Smooth Banach spaces and approximations

If E and F are real Banach spaces let C^p,q(E, F) O ≤ q ≤ p ≤ ∞, denote those maps from E to F which have p continuous Frechet derivatives of which the first q derivatives are bounded. A Banach space E is defined to be C^p,q smooth if C^p,q(E,R) contains a nonzero function with bounded support. This generalizes the standard C^p smoothness classification.

If an L^p space, p ≥ 1, is C^q smooth then it is also C^q,q smooth so that in particular L^p for p an even integer is C^∞,∞ smooth and L^p for p an odd integer is C^p-1,p-1 smooth. In general, however, a C^p smooth B-space need not be C^p,p smooth. C_o is shown to be a non-C^2,2 smooth B-space although it is known to be C^∞ smooth. It is proved that if E is C^p,1 smooth then C_o(E) is C^p,1 smooth and if E has an equivalent C^p norm then c_o(E) has an equivalent C^p norm.

Various consequences of C^p,q smoothness are studied. If f ϵ C^p,q(E,F), if F is C^p,q smooth and if E is non-C^p,q smooth, then the image under f of the boundary of any bounded open subset U of E is dense in the image of U. If E is separable then E is C^p,q smooth if and only if E admits C^p,q partitions of unity; E is C^p,psmooth, p ˂∞, if and only if every closed subset of E is the zero set of some C^P function.

f ϵ C^q(E,F), 0 ≤ q ≤ p ≤ ∞, is said to be C_p,q approximable on a subset U of E if for any ϵ ˃ 0 there exists a g ϵ C^p(E,F) satisfying

sup/xϵU, O≤k≤q ‖ D^k f(x) - D^k g(x) ‖ ≤ ϵ.

It is shown that if E is separable and C^p,q smooth and if f ϵ C^q(E,F) is C_p,q approximable on some neighborhood of every point of E, then F is C_p,q approximable on all of E.

In general it is unknown whether an arbitrary function in C¹(l², R) is C_2,1 approximable and an example of a function in C¹(l², R) which may not be C_2,1 approximable is given. A weak form of C_∞,q, q≥1, to functions in C^q(l², R) is proved: Let {U_α} be a locally finite cover of l² and let {T_α} be a corresponding collection of Hilbert-Schmidt operators on l². Then for any f ϵ C^q(l²,F) such that for all α

sup ‖ D^k(f(x)-g(x))[T_αh]‖ ≤ 1.

xϵU_α,‖h‖≤1, 0≤k≤q

Sequence specific complexation of b DNA at sites containing g,c base pairs

A series of eight related analogs of distamycin A has been synthesized. Footprinting and affinity cleaving reveal that only two of the analogs, pyridine-2- car box amide-netropsin (2-Py N) and 1-methylimidazole-2-carboxamide-netrops in (2-ImN), bind to DNA with a specificity different from that of the parent compound. A new class of sites, represented by a TGACT sequence, is a strong site for 2-PyN binding, and the major recognition site for 2-ImN on DNA. Both compounds recognize the G•C bp specifically, although A's and T's in the site may be interchanged without penalty. Additional A•T bp outside the binding site increase the binding affinity. The compounds bind in the minor groove of the DNA sequence, but protect both grooves from dimethylsulfate. The binding evidence suggests that 2-PyN or 2-ImN binding induces a DNA conformational change.

In order to understand this sequence specific complexation better, the Ackers quantitative footprinting method for measuring individual site affinity constants has been extended to small molecules. MPE•Fe(II) cleavage reactions over a 10^5 range of free ligand concentrations are analyzed by gel electrophoresis. The decrease in cleavage is calculated by densitometry of a gel autoradiogram. The apparent fraction of DNA bound is then calculated from the amount of cleavage protection. The data is fitted to a theoretical curve using non-linear least squares techniques. Affinity constants at four individual sites are determined simultaneously. The distamycin A analog binds solely at A•T rich sites. Affinities range from 10^(6)- 10^(7)M^(-1) The data for parent compound D fit closely to a monomeric binding curve. 2-PyN binds both A•T sites and the TGTCA site with an apparent affinity constant of 10^(5) M^(-1). 2-ImN binds A•T sites with affinities less than 5 x 10^(4) M^(-1). The affinity of 2-ImN for the TGTCA site does not change significantly from the 2-PyN value. At the TGTCA site, the experimental data fit a dimeric binding curve better than a monomeric curve. Both 2-PyN and 2-ImN have substantially lower DNA affinities than closely related compounds.

In order to probe the requirements of this new binding site, fourteen other derivatives have been synthesized and tested. All compounds that recognize the TGTCA site have a heterocyclic aromatic nitrogen ortho to the N or C-terminal amide of the netropsin subunit. Specificity is strongly affected by the overall length of the small molecule. Only compounds that consist of at least three aromatic rings linked by amides exhibit TGTCA site binding. Specificity is only weakly altered by substitution on the pyridine ring, which correlates best with steric factors. A model is proposed for TGTCA site binding that has as its key feature hydrogen bonding to both G's by the small molecule. The specificity is determined by the sequence dependence of the distance between G's.

