How many conformations can a protein remember?

Inverse Design of Self-Folding 3D Shells

Self-folding is an emerging paradigm for the inverse design of three-dimensional structures. While most efforts have concentrated on the shape of the net, our approach introduces a new design dimension-bond specificity between the edges. We transform this design process into a Boolean satisfiability problem to derive solutions for various target structures. This method significantly enhances the yield of the folding process. Furthermore, by linearly combining independent solutions, we achieve designs for shape-shifting nets wherein the dominant structure evolves with varying external conditions. This approach is demonstrated through coarse-grained simulations on two examples of triangular and square nets capable of folding into multiple target shapes.

Design principles of 3D epigenetic memory systems

Cells remember their identities, in part, by using epigenetic marks-chemical modifications placed along the genome. How can mark patterns remain stable over cell generations despite their constant erosion by replication and other processes? We developed a theoretical model that reveals that three-dimensional (3D) genome organization can stabilize epigenetic memory as long as (i) there is a large density difference between chromatin compartments, (ii) modifying "reader-writer" enzymes spread marks in three dimensions, and (iii) the enzymes are limited in abundance relative to their histone substrates. Analogous to an associative memory that encodes memory in neuronal connectivity, mark patterns are encoded in a 3D network of chromosomal contacts. Our model provides a unified account of diverse observations and reveals a key role of 3D genome organization in epigenetic memory.

Self-assembly of emulsion droplets through programmable folding

In the realm of particle self-assembly, it is possible to reliably construct nearly arbitrary structures if all the pieces are distinct^1-3, but systems with fewer flavours of building blocks have so far been limited to the assembly of exotic crystals^4-6. Here we introduce a minimal model system of colloidal droplet chains⁷, with programmable DNA interactions that guide their downhill folding into specific geometries. Droplets are observed in real space and time, unravelling the rules of folding. Combining experiments, simulations and theory, we show that controlling the order in which interactions are switched on directs folding into unique structures, which we call colloidal foldamers⁸. The simplest alternating sequences (ABAB...) of up to 13 droplets yield 11 foldamers in two dimensions and one in three dimensions. Optimizing the droplet sequence and adding an extra flavour uniquely encodes more than half of the 619 possible two-dimensional geometries. Foldamers consisting of at least 13 droplets exhibit open structures with holes, offering porous design. Numerical simulations show that foldamers can further interact to make complex supracolloidal architectures, such as dimers, ribbons and mosaics. Our results are independent of the dynamics and therefore apply to polymeric materials with hierarchical interactions on all length scales, from organic molecules all the way to Rubik's Snakes. This toolbox enables the encoding of large-scale design into sequences of short polymers, placing folding at the forefront of materials self-assembly.

Guiding the folding pathway of DNA origami

DNA origami is a robust assembly technique that folds a single-stranded DNA template into a target structure by annealing it with hundreds of short 'staple' strands. Its guiding design principle is that the target structure is the single most stable configuration. The folding transition is cooperative and, as in the case of proteins, is governed by information encoded in the polymer sequence. A typical origami folds primarily into the desired shape, but misfolded structures can kinetically trap the system and reduce the yield. Although adjusting assembly conditions or following empirical design rules can improve yield, well-folded origami often need to be separated from misfolded structures. The problem could in principle be avoided if assembly pathway and kinetics were fully understood and then rationally optimized. To this end, here we present a DNA origami system with the unusual property of being able to form a small set of distinguishable and well-folded shapes that represent discrete and approximately degenerate energy minima in a vast folding landscape, thus allowing us to probe the assembly process. The obtained high yield of well-folded origami structures confirms the existence of efficient folding pathways, while the shape distribution provides information about individual trajectories through the folding landscape. We find that, similarly to protein folding, the assembly of DNA origami is highly cooperative; that reversible bond formation is important in recovering from transient misfoldings; and that the early formation of long-range connections can very effectively enforce particular folds. We use these insights to inform the design of the system so as to steer assembly towards desired structures. Expanding the rational design process to include the assembly pathway should thus enable more reproducible synthesis, particularly when targeting more complex structures. We anticipate that this expansion will be essential if DNA origami is to continue its rapid development and become a reliable manufacturing technology.

Multifarious assembly mixtures: systems allowing retrieval of diverse stored structures

Self-assembly materials are traditionally designed so that molecular or mesoscale components form a single kind of large structure. Here, we propose a scheme to create "multifarious assembly mixtures," which self-assemble many different large structures from a set of shared components. We show that the number of multifarious structures stored in the solution of components increases rapidly with the number of different types of components. However, each stored structure can be retrieved by tuning only a few parameters, the number of which is only weakly dependent on the size of the assembled structure. Implications for artificial and biological self-assembly are discussed.

