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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.





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.


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.


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. Here experimental results do not correlate with simulation results as strain 3 lies above strain 4. It can be noted that the variability in the distribution of YFP as characterized by FACS demonstrated that Strain-4 had the minimum variability in protein expression indicating lower noise.


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 standard deviation is less as compared to strain 1. We also have a comparison of the results obtained from simulation. The growth rate on lactose of Strain-1 was lower as compared to Strain-4. Further, the growth of Strain-4 was more sensitive to lower concentration than that observed in Strain-1. The variability in the growth rate was lower for strain-4 indicating that the multiple feedback loop yields robust protein expression which translates to stable growth rates. These experiments also demonstrate that the intrinsic noise at the protein expression translates to the phenotypic respose as characterized by growth. Simulations from the mechanistic model could predict the experimental observations accurately, namely, controlled and robust expression of Beta-Galactosidase at different lactose concentration.





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 deviation was about 42% for the Strain-1, while it was only 10% in Strain-4 indicating a robust performance in the growth in strain-4 due to lower variability in protein expression.




1. Detailed Deterministic models: A model was developed to accurately quantify the dynamic behavior of the all the strains. The model was able to predict the dynamic behavior of plasmid copy number as characterized by YFP in Strain-1 and Strain-4. 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. Burden on cell was defined as :



Fig : Burden (B) and Normalized Growth rate (NGR) at various lactose concentration. It can be seen that in strain 4 the burden is lower than the NGR indicating optimal growth. However in strain 1 the burden is always higher at all lactose concentrations indicating excess protein synthesis than necessary.


Fig : Burden (B) and Normalized Growth rate (NGR) at various lactose concentration for 100 uM IPTG. Burden for strain 1 is y=1 for all values of lactose. It can be seen that in strain 4 the burden is now higher than that observed at 0 IPTG, since strain 4 tends to behave like strain 1 due to lower degree of feedback strength. The higher burden in strain 1 is due higher beta-gal production than necessary for a specific lactose concentration due to higher IPTG concentration binding to the excess LacI present in the system.

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 reduced 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. Graph-1.jpg


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.


Fig: Magnitude, phase and sensitivity bode plots for LacI system given in linear model. The green line represents Strain 3 with only C1(s), while blue line represents Strain 4 with both C1(s) and C2(s). The gain margin for both Strain 3 and Strain 4 is ∞. The phase margin is 92.2 degree for Strain 4 and 56 degree for Strain 3. The increased bandwidth from 0.00428 rad/min to 0.0255 rad/min indicates faster response and improved noise rejection. The Strain 3 has higher peak of 2.92 dB while Strain 4 has no peak, again indicating better noise-attentuation.

The bandwidth for strain 4 is 0.0255 rad/min while bandwidth for strain 3 is 0.00428 rad/min.This indicates faster response and better noise attentuation. 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 indicates lower degree of feedback in the system.


Fig: Magnitude, phase and sensitivity bode plots for LacI system with 1000 µM IPTG for linear model The green line represents Strain 3 with only C1(s), while blue line represents Strain 4 with both C1(s) and C2(s). The gain margin for both Strain 3 and Strain 4 is ∞. The phase margin is 70 degree for Strain 4 and 64 degree for Strain 3. The bandwidth increase is not significant for Strain 4 from 0.0061 rad/min to 0.0078 rad/min indicates hardly any difference in noise rejection. The Strain 3 has higher peak of 1.62 dB while Strain 4 has a peak at 0.58 dB indicating a lower peak and a slight better performance in noise attentuation.

The magnitude bode plot for the sensitivity function shows lower bandwidth of 0.0061 rad/min for strain 4 and 0.0078 rad/min. The response for strain 4 is not as fast as compared to the response for strain 4 in absence of IPTG. The response is slower in presence of high IPTG even in multiple feedback system.


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. We were successful in quantifying the gene expression using synthetic networks and correlate the intrinsic noise at the expression level to the phenotypic response of growth. Simulation and control analysis proved conclusively the advantages of multiple feedback to regulate inherent noise in the system. It is therefore, no surprise, that nature has evolved such a design which is observed in systems from bacteria to humans.

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.

Further updates and analysis will be presented here.