Mechanical Patterning in Animal Morphogenesis.

Morphogenesis is one of the most remarkable examples of biological pattern formation. Despite substantial progress in the field, we still do not understand the organizational principles responsible for the robust convergence of the morphogenesis process across scales to form viable organisms under variable conditions. Achieving large-scale coordination requires feedback between mechanical and biochemical processes, spanning all levels of organization and relating the emerging patterns with the mechanisms driving their formation. In this review, we highlight the role of mechanics in the patterning process, emphasizing the active and synergistic manner in which mechanical processes participate in developmental patterning rather than merely following a program set by biochemical signals. We discuss the value of applying a coarse-grained approach that considers the large-scale dynamics and feedback and complements the reductionist approach focused on molecular detail. A central challenge in this approach is identifying relevant coarse-grained variables and developing effective theories that can serve as a basis for an integrated framework toward understanding this remarkable pattern-formation process.

Efficient models of cortical activity via local dynamic equilibria and coarse-grained interactions.

Biologically detailed models of brain circuitry are challenging to build and simulate due to the large number of neurons, their complex interactions, and the many unknown physiological parameters. Simplified mathematical models are more tractable, but harder to evaluate when too far removed from neuroanatomy/physiology. We propose that a multiscale model, coarse-grained (CG) while preserving local biological details, offers the best balance between biological realism and computability. This paper presents such a model. Generally, CG models focus on the interaction between groups of neurons-here termed "pixels"-rather than individual cells. In our case, dynamics are alternately updated at intra- and interpixel scales, with one informing the other, until convergence to equilibrium is achieved on both scales. An innovation is how we exploit the underlying biology: Taking advantage of the similarity in local anatomical structures across large regions of the cortex, we model intrapixel dynamics as a single dynamical system driven by "external" inputs. These inputs vary with events external to the pixel, but their ranges can be estimated a priori. Precomputing and tabulating all potential local responses speed up the updating procedure significantly compared to direct multiscale simulation. We illustrate our methodology using a model of the primate visual cortex. Except for local neuron-to-neuron variability (necessarily lost in any CG approximation) our model reproduces various features of large-scale network models at a tiny fraction of the computational cost. These include neuronal responses as a consequence of their orientation selectivity, a primary function of visual neurons.

A Markovian dynamics for Caenorhabditis elegans behavior across scales.

How do we capture the breadth of behavior in animal movement, from rapid body twitches to aging? Using high-resolution videos of the nematode worm Caenorhabditis elegans, we show that a single dynamics connects posture-scale fluctuations with trajectory diffusion and longer-lived behavioral states. We take short posture sequences as an instantaneous behavioral measure, fixing the sequence length for maximal prediction. Within the space of posture sequences, we construct a fine-scale, maximum entropy partition so that transitions among microstates define a high-fidelity Markov model, which we also use as a means of principled coarse-graining. We translate these dynamics into movement using resistive force theory, capturing the statistical properties of foraging trajectories. Predictive across scales, we leverage the longest-lived eigenvectors of the inferred Markov chain to perform a top-down subdivision of the worm's foraging behavior, revealing both "runs-and-pirouettes" as well as previously uncharacterized finer-scale behaviors. We use our model to investigate the relevance of these fine-scale behaviors for foraging success, recovering a trade-off between local and global search strategies.

The Hill function is the universal Hopfield barrier for sharpness of input-output responses.

The Hill functions, [Formula: see text], have been widely used in biology for over a century but, with the exception of [Formula: see text], they have had no justification other than as a convenient fit to empirical data. Here, we show that they are the universal limit for the sharpness of any input-output response arising from a Markov process model at thermodynamic equilibrium. Models may represent arbitrary molecular complexity, with multiple ligands, internal states, conformations, coregulators, etc, under core assumptions that are detailed in the paper. The model output may be any linear combination of steady-state probabilities, with components other than the chosen input ligand held constant. This formulation generalizes most of the responses in the literature. We use a coarse-graining method in the graph-theoretic linear framework to show that two sharpness measures for input-output responses fall within an effectively bounded region of the positive quadrant, [Formula: see text], for any equilibrium model with [Formula: see text] input binding sites. [Formula: see text] exhibits a cusp which approaches, but never exceeds, the sharpness of [Formula: see text], but the region and the cusp can be exceeded when models are taken away from thermodynamic equilibrium. Such fundamental thermodynamic limits are called Hopfield barriers, and our results provide a biophysical justification for the Hill functions as the universal Hopfield barriers for sharpness. Our results also introduce an object, [Formula: see text], whose structure may be of mathematical interest, and suggest the importance of characterizing Hopfield barriers for other forms of cellular information processing.

