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The heat shock response is a well-conserved defence mechanism against the accumulation of misfolded proteins due to prolonged elevated heat. The cell responds to heat shock by raising the levels of heat shock proteins (hsp), which are responsible for chaperoning protein refolding. The synthesis of hsp is highly regulated at the transcription level by specific heat shock (transcription) factors (hsf). One of the regulation mechanisms is the phosphorylation of hsf's. Experimental evidence shows a connection between the hyper-phosphorylation of hsfs and the transactivation of the hsp-encoding genes. In this paper, we incorporate several (de)phosphorylation pathways into an existing well-validated computational model of the heat shock response. We analyze the quantitative control of each of these pathways over the entire process. For each of these pathways we create detailed computational models which we subject to parameter estimation in order to fit them to existing experimental data. In particular, we find conclusive evidence supporting only one of the analyzed pathways. Also, we corroborate our results with a set of computational models of a more reduced size.