The Robustness of a Signaling Complex to Domain Rearrangements Facilitates Network Evolution

The rearrangement of protein domains is known to have key roles in the evolution of signaling networks and, consequently, is a major tool used to synthetically rewire networks. However, natural mutational events leading to the creation of proteins with novel domain combinations, such as in frame fusions followed by domain loss, retrotranspositions, or translocations, to name a few, often simultaneously replace pre-existing genes. Thus, while proteins with new domain combinations may establish novel network connections, it is not clear how the concomitant deletions are tolerated. We investigated the mechanisms that enable signaling networks to tolerate domain rearrangement-mediated gene replacements. Using as a model system the yeast mitogen activated protein kinase (MAPK)-mediated mating pathway, we analyzed 92 domain-rearrangement events affecting 11 genes. Our results indicate that, while domain rearrangement events that result in the loss of catalytic activities within the signaling complex are not tolerated, domain rearrangements can drastically alter protein interactions without impairing function. This suggests that signaling complexes can maintain function even when some components are recruited to alternative sites within the complex. Furthermore, we also found that the ability of the complex to tolerate changes in interaction partners does not depend on long disordered linkers that often connect domains. Taken together, our results suggest that some signaling complexes are dynamic ensembles with loose spatial constraints that could be easily re-shaped by evolution and, therefore, are ideal targets for cellular engineering.     

From intracellular signaling to population oscillations: bridging size- and time-scales in collective behavior

Collective behavior in cellular populations is coordinated by biochemical signaling networks within individual cells. Connecting the dynamics of these intracellular networks to the population phenomena they control poses a considerable challenge because of network complexity and our limited knowledge of kinetic parameters. However, from physical systems, we know that behavioral changes in the individual constituents of a collectively behaving system occur in a limited number of well-defined classes, and these can be described using simple models. Here, we apply such an approach to the emergence of collective oscillations in cellular populations of the social amoeba Dictyostelium discoideum. Through direct tests of our model with quantitative in vivo measurements of single-cell and population signaling dynamics, we show how a simple model can effectively describe a complex molecular signaling network at multiple size and temporal scales. The model predicts novel noise-driven single-cell and population-level signaling phenomena that we then experimentally observe. Our results suggest that like physical systems, collective behavior in biology may be universal and described using simple mathematical models.     

A proposal for using the ensemble approach to understand genetic regulatory networks

Understanding the genetic regulatory network comprising genes, RNA, proteins and the network connections and dynamical control rules among them, is a major task of contemporary systems biology. I focus here on the use of the ensemble approach to find one or more well-defined ensembles of model networks whose statistical features match those of real cells and organisms. Such ensembles should help explain and predict features of real cells and organisms. More precisely, an ensemble of model networks is defined by constraints on the "wiring diagram" of regulatory interactions, and the "rules" governing the dynamical behavior of regulated components of the network. The ensemble consists of all networks consistent with those constraints. Here I discuss ensembles of random Boolean networks, scale free Boolean networks, "medusa" Boolean networks, continuous variable networks, and others. For each ensemble, M statistical features, such as the size distribution of avalanches in gene activity changes unleashed by transiently altering the activity of a single gene, the distribution in distances between gene activities on different cell types, and others, are measured. This creates an M-dimensional space, where each ensemble corresponds to a cluster of points or distributions. Using current and future experimental techniques, such as gene arrays, these M properties are to be measured for real cells and organisms, again yielding a cluster of points or distributions in the M-dimensional space. The procedure then finds ensembles close to those of real cells and organisms, and hill climbs to attempt to match the observed M features. Thus obtains one or more ensembles that should predict and explain many features of the regulatory networks in cells and organisms.     

Dimensionality of Social Networks Using Motifs and Eigenvalues

We consider the dimensionality of social networks, and develop experiments aimed at predicting that dimension. We find that a social network model with nodes and links sampled from an m-dimensional metric space with power-law distributed influence regions best fits samples from real-world networks when m scales logarithmically with the number of nodes of the network. This supports a logarithmic dimension hypothesis, and we provide evidence with two different social networks, Facebook and LinkedIn. Further, we employ two different methods for confirming the hypothesis: the first uses the distribution of motif counts, and the second exploits the eigenvalue distribution.     

