BACKGROUND:The way mechanical stress is distributed inside and propagated by proteins and other biopolymers largely defines their function. Yet, determining the network of interactions propagating internal strain remains a challenge for both, experiment and theory. Based on molecular dynamics simulations, we developed force distribution analysis (FDA), a method that allows visualizing strain propagation in macromolecules.RESULTS:To be immediately applicable to a wide range of systems, FDA was implemented as an extension to Gromacs, a commonly used package for molecular simulations. The FDA code comes with an easy-to-use command line interface and can directly be applied to every system built using Gromacs. We provide an additional R-package providing functions for advanced statistical analysis and presentation of the FDA data.CONCLUSIONS:Using FDA, we were able to explain the origin of mechanical robustness in immunoglobulin domains and silk fibers. By elucidating propagation of internal strain upon ligand binding, we previously also successfully revealed the functionality of a stiff allosteric protein. FDA thus has the potential to be a valuable tool in the investigation and rational design of mechanical properties in proteins and nano-materials.

Atomistic models of β-sheet crystals with varying number of alanines were mechanically tested in Molecular Dynamics simulations. The gain in mechanical robustness per residue is maximal for strands that are 8 residues long, because the external force is efficiently deflected into neighbouring strands over the distance they span, rendering 8 residues a mechanical optimum for a semi-crystalline material like silk.

Here we decipher the molecular determinants for the extreme toughness of spider silk fibers. Our bottom-up computational approach incorporates molecular dynamics and finite element simulations. Therefore, the approach allows the analysis of the internal strain distribution and load-carrying motifs in silk fibers on scales of both molecular and continuum mechanics. We thereby dissect the contributions from the nanoscale building blocks, the soft amorphous and the strong crystalline subunits, to silk fiber mechanics. We identify the amorphous subunits not only to give rise to high elasticity, but to also ensure efficient stress homogenization through the friction between entangled chains, which also allows the crystals to withstand stresses as high as 2 GPa in the context of the amorphous matrix. We show that the maximal toughness of silk is achieved at 10-40% crystallinity depending on the distribution of crystals in the fiber. We also determined a serial arrangement of the crystalline and amorphous subunits in lamellae to outperform a random or a parallel arrangement, putting forward what we believe to be a new structural model for silk and other semicrystalline materials. The multiscale approach, not requiring any empirical parameters, is applicable to other partially ordered polymeric systems. Hence, it is an efficient tool for the design of artificial silk fibers.

Silk is one of the most resilient fibers in nature. Consisting of an amorphous matrix cross-linked by beta-sheet rich crystalline units, silk is a hierarchically organized material the molecular details of which remain largely unknown. In order to decipher the structural determinants of its mechanical properties, we model silk at different length scales by combining molecular dynamics simulations, force distribution analysis, novel force-based coarse-grain models, and finite element methods. We predict the distinct mechanics of anti-parallel versus parallel silk crystals as force-bearing cross-links [1], and the impact of chain entanglement and crystallinity on fiber mechanics [2]. Our predictions can serve as a guide for the design of artificial silk protein analogues. [1] S. Xiao, W. Stacklies, M. Cetinkaya, B. Markert, F. Gräter. Mechanical Response of Silk Crystalline Units from Force Distribution Analysis. Biophysical Journal, 96(10) 3997-4005 (2009) [2] M. Cetinkaya et al, in preparation

Single-molecule measurements of how proteins respond to applied force can provide valuable clues to their structure-function relationship, particularly for proteins whose role in vivo relies on an ability to resist or sense force. Interpretation of such measurements relies heavily on theoretical and computational modeling; however, the brute-force approach, molecular dynamics simulation at atomic resolution, is only feasible for timescales orders of magnitude shorter than those appropriate to experiments. Thus coarse-graining is essential for accessing experimentally relevant timescales. However, almost all coarse-grained protein models to date have been designed for the explicit purpose of studying protein folding or normal mode flexibility, and are not capable of supplying quantitative predictions about response to large applied forces. We develop a new procedure for using force measurements from all-atom molecular dynamics simulations to parameterize a coarse-grained model specifically designed for studying force response. This model has the novel feature of using the flexible Morse potential as a basis function for describing non-bonded interactions. We test the model by using it to study the kinetics of ubiquitin rupture under quasi-equilibrium forcing, and compare with experimental results.

