Tissue engineering of certain load-bearing parts of the body can be dependent on scaffold adhesion or integration with the surrounding tissue to prevent dislocation. One such area is the regeneration of the intervertebral disc (IVD). In this work, poly(N-isopropylacrylamide) (PNIPAAm) was grafted with chondroitin sulfate (CS) (PNIPAAm-g-CS) and blended with aldehyde-modified CS to generate an injectable polymer that can form covalent bonds with tissue upon contact. However, the presence of the reactive aldehyde groups can compromise the viability of encapsulated cells. Thus, liposomes were encapsulated in the blend, designed to deliver the ECM derivative, gelatin, after the polymer has adhered to tissue and reached physiological temperature. This work is based on the hypothesis that the discharge of gelatin will enhance the biocompatibility of the material by covalently reacting with, or “end-capping”, the aldehyde functionalities within the gel that did not participate in bonding with tissue upon contact. As a comparison, formulations were also created without CS aldehyde and with an alternative adhesion mediator, mucoadhesive calcium alginate particles. Gels formed from blends of PNIPAAm-g-CS and CS aldehyde exhibited increased adhesive strength compared to PNIPAAm-g-CS alone (p<0.05). However, the addition of gelatin-loaded liposomes to the blend significantly decreased the adhesive strength (p<0.05). The encapsulation of alginate microparticles within PNIPAAm-g-CS gels caused the tensile strength to increase two-fold over that of PNIPAAm-g-CS blends with CS aldehyde (p<0.05). Cytocompatibility studies indicate that formulations containing alginate particles exhibit reduced cytotoxicity over those containing CS aldehyde. Overall, the results indicated that the adhesives composed of alginate microparticles encapsulated in PNIPAAm-g-CS have the potential to serve as a scaffold for IVD regeneration.

We study level statistics in ensembles of integrable $N\times N$ matrices linear in a real parameter $x$. The matrix $H(x)$ is considered integrable if it has a prescribed number $n>1$ of linearly independent commuting partners $H^i(x)$ (integrals of motion) $\left[H(x),H^i(x)\right] = 0$, $\left[H^i(x), H^j(x)\right]$ = 0, for all $x$. In a recent work, we developed a basis-independent construction of $H(x)$ for any $n$ from which we derived the probability density function, thereby determining how to choose a typical integrable matrix from the ensemble. Here, we find that typical integrable matrices have Poisson statistics in the $N\to\infty$ limit provided $n$ scales at least as $\log{N}$; otherwise, they exhibit level repulsion. Exceptions to the Poisson case occur at isolated coupling values $x=x_0$ or when correlations are introduced between typically independent matrix parameters. However, level statistics cross over to Poisson at $ \mathcal{O}(N^{-0.5})$ deviations from these exceptions, indicating that non-Poissonian statistics characterize only subsets of measure zero in the parameter space. Furthermore, we present strong numerical evidence that ensembles of integrable matrices are stationary and ergodic with respect to nearest neighbor level statistics.Comment: 18 pages, 26 figures, discussion on number variance added; published versio

We consider energy transport in the classical Toda chain in the presence of an additional pinning potential. The pinning potential is expected to destroy the integrability of the system and an interesting question is to see the signatures of this breaking of integrability on energy transport. We investigate this by a study of the non-equilibrium steady state of the system connected to heat baths as well as the study of equilibrium correlations. Typical signatures of integrable systems are a size-independent energy current, a flat bulk temperature profile and ballistic scaling of equilibrium dynamical correlations, these results being valid in the thermodynamic limit. We find that, as expected, these properties change drastically on introducing the pinning potential in the Toda model. In particular, we find that the effect of a harmonic pinning potential is drastically smaller at low temperatures, compared to a quartic pinning potential. We explain this by noting that at low temperatures the Toda potential can be approximated by a harmonic inter-particle potential for which the addition of harmonic pinning does not destroy integrability.

We construct ensembles of random integrable matrices with any prescribed number of nontrivial integrals and formulate integrable matrix theory (IMT) -a counterpart of random matrix theory (RMT) for quantum integrable models. A type-M family of integrable matrices consists of exactly N − M independent commuting N × N matrices linear in a real parameter. We first develop a rotationally invariant parameterization of such matrices, previously only constructed in a preferred basis. For example, an arbitrary choice of a vector and two commuting Hermitian matrices defines a type-1 family and vice versa. Higher types similarly involve a random vector and two matrices. The basis-independent formulation allows us to derive the joint probability density for integrable matrices, similar to the construction of Gaussian ensembles in the RMT.

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