Compensatory effects in the PI3K/PTEN/AKT signaling network following receptor tyrosine kinase inhibition

Overcoming de novo and acquired resistance to anticancer drugs that target signaling networks is a formidable challenge for drug design and effective cancer therapy. Understanding the mechanisms by which this resistance arises may offer a route to addressing the insensitivity of signaling networks to drug intervention and restore the efficacy of anticancer therapy. Extending our recent work identifying PTEN as a key regulator of Herceptin sensitivity, we present an integrated theoretical and experimental approach to study the compensatory mechanisms within the PI3K/PTEN/AKT signaling network that afford resistance to receptor tyrosine kinase (RTK) inhibition by anti-HER2 monoclonal antibodies. In a computational model representing the dynamics of the signaling network, we define a single control parameter that encapsulates the balance of activities of the enzymes involved in the PI3K/PTEN/AKT cycle. By varying this control parameter we are able to demonstrate both distinct dynamic regimes of behavior of the signaling network and the transitions between those regimes. We demonstrate resistance, sensitivity, and suppression of RTK signals by the signaling network. Through model analysis we link the sensitivity-to-resistance transition to specific compensatory mechanisms within the signaling network. We study this transition in detail theoretically by variation of activities of PTEN, PI3K, AKT enzymes, and use the results to inform experiments that perturb the signaling network using combinatorial inhibition of RTK, PTEN, and PI3K enzymes in human ovarian carcinoma cell lines. We find good alignment between theoretical predictions and experimental results. We discuss the application of the results to the challenges of hypersensitivity of the signaling network to RTK signals, suppression of drug resistance, and efficacy of drug combinations in anticancer therapy.     

Quantifying and analyzing the network basis of genetic complexity

Genotype-to-phenotype maps exhibit complexity. This genetic complexity is mentioned frequently in the literature, but a consistent and quantitative definition is lacking. Here, we derive such a definition and investigate its consequences for model genetic systems. The definition equates genetic complexity with a surplus of genotypic diversity over phenotypic diversity. Applying this definition to ensembles of Boolean network models, we found that the in-degree distribution and the number of periodic attractors produced determine the relative complexity of different topology classes. We found evidence that networks that are difficult to control, or that exhibit a hierarchical structure, are genetically complex. We analyzed the complexity of the cell cycle network of Sacchoromyces cerevisiae and pinpointed genes and interactions that are most important for its high genetic complexity. The rigorous definition of genetic complexity is a tool for unraveling the structure and properties of genotype-to-phenotype maps by enabling the quantitative comparison of the relative complexities of different genetic systems. The definition also allows the identification of specific network elements and subnetworks that have the greatest effects on genetic complexity. Moreover, it suggests ways to engineer biological systems with desired genetic properties.     

Disentangling the Complexity of HGF Signaling by Combining Qualitative and Quantitative Modeling

Signaling pathways are characterized by crosstalk, feedback and feedforward mechanisms giving rise to highly complex and cell-context specific signaling networks. Dissecting the underlying relations is crucial to predict the impact of targeted perturbations. However, a major challenge in identifying cell-context specific signaling networks is the enormous number of potentially possible interactions. Here, we report a novel hybrid mathematical modeling strategy to systematically unravel hepatocyte growth factor (HGF) stimulated phosphoinositide-3-kinase (PI3K) and mitogen activated protein kinase (MAPK) signaling, which critically contribute to liver regeneration. By combining time-resolved quantitative experimental data generated in primary mouse hepatocytes with interaction graph and ordinary differential equation modeling, we identify and experimentally validate a network structure that represents the experimental data best and indicates specific crosstalk mechanisms. Whereas the identified network is robust against single perturbations, combinatorial inhibition strategies are predicted that result in strong reduction of Akt and ERK activation. Thus, by capitalizing on the advantages of the two modeling approaches, we reduce the high combinatorial complexity and identify cell-context specific signaling networks.     

