Thermal versus mechanical unfolding in a model protein.

Force spectroscopy techniques are often used to learn about the free energy landscape of single biomolecules, typically by recovering free energy quantities that, extrapolated to zero force, are compared to those measured in bulk experiments. However, it is not always clear how the information obtained from a mechanically perturbed system can be related to the information obtained using other denaturants since tensioned molecules unfold and refold along a reaction coordinate imposed by the force, which is not likely to be meaningful in its absence. Here, we explore this dichotomy by investigating the unfolding landscape of a model protein, which is unfolded first mechanically through typical force spectroscopy-like protocols and next thermally. When unfolded by nonequilibrium force extension and constant force protocols, we recover a simple two-barrier landscape as the protein reaches the extended conformation through a metastable intermediate. Interestingly, folding-unfolding equilibrium simulations at low forces suggested a totally different scenario, where this metastable state plays little role in the unfolding mechanism, and the protein unfolds through two competing pathways [R. Tapia-Rojo et al., J. Chem. Phys. 141, 135102 (2014)]. Finally, we use Markov state models to describe the configurational space of the unperturbed protein close to the critical temperature. The thermal dynamics is well understood by a one-dimensional landscape along an appropriate reaction coordinate, however it is very different from the mechanical picture. In this sense, the results of our protein model for the mechanical and thermal descriptions provide incompatible views of the folding/unfolding landscape of the system, and the estimated quantities to zero force result are hard to interpret.

A physical picture for mechanical dissociation of biological complexes: from forces to free energies.

Single-molecule force spectroscopy is a powerful technique based on the application of controlled forces to macromolecules. In order to relate the measured response of the molecule to its equilibrium and dynamic properties, a suitable physical picture of the involved process is necessary. In this work, we introduce a plausible model for mechanical unbinding of some molecular complexes, based on a novel free energy profile. We combine two standard theoretical frameworks for analyzing force spectroscopy experiments on two protein:protein complexes, obtaining key magnitudes of the underlying free energy profile, which are only understood within the mentioned model. Additionally, we carry out detailed stochastic dynamics simulations to prove the validity of the analysis protocol and the reliability of the free energy profile. Remarkably, we can compare directly the obtained unbinding free energies with the previously known bulk binding free energies, bridging the gap between bulk and single molecule techniques.

Role of the central cations in the mechanical unfolding of DNA and RNA G-quadruplexes.

Cations are known to mediate diverse interactions in nucleic acids duplexes but they are critical in the arrangement of four-stranded structures. Here, we use all-atom molecular dynamics simulations with explicit solvent to analyse the mechanical unfolding of representative intramolecular G-quadruplex structures: a parallel, a hybrid and an antiparallel DNA and a parallel RNA, in the presence of stabilising cations. We confirm the stability of these conformations in the presence of [Formula: see text] central ions and observe distortions from the tetrad topology in their absence. Force-induced unfolding dynamics is then investigated. We show that the unfolding events in the force-extension curves are concomitant to the loss of coordination between the central ions and the guanines of the G-quadruplex. We found lower ruptures forces for the parallel configuration with respect to the antiparallel one, while the behaviour of the force pattern of the parallel RNA appears similar to the parallel DNA. We anticipate that our results will be essential to interpret the fine structure rupture profiles in stretching assays at high resolution and will shed light on the mechanochemical activity of G-quadruplex-binding machinery.

An integrative approach for modeling and simulation of heterocyst pattern formation in cyanobacteria filaments.

Heterocyst differentiation in cyanobacteria filaments is one of the simplest examples of cellular differentiation and pattern formation in multicellular organisms. Despite of the many experimental studies addressing the evolution and sustainment of heterocyst patterns and the knowledge of the genetic circuit underlying the behavior of single cyanobacterium under nitrogen deprivation, there is still a theoretical gap connecting these two macroscopic and microscopic processes. As an attempt to shed light on this issue, here we explore heterocyst differentiation under the paradigm of systems biology. This framework allows us to formulate the essential dynamical ingredients of the genetic circuit of a single cyanobacterium into a set of differential equations describing the time evolution of the concentrations of the relevant molecular products. As a result, we are able to study the behavior of a single cyanobacterium under different external conditions, emulating nitrogen deprivation, and simulate the dynamics of cyanobacteria filaments by coupling their respective genetic circuits via molecular diffusion. These two ingredients allow us to understand the principles by which heterocyst patterns can be generated and sustained. In particular, our results point out that, by including both diffusion and noisy external conditions in the computational model, it is possible to reproduce the main features of the formation and sustainment of heterocyst patterns in cyanobacteria filaments as observed experimentally. Finally, we discuss the validity and possible improvements of the model.

