Mathematical modelling in cellular biology through compartmentalisation and conservation laws

The aim of this thesis focuses on addressing several open questions in cell biology by using different mathematical approaches and numerical analysis methods to study the evolution of distinct protein families in various cellular phenomena, such as cell polarisation and cytoskeleton remodelling. Our approaches are based on conservation laws and compartmentalisation of proteins within appropriate geometrical subdomains representing different cellular structures, such as the cell membrane and cytosol. The Rho GTPase are proteins responsible of coordinating the cell polarisation response, which is a biological process involving a huge number of different proteins and intricate networks of biochemical reactions. Rho GTPases localise their activity in specific cell regions where they interact with the cell cytoskeleton. Reducing the biological assumptions to a minimal level of complexity, we will present a simple qualitative model for cell polarisation in which proteins cycle between cell membrane and cytosol in an active and inactive form. This is described through a bulk-surface system of two reaction-diffusion equations coupled by the boundary condition. The model supports pattern formation and we will confirm this claim by using both mathematical analysis and simulations. The bulk-surface finite element method is presented and used to solve the model on different geometries. Secondly, we will present a mathematical model for keratin intermediate filament dynamics in resting cells. This model, characterised by a quantitative approach, is a datadriven extension of a pre-existing model, initially introduced by Portet et al. (PlosONE, 2015). We will discuss the new assumptions and modelling ideas, and compare the solution of our model to the experimental data. Part of the biological impact of our model relies in its ability to estimate the amount of assembled and disassembled keratin material as a function of space and time, consistent with the biological model proposed by Windoffer et al. (Journal of Cell Biology, 2011). In the last part we will introduce a second mathematical model for keratin spatiotemporal dynamics in non-resting cells. In this case, the model is derived on two- and three-dimensional geometries and accounts for a more detailed description of the processes involved in the keratin cytoskeleton remodelling process. The evolution of three different forms of keratin is modelled by a system composed of one reaction-diffusion equation and two reaction-advection-diffusion equations. Keratin kinetics are also described by the boundary conditions, which are posed both at the cell membrane and at the nuclear envelope. In solving the model, we will use the Streamline Upwind Petrov Galerkin method, as described in the text. In conclusion, in view of a future estimation of biologically relevant parameters, a simulation is presented, showing consistency of our mathematical model with the biological model proposed by Windoffer et al. (Journal of Cell Biology, 2011). In summary, this thesis presents methods and techniques for data-driven modelling supported by rigorous mathematical analysis and novel numerical methods and simulations. Our approach involving the use of quantitative methods serves as a blue-print for how to study the synergy interplay between mathematics and its applications to experimental sciences

Structure, organization and dynamics of a Sardinian sand dune plant community

Coastal sand dunes have attracted the attention of plant ecologists for over a century, but have largely relied on correlations to explain striking dune plant community organization. We experimentally examined longstanding hypotheses that sand binding, interspecific interactions, abiotic factors and seedling recruitment are drivers of sand dune plant community structure in Sardinia, Italy. Removing foundation species from the fore, middle and back dune over 3 years led to erosion and habitat loss on the fore dune and limited plant recovery that was enhanced with dune elevation. Reciprocal species removals in all zones suggested that interspecific competition is common, but that dominance is transient, particularly due to sand burial disturbance in the middle dune and high summer temperatures in the back dune. A fully factorial 2-year physical factor manipulation of water, nutrient availability and substrate stability revealed no significant proximate response to these abiotic factors in any dune zone. In the fore and middle dune, plant seeds are trapped under adult plants during seed germination, and seedling survivorship and growth generally increase with dune height in spite of increased herbivory in the back dune. Sand and seed erosion lead to limited seed recruitment on the fore dune while high summer temperatures and allelopathy lead to competitive dominance of woody plants in the back dune. Our results suggest that Sardinian sand dune plant communities are hierarchically organized, structured by sand binding foundation species on the fore dune, sand burial in the middle dune and increasingly successful seedling recruitment, growth, competitive dominance and allelopathy in the back dune

