Global warming is exposing plants to more frequent heat stress, with consequent crop yield reduction. Organisms exposed to large temperature increases protect themselves typically with a heat shock response (HSR). To study the HSR in photosynthetic organisms we present here a data driven mathematical model describing the dynamics of the HSR in the model organism Chlamydomonas reinhartii. Temperature variations are sensed by the accumulation of unfolded proteins, which activates the synthesis of heat shock proteins (HSP) mediated by the heat shock transcription factor HSF1. Our dynamical model employs a system of ordinary differential equations mostly based on mass-action kinetics to study the time evolution of the involved species. The signalling network is inferred from data in the literature, and the multiple experimental data-sets available are used to calibrate the model, which allows to reproduce their qualitative behaviour. With this model we show the ability of the system to adapt to temperatures higher than usual during heat shocks longer than three hours by shifting to a new steady state. We study how the steady state concentrations depend on the temperature at which the steady state is reached. We systematically investigate how the accumulation of HSPs depends on the combination of temperature and duration of the heat shock. We finally investigate the system response to a smooth variation in temperature simulating a hot day.