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Nonlinear Control For Algae Growth Models Part II

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Article review by Christine Fanchiang, May 2010.

Article: Mailleret, L., Gouze, J.L., Bernard, O.“Nonlinear control for algae growth models in the chemostat.” Bioprocess & Biosystems Eng., 27(5):319-327, Aug. 2005.

Summary

The purpose of this wiki page is to summarize and interpret the results of the previously described article by Mailleret et al [1]. To best interpret the results of the article, additional sources were reviewed and an analysis was conducted using algorithms developed by the wiki-author to re-create the work done by Mailleret et al. With the analysis, additional parameters of the model were tweaked and analyzed. The results enhance the work done by Mailleret and also bolsters the model developed by his team in having created a more robust model with the nonlinear control.

Background

Purpose of Study

The natural world is a highly complicated system that has intrigued humans since the beginning of time. The past few centuries biologists and mathematicians have done much work trying to capture the natural world with sets of equations that could best model the dynamics we see. Of particular interest is the interactions of the input and outputs of a system. For an enclosed system, the inputs usually involve the flux of energy and nutrients into the system, while the outputs include waste in the form of heat and used or untouched nutrients as seen in Figure 1.

help! the fishy's drowning!
Figure 1. Ecosystem models.

But in the real world there are many confounding factors that can affect an experiment and its measurements, thus biologists devised a controlled laboratory analog called a chemostat in which they could control and manipulate the desired variables to better understand the system. More specifically, chemostats are a bioreactor in which fresh liquid culture is added while outflow (effluent) is removed to maintain a constant volume of the enclosed system.

Scientists are constantly developing analytical models to help characterize systems in the real world. Difficulties arise due to all kinds of environmental factors including changes in climate, multiple species dynamics, multiple nutrient flows, changes in energy input into the system etc. These models can become messy very quickly. Thus the chemostats are a good way of narrowing the variables and selecting specific models to test their accuracy. The ultimate goal is to build upon these models and come-up with an all encompassing model for your ecosystem that is robust enough to handle the range of factors and dynamics within the system.

History of Model Development

Chemostat models have been around since the mid 1900’s, and have been studied extensively. A basic model was developed by Monod in the mid 1940’s and revisited by Lobry et al [2] to justify the model with a mechanistic approach. But in the 1960’s Droop et al discovered that using Monod’s models for algae growth in chemostats gave a poor fit to his data [3]. This led him to create a new model that essentially includes a new dynamic where the cells are modeled with an internal nutrient storage capability. Only nutrients internal to the cell are immediately available for cell growth. By adding this dynamic, it acts essentially like an added time delay to the system, allowing for passage of nutrients from the outside to the inside of a cell. The Droop model is also known as the variable yield model, as it no longer assumes a constant ratio between cell growth and nutrient consumption rate. This is the model Mailleret begins with in his article. The purpose of the paper is to add a nonlinear feedback control to this model to improve its capability of handling varying situations that can occur during laboratory work.

Biological Phenomenon

The phenomenon of interest is a non-linear autonomous variable yield system. Systems suitable for this modeling include a wide class of variable yield models for micro-organism growth in continuous bioreactors. Specifically for this study, they look at growth rates and biomass of phytoplankton algae growing in chemostats. They use a variable yield model to describe the biological phenomenon associated with the flow of medium, uptake of substrates, and the growth of the algae.

Figure 2. Chemostat system.

The dilution rate, D, characterized here is the amount of inflow, Fin, over the volume of the chemostat, V. In this case, the chemostat volume is fixed, but the inflow of medium can be varied. For the different analysis conducted, both variable and fixed values for Fin are used.

The table below describes the variables and parameters used throughout this paper.

Table 1. Variables and parameters used in the article.