General theory for localizing the where and when of entropy production meets single-molecule experiments.

The laws of thermodynamics apply to biophysical systems on the nanoscale as described by the framework of stochastic thermodynamics. This theory provides universal, exact relations for quantities like work, which have been verified in experiments where a fully resolved description allows direct access to such quantities. Complementary studies consider partially hidden, coarse-grained descriptions, in which the mean entropy production typically is not directly accessible but can be bounded in terms of observable quantities. Going beyond the mean, we introduce a fluctuating entropy production that applies to individual trajectories in a coarse-grained description under time-dependent driving. Thus, this concept is applicable to the broad and experimentally significant class of driven systems in which not all relevant states can be resolved. We provide a paradigmatic example by studying an experimentally verified protein unfolding process. As a consequence, the entire distribution of the coarse-grained entropy production rather than merely its mean retains spatial and temporal information about the microscopic process. In particular, we obtain a bound on the distribution of the physical entropy production of individual unfolding events.

Learning hydrodynamic equations for active matter from particle simulations and experiments.

Recent advances in high-resolution imaging techniques and particle-based simulation methods have enabled the precise microscopic characterization of collective dynamics in various biological and engineered active matter systems. In parallel, data-driven algorithms for learning interpretable continuum models have shown promising potential for the recovery of underlying partial differential equations (PDEs) from continuum simulation data. By contrast, learning macroscopic hydrodynamic equations for active matter directly from experiments or particle simulations remains a major challenge, especially when continuum models are not known a priori or analytic coarse graining fails, as often is the case for nondilute and heterogeneous systems. Here, we present a framework that leverages spectral basis representations and sparse regression algorithms to discover PDE models from microscopic simulation and experimental data, while incorporating the relevant physical symmetries. We illustrate the practical potential through a range of applications, from a chiral active particle model mimicking nonidentical swimming cells to recent microroller experiments and schooling fish. In all these cases, our scheme learns hydrodynamic equations that reproduce the self-organized collective dynamics observed in the simulations and experiments. This inference framework makes it possible to measure a large number of hydrodynamic parameters in parallel and directly from video data.

Building and analyzing metacells in single-cell genomics data.

The advent of high-throughput single-cell genomics technologies has fundamentally transformed biological sciences. Currently, millions of cells from complex biological tissues can be phenotypically profiled across multiple modalities. The scaling of computational methods to analyze and visualize such data is a constant challenge, and tools need to be regularly updated, if not redesigned, to cope with ever-growing numbers of cells. Over the last few years, metacells have been introduced to reduce the size and complexity of single-cell genomics data while preserving biologically relevant information and improving interpretability. Here, we review recent studies that capitalize on the concept of metacells-and the many variants in nomenclature that have been used. We further outline how and when metacells should (or should not) be used to analyze single-cell genomics data and what should be considered when analyzing such data at the metacell level. To facilitate the exploration of metacells, we provide a comprehensive tutorial on the construction and analysis of metacells from single-cell RNA-seq data ( https://github.com/GfellerLab/MetacellAnalysisTutorial ) as well as a fully integrated pipeline to rapidly build, visualize and evaluate metacells with different methods ( https://github.com/GfellerLab/MetacellAnalysisToolkit ).

Rigorous Progress in Coarse-Graining.

Low-resolution coarse-grained (CG) models provide remarkable computational and conceptual advantages for simulating soft materials. In principle, bottom-up CG models can reproduce all structural and thermodynamic properties of atomically detailed models that can be observed at the resolution of the CG model. This review discusses recent progress in developing theory and computational methods for achieving this promise. We first briefly review variational approaches for parameterizing interaction potentials and their relationship to machine learning methods. We then discuss recent approaches for simultaneously improving both the transferability and thermodynamic properties of bottom-up models by rigorously addressing the density and temperature dependence of these potentials. We also briefly discuss exciting progress in modeling high-resolution observables with low-resolution CG models. More generally, we highlight the essential role of the bottom-up framework not only for fundamentally understanding the limitations of prior CG models but also for developing robust computational methods that resolve these limitations in practice.

Modeling properties of chromosome territories using polymer filaments in diverse confinement geometries.

