  We consider the low momentum expansion of the four graviton and the two
graviton--two gluon amplitudes in heterotic string theory at one loop in ten
dimensions, and analyze contributions upto the D^2 R^4 interaction from the
four graviton amplitude, and the D^4 R^2 F^2 interaction from the two
graviton--two gluon amplitude. The calculations are performed by obtaining
equations for the relevant modular graph functions that arise in the modular
invariant integrals, and involve amalgamating techniques used in the type II
theory and the calculation of the elliptic genus in the heterotic theory.
