Black box models

The most simple mathematical presentation of cell growth is the so-called black box model, where all the cellular reactions are lumped into a single overall reaction. This implies that the yield of biomass on the substrate (as well as the yield of all other compounds consumed and produced by the cells) is constant. Consequently the specific sub-strate uptake rate can be specified as a function of the specific growth rate of the biomass, simply by rewritingEq. (6.2):

rs= Yxsµ (6.5)

The specific uptake rate of other substrates, e.g. uptake of oxygen, and the formation rate of metabolic products is similarly propor-tional to the specific growth rate. In the black box model, the kinet-ics reduce to a description of the specific growth rate as a function of the variables in the system. In the most simple model description it is assumed that there is only one limiting substrate, typically the carbon source (which is often glucose), and the specific growth rate is therefore specified as a function of the concentration of this sub-strate only. A very general observation for cell growth on a single lim-iting substrate is that at low substrate concentrations (cs) the specific growth rateµ is proportional to cs, but for increasing values there is an upper value for the specific growth rate. This verbal presentation can be described with many different mathematical models, but the most often applied is the Monod model, which states that:

µ = µmax

cs

cs+ Ks

(6.6) whereµmaxrepresents the maximum specific growth rate of the cells and Ks is numerically equal to the substrate concentration at which the specific growth rate is 0.5µmax. The influence of the substrate concentration on the specific growth rate with the Monod model is

Table 6.1 Typical Ks values for different microbial cells growing on different sugars

Species Substrate K_s(mg l⁻¹) Aerobacter aerogenes Glucose 8 Aspergillus oryzae Glucose 5 Escherichia coli Glucose 4 Klebsiella aerogenes Glucose 9 Glycerol 9 Klebsiella oxytoca Glucose 10 Arabinose 50 Fructose 10 Penicillium chrysogenum Glucose 4 Saccharomyces cerevisiae Glucose 180

Table 6.2 Compilation of different unstructured, kinetic models

Model name Kinetic expression Tessier µ = µmax

1− e^−cs/Ks

Moser µ = µmax

c_sⁿ cⁿ_s + Ks

Contois µ = µmax

c_s c_s+ Ksx Blackman µ =

µmax

c_s 2Ks

; c_s≤ 2Ks

µmax; c_s≥ 2Ks

Logistic law µ = µmax

 1− x

Kx



illustrated in Fig. 6.6. The parameter Ks is sometimes interpreted as the affinity of the cells towards the substrates. Since the substrate uptake often is involved in the control of substrate metabolism, the value of Ksis also often in the range of the Kmvalues of the substrate uptake system of the cells. However, K_sis an overall parameter for all the reactions involved in the conversion of the substrate to biomass, and it is therefore completely empirical and has no physical meaning.

Table 6.1summarizes the Ks value for different microbial systems.

The Monod model is not the only kinetic expression that has been proposed to describe the specific growth rate in the black box model.

Many different kinetic expressions have been presented andTable 6.2 compiles some of the most frequently applied models. In the Contois kinetics, an influence of the biomass concentration x is included, i.e.

at high biomass concentrations there is an inhibition on cell growth.

It is unlikely that the biomass concentration as such inhibits cell growth, but there may well be an indirect effect, e.g. by the for-mation of an inhibitor compound by the biomass or high biomass

concentrations may give a very viscous medium that results in mass transfer problems. These different expressions clearly demonstrate the empirical nature of these kinetic models, and it is therefore futile to discuss which model is to be preferred, since they are all simply data fitters, and one should simply choose the model that gives the best description of the system studied.

All the kinetic expressions presented above assume that there is only one limiting substrate, but often more than one substrate con-centration influences the specific growth rate. In these situations com-plex interactions can occur which are difficult to model with unstruc-tured models unless many adjustable parameters are admitted. Sev-eral different multi-parameter, unstructured models for growth on multiple substrates have been proposed, and here one often distin-guishes between whether a second substrate is growth enhancing or also growth limiting. A general kinetic expression that accounts for both types of substrates is:

µ =

where cs,e is the concentration of a growth-enhancing substrate and c_s is the concentration of a substrate essential for growth. The pres-ence of growth-enhancing substrates results in an increased specific growth rate whereas the essential substrates are absolutely necessary for growth to take place. A special case ofEq. (6.7) is the growth in the presence of two essential substrates, i.e.

µ = µmax,1µmax,2c_s,1c_s,2

(c_s,1+ K1) (c_s,2+ Ks) (6.8)

If the concentration of both substrates is at a level where the specific growth rate for each substrate reaches 90% of its maximum value, i.e. cs,i= 0.9 Ki, then the total rate of growth is limited to 81% of the maximum possible value. This is hardly reasonable and several alternatives to Eq. (6.8) have therefore been proposed, and one of these is Growth on two or more substrates, which may substitute for each other, e.g. glucose and lactose, cannot be described by any of the unstructured models described above. Consider, for example, growth of Escherichia coli on glucose and lactose. Glucose is a more favourable substrate and will therefore be metabolised first. The metabolism of lactose will only begin when the glucose is exhausted. The bacterium needs one of the sugars to grow, but in the presence of glucose there is no growth-enhancing effect of lactose. To describe this so-called diauxic growth it is necessary to apply a structured model and, in general, it is advisable to consider a single limiting substrate in black box models only.

In some cases growth is inhibited either by high concentrations of the limiting substrate or by the presence of a metabolic product.

In order to account for these aspects the Monod kinetics is often extended with additional terms. Thus for inhibition by high concen-trations of the limiting substrate we get:

µ = µmax

cs

cs²

Ki+ cs+ Ks

(6.10) and for inhibition by a metabolic product:

µ = µmax

cs

cs+ Ks

1 1+ p/Ki

(6.11) Equations (6.10)and(6.11)may be a useful way of including product or substrate inhibition in a simple model. Extension of the Monod model with additional terms or factors should, however, be done with some hesitation since the result may be a model with a large num-ber of parameters but of little value outside the range in which the experiments were made.