5.3 Biological Systems and Kinetics

Biological processes are central to many chemical engineering applications, particularly in wastewater treatment, food processing, pharmaceuticals, and industrial biotechnology. The FE Chemical exam covers cell structure, cellular growth kinetics, and enzyme kinetics.

Cell Classification and Structure

Organisms are classified into two primary categories based on cellular complexity: prokaryotes and eukaryotes.

• Prokaryotes (e.g., bacteria and archaea) are single-celled organisms that lack a membrane-bound nucleus and membrane-bound organelles. Their genetic material is located in a nucleoid region as a single, circular double-stranded DNA molecule. Most bacteria possess a rigid cell wall composed of peptidoglycan, which provides structural integrity and protects the cell from osmotic lysis.
• Eukaryotes (e.g., yeast, fungi, algae, and protozoa, as well as plant and animal cells) possess a membrane-bound nucleus housing linear chromosomes. They contain specialized membrane-bound organelles that partition biochemical functions.

Key cellular structures and their functions include:

• Cell Membrane: A phospholipid bilayer with embedded proteins that acts as a selectively permeable barrier, controlling the transport of nutrients and wastes via active and passive transport mechanisms.
• Mitochondria: The 'powerhouses' of eukaryotic cells, where aerobic cellular respiration occurs, generating adenosine triphosphate (ATP) via the electron transport chain.
• Ribosomes: Complexes of RNA and protein responsible for translating messenger RNA (mRNA) into polypeptide chains during protein synthesis.
• Cell Wall: A rigid outer layer found in plants (cellulose), fungi (chitin), and bacteria (peptidoglycan) that maintains cell shape and prevents osmotic rupture.

Cell Growth and Decay Kinetics

Bacterial reproduction typically occurs via binary fission, leading to exponential growth under ideal conditions. The rate of cell growth is modeled using the Monod equation, which relates the specific growth rate (\mu) ((T^{-1})) to the concentration of a limiting substrate (S) ((M/L^3)):

[ \mu = \frac{\mu_{\text{max}} S}{K_s + S} ]

where (\mu_{\text{max}}) is the maximum specific growth rate and (K_s) is the half-velocity constant (the substrate concentration at which (\mu = 0.5 \mu_{\text{max}})).

• At high substrate concentrations ((S \gg K_s)), the specific growth rate approaches its maximum value ((\mu \approx \mu_{\text{max}})), resulting in zero-order dependence on substrate and first-order exponential growth of biomass (Malthusian phase).
• At low substrate concentrations ((S \ll K_s)), the specific growth rate is approximately linear with substrate concentration ((\mu \approx \frac{\mu_{\text{max}}}{K_s} S)), reflecting a first-order dependence on substrate.

The rate of biomass growth (r_g) is:

[ r_g = \mu X ]

where (X) is the biomass concentration ((M/L^3)). In real systems, cells also undergo endogenous decay due to maintenance energy requirements and cell death. The net biomass growth rate (r_g') accounts for this decay:

[ r_g' = (\mu - d) X = \left(\frac{\mu_{\text{max}} S}{K_s + S} - d\right) X ]

where (d) is the endogenous decay coefficient ((T^{-1})). The rate of substrate consumption (r_s) is related to biomass growth via the yield coefficient (Y_{X/S}) (mass of cells formed per mass of substrate consumed):

[ r_s = -\frac{r_g}{Y_{X/S}} = -\frac{\mu X}{Y_{X/S}} ]

The doubling time (t_d) of a cell population during the exponential phase (where decay is negligible) is:

[ t_d = \frac{\ln 2}{\mu_{\text{max}}} ]

Chemostat (CSTR) Mass Balance

A chemostat is a continuous stirred-tank reactor (CSTR) used for cell culture. Assuming sterile feed ((X_0 = 0)), a steady-state mass balance on cells yields:

[ \text{Accumulation} = \text{In} - \text{Out} + \text{Generation} ]

[ 0 = 0 - F X + V (\mu - d) X ]

Dividing by (V X) gives:

[ \mu = \frac{F}{V} + d = D + d ]

where (D = F/V) is the dilution rate ((T^{-1})), which is the reciprocal of the space time ((\tau)). This fundamental equation demonstrates that under steady-state conditions, the cell growth rate is locked to the dilution rate and the decay coefficient. If the dilution rate exceeds the maximum specific growth rate ((\mu_{\text{max}})), cells are washed out of the reactor faster than they can reproduce ((X \rightarrow 0)), a condition known as washout.

Enzyme Kinetics and Biocatalysis

Enzymes are specialized protein catalysts that accelerate biochemical reactions by lowering the activation energy barrier. They do not alter the equilibrium of the reaction. The kinetics of single-substrate enzyme-catalyzed reactions are described by the Michaelis-Menten model:

[ v = \frac{V_{\text{max}} [S]}{K_m + [S]} ]

where (v) is the reaction rate, (V_{\text{max}}) is the maximum reaction rate (which equals (k_{\text{cat}} [E]0), where (k{\text{cat}}) is the turnover number and ([E]0) is the total enzyme concentration), and (K_m) is the Michaelis constant, which reflects the affinity of the enzyme for the substrate (a lower (K_m) indicates higher affinity). To determine the kinetic parameters (V{\text{max}}) and (K_m) from experimental data, the Michaelis-Menten equation is linearized into the Lineweaver-Burk (double-reciprocal) plot:

[ \frac{1}{v} = \frac{K_m}{V_{\text{max}}} \frac{1}{[S]} + \frac{1}{V_{\text{max}}} ]

Plotting (1/v) against (1/[S]) yields a straight line where the y-intercept is (1/V_{\text{max}}), the x-intercept is (-1/K_m), and the slope is (K_m / V_{\text{max}}).

Enzyme Inhibition

Inhibitors are molecules that reduce enzyme activity. The three primary types of reversible inhibition are:

1. Competitive Inhibition: The inhibitor competes with the substrate for the active site. Adding more substrate can overcome this inhibition. Thus, (V_{\text{max}}) remains unchanged, but the apparent Michaelis constant increases: (K_{m,\text{app}} = K_m (1 + [I]/K_i)), where ([I]) is the inhibitor concentration and (K_i) is the dissociation constant of the enzyme-inhibitor complex.
2. Non-competitive Inhibition: The inhibitor binds to an allosteric site on the enzyme, regardless of whether substrate is bound. This decreases the concentration of active enzyme. The substrate affinity is unaffected, so (K_m) remains unchanged, but (V_{\text{max}}) decreases: (V_{\text{max,app}} = V_{ ext{max}} / (1 + [I]/K_i)).
3. Uncompetitive Inhibition: The inhibitor binds only to the enzyme-substrate ((ES)) complex. Both (V_{\text{max}}) and (K_m) decrease by the same factor (\alpha = 1 + [I]/K_i), resulting in a parallel shift on the Lineweaver-Burk plot.