arXiv subjects

Johannes G. Borgqvist

Publications and source records attributed to Johannes G. Borgqvist.

6 recordsLinked to original sources

arXiv2026

Analytical characterisation of the Mi- and To-phases in HeMiTo dynamics: exponential growth and logistic saturation of toxic prion-like proteins

Prion-like propagation of misfolded proteins is a key mechanism underlying the progression of neurodegenerative diseases such as Alzheimer's disease. In previous work, we introduced the HeMiTo framework, describing these prion-like dynamics for a class of heterodimer models in terms of three phases: the healthy (He), mixed (Mi), and toxic (To) phases. While the He-phase was characterised analytically, the Mi-phase was described numerically and the To-phase was inferred from linear stability arguments. In this work, we provide a complete analytical characterisation of the Mi- and To-phases for our class of heterodimer models. We derive exact inner solutions governing the Mi-phase and match them with outer solutions from the He-phase, explaining the concave-like behaviour of the healthy species and establishing explicit conditions for exponential growth of the toxic species with a mechanistically interpretable growth rate. Furthermore, we formalise a quasi steady-state reduction near the toxic steady state and show that the dynamics reduce to a logistic growth equation, linking exponential growth to saturation. Together, these results provide a unified and mechanistic description of prion-like dynamics across all phases of disease progression and establish a foundation for predictive modelling of biomarker trajectories.

q-bio.QM↗

arXiv2026

Framing local structural identifiability and observability in terms of parameter-state symmetries

We introduce a subclass of Lie symmetries, called parameter-state symmetries, to analyse the local structural identifiability and observability of mechanistic models consisting of state-dependent ODEs with observed outputs. These symmetries act on parameters and states while preserving observed outputs at every time point. We prove that locally structurally identifiable parameter combinations and locally structurally observable states correspond to universal invariants of all parameter-state symmetries of a given model. We illustrate the framework on four previously studied mechanistic models, confirming known identifiability results and revealing novel insights into which states are observable, providing a unified symmetry-based approach for analysing structural properties of dynamical systems.

math.DS↗

arXiv2024

HeMiTo-dynamics: a characterisation of mammalian prion toxicity using non-dimensionalisation, linear stability and perturbation analyses

Prion-like proteins play crucial parts in biological processes in organisms ranging from yeast to humans. For instance, many neurodegenerative diseases are believed to be caused by the production of prion-like proteins in neural tissue. As such, understanding the dynamics of prion-like protein production is a vital step toward treating neurodegenerative disease. Mathematical models of prion-like protein dynamics show great promise as a tool for predicting disease trajectories and devising better treatment strategies for prion-related diseases. Herein, we investigate a generic model for prion-like dynamics consisting of a class of non-linear ordinary differential equations (ODEs), establishing constraints through a linear stability analysis that enforce the expected properties of mammalian prion-like toxicity. Furthermore, we identify that prion toxicity evolves through three distinct phases for which we provide analytical descriptions using perturbation analyses. Specifically, prion-toxicity is initially characterised by the healthy phase, where the dynamics are dominated by the healthy form of prions, thereafter the system enters the mixed phase, where both healthy and toxic prions interact, and lastly, the system enters the toxic phase, where toxic prions dominate, and we refer to these phases as HeMiTo-dynamics. These findings hold the potential to aid researchers in developing precise mathematical models for prion-like dynamics, enabling them to better understand underlying mechanisms and devise effective treatments for prion-related diseases.

q-bio.QM↗

arXiv2023

Construction of integrable generalised travelling wave models and analytical solutions using Lie symmetries

Certain solutions of autonomous PDEs without any boundary conditions describing the spatiotemporal evolution of a dependent variable in an unbounded spatial domain can be characterised as a travelling wave moving with constant speed. In the simplest case, such PDEs can be reduced to a single autonomous second order ODE with one dependent variable. For certain parameter values it has been shown using perturbations methods in combination with ansätze that numerous such second order ODEs have analytical travelling wave solutions described by a simple sigmoid function. However, this methodology provides no leverage on the problem of finding a generalised class of models possessing such analytical travelling wave solutions. The most efficient methods for both finding analytical solutions and constructing classes of ODEs are based on Lie symmetries which are transformations known as one parameter $\mathcal{C}^{\infty}$ diffeomorphisms mapping solutions to other solutions. Recently, analytical solutions of a second order ODE encapsulating numerous oscillatory models as well as some of the previously mentioned travelling wave models with simple analytical solutions have been found by means of a two dimensional Lie algebra. Based on this Lie algebra, we construct the most general class of integrable autonomous second order ODEs for which these symmetries are manifest. Moreover, we show that a sub-class of second order ODEs has simple analytical travelling wave solutions described by a sigmoid function. Lastly, we characterise the action of the two symmetries in this Lie algebra on these simple analytical travelling wave solutions and we relate our sub-class of ODEs to previously known integrable travelling wave models.

nlin.SI↗

arXiv2023

Energy translation symmetries and dynamics of separable autonomous two-dimensional ODEs

We study symmetries in the phase plane for separable, autonomous two-state systems of ordinary differential equations (ODEs). We prove two main theoretical results concerning the existence and non-triviality of two orthogonal symmetries for such systems. In particular, we show that these symmetries correspond to translations in the internal energy of the system, and describe their action on solution trajectories in the phase plane. In addition, we apply recent results establishing how phase plane symmetries can be extended to incorporate temporal dynamics to these energy translation symmetries. Subsequently, we apply our theoretical results to the analysis of three models from the field of mathematical biology: a canonical biological oscillator model, the Lotka--Volterra (LV) model describing predator-prey dynamics, and the SIR model describing the spread of a disease in a population. We describe the energy translation symmetries in detail, including their action on biological observables of the models, derive analytic expressions for the extensions to the time domain, and discuss their action on solution trajectories.

physics.bio-ph↗

arXiv2022

On the correspondence between symmetries of two-dimensional autonomous dynamical systems and their phase plane realisations

We consider the relationship between symmetries of two-dimensional autonomous dynamical system in two common formulations; as a set of differential equations for the derivative of each state with respect to time, and a single differential equation in the phase plane representing the dynamics restricted to the state space of the system. Both representations can be analysed with respect to the symmetries of their governing differential equations, and we establish the correspondence between the set of infinitesimal generators of the respective formulations. Our main result is to show that every generator of a symmetry of the autonomous system induces a well-defined vector field generating a symmetry in the phase plane and, conversely, that every symmetry generator in the phase plane can be lifted to a generator of a symmetry of the original autonomous system, which is unique up to constant translations in time. The process of lifting requires the solution of a linear partial differential equation, which we refer to as the lifting condition. We discuss in detail the solution of this equation in general, and exemplify the lift of symmetries in two commonly occurring examples; a mass conserved linear model and a non-linear oscillator model.

math.DS↗