arXiv subjects

Benjamin J. Walker

Publications and source records attributed to Benjamin J. Walker.

At least 19 recordsLinked to original sources

arXiv2026

Mathematical modelling of immune persistence and relapse pathways in CAR T-cell therapy for B-ALL

Chimeric antigen receptor (CAR) T-cell therapy has transformed the treatment of B-cell acute lymphoblastic leukaemia (B-ALL). Despite high initial response rates, a substantial fraction of patients relapse, often due to loss of CAR T-cell persistence, antigen escape, or immune-privileged sites that shield tumour cells. Prolonged CAR T-cell persistence is clinically associated with durable remission, but why it is required remains poorly understood. To address this, we develop and analyse the BEAM (Blast, Effector, Activated, Memory) model of CAR T-cell dynamics in B-ALL. BEAM extends predator--prey models with three CAR T-cell states (memory, activated, effector) coupled to a logistic growth equation for the blasts, calibrated against the FELIX trial of obecabtagene autoleucel in adult B-ALL. We find that both memory and effector persistence prevent relapse, but for distinct reasons: memory persistence sustains surveillance against low-burden or slowly proliferating residual disease, while effector persistence clears isolated blasts emerging from immune-privileged sites. The model further predicts a trade-off between immediate cytotoxicity and durable surveillance, and identifies initial tumour burden as a key modifiable factor for reducing antigen-negative relapse. Together, these results offer a framework for designing more durable, individually tailored CAR T-cell therapies.

q-bio.PE↗

arXiv2026

Bacterial adhesion to curved surfaces in fluid flow

Minimising bacterial surface adhesion and subsequent biofilm formation in industrial and medical settings requires understanding how bacteria are transported and adhere to complex surface geometries in the presence of non-uniform flow. In this paper, we consider the transport of a dilute suspension of motile bacteria through a corrugated two-dimensional channel with perfectly adhesive walls. We asymptotically analyse the diffusive boundary layer that forms in high velocity flows using a curvilinear coordinate system based on the fluid streamfunction, presenting a similarity solution to the diffusivity-varying diffusion-type equation that arises. From this solution, we derive an analytical expression for the bacterial adhesion rate as a function of surface arclength and the spatially varying wall shear rate. Our model predicts that bacterial adhesion becomes localised on curved surfaces, with bacteria showing preferential adhesion to wall `peaks' at lower shear rates and preferential adhesion to wall `valleys' at higher shear rates. More broadly, our results highlight how spatially varying flows generated by complex geometries can lead to localised bacterial adhesion, with potential implications for both enhancing and minimising biofilm formation.

cond-mat.soft↗

arXiv2026

Far-field approximations for multi-timescale microswimmers near a boundary

Hydrodynamic interactions with boundaries can significantly affect the trajectories of microscale swimmers. In simple swimmer models, a common assumption is that swimmer shape remains constant, essentially averaging over the rapid oscillations in geometry and associated fluid flows that often are the source of propulsion. Previous work in minimal force-dipole models has shown how the inclusion of time-dependent swimmer changes can lead to a fundamentally wider class of behaviours than for their classic (implicitly averaged) counterparts. However, since force dipole models correspond to the leading-order term in the far-field description of the swimmer-induced flow, they break down as the swimmer approaches a boundary and predictions can become qualitatively inaccurate. Here, we extend the minimal force-dipole model by incorporating higher order flow singularities, systematically accounting for rapid oscillations in shape and singularity strength through a multiscale analysis. We demonstrate that the inclusion of time-dependence into these higher order models significantly expands the reachable parameter space, in particular by increasing its dimensionality. In these extended dynamics, we observe three distinct behaviours: crashing, escaping and hovering. Notably, hovering states are absent from the dynamics predicted by the simplest models, but are observed in more complex models.

physics.flu-dyn↗

arXiv2026

Chemotaxis of cell aggregates: morphology and dynamics of migrating active droplets

