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

Alzheimers disease (AD) is characterized by the accumulation of amyloid-{beta} (A{beta}), yet the specific link between plaque burden and cognitive decline remains a subject of intense investigation. This paper presents a mathematical model that simulates the coupled dynamics of A{beta} monomers, soluble oligomers, and fibrillar species in the brain tissue. By modifying existing moment equations to include a dedicated conservation equation for A{beta} monomers, the model explores how various microscopic processes, such as primary nucleation, surface-catalyzed secondary nucleation, fibril elongation, and fragmentation, contribute to macroscopic disease progression. Central to this study is the concept of "accumulated neurotoxicity" as a surrogate marker of biological age, defined as the time-integrated concentration of soluble A{beta} oligomers. Unlike plaque burden, accumulated neurotoxicity cannot be reversed, and the harm it causes depends critically on the sequence of events that produced it. Numerical results demonstrate that while plaque burden and neurotoxicity both increase over time, their relationship is non-linear and highly sensitive to the efficiency of protein degradation machinery. Specifically, impaired degradation leads to a rapid advancement of biological age relative to calendar age. The model further identifies oligomer dissociation and fibril fragmentation as potential protective mechanisms that can counterintuitively reduce neurotoxic burden by diverting monomers away from the soluble oligomer pool. These findings provide a quantitative framework for understanding why individuals with similar plaque burdens may experience vastly different cognitive outcomes, underscoring the importance of targeting soluble oligomers early in therapeutic interventions.

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.

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.

AO_SCPLOWBSTRACTC_SCPLOWOncolytic viruses provide cancer therapy using replication-competent viruses that selectively infect and lyse tumour cells. Their tumour-specific replication also enables the delivery of targeted, virus-encoded gene products, such as enzymes that activate prodrugs. This dual functionality offers the potential for synergistic effects by combining direct oncolysis with localised drug activation. The interplay between infection, replication, lysis, and gene product delivery remains poorly understood. Here, we introduce a spatially structured, multi-patch model of cancer cells infected by an oncolytic virus engineered to deliver a prodrug-activating enzyme. The spatial system is first represented as a microscopic model and subsequently reduced via spectral dimension reduction techniques. This reduction yields an ordinary differential equation model for a set of coarse-grained variables, which we analyze both without the transgene (OV model) and with the transgene (TOV model). For each case, we compute the basic reproduction number, R0, which determines the conditions for viral spread. Both models exhibit three regimes via transcritical bifurcations: (i) R0 < 0, extinction of both cancer and infected cells; (ii) 0 < R0 [&le;] 1, persistence of cancer cells only; and (iii) R0 > 1, coexistence as a stable node or as a focus. The TOV model, as a difference form the OV model, can undergo periodic oscillations arising from a Hopf-Andronov bifurcation. Notably, oscillation amplitudes can be controlled such that cancer cells largely decrease when drug-induced death is stronger in non-infected cells than in infected ones, enabling effective cancer cells killing while maintaining viral replication and prodrug activation. The qualitative behaviour of the coarse-grained model is shown to be preserved in both the microscopic and spatially explicit models.

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.

The Poisson-Nernst-Planck (PNP) system is an accurate model of electrodiffusion of ionic species. It is commonly used in situations where nanoscale resolution is required, for instance close to ion channels in the membranes of biological cells. The inherent stiffness of the equations has made them challenging to solve and has limited the applicability of the system. In particular, the time step required for stable solutions has typically needed to be very short (nanoseconds), which makes simulations on the time scale of an action potential (milliseconds) difficult. Recently, it has been observed that avoiding operator splitting and instead solving the concentration equations and the electrostatic equation in a coupled manner relaxes the time-step limitation considerably. However, no theoretical explanation of this observation has been provided. Here, we aim to explain why the coupled scheme allows much larger time steps. We illustrate the mechanism by considering special cases that define necessary, but not sufficient, conditions for stability. We also show that these conditions remain relevant for the fully coupled PNP model in 3D.

Functional spinal cord repair in zebrafish is governed by regeneration-favorable biochemical and mechanical cues within the lesion microenvironment. Alterations in extracellular matrix composition and stiffness are closely associated with axon regeneration. However, experimentally dissecting the interplay between mechanical signals and axonal regrowth in vivo remains technically challenging. Here, we present an agent-based modeling framework to simulate stiffness-mediated axonal growth trajectories across the lesion. We use this model to explore potential mechanisms underlying the characteristic growth patterns observed during zebrafish spinal cord regeneration. Computational predictions were qualitatively compared with confocal imaging data obtained from larval zebrafish. These phenomenological comparisons revealed a close agreement between simulated and experimentally observed axon growth, indicating that experimentally observed patterns could be governed by transient changes in the stiffness profile of the spinal cord and lesion microenvironment. Hence, our computational framework provides an in silico platform for investigating the role of mechanical cues in axon regeneration in the injured spinal cord.

