  Life and functioning of higher organisms depends on the continuous supply of
metabolites to tissues and organs. What are the requirements on the transport
network pervading a tissue to provide a uniform supply of nutrients, minerals,
or hormones? To theoretically answer this question, we present an analytical
scaling argument and numerical simulations on how flow dynamics and network
architecture control active spread and uniform supply of metabolites by
studying the example of xylem vessels in plants. We identify the fluid inflow
rate as the key factor for uniform supply. While at low inflow rates
metabolites are already exhausted close to flow inlets, too high inflow flushes
metabolites through the network and deprives tissue close to inlets of supply.
In between these two regimes, there exists an optimal inflow rate that yields a
uniform supply of metabolites. We determine this optimal inflow analytically in
quantitative agreement with numerical results. Optimizing network architecture
by reducing the supply variance over all network tubes, we identify patterns of
tube dilation or contraction that compensate sub-optimal supply for the case of
too low or too high inflow rate.