Bond Type and Discretization of Nonmuscle Myosin II Are Critical for Simulated Contractile Dynamics

Molecular motors drive cytoskeletal rearrangements to change cell shape. Myosins are the motors that move, cross-link, and modify the actin cytoskeleton. The primary force generator in contractile actomyosin networks is nonmuscle myosin II (NMMII), a molecular motor that assembles into ensembles that bind, slide, and cross-link actin filaments (F-actin). The multivalence of NMMII ensembles and their multiple roles have confounded the resolution of crucial questions, including how the number of NMMII subunits affects dynamics and what affects the relative contribution of ensembles’ cross-linking versus motoring activities. Because biophysical measurements of ensembles are sparse, modeling of actomyosin networks has aided in discovering the complex behaviors of NMMII ensembles. Myosin ensembles have been modeled via several strategies with variable discretization or coarse graining and unbinding dynamics, and although general assumptions that simplify motor ensembles result in global contractile behaviors, it remains unclear which strategies most accurately depict cellular activity. Here, we used an agent-based platform, Cytosim, to implement several models of NMMII ensembles. Comparing the effects of bond type, we found that ensembles of catch-slip and catch motors were the best force generators and binders of filaments. Slip motor ensembles were capable of generating force but unbound frequently, resulting in slower contractile rates of contractile networks. Coarse graining of these ensemble types from two sets of 16 motors on opposite ends of a stiff rod to two binders, each representing 16 motors, reduced force generation, contractility, and the total connectivity of filament networks for all ensemble types. A parallel cluster model, previously used to describe ensemble dynamics via statistical mechanics, allowed better contractility with coarse graining, though connectivity was still markedly reduced for this ensemble type with coarse graining. Together, our results reveal substantial tradeoffs associated with the process of coarse graining NMMII ensembles and highlight the robustness of discretized catch-slip ensembles in modeling actomyosin networks.     

Iteration method for predicting essential proteins based on orthology and protein-protein interaction networks

Background

Understanding the information-processing capabilities of signal transduction networks, how those networks are disrupted in disease, and rationally designing therapies to manipulate diseased states require systematic and accurate reconstruction of network topology. Data on networks central to human physiology, such as the inflammatory signalling networks analyzed here, are found in a multiplicity of on-line resources of pathway and interactome databases (Cancer CellMap, GeneGo, KEGG, NCI-Pathway Interactome Database (NCI-PID), PANTHER, Reactome, I2D, and STRING). We sought to determine whether these databases contain overlapping information and whether they can be used to construct high reliability prior knowledge networks for subsequent modeling of experimental data.

Results

We have assembled an ensemble network from multiple on-line sources representing a significant portion of all machine-readable and reconcilable human knowledge on proteins and protein interactions involved in inflammation. This ensemble network has many features expected of complex signalling networks assembled from high-throughput data: a power law distribution of both node degree and edge annotations, and topological features of a ??bow tie?? architecture in which diverse pathways converge on a highly conserved set of enzymatic cascades focused around PI3K/AKT, MAPK/ERK, JAK/STAT, NF??B, and apoptotic signaling. Individual pathways exhibit ??fuzzy?? modularity that is statistically significant but still involving a majority of ??cross-talk?? interactions. However, we find that the most widely used pathway databases are highly inconsistent with respect to the actual constituents and interactions in this network. Using a set of growth factor signalling networks as examples (epidermal growth factor, transforming growth factor-beta, tumor necrosis factor, and wingless), we find a multiplicity of network topologies in which receptors couple to downstream components through myriad alternate paths. Many of these paths are inconsistent with well-established mechanistic features of signalling networks, such as a requirement for a transmembrane receptor in sensing extracellular ligands.

Conclusions

Wide inconsistencies among interaction databases, pathway annotations, and the numbers and identities of nodes associated with a given pathway pose a major challenge for deriving causal and mechanistic insight from network graphs. We speculate that these inconsistencies are at least partially attributable to cell, and context-specificity of cellular signal transduction, which is largely unaccounted for in available databases, but the absence of standardized vocabularies is an additional confounding factor. As a result of discrepant annotations, it is very difficult to identify biologically meaningful pathways from interactome networks a priori. However, by incorporating prior knowledge, it is possible to successively build out network complexity with high confidence from a simple linear signal transduction scaffold. Such reduced complexity networks appear suitable for use in mechanistic models while being richer and better justified than the simple linear pathways usually depicted in diagrams of signal transduction.     