A coupled bulk-surface model for cell polarisation

Several cellular activities, such as directed cell migration, are coordinated by an intricate network of biochemical reactions which lead to a polarised state of the cell, in which cellular symmetry is broken, causing the cell to have a well defined front and back. Recent work on balancing biological complexity with mathematical tractability resulted in the proposal and formulation of a famous minimal model for cell polarisation, known as the wave pinning model. In this study, we present a three-dimensional generalisation of this mathematical framework through the maturing theory of coupled bulk-surface semilinear partial differential equations in which protein compartmentalisation becomes natural. We show how a local perturbation over the surface can trigger propagating reactions, eventually stopped in a stable profile by the interplay with the bulk component. We describe the behaviour of the model through asymptotic and local perturbation analysis, in which the role of the geometry is investigated. The bulk-surface finite element method is used to generate numerical simulations over simple and complex geometries, which confirm our analysis, showing pattern formation due to propagation and pinning dynamics. The generality of our mathematical and computational framework allows to study more complex biochemical reactions and biomechanical properties associated with cell polarisation in multi-dimensions

The extinction time under mutational meltdown driven by high mutation rates.

Mutational meltdown describes an eco-evolutionary process in which the accumulation of deleterious mutations causes a fitness decline that eventually leads to the extinction of a population. Possible applications of this concept include medical treatment of RNA virus infections based on mutagenic drugs that increase the mutation rate of the pathogen. To determine the usefulness and expected success of such an antiviral treatment, estimates of the expected time to mutational meltdown are necessary. Here, we compute the extinction time of a population under high mutation rates, using both analytical approaches and stochastic simulations. Extinction is the result of three consecutive processes: (a) initial accumulation of deleterious mutations due to the increased mutation pressure; (b) consecutive loss of the fittest haplotype due to Muller's ratchet; (c) rapid population decline toward extinction. We find accurate analytical results for the mean extinction time, which show that the deleterious mutation rate has the strongest effect on the extinction time. We confirm that intermediate-sized deleterious selection coefficients minimize the extinction time. Finally, our simulations show that the variation in extinction time, given a set of parameters, is surprisingly small

Integrating actin and myosin II in a viscous model for cell migration

This article presents a mathematical and computational model for cell migration that couples a system of reaction-advection-diffusion equations describing the bio-molecular interactions between F-actin and myosin II to a force balance equation describing the structural mechanics of the actin-myosin network. In eukaryotic cells, cell migration is largely powered by a system of actin and myosin dynamics. We formulate the model equations on a two-dimensional cellular migrating evolving domain to take into account the convective and dilution terms for the biochemical reaction-diffusion equations, with hypothetically proposed reaction-kinetics. We employ the evolving finite element method to compute approximate numerical solutions of the coupled biomechanical model in two dimensions. Numerical experiments exhibit cell polarization through symmetry breaking which are driven by the F-actin and myosin II. This conceptual hypothetical proof-of-concept framework set premises for studying experimentally-driven actin-myosin reaction-kinetic network interactions with generalizations to multi-dimensions

A moving grid finite element method applied to a mechanobiochemical model for 3D cell migration

This work presents the development, analysis and numerical simulations of a biophysical model for 3D cell deformation and movement, which couples biochemical reactions and biomechanical forces. We propose a mechanobiochemical model which considers the actin filament network as a viscoelastic and contractile gel. The mechanical properties are modelled by a force balancing equation for the displacements, the pressure and contractile forces are driven by actin and myosin dynamics, and these are in turn modelled by a system of reaction-diffusion equations on a moving cell domain. The biophysical model consists of highly non-linear partial differential equations whose analytical solutions are intractable. To obtain approximate solutions to the model system, we employ the moving grid finite element method. The numerical results are supported by linear stability theoretical results close to bifurcation points during the early stages of cell migration. Numerical simulations exhibited show both simple and complex cell deformations in 3-dimensions that include cell expansion, cell protrusion and cell contraction. The computational framework presented here sets a strong foundation that allows to study more complex and experimentally driven reaction-kinetics involving actin, myosin and other molecular species that play an important role in cell movement and deformation

