# Bacterial Chemostat Model

<analytics uacct="UA-11196190-1" ></analytics> Authors: Shoko Asei, Brian Byers, Alexander Eng, Nicholas James, Jeffrey Leto

Date Released: September 18, 2007

Stewards: Jeffrey Falta, Taylor Lebeis, Shawn Mayfield, Marc Stewart, Thomas Welch

Date Revised: September 25, 2007

Stewards: Sarah Hebert, Valerie Lee, Matthew Morabito, Jamie Polan

Date Revised: September 27, 2007

## Introduction

Bioreactors are used to grow, harvest, and maintain desired cells in a controlled manner. These cells grow and replicate in the presence of a suitable environment with media supplying the essential nutrients for growth. Cells grown in these bioreactors are collected in order to enzymatically catalyze the synthesis of valuable products or alter the existing structure of a substrate rendering it useful. Other bioreactors are used to grow and maintain various types of tissue cultures. Process control systems must be used to optimize the product output while sustaining the delicate conditions required for life. These include, but are not limited to, temperature, oxygen levels (for aerobic processes), pH, substrate flowrate, and pressure. A bacterial chemostat is a specific type of bioreactor. One of the main benefits of a chemostat is that it is a continuous process (a CSTR), therefore the rate of bacterial growth can be maintained at steady state by controlling the volumetric feed rate. Bacterial chemostats have many applications, a few of which are listed below.

Applications:

Pharmaceuticals: Used to study a number of different bacteria, a specific example being analyzing how bacteria respond to different antibiotics. Bacteria are also used in the production of therapeutic proteins such as insulin for diabetics.

Manufacturing: Used to produce ethanol, the fermentation of sugar by bacteria takes place in a series of chemostats. Also, many different antibiotics are produced in chemostats.

Food Industry: Used in the production of fermented foods such as cheese.

Research: Used to collect data to be used in the creation of a mathematical model of growth for specific cells or organisms.

The following sections cover the information that is needed to evaluate bacterial chemostats.

## Bacterial Chemostat Design

The bacterial chemostat is a continuous stirred-tank reactor (CSTR) used for the continuous production of microbial biomass.

### Chemostat Setup

The chemostat setup consists of a sterile fresh nutrient reservoir connected to a growth chamber or reactor. Fresh medium containing nutrients essential for cell growth is pumped continuously to the chamber from the medium reservoir. The medium contains a specific concentration of growth-limiting nutrient (Cs), which allows for a maximum concentration of cells within the growth chamber. Varying the concentration of this growth-limiting nutrient will, in turn, change the steady state concentration of cells (Cc). Another means of controlling the steady state cell concentration is manipulating the rate at which the medium flows into the growth chamber. The medium drips into culture through the air break to prevent bacteria from traveling upstream and contaminating the sterile medium reservoir.

The well-mixed contents of the vessel, consisting of unused nutrients, metabolic wastes, and bacteria, are removed from the vessel and monitored by a level indicator, in order to maintain a constant volume of fluid in the chemostat. This effluent flow can be controlled by either a pump or a port in the side of the reactor that allows for removal of the excess reaction liquid. In either case, the effluent stream needs to be capable of removing excess liquid faster than the feed stream can supply new medium in order to prevent the reactor from overflowing.

Temperature and pressure must also be controlled within the chemostat in order to maintain optimum conditions for cell growth. Using a jacketed CSTR for the growth chamber allows for easy temperature control. Some processes such as biological fermentation are quite exothermic, so cooling water is used to keep the temperature at its optimum level. As for the reactor pressure, it is controlled by an exit air stream that allows for the removal of excess gas.

For aerobic cultures, purified air is bubbled throughout the vessel's contents by a sparger. This ensures enough oxygen can dissolve into the reaction medium. For anaerobic processes, there generally is not a need for an air inlet, but there must be a gas outlet in order to prevent a build up in pressure within the reactor.

In order to prevent the reaction mixture from becoming too acidic (cell respiration causes the medium to become acidic) or too basic, which could hinder cell growth, a pH controller is needed in order to bring pH balance to the system.

The stirrer ensures that the contents of the vessel are well mixed. If the stirring speed is too high, it could damage the cells in culture, but if it is too low, gradients could build up in the system. Significant gradients of any kind (temperature, pH, concentration, etc.) can be a detriment to cell production, and can prevent the reactor from reaching steady state operation.

