Synchronization of Signaling Systems

Synchronization of Signaling Systems

Hasan Cheema

August 12, 2008

Introduction

Signaling systems are essential for intercommunication. Signaling phenomena are abundant in biology, physics andcryptography. For instance, in the human body the endocrine glands regulate the body with its hormones.We can also find systems such as flashing fireflies,firing neurons, and coupled pendulums operating in synchrony. Synchronization of oscillators leads to some rich behavior. For example, if we couple multiple oscillators we might not even see oscillations. Instead, the system might exhibit quasiperiodic behavior. Leloup and Goldbeter (2008) analyze a computational model for the mammalian circadian clock and discover quasiperiodic oscillations of one of their proteins.

Why should one study synchronization of oscillators? The main motivation behind this project was to generate a robust population of oscillators that would be resistant to noise. Individual oscillators, such as the repressilator, oscillate in a “noisy” fashion. If we put the repressilator in E. coli then we will see that different cells have different period lengths and variation from period to period in a single cell. It should also be pointed out that coupling between oscillators is not a sufficient condition for synchronization. Literature demonstrates that the individual oscillators might actively resist synchronization (Winfree, 1967) or the coupling might be too weak (Nijus et al, 1981).

In this project I analyzed the quorum sensing mechanism for coupling an oscillator (Garcia-Ojalvo et al, 2004) and looked at a genetic circuit that could produce switch like behavior and sustained oscillations with altered connectivity (Atkinson et al, 2003). I proposed a modular approach; I took the quorum sensing mechanism from the Garcia-Ojalvo et al paper and attached it to the end of the genetic circuit built by Atkinson et al.

Models

Model 1:

The repressilator consists of three genes whose protein products inhibit the transcription of each other (see Fig. 1). The gene lacI (from E. coli) produces protein LacI, which inhibits the transcription of gene tetR. The TetR protein then inhibits transcription of the gene cI (from λ phage). The CI protein in turn represses the lacI gene. The quorum-sensing mechanism on the other hand is from the bacterium Vibrio fischeri; it is a bioluminescent organism. The system includes two proteins, LuxI and LuxR. LuxI produces a small autoinducer (AI) molecule that binds to LuxR to activate transcription of the lacI gene. This system was built in E. coli (Atkinson et al, 2003).

The authors of this paper tried many combinations of activations. For example, they tried having the LuxR-AI complex activate transcription of tetR and cI genes. However, activation of the lacI gene produces the best synchronized oscillations.

Model 2:

The second genetic circuit that I studied was by Atkinson et al. They had a basic model in which they linked an activator and a repressor (see Fig. 2). One should note that this is an interesting design because the repressor does not actually bind to the activator protein. Instead, it binds to the activator gene and decreases the rate of transcription. They chose this type of a model because it is easier to build and analyze than one in which the repressor binds to the activator protein.

The symbol X with odd subscripts represents mRNA concentrations while X’s with even subscripts are protein concentrations. For example, X1 is the concentration of NRI mRNA and X4 is the concentration of LacI protein. Also, NRI concentration is considered equivalent to NRI~P concentration under the conditions of their experiment.

My Model:

Upper case letters represent proteins while lower case represents mRNA. A represents NRI~P. B is the concentration of LacI protein and C is LuxI protein concentration. The NRI~P protein binds to the promoter region of glnA to positively regulate its own production. NRI~P also binds to glnK to increase the production of lacI mRNA. LacI protein in turn binds to glnA and inhibits the production of NRI~P and luxI mRNA. LuxI produces a small AI molecule that binds to LuxR to increase the transcription of lacI mRNA. The AI molecule is freely allowed to diffuse in and out of the cell.

