Docking-based long timescale simulation of cell-size protein systems at atomic resolution.

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, thus 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.

Modeling reveals the strength of weak interactions in stacked-ring assembly.

Cells employ many large macromolecular machines for the execution and regulation of processes that are vital for cell and organismal viability. Interestingly, cells cannot synthesize these machines as functioning units. Instead, cells synthesize the molecular parts that must then assemble into the functional complex. Many important machines, including chaperones such as GroEL and proteases such as the proteasome, comprise protein rings that are stacked on top of one another. While there is some experimental data regarding how stacked-ring complexes such as the proteasome self-assemble, a comprehensive understanding of the dynamics of stacked-ring assembly is currently lacking. Here, we developed a mathematical model of stacked-trimer assembly and performed an analysis of the assembly of the stacked homomeric trimer, which is the simplest stacked-ring architecture. We found that stacked rings are particularly susceptible to a form of kinetic trapping that we term "deadlock," in which the system gets stuck in a state where there are many large intermediates that are not the fully assembled structure but that cannot productively react. When interaction affinities are uniformly strong, deadlock severely limits assembly yield. We thus predicted that stacked rings would avoid situations where all interfaces in the structure have high affinity. Analysis of available crystal structures indicated that indeed the majority-if not all-of stacked trimers do not contain uniformly strong interactions. Finally, to better understand the origins of deadlock, we developed a formal pathway analysis and showed that, when all the binding affinities are strong, many of the possible pathways are utilized. In contrast, optimal assembly strategies utilize only a small number of pathways. Our work suggests that deadlock is a critical factor influencing the evolution of macromolecular machines and provides general principles for understanding the self-assembly efficiency of existing machines.

Novel metrics reveal new structure and unappreciated heterogeneity in Caenorhabditis elegans development.

High throughput experimental approaches are increasingly allowing for the quantitative description of cellular and organismal phenotypes. Distilling these large volumes of complex data into meaningful measures that can drive biological insight remains a central challenge. In the quantitative study of development, for instance, one can resolve phenotypic measures for single cells onto their lineage history, enabling joint consideration of heritable signals and cell fate decisions. Most attempts to analyze this type of data, however, discard much of the information content contained within lineage trees. In this work we introduce a generalized metric, which we term the branch edit distance, that allows us to compare any two embryos based on phenotypic measurements in individual cells. This approach aligns those phenotypic measurements to the underlying lineage tree, providing a flexible and intuitive framework for quantitative comparisons between, for instance, Wild-Type (WT) and mutant developmental programs. We apply this novel metric to data on cell-cycle timing from over 1300 WT and RNAi-treated Caenorhabditis elegans embryos. Our new metric revealed surprising heterogeneity within this data set, including subtle batch effects in WT embryos and dramatic variability in RNAi-induced developmental phenotypes, all of which had been missed in previous analyses. Further investigation of these results suggests a novel, quantitative link between pathways that govern cell fate decisions and pathways that pattern cell cycle timing in the early embryo. Our work demonstrates that the branch edit distance we propose, and similar metrics like it, have the potential to revolutionize our quantitative understanding of organismal phenotype.

Machine learning classification can reduce false positives in structure-based virtual screening.

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.

Understanding the separation of timescales in bacterial proteasome core particle assembly.

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 ß, arranged in four stacked seven-member rings (α7ß7ß7α7). While details of CP assembly vary across species, the final step in the assembly process is universally conserved: two half proteasomes (HPs; α7ß7) 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 ß subunit, which is autocatalytically cleaved off after CP formation, plays a key role in regulating this separation of timescales. However, the detailed molecular mechanism of how the propeptide achieves this regulation is unclear. In this work, we used molecular dynamics 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. Furthermore, we found that a propeptide mutant that dimerizes extremely slowly is essentially always in the nondimerizable state, while the wild-type rapidly transitions between the two. Based on these simulations, we designed a propeptide mutant that favored the dimerizable state in molecular dynamics simulations. In vitro assembly experiments confirmed that this mutant dimerizes significantly faster than wild-type. 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.

Robustness and the evolution of length control strategies in the T3SS and flagellar hook.

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.

Crosstalk and ultrasensitivity in protein degradation pathways.

