Protein multi-scale organization through graph partitioning and robustness analysis: application to the myosin–myosin light chain interaction

Mechanism of elastic energy storage of honey bee abdominal muscles under stress relaxation

Energy storage of passive muscles plays an important part in frequent activities of honey bee abdomens due to the muscle distribution and open circulatory system. However, the elastic energy and mechanical properties of structure in passive muscles remain unclear. In this article, stress relaxation tests on passive muscles from the terga of the honey bee abdomens were performed under different concentrations of blebbistatin and motion parameters. In stress relaxation, the load drop with the rapid and slow stages depending on stretching velocity and stretching length reflects the features of myosin-titin series structure and cross-bridge-actin cyclic connections in muscles. Then a model with 2 parallel modules based on the 2 feature structures in muscles was thus developed. The model described the stress relaxation and stretching of passive muscles from honey bee abdomen well for a good fitting in stress relaxation and verification in loading process. In addition, the stiffness change of cross-bridge under different concentrations of blebbistatin is obtained from the model. We derived the elastic deformation of cross-bridge and the partial derivatives of energy expressions on motion parameters from this model, which accorded the experimental results. This model reveals the mechanism of passive muscles from honey bee abdomens suggesting that the temporary energy storage of cross-bridge in terga muscles under abdomen bending provides potential energy for springback during the periodic abdomen bending of honey bee or other arthropod insects. The finding also provides an experimental and theoretical basis for the novel microstructure and material design of bionic muscle.

Allosteric Hotspots in the Main Protease of SARS-CoV-2

Inhibiting the main protease of SARS-CoV-2 is of great interest in tackling the COVID-19 pandemic caused by the virus. Most efforts have been centred on inhibiting the binding site of the enzyme. However, considering allosteric sites, distant from the active or orthosteric site, broadens the search space for drug candidates and confers the advantages of allosteric drug targeting. Here, we report the allosteric communication pathways in the main protease dimer by using two novel fully atomistic graph-theoretical methods: Bond-to-bond propensity, which has been previously successful in identifying allosteric sites in extensive benchmark data sets without a priori knowledge, and Markov transient analysis, which has previously aided in finding novel drug targets in catalytic protein families. Using statistical bootstrapping, we score the highest ranking sites against random sites at similar distances, and we identify four statistically significant putative allosteric sites as good candidates for alternative drug targeting.

Prediction of Protein Allosteric Signalling Pathways and Functional Residues Through Paths of Optimised Propensity

Allostery commonly refers to the mechanism that regulates protein activity through the binding of a molecule at a different, usually distal, site from the orthosteric site. The omnipresence of allosteric regulation in nature and its potential for drug design and screening render the study of allostery invaluable. Nevertheless, challenges remain as few computational methods are available to effectively predict allosteric sites, identify signalling pathways involved in allostery, or to aid with the design of suitable molecules targeting such sites. Recently, bond-to-bond propensity analysis has been shown successful at identifying allosteric sites for a large and diverse group of proteins from knowledge of the orthosteric sites and its ligands alone by using network analysis applied to energy-weighted atomistic protein graphs. To address the identification of signalling pathways, we propose here a method to compute and score paths of optimised propensity that link the orthosteric site with the identified allosteric sites, and identifies crucial residues that contribute to those paths. We showcase the approach with three well-studied allosteric proteins: h-Ras, caspase-1, and 3-phosphoinositide-dependent kinase-1 (PDK1). Key residues in both orthosteric and allosteric sites were identified and showed agreement with experimental results, and pivotal signalling residues along the pathway were also revealed, thus providing alternative targets for drug design. By using the computed path scores, we were also able to differentiate the activity of different allosteric modulators.

