Species interactions through cross-feeding via leakage and uptake of chemicals are important in microbial communities, and play an essential role in the coexistence of diverse species. Here, we study a simple dynamical model of a microbial community in which species interact by competing for the uptake of common metabolites that are leaked by other species. The model includes coupled dynamics of species populations and chemical concentrations in the medium, allowing for a variety of uptake and leakage networks among species. Depending on the structure of these networks, the system exhibits different attractors, including fixed points, limit cycles, low-dimensional chaos, and high-dimensional chaos. In the fixed-point and limit-cycle cases, the number of coexisting species is bounded by the number of exchangeable chemicals, consistent with the well-known competitive exclusion principle. In contrast, in the low-dimensional chaotic regime, the number of coexisting species exhibits noticeable but limited excess over this limit. Remarkably, in the high-dimensional chaotic regime, a much larger number of species beyond this limit coexist persistently over time. In this case, the rank-abundance distribution is broader than exponential, as often observed in real ecosystems. The population dynamics displays intermittent switching among quasi-stationary states, while the chemical dynamics explore most of the high dimensions. We find that such high-dimensional chaos is ubiquitous when the number of uptake chemicals is moderately larger than the number of leaked chemicals. Our results identify high-dimensional chaos with intermittent switching as a generic dynamical mechanism that stabilizes coexistence in interacting systems. We discuss its relevance to sustaining diverse microbial communities with leak-uptake cross-feeding.

Enhanced enzyme diffusion (EED), in which the diffusion coefficient of an enzyme transiently increases during catalysis, has been extensively reported experimentally, although its existence remains under debate. In this Letter, we investigate what macroscopic consequences would arise if EED exists. Through numerical simulations and theoretical analysis, we demonstrate that such enzymes can act as Maxwell's demons: They use their enhanced diffusion as a memory of the previous catalytic reaction, to gain information and drive steady-state chemical concentrations away from chemical equilibrium. Our theoretical analysis identifies the conditions under which this process could operate and discusses its possible biological relevance.

Directional ion transport across membranes maintains living systems in nonequilibrium, which underlies chemiosmotic energy conversion. However, the physical origin of collectively organized ion transport in primitive cellular systems remains unclear. Here, we propose a minimal model in which ion pumps collectively align through feedback between ion transport and electrostatic interactions. In the model, directional ion transport generates a membrane potential, while the resulting electrochemical potential biases pump orientation, leading to self-organized collective alignment. Numerical simulations and mean-field analysis reveal a nonequilibrium transition from a disordered state without net transport to a pump-alignment state with sustained membrane potentials. The critical behavior is consistent with the mean-field Ising universality class; however, the effective field is generated self-consistently by nonequilibrium ion transport. We further show that protocell asymmetry can bias the polarity of the membrane potential. These results provide a generic self-organizing mechanism for the emergence of bioelectricity and a physical route toward chemiosmotic coupling in protocells.

Evolution in changing environments requires both reliable expression of the currently favored phenotype, as quantified by penetrance, and the capacity to reach alternative phenotypes through mutation. Previous studies suggest that high penetrance may restrict such mutational access. However, because these studies focus on evolved genotypes and local mutational neighborhoods, they cannot determine whether this local constraint limits mutational adaptability under environmental change. Such adaptability depends on a genotype's position relative to high-fitness regions for other environments. Addressing this question requires reconstructing the full probability distribution over phenotypes for every genotype and the resulting environment-specific fitness landscapes across genotype space. Such reconstruction is generally infeasible because genotype and phenotype spaces grow combinatorially. Here, an abstract model of stochastic genotype-phenotype mapping, inspired by interacting spins in statistical physics, permits exhaustive reconstruction of the map. We find that high-penetrance genotypes tend to occupy the interior of environment-specific high-fitness regions and are mutationally robust, whereas lower-penetrance genotypes tend to lie near their boundaries and have greater mutational access to high-fitness regions for alternative environments. This global geometry generates a trade-off between penetrance and mutational adaptability. In evolutionary simulations, stronger phenotypic noise in a fixed environment increases the selective advantage of reliable expression, thereby favoring high penetrance and mutational robustness. Frequent environmental change instead favors mutational accessibility at the expense of penetrance. Thus, penetrance and adaptability are opposing consequences of the same global geometry, with environmental conditions determining their evolutionary balance.

Arabidopsis roots show oscillatory growth patterns on homogeneous agar surfaces, whereas other plants, such as maize, do not. Although several explanations have been proposed, a simple and general model that makes testable predictions across species has been lacking. Roots sense gravity and correct their growth direction towards the vertical. Motivated by recent evidence for a time delay in this gravitropic correction, we develop a minimal nonlinear model based on the delay hypothesis that predicts whether a root oscillates or grows vertically downwards. The model identifies a fourfold relation between the delay and time period, robust across different response functions. Analysing images of Arabidopsis, we find that the mode of the oscillatory arc length is not significantly different between inclined and vertical growth conditions. The quantitative agreement between the experimentally measured oscillatory arc length and the arc length estimated from estimated root growth speed and response delay supports this fourfold delay-period rule for delay-driven root oscillations. The simplicity of our model allows for a direct comparison with data from diverse plant species.

