Basic concepts and assumptions

The ODE formalism models the concentrations of RNAs, proteins, and other molecules by time-dependent variables with values contained in the set of nonnegative real numbers. Regulatory interactions take the form of functional and differential relations between the concentration variables. For a typical transcription-translation process, the ODEs modeling approach associates two ODEs with any given gene i; one modeling the rate of change of the concentration of the transcribed mRNA r_i, and the other describing the rate of change of the concentration of its corresponding translated protein p_i. Thus for our network with 3 genes we have:

Where (1) describes transcription, (2) describes translation, and i = 1,…,N. The functions R_i(p_j) describe the dependence of mRNA concentration on protein concentration p_j (If protein p_j has no effect on mRNA r_i, then correspond function is set to zero.) The functional F(·) in (1) is defined in terms of sums and products of functions R_i. Function P_i in (2) describes the translation of the mRNA r_i into a protein p_i. Parameters γ_i, δ_i (i = 1,…,N), represent the degradation parameters of the mRNAs and proteins produced by gene i. As is common, we shall assume that the degradation of proteins or mRNAs is not regulated, namely that it does not depend on the concentrations of other molecules in the cell. Function R_i is assumed to be in the form of Hill function as usual (since our cases are all inhibitors, we shall denote the Hill function h-(p,K,n)), and the function P_i is taken to be a linear term proportional to the concentration of mRNA r_i.

where K_i is the microscopic dissociation constant, and n_i is Hill coefficient, describing cooperativity.

Equations & Parameters

Based on the above, our model could be founded with following ODEs:

Where

Parameters

Variables are defined in following table.

We should note that α and β are two functions of the oxygen concentration, which are determined by the properties of corresponding promoter. For a certain depth of the biofilm, the concentration of oxygen is a constant in our model, so are and . Therefore we could solve these equations at different oxygen concentration and combine all the results to show how this system work.

Because of the nonlinearity of the Hill functions, the solutions of a system of ordinary differential equations of a network of many genes cannot generally be determined by analytical means.

Equilibrium analysis

Notice that we care more about the final state of the bacteria in different depth of the biofilm(thus in different oxygen condition), we assume that the system has reached a steady state. We could calculus this steady state by setting all derivatives with respect to time to zero. That is, let

With (4)-(10), we have:

Clearly, the behavior of this system is determined by the two functions, thus determined by the behavior of promoter vgb & promoter fdhF. On the other hand, the solutions of equations The two functions αand β are hard to determine precisely, but their behavior could be illustrated in follow graph qualitatively.

First case, When the expression of promoter vgb and fdhF could be neglected, that is, we could setαandβto be both zero. The ODEs system become

The results are shown as follows

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Secondly,in the anaerobic condition, that is, when the expression of promoter vgb could be neglected, we setαto be zero, and setβto be bigger than the dissociation constant in (13). Then we have

The results are shown as follow

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Thirdly, since our ODEs model are symmetric respect to YFP & CFP in some sense, therefore the case of microaerobic (where we could setβto be zero) would be the same(symmetric) as in second case.

Results

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