With high-throughput experiments and computing science development, it has become possible to detect protein complexes by computational methods. However, most computational methods are based on either unsupervised learning or supervised learning. Unsupervised learning-based methods do not need training datasets, but they can only detect one or several topological protein complexes. Supervised learning-based methods can detect protein complexes with different topological structures. However, they are usually based on a type of training model, and the generalization of a single model is poor. Therefore, we propose an Ensemble Learning Framework for Detecting Protein Complexes (ELF-DPC) within protein-protein interaction (PPI) networks to address these challenges. The ELF-DPC first constructs the weighted PPI network by combining topological and biological information. Second, it mines protein complex cores using the protein complex core mining strategy we designed. Third, it obtains an ensemble learning model by integrating structural modularity and a trained voting regressor model. Finally, it extends the protein complex cores and forms protein complexes by a graph heuristic search strategy. The experimental results demonstrate that ELF-DPC performs better than the twelve state-of-the-art approaches. Moreover, functional enrichment analysis illustrated that ELF-DPC could detect biologically meaningful protein complexes. The code/dataset is available for free download from  Most complex systems, such as biological systems and human society, can be presented as complex networks in the real world. Social networks, biological networks, brain networks, citation networks, and protein-protein interaction networks are examples of complex networks . CommuniDuring the past decade, various computational methods have been presented to identify protein complexes in PPI networks. We will briefly review the related work from three aspects. The first is identifying protein complexes based on unsupervised learning-based methods. Another type of identifying protein complex methods is based on a model optimization-based method. The last type of identifying protein complex methods is based on supervised learning-based methods.Many researchers hypothesize that subgraphs with different topological structures in PPI networks are factual protein complexes  such as Many local heuristic-based methods have been proposed to identify protein complexes. For instance, Altaf-Ul-Amin et al.  developeλ-module, and clustering coefficients to detect both overlapping and hierarchical protein complexes in PPI networks. PC2P  methods can effectively learn graph structure and node features. For example, Kipf et al.  presenteBased on the related work, assigning weights to the interacting edges by the network embedding method and multiple biological information can effectively improve the accuracy of the detection methods. Meanwhile, some studies have shown that protein complexes have core-attachment structures. Therefore, our ELF-DPC is based on a core-attachment structure, and we constructed a weighted PPI network. Second, we proposed a protein complex core strategy to mine local protein complex cores. We identified global protein complex cores using the CPredictor2.0 method, which endows our ELF-DPC with both global search ability and local search ability. Third, most current methods are based on either unsupervised learning or supervised learning. Unsupervised learning-based methods can detect only one or several topological protein complexes and cannot fully learn the characteristics of known protein complexes. Supervised learning-based methods can learn the characteristics of known protein complexes, detecting protein complexes with different topological structures. Still, current supervised learning-based methods are based on a single base model for training. However, the generalization of a single model is poor. Therefore, we propose an ensemble learning model consisting of a trained voting regression model based on different types of base regression models and structural modularity to detect protein complexes with different topological structures. Finally, we proposed a graph heuristic search strategy to extend each protein complex core to form a protein complex. The results obtained show that ELF-DPC attained superior performances over 12 state-of-the-art methods. Furthermore, functional enrichment analysis results of ELF-DPC showed higher biological relevance by GO enrichment analysis.• We introduce a protein complex core mining strategy based on the core-attachment structure and design a graph heuristic search strategy to search protein complexes.• We propose structural modularity to describe the inherent topological organization of protein complexes.• We present some new topological features and design an ensemble learning model by combining structural modularity and a voting regression model, which quantifies the possibility for a cluster as a protein complex.• We present an ensemble learning framework to identify protein complexes, and it achieves better performance than other competing methods.To summarize, we make the following contributions:The rest of this study is organized as follows. The Materials and methods section introduces the datasets, terminologies, and methods. The Experiments and results section describes evaluation metrics and parameter selection and compare ELF-DPC with the competing methods. Finally, the Conclusion section provides a conclusion and future work.In this paper, we used the four PPI networks for the experiments, i.e., Gavin , Krogan We used two standard protein complexes that were constructed in the literature . Their phttps://downloads.yeastgenome.org. Meanwhile, the gene expression data were obtained from https://www.ncbi.nlm.nih.gov/sites/GDSbrowser. Additionally, subcellular localization data was obtained from https://compartments.jensenlab.org/Downloads.In this study, we used the GO-slim data for describing the functional similarity of interactions, which is available on the link: G = , where V is a set of proteins, E is a set of interactions, and W is a n × n(n = |V|) matrix that represents the reliability of protein pairs in PPI networks. The direct interacting neighbor of node v is defined as Nv = {u| ∈ E, u ∈ V}.Here, we will give some terminologies that are used in this paper. A PPI network is generally described as a weighted graph This work is a novel ensemble learning framework to identify protein complexes from PPI networks. The block diagram of the detection process is shown in The framework of this method is outlined in Algorithm 1. The input to the algorithm is the PPI network, which produces a set of protein complexes as output. Our algorithm consists of five main steps. The first step is to construct a weighted PPI network by combining topological structure, gene expression data, GO annotation data, and subcellular location data in Line 2 (Constructing a weighted PPI network section). The second step is to design a protein complex core mining strategy to identify protein complex cores in the PPI networks (Mining protein complex cores section) in Line 3. The third step is first to construct feature vectors to describe the properties of known and false protein complexes in the PPI networks and train a voting regression model (Training a voting regression model section) to model and represent the protein complex based on supervised learning in Line 5. Then second, we define a quality function called structural modularity to describe the structural modularity of protein complexes. Then we combine the trained voting regression model and structural modularity to obtain an ensemble learning model in Line 6. In the fourth step, based on the ensemble learning model, we propose a graph heuristic search strategy (Forming protein complexes section) to extend each protein complex core for forming protein complexes from the PPI networks in Lines 7–14. Finally, we remove these redundant identified protein complexes in Line 15.Some studies have confirmed that the performance of protein complex detection could be markedly enhanced when the weight of edges is considered . MeanwhiX and Y, their gene expression profiles are X = {x1, x2, … , xi, … , xm} and Y = {y1, y2, … , yi, … , ym}, respectively. The value of their PPC is defined as X and Y at n time points, respectively. The value of PCC ranges from -1 to 1. For convenience, we use  + 1)/2 to replace PCC, which sets the value of PCC in . The value of PCC is higher, and then the coexpression probability of nodes X and Y is larger. At the same time, they could consist of the same protein complex.Generally, for a pair of interacting proteins, their coexpression level can reflect the strength of their interactions. Proteins with coexpressed relationships may also have similar functions  and showFS describe this relationship, which is defined as FS(X)| and |FS(Y)| represent the number of GO-slim annotations for proteins X and Y, respectively. |FS(X) ∩ FS(Y)| denotes the number of common GO-slim annotations for proteins X and Y.From a functional standpoint, we use GO-slim data to reflect the functional similarity of proteins. If a pair of proteins have more common GO-slim annotations, they are more likely to have the same biological function. Even the reliability of interactions between them will become stronger. Here, we let SL, which is defined as SL(X)| and |SL(Y)| denote the number of subcellular localizations of proteins X and Y, respectively. |SL(X) ∩ SL(Y)| represents the number of common subcellular localizations between proteins X and Y.Generally, if two interacting proteins have more exact subcellular locations, the interaction between proteins is more reliable. Here, we define the subcellular location similarity X and Y, their representations are two vectors, namely, X and Y. Meanwhile, the obtained protein embedding vectors by node2vec can reflect the topological structure similarity among proteins, and we use cosine similarity to calculate the similarity of vector representation of proteins X and Y, which is defined as F(X) =  and F(Y) =  is the n dimension of the corresponding vector. TSS indicates the topological structure similarity of two connecting proteins, X and Y.The network embedding method is a representation learning technique for representing the network’s nodes, which can automatically learn topological information from PPI networks. In this study, we use the network embedding method Node2Vec  to learnW is expressed by For each edge, its weighted value According to the constructing a weighted PPI network section, the weight of interactions is weighted using multiple biological properties and its topological structure, so the higher weight the edge has, the more likely it is that two terminate proteins are inside the same protein complex . Furtherv, u), its weight is w, and its neighborhood graph is denoted as NG = , where V* = Nv ∪ Nu ∪ {v, u}. Furthermore, the average weighted degree of NG is denoted as AWD) ) :AWDNGv,w and the average weighted degree of the neighborhood graph of the edge  is defined as As a result, the score function of edge  ∈ E, its edge clustering coefficient ) is defined as the number of triangles to which  belongs, divided by the number of triangles that might potentially include , as shown in Z denotes the number of triangles built on edge , and min(| deg(v)|, | deg(u)|) is the minimum degree of the two terminate proteins.For an edge , and create a protein complex core in Line 6, where neighbors of the complex core are added to both the weight of edge w ≥ Avgedgesweight  is greater than the average edge clustering coefficient ECC of all edges (AvgweightECC), according to the closeness between the seed edge  and its neighbors in Lines 9–17. These two constraints can ensure that the proteins in the protein complex core are correlated in biological relations and closely connected in topological structure. The protein complex core is retained if it contains more than or equals two proteins in Lines 18–20. Meanwhile, the seed edge (including two terminate proteins) would be marked and cannot be used as the seed edge of another cluster in Lines seven and eight. We select the next edge with the highest weight where its two terminal proteins are not included before seed edges, and it is used to form the next protein complex core until the seed queue Q is empty in Lines 6–22.Initially, select the protein with the highest weight edge as the first seed edge (CPrclusters) in Line 23. Next, we combine these local protein complex cores by a graph heuristic search method and global protein complex cores using the CPredictor2.0 method in Line 24.CPredictor2.0  is also Here, Algorithm 2Mining protein complex cores.To obtain the trained regression model, we will follow several steps. First, we collect the known protein complexes and weighted a weighted PPI network based on NA   with knoIn this paper, both known and false protein complexes in the PPI networks are modeled as weighted and unweighted undirected graphs. The weight is calculated based on https://github.com/RongquanWang/ELF-DPC/Methods/Feature_selection.py. Additionally, if the reader discovers other relevant and valid topological features, please use them to represent protein complexes further.On the other hand, we propose some topological features to describe the topological properties of protein complexes. We use 65 topological features to represent protein complexes in the PPI networks. • Accuracy: The individual learner must not be too bad, but it must be accurate.