Monod Kinetics

An analytic solution to the MMM kinetics when microbial biomass significantly affects the response rate has not been tried in either unambiguous or implicit form. The peculiarity is that two average nonlinear differential equations claim to be answered concurrently within the MMM kinetics, one being the Michaelis − Menten equation for the chemicals and the other describing the microbial biomass Monod dynamics. The idea of this study is to deduce and test an implicit logical result of the combined problem of substrate consumption compatible with Michaelis − Menten kinetics and biomass dynamics compatible with Monod kinetics.

Microbial Growth:

Microbial cells use nutrients for growth, energy production, and product formation as indicated in the following expression;

Nutrients + microbial cells > cell growth + energy + reaction products

Consider the operation of the “Batch” system. Let us assume that a container initially contains a known growth substrate concentration S. The container is well mixed. Therefore, the dissolved oxygen concentration O2 does not become a limiting factor for microbial growth. Initially, a known concentration X of viable microbial cells (i.e., inoculum) is added to the container, and, with time, growth substrate S is utilised for cell growth. Therefore, we will observe a decrease in S (negative dS/dt) and a corresponding increase in X (positive dX/dt) over time.

A conceptual plot of microbial cell concentration vs time for the batch system is called a growth curve. 

By plotting the log of viable cell concentration, X, with time, five distinct phases of the growth curve can be identified;

 1) the lag phase occurs immediately after inoculation and persists until the cells have acclimated to their new environment, 

2) exponential growth phase, during which time cell growth proceeds at an exponential rate (indicated by a straight line on the semi-log plot), 

3) a deceleration phase, when essential nutrients are depleted, or toxic products begin to accumulate, 

4) a stationary phase during which time the net cell growth is approximately zero, and

 5) death phase where some cells lose viability or are destroyed by lysis.

Microbial Growth Kinetics

DX/dt and dS/dt are essentially zero during the lag phase. However, as the exponential growth phase begins, it is possible to measure dX/dt and dS/dt values, which are useful for defining important microbial kinetic parameters. Using corresponding observations of dS/dt and dX/dt obtained just after the onset of the exponential growth phase in Figure 2, we can compute the yield coefficient YXS and the specific growth rate µ as:

Yield coefficient

Y=dX/dS – Equation 1

= mass of new cells/mass of substrate consumed

(Dimensionless)

Specific growth rate

𝛍=dX/Xodt – Equation 2

= mass of cells produced/original mass of cells.time {1/time}

The yield coefficient, commonly referred to as the substrate-to-biomass yield, converts between cell growth rate dX/dt and substrate utilisation rate dS/dt. The yield coefficient and the specific growth rate are used to develop three microbial growth kinetic relationships; Monod, first order, and zero-order kinetics.

Monod Kinetics

The batch experiment can be repeated by varying initial substrate concentration S over a wide range of values—resulting in observation of individual µ values which correspond to each substrate concentration. An arithmetic plot of µ vs S will exhibit the general behaviour. 

The most widely used expression for describing specific growth rate as a function of substrate concentration is attributed to Monod. This expression is: 

µ = µ max [(S/(S + Ks)] – Equation 3

Now, conceptually how does the Monod equation fit the observed substrate and specific growth rate data. It is seen that µmax is the maximum specific growth rate observed, and KS  is the substrate concentration corresponding to 1/2 µmax.

By combining equations 2 and 3, we can write the following expression for time-rate-of-change of biomass: 

dX/dt = µXo 

= µmaxXo[S/(S+Ks)] – equation 4

Similarly, by combining equations 1 and 3, we can write an expression for substrate utilisation rate.

dS/dt = µX/Y = µmaxXo/Y[S/(S+Ks)] – Equation 5 

First Order Kinetics

Equation 5 describes the Monod kinetic relationship for substrate utilisation. it can be seen if  S <<  KS, Equation 5 can be approximated  as:

dS/dt = [Xo µmax/Y Kc] S = KoS -Equation 6 

Equation 6 describes the condition where substrate utilisation is proportional to substrate concentration (i.e. first-order concerning S).

Zero Order Kinetics

Likewise, if  S >> KS Equation 5 can be approximated as:

dS/dt = Xo µmax/Y = CONSTANT – equation 7 

Equation 7 describes the condition where substrate utilisation rate is a constant (i.e. zero order concerning S).

Conclusion:

It can be concluded that it is usually impossible, in particular, and surely on a larger scale, to have a substrate concentration throughout the vessel homogeneous enough to get a constant growth rate according to the Monod equation. The growth rate relation with concentration is much more complex, and the critical time is much larger than the Monod-based value. However, somewhere at scale-up, this mechanism can be expected to fail due to increased mixing time with scale. Usually, this will cause a decrease in production yield or result in the production of unwished products.