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Computational methodologies are increasingly addressing modeling of the whole cell at the molecular level. Proteins and their interactions are the key component of cellular processes. Techniques for modeling protein interactions, so far, have included protein docking and molecular simulation. The latter approaches account for the dynamics of the interactions, but are relatively slow, if carried out at all-atom resolution, or are significantly coarse-grained. Protein docking algorithms are far more efficient in sampling spatial coordinates. However, they do not account for the kinetics of the association (i.e., they do not involve the time coordinate). Our proof-of-concept study bridges the two modeling approaches, developing an approach that can reach unprecedented simulation timescales at all-atom resolution. The global intermolecular energy landscape of a large system of proteins was mapped by the pairwise Fast Fourier Transform docking and sampled in space and time by Monte Carlo simulations. The simulation protocol was parametrized on existing data and validated on a number of observations from experiments and molecular dynamics simulations. The simulation protocol performed consistently across very different systems of proteins at different protein concentrations. It recapitulated data on the previously observed protein diffusion rates and aggregation. The speed of calculation allows reaching second-long trajectories of protein systems that approach the size of the cells, at atomic resolution.

A prevailing interpretation of Waddington’s landscape is that distinct cell types with distinct physiologies are produced and stabilized by dynamical attractors in gene expression space. This notion is often applied in the analysis of single-cell omics data, where cells are clustered into groups that represent cell types, prior to downstream analyses like differential gene expression. Until the advent of single-cell measurement technologies, however, it has been impossible to characterize the heterogeneity of cells in the neighborhoods of these attractor states. In this work, we apply graph theory to characterize the distribution of cells in epigenetic space, using data from various tissues and organisms as well as various single-cell omics technologies. Rather than finding distinct clusters of cells that map cell types to specific regions of epigenetic space, we found that cells of very distinct types and lineages occupy the same region of space. Further, we found that the density distribution of cells is approximately power-law, with most cells existing in low-density regions, very far from other cells. This highly heterogeneous density distribution is unexpected, as it is not consistent with the distributions we would expect to see in the neighborhood of an attractor. We found these two observations are universal in single-cell data on epigenetic state of multicellular organisms, regardless of the tissue, organism, measurement technique employed, or the approach used to select the subset of genes on which the analysis was performed. The fact that currently-available single-cell data is inconsistent with the predictions of Waddington’s landscape poses a challenge both for the robust analysis of these data and for our overall understanding of epigenesis in development.

Significance Many potential drug targets have been identified, but development of chemical probes to validate these targets has lagged behind. Computational screening holds promise for providing chemical tools to do so but has long been plagued by high false-positive rates: Many compounds ranked highly against a given target protein do not actually show activity. Machine learning approaches have not solved this problem, which we hypothesize is because models were not trained on sufficiently compelling “decoys.” By addressing this through a unique training strategy, we show that more effective virtual screening is attainable. We expect this insight to enable improved performance across diverse virtual screening pipelines, thus helping to provide chemical probes for new potential drug targets as they are discovered., With the recent explosion in the size of libraries available for screening, virtual screening is positioned to assume a more prominent role in early drug discovery’s search for active chemical matter. In typical virtual screens, however, only about 12% of the top-scoring compounds actually show activity when tested in biochemical assays. We argue that most scoring functions used for this task have been developed with insufficient thoughtfulness into the datasets on which they are trained and tested, leading to overly simplistic models and/or overtraining. These problems are compounded in the literature because studies reporting new scoring methods have not validated their models prospectively within the same study. Here, we report a strategy for building a training dataset (D-COID) that aims to generate highly compelling decoy complexes that are individually matched to available active complexes. Using this dataset, we train a general-purpose classifier for virtual screening (vScreenML) that is built on the XGBoost framework. In retrospective benchmarks, our classifier shows outstanding performance relative to other scoring functions. In a prospective context, nearly all candidate inhibitors from a screen against acetylcholinesterase show detectable activity; beyond this, 10 of 23 compounds have IC50 better than 50 μM. Without any medicinal chemistry optimization, the most potent hit has IC50 280 nM, corresponding to Ki of 173 nM. These results support using the D-COID strategy for training classifiers in other computational biology tasks, and for vScreenML in virtual screening campaigns against other protein targets. Both D-COID and vScreenML are freely distributed to facilitate such efforts.

The 20S proteasome core particle (CP) is a molecular machine that is a key component of cellular protein degradation pathways. Like other molecular machines, it is not synthesized in an active form but rather as a set of subunits that assemble into a functional complex. The CP is conserved across all domains of life and is composed of 28 subunits, 14 α and 14 β, arranged in four stacked seven-member rings (α

The 20S proteasome Core Particle (CP) is a molecular machine that is a key component of cellular protein degradation pathways. Like other molecular machines, it is not synthesized in an active form, but rather as a set of subunits that assemble into a functional complex. The CP is conserved across all domains of life and is composed of 28 subunits, 14 α and 14 β, arranged in four stacked 7-member rings (α7β7β7α7). While details of CP assembly vary across species, the final step in the assembly process is universally conserved: two half proteasomes (HP: α7β7) dimerize to form the CP. In the bacterium Rhodococcus erythropolis, experiments have shown that the formation of the HP is completed within minutes, while the dimerization process takes hours. The N-terminal propeptide of the β subunit, which is autocatalytically cleaved off after CP formation, plays a key role in regulating this separation of time scales. However, the detailed molecular mechanism of how the propeptide achieves this regulation is unclear. In this work, we used Molecular Dynamics (MD) simulations to characterize HP conformations and found that the HP exists in two states: one where the propeptide interacts with key residues in the HP dimerization interface and likely blocks dimerization, and one where this interface is free. We found that a propeptide mutant that dimerizes extremely slowly is essentially always in the non-dimerizable state, while the WT rapidly transitions between the two. Based on these simulations, we designed a propeptide mutant that favored the dimerizable state in MD simulations. In vitro assembly experiments confirmed that this mutant dimerizes significantly faster than WT. Our work thus provides unprecedented insight into how this critical step in CP assembly is regulated, with implications both for efforts to inhibit proteasome assembly and for the evolution of hierarchical assembly pathways.

