Which of the following statements best predicts the effect of increasing the concentration of substrate ethyl alcohol?

EXERCISES

1.

The process described by the Michaelis–Menten equation can be represented by a series of first-order differential equations. These differential equations define the rate of change of each substance to be equal to the rate constant multiplied by the concentration of each molecule in the chemical equation. Develop the four equations and describe the meaning of each term.

Develop the four first-order ordinary differential equations that model the Michaelis–Menten equation when the transition step from ES to E + P is reversible [not assumed to be irreversible].

The characteristics of a sigmoidal curve are dependent upon the constants and the stimulus. Change the constants and observe the effects. What range of values leads to a curve that appears to be hyperbolic?

The MAPK cascade is an important cellular signal transduction pathway. Which diseases are implicated in changes to the MAPK cascade?

In the MAPK cascade example, feedback was not modeled. What cellular mechanisms would allow negative or positive feedback to control the cascade?

The NF-κB signal pathway is active in inflammation. Several models of the pathway have been developed [Hoffmann et al., 2002; Lipniacki et al., 2004]. What are the important proteins that are active in the pathway? How is the pathway controlled? Is there feedback in the pathway? What is the result of the active NF-κB pathway? Which diseases are attributed to the NF-κB pathway?

Cell division models have demonstrated cellular protein oscillations. Which proteins are included in the models? What role does phosphorylation play in the models? How do the proteins regulate the cell cycle?

The combination of stimuli frequency, production, and decay rates determine protein levels within a cell. Modify the protein production code to increase the time of the model by making the following changes to functions simulation and simmodel.

Simulation:

[t,Y] = ode23[@simmodel,[0 1000],[0 0 0],options];

Simmodel:

if mod[t,100] ≤ 2 % Used to model a limited stimulus

s = 5;

else [mod[t,100] > 2]

s = 0;

end

Increase and decrease the stimuli frequency, production, and decay rates and compare the result with Fig. 14.11.

Relating to the phosphorylation cascade example, what are the Goldbeter–Koshland function and ultrasensitivity? How would they affect the example?

In the phosphorylation cascade example, the cell was assumed to be in a high energy state. How is the energy state exhibited in a cell? If the cell was in a low energy state how would the curves change?

The phosphorylation cascade example can also be described by a series of first-order ordinary differential equations, but some of these equations reflect the input stimulus. Develop the ordinary differential equations for each of the molecules in the cascade. [Each phosphorylated molecule is different from a nonphosphorylated molecule.]

Emergent behavior is present in normal cellular and pathologic processes. How does emergent behavior lead to disease?

One of the metabolic enzymes in a cell is phosphofructokinase [PFK-1]. In which pathways is it active? [See the Kyoto Encyclopedia of Genes and Genomes.] Phosphorylation controls PFK-1. How does phosphorylated PFK-1 activate and deactivate metabolic pathways? How does this relate to the energy state of the cell? What other molecules regulate PFK-1 activity and how?

Find example pathways for the following pathway components [Tyson et al., 2003]:

Feed-forward loops

Perfect adaptation

Positive feedback

Negative feedback

Substrate-depletion oscillators

Activator-inhibitor oscillators

Sickle cell anemia is thought to increase the fitness [resistance to specific environmental conditions] of individuals with the mutation. What is the environmental pressure that performs the selection? What other disease also increases the fitness of individuals with the mutations against the same environmental conditions?

8.2.1.2 Michaelis–Menten Equation

A commonly used model for enzymatic reactions is the Michaelis–Menten [MM] equation, which approximates the original dynamics under the assumption that the concentration of the enzyme remains constant. The enzyme interacts with the substrate to form an enzyme–substrate complex, which leads to synthesis of the product and the release of the enzyme:

[8.3]E+S⇌k−1 k1ES⟹k2E+P

where E, S, and P are enzyme, substrate, and product, respectively. The system of differential equations corresponding to the dynamics of these reactions is

[8.4]d[S]dt=−k1[E][S]+k−1[ES]

[8.5]d[ES]dt=k1[E][S]− [k−1+k2][ES]

[8.6]d[E]dt=−k1[E][S]+[k−1+k2][ES]

[8.7]d[P]dt=k2[ES]

This ODE system cannot be solved analytically; therefore, some assumptions have been used to simplify the system. The quasi-steady-state assumption for the enzyme–substrate complex [d[ES]/dt=0], under the premise that the conversion of E and S to ES and vice versa is much faster than the decomposition of ES into E and P leads to

[8.8][ES]=[E][S]KM

with the MM constant

[8.9]KM=k−1+k2 k1

Combining the quasi-steady-state complex concentration approximation and the conservation law for the enzyme [[E]T=[E]+[ES], where [E]Tis the total enzyme concentration], results in

[8.10][ES]=[E]T[S]KM+[S]

This leads to the well-known MM equation:

[8.11]d[P]dt=Vmax [S]KM+[S]

where Vmax=k2[E]T is the maximum reaction rate.

