Intracellular transport relies on the collective behavior of molecular motors, such as Kinesin-II, which must navigate crowded microtubule environments efficiently. While standard exclusion models like the Totally Asymmetric Simple Exclusion Process (TASEP) predict significant velocity reduction at high densities, Kinesin-II exhibits resilience to crowding, suggesting a mechanism of cooperative transport that remains poorly understood in both physical and mathematical theory.

This thesis addresses this gap by combining a rigorous mathematical derivation of hydrodynamic limits with a biologically motivated particle model. Mathematically, we derive the hydrodynamic limit for an active particle system where the active direction of the particles is governed by mean-field Curie-Weiss rates with parameter β for both local and global interactions. We prove that the microscopic stochastic dynamics converge to a macroscopic reaction-diffusion-advection PDE. Through linearization and Fourier-Laplace analysis, we
derive analytical expressions for the velocity and diffusion coefficients, showing significant dependence on β.

Physically, we extend this framework to include exclusion and different interaction ranges σ. Our simulations reveal that exclusion introduces spatial correlation that breaks mean-field assumptions, leading to deviations from the predictions for the global transport coefficients.
We show that for strong coupling β > 1, local interactions lead to the formation of clusters and altered relaxation times. Finally, we validate our model against experimental velocity-density data for Kinesin-II. We show that our mean-field exclusion model provides a statistically more accurate description compared to the standard TASEP-LK model.
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Microorganisms often live in dense, surface-attached communities called biofilms. These biofilms exist in diverse environments, from the inside of your gut and the surfaces of your teeth, to the roots of your house plants and the inside of your coffee machine. These biofilms, composed of various bacteria, fungi, and viruses encased in a self-produced extracellular matrix, exhibit complex behaviours. They also play a dual role in human health and industry, contributing to persistent infections and antibiotic resistance while being crucial for digestion and industrial applications like wastewater treatment. Understanding biofilm formation and function is essential for decreasing their detrimental effects and increasing their potential in biotechnology.
In this thesis, I use individual-based modelling of spherocylindrical particles to learn something about the effects of spatial structure on their mechanical and social interactions.
In chapter 2 we explore the aggregation dynamics of blue-light switchable adhesive E. coli in solution. We aim to understand experimental results that bacteria aggregate more and formed bigger clusters under pulsating light. We simulate a system of particles undergoing Brownian motion, where the cell-cell adhesion can be periodically turned on and off and compare and match our simulations to the experimental data. We show how tuning the light off-period to the decay time of the adhesion leads to increased clustering. We conclude that partial disassembly of the aggregates leads to more effective clustering. In addition, our co-authors show that this increased clustering leads to increased biofilm formation in a laboratory setting. Moreover, it can be used to increase productivity in a bioreactor.
We use what we learnt about cell-cell interactions to simulate growing surface attached systems in chapter 3. We motivate some choices about the interactions between cells and the interaction with the surface. We then show how varying the strengths of these interactions can lead to different microcolony architectures.
We then use this model of growing microcolonies to study cooperator interactions in a spatially structured environment. Where the mechanical interactions occur over short distances, we also assume that metabolic interactions are close range. In chapter 4, we simulate a cross-feeding consortium in the presence of a cheater species by having particles adjust their growth rate based on the cells in their immediate environment. Using simulations and an experimental consortium, we show how the time it takes for cooperators to meet is the determining factor in whether they outcompete their cheating counterparts.
Finally, in chapter 5 we explore the patterning that cooperating particles create by mixing. We show that this cooperator mixing is mostly determined by interaction strength and is robust against variations in size and interaction symmetry. Additionally, we show that in the presence of cheaters, cooperators intermix but cheaters don’t mix with the cooperators and instead remain on the outside. Therefore, we argue that focusing on strong cooperation is a great strategy for cheater exclusion.
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In this thesis we investigate discrete space Markov processes with multiple layers and how these can be applied to physical systems such as RNA transcription. We show that the Markov processes of single particles satisfy an invariance principle (i.e., the limiting behaviour is Brownian with a certain drift) for both homogeneous and random environments. Additionally, we perform an in-depth numerical analysis of multiple particle systems in the context of RNA transcription and reproduce some known phenomena such as RNA polymerase clustering and cooperation. ... Read more