Another concern in reactor design is fouling. Fouling is generally defined as the deposition and accumulation of unwanted materials on the submerged surfaces or surfaces in contact with fluid flow. When the deposited material is biological in nature, it is called biofouling. The fouling or biofouling in a system like this can cause a decrease in the efficiency of heat exchangers or decreased cross-sectional area in pipes. Fouling on heat exchanger surfaces leads to the system not performing optimally, being outside the target range of temperature, or spending excess energy to maintain optimum temperature. Fouling in pipes leads to an increase in pressure drop, which can cause complications down the line. To minimize these effects, industrial chemostat reactors are commonly cylindrical, containing volumes of up to 1300 cubic meters, and are often constructed from stainless steel. The cylindrical shape and smooth stainless steel surface allow for easy cleaning.

### Design Equations

The design equations for contiuous stirred-tank reactors (CSTRs) are applicable to chemostats. Balances have to be made on both the cells in culture and the medium (substrate).

Mass Balance

The mass balance on the microorganisms in a CSTR of constant volume is:

[Rate of accumulation of cells, g/s] = [Rate of cells entering, g/s] – [Rate of cells leaving, g/s] + [Net rate of generation of live cells, g/s]

The mass balance on the substrate in a CSTR of constant volume is:

[Rate of accumulation of substrate, g/s] = [Rate of substrate entering, g/s] – [Rate of substrate leaving, g/s] + [Net rate of consumption of substrate, g/s]

Assuming no cells are entering the reactor from the feed stream, the cell mass balance can be reworked in the following manner:

$(Rate\ Accumulation\ Cells) = V\frac{dC_C}{dt}$
(1)
$(Flow\ Entering) - (Flow\ Leaving) = 0\ - \nu_0C_C$
(2)
$(Rate\ Cell\ Generation) = V(r_g-r_d)\$
(3)

Similarly, the substrate mass balance may be reworked in the following manner:

$(Rate\ Accumulation\ Substrate) = V\frac{dC_S}{dt}$
(4)
$(Flow\ Entering) - (Flow\ Leaving) = \nu_0C_{S0}\ - \nu_0C_S$
(5)
$(Rate\ Substrate\ Consumption) = Vr_S\$
(6)

Putting equations 1, 2, and 3 together gives the design equation for cells in a chemostat:

$V\frac{dC_C}{dt}= 0\ - \nu_0C_{C}\ + V(r_g-r_d)$
(7)

Similarly, equations 4, 5, and 6 together gives the design equation for substrate in a chemostat:

$V\frac{dC_S}{dt}= \nu_0C_{S0}\ - \nu_0C_{S}\ + V(r_g-r_d)$
(8)

Assumptions made about the CSTR include perfect mixing, constant density of the contents of the reactor, isothermal conditions, and a single, irreversible reaction.

Rate Laws

Many laws exist for the rate of new cell growth.

Monod Equation

The Monod equation is the most commonly used model for the growth rate response curve of bacteria.

$r_g\ = \mu C_c\,$
(9)

where rg = cell growth rate

Cc = cell cencentration

μ = specific growth rate

The specific cell growth rate, μ, can be expressed as

$\mu = \mu_{max} \frac{C_s}{K_s+C_s}\,$
(10)

where μmax = a maximum specific growth reaction rate

Ks = the Monod constant

Cs = substrate concentration

Tessier Equation and Moser Equation

Two additional equations are commonly used to describe cell growth rate. They are the Tessier and Moser Equations. These growth laws would be used when they are found to better fit experimental data, specifically at the beginning or end of fermentation.

Tessier Equation:
$r_g\ = \mu_{max} [1 - exp(-\frac{C_s}{k})]C_c\,$
(11)
Moser Equation:
$r_g\ = \frac{\mu_{max} C_s}{1 + k C_s^{-\lambda}}\,$
(12)

where λ and k are empirical constants determined by measured data.

Death Rate

The death rate of cells, rd, takes into account natural death, kd, and death from toxic by-product, kt, where Ct is the concentration of toxic by-product.

$r_d\ = (k_d + k_t C_t) C_c\,$
(13)

Death Phase The death phase of bacteria cell growth is where a decrease in live cell concentration occurs. This decline could be a result of a toxic by-product, harsh environments, or depletion of nutrients.

