Macromolecular Structure Analysis Research Articles

SBILib: a handle for protein modeling and engineering.

The SBILib Python library provides an integrated platform for the analysis of macromolecular structures and interactions. It combines simple 3D file parsing and workup methods with more advanced analytical tools. SBILib includes modules for macromolecular interactions, loops, super-secondary structures, and biological sequences, as well as wrappers for external tools with which to integrate their results and facilitate the comparative analysis of protein structures and their complexes. The library can handle macromolecular complexes formed by proteins and/or nucleic acid molecules (i.e. DNA and RNA). It is uniquely capable of parsing and calculating protein super-secondary structure and loop geometry. We have compiled a list of example scenarios which SBILib may be applied to and provided access to these within the library. SBILib is made available on Github at https://github.com/structuralbioinformatics/SBILib.

Identifying Metal Binding Sites in Proteins Using Homologous Structures, the MADE Approach.

In order to identify the locations of metal ions in the binding sites of proteins, we have developed a method named the MADE (MAcromolecular DEnsity and Structure Analysis) approach. The MADE approach represents an evolution of our previous toolset, the ProBiS H2O (MD) methodology, for the identification of conserved water molecules. Our method uses experimental structures of proteins homologous to a query, which are subsequently superimposed upon it. Areas with a particular species present in a similar location among many homologous protein structures are identified using a clustering algorithm. Dense clusters likely represent positions containing species important to the query protein structure or function. We analyze well-characterized apo protein structures and show that the MADE approach can identify clusters corresponding to the expected positions of metal ions in their binding sites. The greatest advantage of our method lies in its generality. It can in principle be applied to any species found in protein records; it is not only limited to metal ions. We additionally demonstrate that the MADE approach can be successfully applied to predict the location of cofactors in computer-modeled structures, e.g., via AlphaFold. We also conduct a careful protein superposition method comparison and find our methodology robust and the results largely independent of the selected protein superposition algorithm. We postulate that with increasing structural data availability, additional applications of the MADE approach will be possible such as non-protein systems, water network identification, protein binding site elaboration, and analysis of binding events, all in a dynamic manner. We have implemented the MADE approach as a plugin for the PyMOL molecular visualization tool. The MADE plugin is available free of charge at https://gitlab.com/Jukic/made_software.

Shaping the Future of Macromolecular Chemistry: A Successful Path from the Start

Shaping the Future of Macromolecular Chemistry: A Successful Path from the Start

Density Functional Theory and Molecular Modeling Studies of New 4-(Furan-2-yl) Thiazol-2-Amine Derivatives as Cyclooxygenase Inhibitors

Nonsteroidal anti-inflammatory drugs (NSAIDs) involve various of pharmacologically active compounds used in treatment of acute and chronic inflammation, relieve pain and fever, but chronic use of NSAIDs associated with gastrointestinal lesions, hemorrhage and nephrotoxicity. In this present research molecular modeling approach was applied on several derivatives of thiazole bearing Schiff base to design safe and effective compounds. These derivatives were docked inside the crystal structure of cyclooxygenase enzyme (COX-1 and COX-2) to evaluate the binding potency of each one with the active site of enzyme. Also, we used density functional theory (DFT) by applying energies evaluation of the highest occupied molecular orbitals (HOMOs) and lowest unoccupied molecular orbitals (LUMOs) in order to identify the reactivity parameter. Furthermore, root mean square deviation (RMSD) tool was used which shows an important role in the comparison of different conformers of the same ligand and it means a similarity measure vastly utilized in analysis of macromolecular structures and dynamic.

