Team:BIT-China/Modeling

Modeling is a useful and powerful tool, which can reduce the waste of time in experiments and predict a proper direction of our work. Thus, the modeling work is essential dealing to the complexity of E.co-lock.

In E.co-lock, we developed layered AND gates to simulate the construction of electronic lock. At the same time, a common quorum sensing system: Min system is also introduced in E.co-lock. Therefore, we developed an integral and excellent modeling to our system. We will introduce our model to you one by one and at last we have a combination of these model. In particular, we described and solved following questions:

(1) The condition of sensors after adding different inducers.

(2) The effect of quorum control inducer（AHL） to a single cell

(3) The cell density influenced by MIN system

(4) A connection of layered logic gates and MIN system

sRNAs can regulate protein expression by pairing with target mRNAs. We construct an easy model for this process by the equations as follows[1]:

Then we wrote some other equations which describe the normal condition (without sRNA) of mRNA and proteins. We removed the equation for sRNA and the factor related to sRNA in other equations:

The parameters are the same as previous described. We combined (I) and (II) by time parameter. At first we use equations (II) to simulate the normal process of transcription and translation. After some time we use the equations (I) to simulate the repression the process in which sRNAs pair with target mRNAs. The figure 1.1 shows the results.

Because the parameter μ is not a certain number that can be tested by experiment, we use different value of μ to simulate the process. The results shows as figure 1.2:

From our experimental data, the efficiency of sRNAs selected by ourselves are around 55%. According to these data we chose the value of μ that can suit the experimental data best so that we can have better simulation for next big model: Layered AND gates.

The construction strategy we use for this model is the same as sRNA’s. We wrote a series of differential equations associated with each other to simulate the dynamic change of all the materials (including inducers, mRNAs, sRNAs and proteins).In this model, we used the experimental data and the results from sRNA model.

We adopt promoter activity to describe the production of mRNA and sRNA so that we can relate the transcription rate with the concentration of inducers. We use Hill equation to imitate the process when inducers binds with promoters.All the equations and descriptions is from a reference[2].

Hill equation:

where fTL serves as the fraction of transcription factor (TF) bound to inducers, L is the concentration of inducers, KD is the dissociation constant, and n is the cooperativity. By mass balance, the fraction of TF without inducers bound fT is

Each promoter activity we used in our model is given by the following equations[2]:

The parameters for promoter activity equations is given by Table S2 (the parameters are from references[2,4-6] :

The output promoter activity is given by this equation:

And the parameters are described in the following table (the parameters are from reference) [2]:

For line 1, when the first inducer, arabinose, is added, we can give the following three differential equations:

Similarly, for line 2, when IPTG is put into the system, we write another four equations for the materials of line 2:

But the first inducer, arabinose, still exists in the system, so the first three equations are also work at this time. However, because sRNA[ipgC] will bind with mRNA[ipgC], we modified the equaitons for mRNA[ipgC] and sRNA[ipgC]. In other words, we used the sRNA model which we finished before to change these equations. Besides, when the first (Ara) and second (IPTG) inducers exist at the same time, protein[ipgC] and

protein[mxiE] will bind together to form mxiE-ipgC complex so that the transcription from the pipaH* promoter can be activated. So we also wrote two equations to describe the dynamic change of mRNA[sicA] and

protein[sicA] and one equation for mxiE-ipgC complex (px1):

Then we combined these equations to simulate the situation after the IPTG was added. Because ipgC proteins must form a dimmer when binding with mxiE to form the mxiE-ipgC complex, according to mass balance, we minus double px1 in the equations for proteins[ipgC] but just minus single px1 in the equations for proteins[mxiE]. The equations (after IPTG is added) are:

Similarly, sicA proteins must form a dimmer when binding with invF to form the sicA-invF complex. So according to mass balance, we minus double px2 in the equations for proteins[sicA] but just minus single px2 in the equations for proteins[invF]. Therefore when the last inducer, aTc, is added, we can give all the equations for layered AND gates as follows:

Then we use time parameter to combine the three equations (I), (II), (III). We give initial value for all three inducers but make other materials’ initial concentration zero. The simulation result shows as figure2.1 and figure 2.2.

