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Diobeno vreteno je stanična struktura koja je odgovorna za raspodjelu genetskog materijala medu stanicama kćeri, a sastoji se od mikrotubula i pridruženih proteina. Sile i momenti sila igraju ključnu ulogu u funkcioniranju vretena te su odgovorni za pomicanje kromosoma i održavanje njegove grade, a važnu ulogu igraju elastična svojstva mikrotubula. Međutim, sile i momente nije lako odrediti današnjim eksperimentalnim tehnikama. U ovom teorijskom istraživanju uvodimo model za ravnotežu sila i momenata sila u diobenom vretenu. Korištenjem aproksimacije srednjeg polja opisujemo sve mikrotubule u diobenom vretenu, kao i sile i momente sila koje djeluju na njih. Model daje predvidanja za oblik diobenog vretena. Dobivene rezultate planiramo usporediti s eksperimentalno određenim oblicima koje su dobili naši suradnici. Teorijska predviđanja dobivena ovim modelom, zajedno s eksperimentima, trebala bi odgovoriti na pitanje kolike sile djeluju u diobenom vretenu. The mitotic spindle is a cellular structure which is responsible for the partition of genetic material when a cell divides into two daughter cells. It is comprised of microtubules and accompanying proteins. Forces and torques play a key role in the functioning of the spindle, for they are chiefly responsible for the movement of chromosomes and maintaining the overall shape of the spindle. This is facilitated by the elastic properties of microtubules. The forces and torques are, however, hard to determine with contemporary experimental techniques. In this theoretical research we develop a model for the balance of forces and torques in a mitotic spindle. Using a mean-field approximation we describe all microtubules in a mitotic spindle and the forces acting upon them. The model gives a prediction for the shape of the spindle. The results are to be matched to experimental imaging done by our associates, with the aim of answering what magnitude of forces and torques act in mitotic spindles.

U ovom radu smo konstruirali matematički model kako bismo opisali raspodjelu motora na rastućem mikrotubulu. Promatrali smo raspodjelu motora koji hodaju prema plus kraju mikrotubula, kinezin. Raspodjela motora nam je važna kako bismo mogli detaljnije proučavati mehanizme prijenosa tereta u stanicama i promjenu mikrotubula iz faze rasta u fazu smanjivanja, kao posljedicu akumulacije motora na kraju mikrotubula. Raspodjelu molekularnih motora na mikrotubulu smo dobili rješavanjem jednadžbe Laplaceovom metodom. Rješenje smo zatim prokomentirali za različite slučajeve, ovisno o relativnoj razlici brzina rasta mikrotubula i brzine molekularih modela koji hodaju po njemu. U metodičkom dijelu rada pisali smo o modelima u fizici te procesu modeliranja u istraživačkoj nastavi fizike. Ovakav pristup učenju gradiva fizike pokazao se kao jako koristan za sve učenike, neovisno o njihovoj dobi i sposobnostima. In this paper we have constructed a mathematical model to describe the distribution of the molecular motors on growing microtubules. We have observed distribution of molecular motors walking towards the plus end of the microtubule - kinesin. Distribution of the molecular motors is important to us so that we can study in detail the mechanisms of transfer of cargo in the cells and changes from growth phase to shrinking phase of mikroubules, as a result of the accumulation of the engine at the end of microtubules. We got the expression for distribution of molecular motors on microtubules by solving equation with Laplace method. The solution was then observed in three different cases, depending on the relative difference of the growth speed of microtubules and speed of molecular motors walking on it. In the methodological part of the paper we wrote about the models in physics and process of modeling in physics instruction and research. This approach to learning physics proved to be very beneficial for all students, regardless of age and ability.