We present the results of Brownian dynamics simulations of a series of different polymer models which have been used to examine the recent experimental findings of Smith et ad. (1999) who studied the dynamics of a single DNA molecule in steady shear flow. The steady average extension at various Weissenberg numbers (Wi) is shown to be well predicted by multimode nonlinear models. Quite surprisingly, the normalized average extension x/L asymptotes to less than 1/2 even for extremely large Wi and we discuss this result on a physical basis. The probability density function of molecular extension at various values of Wi using the Kramer's chain and the finitely extensible nonlinear elastic dumbbell suggests that the number of internal modes is important in a model designed to capture the dynamics of a real DNA molecule. Three different frequency regimes in the power spectral density observed at finite Wi in the experiments are shown to arise from the coupling of the Brownian fluctuations in the gradient direction and the convection in the streamwise direction. Our simulation results, especially in light of the excellent agreement with experiment, demonstrate the basic physical elements necessary for any rheological model to capture the dynamics of single polymer chains in flow.