We examine through analytical calculations and finite element simulations how the detection efficiency of disk and wire-like biosensors in unmixed fluids varies with size from the micrometer to nanometer scales. Specifically, we determine the total flux of DNA-like analyte molecules on a sensor as a function of time and flow rate for a sensor incorporated into a microfluidic system. In all cases, sensor size and shape profoundly affect the total analyte flux. The calculations reveal that reported femtomolar detection limits for biomolecular assays are very likely an analyte transport limitation, not a signal transduction limitation. We conclude that without directed transport of biomolecules, individual nanoscale sensors will be limited to picomolar-order sensitivity for practical time scales.