INTRODUCTION: Metastasis is a fundamentally physical process in which cells deform through narrow gaps and generate forces to invade surrounding tissues. While it is commonly thought that increased cell deformability is an advantage for invading cells, we previously found that more invasive pancreatic ductal adenocarcinoma (PDAC) cells are stiffer than less invasive PDAC cells. Here we investigate potential mechanisms of the simultaneous increase in PDAC cell stiffness and invasion, focusing on the contributions of myosin II, Arp2/3, and formins. METHOD: We measure cell invasion using a 3D scratch wound invasion assay and cell stiffness using atomic force microscopy (AFM). To determine the effects of actin- and myosin-mediated force generation on cell stiffness and invasion, we treat cells with pharmacologic inhibitors of myosin II (blebbistatin), Arp2/3 (CK-666), and formins (SMIFH2). RESULTS: We find that the activity of myosin II, Arp2/3, and formins all contribute to the stiffness of PDAC cells. Interestingly, we find that the invasion of PDAC cell lines is differentially affected when the activity of myosin II, Arp2/3, or formins is inhibited, suggesting that despite having similar tissue origins, different PDAC cell lines may rely on different mechanisms for invasion. CONCLUSIONS: These findings deepen our knowledge of the factors that regulate cancer cell mechanotype and invasion, and incite further studies to develop therapeutics that target multiple mechanisms of invasion for improved clinical benefit. ELECTRONIC SUPPLEMENTARY MATERIAL: The online version of this article (doi:10.1007/s12195-019-00603-1) contains supplementary material, which is available to authorized users.

A biological reaction network may serve multiple purposes, processing more than one input and impacting downstream processes via more than one output. These networks operate in a dynamic cellular environment in which the levels of network components may change within cells and across cells. Recent evidence suggests that protein concentration variability could explain cell fate decisions. However, systems with multiple inputs, multiple outputs, and changing input concentrations have not been studied in detail due to their complexity. Here, we take a systems biochemistry approach, combining physiochemical modeling and information theory, to investigate how cyclooxygenase-2 (COX-2) processes simultaneous input signals within a complex interaction network. We find that changes in input levels affect the amount of information transmitted by the network, as does the correlation between those inputs. This, and the allosteric regulation of COX-2 by its substrates, allows it to act as a signal integrator that is most sensitive to changes in relative input levels., Complexity of biochemical networks allows flexibly signal transmission Cells respond to a constant stream of intra-and extra-cellular signals as they carry out their many functions. Multiple incoming signals are detected and processed through complex biochemical networks to yield outgoing responses that determine multiple aspects of cell behavior. Although this highly dynamic process is crucial to understanding how cells make decisions, it is poorly understood. Here we present a mechanistic analysis of signal flow in a model biochemical network with multiple incoming and outgoing signals. We find that the preferred path of signal flow between each input and output is dependent on both the levels of the inputs and the relationship between inputs. Our data suggests that the complexity of biochemical networks containing multiple inputs, outputs, and paths allows a cell to flexibly transmit multiple outgoing signals under a range of different incoming signals.

Structural modeling of a cell is an evolving strategic direction in computational structural biology. It takes advantage of new powerful modeling techniques, deeper understanding of fundamental principles of molecular structure and assembly, and rapid growth of the amount of structural data generated by experimental techniques. Key modeling approaches to principal types of macromolecular assemblies in a cell already exist. The main challenge, along with the further development of these modeling approaches, is putting them together in a consistent, unified whole cell model. This opinion piece addresses the fundamental aspects of modeling macromolecular assemblies in a cell, and the state-of-the-art in modeling of the principal types of such assemblies.

Single-cell RNA sequencing (scRNA-seq) aims to characterize how variation in gene expression is distributed across cells in tissues and organisms. Yet, effective comprehension of these extremely high-dimensional datasets remains a critical barrier to progress in biological research. In standard analyses of scRNA-seq data, feature selection steps aim to reduce the dimensionality of the data by focusing on a subset of genes that are the most biologically variable across a set of cells. Ideally, these features provide the genes that are the most informative for partitioning groups of transcriptionally distinct cells, each representing a different cell type or identity. In this work, we propose a simple feature selection model where a binomial sampling process for each mRNA species produces a null model of technical variation. To compare our model to existing methods, we use scRNA-seq data where cell identities have been established a priori for each cell, and characterize whether different feature sets retain biologically varying genes, distort neighborhood structures, and allow popular clustering algorithms to partition groups of cells into their established classes. We find that our model of biological variation, which we term “Differentially Distributed Genes” or DDGs, outperforms existing methods, and enables dimensionality reduction without loss of critical structure within the data set.

