Mapping the topography of a protein energy landscape

Protein energy landscapes are highly complex, yet the vast majority of states within them tend to be invisible to experimentalists. Here, using site-directed mutagenesis and exploiting the simplicity of tandem-repeat protein structures, we delineate a network of these states and the routes between them. We show that our target, gankyrin, a 226-residue 7-ankyrin-repeat protein, can access two alternative (un)folding pathways. We resolve intermediates as well as transition states, constituting a comprehensive series of snapshots that map early and late stages of the two pathways and show both to be polarized such that the repeat array progressively unravels from one end of the molecule or the other. Strikingly, we find that the protein folds via one pathway but unfolds via a different one. The origins of this behavior can be rationalized using the numerical results of a simple statistical mechanics model that allows us to visualize the equilibrium behavior as well as single-molecule folding/unfolding trajectories, thereby filling in the gaps that are not accessible to direct experimental observation. Our study highlights the complexity of repeat-protein folding arising from their symmetrical structures; at the same time, however, this structural simplicity enables us to dissect the complexity and thereby map the precise topography of the energy landscape in full breadth and remarkable detail. That we can recapitulate the key features of the folding mechanism by computational analysis of the native structure alone will help toward the ultimate goal of designed amino-acid sequences with made-to-measure folding mechanisms—the Holy Grail of protein folding

RNA secondary structure design

We consider the inverse-folding problem for RNA secondary structures: for a given (pseudo-knot-free) secondary structure find a sequence that has that structure as its ground state. If such a sequence exists, the structure is called designable. We implemented a branch-and-bound algorithm that is able to do an exhaustive search within the sequence space, i.e., gives an exact answer whether such a sequence exists. The bound required by the branch-and-bound algorithm are calculated by a dynamic programming algorithm. We consider different alphabet sizes and an ensemble of random structures, which we want to design. We find that for two letters almost none of these structures are designable. The designability improves for the three-letter case, but still a significant fraction of structures is undesignable. This changes when we look at the natural four-letter case with two pairs of complementary bases: undesignable structures are the exception, although they still exist. Finally, we also study the relation between designability and the algorithmic complexity of the branch-and-bound algorithm. Within the ensemble of structures, a high average degree of undesignability is correlated to a long time to prove that a given structure is (un-)designable. In the four-letter case, where the designability is high everywhere, the algorithmic complexity is highest in the region of naturally occurring RNA.Comment: 11 pages, 10 figure

Without magic bullets: the biological basis for public health interventions against protein folding disorders

Protein folding disorders of aging like Alzheimer's and Parkinson's diseases currently present intractable medical challenges. 'Small molecule' interventions - drug treatments - often have, at best, palliative impact, failing to alter disease course. The design of individual or population level interventions will likely require a deeper understanding of protein folding and its regulation than currently provided by contemporary 'physics' or culture-bound medical magic bullet models. Here, a topological rate distortion analysis is applied to the problem of protein folding and regulation that is similar in spirit to Tlusty's (2010a) elegant exploration of the genetic code. The formalism produces large-scale, quasi-equilibrium 'resilience' states representing normal and pathological protein folding regulation under a cellular-level cognitive paradigm similar to that proposed by Atlan and Cohen (1998) for the immune system. Generalization to long times produces diffusion models of protein folding disorders in which epigenetic or life history factors determine the rate of onset of regulatory failure, in essence, a premature aging driven by familiar synergisms between disjunctions of resource allocation and need in the context of socially or physiologically toxic exposures and chronic powerlessness at individual and group scales. Application of an HPA axis model is made to recent observed differences in Alzheimer's onset rates in White and African American subpopulations as a function of an index of distress-proneness

