The multivariate adaptive regression splines (MARS) model is one of the well-known, additive non-parametric models that can deal with highly correlated and nonlinear datasets successfully. From our previous analyses, we have seen that lasso-type MARS (LMARS) can be a strong alternative of the Gaussian graphical model (GGM) which is a well-known probabilistic method to describe the steady-state behaviour of the complex biological systems via the lasso regression. In this study, we extend our original LMARS model by taking into account the second-order interaction effects of genes as the representative of the feed-forward loop in biological networks. By this way, we can describe both linear and nonlinear activations of the genes in the same model. We evaluate the performance of our new model under different dimensional simulated and real systems, and then compare the accuracy of the estimates with GGM and LMARS outputs. The results show the advantage of this new model over its close alternatives.