Cellular responses to environmental changes are encoded in the complex temporal patterns of signaling proteins. However, quantifying the accumulation of information over time to direct cellular decision-making remains an unsolved challenge. This is, in part, due to the combinatorial explosion of possible configurations that need to be evaluated for information in time-course measurements. Here, we develop a quantitative framework, based on inferred trajectory probabilities, to calculate the mutual information encoded in signaling dynamics while accounting for cell-cell variability. We use it to understand NFκB transcriptional dynamics in response to different immune threats, and reveal that some threats are distinguished faster than others. Our analyses also suggest specific temporal phases during which information distinguishing threats becomes available to immune response genes; one specific phase could be mapped to the functionality of the IκBα negative feedback circuit. The framework is generally applicable to single-cell time series measurements, and enables understanding how temporal regulatory codes transmit information over time., Understanding how cells discriminate between stimuli is an ongoing challenge. Here, the authors propose a mathematical framework for inferring the mutual information encoded in temporal signaling dynamics and use it to study how information is transmitted over time in response to different stimuli in NFκB, MAPK and p53 signaling pathways.

Summary Proteasomes are multi-subunit protease complexes found in all domains of life. The maturation of the core particle (CP), which harbors the active sites, involves dimerization of two half CPs (HPs) and an autocatalytic cleavage that removes β propeptides. How these steps are regulated remains poorly understood. Here, we used the Rhodococcus erythropolis CP to dissect this process in vitro. Our data show that propeptides regulate the dimerization of HPs through flexible loops we identified. Furthermore, N-terminal truncations of the propeptides accelerated HP dimerization and decelerated CP auto-activation. We identified cooperativity in autocatalysis and found that the propeptide can be partially cleaved by adjacent active sites, potentially aiding an otherwise strictly autocatalytic mechanism. We propose that cross-processing during bacterial CP maturation is the underlying mechanism leading to the observed cooperativity of activation. Our work suggests that the bacterial β propeptide plays an unexpected and complex role in regulating dimerization and autocatalytic activation., Graphical Abstract, Highlights • A flexible region in β propeptide regulates proteasome core particle (CP) formation • Identified pre-holocomplex distinguishes dimerization from autocatalytic activation • Cooperativity found in proteasome core particle (CP) activation • Bacterial CP displayed propeptide cross-cleavage, Biological Sciences; Biochemistry; Microbiology

Bacterial cells construct many structures, such as the flagellar hook and the type III secretion system (T3SS) injectisome, that aid in crucial physiological processes such as locomotion and pathogenesis. Both of these structures involve long extracellular channels, and the length of these channels must be highly regulated in order for these structures to perform their intended functions. There are two leading models for how length control is achieved in the flagellar hook and T3SS needle: the substrate switching model, in which the length is controlled by assembly of an inner rod, and the ruler model, in which a molecular ruler controls the length. Although there is qualitative experimental evidence to support both models, comparatively little has been done to quantitatively characterize these mechanisms or make detailed predictions that could be used to unambiguously test these mechanisms experimentally. In this work, we constructed a mathematical model of length control based on the ruler mechanism and found that the predictions of this model are consistent with experimental data—not just for the scaling of the average length with the ruler protein length, but also for the variance. Interestingly, we found that the ruler mechanism allows for the evolution of needles with large average lengths without the concomitant large increase in variance that occurs in the substrate switching mechanism. In addition to making further predictions that can be tested experimentally, these findings shed new light on the trade-offs that may have led to the evolution of different length control mechanisms in different bacterial species.

As the second of a 3-part series of articles in this issue concerning the development of a mathematical model for comparative characterization of complex mixture drugs using crofelemer (CF) as a model compound, this work focuses on the evaluation of the chemical stability profile of CF. CF is a biopolymer containing a mixture of proanthocyanidin oligomers which are primarily composed of gallocatechin with a small contribution from catechin. CF extracted from drug product was subjected to molecular weight—based fractionation and thiolysis. Temperature stress and metal-catalyzed oxidation were selected for accelerated and forced degradation studies. Stressed CF samples were size fractionated, thiolyzed, and analyzed with a combination of negative-ion electrospray ionization mass spectrometry (ESI-MS) and reversed-phase-HPLC with UV absorption and fluorescence detection. We further analyzed the chemical stability data sets for various CF samples generated from reversed-phase-HPLC-UV and ESI-MS using data-mining and machine learning approaches. In particular, calculations based on mutual information of over 800,000 data points in the ESI-MS analytical data set revealed specific CF cleavage and degradation products that were differentially generated under specific storage/degradation conditions, which were not initially identified using traditional analysis of the ESI-MS results.

Bacteria construct many structures, like the flagellar hook and the type III secretion system, that aid in crucial processes such as locomotion and pathogenesis. Experimental work has suggested two competing mechanisms bacteria could use to regulate length in these structures: the “ruler” mechanism and the “substrate switching” mechanism. In this work, we constructed a mathematical model of length control based on the ruler mechanism, and found that the predictions of this model are consistent with experimental data not just for the scaling of the average length with the ruler protein length, but also the variance. Interestingly, we found that the ruler mechanism allows for the evolution of needles with large average lengths without the concomitant large increase in variance that occurs in the substrate switching mechanism. These findings shed new light on the trade-offs that may have lead to the evolution of different length control mechanisms in different bacterial species.

