Team:WHU-China/Modeling

①

②

while f denotes the concentration of fatty acid outside the bacteria, K_L the Michaelis constant of FadL, and k_L the maximal activity of FadL.

Details for the ODE can be illustrated in Fig 1.

Fig 1 Illustration of the meaning 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

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.

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.

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.

We hope that our E. coslim can modulate gut microbiota therefore have long- term effect to help people lose weight

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. While only a concise description is presented in this page, a detailed support information of this mathematical model can be downloaded here.

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

By carefully assigning values to parameters in the mathematical model of the microbiota regulation, we found that the metabolic capacity of E.coslim should be controlled to a proper level, rather than as great as possible. Although it seems that a greater ability to degrade fatty acid and synthesize cellulose may further reduce the calorie people absorbed, it may also lead to potential healthy risk since the microbiota consititution may change drastically. The metabolic capacity indicated by related parameters also helps us in designing a perfect-functioning E.coslim.

Fig 1. Typical false positive and false negative results of FANCY identification. A. The identification result. B. The original fluorescent image (some cells in the original image are not able to be detected after image transformation).

Fig 2. Comparison of a cell fluorescent image and its identification result by FANCY. A. The cell fluorescent image. B. The identification result, in which objects identified as single cells are dotted with a *.

The false positive and negative rate may decrease with more training data and proper image enhancement.

　 　 All source code programmed in Matlab can be freely downloaded here. For detailed usage please refer to the annotation in the code.