Mathematical Model

The Variable Yield Model

The model begins with a variable yield model (Droop model) that monitors the rate of change of substrate concentration, s, the cell quota, q, and the biomass concentration, x. The basic Droop model equation looks like the following:


Eqn1.png

Though to better exhibit the system of equations, the authors use a change of coordinates to express a new variable z = s + qx, which represents the total amount of intracellular and extracellular nutrient is in the chemostat. The following set of equations is obtained:


Eqn2.png

Discussion from the Mailleret paper [1] walks through the hypotheses that led to their choice of the following equations for the uptake rate and growth rate.

Eqn5.png (3)

To analyze the basic Droop model and debug the developed code, the equations from Equation (2) above were run and simulated for a set of initial conditions. In this case, the dilution rate, D, is set as a constant value. There are two solutions to the Droop model a zero equilibrium state, where the cells washout (population dies), and a steady state where they stabilize to non-trivial steady state solutions. Figure 3 a) and b) shows the two cases of the Droop model equilibrium.

Figure 3. Open-loop Droop model.

The behavior of this system is well studied in additional references [1] and [2]. It’s important to note here that the behavior of the system, whether it goes extinct or not, does depend on the dilution rate, D, and the inflow of substrate, sin.

The purpose then is to create controller that prevents washout from occurring and maintaining the positive equilibrium. Specifically, the biomass concentration must be kept toward positive values. Mailleret et al, used a straightforward gain feedback for their nonlinear controller, as will be discussed in the following section.

The Nonlinear Control Design

To improve robustness of a model, a feedback loop was added to the system of equations. The feedback information used in this paper was changing the dilution rate based on feedback about the uptake rate and the biomass of the system. The equation for the dilution rate with the feedback is characterized as the following equation:

D(.)= γy = γu(q)x where γ > qm/sin

The paper itself delves into a detailed mathematical proof as to why they chose this feedback control. But essentially, they use biological constraints such as initial conditions and additional biological constraints that enforce the solutions must be positive (because you can’t have negative amounts of cells or medium), the authors prove the steady state of the system in Equation (4) is locally stable. Also from their proof, they show that the steady-state solutions are globally attractive, which means the values must be positive globally asymptotically stable equilibrium points for the closed loop system. From these proofs, they show with mathematical rigor that this feedback control law globally stabilizes the system described by Equation (1) toward the positive equilibrium points, determined by the value of γ. Essentially, they go through multiple proofs to show that they can rewrite the system in Equation (2) to include D(.), where D(.) is non-negative:

Eqn3.png

This equation is what the wiki-author used to model the closed-loop system.

Full model with results from the paper

The following section reviews the results documented in Mailleret’s article, but also shows the wiki-author’s re-creation of those results.

Simple Noisy Simulation

They use a simple noisy simulation of the controlled process where the in-flow of substrate is assumed to be equal to 20/sigm.L-1, and the output was corrupt with a relative white noise of 30% amplitude.

The following figures show their simulation of the closed loop system with added noise. The result shows the convergence of the steady-state solutions to their expected value as determined from equation (5).

Figure 4. Noisy simulation of model.

Varying Substrate Input, Sin

To show how time variations in the substrate concentration, sin, do not change the biomass concentration, x, behavior, the authors used a piecewise constant time-dependant sin for half-day time steps. Their simulated results are shown in Figure 3. And again despite these variations, the behavior of variable x remains the same, converging towards its equilibrium.

Figure 5. Varying input substrate concentration.

Periodic Algae Stress

With experimental design, usually, the chemostat medium needs to be changed every five days, this change of medium induces stress to the micro-organisms thus affecting their growth rate. To model these dynamics, they use a time dependant growth rate such that: u(.) = d(t)u(q). They show the difference of how the models react for their closed-loop system versus an open-loop system in Figure 6.

The open-loop system shows the algae population is nearly washed’ out, while the controlled closed-loop system drives the algae concentration back to its desired positive steady state values.

Figure 6. Simulation of closed loop (__) and open loop (--) processes facing periodic algae stress.
Figure 7. Simulation of closed loop (__) and open loop (--) processes facing periodic algae stress.

(See Parameters section for variable definition and units).