High-dimensional data are becoming increasingly common in nearly all areas of science. Developing approaches to analyze these data and understand their meaning is a pressing issue. This is particularly true for the rapidly growing field of single-cell RNA-Seq (scRNA-Seq), a technique that simultaneously measures the expression of tens of thousands of genes in thousands to millions of single cells. The emerging consensus for analysis workflows reduces the dimensionality of the dataset before performing downstream analysis, such as assignment of cell types. One problem with this approach is that dimensionality reduction can introduce substantial distortion into the data; consider the familiar example of trying to represent the three-dimensional earth as a two-dimensional map. It is currently unclear if such distortion affects analysis of scRNA-Seq data sets. Here, we introduce a straightforward approach to quantifying this distortion by comparing the local neighborhoods of points before and after dimensionality reduction. We found that popular techniques like t-SNE and UMAP introduce significant distortion even for relatively simple geometries such as simulated hyperspheres. For scRNA-Seq data, we found the distortion in local neighborhoods was greater than 95% in the 2- and 3-dimensional space typically used for downstream analysis. This high level of distortion can readily introduce important errors into cell type identification, pseudotime ordering, and other analyses that rely on local relationships. We found that principal component analysis can generate accurate embeddings of the data, but only when using dimensionalities that are much higher than typically used in scRNA-Seq analysis. We suggest approaches to take these findings into account and call for a new generation of dimensional reduction algorithms that can accurately embed high dimensional data in its true latent dimension.

All living things have evolved to sense changes in their environment in order to respond in adaptive ways. At the cellular level, these sensing systems generally involve receptor molecules at the cell surface, which detect changes outside the cell and relay those changes to the appropriate response elements downstream. With the advent of experimental technologies that can track signalling at the single-cell level, it has become clear that many signalling systems exhibit significant levels of ‘noise,’ manifesting as differential responses of otherwise identical cells to the same environment. This noise has a large impact on the capacity of cell signalling networks to transmit information from the environment. Application of information theory to experimental data has found that all systems studied to date encode less than 2.5 bits of information, with the majority transmitting significantly less than 1 bit. Given the growing interest in applying information theory to biological data, it is crucial to understand whether the low values observed to date represent some sort of intrinsic limit on information flow given the inherently stochastic nature of biochemical signalling events. In this work, we used a series of computational models to explore how much information a variety of common ‘signalling motifs’ can encode. We found that the majority of these motifs, which serve as the basic building blocks of cell signalling networks, can encode far more information (4–6 bits) than has ever been observed experimentally. In addition to providing a consistent framework for estimating information-theoretic quantities from experimental data, our findings suggest that the low levels of information flow observed so far in living system are not necessarily due to intrinsic limitations. Further experimental work will be needed to understand whether certain cell signalling systems actually can approach the intrinsic limits described here, and to understand the sources and purpose of the variation that reduces information flow in living cells.

Many signaling networks involve scaffold proteins that bind multiple kinases in kinase cascades. While scaffolds play a fundamental role in regulating signaling, few hypotheses regarding their function have been rigorously examined. Here, we used dynamical models of scaffold signaling to investigate the impact scaffolds have on network behavior. We considered two paradigms of scaffold assembly: as either the nucleation point for assembly of discrete multi-subunit proteins (the machine paradigm) or a platform upon which kinases independently associate (the ensemble paradigm). We found that several well-accepted hypotheses regarding the role of scaffolds in regulating signal response either do not hold or depend critically on the assembly paradigm employed. In addition to providing novel insights into the function of scaffold proteins, our work suggests experiments that could distinguish between assembly paradigms. Our findings should also inform attempts to target scaffold proteins for therapeutic intervention and the design of scaffolds for synthetic biology.

Signal transduction networks allow eukaryotic cells to make decisions based on information about intracellular state and the environment. Biochemical noise significantly diminishes the fidelity of signaling: networks examined to date seem to transmit less than 1 bit of information. It is unclear how networks that control critical cell-fate decisions (e.g., cell division and apoptosis) can function with such low levels of information transfer. Here, we use theory, experiments, and numerical analysis to demonstrate an inherent trade-off between the information transferred in individual cells and the information available to control population-level responses. Noise in receptor-mediated apoptosis reduces information transfer to approximately 1 bit at the single-cell level but allows 3–4 bits of information to be transmitted at the population level. For processes such as eukaryotic chemotaxis, in which single cells are the functional unit, we find high levels of information transmission at a single-cell level. Thus, low levels of information transfer are unlikely to represent a physical limit. Instead, we propose that signaling networks exploit noise at the single-cell level to increase population-level information transfer, allowing extracellular ligands, whose levels are also subject to noise, to incrementally regulate phenotypic changes. This is particularly critical for discrete changes in fate (e.g., life vs. death) for which the key variable is the fraction of cells engaged. Our findings provide a framework for rationalizing the high levels of noise in metazoan signaling networks and have implications for the development of drugs that target these networks in the treatment of cancer and other diseases.

There is growing interest in generating physicochemical and biological analytical data sets to compare complex mixture drugs, for example, products from different manufacturers. In this work, we compare various crofelemer samples prepared from a single lot by filtration with varying molecular weight cutoffs combined with incubation for different times at different temperatures. The 2 preceding articles describe experimental data sets generated from analytical characterization of fractionated and degraded crofelemer samples. In this work, we use data mining techniques such as principal component analysis and mutual information scores to help visualize the data and determine discriminatory regions within these large data sets. The mutual information score identifies chemical signatures that differentiate crofelemer samples. These signatures, in many cases, would likely be missed by traditional data analysis tools. We also found that supervised learning classifiers robustly discriminate samples with around 99% classification accuracy, indicating that mathematical models of these physicochemical data sets are capable of identifying even subtle differences in crofelemer samples. Data mining and machine learning techniques can thus identify fingerprint-type attributes of complex mixture drugs that may be used for comparative characterization of products.