Biological tissues have been observed to display emergent fluid-like properties, owing to physical interactions between cells. However, it remains unclear in general how these fluid-like properties affect tissue structure and function. Here, we are motivated by recent experiments in which cell aggregates were observed to behave as active droplets during collective migration along chemical gradients, or chemotaxis. To understand this process, we develop a minimal model of a growing thin active droplet driven by a self-generated chemical gradient. In broad agreement with the experiments, dynamic simulations reveal that chemotacting droplets exhibit proliferation-driven morphological transitions. To fully characterise these transitions, we perform a multiple scales analysis to show that the droplet dynamics follow a sequence of travelling wave solutions defined by a nonlinear eigenvalue problem parametrised by the slowly increasing droplet volume. Our analysis reveals that morphological transitions can occur continuously or through a discontinuous bifurcation. Further asymptotic analysis of the travelling wave problem reveals that these morphological transitions arise from exponentially small ("beyond-all-orders") asymptotic terms that originate from the rear and front contact lines. Moreover, we show that the nature of the transitions is fully determined by two key dimensionless parameters, which quantify the internal stress balance within the droplet and the strength of the coupling between the droplet migration dynamics and the external chemical field. Overall, our results provide a complete characterisation of the morphodynamics of a class of migrating active thin droplets, with implications in a range of biological systems where cell aggregates exhibit fluid-like behaviour.

cond-mat.soft↗

arXiv2026

Tunable asymmetric swimming in biflagellate microswimmers

Many biological microswimmers can modulate their swimming gait to achieve directional control of motility, especially when performing steering towards specific directional cues. This can be achieved without the need for obvious morphological or structural asymmetries in the form of the organism, or in the number or organisation of propulsion-generating appendages such as cilia. In this work, we identify and validate a core principle of asymmetric turning in biflagellate microswimmers: propulsive forces interact constructively to drive translation whilst interacting destructively to drive rotation. We explore the ramifications of this tunable biflagellar swimming mechanism across a range of systems, from a simple, back-of-the-envelope model to a detailed computational representation of an exemplar swimmer. This leads to a markedly general quantitative relation between key drivers of asymmetry, such as ciliary beat frequency, and the curvature of emergent trajectories. We discuss how the model green alga Chlamydomonas reinhardtii, which actuates its two cilia in a symmetric breaststroke for forward swimming, may exploit this feature for phototaxis. Finally, we validate our predictions in a Chlamydomonas-inspired robophysical model, implementing closed-loop control to achieve phototactic turning.

physics.bio-ph↗

arXiv2025

Generative AI performance in core undergraduate mathematics: a curriculum-level case study

Generative artificial intelligence (GenAI) tools such as OpenAI's ChatGPT are transforming the educational landscape, prompting reconsideration of traditional assessment practices. In parallel, universities are exploring alternatives to in-person, closed-book examinations, raising concerns about academic integrity and pedagogical alignment in uninvigilated settings. This study systematically investigates the performance of GenAI on typical mathematics questions from across a first-year mathematics curriculum. Adopting an empirical approach and utilising current examination questions as a proxy for course content, we generate, transcribe, and blind-mark GenAI submissions to eight undergraduate mathematics assessments, spanning the entirety of the first-year curriculum. By combining independent GenAI responses to individual questions, we enable a meaningful evaluation of GenAI performance, both at the level of modules and across the first-year curriculum. We find that GenAI attainment is at the level of a first-class degree, though current performance can vary between modules. Further, we find that GenAI performance is remarkably consistent when viewed across the entire curriculum, significantly more so than that of students in invigilated examinations. Our findings evidence the pressing need for redesigning assessments in mathematics in the era of generative artificial intelligence.

cs.CY↗

arXiv2025

A shear-induced limit on bacterial surface adhesion in fluid flow

Controlling bacterial surface adhesion and subsequent biofilm formation in fluid systems is crucial for the safety and efficacy of medical and industrial processes. Here, we theoretically examine the transport of bacteria close to surfaces, isolating how the key processes of bacterial motility and fluid flow interact and alter surface adhesion. We exploit the disparity between the fluid velocity and the swimming velocity of common motile bacteria and, using a hybrid asymptotic-computational approach, we systematically derive the coarse-grained bacterial diffusivity close to surfaces as a function of swimming speed, rotational diffusivity, and shape. We calculate an analytical upper bound for the bacterial adhesion rate by considering the scenario in which bacteria adhere irreversibly to the surface on first contact. Our theory predicts that maximal adhesion occurs at intermediate flow rates: at low flow rates, increasing flow increases surface adhesion, while at higher flow rates, adhesion is decreased by shear-induced cell reorientation.

