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Comparison between Monod equation and Michaelis-Menton equation

Jacques Monad proposed an empirical kinetic model which relates cell growth with substrate availability. The Monad’s equation is similar to the Michaelis-Menten kinetics in terms of its form but they are conceptually quite different. Michaelis-Menten kinetics is a mechanistic model for enzyme kinetics. The conjecture on the similarity and relationship between the two equations is long standing and still not concluded.

Arriving at Monad equation

A typical bacterial growth curve

Batch culture of microbes in a bioprocess is a type of closed culture. Since the culture is closed, the nutrient availability is minimal. Post inoculation, the growth curve is sigmoidal as the organism grows experiencing a lag phase initially when the cell undergo maturation prior to division followed by a log phase where the cells divide and increase in number exponentially and finally a stationary phase when the nutrient reserve is exhausted. Further observation reveled a death phase when the growth starts to decline on cessation of supply of nutrients for prolonged period of time.

The Lag phase

The lag phase is accounted for growth and maturation of cell as well as its adaptation to the media. Reduced lag phase increases the efficiency of the bioprocess and is a commercial success.

The Log phase

The cell growth rate gradually increases, until it reaches a maximum growth rate at log or exponential phase. During this phase an increase in biomass is proportional to the initial biomass concentration available.

$dx/dt\propto x$

Where,

x is concentration of microbial biomass in g/dm³

t is time in hours

d is a small change

We can eliminate the ambiguity due to proportionality by introducing the specific growth rate constant

$dx/dt=\mu x$

Here µ is specific growth rate, its the biomass produced per unit of biomass and takes the unit per hours

On integrating the above equation with respect to the time t,

$x_{t}=x_{0}e^{\mu t}$

Where,

x₀ is the original biomass

x_t is the biomass concentration after time t hours

Taking natural logarithms of this equation

$\ln x_{t}=\ln x_{0}+\mu t$

This plot of natural logarithm of biomass concentration versus time yields a straight line with the slope µ.

Maximum specific growth rates of selected microorganisms

At exponential growth phase the nutrients are in excess and the organism grows at maximum specific growth rate µ_max which will be affected by medium composition, pH, and temperature.

The table shows µ_max values for a few microbial species.

Exponential growth of animal and plant cell in suspension and mycelia cells of hyphae apices is similar to unicellular microbes growing by binary fission. Plomley (1959) believed that filamentous fungi have a growth unit replicating at constant rate and composed of hyphal apex. Later Trinci (1973) showed that Branch initiated when a certain hyphal length is reached. Robinson and Smith (1979) found out that branch initiation factor is volume of fungal hyphae and not its length. According to this idea, Fungal hyphae branches when its biomass reaches a certain critical growth unit level similar to single celled organism that divides when the cell reaches a certain critical biomass.

Thus one can mathematically express this as,

$dx/dt=\mu x$ x is concentration of biomass

$dH/dt=\mu H$ H is total hyphal length

$dA/dt=\mu A$ A is number of growing tips

An exception to this model of growth kinetics is growth rate of a mycelial  biomass pellet in a submerged liquid culture which may be exponential only until density of the pellet results in diffusion limitation. After this limitation, the central biomass will not receive nutrient supply and added to this, potentially toxic products don’t diffuse out. Incorporating this limitation, Pirt (1975) gave his equation

$M^{1/3}=kt+M_{0}^{1/3}$

M₀ and M are mycelia biomass at time 0 and t.

Plot of cube root of mycelial biomass versus time will be a straight line with slope k.

However, this is not always consistent as new pellets may be generated from fragmentation of the old pellets. In that case behavior of the culture will be an intermediate between exponential and cube root growth

The Stationary phase

All of the above equations depict that growth is indefinite while in reality the growth starts to decrease and finally cease after a time period. This cessation of growth due to substrate limitation or toxin limitation

Monod Equation

Monad’s equation can be derived to explain the substrate limitation. Growth occurs in a range of substrate concentration. If the biomass produced is plotted against this substrate concentration we get the plot as shown on the below.  

In the zone A to B, Substrate concentration is proportional to biomass produced. Hence the function for this zone is,

$x=Y{(S_{R}-S)}$

Where,

x is concentration of biomass

Y is yield factor (biomass produced per substrate consumed)

S_R initial substrate concentration

S is residual substrate concentration

Clearly S = 0 at point of cessation of growth

In the zone B to C, substrate concentration is not proportional to increase in biomass. This is because of exhaustion of another substrate or accumulation of toxins.

Yield factor Y is measure of efficiency of conversion of any one substrate in to the required biomass. It’s used to predict substrate concentration required to produce given amount of biomass. Y depends on growth rate, pH, temperature, limiting substrate, conc. of substrates in excess

The decrease in growth rate or its cessation is explained by the monad equation as below,

$\mu ={(\mu _{max}s)/(K_{s}+s)}$

s is substrate concentration in presence of organism

K_s or substrate utilization constant is the measure of the affinity of the organism for its substrate. K_s is numerically equal to substrate concentration when $\mu =\mu _{max}/2$ 

Michaelis-Menton equation

Michaelis-Menton kinetics is a mechanistic model describing enzyme kinetics. Unlike the empirical derivation of Monod equation, Michaelis-Menton equation is derived as follows-

For the reaction of substrate E converting to product P assisted by the enzyme E which forms the enzyme substrate complex ES with the rate constant k₁ which degrades to produce the product with the rate constant k₂ or revert back to reactants with the rate k₀.