Here ai, bi, and ci are concentrations of NRI mRNA, lacI mRNA, and luxImRNA, respectively, in cell i. The concentrations of the corresponding proteins are represented by Ai, Bi, and Ci. The dynamics of intracellular AI is given by:

The protein dynamics are given by:

Note that the genetic circuit portrayed in My Model can actually be built in the lab and tested because the Atkinson et al. paper has a detailed description of their genetic circuitry design. The only difference is that we would not use the genes needed to produce LacZ. Instead, we would substitute these genes with the ones needed to create LuxI protein. The bio bricks are available online (

Model Analysis

Before we move on to representing this genetic circuit via differential equations, we need to go over the assumptions. In modeling this genetic network I:

• did not take cell growth and division into account
• assumed uniform AI concentration through the cell culture (i.e. no clumps of AI)
• These assumptions are actually valid because these conditions are found in a well controlled chemostat
• let [NRI]=[NRI~P]
• assumed a certain amount of cooperativity in the repression mechanism represented by the Hill coefficient n=4 and in the activation mechanism represented by the Hill coefficient n=6
• assumed equal lifetime of TetR and LuxI protein

Simulations (Single Cell)

We begin by analyzing single cell behavior first. For a single cell, it was easy to see oscillations. However, the degradation of the lacI gene, which is represented by deg, had to be set really high.

As can be seen by comparing figures 5 and 6, small changes in deg lead to drastically different behaviors. In Fig 4. deg is set to 17.13 and if we decrease it by .01 we do not see any oscillations (Fig. 5).

I was also able to see damped oscillations if I increased the parameter deg from 17.13 to 20 (Fig. 6). This demonstrates that there is a narrow range of deg that will yield oscillations. Therefore, it can be concluded that oscillations are sensitive to the degradation of the activator (LacI) mRNA.

Simulations (Multiple Cells)

In order to see if my set of differential equations works I simulated a hundred cells and set the initial conditions and parameters to be identical for all the cells. We would expect to see identical overlapping oscillations and that is what I got (Fig. 7). The parameters for the simulation were α=100, β=1.0, κ=100, γ=1, ks0=1, ks1=0.01, η=1.0, Q=1, and deg=18.

Now I wanted to introduce some cell-cell variation. I let β vary for each cell. It was chosen from a random Gaussian distribution with a mean of 1.0 and standard deviation 0.05. This produced oscillations in the different cells that never ended up synchronizing (Fig. 8). Even though the cells started out initially synchronized, as time progressed they kept desynchronizing until there were no clear patters. As another way to vary cell-cell dynamics I chose different initial conditions for each cell. Instead of having a constant for each parameter (X[0]==15) I introduced an expression to represent variation (X[0]==15 + 10*Random[] ). Doing so produced the results seen in Fig. 9.

I attempted to replicate the same behaviour that I saw in a single cell in the entire population. First I tried to replicate damped oscillations. This result is displayed in Fig. 10. I was successfully able to replicate it. I noticed that increasing the degradation, deg, from 20 to 21 produced better synchronization. However, it seems like 21 is really high to be biologically plausible. Every other degradation constant is close to 1 while deg is twenty times larger.

Another interesting choice of parameters for deg was to set it equal to 17.9. This resulted in some cells oscillating while others showed steady state dynamics. For example, in one simulation it was found that four out of the five cells that were plotted displayed oscillations while the fifth one simply approached zero (Fig. 11). This is likely due to the fact that each cell has different values of beta.

Synchronization

It is time to introduce a parameter that will measure synchrony. The Atkinson et al paper introduced R. R is defined as the ratio of the standard deviation of the time series of M(t) to the standard deviation of bi averaged over i:

where <.> denotes time average, and … denotes average over all cells. Fig. 12 shows a nice transition from desynchronization (R=0) to synchronization (R=1) as we increase Q. This transition is quick. R increases quickly after a certain value of Q is reached.

When I tried to incorporate R into my simulations I saw similar results. Figure 13 displays the effect of Q on the synchronization of 100 cells. As Q is increased from 0.1 to 1 the cells go from being completely desynchronized to being fully synchronized. However, there is interesting behavior in the transition states. For instance, when Q=0.75 we see quasiperiodic behavior in the range 500<t<1500.

