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The Michaelis constant K M describes the affinity of an enzyme for a specific substrate and is a central parameter in studies of enzyme kinetics and cellular physiology. As measurements of K M are often difficult and time-consuming, experimental estimates exist for only a minority of enzyme–substrate combinations even in model organisms. Here, we build and train an organism-independent model that successfully predicts K M values for natural enzyme–substrate combinations using machine and deep learning methods. Predictions are based on a task-specific molecular fingerprint of the substrate, generated using a graph neural network, and on a deep numerical representation of the enzyme’s amino acid sequence. We provide genome-scale K M predictions for 47 model organisms, which can be used to approximately relate metabolite concentrations to cellular physiology and to aid in the parameterization of kinetic models of cellular metabolism.

For most proteins annotated as enzymes, it is unknown which primary and/or secondary reactions they catalyze. Experimental characterizations of potential substrates are time-consuming and costly. Machine learning predictions could provide an efficient alternative, but are hampered by a lack of information regarding enzyme non-substrates, as available training data comprises mainly positive examples. Here, we present ESP, a general machine-learning model for the prediction of enzyme-substrate pairs with an accuracy of over 91% on independent and diverse test data. ESP can be applied successfully across widely different enzymes and a broad range of metabolites included in the training data, outperforming models designed for individual, well-studied enzyme families. ESP represents enzymes through a modified transformer model, and is trained on data augmented with randomly sampled small molecules assigned as non-substrates. By facilitating easy in silico testing of potential substrates, the ESP web server may support both basic and applied science. For many enzymes, it is unknown which primary and/or secondary reactions they catalyze. Here, the authors use machine and deep learning to develop a general model for the prediction of enzyme-small molecule substrate pairs and make the resulting model available through a webserver.

The turnover number k cat , a measure of enzyme efficiency, is central to understanding cellular physiology and resource allocation. As experimental k cat estimates are unavailable for the vast majority of enzymatic reactions, the development of accurate computational prediction methods is highly desirable. However, existing machine learning models are limited to a single, well-studied organism, or they provide inaccurate predictions except for enzymes that are highly similar to proteins in the training set. Here, we present TurNuP, a general and organism-independent model that successfully predicts turnover numbers for natural reactions of wild-type enzymes. We constructed model inputs by representing complete chemical reactions through differential reaction fingerprints and by representing enzymes through a modified and re-trained Transformer Network model for protein sequences. TurNuP outperforms previous models and generalizes well even to enzymes that are not similar to proteins in the training set. Parameterizing metabolic models with TurNuP-predicted k cat values leads to improved proteome allocation predictions. To provide a powerful and convenient tool for the study of molecular biochemistry and physiology, we implemented a TurNuP web server. The turnover numbers of most enzyme-catalyzed reactions are unknown. Kroll et al. developed a general model that can predict turnover numbers even for enzymes dissimilar to those used for training, outperforming existing models.

A substantial fraction of the bacterial cytosol is occupied by catalysts and their substrates. While a higher volume density of catalysts and substrates might boost biochemical fluxes, the resulting molecular crowding can slow down diffusion, perturb the reactions’ Gibbs free energies, and reduce the catalytic efficiency of proteins. Due to these tradeoffs, dry mass density likely possesses an optimum that facilitates maximal cellular growth and that is interdependent on the cytosolic molecule size distribution. Here, we analyze the balanced growth of a model cell, accounting systematically for crowding effects on reaction kinetics. Its optimal cytosolic volume occupancy depends on the nutrient-dependent resource allocation into large ribosomal vs. small metabolic macromolecules, reflecting a tradeoff between the saturation of metabolic enzymes, favoring larger occupancies with higher encounter rates, and the inhibition of the ribosomes, favoring lower occupancies with unhindered diffusion of tRNAs. Our predictions across growth rates are quantitatively consistent with the experimentally observed reduction in volume occupancy on rich media compared to minimal media in E . coli . Strong deviations from optimal cytosolic occupancy only lead to minute reductions in growth rate, which are nevertheless evolutionarily relevant due to large bacterial population sizes. In sum, cytosolic density variation in bacterial cells appears to be consistent with an optimality principle of cellular efficiency.

The biological fitness of microbes is largely determined by the rate with which they replicate their biomass composition. Mathematical models that maximize this balanced growth rate while accounting for mass conservation, reaction kinetics, and limits on dry mass per volume are inevitably non-linear. Here, we develop a general theory for such models, termed Growth Balance Analysis (GBA), which provides explicit expressions for protein concentrations, fluxes, and growth rates. These variables are functions of the concentrations of cellular components, for which we calculate marginal fitness costs and benefits that are related to metabolic control coefficients. At maximal growth rate, the net benefits of all concentrations are equal. Based solely on physicochemical constraints, GBA unveils fundamental quantitative principles of cellular resource allocation and growth; it accurately predicts the relationship between growth rates and ribosome concentrations in E. coli and yeast and between growth rate and dry mass density in E. coli . Genome-scale models of microbial metabolism largely ignore reaction kinetics. Here, the authors develop a general mathematical framework for modeling cellular growth with explicit non-linear reaction kinetics and use it to glean insights into the principles of cellular resource allocation and growth.