Mechanical force is routinely applied to proteins in force probing experiments and simulations to observe a protein's response to external stress. A yet unanswered question is how force propagates through proteins. How do perturbations like an external force flow through protein scaffolds and how is this related to protein stability and function? We elucidate force distribution in the titin 127 domain by monitoring alterations in forces between pairs of atoms in the folded state upon pulling the protein with a constant force. We find forces to be a more direct measure for internal strain than the only minor changes in atomic coordinates. We observe that the externally applied force is anisotropically distributed throughout the protein scaffold highlighting three prominent regions that contribute most of the protein's mechanical resistance. The functional relevance of the force distribution network is highlighted by additional mutant simulations. The method can easily be extended to other types of perturbation including allosteric signals, such as ligand binding, phosphorylation, or a change in protonation state.

Many fundamental cellular processes such as gene expression are tightly regulated by protein allostery. Allosteric signal propagation from the regulatory to the active site requires long-range communication, the molecular mechanism of which remains a matter of debate. A classical example for long-range allostery is the activation of the methionine repressor MetJ, a transcription factor. Binding of its co-repressor SAM increases its affinity for DNA several-fold, but has no visible conformational effect on its DNA binding interface. Our molecular dynamics simulations indicate correlated domain motions within MetJ, and quenching of these dynamics upon SAM binding entropically favors DNA binding. From monitoring conformational fluctuations alone, it is not obvious how the presence of SAM is communicated through the largely rigid core of MetJ and how SAM thereby is able to regulate MetJ dynamics. We here directly monitored the propagation of internal forces through the MetJ structure, instead of relying on conformational changes as conventionally done. Our force distribution analysis successfully revealed the molecular network for strain propagation, which connects collective domain motions through the protein core. Parts of the network are directly affected by SAM binding, giving rise to the observed quenching of fluctuations. Our results are in good agreement with experimental data. The force distribution analysis suggests itself as a valuable tool to gain insight into the molecular function of a whole class of allosteric proteins.

BACKGROUND:To avoid pathological platelet aggregation by von Willebrand factor (VWF), VWF multimers are regulated in size and reactivity for adhesion by ADAMTS13-mediated proteolysis in a shear flow dependent manner.OBJECTIVE AND METHODS:We examined whether tensile stress in VWF under shear flow activates the VWF A2 domain for cleavage by ADAMTS13 using molecular dynamics simulations. We generated a full length mutant VWF featuring a homologous disulfide bond in A2 (N1493C and C1670S), in an attempt to lock A2 against unfolding.RESULTS:We indeed observed stepwise unfolding of A2 and exposure of its deeply buried ADAMTS13 cleavage site. Interestingly, disulfide bonds in the adjacent and highly homologous VWF A1 and A3 domains obstruct their mechanical unfolding. We find this mutant A2 (N1493C and C1670S) to feature ADAMTS13-resistant behavior in vitro.CONCLUSIONS:Our results yield molecular-detail evidence for the force-sensing function of VWF A2, by revealing how tension in VWF due to shear flow selectively exposes the A2 proteolysis site to ADAMTS13 for cleavage while keeping the folded remainder of A2 intact and functional. We find the unconventional 'knotted' Rossmann fold of A2 to be the key to this mechanical response, tailored for regulating VWF size and activity. Based on our model we discuss the pathomechanism of some natural mutations in the VWF A2 domain that significantly increase the cleavage by ADAMTS13 without shearing or chemical denaturation, and provide with the cleavage-activated A2 conformation a structural basis for the design of inhibitors for VWF type 2 diseases.

