Abstract Gene regulatory networks are essential for understanding the mechanisms underlying genetic disorders. However, traditional models based on deterministic Hill functions (without fluctuations) often fail to capture critical stochastic effects, particularly in small systems in which molecular fluctuations play a significant role. In this study, we introduced a modified Hill function derived from the stochastic dynamics of ligand–receptor interactions. By performing a second-order expansion of the chemical master equation, we derived a set of ordinary differential equations governing the mean concentrations and covariances. This framework allows the identification of effective reaction rates using Hill functions that include stochastic correction terms that account for fluctuations in ligand–receptor binding. We applied this approach to canonical models, such as the Toggle Switch, Repressilator, and Activator–Repressor Clock, and showed that the Hill function with stochastic corrections provides a more accurate description of system dynamics with intrinsic fluctuations. Moreover, the proposed method offers substantial computational advantages over direct stochastic simulations using the Gillespie algorithm. Our results demonstrate that this approximation captures the essential features of stochastic biochemical systems more accurately than models that use deterministic Hill functions, offering a powerful tool for studying the role of fluctuations in complex reaction networks and uncovering potential emergent behaviors driven by intrinsic fluctuations in the system.

Abstract We develop a theoretical framework based on the cavity and replica methods to analyze the spectral properties of sparse asymmetric correlation matrices of the form F = ( X Y ⊤ + ω Y X ⊤ ) / 2 T , where X and Y are adjacency matrices of weighted Erdős–Rényi random graphs. We examine how the spectral density evolves as the asymmetry parameter ω varies from 0 < ω < 1 (nearly symmetric matrices) to − 1 < ω ⩽ 0 (nearly antisymmetric matrices). Analytical predictions are validated through exact numerical diagonalization, showing excellent agreement with theoretical results in the thermodynamic limit.

Abstract Zero forcing is a graph coloring process that is used to model spreading phenomena in real-world scenarios. It can also be viewed as a single-player combinatorial game on a graph, where the player’s goal is to select a subset of vertices of minimum cardinality that eventually leads to all vertices of the graph being colored. A variant of this game, called the q -analogue of zero forcing, was later introduced. In this version, the player again seeks to choose the smallest number of vertices that will eventually color the entire graph, while an oracle attempts to force the player to select a larger subset. In this paper, we exploit the structural properties of several graph classes in order to both derive algorithms to compute the exact value of Z q , and to establish bounds and exact values of the parameter for these graph classes. In particular, we present a SAT-based algorithm to compute the q -analogue zero forcing number, offering optimal strategies for both the player and the oracle. Additionally, we propose a polynomial-time algorithm for computing the q -analogue zero forcing number for q = 1 of cactus graphs. Lastly, we prove the exact value of this parameter for several graph classes such as block graphs. Our work extends previous results about trees by Butler et al (2020 Graphs Comb. 36 1401–19) and Blanco et al (2024 arXiv:2305.11748).

Abstract The relationship between computational models and dynamics has captivated mathematicians and computer scientists since the earliest conceptualizations of computation. Recently, this connection has gained renewed attention, fuelled by T. Tao’s programme aiming to discover blowing-up solutions of the Navier–Stokes equations using an embedded computational model. In this survey paper, we review some of the recent works that introduce novel and exciting perspectives on the representation of computability through dynamical systems. Starting from dynamical universality in a classical sense, we shall explore the modern notions of Turing universality in fluid dynamics and Topological Kleene field theories as a systematic way of representing computable functions by means of dynamical bordisms. Finally, we will discuss some important open problems in the area.

Abstract Recently the global average temperature has temporarily exceeded the 1.5°C goal of the Paris Agreement, and so an overshoot of various climate tipping elements becomes increasingly likely. In this study we analyze the physical processes of an overshoot of the Atlantic Meridional Overturning Circulation (AMOC), one of the major tipping elements, using a conceptual box model. Here either the atmospheric temperature above the North Atlantic, or the freshwater forcing into the North Atlantic overshoot their respective critical boundaries. In both cases a higher forcing rate can prevent a collapse of the AMOC, since a higher rate of forcing causes initially a fresher North Atlantic, which in turn results in a higher northward transport by the subtropical gyre supplementing the salinity loss in time. For small exceedance amplitudes the AMOC is still resilient as the forcing rates can be low and so other state variables outside of the North Atlantic can adjust. Contrarily, for larger overshoots the trajectories are dynamically similar and we find a lower limit in volume and exceedance time for respectively freshwater and temperature forcing in order to prevent a collapse. Moreover, for a large overshoot an increased air-sea temperature coupling has a destabilizing effect, while the reverse holds for an overshoot close to the tipping point. The understanding of the physics of the AMOC overshoot behavior is important for interpreting results of Earth System Models and for evaluating the effects of mitigation and intervention strategies.

