Difference between revisions of "Team:ECUST China/Model"

To guide the experiment, systematic modeling and analysis were conducted, including micro dynamics, macro fermentation process, genome-scale metabolic network, macro color rendering model, economic income accounting and fermentation plant CAD. These models presented the feasibility and prospect of our project.

In model 1, we first use the gray box model for mixed mechanism modeling. In order to make the fermentation conditions more suitable for our phycocyanin yeast fermentation process, we organically combine the kinetic mechanism modeling with parameter estimation, and improve each mechanism model to meet the requirements of the project. Then, the modeling and analysis of process dynamic characteristics, transmission characteristics and biochemical reaction characteristics are carried out, which are finally presented in cell growth model, matrix consumption model and product generation model, which can guide the quantification of phycocyanin fermentation process in the later stage of the project and bring convenience to strategy control; In model 2, we established a micro reaction molecular mechanism model. Based on the yeast genome-scale metabolic network model, through metabolic flow control and analysis, we obtained the metabolic pathway impact of our project route in yeast cells from the metabolic network, and obtained that the maximum threshold of phycocyanin yield was 1.0 ×10-3 mmol / (g DW· h),when the specific growth rate was about 0.083×10-3 mmol / (g DW· h). With the 110h fermentation time of yeast, the theoretical production of phycocyanin is 0.110mmol/g DW, which provides us with an ideal estimation of phycocyanin production, which can effectively guide the regulation of fermentation process, make the actual production closer to the theoretical production, and provide an effective estimation of phycocyanin production for the project; Model 3 is the yeast color development model. Photon simulation method is used to simulate the propagation of electromagnetic wave in the cell wall, and combined with the absorption spectrum of phycocyanin to obtain the reflection spectrum of yeast cells. And we use the phycocyanin concentration 0.7mg/L ,which was calculated in model 2, and predict the reflection spectrum of the cell. CIE1931 chromaticity calculation method is used to convert the spectrum into the corresponding chromaticity, to obtain the corresponding relationship between the production of phycocyanin and the color of yeast cells, which can provide a reference basis for our subsequent factory application and guide the downstream application of the product.

The innovation of our model is mainly reflected in the third model, yeast color-rendering model. According to the literature we know, there is no relevant research on cell color-rendering, taking into account both the optical effect of cell wall structures and the light path of pigments. In our model, we explore the influence of the fungal cell wall as a modulator of the light that reaches the inner part of the cell, by considering it as a photonic structure. The computer simulation method is simple and operable. The results obtained are very satisfactory.

During this experiment, yeast was used as the expression vector of phycocyanin. We could study the fermentation process through fermentation kinetics, and describe different indexes in the fermentation process through mathematical modeling.

Due to the fact that actual data of phycocyanin fermentation was unavailable, we utilized the experimental data of glutathione fermentation from the related literature ,whose basic principle is similar to the phycocyanin production, to fit the first three models and judge the applicability.

The charts below illustrated the relevant data of glutathione fermentation.

The first model we built is the strain growth model. The most commonly used models are Monod equation and logistic equation. As Monod equation is an idealized model, it has certain limitations. However, logistic equation is a typical S-shaped curve, which could well reflect the inhibition of strain concentration increase on its own growth in batch fermentation.As a result, we used logistic equation to describe the process of strain growth.

Below is the basic form of logistic equation:

The general solution of the equation is as follows:

In order to facilitate fitting, we properly deform the equation and introduce three parameters a, b and c.

Finally, we can obtain the simplified equation, which could be employed in the curve fitting.

The fitted image is shown in the following figure:

We found that R2 = 0.9947, which indicated the fitting effect is ideal. Through fitting the curve, we get the values of three parameters a, b and c, and then xm, x0, μm are also clear by calculating. Furthermore, the specific expression of strain growth model could be got.

The product accumulation model is the second model. In accordance to the fact that phycocyanin fermentation is batch fermentation, we decided to use piret equation to describe product accumulation after consulting the relevant literature.

Owing to the fact that the desired product phycocyanin and the obtained data product glutathione are amino acid compounds, the accumulation of the product is partially related to the growth of the strain. Therefore, none of α、β in this equation is 0.

The concrete solution can be known by integrating the equation.

We can replace the two complex parts of this expression with φ(t) and Φ(t). In accordance to the previously obtained x_m, x₀, μ_m , the concrete form of φ(t) and Φ(t) is known.

After software fitting, it is found that R2 = 0.9995, indicating the fitting effect is excellent. Obtaining α、β ,the specific form of the product accumulation model could be written.