Q / Plot of sum(bi) / Plot of 10 bi’s
0.1 / /
0.25 / /
0.5 / /
0.75 / /
0.8 / /
1 / /

Therefore, I tried to simulate the 100 cells with varying initial conditions while setting Q to 0.8 for every cell. I changed the initial concentration of the protein S (S[0]) from 0 to 50. In the end I chose a random concentration of protein S for each cell. As I increased S[0] I noticed that the population of cells kept becoming synchronized and desynchronized. For example, for S[0]=0 and S[0]=5 the cells are more synchronized than when S[0]=1 and S[0]=15. This system appears to behave in a chaotic manner because small changes of the IC from 0 to 1 lead to large variations in the behavior of the system. Quasiperiodic behavior was also present for most of the IC’s.

IC / Plot of sum(bi) / Plot of 10 bi’s
0 / /
1 / /
5 / /
15 / /
50 / /
15+10*rand / /

Another interesting aspect of this model is that it is capable of generating different waveforms. When we vary from 0.2 to 0.24 we get the results in figure 15. If we decrease below 0.145 or increase it beyond 0.353 we do not get any oscillations.

Delta1=0.2 /
Delta1=0.21 /
Delta1=0.22 /
Delta1=0.23 /
Delta1=0.24 /

We can also see from figure 16 that not only can we change the waveform of our solution but the period is also sensitive to .For each cell was chosen from a normal distribution with mean 0.25 and standard deviation 0.03. There were 100 cells and they were uncoupled.

Discussion

The dynamic modeling of coupled cells allows us to observe a transition from periodicity to quasiperiodicity, and chaos. An earlier theoretical study showed that biochemical systems can transition from a stable steady state to quasiperiodicity, and chaos (Fuente et al, 1996).We have proposed a modular coupling mechanism using quorum sensing that leads to synchronization after the coupling strength is significant. Due to its design, the quorum sensing mechanism can be directly added to the genetic circuit designed by Atkinson et al.

The synchrony observed herewas also present in a heterogeneous population of oscillators. Varying production rate of protein from mRNA, , and degradation of protein for each cell, , still led to synchrony.Slight changes in the degradation parameter also seemed to alter the period and shape of oscillations for each cell. This gives us freedom in manipulating a population of coupled oscillators that were originally independent sloppy clocks.

Finally,our model might not be representative of the various coupling mechanisms present in other systems such as the circadian pacemaker; the results only hold true for similar diffusive mechanisms.Also, the model was developed without a complete set of biologically plausible values for some parameters (,,) i.e. values were chosen to get the desired results and no biological reference for the parameters was made.

Acknowledgements

Thank you to Professor Daniel Forger who made this project possible with his time

spent and patience in discussing the literature.

References

Atkinson, M. R. , Savageau, M. A. , Myers, J. T. , & Ninfa, A. J. (2003). Development of Genetic Circuitry

Exhibiting Toggle Switch or Oscillatory Behavior in Escherichia coli. Cell, 113, 597-607.

Fuente, M., Martinez, L., Veguillas, J., & Aguirregabiria, J. M. (1996). Quasiperiodicity Route to Chaos in a

Biochemical System. Biophysical Journal, 71, 2375-2379.

Garcia-Ojalvo, J., Elowitz, M. B., & Strogatz, S. H. (2004). Modeling a synthetic multicellular clock: Repressilators

coupled by quorum sensing. PNAS. 101, 10955-10960.

Leloup, J.C., & Goldbeter, A. (2008). Modeling the circadian clock: from molecular mechanism to physiological

disorders. BioEssays. 30, 590-600.

Leloup, J.C., & Goldbeter, A. (2004). Modeling the mammalian circadian clock: Sensitivity analysis and

multiplicity of oscillatory mechanisms. Journal of Theoretical Biology. 230, 541-562.

Pikovsky, A., Rosenblum, M., & Kurths, J. (2003). Synchronization A universal concept in nonlinear sciences.

Cambridge: Cambridge University Press.