We established two kinds of model to describe the principles of MIN system, one in single cell, and the other one for population.

We constructed the first MIN system model which simulate the system in single cell. Also a series of differential equations were written to construct the model. The construction strategy is still dynamic simulation method. We assume that the initial concentration in every single cell is the same. And we ignore the diffusion of AHL because the consumption of AHL will be very quick due to high E.coli population. With AHL gradually decreases, the transcription rate of MinC increases. We use kM(1-I[AHL]/I0[AHL]) to simulate the process. So four equations are used to describe the dynamic change of some important materials:

We give an initial value for AHL, and the simulation result shows as figure 3.1

We also designed a model for E.colis’ population.

At the beginning, we use a classical model, density-dependent growth model, for our population growing simulation. An equation, dN/dt=k1N(1-N/(Nm-M)), were used to describe the change of cell density. But we find it just cannot describe the function of our system. Because if we use this model, however we adjust the parameters or change the initial value of AHL, the cell density will finally come to the maximum Nm. The Nm is set according to normal condition. But in our MIN system, MinC and MinD can bind with each other to form a MinCD division inhibitor. The inhibitor can prevent cell division process. After some time, the number of cells that are still able to make binary fission will decrease. Eventually all the cells lose the ability to multiply themselves. So the cell density should not come to the maximum. What’s more, in the first model (the classical one), the cells grows with limited resources and space. But at the beginning of E.coli’s growth, the cell density is not high, and there are ample supply of food and living space. Therefore, we change the equation for cell density and give the following equations to construct the model for E.colis’ population:

The simulation result goes as figure 3.2 shows:

We also changed the initial value of AHL in our model. The results is given by figure 3.3:

We combined the MIN system model (in single cell) with the model for layered AND gates so that we can simulate the whole process of our E.co-Lock. We add the equations in MIN system into the layered AND gates model:

We use the whole equations to give simulations to E.co-Lock. When combined, the dynamic change of AND gates will affect the MinC proteins concentration, and finally make a difference on cell division. The results shows as figure 4.1:

Then we also wrote equations for wrong order, for example: 3 2 1 (first: aTc, second: IPTG, last: Ara).

All the parameters’ value are the same as the equations for 1 2 3. And if we use the equations for 3 2 1 to simulate, the result goes as figure4.2. You can see the huge difference.

Also we simulate the situation if you add the inducers by order 2 3 1 (IPTG, aTc, Ara). The result goes as figure4.3 (The equations for 2 3 1 are not written down here).

From the above figures, the model tells us that if you put the inducers in wrong order, you cannot unlock the AND gates and make the E.coli recover to the normal condition.

Besides, we developed a program in matlab. You can enter the kind of inducers as well as their concentration, the order you want to add the inducers, and the time interval between two inducers to see the results.

References:

[1] Mehta P, Goyal S, Wingreen N S. A quantitative comparison of sRNA‐based and protein‐based gene regulation[J]. Molecular systems biology, 2008, 4(1).

[2] Moon T S, Lou C, Tamsir A, et al. Genetic programs constructed from layered logic gates in single cells[J]. Nature, 2012, 491(7423): 249-253.

[3] Wang B, Kitney R I, Joly N, et al. Engineering modular and orthogonal genetic logic gates for robust digital-like synthetic biology[J]. Nature communications, 2011, 2: 508.

[4] Collins, C.H., Arnold, F.H. & Leadbetter, J.R. Directed evolution of Vibrio fischeri LuxR for increased sensitivity to a broad spectrum of acyl-homoserine lactones. Mol Microbiol 55, 712-723 (2005).

[5] Grigorova, I.L., Phleger, N.J., Mutalik, V.K. & Gross, C.A. Insights into transcriptional regulation and sigma competition from an equilibrium model of RNA polymerase binding to DNA. Proceedings of the National Academy of Sciences of the United States of America 103, 5332-5337 (2006).

[6] Passador, L. et al. Functional analysis of the Pseudomonas aeruginosa autoinducer PAI. Journal of bacteriology 178, 5995-6000 (1996).