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Mathematical Model on Fatty Acid Degradation Device

　　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.

The Ordinary Differential Equations of the Model

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

…………………………①

　　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,

…………………………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

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

…………………………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 high while that of gene fadX is relatively low, 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.