Protein folding in high-dimensional spaces: hypergutters and the role of nonnative interactions

Capacities and efficient computation of first-passage probabilities

A reversible diffusion process is initialized at position x_{0} and run until it first hits any of several targets. What is the probability that it terminates at a particular target? We propose a computationally efficient approach for estimating this probability, focused on those situations in which it takes a long time to hit any target. In these cases, direct simulation of the hitting probabilities becomes prohibitively expensive. On the other hand, if the timescales are sufficiently long, then the system will essentially "forget" its initial condition before it encounters a target. In these cases the hitting probabilities can be accurately approximated using only local simulations around each target, obviating the need for direct simulations. In empirical tests, we find that these local estimates can be computed in the same time it would take to compute a single direct simulation, but that they achieve an accuracy that would require thousands of direct simulation runs.

Mechanobiology: protein refolding under force

The application of direct force to a protein enables to probe wide regions of its energy surface through conformational transitions as unfolding, extending, recoiling, collapsing, and refolding. While unfolding under force typically displayed a two-state behavior, refolding under force, from highly extended unfolded states, displayed a more complex behavior. The first recording of protein refolding at a force quench step displayed an initial rapid elastic recoil, followed by a plateau phase at some extension, concluding with a collapse to a final state, at which refolding occurred. These findings stirred a lively discussion, which led to further experimental and theoretical investigation of this behavior. It was demonstrated that the polymeric chain of the unfolded protein is required to fully collapse to a globular conformation for the maturation of native structure. This behavior was modeled using one-dimensional free energy landscape over the end-to-end length reaction coordinate, the collective measured variable. However, at low forces, conformational space is not well captured by such models, and using two-dimensional energy surfaces provides further insight into the dynamics of this process. This work reviews the main concepts of protein refolding under constant force, which is essential for understanding how mechanotransducing proteins operate in vivo.

A non-equilibrium approach to allosteric communication

While the theory of protein folding is well developed, including concepts such as rugged energy landscape, folding funnel, etc., the same degree of understanding has not been reached for the description of the dynamics of allosteric transitions in proteins. This is not only due to the small size of the structural change upon ligand binding to an allosteric site, but also due to challenges in designing experiments that directly observe such an allosteric transition. On the basis of recent pump-probe-type experiments (Buchli et al. 2013 Proc. Natl Acad. Sci. USA110, 11 725-11 730. (doi:10.1073/pnas.1306323110)) and non-equilibrium molecular dynamics simulations (Buchenberg et al. 2017 Proc. Natl Acad. Sci. USA114, E6804-E6811. (doi:10.1073/pnas.1707694114)) studying an photoswitchable PDZ2 domain as model for an allosteric transition, we outline in this perspective how such a description of allosteric communication might look. That is, calculating the dynamical content of both experiment and simulation (which agree remarkably well with each other), we find that allosteric communication shares some properties with downhill folding, except that it is an 'order-order' transition. Discussing the multiscale and hierarchical features of the dynamics, the validity of linear response theory as well as the meaning of 'allosteric pathways', we conclude that non-equilibrium experiments and simulations are a promising way to study dynamical aspects of allostery.This article is part of a discussion meeting issue 'Allostery and molecular machines'.

Computational Exploration of Conformational Transitions in Protein Drug Targets

Protein drug targets vary from highly structured to completely disordered; either way dynamics governs function. Hence, understanding the dynamical aspects of how protein targets function can enable improved interventions with drug molecules. Computational approaches offer highly detailed structural models of protein dynamics which are becoming more predictive as model quality and sampling power improve. However, the most advanced and popular models still have errors owing to imperfect parameter sets and often cannot access longer timescales of many crucial biological processes. Experimental approaches offer more certainty but can struggle to detect and measure lightly populated conformations of target proteins and subtle allostery. An emerging solution is to integrate available experimental data into advanced molecular simulations. In the future, molecular simulation in combination with experimental data may be able to offer detailed models of important drug targets such that improved functional mechanisms or selectivity can be accessed.

Simulated Force Quench Dynamics Shows GB1 Protein Is Not a Two State Folder

Single molecule force spectroscopy is a useful technique for investigating mechanically induced protein unfolding and refolding under reduced forces by monitoring the end-to-end distance of the protein. The data is often interpreted via a "two-state" model based on the assumption that the end-to-end distance alone is a good reaction coordinate and the thermodynamic behavior is then ascribed to the free energy as a function of this one reaction coordinate. In this paper, we determined the free energy surface (PMF) of GB1 protein from atomistic simulations in explicit solvent under different applied forces as a function of two collective variables (the end-to-end-distance, and the fraction of native contacts ρ). The calculated 2-d free energy surfaces exhibited several distinct states, or basins, mostly visible along the ρ coordinate. Brownian dynamics (BD) simulations on the smoothed free energy surface show that the protein visits a metastable molten globule state and is thus a three state folder, not the two state folder inferred using the end-to-end distance as the sole reaction coordinate. This study lends support to recent experiments that suggest that GB1 is not a two-state folder.