Macro and nano scale modelling of water-water interactions at ambient and low temperature: relaxation and residence times.

The decay dynamics of ambient and low temperature liquid water has been investigated through all-atom molecular dynamics simulations, residence times calculations and time correlation functions from 300 K down to 243 K. Those simulations replicate the experimental value of the self-diffusion constant as a function of temperature by tuning the damping factor of the Langevin equation of motion. A stretched exponential function exp[-(t/τ)(ß)] has been found to properly describe the relaxation of residence times calculated at different temperatures for solvent molecules in a nanodrop of free water modelled as a sphere of nanometric dimensions. As the temperature goes down the decay time τ increases showing a divergence at Ts = 227 ± 3 K. The temperature independence of the dimensionless stretched exponent ß = 0.59 ± 0.01 suggests the presence of, not a characteristic relaxation time (since ß≠ 1), but a distribution of decay times that also holds at low temperature. An explanation for such heterogeneity can be found at the nanoscopic level. Moreover it can be concluded that the distribution of times already reported for the dynamics of water surrounding proteins (ß≤ 0.5) can not be exclusively due to the presence of the biomolecule itself since isolated water also exhibits such behaviour. The above reported Ts and ß values quantitatively reproduce experimental data.

Mesoscopic model and free energy landscape for protein-DNA binding sites: analysis of cyanobacterial promoters.

The identification of protein binding sites in promoter sequences is a key problem to understand and control regulation in biochemistry and biotechnological processes. We use a computational method to analyze promoters from a given genome. Our approach is based on a physical model at the mesoscopic level of protein-DNA interaction based on the influence of DNA local conformation on the dynamics of a general particle along the chain. Following the proposed model, the joined dynamics of the protein particle and the DNA portion of interest, only characterized by its base pair sequence, is simulated. The simulation output is analyzed by generating and analyzing the Free Energy Landscape of the system. In order to prove the capacity of prediction of our computational method we have analyzed nine promoters of Anabaena PCC 7120. We are able to identify the transcription starting site of each of the promoters as the most populated macrostate in the dynamics. The developed procedure allows also to characterize promoter macrostates in terms of thermo-statistical magnitudes (free energy and entropy), with valuable biological implications. Our results agree with independent previous experimental results. Thus, our methods appear as a powerful complementary tool for identifying protein binding sites in promoter sequences.

Organization of two kinesins in a two-dimensional microtubule network.

In intracellular active transport, molecular motors are responsible for moving biological cargo along networks of microtubules that serve as scaffolds. Cargo dynamics can be modified by different features of microtubule networks such as geometry, density, orientation modifications. Also, the dynamical behaviour of the molecular motors is determined by the microtubule network and by the individual and/or collective action of the motors. For example, unlike single kinesins, the mechanistic behavior of multiple kinesins varies from one experiment to another. However, the reasons for this experimental variability are unknown. Here we show theoretically how non-radial and quasi-radial microtubule architectures modify the collective behavior of two kinesins attached on a cargo. We found out under which structural conditions transport is most efficient and the most likely way in which kinesins are organized in active transport. In addition, with motor activity, mean intermotor distance and motor organization, we determined the character of the collective interaction of the kinesins during transport. Our results demonstrate that two-dimensional microtubule structures promote branching due to crossovers that alter directionality in cargo movement and may provide insight into the collective organization of the motors. Our article offers a perspective to analyze how the two-dimensional network can modify the cargo-motor dynamics for the case in which multiple motors move in different directions as in the case of kinesin and dynein.

DNA overstretching transition induced by melting in a dynamical mesoscopic model.