• Diversity: The output of individual learners should be different from each other.Ensemble learning combines multiple individual learners with certain strategies to form a learning committee, so that the overall generalization performance is greatly improved. In general, the generalization capability of an ensemble learner model is much greater than the generalization capability of a single learner model. Meanwhile, we know that there is a barrel theory so we focus on two major standards: accuracy and diversity:https://scikit-learn.org/stable/supervised_learning.html#supervised-learning.Therefore, producing and combining “good but different” individual learners is the core of ensemble learning. The VotingRegressor model is one of the most efficient ensemble learning techniques to reduce the variance and improve detection accuracy. In this paper, we use a VotingRegressor model based on several base models for training. A VotingRegressor is an ensemble meta-estimator that fits several base estimators and averages the individual predictions to form a final prediction. Here, linear regression, BayesianRidge, DecisionTreeRegressor, and SVM. SVR (kernel = “linear”) are used as the base estimators to build the VotingRegressor model. We select the VotingRegressor model due to its reduced variance in individual base estimators and better generalization capabilities, and the Voting Regressor model has more robustness than a single estimator. In this study, the VotingRegressor model and base estimators use default parameters. These models are a freely available machine learning tool used on scikit-learn , and theAs a result, a trained VotingRegressor model could be used to estimate the probability of a subgraph being a natural protein complex from a supervised learning perspective to detect protein complexes with various topological structures. The score of the VotingRegressor is based on the higher probability that it is an actual protein complex. The VotingRegressor is defined as C =  being a protein complex in the PPI network. We introduce a structural modularity (SM) model to estimate the likelihood of a cluster C =  being a protein complex, which can detect both dense and sparser protein complexes in PPI networks. First, structural modularity (SM) is combined by Cohesion(C) and Coupling(C), and Cohesion(C) is defined as Coupling(C) is defined as C, and |C| is the number of nodes in the cluster C. Cohesion(C) could estimate a protein complex with a community structure having dense connections among its nodes. Here, Cohesion(C) is based on the definition of density of a cluster C by density multiplied by the square root of the size of cluster C to quantify the likelihood that a cluster is a protein complex. The idea of Cohesion(C) is that a protein complex in the PPI network is usually relatively sparse, so Cohesion(C) is used to adopt density as the quality function, and it may be more appropriate.Wout(C) = ∑v∈C,u∉Cw represents the total weight of the boundary edges that connect the cluster C with the rest of the PPI network, and it can measure that the cluster C has sparse connections with its neighbor nodes.Based on the within-module and between module edges of subgraphs and the size of the subgraph, we present a new formal definition of protein complexes in PPI networks . Given tFinally, Structural Modularity (SM) is calculated as SM(C) when it has a high adapting density and is well separated from the rest of the network. SM(C) can identify protein complexes with cohesion and separation topological properties. This shows that proteins in a protein complex displayed intense and frequent connections within the protein complex and weak and rare connections to proteins outside of the protein complex.In this work, a protein complex will be assigned a higher value of C =  being a candidate protein complex to guide the identification of protein complex processes. An ensemble learning model can improve the robustness and stability of the clusterings by combining the output of several models, thus improving the overall accuracy. For a cluster C, its ensemble learning model is defined as In this paper, we propose an ensemble learning model that combines the VotingRegressor model and structural modularity (SM) to quantify the likelihood of a cluster Based on the ensemble learning model, we will introduce a graph heuristic search strategy by using the ensemble learning model to form protein complexes.Based on the fact that a protein complex core and attachment proteins form a protein complex, we obtain some protein complex cores. Next, we extract the attachment proteins of each protein complex core and select reliable attachments cooperating with its protein complex core to form a protein complex. We design a graph heuristic search strategy for each protein complex core to extend the protein complex core to form a whole protein complex. First, it starts with a protein complex core, which iteratively inserts neighboring proteins into the protein complex core and then removes proteins from the protein complex core to search for a locally optimal cluster. In this paper, each protein complex core is subjected to a graph heuristic search strategy and an ensemble learning model to form a protein complex. The basic idea of a graph heuristic search strategy for a protein complex core is iteratively extended and corrected to form a protein complex by maximizing the score of the ensemble learning model (please see Obtaining an ensemble learning model section).i Input a protein complex core.ii Adding outer boundary proteins process in Lines 3–12: First, for the current protein complex core, we construct its outer boundary proteins set. We first obtain all directly connected neighbor proteins of the current protein complex core, and then we rank these neighbor proteins according to the number of shared proteins between the neighbor of the neighbor protein and current protein complex core. We discard the neighboring proteins with fewer than two common proteins to select high-quality candidate neighboring proteins. Then we select only half of the neighboring protein set reserved according to the sorting results as the outer boundary proteins set in Line 3. Second, we calculate the ensemble learning model score for the current protein complex core when each outer boundary protein is temporarily added. The outer boundary protein that allows the ensemble learning model score to reach a maximum will be inserted into the protein complex core in Lines 5–11. This process is repeated until the ensemble learning model score of the protein complex core is not increased, or the size of the outer boundary nodes is zero in Lines 10 and 4.iii First, for the current protein complex core, inner boundary proteins are the set of proteins that belong to the protein complex core and connect at least one other protein in the PPI networks in Line 16. Second, we calculate the score of the ensemble learning model after each inner boundary node is temporarily removed from the protein complex core. The inner boundary protein that increases the ensemble learning model score is determined, and it will be eliminated from the protein complex core in Lines 19–21. This process is continued until the ensemble learning model score of the protein complex core reaches a maximum or the size of the inner boundary protein set is zero, and the number of current protein complex cores is less than or equal to 2 in Lines 22–23 and 17.Fitness(SG) of the protein complex core in Lines 27–30, the current protein complex core is considered to be formed as a locally optimal cluster in Line 2–31, and then output it as a detected protein complex in Line 32.iv We repeat ii) and iii) until the protein complex core is no longer changed or no increment in the The pseudocode of the graph heuristic search strategy is shown in Finally, we select the next protein complex core. Then we repeat this process using a graph heuristic search strategy  to extenAlgorithm 3A graph heuristic search strategyELF-DPC was implemented in Python three and was successfully executed on a PC with an Intel i7-4790 CPU @3.60 GHz and 80 GB RAM.S denote the known protein complexes, and D denote the protein complexes identified by a detection method.In this study, to evaluate the proposed method, we need to compare the performance of our method against the compared methods by some statistical metrics. For this purpose, we used the neighborhood affinity, F-measure, CR, ACC, MMR, and Jaccard criteria to evaluate the protein complex detection algorithms. Let Si is a standard protein complex in S, and Dj is a discovered protein complex D. Their neighborhood affinity score ) )  can descNA is larger than or equal to 0.2, protein complexes Si and Dj are regarded as matching protein complexes  ≥ ω}| and Nim be the number of detected protein complexes that match at least one standard protein complex, i.e., Nim = |{d|d ∈ D, ∃s ∈ S, NA ≥ ω}|, where ω is a predefined threshold and is usually 0.20. Recall and precision are defined as Let Tij be the number of proteins that are included in both standard protein complex Si and detected protein complex Dj, and let Ni be the number of proteins that are included in standard protein complexes S. Meanwhile, Sn and PPV are calculated by Let S and D, and then each standard protein complex Si ∈ S and detected protein complex Dj ∈ D are connected by the weight W edge. Next, we select disjoint edges from the bipartite graph to maximize the sum of their weights; Finally, the MMR is the sum of the weights of all selected edges divided by |S|, which is denoted by We used the third metric, the maximum matching ratio (MMR)  based onS and the detected protein complexes D are given, the |S|×|D| matrix T is constructed, where each element max{Tij} is the most significant number of shared proteins between the ith standard protein complex, and the jth detected protein complex. The coverage rate is calculated by Ni is the number of proteins in the ith standard complex.The coverage rate (CR) was used to assess how many proteins in the standard protein complexes could be covered by the identified complexes. When the standard protein complexes Si ∈ S, and a discovered protein complex is Dj ∈ D. Then, their Jaccard is Dj, its Jaccard is Si, its Jaccard is D, the average of the weighted Jaccard is S, its JaccardS is defined by Jaccard is the final method for measuring the clustering methods . Here, ap-value. Generally, a detected protein complex possesses biological significance if its p-value is less than 0.01. In this paper, we used the fast tool LAGO , and the fluctuations of the Total score are not significant. Therefore, the value of ratio is set as 5 by the default value in this study.To study the effect of parameter We obtained the software implementations for all the compared methods, and their parameters are shown in In this section, we tested ELF-DPC on four original PPI networks, i.e., Gavin and Krogan core, DIP, and MIPS, and two known protein complexes were used for training and assessing the performance of ELF-DPC. We used six computational metrics, the F-measure, CR, ACC, MMR, Jaccard, and total score, to evaluate the performance. Here, we define the sum of the top five measures as the Total score. Note that the number of identified protein complexes (Num) was counted by each method. To illustrate the performance of ELF-DPC, we selected ten representative unsupervised methods, including DPClus , CMC Li, ClusterAs shown in We used standard protein complexes 1 as the positive training set and standard protein complexes 2 as the test set. The results are presented in p-value of the identified proteins in GO (Gene Ontology) databases, which cover three domains: biological process, molecular function, and cellular component. Since the p-values of identified protein complexes are closely related to their size  as 10.16. The AS of our ELF-DPC is 9.55 and 9.13, respectively. Generally, the p-value of an identified protein complex is closely associated with the size of the identified protein complex. Then the p-value decreases gradually when the size of the detected protein complexes increases , we will consider integrating other data sources  to ident


