Cells are complex biological units with advanced signalling systems, a dynamic capacity to adapt to its environment, and the ability to divide and grow. In fact, they are of such high level of complexity that it has deemed extremely difficult or even impossible to completely understand cells as complete units. The search for comprehending the cell has instead been divided into small, relatively isolated research fields, in which simplified models are used to explain cell biology. The result produced through these reductionistic investigations is integral for our current description of biology. However, there comes a time when it is possible to go beyond such simplifications and investigate cell biology at a higher level of complexity. That time is now. This thesis describes the development of mathematical tools to investigate intricate biological systems, with focus on heterogeneous protein interactions. By the use of simulations, real-time measurements and kinetic fits, standard assays for specificity measurements and receptor quantification were scrutinized in order to find optimal experimental settings and reduce labour time as well as reagent cost. A novel analysis platform, called Interaction Map, was characterized and applied on several types of interactions. Interaction Map decomposes a time-resolved binding curve and presents information on the kinetics and magnitude of each interaction that contributed to the curve. This provides a greater understanding of parallel interactions involved in the same biological system, such as a cell. The heterogeneity of the epidermal growth factor receptor (EGFR) system was investigated with Interaction Map applied on data from the instrument LigandTracer, together with complementing manual assays. By further introducing disturbances to the system, such as tyrosine kinase inhibitors and variation in temperature, information was obtained about dimerization, internalization and degradation rates. In the long term, analysis of binding kinetics and combinations of parallel interactions can improve the understanding of complex biomolecular mechanisms in cells and may explain some of the differences observed between cell lines, medical treatments and groups of patients.

The time-course of interaction formation or breaking can be studied using LigandTracer, and the data obtained from an experiment can be analyzed using a model of ligand-target binding kinetics. There are different kinetic models, and the choice of model is currently motivated by knowledge about the interaction, which is problematic when the knowledge about the interaction is unsatisfactory. In this project, a Bayesian model selection procedure was implemented to motivate the model choice using the data obtained from studying a biological system. The model selection procedure was implemented for four kinetic models, the 1:1 model, the 1:2 model, the bivalent model and a new version of the bivalent model.Bayesian inference was performed on the data using each of the models to obtain the posterior distributions of the parameters. Afterwards, the Bayes factor was approximated from numerical calculations of the marginal likelihood. Four numerical methods were implemented to approximate the marginal likelihood, the Naïve Monte Carlo estimator, the method of Harmonic Means of the likelihood, Importance Sampling and Sequential Monte Carlo. When tested on simulated data, the method of Importance Sampling seemed to yield the most reliable prediction of the most likely model. The model selection procedure was then tested on experimental data which was expected to be from a 1:1 interaction and the result of the model selection procedure did not agree with the expectation on the experimental test dataset. Therefore no reliable conclusion could be made when the model selection procedure was used to analyze the interaction between the anti-CD20 antibody Rituximab and Daudi cells. / Interaktioner kan analyseras med hjälp av LigandTracer. Data från ett LigandTracer experiment kan sedan analyseras med avseende på en kinetisk modell. Det finns olika kinetiska modeller, och modellvalet motiveras vanligen utifrån tidigare kunskap om interaktionen, vilket är problematiskt när den tillgängliga informationen om en interaktion är otillräcklig. I det här projektet implementerades en Bayesiansk metod för att motivera valet av modell utifrån data från ett LigandTracer experiment. Modellvalsmetoden implementerades för fyra kinetiska modeller, 1:1 modellen, 1:2 modellen, den bivalenta modellen och en ny version av den bivalenta modellen. Bayesiansk inferens användes för att få fram aposteriorifördelningarna för de olika modellernas parametrar utifrån den givna datan. Sedan beräknades Bayes faktor utifrån numeriska approximationer av marginalsannolikeheten. Fyra numeriska metoder implementerades för att approximera marginalsannolikheten; Naïve Monte Carlo estimator, det harmoniska medelvärdet av likelihood-funktionen, Importance Sampling och Sekventiell Monte Carlo. När modellvalsmetoden testades på simulerad data gav metoden Importance Sampling den mest tillförlitliga förutsägelsen om vilken modell som generade datan. Metoden testades också på experimentell data som förväntades följa en 1:1 interaktion och resultatet avvek från det förväntade resultatet. Följaktligen kunde ingen slutsas dras av resultet från modelvalsmetoden när den sedan används för att analysera interaktionen mellan anti-CD antikroppen Rituximab och Daudi-celler.