Team:ZJU-China/Modeling/Biobrick

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.

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 follow.

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

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.

and results:

Robustness analysis

  Robustness is very important to a genetic circuit. Scientists hope their artificial system have high tolerance to the variety of environment or system parameters. In our genetic circuit, and are directly represent the properties of Vgb and Fdhf, so we are interested in to what degree these two parameters could influence the behavior of Ptet.

  We could show the contribution of the two parameters and by fix one of them, and let the other one varies linearly. When fix at 1e-9, and changing from 0 to 1e-7 continuously, the production of each protein is shown in following graphs.

Cfp.figure

Yfp.figure

Rfp.figure

Also, we could let both and to vary linearly (from 0 to 1e-11) to see how protein CFP changes. The actual curve with respect to depth of biofilm for CFP production is a curve on this surface.

Results

combine Data acquired in part characterization and modeling of biofilm and biobrick, we can get the relationship between distribution of oxygen in the biofilm and the expression rate of CFP RFP YFP as follow:

YFP:

RFP:

From the fitting results we know that: The PoPS of YFP could be regard as zero since PPO of 10%, and approximately linearly increasing from 0% to 2.5%, and decreasing from 2.5% to 10%. The PoPS of RFP could be regard as zero since PPO of 2%, and approximately linear from 0% to 2%. Together with the results of former modeling sections, we could show the stratified biofilm in following graph.