Team:FAFU-CHINA/Model/Growth Model

Team FAFU-CHINA

Growth Model of Chlamydomonas reinhardtii – Supported by BNU-CHINA

1.Introduction

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.

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.

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.

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. 　

2.Hypothesis of Model

The fundamental way that the algae grow is through photosynthesis, in which the inorganic carbon in the water (carbon dioxide) can be transformed into the organic carbon (carbohydrate). However, the photosynthesis of the algae, which is fixed in the water, is influenced and limited by lots of factor.

Not only in the laboratory but also the factory, Culturing algae in the stirred-well mixed culturing vessel with fixed volume is a common way. Under the circumstances, we believe that those important parameters, which is related to the growth of the algae, are all isotropy. In another word, by only considering the result which is dependent on time, we can meet the requirements of the experiments, or even the production.

Thus, after weighing up the actual conditions, based on the combination of the existing model of the first order ordinary differential equations, we set up some second order ordinary differential equations to simulate the growth velocity - the algae’s dry weight of the time derivative. Furthermore, the results, which the model has simulated the growth condition of the algae in limited light and nutritive substance, quite tally with the actual situation. In some limiting cases, such as limited light with unlimited nutritive substance or unlimited light with limited nutritive substance, these equations can be simplified into some common first order ordinary differential equations.

Figure1: The logical relationships among four parameters

Firstly, according to the flow figure below, we explain the logical relationship of the four important parameters of the model.

The growing speed of the alga in the incubator, A, which means the change of dry weight of the alga in unit interval is equal to the increase minus the decrease of the organic substance in the cells. The decrease is mainly based on two ways. One is the natural death and the other is the artificial separation of the useful mature alga.

The carbon dioxide in the water area is converted to saccharides by the photosynthesis of alga and then stored in the cells. This also shows the conclusion that the growth of the population density of the alga will accelerate the speed of the decrease of carbon dioxide. Hence the content of carbon is equal to the inflow minus the part which are converted by photosynthesis.

Similarly, the content of mineral substance in the water area M is also a factor which has its influence on photosynthesis. It will decrease faster when the amount of the alga is increasing as well.

In a certain space, the sum of the carbohydrate, S (exclusive that in the cell), can be considered as the result which the fixed sum of the photosynthesis subtracts the total sum of the increment of the dry weight and the decrement of the dry weight in the culturing medium. (death, artificial extraction).

Figure2: The quantitative relationships among four parameters

3.Mathematic Formulation

According to the expression of photosynthesis as following:

Figure3: The reaction of photosynthesis

We know that the coefficient of the carbon dioxide which is consumed by carbohydrates is 44/33g [CO2] g[CH2O]^-1. If the increment of the carbohydrates’ dry weight is influenced only by the content of the organic matters and mineral substance, and the proportion of the two is 1:9 approximately, the coefficient of the consumed mineral substance, k₂, is:

Figure4: The equation of k₂

The coefficient of the consumed organic matters, k3, is:

Figure5: The equation of k₃

From the above, what can be seen is that the main idea of these ordinary differential equations is element conservation. They work out the growth velocity indirectly by analyzing the transform of the substances in the fixed space.

4.Result

Figure 6. The model result of Algae concentration

Reference

[1] Jayaraman S K, Rhinehart R R. Modeling and Optimization of Algae Growth[J]. Industrial & Engineering Chemistry Research, 2010, 54(33).

Acknowledgement

Zhiyao Chen and BNU-CHINA team.

Written by Zhiyao Chen (BNU-CHINA team member) and Junhao Lu

Demo Model

Abstract

Demo model can be divided into two main parts, the first part is the establishment of algae growth model, the second part is to describe the changes in the number of larvae. In the first part, we model and the transformation of material based on the idea of the establishment of changes in the number of differential equations of algae on time, then consider the distribution of the number of algae in space, to obtain more accurate results. The results showed that the algal growth curve was similar to that of the S type curve, which was consistent with our expectations.

