Pulse Fed batch

The Pulse Fed-batch fermentation app calculates the key parameters of the fermentation over time. The app has a series of inputs. These inputs can be categorized into inputs depending on the micro-organism used, starting concentrations, and the bioreactor design.

Micro-organism dependent inputs

The Yield (Y_x/s)

The yield is the ratio between the amount of biomass produced over the amount of substrate consumed in kgX/kgS. The value is between 0 and 1, and the higher the value, the more efficient the micro-organism is using the available substrate for growth. This is a value that depends on the micro-organism and should be calculated with an experiment.

The maximum growth rate (µ_max)

The maximum growth rate is the maximum growth the micro-organism achieves during exponential growth (no nutrient limitation in ideal circumstances). The higher this value, the shorter the doubling time of the micro-organism is. The value depends on the micro-organism and should be calculated with an experiment.

The substrate specificity constant (K_S)

The substrate specifity constant is the substrate concentration where the growth rate is exactly half of the µ_max. This is a micro-organism specific constant and should be obtained from experiments. The units of the constant is kgS/m³.

The cell specific death rate (K_d)

While organisms have a cell specific growth rate (µ), they also have a cell specific death rate. When the cell specific growth and death rate are equal, the organism enters the stationary phase.

The product yield constant acetate(Y_p,acetate/s)

In anaerobic fermentations, micro-organisms can not fully combust their substrates. Meaning that there are some catabolic products. While in general, products are not interesting for N.I.N.O., some products have an inhibitory effect on the growth rate depending on their concentration. Acetic acid/acetate is a product that is known to have an inhibitory effect on the growth rate^10,11.

The product yield constant ethanol(Y_p,ethanol/s)

Another product of the fermentation that has an inhibitory effect is ethanol¹⁵. The yield is necessary to know to calculate the effect it has on the growth rate.

The product specific inhibitory constant acetate (K_P,ace)

Acetate has an inhibitory effect on the growth of micro-organisms. The specific inhibitory constant is necessary to calculate this effect.

The product specific inhibitory constant ethanol (K_P,eth)

Ethanol has an inhibitory effect on the growth of micro-organisms. The specific inhibitory constant is necessary to calculate this effect.

The product yield constant CO₂(Y_P,CO2/S)

The volume of gas produced during the fermentation is an important design parameter. The size of the off gas container depends on it. To calculate the volume of gas produced, you need the yield of CO₂ over substrate.

Starting concentrations

Substrate concentration at t=0 (C_S0)

The starting concentration of substrate is important for the growth curve and rate of the organism. If the concentration is very high, the organism can grow at its maximum growth rate for a longer time and additional feeding is not necessary. If it is zero, the micro-organisms will not grow unless feed containing substrate is added.

Substrate concentration in the feed (C_s,in)

The substrate concentration in the feed together with the feed volume determines the amount of substrate entering the reactor when the feed button is activated.

Biomass concentration at t=0 (C_X0)

The initial biomass concentration is important, because a high concentration will cause the game to end earlier. This amount is therefore dependent on the designed t_final (max total time of the game).

Starting volume in the bioreactor (V₀)

The starting volume of the bioreactor is important because it determines the left-over volume to be fed into the reactor.

Activation interval (h)

The activation interval is the time between two activations of the feed button. The shorter the interval the more the reactor is fed. Feeding every 10 hours might be a bit boring, while feeding every 5 minutes is too much.

Bioreactor design dependent inputs

The maximum bioreactor volume (V_max)

The maximum volume of the bioreactor is the limited factor of the fed-batch principle of the bioreactor.

Feed volume per activation (F_act)

The amount of added volume when a button/pump is activated.

Euler steps model explanation

The calculations described below consist mainly of derivatives. The easiest way to solve these derivatives is to use the Euler’s method or Euler steps. It is a numerical method to approximate the actual outcome. The Euler method takes small time steps and evaluates the direction of the point in the graph. In this case, the X-axis is the time and the Y-axis the concentration of a certain compound. Subsequently, the direction is the rate of change of concentration of this compound. When you take this direction/slope at a time point X, you can approximate the Y value at X+1. The smaller steps you take, the more accurate the model calculates. In this case, we use a step of 1 second. On a whole day (86400 seconds), these steps are quite small. Therefore, the approximation is also quite good.

