This thesis consists of five results sections, in which biological systems are examined through mathematical modelling. The first section examines how the transcription factor NF-B is aected by oscillations in the ligand TNF. Since the presence of a constant level of TNF induces oscillations in NF-B, they create a system of two coupled oscillators that can lead to entrainment depending on the coupling strength between them and the ratio between their original frequencies. For a range of parameters, this led to two stable limit cycles, and in the presence of noise transitions between the two cycles occurred and we termed this modehopping. We measured the distribution of transition times, and found this to be the sum of two exponentials we described by a simple 1D model. Next we considered how this affects downstream genes, and constructed a model that separates genes depending on the affinity and cooperativity of the NF-B binding to the promoter region of the gene. We found that the transitions in NF-B oscillations creates multiplexing between different families of genes. Then we increased the amplitude of TNF further and observed chaotic dynamics in NF-B, with statistical properties similar to the trends found in modehopping. The chaotic dynamics created a variety of different amplitudes, and we realized that this was a mechanism to enhance low affinity genes. We found that this led to a signicant raise in protein complex formation and that chaos enhanced both the efficiency and economy of this process. Finally we found that chaotic dynamics creates a population of heterogeneous cells that individually changes state in time. This was found to increase the survival rate in various toxic environments. The second project investigates the dynamics of another transcription factor, p53, following depletion of the protein Mdmx. The dynamics of p53 is believed to be important for the cellular control of processes as division and DNA repair. and previous reports have shown that p53 starts to oscillate following -radiation. Here we analyzed experimental data of p53 before and after Mdmx depletion, which revealed a typical response characterized by a large transient pulse followed by sustained oscillations. We used these experimental markers as guidelines to construct a simple mathematical model, and considered the different hypotheses by introducing impact parameters to represent each individual hypothesis. In this way we found that the main results was explained by an enhanced degradation of p53 caused by Mdmx. We then used the model to predict how cells depleted of Mdmx would respond to UV exposure in terms of p53 dynamics. By comparing the predictions to experimental results, we found a strong agreement between the two. The third project investigates how the dynamics of the membrane potential in neurons are affected by changes in extracellular ion concentrations. Inspired by previous experimental findings, showing different ion compositions in sleep than in awake, we extended an existing model to include extra-cellular ion concentrations. We then examined how the dynamics changed, if previously reported ion concentrations from sleep and awake were inserted into the model. By solely changing the ion concentrations a state transitions did not occur, but by changing ion concentrations accompanied by a perturbation in one of the gating channels, a transition occurred. We examined this further, by introducing an active ion composition, and found that this was enough to drive the neurons to a third state that we termed active awake. We argue that while the transition between sleep and awake is robust and needs perturbations in both ions and the gating channel, the transition between quiet awake and active awake does only need ion changes and can thus be done more quickly. Finally, we investigated the nature of the dynamics in the observed states and found that the quiet awake state was governed by chaotic dynamics whereas the sleep state was a closed limit cycle which allows the possibility of neuronal synchronization. The fourth project investigates how the dynamics of extracellular dopamine in the human brain is affected when the dopaminergic neurons are dying. It is well known that the correct stimulation of dopamine is fundamental for processes as memory, learning and movement, and it has been shown that at the onset of Parkinsons disease, a large fraction of all dopaminergic neurons are gone. We started by considering a previously published model for extracellular dopamine in a small subspace of striatum, and examined the dynamical properties as the density of neurons decreased. We then constructed a stochastic model for neuronal ring which were included in the dynamical model. Inspired by experimental findings, we introduced three different compensation mechanisms, and found many measures to differ, but for all models the signal to noise ratio was signicantly lowered. From this examined the landscape of remaining neurons in the entire striatum, after severe levels of denervation. Again we used previously suggested hypotheses from the literature as inspiration and based on this we constructed three different models for dopamine neuron denervation in a network. We found that these models can give rise to completely different signal transmission properties, but at severe levels it was a common feature that the network was divided into small communication classes and regions completely deprived of dopamine started to exist. The fifth project investigates how the protein production of GFP in bacteria is affected by incorporation of a strong promoter, a strong RBS and non canonical amino acids. Since transcription and translation in bacteria occurs simultaneously, we hypothesized that there could be limits to the transcriptional-translational density. In experiments we found that a system with a strong promoter and a strong RBS had almost no protein expression, but this expression was rescued if a codon for a non canonical amino acid was inserted early in the sequence of the gene. From this we constructed a model that allowed interaction of ribosomes from neighbouring mRNA strands during the time of transcription, in a process we termed Density Induced Translation Arrest. From the model we could reproduce the trends in the data, and predict how the rescue effect would disappear if the mutation was placed in the late part of the genome. This prediction was conrmed from the experiments. We then predicted an increased production if either the promoter or the RBS was decreased, which was also conrmed in experiments. Finally we used the model to predict the production from other genes depending on their length and sequence, which was again found to match what was found in the experiments.