Staphylococcus aureus and Pseudomonas aeruginosa are two species of bacteria that are involved in numerous conditions, including lung infections and chronic wound infections. The aim of this project was to study the short-term interactions that occur when P. aeruginosa first encounters an established S. aureus colony, which it then seeks to break apart whilst mixing with S. aureus. Limoli et al. have studied these interactions using experiments, and have thus identified several key aspects involved in these interactions, such as the mechanisms that P. aeruginosa employs to approach the S. aureus colony. The means by which we intended to study interactions between S. aureus and P. aeruginosa is a model that was made by previous members of the Idema group and that was based on the experiments by Limoli et al. In this report, we discuss this model and the biological background relevant to it. We also document the problems that we encountered while trying to run simulations using an existing implementation of this model. ... Read more

All organisms are built out of cellular tissue. Being able to recognise abnormalities in these tissues could be useful in recognizing cancerous cells. In this thesis we construct a mathematical model for cellular tissue based on its spatial structure. We consider cells as elements of the network. Touching cells are considered connected. Cells grow at different growth speeds. We determine the point in time when this network is fully connected, meaning there is a path between every pair of cells through touching cells. This point indicates the start of the last phase of the cellular growth, where friction restricts cell movement. We first use a Poisson point process to generate the locations of the cells. To make the model more similar to cellular tissue, we introduce determinantal point processes which have short-ranged repulsion, meaning points repel each other and thus spread. We compare the repulsion of Poisson and Determinantal point processes with real cellular tissue. We conclude that determinantal point processes have significantly higher repulsion than Poisson point processes. We also conclude that the repulsion in cellular tissue is higher than both point processes. Using simulations, we show that the model with determinantal point processes reaches connectivity significantly earlier than the Poisson model. We conclude that the analytically derived connectivity time point from the Poisson model can be used as an upper bound for the determinantal model. ... Read more

“From chaos, order can emerge”, a counterintuitive statement but also one laying the foundation for complex living systems. Individually acting agents can collectively produce organised structures at a larger scale. Possibly the most ubiquitous example of this phenomenon are bacterial colonies.
Everywhere around us, a pandemonium of pushing and pulling produces complex structures. The most researched bacterium is E. coli and partially due to its excellent swimming capabilities, the internal structure of its colony is predominantly shaped through mechanical repulsion coupled with
individual motility. In this thesis, we are interested in the emergent dynamics within a bacterial colony. Our focus will lie on how the colony's density \rho affects the internal structure and motility.
To study the colony's interior, we built a three-dimensional individual-based model (IBM) with self-propelling sphero-cylindrical agents representing E. coli bacteria and governed by mechanical interactions. A downside of IBM is its computational costliness, posing an optimisation challenge which will also be covered in this thesis.
A phase transition spontaneously occurs over time from isotropic to an aligned nematic phase. This transition takes longer for higher-density systems. We found a linear relation between the density and the local order for a colony in a quasi-infinite domain. Furthermore, after equilibration, the particles initially behave ballistically. However, this changes to diffusive behaviour in a later stadium. The Reynolds number Re ~ 10^-3 is two orders of magnitude larger than expected for E. coli, possibly due to underestimating the viscosity.
On a final note, the method to determine the moment of equilibrium tequi gives an underestimation in the case of a two-step phase transition; an improved method is proposed.
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In this thesis, we study the asymptotic behaviour and the dynamics of a one-dimensional active particle model with excluded volume interactions. The model is a version of run-and-tumble motion, where a particle performs both symmetric random walks and active transport. The direction and the speed of the transport are governed by an internal state process. We show that this motion converges to Brownian motion upon diffusive scaling and determine the limiting diffusion coefficient. The internal state converges to a stationary distribution, by which it manifests itself in the diffusion coefficient. Furthermore, we prove that the active particle satisfies the large deviation principle. This allows us to derive an implicit expression for the rate by which the probability of rare events tends to zero. Numerically, we investigate the influence of excluded volume interactions on the diffusion coefficient and the average velocity. We find that the velocity converges exponentially to its theoretical value as the number of particles allowed per position increases. In addition, this exclusion number strongly influences the manner in which the velocity decreases for high particle densities. Predictions for the velocity as a function of particle density based on the model are compared to experimental data of the molecular motor kinesin-II. We find that the model is not adequate for approximating the velocity of molecular motors in crowded environments and extensions in the form of Langmuir kinetics are suggested. ... Read more

