Blooms of the seaweed <em>Sargassum</em> have been reported in the Tropical Atlantic Ocean and Caribbean Sea since 2011. Large-scale inundation events of this seaweed in coastal regions have a negative impact on both the economy and the environment. To predict the timing, location and quantity of <em>Sargassum</em> strandings, model frameworks that link open ocean distributions to coastal regions are required. Here, we develop an open-source customizable Lagrangian simulation framework and growth model for <em>Sargassum</em> that combines transport by currents, wind and waves with a biological modelling framework that includes dynamic growth limitation depending on temperature, nitrate availability and salinity. The framework can use satellite detections of the <em>Sargassum</em> Watch System (SaWS) to initialise virtual <em>Sargassum</em> particles in the Tropical Atlantic Ocean. We demonstrate that the combination of physical transport and biological growth strongly affects <em>Sargassum</em> distribution, with substantial variability on spatio-temporal scales. We show that temperature is the strongest growth-limiting parameter, and in particular that elevated surface temperatures, together with low salinity from the Amazon River plume, play a crucial role in <em>Sargassum</em> decline.