There is an impressive impact of establishing mathematic modeling in population of algae.</p> |
| − | <figure class="text-center">
| + | <p> They helped us to establish a mathematic model to illustrate the whole temporal change of algae population. In general, it’s an original differential equations based on light intensity, mineral nutrient, organism and carbon dioxide, which are four main parameters in that. As for the temporal changing rate of population of algae growing in ideal conditions, there has been a lot of methods to solve this question. They referred to Huisman model and combined with practice factors. Then we got our deducted model. This model has a few parameters and it’s easy to get the solution.</p> |
| − | <img src="https://static.igem.org/mediawiki/2016/e/e9/Modeling_steps.png" alt="this is a pic" width="60%">
| + | <p> We provided the data of wet laboratory for us, and they run the model to get result. Finally, these results can help us to complete experiment. </p> |
| − | <figcaption>
| + | |
| − | Fig.1 Our process
| + | |
| − | </figcaption>
| + | |
| − | </figure>
| + | |
| − | <p>First we proved that taxol has a significant influence on microtubule growth by using the data which are provided from wet parts. Then we presented two differential equations model. One is an original differential equation model that can predict taxol’s influence on amount of microtubule. The other is a partial differential equation model expounding analytic solution. Since it’s hard to understand if someone lacks of necessary math skills, we build a program in MATLAB<sup>@</sup> to visualize the kinetics in microtubule disaggregation.</p>
| + | |
| − | <h2>Modeling theory</h2>
| + | |
| − | <p>Microtubule is made up of 13 protofilaments. Now there is a widely accepted feature about the microtubule that microtubule has highly complicated dynamic instability. Under vitro cultures conditions, on the one hand, subunits will polymerize automatically to form fine structure when subunits condition is above the critical concentration; on the other hand, the microtubule will depolymerize into subunits when subunits condition is under the critical concentration. Apart from that, the single microtubule will always in the stage of aggregation and disaggregation.</p>
| + | |
| − | <p>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 aggregated 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.
| + | |
| − | </p>
| + | |
| − | | + | |
| − | <p>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 disaggregate.</p>
| + | |
| − | <figure class="text-center">
| + | |
| − | <img src="https://static.igem.org/mediawiki/2016/1/1d/Modeling_theory1.png" alt="this is a pic" width="60%">
| + | |
| − | <figcaption>
| + | |
| − | Fig.2 Microtubule dynamic instability
| + | |
| − | </figcaption>
| + | |
| − | </figure>
| + | |
| − | <p>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.</p>
| + | |
| − | <ol>
| + | |
| − | <li>
| + | |
| − | GTP-tubulin dimer in endpoint can aggregate new GTP to make the single protofilament grow, and microtubules extend.
| + | |
| − | </li>
| + | |
| − | <li>
| + | |
| − | At the same time, the endpoint GTP may also be made off, thereby protofilaments shorter.
| + | |
| − | </li>
| + | |
| − | <li>
| + | |
| − | Any place of GTP (in addition to the right endpoints of the GTP) made made random hydrolyzed to GDP have a chance.
| + | |
| − | </li>
| + | |
| − | </ol>
| + | |
| − |
| + | |
| − | <h2>Single Factor Analysis of Variance</h2>
| + | |
| − | <p>In order to verify the effect of taxol on the length of microtubule formation, we used descriptive statistical analysis and single factor analysis of variance (ANOVA) to extract the experimental data.</p>
| + | |
| − | <h3>1.0 - Descriptive statistical analysis</h3>
| + | |
| − | <p>Before we use the one-way analysis of variance to deal with the data, we need to conduct some simple descriptive statistical analysis. This analysis can help us gain the overall understanding of the data, find the abnormal data, and then guide our next work.</p>
| + | |
| − | <p>Now that the interpreted variable is discrete, we use the Box Plot to describe the light absorption value OD350 under the different concentration of the Tax.</p>
| + | |
| − | <h3>2.0 - Theory of the one-way analysis of variance</h3>
| + | |
| − | <p>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.</p>
| + | |
| − | <h3>To determine whether the data suitable for single factor analysis of variance</h3>
| + | |
