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This paper presents an overview of modeling fractal media by continuum mechanics using the method of dimensional regularization. The basis of this method is to express the balance laws for fractal media in terms of fractional integrals and, then, convert them to integer-order integrals in conventional (Euclidean) space. Following an account of this method, we develop balance laws of fractal media (continuity, linear and angular momenta, energy, and second law) and discuss wave equations in several settings (1d and 3d wave motions, fractal Timoshenko beam, and elastodynamics under finite strains). We then discuss extremum and variational principles, fracture mechanics, and equations of turbulent flow in fractal media. In all the cases, the derived equations for fractal media depend explicitly on fractal dimensions and reduce to conventional forms for continuous media with Euclidean geometries upon setting the dimensions to integers. We also point out relations and potential extensions of dimensional regularization to other models of microscopically heterogeneous physical systems.

In this review, recent advances on the measurement and modeling of elastic properties of cortical and trabecular bone are presented. Bone is a multifunctional material which among its other functions serves as a support for other tissues in the body. As a structural material it is stiff, strong, tough, lightweight and is adaptable. Its excellent mechanical properties are due to its complex, composite and hierarchical structure. In this paper, we outline the experimental approaches that have been used to characterize bone’s structure, composition and elastic properties at several different length scales. Then, we discuss different modeling approaches that have been employed to compute bone’s elastic moduli. We conclude by discussing the challenges and open issues in this area. Analysis of bone is of importance in orthopedics. Also, gained knowledge on bone can be used by engineers to design new bioinspired composite materials for a wide range of engineering applications

FOXA2, a member of the forkhead family of transcription factors, plays essential roles in liver development and bile acid homeostasis. In this study, we report a 2.8 Å co-crystal structure of the FOXA2 DNA-binding domain (FOXA2-DBD) bound to a DNA duplex containing a forkhead consensus binding site (GTAAACA). FOXA2-DBD adopts the canonical winged-helix fold, with helix H3 and wing 1 regions mainly mediating the DNA recognition. Although the wing 2 region was not defined in the structure, isothermal titration calorimetry (ITC) assays suggested that this region was required for optimal DNA binding. Structure comparison with the FOXA3-DBD bound to DNA revealed more major groove contacts and less minor groove contacts in the FOXA2 structure compared to the FOXA3 structure. Structure comparison with the FOXO1-DBD bound to DNA showed that different forkhead proteins could induce different DNA conformations upon binding to identical DNA sequences. Our findings provide the structural basis for FOXA2 protein binding to a consensus forkhead site and elucidate how members of the forkhead protein family bind different DNA sites.

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Aberrant FGFR4 signaling has been documented abundantly in various human cancers. The majority of FGFR inhibitors display significantly reduced potency toward FGFR4 compared to FGFR1-3. However, LY2874455 has similar inhibition potency for FGFR1-4 with IC50 less than 6.4 nM. To date, there is no published crystal structure of LY2874455 in complex with any kinase. To better understand the pan-FGFR selectivity of LY2874455, we have determined the crystal structure of the FGFR4 kinase domain bound to LY2874455 at a resolution of 2.35 Å. LY2874455, a type I inhibitor for FGFR4, binds to the ATP-binding pocket of FGFR4 in a DFG-in active conformation with three hydrogen bonds and a number of van der Waals contacts. After alignment of the kinase domain sequence of 4 FGFRs, and superposition of the ATP binding pocket of 4 FGFRs, our structural analyses reveal that the interactions of LY2874455 to FGFR4 are largely conserved in 4 FGFRs, explaining at least partly, the broad inhibitory activity of LY2874455 toward 4 FGFRs. Consequently, our studies reveal new insights into the pan-FGFR selectivity of LY2874455 and provide a structural basis for developing novel FGFR inhibitors that target FGFR1-4 broadly.