Editorial on the Research TopicInteractions Between Proteins and Biomacromolecules: Tools and ApplicationsAs the protagonist in the symphony of life, proteins perform fundamental functions by interacting with other biomacromolecules including nucleic acids, and carbohydrates. Therefore, the interactions between proteins and biomacromolecules have always been the pivotal issue in biomedical research. In the past 2 decades, advances in technologies including mass spectrometry (MS), next-generation DNA sequencing, and bioinformatics give birth to a new field: interactomics .In this Special Issue, we selected a series of articles that highlight technological advances for studying interactomics and the interactomics of protein-glycosaminoglycans (GAGs) and protein-miRNA. We hope that this Special Issue will instigate novel questions in the minds of our readers and will be helpful in facilitating the development of the field.Shi et al. A more specific review, focused on NMR characterization of GAGs and proteins, was provided by Bu and Jin. With the rapid accumulation of omics data, how to fully utilize these high-throughput data to uncover interactions between proteins and biomacromolecules has become a highly attractive research area  and bio-layer interferometry. The advances of structural characterization technologies, such as MS and nuclear magnetic resonance (NMR) spectroscopy, make it possible to directly analyze the complexes formed by proteins and biomacromolecules. In this Special Issue, the technological advances for studying interactomics has been reviewed by rch area . Ding etGAGs are a family of highly negatively charged linear polysaccharides including heparin/heparan sulfate (HS), chondroitin sulfate (CS)/dermatan sulfate (DS), and keratan sulfate (KS). GAGs play vital roles in many pathological and physiological processes, such as embryonic development, extracellular matrix assembly, inflammation, cancer, and cardiovascular diseases, by interacting with numerous proteins. The study of protein-GAG interactions is an important theme in glycobiology, resulting in many therapeutic implications. Recently, the first comprehensive draft of GAG interactome was reported, which composes of 932 protein-GAG interactions .Shi et al. present an excellent review on the progress of protein-GAG interactions research. Bu and Jin review the application of NMR spectroscopy on the characterization of protein-GAG interactions. Since the COVID-19 pandemic, it was reported that the interactions between cellular HS and S-protein of SARS-CoV-2 are critical for the viral infection  research, protein Tau (tau) and related tau pathology have been a hot research area since the interactions between tau and HS proteoglycans (HSPGs) are key facilitator in each stage of the prion-like propagation of pathology. Mah et al. review the sulfation code of HSPGs in tauopathies. Additionally, Kim et al. report the kinetics and structural features of heparin and its interactions with cellular prion protein measured by SPR.In this Special Issue, nfection . Yue et Guo et al. discovered that MSI2 could bind to miR-301a-3p and facilitate its distribution in mitochondria using affinity purification and non-labeling proteomic techniques. This study provided valuable insight into the mechanism of mitochondrial distribution of miRNAs.MiRNAs are a class of RNA molecules with important regulatory functions , but theThis collection of articles highlights the different approaches for studying interactomics and related new analytical techniques. Indeed, the breakthrough in analytical tools (ultra-highly sensitive and automatic level) and approaches (such as machination learning) during the last decade has facilitated a greater understanding of the importance of interactions between biomacromolecules and their roles in diseases. We expect the more outcomes from this interdisciplinary research will accelerate the identification of new biomarkers for diagnostics and drug discovery.


Are all protein interactions fully optimized? Do suboptimal interactionscompromise specificity? What is the functional impact of frustration?Why does evolution not optimize some contacts? Proteins and theircomplexes are best described as ensembles of states populating anenergy landscape. These ensembles vary in breadth from narrow ensemblesclustered around a single average X-ray structure to broader ensemblesencompassing a few different functional “taxonomic”states on to near continua of rapidly interconverting conformations,which are called “fuzzy” or even “intrinsicallydisordered”. Here we aim to provide a comprehensive frameworkfor confronting the structural and dynamical continuum of proteinassemblies by combining the concepts of energetic frustration andinteraction fuzziness. The diversity of the protein structural ensemblearises from the frustrated conflicts between the interactions thatcreate the energy landscape. When frustration is minimal after folding,it results in a narrow ensemble, but residual frustrated interactionsresult in fuzzy ensembles, and this fuzziness allows a versatile repertoireof biological interactions. Here we discuss how fuzziness and frustrationplay off each other as proteins fold and assemble, viewing their significancefrom energetic, functional, and evolutionary perspectives.Wedemonstrate, in particular, that the common physical originof both concepts is related to the ruggedness of the energy landscapes,intramolecular in the case of frustration and intermolecular in thecase of fuzziness. Within this framework, we show that alternativesets of suboptimal contacts may encode specificity without achievinga single structural optimum. Thus, we demonstrate that structuredcomplexes may not be optimized, and energetic frustration is realizedvia different sets of contacts leading to multiplicity of specificcomplexes. Furthermore, we propose that these suboptimal, frustrated,or fuzzy interactions are under evolutionary selection and expandthe biological repertoire by providing a multiplicity of biologicalactivities. In accord, we show that non-native interactions in foldingor interaction landscapes can cooperate to generate diverse functionalstates, which are essential to facilitate adaptation to differentcellular conditions. Thus, we propose that not fully optimized structuresmay actually be beneficial for biological activities of proteins viaan alternative set of suboptimal interactions. The importance of suchvariability has not been recognized across different areas of biology.This account provides a modern view on folding, function, and assemblyacross the protein universe. The physical framework presented hereis applicable to the structure and dynamics continuum of proteinsand opens up new perspectives for drug design involving not fullystructured, highly dynamic protein assemblies. FuxreiterM.Fuzziness in ProteinInteractions-A Historical Perspective. J.Mol. Biol.2018, 430, 2278–2287.29477337 and disorder (high frustration).44Globular proteins usually attain a low degree offrustration finallyby having a well-defined tertiary structure. However, increasing thefrustration level promotes the population of new conformational substates,which results in more heterogeneous ensembles. Under such conditions,protein function can exploit then the ability to visit many conformationalsubstates whose population can be modulated by other partners. Structuraldiversity can be manifested in many different ways: having differentsecondary structure conformations with similar probabilities;46 the transition state of the folding reaction.49 Thus, the transition state of templated folding isliable to change with the experimental or cellular conditions.In principle, through interaction of intrinsicallydisordered regionswith other partner proteins, proteins may decrease their local frustration.In a binding-induced folding process, highly frustrated disorderedproteins adopt a well-defined conformation once bound46 1. In con51 alpha 3D,52 andfrataxin.34 In these cases, large roughnessof the folding energy landscape leads to multiple qualitatively differentcompeting pathways. Different binding partners may tune the foldingtrajectories, so the distributions of the nucleation may differ. Consequently,although templated folding results in well-defined structures in thebound state, there is often considerable variability in the binding-inducedfolding process.A typical signature of frustration in protein folding lies in thepresence of transient non-native interactions along the major foldingpathway. This effect has been observed in several different proteinsystems including the aforementioned Im7, the designed and overstabilizedTop7,553 Intrinsically disorderedproteins, in particular, may adopt folded structures in some of theircomplexes, while they remain disordered when interacting with otherpartners,54 a phenomenon that is termedconditional folding . Thus, the frustrated conformationalenergy landscape in the bound state seems to provide a functionaladvantage by allowing an increased interaction capacity with a limitedset of partners.Recently, theenergy landscape theory was applied to assess thelevel of local frustration in protein complexes, which are formedby templated folding.60 Incontrast, protein motifs similar to their embedding regions likelyremain disordered in the bound state.60 Within this framework, protein complexes sample a dynamical continuumfrom well-defined, ordered bound states to highly heterogeneous, disorderedbound states, and the different modes of binding can be quantifiedby the conformational entropy. Frustration of the energy landscapein the bound state leads to variation between these different bindingmodes. Based on the local sequence biases with different possiblebinding sites, the pool of possible binding modes leading to fuzzinesscan also be quantified.53 Frustration inthe binding landscape also may stem from non-native interactions asobserved for c-Myb/KIX63 or ACTR/NCBD.65The sequence properties that lead to the emergenceof fuzzy bindinghave been recently elaborated. Interaction motifs, with distinguishedcomposition and physicochemical features as compared to their flankingregions, will undergo disorder-to-order transition upon binding andfold in a partner induced manner.66All these alternativebinding modes expand the functional repertoireof proteins, as the relative populations of the different conformationalsubstates can now be shifted according to the cellular milieu.667 The evolutionary historyof the interaction between a longer disordered region called CID fromthe transcriptional coactivator ACTR and the NCBD domain of CBP/p300has recently been investigated.65 Here, a likely evolutionaryscenario is that the ancestral interaction involved a weaker bindingthat was under positive selection and that a few key mutations inNCBD finally yielded a functional affinity. Interestingly, structuraldata4 suggested that the plasticity ofthe disordered CID could relieve frustration and compensate for apparentlydetrimental point mutations.The possibility of expanding the functionalrepertoire throughfrustrated interactions suggests an interesting evolutionary mechanism.The ability of any system to respond to a selection pressure demandsits constituents to be mutable and malleable. In the case of a cell,optimization of