Universal observations in Biology are sometimes described as “laws”. In E. coli, experimental studies performed over the past six decades have revealed major growth laws relating ribosomal mass fraction and cell size to the growth rate. Because they formalize complex emerging principles in biology, growth laws have been instrumental in shaping our understanding of bacterial physiology. Here, we discovered a novel size law that connects cell size to the inverse of the metabolic proteome mass fraction and the active fraction of ribosomes. We used a simple whole-cell coarse-grained model of cell physiology that combines the proteome allocation theory and the structural model of cell division. This integrated model captures all available experimental data connecting the cell proteome composition, ribosome activity, division size and growth rate in response to nutrient quality, antibiotic treatment and increased protein burden. Finally, a stochastic extension of the model explains non-trivial correlations observed in single cell experiments including the adder principle. This work provides a simple and robust theoretical framework for studying the fundamental principles of cell size determination in unicellular organisms. Bacteria respond to environmental changes by adjusting their molecular composition, cell size and growth rate. This plasticity is thought to result from years of evolution and to be at least in part optimal for bacterial physiology. Over the past decades, quantitative studies of bacterial growth have revealed simple phenomenological relationships, called “growth laws”, which link cell size and cell composition to the growth rate. Simplified mathematical models of cell physiology are useful tools to gain quantitative understanding of the molecular mechanisms that underlie growth laws. For instance, these models helped explaining how optimal allocation of cellular resource to physiological processes and pathways governs the cell molecular composition in response to specific environmental conditions. In this study, we have extended and integrated existing mathematical models and used experimental data from several recent studies to understand the co-regulation of cell composition, cell size and the cellular growth rate. The model predictions uncovered a novel “size law” that links cell size to the levels of metabolic proteins and the fraction of active ribosomes present in the cell. This work provides a useful theoretical tool and a quantitative basis for understanding mechanistically bacterial physiology as a function of external conditions.

We point out an instantiation of Simpson's paradox in COVID-19 case fatality rates ( cfr s): comparing a large-scale study from China (February 17) with early reports from Italy (March 9), we find that cfr s are lower in Italy for every age group, but higher overall. This phenomenon is explained by a stark difference in case demographic between the two countries. Using this as a motivating example, we introduce basic concepts from mediation analysis and show how these can be used to quantify different direct and indirect effects when assuming a coarse-grained causal graph involving country, age, and case fatality. We curate an age-stratified cfr dataset with $>$ 750 k cases and conduct a case study, investigating total, direct, and indirect (age-mediated) causal effects between different countries and at different points in time. This allows us to separate age-related effects from others unrelated to age and facilitates a more transparent comparison of cfr s across countries at different stages of the COVID-19 pandemic. Using longitudinal data from Italy, we discover a sign reversal of the direct causal effect in mid-March, which temporally aligns with the reported collapse of the healthcare system in parts of the country. Moreover, we find that direct and indirect effects across 132 pairs of countries are only weakly correlated, suggesting that a country's policy and case demographic may be largely unrelated. We point out limitations and extensions for future work, and finally, discuss the role of causal reasoning in the broader context of using AI to combat the COVID-19 pandemic. Impact Statement—During a global pandemic, understanding the causal effects of risk factors such as age on COVID-19 fatality is an important scientific question. Since randomised controlled trials are typically infeasible or unethical in this context, causal investigations based on observational data—such as the one carried out in this article—will, therefore, be crucial in guiding our understanding of the available data. Causal inference, in particular mediation analysis, can be used to resolve apparent statistical paradoxes; help educate the public and decision-makers alike; avoid unsound comparisons; and answer a range of causal questions pertaining to the pandemic, subject to transparently stated assumptions. Our exposition helps clarify how mediation analysis can be used to investigate direct and indirect effects along different causal paths and thus serves as a stepping stone for future studies of other important risk factors for COVID-19 besides age.