Applications of gradient magnetic fields in studies of biological macromolecules

As a kind of physical environment, magnetic fields are widely used in many different fields, ranging from scientific research to industrial production. With the rapid development of magnetic technology, the importance of magnetic fields in scientific research and practical applications has become increasingly prominent. One notable research field is the application of magnetic fields in the research of biological macromolecules. In this research field, gradient magnetic fields, as one kind of magnetic fields, have attracted much attention and have been widely used in various research fields because they can induce effects caused not only by a magnetic field but also by a magnetic field gradient. This feature makes magnetic fields applicable in many research areas, except for those directly related to the magnetic field itself. So far, the application-based research of gradient magnetic fields involving biological macromolecules has mainly focused on the crystallization, separation and purification as well as the self-assembly of biological macromolecules. Since biological macromolecules usually need to complete various processes in their solution state, the influence of gradient magnetic fields on biological macromolecules is mainly realized through their influence on the solution system. It has been found that a gradient magnetic field can affect convection in biological macromolecular solutions through Lorentz forces and magnetization forces; in practice, partial or even complete suppression of convection can often be observed. In addition, other conditions or parameters of solutions are significantly affected by the gradient magnetic field. For example, in a gradient magnetic field, the following phenomena may occur: (1) superparamagnetic particles in the solution are strongly attracted by the magnetization force, and quick clustering of these particles can happen; (2) the solution itself may have some kind of solution structure; (3) the aggregation of biological macromolecules may show a preferred orientation in the magnetic field direction; (4) the diffusion coefficient of biological macromolecules in the solution will decrease; (5) the solutions viscosity will increase, and so on. As a result of these effects, the following phenomena can be observed: during the crystallization of biological macromolecules, a preferred orientation of the crystals and crystal quality improvement of the biological macromolecules can happen; in the process of separation and purification of biological macromolecules, high-purity and highly bioactive macromolecules can be obtained rapidly and efficiently; and, in the process of the self-assembly of biological macromolecules, abnormal behaviours can occur. Based on these specific effects, the utilization of gradient magnetic fields can be further explored in more professional applications. Currently, the most common applications include the utilization of gradient magnetic fields for obtaining high- quality protein crystals and the achievement of highly efficient and low-cost separation and purification of biological macromolecules. It is obvious that gradient magnetic fields have very important application value in biological macromolecular structure analysis technology, biological drug preparation and preservation technology, theoretical studies involving biomolecular phase separation and nucleation processes, and many other directions. In this paper, the fundamental effects of the gradient magnetic field on the solution of biological macromolecules are reviewed, the applications based on the effects are discussed, and the future prospects in this field are anticipated.

MMTF-An efficient file format for the transmission, visualization, and analysis of macromolecular structures.

Recent advances in experimental techniques have led to a rapid growth in complexity, size, and number of macromolecular structures that are made available through the Protein Data Bank. This creates a challenge for macromolecular visualization and analysis. Macromolecular structure files, such as PDB or PDBx/mmCIF files can be slow to transfer, parse, and hard to incorporate into third-party software tools. Here, we present a new binary and compressed data representation, the MacroMolecular Transmission Format, MMTF, as well as software implementations in several languages that have been developed around it, which address these issues. We describe the new format and its APIs and demonstrate that it is several times faster to parse, and about a quarter of the file size of the current standard format, PDBx/mmCIF. As a consequence of the new data representation, it is now possible to visualize structures with millions of atoms in a web browser, keep the whole PDB archive in memory or parse it within few minutes on average computers, which opens up a new way of thinking how to design and implement efficient algorithms in structural bioinformatics. The PDB archive is available in MMTF file format through web services and data that are updated on a weekly basis.

How Molecular Size Impacts RMSD Applications in Molecular Dynamics Simulations.

The root-mean-square deviation (RMSD) is a similarity measure widely used in analysis of macromolecular structures and dynamics. As increasingly larger macromolecular systems are being studied, dimensionality effects such as the "curse of dimensionality" (a diminishing ability to discriminate pairwise differences between conformations with increasing system size) may exist and significantly impact RMSD-based analyses. For such large bimolecular systems, whether the RMSD or other alternative similarity measures might suffer from this "curse" and lose the ability to discriminate different macromolecular structures had not been explicitly addressed. Here, we show such dimensionality effects for both weighted and nonweighted RMSD schemes. We also provide a mechanism for the emergence of the "curse of dimensionality" for RMSD from the law of large numbers by showing that the conformational distributions from which RMSDs are calculated become increasingly similar as the system size increases. Our findings suggest the use of weighted RMSD schemes for small proteins (less than 200 residues) and nonweighted RMSD for larger proteins when analyzing molecular dynamics trajectories.

UQlust: combining profile hashing with linear-time ranking for efficient clustering and analysis of big macromolecular data.