fitness is a constantly ongoing process in which conditionschange over both evolutionary and ecological time scales and wheresurviving lineages manage to adapt via changes in expression levelsof proteins as well as the protein sequences themselves. From a proteinstructure point of view, it is clear that both extremely well-foldedas well as highly dynamic proteins are vital for life. It is not clearto what extent ‘new’ folded proteins arise during evolution,but it is likely that the majority of folded proteins today are descendantsof ancient folded structures. On the other hand, short linear or peptideinteraction motifs consisting of a few up to a dozen amino acid residuesrecurrently arise de novo, as demonstrated for virus-host protein–proteininteractions.69 Coevolutionaryanalysis provides a powerful method to identify intermolecular interactionpartners as well as long-range intramolecular contacts within a protein.Analysis of coevolving residues has demonstrated that the supramodularstructure of the nonreceptor tyrosine kinase c-Src involves intramolecularcontacts between two domains (Unique and SH3),70 an interaction, which is structurally heterogeneous asshown by biophysical methods (71 Thus, in a cell, disordered protein regionsare constantly being exposed to various other proteins and engagein numerous promiscuous binding events, also referred to as quinaryinteractions.72 Whenever such interactionsare advantageous to the organism, they might become fixed in the populationby positive selection. The original interaction was likely frustratedand explored multiple conformations, but then depending on the evolutionarydice and functional restraints, further point mutations might leadthe complex to evolve into a more minimally frustrated and structurallywell-defined complex, like that between CID and NCBD. Alternatively,the complex may remain locally frustrated and structurally heterogeneous,fuzzy, as seen for Unique/SH3. It is clear that selection has generateda wide spectrum of local frustration levels in natural protein–proteininteractions, resulting in well-defined complexes as well as sometimesstructurally heterogeneous complexes.The idea that structural pliabilitygives an evolutionary edgewith respect to the emergence of new protein–protein interactionsis appealing. Along these lines, it is likely that initial promiscuousinteractions between a disordered region and a protein domain mayexplore alternative conformations. While many solved complexes betweendisordered regions and folded domains are structurally well-defined,like that for CID/NCBD, there are examples of positive selection forinteractions with fuzzy complexes. methods 3. Anothe774Frustration implies deviations from the perfectfunnel-like energylandscape, often resulting in conformational heterogeneity of nativeproteins. Ruggedness of the landscape is caused by the suboptimalinteractions that take place through non-native contacts, and thuswe use it in thermodynamic terms. Multiple, minimalistic motifs coupledto non-native interactions may compromise folding. Ruggedness of thebinding energy landscape can also arise from suboptimal, often redundantinteraction motifs, for example, in low-complexity protein regions.Frustration results in alternative binding modes, which imparts plasticityto the templated folding mechanism and can enhance adaptability toa range of partners in a versatile form. Thus, ruggedness of the bindinglandscape leads to fuzzy binding and variations between many differentinteraction modes. Similar to frustration in free proteins, fuzzinessof protein complexes also expands the functional repertoire to achievefine-tuned, context-dependent regulation.70 Residues mediating fuzzy interactions are well-conservedand often display coevolutionary signatures. This example illustratesthe way fuzziness originating in frustrated contacts can control signalingspecificity.Here we have shown that frustration and fuzziness are parallelconcepts, aspects of suboptimal interactions of the energy landscape1. This c8Severalobservations suggest that the contrasting requirementsof physical folding and biological function may give rise to energeticfrustration in natural proteins. Conformational heterogeneity promotesadaptability which itself may often be under evolutionary selection.Landscape ruggedness can be modulated by interactions with physiologicalpartners. Suboptimal contacts with the partner often lead to fuzzybinding and give protein complexes with partner-specific, alternativebinding modes. Although the concept of local frustration has beenapplied to folding and function of globular proteins, while fuzzinesshas mostly been discussed in the area of protein interactions andassemblies, these are really two sides of the same coin. Both frustrationand fuzziness emerge from common physical principles of the energylandscape and can be described by a common formalism. We feel thisrealization can open new perspectives unifying protein folding andassembly with biological function.


Predicting protein side-chains is important for both protein structure prediction and protein design. Modeling approaches to predict side-chains such as SCWRL4 have become one of the most widely used tools of its type due to fast and highly accurate predictions. Motivated by the recent success of AlphaFold2 in CASP14, our group adapted a 3D equivariant neural network architecture to predict protein side-chain conformations, specifically within a protein-protein interface, a problem that has not been fully addressed by AlphaFold2. These constraints ensure that network predictions are equivariant with respect to the global roto-translational transformations of the input point cloud, thus improving the robustness and overall performance. In our case, the point cloud represents the CA atoms of the protein, and we train the neural network to predict the key atom positions of the residue side chains given the positions of the neighboring backbone atoms.2.−2 distal atom or the functional end-group atom) of the central residue. Finally, a full atomic representation of the side-chain orientation is constructed by identifying a rotamer from a library having a sidechain atom that is nearest to the prediction(s).The side-chain prediction method explored in this work is schematically represented in 2.1.N residues is represented as a collection of neighborhoods, Ni ∈ {1, …, N}, where each neighborhood Ni is centered on a residue i and is defined based on plausible interactions between residue i and any residue j within a local environment. Specifically, a pair of residues, i and j, are deemed to have a plausible interaction if any pair of backbone-dependent rotamers of i and j result in an interatomic distance of less than 5Å. In the context of the attention mechanism described below, considering local neighborhoods allows reducing the computational complexity from quadratic to linear in the number of residues.A protein containing 2.2.In this work, we rely on the SE(3)-Transformer architecture by attention mechanism  group of roto-translational transformations. These two properties represent important symmetries of a problem, and while a general neural network can learn to respect these symmetries, explicitly incorporating symmetry constraints into the neural network can be more efficient with respect to the number of learnable parameters and amount of data required for training. One successful example of such symmetryaware architecture is the Tensor Field Network , which mf : R3 → Rd defined on a discrete set of points in space - point cloud:xj} are the 3D point coordinates, and fj ∈ Rd is a concatenation of vectors l of SO(3)-group irreducible representations: Recently, an expansion of this architecture incorporating the attention mechanism was implemented in the SE(3)-Transformer. In this architecture, the input is a feature vector field aij is a scalar attention weight, and k to degree l, which can be written as:k to degree l,  is a spherical harmonic, and A learnable attention-based transformation of this vector field satisfying the SE(3) equivariance can be expressed as:Wlk (x) is expressed as a linear combination of the non-learnable angular kernels k vector to degree l vector.The equivariance of the transformation is ensured by the fact that the learnable weight kernel i and a set of key vectors ij in the neighborhood Ni of node i. This architecture can be easily generalized by introducing several channels per representation degree.Furthermore, invariant attention weights are achieved via a dot-product attention structure, as shown below:2.3.l=1) input feature vectors to represent the relative positions of the C and N atoms of the same residue with respect to the CA atom. In addition, we use 20 scalar (degree l=0) input features to represent one-hot encoding of the residue type.Our goal was to predict the positions of specific side-chain atoms given the positions of the backbone atoms of the neighboring residues, the so-called side chain packing problem In our implementation, each point of the 3D cloud represents a single residue, more specifically, the coordinate of the CA atom of that residue. We use two 3D vectors (i.e. degree 2.4.si and one 3D vector (degree l=1) feature e vi per node. The final prediction p of the specific side chain atom position is made by performing a global pooling over the neighborhood nodes using scalar features as weights for the vector features: xi is the coordinate of the CA atom of the residue represented by node i.The final layer of the SE(3)-Transformer is set to produce one scalar (degree l=0) feature 2.5.−2 distal atom and the functional end-group atom, we reconstruct the side-chain conformation by selecting the rotamer with the lowest 2-atom RMSD from a library of backbone-dependent rotamers .2.6.The deep-learning model was trained on a PISCES PDB  corpus, −2 atom and (2) the functional end-group atom (The trained deep-learning model was tested based on four types of experiments for side-chain prediction: (1) unbound proteins at a protein-protein interface, (2) bound proteins at a protein-protein interface, (3) unbound protein at a protein-protein interface without its binding partner, and (4) bound protein at a protein-protein interface without its binding partner. For the unbound cases, (1) and (3), the unbound proteins were superimposed onto the respective bound cases. Furthermore, the trained model was used to predict at least one side-chain coordinate: (1) the distal χoup atom .Subsequent to predicting the above-identified atomic coordinates for each sidechain, the sidechain was reconstructed based on a backbone dependent rotamer library . The rotFinally, side-chain predictions were carried out for the same four experiments using SCWRL4, and are presented in the 3.The performance of the trained model on a test set and its comparison with the performance of SCWRL is presented in −2 atom prediction, ‘EG’ refers to the functional end-group atom prediction, ‘WP’ refers to the prediction with the protein partner included in as part of the input, and ‘NP’ refers the prediction without the protein partner included as part of the input.In all figures, the notation ‘LEARN’ refers to the SE(3)-Transformer results, ‘SCWRL’ refers to the SCWRL4 prediction, ‘PROJ’ refers to the projection prediction based on the side-chain reconstruction, ‘C2’ refers to the distal χ4.This manuscript describes our attempt to apply the SE(3)-equivariant transformer architecture to the problem of predicting the protein residue side chain orientations. The classical methods attempting to solve the same problem usually approach it by performing a combinatorial search over the side-chain orientation space and trying to find a minimal energy solution. Here, we demonstrate that a novel SE(3)-equivariant transformer architecture can be straightforwardly used to learn a solution to the same problem given enough training data. We test the approach on a task of reconstructing the side-chains in protein interfaces, using both bound and unbound subunit structures. The quality of the resulting predictions is comparable to that of the best-in-class classical methods. We suggest that this type of approach could be used as a part of larger network architectures dedicated to solving problems related to protein structure, in particular those used for prediction and design of protein complexes.


