Liquid-liquid phase separation and static light scattering of concentrated ternary mixtures of bovine alpha and gammaB crystallins

We have used light scattering, turbidimetry, and thermodynamic analysis to study the phase diagram of concentrated aqueous mixtures of the bovine lens proteins, gammaB crystallin, and alpha crystallin. We find that dilute alpha crystallin raises the phase separation temperature of concentrated gammaB crystallin, while more concentrated alpha crystallin suppresses phase separation. Very concentrated alpha/gammaB mixtures can reversibly cloud above 37 degrees C, even though gammaB alone phase separates only below temperatures near 0 degrees C, and alpha does not phase separate. At the scattering vector magnitude used, high-concentration alpha/gammaB mixtures scatter less light than the weighted average of their component alpha and gammaB solutions, while low-concentration alpha/gammaB mixtures scatter more than such a weighted average. We use a mean-field thermodynamic analysis of such ternary mixtures to show that the observed light scattering and phase boundaries of alpha and gammaB crystallin mixtures give evidence for prominent local fluctuations of relative protein composition. In the single phase, these fluctuations scatter comparatively little light, but are associated with enhanced thermodynamic instability. By applying this analysis to the experimental tie lines we estimate the magnitude of the saddlelike component of the free energy near the aqueous-gammaB critical point.     

Mathematical and computational aspects of quaternary liquid mixing free energy measurement using light scattering

We provide a mathematical and computational analysis of light scattering measurement of mixing free energies of quaternary isotropic liquids. In previous work, we analyzed mathematical and experimental design considerations for the ternary mixture case [D. Ross, G. Thurston, and C. Lutzer, J. Chem. Phys. 129, 064106 (2008); C. Wahle, D. Ross, and G. Thurston, J. Chem. Phys. 137, 034201 (2012)]. Here, we review and introduce dimension-free general formulations of the fully nonlinear partial differential equation (PDE) and its linearization, a basis for applying the method to composition spaces of any dimension, in principle. With numerical analysis of the PDE as applied to the light scattering implied by a test free energy and dielectric gradient combination, we show that values of the Rayleigh ratio within the quaternary composition tetrahedron can be used to correctly reconstruct the composition dependence of the free energy. We then extend the analysis to the case of a finite number of data points, measured with noise. In this context the linearized PDE describes the relevant diffusion of information from light scattering noise to the free energy. The fully nonlinear PDE creates a special set of curves in the composition tetrahedron, collections of which form characteristics of the nonlinear and linear PDEs, and we show that the information diffusion has a time-like direction along the positive normals to these curves. With use of Monte Carlo simulations of light scattering experiments, we find that for a modest laboratory light scattering setup, about 100-200 samples and 100 s of measurement time are enough to be able to measure the mixing free energy over the entire quaternary composition tetrahedron, to within an L(2) error norm of 10(-3). The present method can help quantify thermodynamics of quaternary isotropic liquid mixtures.     

The Size of Spanning Disks for Polygonal Curves

Abstract. For each integer n ≥ 0 , there is a closed, unknotted, polygonal curve K _n in R ³ having less than 10n+9 edges, with the property that any piecewise-linear triangulated disk spanning the curve contains at least 2 ^n-1 triangles.     

The computational complexity of knot genus and spanning area

We show that the problem of deciding whether a polygonal knot in a closed three-dimensional manifold bounds a surface of genus at most is NP-complete. We also show that the problem of deciding whether a curve in a PL manifold bounds a surface of area less than a given constant is NP-hard.

     

Highly efficient methods for the one-pot synthesis of beta-substituted enones

A mild and practically-convenient one-pot procedure for the direct beta-substitution of enones has been developed using a conjugate addition-oxidation strategy with a full range of copper-based reagents and N-tert-butylphenylsulfinimidoyl chloride; alkyl- and aryl-substituted enones are delivered in good to excellent yields.