Connecting the Sequence-Space of Bacterial Signaling Proteins to Phenotypes Using Coevolutionary Landscapes.

Two-component signaling (TCS) is the primary means by which bacteria sense and respond to the environment. TCS involves two partner proteins working in tandem, which interact to perform cellular functions whereas limiting interactions with non-partners (i.e., cross-talk). We construct a Potts model for TCS that can quantitatively predict how mutating amino acid identities affect the interaction between TCS partners and non-partners. The parameters of this model are inferred directly from protein sequence data. This approach drastically reduces the computational complexity of exploring the sequence-space of TCS proteins. As a stringent test, we compare its predictions to a recent comprehensive mutational study, which characterized the functionality of 204 mutational variants of the PhoQ kinase in Escherichia coli We find that our best predictions accurately reproduce the amino acid combinations found in experiment, which enable functional signaling with its partner PhoP. These predictions demonstrate the evolutionary pressure to preserve the interaction between TCS partners as well as prevent unwanted cross-talk. Further, we calculate the mutational change in the binding affinity between PhoQ and PhoP, providing an estimate to the amount of destabilization needed to disrupt TCS.

Self-regulating gene: an exact solution.

An exact steady-state solution of the stochastic equations governing the behavior of a gene regulated by a self-generated proteomic atmosphere is presented. The solutions depend on an adiabaticity parameter measuring the relative rate of DNA-protein unbinding and protein degradation. The steady-state solution reveals deviations from the commonly used Ackers et al approximation based on the equilibrium law of mass action, allowing anticooperative behavior in the "nonadiabatic" limit of slow binding and unbinding rates. Noise from binding and unbinding events dominates the shot noise of protein synthesis and degradation up to quite high values of the adiabaticity parameter.

Solvation in protein folding analysis: combination of theoretical and experimental approaches.

An effort to combine theoretical analyses and protein engineering methods has been made to probe the folding mechanism of SH3 by using Energy Landscape Theory and a phi-value analysis. Particular emphasis was given to core residues and the effect of desolvation during the folding event by replacing the core valines with isosteric threonines. These mutations have the advantage of keeping the core structurally invariant while affecting core stability relative to the unfolded state. Although the valines that form the core appear spatially invariant, the folding kinetics of their threonine mutants varies, indicating their different extent of solvation in the transition-state ensemble. Theoretical studies predicted the distribution of folding kinetics of threonine mutants without previous knowledge of the measured rates. This initial success encourages further investigations of the molecular details behind these macroscopic phenomena and of the role of solvation in the folding mechanism.

Nonlinear elasticity, proteinquakes, and the energy landscapes of functional transitions in proteins.

Large-scale motions of biomolecules involve linear elastic deformations along low-frequency normal modes, but for function nonlinearity is essential. In addition, unlike macroscopic machines, biological machines can locally break and then reassemble during function. We present a model for global structural transformations, such as allostery, that involve large-scale motion and possible partial unfolding, illustrating the method with the conformational transition of adenylate kinase. Structural deformation between open and closed states occurs via low-frequency modes on separate reactant and product surfaces, switching from one state to the other when energetically favorable. The switching model is the most straightforward anharmonic interpolation, which allows the barrier for a process to be estimated from a linear normal mode calculation, which by itself cannot be used for activated events. Local unfolding, or cracking, occurs in regions where the elastic stress becomes too high during the transition. Cracking leads to a counterintuitive catalytic effect of added denaturant on allosteric enzyme function. It also leads to unusual relationships between equilibrium constant and rate like those seen recently in single-molecule experiments of motor proteins.

Different scenarios for inter-protein electron tunneling: the effect of water-mediated pathways.

Recent theoretical developments now allow for reliable calculation oftunneling matrix elements in unimolecular biological electron transferreactions that have been tested experimentally. Most biological ETprocesses, however, are bimolecular, or involve large-scale proteindomain motions. In this paper, initial advances in this direction bystudying the inter-protein electron transfer between cytochrome c(2)andthe photosynthetic reaction center. Utilizing an approach that integratesmolecular dynamics and the Pathways method, we have observed that theensemble dominant tunneling pathways in this reaction go though thetyrosine 162 or are water mediated.

Prediction of folding mechanism for circular-permuted proteins.

