A Finite State Machine-based Biological Reactor

Recently a colleague of mine told me about a problem he was having with algae cultures. With other fellows, he was studying new processes to extract bio-fuel from algae strains and to do so he needed regular supplies of cultures. To avoid relying on some external people, he decided to cultivate algae himself which required constant care and regular transplantations of the biomass to fresh medium in order to keep his algae in good conditions.

While I was listening to his problem, one idea came directly to me: “why doesn’t he build a system which would automatically transplant the culture to fresh medium when the algae biomass would reach a defined threshold?”. When I got back home, I decided to write down a few equations to see what could actually be done. As I’m not familiar with algae cultures, I designed and tested the model on a simulation of yeast fermentation but the basic is the same.

The overall idea is to consider the culture as an autocatalytic black-box model concerning biomass X which turns a substrate into products (e.g.: a protein or a desired chemical) and more biomass with a conversion ratio of biomass over substrate written Y_X/S. The kinetics and molar ratios are considered unknown and time-changing although a restriction is put concerning the speed of the change as it should be smaller than the characteristic doubling-time of the biomass. Put in equations this becomes:

The culture is operated in a continuously stirred tank reactor with a substrate input feed rate F={0,F_max} which can either be null (“off” state) or maximum (F_max – “on” state). The output flow rate is equal to the input flow rate; so fluid is taken out of the reactor to be processed (distilled, filtered or any other post-process) at the same time substrate is fed in. I have considered that the only available quantity about the culture is a biomass sensor which outputs a value that is proportional to the biomass concentration. It is not required that the sensor function is known as long as f(X₁)>f(X₂) for any concentration X₁, X₂ where X₁>X₂. Other sensors may be used but they are not mandatory in the context of the model presented here (although they might enhance the performance by being more accurate). A classical example of biomass sensor that could be used is an optical density sensor which measures the attenuation of light due to the presence of biomass. The sensor response time is written ∆t and is assumed to be the same order of magnitude than the homogenisation time of the reactor through the stirring motion.

The model should then determinate, based on the sensor output, when to turn on the feeding pump and when to turn it off in order to maximize the amount of product (or biomass) produced over time and to protect the reactor from wash outs which occur when all the biomass is drained away by too fast feed rate. The idea is to build the model as a finite state machine (FSM) which is applied at regular ∆t intervals to determinate the state of the pump (on or off). The following indices notation will be used:

The basic FSM is relatively simple and only checks if the biomass is above or below a given threshold T in which case the pump is either turned on or off respectively. The FSM is represented on Figure 1. The threshold is slightly tuned by a small value δ for each case to create a hysteresis pattern and prevent the pump from being turned on and off constantly. During execution, the pump is turned off until the biomass reaches a level T+δ where the feeding pump is turned on. As feeding occurs, the culture medium is diluted and the biomass gradually lowers. The pump is stopped when the biomass reach the level T-δ where the cells are left growing on the new substrate added. The biomass should then settle around the threshold T with small oscillations due to the hysteresis pattern.

However, if the threshold is fixed above a critical value T_opt (I will show that this critical value is also the optimal one), there will not be enough substrate added by the feeding to allow the biomass to grow above T+δ and so the FSM will never enter the “on” state again, leaving the culture with no more substrate. That way, the culture will quickly enter stationary state or simply die. On the other hand, if the threshold is fixed well below the critical value, the pump will start diluting the culture with remaining substrate left in the medium which would impair the productivity. This is why the critical value is also the optimal one; it is the value at which all the substrate added previously will be consumed.

We know that every feed over a time t_dil will give a finite amount of substrate which will be converted to biomass. With a substrate concentration S_in in the feeding, this lead to a biomass change of t_dil*F_max*S_in*Y_X/S. For the reactor to be stable, this amount should be greater or equal to the difference between the high and low thresholds T+δ and T-δ (that is, 2δ):

On the other hand, we know that the time required to dilute the medium from a biomass concentration equal to T+δ to T-δ is bounded to:

where V is the volume of the reactor.

After a few re-arrangements, we may express the inequality:

And so the high-threshold (T+δ) is bounded by the product of the reactor volume, the feed concentration in substrate and the biomass over substrate conversion ratio. The optimal/critical threshold value is then:

The major problem is that the biomass over substrate conversion ratio is unknown and may change with time. And even if we could measure or approximate its value, it would not be safe to rely on it as any variation on the system (such as a change in temperature or feed rate due to pump failure) would risk the stability of the model.

The second part of the FSM model idea is to use a time varying threshold, T_n, which is updated dynamically to adapt to the real conditions and to keep the production as high as possible. T_n should then try to match the optimal value at any time, independently of the system conditions and without any assumptions about the strain used. From the previous paragraphs, we know that the value of T_n will be over-estimated when the biomass stop growing. On the other hand and as long as the biomass grows, we may postulate that the current threshold is too low.

The FSM is modified as presented on Figure 2. During the off state, the growth rate of the biomass is checked. If it is above a growth-threshold µ* (biomass growing), the value of T_n is increased by an amount γ∆t. On the other hand, if it is below the growth-threshold (biomass at rest) then the value of T_n is decreased by the same amount. The procedure should only be applied during off states because it is difficult to assert the growth rate of a volume changing culture with cheap DIY sensors.

In the description, I have considered the growth rate which can be approximated during an off state as being related to the ratio of the current and previous biomass concentration:

However, as for the biomass sensor, we do not need the actual growth rate of the culture since all we need to know is if the biomass is at rest or not. For that purpose, we may simply use the derivative of the biomass sensor:

Finally, the model was tested on a simulated Saccharomyces cerevisiae fermentation reaction (Sonnleitner/Käppelli model) with an initial biomass inside a 1 litre reactor of 10 g/l with 20 g/l substrate initially present. The maximum feed rate is fixed to 1 l/h, δ to 0.25 g/l, µ* to 0.05 h^-1 (growth rate version) and γ to 1 g/l. The initial threshold T₀ was fixed well above the optimal value (calculated to be about 10 g/l) to 1 g/l. The simulation was run for a 300 hours scenario and the results are presented on Figure 3 with ∆t steps of 10 minutes.

The FSM first obey the dilution rule because the initial biomass amount is above the threshold T₀. Once the system reaches 1 g/l (T₀), it starts increasing slowly the threshold as long as the culture grows. It reaches the optimum value after about 20 hours where it settles for a few hours due to an overshoot in the threshold value (up to 17 g/l which is not possible for the culture given the operating conditions). The model behaves relatively good in terms of identifying the optimal threshold and operates quite well after that. The initial overshoot can be dramatically reduced by using a better approximation for the initial threshold value.

Concerning the productivity, it reached 60% of the best theoretical value for chemostat models but it was found to be highly dependent on the value of ∆t. Using smaller steps, such as 1 minute, allowed increasing the productivity above 85% of the theoretical value. Clearly, some good value can be taken out of this extremely simple model without making lots of assumption about the culture itself.

The next step would be to try this out on a real reactor but I did not have the time to build it yet... But keep updated because I will certainly try it some days!