cond-mat.soft↗

arXiv2025

Motility and rotation of multi-timescale microswimmers in linear background flows

Microswimming cells and robots exhibit diverse behaviours due to both their swimming and their environment. One of the core environmental features impacting inertialess swimming is background flows. While the influence of select flows, particularly shear flows, have been extensively investigated, these are special cases. Here, we examine inertialess swimmers in more general flows, specifically general linear planar flows that may also possess rapid oscillations. Relatively weak symmetry constraints are imposed on the swimmer to ensure planarity and to reduce complexity. A further constraint reflecting common observation is imposed, namely that the swimmer is inefficient, which we suitably define. This introduces two separate timescales: a fast timescale associated with swimmer actuation, and a second timescale associated with net swimmer movement, with inefficiency dictating that this latter timescale is much slower, allowing for a multiple timescale simplification of the governing equations. With the exception of mathematically precise edge cases, we find that the behaviour of the swimmer is dictated by two parameter groupings, both of which measure balances between the angular velocity and rate of strain of the background flow. While the measures of flow angular velocity and strain rates that primarily govern the rotational dynamics are modulated by swimmer properties, the primary features of the translational motion are determined solely by a ratio of flow angular velocity to strain rate. Hence, a simple classification of the swimmer dynamics emerges. For example, this illustrates the limited extent to which, and how, microswimmers may control their orientations and trajectories in flows.

physics.flu-dyn↗

arXiv2024

On discretely structured growth models and their moments

The logistic equation is ubiquitous in applied mathematics as a minimal model of saturating growth. Here, we examine a broad generalisation of the logistic growth model to discretely structured populations, motivated by examples that range from the ageing of individuals in a species to immune cell exhaustion by cancerous tissue. Through exploration of a range of concrete examples and a general analysis of polynomial kinetics, we derive necessary and sufficient conditions for the dependence of the kinetics on structure to result in closed, low-dimensional moment equations that are exact. Further, we showcase how coarse-grained moment information can be used to elucidate the details of structured dynamics, with immediate potential for model selection and hypothesis testing. This paper belongs to the special collection: Problems, Progress and Perspectives in Mathematical and Computational Biology.

q-bio.PE↗

arXiv2024

Minimal design of a synthetic cilium

We study a slender filament beating in a viscous fluid with novel curvature-dependent bending stiffness. Our numerical and experimental investigations reveal that such differential stiffness can sustain planar bending waves far along flexible filaments, in stark contrast to the uniform-stiffness case which requires more sophisticated control. In particular, we establish basal actuation as a viable, parsimonious mechanism for generating high-amplitude planar bending waves. Moreover, the resulting beat patterns closely resemble the power-and-recovery strokes of propulsive biological filaments such as cilia, suggesting extensive applications in robotic and engineered systems.

cond-mat.soft↗

arXiv2023

Turing instabilities are not enough to ensure pattern formation

Symmetry-breaking instabilities play an important role in understanding the mechanisms underlying the diversity of patterns observed in nature, such as in Turing's reaction--diffusion theory, which connects cellular signalling and transport with the development of growth and form. Extensive literature focuses on the linear stability analysis of homogeneous equilibria in these systems, culminating in a set of conditions for transport-driven instabilities that are commonly presumed to initiate self-organisation. We demonstrate that a selection of simple, canonical transport models with only mild multistable non-linearities can satisfy the Turing instability conditions while also robustly exhibiting only transient patterns. Hence, a Turing-like instability is insufficient for the existence of a patterned state. \ak{While it is known that linear theory can fail to predict the formation of patterns, we demonstrate that such failures can appear robustly in systems with multiple stable homogeneous equilibria.} Given that biological systems \ak{such as} gene regulatory networks and spatially distributed ecosystems often exhibit a high degree of multistability and nonlinearity, this raises important questions of how to analyse prospective mechanisms for self-organisation.