The growth rate (µ) is dependent on the maximum growth rate (µ_max), the concentration of substrate (C_S and K_S, blue part of the equation), the concentration of acetate (C_P,ace and K_P,ace, red part of the equation), the concentration of ethanol (C_P,eth and K_P,eth, pink part of the equation). The formula is depicted below this paragraph.

The actual rate of change in biomass due to growth depends not only on the growth rate but also the concentration of biomass available at that point in time (C_X). This rate is later used to determine the amount of biomass at a certain time point.

Next to the growth rate of the bacteria, there is also a death rate (K_d) that is always there. The actual change of biomass due to this deathrate depends also on the available biomass at a certain time point (C_X).

Subsequently, the actual rate at which biomass changes (r_X) is the change due to growth minus the change due to death.

The change in substrate (r_S) is calculate using the yield of biomass on substrate (Y_X/S) and the rate of biomass growth (r_X,growth), since the energy of the substrate is used to grow biomass.

Because the growth is anaerobic, the sugar can not be combusted, but is changed into a variety of other catabolic products. Two of these products, acetic acid and ethanol, are interesting because they have an inhibitory effect on the growth of the micro-organism. The next two equation show the calculation of the rate, which is based on the yield of acetate on the substrate and likewise for ethanol.

Another catabolic product of interest is CO₂. This is a gas and its volume is important to know because it determines the size of the off gass reservoir. Because it is a catabolic product the rate of production is depend on the substrate rate and the yield of CO2 over the substrate.

Now that all rates are defined it is possible to calculate the mass balances over time for the biomass, the substrate, and three two products. First the substrate, the amount of substrate over time in the bioreactor is depending on two parts, the amount of substrate entering the reactor at that moment and the consumption of the substrate already in the reactor. This yields the following equation:

Next up is the biomass, this mass balance also consists of the two parts, the increase of biomass and the income concentration of biomass. Since, the inflow does not contain any biomass a flow pulse decreases the concentration of biomass.

The mass balances of acetate and ethanol are produced in a similar way because they as well increase over time.

Because CO₂ is a gas and the assumption is made that it is not dissolving in the solution. The mass balance is equal to the rate of CO₂ production.

To calculate the amount of off gas produced Henry’s law is used. Henry’s law states that the concentration of a gas in a solution is proportional to the partial pressure of this gas and the concentration in the liquid. This proportion is summarized by the henry’s constant (K_H).

The K_H of CO₂ is 0.037 mol/L*atmosphere. It is assumed that only CO₂ is present and that the N.I.N.O. is under atmospheric pressure. This means that the partial pressure is 1 atmosphere. This means that the concentration of CO₂ in the bioreactor needs to be >0.61 g/l before off gas is produced.

Showcasing the model

The first step is to characterize the micro-organism used. In this case the values used will come from the characterization of W3110 e. coli ‘ Wildtype’ strain¹³ .Table 1 shows the parameters that are used as input for the growth calculator. The italic parameters are easy adjustable, the non-talic parameters are inherent to the used micro-organism and its environment (temperature, type of medium, etc.).

Table 1: Key input parameters of the growth calculator and their values.

*Values come from table 3 and 4 of REFXXX

** Value is not in the report and is made up

When the values of are used as an input for the growth calculator the following graphs are returned (see figure 3). The top left panel shows the concentrations of glucose, biomass, acetate, and ethanol. At first the concentration of glucose is increasing because the rate of consumption is lower than the amount of glucose added by the feed. After approximately 6 hours this starts to change and consumption goes faster than the growth. This has to do with the increased biomass concentration, more cells means faster consumption of glucose. After 12 hours all glucose gets consumed before new glucose is fed into the reactor. This immediately has an effect on the growth rate (bottom left panel). At first the growth rate sky rockets to 80% of the maximum. When the glucose runs out the growth rate drops to zero until new glucose is fed into the reactor. The effect of this is also seen in the biomass concentration. It shows that the population is going into a stationary phase (plateauing).

Figure 3: Resulting graphs after running the growth calculator with the inputs shown in table 1.

The next thing you do is turn some knobs of the design to see whether you can make the game playable. With a maximum growth rate of 0.44 it is probably difficult to make a game last 72 hours without getting into stationary phase. So we limit ourselves to 1.5 days (36 hours). We will also increase the amount of sugar in the feed by increasing the feed volume per activation from 0.5 to 1ml, and decreasing the start volume to 10ml. This yields the following figure.