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Coupled-channel three-body calculations of an $I=1/2$, $J^{\pi}=0^-$$\bar{K}NN$ quasi-bound state in the $\bar{K}NN - \pi \Sigma N$ system wereperformed and the dependence of the resulting three-body energy on the two-body$\bar{K}N - \pi \Sigma$ interaction was investigated. Earlier results ofbinding energy $B_{K^-pp} \sim 50 -70$ MeV and width $\Gamma_{K^-pp} \sim 100$MeV are confirmed [N.V. Shevchenko {\it et al.}, Phys. Rev. Lett. {\bf 98},082301 (2007)]. It is shown that a suitably constructed energy-independentcomplex $\bar{K}N$ potential gives a considerably shallower and narrowerthree-body quasi-bound state than the full coupled-channel calculation.Comparison with other calculations is made.

$\bar{K}NN$ quasi-bound state and the $\bar{K}N$ interaction: coupled-channel Faddeev calculations of the $\bar{K}NN - \pi \Sigma N$ system