Abstract The coupling of microscopic traffic simulation models with emission models offers a powerful tool for assessing and optimising traffic control strategies to reduce fuel consumption and vehicle emissions. Although many studies use traffic simulation for emission analysis and designing traffic control measures, most focus on calibrating a selected traffic model to replicate observed traffic flow. This raises a critical question: are the resulting optimal emission control strategies adequately designed to account for the sensitivity of traffic models in capturing vehicle dynamics and emissions? To address this issue, we compared three car-following models—the Krauss model, the Intelligent Driver Model (IDM), and the Wiedemann model—each rooted in distinct theoretical frameworks to understand traffic dynamics. We evaluated their performance in optimising road speed limits to minimise (PMx) emissions in a school case study. A school was selected as the case because children are highly vulnerable and particularly exposed to pollutants during their school commute, and their exposure can be mitigated through optimal traffic control. Our findings reveal that, even when tuned to achieve comparable levels of traffic flow, the models displayed significant differences in their objective functions for traffic control optimisation. These discrepancies stemmed from variations in fuel consumption and particulate matter (PMx) emission patterns resulting from the traffic dynamics captured by the selected traffic model. At a macroscopic level (e.g., average speed, flow, and density), the models exhibited minimal differences. However, at a microscopic level (e.g., acceleration, deceleration rates, and deviations from the mean), pronounced differences became evident. These results highlight that while certain traffic control strategies appeared less effective, revisiting and critically examining the limitations of the models is essential to ensure robust and tailored solutions for emission reduction.

Khovanov homology has been the subject of much study in knot theory and low dimensional topology since 2000. This work introduces a Khovanov Laplacian and a Khovanov Dirac to study knot and link diagrams. The harmonic spectrum of the Khovanov Laplacian or the Khovanov Dirac retains the topological invariants of Khovanov homology, while their non-harmonic spectra reveal additional information that is distinct from Khovanov homology.

Abstract Triadic interactions in the brain are general mechanisms by which a node, e.g. a neuron or a glia cell such as the astrocyte, can regulate directly the link, e.g. synapse between other two nodes. The regulation takes place in a familiar way by either depressing or facilitating synaptic transmission. Such interactions are ubiquitous in neural systems, accounting both for axo-axonic and tripartite synapses mediated by astrocytes, for instance, and have been related to neuronal and synaptic processes at different time-scales, including short- and long-term synaptic plasticity. In the field of network science, triadic interactions have been shown to produce complex spatio-temporal patterns of connectivity. Here, we investigate the emergent behavior of an in silico neural medium constituted by a population of leaky integrate-and-fire neurons with triadic interactions. We observe that, depending on relevant parameters defining triadic interactions, different activity patterns emerge. These include (i) a silent phase, (ii) a low-activity phase in which complex spatio-temporal patterns of low neuronal firing rate emerge that propagate through the medium, (iii) a high-activity phase characterized by complex spatio-temporal patterns of high neuronal firing rate that propagate through the neural medium as waves of high firing activity over a bulk of low activity neurons, and (iv) a pseudo-blinking phase in which the neural medium switches between high and low activity states. Here we analyze in depth the features of such patterns and relate our findings to the recently proposed model of triadic percolation.

Abstract Cities serve as the main platform for human socio-economic activities, where large crowds, products and information converge, forming complex societal interactions and the space for human to interact with each other. Like traditional geographical space, in cities, cyberspace and societal space also offer diverse channels for human interactions. While urban mobility in geographical space has been extensively studied across disciplines such as geography, economics, transportation and human behavior, mobility within societal space in cities has rarely been examined in depth. This paper focuses on the self-organized consolidation of mobility within urban societal spaces. Based on the rational actor assumption and Dunbar’s number, we construct a simple model to explain why urban societal classes spontaneously emerge in the evolution of societal space. We also find that the mobility within urban societal space decreases spontaneously over time and eventually outstanding individuals potentially can be trapped in lower classes, resulting in an inefficient matching of urban resources.