  We show that the Mellin transform of an $n$-point tree level MHV gluon
scattering amplitude, also known as the celestial amplitude in pure Yang-Mills
theory, satisfies a system of $(n-2)$ linear first order partial differential
equations corresponding to $(n-2)$ positive helicity gluons. Although these
equations closely resemble Knizhnik-Zamolodchikov equations for $SU(N)$ current
algebra there is also an additional "correction" term coming from the
subleading soft gluon current algebra. These equations can be used to compute
the leading term in the gluon-gluon OPE on the celestial sphere. Similar
equations can also be written down for the momentum space tree level MHV
scattering amplitudes. We also propose a way to deal with the non closure of
subleading current algebra generators under commutation. This is then used to
compute some subleading terms in the mixed helicity gluon OPE and our results
match with those obtained from an explicit calculation using the Mellin MHV
amplitude.
