On the complements of union of open balls of fixed radius in the Euclidean space

Let an $R$-body be the complement of the union of open balls of radius $R$ in $\mathbb{E}^d$. The $R$-hulloid of a closed not empty set $A$, the minimal $R$-body containing $A$, is investigated; if $A$ is the set of the vertices of a simplex, the $R$-hulloid of $A$ is completely described (if $d=2$) and if $d> 2$ special examples are studied. The class of $R$-bodies is compact in the Hausdorff metric if $d =2$, but not compact if $d>2$.Comment: 1 figur

Estimation of interdomain flexibility of N-terminus of factor H using residual dipolar couplings

Characterization of segmental flexibility is needed to understand the biological mechanisms of the very large category of functionally diverse proteins, exemplified by the regulators of complement activation, that consist of numerous compact modules or domains linked by short, potentially flexible, sequences of amino acid residues. The use of NMR-derived residual dipolar couplings (RDCs), in magnetically aligned media, to evaluate interdomain motion is established but only for two-domain proteins. We focused on the three N-terminal domains (called CCPs or SCRs) of the important complement regulator, human factor H (i.e. FH1-3). These domains cooperate to facilitate cleavage of the key complement activation-specific protein fragment, C3b, forming iC3b that no longer participates in the complement cascade. We refined a three-dimensional solution structure of recombinant FH1-3 based on nuclear Overhauser effects and RDCs. We then employed a rudimentary series of RDC datasets, collected in media containing magnetically aligned bicelles (disk-like particles formed from phospholipids) under three different conditions, to estimate interdomain motions. This circumvents a requirement of previous approaches for technically difficult collection of five independent RDC datasets. More than 80% of conformers of this predominantly extended three-domain molecule exhibit flexions of < 40 °. Such segmental flexibility (together with the local dynamics of the hypervariable loop within domain 3), could facilitate recognition of C3b via initial anchoring and eventual reorganization of modules to the conformation captured in the previously solved crystal structure of a C3b:FH1-4 complex

Mapping the encounter state of a transient protein complex by PRE NMR spectroscopy

Many biomolecular interactions proceed via a short-lived encounter state, consisting of multiple, lowly-populated species invisible to most experimental techniques. Recent development of paramagnetic relaxation enhancement (PRE) nuclear magnetic resonance (NMR) spectroscopy has allowed to directly visualize such transient intermediates in a number of protein-protein and protein-DNA complexes. Here we present an analysis of the recently published PRE NMR data for a protein complex of yeast cytochrome c (Cc) and cytochrome c peroxidase (CcP). First, we describe a simple, general method to map out the spatial and temporal distributions of binding geometries constituting the Cc-CcP encounter state. We show that the spatiotemporal mapping provides a reliable estimate of the experimental coverage and, at higher coverage levels, allows to delineate the conformational space sampled by the minor species. To further refine the encounter state, we performed PRE-based ensemble simulations. The generated solutions reproduce well the experimental data and lie within the allowed regions of the encounter maps, confirming the validity of the mapping approach. The refined encounter ensembles are distributed predominantly in a region encompassing the dominant form of the complex, providing experimental proof for the results of classical theoretical simulations