Designed folding pathway of modular coiled-coil-based proteins.

Natural proteins are characterised by a complex folding pathway defined uniquely for each fold. Designed coiled-coil protein origami (CCPO) cages are distinct from natural compact proteins, since their fold is prescribed by discrete long-range interactions between orthogonal pairwise-interacting coiled-coil (CC) modules within a single polypeptide chain. Here, we demonstrate that CCPO proteins fold in a stepwise sequential pathway. Molecular dynamics simulations and stopped-flow Förster resonance energy transfer (FRET) measurements reveal that CCPO folding is dominated by the effective intra-chain distance between CC modules in the primary sequence and subsequent folding intermediates, allowing identical CC modules to be employed for multiple cage edges and thus relaxing CCPO cage design requirements. The number of orthogonal modules required for constructing a CCPO tetrahedron can be reduced from six to as little as three different CC modules. The stepwise modular nature of the folding pathway offers insights into the folding of tandem repeat proteins and can be exploited for the design of modular protein structures based on a given set of orthogonal modules.

Design of coiled-coil protein-origami cages that self-assemble in vitro and in vivo.

Polypeptides and polynucleotides are natural programmable biopolymers that can self-assemble into complex tertiary structures. We describe a system analogous to designed DNA nanostructures in which protein coiled-coil (CC) dimers serve as building blocks for modular de novo design of polyhedral protein cages that efficiently self-assemble in vitro and in vivo. We produced and characterized >20 single-chain protein cages in three shapes-tetrahedron, four-sided pyramid, and triangular prism-with the largest containing >700 amino-acid residues and measuring 11 nm in diameter. Their stability and folding kinetics were similar to those of natural proteins. Solution small-angle X-ray scattering (SAXS), electron microscopy (EM), and biophysical analysis confirmed agreement of the expressed structures with the designs. We also demonstrated self-assembly of a tetrahedral structure in bacteria, mammalian cells, and mice without evidence of inflammation. A semi-automated computational design platform and a toolbox of CC building modules are provided to enable the design of protein cages in any polyhedral shape.

Designed Protein Origami.

Proteins are highly perfected natural molecular machines, owing their properties to the complex tertiary structures with precise spatial positioning of different functional groups that have been honed through millennia of evolutionary selection. The prospects of designing new molecular machines and structural scaffolds beyond the limits of natural proteins make design of new protein folds a very attractive prospect. However, de novo design of new protein folds based on optimization of multiple cooperative interactions is very demanding. As a new alternative approach to design new protein folds unseen in nature, folds can be designed as a mathematical graph, by the self-assembly of interacting polypeptide modules within the single chain. Orthogonal coiled-coil dimers seem like an ideal building module due to their shape, adjustable length, and above all their designability. Similar to the approach of DNA nanotechnology, where complex tertiary structures are designed from complementary nucleotide segments, a polypeptide chain composed of a precisely specified sequence of coiled-coil forming segments can be designed to self-assemble into polyhedral scaffolds. This modular approach encompasses long-range interactions that define complex tertiary structures. We envision that by expansion of the toolkit of building blocks and design strategies of the folding pathways protein origami technology will be able to construct diverse molecular machines.

Design principles for rapid folding of knotted DNA nanostructures.

Knots are some of the most remarkable topological features in nature. Self-assembly of knotted polymers without breaking or forming covalent bonds is challenging, as the chain needs to be threaded through previously formed loops in an exactly defined order. Here we describe principles to guide the folding of highly knotted single-chain DNA nanostructures as demonstrated on a nano-sized square pyramid. Folding of knots is encoded by the arrangement of modules of different stability based on derived topological and kinetic rules. Among DNA designs composed of the same modules and encoding the same topology, only the one with the folding pathway designed according to the 'free-end' rule folds efficiently into the target structure. Besides high folding yield on slow annealing, this design also folds rapidly on temperature quenching and dilution from chemical denaturant. This strategy could be used to design folding of other knotted programmable polymers such as RNA or proteins.

Protein alignment: Exact versus approximate. An illustration.

We illustrate solving the protein alignment problem exactly using the algorithm VESPA (very efficient search for protein alignment). We have compared our result with the approximate solution obtained with BLAST (basic local alignment search tool) software, which is currently the most widely used for searching for protein alignment. We have selected human and mouse proteins having around 170 amino acids for comparison. The exact solution has found 78 pairs of amino acids, to which one should add 17 individual amino acid alignments giving a total of 95 aligned amino acids. BLAST has identified 64 aligned amino acids which involve pairs of more than two adjacent amino acids. However, the difference between the two outputs is not as large as it may appear, because a number of amino acids that are adjacent have been reported by BLAST as single amino acids. So if one counts all amino acids, whether isolated (single) or in a group of two and more amino acids, then the count for BLAST is 89 and for VESPA is 95, a difference of only six.