We present a phenomenological dynamical model describing the force induced melting as responsible for the DNA overstretching transition. The denaturation mechanism is developed under the framework of the mesoscopic one-dimensional Peyrard-Bishop-Dauxois (PBD) picture which models the melting features of a polymer chain by means of a Morse potential and the stacking interaction. We find a good agreement with both the experimental overstretching curve and the asymmetric hysteretic properties with different simulation times. The comparison of the standard PBD model with a modification of the Morse potential which takes into account the interaction with the solvent has been also successfully investigated.

Elastic traits of the extensible discrete wormlike chain model.

Polymer models play the special role of elucidating the elementary features describing the physics of long molecules and become essential to interpret the measurements of their magnitudes. In this work the end-to-end distance of an extensible discrete wormlike chain polymer as a function of the applied force has been calculated both numerically and analytically, the latter as an effective approximation. The numerical evaluation uses the transfer matrix formalism to obtain an exact calculation of the partition function, while the analytic derivations generalize the simple phenomenological formulas largely used up to now. The obtained formulas are simple enough to be implemented in the fit analysis of experimental data of semiflexible extensible polymers, with the result that the elastic parameters obtained are compatible with previous measurements, and more, their accuracy strongly improves in a large range of chain extensibility.

Tug of war of molecular motors: the effects of uneven load sharing.

We analyze theoretically the problem of cargo transport along microtubules by motors of two species with opposite polarities. We consider two different one-dimensional models previously developed in the literature: a quite widespread model which assumes equal force sharing, here referred to as the mean field model (MFM), and a stochastic model (SM) which considers individual motor-cargo links. We find that in generic situations, the MFM predicts larger cargo mean velocity, smaller mean run time and less frequent reversions than the SM. These phenomena are found to be the consequences of the load sharing assumptions and can be interpreted in terms of the probabilities of the different motility states. We also explore the influence of the viscosity in both models and the role of the stiffness of the motor-cargo links within the SM. Our results show that the mean cargo velocity is independent of the stiffness, while the mean run time decreases with such a parameter. We explore the case of symmetric forward and backward motors considering kinesin-1 parameters, and the problem of transport by kinesin-1 and cytoplasmic dyneins considering two different sets of parameters previously proposed for dyneins.

The influence of direct motor-motor interaction in models for cargo transport by a single team of motors.

We analyze theoretically the effects of excluded-volume interactions between motors on the dynamics of a cargo driven by multiple motors. The model considered shares much in common with others recently proposed in the literature, with the addition of direct interaction between motors and motor back steps. The cargo is assumed to follow a continuum Langevin dynamics, while individual motors evolve following a Monte Carlo algorithm based on experimentally accessible probabilities for discrete forward and backward jumps, and attachment and detachment rates. The links between cargo and motors are considered as nonlinear springs. By means of numerical simulations we compute the relevant quantities characterizing the dynamical properties of the system, and we compare the results to those for noninteracting motors. We find that interactions lead to quite relevant changes in the force-velocity relation for cargo, with a considerable reduction of the stall force, and also cause a notable decrease of the run length. These effects are mainly due to traffic-like phenomena in the microtubule. The consideration of several parallel tracks for motors reduces such effects. However, we find that for realistic values of the number of motors and the number of tracks, the influence of interactions on the global parameters of transport of cargo are far from being negligible. Our studies also provide an analysis of the relevance of motor back steps on the modeling, and of the influence of different assumptions for the detachment rates. In particular, we discuss these two aspects in connection with the possibility of observing processive back motion of cargo at large load forces.

Exploring the free energy landscape: from dynamics to networks and back.

Knowledge of the Free Energy Landscape topology is the essential key to understanding many biochemical processes. The determination of the conformers of a protein and their basins of attraction takes a central role for studying molecular isomerization reactions. In this work, we present a novel framework to unveil the features of a Free Energy Landscape answering questions such as how many meta-stable conformers there are, what the hierarchical relationship among them is, or what the structure and kinetics of the transition paths are. Exploring the landscape by molecular dynamics simulations, the microscopic data of the trajectory are encoded into a Conformational Markov Network. The structure of this graph reveals the regions of the conformational space corresponding to the basins of attraction. In addition, handling the Conformational Markov Network, relevant kinetic magnitudes as dwell times and rate constants, or hierarchical relationships among basins, completes the global picture of the landscape. We show the power of the analysis studying a toy model of a funnel-like potential and computing efficiently the conformers of a short peptide, dialanine, paving the way to a systematic study of the Free Energy Landscape in large peptides.