Recent theoretical and experimental studies have suggested that real proteins have sequences with sufficiently small energetic frustration that topological effects are central in determining the folding mechanism. A particularly interesting and challenging framework for exploring and testing the viability of these energetically unfrustrated models is the study of circular-permuted proteins. Here we present the results of the application of a topology-based model to the study of circular permuted SH3 and CI2, in comparison with the available experimental results. The folding mechanism of the permuted proteins emerging from our simulations is in very good agreement with the experimental observations. The differences between the folding mechanisms of the permuted and wild-type proteins seem then to be strongly related to the change in the native state topology.

Dynamically controlled protein tunneling paths in photosynthetic reaction centers.

Marcus theory has explained how thermal nuclear motions modulate the energy gap between donor and acceptor sites in protein electron transfer reactions. Thermal motions, however, may also modulate electron tunneling between these reactions. Here we identify a new mechanism of nuclear dynamics amplification that plays a central role when interference among the dominant tunneling pathway tubes is destructive. In these cases, tunneling takes place in protein conformations far from equilibrium that minimize destructive interference. As an example, we demonstrate how this dynamical amplification mechanism affects certain reaction rates in the photosynthetic reaction center and therefore may be critical for biological function.

Investigation of routes and funnels in protein folding by free energy functional methods.

We use a free energy functional theory to elucidate general properties of heterogeneously ordering, fast folding proteins, and we test our conclusions with lattice simulations. We find that both structural and energetic heterogeneity can lower the free energy barrier to folding. Correlating stronger contact energies with entropically likely contacts of a given native structure lowers the barrier, and anticorrelating the energies has the reverse effect. Designing in relatively mild energetic heterogeneity can eliminate the barrier completely at the transition temperature. Sequences with native energies tuned to fold uniformly, as well as sequences tuned to fold reliably by a single or a few routes, are rare. Sequences with weak native energetic heterogeneity are more common; their folding kinetics is more strongly determined by properties of the native structure. Sequences with different distributions of stability throughout the protein may still be good folders to the same structure. A measure of folding route narrowness is introduced that correlates with rate and that can give information about the intrinsic biases in ordering arising from native topology. This theoretical framework allows us to investigate systematically the coupled effects of energy and topology in protein folding and to interpret recent experiments that investigate these effects.

How native-state topology affects the folding of dihydrofolate reductase and interleukin-1beta.

The overall structure of the transition-state and intermediate ensembles observed experimentally for dihydrofolate reductase and interleukin-1beta can be obtained by using simplified models that have almost no energetic frustration. The predictive power of these models suggests that, even for these very large proteins with completely different folding mechanisms and functions, real protein sequences are sufficiently well designed, and much of the structural heterogeneity observed in the intermediates and the transition-state ensembles is determined by topological effects.

Topological and energetic factors: what determines the structural details of the transition state ensemble and "en-route" intermediates for protein folding? An investigation for small globular proteins.

Recent experimental results suggest that the native fold, or topology, plays a primary role in determining the structure of the transition state ensemble, at least for small, fast-folding proteins. To investigate the extent of the topological control of the folding process, we studied the folding of simplified models of five small globular proteins constructed using a Go-like potential to retain the information about the native structures but drastically reduce the energetic frustration and energetic heterogeneity among residue-residue native interactions. By comparing the structure of the transition state ensemble (experimentally determined by Phi-values) and of the intermediates with those obtained using our models, we show that these energetically unfrustrated models can reproduce the global experimentally known features of the transition state ensembles and "en-route" intermediates, at least for the analyzed proteins. This result clearly indicates that, as long as the protein sequence is sufficiently minimally frustrated, topology plays a central role in determining the folding mechanism.

Landscape approaches for determining the ensemble of folding transition states: success and failure hinge on the degree of frustration.

We present a method for determining structural properties of the ensemble of folding transition states from protein simulations. This method relies on thermodynamic quantities (free energies as a function of global reaction coordinates, such as the percentage of native contacts) and not on "kinetic" measurements (rates, transmission coefficients, complete trajectories); consequently, it requires fewer computational resources compared with other approaches, making it more suited to large and complex models. We explain the theoretical framework that underlies this method and use it to clarify the connection between the experimentally determined Phi value, a quantity determined by the ratio of rate and stability changes due to point mutations, and the average structure of the transition state ensemble. To determine the accuracy of this thermodynamic approach, we apply it to minimalist protein models and compare these results with the ones obtained by using the standard experimental procedure for determining Phi values. We show that the accuracy of both methods depends sensitively on the amount of frustration. In particular, the results are similar when applied to models with minimal amounts of frustration, characteristic of rapid-folding, single-domain globular proteins.

