Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

Imagine you walk in a plane. You move by making a step of a certain length per time interval in a chosen direction. Repeating this process by randomly sampling step length and turning angle defines a two-dimensional random walk in what we call comoving frame coordinates. This is precisely how Ross and Pearson proposed to model the movements of organisms more than a century ago. Decades later their concept was generalised by including persistence leading to a correlated random walk, which became a popular model in Movement Ecology. In contrast, Langevin equations describing cell migration and used in active matter theory are typically formulated by position and velocity in a fixed Cartesian frame. In this article, we explore the transformation of stochastic Langevin dynamics from the Cartesian into the comoving frame. We show that the Ornstein-Uhlenbeck process for the Cartesian velocity of a walker can be transformed exactly into a stochastic process that is defined self-consistently in the comoving frame, thereby profoundly generalising correlated random walk models. This approach yields a general conceptual framework how to transform stochastic processes from the Cartesian into the comoving frame. Our theory paves the way to derive, invent and explore novel stochastic processes in the comoving frame for modelling the movements of organisms. It can also be applied to design novel stochastic dynamics for autonomously moving robots and drones.

The European Unions Safety Gate Rapid Alert System (RAPEX) requires Hazard and Risk Assessment Methodology (HARM) evaluations addressing both injury lethality and long-term consequences (LTC). This paper developed a post-processing method to use AIS 2015-coded trauma data directly for RAPEX HARM assessments. AIS 2015 utilizes two metrics: the AIS Code (AIS-CD) for threat to life and the predicted Functional Capacity Index (pFCI) for LTC. While the AIS-CD has been validated on numerous datasets, the pFCI values are based on a theoretical framework that is pending validation. To counter coding variability and poor alignment with clinical diagnoses, initial AIS identifiers (AIS-IDs) were aggregated to a robust level of detail for both metrics. Individual injury severities (AIS-CD/FCI-CD) were aggregated to the person level using a conversion derived from the three most severe injuries (triples), mirroring the concept of the New Injury Severity Score (NISS). The final HARM Level is the most severe outcome derived independently from either the AIS-CD or FCI-CD triple conversion. Analysis showed over 70% of injuries in the aggregated codebook had no LTC. While AIS-CD dominated lower HARM scores, LTC became more defining with increasing HARM severity for the GIDAS sample. At HARM 4 (highest severity), AIS-CD accounted for 53% of cases, FCI-CD accounted for 16%, and both were equally severe in 31% of cases. This method successfully assigns HARM values to AIS 2015 injuries, providing a more holistic severity measure than the current AIS-CD-only approach. HighlightsO_LINovel method assigns RAPEX HARM values to AIS 2015-coded injuries. C_LIO_LICombines lethality (AIS-code) and long-term consequences (FCI-code) for injury severity assessment. C_LIO_LIAggregates injury severity using the three most severe injuries per person. C_LIO_LILong-term consequences account for 13% and 16% of the highest two HARM ratings, respectively. C_LI

AA amyloidosis is a severe complication of chronic inflammatory diseases characterized by fibrillar protein deposition in the kidneys, leading to progressive organ failure. This study presents a mathematical model coupling SAA-HDL binding dynamics with renal amyloid aggregation kinetics to elucidate disease pathogenesis. Under normal conditions, Serum Amyloid A (SAA) circulates bound to high-density lipoprotein (HDL), which acts as a molecular chaperone preventing misfolding. However, during chronic inflammation, SAA production exceeds HDL binding capacity, resulting in free SAA that undergoes renal filtration. The model calculates free SAA concentration from reversible binding equilibrium and incorporates renal filtration, mesangial accumulation, and conversion to amyloid fibrils through primary nucleation and autocatalytic growth mechanisms. A central contribution of this work is quantifying accumulated nephrotoxicity arising from AA oligomers, which inflict cumulative cytotoxic damage to mesangial and tubular cells over time. Because oligomers are continuously generated during ongoing aggregation, their toxic burden integrates across the entire duration of the disease. Combined nephrotoxicity, encompassing both oligomer-mediated cellular injury and fibril-driven mechanical disruption of renal architecture, therefore reflects not merely the current disease state but the full inflammatory trajectory of the patient. This cumulative damage defines renal biological age, a measure of functional deterioration whose portion attributable to accumulated nephrotoxicity is irreversible. Renal biological age is also path-dependent: two patients with identical present-day SAA levels may carry different renal damage burdens depending on the duration, timing, and severity of their prior inflammatory episodes. Sensitivity analysis reveals that HDL concentration and SAA cleavage rate are critical determinants of amyloid burden.

