An extension of the class of piecewise linear (PL) systems is proposed to model gene expression dynamics dependent on dilution due to cell growth rate. The growth rate is modeled as the weighted minimum of two or more limiting gene products responsible for bacterial growth. The production terms are still piecewise constant, but now the degradation terms are piecewise quadratic (PQ). This new mathematical formalism exhibits continuous switches between PQ modes. We first study the novel dynamical behavior generated by the nonlinear terms at the regions of discontinuity of the vector fields, showing that the sliding motion configurations occurring in PL systems can further lead to damped convergent oscillations or periodic behavior in PQ systems. As an application, a core model of the bacterial gene expression machinery is studied with the goal of externally tuning the growth rate of cells. This system may exhibit several behaviors including bi-mode bistability or damped oscillatory behavior.