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.

This work is concerned with the sequence of events taking place during the first stages of bone fracture healing, from bone breakup until the formation of early fibrous callus (EFC). The latter provides a scaffold over which subsequent remodeling processes will eventually result in successful bone repair. Specifically, some mathematical models are proposed to estimate the time required for (1) the formation immediately after fracture of a fibrin clot, described in terms of a phase transition in a polymerization process, and (2) the onset of EFC which is produced when fibroblasts arising from differentiation of chemotactically recruited mesenchymal stem cells remodel a previous fibrin clot by releasing a collagen matrix over it. An attempt has been made to keep models as simple as possible, so that a explicit dependence of the estimates obtained on relevant biochemical parameters involved is obtained.

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.

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PLOS1/25 Author summaryOur 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. 2 supporting framework for the body. It acts as a shield to protect internal organs and 3 plays a crucial role in locomotion by mediating the force arising from muscle 4 contraction. In spite of its inert appearance, bone is an extremely dynamic tissue that is 5 continuously being remodelled to adapt to changing mechanical demands. Such 6 remodeling, which is carried out at a microscopic scale, consists in the removal of 7 low-performing bone and its replacement by new, fully functional one. This task is 8 fulfilled by suitable agents called for that purpose, as described below. 9 Bone tissue is formed by a mineralized matrix that has been hardened to provide a 10 supporting function. There are three key cell types that are responsible for matrix 11 production, maintenance and remodeling: osteoclasts, osteoblasts and osteocytes which 12 perform different homeostatic functions [1-3]. Osteoclasts, recruited when needed from 13 their cell precursors, are in charge of degrading dysfunctional bone, whereas synthesis of 14 new bone to replace the former is carried out by osteoblasts. Osteocytes, the most 15 abundant bone cells, form a three-dimensional interconnected network throughout the 16 bones. They act as mechanosensors that monitor mechanical stress within bone tissues , 17 and react to changes in both the amount and the direction of loading applied on bones. 18 A key event that triggers bone remodeling is osteocyte cell death (apoptosis) 19 occurring over comparatively small length scales and resulting, for instance, from 20 unusual mechanical load or from micro-fractures induced by physical exercise. 21Concerning the first situation, it should be mentioned that the relation between 22 osteocyte apoptosis and load applied is known to be U-shaped. This means that 23 mechanical stresses within a normal physiological range prevent it, whereas those above 24 or below this range induce it [4][5][6]. In traumatic bone fracture...