Variability in gene expression among single cells and growing cell populations can arise from the stochastic nature of protein synthesis, which often occurs in random bursts. This study investigates the variability in the expression of a growth-sustaining protein, whose concentration is regulated by a negative feedback loop due to cell growth-induced dilution. We model the distribution of protein concentration using a Chapman-Kolmogorov equation for single cells and a population balance equation for growing cell populations. For single cells, we derive an explicit solution for the protein concentration distribution in state space and represent it as a Bessel function in Laplace space. For growing populations, we find that the distribution satisfies a Heun differential equation with singular boundary conditions. By addressing the central connection problem for the Heun equation, we quantify the population-level protein distribution and determine the Mathusian parameter, which characterizes population growth. This work provides a comprehensive analytical framework for understanding how stochastic protein synthesis impacts gene expression variability and population dynamics.

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.

Studies on active and sedated fish passing through turbines and pumps show different mortality and injury rates for both cases. Consequently, fish behavior appears to play a substantial role in these outcomes. However, direct behavioral observations in hydraulic machines using quantitative parameters to draw conclusions about the underlying mechanisms are hardly possible and remain understudied. In this study, we examined the behavior of adult brown trout (Salmo trutta) in an experimental flume under hydraulic conditions characterized by strong flow acceleration and high velocities typical of turbine and pump intakes. Fish movement behavior was analyzed based on a quantitative approach to enable the analysis of swimming behavior even in flow velocities exceeding the sprint swimming speed of fish. The application of Hidden Markov Models (HMM) to analyze activity states and movement modes of fish from video tracking data demonstrated significant effects of the spatial velocity gradient (SVG) and flow velocity on fish behavior. Notably, SVG emerged as the primary trigger for avoidance reactions when exceeding a threshold of [Formula]. Fish exhibited distinct movement patterns under dark and daylight conditions, with more avoidance reactions in darkness. Whereas a considerable proportion of fish in daylight increased their swimming activity in the zone were flow velocity exceeded sprint swimming speed, in dark conditions no activity peak occurred in the same zone. The results illustrate how hydraulic conditions and lighting influence fish behavior. Integrating the behavioral rules identified in this study into numerical mortality-risk models could substantially improve their predictive accuracy. Thus, the findings allow for the development of less fish harming engineering solutions for hydropower facilities and pumping stations.

As the duration of space flights increases, so does the need to optimize off-planet microbial growth. Microbes can both be unintentionally brought into space and cause human disease or be intentionally harnessed for on-site bioengineering functions. However, optimizing microbial growth is challenging due to an insufficient understanding of how microbial communities are affected by the extraterrestrial environment. To address this gap, we have modified a previously developed model for cell growth in microgravity. By improving the functional form used for cell growth as well as the code usability, we enable further research into how microbial communities are influenced by gravity. Applying this model to isolate individual effects of gravity on cell growth indicates that a lack of gravity-driven flow decreases cell growth in microgravity, while the absence of sedimentation increases cell growth in microgravity. These opposite effects likely contribute to the system-dependent effects of microgravity observed experimentally.

Finite element (FE) head models are valuable tools for investigating brain injury mechanics, with their reliability critically dependent on accurate material modelling. White matter (WM) is often considered mechanically anisotropic due to its aligned axonal fiber architecture and is commonly represented using fiber-reinforced hyperelastic formulations such as the Gasser-Ogden-Holzapfel (GOH) model. A fundamental assumption of the GOH model is that fibers contribute only in tension and not in compression, requiring the use of tension-compression switches. However, inconsistencies were noted in the formulation of tension-compression switches with the influence on computational biomechanics unknown. To address this knowledge gap, three commonly used switching schemes - differing in both the switching parameter and the treatment of compressed fibers - were theoretically elaborated and numerical implementation within the GOH framework to simulate the mechanical anisotropy of WM in impact simulations. Results from the case-based and group-level analyses demonstrated that both the switching parameter and the treatment of compressed fibers affected WM deformation. Significant cross-scheme strain differences were noted in the first principal strain at the element level and fiber strain at the fiber level. These findings highlighted the mechanical role of tension-compression switch in the GOH-based brain modelling and advocated the adoption of fiber stretch itself as the switching parameter to discriminate the tensile and compressive fibers. The current study provides important guidance for the anisotropic constitutive models in brain tissue and calls for direct verification of the tension-compression switch hypothesis in axonal fibers.

