On Quadrirational Yang-Baxter Maps

We use the classification of the quadrirational maps given by Adler, Bobenko and Suris to describe when such maps satisfy the Yang-Baxter relation. We show that the corresponding maps can be characterized by certain singularity invariance condition. This leads to some new families of Yang-Baxter maps corresponding to the geometric symmetries of pencils of quadrics.Comment: Proceedings of the workshop "Geometric Aspects of Discrete and Ultra-Discrete Integrable Systems" (Glasgow, March-April 2009

Boussinesq-like multi-component lattice equations and multi-dimensional consistency

We consider quasilinear, multi-variable, constant coefficient, lattice equations defined on the edges of the elementary square of the lattice, modeled after the lattice modified Boussinesq (lmBSQ) equation, e.g., $\tilde y z=\tilde x-x$. These equations are classified into three canonical forms and the consequences of their multidimensional consistency (Consistency-Around-the-Cube, CAC) are derived. One of the consequences is a restriction on form of the equation for the $z$ variable, which in turn implies further consistency conditions, that are solved. As result we obtain a number of integrable multi-component lattice equations, some generalizing lmBSQ.Comment: 24 page

Infinitely many symmetries and conservation laws for quad-graph equations via the Gardner method

The application of the Gardner method for generation of conservation laws to all the ABS equations is considered. It is shown that all the necessary information for the application of the Gardner method, namely B\"acklund transformations and initial conservation laws, follow from the multidimensional consistency of ABS equations. We also apply the Gardner method to an asymmetric equation which is not included in the ABS classification. An analog of the Gardner method for generation of symmetries is developed and applied to discrete KdV. It can also be applied to all the other ABS equations

Microfluidic Fabrication of Microcarriers with Sequential Delivery of VEGF And BMP-2 For Bone Regeneration

Wound instability and poor functional vascularization in bone tissue engineering lead to lack of tissue integration and ultimate failure of engineered grafts. In order to harness the regenerative potential of growth factors and stimulate bone healing, present study aims to design multifunctional cell therapy microcarriers with the capability of sequential delivery of essential growth factors, bone morphogenetic protein 2 (BMP-2) and vascular endothelial growth factor (VEGF). An on-chip double emulsion method was implemented to generate monodisperse VEGF encapsulated microcarriers. Bio-inspired poly(3,4-dihydroxyphenethylamine) (PDA) was then functionalized to the microcarriers surface for BMP-2 conjugation. The microcarriers were seeded with mesenchymal stem cells (MSCs) using a dynamic culture technique for cells expansion. Finally, the microcarriers were incorporated into an injectable alginate-RGD hydrogel laden with endothelial cells (ECs) for further analysis. The DNA and calcium content, as well as ALP activity of the construct were analyzed. The confocal fluorescent microscopy was employed to monitor the MSCs and tunneling structure of ECs. Eventually, the capability of developed microcarriers for bone tissue formation was examined in vivo. Microfluidic platform generated monodisperse VEGF-loaded PLGA microcarriers with size-dependent release patterns. Microcarriers generated with the on-chip technique showed more sustained VEGF release profiles compared to the conventional bulk mixing method. The PDA functionalization of microcarriers surface not only provided immobilization of BMP-2 with prolonged bioavailability, but also enhanced the attachment and proliferation of MSCs. Dynamic culturing of microcarriers showcased their great potential to boost MSCs population required for stem cell therapy of bone defects. ALP activity and calcium content analysis of MSCs-laden microcarriers loaded into injectable hydrogels revealed their capability of tunneling formation, vascular cell growth and osteogenic differentiation. The in vivo histology and real-time polymerase chain reaction analysis revealed that transplantation of MSC-laden microcarriers supports ectopic bone formation in the rat model. The presented approach to design bioactive microcarriers offer sustained sequential delivery of bone ECM chemical cues and offer an ideal stabilized 3D microenvironment for patient-specific cell therapy applications. The proposed methodology is readily expandable to integrate other cells and cytokines in a tuned spatiotemporal manner for personalized regenerative medicine

Yang-Baxter maps and symmetries of integrable equations on quad-graphs

A connection between the Yang-Baxter relation for maps and the multi-dimensional consistency property of integrable equations on quad-graphs is investigated. The approach is based on the symmetry analysis of the corresponding equations. It is shown that the Yang-Baxter variables can be chosen as invariants of the multi-parameter symmetry groups of the equations. We use the classification results by Adler, Bobenko and Suris to demonstrate this method. Some new examples of Yang-Baxter maps are derived in this way from multi-field integrable equations.Comment: 20 pages, 5 figure