This paper investigates how to compensate for curvature response mismatch in lateral Model Predictive Control (MPC) of an autonomous vehicle. The standard kinematic bicycle model does not describe accurately the vehicle yaw-rate dynamics, leading to inaccurate motion prediction when used in MPC. Therefore, the standard model is extended with a nonlinear function that maps the curvature response of the vehicle to a given request. Experimental data shows that a two Gaussian functions approximation gives an accurate description of this mapping. Both simulation and experimental results show that the corresponding modified model significantly improves the control performance when using Reference Aware MPC for autonomous driving of a Scania heavy-duty construction truck.