BackgroundAdvances in computing have enabled current protein and RNA structure prediction and molecular simulation methods to dramatically increase their sampling of conformational spaces. The quickly growing number of experimentally resolved structures, and databases such as the Protein Data Bank, also implies large scale structural similarity analyses to retrieve and classify macromolecular data. Consequently, the computational cost of structure comparison and clustering for large sets of macromolecular structures has become a bottleneck that necessitates further algorithmic improvements and development of efficient software solutions.ResultsuQlust is a versatile and easy-to-use tool for ultrafast ranking and clustering of macromolecular structures. uQlust makes use of structural profiles of proteins and nucleic acids, while combining a linear-time algorithm for implicit comparison of all pairs of models with profile hashing to enable efficient clustering of large data sets with a low memory footprint. In addition to ranking and clustering of large sets of models of the same protein or RNA molecule, uQlust can also be used in conjunction with fragment-based profiles in order to cluster structures of arbitrary length. For example, hierarchical clustering of the entire PDB using profile hashing can be performed on a typical laptop, thus opening an avenue for structural explorations previously limited to dedicated resources. The uQlust package is freely available under the GNU General Public License at https://github.com/uQlust.ConclusionuQlust represents a drastic reduction in the computational complexity and memory requirements with respect to existing clustering and model quality assessment methods for macromolecular structure analysis, while yielding results on par with traditional approaches for both proteins and RNAs.Electronic supplementary materialThe online version of this article (doi:10.1186/s12859-016-1381-2) contains supplementary material, which is available to authorized users.

Predicting X-ray diffuse scattering from translation-libration-screw structural ensembles.

Identifying the intramolecular motions of proteins and nucleic acids is a major challenge in macromolecular X-ray crystallography. Because Bragg diffraction describes the average positional distribution of crystalline atoms with imperfect precision, the resulting electron density can be compatible with multiple models of motion. Diffuse X-ray scattering can reduce this degeneracy by reporting on correlated atomic displacements. Although recent technological advances are increasing the potential to accurately measure diffuse scattering, computational modeling and validation tools are still needed to quantify the agreement between experimental data and different parameterizations of crystalline disorder. A new tool, phenix.diffuse, addresses this need by employing Guinier's equation to calculate diffuse scattering from Protein Data Bank (PDB)-formatted structural ensembles. As an example case, phenix.diffuse is applied to translation-libration-screw (TLS) refinement, which models rigid-body displacement for segments of the macromolecule. To enable the calculation of diffuse scattering from TLS-refined structures, phenix.tls_as_xyz builds multi-model PDB files that sample the underlying T, L and S tensors. In the glycerophosphodiesterase GpdQ, alternative TLS-group partitioning and different motional correlations between groups yield markedly dissimilar diffuse scattering maps with distinct implications for molecular mechanism and allostery. These methods demonstrate how, in principle, X-ray diffuse scattering could extend macromolecular structural refinement, validation and analysis.

Macromolecular chain structure design, synthesis and analysis of poly(l-lactide) linking ultraviolet absorbing groups

Poly(L-lactide) (PLA) with different chain structures was designed, and successfully synthesized by ring opening polymerization using 2-hydroxy-4-(3-methacryloxy-2-hydroxylpropoxy) benzophenone (BPMA), 2,2′-dihydroxy-4,4′-(2-hydroxylpropoxy) dibenzophenone (DHDBP) and 2-hydroxy-4-(2,3-dihydroxylpropoxy) benzophenone (HPBP) as initiators, respectively, to produce corresponding PLA-B (end-capped with BPMA), PLA-DB (end-capped with DHDBP) and PLA-HB-PLA (blocked with HPBP in the middle). High-molecular-weight PLA-DB400 with good visible light transparency and UV opacity was prepared. The chemical structures of the samples were characterized, the crystallization behavior, thermal stability, UV absorption properties and transmittance of PLA were investigated and analyzed. Results of GPC and DSC reveal that when the number average molecular weights (Mn) of PLA are around 4000, termination of a UV absorbing group like DHDBP greatly restricts the crystallization of PLA due to the larger volume and rigidity of the end-capping group, but increasing Mn to about 12 000 or higher weakens the hindering effect, resulting in a similar degree of crystallization (Xc). TG results show that PLA-DB400 has the best thermal stability due to the highest molar mass. When the UV absorber is blocked in the PLA chain, the restriction and steric hindrance of the absorber are much stronger than that in the end-capped material, making Xc significantly reduced. UV absorbance of the PLA solutions reveals that the introduction of UV absorbing groups gives an absorption to UV light below 350 nm and the position of the introduced groups has no influence on the UV absorption properties although the content of the UV group does enhance the UV absorbance, of which PLA-DB has the highest since DHDBP bears two 2-dihydroxybenzophenone groups. The transparency of PLA films is neither affected by position nor content of the introduced UV absorbing groups, which is very beneficial for the application of PLA in packaging materials that require high transparency.