Crofelemer is a botanical polymeric proanthocyanidin that inhibits chloride channel activity and is used clinically for treating HIV-associated secretory diarrhea. Crofelemer lots may exhibit significant physicochemical variation due to the natural source of the raw material. A variety of physical, chemical, and biological assays were used to identify potential critical quality attributes (CQAs) of crofelemer, which may be useful in characterizing differently sourced and processed drug products. Crofelemer drug substance was extracted from tablets of one commercial drug product lot, fractionated, and subjected to accelerated thermal degradation studies to produce derivative lots with variations in chemical and physical composition potentially representative of manufacturing and raw material variation. Liquid chromatography, UV absorbance spectroscopy, mass spectrometry, and nuclear magnetic resonance analysis revealed substantial changes in the composition of derivative lots. A chloride channel inhibition cell-based bioassay suggested that substantial changes in crofelemer composition did not necessarily result in major changes to bioactivity. In 2 companion papers, machine learning and data mining approaches were applied to the analytical and biological data sets presented herein, along with chemical stability data sets derived from forced degradation studies, to develop an integrated mathematical model that can identify CQAs which are most relevant in distinguishing between different populations of crofelemer.

Signaling networks have evolved to transduce external and internal information into critical cellular decisions such as growth, differentiation, and apoptosis. These networks form highly interconnected systems within cells due to network crosstalk, where an enzyme from one canonical pathway acts on targets from other pathways. It is currently unclear what types of effects these interconnections can have on the response of networks to incoming signals. In this work, we employ mathematical models to characterize the influence that multiple substrates have on one another. These models build off of the atomistic motif of a kinase/phosphatase pair acting on a single substrate. We find that the ultrasensitive, switch-like response these motifs can exhibit becomes transitive: if one substrate saturates the enzymes and responds ultrasensitively, then all substrates will do so regardless of their degree of saturation. We also demonstrate that the phosphatases themselves can induce crosstalk even when the kinases are independent. These findings have strong implications for how we understand and classify crosstalk, as well as for the rational development of kinase inhibitors aimed at pharmaceutically modulating network behavior.

Most cellular processes rely on large multiprotein complexes that must assemble into a well-defined quaternary structure in order to function. A number of prominent examples, including the 20S core particle of the proteasome and the AAA+ family of ATPases, contain ring-like structures. Developing an understanding of the complex assembly pathways employed by ring-like structures requires a characterization of the problems these pathways have had to overcome as they evolved. In this work, we use computational models to uncover one such problem: a deadlocked plateau in the assembly dynamics. When the molecular interactions between subunits are too strong, this plateau leads to significant delays in assembly and a reduction in steady-state yield. Conversely, if the interactions are too weak, assembly delays are caused by the instability of crucial intermediates. Intermediate affinities thus maximize the efficiency of assembly for homomeric ring-like structures. In the case of heteromeric rings, we find that rings including at least one weak interaction can assemble efficiently and robustly. Estimation of affinities from solved structures of ring-like complexes indicates that heteromeric rings tend to contain a weak interaction, confirming our prediction. In addition to providing an evolutionary rationale for structural features of rings, our work forms the basis for understanding the complex assembly pathways of stacked rings like the proteasome and suggests principles that would aid in the design of synthetic ring-like structures that self-assemble efficiently.

The proteasome is a large protease complex which degrades ubiquitinated proteins. It consists of a core particle (CP) that contains the proteolytic active sites and a regulatory particle (RP) that ...

The determination of the phylogenetic relationships among microorganisms has long relied primarily on gene sequence information. Given that prokaryotic organisms often lack morphological characteristics amenable to phylogenetic analysis, prokaryotic phylogenies, in particular, are often based on sequence data. In this work, we explore a new source of phylogenetic information, the distribution of protein structural domains within fully sequenced prokaryotic genomes. The evolution of the structural domains we use has been studied extensively, allowing us to base our phylogenetic methods on testable theoretical models of structural evolution. We find that the methods that produce reasonable phylogenetic relationships are indeed the methods that are most consistent with theoretical evolutionary models. This work represents, to our knowledge, the first such theoretically motivated phylogeny, as well as the first application of structural information to phylogeny on this scale. Our results have strong implications for the phylogenetic relationships among prokaryotic organisms and for the understanding of protein evolution as a whole.

Understanding the observed variability in the number of homologs of a gene is a very important unsolved problem that has broad implications for research into coevolution of structure and function, gene duplication, pseudogene formation, and possibly for emerging diseases. Here, we attempt to define and elucidate some possible causes behind the observed irregularity in sequence space. We present evidence that sequence variability and functional diversity of a gene or fold family is influenced by quantifiable characteristics of the protein structure. These characteristics reflect the structural potential for sequence plasticity, i.e., the ability to accept mutation without losing thermodynamic stability. We identify a structural feature of a protein domain—contact density—that serves as a determinant of entropy in sequence space, i.e., the ability of a protein to accept mutations without destroying the fold (also known as fold designability). We show that (log) of average gene family size exhibits statistical correlation (R2 > 0.9.) with contact density of its three-dimensional structure. We present evidence that the size of individual gene families are influenced not only by the designability of the structure, but also by evolutionary history, e.g., the amount of time the gene family was in existence. We further show that our observed statistical correlation between gene family size and contact density of the structure is valid on many levels of evolutionary divergence, i.e., not only for closely related sequence, but also for less-related fold and superfamily levels of homology.