Gene expression in cells is stochastic, yet differentiation is robust. We propose a mechanism in which frustrated genes with weakly stable intermediate expression undergo noise-driven switching between basins of attraction, followed by irreversible fate fixation through slow epigenetic feedback. Regulatory interactions amplify effective noise and promote differentiation. We derive analytic expression for the logarithmic dependence of differentiation time on noise strength and input-dependent cell-fate selection, and demonstrate homeorhesis, the dynamical robustness of the epigenetic landscape.

In multicellular organisms, cells differentiate into multiple types as they divide. States of these cell types, as well as their numbers, are known to be robust to external perturbations, as conceptualized by Waddington’s epigenetic landscape where cells embed themselves in valleys corresponding to final cell types. How is such robustness achieved by developmental dynamics and evolution? To address this question, we consider a model of cells with gene expression dynamics and epigenetic feedback, governed by a gene regulation network. By evolving the network to achieve more cell types, we identified three major differentiation processes exhibiting different properties regarding their variance, attractors, stability, and robustness. The first of these, type A, exhibits chaos and long-lived oscillatory dynamics that slowly transition until reaching a steady state. The second, type B, follows a channeled annealing process where the epigenetic changes in combination with noise shift the cells toward varying final cell states that increase the stability. Finally, type C exhibits a quenching process where cell fate is quickly decided by falling into preexisting fixed points while cell trajectories are separated through periodic attractors or saddle points. We find types A and B to correspond well with Waddington’s landscape while being robust. Finally, the dynamics of type B demonstrate a differentiation process that uses a directed shifting of fixed points, visualized through the dimensional reduction of gene expression states. Correspondence with the experimental data of gene expression variance through differentiation is also discussed.

Biological systems are generally complicated and/or complex. In the former approach, one sets up a model with a large number of parameters to describe the system in detail. The latter approach focuses on understanding the universal aspects of biological systems. In this case, an appropriate simple model represents a universality class. The extraction of universal properties is supported by evolutionary robustness and the reduction of dimensionality in high-dimensional states. Integrating the data-driven omics approach with the universality approach is an important step in systems biology.

Crises such as starvation pose a serious threat to microbial populations, prompting cells to adopt survival strategies, such as cooperation or competition. Although cooperation among clones is common, recent studies have shown that yeast cells can kill clonal cells under glucose depletion by secreting autotoxins. Adapted cells survive, whereas non-adapted latecomers are eliminated. Remarkably, this toxin-adaptation (TA) system, which uses the same set of autotoxins, is conserved across distantly related yeast species. This is puzzling because conventional toxin-immunity (TI) systems are prone to exploitation by 'cheaters', cells that benefit from immunity without producing toxins, and typically diverge in an evolutionary arms race. To investigate how this system is maintained, we analysed its evolutionary stability using population dynamics modelling. The system does not evolve in constant environments: cheaters outcompete adaptive cells during continuous starvation, while sensitive cells that produce neither toxin nor immunity dominate in continuous nutrient-rich conditions. However, when the environment switches stochastically between starvation and nutrient-rich phases, with short starvation periods and long nutrient-rich periods, the TA system becomes evolutionarily stable. These findings suggest that fluctuating environments can promote the emergence and long-term maintenance of the TA system, highlighting the critical role of environmental switching in shaping microbial survival strategies.

Embryonic development in multicellular organisms exhibits diverse morphogenetic patterns, which can generally be categorized into fundamental types such as monolayer and multilayer spheres, as well as cell masses. Furthermore, we identify two distinct processes for the formation of spherical structures. These basic patterns are thought to be governed by the microscopic properties of intercellular adhesion. However, the specific mechanisms linking the microscopic factors to the emergence of distinct macroscopic morphogenetic patterns remain poorly understood. In this study, we explore how different morphogenetic patterns arise by employing a computational model that incorporates intercellular adhesion and polarity. Our results demonstrate that all fundamental morphogenetic patterns can be generated through the interplay of two key parameters: the polarity strength of the cell and the regulation of polarity via mechanical signals. Furthermore, analytical considerations reveal key mechanisms underlying the formation of these patterns. These findings highlight the critical role of physical constraints in morphogenesis and suggest potential applications to the design of artificial tissues and organoids.

Living systems consist of diverse components and constitute a hierarchy, from molecules to cells to organisms, which adapt to external perturbations and reproduce stably. This book describes the statistical and physical principles governing cell growth and reproduction, and the mechanisms for adaptation through noise, kinetic memory, and robust cell differentiation through cell to cell interaction and epigenetics. The laws governing rate, direction, and constraints of phenotypic evolution are examined from the perspective of microscopic units (molecules) and macroscopic states (cells), with a focus on maintaining consistency between these length and temporal scales. By integrating theoretical, computational, and experimental approaches, this book offers novel insights into biology from a physicist's perspective and provides a detailed picture of the universal characteristics of living systems. It is indispensable for students and researchers in physics, biology and mathematics interested in understanding the nature of life and the physical principles it is based upon.