Structural Interpretation of Paramagnetic Relaxation Enhancement-Derived Distances for Disordered Protein States

Paramagnetic relaxation enhancement (PRE) is a powerful technique for studying transient tertiary organizations of unfolded and partially folded proteins. The heterogeneous and dynamic nature of disordered protein states, together with the r⁻⁶ dependence of PRE, presents significant challenges for reliable structural interpretation of PRE-derived distances. Without additional knowledge of accessible conformational substates, ensemble-simulation-based protocols have been used to calculate structure ensembles that appear to be consistent with the PRE distance restraints imposed on the ensemble level with the proper r⁻⁶ weighting. However, rigorous assessment of the reliability of such protocols has been difficult without intimate knowledge of the true nature of disordered protein states. Here we utilize sets of theoretical PRE distances derived from simulated structure ensembles that represent the folded, partially folded and unfolded states of a small protein to investigate the efficacy of ensemble-simulation-based structural interpretation of PRE distances. The results confirm a critical limitation that, due to r⁻⁶ weighting, only one or a few members need to satisfy the distance restraints and the rest of the ensemble are essentially unrestrained. Consequently, calculated structure ensembles will appear artificially heterogeneous no matter whether the PRE distances are derived from the folded, partially unfolded or unfolded state. Furthermore, the nature of the heterogeneous ensembles is largely determined by the protein model employed in structure calculation and reflects little on the true nature of the underlying disordered state. These findings suggest that PRE measurements on disordered protein states alone generally do not contain enough information for a reliable structural interpretation and that the latter will require additional knowledge of accessible conformational substates. Interestingly, when a very large number of PRE measurements is available, faithful structural interpretation might be possible with intermediate ensemble sizes under ideal conditions.     

A markov classification model for metabolic pathways

Background  

This paper considers the problem of identifying pathways through metabolic networks that relate to a specific biological response. Our proposed model, HME3M, first identifies frequently traversed network paths using a Markov mixture model. Then by employing a hierarchical mixture of experts, separate classifiers are built using information specific to each path and combined into an ensemble prediction for the response.     

Validation of X-ray Crystal Structure Ensemble Representations of SARS-CoV-2 Main Protease by Solution NMR Residual Dipolar Couplings

Considerable debate has focused on whether sampling of molecular dynamics trajectories restrained by crystallographic data can be used to develop realistic ensemble models for proteins in their natural, solution state. For the SARS-CoV-2 main protease, M^pro, we evaluated agreement between solution residual dipolar couplings (RDCs) and various recently reported multi-conformer and dynamic-ensemble crystallographic models. Although Phenix-derived ensemble models showed only small improvements in crystallographic R_free, substantially improved RDC agreement over fits to a conventionally refined 1.2-Å X-ray structure was observed, in particular for residues with above average disorder in the ensemble. For a set of six lower resolution (1.55–2.19 Å) M^pro X-ray ensembles, obtained at temperatures ranging from 100 to 310 K, no significant improvement over conventional two-conformer representations was found. At the residue level, large differences in motions were observed among these ensembles, suggesting high uncertainties in the X-ray derived dynamics. Indeed, combining the six ensembles from the temperature series with the two 1.2-Å X-ray ensembles into a single 381-member “super ensemble” averaged these uncertainties and substantially improved agreement with RDCs. However, all ensembles showed excursions that were too large for the most dynamic fraction of residues. Our results suggest that further improvements to X-ray ensemble refinement are feasible, and that RDCs provide a sensitive benchmark in such endeavors. Remarkably, a weighted ensemble of 350 PDB M^pro X-ray structures provided slightly better cross-validated agreement with RDCs than any individual ensemble refinement, implying that differences in lattice confinement also limit the fit of RDCs to X-ray coordinates.     

Structural Properties and Complexity of a New Network Class: Collatz Step Graphs

In this paper, we introduce a biologically inspired model to generate complex networks. In contrast to many other construction procedures for growing networks introduced so far, our method generates networks from one-dimensional symbol sequences that are related to the so called Collatz problem from number theory. The major purpose of the present paper is, first, to derive a symbol sequence from the Collatz problem, we call the step sequence, and investigate its structural properties. Second, we introduce a construction procedure for growing networks that is based on these step sequences. Third, we investigate the structural properties of this new network class including their finite scaling and asymptotic behavior of their complexity, average shortest path lengths and clustering coefficients. Interestingly, in contrast to many other network models including the small-world network from Watts & Strogatz, we find that CS graphs become ‘smaller’ with an increasing size.     