Stoichiometry

In order to model the amount of substrate and product being consumed/produced in following equations, yield coefficients are utilized. Ysc and Ypc are the yield coefficients for substrate-to-cells and product-to-cells, respectively. Yield cofficients have the units of g variable/g cells. Equation (14) represents the depletion rate of substrate:

$-r_s\ = Y_{sc}r_g+mC_c\,$
(14)

Equation (15) represents the rate of product formation:

$r_p\ = Y_{pc}r_g\,$
(15)

## Control Factors

The growth and survival of bacteria depend on the close monitoring and control of many conditions within the chemostat such as the pH level, temperature, dissolved oxygen level, dilution rate, and agitation speed.

As expected with CSTRs, the pumps delivering the fresh medium and removing the effluent are controlled such that the fluid volume in the vessel remains constant.

pH level

Different cells favor different pH environments. The operators need to determine an optimal pH and maintain the CSTR at it for efficient operation. Controlling the pH at a desired value during the process is extremely important because there is a tendency towards a lower pH associated with cell growth due to cell respiration (carbon dioxide is produced when cells respire and it forms carbonic acid which in turn causes a lower pH). Under extreme pH conditions, cells cannot grow properly, therefore appropriate action needs to be taken to restore the original pH (i.e. adding acid or base).

Temperature

Controlling the temperature is also crucial because cell growth can be significantly affected by environmental conditions. Choosing the appropriate temperature can maximize the cell growth rate as many of the enzymatic activates function the best at its optimal temperature due to the protein nature of enzymes.

Dilution rate

One of the important features of the chemostat is that it allows the operator to control the cell growth rate. The most common way is controlling the dilution rate, although other methods such as controlling temperature, pH or oxygen transfer rate can be used. Dilution rate is simply defined as the volumetric flow rate of nutrient supplied to the reactor divided by the volume of the culture (unit: time-1). While using a chemostat, it is useful to keep in mind that the specific growth rate of bacteria equals the dilution rate at steady state. At this steady state, the temperature, pH, flow rate, and feed substrate concentration will all remain stable. Similarly, the number of cells in the reactor, as well as the concentration of reactant and product in the effluent stream will remain constant.

Negative consequences can occur if the dilution rate exceeds the specific growth rate. As can be seen in Equation (16) below, when the dilution rate is greater than the specific growth rate (D > μ), the dCC/dt term becomes negative.

$\frac{dC_C}{dt}= (\mu - D)C_C$
(16)

This shows that the concentration of cells in the reactor will decrease and eventually become zero. This is called wash-out, where cells can no longer maintain themselves in the reactor. Equation (17) represents the dilution rate at which wash-out will occur.

$D_{max} =\frac{\mu_{max}C_{s0}}{K_s+C_{s0}}$
(17)

In general, increasing the dilution rate will increase the growth of cells. However, the dilution rate still needs to be controlled relative to the specific growth rate to prevent wash-out. The dilution rate should be regulated so as to maximize the cell production rate. Figure 1 below shows how the dilution rate affects cell production rate(DCC), cell concentration (CC), and substrate concentration (CS).

Figure 1: Cell concentration, cell production, and substrate concentration as a function of dilution rate

Initially, the rate of cell production increases as dilution rate increases. When Dmaxprod is reached, the rate of cell production is at a maximum. This is the point where cells will not grow any faster. D = μ (dilution rate = specific growth rate) is also established at this point, where the steady-state equilibrium is reached. The concentration of cells (CC) starts to decrease once the dilution rate exceeds the Dmaxprod. The cell concentration will continue to decrease until it reaches a point where all cells are washed out. At this stage, there will be a steep increase in substrate concentration because fewer and fewer cells are present to consume the substrate.

Oxygen transfer rate

Since oxygen is an essential nutrient for all aerobic growth, maintaining an adequate supply of oxygen during aerobic processes is crucial. Therefore, in order to maximize the cell growth, optimization of oxygen transfer between the air bubbles and the cells becomes extremely important. The oxygen transfer rate (OTR) tells us how much oxygen is consumed per unit time when given concentrations of cells are cultured in the bioreactor. This relationship is expressed in Equation (18) below.

Oxygen Transfer Rate (OTR) = QO2CC
(18)

Where CC is simply the concentration of cell in the reactor and QO2 is the microbial respiration rate or specific oxygen uptake rate. The chemostat is a very convenient tool to study the growth of specific cells because it allows the operators to control the amount of oxygen supplied to the reactor. Therefore it is essential that the oxygen level be maintained at an appropriate level because the cell growth can be seriously limited if inadequate oxygen is supplied.