In a molecular network, many cellular processes with no cooperative interactions among molecules [e.g., gene regulation] can be approximated by the MM equation. Gene expression is regulated by transcription factors [TFs] that bind to specific sites in the promoters of the regulated genes. TFs are considered activators if they increase the transcription rate of a gene and repressors if they reduce the transcription rate. Considering the binding of a repressor protein P to an inducer I to form a complex PI:

[8.12]P+I⇌ k−1k1PI

The mass action kinetic equation is

[8.13] d[PI]dt=k1[P][I]−k−1[PI]

At steady state, d[PI]/dt=0 , and assuming the conservation of total repressor [[PT]=[P]+[PI]], we arrive at the same MM equation used in the context of enzyme kinetics:

[8.14][PI]=[PT][I]Keq+[I]

where Keq=k−1/k1 is the dissociation constant.

Steady-State Rate Equations for Enzymes with Two Substrates and Two Products

Many enzymes with two or more substrates obey the Michaelis-Menten equation with respect to one substrate at constant concentrations of the other substrates. The velocity of the enzyme-catalyzed reaction, where one substrate is varied and the others are held constant, is represented by a rectangular hyperbola. Enzymes with two substrates [A and B] that exhibit random-ordered or compulsory-ordered reactions with the formation of a ternary complex obey the following rate equation:

[27]v=VmaxAB/KiaKb+KbA+KaB+AB

where Vmax is the maximal velocity with both A and B saturating, Ka is the concentration of A that gives 1/2 Vmax when B is saturating, Kb is the concentration of B that gives 1/2 Vmax when A is saturating, and Kia is the dissociation constant of the enzyme for A [[EA] Φ [E] + [A]].

The corresponding Lineweaver-Burk double reciprocal equation is given by the following formulation:

[28]1/v=Ka/Vmax1+KiaKb/KaB×1/A+1/Vmax1+K b/B

When double reciprocal plots [1/v versus 1/[A] at several fixed concentrations of B] of data for random ordered or compulsory ordered reactions are examined, a series of lines that intersect to the left of the x-axis result [Figure 3]. The lines can intersect above, on, or below the x-axis. To determine whether the reaction is ordered or random requires additional studies using product inhibition and other ancillary work.

Secondary plots of the slope and intercept [Figure 4] of the data depicted in Figure 3 provide a means to determine the Vmax [both [A] and [B] saturating] and other kinetic constants [Ka, Kb, Kia] describing sequential enzyme kinetics.

For enzymes that obey ping-pong [double-displacement] kinetics, the corresponding rate is given by the following equation:

[29]v=VmaxAB/KbA+KaB+AB

The corresponding Lineweaver-Burk equations are given by the following formulations:

[30]1/v=Ka/Vmax×1/A+1/Vmax×1+Kb/B

[31]1/v=Kb/Vmax×1/B+1/V max×1+Ka/A

When double reciprocal plots [1/v versus 1/[A] at several fixed concentrations of B] of data for ping-pong reactions are examined, a series of parallel lines is observed. Provided the lines are parallel and do not intersect far to the left of the y-axis, a diagnosis of a ping-pong mechanism can be made based on the finding of such parallel line plots [Figure 5[a]]. Double reciprocal plots of 1/v versus 1/[B] at several fixed concentrations of B yield similar data [Figure 5[b]].

Ka, Kb, and Vmax can be determined by secondary plots of the reciprocal of the substrate concentration versus the intercepts, as illustrated in Figure 6.

