Cell motility represents one of the most fundamental function in mechanobiology. Cell motility is directly implicated in development, cancer or tissue regeneration, but it also plays a key role in the future of tissue and biomedical engineering. Here, we derived a computational model of cell motility that incorporates the most important mechanisms toward cell motility: cell protrusion, polarization and retrograde flow. We first validate our model to explain two important types of cell migration, i.e. confined and ameboid cell migration, as well as all phases of the latter cell migration type, i.e. symmetric cell spreading, cell polarization and latter migration. Then, we use our model to investigate durotaxis and chemotaxis. The model predicts that chemotaxis alone induces larger migration velocities than durotaxis and that durotaxis is activated in soft matrices but not in stiff ones. More importantly, we analyze the competition between chemical and mechanical signals. We show that chemotaxis rules over durotaxis in most situations although durotaxis diminishes chemotaxis. Moreover, we show that inhibiting the effect of GTPases in actin polymerization at the cell front may allow durotaxis to take control over chemotaxis in soft substrates. Understanding how the main forces in cell motility cooperate, and how a precise manipulation of external cues may control directed cell migration is not only key for a fundamental comprehension of cell biology but also to engineer better biomimetic tissues. To this end, we provide a freely-available platform to predict all phases and modes of cell motility analyzed in this work.

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