A new generation of CCP4 monomer library based on Crystallography Open Database

The use of prior chemical knowledge such as bond lengths, bond angles about constituent blocks of macromolecules and ligands is an essential part of macromolecular crystal structure analysis. One of the reliable sources of such chemical knowledge is small molecule database where small molecule crystal structures have been analysed against high-resolution, high-quality experimental data. Furthermore, vast amount of data in small molecule database provide comprehensive coverage of flexible chemical environment and enable proper statistical analysis to avoid biased representation of those chemical properties. This presentation describes our work on organization of the data from open-access and daily-updated small molecule database, Crystallography Open Database(COD) [1], into a new generation of CCP4 monomer library (Dictionary), a container of prior chemical knowledge [2]. In order to describe specific environment atoms are in, they are classified into different atomic types based on local graphs and some basic chemical properties of atoms. This scheme can be applied to any small molecule databases. The atom types, and values of bond lengths and bond associated with them, are further clustered into a hierarchical tree and an isomorphism-mapping algorithm is implemented to facilitate fast search among a large number of atom types (typically several millions). This also provides a mechanism to derive reliable values for bond lengths and angles of novel ligands. Metal and non-organic atoms are treated differently with organic ones. The original data in COD are curated using several criteria and further statistical analysis on derived values of bond lengths and angles are allow to extract reliable chemical information from such databanks as COD. There are several software tools associated with new dictionary including 1) generate "ideal" bond lengths and angles for unknown ligand; 2) generate starting coordinates to represent one of the conformation of the ligand under consideration.

Conformation-independent structural comparison of macromolecules with ProSMART

Structural comparison often aids insight into the function and mechanics of biological macromolecules. To make such analyses more accessible, we present the Procrustes Structural Matching Alignment and Restraints Tool (ProSMART), which is designed to allow fast but detailed comparative analysis of macromolecular structures despite potential dissimilarities in global arrangement, such as domain motion and distortion. Whilst obtaining a residue alignment between structures is a prerequisite for comparative analysis, conventional alignment methods may fail in cases where conformational differences are dramatic. However, ProSMART achieves a conformation-independent structural alignment by focusing purely on local dissimilarities, rather than enforcing chain/domain rigidity. This allows the sensible comparison of protein (or DNA/RNA) structures in the presence of conformational change. ProSMART allows analysis of the structural conservation of local backbone and side chains in a wide variety of scenarios - the method is sensitive enough to allow identification of subtle dissimilarities between structures sharing high sequence homology, whilst being versatile enough to allow identification of local similarities between more distantly-related structures. In addition, ProSMART can be used for the identification of conserved rigid substructures, which may or may not represent functional domains. ProSMART is also used for the generation of external restraints for use in crystallographic refinement. Results from ProSMART can be visualised in either CCP4mg or PyMOL. All residue-based scores are illustrated using intuitive colour gradients, allowing easy visual assessment of local backbone and side chain conservation. Complementary structural comparison tools such as ProSMART can help break the complexity of the constantly growing pool of available structural data into a more readily accessible form, and consequently may aid biological insight into macromolecular structures.