Protein turnover is vital to cellular homeostasis. Many proteins are degraded efficiently only after they have been post-translationally "tagged" with a polyubiquitin chain. Ubiquitylation is a form of Post-Translational Modification (PTM): addition of a ubiquitin to the chain is catalyzed by E3 ligases, and removal of ubiquitin is catalyzed by a De-UBiquitylating enzyme (DUB). Nearly four decades ago, Goldbeter and Koshland discovered that reversible PTM cycles function like on-off switches when the substrates are at saturating concentrations. Although this finding has had profound implications for the understanding of switch-like behavior in biochemical networks, the general behavior of PTM cycles subject to synthesis and degradation has not been studied. Using a mathematical modeling approach, we found that simply introducing protein turnover to a standard modification cycle has profound effects, including significantly reducing the switch-like nature of the response. Our findings suggest that many classic results on PTM cycles may not hold in vivo where protein turnover is ubiquitous. We also found that proteins sharing an E3 ligase can have closely related changes in their expression levels. These results imply that it may be difficult to interpret experimental results obtained from either overexpressing or knocking down protein levels, since changes in protein expression can be coupled via E3 ligase crosstalk. Understanding crosstalk and competition for E3 ligases will be key in ultimately developing a global picture of protein homeostasis.

Fundamental trade-offs between information flow in single cells and cellular populations.

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.

Mathematical Model for Length Control by the Timing of Substrate Switching in the Type III Secretion System.

Type III Secretion Systems (T3SS) are complex bacterial structures that provide gram-negative pathogens with a unique virulence mechanism whereby they grow a needle-like structure in order to inject bacterial effector proteins into the cytoplasm of a host cell. Numerous experiments have been performed to understand the structural details of this nanomachine during the past decade. Despite the concerted efforts of molecular and structural biologists, several crucial aspects of the assembly of this structure, such as the regulation of the length of the needle itself, remain unclear. In this work, we used a combination of mathematical and computational techniques to better understand length control based on the timing of substrate switching, which is a possible mechanism for how bacteria ensure that the T3SS needles are neither too short nor too long. In particular, we predicted the form of the needle length distribution based on this mechanism, and found excellent agreement with available experimental data from Salmonella typhimurium with only a single free parameter. Although our findings provide preliminary evidence in support of the substrate switching model, they also make a set of quantitative predictions that, if tested experimentally, would assist in efforts to unambiguously characterize the regulatory mechanisms that control the growth of this crucial virulence factor.

Crosstalk and the evolution of specificity in two-component signaling.

Two-component signaling (TCS) serves as the dominant signaling modality in bacteria. A typical pathway includes a sensor histidine kinase (HK) that phosphorylates a response regulator (RR), modulating its activity in response to an incoming signal. Most HKs are bifunctional, acting as both kinase and phosphatase for their substrates. Unlike eukaryotic signaling networks, there is very little crosstalk between bacterial TCS pathways; indeed, adding crosstalk to a pathway can have disastrous consequences for cell fitness. It is currently unclear exactly what feature of TCS necessitates this degree of pathway isolation. In this work we used mathematical models to show that, in the case of bifunctional HKs, adding a competing substrate to a TCS pathway will always reduce response of that pathway to incoming signals. We found that the pressure to maintain cognate signaling is sufficient to explain the experimentally observed "kinetic preference" of HKs for their cognate RRs. These findings imply a barrier to the evolution of new HK-RR pairs, because crosstalk is unavoidable immediately after the duplication of an existing pathway. We characterized a set of "near-neutral" evolutionary trajectories that minimize the impact of crosstalk on the function of the parental pathway. These trajectories predicted that crosstalk interactions should be removed before new input/output functionalities evolve. Analysis of HK sequences in bacterial genomes provided evidence that the selective pressures on the HK-RR interface are different from those experienced by the input domain immediately after duplication. This work thus provides a unifying explanation for the evolution of specificity in TCS networks.

Curvature in metabolic scaling.

For more than three-quarters of a century it has been assumed 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 taxonomy. 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 model 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.

Phosphatase specificity and pathway insulation in signaling networks.

