In this thesis, research was done in the area of interacting particle systems. Especially, the symmetric exclusion process with local perturbations was investigated. These perturbations, were in the form of sinks and sources, which add or take away particles at certain rates. Moreover, simulations were done for the asymmetric exclusion process. This process took place on a ring, with the addition of a source. For the symmetric exclusion process with sinks and sources, certain expressions were proven for the expected occupancy of a site. For the simulations, the main goal was to find out how the jumping rates and starting density, influenced the time to get to the fully occupied state, at different source rates. The first proof was for a source at an arbitrary site. From this expression, one could see that if the rates were recurrent, the system converged to the fully occupied state. If, however, the rates were transient, the system had a limiting density. Thereafter, it was shown that if a sink and source are placed in the same arbitrary site, the system always converged to a density, which under the Bernoulli measure, was not equal to the fully occupied state. The fact that the sink and source were in the same site, was an indispensable condition. Subsequently, the case of countably many sources was investigated. For which it was also shown that recurrent rates always yield a fully occupied state, as time tends to infinity. Whereas transient rates, once again, caused a limiting density. Moreover, the special case of a simple random walk in three dimensions or higher was investigated. If, for distances far away from the origin, the source rates could be bounded above by a certain function, then the system would not converge to the fully occupied state. Also, another proof showed that a recurrent set of source sites would always let the process converge to a fully occupied state. Lastly, similar conditions for one time dependent source at the origin were proven. Namely, it was shown for recurrent rates, that if the source dies out quick enough, as t tends to infinity, the system did not converge to the fully occupied state. For the simulations, it was found that the expected time to fill up was influenced by the starting density, both at small and large source rates, in a linear way. This conclusion could not be drawn for the rate difference: p-q. Due to the large error of the fit, a linear relation was not clearly seen.  Finally, some concluding remarks were made on both aspects of the thesis, along with some recommendations for future projects related to this topic.

Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) is a respiratory virus capable of infecting humans and causing mild to severe complications during and after infection, which collectively comprise coronavirus disease 2019 (COVID-19). After an initial local epidemic in late 2019, the virus quickly spread on a global scale and was declared a pandemic shortly after. Many research efforts were devoted to SARS-CoV-2 leading to the quick discovery and communication of crucial data, resulting in improved countermeasures and accelerated development and deployment of new vaccines. These achievements notwithstanding, a number of unknowns of the working mechanisms of the virus and its replication inside of a host still remain. The exact function and structure of the nucleocapsid (N) protein have not been clearly established despite its recognized importance throughout the virus’ life cycle. N is involved in both the replication and translation of viral RNA, although its most prominent function is the formation of ribonucleoprotein (RNP) for the stable and compact storage of genomic RNA inside virions. Here we present a course-grained model of N for molecular dynamics simulations. Employing simple rules, the dimerisation and RNA-binding capacities of N have been emulated. A range of parameters giving rise to structures resembling those found in vivo by balancing attractive and repulsive properties are analysed and described. Lastly, by implementing an RNA substitute, conformations have been found which could give an indication of the structure of the RNP found inside SARS-CoV-2 virions. The model described in this work can be used as a basis for more extensive simulations incorporating further elements to gain a better understanding of the behaviour of this and similar viruses, contributing to the prevention of future large-scale disease outbreaks.

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.

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.