Sea-ice ecosystems are characterized by micro-algae living in brine inclusions. The growth rate of ice algae depends on light and nutrient supply. Here, the interactions between nutrients and brine dynamics under the influence of algae are investigated using a one-dimensional model. The model includes snow and ice thermodynamics with brine physics and an idealized sea-ice biological component characterized by one nutrient, namely dissolved silica (DSi). In the model, DSi follows brine motion and is consumed by ice algae. Depending on physical ice characteristics, the brine flow is advective, diffusive or turbulent. The vertical profiles of ice salinity and DSi concentration are solutions of advection–diffusion equations. The model is configured to simulate the typical thermodynamic regimes of first-year Antarctic pack ice. The simulated vertical profiles of salinity and DSi qualitatively reproduce observations. Analysis of results highlights the role of convection in the lowermost 5–10 cm of ice. Convection mixes saline nutrient-poor brine with comparatively fresh nutrient-rich sea water. This implies a rejection of salt to the ocean and a flux of DSi to the ice. In the presence of growing algae, the simulated ocean-to-ice DSi flux increases by 0–115% compared with an abiotic situation. In turn, primary production and brine convection act in synergy to form a nutrient pump. The other important processes are the flooding of the surface by sea water and the percolation of meltwater. The former refills nutrients near the ice surface in spring. The latter, if present, tends to expell nutrients from the ice in summer.