| − | <p>We use the SPSS to do the homogeneity test of variance with the data we got, the outcome is shown in the figure below:</p> | + | |
| − | <figure class="text-center">
| + | |
| − | <img src="../img/paper/modeling/2.png" width="60%">
| + | |
| − | <figcaption>
| + | |
| − | Fig.3 The figure of the data’s homogeneity test of variance
| + | |
| − | </figcaption>
| + | |
| − | </figure>
| + | |
| − | <p>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.</p>
| + | |
| − | <h3>3.0 - Construct the F-test statistics</h3>
| + | |
| − | <p>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:</p>
| + | |
| − | <p>
| + | |
| − | $$ y = u_i + \varepsilon_{ij} \tag{1} $$
| + | |
| − | </p>
| + | |
| − | <p>In the equation (1), \(y\) is the dependent variable, the change of the microtubule's length. \(y_{ij}\) 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 \(N(0, \sigma_i ^2)\)</p>
| + | |
| − | <p>Then we construct the F test statistics. First, we define the quadratic sum of the residual:</p>
| + | |
| − | <p>
| + | |
| − | $$ SSE = \sum_{i=1}^k \sum_{j=1}^{n_i} (y_{ij}-\overline y_1)^2 \tag{2}$$
| + | |
| − | </p>
| + | |
| − | <p>
| + | |
| − | And the quadratic sum of the elements:
| + | |
| − | </p>
| + | |
| − | <p>
| + | |
| − | $$ SSA = \sum_{i=1}^k n_i (\overline y_{1}-\overline y)^2 \tag{3}$$
| + | |
| − | </p>
| + | |
| − | <p>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 taxol into the test tube.</p>
| + | |
| − | <p>On the basis of the theory, our F test statistics is:</p>
| + | |
| − | <p>
| + | |
| − | $$ F = \frac{SSA/(n-k)}{SSE/(k-1)} \sim F(n-k, k-1) \tag{4}$$
| + | |
| − | </p>
| + | |
| − | <p>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.</p>
| + | |
| − | | + | |
| − | <h3>4.0 - The F-test on the data</h3>
| + | |
| − | <figure class="text-center">
| + | |
| − | <img src="../img/paper/modeling/5.png" width="60%">
| + | |
| − | <figcaption>
| + | |
| − | Fig.4 Outcome of the F-test about the data
| + | |
| − | </figcaption>
| + | |
| − | </figure>
| + | |
| − | <P>Our F-test’s null hypothesis is:</P>
| + | |
| − | <p>$$ H_0:μ_1=μ_2=⋯=μ_k \quad vs \quad H_1=\quad not \quad H_0 \tag{5}$$</p>
| + | |
| − | <p>
| + | |
| − | \(H_0\) stands for that different values of the independent variable（the added taxol‘s concentration) make no difference to the mean value of the dependent variable(the light absorption value OD350), 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:
| + | |
| − | </p>
| + | |
| − | <figure class="text-center">
| + | |
| − | <img src="../img/paper/modeling/5.png" width="60%">
| + | |
| − | <figcaption>
| + | |
| − | Fig.5 Outcome of the F-test about the data
| + | |
| − | </figcaption>
| + | |
| − | </figure>
| + | |
| − | <p>From the outcome, \(F’s\) value is XXX. Therefore, we can think the independent variable make distinct influence on the dependent variable.</p>
| + | |
| − | <h2>Differential Equation Model</h2>
| + | |
| − | <p>For studying the dynamic progress of tubulin assembling under the influence of Taxol, we build the differential equations to describe such progress numerically. </p>
| + | |
| − | <p>First we have to consider what happened in the tubulin solution. Protein filaments will assemble into the tubulins automatically by arranging dimers. Specifically, the GTP-cap can aggregate or disaggregate at one side of the protein filament. Besides, there are chances that the GTP can hydrolysis to GDP, which can no longer combine the new GTP. In this process, the length and amount of the tubulin are in the dynamic equilibrium or changing. </p>
| + | |
| − | <p> We know that the differential equations can describe the numerical relationships and patterns between related promoters. Based on these studies, we can make the better plan, central or prediction in the experiments. </p> | + | |
| − | <p>We consider the tubulin length is the function of time, assembling rate, disaggregation and hydrolysis rate. According to the literature, there are some relationships between these promoters.</p>
| + | |
| − | <h3>Details of Partial Differential Equation Model</h3>
| + | |
| − | <p>We know that the movement of any material is ruled by the certain laws of nature (physical and chemical laws). We establish some mathematical model including partial differential equations to describe these rules especially the mechanism of numerical relationships in the tubulin. </p>
| + | |
| − | <p>We build the partial differential equations and definition conditions based on the conservation laws.</p>
| + | |
