Team:WHU-China/Modeling

To date, we have built all our three devices and tested the function and regulation of each biobrick. However, our aim is far more than just making a toy in the laboratory. We aim to create a product that can be widely applied in clinical and other areas.

First, we will amalgamate the three devices, namely, the Fatty Acid Degradation, the Cellulose Synthesis, and the Colonization, into one whole system. Since there are tens of genes and regulation elements in all and their total length may exceed 30kb, a larger vector like λ phage rather than a plasmid may be adopted. It may also be a good choice to integrate the three devices into the chromosome.

Second, the Escherichia coli may be a good model for molecule cloning operations, but not suit for pharmacy since its possible risk in infection and diseases. We propose Lactobacillus as a better model because not only its safety has been well demonstrated in food industry, but also the yogurt made from the genetically modified Lactobacillus will possess the property of making you slim!

The function of the whole system in Escherichia coli or Lactobacillus will be tested both in vitro and in vivo to confirm the effectiveness of our project. The gut microbiota and E.coslim will be inoculated in a glass tube, through which plasma made from different kinds of food will flow. We will test the changes of the microbe community inoculated. Furthermore, a gut microbiota transplantation experiment on mice may also be conducted for further confirmation.

Our product will be finally packaged into two capsules. Capsule A is E.coslim, which is able to influent the human body's absorption to high energy contained nutrients, regulate the gut microbiota and make you slimmer day by day. And Capsule B is xylose, which can induce the Death Device in E.coslim and avoid people from malnutrition after taking in E.coslim for a long enough time.

Last but not least, we not only create a new microbe that can make people slim, but also provide tools to sense the fatty acid, the glucose and the xylose in other circumstances. So the new biobricks we submit may be applied in a variety of areas, such as the degradation of waste oil, urine sugar test for diabetes patients, safety control of genetic engineering, and so on. We are looking forward the day when E.coslim is truly applied in people's life, in clinical and more widely in other areas beyond our imagination.

Model I: Fatty Acid Degradation

The Fatty Acid Degradation Device may be the most complicated part in our project, along with its great importance. The antagonistic relationship between gene fadR and those related to β oxidation -- the fadL, fadD, etc, -- makes it regulatable to the concentration of fatty acid in the environment. Thus, it is necessary to explore the quantitative response corresponding to the concentration change of fatty acid. We build an ordinary differential equations-based mathematical model to describe the device and find a proper set of parameters under which the proportion of the steady expression level of fadL to fadR changes broadly from 0.2 to 3.5. The model mathematically demonstrates the effectiveness of the Fatty Acid Degradation Device and also provides meaningful clues for the optimization of the device in experiments.

We conduct an evaluation by mathematical modeling and build the ordinary differential equations (ODE) as follows:

For simplicity, all genes with a promoter PfadR and equally regulated by FadR are deemed as a whole and represented as FadX, i.e., FadX refers to FadL, FadD, FadE, FadA, FadB, FadI, FadJ. And the Complex, or variable x7, refers to the Fatty Acyl-CoA-FadR Complex.

Parameters in the ODEs:

　　① E denotes the constitutive expression rate of FadR, and D the degradation rates of FadR, FadX and Complex, which is assumed equal.

　　② a denotes the affinity of FadR to the promoter PfadR, and V denotes the background expression rate of related genes.

　　③ k₁ and k₂ denote the forward and reverse reaction rate coefficients, respectively. k₃ to k₆ are parameters related to enzyme-catalyzed reactions based on the Michaelis-Menten Equation. Specially,

Fig 1 Illustration of the meaning of the ODE

Analysis on the Steady State of the ODE

By setting the right side of the equations to zeros, we get algebra equations about the five variables at the steady state. And after elimination we obtain the cubit equation

(ak₁D³)x³ + (D³k₁ - ak₁D²E)x² + (k₆D²V - k₁D²E + k₁k₃DV + k₂k₆DV)x - k₆DEV - k₂k₆EV = 0 ③

And the value of each variable in its steady state (the balanced point) is

④

The ODE (1) is highly complicated and we adopt numerical methods to analyze its properties. First, we generate 100000 sets of parameters stochastically (all in interval [0,10], and this setting keeps unchanged without special statement) to see the root distribution of equation (3). The results show that there is only one real positive root in 99890 cases and 3 in the rest cases. No cases when the real positive root doesn't exist are found.