Biocatalyst membrane reactors

Mathematical models of enzymatic membrane bioreactors are mostly derived from Michaelis - Menten equation, Fick’s diffusion law, chemical equilibrium and material balance, and other affecting factors whose interactions are revealed by response surface methodology, in order to optimize the parameters of enzymatic membrane bioreactor [Xia and Ying, 2011].The diffusive and convective mass transport through biocatalytic membrane layer and with biochemical reactions pertaining to enzymatic membrane reactors used in food and pharmaceutical industries for hydrolysis of mac romolecules [proteins, poly- and oligo-saccharides, lipids, etc.] have been investigated [Nagy, 2009; Nagy and Kulcsar, 2009; Noworyta and Trusek- Holownia, 2004; Trusek-Holownia and Noworyta, 2004].However, fewer models on enzymatic membrane reactors particularly utilized for emerging pollutant degradation have been reported.López et al. [2007] developed a kinetic model to optimize an enzymatic membrane reactor system consisting of a stirred tank coupled to an ultrafiltration membrane, set up for the enzymatic oxidation of dye. The reaction kinetics were defined using a Michaelis–Menten model with respect to the dye concentration and a first-order linear dependence relative to the H2O2 addition rate. A dynamic model, taking into account both the kinetics and the hydraulics of the system, was validated by comparing the experimental results in continuous operation under steady and non-steady state to model predictions.Calabrò et al. [2009] developed a transport model to characterize the behavior of a tyrosinase-immobilized membrane reactor for oxidation of polyphenols. Gallifuoco et al. [2001] proposed a modified model of enzymatic depolymerization of polymers [e.g., of polygalacturonic acid] in an ultrafiltration membrane reactor.

2.05.2.3 Experimental Determination of Kinetic Parameters in the Michaelis–Menten Equation

To experimentally determine kinetic parameters Vm and Km, Michaelis–Menten equation is modified as described in Figure 3. Each equation derived from Michaelis–Menten equation has been suggested to determine these kinetic parameters with high precision [4]. That is, suitable method should be used to minimize the effect of the error of data. Experimental data obtained from enzymatic reaction experiment in a batch reactor are subject to include the error. Usually, Lineweaver–Burk plot [L–B plot] plot is applied to determine the kinetic parameters due to its simplicity [11]. However, we must pay attention to accuracy of parameter in applying least-square analysis to data points when the variability of initial rate is not negligible at low substrate concentrations. In such a case, Hanes–Woolf plot or Eadie–Hofstee plot is applied to determine the parameters more accurately rather than L–B plot. It is highly possible that the error in initial rate is observed in measuring the change of concentrations. Therefore, it is important not to expand the error by applying the initial rate to arithmetic operation. The Eadie–Hofstee plot does not emphasize the plots at low substrate concentrations, therefore recommendable as a suitable method.

The Eisenthal–Cornish–Bowden [ECB] plot [6, 7] is the method that there are not any calculations to determine parameters. This method can also exclude outliers and provide unbiased estimates of the kinetic parameters because each pair of experimental data [v, [S]] does not interfere with other pairs. When it is an ideal case without any data error, all lines intersect at one fixed point, giving Vm and Km. When the data include errors, individual intersections of plot are dispersed around the fixed point. Therefore, in the ECB plot, the error of data can be simultaneously evaluated besides the estimation of the kinetic parameters.

Instead of the above four types of plots, kinetic parameters can be determined by plotting time-course data of [[S0]–[S]]/t versus [ln[[S0]/[S]]/t. However, the quality of data in this integration method is different from those of other plots. The data in this method contain the reaction time and substrate concentration, while the initial reaction rate and substrate concentration are used in other four plots.

6.2.2 Uncompetitive inhibition kinetics

Uncompetitive inhibition results from the binding of an inhibitor with the enzyme–substrate complex to form an enzyme–substrate–inhibitor complex [ESI], according to Equation 6.12, where the rate constants k4′ and k5′ refer to the forward and reverse reactions respectively. Since the rate of change of ESI is negligible, the relationship between the rates of formation and dissociation of the enzyme–substrate–inhibitor complex can be equated to give KI′, the dissociation constant [Equation 6.13]. Again, a small KI′ indicates little dissociation of the complex with correspondingly high inhibition [and vice versa].

[6.12]ES+I k4′k5′↔ESI

[6.13]ESIESI=k5′k4′=KI′

The maximum velocity, again expressed in terms of the given quantity of enzyme, now includes both the enzyme–substrate and enzyme–substrate–inhibitor complexes according to Equation 6.14.