The 6th Computational Structural Bioinformatics Workshop

The rapid accumulation of macromolecular structures presents a unique set of challenges and opportunities in the analysis, comparison, modeling, and prediction of macromolecular structures and interactions. The 6th Computational Structural Bioinformatics Workshop (CSBW) was held in Philadelphia on October 4, 2012. This issue includes eleven papers selected from the work presented at the CSBW of 2012. In "Four-body atomic potential for modeling protein-ligand binding affinity: application to enzyme-inhibitor binding energy prediction", a predictive model of free-energy was built to evaluate the binding between a protein and a ligand. In "Unbiased, scalable sampling of protein loop conformations from probabilistic priors", a new Markov chain Monte Carlo algorithm was proposed to generate unbiased conformations of closed protein loops from probabilistic priors. In "Enhancement of accuracy and efficiency for RNA secondary structure prediction by sequence segmentation and MapReduce", authors demonstrate an enhanced method by cutting a long RNA sequence into smaller chunks at strategically selected points, and distributing the tasks to multiple processors. In "A population-based evolutionary search approach to the multiple minima problem in de novo protein structure prediction", the authors present an evolutionary search algorithm to obtain a discrete representation of the protein energy surface in terms of an ensemble of conformations representing local energy minima. In "Estimating loop length from Cryo-EM images at medium resolutions", the authors developed a computational geometry method to simplify the points along the skeleton to measure loop length in 3D images. The paper "A conservation and rigidity based method for detecting critical protein residues" presents a method that combines the rigidity and the evolutionary conservation in detection of the critical residues. In "A conservation and biophysics guided stochastic approach to refining docked multimeric proteins", the authors introduce a refinement method that accepts complexes consisting of any number of monomeric units. In "Elucidating the ensemble of functionally-relevant transitions in protein systems with a robotics-inspired method," the authors present a robotics-inspired tree-based method to sample energetically-credible conformational pathways connecting diverse functional states in multimodal proteins. The paper "An aggregate analysis of many predicted structures to reduce errors in protein structure comparison caused by conformational flexibility" applies protein structure prediction algorithms to enhance the classification of homologous proteins according to their binding preferences. In "DINC: A new AutoDock-based protocol for docking large ligands", an enhanced method was demonstrated for docking large ligands. The paper "Modeling protein conformational transitions by a combination of coarse-grained normal mode analysis and robotics-inspired methods" presents an efficient approach involving robotics concepts.

Low-resolution refinement tools inREFMAC5

Two aspects of low-resolution macromolecular crystal structure analysis are considered: (i) the use of reference structures and structural units for provision of structural prior information and (ii) map sharpening in the presence of noise and the effects of Fourier series termination. The generation of interatomic distance restraints by ProSMART and their subsequent application in REFMAC5 is described. It is shown that the use of such external structural information can enhance the reliability of derived atomic models and stabilize refinement. The problem of map sharpening is considered as an inverse deblurring problem and is solved using Tikhonov regularizers. It is demonstrated that this type of map sharpening can automatically produce a map with more structural features whilst maintaining connectivity. Tests show that both of these directions are promising, although more work needs to be performed in order to further exploit structural information and to address the problem of reliable electron-density calculation.

Low-resolution refinement tools in REFMAC5.

Two aspects of low-resolution macromolecular crystal structure analysis are considered: (i) the use of reference structures and structural units for provision of structural prior information and (ii) map sharpening in the presence of noise and theeffects of Fourier series termination. The generation of interatomic distance restraints by ProSMART and their subsequent application in REFMAC5 is described. It is shown that the use of such external structural information can enhance the reliability of derived atomic models and stabilize refinement. The problem of map sharpening is considered as an inverse deblurring problem and is solved using Tikhonov regularizers. It is demonstrated that this type of map sharpening can automatically produce a map with more structural features whilst maintaining connectivity. Tests show that both of these directions are promising, although more work needs to be performed in order to further exploit structural information and to address the problem of reliable electron-density calculation.

Filter Diagonalization Method-Based Mass Spectrometry for Molecular and Macromolecular Structure Analysis

Molecular and macromolecular structure analysis by high resolution and accurate mass spectrometry (MS) is indispensable for a number of fundamental and applied research areas, including health and energy domains. Comprehensive structure analysis of molecules and macromolecules present in the extremely complex samples and performed under time-constrained experimental conditions demands a substantial increase in the acquisition speed of high resolution MS data. We demonstrate here that signal processing based on the filter diagonalization method (FDM) provides the required resolution for shorter experimental transient signals in ion cyclotron resonance (ICR) MS compared to the Fourier transform (FT) processing. We thus present the development of a FDM-based MS (FDM MS) and demonstrate its implementation in ICR MS. The considered FDM MS applications are in bottom-up and top-down proteomics, metabolomics, and petroleomics.

