9. Controller - design for bioreactors

9.1 General considerations for control of bioreactors
 
The following features distinguish the biochemical reactors from the chemical reactors:²²
1. Several of the crucial variables can not be directly measured quickly or easily. Quite often a time delay, which is larger than the system time-constants, is associated with their measurement. Therefore, a mathematical model must be used in place of feedback information.
2. Linear system analysis is mostly not applicable in case of bioreactors, especially for evaluation of long term response, since biochemical reaction systems usually are nonlinear. Hence, numerical solution of differential equations is required.
3. As already pointed out, the fed-batch reactor operation does not have a true steady state. In this case, evaluation of the state variables will locate the position of the system on a trajectory through the operational cycle. Since these state variables can not be measured online, the estimation of state becomes an important element of optimization and control of the reactor.

9.2 Instrumentation for bioprocess control

As discussed in section 2, the complete state of a bioreactor can be assessed by estimating physical parameters, chemical parameters, biochemical parameters and biological parameters. Various methods to determine physical properties can be found out in Biochemical Engineering and Biotechnology Handbook by Atkinson and Mavituna.³

Analysis of chemical (extracellular) parameters can be carried out either on – line or off – line. Assays to monitor various carbohydrates, proteins, phosphate, lipids, and enzyme activities have been standardized. Biochemical assays for ATP, NADH, cell vitality, and cell viability are well – developed. Amongst the various biological parameters, optical density and cell dry weight are routinely monitored. There has been a move towards more scientific quantification of biomass in recent years. Haemacytometry is gaining popularity in this aspect.³

9.3 Potential problems in bioprocess - control realization

The difficulty of implementing a feedback control is three fold. First, response of oxygen sensors tends to be slower than many of the processes they monitor. Second, sensors are generally not available for measurement of substrate with rapid dynamics for feedback application. And third, there is also generally not available a sensor for the biomass concentration without which the state of the system can not be estimated.²⁷

9.4 Case study: Implementation of a control process to maximize the enzyme (protease) production by a marine bacterium, Teredinobacter Turnirae in a high cell density fed-batch fermentation process

Objective : To design a control algorithm to maximize the protease productivity by Teredinobacter Turnirae in a fed-batch fermentation process.

System description:
1. The cells are grown aerobically in the presence of a single carbon source, i.e., sucrose.
2. The protease is an extracellular and growth - associated product.
3. Preliminary experiments show that the enzyme production is repressed by the presence of excess sucrose.

Assumptions:
1. The cells exhibit balanced growth.
2. The reactor is considered a well - mixed system.
3. Sucrose and oxygen are limiting nutrients.
4. Monod's equation is valid to represent the dependency of m on S.
5. There are no nutritional requirements for the maintenance of the cells.
6. Y_x/s, Y_x/o, Y_P/S, Y_P/O, K_s, K_o, m_max, and a are constant.
7. The critical concentration of oxygen, i.e., below which oxygen becomes limiting is 10%.

Mathematical Description:

The growth of the cells can be represented by a model of Monod's type for two limiting substrates, i.e. sucrose and oxygen in this case. According to this model, the specific growth rate of the bacterium can be written as a function of the concentrations of sucrose (S) and oxygen (C_L):

m = m_max (S/(S + K_s))(C_L/(C_L + K_o))                                                           (25)
where m = specific growth rate (hr^-1)
m_max = maximum specific growth rate (hr^-1)
S = substrate concentration (g/l)
K_s = saturation constant for substrate (g/l)
C_L = oxygen concentration (mg/l)
K_o= saturation constant for oxygen (mg/l)

Neglecting the endogenous metabolism, the expressions for biomass production rate (r_x(g/l-hr)), substrate consumption rate (r_s(g/l-hr)), oxygen consumption rate (r_o(g/l-hr)), and enzyme production rate (r_p(g/l-hr)) are given by

r_x = mX                                                                                                            (26)

r_s = - (mX)/Y_x/s - amX/Y_P/S                                                                                              (27)

r_o= - (mX)/Y_x/o - amX/Y_P/O                                                                                             (28)

r_p = amX                                                                                                          (29)

where, X = cell concentration (g/l)
Y_x/s = gms of cells produced/gms of substrate consumed
Y_P/S = gms of enzyme produced/gms of substrate consumed
Y_x/o = gms of cells produced/gms of oxygen consumed
Y_P/O = gms of enzyme produced/gms of oxygen consumed
a = gms of enzyme produced/gms of cells produced

Eq. (29) results from the fact that the enzyme is a growth - associated product.

For a fed batch reactor in which a substrate at concentration C_s (g/l) is introduced at a rate of F (l/hr), the dynamic mass balance equations for biomass (X), substrate (S), oxygen (C_L), enzyme (P(g/l)), and volume of liquid (V(l)) in bioreactor are given by

dX/dt = r_x - (F/V)X                                                                                         (30)

dS/dt = r_s + F/V(C_s - S)                                                                                   (31)

dC_L/dt = r_o + K_La(C_L^* - C_L)                                                                         (32)

dP/dt = r_p - (F/V)P                                                                                          (33)

dV/dt = F                                                                                                          (34)

where, K_La = volumetric gas - liquid mass transfer coefficient

C_L^* = equilibrium solubility of oxygen in broth

Using eqs. 26, 30, 34, and using the relation for the total cell mass, i.e., M(g) = XV, the following equation can be derived

dM/dt = mM                                                                                                     (35)

If the cells grow at a constant m during the fed - batch fermentation, then the expression for M is given by

M = M_oexp(mt)                                                                                                (36)

where, M_o = total cell mass before starting the feed (g) = X_oV_o
where, X_o = biomass concentration before starting the feed (g/l)
V_o = volume of the fermentor before starting the fermentor (l)
t = elapsed time since beginning of the feed

Similar expressions to eqs. 35 and 36 for total enzyme mass (Mp) can be derived by using eqs. 29, 33, and 34, which are given by

dM_p/dt = amM_p                                                                                                (37)

M_p = M_poexp(mt)                                                                                            (38)

where, M_po = total cell mass before starting the feed (g)

Hence, m should be kept as high as possible to maximize enzyme productivity (g/l-hr). The desired value of m which should be maintained in the reactor is m_max, which is the physiological high limit for the specific growth rate of the cell. The high specific growth rate of cells, however, has to be constrained due to the limited oxygen transfer capacity in fermentation system and the reduced enzyme production at high m. The latter constraint arises because of the fact that high concentration of sucrose results in reduced enzyme production. The value of the desired m can be found out by doing the shake - flask experiments at different substrate concentrations. The value of m corresponding to optimum substrate concentration for the enzyme production would be the desired value of m to be maintained in the fed - batch operation.

An optimal cycle for this fed batch fermentation would involve the initial feed profile to maintain the constant m. The flow rate of sugar solution, F, required for the growth of cells at a desired specific growth rate (m) can be calculated by solving eqs. 27, 31, and 36, and is given as

F = (mX_oV_oexp(mt)/C_s)((1/Y_x/s) + (a/Y_P/S))                                                 (39)

The following assumptions are made while deriving the above relationship

1. Quasi - steady state w.r.t substrate is assumed to exist throughout the fed - batch fermentation span where cells grow at constant m, that leads to the derivative dS/dt reducing to 0.
2. Cs >> S.

This profile should continue till the system reaches the critical oxygen concentration. Afterwards, the oxygen concentration would be controlled at 10% of its maximum solubilty. The feed profile now would be changed accordingy to control oxygen concentration at this desired level. This feed profile can be numerically calculated by using eqs. 26,27,28,31,32, and 34.