= Solution
Write $A=V/D_3$ and $B=L/D_1$. The given mean-search scaling is $\tau=A/\lambda+B\lambda$. Three-dimensional relocation sees an effective capture length of order $\lambda$, giving a bulk encounter time $V/(D_3\lambda)$. A <Facilitated target search>[sliding] excursion explores length of order $\lambda$ in time of order $\lambda^2/D_1$; sampling order $L/\lambda$ contour segments gives a total <Facilitated target search>[sliding] cost of order $L\lambda/D_1$. Prefactors, correlations and repeated visits are coarse-grained in this scaling model.
Small <Facilitated target search>[sliding] lengths make relocation inefficient; very long <Facilitated target search>[sliding] wastes time revisiting a local <DNA> region rather than relocating to another. The optimum balances both contributions:
$$
\frac{d\tau}{d\lambda}=-\frac{V}{D_3\lambda^2}+\frac L{D_1}=0,
\qquad
\boxed{\lambda_* =\sqrt{\frac{VD_1}{LD_3}},\qquad
\tau_{\min}=2\sqrt{\frac{VL}{D_1D_3}}.}
$$
Since $d^2\tau/d\lambda^2=2V/(D_3\lambda^3)>0$, it is the unique minimum for unrestricted positive $\lambda$. At the optimum, bulk and <Facilitated target search>[sliding] times are equal. With $s=\lambda/\lambda_*$,
$$
\boxed{\tau/\tau_{\min}=\tfrac12(s^{-1}+s).}
$$
\Image[/past-exam-of-the-biological-physics-course-of-the-university-of-oxford/2017-diffusive-search-balance.png]
{title=Bulk relocation and local sliding impose opposing search-time costs}
{description=The original dimensionless plot shows a decreasing inverse-sliding-length contribution and an increasing sliding contribution. Their sum has a minimum where the two costs agree, at a sliding length equal to the analytic optimum.}
The model is meaningful only over an accessible range between molecular steps and the available chromosome length. An optimum outside that range is replaced by a boundary choice; the unphysical divergences at arbitrarily small or large $\lambda$ describe limits of the scaling expression. A common multiplicative searcher-number factor would not change the optimum, but an exact many-searcher first-arrival mean is not generally obtained by dividing an arbitrary single-searcher mean by their number without a waiting-time assumption.
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