Interphase chromosomes reside within distinct nuclear regions known as chromosome territories (CTs). Recent observations from Hi-C analyses, a method mapping chromosomal interactions, have revealed varied decay in contact probabilities among different chromosomes. Our study explores the relationship between this contact decay and the particular shapes of the chromosome territories they occupy. For this, we employed molecular dynamics (MD) simulations to examine how confined polymers, resembling chromosomes, behave within different confinement geometries similar to chromosome territory boundaries. Our simulations unveil so far unreported relationships between contact probabilities and end-to-end distances varying based on different confinement geometries. These findings highlight the crucial impact of chromosome territories on shaping the larger-scale properties of 3D genome organization. They emphasize the intrinsic connection between the shapes of these territories and the contact behaviors exhibited by chromosomes. Understanding these correlations is key to accurately interpret Hi-C and microscopy data, and offers vital insights into the foundational principles governing genomic organization.

Development of a bottom-up coarse-grained model for interactions of lipids with TiO 2 nanoparticles.

Understanding interactions of inorganic nanoparticles with biomolecules is important in many biotechnology, nanomedicine, and toxicological research, however, the size of typical nanoparticles makes their direct modeling by atomistic simulations unfeasible. Here, we present a bottom-up coarse-graining approach for modeling titanium dioxide (TiO 2 ) nanomaterials in contact with phospholipids that uses the inverse Monte Carlo method to optimize the effective interactions from the structural data obtained in small-scale all-atom simulations of TiO 2 surfaces with lipids in aqueous solution. The resulting coarse-grained models are able to accurately reproduce the structural details of lipid adsorption on different titania surfaces without the use of an explicit solvent, enabling significant computational resource savings and favorable scaling. Our coarse-grained simulations show that small spherical TiO 2 nanoparticles ( r = 2 nm) can only be partially wrapped by a lipid bilayer with phosphoethanolamine headgroups, however, the lipid adsorption increases with the radius of the nanoparticle. The current approach can be used to study the effect of the size and shape of TiO 2 nanoparticles on their interactions with cell membrane lipids, which can be a determining factor in membrane wrapping as well as the recently discovered phenomenon of nanoquarantining, which involves the formation of layered nanomaterial-lipid structures.

Martini 3 Coarse-Grained Model for the Cofactors Involved in Photosynthesis.

As a critical step in advancing the simulation of photosynthetic complexes, we present the Martini 3 coarse-grained (CG) models of key cofactors associated with light harvesting (LHCII) proteins and the photosystem II (PSII) core complex. Our work focuses on the parametrization of beta-carotene, plastoquinone/quinol, violaxanthin, lutein, neoxanthin, chlorophyll A, chlorophyll B, and heme. We derived the CG parameters to match the all-atom reference simulations, while structural and thermodynamic properties of the cofactors were compared to experimental values when available. To further assess the reliability of the parameterization, we tested the behavior of these cofactors within their physiological environments, specifically in a lipid bilayer and bound to photosynthetic complexes. The results demonstrate that our CG models maintain the essential features required for realistic simulations. This work lays the groundwork for detailed simulations of the PSII-LHCII super-complex, providing a robust parameter set for future studies.

Integrating Explicit and Implicit Fullerene Models into UNRES Force Field for Protein Interaction Studies.

Fullerenes, particularly C60, exhibit unique properties that make them promising candidates for various applications, including drug delivery and nanomedicine. However, their interactions with biomolecules, especially proteins, remain not fully understood. This study implements both explicit and implicit C60 models into the UNRES coarse-grained force field, enabling the investigation of fullerene-protein interactions without the need for restraints to stabilize protein structures. The UNRES force field offers computational efficiency, allowing for longer timescale simulations while maintaining accuracy. Five model proteins were studied: FK506 binding protein, HIV-1 protease, intestinal fatty acid binding protein, PCB-binding protein, and hen egg-white lysozyme. Molecular dynamics simulations were performed with and without C60 to assess protein stability and investigate the impact of fullerene interactions. Analysis of contact probabilities reveals distinct interaction patterns for each protein. FK506 binding protein (1FKF) shows specific binding sites, while intestinal fatty acid binding protein (1ICN) and uteroglobin (1UTR) exhibit more generalized interactions. The explicit C60 model shows good agreement with all-atom simulations in predicting protein flexibility, the position of C60 in the binding pocket, and the estimation of effective binding energies. The integration of explicit and implicit C60 models into the UNRES force field, coupled with recent advances in coarse-grained modeling and multiscale approaches, provides a powerful framework for investigating protein-nanoparticle interactions at biologically relevant scales without the need to use restraints stabilizing the protein, thus allowing for large conformational changes to occur. These computational tools, in synergy with experimental techniques, can aid in understanding the mechanisms and consequences of nanoparticle-biomolecule interactions, guiding the design of nanomaterials for biomedical applications.