Protein-protein interactions (PPIs) contribute to regulate many aspects of cell physiology and metabolism. Protein domains involved in PPIs are important building blocks for engineering genetic circuits through synthetic biology. These domains can be obtained from known proteins and rationally engineered to produce orthogonal scaffolds, or computationally designed de novo thanks to recent advances in structural biology and molecular dynamics prediction. Such circuits based on PPIs (or protein circuits) appear of particular interest, as they can directly affect transcriptional outputs, as well as induce behavioral/adaptational changes in cell metabolism, without the need for further protein synthesis. This last example was highlighted in recent works to enable the production of fast-responding circuits which can be exploited for biosensing and diagnostics. Notably, PPIs can also be engineered to develop new drugs able to bind specific intra- and extra-cellular targets. In this review, we summarize recent findings in the field of protein circuit design, with particular focus on the use of peptides as scaffolds to engineer these circuits. Living organisms, both unicellular and multicellular, respond to exogenous and endogenous signals that trigger appropriate responses, within individual cells, by using a complex system of receptors and signal-transducing proteins ,2. SignaIn the last decade, synthetic biology has emerged as a promising field with possible applications in many aspects of human life . Its appSo far, many synthetic genetic circuits have relied on activation of target gene expression under desired circumstances . FeedbacA speedier method, which allows cells to relay an external stimulus, and directly elicits a cellular response, is the employment of reversible protein-protein interactions (PPIs), thus bypassing the need for transcriptional induction and translation of the signal transducer, which can be already expressed in the cell a priori and activated only under desired circumstances. PPIs can directly trigger a cellular response by sequestration of enzymes, degradation of target proteins or other mechanisms . PPIs reGenetic circuits with transcriptional control as final outputs are constructed and tested using DNA pieces stacked together. Synthetic genetic circuits based on PPIs used to modulate cellular responses  are less developed because the information and rules for PPIs are still poorly understood, especially when compared to technologies that allow the manipulation of the genetic code. For these reasons, the first PPI circuits used to build new biological tools and devices were devised to work with known proteins, as shown by the well-known example of the development of the yeast-two hybrid assay .Today, computational methods, and the increasing importance of structural data, allow the design of fully synthetic and orthogonal PPI-scaffolds and linkers which can be employed for protein circuits design ,21. A wiIn this review we explore new and old aspects of protein circuits design, encompassing novel and/or established strategies to develop fast-responding, orthogonal and non-orthogonal synthetic circuits and devices based solely on PPIs. These systems can be developed both to drive a new transcriptional program in the cell or to directly give an adaptive response to the environment. Moreover, recent advances in the field of structural biology and molecular modeling are also discussed, as these can greatly improve the development of this field of synthetic biology. Emphasis will be laid also on the use of combinatorial libraries to isolate new scaffolds or drugs, and on the use of small peptides as convenient mediator of PPIs. The synthetic circuits discussed in this review are summarized in PPIs are a cornerstone of every single biological process, such as signal transduction, differentiation, and many others. Protein interaction motifs and binding regions are extremely versatile as building blocks to construct synthetic cellular networks , and theE. coli RNA polymerase and DNA-binding proteins, and trigger transcription . A. A36]. AAlternatively, peptides have been used to produce an ultrasensitive protein switch based on the autoinhibition mediated by the cooperative binding to the neural Wiskott-Aldrich syndrome protein (N-WASP). N-WASP output domain was linked to Src Homology 3 (SH3) domains and to their peptide ligand. The sensitivity of such a system can be modulated varying the number of SH3 interaction modules F 39]..39].Peptides can be rationally engineered to drive complex reactions inside cells and exploited to process complex signals. For instance, they could be used as biosensors for the detection of specific proteins, like in the case of the neurodegenerative-related protein Tau . By mimiAdditionally, rational design and assembling of peptides enabled their usage as modules to control the activity of viral proteases and modulate downstream signals, as demonstrated by Stein and Alexandrov (2014). Here, a modular design strategy was employed to develop a transducer protease connected to an auto-inhibitory (AI) peptide and containing different functional units able to modulate the activity of the auto-inhibited enzyme . An AI dBesides rational design, de novo proteins and peptides proved the possibility to generate an unlimited number of orthogonal and composable modules. α-helical coiled coils (CCs) are probably the most used scaffolds for engineering synthetic PPIs, as the H-bonding patterning can be designed in a predictable fashion. Noteworthy, freely-available toolboxes, like Pcomp and SYNZIP, provide several validated CC pairs which can be used to compose PPI networks ,45. PrevE. coli cytoplasm. Pyruvate decarboxylase and alcohol dehydrogenase were targeted to this cyto-scaffold leading to enhanced ethanol production thanks to enzyme co-location (Another system named CIPHR (cooperatively inducible protein heterodimer) used de novo design of heterodimers to regulate the association among proteins of choice, as split enzymes and transcription machines, allowing the execution of logic functions in vitro and in vivo C 47]. P. P47]. Plocation H 51]..E. coli Peptides were already shown to be highly selective in the engagement of interactions with endogenous components. This ability was exploited to trigger protein degradation through the proteolysis-targeting chimera (PROTAC) technology, where a small molecule can be isolated and used to target endogenous proteins for degradation, through the linkage with different E3 ubiquitin ligase D 52]..52].Protac-1 was composed by a β-TRCP E3-recruiting peptide linked to ovalicin, which targets the MetAP-2 enzyme, causing its ubiquitination and degradation in vitro . SuccessDue to the inaccessibility of some extracellular targets, lysosomal-targeting chimeras (LYTACs) were also developed G. SignalAdditionally, peptide-mediated protein degradation was used in hydrophobic protein tagging (HyT). Hydrophobic groups are guided and attached to a POI by POI-interacting peptides , to induce permanent chaperone binding, and to exploit natural surveillance mechanisms which degrades misfolded proteins. In fact, some chaperones (Hsp70 and Hsp90) could recruit the co-chaperones E3 ligase CHIP which mediates protein ubiquitination and degradation ,59. ThisCombinatorial libraries have been developed to identify sequences in short times, which can be further screened using different assays . PeptideE. coli through cell-cell adhesion. The library relies on peptide nanobodies and antigens displayed on the outer bacterial membrane  of chimera proteins. Grawe and collaborators (2020) developed a new DNA assembly strategy enabling the efficient fusion of protein domains with combinatorial linker sequences. This DNA assembly system, named iterative functional linker cloning (iFlinkC), relies just on the combined action of restriction enzymes and T4 DNA ligase and enable the sequential assembly of fusion proteins. iFLinkC functionality was demonstrated through the construction of synthetic protease switches. Linker space was highly dynamic as the induction of protease activity can vary by several folds depending on the linker sequence and lenght . iFLinkCThe field of protein circuits is quite vast and comprises several approaches for the isolation and characterization of protein scaffolds suitable for synthetic circuit design, reflecting great interest in the development of this area of synthetic biology.Synthetic proteins/peptides scaffolds are convenient for protein circuit design for several reasons: (i) the combination of 20 different amino acids in protein or peptide folds allows a high number of possible combinations in terms of interactions, which are also expanded by the use of non-canonical amino acid or post-translational modifications ; (ii) thImportantly, protein circuits can be based solely on components which are constitutively expressed, and interact with each other only under desired circumstances to produce a fast cell response, without the need for downstream transcriptional activation, even if this area of research is still at the beginning.http://crdd.osdd.net/raghava/thpdb/; access date 25 October 2021) database curated by the Food and Drug Administration (FDA) [Synthetic proteins and peptides are also remarkable candidates for developing new drugs, as they can be used to deliver functional domains to a POI as seen for PROTACs. As mRNA-based drugs seem to be riding the wave as human therapeutics, many companies and research laboratories are clearly investing in protein-based therapies, as highlighted in the THPdb (on (FDA) . This sh