Prediction of allosteric sites and signaling: Insights from benchmarking datasets

Allostery is a pervasive mechanism that regulates protein activity through ligand binding at a site different from the orthosteric site. The universality of allosteric regulation complemented by the benefits of highly specific and potentially non-toxic allosteric drugs makes uncovering allosteric sites invaluable. However, there are few computational methods to effectively predict them. Bond-to-bond propensity analysis has successfully predicted allosteric sites in 19 of 20 cases using an energy-weighted atomistic graph. We here extended the analysis onto 432 structures of 146 proteins from two benchmarking datasets for allosteric proteins: ASBench and CASBench. We further introduced two statistical measures to account for the cumulative effect of high-propensity residues and the crucial residues in a given site. The allosteric site is recovered for 127 of 146 proteins (407 of 432 structures) knowing only the orthosteric sites or ligands. The quantitative analysis using a range of statistical measures enables better characterization of potential allosteric sites and mechanisms involved.

Spectral clustering based on structural magnetic resonance imaging and its relationship with major depressive disorder and cognitive ability

There is increasing interest in using data-driven unsupervised methods to identify structural underpinnings of common mental illnesses, including major depressive disorder (MDD) and associated traits such as cognition. However, studies are often limited to severe clinical cases with small sample sizes and most do not include replication. Here, we examine two relatively large samples with structural magnetic resonance imaging (MRI), measures of lifetime MDD and cognitive variables: Generation Scotland (GS subsample, N = 980) and UK Biobank (UKB, N = 8,900), for discovery and replication, using an exploratory approach. Regional measures of FreeSurfer derived cortical thickness (CT), cortical surface area (CSA), cortical volume (CV) and subcortical volume (subCV) were input into a clustering process, controlling for common covariates. The main analysis steps involved constructing participant K-nearest neighbour graphs and graph partitioning with Markov stability to determine optimal clustering of participants. Resultant clusters were (1) checked whether they were replicated in an independent cohort and (2) tested for associations with depression status and cognitive measures. Participants separated into two clusters based on structural brain measurements in GS subsample, with large Cohen's d effect sizes between clusters in higher order cortical regions, commonly associated with executive function and decision making. Clustering was replicated in the UKB sample, with high correlations of cluster effect sizes for CT, CSA, CV and subCV between cohorts across regions. The identified clusters were not significantly different with respect to MDD case-control status in either cohort (GS subsample: p_FDR  = .2239-.6585; UKB: p_FDR  = .2003-.7690). Significant differences in general cognitive ability were, however, found between the clusters for both datasets, for CSA, CV and subCV (GS subsample: d = 0.2529-.3490, p_FDR  < .005; UKB: d = 0.0868-0.1070, p_FDR  < .005). Our results suggest that there are replicable natural groupings of participants based on cortical and subcortical brain measures, which may be related to differences in cognitive performance, but not to the MDD case-control status.

ProteinLens: a web-based application for the analysis of allosteric signalling on atomistic graphs of biomolecules

The investigation of allosteric effects in biomolecular structures is of great current interest in diverse areas, from fundamental biological enquiry to drug discovery. Here we present ProteinLens, a user-friendly and interactive web application for the investigation of allosteric signalling based on atomistic graph-theoretical methods. Starting from the PDB file of a biomolecule (or a biomolecular complex) ProteinLens obtains an atomistic, energy-weighted graph description of the structure of the biomolecule, and subsequently provides a systematic analysis of allosteric signalling and communication across the structure using two computationally efficient methods: Markov Transients and bond-to-bond propensities. ProteinLens scores and ranks every bond and residue according to the speed and magnitude of the propagation of fluctuations emanating from any site of choice (e.g. the active site). The results are presented through statistical quantile scores visualised with interactive plots and adjustable 3D structure viewers, which can also be downloaded. ProteinLens thus allows the investigation of signalling in biomolecular structures of interest to aid the detection of allosteric sites and pathways. ProteinLens is implemented in Python/SQL and freely available to use at: www.proteinlens.io.

HCGA: Highly comparative graph analysis for network phenotyping

Networks are widely used as mathematical models of complex systems across many scientific disciplines. Decades of work have produced a vast corpus of research characterizing the topological, combinatorial, statistical, and spectral properties of graphs. Each graph property can be thought of as a feature that captures important (and sometimes overlapping) characteristics of a network. In this paper, we introduce HCGA, a framework for highly comparative analysis of graph datasets that computes several thousands of graph features from any given network. HCGA also offers a suite of statistical learning and data analysis tools for automated identification and selection of important and interpretable features underpinning the characterization of graph datasets. We show that HCGA outperforms other methodologies on supervised classification tasks on benchmark datasets while retaining the interpretability of network features. We exemplify HCGA by predicting the charge transfer in organic semiconductors and clustering a dataset of neuronal morphology images.