Scaffold proteins are key components of signaling networks, recruiting and organizing other signaling proteins. Certain scaffolds have been extensively characterized and a number of hypotheses exist regarding general scaffold functions, including the linearization of dose-response behavior and the mediation of feedback. Despite intense investigation, however, we lack a complete understanding of how scaffolds might influence signal transduction. The goal of this work is to systematically determine how various properties of signaling cascades are impacted in the presence or absence of scaffold proteins. Using rule-based modeling languages, we constructed three basic models of signaling based loosely on the network structure of the yeast pheromone signaling cascade. The first model lacks a scaffold; this solution cascade involves a series of Goldbeter-Koshland covalent modification cycles where each activated substrate is the kinase for a subsequent substrate. The second, machine, model contains a scaffold that acts as a nucleation point for the hierarchical assembly of kinases into a discrete multi-subunit signaling complex. Finally, the ensemble model contains a scaffold to which kinases bind independently. This ensemble network therefore exhibits considerably more combinatorial complexity in terms of the numbers of possible signaling species compared to the others. We performed a comparative analysis of these three paradigms, and found significant phenotypic differences. Notably, scaffold-based models exhibited far less noise as cascade depth increases. Furthermore, only machine-like models are capable of preventing crosstalk in instances where two pathways share components. More generally, we observed that a machine-like cascade generally offers greater functional robustness, though at the cost of evolutionary plasticity and time to signal response. The evolution of scaffold proteins, and the rules that govern the assembly of signaling complexes, has likely been driven by the distinct functional roles that scaffold-based signaling complexes can play.

Proteasome assembly is a complex process, requiring 66 subunits distributed over several subcomplexes to associate in a coordinated fashion. Ten proteasome-specific chaperones have been identified that assist in this process. For two of these, the Pba1-Pba2 dimer, it is well established that they only bind immature core particles (CPs) in vivo. In contrast, the regulatory particle (RP) utilizes the same binding surface but only interacts with the mature CP in vivo. It is unclear how these binding events are regulated. Here, we show that Pba1-Pba2 binds tightly to the immature CP, preventing RP binding. Changes in the CP that occur on maturation significantly reduce its affinity for Pba1-Pba2, enabling the RP to displace the chaperone. Mathematical modelling indicates that this 'affinity switch' mechanism has likely evolved to improve assembly efficiency by preventing the accumulation of stable, non-productive intermediates. Our work thus provides mechanistic insights into a crucial step in proteasome biogenesis.

Despite the importance of intracellular signaling networks, there is currently no consensus regarding the fundamental nature of the protein complexes such networks employ. One prominent view involves stable signaling machines with well-defined quaternary structures. The combinatorial complexity of signaling networks has led to an opposing perspective, namely that signaling proceeds via heterogeneous pleiomorphic ensembles of transient complexes. Since many hypotheses regarding network function rely on how we conceptualize signaling complexes, resolving this issue is a central problem in systems biology. Unfortunately, direct experimental characterization of these complexes has proven technologically difficult, while combinatorial complexity has prevented traditional modeling methods from approaching this question. Here we employ rule-based modeling, a technique that overcomes these limitations, to construct a model of the yeast pheromone signaling network. We found that this model exhibits significant ensemble character while generating reliable responses that match experimental observations. To contrast the ensemble behavior, we constructed a model that employs hierarchical assembly pathways to produce scaffold-based signaling machines. We found that this machine model could not replicate the experimentally observed combinatorial inhibition that arises when the scaffold is overexpressed. This finding provides evidence against the hierarchical assembly of machines in the pheromone signaling network and suggests that machines and ensembles may serve distinct purposes in vivo. In some cases, e.g. core enzymatic activities like protein synthesis and degradation, machines assembled via hierarchical energy landscapes may provide functional stability for the cell. In other cases, such as signaling, ensembles may represent a form of weak linkage, facilitating variation and plasticity in network evolution. The capacity of ensembles to signal effectively will ultimately shape how we conceptualize the function, evolution and engineering of signaling networks., Author Summary Intracellular signaling networks are central to a cell's ability to adapt to its environment. Developing the capacity to effectively manipulate such networks would have a wide range of applications, from cancer therapy to synthetic biology. This requires a thorough understanding of the mechanisms of signal transduction, particularly the kinds of protein complexes that are formed during transmission of extracellular information to the nucleus. Traditionally, signaling complexes have been largely perceived (albeit often implicitly) as machine-like structures. However, the number of molecular complexes that could theoretically be formed by complex signaling networks is astronomically large. This has led to the pleiomorphic ensemble hypothesis, which posits that diverse and rapidly changing sets of transient protein complexes can transmit and process information. Our goal was to use computational approaches, specifically rule-based modeling, to test these hypotheses. We constructed a model of the prototypical yeast mating pathway and found significant ensemble-like behavior. Our results thus demonstrated that ensembles can in fact transmit extracellular signals with minimal noise. Additionally, a comparison of this model with one tailored to generate machine-like complexes displayed notable phenotypic differences, revealing potential advantages for ensemble-like signaling. Our demonstration that ensembles can function effectively will have a significant impact on how we conceptualize signaling and other processes inside cells.

International audience; The assembly of molecular machines and transient signaling complexes does not typically occur under circumstances in which the appropriate proteins are isolated from all others present in the cell. Rather, assembly must proceed in the context of large-scale protein-protein interaction (PPI) networks that are characterized both by conflict and combinatorial complexity. Conflict refers to the fact that protein interfaces can often bind many different partners in a mutually exclusive way, while combinatorial complexity refers to the explosion in the number of distinct complexes that can be formed by a network of binding possibilities. Using computational models, we explore the consequences of these characteristics for the global dynamics of a PPI network based on highly curated yeast two-hybrid data. The limited molecular context represented in this data-type translates formally into an assumption of independent binding sites for each protein. The challenge of avoiding the explicit enumeration of the astronomically many possibilities for complex formation is met by a rule-based approach to kinetic modeling. Despite imposing global biophysical constraints, we find that initially identical simulations rapidly diverge in the space of molecular possibilities, eventually sampling disjoint sets of large complexes. We refer to this phenomenon as ''compositional drift". Since interaction data in PPI networks lack detailed information about geometric and biological constraints, our study does not represent a quantitative description of cellular dynamics. Rather, our work brings to light a fundamental problem (the control of compositional drift) that must be solved by mechanisms of assembly in the context of large networks. In cases where drift is not (or cannot be) completely controlled by the cell, this phenomenon could constitute a novel source of phenotypic heterogeneity in cell populations.