${\ce {E + S <=>[k_1][k_0] ES ->[k_2] E + P}}$

Rate of formation of product by law of mass action is,

$r={d[P] \over dt}=k_{2}[ES]$

Similarly the rate of the reverse reaction is,

$r={d[ES] \over dt}=k_{1}[E][S]$

Since the enzyme substrate complex ES is a transient entity, applying steady-state approximation for this model we get,

$k_{2}[ES]+k_{0}[ES]=k_{1}[E][S]$

On rearranging this equation we get,

${[E]+[S] \over [ES]}={k_{0}+k_{2} \over k_{1}}$

Michaelis Menton plot

Here the right hand side of the equation is a constant which could be substituted with K_m,g which is the Michaelis-Menton Constant.

After further simplification and rearrangement of the above equation we get,

$V={V_{M}[S] \over K_{m,g}+[S]}$

This is the Michaelis-Menton equation. It can be plotted as shown in the graph.

Comparison between Monad’s equation and Michaelis-Menton equation

Difference in their significance and derivation

Although they both share the same form and physical dimensions, the primary description of both the equations is vastly different. Monad’s equation explains the variation of specific growth rate with substrate concentration and plots the descending growth rate of biomass production due to substrate limitation while Michaelis-Menton equation describes the rate of enzyme catalysis and plots its variation with substrate concentration.

Another interesting deviation between the two lies in its very foundation, while Monad’s equation is derived empirically, Michaelis-Menton equation is mechanistic.

Mechanistic interpretation of empirical Monod kinetics links it to Michaelis-Menton kinetics

A recent study by Zeng and Yang used flux balance analysis (FBA) model by integrating the proteome allocation principles in Escherichia coli which was quantitatively analyzed to predict potential mechanisms that underlay the Monod parameters. This revealed the mechanistic insights into Monad’s equation bringing it closer to the Michaelis-Menton equation. The study showed that Monod constant K_s was related to the Michaelis constant for substrate transport K_m,g. The link between the two constants was dependent on the cell’s metabolic strategy and mechanism.

The occurrence of Michaelis-Menton kinetics could be thought to occur prior to and cause the occurrence of Monad kinetics. This is because the enzymatic kinetics of substrate intake can be correlated to overall growth phenotype. Hence by determining the mechanistic basis of the empirical Monod kinetics, one can couple this model of growth kinetics to the Michaelis-Menton model of enzyme kinetics.

Kinetic parameters of Monod equation

The two underlying kinetic parameters of Monod’s equation to correlate specific growth rate µ with the extra cellular substrate concentration S are the maximum specific growth rate µ_max and the Monod constant K_s. Surprisingly recent studies clearly indicates that these growth kinetic parameters are functions of the culture’s history and background (For instance the sludge’s age) as well as the community composition.  In early 2000, Kover and Egli stated that growth kinetics of the cells undergo adaptation and continuously varies. This highlights a drawback in considering a fixed set of kinetic parameters in a growth rate model since the all the properties are continuously adapting and variable. Several experiment showing the evolution of Ks, Intrinsic and Extant µ_max, as well as the K_s-µ relationship also clearly indicate that growth kinetics is continuously variable.

Link between K_S and K_m,g

While the interpretation of µ_max as highest specific growth rate is unambiguous, the significance and biological interpretation of K_S still not completely clear. Hence to satisfy the explanation, K_S is considered as a mathematical analogy to Michaelis-Menton enzyme kinetics. The Monod constant K_S also shares significant link with Michaelis constant K_m,g. This has also often led to misterming Monod growth kinetics as Michaelis-Menton kinetics, even though it describes a bioprocess.

Another interpretation of physical meaning of K_S is that 1/K_S relates to the overall affinity of a cell to the substrate. This further brings it closely dependent to K_m,g. K_S is also considered as a function of overall change of free energy during a microbial growth bioprocess.

Monod himself commented on the similarity between K_S and K_m,g quoting the following in his paper.

(i)                ‘the value of Ks should be expected to bear some more or less distant relation to the apparent dissociation constant of the enzyme involved in the first step of breakdown of a given compound’

(ii)              ‘since a change of conditions affecting primarily the velocity of only one rate-determining step will, in general (but not necessarily), be only partially reflected in the overall rate, one might expect Ks values to be lower than the corresponding values of the Michaelis constant of the enzyme catalyzing the reaction’

The above two comments clearly suggests that K_S is a function of K_m,g and generally, K_S is smaller than K_m,g.

Finally, a recent study also demonstrated a direct relationship between K_S and K_m,g, on the basis of control of transport step on specific growth rate as,

$K_{S}=\delta K_{m,g}$

Here $\delta ={\begin{cases}P_{c}/\Sigma _{i}P_{i}<1,&{\text{if }}\mu <\mu _{ac}\\1,&{\text{if }}\mu \geq \mu _{ac}\end{cases}}$