A unique and extensive data set of cirrus properties collected on 13 July 2002 during CRYSTAL-FACE provides the framework for simulations using cloud models to interpret the observations and to develop recommendations for microphysical parameterizations in large-scale models. Several outstanding issues in the simulations of cirrus clouds are addressed using detailed bin-resolving and bulk microphysics models. A new heterogeneous ice nucleation formulation based on extended classical theory with simultaneous dependence on temperature and saturation ratio is applied for the first time to thin tropopause cirrus. The simulated cloud microphysical properties are similar to observations, suggesting that tropopause cirrus may potentially form as a result of heterogeneous immersion freezing of internally mixed aerosols serving as ice nuclei (IN). The potential for mixed aerosols to serve as IN in tropopause cirrus is consistent with measurements of comparable amounts of soluble and insoluble material in cirrus residues and aerosols during CRYSTAL-FACE. Simulations using homogeneous nucleation theory are also able to produce comparable microphysical properties if the heterogeneous mode is turned off; hence the homogeneous mode cannot be excluded if insoluble material capable of serving as IN is not available. The calculated critical ice supersaturation for the onset of heterogeneous nucleation at these cold temperatures (~200 K) was 70--80% (for the assumed aerosol nucleation parameters) and 15--20% higher for homogeneous nucleation. The calculated supersaturation relaxation time ranged from ~1--2 hours in the center of the cloud to 3--6 hours near the boundaries, which may explain the high values of ice supersaturation (30--80%) observed in this cloud. Analysis of the supersaturation budget showed that supersaturation was generally nonequilibrium, and relaxation from the initial critical values to near equilibrium occurred only after several hours. The bulk model was able to simulate this case and in particular the slow crystal growth and large supersaturation because of its detailed treatment of ice nucleation and supersaturation. The fraction of condensed ice relative to excess vapor predicted by both models was 40--60% for several hours, indicating that bulk models with zero supersaturation (instantaneous condensation of all excess vapor) would substantially overpredict the ice water path and optical thickness.