nlin.PS↗

arXiv2023

Generalised Jeffery's equations for rapidly spinning particles. Part 2: Helicoidal objects with chirality

In this two-part study, we investigate the motion of rigid, active objects in shear Stokes flow, focusing on bodies that induce rapid rotation as part of their activity. In Part 2, we derive and analyse governing equations for rapidly spinning complex-shaped particles - general helicoidal objects with chirality. Using the multiscale framework we develop in Part 1 (arXiv:2301.11311), we systematically derive emergent equations of motion for the angular and translational dynamics of these chiral spinning objects. We show that the emergent dynamics due to rapid rotation can be described by effective generalised Jeffery's equations, which differ from the classic versions via the inclusion of additional terms that account for chirality and other asymmetries. Furthermore, we use our analytic results to characterise and quantify the explicit effect of rotation on the effective hydrodynamic shape of the chiral objects, significantly expanding the scope of Jeffery's seminal study.

physics.flu-dyn↗

arXiv2023

Generalised Jeffery's equations for rapidly spinning particles. Part 1: Spheroids

The observed behaviour of passive objects in simple flows can be surprisingly intricate, and is complicated further by object activity. Inspired by the motility of bacterial swimmers, in this two-part study we examine the three-dimensional motion of rigid active particles in shear Stokes flow, focusing on bodies that induce rapid rotation as part of their activity. Here, in Part 1, we develop a multiscale framework to investigate these emergent dynamics and apply it to simple spheroidal objects. In Part 2 (arXiv:2301.11032), we apply our framework to understand the emergent dynamics of more complex shapes; helicoidal objects with chirality. Via a multiple-scales asymptotic analysis for nonlinear systems, we systematically derive emergent equations of motion for long-term trajectories that explicitly account for the strong (leading-order) effects of fast spinning. Supported by numerical examples, we constructively link these effective dynamics to the well-known Jeffery's orbits for passive spheroids, deriving an explicit closed-form expression for the effective shape of the active particle, broadening the scope of Jeffery's seminal study to spinning spheroids.

physics.flu-dyn↗

arXiv2023

VisualPDE: rapid interactive simulations of partial differential equations

Computing has revolutionised the study of complex nonlinear systems, both by allowing us to solve previously intractable models and through the ability to visualise solutions in different ways. Using ubiquitous computing infrastructure, we provide a means to go one step further in using computers to understand complex models through instantaneous and interactive exploration. This ubiquitous infrastructure has enormous potential in education, outreach and research. Here, we present VisualPDE, an online, interactive solver for a broad class of 1D and 2D partial differential equation (PDE) systems. Abstract dynamical systems concepts such as symmetry-breaking instabilities, subcritical bifurcations and the role of initial data in multistable nonlinear models become much more intuitive when you can play with these models yourself, and immediately answer questions about how the system responds to changes in parameters, initial conditions, boundary conditions or even spatiotemporal forcing. Importantly, VisualPDE is freely available, open source and highly customisable. We give several examples in teaching, research and knowledge exchange, providing high-level discussions of how it may be employed in different settings. This includes designing web-based course materials structured around interactive simulations, or easily crafting specific simulations that can be shared with students or collaborators via a simple URL. We envisage VisualPDE becoming an invaluable resource for teaching and research in mathematical biology and beyond. We also hope that it inspires other efforts to make mathematics more interactive and accessible.

physics.ed-ph↗

arXiv2023

Optimal face-to-face coupling for fast self-folding kirigami

Kirigami-inspired designs can enable self-folding three-dimensional materials from flat, two-dimensional sheets. Hierarchical designs of connected levels increase the diversity of possible target structures, yet they can lead to longer folding times in the presence of fluctuations. Here, we study the effect of rotational coupling between levels on the self-folding of two-level kirigami designs driven by thermal noise in a fluid. Naturally present due to hydrodynamic resistance, we find that optimization of this coupling as control parameter can significantly improve a structure's self-folding rate and yield.