| − | <p>The tubulin solution area is thought to be a kind of fluid motion. In the solution there are many assembling of tubulins, the hydrolysis, aggregations and disaggregation of the protein filaments. Although the mechanisms of these biological phenomena are still uncertain but we can build the equations based on conservation laws, which certainly rule the biomass.</p>
| + | |
| − | <p>We take a solution area as our study object. The quantity in the solution area is certainly ruled by conservation laws no matter what shape of the area is. Our models are based on conservation of mass.</p>
| + | |
| − | <p>We choose a solution area in the . Considering in the time range , according to the mass conservation:</p>
| + | |
| − |
| + | |
| − | <figure class="text-center">
| + | |
| − | <img src="../img/paper/modeling/5.png" width="60%">
| + | |
| − | <figcaption>
| + | |
| − | Fig.6
| + | |
| − | </figcaption>
| + | |
| − | </figure>
| + | |
| − |
| + | |
| − | <p> is the velocity of assembling of the solution which flow into the . can be regarded as the product of the velocity and the assembling ratio . However, there is no need to consider the and . The inflow mass during through any of the can be given:</p>
| + | |
| − |
| + | |
| − | <p>$$ $$</p>
| + | |
| − |
| + | |
| − | <p> is the unit normal vector toward outside so there is a minus in formula. is the density of the tubulin solution in . In a similar way£¬ is the velocity of the disaggregation mass which flow out the and can be considered as the product of outflow velocity and disaggregation ratio £¬there is no need to consider the and . So the outflow mass during through any of the can be given:</p>
| + | |
| − |
| + | |
| − | <p>$$ $$</p>
| + | |
| − |
| + | |
| − | <p>According to the mass conservation we can get the equation:</p>
| + | |
| − |
| + | |
| − | <p>$$ $$</p>
| + | |
| − |
| + | |
| − | <p>Assuming that ， ， are all continuously differentiable, according to Gauss Formula we can get that:</p>
| + | |
| − |
| + | |
| − | <p>$$ $$</p>
| + | |
| − |
| + | |
| − | <p>Because of the continuity of integrand in and the randomicity of and , we can get that:</p>
| + | |
| − |
| + | |
| − | <p>$$ $$</p>
| + | |
| − |
| + | |
| − | <p>And the definition conditions:</p>
| + | |
| − |
| + | |
| − | <p>$$ $$</p>
| + | |
| − | <p>$$ $$</p>
| + | |
| − |
| + | |
| − | <p> ， can be measured directly in the experiment. For the ， , we can estimate the value range based on our experiments and determine the the most appropriate value according to the simulated annealing algorithm. Then the can be calculate according to the equation and definition conditions. Finally, the variation tendency can be described through the integration </p>
| + | |
| − |
| + | |
| − | <h2>Visual Simulation</h2>
| + | |
| − | <p>We applied to programing visualization in this complex process based on certain laws of Microtubule dynamic instability.</p>
| + | |
| − | <p>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.</p>
| + | |
| − | <p>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.</p>
| + | |
| − | <p>Caplow M<sup><a href="#ref-1">[1]</a></sup> 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.</p>
| + | |
| − | <figure class="text-center">
| + | |
| − | <img src="../img/paper/modeling/3.png" width="60%">
| + | |
| − | <figcaption>Fig.6 Microtubule dynamic instability</figcaption>
| + | |
| − | </figure>
| + | |
| − | <p>
| + | |
| − | 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:
| + | |
| − | </p>
| + | |
| − | <ol>
| + | |
| − | <li>
| + | |
| − | GTP-tubulin dimer in endpoint can aggregate new GTP to make the single protofilament grow, and microtubules extend.
| + | |
| − | </li>
| + | |
| − | <li>
| + | |
| − | At the same time, the endpoint GTP may also be made off, thereby protofilaments shorter.
| + | |
| − | </li>
| + | |
| − | <li>
| + | |
| − | Any place of GTP (in addition to the right endpoints of the GTP) made made random hydrolyzed to GDP have a chance.
| + | |
| − | </li>
| + | |
| − | </ol>
| + | |
| − | | + | |
| − | 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.
| + | |
| − | </p>
| + | |
| − | <p>
| + | |
| − | 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.
| + | |
| − | </p>
| + | |
| − | <figure class="text-center">
| + | |
| − | <img src="../img/paper/modeling/4.png" width="60%">
| + | |
| − | <figcaption>Fig.7 Parameters of GTP-tubulin dimer hydrolysis</figcaption>
| + | |
| − | </figure>
| + | |
| | </article> | | </article> |
| | </div> | | </div> |