Small Conformational Changes Underlie Evolution of Resistance to NNRTI in HIV Reverse Transcriptase

Despite achieving considerable success in reducing the number of fatalities due to acquired immunodeficiency syndrome, emergence of resistance against the reverse transcriptase (RT) inhibitor drugs remains one of the biggest challenges of the human immunodeficiency virus antiretroviral therapy (ART). Non-nucleoside reverse transcriptase inhibitors (NNRTIs) form a large class of drugs and a crucial component of ART. In NNRTIs, even a single resistance mutation is known to make the drugs completely ineffective. Additionally, several inhibitor-bound RTs with single resistance mutations do not exhibit any significant variations in their three-dimensional structures compared with the inhibitor-bound RT but completely nullify their inhibitory functions. This makes understanding the structural mechanism of these resistance mutations crucial for drug development. Here, we study several single resistance mutations in the allosteric inhibitor (nevirapine)-bound RT to analyze the mechanism of small structural changes leading to these large functional effects. In this study, we have shown that in absence of significant conformational variations in the inhibitor-bound wild-type RT and RT with single resistance mutations, the protein contact network analysis of their static structures, along with molecular dynamics simulations, can be a useful approach to understand the functional effect of small local conformational variations. The simple network analysis exposes the localized contact changes that lead to global rearrangement in the communication pattern within RT. Furthermore, these conformational changes have implications on the overall dynamics of RT. Using various measures, we show that a single resistance mutation can change the network structure and dynamics of RT to behave more like unbound RT, even in the presence of the inhibitor. This combined coarse-grained contact network and molecular dynamics approach promises to be a useful tool to analyze structure-function studies of proteins that show large functional changes with negligible variations in their overall conformation.

Multiscale dynamical embeddings of complex networks

Complex systems and relational data are often abstracted as dynamical processes on networks. To understand, predict, and control their behavior, a crucial step is to extract reduced descriptions of such networks. Inspired by notions from control theory, we propose a time-dependent dynamical similarity measure between nodes, which quantifies the effect a node-input has on the network. This dynamical similarity induces an embedding that can be employed for several analysis tasks. Here we focus on (i) dimensionality reduction, i.e., projecting nodes onto a low-dimensional space that captures dynamic similarity at different timescales, and (ii) how to exploit our embeddings to uncover functional modules. We exemplify our ideas through case studies focusing on directed networks without strong connectivity and signed networks. We further highlight how certain ideas from community detection can be generalized and linked to control theory, by using the here developed dynamical perspective.

Allostery and cooperativity in multimeric proteins: bond-to-bond propensities in ATCase

Aspartate carbamoyltransferase (ATCase) is a large dodecameric enzyme with six active sites that exhibits allostery: its catalytic rate is modulated by the binding of various substrates at distal points from the active sites. A recently developed method, bond-to-bond propensity analysis, has proven capable of predicting allosteric sites in a wide range of proteins using an energy-weighted atomistic graph obtained from the protein structure and given knowledge only of the location of the active site. Bond-to-bond propensity establishes if energy fluctuations at given bonds have significant effects on any other bond in the protein, by considering their propagation through the protein graph. In this work, we use bond-to-bond propensity analysis to study different aspects of ATCase activity using three different protein structures and sources of fluctuations. First, we predict key residues and bonds involved in the transition between inactive (T) and active (R) states of ATCase by analysing allosteric substrate binding as a source of energy perturbations in the protein graph. Our computational results also indicate that the effect of multiple allosteric binding is non linear: a switching effect is observed after a particular number and arrangement of substrates is bound suggesting a form of long range communication between the distantly arranged allosteric sites. Second, cooperativity is explored by considering a bisubstrate analogue as the source of energy fluctuations at the active site, also leading to the identification of highly significant residues to the T ↔ R transition that enhance cooperativity across active sites. Finally, the inactive (T) structure is shown to exhibit a strong, non linear communication between the allosteric sites and the interface between catalytic subunits, rather than the active site. Bond-to-bond propensity thus offers an alternative route to explain allosteric and cooperative effects in terms of detailed atomistic changes to individual bonds within the protein, rather than through phenomenological, global thermodynamic arguments.