[6.14]vmax=k3E+ES+ESI

Combining Equations 6.2, 6.3, 6.13 and 6.14 yields the Michaelis–Menten equation for prediction of the reaction velocity under conditions of uncompetitive inhibition, expressed in terms of the Michaelis–Menten constant and the inhibition constant [Equation 6.15]. As with competitive inhibition, the velocity of product formation is decreased in the presence of an inhibitor, especially for small KI11 and higher inhibitor concentrations.12

[6.15]v=vmaxS Km+S+SIKI′=vmaxSKm +S1+IKI′

Unlike competitive inhibition, in uncompetitive inhibition, the maximum velocity for a given quantity of enzyme is lowered, even at high substrate concentration. The bonding of the enzyme–substrate complex with the inhibitor reduces the effective concentration of the enzyme–substrate complex, ultimately resulting in a decrease of vmax. Nevertheless, the reduction in the concentration of the enzyme–substrate complex also increases the enzyme’s affinity for the substrate according to Le Chatelier’s Principle13 and Km is lowered.

The constants can be evaluated from a Lineweaver-Burk relationship. A gradient of Kmvmax , y-intercept 1+IKI′vmax and an x-intercept of −1+IKI′Km , yields the constants Km, vmax and KI′.

12.3.3 Michaelis–Menten Kinetics

The kinetics of most enzyme reactions are reasonably well represented by the Michaelis–Menten equation:

[12.36]rA=vmaxCAKm+CA

where rA is the volumetric rate of reaction with respect to reactant A, CA is the concentration of A, vmax is the maximum rate of reaction, and Km is the Michaelis constant. vmax has the same dimensions as rA; Km has the same dimensions as CA. Typical units for vmax are mol m−3 s−1; typical units for Km are mol m−3. As defined in Eq. [12.36], vmax is a volumetric rate that is proportional to the amount of active enzyme present. The Michaelis constant Km is equal to the reactant concentration at which rA=vmax/2. Km is independent of enzyme concentration but varies from one enzyme to another and with different substrates for the same enzyme. Values of Km for some enzyme–substrate systems are listed in Table 12.3. Km and other enzyme properties depend on the source of the enzyme.

Table 12.3. Michaelis Constants for Some Enzyme–Substrate Systems

From B. Atkinson and F. Mavituna, 1991, Biochemical Engineering and Biotechnology Handbook, 2nd ed., Macmillan, Basingstoke.

If we adopt conventional symbols for biological reactions and call reactant A the substrate, Eq. [12.36] can be rewritten in the familiar form:

[12.37]v=vmaxsKm+s

where v is the volumetric rate of reaction and s is the substrate concentration. The biochemical basis of the Michaelis–Menten equation will not be covered here; discussion of enzyme reaction models and the assumptions involved in derivation of Eq. [12.37] can be found elsewhere [2, 3]. Suffice it to say here that the simplest reaction sequence that accounts for the kinetic properties of many enzymes is:

[12.38]E+S⇄k−1k1 ES→k2E+P

where E is enzyme, S is substrate, and P is product. ES is the enzyme–substrate complex. As expected in catalytic reactions, enzyme E is recovered at the end of the reaction. Binding of substrate to the enzyme in the first step is considered reversible with forward reaction constant k1 and reverse reaction constant k−1. Decomposition of the enzyme–substrate complex to give the product is an irreversible reaction with rate constant k2; k2 is known as the turnover number as it defines the number of substrate molecules converted to product per unit time by an enzyme saturated with substrate. The turnover number is sometimes referred to as the catalytic constant kcat. The dimensions of kcat are T−1 and the units are, for example, s−1. Analysis of the reaction sequence yields the relationship:

[12.39] vmax=kcatea

where ea is the concentration of active enzyme and vmax and ea are expressed in Eq. [12.39] using molar units. Values of kcat range widely for different enzymes from about 50 min−1 to 107 min−1.

The definition of Km as the substrate concentration at which v=vmax/2 is equivalent to saying that Km is the substrate concentration at which half of the enzyme’s active sites are saturated with substrate. Km is therefore considered a relative measure of the substrate binding affinity or the stability of the enzyme–substrate complex: lower Km values imply higher enzyme affinity for the substrate. The catalytic efficiency of an enzyme is defined as the ratio kcat/Km with units of, for example, mol−1 l s−1. Catalytic efficiency is often used to compare the utilisation of different substrates by a particular enzyme and is a measure of the substrate specificity or relative suitability of a substrate for reaction with the enzyme. Substrates with higher catalytic efficiency are more favourable.

