The Fundamental Plane of Open Clusters

We utilize the data from the Apache Point Observatory Galactic Evolution Experiment-2 (APOGEE-2) in the fourteenth data release of the Sloan Digital Sky Survey (SDSS) to calculate the line-of-sight velocity dispersion $\sigma_{1D}$ of a sample of old open clusters (age larger than 100\,Myr) selected from the Milky Way open cluster catalog of Kharchenko et al. (2013). Together with their $K_s$ band luminosity $L_{K_s}$, and the half-light radius $r_{h}$ of the most probable members, we find that these three parameters show significant pairwise correlations among each other. Moreover, a fundamental plane-{\it like} relation among these parameters is found for the oldest open clusters (age older than 1\,Gyr), $L_{K_s}\propto\sigma_{1D}^{0.82\pm0.29}\cdot r_h^{2.19\pm0.52}$ with $rms \sim\, 0.31$\,mag in the $K_s$ band absolute magnitude. The existence of this relation, which deviates significantly from the virial theorem prediction, implies that the dynamical structures of the old open clusters are quite similar, when survived from complex dynamical evolution to age older than 1 Gyr.Comment: accepted publication for ApJ lette

Counterexamples in Scale Calculus

We construct counterexamples to classical calculus facts such as the Inverse and Implicit Function Theorems in Scale Calculus -- a generalization of Multivariable Calculus to infinite dimensional vector spaces in which the reparameterization maps relevant to Symplectic Geometry are smooth. Scale Calculus is a cornerstone of Polyfold Theory, which was introduced by Hofer-Wysocki-Zehnder as a broadly applicable tool for regularizing moduli spaces of pseudoholomorphic curves. We show how the novel nonlinear scale-Fredholm notion in Polyfold Theory overcomes the lack of Implicit Function Theorems, by formally establishing an often implicitly used fact: The differentials of basic germs -- the local models for scale-Fredholm maps -- vary continuously in the space of bounded operators when the base point changes. We moreover demonstrate that this continuity holds only in specific coordinates, by constructing an example of a scale-diffeomorphism and scale-Fredholm map with discontinuous differentials. This justifies the high technical complexity in the foundations of Polyfold Theory.Comment: published in PNAS, final versio

An Apparent Redshift Dependence of Quasar Continuum: Implication for Cosmic Dust Extinction?

We investigate the luminosity and redshift dependence of the quasar continuum by means of composite spectrum using a large non-BAL radio-quiet quasar sample drawn from the Sloan Digital Sky Survey. Quasar continuum slopes in the UV-Opt band are measured at two different wavelength ranges, i.e., $\alpha_{\nu12}$ ($1000\sim 2000 \rm\AA$) and $\alpha_{\nu24}$ ($2000 \sim 4000 \rm\AA$) derived from power law fitting. Generally, the UV spectra slope becomes harder (higher $\alpha_{\nu}$) towards higher bolometric luminosity. On the other hand, when quasars are further grouped into luminosity bins, we find both $\alpha_{\nu12}$ and $\alpha_{\nu24}$ show significant anti-correlation with redshift (i.e., quasar continuum becomes redder towards higher redshift). We suggest that the cosmic dust extinction is very likely the cause of this observed $\alpha_\nu-z$ relation. We build a simple cosmic dust extinction model to quantify the observed reddening tendency and find an effective dust density $n\sigma_v \sim 10^{-5}h~\rm Mpc^{-1}$ at $z<1.5$. The other possibilities that could produce such a reddening effect have also been discussed.Comment: 6 pages, 5 figures; published in ApJ

Diversity of Receptor Tyrosine Kinase Signaling Mechanisms

Receptor tyrosine kinases (RTKs) are a family of 58 transmembrane proteins in humans that play crucial roles in many biological processes and diseases. Different RTKs utilize subtly (but importantly) distinct molecular mechanisms for transmembrane signaling, and understanding these differences is crucial for devising new ways to intervene pharmacologically when aberrant RTK signaling causes cancer and other diseases. In this thesis, I focus on three RTK families: the ALK/LTK family, the Wnt-binding RTKs, and the EGF receptor – where I concentrate on efforts to understand its C-terminal regulatory region. My studies of ALK, for anaplastic lymphoma kinase, were motivated by the fact that this RTK sub-group has a unique domain architecture in its extracellular region. Little is known about the mechanisms of ligand binding to – and activation of – ALK, and the nature of its ligand(s) is(are) still not completely clear. Using biochemical, biophysical and structural biology approaches, I characterized the low-resolution structure of the ALK extracellular region. I further identified the binding mode of ALK binding to heparin, a recently discovered modulatory ligand for ALK. Based on a low-resolution structural analysis of ALK/heparin complex, I propose a model for ligand-induced ALK dimerization and activation. Ryk is one of the five RTKs that are now known to be Wnt receptors. In this thesis, I studied the Drosophila homolog of Ryk, Derailed (Drl), and its binding to ligand DWnt5. We were able to express and purify milligrams of active DWnt5 – thus overcoming a major obstacle in this field. We further characterized Drl/DWnt5 interactions. Using hydrogen/deuterium exchange approaches, I identified the DWnt5-binding interface on Drl. My efforts to understand the molecular mechanisms of Drl/DWnt5 binding using experimental and computational approaches suggest that DWnt5 may interact with Drl through a binding mode that differs from Wnt binding to other receptors. Across the RTK family, many receptors contain a long carboxy-terminal tail (C-tail) that harbors autophosphorylation sites for docking of downstream signaling molecules. This region is generally considered to be intrinsically disordered. I studied the dynamics of the EGFR C-tail, and showed that it is highly unstructured – but contains some somewhat ‘structured’ regions. I also showed that phosphorylation of the EGFR C-tail promotes receptor dimerization. Using hydrogen exchange, I identified possible C-tail docking sites on the kinase domain that may be responsible for this effect. I also studied binding of downstream SH2 domain-containing molecules to the EGFR C-tail, with results that indicate that not all features of SH2 domain binding to the C-tail can be recapitulated by simple phosphopeptides; binding of SH2 domains to the C-tail exhibits binding affinities and stoichiometries that are not captured by simple peptide-level studies. Moreover, my binding competition assays suggest that there may be cooperativity in binding of multiple SH2 domains to a single phosphorylated C-tail