Mimicking Polymer Processing Conditions on the Meso-Scale: Relaxation and Crystallization in Polyethylene Systems after Uni- and Biaxial Stretching.

Highly varying process conditions drive polymers into nonequilibrium molecular conformations. This has direct implications for the resulting structural and mechanical properties. This study rigorously investigated processing-property relations from a microscopic perspective. The corresponding models use a mesoscale molecular dynamics (MD) approach. Different loading conditions, including uniaxial and biaxial stretching, along with various cooling conditions, were employed to mimic process conditions on the micro-scale. The resulting intricate interplay between equi-biaxial stretching, orientation, and crystallization behavior in long polyethylene chains was reviewed. The study reveals notable effects depending on different cooling and biaxial stretching procedures. The findings emphasize the significance of considering distributions and directions of chain ordering. Local inspections of trajectories unveil that crystal growth predominantly occurs in regions devoid of entanglements.

Correlation change analysis and NDVI prediction in the Yellow River Basin of China using complex networks and GRNN-PSRLSTM.

The Normalized Difference Vegetation Index (NDVI) is affected by various environmental factors, and its relationship with these factors is complex. In order to explore the complex relationship between NDVI and environmental factors, this paper adopts the complex network method to construct a correlation fluctuation network and analyze the interaction between them. It is found that temperature, precipitation, soil moisture, sunshine duration, and PM2.5 are all correlated with NDVI to varying degrees, and their combined correlation with NDVI varies over time. The correlation typically takes 3 to 6 months to change, and it tends to persist to some extent. Moreover, we fuse a generalized regression neural network (GRNN) with a long-short-term memory (LSTM) network combining phase space reconstruction (PSR) to propose a GRNN-PSRLSTM prediction model. The model achieves the prediction of monthly NDVI using the five environmental factors of the fluctuation network. The results indicate that the averages of root mean squared error (RMSE) and mean absolute percentage error (MAPE) predicted by the GRNN-PSRLSTM model in the nine provinces are 0.0232 and 0.0564 respectively. This model performs better in the assessment metrics for monthly NDVI forecasts. These findings are significant for evaluating vegetation changes and have some theoretical value for the ecological protection of the Yellow River Basin.

An Exact Theory of Causal Emergence for Linear Stochastic Iteration Systems.

After coarse-graining a complex system, the dynamics of its macro-state may exhibit more pronounced causal effects than those of its micro-state. This phenomenon, known as causal emergence, is quantified by the indicator of effective information. However, two challenges confront this theory: the absence of well-developed frameworks in continuous stochastic dynamical systems and the reliance on coarse-graining methodologies. In this study, we introduce an exact theoretic framework for causal emergence within linear stochastic iteration systems featuring continuous state spaces and Gaussian noise. Building upon this foundation, we derive an analytical expression for effective information across general dynamics and identify optimal linear coarse-graining strategies that maximize the degree of causal emergence when the dimension averaged uncertainty eliminated by coarse-graining has an upper bound. Our investigation reveals that the maximal causal emergence and the optimal coarse-graining methods are primarily determined by the principal eigenvalues and eigenvectors of the dynamic system's parameter matrix, with the latter not being unique. To validate our propositions, we apply our analytical models to three simplified physical systems, comparing the outcomes with numerical simulations, and consistently achieve congruent results.

Optimization of parallel implementation of UNRES package for coarse-grained simulations to treat large proteins.

We report major algorithmic improvements of the UNRES package for physics-based coarse-grained simulations of proteins. These include (i) introduction of interaction lists to optimize computations, (ii) transforming the inertia matrix to a pentadiagonal form to reduce computing and memory requirements, (iii) removing explicit angles and dihedral angles from energy expressions and recoding the most time-consuming energy/force terms to minimize the number of operations and to improve numerical stability, (iv) using OpenMP to parallelize those sections of the code for which distributed-memory parallelization involves unfavorable computing/communication time ratio, and (v) careful memory management to minimize simultaneous access of distant memory sections. The new code enables us to run molecular dynamics simulations of protein systems with size exceeding 100,000 amino-acid residues, reaching over 1 ns/day (1 µs/day in all-atom timescale) with 24 cores for proteins of this size. Parallel performance of the code and comparison of its performance with that of AMBER, GROMACS and MARTINI 3 is presented.

Long-time scale simulations of virus-like particles from three human-norovirus strains.

