= Solution
Write $A=V/D_3$ and $B=L/D_1$. The search time is $A/\lambda+B\lambda$: it decreases from infinity at small positive $\lambda$, reaches one minimum, and increases without bound at large $\lambda$ within this idealized model.
During a <facilitated diffusion> search, roughly $L/\lambda$ separate <DNA> landing-and-sliding episodes sample a chromosome. A three-dimensional encounter with any chromosomal landing region has characteristic time $V/(D_3L)$, so the total bulk-search contribution is $(L/\lambda)V/(D_3L)=V/(D_3\lambda)$. Each one-dimensional <diffusion> excursion lasts order $\lambda^2/D_1$; multiplying by $L/\lambda$ gives $L\lambda/D_1$. Longer sliding therefore reduces the number of bulk relocations but increases time spent repeatedly exploring local <DNA>. Numerical constants and overlap corrections are already omitted in the given model.
Differentiating and using the positive second <derivative> gives
$$
-\frac{V}{D_3\lambda^2}+\frac{L}{D_1}=0,
\qquad
\boxed{\lambda_* =\sqrt{\frac{VD_1}{LD_3}},\qquad
\tau_{\min}=2\sqrt{\frac{VL}{D_1D_3}}}.
$$
At $\lambda_*$ the two time contributions are equal. The optimum is meaningful only if $\lambda_*$ lies between the microscopic capture scale and the available chromosomal contour length.
A decrease in <DNA> concentration means a decrease in $L/V$, but its effect on the time is not determined by that ratio alone. At fixed chromosome length $L$, dilution by increasing $V$ increases $\tau$ at fixed $\lambda$ and increases $\tau_{\min}$ as $\sqrt V$; bulk relocation becomes slower. If instead $L$ decreases at fixed $V$, while the target is retained, the stated model predicts a smaller sliding contribution and a smaller optimized time, proportional to $\sqrt L$. In both cases $\lambda_*\propto(L/V)^{-1/2}$. This distinction prevents attributing a unique time trend to concentration without specifying how it changes.
\Image[/past-exam-of-the-biological-physics-course-of-the-university-of-oxford/2022-search-and-motor.png]
{title=Search-time optimum and position-dependent <molecular motor> binding and <force>}
The left panel normalizes the search law to $\tau/\tau_{\min}=(u+u^{-1})/2$, $u=\lambda/\lambda_*$. The other panels illustrate <4/b> and <4/c>.
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