The third model is the substrate consumption model. Since substrate consumption during the phycocyanin fermentation includes strain growth consumption, strain metabolism consumption and product accumulation consumption, we can employ Pirt equation to describe the whole process of substrate consumption.

More complex solutions are obtained by integration.

To make the curve fitting easier, we substitute the expression of the product accumulation model, and simplify the expression of the substrate consumption model by introducing three parameters L, M and N and the previously assumed φ(t) and Φ(t).

After simplifying to this clear form, we can carry out the curve fitting to obtain the values of L, M and N.

Substituting the values of S₀、 x₀、 α、β and solving the equations , the specific values of the three coefficients can be obtained. Moreover, the specific expression of the product accumulation model can also be clear.

From the above fitting operations, we can find that the three models we built have good applicability. As a result, once we get relevant experimental statistics, we can fit the curves by using these three models.Additionally, we can quantify the whole process of phycocyanin fermentation and make the strategic control of phycocyanin fermentation more convenient.

The second model is Genome-scale metabolic model ,which use the data of gene, metabolites, reactions to construct network.

Related studies have made great progress in recent years, for example, the paper, A consensus S. cerevisiae metabolic model Yeast8 and its ecosystem for comprehensively probing cellular metabolism, constructed a network which forms the basis of the ecosystem. And based on these papers, we carried out our work.

Firstly, as presented in the experiment part, new genes are introduced in S288c, so we add the involved reactions and important metabolites to the original network. We use MATLAB to construct the new network.

Secondly, we found that cofactors, such as carbon monoxide and ferredoxin, are not considered in original network. So we ignored those cofactors.

Thirdly, drawing upon the visualization tool fluxer, we mapped the new metabolic networks and present the flux.

Finally, we use the FBA method to calculate the growth rate and discuss the robustness analysis. As shown in the figure, growth rate gradually decreases with the increase of phycocyanin synthesis rate. When the synthesis rate of phycocyanin reaches the order of 1 micro molar, the growth rate decreases sharply. This means that the maximum threshold of phycocyanin synthesis rate is 10×10^-3 mmol/g DW/h, This conclusion is consistent with the self regulation of heme, which verifies the correctness of our metabolic network model to a certain extent.

With the 110h fermentation time of yeast, the theoretical production of phycocyanin p is 0.110 mmol/g DW.

Correspondingly, intracellular phycocyanin concentration is about 0.49798 mg/L. The calculation formula is as follows:

where M_phycocyanin = 20kd, which is the assumed molecular weight of phycocyanin, N = 2×10¹⁰g^-1 ,which means that It takes twenty billion yeast cells to weigh one gram , r = 3.75×10^-5dm , which is the radius of yeast cell.

The third model is color-rendering model for Yeast cell, which aims to study the corresponding relationship between intracellular phycocyanin concentration and cell color.

Yeast cell wall is a micro nano biological structure. Because micro nanostructures can affect the color rendering of light, we first need to study the optical properties of yeast cell wall. In the paper Photonics of fungal cell wall, there's much in their method that we can use. fungal cell wall behaves as a photonic structure that presents an optical response similar to that of an inhomogeneous thin film.

Firstly, we selected a typical TEM image of Saccharomyces cerevisiae S288C and intercepted a fragment of cell wall.

Based on TEM image of the cell wall, we use photonic simulation to reproduces the propagation of electromagnetic waves. We call this bitmap M and its pixel represents the position of the particle. Three matrices Mphys, Dphys and Ephys containing the physical values of mass, damping constant and externally applied force are defined. These matrices are related to M, D and E as follows:

Where m₀ is a ground level mass, the proportionality constant m_p has units of (kg/grey level) , the constant μ_p has units of (N s/m/grey level, and r_p is a proportionality constant with units of (N/grey level) that converts the value of grey level provided by the bitmap E to a value of force. According to the parameters in the paper, combining with the attempt of programming, we set parameters as follows:

The figure on the left is bitmap E that indicates the masses to be excited harmonically, which in this case are those contained in a vertical line on the left side of the bitmap, in white. The bitmap D showing in grey levels the region with damping constant is shown on the right .

The simulation consists in an algorithm that begins by sweeping all the elements of the matrices.

We suppose that the applied force on each mass determined by the bitmap E varies harmonically over time so that

The frequency of the harmonic excitation was set to ω 250 rad s−1 and ϕ was set to 0. The adapting constant was set to τ 1 ms/loop cycle.

The total force for the mass located at (i, j) can be expressed as:

The damping force is given by

where the matrix multiplication is a point-to-point multiplication, each element of Dphys being multiplied by its corresponding element of V .

