Team:IIT Bombay India/CAM

Control Theory Approach to Study Multiple Feedbacks on Lac-operon

Objectives

1. Characterize the system.

2. Linearize the system around a set-point on LacI.

3. Obtain a linear model in transfer-function (s) domain.

4. Frequency response analysis using magnitude and phase bode plots.

5. Sensitivity analysis using magnitude bode plot for sensitivity function.

6. Steps 2-5 for 1000μM IPTG.

7. Add external noise in the system and tried to determine the reduction in the noise for the system with multiple feedbacks and open-loop system.

Methodology

We have 2 control levels. By combination, we have 4 different control loops or structures possible, expressed in 4 different strains. They are as follows:-

Strain 1 (Open loop) with plasmid (BBa_K255004)

It has got open loop without any feedback.re there is constitutive expression of LacI.

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 a single negative feedback loop on the feedback 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.

The dynamic model for the system could be represented as given below:

We linearize the system around a set-point on LacI and try to obtain a linear equation model around the setpoint. This enables us to separate the controllers from the system of equations. The controllers are designed as proportional-integral (PI) controllers. The process and controller parameters for the system were tuned in a manner as to obtain steady state and dynamic characteristics that closely match with experimental data. The utility of the multiple feedbacks was analysed using the frequency response tools of control systems’ theory using functions in MATLAB 7.8. We use bode plots to obtain the frequency response analysis for the multiple feedback and single feedback system. Further, we do frequency response analysis for high IPTG concentrations.

The linearized system in transfer-function (s) domain is as given below:

We add external noise in the system using random noise block in SIMULINK in each of the differential equation blocks individually or together and compare the normalized standard deviations in steady-state LacI production for system with multiple feedbacks and open-loop system. The noise was given in relation to the steady-state value of copy number or LacI values such that standard deviation/steady-state value is constant for open loop and multiple-feedback systems.. With this we try to see whether external noise is attenuated in the system with multiple feedbacks.

Results

The magnitude and phase bode plots for the system is given below:

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.

1. The phase margin for a distributed, multiple feedback system (DFS) is 92.2 degree, while it is 56 degree for a single, conventional feedback system (CFS).

2. The bandwidth increases from 0.00428 rad/min to 0.0255 rad/min for Strain 3 to Strain 4.

For system with IPTG concentration of 1000μM,

Fig: Magnitude, phase and sensitivity bode plots for LacI system with 1000 µM IPTG for linear model given in Fig 2. 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.

1. The phase margin for Strain 3 and Strain 4 are 64 degree and 70 degree respectively.

2. The bandwidth for Strain 3 and Strain 4 are 0.0061 rad/min and 0.0078 rad/min respectively.

Fig: Simulink block model for LacI system with external noise. For noise in replication of plasmid copy number, mean is 0, and variance is 10 for multiple feedback and 62.5 for open-loop systems respectively. For noise in production of plasmid copy number, mean is 0, and variance is 10 for multiple feedback and 18779 for open-loop systems respectively. The standard-deviation/mean value of the LacI is used to characterize the noise at the output.

With external noise in the replication of copy number the normalised standard deviation is 43.67% for multiple-feedback system and 82.28% for open-loop system in terms of external white noise.

With external noise in the production of LacI the normalised standard deviation is 136.5% for multiple-feedback system and 151.78% for open-loop system.

With external noise in the production of LacI and the replication of copy number the normalised standard deviation is 44.59% for multiple-feedback system and 83.18% for open-loop system.

Interpretation

1. The increased phase margin for Strain 4 indicates that Strain 4 can take care of delays in production LacI directly and by virtue of production of multiple plasmid copies better than the Strain 3 which has regulation only on the plasmid copy number.

2. This indicates faster expression of the protein LacI in the system with low noise.

3. The increased bandwidth nearly 6 times for Strain 4 indicates a faster response and a better noise rejection over a wide range of frequencies indicating a far robust response as compared to Strain 3.

4. For system with higher IPTG concentrations, IPTG takes away LacI, and thus acting as an inducer. This makes the system resemble open loop system more as compared to IPTG at lower concentrations.

5. The phase margin of 70 degree and 64 degree for Strain 4 and Strain 3 respectively indicates the difference in ability to take care of delays in the two systems has reduced. The bandwidth increase for Strain 4 is not high as compared Strain 3, with IPTG concentration of 1000μM. Also, the bandwidth for Strain 4 with1000μM IPTG is far lower as compared to the bandwidth of Strain 4 with no IPTG.

6. In presence of external noise, the multiple-feedback system attenuates noise at the output better than open-loop system.

The detailed methodology, system equations, results and discussion can be seen here.