A DYNAMIC-MODEL FOR THE MORPHOGENESIS OF THE LATE VERTEBRATE LENS

A mathematical model is presented for the morphogenesis of the post-ve sicular vertebrate lens with an umbilical suture. The lens is modeled as having four compartments: anterior epithelium (germinative and cent ral anterior zones), recruitment zone (transitional zone), cortex (dis crete 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 epitheli um cells into the cortex is described as resulting from an overproduct ion of epithelial basal (capsular) surface in the anterior epithelium. The equations contain three dimensionless numbers determined by the p hysiology of the epithelium and cortex cells. Solutions are stable att ractors in a morphological space. All solutions entail exponential gro wth of the lens diameter; a portion of parameter space corresponds to exponential growth superimposed on large amplitude oscillations in len s shape. Emergent time-scales for increase in lens size and oscillatio n period are an order of magnitude longer than the cellular growth tim e-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 d ynamics described are seen as exemplifying an auto-regulatory morphoge nesis process wherein a system passes through a sequence of developmen tal stages. Each stage is characterized by its own fixed informing geo metry (a set of defining spatial relationships), within which a growth process unfolds autonomously, generating a dynamically stable structu re. The developing system invokes a means of forgetting dated structur al information; this dissipation is necessary to the pattern formation process. (C) 1997 Academic Press Limited.