Detailed Balance Condition and Effective Free Energy in the Primitive Chain Network Model

We consider statistical mechanical properties of the primitive chain network (PCN) model for entangled polymers from its dynamic equations. We show that the dynamic equation for the segment number of the PCN model does not reduce to the standard Langevin equation which satisfies the detailed balance condition. We propose heuristic modifications for the PCN dynamic equation for the segment number, to make it reduce to the standard Langevin equation. We analyse some equilibrium statistical properties of the modified PCN model, by using the effective free energy obtained from the modified PCN dynamic equations. The PCN effective free energy can be interpreted as the sum of the ideal Gaussian chain free energy and the repulsive interaction energy between slip-links. By using the single chain approximation, we calculate several distribution functions of the PCN model. The obtained distribution functions are qualitatively different from ones for the simple slip-link model without any direct interactions between slip-links.Comment: 29 pages, 7 figures, to appeare in J. Chem. Phy

Plateau Moduli of Several Single-Chain Slip-Link and Slip-Spring Models

We calculate the plateau moduli of several single-chain slip-link and slip-spring models for entangled polymers. In these models, the entanglement effects are phenomenologically modeled by introducing topological constraints such as slip-links and slip-springs. The average number of segments between two neighboring slip-links or slip-springs, $N_{0}$, is an input parameter in these models. To analyze experimental data, the characteristic number of segments in entangled polymers $N_{e}$ estimated from the plateau modulus is used instead. Both $N_{0}$ and $N_{e}$ characterize the topological constraints in entangled polymers, and naively $N_{0}$ is considered to be the same as $N_{e}$. However, earlier studies showed that $N_{0}$ and $N_{e}$ (or the plateau modulus) should be considered as independent parameters. In this work, we show that due to the fluctuations at the short time scale, $N_{e}$ deviates from $N_{0}$. This means that the relation between $N_{0}$ and the plateau modulus is not simple as naively expected. The plateau modulus (or $N_{e}$) depends on the subchain-scale details of the employed model, as well as the average number of segments $N_{0}$. This is due to the fact that the subchain-scale fluctuation mechanisms depend on the model rather strongly. We theoretically calculate the plateau moduli for several single-chain slip-link and slip-spring models. Our results explicitly show that the relation between $N_{0}$ and $N_{e}$ is model-dependent. We compare theoretical results with various simulation data in the literature, and show that our theoretical expressions reasonably explain the simulation results.Comment: 26 pages, 3 figures, to appear in Macromolecule

FGF23 and Hypophosphatemic Rickets/Osteomalacia

Purpose of review X-linked hypophosphatemia and tumor-induced osteomalacia are diseases characterized by hypophosphatemia with impaired proximal tubular phosphate reabsorption. Complete resection of responsible tumors is the first line therapy for patients with tumor-induced osteomalacia. In contrast, phosphate and active vitamin D have been used for patients with X-linked hypophosphatemia and inoperable ones with tumor-induced osteomalacia. The purpose of this review is to summarize the pathogenesis of these diseases and discuss about the new treatment. Recent findings Excessive FGF23 production has been shown to underline several kinds of hypophosphatemic rickets/osteomalacia including X-linked hypophosphatemia and tumor-induced osteomalacia. Burosumab, an anti-FGF23 monoclonal antibody, was approved for clinical use while the indications of burosumab are different depending on countries. Summary The inhibition of excessive FGF23 activity has been approved as a new therapy for several kinds of hypophosphatemic diseases. Further studies are necessary to clarify the long-term effects and safety of burosumab