Multiresolution Consensus Clustering in Networks

Networks often exhibit structure at disparate scales. We propose a method for identifying community structure at different scales based on multiresolution modularity and consensus clustering. Our contribution consists of two parts. First, we propose a strategy for sampling the entire range of possible resolutions for the multiresolution modularity quality function. Our approach is directly based on the properties of modularity and, in particular, provides a natural way of avoiding the need to increase the resolution parameter by several orders of magnitude to break a few remaining small communities, necessitating the introduction of ad-hoc limits to the resolution range with standard sampling approaches. Second, we propose a hierarchical consensus clustering procedure, based on a modified modularity, that allows one to construct a hierarchical consensus structure given a set of input partitions. While here we are interested in its application to partitions sampled using multiresolution modularity, this consensus clustering procedure can be applied to the output of any clustering algorithm. As such, we see many potential applications of the individual parts of our multiresolution consensus clustering procedure in addition to using the procedure itself to identify hierarchical structure in networks.

Multi-scale brain networks

The network architecture of the human brain has become a feature of increasing interest to the neuroscientific community, largely because of its potential to illuminate human cognition, its variation over development and aging, and its alteration in disease or injury. Traditional tools and approaches to study this architecture have largely focused on single scales-of topology, time, and space. Expanding beyond this narrow view, we focus this review on pertinent questions and novel methodological advances for the multi-scale brain. We separate our exposition into content related to multi-scale topological structure, multi-scale temporal structure, and multi-scale spatial structure. In each case, we recount empirical evidence for such structures, survey network-based methodological approaches to reveal these structures, and outline current frontiers and open questions. Although predominantly peppered with examples from human neuroimaging, we hope that this account will offer an accessible guide to any neuroscientist aiming to measure, characterize, and understand the full richness of the brain's multiscale network structure-irrespective of species, imaging modality, or spatial resolution.

Prediction of allosteric sites and mediating interactions through bond-to-bond propensities

Allostery is a fundamental mechanism of biological regulation, in which binding of a molecule at a distant location affects the active site of a protein. Allosteric sites provide targets to fine-tune protein activity, yet we lack computational methodologies to predict them. Here we present an efficient graph-theoretical framework to reveal allosteric interactions (atoms and communication pathways strongly coupled to the active site) without a priori information of their location. Using an atomistic graph with energy-weighted covalent and weak bonds, we define a bond-to-bond propensity quantifying the non-local effect of instantaneous bond fluctuations propagating through the protein. Significant interactions are then identified using quantile regression. We exemplify our method with three biologically important proteins: caspase-1, CheY, and h-Ras, correctly predicting key allosteric interactions, whose significance is additionally confirmed against a reference set of 100 proteins. The almost-linear scaling of our method renders it suitable for high-throughput searches for candidate allosteric sites.

Using higher-order Markov models to reveal flow-based communities in networks

Complex systems made of interacting elements are commonly abstracted as networks, in which nodes are associated with dynamic state variables, whose evolution is driven by interactions mediated by the edges. Markov processes have been the prevailing paradigm to model such a network-based dynamics, for instance in the form of random walks or other types of diffusions. Despite the success of this modelling perspective for numerous applications, it represents an over-simplification of several real-world systems. Importantly, simple Markov models lack memory in their dynamics, an assumption often not realistic in practice. Here, we explore possibilities to enrich the system description by means of second-order Markov models, exploiting empirical pathway information. We focus on the problem of community detection and show that standard network algorithms can be generalized in order to extract novel temporal information about the system under investigation. We also apply our methodology to temporal networks, where we can uncover communities shaped by the temporal correlations in the system. Finally, we discuss relations of the framework of second order Markov processes and the recently proposed formalism of using non-backtracking matrices for community detection.