The forces located at (i, j) due to the neighbour masses located at are calculated in terms of the previously defined matrices as

By means of Newton’s second law, we calculate the acceleration matrix A determining the acceleration of each mass. This matrix is computed as

where the matrix division is a point-to-point division, each element of F being divided by its corresponding element of M_phys.

The speed matrix V is obtained by integrating the acceleration.

The new displacement matrix H is also obtained by refreshing as

The intensity is calculated by integrating the square of the displacement matrix as

We repeated the simulation cycle. And for a certain wavelength, we can calculate the transmittance as:

The above is the process of computer simulation. We use computer simulation to calculate the reflectance as a function of the optical wavelength (λ).

After simulating the optical properties of cell wall, we also need to obtain the optical curve of phycocyanin. In the laboratory, we also measured the absorbance of pure phycocyanin. Combined with the literature, we obtained the following absorbance spectra. The ordinate is the absorption coefficient with units of mL / (mg · cm).

Next, we consider the light path in cells. When the external light enters the cell, it first transmits through the cell wall, and transmits in the solution. The following step is the reflection of the cell wall, then transmits again in the solution, and finally transmits through the cell wall and leaves the cell.

Because the cells are very small, about 5-10 microns in diameter, the color we see should be the accumulation of the colors of thousands of cells. So we overlap the cells and analyze their light path.

We assume that yeast cells overlap into spheres, as shown in the left, and the light path is shown on the right. To simplify the calculation, we ignore the loss of light transmission between cells and diffuse reflection and refraction of cells.

We also assume that the volume of yeast cell clusters is 1 cubic millimeter, which can just be seen by human eyes. According to the volume formula, the diameter of yeast cells d can be calculated to be about 1.2407 mm. However, due to the existence of cell wall and gap between cells, the optical path of direct light in phycocyanin solution is less than d, and we assume a constant σ = 0.95. So cumulative optical distance can be calculated:

b = d · σ

According to Lambert-Beer law, we can calculate the absorbance a under the condition of a certain optical distance b and a certain phycocyanin concentration c

A = abc

Combined with the results of photon simulation, the reflectivity of yeast cell population for a certain wavelength λ can be calculated. The calculation formula is as follows:

R = T * (1 - A) * (1 - T) * (1 - A) * T

The reflection spectrum can be obtained by calculating the corresponding reflectivity for the wavelength from 380nm to 760nm.

When the phycocyanin concentration is 0.49798 mg/L which is the concentration predicted by model 2, the synthetic reflection spectrum is as follows:

According to the chromaticity calculation theory of CIE1931, we can use the following formula to convert the reflection spectrum into the corresponding chromaticity coordinates, where the color matching function of each spectrum,(λ)、(λ)、(λ) and power distribution of reference S(λ).

Corresponding chromaticity coordinates are :

In addition, by changing the phycocyanin concentration, we can get different spectra and color coordinates. In the following table, we list the phycocyanin concentrations c and b*c, which is the product of optical path and phycocyanin concentration, the color coordinates of pure phycocyanin solution and the color coordinates of yeast cells.

We draw the color coordinates on the chromaticity diagram and compare the results of solution and cell. We find that after the influence of cell wall, the color temperature of color decreases, but the chromaticity deviates from cyan and is closer to the blue gamut.

Reference:

[1] A consensus S. cerevisiae metabolic model Yeast8 and its ecosystem for comprehensively probing cellular metabolism
[2] Computational Analysis of Reciprocal Association of Metabolism and Epigenetics in the Budding Yeast: A Genome-Scale Metabolic Model (GSMM) Approach
[3] Song, Yanqun, Rongfeng Zhu, and Peng Chen. "Physiological distribution and regulation of heme." SCIENTIA SINICA Chimica 45.11 (2015): 1194-1205.
[4] Hari, Archana, and Daniel Lobo. "Fluxer: a web application to compute, analyze and visualize genome-scale metabolic flux networks." Nucleic Acids Research 48.W1 (2020): W427-W435.
[5] Dolinko, Andrés E., and Diana C. Skigin. "A simulation method for determining the optical response of highly complex photonic structures of biological origin." arXiv preprint arXiv:1301.0754 (2013).
[6] Zakhartsev, Maksim, and Matthias Reuss. "Cell size and morphological properties of yeast Saccharomyces cerevisiae in relation to growth temperature." FEMS yeast research 18.6 (2018): foy052.
[7] Dolinko, A. E. "From Newton's second law to Huygens's principle: visualizing waves in a large array of masses joined by springs." European journal of physics 30.6 (2009): 1217.