Protein folding as a complex reaction: a two-component potential for the driving force of folding and its variation with folding scenario

The Helmholtz decomposition of the vector field of probability fluxes in a two-dimensional space of collective variables makes it possible to introduce a potential for the driving force of protein folding [Chekmarev, J. Chem. Phys. 139 (2013) 145103]. The potential has two components: one component (Φ) is responsible for the source and sink of the folding flow, which represent, respectively, the unfolded and native state of the protein, and the other (Ψ) accounts for the flow vorticity inherently generated at the periphery of the flow field and provides the canalization of the flow between the source and sink. Both components obey Poisson’s equations with the corresponding source/sink terms. In the present paper, we consider how the shape of the potential changes depending on the scenario of protein folding. To mimic protein folding dynamics projected onto a two-dimensional space of collective variables, the two-dimensional Müller and Brown potential is employed. Three characteristic scenarios are considered: a single pathway from the unfolded to the native state without intermediates, two parallel pathways without intermediates, and a single pathway with an off-pathway intermediate. To determine the probability fluxes, the hydrodynamic description of the folding reaction is used, in which the first-passage folding is viewed as a steady flow of the representative points of the protein from the unfolded to the native state. We show that despite the possible complexity of the folding process, the Φ-component is simple and universal in shape. The Ψ-component is more complex and reveals characteristic features of the process of folding. The present approach is potentially applicable to other complex reactions, for which the transition from the reactant to the product can be described in a space of two (collective) variables.

The role of high-dimensional diffusive search, stabilization, and frustration in protein folding

Proteins are polymeric molecules with many degrees of conformational freedom whose internal energetic interactions are typically screened to small distances. Therefore, in the high-dimensional conformation space of a protein, the energy landscape is locally relatively flat, in contrast to low-dimensional representations, where, because of the induced entropic contribution to the full free energy, it appears funnel-like. Proteins explore the conformation space by searching these flat subspaces to find a narrow energetic alley that we call a hypergutter and then explore the next, lower-dimensional, subspace. Such a framework provides an effective representation of the energy landscape and folding kinetics that does justice to the essential characteristic of high-dimensionality of the search-space. It also illuminates the important role of nonnative interactions in defining folding pathways. This principle is here illustrated using a coarse-grained model of a family of three-helix bundle proteins whose conformations, once secondary structure has formed, can be defined by six rotational degrees of freedom. Two folding mechanisms are possible, one of which involves an intermediate. The stabilization of intermediate subspaces (or states in low-dimensional projection) in protein folding can either speed up or slow down the folding rate depending on the amount of native and nonnative contacts made in those subspaces. The folding rate increases due to reduced-dimension pathways arising from the mere presence of intermediate states, but decreases if the contacts in the intermediate are very stable and introduce sizeable topological or energetic frustration that needs to be overcome. Remarkably, the hypergutter framework, although depending on just a few physically meaningful parameters, can reproduce all the types of experimentally observed curvature in chevron plots for realizations of this fold.

Free energy landscape of protein-like chains with discontinuous potentials

In this article the configurational space of two simple protein models consisting of polymers composed of a periodic sequence of four different kinds of monomers is studied as a function of temperature. In the protein models, hydrogen bond interactions, electrostatic repulsion, and covalent bond vibrations are modeled by discontinuous step, shoulder, and square-well potentials, respectively. The protein-like chains exhibit a secondary alpha helix structure in their folded states at low temperatures, and allow a natural definition of a configuration by considering which beads are bonded. Free energies and entropies of configurations are computed using the parallel tempering method in combination with hybrid Monte Carlo sampling of the canonical ensemble of the discontinuous potential system. The probability of observing the most common configuration is used to analyze the nature of the free energy landscape, and it is found that the model with the least number of possible bonds exhibits a funnel-like free energy landscape at low enough temperature for chains with fewer than 30 beads. For longer proteins, the free landscape consists of several minima, where the configuration with the lowest free energy changes significantly by lowering the temperature and the probability of observing the most common configuration never approaches one due to the degeneracy of the lowest accessible potential energy.