TOPOFOLD, the designed modular biomolecular folds: polypeptide-based molecular origami nanostructures following the footsteps of DNA.

Biopolymers, the essential components of life, are able to form many complex nanostructures, and proteins in particular are the material of choice for most cellular processes. Owing to numerous cooperative interactions, rational design of new protein folds remains extremely challenging. An alternative strategy is to design topofolds-nanostructures built from polypeptide arrays of interacting modules that define their topology. Over the course of the last several decades DNA has successfully been repurposed from its native role of information storage to a smart nanomaterial used for nanostructure self-assembly of almost any shape, which is largely because of its programmable nature. Unfortunately, polypeptides do not possess the straightforward complementarity as do nucleic acids. However, a modular approach can nevertheless be used to assemble polypeptide nanostructures, as was recently demonstrated on a single-chain polypeptide tetrahedron. This review focuses on the current state-of-the-art in the field of topological polypeptide folds. It starts with a brief overview of the field of structural DNA and RNA nanotechnology, from which it draws parallels and possible directions of development for the emerging field of polypeptide-based nanotechnology. The principles of topofold strategy and unique properties of such polypeptide nanostructures in comparison to native protein folds are discussed. Reasons for the apparent absence of such folds in nature are also examined. Physicochemical versatility of amino acid residues and cost-effective production makes polypeptides an attractive platform for designed functional bionanomaterials.

Novel graph distance matrix.

We have introduced novel distance matrix for graphs, which is based on interpretation of columns of the adjacency matrix of a graph as a set of points in n-dimensional space, n being the number of vertices in the graph. Numerical values for the distances are based on the Euclidean distance between n points in n-dimensional space. In this way, we have combined the traditional representation of graphs (drawn as 2D object of no fixed geometry) with their representation in n-dimensional space, defined by a set of n-points that lead to a representation of definite geometry. The novel distance matrix, referred to as natural distance matrix, shows some structural properties and offers novel graph invariants as molecular descriptors for structure-property-activity studies. One of the novel graph descriptors is the modified connectivity index in which the bond contribution for (m, n) bond-type is given by 1/ radical(m + n), where m and n are the valence of the end vertices of the bond. The novel distance matrix (ND) can be reduced to sparse distance-adjacency matrix (DA), which can be viewed as specially weighted adjacency matrix of a graph. The quotient of the leading eigenvalues of novel distance-adjacency matrix and novel distance matrix, as illustrated on a collection of graphs of chemical interest, show parallelism with a simple measure of graph density, based on the quotient of the number of edges in a graph and the maximal possible number of edges for graphs of the same size.

HOMO-LUMO Maps for Fullerenes.

The HOMO-LUMO map is found to be a useful tool for classifying p-electron configurations of fullerenes and identifying research questions about their adjacency spectra.

Graphical representation of proteins as four-color maps and their numerical characterization.

We put forward a novel compact 2-D graphical representation of proteins based on the concept of virtual genetic code and a four-color map. The novel graphical representation uniquely represents proteins and allows one to easily and quickly visually observe and inspect similarity/dissimilarity between them. It also leads to a novel protein descriptor, a 10-dimensional vector derived from a novel structure matrix S associated with the map. The introduced numerical characterization of proteins is not only useful for their comparative study, but also for cataloguing information on a single protein. The approach is illustrated with the A chain of human insulin and the A chain of human insulin analogue glargine.

Growth in catacondensed benzenoid graphs.

The generating function of the sequence counting the number of graph vertices at a given distance from the root is called the spherical growth function of the rooted graph. The vertices farthest from the root form an induced subgraph called the distance-residual graph. These mathematical notions are applied to benzenoid graphs which are used in graph theory to represent benzenoid hydrocarbons. An algorithm for calculating the growth in catacondensed benzenoids is presented, followed by some examples.

Characterization of complex biological systems by matrix invariants.

One direction in exploring similarities among biological sequences (such as DNA, RNA, and proteins), is to associate with such systems ordered sets of sequence invariants. These invariants represent selected properties of mathematical objects, such as matrices, that one can associate with biological sequences. In this article, we are exploring properties of recently introduced Line Distance matrices, and in particular we consider properties of their eigenvalues. We prove that Line Distance matrices of size n have one positive and n - 1 negative eigenvalues. Visual representation of Cauchy's interlacing property for Line Distance matrices is considered. Matlab programs for line distance matrices and examples are available on the following website: www.fmf.uni-lj.si/ approximately jaklicg/ldmatrix.html.