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, in which 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 to faithfully capture 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. Author summaryCells often migrate through very narrow spaces in tissues, a process critical for cancer invasion, immune surveillance, and tissue development. In particular, the stiffness of the nucleus, the largest and most rigid organelle, can limit migration through tight pores. In this study, we present a mathematical model describing the motion of a cell and its nucleus through a microchannel during cell translocation, using a geometric formulation based on surface partial differential equations. The model is general and applicable to a variety of scenarios involving confined cell transport. The model is validated by reproducing key experiments on cell translocation through narrow microchannels. The framework incorporates essential surface features, including mechanical responses, bending rigidity, and surface tension. Sensitivity analysis highlights surface tension and channel geometry as the parameters that most strongly influence translocation. Overall, the model provides new insights into the mechanics of confined cell transport, grants access to cellular quantities that are difficult to measure experimentally, such as cell and nucleus areas, perimeters, and stresses, and establishes a foundation for future extensions incorporating more complex biochemical processes.

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.

Stratified epithelial tissues such as the skin epidermis maintain barrier integrity during development and homeostasis through the coordinated action of cell proliferation, differentiation, delamination, and tissue-scale mechanical forces. During development, the orientation of cell division within the basal layer plays a pivotal role in tissue stratification; however, the mechanical principles linking the orientation of the division plane to these processes across developmental stages remain poorly understood. Here, we expand a recently developed three-dimensional vertex model for stratified epithelia, composed of the basement membrane, basal, and suprabasal layers, to study the mechanical and structural impact of cell divisions with a wider range of orientations. The model integrates developmental stage via specific changes in heterotypic interfacial tensions (arising from actomyosin cortical contractility and adhesion molecules at the basal-suprabasal interface) and tissue stiffness that have been quantified previously in experiments. By systematically varying background mechanical parameters, we investigate how heterotypic tension, division orientation, and tissue fluidity collectively influence the outcome of cell division. Our goal is to uncover the strategies that the embryo may employ to generate stratified phenotypes at different developmental stages, recognizing that these strategies might evolve over time. Although our focus is on the embryonic developmental stages of the epidermis, this framework may also be extended to investigate transformed cells, such as in cancer, to explore how altered division orientation contributes to precancerous or transformed phenotypes.

Cells communicate via extracellular ligands, such as hormones, which bind to plasma membrane receptors and trigger intracellular signaling cascades. G Protein-Coupled Receptors (GPCRs) exemplify this mechanism by initiating signaling both at the cell surface and, from intracellular compartments such as endosomes. The kinetics and spatial localization of these signals are critical determinants of cellular responses, yet receptor trafficking-including internalization, endosomal sorting, and recycling-remains a pivotal but often overlooked component of theoretical GPCR models. In this study, we present a mathematical framework that integrates receptor trafficking and signaling compartmentalization into generic GPCR dynamic models. Using a compartmentalized approach based on systems of ordinary differential equations (Chemical Reaction Networks), we analyze how receptor internalization and recycling modulate ligand-induced responses. Our results show that the balance between plasma membrane and endosomal signaling can significantly enhance or diminish ligand efficacy. Calibrated with high-throughput kinetic data, our model offers a refined tool for ligand pharmacological characterization and advances the understanding of GPCR signaling spatial organization.

Bacterial communities grow as dynamic populations that respond to their environments. A clinically relevant example is the inactivation of beta-lactam antibiotics by intracellular beta-lactamase in E. faecalis resistant strains. In these populations, resistant bacteria act as antibiotic sinks, detoxifying the environment and allowing sensitive bacteria to survive treatment through a cooperative interaction. In this work, we study strongly coupled planktonic and biofilm populations of mixed sensitive-resistant E. faecalis bacteria under antibiotic stress using fluorescent microscopy. The presence of resistant bacteria in the system benefits both resistant and sensitive cells, leading to mixed planktonic and biofilm populations at super-inhibitory drug concentrations. We show that a beta-lactam antibiotic with or without the addition of a beta-lactam inhibitor can lead to a population inversion effect, characterized by a non-monotonic relation between initial and final fractions of resistant bacteria. The effect is observed in both the planktonic and biofilm populations and is modulated by the total initial cell density. A well-mixed model with competition mediated by resource sharing and cooperation from global degradation of toxins predicts the experimentally observed behavior. These observations suggest underlying population-level mechanisms that are largely independent of biofilm spatial structure.