AO_SCPLOWBSTRACTC_SCPLOWWe study an extension of an environmentally mediated epidemiological model that incorporates direct human-to-human transmission. While the original formulation accounted for environmental exposure, it did not include direct transmission between individuals. Allowing both transmission routes to interact leads to significant qualitative changes in the system dynamics. The analysis reveals multiple dynamical regimes governed by environmental and combined threshold quantities. The stability of the disease-free equilibrium is controlled by an environmental threshold, whereas a combined reproduction number determines the onset of multistability. For certain parameter ranges, endemic equilibria coexist with the disease-free equilibrium, giving rise to backward-type bifurcation behavior and sensitivity to initial conditions. Moreover, the direct transmission rate acts as an organizing parameter by inducing the emergence of an environmental-free equilibrium when exceeding its classical threshold. These results highlight how environmentally coupled transmission mechanisms can generate rich dynamics in low-dimensional models.

Cerebrospinal fluid (CSF) is a Newtonian fluid that bathes the brain and spinal cord and oscillates in response to the physiological periodic changes in brain volume, of which the cardiac cycle is a major driver. Understanding this motion is essential for clarifying its contribution to solute transport, waste clearance, and drug delivery. In this work, we study oscillatory and steady streaming flow in the cranial subarachnoid space using a lubrication-based theoretical framework. The model represents the cranial CSF compartment as a thin fluid layer bounded internally by the brain surface and externally by the dura, driven by time-dependent brain surface displacements. We first derive simplified governing equations for flow over an arbitrary smooth sphere-like brain surface and obtain analytical solutions for an idealised spherical geometry with uniform displacements. We then incorporate realistic displacement fields reconstructed from MRI measurements in healthy subjects and solve the reduced equations numerically. The results show that oscillatory forcing produces a steady streaming component that may enhance solute transport compared with diffusion alone. This work provides a mechanistic description of the flow generated by physiological brain motion and highlights the potential presence of steady streaming in cranial subarachnoid fluid dynamics.

Cytokine-mediated communication is a central mechanism by which immune cells coordinate activation, differentiation and proliferation. While mechanistic reaction-diffusion models provide detailed descriptions of cytokine secretion and uptake at the cellular scale, their computational cost limits their applicability to large and densely packed cell populations. Previously employed approximations of cytokine diffusion fields rely on assumptions that neglect the influence of cellular geometry and volume exclusion. In this work, we study a macroscopic description of cytokine diffusion and reaction dynamics based on homogenization techniques, rigorously linking microscopic reaction-diffusion formulations to effective continuum models. The resulting homogenized equations replace discrete responder cells with a continuous density, while retaining essential features of cellular uptake and excluded-volume effects. Further, we show that in regimes with approximate radial symmetry, classical Yukawa-type solutions emerge as limiting cases of the homogenized model, provided appropriate correction factors are included. Overall, our approach allows efficient multiscale modeling of cytokine signaling in complex immune-cell environments.

Biological tissues grow at rates that depend on the geometry of the supporting tissue substrate. In this study, we present a novel discrete mathematical model for simulating biological tissue growth in a range of geometries. The discrete model is deterministic and tracks the evolution of the tissue interface by representing it as a chain of individual cells that interact mechanically and simultaneously generate new tissue material. To describe the collective behaviour of cells, we derive a continuum limit description of the discrete model leading to a reaction-diffusion partial differential equation governing the evolution of cell density along the evolving interface. In the continuum limit, the mechanical properties of discrete cells are directly linked to their collective diffusivity, and spatial constraints introduce curvature dependence that is not explicitly incorporated in the discrete model. Numerical simulations of both the discrete and continuum models reproduce the smoothing behaviour observed experimentally with minimal discrepancies between the models. The discrete model offers further individual-level details, including cell trajectory data, for any restoring force law and initial geometry. Where applicable, we discuss how the discrete model and its continuum description can be used to interpret existing experimental observations.

One of the most critical steps in human reproduction is the selection of the dominant follicle when a single follicle is chosen from a large group of follicles to ovulate. Although this process involves complex hormonal regulation, the complete microscopic picture of unique selectivity remains unclear. We propose a novel stochastic mechanism for dominant follicle selection that incorporates the actions of the most relevant hormones, follicle-stimulating hormone (FSH) and estradiol. Our theoretical picture suggests the following sequence of events. As soon as the FSH concentration reaches the critical threshold, one of the available follicles is randomly selected, which immediately stimulates the production of estradiol, which, via a negative feedback mechanism, suppresses further FSH production, lowering its concentration below the critical threshold. This suppression limits the time window for the possible second follicle selection event, allowing only a single follicle to be selected. Based on this picture, a minimal quantitative theoretical model of dominant follicle selection is developed and analyzed using analytical calculations and computer simulations. Theoretical analysis shows how the interplay between different parameters that govern follicle selection leads to high selectivity. Our theoretical approach can explain some key known observations, providing a quantitative tool for analyzing biological reproduction phenomena.