Screening of antibiotic susceptibility to β-lactam-induced elongation of Gram-negative bacteria based on dielectrophoresis

We demonstrate a rapid antibiotic susceptibility test (AST) based on the changes in dielectrophoretic (DEP) behaviors related to the β-lactam-induced elongation of Gram-negative bacteria (GNB) on a quadruple electrode array (QEA). The minimum inhibitory concentration (MIC) can be determined within 2 h by observing the changes in the positive-DEP frequency (pdf) and cell length of GNB under the cefazolin (CEZ) treatment. Escherichia coli and Klebsiella pneumoniae and the CEZ are used as the sample bacteria and antibiotic respectively. The bacteria became filamentous due to the inhibition of cell wall synthesis and cell division and cell lysis occurred for the higher antibiotic dose. According to the results, the pdfs of wild type bacteria decrease to hundreds of kHz and the cell length is more than 10 μm when the bacterial growth is inhibited by the CEZ treatment. In addition, the growth of wild type bacteria and drug resistant bacteria differ significantly. There is an obvious decrease in the number of wild type bacteria but not in the number of drug resistant bacteria. Thus, the drug resistance of GNB to β-lactam antibiotics can be rapidly assessed. Furthermore, the MIC determined using dielectrophoresis-based AST (d-AST) was consistent with the results of the broth dilution method. Utilizing this approach could reduce the time needed for bacteria growth from days to hours, help physicians to administer appropriate antibiotic dosages, and reduce the possibility of the occurrence of multidrug resistant (MDR) bacteria.

A circumventing role for the non-native intermediate in the folding of β-lactoglobulin

Folding experiments have suggested that some proteins have kinetic intermediates with a non-native structure. A simple G ̅o model does not explain such non-native intermediates. Therefore, the folding energy landscape of proteins with non-native intermediates should have characteristic properties. To identify such properties, we investigated the folding of bovine β-lactoglobulin (βLG). This protein has an intermediate with a non-native α-helical structure, although its native form is predominantly composed of β-structure. In this study, we prepared mutants whose α-helical and β-sheet propensities are modified and observed their folding using a stopped-flow circular dichroism apparatus. One interesting finding was that E44L, whose β-sheet propensity was increased, showed a folding intermediate with an amount of β-structure similar to that of the wild type, though its folding took longer. Thus, the intermediate seems to be a trapped intermediate. The high α-helical propensity of the wild-type sequence likely causes the folding pathway to circumvent such time-consuming intermediates. We propose that the role of the non-native intermediate is to control the pathway at the beginning of the folding reaction.

Protein folded states are kinetic hubs

Understanding molecular kinetics, and particularly protein folding, is a classic grand challenge in molecular biophysics. Network models, such as Markov state models (MSMs), are one potential solution to this problem. MSMs have recently yielded quantitative agreement with experimentally derived structures and folding rates for specific systems, leaving them positioned to potentially provide a deeper understanding of molecular kinetics that can lead to experimentally testable hypotheses. Here we use existing MSMs for the villin headpiece and NTL9, which were constructed from atomistic simulations, to accomplish this goal. In addition, we provide simpler, humanly comprehensible networks that capture the essence of molecular kinetics and reproduce qualitative phenomena like the apparent two-state folding often seen in experiments. Together, these models show that protein dynamics are dominated by stochastic jumps between numerous metastable states and that proteins have heterogeneous unfolded states (many unfolded basins that interconvert more rapidly with the native state than with one another) yet often still appear two-state. Most importantly, we find that protein native states are hubs that can be reached quickly from any other state. However, metastability and a web of nonnative states slow the average folding rate. Experimental tests for these findings and their implications for other fields, like protein design, are also discussed.

Dielectrophoresis as a cell characterisation tool

Dielectrophoresis (DEP) is a technique which offers label-free measurement of cell electrophysiology by monitoring its movement in non-uniform electric fields. In this chapter, the theory underlying DEP is explored, as are the implications of the development of equipment for taking such measurements. Practical considerations such as the selection of a suspending medium are also discussed.

New dynamical window onto the landscape for forced protein unfolding

The unfolding of a protein by the application of an external force pulling two atoms of the protein can be detected by atomic force and optical tweezers technologies as have been broadly demonstrated in the past decade. Variation of the applied force results in a modulation of the free-energy barrier to unfolding and thus, the rate of the process, which is often assumed to have single exponential kinetics. It has been recently shown that it is experimentally feasible, through the use of force clamps, to estimate the distribution of unfolding times for a population of proteins initially in the native state. In this Letter we show how the analysis of such distributions under a range of forces can provide unique information about the underlying free-energy surface such as the height of the free-energy barrier, the preexponential factor and the force dependence of the unfolding kinetics without resorting to ad hoc kinetic models.

