Pierpaolo Bruscolini

Exact solution of the Muñoz-Eaton model for protein folding

A transfer-matrix formalism is introduced to evaluate exactly the partition function of the Muñoz-Eaton model, relating the folding kinetics of proteins of known structure to their thermodynamics and topology. This technique can be used for a generic protein, for any choice of the energy and entropy parameters, and in principle allows the model to be used as a first tool to characterize the dynamics of a protein of known native state and equilibrium population. Applications to a beta-hairpin and to protein CI-2, with comparisons to previous results, are also shown.

Dynamic interplay between catalytic and lectin domains of GalNAc-transferases modulates protein O -glycosylation

Protein O-glycosylation is controlled by polypeptide GalNAc-transferases (GalNAc-Ts) that uniquely feature both a catalytic and lectin domain. The underlying molecular basis of how the lectin domains of GalNAc-Ts contribute to glycopeptide specificity and catalysis remains unclear. Here we present the first crystal structures of complexes of GalNAc-T2 with glycopeptides that together with enhanced sampling molecular dynamics simulations demonstrate a cooperative mechanism by which the lectin domain enables free acceptor sites binding of glycopeptides into the catalytic domain. Atomic force microscopy and small-angle X-ray scattering experiments further reveal a dynamic conformational landscape of GalNAc-T2 and a prominent role of compact structures that are both required for efficient catalysis. Our model indicates that the activity profile of GalNAc-T2 is dictated by conformational heterogeneity and relies on a flexible linker located between the catalytic and the lectin domains. Our results also shed light on how GalNAc-Ts generate dense decoration of proteins with O-glycans.

Downhill versus two-state protein folding in a statistical mechanical model

The authors address the problem of downhill protein folding in the framework of a simple statistical mechanical model, which allows an exact solution for the equilibrium and a semianalytical treatment of the kinetics. Focusing on protein 1BBL, a candidate for downhill folding behavior, and comparing it to the WW domain of protein PIN1, a two-state folder of comparable size, the authors show that there are qualitative differences in both the equilibrium and kinetic properties of the two molecules. However, the barrierless scenario which would be expected if 1BBL were a true downhill folder is observed only at low enough temperature.

Computational protein design with side-chain conformational entropy

Recent advances in modeling protein structures at the atomic level have made it possible to tackle “de novo” computational protein design. Most procedures are based on combinatorial optimization using a scoring function that estimates the folding free energy of a protein sequence on a given main‐chain structure. However, the computation of the conformational entropy in the folded state is generally an intractable problem, and its contribution to the free energy is not properly evaluated. In this article, we propose a new automated protein design methodology that incorporates such conformational entropy based on statistical mechanics principles. We define the free energy of a protein sequence by the corresponding partition function over rotamer states. The free energy is written in variational form in a pairwise approximation and minimized using the Belief Propagation algorithm. In this way, a free energy is associated to each amino acid sequence: we use this insight to rescore the results obtained with a standard minimization method, with the energy as the cost function. Then, we set up a design method that directly uses the free energy as a cost function in combination with a stochastic search in the sequence space. We validate the methods on the design of three superficial sites of a small SH3 domain, and then apply them to the complete redesign of 27 proteins. Our results indicate that accounting for entropic contribution in the score function affects the outcome in a highly nontrivial way, and might improve current computational design techniques based on protein stability. Proteins 2009.

Lattice model for cold and warm swelling of polymers in water

We define a lattice model for the interaction of a polymer with water. We solve the model in a suitable approximation. In the case of a non-polar homopolymer, for reasonable values of the parameters, the polymer is found in a non-compact conformation at low temperature; as the temperature grows, there is a sharp transition towards a compact state, then, at higher temperatures, the polymer swells again. This behaviour closely reminds that of proteins, that are unfolded at both low and high temperatures.