Flexible modulation of temporal dynamics in neural sequences underlies many cognitive processes. For instance, we can adaptively change the speed of motor sequences and speech. While such flexibility is influenced by various factors such as attention and context, the common neural mechanisms responsible for this modulation remain poorly understood. We developed a biologically plausible neural network model that incorporates neurons with multiple timescales and Hebbian learning rules. This model is capable of generating simple sequential patterns as well as performing delayed match-to-sample (DMS) tasks that require the retention of stimulus identity. Fast neural dynamics establish metastable states, while slow neural dynamics maintain task-relevant information and modulate the stability of these states to enable temporal processing. We systematically analyzed how factors such as neuronal gain, external input strength (contextual cues), and task difficulty influence the temporal properties of neural activity sequences - specifically, dwell time within patterns and transition times between successive patterns. We found that these factors flexibly modulate the stability of metastable states. Our findings provide a unified mechanism for understanding various forms of temporal modulation and suggest a novel computational role for neural timescale diversity in dynamically adapting cognitive performance to changing environmental demands.

The human capacity for sudden insight, often marked by abrupt and illuminating “eureka” moments, has long intrigued neuroscientists seeking to understand its elusive neural basis[1][1]–[6][2]. Mooney image recognition offers a compelling behavioral paradigm of this phenomenon, in which ambiguous black-and-white patterns are abruptly perceived as coherent objects[7][3]–[13][4]. Whole-brain cortical involvement, which is mainly revealed by the contrast between recognition and nonrecognition, is well established[8][5],[14][6]–[27][7]. However, the neural dynamics underlying the transition from nonrecognition to recognition, particularly the contributions of noncortical structures, have remained largely unknown. Here, we show that the transition in whole-brain dynamics can be effectively characterized by three large-scale activity patterns and that the superior colliculus (SC) emerges as a structure with a distinct temporal profile that is potentially critical for insight. Using functional clustering of functional magnetic resonance imaging data comprising 41,446 time points from 14 human participants performing this insight task, we identified three distinct functional clusters exhibiting stimulus-driven activation, suppression, and recognition-associated patterns. Notably, the SC in the third cluster displayed a distinct activation peak immediately before the recognition response. Furthermore, empirical dynamic modeling revealed that the SC was the only region in the whole brain that exhibited mutual positive directed interactions with all three clusters and was functionally embedded within higher-order cortical interactions mediating the transformation from stimulus input to motor output. Our findings provide compelling experimental evidence that the SC plays a critical role in orchestrating whole-brain dynamics for sudden insight, calling for a reappraisal of this evolutionarily conserved structure as a key player supporting unconscious but high-level cognitive processing. ### Competing Interest Statement The authors have declared no competing interest. Japan Society for the Promotion of Science, JP24K15188, JP25H01365 Ministry of Internal Affairs and Communications, https://ror.org/00vs1pz50, JPMI00316 [1]: #ref-1 [2]: #ref-6 [3]: #ref-7 [4]: #ref-13 [5]: #ref-8 [6]: #ref-14 [7]: #ref-27

Learning speed depends on both task structure and neural dynamics prior to learning, yet a theory connecting them has been missing. Inspired by the fluctuation-response relation, we derive two formulae linking neural dynamics to learning. Initial learning speed is proportional to the covariance between pre-learning spontaneous activity and network's input-evoked response, independent of the learning rule. For Hebb-type learning, initial speed scales with the variance of activity along target and input directions. These results apply across tasks including input-output mapping and time-series generation. Numerical simulations across diverse models validate the formulae beyond the theoretical-derivation's assumptions. Although derived for early learning, the formulae predict total learning time. A straightforward implication is learning is faster when task-relevant directions align with high-variance spontaneous activities, consistent with empirical findings. Our framework establishes how the geometrical relationship between pre-learning dynamics and task directions governs learning speed, independent of details of tasks.

Soil is a complex, dynamic material with physical properties that depend on its biological content. We propose a cellular automaton model for self-organizing soil structure, where soil aggregates serve as food for microbial species. These, in turn, produce nutrients that facilitate self-amplification, establishing a cyclical dynamic of consumption and regeneration. Our model explores the spatial interactions between these components and their role in the sustainability of a balanced ecosystem. The main results demonstrate that (1) spatial structure supports a stable living state, preventing population collapse or uncontrolled growth; (2) the spatial model allows for the coexistence of parasitic species, which exploit parts of the system without driving it to extinction; and (3) optimal growth conditions for microbes are associated with diverse length scales in the soil structure, suggesting that heterogeneity is key to ecosystem resilience. These findings highlight the importance of spatiotemporal dynamics of life in soil ecology.