Padé approximants on Riemann surfaces and KP tau functions

The paper has two relatively distinct but connected goals; the first is to define the notion of Padé approximation of Weyl–Stiltjes transforms on an arbitrary compact Riemann surface of higher genus. The data consists of a contour in the Riemann surface and a measure on it, together with the additional datum of a local coordinate near a point and a divisor of degree g. The denominators of the resulting Padé-like approximation also satisfy an orthogonality relation and are sections of appropriate line bundles. A Riemann–Hilbert problem for a square matrix of rank two is shown to characterize these orthogonal sections, in a similar fashion to the ordinary orthogonal polynomial case. The second part extends this idea to explore its connection to integrable systems. The same data can be used to define a pairing between two sequences of line bundles. The locus in the deformation space where the pairing becomes degenerate for fixed degree coincides with the zeros of a “tau” function. We show how this tau function satisfies the Kadomtsev–Petviashvili hierarchy with respect to either deformation parameters, and a certain modification of the 2-Toda hierarchy when considering the whole sequence of tau functions. We also show how this construction is related to the Krichever construction of algebro-geometric solutions.

Algebro-geometric integration of a modified shallow wave hierarchy

By introducing two sets of Lenard recursion relations, we derive a hierarchy of modified shallow wave equations associated with a 3 × 3 matrix spectral problem with three potentials from the zero-curvature equation. The Baker–Akhiezer function and two meromorphic functions are defined on the trigonal curve which is introduced by utilizing the characteristic polynomial of the Lax matrix. Analyzing the asymptotic properties of the Baker–Akhiezer function and two meromorphic functions at two infinite points, we arrive at the explicit algebro-geometric solutions for the entire hierarchy in terms of the Riemann theta function by showing the explicit forms of the normalized Abelian differentials of the third kind.

Algebro-Geometric Solutions of the Coupled Chaffee-Infante Reaction Diffusion Hierarchy

The coupled Chaffee-Infante reaction diffusion (CCIRD) hierarchy associated with a 3 × 3 matrix spectral problem is derived by using two sets of the Lenard recursion gradients. Based on the characteristic polynomial of the Lax matrix for the CCIRD hierarchy, we introduce a trigonal curve K m − 2 of arithmetic genus m − 2 , from which the corresponding Baker-Akhiezer function and meromorphic functions on K m − 2 are constructed. Then, the CCIRD equations are decomposed into Dubrovin-type ordinary differential equations. Furthermore, the theory of the trigonal curve and the properties of the three kinds of Abel differentials are applied to obtain the explicit theta function representations of the Baker-Akhiezer function and the meromorphic functions. In particular, algebro-geometric solutions for the entire CCIRD hierarchy are obtained.

Generation of ordered protein assemblies using rigid three-body fusion

Protein nanomaterial design is an emerging discipline with applications in medicine and beyond. A long-standing design approach uses genetic fusion to join protein homo-oligomer subunits via α-helical linkers to form more complex symmetric assemblies, but this method is hampered by linker flexibility and a dearth of geometric solutions. Here, we describe a general computational method for rigidly fusing homo-oligomer and spacer building blocks to generate user-defined architectures that generates far more geometric solutions than previous approaches. The fusion junctions are then optimized using Rosetta to minimize flexibility. We apply this method to design and test 92 dihedral symmetric protein assemblies using a set of designed homodimers and repeat protein building blocks. Experimental validation by native mass spectrometry, small-angle X-ray scattering, and negative-stain single-particle electron microscopy confirms the assembly states for 11 designs. Most of these assemblies are constructed from designed ankyrin repeat proteins (DARPins), held in place on one end by α-helical fusion and on the other by a designed homodimer interface, and we explored their use for cryogenic electron microscopy (cryo-EM) structure determination by incorporating DARPin variants selected to bind targets of interest. Although the target resolution was limited by preferred orientation effects and small scaffold size, we found that the dual anchoring strategy reduced the flexibility of the target-DARPIN complex with respect to the overall assembly, suggesting that multipoint anchoring of binding domains could contribute to cryo-EM structure determination of small proteins.

Generation of ordered protein assemblies using rigid three-body fusion

Protein nanomaterial design is an emerging discipline with applications in medicine and beyond. A longstanding design approach uses genetic fusion to join protein homo-oligomer subunits via α-helical linkers to form more complex symmetric assemblies, but this method is hampered by linker flexibility and a dearth of geometric solutions. Here, we describe a general computational method that performs rigid three-body fusion of homo-oligomer and spacer building blocks to generate user-defined architectures, while at the same time significantly increasing the number of geometric solutions over typical symmetric fusion. The fusion junctions are then optimized using Rosetta to minimize flexibility. We apply this method to design and test 92 dihedral symmetric protein assemblies from a set of designed homo-dimers and repeat protein building blocks. Experimental validation by native mass spectrometry, small angle X-ray scattering, and negative-stain single-particle electron microscopy confirms the assembly states for 11 designs. Most of these assemblies are constructed from DARPins (designed ankyrin repeat proteins), anchored on one end by α-helical fusion and on the other by a designed homo-dimer interface, and we explored their use for cryo-EM structure determination by incorporating DARPin variants selected to bind targets of interest. Although the target resolution was limited by preferred orientation effects, small scaffold size, and the low-order symmetry of these dihedral scaffolds, we found that the dual anchoring strategy reduced the flexibility of the target-DARPIN complex with respect to the overall assembly, suggesting that multipoint anchoring of binding domains could contribute to cryo-EM structure determination of small proteins.

Thermal Stress in Concrete Slab of the Airfield Pavement

Stress occurring in concrete pavement slab changes on daily and annual basis. Daily changes of air temperature in between environments depend on the following: latitude where the structure is located and season. Change of air temperature causes the variable thermal stress of a slab and consequently change of its linear dimensions (slab extension or shortening. “Thermal balance of pavement” prepared for individual structure solutions, should be the basis for construction design and geometric solutions concerning air field structure. This publication refers only to natural changes of temperature conditions of the environment. Other phenomena occurring on concrete airfield pavement under the influence of imposed thermal loads will be the subject of another publication in this regard.