In the second part, we establish the relationship between the number of larvae change on time using predator-prey model, and extract the parameters required by the model from the real experimental data, to predict the change of the number of larvae. This model can predict algae in different experiments used to kill mosquito larvae.

Basic assumption

1.The distribution of algae is spatially isotropic.

2.The existence of water and algae can make the transmission of light intensity attenuation

3.The death rate of the algae and the concentration of the algae were fixed proportion.

4.the external incident light intensity changes in 24h cycle

Algae growth model

Model One

First, we based on the logical relationship between the four important parameters in the flow chart and built our model. This part of the basic theory by BNU-CHINA students as we provide, we added a richer explanation for their model.

We define A as the concentration of algae, so the A derivative to time is the growth rate of algae. The content of mineral substance in the water area is M. S, C represent the carbohydrate and the carbon dioxide in the water respectively. Suppose that the pool is spatially isotropic, and let the input of mineral substance and co2 is <img title="" alt="本地图片，请重新上传">，<img title="" alt="本地图片，请重新上传">。

The Natural algae mortality is <img title="" alt="本地图片，请重新上传">, which will decrease the amount of A, and will also take away the sugar stored in algae. The sugar will be converted from a part of carbon dioxide, and we put it as <img title="" alt="本地图片，请重新上传">, the cost of C is <img title="" alt="本地图片，请重新上传">.

From the photosynthetic reaction, we can easily obtain the coefficient K1

so <img title="" alt="本地图片，请重新上传">

New algae are produced inside the existing algae at a rate <img title="" alt="本地图片，请重新上传"> from nutrients and sugar, where <img title="" alt="本地图片，请重新上传">is the rate constant and f m (M) denotes the concentration of nutrients inside the cells. This depletes nutrients and sugar by a rate of <img title="" alt="本地图片，请重新上传"> and <img title="" alt="本地图片，请重新上传">, respectively. The ratio is about 1: 9

So far, we can list a set of differential equations to describe the changes in the amount of algae, mineral substance, sugar and carbon dioxide

The model also obeys the following conservation law

There are many coefficients in the above equation that have not been explained, then we will derive the specific expression of each coefficient and its value.

<img title="" alt="本地图片，请重新上传"> is supposed to be a parameter that relate with the current concentration of algae. We borrowed from the Logistic model as the following form:

<img title="" alt="本地图片，请重新上传"> represent the max algae that the pool can support, k4 is a proportional coefficient

The form of <img title="" alt="本地图片，请重新上传"> is similar

As for <img title="" alt="本地图片，请重新上传">, we think It should be a function of temperature , light , A and C. Let us suppose that

The Influence of Light Intensity on this coefficient are borrowed from Huisman model:

This function considered the influence that comes from A, the depth of the pool d. H is half saturation coefficient,<img title="" alt="本地图片，请重新上传"> and <img title="" alt="本地图片，请重新上传"> is the light absorption constants of algae and background. We use a simple fractional expression to describe The effect of temperature

Similarly, we also deal with the effects of C on the parameters in the same form:

Now we update the terms and rewrite the equations as follows:

<img title="" alt="本地图片，请重新上传"> When solving the equations, the value of each parameter is taken from the real environment of the experiment. Some of them is inconvenient to measure, and we found them from the Internet and the literature t. The two terms that represent the external input we treat as follows:

In order to maintain the concentration of carbon dioxide in the water to maintain near the ideal value, we set the input amount

And the growth curve of algae could be solved by MATLAB.

As we can see from the image, it is a curve that is similar to the S type, which is in line with our expectations.

Model 2

We only consider the relationship between time and algae’ concentration in model 1. In this part, we add the factor of space, and combine it with certain time.

The biomass of algae in the water body is related to the growth rate (which Is largely decided by the rate of photosynthesis) and the process of mixing and death. In this part, we mainly focus on the influence of availability of light to the photosynthesis rate. The death rate includes both the rate of being token in as well as the natural death rate of the algae. Advection is assumed to be absent which corresponds to the still water body. In the horizontal plane, we consider no variation and hence, the growth rate is independent of x and y coordinates. The depth in the water body is denoted by z.