The re-created outputs were not exactly the same as the article. This can be attributed to coding errors, which were not exactly defined as those in the article. But the general trend can be seen where the basic result is a stabilized population with the closed-loop simulation, and an extinction or washout occurs in the open-loop system.

Additional Analysis

Additional analysis was done for the model to better understand how the different parameters affect the final system outputs.

Varying Initial Conditions

The first tweak was to change the initial values for the biomass and intracellular substrate.

Figure 8. Varying values for the initial biomass, x.

As the initial value of the biomass was increased, it still tended towards the steady-state value of the inverse of the feedback gain, 1/γ. Additionally, the initial biomass and intracellular substrate amount was varied simultaneously to see any affects to the system.

Figure 9. Varying both the biomass and intracellular substrate.

Varying Feedback Gain

The gain of the feedback control was varied to achieve a better understanding of how this controller affected the dynamics of the system.

Figure 9. Variations of feedback gain.

As the value of the feedback gain, γ, was increased, the faster the system approached the steady state values, and the opposite was also true, as the gain decreased, it took longer for the system to achieve its equilibrium state.

Varying Maximum Cell Uptake Rate

Varying the max uptake rate for the cell allows changes to intracellular substrate values, q to change. If the cell can accommodate more substrate, than the expectation is that there will be less substrate in the extracellular fluid. And these plots show such a dynamic.

Figure 10. Varying the maximum cell uptake rate.

With these changes, it can also be seen that it takes longer to get to the steady-state value of the system for biomass. This makes sense as it will take longer for the substrates to be moved from the extracellular medium into the cell, and thus fewer cells will be produced in that same time frame.

Noisy Cell Uptake Rates

Cell dynamics can at times be variable as it grows. The cell uptake rate is currently modeled with Equation (3), but in the natural world, these values will fluctuate through the cells life. To model these changes, additional noise was added for the uptake rate parameter.

Figure 21. Noisy cell uptake rates.

The figures above show that even with variable noise in the uptake rate, the model is still robust enough to maintain a stable cell population.

Conclusions

The Droop model is a very useful tool for describing and quantifying steady state equilibrium of chemostat models. And through the work of Mailleret et al, they have shown that adding a feedback controller into this system will add robustness to the Droop Model. The additional simulations and test cases done by the wiki-author also show that even with widely varying parameter changes, the model can still maintain its stability, and allow the system to maintain the steady-state equilibrium without washout.

This is useful for future work that use chemostat growth chambers. In labs, it is found that cell growth can be very robust even with large swings in inputs like feeding times, dilution rates of the medium, etc. Thus using the Droop model to represent these changes will show a final extinction of the population, which may not be the case. Of course this may be very dependent on the cell line itself, as some are more fragile than others, but in the case that there is a persistent population, it is important to have Mailleret’s nonlinear control model ready in hand.

References

[1] Mailleret, L., Gouze, J.L., Bernard, O.“Nonlinear control for algae growth models in the chemostat.” Bioprocess & Biosystems Eng., 27(5):319-327, Aug. 2005.

[2] Lobry, J.R., Flandrois, J.P., Carret, G., & Pave, A. “Monod’s bacterial growth model revisited,” Bulletin of Mathematical Biology Vol 54 No 1: 117-122, 1992.

[3] Lange, K. , Oyarzun, F.J. “The attractiveness of the Droop equations,” Math Biosci 111:61–278, 1992.

[4] O. Bernard, J.L. Gouze. “Transient behavior of biological loop models with application to the Droop model,” Math Biosci.127:19–43, 1995.

[5] J. P. Gauthier, H. Hammouri, and S. Othman, “A simple observer for nonlinear systems applications to bioreactors,” IEEE Trans. Automat. Contr., vol. 37, pp. 875-880, 1995. http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=256352

Matlab Code

The runs were using ode45, so there are pieces of the code, the .m file for the function, and the script file.

Function for finding biomass,x, intracellular substrate concentrations,s, and total substrate concentration,z, here.

The script file for running the different cases is here. (Updates are necessary when changing parameters or showing different cases).