A new multi-scale method to reveal hierarchical modular structures in biological networks

Biological networks are effective tools for studying molecular interactions.

Diffusion on networked systems is a question of time or structure

Network science investigates the architecture of complex systems to understand their functional and dynamical properties. Structural patterns such as communities shape diffusive processes on networks. However, these results hold under the strong assumption that networks are static entities where temporal aspects can be neglected. Here we propose a generalized formalism for linear dynamics on complex networks, able to incorporate statistical properties of the timings at which events occur. We show that the diffusion dynamics is affected by the network community structure and by the temporal properties of waiting times between events. We identify the main mechanism--network structure, burstiness or fat tails of waiting times--determining the relaxation times of stochastic processes on temporal networks, in the absence of temporal-structure correlations. We identify situations when fine-scale structure can be discarded from the description of the dynamics or, conversely, when a fully detailed model is required due to temporal heterogeneities.

Revealing cell assemblies at multiple levels of granularity

BACKGROUND

Current neuronal monitoring techniques, such as calcium imaging and multi-electrode arrays, enable recordings of spiking activity from hundreds of neurons simultaneously. Of primary importance in systems neuroscience is the identification of cell assemblies: groups of neurons that cooperate in some form within the recorded population.

NEW METHOD

We introduce a simple, integrated framework for the detection of cell-assemblies from spiking data without a priori assumptions about the size or number of groups present. We define a biophysically-inspired measure to extract a directed functional connectivity matrix between both excitatory and inhibitory neurons based on their spiking history. The resulting network representation is analyzed using the Markov Stability framework, a graph theoretical method for community detection across scales, to reveal groups of neurons that are significantly related in the recorded time-series at different levels of granularity.

RESULTS AND COMPARISON WITH EXISTING METHODS

Using synthetic spike-trains, including simulated data from leaky-integrate-and-fire networks, our method is able to identify important patterns in the data such as hierarchical structure that are missed by other standard methods. We further apply the method to experimental data from retinal ganglion cells of mouse and salamander, in which we identify cell-groups that correspond to known functional types, and to hippocampal recordings from rats exploring a linear track, where we detect place cells with high fidelity.

CONCLUSIONS

We present a versatile method to detect neural assemblies in spiking data applicable across a spectrum of relevant scales that contributes to understanding spatio-temporal information gathered from systems neuroscience experiments.

Constant communities in complex networks

Identifying community structure is a fundamental problem in network analysis. Most community detection algorithms are based on optimizing a combinatorial parameter, for example modularity. This optimization is generally NP-hard, thus merely changing the vertex order can alter their assignments to the community. However, there has been less study on how vertex ordering influences the results of the community detection algorithms. Here we identify and study the properties of invariant groups of vertices (constant communities) whose assignment to communities are, quite remarkably, not affected by vertex ordering. The percentage of constant communities can vary across different applications and based on empirical results we propose metrics to evaluate these communities. Using constant communities as a pre-processing step, one can significantly reduce the variation of the results. Finally, we present a case study on phoneme network and illustrate that constant communities, quite strikingly, form the core functional units of the larger communities.

Structure of complex networks: Quantifying edge-to-edge relations by failure-induced flow redistribution

The analysis of complex networks has so far revolved mainly around the role of nodes and communities of nodes. However, the dynamics of interconnected systems is often focalized on edge processes, and a dual edge-centric perspective can often prove more natural. Here we present graph-theoretical measures to quantify edge-to-edge relations inspired by the notion of flow redistribution induced by edge failures. Our measures, which are related to the pseudo-inverse of the Laplacian of the network, are global and reveal the dynamical interplay between the edges of a network, including potentiallynon-localinteractions. Our framework also allows us to define the embeddedness of an edge, a measure of how strongly an edge features in the weighted cuts of the network. We showcase the general applicability of our edge-centric framework through analyses of the Iberian power grid, traffic flow in road networks, and theC. elegansneuronal network.