cond-mat.soft↗

arXiv2023

Viscous tubular-body theory for plane interfaces

Filaments are ubiquitous within the microscopic world. They occur frequently in both biological and industrial environments and display varied and rich dynamics. Their wide range of applications has spurred the development of a special branch of asymptotics focused on the behaviour of filaments, called slender-body theory. Slender-body theories are typically computationally efficient and focus on the mechanics of an isolated fibre that is not too curved. However, slender-body theories that work beyond these standard limits are needed to explore more complex systems. Recently, we developed tubular-body theory for slow viscous flows, an approach similar to slender-body theory that allows the hydrodynamic traction on any isolated cable-like body in a highly viscous fluid to be determined exactly. In this paper, we extend tubular-body theory to model filaments near plane interfaces by performing an similar expansion on the single-layer boundary integral equations for bodies by a plane interface. In the derivation of the new theory, called tubular-body theory for interfaces, we established a criteria for the convergence of the tubular-body theory series representation, before comparing the result to boundary integral simulations for a prolate spheroid by a wall. The tubular-body theory for interfaces simulations are found to capture the lubrication effects when close to the plane wall. Finally we simulate the hydrodynamics of a helix beneath a free interface and a plane wall to demonstrate the broad applicability of the technique.

physics.flu-dyn↗

arXiv2023

A hydrodynamic slender-body theory for local rotation at zero Reynolds number

Slender objects are commonplace in microscale flow problems, from soft deformable sensors to biological filaments such as flagella and cilia. Whilst much research has focussed on the local translational motion of these slender bodies, relatively little attention has been given to local rotation, even though it can be the dominant component of motion. In this study, we explore a classically motivated ansatz for the Stokes flow around a rotating slender body via superposed rotlet singularities, which leads us to pose an alternative ansatz that accounts for both translation and rotation. Through an asymptotic analysis that is supported by numerical examples, we determine the suitability of these flow ansatzes for capturing the fluid velocity at the surface of a slender body, assuming local axisymmetry of the object but allowing the cross-sectional radius to vary with arclength. In addition to formally justifying the presented slender-body ansatzes, this analysis reveals a markedly simple relation between the local angular velocity and the torque exerted on the body, which we term resistive torque theory. Though reminiscent of classical resistive force theories, this local relation is found to be algebraically accurate in the slender-body aspect ratio, even when translation is present, and is valid and required whenever local rotation contributes to the surface velocity at leading asymptotic order.

physics.flu-dyn↗

arXiv2023

Concentration-Dependent Domain Evolution in Reaction-Diffusion Systems

Pattern formation has been extensively studied in the context of evolving (time-dependent) domains in recent years, with domain growth implicated in ameliorating problems of pattern robustness and selection, in addition to more realistic modelling in developmental biology. Most work to date has considered prescribed domains evolving as given functions of time, but not the scenario of concentration-dependent dynamics, which is also highly relevant in a developmental setting. Here, we study such concentration-dependent domain evolution for reaction-diffusion systems to elucidate fundamental aspects of these more complex models. We pose a general form of one-dimensional domain evolution, and extend this to $N$-dimensional manifolds under mild constitutive assumptions in lieu of developing a full tissue-mechanical model. In the 1D case, we are able to extend linear stability analysis around homogeneous equilibria, though this is of limited utility in understanding complex pattern dynamics in fast growth regimes. We numerically demonstrate a variety of dynamical behaviours in 1D and 2D planar geometries, giving rise to several new phenomena, especially near regimes of critical bifurcation boundaries such as peak-splitting instabilities. For sufficiently fast growth and contraction, concentration-dependence can have an enormous impact on the nonlinear dynamics of the system both qualitatively and quantitatively. We highlight crucial differences between 1D evolution and higher dimensional models, explaining obstructions for linear analysis and underscoring the importance of careful constitutive choices in defining domain evolution in higher dimensions. We raise important questions in the modelling and analysis of biological systems, in addition to numerous mathematical questions that appear tractable in the one-dimensional setting, but are vastly more difficult for higher-dimensional models.

nlin.PS↗