Common conformational changes in flavodoxins induced by FMN and anion binding: the structure of Helicobacter pylori apoflavodoxin.

Flavodoxins, noncovalent complexes between apoflavodoxins and flavin mononucleotide (FMN), are useful models to investigate the mechanism of protein/flavin recognition. In this respect, the only available crystal structure of an apoflavodoxin (that from Anabaena) showed a closed isoalloxazine pocket and the presence of a bound phosphate ion, which posed many questions on the recognition mechanism and on the potential physiological role exerted by phosphate ions. To address these issues we report here the X-ray structure of the apoflavodoxin from the pathogen Helicobacter pylori. The protein naturally lacks one of the conserved aromatic residues that close the isoalloxazine pocket in Anabaena, and the structure has been determined in a medium lacking phosphate. In spite of these significant differences, the isoallozaxine pocket in H. pylori apoflavodoxin appears also closed and a chloride ion is bound at a native-like FMN phosphate site. It seems thus that it is a general characteristic of apoflavodoxins to display closed, non-native, isoalloxazine binding sites together with native-like, rather promiscuous, phosphate binding sites that can bear other available small anions present in solution. In this respect, both binding energy hot spots of the apoflavodoxin/FMN complex are initially unavailable to FMN binding and the specific spot for FMN recognition may depend on the dynamics of the two candidate regions. Molecular dynamics simulations show that the isoalloxazine binding loops are intrinsically flexible at physiological temperatures, thus facilitating the intercalation of the cofactor, and that their mobility is modulated by the anion bound at the phosphate site.

Dynamical model for the full stretching curve of DNA.

We present a phenomenological dynamical model able to describe the stretching features of the curve of DNA length vs applied force. As concerns the chain, the model is based on the discrete wormlike chain model with elastic modifications, which properly describes the elongation features at low and intermediate forces. The dynamics is developed under a double-well potential with a linear term, which, at high forces, accounts for the narrow transition present in the DNA elongation (overstretching). A quite good agreement between simulation and experiment is obtained.

Translocation time of periodically forced polymer chains.

In this work we study the presence of both a minimum and clear oscillations in the frequency dependence of the translocation time of a polymer described as a unidimensional Rouse chain driven by a spatially localized oscillating linear potential. The observed oscillations of the mean translocation time arise from the synchronization between the very mean translocation time and the period of the external force. We have checked the robustness of the frequency value for the minimum translocation time by changing the damping parameter, finding a very simple relationship between this frequency and the correspondent translocation time. The translocation time as a function of the polymer length has been also evaluated, finding a precise L2 scaling. Furthermore, the role played by the thermal fluctuations described as a gaussian uncorrelated noise has been also investigated, and the analogies with the resonant activation phenomenon are commented.

A double-deletion method to quantifying incremental binding energies in proteins from experiment: example of a destabilizing hydrogen bonding pair.

The contribution of a specific hydrogen bond in apoflavodoxin to protein stability is investigated by combining theory, experiment and simulation. Although hydrogen bonds are major determinants of protein structure and function, their contribution to protein stability is still unclear and widely debated. The best method so far devised to estimate the contribution of side-chain interactions to protein stability is double mutant cycle analysis, but the interaction energies so derived are not identical to incremental binding energies (the energies quantifying net contributions of two interacting groups to protein stability). Here we introduce double-deletion analysis of 'isolated' residue pairs as a means to precisely quantify incremental binding. The method is exemplified by studying a surface-exposed hydrogen bond in a model protein (Asp96/Asn128 in apoflavodoxin). Combined substitution of these residues by alanines slightly destabilizes the protein due to a decrease in hydrophobic surface burial. Subtraction of this effect, however, clearly indicates that the hydrogen-bonded groups in fact destabilize the native conformation. In addition, molecular dynamics simulations and classic double mutant cycle analysis explain quantitatively that, due to frustration, the hydrogen bond must form in the native structure because when the two groups get approximated upon folding their binding becomes favorable. We would like to remark that 1), this is the first time the contribution of a specific hydrogen bond to protein stability has been measured by experiment; and 2), more hydrogen bonds need to be analyzed to draw general conclusions on protein hydrogen bond energetics. To that end, the double-deletion method should be of help.