  We investigate the connection of lattice calculations of moments of isovector
parton distributions to the physical regime through extrapolations in the quark
mass. We consider the one pion loop renormalisation of the nucleon matrix
elements of the corresponding operators and thereby develop formulae with which
to extrapolate the moments of the unpolarised, helicity and transversity
distributions. These formulae are consistent with chiral perturbation theory in
the chiral limit and incorporate the correct heavy quark limits. In the
polarised cases, the inclusion of intermediate states involving the $\Delta$
isobar is found to be very important. The results of our extrapolations are in
general agreement with the phenomenological values of these moments where they
are known, and for the first time we perform an extrapolation of the low
moments of the isovector transversity distribution which is consistent with
chiral symmetry.