Adhesion of spherical nanoparticles or virus-like particles to membranes can lead to membrane tubules in which linear chains of adhering particles are cooperatively wrapped by the membrane. This cooperative wrapping of spherical particles in tubules is energetically favourable compared to the individual wrapping of the particles because of a favourable interplay of bending and adhesion energies in the contact regions in which the membrane detaches from the particles, and because a particle in a tubule has two such contact regions in the membrane necks that connect the particle to the neighbouring particles, whereas an individually wrapped particle has only one contact region to the surrounding membrane. The energetic gain of cooperative wrapping strongly depends on the range of the particle-membrane adhesion potential, which determines the size of the contact regions. At sufficiently large adhesion energies for wrapping, the energy gain {Delta}E per particle is only weakly affected by the membrane tension{tau} as long as the characteristic length [Formula] of the membranes with bending rigidity{kappa} is clearly larger than the contact regions. For large particle adhesion energies at which the particles are fully wrapped, however, {Delta}E can be limited by the minimum possible radius of the membrane necks, depending on the adhesion potential range.

Cerebrospinal fluid (CSF) circulates around and through the brain, supporting neural homeostasis by regulating the extracellular chemical environment. Yet the physical mechanisms governing CSF-driven solute transport remain poorly understood, limiting the design of diagnostic and therapeutic strategies targeting brain clearance and drug delivery. Pulsatile CSF flow in the cranial subarachnoid space (cSAS), is driven by cardiac, respiratory, and sleep-related vasomotion. Over longer timescales weaker steady flows, such as inertial steady streaming, Stokes drift, and production-drainage flow, may contribute to solute transport, but their role and relative importance remain unclear. Here, we develop a simplified two-dimensional model of CSF flow and solute transport in the cSAS using lubrication theory. Through multiple-timescale and asymptotic analyses, we derive a reduced long-time transport equation in which advection is governed by the Lagrangian mean velocity, incorporating steady streaming, production-drainage flow, and Stokes drift. Analysing three physiologically relevant case studies, we show that steady flows can substantially reshape concentration profiles, enhance dispersion, and alter clearance efficiency. Our results clarify the mechanisms underlying CSF-mediated transport, predict distinct regimes in humans and mice, and highlight the importance of subject-specific physiological parameters when interpreting contrast-agent and intrathecal drug-delivery studies.

Dynamical processes on complex networks have a long history of study with increasing applications across many fields. While epidemics in heterogeneous networks have received much attention in terms of how connectivity patterns drive epidemic outbreaks, affect critical thresholds, timescales, final outbreak size and immunization efforts, less attention has been devoted to endemic multi-strain scenarios and questions of selection and coexistence dynamics. Here, we provide an SIS framework for multiple co-circulating strains and co-infection, which can be reduced to a replicator dynamics on a host contact network. Using the analytical tractability of the replicator formalism, we study how network heterogeneity affects multi-strain dynamics, and compare its effects relative to the homogeneous contact distribution, identifying key relevant metrics for comparison. In particular, the pairwise invasion fitness matrix comparison reveals that higher network heterogeneity acts to increase the speed of multi-strain dynamics and typically tends to have stabilizing effects that reduce the number of coexisting strains. While many aspects of the replicator dynamics remain complex to study, especially for high number of strains, the advantage of this model representation lies in the dimensional reduction of a huge system, enabling general, more direct and efficient numerical computations. Furthermore the explicit bottom-up constitution of crucial parameters yields biological and epidemiological insight for critical system transitions across macroscopic gradients and can be used to guide interventions.

Foraging is a central decision-making behavior performed by all animals, essential to garnishing enough energy for an organism to survive. Similarly, mating is crucial for evolutionary continuity and offspring production. Mate choice is one of the central tenets of sexual selection, driving major evolutionary processes, and can be regarded as a decision-making process between potential mating partners. Often researchers have used coarse-grained models to describe macroscopic phenomenology pertaining to mate choice without detailed quantitative mechanisms of how animals use individual and environmental signals to guide their mating decisions. In this letter, we show that mate choice can be cast as a foraging problem, and we present an analytically tractable optimal foraging-inspired mechanistic theory of decision-making underlying mate choice. We begin from the premise that deciding upon which partner with which to mate is at its core a stochastic decision-making process. Agents adopt a variety of decision strategies, tuned by decision thresholds for leaving or committing to a mate. We find that sensitive leaving thresholds are favored independently of signal availability in the population. By contrast, optimal thresholds for committing to a mate depend upon signal availability in the population, with signal-rich populations generally favoring less eager strategies compared to signal-poor populations.

