**Abstract**

Free intracellular cholesterol is a fundamental structural component for the membrane formation of organelles (such as mitochondria and endoplasmic reticulum) and plasma membrane, participates in the synthesis (such as cortisol, estradiol and testosterone) and participates in intracellular signaling pathways, being therefore crucial for the maintenance of the human organism. On the other hand, although it is an essential component of these structures, high concentrations of this component leads to changes in the physicochemical properties of these membranes, being toxic for the cells and for the organism as a whole. High cholesterol levels can lead to clogged arteries that come from a process known as atherosclerosis, or hardening of the arteries, which triggers a heart attack or a stroke, being one of the leading causes of deaths nowadays, with a total annual of 13 million deaths worldwide. Therefore, the control of intracellular cholesterol metabolism, i.e. the maintenance of its level within of a narrow range of values, is crucial for the survival of the organism. Intracellular cholesterol has two distinct origins: endogenous (de novo synthesis by the cell) or exogenous (plasma circulation). All mammalian cells have the ability to synthesize cholesterol, although this production is more pronounced in some cell types such as hepatocytes. We propose, in this project of two years, the mathematical modeling of the mechanisms that control the metabolism of intracellular cholesterol and the possible causes of its rupture through the notions of ``homeostasis'' and ``escape from homeostasis''. The rationale is that a biological system is considered homeostatic when there is a ``regulated variable'', which is maintained by the action of effector agents within a narrow range of values. However, homeostatic systems are never perfect and can be destabilized by mutations that alter the kinetics of the underlying mechanisms, placing an organism closer to the limits of the homeostatic region and predisposing it to deleterious effects caused by environmental changes, called escape from homeostasis. Considerations about mutations, environment, homeostasis and escape from homeostasis help to explain why the etiology of so many diseases is complex. The mathematical modeling of intracellular cholesterol metabolism has already been considered in the literature, but there are no studies characterizing mathematically the intracellular metabolism of cholesterol as a homeostatic system. From the mathematical point of view, homeostasis is defined in the context of network dynamical systems. Let x_0 be a stable equilibrium of a system x'=f(x,l), where l is real. Stability implies that there is a family of equilibria x(l)=(x^1(l),...,x^n(l)), with x_0=x(l_0). Homeostasis occurs if the regulated variable z, which depends on the equilibrium family x(l) is approximately constant when l varies in an open neighborhood of l_0. The mapping z(») is called the input-output function. Typically, z(l)=x^j(l), for some coordinate x^j. A parameter value l_0 is called a point of (simple) infinitesimal homeostasis if z_l(l_0)=0. Here, the indices denote derivatives with respect to the respective variables. These notions can be generalized to multivariate input-output functions. It is usually easier to use implicit differentiation to find points of infinitesimal homeostasis than to find intervals over which z(l) is approximately constant. The central idea is to formulate a mathematical model for intracellular cholesterol metabolism and look for points of infinitesimal homeostasis as a function of one or more input parameters of their model, where regulated variable is given by the of free intracellular cholesterol. As a future perspective, we propose to map the homeostasis region to the phenotypic variations associated with polymorphisms in human populations. (AU)