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The dilution in the Bradford assay is $1000/5=200$. Read the fitted line rather than assuming it passes through the origin: the two absorbances correspond approximately to $4.3$ and $15\mu \mathrm{g}\mathrm{m}{\mathrm{L}}^{-1}$ in the assay mixture. ThereforeMultiplying by the $0.500\mathrm{mL}$ eluate volume gives about $0.43\mathrm{mg}$ of TEM-1 and $1.5\mathrm{mg}$ of TEM-52. Only TEM-52 meets the requirement.

Assuming the same recovery per culture volume, the required volumes areUse approximately $120\mathrm{mL}$ or more for TEM-1; the existing $50\mathrm{mL}$ is sufficient for TEM-52. Rounding the former up to $125\mathrm{mL}$ is sensible given graphical precision. Reading the lower standard as $4$–$4.5\mu \mathrm{g}\mathrm{m}{\mathrm{L}}^{-1}$ gives a corresponding requirement of roughly $110$–$125\mathrm{mL}$, not an exact integer volume. These estimates assume comparable dye response to bovine serum albumin, suitable buffer blanks and that recovered protein is the purified target; contaminants would inflate the inferred target yield.

For the Hanes-Woolf plot, the slope is $1/V_{\max }$, the vertical intercept is $K_m/V_{\max }$, and the horizontal intercept is $-K_m$. The lines have approximately equal slopes, $0.50\min \mu {\mathrm{M}}^{-1}$. The dashed TEM-1 line intercepts the vertical axis at about $13\min$, while the dotted TEM-52 line intercepts it at about $2.5\min$. ThusThe negative-axis intercepts independently give approximately $-26$ and $-5\mu \mathrm{M}$. Allow normal graph-reading uncertainty, especially at the smaller intercept. The limiting rates are similar; the Michaelis constant is substantially lower for TEM-52. A Michaelis constant is not generally a binding dissociation constant, and calculating a turnover number additionally requires the concentration of active enzyme.

The displayed separation between the two entrance landmarks increases from approximately $6.2$ to $7.7\text{A˚}$ in TEM-52, indicating a wider opening. The Gly238-to-Ser substitution can alter backbone packing and loop geometry; its effect is not simply the steric size of the new side chain. Increased accessibility or altered productive positioning could increase the effective association rate for penicillin G, producing a lower apparent Michaelis constant without changing the limiting catalytic step. In a simple Michaelis-Menten kinetics scheme, $K_m=(k_{-1}+k_{\mathrm{cat}})/k_1$, so increasing $k_1$ can lower $K_m$.

The structural change is consistent with improved access and the lower measured $K_m$, while similar $V_{\max }$ suggests similar limiting turnover under these conditions. It does not uniquely establish the mechanism or prove tighter equilibrium binding. TEM-52 contains additional substitutions, so a matched single-substitution series would be needed to isolate the contribution of residue 238. Structural work on TEM-52 independently supports the importance of remodeling access to its active site.^[ref]

The bulky substituents of ceftazidime can hinder productive entry into the more restricted TEM-1 active site. A wider entrance in TEM-52 could accommodate this substrate and allow its beta-lactam ring to be hydrolyzed. The prediction is reduced susceptibility of TEM-52-expressing bacteria compared with matched TEM-1-expressing bacteria. This is a mechanistic prediction, not a quantitative conclusion from the penicillin G data: entry alone does not guarantee efficient catalytic chemistry for a different substrate. Test ceftazidime hydrolysis directly and measure its minimum inhibitory concentration in an otherwise matched bacterial background, controlling enzyme expression, growth, drug entry and other resistance mechanisms.