Over the last decades, studies have shown that the gene regulation of a wide range of organisms can be described with networks [Jeong H., Tombor B., Albert R., Ottval Z.N., Barab´asi A.L., 2000] [Jeong H., Mason S.P., Barab´asi A.L., Oltvai Z.N., 2001]. The interactions between the mRNA strands and proteins form links in the network, while these molecules form the nodes of the network. Numerical models for the dynamics of such a network, through solving a minimally nonlinear stochastic differential equation show nontrivial dynamics. This behavior is caused by non linearity introduced by the positivity condition, this is due to the fact that molecular concentrations cannot cannot be negative. Whether these dynamics are stable, oscillatory or chaotic seems to depend on the average connectivity of the network. There appears to be a region of networks that can transition from chaotic behavior to stable behavior. For a simple differential equation this would not be a region, but just one value. This region, referred to as the “Edge-of-Chaos” region, is therefore rather interesting. In this report we analyse the Lyapunov spectrum in order to quantify these dynamics. This is a way to measure the stability of the solution of out differential equation. We look at how different parameters in the minimally nonlinear differential equation and the generation of the network affect these dynamics. The investigated parameters include different edge weights, different noise levels, different equilibrium values and different network types. After studying the dynamics of a single network, we study the dynamics of populations of networks using an evolutionary algorithm and a co-evolutionary algorithm. We found that an edge of chaos region exists universally, that is for all types of networks we looked at, for all feasible noise levels, for all equilibrium values and for all network sizes. Whether or not a network lies in this region is determined by how much self interaction its nodes have compared to the strength of the interactions between the nodes. Additionally, the size of the network determines how well this “Edge-of-Chaos” plateau forms. Larger networks in this region behave more stable than smaller ones. We also came to the conclusion that both the evolutionary and co-evolutionary algorithm, using the Kullback-Leibler divergence in accordance with a probability distribution defined by a Hamiltonian to compare the fitness of individuals, do not yield the expected results. That is, they do not show any correlation between the connectivity of a network and the development of the population over time. ... Read more

Collective dynamics is something that can be found in nature on macro and micro scale. Since the 90s of the previous century researchers have been interested in finding a model for this group behaviour. The dynamics of a group as a whole is only determined by shortrange interactions of the individuals. To better understand the working of this process, we make a model of this system with soft two-dimensional spheres with a active selfpropulsion force. Furthermore there are repulsion, alignment and noise interactions, all depending exclusively on nearest neighbours. We focus on binary systems with particles of two different sizes. Migrating and rotating states are typically found in systems with homogeneous sizes. We show that these states are also found in binary systems. The migrating state has circulation of particles and this leads to segregation of the small and big particles. Small particles are more likely to be found in the tip of the group, while big particles accumulate at the tail. Active noise in the system plays a role in the degree of segregation. The lower the noise is, the more segregated the system gets in the end. ... Read more

We consider adaptive network models with discrete and continuous state sets obeying dynamical rules that enable application to swarming systems. The 2-state adaptive network contains a supercritical pitchfork bifurcation in the transition between ordered and disordered stationary solutions. We derive an adaptive network model that works on a continuous state set and apply it to swarming motion in both a mean field and a moment closure approximation. In numerical solutions of the mean field approximation the relation between the variance of the ordered stationary distributions and the system parameters is given by a square root function. Cauchy distributions form a good fit to these steady state distributions, although they are not the analytic stationary solutions. We show that in numerical solutions of the moment closure approximation a bistable region is formed, in which the initial condition determines if the system ends up in an ordered or a disordered state. Further research could focus on finding the exact details of the corresponding subcritical pitchfork and saddle-node bifurcations and comparing the derived models to real-life swarming systems. ... Read more

In biological cells, information from the external environment of the cell is used to make survival related decisions. For these decisions, it is important that signals are accurately transduced from the outside to the inside of the cell. In \textit{Dictyostelium discoideum}, two opposing mechanisms using G-protein coupled receptors are used for this signalling: the precoupling mechanism, where second messenger molecules bind to the receptor before a ligand binds to it, and the collision coupling mechanism, in which the ligand binding comes first. In this paper, we investigated both models by analyzing how accurately they detect ligand bindings when different receptors are able to interact with each other. A similar analysis is done for returning messenger molecules. We found that the influence of receptors upon each other is low if the receptors operate under the same conditions. However, when the conditions are heterogeneous, the influence of receptors on each other is huge. The main reason for this influence is that ligand binding receptors which are more likely to get detected by messenger molecules will receive more of those messengers, because of their high diffusion rate. The effect of returning messenger molecules on the receptor signal was that ligand detection became possible at times where they otherwise would not be able to be detected anymore, given a replace rate which is low compared to the rate of binding and unbinding of a ligand to a receptor. ... Read more

In this thesis, we investigate interactions between conical inclusions in a lipid bilayer membrane and make predictions about the patterns they form. To find these patterns, we derive an expression for the energy of a membrane as a function of the inclusion locations and search numerically for the pattern
that gives minimum energy.

