In the 1960s French mathematician George de Rham found a relationship between two invariants of knots. He found that there exist representations of the fundamental group of a knot into a group G of upper right triangular matrices in C with determinant one that is described exactly by the roots of the Alexander polynomial. I extended this result to find that the representations of the fundamental group of a link into G are described by the multivariable Alexander polynomial of the link.

Dimensional Analysis is a technique that has allowed engineering evaluation of complex objects by scaling analysis results of representative simpler models. The original premise of the procedure stems from the idea of developing non-dimensional parameters to relate physical events and underlying analytical basis. Extending the process to incorporate non-linear and time variant behavior has led to development of a novel process of similitude called the Empirical Similitude Method (ESM) where experimental data of test specimen is combined to produce the required prediction values. Using the original motivation and hypothesis of ESM, this research has expanded the experimental similitude process by using adapted matrix representations and continuous functional mapping of test results. This new approach has provided more rigorous mathematical definitions for similarity and prediction estimations based on an innovative error minimization algorithm. Shape factors are also introduced and integrated into ESM to obtain comprehensive evaluation of specimen choices. A detailed overview is provided summarizing methods, principles and laws of traditional similitude (TSM) and systems that satisfy extension into ESM. Applicability of ESM in different systems is described based on the limitations of TSM in the evaluation of complex structures. Several examples and ideas spanning aerodynamic, thermal, mechanical and electro-magnetic domains are illustrated to complement inherent technical analysis. For example, the new ESM procedure is shown to be considerably more accurate than earlier methods in predicting the values of drag coefficient of an airfoil. A final foray into the regime of \design evaluation by similarity" is made to elucidate applicability and efficiency of developed techniques in practical systems and products. A thorough methodology is also presented highlighting pertinent procedures and processes in usage of this method. / text

<p>Gene expression is regulated in response to metabolic necessities and environmental changes throughout the life of a cell.</p><p>A major part of this regulation is governed at the level of transcription, deciding whether messengers to specific genes are produced or not.</p><p>This decision is triggered by the action of transcription factors, proteins which interact with specific sites on DNA and thus influence the rate of transcription of proximal genes.</p><p>Mapping the organisation of these transcription factor binding sites sheds light on potential causal relations between genes and is the key to establishing networks of genetic interactions, which determine how the cell adapts to external changes.</p><p>In this work I review briefly the basics of genetics and summarise popular approaches to describe transcription factor binding sites, from the most straight forward to finally discuss a biophysically motivated representation based on the estimation of free energies of molecular interactions.</p><p>Two articles on transcription factors are contained in this thesis, one published (Aurell, Fouquier d'Hérouël, Malmnäs and Vergassola, 2007) and one submitted (Fouquier d'Hérouël, 2008).</p><p>Both rely strongly on the representation of binding sites by matrices accounting for the affinity of the proteins to specific nucleotides at the different positions of the binding sites.</p><p>The importance of non-specific binding of transcription factors to DNA is briefly addressed in the text and extensively discussed in the first appended article:</p><p>In a study on the affinity of yeast transcription factors for their binding sites, we conclude that measured in vivo protein concentrations are marginally sufficient to guarantee the occupation of functional sites, as opposed to unspecific emplacements on the genomic sequence.</p><p>A common task being the inference of binding site motifs, the most common statistical method is reviewed in detail, upon which I constructed an alternative biophysically motivated approach, exemplified in the second appended article.</p>

Gene expression is regulated in response to metabolic necessities and environmental changes throughout the life of a cell. A major part of this regulation is governed at the level of transcription, deciding whether messengers to specific genes are produced or not. This decision is triggered by the action of transcription factors, proteins which interact with specific sites on DNA and thus influence the rate of transcription of proximal genes. Mapping the organisation of these transcription factor binding sites sheds light on potential causal relations between genes and is the key to establishing networks of genetic interactions, which determine how the cell adapts to external changes. In this work I review briefly the basics of genetics and summarise popular approaches to describe transcription factor binding sites, from the most straight forward to finally discuss a biophysically motivated representation based on the estimation of free energies of molecular interactions. Two articles on transcription factors are contained in this thesis, one published (Aurell, Fouquier d'Hérouël, Malmnäs and Vergassola, 2007) and one submitted (Fouquier d'Hérouël, 2008). Both rely strongly on the representation of binding sites by matrices accounting for the affinity of the proteins to specific nucleotides at the different positions of the binding sites. The importance of non-specific binding of transcription factors to DNA is briefly addressed in the text and extensively discussed in the first appended article: In a study on the affinity of yeast transcription factors for their binding sites, we conclude that measured in vivo protein concentrations are marginally sufficient to guarantee the occupation of functional sites, as opposed to unspecific emplacements on the genomic sequence. A common task being the inference of binding site motifs, the most common statistical method is reviewed in detail, upon which I constructed an alternative biophysically motivated approach, exemplified in the second appended article. / QC 20101110