Synergistic effects of repair, resilience and retention of damage determine the conditions for replicative ageing

Accumulation of damaged proteins is a hallmark of ageing, occurring in organisms ranging from bacteria and yeast to mammalian cells. During cell division in Saccharomyces cerevisiae, damaged proteins are retained within the mother cell, resulting in an ageing mother while a new daughter cell exhibits full replicative potential. The cell-specific features determining the ageing remain elusive. It has been suggested that the replicative ageing is dependent on the ability of the cell to repair and retain pre-existing damage. To deepen the understanding of how these factors influence the life of individual cells, we developed and experimentally validated a dynamic model of damage accumulation accounting for replicative ageing on the single cell level. The model includes five essential properties: cell growth, damage formation, damage repair, cell division and cell death, represented in a theoretical framework describing the conditions allowing for replicative ageing, starvation, immortality or clonal senescence. We introduce the resilience to damage, which can be interpreted as the difference in volume between an old and a young cell. We show that the capacity to retain damage deteriorates with high age, that asymmetric division allows for retention of damage, and that there is a trade-off between retention and the resilience property. Finally, we derive the maximal degree of asymmetry as a function of resilience, proposing that asymmetric cell division is beneficial with respect to replicative ageing as it increases the lifespan of a given organism. The proposed model contributes to a deeper understanding of the ageing process in eukaryotic organisms

Symmetry structures in dynamic models of biochemical systems

Understanding the complex interactions of biochemical processes underlying human disease represents the holy grail of systems biology. When processes are modelled in ordinary differential equation (ODE) fashion, the most common tool for their analysis is linear stability analysis where the long-term behaviour of the model is determined by linearizing the system around its steady states. However, this asymptotic behaviour is often insufficient for completely determining the structure of the underlying system. A complementary technique for analysing a system of ODEs is to consider the set of symmetries of its solutions. Symmetries provide a powerful concept for the development of mechanistic models by describing structures corresponding to the underlying dynamics of biological systems. To demonstrate their capability, we consider symmetries of the nonlinear Hill model describing enzymatic reaction kinetics and derive a class of symmetry transformations for each order of the model. We consider a minimal example consisting of the application of symmetry-based methods to a model selection problem, where we are able to demonstrate superior performance compared to ordinary residual-based model selection. Moreover, we demonstrate that symmetries reveal the intrinsic properties of a system of interest based on a single time series. Finally, we show and propose that symmetry-based methodology should be considered as the first step in a systematic model building and in the case when multiple time series are available it should complement the commonly used statistical methodologies

The synergy of damage repair and retention promotes rejuvenation and prolongs healthy lifespans in cell lineages

Damaged proteins are inherited asymmetrically during cell division in the yeast Saccharomyces cerevisiae, such that most damage is retained within the mother cell. The consequence is an ageing mother and a rejuvenated daughter cell with full replicative potential. Daughters of old and damaged mothers are however born with increasing levels of damage resulting in lowered replicative lifespans. Remarkably, these prematurely old daughters can give rise to rejuvenated cells with low damage levels and recovered lifespans, called second-degree rejuvenation. We aimed to investigate how damage repair and retention together can promote rejuvenation and at the same time ensure low damage levels in mother cells, reflected in longer health spans. We developed a dynamic model for damage accumulation over successive divisions in individual cells as part of a dynamically growing cell lineage. With detailed knowledge about single-cell dynamics and relationships between all cells in the lineage, we can infer how individual damage repair and retention strategies affect the propagation of damage in the population. We show that damage retention lowers damage levels in the population by reducing the variability across the lineage, and results in larger population sizes. Repairing damage efficiently in early life, as opposed to investing in repair when damage has already accumulated, counteracts accelerated ageing caused by damage retention. It prolongs the health span of individual cells which are moreover less prone to stress. In combination, damage retention and early investment in repair are beneficial for healthy ageing in yeast cell populations

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.Comment: 22 pages, 7 figure

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.Comment: 18 pages, 3 figure

Cell polarisation in a bulk-surface model can be driven by both classic and non-classic Turing instability

The GTPase Cdc42 is the master regulator of eukaryotic cell polarisation. During this process, the active form of Cdc42 is accumulated at a particular site on the cell membrane called the pole. It is believed that the accumulation of the active Cdc42 resulting in a pole is driven by a combination of activation–inactivation reactions and diffusion. It has been proposed using mathematical modelling that this is the result of diffusion-driven instability, originally proposed by Alan Turing. In this study, we developed, analysed and validated a 3D bulk-surface model of the dynamics of Cdc42. We show that the model can undergo both classic and non-classic Turing instability by deriving necessary conditions for which this occurs and conclude that the non-classic case can be viewed as a limit case of the classic case of diffusion-driven instability. Using three-dimensional Spatio-temporal simulation we predicted pole size and time to polarisation, suggesting that cell polarisation is mainly driven by the reaction strength parameter and that the size of the pole is determined by the relative diffusion