Phosphatases play an important role in cellular signaling networks by regulating the phosphorylation state of proteins. Phosphatases are classically considered to be promiscuous, acting on tens to hundreds of different substrates. We recently demonstrated that a shared phosphatase can couple the responses of two proteins to incoming signals, even if those two substrates are from otherwise isolated areas of the network. This finding raises a potential paradox: if phosphatases are indeed highly promiscuous, how do cells insulate themselves against unwanted crosstalk? Here, we use mathematical models to explore three possible insulation mechanisms. One approach involves evolving phosphatase KM values that are large enough to prevent saturation by the phosphatase's substrates. Although this is an effective method for generating isolation, the phosphatase becomes a highly inefficient enzyme, which prevents the system from achieving switch-like responses and can result in slow response kinetics. We also explore the idea that substrate degradation can serve as an effective phosphatase. Assuming that degradation is unsaturatable, this mechanism could insulate substrates from crosstalk, but it would also preclude ultrasensitive responses and would require very high substrate turnover to achieve rapid dephosphorylation kinetics. Finally, we show that adaptor subunits, such as those found on phosphatases like PP2A, can provide effective insulation against phosphatase crosstalk, but only if their binding to substrates is uncoupled from their binding to the catalytic core. Analysis of the interaction network of PP2A's adaptor domains reveals that although its adaptors may isolate subsets of targets from one another, there is still a strong potential for phosphatase crosstalk within those subsets. Understanding how phosphatase crosstalk and the insulation mechanisms described here impact the function and evolution of signaling networks represents a major challenge for experimental and computational systems biology.

Optimizing ring assembly reveals the strength of weak interactions.

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.

Machines vs. ensembles: effective MAPK signaling through heterogeneous sets of protein complexes.

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.

Structural properties of non-traditional drug targets present new challenges for virtual screening.

Traditional drug targets have historically included signaling proteins that respond to small molecules and enzymes that use small molecules as substrates. Increasing attention is now being directed toward other types of protein targets, in particular those that exert their function by interacting with nucleic acids or other proteins rather than small-molecule ligands. Here, we systematically compare existing examples of inhibitors of protein-protein interactions to inhibitors of traditional drug targets. While both sets of inhibitors bind with similar potency, we find that the inhibitors of protein-protein interactions typically bury a smaller fraction of their surface area upon binding to their protein targets. The fact that an average atom is less buried suggests that more atoms are needed to achieve a given potency, explaining the observation that ligand efficiency is typically poor for inhibitors of protein-protein interactions. We then carried out a series of docking experiments and found a further consequence of these relatively exposed binding modes is that structure-based virtual screening may be more difficult: such binding modes do not provide sufficient clues to pick out active compounds from decoy compounds. Collectively, these results suggest that the challenges associated with such non-traditional drug targets may not lie with identifying compounds that potently bind to the target protein surface, but rather with identifying compounds that bind in a sufficiently buried manner to achieve good ligand efficiency and, thus, good oral bioavailability. While the number of available crystal structures of distinct protein interaction sites bound to small-molecule inhibitors is relatively small at present (only 21 such complexes were included in this study), these are sufficient to draw conclusions based on the current state of the field; as additional data accumulate it will be exciting to refine the viewpoint presented here. Even with this limited perspective however, we anticipate that these insights, together with new methods for exploring protein conformational fluctuations, may prove useful for identifying the "low-hanging fruit" among non-traditional targets for therapeutic intervention.

Novel metrics reveal new structure and unappreciated heterogeneity in C. elegans development.

High throughput experimental approaches are increasingly allowing for the quantitative description of cellular and organismal phenotypes. Distilling these large volumes of complex data into meaningful measures that can drive biological insight remains a central challenge. In the quantitative study of development, for instance, one can resolve phenotypic measures for single cells onto their lineage history, enabling joint consideration of heritable signals and cell fate decisions. Most attempts to analyze this type of data, however, discard much of the information content contained within lineage trees. In this work we introduce a generalized metric, which we term the branch distance, that allows us to compare any two embryos based on phenotypic measurements in individual cells. This approach aligns those phenotypic measurements to the underlying lineage tree, providing a flexible and intuitive framework for quantitative comparisons between, for instance, Wild-Type (WT) and mutant developmental programs. We apply this novel metric to data on cell-cycle timing from over 1300 WT and RNAi-treated Caenorhabditis elegans embryos. Our new metric revealed surprising heterogeneity within this data set, including subtle batch effects in WT embryos and dramatic variability in RNAi-induced developmental phenotypes, all of which had been missed in previous analyses. Further investigation of these results suggests a novel, quantitative link between pathways that govern cell fate decisions and pathways that pattern cell cycle timing in the early embryo. Our work demonstrates that the branch distance we propose, and similar metrics like it, have the potential to revolutionize our quantitative understanding of organismal phenotype.