Then note that the balanced point in (4) may not be authentic when k₄＜k₃ and x₃^* becomes negative, which is impossible to occur. So we stochastically generate 100 parameters in which k₄＜k₃, and after solve the ODE numerically we find that all variables tend to an asymptotic steady state except for x₃, which approaches +∞ as t →+∞ (Fig 2). Interestingly, we find that the authentic balanced point of x₁ and x₂ calculated by directly solving the ODE (1) numerically is very close to that calculated by formula (4). For example, when E=4.5249, a=8.0649, V=2.5906, D=1.6831, k₁=5.2315, k₂=8.6560, k₃=8.7696, k₄=1.0092, k₅=6.9635, k₆=9.3253, x₁ and x₂ finally approach to 2.6599 and 0.0686, respectively, while x₁^*=2.3952 and x₂^*=0.0758 (Fig 2). The term close may not be mathematically strict, but it plays an important role in the later discussion.

Fig 2 Numerical simulation when k₄＜k₃.

Besides, x₄^* and x₅^* may also be negative. We also generate 100000 sets of parameters stochastically (without the limitation of k₄＜k₃) to see how frequently x₄^* or x₅^* will be negative and what will it be like. However, it turns out that in no case will x₄^* orx₅^* be negative. So we may draw a fuzzy conclusion according to all the results above that under most conditions (99.89%), we can obtain a balanced point of ODE (1) which may not be authentic by formula (4). Fuzzy as the conclusion is, it is still useful to serve as an indicator for searching for a proper set of parameters, under which the Fatty Acid Degradation Device is highly regulatable.

Parameter Screening

It is expected that when the concentration of fatty acid in the environment is high, the expression level of gene fadR is relatively low while that of gene fadX is relatively high, and vice versa. And among the 10 parameters in ODE (1), k₃ is positively related to the concentration of fatty acid outside the bacteria according to formula (2). So we assume that the device is thought to be regulatable when k₃ is equal to 0.5 and 1.5, the ratio of the expression level of fadX to fadR at steady state rises but is still lower than 0.5, and when k₃ is equal to 8.5 and 9.5, the ratio also rises and is both greater than 2.0.

We take advantage of the simplicity in calculating complexity of formula (4) to calculate the steady expression levels of fadR and fadX despite its possible errors. 10000 random parameters (k₃ excluded) are generated and for each k₃ in [0.5, 1.5, 8.5, 9.5], the balanced point of ODE (1) is calculated according to formula (4), respectively. Then compare the ratio of the expression levels of fadX to fadR at the balanced point and save it if it meets the condition above. We verify all the parameters saved by directly solving the ODE (1) numerically to see if it really meets the condition. 182 out of 10000 sets of parameters are saved and 57 of them remain after the verification. A typical example below (Fig 3) illustrates the change of the expression levels of fadR and fadX and their ratio corresponding to k₃. As k₃ increases, the ratio rises smoothly from 0.2 to 3.5, while the expression level of fadX rises from 0.6 to 2.0, and that of fadR decreases from 2.6 to 0.6.

Fig 3 The change of the expression levels of fadR and fadX and their ratio corresponding to k₃.

Conclusion

To evaluate the response of gene expression levels to the concentration of fatty acid in the environment quantitatively, we build a mathematical model based on ODE and demonstrate that the antagonistic relationship between fadR and fadX serves as a linear regulator to the gene expression. This is important for the function of Fatty Acid Degradation Device because the model suggests that the Device can adjust itself to an appropriate state when induced by fatty acid and function properly rather than changes drastically. So the Device is implied mathematically to possess a great potential of applications in human being.

Perhaps the most challenging part of our idea is that the E.coslim will make a difference not only by influence the metabolism but also interact with our inner ecosystem --- the Human Gut Microbiota. Collectively, the microbial associates that reside in and on the human body constitute our microbiota, and the genes they encode is known as our microbiome. Containing at least 100 trillion of cells, the human gut microbiota is of high complexity and diversity, with taxa across the tree of life, bacteria, eukaryotes, viruses and archaeons. As sometimes referred to as our "forgotten organ", it plays a major role in health and diseases in human, including obesity and diabetes. And maybe even more importantly, it interacts with the immune system, providing signals to promote the maturation of immune cells and the normal development of immune functions.