The distribution of cellular resources across bacterial proteins has been quantified through phenomenological growth laws. Here, we describe a complementary bacterial growth law for RNA composition, emerging from optimal cellular resource allocation into ribosomes and ternary complexes. The predicted decline of the tRNA/rRNA ratio with growth rate agrees quantitatively with experimental data. Its regulation appears to be implemented in part through chromosomal localization, as rRNA genes are typically closer to the origin of replication than tRNA genes and thus have increasingly higher gene dosage at faster growth. At the highest growth rates in E . coli , the tRNA/rRNA gene dosage ratio based on chromosomal positions is almost identical to the observed and theoretically optimal tRNA/rRNA expression ratio, indicating that the chromosomal arrangement has evolved to favor maximal transcription of both types of genes at this condition.

Much recent progress has been made to understand the impact of proteome allocation on bacterial growth; much less is known about the relationship between the abundances of the enzymes and their substrates, which jointly determine metabolic fluxes. Here, we report a correlation between the concentrations of enzymes and their substrates in Escherichia coli . We suggest this relationship to be a consequence of optimal resource allocation, subject to an overall constraint on the biomass density: For a cellular reaction network composed of effectively irreversible reactions, maximal reaction flux is achieved when the dry mass allocated to each substrate is equal to the dry mass of the unsaturated (or “free”) enzymes waiting to consume it. Calculations based on this optimality principle successfully predict the quantitative relationship between the observed enzyme and metabolite abundances, parameterized only by molecular masses and enzyme–substrate dissociation constants ( K m ). The corresponding organizing principle provides a fundamental rationale for cellular investment into different types of molecules, which may aid in the design of more efficient synthetic cellular systems.

The activities of most enzymes and drugs depend on interactions between proteins and small molecules. Accurate prediction of these interactions could greatly accelerate pharmaceutical and biotechnological research. Current machine learning models designed for this task have a limited ability to generalize beyond the proteins used for training. This limitation is likely due to a lack of information exchange between the protein and the small molecule during the generation of the required numerical representations. Here, we introduce ProSmith, a machine learning framework that employs a multimodal Transformer Network to simultaneously process protein amino acid sequences and small molecule strings in the same input. This approach facilitates the exchange of all relevant information between the two molecule types during the computation of their numerical representations, allowing the model to account for their structural and functional interactions. Our final model combines gradient boosting predictions based on the resulting multimodal Transformer Network with independent predictions based on separate deep learning representations of the proteins and small molecules. The resulting predictions outperform recently published state-of-the-art models for predicting protein-small molecule interactions across three diverse tasks: predicting kinase inhibitions; inferring potential substrates for enzymes; and predicting Michaelis constants K M . The Python code provided can be used to easily implement and improve machine learning predictions involving arbitrary protein-small molecule interactions.

The physiology of biological cells evolved under physical and chemical constraints, such as mass conservation across the network of biochemical reactions, nonlinear reaction kinetics, and limits on cell density. For unicellular organisms, the fitness that governs this evolution is mainly determined by the balanced cellular growth rate. We previously introduced growth balance analysis (GBA) as a general framework to model and analyze such nonlinear systems, revealing important analytical properties of optimal balanced growth states. It has been shown that at optimality, only a minimal subset of reactions can have nonzero flux. However, no general principles have been established to determine if a specific reaction is active at optimality. Here, we extend the GBA framework to study the optimality of each biochemical reaction, and we identify the mathematical conditions determining whether a reaction is active or not at optimal growth in a given environment. We reformulate the mathematical problem in terms of a minimal number of dimensionless variables and use the Karush-Kuhn-Tucker (KKT) conditions to identify fundamental principles of optimal resource allocation in GBA models of any size and complexity. Our approach helps to identify from first principles the economic values of biochemical reactions, expressed as marginal changes in cellular growth rate; these economic values can be related to the costs and benefits of proteome allocation into the reactions’ catalysts. Our formulation also generalizes the concepts of Metabolic Control Analysis to models of growing cells. We show how the extended GBA framework unifies and extends previous approaches of cellular modeling and analysis, putting forward a program to analyze cellular growth through the stationarity conditions of a Lagrangian function. GBA thereby provides a general theoretical toolbox for the study of fundamental mathematical properties of balanced cellular growth.

Knowing the catalytic turnover numbers of enzymes is essential for understanding the growth rate, proteome composition, and physiology of organisms, but experimental data on enzyme turnover numbers is sparse and noisy. Here, we demonstrate that machine learning can successfully predict catalytic turnover numbers in Escherichia coli based on integrated data on enzyme biochemistry, protein structure, and network context. We identify a diverse set of features that are consistently predictive for both in vivo and in vitro enzyme turnover rates, revealing novel protein structural correlates of catalytic turnover. We use our predictions to parameterize two mechanistic genome-scale modelling frameworks for proteome-limited metabolism, leading to significantly higher accuracy in the prediction of quantitative proteome data than previous approaches. The presented machine learning models thus provide a valuable tool for understanding metabolism and the proteome at the genome scale, and elucidate structural, biochemical, and network properties that underlie enzyme kinetics. Experimental data on enzyme turnover numbers is sparse and noisy. Here, the authors use machine learning to successfully predict enzyme turnover numbers for E. coli , and show that using these to parameterize mechanistic genome-scale models enhances their predictive accuracy.