Altered physical phenotypes of leukemia cells that survive chemotherapy treatment.

The recurrence of cancer following chemotherapy treatment is a major cause of death across solid and hematologic cancers. In B-cell acute lymphoblastic leukemia (B-ALL), relapse after initial chemotherapy treatment leads to poor patient outcomes. Here we test the hypothesis that chemotherapy-treated versus control B-ALL cells can be characterized based on cellular physical phenotypes. To quantify physical phenotypes of chemotherapy-treated leukemia cells, we use cells derived from B-ALL patients that are treated for 7 days with a standard multidrug chemotherapy regimen of vincristine, dexamethasone, and L-asparaginase (VDL). We conduct physical phenotyping of VDL-treated versus control cells by tracking the sequential deformations of single cells as they flow through a series of micron-scale constrictions in a microfluidic device; we call this method Quantitative Cyclical Deformability Cytometry. Using automated image analysis, we extract time-dependent features of deforming cells including cell size and transit time (TT) with single-cell resolution. Our findings show that VDL-treated B-ALL cells have faster TTs and transit velocity than control cells, indicating that VDL-treated cells are more deformable. We then test how effectively physical phenotypes can predict the presence of VDL-treated cells in mixed populations of VDL-treated and control cells using machine learning approaches. We find that TT measurements across a series of sequential constrictions can enhance the classification accuracy of VDL-treated cells in mixed populations using a variety of classifiers. Our findings suggest the predictive power of cell physical phenotyping as a complementary prognostic tool to detect the presence of cells that survive chemotherapy treatment. Ultimately such complementary physical phenotyping approaches could guide treatment strategies and therapeutic interventions. Insight box Cancer cells that survive chemotherapy treatment are major contributors to patient relapse, but the ability to predict recurrence remains a challenge. Here we investigate the physical properties of leukemia cells that survive treatment with chemotherapy drugs by deforming individual cells through a series of micron-scale constrictions in a microfluidic channel. Our findings reveal that leukemia cells that survive chemotherapy treatment are more deformable than control cells. We further show that machine learning algorithms applied to physical phenotyping data can predict the presence of cells that survive chemotherapy treatment in a mixed population. Such an integrated approach using physical phenotyping and machine learning could be valuable to guide patient treatments.

Crosstalk and competition in signaling networks.

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.

Cooperativity in Proteasome Core Particle Maturation.

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 ß 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 ß propeptide plays an unexpected and complex role in regulating dimerization and autocatalytic activation.

Sizing up allometric scaling theory.

Metabolic rate, heart rate, lifespan, and many other physiological properties vary with body mass in systematic and interrelated ways. Present empirical data suggest that these scaling relationships take the form of power laws with exponents that are simple multiples of one quarter. A compelling explanation of this observation was put forward a decade ago by West, Brown, and Enquist (WBE). Their framework elucidates the link between metabolic rate and body mass by focusing on the dynamics and structure of resource distribution networks-the cardiovascular system in the case of mammals. Within this framework the WBE model is based on eight assumptions from which it derives the well-known observed scaling exponent of 3/4. In this paper we clarify that this result only holds in the limit of infinite network size (body mass) and that the actual exponent predicted by the model depends on the sizes of the organisms being studied. Failure to clarify and to explore the nature of this approximation has led to debates about the WBE model that were at cross purposes. We compute analytical expressions for the finite-size corrections to the 3/4 exponent, resulting in a spectrum of scaling exponents as a function of absolute network size. When accounting for these corrections over a size range spanning the eight orders of magnitude observed in mammals, the WBE model predicts a scaling exponent of 0.81, seemingly at odds with data. We then proceed to study the sensitivity of the scaling exponent with respect to variations in several assumptions that underlie the WBE model, always in the context of finite-size corrections. Here too, the trends we derive from the model seem at odds with trends detectable in empirical data. Our work illustrates the utility of the WBE framework in reasoning about allometric scaling, while at the same time suggesting that the current canonical model may need amendments to bring its predictions fully in line with available datasets.