The saltation–abrasion model predicts rates of river incision into bedrock as an explicit function of sediment supply, grain size, boundary shear stress and rock strength. Here we use this experimentally calibrated model to explore the controls on river longitudinal profile concavity and relief for the simple but illustrative case of steady-state topography. Over a wide range of rock uplift rates we find a characteristic downstream trend, in which upstream reaches are close to the threshold of sediment motion with large extents of bedrock exposure in the channel bed, while downstream reaches have higher excess shear stresses and lesser extents of bedrock exposure. Profile concavity is most sensitive to spatial gradients in runoff and the rate of downstream sediment fining. Concavity is also sensitive to the supply rate of coarse sediment, which varies with rock uplift rate and with the fraction of the total sediment load in the bedload size class. Variations in rock strength have little influence on profile concavity. Profile relief is most sensitive to grain size and amount of runoff. Rock uplift rate and rock strength influence relief most strongly for high rates of rock uplift. Analysis of potential covariation of grain size with rock uplift rate and rock strength suggests that the influence of these variables on profile form could occur in large part through their influence on grain size. Similarly, covariation between grain size and the fraction of sediment load in the bedload size class provides another indirect avenue for rock uplift and strength to influence profile form. Copyright © 2008 John Wiley & Sons, Ltd.