One derivative of 2-PyN exhibits pH dependent sequence specificity. At low pH, 4-dimethylaminopyridine-2-carboxamide-netropsin binds tightly to A•T sites. At high pH, 4-Me_(2)NPyN binds most tightly to the TGTCA site. In aqueous solution, this compound protonates at the pyridine nitrogen at pH 6. Thus presence of the protonated form correlates with A•T specificity.

The binding site of a class of eukaryotic transcriptional activators typified by yeast protein GCN4 and the mammalian oncogene Jun contains a strong 2-ImN binding site. Specificity requirements for the protein and small molecule are similar. GCN4 and 2-lmN bind simultaneously to the same binding site. GCN4 alters the cleavage pattern of 2-ImN-EDTA derivative at only one of its binding sites. The details of the interaction suggest that GCN4 alters the conformation of an AAAAAAA sequence adjacent to its binding site. The presence of a yeast counterpart to Jun partially blocks 2-lmN binding. The differences do not appear to be caused by direct interactions between 2-lmN and the proteins, but by induced conformational changes in the DNA protein complex. It is likely that the observed differences in complexation are involved in the varying sequence specificity of these proteins.

Computational Predictions of G Protein-Coupled Receptor Structures and Binding Sites

G protein-coupled receptors (GPCRs) are the largest family of proteins within the human genome. They consist of seven transmembrane (TM) helices, with a N-terminal region of varying length and structure on the extracellular side, and a C-terminus on the intracellular side. GPCRs are involved in transmitting extracellular signals to cells, and as such are crucial drug targets. Designing pharmaceuticals to target GPCRs is greatly aided by full-atom structural information of the proteins. In particular, the TM region of GPCRs is where small molecule ligands (much more bioavailable than peptide ligands) typically bind to the receptors. In recent years nearly thirty distinct GPCR TM regions have been crystallized. However, there are more than 1,000 GPCRs, leaving the vast majority of GPCRs with limited structural information. Additionally, GPCRs are known to exist in a myriad of conformational states in the body, rendering the static x-ray crystal structures an incomplete reflection of GPCR structures. In order to obtain an ensemble of GPCR structures, we have developed the GEnSeMBLE procedure to rapidly sample a large number of variations of GPCR helix rotations and tilts. The lowest energy GEnSeMBLE structures are then docked to small molecule ligands and optimized. The GPCR family consists of five subfamilies with little to no sequence homology between them: class A, B1, B2, C, and Frizzled/Taste2. Almost all of the GPCR crystal structures have been of class A GPCRs, and much is known about their conserved interactions and binding sites. In this work we particularly focus on class B1 GPCRs, and aim to understand that family’s interactions and binding sites both to small molecules and their native peptide ligands. Specifically, we predict the full atom structure and peptide binding site of the glucagon-like peptide receptor and the TM region and small molecule binding sites for eight other class B1 GPCRs: CALRL, CRFR1, GIPR, GLR, PACR, PTH1R, VIPR1, and VIPR2. Our class B1 work reveals multiple conserved interactions across the B1 subfamily as well as a consistent small molecule binding site centrally located in the TM bundle. Both the interactions and the binding sites are distinct from those seen in the more well-characterized class A GPCRs, and as such our work provides a strong starting point for drug design targeting class B1 proteins. We also predict the full structure of CXCR4 bound to a small molecule, a class A GPCR that was not closely related to any of the class A GPCRs at the time of the work.

Structural Characterization of Pro-inflammatory and Anti-inflammatory Immunoglobulin G Fc Proteins

Immunoglobulin G (IgG) is central in mediating host defense due to its ability to target and eliminate invading pathogens. The fragment antigen binding (Fab) regions are responsible for antigen recognition; however the effector responses are encoded on the Fc region of IgG. IgG Fc displays considerable glycan heterogeneity, accounting for its complex effector functions of inflammation, modulation and immune suppression. Intravenous immunoglobulin G (IVIG) is pooled serum IgG from multiple donors and is used to treat individuals with autoimmune and inflammatory disorders such as rheumatoid arthritis and Kawasaki’s disease, respectively. It contains all the subtypes of IgG (IgG1-4) and over 120 glycovariants due to variation of an Asparagine 297-linked glycan on the Fc. The species identified as the activating component of IVIG is sialylated IgG Fc. Comparisons of wild type Fc and sialylated Fc X-ray crystal structures suggests that sialylation causes an increase in conformational flexibility, which may be important for its anti-inflammatory properties.

Although glycan modifications can promote the anti-inflammatory properties of the Fc, there are amino acid substitutions that cause Fcs to initiate an enhanced immune response. Mutations in the Fc can cause up to a 100-fold increase in binding affinity to activating Fc gamma receptors located on immune cells, and have been shown to enhance antibody dependent cell-mediated cytotoxicity. This is important in developing therapeutic antibodies against cancer and infectious diseases. Structural studies of mutant Fcs in complex with activating receptors gave insight into new protein-protein interactions that lead to an enhanced binding affinity.

Together these studies show how dynamic and diverse the Fc region is and how both protein and carbohydrate modifications can alter structure, leading to IgG Fc’s switch from a pro-inflammatory to an anti-inflammatory protein.