The dynamics of the virus like particles (VLPs) corresponding to the GII.4 Houston, GII.2 SMV, and GI.1 Norwalk strains of human noroviruses (HuNoV) that cause gastroenteritis was investigated by means of long-time (about 30 µs in the laboratory timescale) molecular dynamics simulations with the coarse-grained UNRES force field. The main motion of VLP units turned out to be the bending at the junction between the P1 subdomain (that sits in the VLP shell) and the P2 subdomain (that protrudes outside) of the major VP1 protein, this resulting in a correlated wagging motion of the P2 subdomains with respect to the VLP surface. The fluctuations of the P2 subdomain were found to be more pronounced and the P2 domain made a greater angle with the normal to the VLP surface for the GII.2 strain, which could explain the inability of this strain to bind the histo-blood group antigens (HBGAs).

Exact and Soft Successive Refinement of the Information Bottleneck.

The information bottleneck (IB) framework formalises the essential requirement for efficient information processing systems to achieve an optimal balance between the complexity of their representation and the amount of information extracted about relevant features. However, since the representation complexity affordable by real-world systems may vary in time, the processing cost of updating the representations should also be taken into account. A crucial question is thus the extent to which adaptive systems can leverage the information content of already existing IB-optimal representations for producing new ones, which target the same relevant features but at a different granularity. We investigate the information-theoretic optimal limits of this process by studying and extending, within the IB framework, the notion of successive refinement, which describes the ideal situation where no information needs to be discarded for adapting an IB-optimal representation's granularity. Thanks in particular to a new geometric characterisation, we analytically derive the successive refinability of some specific IB problems (for binary variables, for jointly Gaussian variables, and for the relevancy variable being a deterministic function of the source variable), and provide a linear-programming-based tool to numerically investigate, in the discrete case, the successive refinement of the IB. We then soften this notion into a quantification of the loss of information optimality induced by several-stage processing through an existing measure of unique information. Simple numerical experiments suggest that this quantity is typically low, though not entirely negligible. These results could have important implications for (i) the structure and efficiency of incremental learning in biological and artificial agents, (ii) the comparison of IB-optimal observation channels in statistical decision problems, and (iii) the IB theory of deep neural networks.

A Differential-Geometric Approach to Quantum Ignorance Consistent with Entropic Properties of Statistical Mechanics.

In this paper, we construct the metric tensor and volume for the manifold of purifications associated with an arbitrary reduced density operator ρS. We also define a quantum coarse-graining (CG) to study the volume where macrostates are the manifolds of purifications, which we call surfaces of ignorance (SOI), and microstates are the purifications of ρS. In this context, the volume functions as a multiplicity of the macrostates that quantifies the amount of information missing from ρS. Using examples where the SOI are generated using representations of SU(2), SO(3), and SO(N), we show two features of the CG: (1) A system beginning in an atypical macrostate of smaller volume evolves to macrostates of greater volume until it reaches the equilibrium macrostate in a process in which the system and environment become strictly more entangled, and (2) the equilibrium macrostate takes up the vast majority of the coarse-grained space especially as the dimension of the total system becomes large. Here, the equilibrium macrostate corresponds to a maximum entanglement between the system and the environment. To demonstrate feature (1) for the examples considered, we show that the volume behaves like the von Neumann entropy in that it is zero for pure states, maximal for maximally mixed states, and is a concave function with respect to the purity of ρS. These two features are essential to typicality arguments regarding thermalization and Boltzmann's original CG.

Metacells untangle large and complex single-cell transcriptome networks.

BACKGROUND: Single-cell RNA sequencing (scRNA-seq) technologies offer unique opportunities for exploring heterogeneous cell populations. However, in-depth single-cell transcriptomic characterization of complex tissues often requires profiling tens to hundreds of thousands of cells. Such large numbers of cells represent an important hurdle for downstream analyses, interpretation and visualization. RESULTS: We develop a framework called SuperCell to merge highly similar cells into metacells and perform standard scRNA-seq data analyses at the metacell level. Our systematic benchmarking demonstrates that metacells not only preserve but often improve the results of downstream analyses including visualization, clustering, differential expression, cell type annotation, gene correlation, imputation, RNA velocity and data integration. By capitalizing on the redundancy inherent to scRNA-seq data, metacells significantly facilitate and accelerate the construction and interpretation of single-cell atlases, as demonstrated by the integration of 1.46 million cells from COVID-19 patients in less than two hours on a standard desktop. CONCLUSIONS: SuperCell is a framework to build and analyze metacells in a way that efficiently preserves the results of scRNA-seq data analyses while significantly accelerating and facilitating them.