Though highly complicated, the gut microbiota is typically dominated by bacteria and specifically by members of the divisions Bacteriodetes and Firmicutes. And interestingly, it has been found that in animal model of obesity, the interplay of the two phyla is shifted with a significant reduction of Bacteriodetes and a corresponding increase of Firmicutes. This results in an increased capacity for harvesting energy from food and produces low-level inflammation. And more importantly, the imbalanced microbiota may be related to the recurrence of obesity after the treatment (whatever drugs, diets or exercises) stops.

Let us refocus on our versatile E.coslim. When establishing itself in the gut, E.coslim will interact with the microbiota and thus regulate them. Considering the importance of microbiota in human health, it is necessary to know what effects the E.coslim has on the gut microbiota. On one hand, it may be a great disaster for people if the establishment of E.coslim leads to the extinction of some major bacteria taxa. And on the other hand, it may be proven noneffective if E.coslim died gradually after intake.

However, the metagenomic sequencing may be required to conduct the experiment to see the change of microbiota caused by E.coslim. Due to the limitation of time and funds, we as undergraduate students are not able to perform such experiments. Instead, we use a mathematical model to predict the results of the interaction between E.coslim and the gut microbiota.

Consistent with previous discoveries and without particularity, we assume that the gut microbiota is constituted by two populations of bacteria, the Firmicutes (x1) and the Bacteriodetes (x2). Bacterium within each population has the same properties, which is reflected in the equality of the corresponding mathematical parameters. And it is also assumed that there are two kinds of resources, glucose (S) and fatty acid (R), which are perfectly substitutable for both populations. We simplify this situation as an exploitative competition in a chemostat, and the ODEs are:

　　1. S(t) and R(t): concentrations of glucose and fatty acid, respectively.

　　2. x_i: biomass of the competing populations at time t

　　3. S⁰ and R⁰: concentrations of resource S and R in the feed bottle

　　4. D: dilution rate

　　[The specific death rates of the microorganisms are assumed to be insignificant compared to this dilution rate D]

　　5.S_i and R_i: the rate of conversion of nutrient S to biomass of population x_i

　　[if the conversion of nutrient to biomass is proportional to the amount of nutrient consumed, the consumption rate of resource S per unit of　competitor x_i is denoted S_i[S(t),R(t)]/ξ_i where ξ_i is the respective growth yield constant. ]

　　6. G_i: the rate of conversion of nutrient to biomass of population x_i

　　[Since perfectly substitutable resources are alternative sources of the same essential nutrient, the rate of conversion of nutrient to biomass of population x_i is made up of a contribution from the consumption of resource S as well as R:

　　Here we choose

　　And let

They denote the maximal growth rate of population x_i on resource S(R) when none of the other resource is available.

A model is built to describe the quantitative relationship between Firmicutes and Bacteriodetes in obese people's intestines.

Parameters m_Si(m_Si) can be assigned values to Firmicutes and Bacteriodetes so as to simulate the ability to utilize glucose and fatty acid. If their ability to use nutrient are given as follow:

In this situation, the ratio N(Firmicutes)/N(Bacteriodetes) is rather high, usually achieving a value 8.0, and each of their absolute number, or concentration, is stable.

Situation After the Establishment of E. coslim Flora

We try to add the E.coslim into system. E.coslilm consumes glucose as well as fatty acid, thus makes itself a competitor to Firmicutes and Bacteriodetes. While it is reproducing in intestines, the competition among these three types of bacteria makes the number change gradually. And we want to know the results of the establishment of the GEB in the gut.

The key point is to find out how competitive our E.coslim should be. In other words, we have to point out its ability to consume glucose and fatty acid---to study new parameters (m_S3 ,m_R3). For example, if m_S3＞m_S1, then we conclude that E.coslim has stronger ability to consume glucose than Firmicutes. We try to find out an appropriate pair of (m_S3 ,m_R3). And next we will discuss different situations of (m_S3 ,m_R3), respectively.

Situation 1. (m_S3 ,m_R3)=(2.5, 2.1)

The number of "+" in the table below qulitatively represents the ability of the corresponding bacteria to consume the corresponding resource. "+" represents "low", "++" moderate, and "+++" strong, etc.