Many macromolecular machines inside the cell exhibit a “stacked ring” architecture, with multiple uniform-length rings of protein subunits bound to one another. The majority of these machines must adopt their fully assembled quaternary structure in order to function, making the assembly process vital for cellular function and survival. The assembly of protein complexes containing stable substructures has been shown to suffer from a type of kinetic trapping that we term “assembly deadlock,” which occurs when smaller intermediates are exhausted from the system before all of the fully functional structures have formed. Deadlock can result in plateaus in the assembly dynamics, leading to delays in reaching maximum complex assembly and a reduced final complex concentration. While these plateaus have been extensively studied for simple rings, the effect of assembly deadlock on more general structures like stacked rings remains to be fully investigated. In this work, we focused on the case of a stacked homotrimer; this structure contains both three- and four-member rings as substructures, but is simple enough to allow for extensive investigation. Our mathematical models revealed that this structure could suffer from extreme deadlock that significantly reduces the efficiency of assembly. Using a computationally efficient simulation approach, we exhaustively analyzed the parameter space of self-assembly for this case, and found that the number and duration of plateaus in the assembly dynamics depended strongly on the pattern of affinities in this structure. Since these complexes are generally only functional when fully assembled, we hypothesized that existing stacked ring architectures would evolve to utilize the most efficient assembly pathways predicted by our models. Analysis of interfaces in solved crystal structures of stacked homotrimers confirmed this prediction. Our findings have important implications for understanding how assembly dynamics have influenced the structural evolution of large macromolecular machines.

Proteins, by virtue of their central role in most biological processes, represent one of the key subjects of the study of molecular evolution. Inherent in the indispensability of proteins for living cells is the fact that a given protein can adopt a specific three-dimensional shape that is specified solely by the protein's sequence of amino acids. Over the past several decades, structural biologists have demonstrated that the array of structures that proteins may adopt is quite astounding, and this has lead to a strong interest in understanding how protein structures change and evolve over time. In this review we consider a large body of recent work that attempts to illuminate this structure-centric picture of protein evolution. Much of this work has focused on the question of how completely new protein structures (i.e., new folds or topologies) are discovered by protein sequences as they evolve. Pursuant to this question of structural innovation has been a desire to describe and understand the observation that certain types of protein structures are far more abundant than others and how this uneven distribution of proteins implicates on the process through which new shapes are discovered. We consider a number of theoretical models that have been successful at explaining this heterogeneity in protein populations and discuss the increasing amount of evidence that indicates that the process of structural evolution involves the divergence of protein sequences and structures from one another. We also consider the topic of protein designability, which concerns itself with understanding how a protein's structure influences the number of sequences that can fold successfully into that structure. Understanding and quantifying the relationship between the physical feature of a structure and its designability has been a long-standing goal of the study of protein structure and evolution, and we discuss a number of recent advances that have yielded a promising answer to this question. Finally, we review the relatively new field of protein structural phylogeny, an area of study in which information about the distribution of protein structures among different organisms is used to reconstruct the evolutionary relationships between them. Taken together, the work that we review presents an increasingly coherent picture of how these unique polymers have evolved over the course of life on Earth.

The relationship between an organism's metabolic rate and its body mass has fascinated biologists since Max Kleiber first proposed, in 1932, that metabolic rate scales across species with body mass raised to the power of 3/4. This 'scaling exponent' has been recalculated many times since, with some estimating the exponent to be close to 2/3 and others nearer to 3/4. A new analysis suggests that the relationship does not follow a straight line on a logarithmic scale, so is not a pure power law at all. Attempts to fit a straight line to what is really a curve produce 'scaling exponents' that are highly dependent on the data used. Data sets dominated by small organisms tend to produce exponents of 2/3 while those dominated by large organisms will produce 3/4. It has been thought that the basal metabolic rate of organisms increases as body mass is raised to some power, p. But the value of p has proved controversial, with both 2/3 and 3/4 being proposed. It is found here that the relationship between mass and metabolic rate does not follow a pure power law at all, and requires a quadratic term to account for curvature. Taking temperature and phylogeny into account, this explains why different data sets have produced different exponents when a power law has been fitted. For more than three-quarters of a century it has been assumed1 that basal metabolic rate increases as body mass raised to some power p. However, there is no broad consensus regarding the value of p: whereas many studies have asserted that p is 3/4 (refs 1–4; ‘Kleiber’s law’), some have argued that it is 2/3 (refs 5–7), and others have found that it varies depending on factors like environment and taxonomy6,8,9,10,11,12,13,14,15,16. Here we show that the relationship between mass and metabolic rate has convex curvature on a logarithmic scale, and is therefore not a pure power law, even after accounting for body temperature. This finding has several consequences. First, it provides an explanation for the puzzling variability in estimates of p, settling a long-standing debate. Second, it constitutes a stringent test for theories of metabolic scaling. A widely debated model17 based on vascular system architecture fails this test, and we suggest modifications that could bring it into compliance with the observed curvature. Third, it raises the intriguing question of whether the scaling relation limits body size.

