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From 2012.igem.org

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 0.02 to 50. 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, KL the Michaelis constant of FadL, and kL 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 (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* or x5* 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