Effects of noise on neural signal transmission: analysis of some simple models

Neurons transmit information with each other through spike trains, that are sequences of action potentials, mathematically described as Cox processes. We study a neural population encoding a Gaussian signal received as an input. We estimate the amounts of information transmitted about the signal, when such population is subject to independent noise sources, and we see that noise actually may have a beneficial role in information transmission. With this aim, we introduce some concepts of neuroscience, which are useful for a first approach to the subject. We describe the main concepts about the theory of point processes and random measures, focusing on the Cox processes. Then, an overview of the analysis of random signals is provided, which is necessary to quantify the amounts of information encoded by a neural population. We then show two models, studied in "Shifting Spike Times or Adding and Deleting Spikes" (S. Voronenko, W. Stannat, B. Lindner, 2015) and replay their numerical experiments. The models describe two different ways in which the noise shapes the population response to the stimulus, increasing the amounts of information transmitted in both cases. Finally, we present several new models combining the previous models and their respective numerical results

The Role of disturbance in promoting the spread of the invasive seaweed <i>Caulerpa racemosa</i> in seagrass meadows

Human disturbances, such as anchoring and dredging, can cause physical removal of seagrass rhizomes and shoots, leading to the fragmentation of meadows. The introduced green alga, Caulerpa racemosa, is widely spread in the North-West Mediterranean and, although it can establish in both degraded and pristine environments, its ability to become a dominant component of macroalgal assemblages seems greater in the former. The aim of this study was to estimate whether the spread of C. racemesa depends on the intensity of disturbance to the canopy structure of Posidonia oceanic. A field experiment was started in July 2010 when habitat complexity of a P. oceanica meadow was manipulated to simulate mechanical disturbances of different intensity: rhizome damage (High disturbance intensity = H), leaf removal (Low disturbance intensity = L), and undisturbed (Control = C). Disturbance was applied within plots of different size (40 × 40 cm and 80 × 80 cm), both inside and at the edge of the P. oceanica meadow, according to an orthogonal multifactorial design. In November 2011 (16 months after the start of the experiment), no C. racemosa was found inside the seagrass meadow, while, at the edge, the cover of the seaweed was dependent on disturbance intensity, being greater where the rhizomes had been damaged (H) than in leaf removal (L) or undisturbed (C) plots. The results of this study indicate that physical disturbance at the margin of seagrass meadows can promote the spread of C. racemosa

The role of disturbance in promoting the spread of the invasive seaweed Caulerpa racemosa in seagrass meadows

Human disturbances, such as anchoring and dredging, can cause physical removal of seagrass rhizomes and shoots, leading to the fragmentation of meadows. The introduced green alga, Caulerpa racemosa, is widely spread in the North-West Mediterranean and, although it can establish in both degraded and pristine environments, its ability to become a dominant component of macroalgal assemblages seems greater in the former. The aim of this study was to estimate whether the spread of C. racemosa depends on the intensity of disturbance to the canopy structure of Posidonia oceanica. A field experiment was started in July 2010 when habitat complexity of a P. oceanica meadow was manipulated to simulate mechanical disturbances of different intensity: rhizome damage (High disturbance intensity = H), leaf removal (Low disturbance intensity = L), and undisturbed (Control = C). Disturbance was applied within plots of different size (40 x 40 cm and 80 x 80 cm), both inside and at the edge of the P. oceanica meadow, according to an orthogonal multifactorial design. In November 2011 (16 months after the start of the experiment), no C. racemosa was found inside the seagrass meadow, while, at the edge, the cover of the seaweed was dependent on disturbance intensity, being greater where the rhizomes had been damaged (H) than in leaf removal (L) or undisturbed (C) plots. The results of this study indicate that physical disturbance at the margin of seagrass meadows can promote the spread of C. racemosa