Rate of loop formation in peptides: a simulation study

Experimental techniques with high temporal and spatial resolution extend our knowledge of how biological macromolecules self-organise and function. Here, we provide an illustration of the convergence between simulation and experiment made possible by techniques such as triplet-triplet energy transfer and fluorescence quenching with long-lifetime and fast-quenching fluorescent probes. These techniques have recently been used to determine the average time needed for two residues in a peptide or protein segment to form a contact. The timescale of this process is accessible to computer simulation, providing a microscopic interpretation of the data and yielding new insight into the disordered state of proteins. Conversely, such experimental data also provide a test of the validity of alternative choices for the molecular models used in simulations, indicating their possible deficiencies. We carried out simulations of peptides of various composition and length using several models. End-to-end contact formation rates and their dependence on peptide length agree with experimental estimates for some sequences and some force fields but not for others. The deviations are due to artefactual structuring of some peptides, which is not observed when an atomistic model for the solvation water is used. Simulations show that the observed experimental rates are compatible with considerably different distributions of the end-to-end distance; for realistic models, these are never Gaussian but indicative of a rugged energy landscape.

The Effect of Increasing the Stability of Non-native Interactions on the Folding Landscape of the Bacterial Immunity Protein Im9

How stabilising non-native interactions influence protein folding energy landscapes is currently not well understood: such interactions could speed folding by reducing the conformational search to the native state, or could slow folding by increasing ruggedness. Here, we examine the influence of non-native interactions in the folding process of the bacterial immunity protein Im9, by exploiting our ability to manipulate the stability of the intermediate and rate-limiting transition state (TS) in the folding of this protein by minor alteration of its sequence or changes in solvent conditions. By analysing the properties of these species using Phi-value analysis, and exploration of the structural properties of the TS ensemble using molecular dynamics simulations, we demonstrate the importance of non-native interactions in immunity protein folding and demonstrate that the rate-limiting step involves partial reorganisation of these interactions as the TS ensemble is traversed. Moreover, we show that increasing the contribution to stability made by non-native interactions results in an increase in Phi-values of the TS ensemble without altering its structural properties or solvent-accessible surface area. The data suggest that the immunity proteins fold on multiple, but closely related, micropathways, resulting in a heterogeneous TS ensemble that responds subtly to mutation or changes in the solvent conditions. Thus, altering the relative strength of native and non-native interactions influences the search to the native state by restricting the pathways through the folding energy landscape.

Intermediates: ubiquitous species on folding energy landscapes?

Although intermediates have long been recognised as fascinating species that form during the folding of large proteins, the role that intermediates play in the folding of small, single-domain proteins has been widely debated. Recent discoveries using new, sensitive methods of detection and studies combining simulation and experiment have now converged on a common vision for folding, involving intermediates as ubiquitous stepping stones en route to the native state. The results suggest that the folding energy landscapes of even the smallest proteins possess significant ruggedness in which intermediates stabilized by both native and non-native interactions are common features.

Errorproof programmable self-assembly of DNA-nanoparticle clusters

We study theoretically a generic scheme of programmable self-assembly of nanoparticles into clusters of desired geometry. The problem is motivated by the feasibility of highly selective DNA-mediated interactions between colloidal particles. By analyzing both a simple generic model and a more realistic description of a DNA-colloidal system, we demonstrate that it is possible to suppress the glassy behavior of the system, and to make the self-assembly nearly errorproof. This regime requires a combination of stretchable interparticle linkers (e.g., sufficiently long DNA), and a soft repulsive potential. The jamming phase diagram and the error probability are computed for several types of clusters. The prospects for the experimental implementation of our scheme are also discussed.

A multi-objective evolutionary approach to the protein structure prediction problem

The protein structure prediction (PSP) problem is concerned with the prediction of the folded, native, tertiary structure of a protein given its sequence of amino acids. It is a challenging and computationally open problem, as proven by the numerous methodological attempts and the research effort applied to it in the last few years. The potential energy functions used in the literature to evaluate the conformation of a protein are based on the calculations of two different interaction energies: local (bond atoms) and non-local (non-bond atoms). In this paper, we show experimentally that those types of interactions are in conflict, and do so by using the potential energy function Chemistry at HARvard Macromolecular Mechanics. A multi-objective formulation of the PSP problem is introduced and its applicability studied. We use a multi-objective evolutionary algorithm as a search procedure for exploring the conformational space of the PSP problem.

Diffusive searches in high-dimensional spaces and apparent 'two-state' behaviour in protein folding

We extend a simple model for protein folding as a high-dimensional diffusive search. By solving a steady-state diffusion equation on a hypersphere centred on an absorbing 'native state' we find the general property that the kinetics of such a search will always be nearly single exponential. This explains the common observation of such simple 'two-state' folding kinetics in models that contain considerable intermediate structure. It also suggests that the experimental signature of single-exponential folding kinetics does not imply a simple two-state structure to the folding space.