Organisms must manage a trade-off between robustness and flexibility as they enact adaptive behaviors. One way organisms achieve this is by navigating a network of quasi-stable behavioral states. Evidence for such behavioral states has been observed in many organisms, and new methods for detecting these states have taken on a prominent research focus. Although dynamical models demonstrating behavioral switching have been developed significantly over the past few decades, theories of the similarities and differences among these models, necessary for advancing empirical modeling, have not yet been fully elaborated. Here, we consider behavioral switching in two different classes of dynamical models of the forward-reversal behavioral transition in C. elegans. We first show how fundamentally different models can give rise to similar phenomena under noisy conditions. We then analyze the deterministic aspects of these models to expand on their differences, clarifying the theoretical relationship between them. Finally, we demonstrate how sequence models can be further extended to incorporate dwell times for behavioral states. Our work contributes toward a broader theoretical understanding of behavioral switching in adaptive systems.

Cancer-associated fibroblasts (CAFs) are a central component of the tumor microenvironment that facilitate a supportive niche for cancer progression and metastasis. Experimental evidence suggests that CAFs can facilitate estrogen-independent tumor growth, thereby reducing the efficacy of anti-hormonal therapies. Understanding and quantifying the complex interactions between tumor cells, hormonal signalling, and the microenvironment are crucial for designing more effective and individualized treatment strategies. We propose a mathematical framework to explore the influence of CAFs on ER+ breast cancer progression and to evaluate strategies to mitigate their impact. We develop a system of nonlinear ordinary differential equations that substantiates the experimental observations by providing a mechanistic basis for the role of CAFs in regulating estrogen-independent growth dynamics. We then employ optimal control theory to evaluate distinct therapeutic approaches involving monotherapy or combinations of: (i) inhibition of tumor-to-CAF signaling, (ii) inhibition of CAF-to-tumor proliferative signaling, and (iii) endocrine therapy. Taken together, our results demonstrate that CAF-targeted strategies can enhance treatment efficacy across various estrogen dosing regimens. Our study provides new insights into the potential of CAF as a therapeutic target that could help to improve existing approaches for endocrine therapies.

Movement is costly, and animals are under strong selective pressure to move efficiently, yet, in patchy, dynamic landscapes, decision-making is inherently uncertain. We quantify the energetic savings achieved by using up-to-date information presented within social cues for reducing movement costs. We use an agent-based model, founded on realistic aeronautical rules and parametrised on the Andean condor (Vultur gryphus), to study movement in patchy landscapes. By explicitly considering altitude, flight results in a sequence of soaring and gliding in the 3D space. We investigate how the cost of movement to an overall goal varies when birds use social information from others that are either fixed in space or moving collectively to the common goal, and under different risk-taking speed strategies, from slow and cautious to fast and risky. The value of social information is operationalised as energetic savings in units of basal metabolic rate. Under low predictability, agents with intermediate risk and high social-information use exhibit lowest movement costs, with up to 41% energy savings over asocial movement. By extending classical aeronautical theory to social and variable environments we demonstrate the adaptive value of social information for efficient movement in patchy, unpredictable landscapes.

Understanding how cells migrate through confined environments is crucial for elucidating fundamental biological processes, including cancer invasion, immune surveillance, and tissue morphogenesis. The nucleus, as the largest and stiffest cellular organelle, often limits cellular deformability, making it a key factor in migration through narrow pores or highly constrained spaces. In this work, we introduce a geometric surface partial differential equation (GS-PDE) model in which the cell plasma membrane and nuclear envelope are described as evolving energetic closed surfaces governed by force-balance equations. We replicate the results of a biophysical experiment, where a microfluidic device is used to impose compressive stresses on cells by driving them through narrow microchannels under a controlled pressure gradient. The model is validated by reproducing cell entry into the microchannels. A parametric sensitivity analysis highlights the dominant influence of specific parameters, whose accurate estimation is essential for faithfully capturing the experimental setup. We found that surface tension and confinement geometry emerge as key determinants of translocation efficiency. Although tailored to this specific setup for validation purposes, the framework is sufficiently general to be applied to a broad range of cell mechanics scenarios, providing a robust and flexible tool for investigating the interplay between cell mechanics and confinement. It also offers a solid foundation for future extensions integrating more complex biochemical processes such as active confined migration.

