We determine the shift and line shape of the amide I band of a model AK peptide from molecular dynamics (MD) simulations of the peptide dissolved in methanol/water mixtures with varying composition. The IR spectra are determined from a transition dipole coupling exciton model. A simplified empirical model Hamiltonian is employed, which takes into account both the effect of hydrogen bonding and the intramolecular vibrational coupling. We consider a single isolated AK peptide in a mostly helical conformation, while the solvent is represented by 2600 methanol or water molecules, simulated for a pressure of 1 bar and a temperature of 300 K. Over the course of the simulations, minor reversible conformational changes at the termini are observed, which are found to only slightly affect the calculated spectral properties. Over the entire composition range, which varies from pure water to the pure methanol solvent, a monotonous shift towards higher frequency of the IR amide I band of about 8 wavenumbers is observed. This shift towards higher frequency is comparable to the shift found in preliminary experimental data also presented here on the amide I' band. The shift is found to be caused by two counter-compensating effects. An intramolecular red shift of about 1.2 wavenumbers occurs, due to stronger intramolecular hydrogen bonding in a methanol-rich environment. Dominating, however, is the intermolecular solvent-dependent shift towards higher frequency of about 10 wavenumbers, which is attributed to the less effective hydrogen-bond-donor capabilities of methanol compared to water. The importance of the solvent contribution to the IR shift, as well as the significantly different hydrogen formation capabilities of water and methanol, makes the amide I band sensitive to composition changes in the local environment close to the peptide/solvent interface. This allows, in principle, an experimental determination of the composition of the solvent in close proximity to the peptide surface. For the AK peptide case, we observe at low methanol concentrations a significantly enhanced methanol concentration at the peptide/solvent interface, supposedly promoted by the partially hydrophobic character of the AK peptide's solvent-accessible surface.