= Solution
The term $V/(D_3\lambda)$ is the bulk-<diffusion> search cost: increasing the effective length of <DNA> searched after each encounter enlarges the useful capture region and reduces this contribution. The term $L\lambda/D_1$ is the accumulated sliding cost. Approximately $L/\lambda$ <DNA> landing-and-sliding episodes, each taking order $\lambda^2/D_1$, give $L\lambda/D_1$. These are scaling estimates; geometric constants and correlated revisits are omitted by the supplied model.
Thus the <facilitated target search> is slow as $\lambda\to0$, because too little sequence is examined per encounter, and slow at very large $\lambda$, because sliding and repeated local exploration take too long. A finite intermediate minimum balances 1D scanning against 3D relocation. The right panel of the figure in <past-exam-of-the-biological-physics-course-of-the-university-of-oxford/2019/1/a/solution> shows the two contributions and their sum. The model requires a useful range of sliding lengths below the full <chromosome> length, finite positive <diffusion coefficients> and effective relocation between episodes; it is not a claim that arbitrarily large $\lambda$ is physically available.
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