The construction, analysis and validation of mechanistic mathematical models of protein kinetics in the context of replicative ageing in budding yeast

Mathematical modelling constitutes a forceful tool for elucidating properties of biological systems. Using theoretical approaches in combination with experimental techniques it is possible to study specific molecular aspects of phenomena such as the ageing of human beings. In fact, as many processes are similar in simpler organisms such as the budding yeast \textit{Saccharomyces cerevisiae} it is possible to experimentally investigate for instance the accumulation of damaged proteins due to ageing in these biological systems. The aim of this thesis is to construct, analyse and validate mathematical mechanistic models of protein kinetics consisting of both ordinary and partial differential equations in the context of ageing. This is done both on a large time scale corresponding to the entire life span of cells and a short time scale corresponding to an isolated part of the cell division. The focus of the work on the large time scale is twofold, firstly the life span of individual yeast cells is modelled (Paper II) and secondly the life spans of vast numbers of cells in numerous populations are simulated (Paper III). Using a model of the accumulation of damage involving the forces cell growth, formation and repair of damage as well as the cell division, the impact of these individual parts on the overall fitness of individual cells and entire populations is investigated. On the short time scale, a more detailed model of a single protein called Cdc42 involved in the cell division is presented (Paper IV) and this theoretical framework has a high level of detail as it describes the spatial movement of the protein of interest within the cell over time. Given this precise description of the geometry of an individual cell, the mathematical properties of the model is analysed and these theoretical results are used to conduct numerical simulations of the activity of this protein. Lastly, an overall theme of the thesis is the difficulty of validating mechanistic models even in the presence of data. More precisely, as numerous and sometimes mutually exclusive models can describe a system equally well it is currently very hard, even by calibrating the models to experimental data using statistical methods, to differentiate between various models. To this end, a mathematical tool called symmetry methods is introduced as a potential remedy to this problem, and using this methodology it is possible to extract information in the data as well as in the model that is not available using standard approaches. To showcase the power of symmetries, a minimal example of the usage of these methods in the context of enzyme kinetics is presented (Paper V). In conclusion, this work suggests that novel analytical tools such as symmetry methods could complement and assist the current standard approaches for modelling protein kinetics where the purpose is to deduce the underlying mechanisms of biological systems

Systems biology of aging

Mathematical modeling has emerged as a powerful descriptive and predictive tool to analyze complex biological systems. It is deeply embedded in the systems biology cycle, providing the means to deliver predictive quantitative models. Aging is a highly complex, irreversible process that arises from interactions of many different components. It is characterized by the accumulation of harmful molecules that damage the cell over the course of time coupled with progressive functional decline, inevitably culminating in death. This underpins the universal hallmark of aging - the accumulation and segregation of aging factors. Integrating mathematical modeling and experimental work may prove to be a powerful way to address certain evolutionary questions that might have profound implications for the whole study of aging.This systems biology approach may reveal the underlying mechanisms that cause the functions of the cell to deteriorate over the course of time and predict optimal division strategies that will lead to increased fitness and prolonged lifespan. In this chapter, we provide an overview of the biology of the aging process including several aging theories and the current state of mathematical models in aging research, together with a case study illustrating damage accumulation theory

Turing pattern formation on the sphere is robust to the removal of a hole

The formation of buds on the cell membrane of budding yeast cells is thought to be driven by reactions and diffusion involving the protein Cdc42. These processes can be described by a coupled system of partial differential equations known as the Schnakenberg system. The Schnakenberg system is known to exhibit diffusion-driven pattern formation, thus providing a mechanism for bud formation. However, it is not known how the accumulation of bud scars on the cell membrane affect the ability of the Schnakenberg system to form patterns. We have approached this problem by modelling a bud scar on the cell membrane with a hole on the sphere. We have studied how the spectrum of the Laplace–Beltrami operator, which determines the resulting pattern, is affected by the size of the hole, and by numerically solving the Schnakenberg system on a sphere with a hole using the finite element method. Both theoretical predictions and numerical solutions show that pattern formation is robust to the introduction of a bud scar of considerable size, which lends credence to the hypothesis that bud formation is driven by diffusion-driven instability