Agitation speed

A stirrer, usually automated and powered with a motor, mixes the contents of the chemostat to provide a homogeneous suspension. This enables individual cells in the culture to come into contact with the growth-limiting nutrient and to achieve optimal distribution of oxygen when aerobic cultures are present. Faster, more rigorous stirring expedites cell growth. Stirring may also be required to break agglutinations of bacterial cells that may form.

## Q&A

Q1: Why is a chemostat called a chemostat?

A1: Because the chemical environment is static, or at steady state. The fluid volume, concentration of nutrients, pH, cell density, and other parameters all are assumed to remain constant throughout the operation of the vessel.

Q2: What are some concerns regarding chemostats?

A2: a) Foaming results in overflow so the liquid volume will not be constant. b) Changing pumping rate by turning the pump on/off over short time periods may not work. Cells respond to these changes by altering rates. A very short interval is needed for it to respond correctly. c) Fragile and vulnerable cells can be damaged/ruptured when they are caught between magnetic stirring bar and vessel glass. d)Bacteria contamination occurs because bacteria travel upstream easily and contaminate the sterile medium. This can be solved by interrupting the liquid path with an air break.

Q3: The Monod equation uses a Michaelis-Menten relationship which is based on a quasi-state assumption. (T/F)

A3: T

Q4: An important feature of chemostat is the dilution rate. Define dilution rate.

A4: Dilution Rate = volume of nutrient medium supplied per hour divided by the volume of the culture.

A5: Advantages: 1. A chemostat has better productivity than a batch reactor. There is a higher rate of product per time per volume. A batch process wastes time. 2. A chemostat is operated at steady state, therefore it has better control maintaining the same conditions for all product produced.

Disadvantages: 1. A chemostat is less flexible than a batch reactor. A batch reactor can be used to make more than one product. 2. It is harder to maintain a sterile system in a chemostat. A batch reactor is easier to clean.

Q6: What is the physical meaning of the Monod constant?

A6: The Monod constant is a substrate concentration at which the growth rate of the biomass of microbial cells participating in the reaction is half the maximum growth rate.

## Worked out Example 1

Note: The context and values given in this problem are not factual.

Researchers at the University of Michigan are using a bacterial chemostat to model the intestinal tract of a pig in order to study the metabolism of E. Coli bacteria in that particular environment. The growth chamber of the chemostat has a volume of 500 dm3.

The initial concentration of E. Coli bacteria inoculated in the chemostat growth chamber is 1 g/dm3. A 100g/dm3 substrate feed is fed to the chemostat at a volumetric flow rate of 20 dm3/hr. How much time is required for this biochemical process to reach steady rate from the point of startup? Assume the growth rate is the Monod equation for bacteria bacterial cell growth, shown above.

Additional data pertaining to the problem is given: μmax = 0.8; Ks = 1.7 g/dm3; Ys/c = 8; Yp/c = 5; m = 0; rd = 0;

Schematic Drawing

Solution:

The Chemostat was modeled in Excel using the design equations above and Euler's Method. A graph of Cell Concentration (g/dm3) vs Time(hr) was then plotted. When the Cell Concentration become stable, steady state has been reached and the time can be read off the graph. Below is a screen shot of the model and the graph created.

Excel Model Screen Shot

Excel Graph

This graph clearly shows that steady state is reached 3.7 hours after start up.

## Worked out Example 2

Note: The context and values given in this problem are not factual.

After calculating the time required to reach steady state, the researchers decide to start up the chemostat. While do so, the control valve for the inlet substrate feed malfunctions. The flow rate of substrate into the chemostat is accelerating at 40 dm3/hr2. Determine how long they have to correct the problem before wash-out occurs and all of the bacteria in the chemostat is lost.

Schematic Drawing

Modeling the Malfunction
$\frac{d\nu_0}{dt}= kvalve = 40$

Solution:

The Chemostat was modeled in Excel using the design equations above and Euler's Method. A graph of Cell Concentration (g/dm3) vs Time(hr) was then plotted. When the Cell Concentration becomes zero wash-out of the bacteria took place. Below is a screen shot of the model and the graph created.

Excel Model Screen Shot

Excel Graph

This graph clearly shows wash-out occurs 20 hours after start up. We can see in example that process controls are extremely important for Bacterial Chemostats.

The template model used for both Worked Out Example 1 and 2 can be downloaded here Media: Bacterial Chemostat Template.xls

## Sage's Corner

Bacterial Chemostats