An essential feature of Michaelis–Menten kinetics is that the catalyst becomes saturated at high substrate concentrations. Figure 12.7 shows the form of Eq. [12.37]; the reaction rate v does not increase indefinitely with substrate concentration but approaches a limit, vmax. When v=vmax, all the enzyme is bound to substrate in the form of the enzyme–substrate complex. At high substrate concentrations s ≫ Km, Km in the denominator of Eq. [12.37] is negligibly small compared with s so we can write:

[12.40]v≈vmaxss

or

[12.41]v≈vmax

Therefore, at high substrate concentrations, the reaction rate approaches a constant value independent of substrate concentration; in this concentration range, the reaction is essentially zero order with respect to substrate. On the other hand, at low substrate concentrations s ≪ Km, the value of s in the denominator of Eq. [12.37] is negligible compared with Km, and Eq. [12.37] can be simplified to:

[12.42]v≈vmaxKm s

The ratio of constants vmax/Km is, in effect, a first-order rate coefficient for the reaction. Therefore, at low substrate concentrations there is an approximate linear dependence of reaction rate on s; in this concentration range, Michaelis–Menten reactions are essentially first order with respect to substrate.

The rate of enzyme reactions depends on the amount of enzyme present as indicated by Eq. [12.39]. However, enzymes are not always available in pure form so that ea may be unknown. In this case, the amount of enzyme can be expressed as units of activity; the specific activity of an enzyme–protein mixture could be reported, for example, as units of activity per mg of protein. The international unit of enzyme activity, which is abbreviated IU or U, is the amount of enzyme required to convert 1 μmole of substrate into products per minute under standard conditions. Alternatively, the SI unit for enzyme activity is the katal, which is defined as the amount of enzyme required to convert 1 mole of substrate per second. The abbreviation for katal is kat. Enzyme concentration can therefore be expressed using units of, for example, U ml−1 or kat l−1.

The Michaelis–Menten equation is a satisfactory description of the kinetics of many industrial enzymes, although there are exceptions such as glucose isomerase and amyloglucosidase. Procedures for checking whether a particular reaction follows Michaelis–Menten kinetics and for evaluating vmax and Km from experimental data are described in Section 12.4. More complex kinetic equations must be applied if there are multiple substrates [2–4]. Modified kinetic expressions for enzymes subject to inhibition and other forms of regulation are described in Section 12.5.

Saturation of Elimination

Saturation kinetics is also called zero-order kinetics or Michaelis–Menten kinetics. The Michaelis–Menten equation is mainly used to characterize the enzymatic rate at different substrate concentrations, but it is also widely applied to characterize the elimination of chemical [the first-order kinetics] compounds from the body. The substrate concentration that produces half-maximal velocity of an enzymatic reaction, termed Km value or Michaelis–Menten constant, can be determined experimentally by graphing vi as a function of substrate concentration, [S].

The Michaelis–Menten equation is written

[5.53]vi=vmax[S]Km+[S]

where vi is the measured initial velocity of an enzymatic reaction, vmax is the maximal velocity of the enzymatic reaction, and Km is the Michaelis–Menten constant. Note that when [S] far exceeds the Km, the initial velocity, vi, is close to the maximal velocity, vmax.

In zero-order kinetics, a constant amount of a chemical compound is excreted per unit of time. In most cases this phenomenon is caused by the saturation of a rate-limiting enzyme, and the enzyme commonly functions at its maximal rate, that is, a constant amount of a chemical compound is metabolized per unit time. A good example is ethyl alcohol; alcohol dehydrogenase becomes saturated with normal doses of alcohol beverages; for example, 0.5‰ blood concentration is more than 20 times higher than the Km-value of ethanol for alcohol dehydrogenase. Because of this saturation, ethyl alcohol is eliminated at a constant rate about 0.1 g/h/kg or 7 g/h in human. However, the reason is not always an enzyme; any system that becomes saturated follows zero-order kinetics. When the concentration of a chemical compound decreases below the saturation concentration, it returns to the first-order kinetics.

From a practical point of view, saturation of elimination has important consequences. If the metabolism becomes saturated, the duration of the action of the compound is prolonged and half-life cannot be determined. In such a case, correct timing for collection of biological monitoring samples also becomes difficult to assess. Furthermore saturation of metabolism may also have qualitative effects. For example, it has been argued [but not yet proved] that arsenic compounds cause cancer at high doses at which methylation of inorganic arsenic becomes saturated.