The developed classifier predicts what proteins adsorb to nanoparticles and what protein features drive these interactions. Engineered nanoparticles are advantageous for biotechnology applications including biomolecular sensing and delivery. However, testing compatibility and function of nanotechnologies in biological systems requires a heuristic approach, where unpredictable protein corona formation prevents their effective implementation. We develop a random forest classifier trained with mass spectrometry data to identify proteins that adsorb to nanoparticles based solely on the protein sequence . We model proteins that populate the corona of a single-walled carbon nanotube (SWCNT)–based nanosensor and study the relationship between the protein’s amino acid–based properties and binding capacity. Protein features associated with increased likelihood of SWCNT binding include high content of solvent-exposed glycines and nonsecondary structure–associated amino acids. To evaluate its predictive power, we apply the classifier to identify proteins with high binding affinity to SWCNTs, with experimental validation. The developed classifier provides a step toward undertaking the otherwise intractable problem of predicting protein-nanoparticle interactions. Engineered nanoparticles are poised to transform how we undertake biological sensing  and the brain environment [cerebrospinal fluid (CSF)] 15-SWCNTs in CSF, the total set with biofluid labels, and the total set naïve of biofluid labels. The biofluid label refers to the knowledge of where the protein originated (blood plasma or CSF).The protein corona dataset was experimentally generated from a selective adsorption assay that quantified protein amounts present on nanoparticles via liquid chromatography–tandem MS (LC-MS/MS) characterization  relative change and (ii) an abundance threshold. First, if a protein was more abundant in the nanoparticle-bound case than it was in the control solution of the native biofluid without any nanoparticles present , then it was classified as in the corona. Second, the remaining proteins were ordered by abundance and fit to an exponential distribution. Increasing the power of the exponential leads to a higher in-corona threshold, placing fewer proteins in the corona. This thresholding approach reflects that lower abundance of a protein in the corona relative to its abundance in the biofluid  does not necessarily mean that protein is out of the corona; a protein that is substantially depleted can still be present in the corona with a high absolute quantity. The thresholding method that we have developed is discussed further in Materials and Methods.15-SWCNT nanoparticles. Although we initially focus on protein corona characterization with one nanoparticle type, SWCNTs, these classifiers do not require any information regarding the nanoparticle itself. We chose to pursue an RFC because this is an ensemble method with a well-known ability to be resistant to overfitting by using several weak learners that fit to different parameters (We implemented an RFC to classify corona proteins on (GT)n trials used in this verification step.During development, stratified shuffle split validation was used to check the success of our classifier with respect to accuracy, AUC, precision, and recall. The dataset was divided into a training and test set at the beginning of each split, and then the training set was fit to an untrained classifier. Next, predictions were made on the test set and compared with our true answers. The results from this classifier were saved, and the process was repeated with the classifier naïve at the beginning of each iteration, as graphically depicted in Using an RFC, classification tests were run on the total naïve dataset of proteins marked as being in or out of the corona with the aforementioned thresholding method. The classifier performance was scored for a range of thresholding powers . The claTo reconcile the imbalance in our LC-MS/MS experimental dataset , we up-sampled our minority class . This up-sampling ensures that each time the classifier was trained, we were able to recover an appropriate amount of the minority class. For this reason, the classifier was validated using a stratified shuffle split repeated 100 times. Moreover, we noticed that generalization of this classifier could be improved, especially when considering that the recall was below 0.6. To address this issue, a synthetic minority oversampling technique (SMOTE) 15-SWCNTs from the other biofluid . This discrepancy occurs because our classifier has no context of which proteins are in the corona formed from which biofluid, and thus, there is no method of adjusting for proteins displaying contradictory adsorptive behavior across biofluids. However, this classification discrepancy only occurs for a few proteins . Including the additional feature of biofluid label did not resolve this problem , AUC , and precision  than the classifier trained on the CSF dataset . However problem , indicatWe briefly note that both the train/test classification workflow and the classifier itself are applicable to other nanoparticle-corona systems. To demonstrate this, we used quantitative MS-based protein corona datasets for a model nanoparticle, PNPs, in blood plasma and CSF . Each feature was examined for the extent of contribution to the overall classification ability of the system using an analysis of variance (ANOVA) test . This pr15-SWCNTs were identified entified . Because15-SWCNT nanoparticles. These proteins represent a broad class of functions and subcellular locations and are distinct from those present in the plasma and CSF training datasets. Protein binding propensity was determined with associated binding probabilities, as summarized in table S3. We then implemented a corona exchange assay to measure real-time, in-solution protein binding dynamics on the nanotube surface, as described previously , l-lactate dehydrogenase A chain (LDH-A), and glutathione S-transferase (GST) that were predicted to not adsorb to (GT)15-SWCNTs (classifier results listed in table S3). Protein adsorption based on the end-state fluorescence values predominantly matched classifier predicted outcomes of in versus out of the corona: Addition of CD44 antigen and TDP-43 both resulted in sizeable ssDNA desorption from the SWCNT surface, whereas all proteins predicted to be out of the corona produced less ssDNA desorption (The corona exchange assay was used to test a panel of proteins predicted to be in the corona (probability ≥ 0.5) versus out of the corona (probability < 0.5). Specifically, we tested the protein panel: CD44 antigen, transgelin, and TAR DNA binding protein 43 (TDP-43) that were predicted to adsorb to (GT)sorption . Transgesorption , in comp6-SWCNTs, as this shorter ssDNA oligomer is displaced more readily from the SWCNT surface and thus displays a greater spread in desorption dynamics between protein species (6-SWCNTs was highly similar to that on (GT)15-SWCNTs (6-SWCNTs in plasma and (GT)15-SWCNTs in CSF [the equivalent study of (GT)6-SWCNTs in CSF has not been performed because of the limited availability of CSF]. The resultant protein panel binding order was largely the same as that of Cy5-(GT)15-SWCNTs, with a slightly higher Spearman’s correlation coefficient of 0.762 (6 ssDNA merely desorbs to a greater extent and thus yields more available SWCNT surface area for protein attachment. This end-state agreement between (GT)6- and (GT)15-SWCNT datasets further accounts for any mechanistic binding differences in whether protein adsorption causes full or partial ssDNA displacement from the SWCNT surface, where the latter case may occur for longer ssDNA strands 15, with the exception of RNase A (fig. S7).Experimental validation was repeated for the protein panel with Cy5-(GT) species  (28). Moof 0.762 . These r15-SWCNTs, as inferred by high glycine content and less strict secondary structural domains. This result is in agreement with previous experimental work demonstrating that peptides and small-molecule ligands with more conformational flexibility bind more readily to carbon nanotubes  for each residue. B factors incorporate the dependence on neighboring amino acids with a nine-residue sliding window averaging approach 15-SWCNTs. For physical context, (GT)15 ssDNA is observed to wrap helically around SWCNTs based on both experiment The reasoning for the highly ranked, inverse relationship between leucine-rich proteins and SWCNT binding may be that leucine content is a proxy for hydrophobic core stability, such as in α-helix motifs, and that more internally stable proteins are less likely to reorient and bind to nearby nanoparticles. This analysis is further supported by high grand average hydropathy (GRAVY) score  appearing as the third most important feature for proteins that do not enter the corona phase. Also in line with this protein reorientation argument, the finding that high content of amino acids associated with β-sheet structures leads to low protein adsorption indicates the difficulty for planar protein secondary structures to adapt to the highly curved SWCNT surface. For context, each SWCNT has an extremely high aspect ratio, with an average diameter of 1 nm and length of 500 nm. Our result is in agreement with previous work demonstrating that the high curvature of carbon nanotubes must be aligned at the amino acid level of proteins, less the secondary structure level  against physicochemical protein properties to understand protein features that govern corona formation  and cationic amino acids (arginine and lysine) are hypothesized to drive adsorption  leveraging quantitative protein corona data  greater than the control of protein abundances in biofluids alone (Abiofluid) were assigned as in the corona . Second, an exponential decay, n = n0 exp(− kA), was fit to the distribution of abundances for the remaining proteins, where n0 and k are fitting parameters. An abundance threshold (Athreshold) was selected at a value where the exponential decay fell to a value of n0 exp(− p), or Athreshold = p/k, where p is an optimization parameter. Proteins with an abundance greater than Athreshold were assigned as being in the corona. We varied p between 0 and 3.5 and chose the value 2.25, which optimized the performance of the classifier following training  was next validated using a stratified shuffle split (100 repeats) validation to ensure consistent levels of the minority class. The minority class here is the in-corona class, which has fewer proteins than the out-of-corona class. The shuffle split retained 10% of the dataset for corona validation. The training split was augmented with entries developed from SMOTE (minority/majority ratio of 0.7:1 with 12 k-neighbors), as detailed in the main text. Results were collected for each fold. For cross-biofluid tests, the percentage of proteins in the test set was varied to keep the same number of proteins in the test set equal to 10% of the total number of proteins used for mixed biofluid cases. The adjusted value was set by scaling 10% by a factor of the total number of proteins divided by the number of proteins in the test biofluid .Using Scikit-Learn’s GridSearchCV  on the total system, features were ranked using Scikit-Learn’s SelectKBest function 15-SWCNTs in plasma and CSF as the training and verification data. Predictions were recorded at the end of each fold. When protein targets were run, all available data were used to train the classifier. This last classifier then provided predictions on the test proteins, as listed in table S3.The classifier was tested against a list of 996 cytoplasmic proteins and 999 nuclear proteins [available for batch download through the UniProt database (15 or Cy5-(GT)6] were prepared with 0.2 mg of mixed-chirality SWCNTs  and 20 μM ssDNA  added in 1 ml total volume of 0.1× phosphate-buffered saline  (910nm = 0.02554 liter mg−1 cm−1 (−1 in 1× PBS ≥2 hours before use.Suspensions of SWCNTs with fluorophore-labeled ssDNA [Cy5-(GT)Proteins were sourced as listed in Corona dynamics were measured as described previously (