Measurements of systems taken along a continuous functional dimension, such as time or space, are ubiquitous in many fields, from the physical and biological sciences to economics and engineering. Such measurements can be viewed as realisations of an underlying smooth process sampled over the continuum. However, traditional methods for independence testing and causal learning are not directly applicable to such data, as they do not take into account the dependence along the functional dimension. By using specifically designed kernels, we introduce statistical tests for bivariate, joint, and conditional independence for functional variables. Our method not only extends the applicability to functional data of the Hilbert–Schmidt independence criterion (hsic) and its d-variate version (d-hsic), but also allows us to introduce a test for conditional independence by defining a novel statistic for the conditional permutation test (cpt) based on the Hilbert–Schmidt conditional independence criterion (hscic), with optimised regularisation strength estimated through an evaluation rejection rate. Our empirical results of the size and power of these tests on synthetic functional data show good performance, and we then exemplify their application to several constraint- and regression-based causal structure learning problems, including both synthetic examples and real socioeconomic data.

Two-sample and independence tests with the kernel-based mmd and hsic have shown remarkable results on i.i.d. data and stationary random processes. However, these statistics are not directly applicable to nonstationary random processes, a prevalent form of data in many scientific disciplines. In this work, we extend the application of mmd and hsic to nonstationary settings by assuming access to independent realisations of the underlying random process. These realisations—in the form of nonstationary time-series measured on the same temporal grid—can then be viewed as i.i.d. samples from a multivariate probability distribution, to which mmd and hsic can be applied. We further show how to choose suitable kernels over these high-dimensional spaces by maximising the estimated test power with respect to the kernel hyperparameters. In experiments on synthetic data, we demonstrate superior performance of our proposed approaches in terms of test power when compared to current state-of-the-art functional or multivariate two-sample and independence tests. Finally, we employ our methods on a real socioeconomic dataset as an example application.

Early on during a pandemic, vaccine availability is limited, requiring prioritisation of different population groups. Evaluating vaccine allocation is therefore a crucial element of pandemics response. In the present work, we develop a model to retrospectively evaluate age-dependent counterfactual vaccine allocation strategies against the COVID-19 pandemic. To estimate the effect of allocation on the expected severe-case incidence, we employ a simulation-assisted causal modelling approach which combines a compartmental infection-dynamics simulation, a coarse-grained, data-driven causal model and literature estimates for immunity waning. We compare Israel's implemented vaccine allocation strategy in 2021 to counterfactual strategies such as no prioritisation, prioritisation of younger age groups or a strict risk-ranked approach; we find that Israel's implemented strategy was indeed highly effective. We also study the marginal impact of increasing vaccine uptake for a given age group and find that increasing vaccinations in the elderly is most effective at preventing severe cases, whereas additional vaccinations for middle-aged groups reduce infections most effectively. Due to its modular structure, our model can easily be adapted to study future pandemics. We demonstrate this flexibility by investigating vaccine allocation strategies for a pandemic with characteristics of the Spanish Flu. Our approach thus helps evaluate vaccination strategies under the complex interplay of core epidemic factors, including age-dependent risk profiles, immunity waning, vaccine availability and spreading rates.

Measurements of systems taken along a continuous functional dimension, such as time or space, are ubiquitous in many fields, from the physical and biological sciences to economics and engineering. Such measurements can be viewed as realisations of an underlying smooth process sampled over the continuum. However, traditional methods for independence testing and causal learning are not directly applicable to such data, as they do not take into account the dependence along the functional dimension. By using specifically designed kernels, we introduce statistical tests for bivariate, joint, and conditional independence for functional variables. Our method not only extends the applicability to functional data of the Hilbert-Schmidt independence criterion (HSIC) and its d-variate version (d-HSIC), but also allows us to introduce a test for conditional independence by defining a novel statistic for the conditional permutation test (CPT) based on the Hilbert-Schmidt conditional independence criterion (HSCIC), with optimised regularisation strength estimated through an evaluation rejection rate. Our empirical results of the size and power of these tests on synthetic functional data show good performance, and we then exemplify their application to several constraint- and regression-based causal structure learning problems, including both synthetic examples and real socio-economic data.