Enzyme immobilisation and diffusion effects

$z$ = signed charge number of the substrate, $\psi$ = signed electrostatic potential of the support (V), $F=96485$ C/mol, $R_g=8.314$ J mol${}^{-1}$ K${}^{-1}$, $T$ in K. Assumes a uniform potential in the enzyme's microenvironment and Boltzmann partitioning; the effect is largest at low ionic strength, because added salt screens the charge. Same signs ($z\psi >0$): substrate repelled, $K_{m,app}>K_m$. Opposite signs: substrate attracted, $K_{m,app}<K_m$. Partitioning alone leaves $V_{max}$ unchanged. More generally, any partition coefficient $K_p=S_{local}/S_b$ (electrostatic or hydrophobic) gives $K_{m,app}=K_m/K_p$ with $V_{max}$ unchanged.

The same Boltzmann partitioning applied to ${\mathrm{H}}^{+}$ ($z=+1$), with $0.434={\log }_{10}e$ and the same assumptions. A negatively charged (polyanionic) support has $\psi <0$, so the local pH is lower than the bulk pH.

Steady state; enzyme on the external surface only, no internal diffusion. Solve for $S_s$; the observed rate per unit area is either side of the equation.

Dimensionless. The maximum mass-transfer rate is $k_L{S}_b$, reached when $S_s=0$. Large $\mathrm{Da}$ favours film control, small $\mathrm{Da}$ favours reaction control (see the criteria below).

Exact for Michaelis–Menten kinetics. It comes from the quadratic $x^2+(\mathrm{Da}+\kappa -1)x-\kappa =0$; take the positive root ($0<x<1$).

Referred to the bulk concentration. Always $\le 1$ for Michaelis–Menten kinetics, because $S_s\le S_b$.

$k_s=V_{max}^{\prime }/K_m$ (m/s) is the surface first-order rate constant. Film and reaction act as resistances in series. Note ${\mathrm{Da}}_1=\mathrm{Da}/\kappa$: it uses $K_m$ in place of $S_b$.

Shell balance on a thin spherical shell with constant $D_e$; symmetry gives zero gradient at the centre. For a slab the left side is $D_e{d}^{2}S/d{z}^2$ with $z$ measured from the mid-plane.

Referred to the surface concentration $S_s$, not the bulk. $0<\eta \le 1$ for first-order and Michaelis–Menten kinetics.

${\varphi }^2$ compares the characteristic reaction rate with the characteristic diffusion rate. For a slab, $L$ is the half-thickness when both faces are exposed (the full thickness when one face is sealed). $R$ is a radius, not a diameter.

$L$ = half-thickness with both faces exposed. Exact for isothermal first-order kinetics.

$R$ = particle radius. Exact for isothermal first-order kinetics.

Gives identical numbers; only the modulus is defined without the factor 1/3. In this convention the large-modulus limit is $\eta \approx 3/{\varphi }_s$.

Small $\varphi$: reaction is slow compared with diffusion and the whole particle sees $S_s$. Large $\varphi$: substrate is consumed in a thin shell near the surface and the core is starved. For a slab $\tanh \varphi /\varphi$ is within 1% of $1/\varphi$ once $\varphi \ge 2.65$ ($\tanh 3=0.995$). A sphere converges more slowly — at $\varphi =3$, $1/\varphi$ overestimates $\eta$ by about 12% — hence the correction term, which is within 1% of the exact value for $\varphi \ge 1$.

Constant $D_e$, isothermal particle. Reduces to $\varphi =(V_p/A_p)\sqrt{k_1/D_e}$ for first-order kinetics. The lower limit of the integral is 0 for an irreversible reaction.

${\varphi }_1$ is the first-order modulus; $\ln$ is the natural logarithm. The factor multiplying ${\varphi }_1$ tends to 1 as $\beta \to 0$ and to $1/\sqrt{2\beta }$ as $\beta \to \infty$, and decreases steadily in between, so $\Phi$ falls as $S_s$ rises.

The zero-order limit of Michaelis–Menten kinetics ($\beta =S_s/K_m\gg 1$); the slab result is exact for truly zero-order kinetics. For ${\Phi }_0>1$ substrate is exhausted at a penetration depth $\delta =\sqrt{2D_e{S}_s/V_{max}}=L/{\Phi }_0$ from each face, and the enzyme beyond it is idle. For a real Michaelis–Menten enzyme the rate tapers off smoothly near that front, so there the result is an approximation.

$v_{obs}$ = measured rate per unit particle volume. The identity ${\Phi }_W=\eta {\varphi }^2$ holds for first-order kinetics with $\varphi$ based on $V_p/A_p$.

From ${\Phi }_W=\eta {\varphi }^2$ with the exact sphere $\eta$. For large $\varphi$, ${\Phi }_W\approx \varphi -\frac{1}{3}$.

Rate per unit particle volume, from $k_L{a}_p(S_b-S_s)=\eta k_1{S}_s$. For isothermal first-order kinetics $\eta$ does not depend on $S_s$, so the two resistances simply add.

First order, with $\varphi$ based on $V_p/A_p$ and ${\eta }_o$ referred to $S_b$. ${\mathrm{Bi}}_m$ compares film transport with internal diffusion; the identity $k_1/(k_L{a}_p)={\varphi }^2/{\mathrm{Bi}}_m$ turns the series-resistance result into this form. If a question defines $\mathrm{Bi}=k_{L}R/D_e$ for a sphere together with ${\varphi }_s=R\sqrt{k_1/D_e}$, the second term is ${\varphi }_s^2/(3\mathrm{Bi})$. Large ${\mathrm{Bi}}_m$: the film is negligible and ${\eta }_o\to \eta$.

Referred to the bulk concentration. It is the product of the external and internal factors because $v_{obs}=\eta v(S_s)$.

Valid for $\varphi \gg 1$. The observed rate scales with $\sqrt{k_1}$, hence with the square root of the enzyme loading (uniformly spread enzyme), and inversely with particle size.

Strong internal diffusion limitation ($\varphi \gg 1$) in an isothermal particle, where $v_{obs}\propto \sqrt{k_1{D}_e}$. $E$ = intrinsic activation energy of the enzyme reaction; $E_D$ = activation energy of diffusion, usually small. Under external film control the apparent value falls further, to the low value of liquid-film mass transfer.

Strong internal limitation ($\varphi \gg 1$), isothermal particle, intrinsic power-law order $n$ ($n>-1$): first order stays first order and zero order appears as half order. A Michaelis–Menten enzyme that is saturated at the surface ($\beta \gg 1$) under strong limitation therefore shows $v_{obs}\propto S_s^{1/2}$.

Surface-bound enzyme with Michaelis–Menten kinetics and a liquid film. The plateau is still $V_{max}^{\prime }$; half of it needs $S_s=K_m$ at the surface, and the film must then carry $V_{max}^{\prime }/2$, so $k_L(S_b-K_m)=V_{max}^{\prime }/2$. This bulk half-saturation concentration is what the measured curve shows as its apparent $K_m$. Because the whole curve is not exactly Michaelis–Menten, a $K_m$ fitted by linearisation depends on the substrate range used.