The energy of a membrane with conical inclusions can be derived using the point particles model with corresponding formalism developed by Dommersnes and Fournier [1]. In this thesis, we apply this formalism to the finite size particles model described by Weikl et al. [2]. We compare the results of both
models for a system of three inclusions, to validate the point particles model’s ability to accurately predict equilibrium patterns for conical inclusions. For most non-conical inclusions, however, the point particles model proves inadequate, leaving only the computationally intensive finite size particles model to be used for more complex inclusions.

We develop a new numerical method for finding equilibrium patterns: the gradient descent method. This method is several hundred times faster than the standard Metropolis algorithm, and gives acceptable results. For large systems of inclusions, the method is very sensitive to local minima and has difficulties
merging small groups. The addition of noise in the Brownian motion method proves to be unable to resolve the local minima sensitivity, but we speculate that small bursts of high noise or grouping stable inclusions structures and moving the groups as a whole may be more effective.

Using the point particles model, we found that four-inclusion square-shaped structures and six-inclusion butterfly-shaped structures are favored in all systems with more than six inclusions.
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Biological membranes are selective soft barriers that compartmentalize internal structure of a cell into organelles and separate them as a whole from the external environment. Due to their innate feature of being able to undergo constant reshaping, cellular membranes spatially attain diverse shapes ranging from simple spherical vesicles to more peculiar structures like the interconnected network of tubes found in the endoplasmic reticulum. Membranes are not only composed of lipids, but also host an enormous number of inclusions like proteins. Recent studies of biological membranes have revealed that such inclusions play a key role in diverse biological processes through either sensing or inducing perturbations to the membrane shape. In this dissertation, we studied the interplay between the shape of membrane and the spatial organization of attached curvature inducing objects using mathematical tools and numerical simulations in highly curved spherical and cylindrical geometries.
First, we investigated the interaction between inclusions of different shapes embedded in/adhered to tubular membranes. Our combined theoretical analysis and numerical simulation results evinced that tubular membranes, in contrast to their planar counterpart, transmit an attractive force between inclusions, stemming from their closed and curved geometry. We then elucidated that collective interaction between proteins results in the formation of line-like and ring-like clusters, depending on the their intrinsic shape (Chapters 2–4). We further showed how curvature sensing crescent-like proteins in high densities can constrict tubular membranes and facilitate their splitting, demonstrating that both the curvature-sensing and curvature-inducing property of proteins are two sides of the same coin. Moreover, we used our simulation results to explain how mitochondorial machinery triggers, facilitates and drives membrane fission in its tubular network to avoid entanglements (Chapter 3).
Next, we examined the interaction of spherical proteins adhered to closed vesicles. Our simulation results – supported by recent experimental evidence – revealed membrane curvature as a common physical origin for interactions between any membrane deforming objects, from nanometre-sized proteins to micrometre-sized particles (Chapter 5). Our further simulations unraveled how introducing curvature variation on the surface of a closed vesicle can be exploited by inanimate particles to regulate their pattern formation (Chapter 6).
Finally, through theoretical calculations,we analyzed the interplay between the shape of a cell and the rearrangement of attached microtubules (Chapter 7). Our results particularly suggested that the commonly reported parallel structure and bundling of microtubules can be induced by membrane mediated interactions. ... Read more

In this thesis, we model the diusion of a tracer polymer inside of a gel network and simulate it, hoping to nd a connection between the diusion of the polymer and the strength of the gel network. This model is made by using the Rouse model for the gel network and the tracer polymer.
The overdamped Langevin equation is then used to nd a set of coupled stochastic dierential equations for the motion of a single tracer bead and the Fourier modes of the gel particles. The single particle system is then analyzed using three dierent numerical methods: The Euler forward method, the Metropolis Monte Carlo method and the Gillespie algorithm. The Gillespie algorithm is then used to expand the single particle model to a model which again includes a tracer polymer instead of a single tracer bead. The simulations of the tracer polymer suggest that the motion of the tracer polymer is superdiusive. This contradicts the theory and the measurements of the single
tracer particle, which suggest that the simulation of the polymer should result in subdiusion. This contradiction seems to be caused by an error in the implementation of the interaction between the dierent beads that make up the tracer polymer, as it creates a tendency for the polymer to move away from its original position. This possible error is hinted at by a simulation of the system with the tracer polymer where the gel is considered stationary. The simulation implies superdiusion as well, which means that the superdiusion is not caused by the gel network. In fact, the simulation with the frozen gel network is much further away from subdiusion that the simulation with the gel network intact, which does seem to imply that the motion would be subdiusive if the model
was implemented correctly, but it is not conclusive. ... Read more