3.5.3 Khalifah Stopped-Flow Colorimetric Method

Khalifah's44 pH Indicator Stopped-Flow method is cited by lead researchers111 for measurement of Michaelis–Menten kinetic parameters. The Michaelis–Menten equation [Eqn [4]] is the rate equation for a one-substrate enzyme-catalyzed reaction.38 This equation relates the initial reaction rate [v0], the maximum reaction rate [Vmax], and the initial substrate concentration [S] through the Michaelis constant KM—a measure of the substrate-binding affinity. The reactions catalyzed by CA are found to fit the Michaelis–Menten model, and an alignment between the generalized Michaelis–Menten mechanism and the mechanistic steps in α-CA catalysis is shown in Table 3. Standard biochemistry texts can be consulted for more details on the kinetic model assumptions.

Table 3. Carbonic anhydrase mechanism aligned with generalized Michaelis–Menten kinetic model

[4]v0=Vmax[S][KM+[S]]

Conducting the method requires a stopped-flow spectrophotometer, but otherwise is similar in principle to the Test Tube method. A pH colorimetric indicator is used to monitor the release of protons as CO2 is converted to bicarbonate. Enzyme plus buffer with indicator and CO2–water substrate are kept isolated until forced rapidly and simultaneously into a mixing zone designed to minimize diffusion limitations as enzyme and substrate combine. Mixing liquid flow is “stopped” in a detection cell instrumented to follow the rapid reaction progress. Although the stopped-flow approach allows kinetic quantification and offers much versatility, the method to quantify CO2 reaction kinetics needs to take into account the relationship between change in absorbance with time [dA/dt] and the rate of proton release [dx/dt].44 Therefore, it is necessary to choose buffer–indicator pairs with closely matching ionization constants [pKa], work in solutions with good buffering capacity, and determine the buffer factor [Q] needed to satisfy the relationship shown in Eqn [5].44

[5]dAdt=[∂A∂x]t[ ∂x∂t]=[1Q][dx dt]

With modifications as described by Chirica,112 initial rates of CO2 hydration have been measured at room temperature [25 °C] using a stopped-flow apparatus. In this approach, for example, buffers are prepared according to Table 4, keeping the total ionic strength at 100 mM by addition of Na2SO4.

Table 4. Example buffer–indicator pairs for conducting stopped-flow assay

Uncatalyzed rates, measured in the absence of enzyme, are subtracted from the catalyzed rates to give corrected rate data for determining the kinetic parameters. This can be done using a software program99 or manually determined by plotting the initial rates [v0] versus v0/[S] at several different substrate concentrations [S],44 to generate a graphical representation of Eqn [6] commonly known in biochemistry33 and bioprocess engineering texts38 as an Eadie–Hofstee plot. By this method, the y-intercept gives the value for Vmax, the slope equals the negative of the constant KM, and the x-intercept equals Vmax/KM. The catalytic constant, or ‘turnover number,’ for the enzyme, kcat, is calculated from Vmax according to Eqn [7], in which [ET] is the concentration of enzyme, under the condition that the concentration of substrate [S], in this case [CO2], is greater than [ET] by at least 100-fold44 to ensure the enzyme is saturated with substrate, a fundamental assumption of the model.113 The ratio of kcat/KM is called the catalytic efficiency and has the units of a second-order rate constant because the reaction depends on both the concentration of substrate and the concentration of enzyme. Values of kcat, KM, and kcat/KM for a number of CAs are provided in Table 1. For context, in the absence of catalyst, the first-order rate constant for reaction of CO2 with H2O [Reaction [1]] in certain pH 5.8–8.8 buffers at 25 °C is in the range 0.03–0.041 s−1,44,51,114 whereas the lowest kcat reported in Table 1 is 1.7 × 104 s−1 for Cab from Methanobacterium thermoautotrophicum. It has been proposed that differences in KM values for different CAs could be used to optimize CO2 gas-scrubber design by placing CAs with high KM near the absorber gas inlet where CO2 concentration is high and placing CAs with low KM near the absorber outlet where CO2 concentration is low,115 with similar reasoning for bicarbonate KM in the desorber. Note that measurement conditions can impact CA kinetic values and closer inspection of the literature and further experimentation is advised for making detailed comparisons [see for example, Supporting Information regarding hCAII in Mikulski61].

[6]v0=Vmax−KMv0[ S]

[7]kcat=Vmax[E T]

11.4.2.2 Continuous Stirred Tank Reactor

1.