Situation 2. (m_S3 ,m_R3)=(2.1,0.4)

Situation 3. (m_S3 ,m_R3)=(2.1,2.11)

The simulation result shows that only in this situation will the E.coslim establish itself in the gut successfully without leading to the extinction of Firmicutes or Bacteriodetes. And surprisingly, the proportion of Firmicutes to Bacteriodetes also declines, as consistent to the condition in normal people. We may predict that E.coslim which suits Situation 4 can not only influence the human metabolism but also regulate the gut microbiota constitution, thus preventing the recurrence of obesity.

We also map the change of the consuming rate on each resource between the situation before and after the establishment of E.coslim (Situation 4 is used), as shown in Fig 7. And it is illustrated that the consuming rates of Bacteriodetes on both of the resources decline slightly, while that of Firmicutes is down-regulated acompanied with the increase of that of E.coslim.

Fig 7

Gene Expression Measurement by FANCY

Abstract

The quantitative measurement of gene expression levels is of great importance in the research of molecule biology, but existing measuring methods are either semi-quantitative (Western Blotting) or require costly instruments and agents (Flow Cytometry). We invent a novel quantitative and inexpensive method, FANCY, which is short for Fluorescent ANalysis of CYto-imaging (also in the honor of a wise and pretty girl Fan Cheng in our team) and based on cell fluorescence imaging and Support Vector Machine, to measure the gene expression in our project. FANCY contains three sub-programme: FANCYSelector to select single cells manually and calculate their properties, FANCYTrainer to train an SVM, and FANCYScanner to identify single cells by SVM trained by FANCYTrainer. The application of FANCY to 5 typical images proves the effectiveness and the error rate is acceptable. All our programs are freely available online and we suggest FANCY more widely applied in iGEM and other further researches.

　　Preparation of fluorescent images

　　1. Connect a red fluorescent protein gene (gene rfp, or biobrick 1-22O) with the objective gene under the same promoter in a plasmid by molecule cloning, i.e., make a polycistron which can encode both the objective gene product and RFP, thus making the expression levels of the two genes the same. Other fluorescent genes such as gfp or yfp may be also applicable.

　　2. Transform the plasmid into E.coli and incubate it until a proper density.

　　3. Prepare a slide sample of the transformed E.coli and dry it in the air. Cover the bacteria with anti-fluorescence quencher to maintain the fluorescence. The bacteria may be diluted to avoid cell mass formation on the slide during the preparation.

　　4. Take photos of the slide by fluorescent microscopy. Select visual fields with as many single cells and little cell masses as possible, and avoid photographing for too long a time since the fluorescent density may decrease under the exciting light.

　　Support vector machine training (FANCYSelector and FANCYTrainer)

　　1. 10 typical images are selected as the training images.

　　2. Transfer the original RGB images to gray images G and binary images B. No image enhancement was processed in our project, but it is suggested to be done in other conditions if needed.

　　3. For each object detected in the binary image B, ask the user to classify whether it is a single cell or a cell mass, and after that the object will be marked TRUE or FALSE, respectively. Then calculate the area A, perimeter P, Euler number E, the maximal length of any two points in the object L and the fluorescence F. The fluorescence F is calculated by the numerical double integral of the gray level of the image G on the region identified by the current object in the image B.

　　4. Record all the data obtained in Step 3 and save them in an ASCII or .mat file (optional).

　　5. Use the data A, P, E and L as the training data and the user's classification result as the group data to train the support vector machine (SVM). The Gaussian radial basis function with a scaling factor of 1 is selected as the kennel function of the SVM.

　　Single cell identification and fluorescence calculation (FANCYScanner)

　　1. For all the images taken from the slide, transfer them to gray and binary images respectively, calculate the A, P, E, L and F of each object in the binary images (the same as the former part) and classify them as single cells or cell masses using the SVM trained. All objects classified as single cells will be marked with * in the binary image B. And B will be saved for possible performance evaluation.

2. All data from objects classified as single cells will be recorded in a ASCII file. And the average and the standard variance of the fluorescent density (fluorescence F divided by area A) is calculated as the quantitative description of the objective gene expression level.

We randomly selected and apply both FANCY and manual identification to 5 other images to evaluate the performance of FANCY. The false positive and false negative rate of FANCY are calculated in comparison to manual identification, which defaults to be absolutely correct. 179 positive and 122 negative objects in all are identified by FANCY and the evaluation result is shown in Table 1.