Multi-scale community organization of the human structural connectome and its relationship with resting-state functional connectivity

The human connectome has been widely studied over the past decade. A principal finding is that it can be decomposed into communities of densely interconnected brain regions. Past studies have often used single-scale modularity measures in order to infer the connectome's community structure, possibly overlooking interesting structure at other organizational scales. In this report, we used the partition stability framework, which defines communities in terms of a Markov process (random walk), to infer the connectome's multi-scale community structure. Comparing the community structure to observed resting-state functional connectivity revealed communities across a broad range of scales that were closely related to functional connectivity. This result suggests a mapping between communities in structural networks, models of influence-spreading and diffusion, and brain function. It further suggests that the spread of influence among brain regions may not be limited to a single characteristic scale.

Uncovering allosteric pathways in caspase-1 using Markov transient analysis and multiscale community detection

Atomistic graph–theoretical analysis of caspase-1 reveals details of intra-protein communication pathways.

Structure and dynamics of molecular networks: a novel paradigm of drug discovery: a comprehensive review

Despite considerable progress in genome- and proteome-based high-throughput screening methods and in rational drug design, the increase in approved drugs in the past decade did not match the increase of drug development costs. Network description and analysis not only give a systems-level understanding of drug action and disease complexity, but can also help to improve the efficiency of drug design. We give a comprehensive assessment of the analytical tools of network topology and dynamics. The state-of-the-art use of chemical similarity, protein structure, protein-protein interaction, signaling, genetic interaction and metabolic networks in the discovery of drug targets is summarized. We propose that network targeting follows two basic strategies. The "central hit strategy" selectively targets central nodes/edges of the flexible networks of infectious agents or cancer cells to kill them. The "network influence strategy" works against other diseases, where an efficient reconfiguration of rigid networks needs to be achieved by targeting the neighbors of central nodes/edges. It is shown how network techniques can help in the identification of single-target, edgetic, multi-target and allo-network drug target candidates. We review the recent boom in network methods helping hit identification, lead selection optimizing drug efficacy, as well as minimizing side-effects and drug toxicity. Successful network-based drug development strategies are shown through the examples of infections, cancer, metabolic diseases, neurodegenerative diseases and aging. Summarizing >1200 references we suggest an optimized protocol of network-aided drug development, and provide a list of systems-level hallmarks of drug quality. Finally, we highlight network-related drug development trends helping to achieve these hallmarks by a cohesive, global approach.

The effect of edge definition of complex networks on protein structure identification

The main objective of this study is to explore the contribution of complex network together with its different definitions of vertexes and edges to describe the structure of proteins. Protein folds into a specific conformation for its function depending on interactions between residues. Consequently, in many studies, a protein structure was treated as a complex system comprised of individual components residues, and edges were interactions between residues. What is the proper time for representing a protein structure as a network? To confirm the effect of different definitions of vertexes and edges in constructing the amino acid interaction networks, protein domains and the structural unit of proteins were described using this method. The identification performance of 2847 proteins with domain/domains proved that the structure of proteins was described well when R_Cα was around 5.0–7.5 Å, and the optimal cutoff value for constructing the protein structure networks was 5.0 Å (C_α-C_α distances) while the ideal community division method was community structure detection based on edge betweenness in this study.

Regulation of the Plasmodium motor complex: phosphorylation of myosin A tail-interacting protein (MTIP) loosens its grip on MyoA

The interaction between the C-terminal tail of myosin A (MyoA) and its light chain, myosin A tail domain interacting protein (MTIP), is an essential feature of the conserved molecular machinery required for gliding motility and cell invasion by apicomplexan parasites. Recent data indicate that MTIP Ser-107 and/or Ser-108 are targeted for intracellular phosphorylation. Using an optimized MyoA tail peptide to reconstitute the complex, we show that this region of MTIP is an interaction hotspot using x-ray crystallography and NMR, and S107E and S108E mutants were generated to mimic the effect of phosphorylation. NMR relaxation experiments and other biophysical measurements indicate that the S108E mutation serves to break the tight clamp around the MyoA tail, whereas S107E has a smaller but measurable impact. These data are consistent with physical interactions observed between recombinant MTIP and native MyoA from Plasmodium falciparum lysates. Taken together these data support the notion that the conserved interactions between MTIP and MyoA may be specifically modulated by this post-translational modification.