It summarizes complex composition, topology and function along with links to a large range of domain-specific resources . Since the last update in 2019, we have produced a first draft complexome for Escherichia coli, maintained and updated that of Saccharomyces cerevisiae, added over 40 coronavirus complexes and increased the human complexome to over 1100 complexes that include approximately 200 complexes that act as targets for viral proteins or are part of the immune system. The display of protein features in ComplexViewer has been improved and the participant table is now colour-coordinated with the nodes in ComplexViewer. Community collaboration has expanded, for example by contributing to an analysis of putative transcription cofactors and providing data accessible to semantic web tools through Wikidata which is now populated with manually curated Complex Portal content through a new bot. Our data license is now CC0 to encourage data reuse. Users are encouraged to get in touch, provide us with feedback and send curation requests through the ‘Support’ link.The Complex Portal ( Protein complexes, stable functional assemblies consisting of two or more associated polypeptide chains, are responsible for driving and regulating many cellular processes. Multi-chain assemblies perform many functions, including (a) positioning molecules involved in the same process in close proximity (b) bringing structure to disordered regions of proteins and (c) creating novel substrate binding sites at subunit interfaces. These assemblies can contain additional molecules, such as nucleic acids and small molecules. In budding yeast , around www.ebi.ac.uk/complexportal) was created to meet this unmet need: It is a manually curated, encyclopaedic resource that provides stable identifiers and summarizes compositional, topological and functional aspects of stable macromolecular complexes from a selection of model organisms and organisms of special interest. It enables protein complex identification in large-scale data analyses, contributing to the study of complex evolution and increasing our understanding of cell biology  for which there is experimental or inferred evidence that these molecules stably interact with each other. As of release 241 (18 October 2021), 3572 complexes from 26 species have been curated and released. Each entry is species-specific, describing the complex composition, topology and function and linking out to external databases that provide further domain-specific information, such as structural details from wwPDB  or EMDB Since our last update three years ago, we have focused on a number of curation priorities:E. coli K12 (NCBI reference taxon: 83333) complexome currently consisting of 321 complexes. This work was based on extensive literature mining and subsequent comparison with existing resources, in this case primarily UniProt KB and EcoCyc . 95% (305) of complexes contain 5 or fewer proteins (median = 3)  and heat-labile enterotoxin IIB complex (CPX-2304).As with the recently completed  in phnE , therefoIn response to the COVID-19 pandemic we have curated coronavirus complexes as well as human targets of viral proteins . To dateS. cerevisiae are conserved in multicellular organisms, but human complexes often contain additional protein components or the existence of paralogous proteins lead to increased numbers of complex variants. One simple example of the latter case is the replication protein A complex, a single heterotrimer in yeast (CPX-21), but in humans one protein (P15927/Q13156) has been duplicated resulting in two heterotrimeric complex variants (CPX-1878/CPX-1879). Additionally, there will be an appreciable number of transmembrane and extra-cellular complexes with a role in the immune response, inter- and intracellular communication and signalling. A data mining exercise, undertaken in collaboration with the UniProt group, of information embedded in UniProtKB records has suggested that there may be at least 4000 different assemblies in the human complexome. To date, we have released 1255 human complexes including almost 200 that are either targets of viral proteins or play a role in the immune system, including the B-cell and T-cell receptor complexes, the interferon-receptor family of complexes and the complete complement cascade of the innate immune system. We are also focusing on dimeric transcription factor complexes, complementing a recent revision of the human transcription factor proteins undertaken by members of the Gene Ontology and Gene Regulation Consortia . If a complex has been merged into an existing entry, the accession number of the obsoleted complex is added to the complex it has been merged into as a secondary identifier, allowing an external user to still retrieve an entry. Secondary identifiers are available in all download files, while the website currently only displays the primary accession number (e.g. CPX-3042 is now part of CPX-2161). This enables us to adhere to the FAIR principle of data Findability. More minor updates, for example the identification of an additional participant, are indicated by entry versioning, as previously described .- Binding features in subcomplexes: binding features are displayed between any participants in a complex, which now includes participants which are part of a subcomplex and bind another molecule outside the subcomplex, e.g. the binding of human ACE2 receptor to the SARS-CoV-2 Spike S1 protein chain (CPX-5683) .- Multiple feature types and ranges: are displayed as separate tracks in the bar representation of a protein and separate wedges or circles within the circular icon (e.g. CPX-1919 or CPX-1003). Different feature types can be turned on and off using a new set of buttons below the viewer window.- Additionally, the participant table is now colour-coordinated, matching the node colours of proteins or background colours of complexes as participants.We have refreshed and updated the Home, About and Documentation pages, including adding more information about our curation practices, handling of edge cases and documentation about our file formats. These new pages are also linked to our GitHub repository which allows us to make any updates easier and quicker.Complexes may contain molecules other than proteins, such as nucleic acids or small molecules, but some users, for example mass spectrometry proteomics scientists, are only interested in the protein components. Protein complexes can be subcomponents of larger assemblies, and users previously needed to parse the files separately to retrieve the participant of these subcomplexes. In response to requests from this user community, we have added a new column to the tab-delimited format, ComplexTab, that contains a list of UniProt accession numbers (including isoform or post-processes chain extensions) in pipe-separated style with stoichiometry in parentheses.https://creativecommons.org/publicdomain/zero/1.0/). This applies to all Complex Portal data, i.e. PSI-MI XML3.0  License  is part of the infrastructure provided by the Wikimedia Foundation and a sister project of Wikipedia; it initially provided a semantic web infrastructure for encyclopedic knowledge to be used in Wikipedia . During this hackathon we added Wikidata entities for these eleven complexes using a semi-automated curation pipeline using OpenRefine (https://openrefine.org/) and the Wikidata Integrator Python module (https://www.wikidata.org/wiki/Property:P7718) for SARS-CoV-2 complexes as well as selected human complexes, which had already been manually added to Wikidata, were immediately used in the WikiPathways COVID-19 Pathways Collection (http://covid.wikipathways.org/) (https://covid19map.elixir-luxembourg.org/minerva/)  and approved after an online vetting process, where the rationale and the software was assessed. The open source code of the bot can be found at https://github.com/lubianat/complex_bot. Currently, the bot runs manually; after each Complex Portal update a request is sent to the bot developer to run the bot on the recent update. We are working towards an automatic integration of the two processes, similar to the continuous integration applied elsewhere (https://scholia.toolforge.org/complex/Q104836061 shows information for the SARS-CoV-2 polymerase complex (CPX-5742). Similarly, protein pages show in which complexes they participate (e.g. https://scholia.toolforge.org/protein/Q90038963 for SARS-CoV-2 NSP7).After we released the first eleven SARS-CoV-2 complexes in April 2020 we joined the COVID19 Virtual Elixir BioHackathon 2020 (n module . Throughys.org/)  which isinerva/)  that alsinerva/) . This enlsewhere . Anotherlsewhere . For exaSaccharomyces Genome Database  who is studying the relation between chromatin structure and transcription and is providing extensive lists of important curation targets. The group is developing a pipeline to identify additional potential protein complexes with a role in cotranscription which will be further evaluated by manual curation in the Complex Portal.We are developing an import pipeline for heteromeric structures from PDBe which will speed up the manual part of the curation process by populating all standardized, structured fields directly from the PDB files. We are also working with the group of Colin Logie  or our REST API (https://www.ebi.ac.uk/intact/complex-ws/).Data can be accessed either via our ftp site .Lists of putative complexes on our ‘watch lists’ are available on request (https://github.com/SuLab/WikidataIntegrator.Wikidata Integrator python module: https://github.com/lubianat/complex_bot.Wikidata bot:


Since 2020, the receptor-binding domain (RBD) of the spike protein of the novel severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) has been constantly mutating, producing most of the notable missense mutations in the context of “variants of concern”, probably in response to the vaccine-driven alteration of immune profiles of the human population. The Delta variant, in particular, has become the most prevalent variant of the epidemic, and it is spreading in countries with the highest vaccination rates, causing the world to face the risk of a new wave of the contagion. Understanding the physical mechanism responsible for the mutation-induced changes in the RBD’s binding affinity, its transmissibility, and its capacity to escape vaccine-induced immunity is the “urgent challenge” in the development of preventive measures, vaccines, and therapeutic antibodies against the coronavirus disease 2019 (COVID-19) pandemic. In this study, entropy–enthalpy compensation and the Gibbs free energy change were used to analyze the impact of the RBD mutations on the binding affinity of SARS-CoV-2 variants with the receptor angiotensin converting enzyme 2 (ACE2) and existing antibodies. Through the analysis, we found that the existing mutations have already covered almost all possible detrimental mutations that could result in an increase of transmissibility, and that a possible mutation in amino-acid position 498 of the RBD can potentially enhance its binding affinity. A new calculation method for the binding energies of protein–protein complexes is proposed based on the entropy–enthalpy compensation rule. All known structures of RBD–antibody complexes and the RBD–ACE2 complex comply with the entropy–enthalpy compensation rule in providing the driving force behind the spontaneous protein–protein docking. The variant-induced risk of breakthrough infections in vaccinated people is attributed to the L452R mutation’s reduction of the binding affinity of many antibodies. Mutations reversing the hydrophobic or hydrophilic performance of residues in the spike RBD potentially cause breakthrough infections of coronaviruses due to the changes in geometric complementarity in the entropy–enthalpy compensations between antibodies and the virus at the binding sites. The coronavirus disease 2019 (COVID-19) has spread worldwide, with more than 230 million confirmed cases, and has led to an ongoing pandemic. In response to this once-in-a-century, sustained, worldwide pandemic, unprecedented amounts of funds and manpower have been invested to develop vaccines against the disease by governments, corporations, university