A simple simulation model can reproduce the thermodynamic folding intermediate of apoflavodoxin

Flavodoxins are single domain proteins with an α/β structure, whose function and folding have been well studied. Detailed experiments have shown that several members of this protein family present a stable intermediate, which accumulates along the folding process. In this work, we use a coarse‐grained model for protein folding, whose interactions are based on the topology of the native state, to analyze the thermodynamic characteristics of the folding of Anabaena apoflavodoxin. Our model shows evidence for the existence of a thermodynamic folding intermediate, which reaches a significant population along the thermal transition. According to our simulation results, the intermediate is compact, well packed, and involves distortions of the native structure similar to those experimentally found. These mainly affect the long loop in the protein surface comprising residues 120–139. Although the agreement between simulation and experiment is not perfect, something impossible for a crude model, our results show that the topology of the native state is able to dictate a folding process which includes the presence of an intermediate for this protein. Proteins 2010.

Lattice model for polymer hydration: Collapse of poly(N-isopropylacrylamide)

Poly(N-isopropylacrylamide) (PNIPAM) in dilute aqueous solution undergoes a collapse transition from coil to globule on increasing temperature. Such coil-to-globule collapse is usually considered analogous to the cold renaturation of small globular proteins. In this paper we propose a theoretical approach that is able to reproduce, in a semi-quantitative way, the unusual behavior of PNIPAM, and the observed thermodynamic properties. The procedure is based on two main steps: (i) the characterization of single monomer hydration thermodynamics, interpreted by a balance between the removal of monomer-monomer interactions and the addition of water-monomer interactions, and (ii) a simplified analysis of a lattice self-avoiding walk (SAW) model, which allows to account for the configurational entropy in a controlled way, and hence to relate the microscopic interactions to the macroscopic behavior of the polymer chain. The results show that the temperature dependence and magnitude of the interaction parameters that best fit experimental data validate a recently proposed qualitative interpretation of the mechanism of collapse transition for PNIPAM. The latter result turns out to be relevant to support the analogy with the cold renaturation of small globular proteins, and to clarify some important aspects of protein thermodynamics.

Sequence determinants of protein folding rates: Positive correlation between contact energy and contact range indicates selection for fast folding

In comparison with intense investigation of the structural determinants of protein folding rates, the sequence features favoring fast folding have received little attention. Here, we investigate this subject using simple models of protein folding and a statistical analysis of the Protein Data Bank (PDB). The mean‐field model by Plotkin and coworkers predicts that the folding rate is accelerated by stronger‐than‐average interactions at short distance along the sequence. We confirmed this prediction using the Finkelstein model of protein folding, which accounts for realistic features of polymer entropy. We then tested this prediction on the PDB. We found that native interactions are strongest at contact range l = 8. However, since short range contacts tend to be exposed and they are frequently formed in misfolded structures, selection for folding stability tends to make them less attractive, that is, stability and kinetics may have contrasting requirements. Using a recently proposed model, we predicted the relationship between contact range and contact energy based on buriedness and contact frequency. Deviations from this prediction induce a positive correlation between contact range and contact energy, that is, short range contacts are stronger than expected, for 2/3 of the proteins. This correlation increases with the absolute contact order (ACO), as expected if proteins that tend to fold slowly due to large ACO are subject to stronger selection for sequence features favoring fast folding. Our results suggest that the selective pressure for fast folding is detectable only for one third of the proteins in the PDB, in particular those with large contact order. Proteins 2012;

Mean-field approach for a statistical mechanical model of proteins

We study the thermodynamical properties of a topology-based model proposed by Galzitskaya and Finkelstein for the description of protein folding. We devise and test three different mean-field approaches for the model, that simplify the treatment without spoiling the description. The validity of the model and its mean-field approximations is checked by applying them to the β-hairpin fragment of the immunoglobulin-binding protein (GB1) and making a comparison with available experimental data and simulation results. Our results indicate that this model is a rather simple and reasonably good tool for interpreting folding experimental data, provided the parameters of the model are carefully chosen. The mean-field approaches substantially recover all the relevant exact results and represent reliable alternatives to the Monte Carlo simulations.

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.