New alphabet-dependent morphological transition in random RNA alignment

We study the fraction f of nucleotides involved in the formation of a cactuslike secondary structure of random heteropolymer RNA-like molecules. In the low-temperature limit, we study this fraction as a function of the number c of different nucleotide species. We show, that with changing c, the secondary structures of random RNAs undergo a morphological transition: f(c)→1 for c≤c(cr) as the chain length n goes to infinity, signaling the formation of a virtually perfect gapless secondary structure; while f(c)<1 for c>c(cr), which means that a nonperfect structure with gaps is formed. The strict upper and lower bounds 2≤c(cr)≤4 are proven, and the numerical evidence for c(cr) is presented. The relevance of the transition from the evolutional point of view is discussed.

Computing spatial information from Fourier coefficient distributions

The spatial relationships between molecules can be quantified in terms of information. In the case of membranes, the spatial organization of molecules in a bilayer is closely related to biophysically and biologically important properties. Here, we present an approach to computing spatial information based on Fourier coefficient distributions. The Fourier transform (FT) of an image contains a complete description of the image, and the values of the FT coefficients are uniquely associated with that image. For an image where the distribution of pixels is uncorrelated, the FT coefficients are normally distributed and uncorrelated. Further, the probability distribution for the FT coefficients of such an image can readily be obtained by Parseval's theorem. We take advantage of these properties to compute the spatial information in an image by determining the probability of each coefficient (both real and imaginary parts) in the FT, then using the Shannon formalism to calculate information. By using the probability distribution obtained from Parseval's theorem, an effective distance from the uncorrelated or most uncertain case is obtained. The resulting quantity is an information computed in k-space (kSI). This approach provides a robust, facile and highly flexible framework for quantifying spatial information in images and other types of data (of arbitrary dimensions). The kSI metric is tested on a 2D Ising model, frequently used as a model for lipid bilayer; and the temperature-dependent phase transition is accurately determined from the spatial information in configurations of the system.

A hypothesis for bacteriophage DNA packaging motors

The hypothesis is presented that bacteriophage DNA packaging motors have a cycle comprised of bind/release thermal ratcheting with release-associated DNA pushing via ATP-dependent protein folding. The proposed protein folding occurs in crystallographically observed peptide segments that project into an axial channel of a protein 12-mer (connector) that serves, together with a coaxial ATPase multimer, as the entry portal. The proposed cycle begins when reverse thermal motion causes the connector’s peptide segments to signal the ATPase multimer to bind both ATP and the DNA molecule, thereby producing a dwell phase recently demonstrated by single-molecule procedures. The connector-associated peptide segments activate by transfer of energy from ATP during the dwell. The proposed function of connector/ATPase symmetry mismatches is to reduce thermal noise-induced signaling errors. After a dwell, ATP is cleaved and the DNA molecule released. The activated peptide segments push the released DNA molecule, thereby producing a burst phase recently shown to consist of four mini-bursts. The constraint of four mini-bursts is met by proposing that each mini-burst occurs via pushing by three of the 12 subunits of the connector. If all four mini-bursts occur, the cycle repeats. If the mini-bursts are not completed, a second cycle is superimposed on the first cycle. The existence of the second cycle is based on data recently obtained with bacteriophage T3. When both cycles stall, energy is diverted to expose the DNA molecule to maturation cleavage.

Simplicial edge representation of protein structures and alpha contact potential with confidence measure

Protein representation and potential function are two important ingredients for studying protein folding, equilibrium thermodynamics, and sequence design. We introduce a novel geometric representation of protein contact interactions using the edge simplices from the alpha shape of the protein structure. This representation can eliminate implausible neighbors that are not in physical contact, and can avoid spurious contact between two residues when a third residue is between them. We developed statistical alpha contact potential using an odds-ratio model. A studentized bootstrap method was then introduced to assess the 95% confidence intervals for each of the 210 propensity parameters. We found, with confidence, that there is significant long-range propensity (>30 residues apart) for hydrophobic interactions. We tested alpha contact potential for native structure discrimination using several sets of decoy structures, and found that it often performs comparably with atom-based potentials requiring many more parameters. We also show that accurate geometric representation is important, and that alpha contact potential has better performance than potential defined by cutoff distance between geometric centers of side chains. Hierarchical clustering of alpha contact potentials reveals natural grouping of residues. To explore the relationship between shape and physicochemical representations, we tested the minimum alphabet size necessary for native structure discrimination. We found that there is no significant difference in performance of discrimination when alphabet size varies from 7 to 20, if geometry is represented accurately by alpha simplicial edges. This result suggests that the geometry of packing plays an important role, but the specific residue types are often interchangeable.

Protein design depends on the size of the amino acid alphabet

We consider the design of proteins to be simultaneously thermodynamically stable in multiple independent and correlated conformations. We first show that a protein can be trained to fold to multiple independent conformations and calculate its capacity. The number of configurations that it can remember is proportional to the logarithm of the number of amino acid species A, independent of chain length. Next we investigate the recognition of correlated conformations, which we apply to funnel design around a single configuration. The maximum basin of attraction, as parametrized in our model, also depends on the number of amino acid species as ln A. We argue that the extent to which the protein energy landscape can be manipulated is fixed, effecting a trade off between well breadth, well depth, and well number. This emerging picture motivates a clearer understanding of the scope and limits of protein and heteropolymer function.