research groups, and international health organizations . At presFour Variants of Concern (VOCs) and five Variants of Interest (VOIs) are recognized by the WHO. The increased transmissibility and breakthrough infections caused by the VOCs are believed to be due to mutations in the structure of the spike (S) proteins . There hAs a typical spontaneous reaction, protein–protein docking  must release Gibbs free energy as it proceeds, namely, the change of Gibbs free energy ntaneity . When usntaneity . Water mntaneity ,34,35 bentaneity ,41,42,43ntaneity ,44. The ntaneity ,34,35,45Using biophysical assays, Wrapp et al. found that the binding affinity of SARS-CoV-2 S RBD to ACE2 is more than 10-fold higher than that of the corresponding spike protein of SARS-CoV to the host cell receptor . The hydAt the interface of a protein–protein complex, interactions among the surface areas of proteins can be classified into four types: hydrophobic–hydrophobic (Ho-Ho) interaction, hydrophobic–hydrophilic (Ho-Hi) interaction, attractive dipole–dipole (ADD) interaction, and repulsive dipole–dipole (RDD) interaction. The four types of surface interactions between the SARS-CoV-2 S RBD and the ACE2 are illustrated in At the interface of the RBD–ACE2 complex, because of the repulsion between the hydrophobic groups and the hydrophilic groups in water, when a local hydrophobic surface area of the RBD attaches to another hydrophilic surface area of the ACE2, the interactions play a role in decreasing the binding affinity between the two proteins. At the interface, seven cases of the Ho-Hi surface interactions are identified at the amino-acid positions L452, K417, Y453, E484, N501, F490, and Q498 of the RBD, as illustrated in Entropy–enthalpy compensation and the Gibbs free energy change can be used to analyze the spontaneous nature of protein–protein docking. The change of Gibbs free energy 2 [2 hydrophobic surfaces areas can expel approximately 4 ordered water molecules from the hydration shell of the hydrophobic regions into a liquid water solvent; the entropy increment for the Ho-Ho binding is about 35,800 J/mol, that is, the A characteristic of hydration shell water molecules is that their hydrogen bonding network is much more ordered than that of free liquid water molecules; that is, their entropy is lower (less entropy in the system) ,55,56,572 . At the 2. Thus, the docking-induced attachment between a 10 Å2 hydrophobic surface area and a 10 Å2 hydrophilic surface area can expel approximately 2 ordered water molecules from the hydration shell, and break 2 H-bonds between the protein surface hydrophilic groups and the water molecules. The entropy increment is about 17,900 J/mol and the enthalpy increment is about 24,000 J/mol. As a result, the In the case of a hydrophobic surface area attaching to a hydrophilic surface area at the interface of a protein–protein complex, the hydrophobic surface area loses its hydration shell  due to the docking, whereas the hydrophilic surface area loses its hydrogen-bonded water molecules due to the docking. The hydrophilicity of hydrophilic side chains or main chain is normally expressed by CO or NH groups at their ends. These hydrophilic CO and NH groups are hydrogen-bonded with the surrounding water molecules in a hydration shell before the docking. The binding energy of the H-bond between the NH group and a water molecule is about −7.65 kcal/mol , and the binding bond energy of the H-bond between the CO group and a water molecule is about −4.66 kcal/mol  ,40,41,422. Thus, for the docking-induced attractive dipole–dipole interaction between two 10 Å2 hydrophilic surface areas, the Gibbs free energy change is Intermolecular H-bonds in protein–protein complexes are normally the dipole–dipole attractions between the CO groups and NH groups of the protein chains, acting as the pairing of the H-bond donors and acceptors . The bin2. Thus, in the case of the docking of the repulsive dipole–dipole between two 10 Å2 hydrophilic surface areas, the change in Gibbs free energy is Taking the average as about Ho-Ho, SHo-Hi, SADD, and SRDD. Among the four types of interactions between surface areas during the protein–protein docking, only the Ho-Ho interaction causes the Experimentally determined structures of RBD–antibody complexes and the RBD–ACE2 complex are stored in the protein data bank (PDB) archives. The corresponding total area for the Ho-Ho, Ho-Hi, ADD, and RDD interactions at the interface of these protein–protein complexes can be easily measured by using these PDB files, denoted as SIt should be noted that Equation (1) does not possess an ab initio physico-mathematical derivation. Equation (1) is more appropriate for binding affinity estimation when proteins are in close contact with each other in the complex. For the cases where proteins are not in close contact with each other in the complex, the equation simply cannot be used.Ho-Ho, SHo-Hi, SADD, and SRDD areas at the interface of the RBD–ACE2 and RBD–antibody protein–protein complexes. The proportions of SHo-Ho, SHo-Hi, SADD, and SRDD at the interface of the complexes are calculated and illustrated in To realize the spontaneity of protein–protein binding, the total area of the Ho-Ho interaction should normally be larger than that of the three other types of interactions. For example, 22 structures of the RBD–antibody complexes demonstrate the characteristic that the hydrophobic surface areas fully collapsing together in between the proteins of the complexes see . ProteinThe results show that all known structures of the RBD–antibody complexes, the RBD–ACE2 complex (PDBID:7KMB), and the SARS-COV-S-RBD–ACE2 complex (PDBID:3D0H) confirm the entropy–enthalpy compensation rule for providing the driving force behind the spontaneous protein–protein docking, namely, all the calculated values of the tive see . The docplex see ; it is tplex see d. This mplex see d,e. It iRS-CoV-2 .When a mutation reverses the hydrophobic or hydrophilic performance of a residue in the RBD, it potentially changes the type of surface interactions in the complexes at this amino acid position. This means that a mutation can change the binding affinity of the RBD with ACE2 or antibodies due to the changes in geometric complementariness for the entropy–enthalpy compensations between proteins. For each variant, the oV-2 see . The calUsing the same method, we evaluated the impact of these mutations on the RBD–antibodies binding affinities. We analyzed the mutation-induced surface hydrophobic and hydrophilic area changes at the binding sites of the complexes of the SARS-CoV-2 variants and 22 existing antibodies ,64,65,66In this study, many experimentally determined native structures of proteins are used to study the mechanism triggering the docking of SARS-CoV-2 variants to ACE2 and the antibodies. All the three-dimensional (3D) structure data of protein molecules were sourced from the PDB database, including the experimentally determined RBD of SARS-CoV-2 S, ACE2, antibodies, and their complexes. The IDs of these proteins, according to the PDB database, are marked in all the figures. In order to show the distribution of the hydrophobic areas on the surface of the SARS-CoV-2 RBD, ACE2, antibodies, and their complexes at the binding sites in these figures, we used the structural biology visualization software PyMOL to display the protein hydrophobic surface areas. The distribution of the hydrophobic surface areas is highlighted in green (carbon) and yellow (sulfur). The distribution of the hydrophilic surface areas is highlighted in red (oxygen) and blue (nitrogen) since the hydrophilicity of the protein at the binding site is normally expressed by the CO or NH groups at the ends of the hydrophilic side chains or the main chain.Ho-Ho, SHo-Hi, SADD, and SRDD in the structures of the protein complexes, we sketched the interfaces between the proteins in the complexes as the basis for the calculations of SHo-Ho, SHo-Hi, SADD, and SRDD. According to Equation (1), the binding affinity of RBD and ACE2/antibodies can be characterized by calculating the size of the hydrophobic and hydrophilic contact area in the complex structures. We used the molecular 3D structure display software PyMOL to draw the hydrophobic and hydrophilic surface areas at the binding sites of these complex structures. We calculated the hydrophobic attraction surface areas and the hydrophobic repulsion surface areas involved in the docking among the RBD, ACE2, and antibodies in this study by using integration. With regard to the calculation of S2) is a conservative estimate. At the human body temperature, the docking-induced hydrophobic interaction between two 10 Å2 hydrophobic surface areas may expel approximately more than 4 ordered water molecules from the hydration shell of hydrophobic regions into a liquid water solvent, while the entropy increment for the Ho-Ho binding may be larger than 35,800 J/mol. The surface density of hydrogen bonds between water molecules and the hydrophilic groups  was estimated from a survey of the PDB file (PDBID: 6LZG). Experimental results show that water molecules slow down greatly when they encounter the hydrophobic areas of a protein, and the speed is reduced by 99% . TherefoBased on the entropy–enthalpy compensation rule, a simple measurement calculation of the binding energies of protein–protein complexes was developed. As a typical spontaneous reaction, the RBD–ACE2 docking or the RBD–antibody docking must release Gibbs free energy according the second law of thermodynamics. All the four types of interactions between the surface areas of the docking proteins changed the values of entropy and enthalpy of the system, and affected the spontaneity of protein–protein docking. The entropy–enthalpy compensation involved in the surface area interactions during the docking can be approximately calculated through a survey of these hydrophobic and hydrophilic surface areas, thus enabling the calculation of the binding energy. With these calculations, we showed that all known structures of the RBD–antibody and RBD–ACE2 complexes have strong binding affinities, and enable the hydrophobic surface areas to fully collapse together in between the proteins of the complexes. By means of this analysis, we found that the existing mutations have already covered almost all possible detrimental mutations that could result from the increase in the RBD–ACE2 binding affinity, and a possible mutation at amino-acid position 498 of the RBD can potentially enhance the binding affinity. Variant-induced risk of breakthrough infections among vaccinated people was attributed to the L452R mutations, decreasing the binding affinity of many antibodies. Mutations reversing the hydrophobic or hydrophilic performance of residues in the RBD of coronaviruses potentially cause breakthrough infections due to the changes in geometric complementariness of the entropy–enthalpy compensations between antibodies and the virus at the binding sites. Even though the stronger binding affinity could promote faster entry kinetics for the virus entering the host, this may not equate to the faster spread of the virus in the population; the calculation results match the findings on the Delta variant being more transmissible and spreading more easily in the most vaccinated countries.