Encoding dynamics for multiscale community detection: Markov time sweeping for the map equation

The detection of community structure in networks is intimately related to finding a concise description of the network in terms of its modules. This notion has been recently exploited by the map equation formalism [Rosvall and Bergstrom, Proc. Natl. Acad. Sci. USA 105, 1118 (2008)] through an information-theoretic description of the process of coding inter- and intracommunity transitions of a random walker in the network at stationarity. However, a thorough study of the relationship between the full Markov dynamics and the coding mechanism is still lacking. We show here that the original map coding scheme, which is both block-averaged and one-step, neglects the internal structure of the communities and introduces an upper scale, the "field-of-view" limit, in the communities it can detect. As a consequence, map is well tuned to detect clique-like communities but can lead to undesirable overpartitioning when communities are far from clique-like. We show that a signature of this behavior is a large compression gap: The map description length is far from its ideal limit. To address this issue, we propose a simple dynamic approach that introduces time explicitly into the map coding through the analysis of the weighted adjacency matrix of the time-dependent multistep transition matrix of the Markov process. The resulting Markov time sweeping induces a dynamical zooming across scales that can reveal (potentially multiscale) community structure above the field-of-view limit, with the relevant partitions indicated by a small compression gap.

Markov dynamics as a zooming lens for multiscale community detection: non clique-like communities and the field-of-view limit

In recent years, there has been a surge of interest in community detection algorithms for complex networks. A variety of computational heuristics, some with a long history, have been proposed for the identification of communities or, alternatively, of good graph partitions. In most cases, the algorithms maximize a particular objective function, thereby finding the 'right' split into communities. Although a thorough comparison of algorithms is still lacking, there has been an effort to design benchmarks, i.e., random graph models with known community structure against which algorithms can be evaluated. However, popular community detection methods and benchmarks normally assume an implicit notion of community based on clique-like subgraphs, a form of community structure that is not always characteristic of real networks. Specifically, networks that emerge from geometric constraints can have natural non clique-like substructures with large effective diameters, which can be interpreted as long-range communities. In this work, we show that long-range communities escape detection by popular methods, which are blinded by a restricted 'field-of-view' limit, an intrinsic upper scale on the communities they can detect. The field-of-view limit means that long-range communities tend to be overpartitioned. We show how by adopting a dynamical perspective towards community detection [1], [2], in which the evolution of a Markov process on the graph is used as a zooming lens over the structure of the network at all scales, one can detect both clique- or non clique-like communities without imposing an upper scale to the detection. Consequently, the performance of algorithms on inherently low-diameter, clique-like benchmarks may not always be indicative of equally good results in real networks with local, sparser connectivity. We illustrate our ideas with constructive examples and through the analysis of real-world networks from imaging, protein structures and the power grid, where a multiscale structure of non clique-like communities is revealed.

Optimal number of coarse-grained sites in different components of large biomolecular complexes

The computational study of large biomolecular complexes (molecular machines, cytoskeletal filaments, etc.) is a formidable challenge facing computational biophysics and biology. To achieve biologically relevant length and time scales, coarse-grained (CG) models of such complexes usually must be built and employed. One of the important early stages in this approach is to determine an optimal number of CG sites in different constituents of a complex. This work presents a systematic approach to this problem. First, a universal scaling law is derived and numerically corroborated for the intensity of the intrasite (intradomain) thermal fluctuations as a function of the number of CG sites. Second, this result is used for derivation of the criterion for the optimal number of CG sites in different parts of a large multibiomolecule complex. In the zeroth-order approximation, this approach validates the empirical rule of taking one CG site per fixed number of atoms or residues in each biomolecule, previously widely used for smaller systems (e.g., individual biomolecules). The first-order corrections to this rule are derived and numerically checked by the case studies of the Escherichia coli ribosome and Arp2/3 actin filament junction. In different ribosomal proteins, the optimal number of amino acids per CG site is shown to differ by a factor of 3.5, and an even wider spread may exist in other large biomolecular complexes. Therefore, the method proposed in this paper is valuable for the optimal construction of CG models of such complexes.