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In this paper, we first study a mapping problem between indefinite hyperbolic spaces by employing the work established earlier by the authors. In particular, we generalize certain theorems proved by Baouendi-Ebenfelt-Huang [Amer. J. Math. 133 (2011), pp. 1633–1661] and Ng [Michigan Math. J. 62 (2013), pp. 769–777; Int. Math. Res. Not. IMRN 2 (2015), pp. 291–324]. Then we use these results to prove a rigidity result for proper holomorphic mappings between type I classical domains, which confirms a conjecture formulated by Chan [Int. Math. Res. Not., doi.org/10.1093/imrn/rnaa373] after the work of Zaitsev-Kim [Math. Ann. 362 (2015), pp. 639-677], Kim [ Proper holomorphic maps between bounded symmetric domains , Springer, Tokyo, 2015, pp. 207–219] and himself. 

Interactions between biomolecules are characterized by where they occur and how they are organized, e.g., the alignment of lipid molecules to form a membrane. However, spatial and angular information are mixed within the image of a fluorescent molecule–the microscope’s dipole-spread function (DSF). We demonstrate the pixOL algorithm to simultaneously optimize all pixels within a phase mask to produce an engineered Green’s tensor–the dipole extension of point-spread function engineering. The pixOL DSF achieves optimal precision to simultaneously measure the 3D orientation and 3D location of a single molecule, i.e., 4.1° orientation, 0.44 sr wobble angle, 23.2 nm lateral localization, and 19.5 nm axial localization precisions in simulations over a 700 nm depth range using 2500 detected photons. The pixOL microscope accurately and precisely resolves the 3D positions and 3D orientations of Nile red within a spherical supported lipid bilayer, resolving both membrane defects and differences in cholesterol concentration in six dimensions. 

Abstract Proteins gain optimal fitness such as foldability and function through evolutionary selection. However, classical studies have found that evolutionarily designed protein sequences alone cannot guarantee foldability, or at least not without considering local contacts associated with the initial folding steps. We previously showed that foldability and function can be restored by removing frustration in the folding energy landscape of a model WW domain protein, CC16, which was designed based on Statistical Coupling Analysis (SCA). Substitutions ensuring the formation of five local contacts identified as “on‐path” were selected using the closest homolog native folded sequence, N21. Surprisingly, the resulting sequence, CC16‐N21, bound to Group I peptides, while N21 did not. Here, we identified single‐point mutations that enable N21 to bind a Group I peptide ligand through structure and dynamic‐based computational design. Comparison of the docked position of the CC16‐N21/ligand complex with the N21 structure showed that residues at positions 9 and 19 are important for peptide binding, whereas the dynamic profiles identified position 10 as allosterically coupled to the binding site and exhibiting different dynamics between N21 and CC16‐N21. We found that swapping these positions in N21 with matched residues from CC16‐N21 recovers nature‐like binding affinity to N21. This study validates the use of dynamic profiles as guiding principles for affecting the binding affinity of small proteins.