Studies of the role of sex in evolution typically involve a longitudinal comparison of a single ancestor to several intermediate descendants and to one terminally evolved descendant after many generations of adaptation under a given selective regime. Here we take a complementary, statistical approach to sex in evolution, by describing the distribution of phenotypic similarity in a population of yeast F 1 meiotic recombinants. By applying graph theory to fitness measurements of thousands of Saccharomyces cerevisiae recombinants treated with 10 mechanistically distinct, growth-inhibitory small-molecule perturbagens (SMPs), we show that the network of phenotypic similarity among F 1 recombinants exhibits a scale-free degree distribution. F 1 recombinants are often phenotypically unique and sometimes exceptional, and their fitness strengths are unevenly distributed across the 10 compound treatments. By contrast, highly phenotypically similar F 1 recombinants constitute failing hubs that display below-average fitness across all compound treatments and are candidate substrates for purifying selection. Comparison of the F 1 generation with the parental strains reveals that ( i ) there is a specialist more fit in any given single condition than any of the parents but ( ii ) only rarely are there generalists that exhibit greater fitness than both parental strains across a majority of conditions. This analysis allows us to evaluate and to gain better theoretical understanding of the costs and benefits of sex in the F 1 generation.

The capacity of proteins to interact specifically with one another underlies our conceptual understanding of how living systems function. Systems-level study of specificity in protein–protein interactions is complicated by the fact that the cellular environment is crowded and heterogeneous; interaction pairs may exist at low relative concentrations and thus be presented with many more opportunities for promiscuous interactions compared with specific interaction possibilities. Here we address these questions by using a simple computational model that includes specifically designed interacting model proteins immersed in a mixture containing hundreds of different unrelated ones; all of them undergo simulated diffusion and interaction. We find that specific complexes are quite robust to interference from promiscuous interaction partners only in the range of temperatures T design > T > T rand . At T > T design , specific complexes become unstable, whereas at T < T rand , formation of specific complexes is suppressed by promiscuous interactions. Specific interactions can form only if T design > T rand . This condition requires an energy gap between binding energy in a specific complex and set of binding energies between randomly associating proteins, providing a general physical constraint on evolutionary selection or design of specific interacting protein interfaces. This work has implications for our understanding of how the protein repertoire functions and evolves within the context of cellular systems.

Proteins, by virtue of their central role in most biological processes, represent one of the key subjects of the study of molecular evolution. Inherent in the indispensability of proteins for living cells is the fact that a given protein can adopt a specific three-dimensional shape that is specified solely by the protein's sequence of amino acids. Over the past several decades, structural biologists have demonstrated that the array of structures that proteins may adopt is quite astounding, and this has lead to a strong interest in understanding how protein structures change and evolve over time. In this review we consider a large body of recent work that attempts to illuminate this structure-centric picture of protein evolution. Much of this work has focused on the question of how completely new protein structures (i.e., new folds or topologies) are discovered by protein sequences as they evolve. Pursuant to this question of structural innovation has been a desire to describe and understand the observation that certain types of protein structures are far more abundant than others and how this uneven distribution of proteins implicates on the process through which new shapes are discovered. We consider a number of theoretical models that have been successful at explaining this heterogeneity in protein populations and discuss the increasing amount of evidence that indicates that the process of structural evolution involves the divergence of protein sequences and structures from one another. We also consider the topic of protein designability, which concerns itself with understanding how a protein's structure influences the number of sequences that can fold successfully into that structure. Understanding and quantifying the relationship between the physical feature of a structure and its designability has been a long-standing goal of the study of protein structure and evolution, and we discuss a number of recent advances that have yielded a promising answer to this question. Finally, we review the relatively new field of protein structural phylogeny, an area of study in which information about the distribution of protein structures among different organisms is used to reconstruct the evolutionary relationships between them. Taken together, the work that we review presents an increasingly coherent picture of how these unique polymers have evolved over the course of life on Earth.

It has long been known that a protein's amino acid sequence dictates its native structure. However, despite significant recent advances, an ensemble description of how a protein achieves its native conformation from random coil under physiologically relevant conditions remains incomplete. Here we present a detailed all-atom model with a transferable potential that is capable of ab initio folding of entire protein domains using only sequence information. The computational efficiency of this model allows us to perform thousands of microsecond-time scale-folding simulations of the engrailed homeodomain and to observe thousands of complete independent folding events. We apply a graph-theoretic analysis to this massive data set to elucidate which intermediates and intermediary states are common to many trajectories and thus important for the folding process. This method provides an atomically detailed and complete picture of a folding pathway at the ensemble level. The approach that we describe is quite general and could be used to study the folding of proteins on time scales orders of magnitude longer than currently possible.

A generalized computational method for folding proteins with a fully transferable potential and geometrically realistic all-atom model is presented and tested on seven different helix bundle proteins. The protocol, which includes graph-theoretical analysis of the ensemble of resulting folded conformations, was systematically applied and consistently produced structure predictions of approximately 3 Angstroms without any knowledge of the native state. To measure and understand the significance of the results, extensive control simulations were conducted. Graph theoretic analysis provides a means for systematically identifying the native fold and provides physical insight, conceptually linking the results to modern theoretical views of protein folding. In addition to presenting a method for prediction of structure and folding mechanism, our model suggests that a accurate all-atom amino acid representation coupled with a physically reasonable atomic interaction potential (that does not require optimization to the test set) and hydrogen bonding are essential features for a realistic protein model., submitted to PNAS 2005-03-16