Cell-cell adhesion is a key regulator of cancer invasion. In this work, we extend a pre-existing individual based cancer invasion model by introducing a stochastic representation of N-cadherin-mediated adhesion, where the lifetime of a cell-cell bond depends on the pulling force acting on the bond. Using experimental data, we derive expressions for the mean and standard deviation of N-cadherin bond lifetimes and fit them to Gamma distributions, enabling their treatment as force-dependent random variables. These distributions are then used to modify the diffusion coefficient of mesenchymal cancer cells. The model predicts reduced random motility with increasing adhesion and incorporates a dynamic transition between catch- and slip-bond behaviour. Along with this model for cell motility, we propose a preliminary physical framework, that can be used to model pattern formation as a result of the new adhesion mechanic.

Infectious disease dynamics are strongly shaped by human mobility, social structure, and heterogeneous contact patterns, yet many epidemic models do not jointly capture these features. This study develops a spatial metapopulation epidemic model incorporating recurrent group-switch interactions to represent real-world transmission processes. Building on the Movement-Interaction-Return framework, the model integrates household structure, age-stratified contacts, and mobility between locations within a single SEIR framework. Using UK demographic, mobility, and social contact data, the model quantifies how within- and between-group interactions, mobility rates, and location connectivity influence epidemic spread. Both deterministic and stochastic simulations are implemented to analyse outbreak dynamics, variability, and fade-out probabilities for COVID-19-like and Ebola-like infections. Results shows that highly connected locations drive faster transmission, earlier epidemic peaks, and greater difficulty in containment, whereas larger but less connected locations tend to produce slower, more localised outbreaks despite their population size. Comparative analysis reveals that COVID-19-like infections spread rapidly and remain difficult to control even under interventions, while Ebola-like infections exhibit slower dynamics and are more effectively contained, particularly under targeted measures. Non-pharmaceutical interventions, particularly widespread closures, substantially reduce infections, hospitalisations, and deaths, although effectiveness depends on timing and pathogen characteristics. These findings highlight the importance of integrating mobility, clustering, and demographic heterogeneity to inform targeted and effective epidemic control strategies.

The synaptic cleft between neighboring neurons is the site of neurotransmitter-mediated communication that underlies normal brain function, including learning and memory. When an action potential reaches the presynaptic terminal, released neurotransmitters cross the cleft under the combined influence of diffusion and electrical forces to activate postsynaptic receptors. Despite this, synaptic-cleft transport is commonly modeled using a pure diffusion model, neglecting electrical drift. Here, we quantify the relative contributions of diffusion and electrical terms in the Poisson-Nernst-Planck (PNP) framework and assess whether the pure diffusion approximation is adequate. We solve the full PNP system in a three-dimensional computational model of the synaptic cleft at nanometer-scale resolution, tracking five ionic species (Na+, K+, Ca2+, Cl-, Glu-) with full spatial and temporal detail. Solutions are compared directly with those of the pure diffusion (D) model. The D and PNP models produce markedly different ionic concentration fields. Analysis of ionic fluxes confirms that diffusive and electrical contributions are of comparable magnitude across all species. These discrepancies are robust across parameter variations, including the number of AMPA receptors, the amount of released glutamate, the cleft height, and the cleft diffusion coefficient, and are amplified as the number of AMPA receptors increases, the cleft becomes narrower or diffusion more restricted. The quantitative and qualitative differences between the pure D model and the full PNP model demonstrate that neglecting electrical forces in the synaptic cleft has consequences. These discrepancies are large enough to alter the predicted dynamics and biological interpretation of synaptic transmission, establishing that accurate computation of ionic concentrations in the synaptic cleft requires the full PNP equations.

It has been broadly recognized that the crosstalk between cells and their extracellular matrix (ECM) is crucial for the proper function of biological tissues. Relatively recently the role of ECM came in focus in the context of neuronal development and regeneration, where the effects of the ECM mechanics on the migration of neurons and neurite growth are still incompletely understood. Here we present an in silico twin framework for neurite growth focusing on its biophysical interactions with the ECM. This coarsegrained model accounts for viscoelastic liquid- and solid-like ECMs and neurite growth by ECM-mediated traction forces. Resulting growth trajectories can be rationalized based on the theory of random walks and polymer physics. To critically assess models predictive power, we performed experiments on neurites of hippocampal rat neurons growing in 3D collagen gels and observed a more persistent axon outgrowth in denser matricies. The model fully recapitulated the effect, thereby underpinning the central role of mechanical interactions with ECM as guiding principle of axonal growth. We argue that a combination our model with optical microscopy may provide an is silico twin helping to disentangle the contributions of "passive" physics from more complex effects of chemical queues or an apparent mechanosensing.