<img title="" alt="本地图片，请重新上传">is a constant related to diffusion efficiency,<img title="" alt="本地图片，请重新上传">、<img title="" alt="本地图片，请重新上传">、<img title="" alt="本地图片，请重新上传">、<img title="" alt="本地图片，请重新上传"> are correlated variables related to growth rate: light intensity, nutrition (phosphorus and nitrogen) and the concentration of carbon dioxide. <img title="" alt="本地图片，请重新上传">means this equation has considered the decreased rate because of being token and natural death.

For light intensity, we adopt dependency relationship in the form of Monod:

<img title="" alt="本地图片，请重新上传">lm is the efficiency for algae to absorb light. HL is semi-saturation concentration. This equation could guarantee that during less light irradiation, growth rate is near linear, and when light is intensified, growth rate is limited by the border of μ0. The light absorbed by algae is not consistent. Light intensity is affected by two aspects: the existence of algae (upper layer) and the quantity of water (upper layer). The following equation describes the light-tight phenomenon caused by algae, with the decrease of light due to the increase of depth

I0(t) represents light intensity from the outside related to time (for example the cycle of day and night). k is the constant represents the decrease of water towards light. rs is correlated constant of light decrease with the existence of algae.

Through the operator after discretization, the partial differential equation PDE with boundary conditions can be changed into ordinary differential equation ODE:

After approximation:

Use the accumulation of small quantity to replace integral operation.

According to the parameters given in the essay, we draw a sketch after several adjustments and completion:

In this equation, when <img title="" alt="本地图片，请重新上传"> 、<img title="" alt="本地图片，请重新上传"> we do discretization and solving, and the time period is 3 days; the quantity of mineral substance is constant with nitrogen of 3.64 · 10−10 [mol/(l · s)] and phosphorus of 2.78 · 10−10 [mol/(l · s)], without extra adding of carbon dioxide. Seeing from the figure we can find the periodicity of algae density, which is caused by the periodically changed light intensity<img title="" alt="本地图片，请重新上传">. Similar to our expectation, the concentration of algae is lower in the deep. Because of the parameter, the growth cannot be guaranteed at the beginning of simulation due to the lack of light. After a day, at night, algae density falls below the initial value, and then decrease gradually. Under certain depth, algae can no longer continue growth, and will be token or die naturally.

With different nutrition, light intensity and other boundary conditions, we will get different response system.

Model of wriggler quantity

We have analyzed the relationship among algae concentration, space and time. In order to predict the effect of algae to kill pests, we construct a new model to describe this. Our goal is to establish a system to predict the killing effect under different circumstances.

In this part, we use “predators and prey” to describe algae and wriggler. We use prey model to describe their mutual effect. In this model, W represents the quantity of wriggler, which is predators. Because they may die due to their own reasons (we suppose <img title="" alt="本地图片，请重新上传">as proportionality factor). Therefore, we should add an item on the right side of the equation,<img title="" alt="本地图片，请重新上传">.<img title="" alt="本地图片，请重新上传">represents the change of algae due to time. For simplification, we assume the denser the algae concentration, the better effect towards kill wrigglers. We use to<img title="" alt="本地图片，请重新上传">represent proportionality factor of inhibiting ability, which is decided by the principle to kill wrigglers. So, we could get:

From the experiment data,

when t=5, wrigglers begin to die

when t=24, livability is 0.04885, mortality is 95.12%

when t=48, livability <img title="" alt="本地图片，请重新上传">

We think when t=5, toxicity begins to appear, so we choose t=5 as our time starting point. Following is drawn by MATLAB. Apart from this, we can get the equation’s analytic solution using MATLAB, and the parameters are decided by data. Through this way, we can have a better prediction that under different circumstances, the quantity of wriggler will change in which way.

The change curve of wriggler quantity is:

The equation’s analytic solution is:

Team:FAFU-CHINA/Model/Growth Model

Contents

3.Mathematic Formulation

4.Result

Reference