Adapting Visual‐Analytical Tools for the Exploration of Structural and Dynamical Features of Polymer Conformations

AbstractConformational analysis of macromolecular structures reveals interesting higher‐order spatial arrangements. Analyzing these features as a function of time provides insights into the dynamical behavior of these systems and the identification of relevant subdomains. We present some visual‐analytic methods that we devised to explore the spatial‐temporal properties from molecular dynamics simulation data. These methods automatically detect common features and connect them to properties of interest. These methods yield physical insights that are not easily obtainable with existing methods for particle simulation data, as illustrated for polyacetylene interacting with a carbon nanotube. magnified image

Generalized X-ray and neutron crystallographic analysis: more accurate and complete structures for biological macromolecules

X-ray and neutron crystallographic techniques provide complementary information on the structure and function of biological macromolecules. X-ray and neutron (XN) crystallographic data have been combined in a joint structure-refinement procedure that has been developed using recent advances in modern computational methodologies, including cross-validated maximum-likelihood target functions with gradient-based optimization and simulated annealing. The XN approach for complete (including hydrogen) macromolecular structure analysis provides more accurate and complete structures, as demonstrated for diisopropyl fluorophosphatase, photoactive yellow protein and human aldose reductase. Furthermore, this method has several practical advantages, including the easier determination of the orientation of water molecules, hydroxyl groups and some amino-acid side chains.

Position of the non-template DNA in the Polymerase II elongation complex revealed using single-molecule Fluorescence Resonance Energy Transfer

Single Molecule Fluorescence Resonance Energy Transfer (FRET) was used as a quantitative tool to localize non-template DNA in the RNA Polymerase II (Pol II) elongation complex.We measured the FRET efficiencies between ‘antenna dye molecules’ (ADMs) attached to the non-template DNA and several ‘satellite dye molecules’ (SDMs) attached to known positions within the Pol II elongation complex using the recently developed Nano Positioning System (NPS) (1). NPS combines x-ray crystallographic information, single-molecule FRET data and bayesian parameter estimation, a probability based analysis method, to compute the three dimensional probability density for the position of ADMs. Within the bayesian framework, we were able to account for errors in the determined Forster radii, errors in the measured FRET efficiencies and uncertainties in the SDM positions (due to the attachment of the SDMs via flexible linkers) as well as for geometric constraints.We determined positions of the non-template DNA at +1, -2, -4, -7 (within the transcription bubble) and -12, -15, -18 of the up-stream DNA. Together with the high resolution x-ray structure and our earlier studies (2) determining the position of the nascent RNA this study completes the picture of the Pol II elongation complex.1. A. Muschielok, J. Andrecka, A. Jawhari, F. Bruckner, P. Cramer and J. Michaelis, A nano-positioning system for macromolecular structural analysis. Nature Methods, accepted.2. J. Andrecka, R. Lewis, F. Brueckner, E. Lehmann, P. Cramer and J. Michaelis, Tracking the position of mRNA in RNA polymerase II elongation complexes. PNAS 105, 135-140 (2008).

Comparison of conventional and controlled bulk polymerization of styrene by N-methyl-2-pyrrolidone and 1-dodecanethiol

AbstractControlled radical polymerization of styrene was initiated using Nmethyl- 2-pyrrolidone (NM2P) and 1-dodecanethiol (1DT). This polymerization system exhibited “living” characteristics, namely the molecular weight averages of resulting polymers increased linearly with conversion, which has been determined by gel permeation chromatography (GPC) analysis. The polymer was additionally characterized with Fourier transform infrared spectroscopy (FTIR) and proton nuclear magnetic resonance (1H NMR) spectroscopy. Kinetics of controlled radical polymerization has been studied and the temperature dependency of overall polymerization rate constant has been determined. A comparison of conventional bulk radical and controlled bulk radical polymerization of styrene has been explored. Conventional radical polymerization was initiated with 2,2’-azobis(2- methylpropionitrile) (AIBN), whereas 1-dodecanethiol was applied as a chain transfer agent. The reaction rate of controlled radical polymerization was slower than the reaction rate of conventional radical polymerization initiated with AIBN. Consequentially, the activation energy and the pre-exponential factor were lower in the case of controlled polymerization in comparison to the ones observed in the conventional - AIBN initiated - system. Furthermore, by comparing the controlled to the conventional radical polymerization lower polydispersities were observed. Macromolecular structure analysis suggested a more linear chain structure in the controlled radical polymerization system. The thermal properties of polymer products have also been studied and the corresponding glass transition temperatures were higher upon comparison of the conventional and the AIBN initiated system, respectively