Mathematical Models of Moving Boundary Problems in Developmental Biology and Cell Motility

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Dillon0109957 The principal investigator and his colleagues developmathematical models and computational methods for analyzing themotion of eucaryotic cilia and flagella, the growth anddevelopment of the vertebrate limb bud and tumor growth. Theobjective of the cilia modeling is to develop three-dimensionalmodels for an individual cilium or flagellum and forthree-dimensional arrays of cilia. These models incorporatediscrete representations of the axoneme's ultrastructureincluding the individual dyneins and microtubules, as well as acontinuous description of the fluid mechanics using the immersedboundary method. Vertebrate limb development is a widely-usedmodel system for the study of development. The investigator andhis collaborators have developed a 2D model for vertebrate limbdevelopment that describes limb bud outgrowth and predicts thespatio-temporal distribution of growth factors and morphogens.The major goals of the limb development modeling are to furthervalidate the 2D model, to extend it to three dimensions in orderto determine what signals cells see as a function of theirposition in the three-dimensional limb bud and developmentalhistory and, from this, to suggest how the signals can beintegrated by cells to produce the observed spatial patterning.Limb development and tumor growth are moving boundary problemsthat involve tissue growth, the transport of growth factors andcontrol of gene activity. The investigators are developing animmersed boundary model for tumor growth along with the limbdevelopment model. These mathematical models focus on the complex biologicalsystems involved in cell motility, vertebrate development, andtumor growth. An important aspect is the development ofcomputational tools for the study of these processes. Thus, theciliary model is used for the computational study of dyneinactivation mechanisms in cilia and flagella and the multiciliarymodel for the study of mucociliary interaction. The vertebratelimb development model may be used as an aid in interpretingexperimental results, to help identify new experiments, and totest hypotheses that may be difficult to test experimentally.Because each of the models described in this proposal includes afluid-mechanical representation, there is a unifying element intheir mathematical description. The computational methods arebased in part on the immersed boundary method, and the computercodes share many of the major subroutines. An important aspect ofthis proposal is the development of parallel numerical methodsfor the solution of these problems; advances in this area can beused in a variety of problems in the field of computationalbiofluids.