Team:IIT Bombay India/Project

Analysis of multiple feed-backs in biological systems

In our project, we wanted to quantify how the protein expression in a cell changes due to presence of multiple feedback loops. We developed 4 mutant strains of E. coli. Beta-galactosidase responsible for growth on lactose medium was then used to characterize the phenotypic property of specific growth rate. Experimental results were used to verify the simulated models.

Our Constructs

Strain 1 (Open Loop) with plasmid (BBa_K255004). It has got open loop without any feedback. Here there is constitutive expression of lacI. Here the copy number of the plasmid is fixed.

Strain 2 (Single Input Single Output with regulation on LacI [SISO_LacI]) with plasmid (BBa_K255003). It has got a single negative feedback loop. So the expression of lacI is under regulation. Here also the copy number of the plasmid is fixed.

Strain 3 (Single Input Single Output with regulation on copy number [SISO_CN]) with plasmid(BBa_K255002). It has got single negative feedback loop on the plasmid copy number . Here there is no control on the LacI expression.

Strain 4 (Multiple Input Multiple Output with regulation on copy number and LacI [MIMO]) with plasmid (BBa_K255001). It has dual negative feedback loop one on the plasmid copy number and second on the LacI expression.

Methodology

Experimental

We transformed the four constructs in an E. coli strain with no intrinsic LacI. Using the 4 strains we characterized YFP expression, beta-galactosidase and Growth rate. We conducted our experiments in two parts, first growth on other medium and then growth on lactose media keeping IPTG constant.

Simulation

In simulation we applied a gamut of simulation techniques to quantify our model. We developed a kinetics based model for the 4 strains and used it to quantify the dynamic and steady state profiles. Using Langevin approach, we applied stochastic model to simplified logistics equations. We linearised the dynamic model around a set point and converted the dynamic model to transfer function domain (s), we then did frequency response from control theory and generated the magnitude and phase Bode plot.

Results

Experimental results

1. Steady state value of YFP v/s IPTG: For Strain1 and 2, increase in IPTG, does not affect the YFP. For Strain 3 and 4, increase in IPTG increases YFP expressions. Due to lack of accurate rate constants the experimental results do not correlate with simulation results for strain 3 and strain 4 as expression profile in Strain-3 lies above that in strain-4.

2. Growth on Lactose for Strain 1 and 4: We obtained specific growth rate and normalized beta-gal expression on lactose. In strain 4, we observe that the standard deviation is less for both in specific growth rate and in Beta-galactosidase expression as compared to that observed in strain 1. The simulations could correlate the experimental observations accuratly.

3. Agar plate experiment: Strain-4 demonstrated higher colony forming unit as compared to strain-1. The increase was about 40%. The interesting fact was that the deviation observed in Strain-4 was less compared to that of Strain-1, reiterating the fact that the noise at the protein level was translated to the phenotypic level. The percentage deviation was in the order of 35% in strain-1 as compared to only 10% in strain-4.

Simulations

1. Detailed Deterministic models: A model was developed to accurately quantify the dynamic behavior of the all the strains. The specific growth on lactose, was able to correlate with the experimental data as shown above. Also a concept of burden was introduced and it was shown that strain 4 has optimized its protein concentration for maximizing growth.

2. Stochastic modeling A phenomenological model was developed for expression of LacI and replication of copy number and stochasticity was introduced in the system using Langevin approach. The main aim of this exercise was to characterize the inherent noise present in the system, which was shown to greatly reduce for strain-4 as compared to strain-1. In the diagram shown the error bars at the end show how the noise varies dynamically in all the 4 strains.

3. Control theory approach We have done frequency response analysis on the linear system using magnitude and phase Bode plots.

The phase margin for strain 4 is 92.2 deg while phase margin for strain 3 is 56 deg. This indicates better ability in response to delays in production of LacI and replication of copy number. Further, we have done magnitude bode plot for the sensitivity function to determine sensitivity of the system.

The bandwidth for strain 4 is 0.0255 rad/min while bandwidth for strain 3 is 0.00428 rad/min. For system with 1000 µM IPTG, the magnitude and phase bode plots are given as below:

The phase margin difference is not that significant. It is 70 deg for strain 4 and 64 deg for strain 3.

The magnitude bode plot for the sensitivity function shows lower bandwidth of 0.0061 rad/min for strain 4 and 0.0078 rad/min.

Conclusions

We characterized a phenotypic property of a cell (growth) with the help of synthetic genetic circuits. We proved that the specific growth rate on lactose was optimized in the mutant strain containing multiple feedbacks. The noise or variance associated with the protein expression of a MIMO strain was comparatively lower than that of Open loop strain containing zero feedbacks.

Future work

1. Experimental analysis of CFP: Due to unavailability of cyan laser filter for FACs we were unable to do CFP expressions. By December 2009, we aim to generate CFP expression profiles for the four strains.

2. More experimentation for different values of lactose and IPTG, to get more data for beta-gal expressions and growth rate for all 4 strains

3. Detailed model: Accurately finding the kinetic constants from literature and utilizing them to accurately correlate with the experimental results;

4. Stochastic Analysis:- To develop a simplified model for growth and to introduce stochasticity in the same and characterize the inherent noise in the system.

5. Control analysis: We have done the analysis for only one kind of feedback system. We could use different feedback systems characterized by different Hill-Coefficients and try to do further analysis on the same lines as the analysis done here.