On Size and Growth of Business Firms

We study size and growth distributions of products and business firms in the context of a given industry. Firm size growth is analyzed in terms of two basic mechanisms, i.e. the increase of the number of new elementary business units and their size growth. We find a power-law relationship between size and the variance of growth rates for both firms and products, with an exponent between -0.17 and -0.15, with a remarkable stability upon aggregation. We then introduce a simple and general model of proportional growth for both the number of firm independent constituent units and their size, which conveys a good representation of the empirical evidences. This general and plausible generative process can account for the observed scaling in a wide variety of economic and industrial systems. Our findings contribute to shed light on the mechanisms that sustain economic growth in terms of the relationships between the size of economic entities and the number and size distribution of their elementary components.Firm Growth; Power Laws, Gibrat's Law; Economic Growth; Pharmaceutical Industry

TensorNet: Cartesian Tensor Representations for Efficient Learning of Molecular Potentials

The development of efficient machine learning models for molecular systems representation is becoming crucial in scientific research. We introduce TensorNet, an innovative O(3)-equivariant message-passing neural network architecture that leverages Cartesian tensor representations. By using Cartesian tensor atomic embeddings, feature mixing is simplified through matrix product operations. Furthermore, the cost-effective decomposition of these tensors into rotation group irreducible representations allows for the separate processing of scalars, vectors, and tensors when necessary. Compared to higher-rank spherical tensor models, TensorNet demonstrates state-of-the-art performance with significantly fewer parameters. For small molecule potential energies, this can be achieved even with a single interaction layer. As a result of all these properties, the model's computational cost is substantially decreased. Moreover, the accurate prediction of vector and tensor molecular quantities on top of potential energies and forces is possible. In summary, TensorNet's framework opens up a new space for the design of state-of-the-art equivariant models.Comment: NeurIPS 2023, camera-ready versio

Top-down machine learning of coarse-grained protein force-fields

Developing accurate and efficient coarse-grained representations of proteins is crucial for understanding their folding, function, and interactions over extended timescales. Our methodology involves simulating proteins with molecular dynamics and utilizing the resulting trajectories to train a neural network potential through differentiable trajectory reweighting. Remarkably, this method requires only the native conformation of proteins, eliminating the need for labeled data derived from extensive simulations or memory-intensive end-to-end differentiable simulations. Once trained, the model can be employed to run parallel molecular dynamics simulations and sample folding events for proteins both within and beyond the training distribution, showcasing its extrapolation capabilities. By applying Markov State Models, native-like conformations of the simulated proteins can be predicted from the coarse-grained simulations. Owing to its theoretical transferability and ability to use solely experimental static structures as training data, we anticipate that this approach will prove advantageous for developing new protein force fields and further advancing the study of protein dynamics, folding, and interactions

The pathway of ligand entry from the membrane bilayer to a lipid G protein-coupled receptor

The binding process through the membrane bilayer of lipid-like ligands to a protein target is an important but poorly explored recognition process at the atomic level. In this work we succeeded in resolving the binding of the lipid inhibitor ML056 to the sphingosine-1-phosphate receptor 1 (S1PR) using unbiased molecular dynamics simulations with an aggregate sampling of over 800 μs. The binding pathway is a multi-stage process consisting of the ligand diffusing in the bilayer leaflet to contact a "membrane vestibule" at the top of TM 7, subsequently moving from this lipid-facing vestibule to the orthosteric binding cavity through a channel formed by TMs 1 and 7 and the N-terminal of the receptor. Unfolding of the N-terminal alpha-helix increases the volume of the channel upon ligand entry, helping to reach the crystallographic pose that also corresponds to the predicted favorable pose. The relaxation timescales of the binding process show that the binding of the ligand to the "membrane vestibule" is the rate-limiting step in the multi microseconds timescale. We comment on the significance and parallels of the binding process in the context of other binding studies