The human skeleton undergoes constant remodeling throughout the lifetime. Processes occurring on microscopic and molecular scales degrade bone and replace it with new, fully functional tissue. Multiple bone remodeling events occur simultaneously, continuously and independently throughout the body, so that the entire skeleton is completely renewed about every ten years.Bone remodeling is performed by groups of cells called Bone Multicellular Units (BMU). BMUs consist of different cell types, some specialized in the resorption of old bone, others encharged with producing new bone to replace the former. These processes are tightly regulated so that the amount of new bone produced is in perfect equilibrium with that of old bone removed, thus maintaining bone microscopic structure.To date, many regulatory molecules involved in bone remodeling have been identified, but the precise mechanism of BMU operation remains to be fully elucidated. Given the complexity of the signaling pathways already known, one may question whether such complexity is an inherent requirement of the process or whether some subset of the multiple constituents could fulfill the essential role, leaving functional redundancy to serve an alternative safety role. We propose in this work a minimal model of BMU function that involves a limited number of signals able to account for fully functional BMU operation. Our main assumptions were i) at any given time, any cell within a BMU can select only one among a limited choice of decisions, i.e. divide, die, migrate or differentiate, ii) this decision is irreversibly determined by depletion of an appropriate internal inhibitor and iii) the dynamics of any such inhibitor are coupled to that of specific external mediators, such as hormones, cytokines, growth factors. It was thus shown that efficient BMU operation manifests as an emergent process, which results from the individual and collective decisions taken by cells within the BMU unit in the absence of any external planning.

The mechanism underlying the divergence of perturbation theory is exposed. This is done through a detailed study of the violation of the hypothesis of the Dominated Convergence Theorem of Lebesgue using familiar techniques of Quantum Field Theory. That theorem governs the validity (or lack of it) of the formal manipulations done to generate the perturbative series in the functional integral formalism. The aspects of the perturbative series that need to be modified to obtain a convergent series are presented. Useful tools for a practical implementation of these modifications are developed. Some resummation methods are analyzed in the light of the above mentioned mechanism.

Human skeleton undergoes constant remodeling during the whole life. By means of such process, which occurs at a microscopic scale, worn out bone is replaced by new, fully functional one. Multiple bone remodeling events occur simultaneously and independently throughout the body, so the whole skeleton is completely renewed about every ten years. Bone remodeling is performed by groups of cells called Bone Multicellular Units (BMU). BMUs consist of different cell types; some are specialized in destroying old bone, whereas others produce new bone to replace the former. The whole process is tightly regulated so that the amount of new bone produced exactly balances that of old one removed and bone microscopic structure is maintained. To date, many regulatory molecules involved in bone remodeling have been identified, but the precise mechanism of BMU operation remains to be fully elucidated. Given the complexity of the signaling pathways already known, the question arises of ascertaining whether such complexity is an inherent requirement of the process, or a consequence of operational redundancy. In this work we propose a minimal model of BMU function which involves a small number of signals and accounts for fully functional BMU operation. Our main assumptions are i) at any given time, any cell within a BMU can select only one among a reduced choice of decisions: divide, die, migrate or differentiate, ii) such decision is irreversibly determined by depletion of an appropriate internal inhibitor and iii) the dynamics of any such inhibitor is coupled to that of a few external mediators. It is shown that efficient BMU operation then unfolds as an emergent property, which results from individual decisions taken by cells in the BMU unit in the absence of any external planning. Author summary: Our skeleton is a living organ that is being renewed throughout our life. This task is accomplished by teams of bone cells termed as Bone Multicellular Units (BMUs) that are recruited when and where needed, to operate at places where bone has lost functionality either for an excess of mechanical stress or because loss of activity. Once assembled, BMU remove old bone and replace it by new one, and disband as soon as their mission has been accomplished. No single bone evades BMU screening, so that the whole human skeleton is completely renewed approximately every ten years. It is natural to wonder how such robust and fascinating process is regulated. Many signaling pathways involved in bone remodeling have been identified so far, but whether all of them are necessary for BMU operation remains unclear. In this work we show that just a reduced number of such signals could suffice for that purpose. This suggests that a large degree of redundancy might have been kept in place, perhaps as a consequence of different convergent strategies developed in the course of evolution.