Mechanical model of folding of tubular epithelia

V tem delu preučujemo mehanske lastnosti epitelijskega tkiva z dvodimenzionalnim ogliščnim modelom, osnovanim na apikobazalni polarizaciji. Znotraj tega modela, ki obravnava prečni presek tkiva, se posvetimo vprašanju nastanka vzdolžnih gub v cevastih epitelijih, ki jih najdemo npr. v različnih delih prebavnega trakta in ki so v dobrem približku značilna tudi za zgodnje stopnje razvoja vinske mušice kot pomembnega modelskega organizma. Najprej določimo osnovne oblike modelskega tkiva, tj. oblike z najnižjo energijo, v odvisnosti od mehanske polarizacije tkiva, ki jo zajamemo z razliko površinskih napetosti apikalne in bazalne stranice, in od polmera toge ograditve. V faznem diagramu najdemo dve kvalitativno različni območji. Nad kritičnim polmerom ograditve, ki je odvisen od razlike napetosti, število gub v tkivu narašča s to razliko v skokih po 1, pod kritično ograditvijo pa se oblikujeta ločena režima majhnega in velikega števila gub. V nadaljevanju se posvetimo vprašanju gubanja tkiva, do katerega pride zaradi rasti oziroma celične delitve. Preučimo dva modela: kvazistatično rast, pri kateri se tkivo pri vsakem številu celic nahaja v globalnem minimumu energije, in dinamično rast, pri kateri relaksacijo energije določa nadkritično dušena enačba gibanja. Pri kvazistatični rasti število gub v tkivu hitro narašča tako z razliko površinskih napetosti apikalne in bazalne stranice kot s številom celic. Nasprotno se pri dinamični rasti število gub v dobrem približku ne spreminja s časom, oblikujejo pa se trije režimi razvoja tkiva. V enem je število gub odvisno le od hitrosti delitve, v drugem le od razlike površinskih napetosti apikalne in bazalne stranice, v vmesnem pa je vpliv obeh dejavnikov primerljiv. Ti rezultati nudijo svež teoretični vpogled v morfogenezo cevastih epitelijev, kakršen je npr. črevesni epitelij v piščančjem zarodku.In this Thesis, we theoretically study the mechanical properties of epithelial tissues using a two-dimensional vertex-based model building on apicobasal polarization. Within this model, which focuses on tissue cross-section, we investigate the formation of lengthwise folds in tubular epithelia, which are found, e.g., in the gastrointestinal tract and which are to a good approximation characteristic of the early stages of development in the embryo of the fruit fly as an important model organism. We first compute the energy-minimizing shapes as a function of the mechanical apico-basal polarity defined as the difference between the apical and the basal surface tension, and as a function of the radius of the external rigid cylindrical constraint. In the phase diagram, we find two qualitatively different regions. If the radius of the constraint exceeds a critical, polarity-dependent value, the number of folds in the tissue increases with differential tension in steps of 1. Below the critical radius, separate few- and many-lobes regions appear. We also focus on tissue folding due to growth caused by cell division. We analyze two models: Quasi-stationary growth, in which the tissue takes the energy-minimizing shape for each separate number of cells, and dynamic growth, in which energy relaxation is determined by the overdamped equation of motion. In the quasi-stationary growth mode, the number of lobes in the tissue increases quickly with the difference between the apical and the basal surface tension as well as with the number of cells. Conversely, in the dynamic growth mode the number of lobes does not change with time to a good approximation, and three regimes of tissue growth exist. In the first one, the number of lobes depends solely on the rate of cell divisionin the second one it depends solely on the difference between the apical and the basal surface tensionand in the intermediate regime the effects of both factors are comparable. These results provide a fresh theoretical insight into the morphogenesis of tubular epithelia such as the intestinal epithelium in the chick embryo