Eric J. Deeds

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.

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.

Understanding ensemble protein folding at atomic detail.

It has long been known that a proteins 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.

Robust protein–protein interactions in crowded cellular environments

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 Tdesign > T > Trand. At T > Tdesign, specific complexes become unstable, whereas at T < Trand, formation of specific complexes is suppressed by promiscuous interactions. Specific interactions can form only if Tdesign > Trand. 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.

High-resolution protein folding with a transferable potential

A generalized computational method for folding proteins with a fully transferable potential and geometrically realistic all-atom model is presented and tested on seven 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 A 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 an accurate all-atom amino acid representation coupled with a physically reasonable atomic interaction potential and hydrogen bonding are essential features for a realistic protein model.

Combinatorial Complexity and Compositional Drift in Protein Interaction Networks

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.

Crosstalk and the evolution of specificity in two-component signaling

Significance The global architectures of signaling networks in bacteria and eukaryotes are remarkably different: crosstalk between pathways is very common in eukaryotes but is very limited in bacteria. Bacteria use two-component signaling (TCS) to transduce information, relying on a single enzyme to act as both kinase and phosphatase for targets. We used mathematical models to show that introducing crosstalk in TCS always decreases system performance. This indicates that the large-scale differences between eukaryotic and bacterial networks likely derive from differences in the dynamics of the fundamental motifs from which the networks themselves are constructed. We further demonstrated that the pressure to avoid crosstalk has influenced the evolution of new TCS pairs, driving rapid sequence divergence in protein interaction interfaces immediately postduplication. 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.

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.

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.

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.