A dynamic model for the morphogenesis of the late vertebrate lens

A mathematical model is presented for the morphogenesis of the post-vesicular vertebrate lens with an umbilical suture. The lens is modeled as having four compartments: anterior epithelium (germinative and central anterior zones), recruitment zone (transitional zone), cortex (discrete concentric cohorts of secondary cortex fiber cells, each cohort treated individually), and nucleus. Equations are written to describe the time evolution of the cohorts; their shapes collectively determine the shape of the lens. The growth of cell volume is exponential, with different rates in the cortex and epithelium; recruitment of epithelium cells into the cortex is described as resulting from an overproduction of epithelial basal (capsular) surface in the anterior epithelium. The equations contain three dimensionless numbers determined by the physiology of the epithelium and cortex cells. Solutions are stable attractors in a morphological space. All solutions entail exponential growth of the lens diameter; a portion of parameter space corresponds to exponential growth superimposed on large amplitude oscillations in lens shape. Emergent time-scales for increase in lens size and oscillation period are an order of magnitude longer than the cellular growth time-scales. The lens shapes tend to a family of stable scaling solutions, the shapes of which remain unchanged as the lens grows. The model is applied to morphological data for the chick and lamprey lenses. The dynamics described are seen as exemplifying an auto-regulatory morphogenesis process wherein a system passes through a sequence of developmental stages. Each stage is characterized by its own fixed informing geometry (a set of defining spatial relationships), within which a growth process unfolds autonomously, generating a dynamically stable structure. The developing system invokes a means of forgetting dated structural information; this dissipation is necessary to the pattern formation process.