Pressure-induced protein-folding/unfolding kinetics.

We use an off-lattice minimalist model to describe the effects of pressure in slowing down the folding/unfolding kinetics of proteins when subjected to increasingly larger pressures. The potential energy function used to describe the interactions between beads in the model includes the effects of pressure on the pairwise interaction of hydrophobic groups in water. We show that pressure affects the participation of contacts in the transition state. More significantly, pressure exponentially decreases the chain reconfigurational diffusion coefficient. These results are consistent with experimental results on the kinetics of pressure-denaturation of staphylococcal nuclease.

Exploring the origins of topological frustration: design of a minimally frustrated model of fragment B of protein A.

Topological frustration in an energetically unfrustrated off-lattice model of the helical protein fragment B of protein A from Staphylococcus aureus was investigated. This G-type model exhibited thermodynamic and kinetic signatures of a well-designed two-state folder with concurrent collapse and folding transitions and single exponential kinetics at the transition temperature. Topological frustration is determined in the absence of energetic frustration by the distribution of Fersht phi values. Topologically unfrustrated systems present a unimodal distribution sharply peaked at intermediate phi, whereas highly frustrated systems display a bimodal distribution peaked at low and high phi values. The distribution of phi values in protein A was determined both thermodynamically and kinetically. Both methods yielded a unimodal distribution centered at phi = 0.3 with tails extending to low and high phi values, indicating the presence of a small amount of topological frustration. The contacts with high phi values were located in the turn regions between helices I and II and II and III, intimating that these hairpins are in large part required in the transition state. Our results are in good agreement with all-atom simulations of protein A, as well as lattice simulations of a three- letter code 27-mer (which can be compared with a 60-residue helical protein). The relatively broad unimodal distribution of phi values obtained from the all-atom simulations and that from the minimalist model for the same native fold suggest that the structure of the transition state ensemble is determined mostly by the protein topology and not energetic frustration.

Stretching lattice models of protein folding.

A new class of experiments that probe folding of individual protein domains uses mechanical stretching to cause the transition. We show how stretching forces can be incorporated in lattice models of folding. For fast folding proteins, the analysis suggests a complex relation between the force dependence and the reaction coordinate for folding.

The folding funnel landscape for the peptide Met-enkephalin.

We study the free energy landscape of the small peptide Met-enkephalin. Our data were obtained from a generalized-ensemble Monte Carlo simulation taking the interactions among all atoms into account. We show that the free energy landscape resembles that of a funnel, indicating that this peptide is a good folder. Our work demonstrates that the energy landscape picture and folding concept, developed in the context of simplified protein models, can also be used to describe the folding in more realistic models.

Proposed mechanism for stability of proteins to evolutionary mutations.

It is shown that the sequence-ordering tendencies induced by design into different fast-folding, thermally stable native structures interfere. This interference results in a type of quasiorthogonality between optimal native structures, which divides sequence space into fast-folding, thermally stable families surrounded by slow-folding, low stability shells. A concrete example of this effect is provided by using a simple alpha carbon type model in which a complete correspondence is established between sequence and structure. It is speculated that gaps can occur in the space of protein-like sequences separating the sequence families and resulting in a mechanism for stability and diversity of protein sequence information.

Protein folding mechanisms and the multidimensional folding funnel.

An important idea that emerges from the energy landscape theory of protein folding is that subtle global features of the protein landscape can profoundly affect the apparent mechanism of folding. The relationship between various characteristic temperatures in the phase diagrams and landmarks in the folding funnel at fixed temperatures can be used to classify different folding behaviors. The one-dimensional picture of a folding funnel classifies folding kinetics into four basic scenarios, depending on the relative location of the thermodynamic barrier and the glass transition as a function of a single-order parameter. However, the folding mechanism may not always be quantitatively described by a single-order parameter. Several other order parameters, such as degree of secondary structure formation, collapse and topological order, are needed to establish the connection between minimalist models and proteins in the laboratory. In this article we describe a simple multidimensional funnel based on two-order parameters that measure the degree of collapse and topological order. The appearance of several different "mechanisms" is illustrated by analyzing lattice models with different potentials and sequences with different degrees of design. In most cases, the two-dimensional analysis leads to a classification of mechanisms totally in keeping with the one-dimensional scheme, but a topologically distinct scenario of fast folding with traps also emerges. The nature of traps depends on the relative location of the glass transition surface and the thermodynamic barrier in the multidimensional funnel.