Modeling the bistable transition between cell phenotypes during limb precartilage condensation
The tetrapod limb skeleton is initiated in unpatterned limb bud mesenchyme by the formation of precartilage condensations. Here, based on time-lapse videographic analysis of a forming condensation in a high-density culture of chicken limb bud mesenchyme, we observe a phase transition to a more fluidized state for cells within spatial compacted foci (protocondensations that will progress to condensations), as reflected in their spatial confinement, cell-substratum interaction and speed of motion. Previous work showed that galectin-8 and galectin-1A, two proteins of the galactoside-binding galectin family, are the earliest determinants of this process in the chicken limb bud, and that their interactions in forming skeletogenic patterns of condensations can be interpreted mathematically through a reaction-diffusion-adhesion framework. Based on this framework, we use an ordinary differential equation-based approach to analyze the core switching modality of the galectin reaction network and characterize the states of the network independent of the diffusive and adhesive arms of the patterning mechanism. We identify two steady states where the concentrations of both galectins are respectively, negligible, and very high. An explicit Lyapunov function shows that there are no periodic solutions. For sigmoidal galectin production terms, the model exhibits a bistable switch that arises from a monostable state via saddle-node bifurcation. Our model therefore predicts that the galectin network exists in low and high expression states separated in space or time without any intermediate states. This provides a causal basis for the observed outside vs. inside transition observed in the in vitro video data. We performed a quantitative analysis of the distribution of galectin-1A in cultures of condensing chick limb mesenchymal cells and found that the interior of the protocondensations had concentrations of this protein (compared to the immediate exterior) over and above that expected from its higher cell density, consistent with the models predictions. The galectin-based patterning network is thus suggested, on theoretical grounds, to incorporate a core switch independent of any spatial or temporal dynamics, that drives the chondrogenic cell state transition in limb skeletogenesis.