Protein folding tames chaos

Protein folding produces characteristic and functional three-dimensional structures from unfolded polypeptides or disordered coils. The emergence of extraordinary complexity in the protein folding process poses astonishing challenges to theoretical modeling and computer simulations. The present work introduces molecular nonlinear dynamics (MND), or molecular chaotic dynamics, as a theoretical framework for describing and analyzing protein folding. We unveil the existence of intrinsically low dimensional manifolds (ILDMs) in the chaotic dynamics of folded proteins. Additionally, we reveal that the transition from disordered to ordered conformations in protein folding increases the transverse stability of the ILDM. Stated differently, protein folding reduces the chaoticity of the nonlinear dynamical system, and a folded protein has the best ability to tame chaos. Additionally, we bring to light the connection between the ILDM stability and the thermodynamic stability, which enables us to quantify the disorderliness and relative energies of folded, misfolded and unfolded protein states. Finally, we exploit chaos for protein flexibility analysis and develop a robust chaotic algorithm for the prediction of Debye-Waller factors, or temperature factors, of protein structures

The difficulty of folding self-folding origami

Why is it difficult to refold a previously folded sheet of paper? We show that even crease patterns with only one designed folding motion inevitably contain an exponential number of `distractor' folding branches accessible from a bifurcation at the flat state. Consequently, refolding a sheet requires finding the ground state in a glassy energy landscape with an exponential number of other attractors of higher energy, much like in models of protein folding (Levinthal's paradox) and other NP-hard satisfiability (SAT) problems. As in these problems, we find that refolding a sheet requires actuation at multiple carefully chosen creases. We show that seeding successful folding in this way can be understood in terms of sub-patterns that fold when cut out (`folding islands'). Besides providing guidelines for the placement of active hinges in origami applications, our results point to fundamental limits on the programmability of energy landscapes in sheets.Comment: 8 pages, 5 figure

Buttressing a new paradigm in protein folding: experimental tools to distinguish between downhill and multi-state folding mechanisms

Tesis doctoral inédita leída en la Universidad Autónoma de Madrid, Facultad de Ciencias, Departamento de Biologíoa Molecular. Fecha de lectura: 15-07-2014Many single-domain proteins fold in milliseconds or longer. However, the advent of fast folding kinetic techniques has permitted to identify many other proteins that fold in the order of (few) microseconds and thus very closely to the folding speed limit. This suggests that the proteins that fold in microsecond timescale either cross a marginal single free energy barrier, multiple very small barriers (multi-state), or no barrier at all (downhill). This results in the potential observation of broad complex unfolding transitions in these ultrafast folding proteins (in contrast to simple two-state behavior). Many of the ultrafast folding proteins have small size and fold into simple alpha helix-bundle topologies. Theoretical studies support the size scaling of protein folding barriers. Engrailed homeodomain, a 61-residue α-helical domain with a helixturn- helix topology folds in microseconds and exhibits an apparently complex (un)folding process. The observed complexity in the (un)folding behavior of engrailed homeodomain rules out a simple two-state model, but the folding mechanism of this protein has been interpreted with a conventional three-state model. The current work aims to develop a set of experimental and analytical methods that can determine unambiguously whether an apparently complex folding process of a fast folding protein is downhill or multi-state using engrailed homeodomain as a model. A large-scale multiple probe approach that combines equilibrium, fast-folding measurement and single molecule measurements has been used to provide critical information to unravel the mechanistic details of the folding mechanism of this protein. Double perturbation measurement on engrailed, in which the protein was unfolded by both chemical denaturant and temperature, showed complex results. Multi-probe equilibrium thermal and chemical unfolding measurements on engrailed revealed differences in the melting temperature and chemical denaturation midpoints respectively. All these signatures conformed to downhill folding mechanism or existence of low-barrier(s). The estimated overall barrier height was ~ 0.5 RT near Tm, by globally fitting the entire equilibrium thermal unfolding data to Mean Field Model. Multi-probe temperature jump studies resulted in single exponential relaxations by infrared and non-exponential relaxations by fluorescence and probe-dependent kinetic amplitudes for the slow rates. This result could still be explained by a downhill behavior by globally fitting both the equilibrium and the kinetic data using the same model. Single molecule FRET measurements explored the transition path of engrailed near Cm and further confirmed the existence of downhill behavior with the estimated marginal barrier of < 1 RT. These results emphasize the importance of multi-probe measurements and appropriate utilization of statistical mechanical for analysis for fast-folding protein