Data-enhanced physics-based models that are based on the Navier-Stokes equations have been shown to capture various structural and statistical features of turbulent flows. In this study, we evaluate the predictive capability of the stochastic dynamical models proposed by Zare, Jovanovic, and Georgiou (J. Fluid Mech., vol. 812, 2017) in capturing spatio-temporal features of the near-wall-cycle. These models are formed by systematically introducing low-rank dynamical modifications to the linearized Navier-Stokes operator allowing them to not only match the one-dimensional energy spectrum, but to provide good estimates of two-point correlations of the turbulent velocity field. For a turbulent channel flow at a friction Reynolds number of 186, we show that the quality of predicting two-point velocity correlations and the spatio-temporal frequency response depends on the choice of a low-rank inducing parameter that alters the structure of dynamical modifications to the linearized operator. We fine tune this parameter to better predict signatures of the near-wall cycles and compare the stochastic and optimal harmonic response of the resulting data-enhanced model with that of the eddy-viscosity-enhanced linearized Navier-Stokes equations.