Enzyme-catalyzed reaction. In a CSTR, if the reaction is controlled by enzyme catalysis and the kinetics equation follows the Michaelis–Menten equation, substituting the kinetics equation directly into Eq. [3.42] gives the residence time

[11.46]τ=VrQ0 =[cs0−cs][Km+cs]rmaxcs=[cs0−cs][Km+cs]k2 cE0cs

Substituting cs=cs0[1−Xs] into the above equation and simplifying gives

[11.47]τ =1k2cE0[cs0Xs+KmXs1−Xs]

When inhibitor is present, we need to substitute the corresponding kinetics equation into Eq. [3.42].

Microorganism reaction. In a CSTR, assuming there is no biomass in the feed material, then at steady-state, biomass growth rate in the reactor equals biomass outflow rate from the reactor, i.e.,

[11.48]Q0cx=rxVr=μcxV τ

The ratio of feed volumetric flow rate to culture solution volume is defined as dilution rate, i.e., D=Q0/Vt. Substituting it into Eq. [11.48] yields

[11.49]µ=D

D represents the extent that feed material was diluted in the reactor, and the dimension is [time]−1.

It can be inferred from Eq. [11.49] that if a cell is cultured in a CSTR, at steady-state, cell-specific growth rate equals the dilution rate of the reactor. This is an important feature for cell cultures in CSTRs. We can use this feature to change cell-specific growth rate at steady-state by controlling the feeding rate of the substrate. Therefore, CSTR is also called chemostat when it is used for cell culture. Cell growth characteristics can be conveniently investigated with a chemostat.

In a CSTR, the limiting substrate concentration and biomass concentration are related to the dilution rate. If biomass growth follows the Michaelis–Menten equation, then

[11.50]D=μ= μmaxcsKs+cs

Therefore, in the reactor substrate concentration and dilution rate has the following relationship

[11.51]cs=Ks Dμmax−D

Assuming the limiting substrate is only used for cell growth, then at steady-state

[11.52]Q0 [cs0−cs]=rs ⋅Vr

While

[11.53]rs=r xYx/s=μ⋅csYx/s

Substituting Eq. [11.52] into the above equation and combining with Eq. [11.49], we get the cell concentrations in the reactor

[11.54] cx=Yx/s[cs0−cs]

Substituting Eq. [11.51] gives the relationship between cell concentration and dilution rate, i.e.,

[11.55]cx=Yx/s[cs0− Ks⋅Dμmax−D]

From Eq. [11.51] we can see that, as D increases, cs in reactor also increases; when D is large enough to make cs=cs0, the dilution rate becomes the critical dilution rate, i.e.,

[11.56]Dc=μ c=μmaxcs0Ks+cs0

The dilution rate of the reactor must be less than the critical dilution rate. When D>Dc, cell concentration in the reactor will become smaller and smaller, and the cell will be finally “washed” out from reactor, which is definitely not allowed.

Cell yield rate Px is also cell’s growth rate, i.e.,

[11.57]Px=rx=μcx=DCx=DYx/s [cs0−KsDμ max−D]

Fig. 11.16 shows cell concentration, limiting substrate concentration and cell formation rate as a function of dilution rate at cs0=10 g/L, µmax=1 h−1, Yx/s=0.5, and Ks=0.2 g/L. There is a maximum value in the cell formation curve. Setting dPx/dD=0 gives the optimal dilution rate Dopt [Fig. 11.17].

[11.58]Dopt=μmax[1−Ks[Ks+cs0]]

Then, cell concentration in the reactor is

[11.59]cx=Yx/s[cs0 +Ks−Ks[Ks +cs0]]

The maximum cell formation rate, Px,max is

[11.60]Px ,max=Yx/sμmaxcs0[1+Kscs0−Kscs0 ]2

When cs0>>Ks

[11.61]Dopt≈μmax

[11.62]Px,max≈Yx/sμ maxcs0

In a CSTR, the relationship between the product formation rate and dilution rate can be derived by performing material balance over the reactor based on the type of product formation and kinetics equation.

If product formation belongs to type I, i.e., growth-associated product formation, performing material balance over product,

[11.63]Q0cp−Q0cp0=rp⋅VL

Because

[11.64] rp=qp⋅cx=Yp/xμcx

Substituting into Eq. [11.63] and assuming there is no product in the feeding material, rearranging Eq. [11.63] gives product concentration

[11.65]cp=qp⋅cxD