The outstanding mechanical toughness of silk fibers is thought to be caused by embedded crystalline units acting as cross links of silk proteins in the fiber. Here, we examine the robustness of these highly ordered beta-sheet structures by molecular dynamics simulations and finite element analysis. Structural parameters and stress-strain relationships of four different models, from spider and Bombyx mori silk peptides, in antiparallel and parallel arrangement, were determined and found to be in good agreement with x-ray diffraction data. Rupture forces exceed those of any previously examined globular protein many times over, with spider silk (poly-alanine) slightly outperforming Bombyx mori silk ((Gly-Ala)(n)). All-atom force distribution analysis reveals both intrasheet hydrogen-bonding and intersheet side-chain interactions to contribute to stability to similar extent. In combination with finite element analysis of simplified beta-sheet skeletons, we could ascribe the distinct force distribution pattern of the antiparallel and parallel silk crystalline units to the difference in hydrogen-bond geometry, featuring an in-line or zigzag arrangement, respectively. Hydrogen-bond strength was higher in antiparallel models, and ultimately resulted in higher stiffness of the crystal, compensating the effect of the mechanically disadvantageous in-line hydrogen-bond geometry. Atomistic and coarse-grained force distribution patterns can thus explain differences in mechanical response of silk crystals, opening up the road to predict full fiber mechanics.

The role of mechanical force in cellular processes is increasingly revealed by single molecule experiments and simulations of force-induced transitions in proteins. How the applied force propagates within proteins determines their mechanical behavior yet remains largely unknown. We present a new method based on molecular dynamics simulations to disclose the distribution of strain in protein structures, here for the newly determined high-resolution crystal structure of I27, a titin immunoglobulin (IG) domain. We obtain a sparse, spatially connected, and highly anisotropic mechanical network. This allows us to detect load-bearing motifs composed of interstrand hydrogen bonds and hydrophobic core interactions, including parts distal to the site to which force was applied. The role of the force distribution pattern for mechanical stability is tested by in silico unfolding of I27 mutants. We then compare the observed force pattern to the sparse network of coevolved residues found in this family. We find a remarkable overlap, suggesting the force distribution to reflect constraints for the evolutionary design of mechanical resistance in the IG family. The force distribution analysis provides a molecular interpretation of coevolution and opens the road to the study of the mechanism of signal propagation in proteins in general.

Mechanical force is routinely applied to proteins in force probing experiments and simulations to observe a protein's response to external stress. A yet unanswered question is how force propagates through proteins. How do perturbations like an external force flow through protein scaffolds and how is this related to protein stability and function? We here present a new method based on molecular dynamics simulations that allows visualizing stress propagation in proteins, resembling finite element analysis for macroscopic structures. Using this method we elucidate force distribution in I27, an immunoglobulin domain from human muscle titin and one of the most stable proteins known. Hereto we monitor alterations in forces between pairs of atoms in the folded state upon pulling the protein with a constant force. We find forces to be a more direct measure for internal strain than the only minor changes in atomic coordinates. We observe that the externally applied force is anisotropically distributed throughout the protein scaffold highlighting three prominent regions that contribute most of the protein's mechanical resistance. The functional relevance of the force distribution network is highlighted by unfolding simulations of in-silico mutants and, interestingly, by comparison with a network of coevolved residues found in the titin immunoglobulin family. Both networks show a remarkable overlap thereby suggesting that the force distribution pattern reflects evolutionary constraints used to render I27 a mechanically robust protein. We also show that the method can easily be extended to other types of perturbation including point mutations and allosteric signals, such as ligand binding or phosphorylation.

UNLABELLED pcaMethods is a Bioconductor compliant library for computing principal component analysis (PCA) on incomplete data sets. The results can be analyzed directly or used to estimate missing values to enable the use of missing value sensitive statistical methods. The package was mainly developed with microarray and metabolite data sets in mind, but can be applied to any other incomplete data set as well. AVAILABILITY http://www.bioconductor.org

Overview The pcaMethods package [1] provides a set of di(cid:27)erent PCA implementations, together with tools for cross validation and visualisation of the results. The methods basically allow to perform PCA on incomplete data and thus may also be used for missing value estimation