Exhaustive analysis of the modular structure of the spliceosomal assembly network: a petri net approach

Spliceosomes are macro-complexes involving hundreds of proteins with many functional interactions. Spliceosome assembly belongs to the key processes that enable splicing of mRNA and modulate alternative splicing. A detailed list of factors involved in spliceosomal reactions has been assorted over the past decade, but, their functional interplay is often unknown and most of the present biological models cover only parts of the complete assembly process. It is a challenging task to build a computational model that integrates dispersed knowledge and combines a multitude of reaction schemes proposed earlier.Because for most reactions involved in spliceosome assembly kinetic parameters are not available, we propose a discrete modeling using Petri nets, through which we are enabled to get insights into the system's behavior via computation of structural and dynamic properties. In this paper, we compile and examine reactions from experimental reports that contribute to a functional spliceosome. All these reactions form a network, which describes the inventory and conditions necessary to perform the splicing process. The analysis is mainly based on system invariants. Transition invariants (T-invariants) can be interpreted as signaling routes through the network. Due to the huge number of T-invariants that arise with increasing network size and complexity, maximal common transition sets (MCTS) and T-clusters were used for further analysis. Additionally, we introduce a false color map representation, which allows a quick survey of network modules and the visual detection of single reactions or reaction sequences, which participate in more than one signaling route. We designed a structured model of spliceosome assembly, which combines the demands on a platform that i) can display involved factors and concurrent processes, ii) offers the possibility to run computational methods for knowledge extraction, and iii) is successively extendable as new insights into spliceosome function are reported by experimental reports. The network consists of 161 transitions (reactions) and 140 places (reactants). All reactions are part of at least one of the 71 T-invariants. These T-invariants define pathways, which are in good agreement with the current knowledge and known hypotheses on reaction sequences during spliceosome assembly, hence contributing to a functional spliceosome. We demonstrate that present knowledge, in particular of the initial part of the assembly process, describes parallelism and interaction of signaling routes, which indicate functional redundancy and reflect the dependency of spliceosome assembly initiation on different cellular conditions. The complexity of the network is further increased by two switches, which introduce alternative routes during A-complex formation in early spliceosome assembly and upon transition from the B-complex to the C-complex. By compiling known reactions into a complete network, the combinatorial nature of invariant computation leads to pathways that have previously not been described as connected routes, although their constituents were known. T-clusters divide the network into modules, which we interpret as building blocks in spliceosome maturation. We conclude that Petri net representations of large biological networks and system invariants, are well-suited as a means for validating the integration of experimental knowledge into a consistent model. Based on this network model, the design of further experiments is facilitated.     

Dynamic patterns of correlated activity in the prefrontal cortex encode information about social behavior

New technologies make it possible to measure activity from many neurons simultaneously. One approach is to analyze simultaneously recorded neurons individually, then group together neurons which increase their activity during similar behaviors into an “ensemble.” However, this notion of an ensemble ignores the ability of neurons to act collectively and encode and transmit information in ways that are not reflected by their individual activity levels. We used microendoscopic GCaMP imaging to measure prefrontal activity while mice were either alone or engaged in social interaction. We developed an approach that combines a neural network classifier and surrogate (shuffled) datasets to characterize how neurons synergistically transmit information about social behavior. Notably, unlike optimal linear classifiers, a neural network classifier with a single linear hidden layer can discriminate network states which differ solely in patterns of coactivity, and not in the activity levels of individual neurons. Using this approach, we found that surrogate datasets which preserve behaviorally specific patterns of coactivity (correlations) outperform those which preserve behaviorally driven changes in activity levels but not correlated activity. Thus, social behavior elicits increases in correlated activity that are not explained simply by the activity levels of the underlying neurons, and prefrontal neurons act collectively to transmit information about socialization via these correlations. Notably, this ability of correlated activity to enhance the information transmitted by neuronal ensembles is diminished in mice lacking the autism-associated gene Shank3. These results show that synergy is an important concept for the coding of social behavior which can be disrupted in disease states, reveal a specific mechanism underlying this synergy (social behavior increases correlated activity within specific ensembles), and outline methods for studying how neurons within an ensemble can work together to encode information.

Behaviorally-specific patterns of correlated activity between prefrontal neurons normally enhance the information that neuronal ensembles transmit about social behavior. This study shows that in a mouse model of autism, individual neurons continue to encode social information, but this additional information carried by patterns of correlated activity is lost.