Deligne pairing and determinant bundle

Let $X\to S$ be a smooth projective surjective morphism, where $X$ and $S$ are integral schemes over complex numbers. Let L_0, L_1, … L_{n-1}, L_{n} be line bundles over $X$. There is a natural isomorphism of the Deligne pairing $<L_0,\mathrm{.}\mathrm{.}\mathrm{.},L_n>$ with the determinant line bundle ${\rm Det}(\otimes_{i=0}^{n} (L_i- {\mathcal O}_{X}))$.