how do plants take in a higher concentration of nutrients than is present in solution?

This is not strictly an aquaponics question, but it came to me while thinking about aquaponics, and an aquaponic system makes it easier to think about.

Aqupaonics recycles water, running it through a biofilter (grow bed) full of plants. In the grow bed, bacteria convert nitrogen from one form to another, and plants remove nutrients. They also remove some water. Obviously, though, the plants must remove proportionally more nutrients then would be dissolved in a given amount of water, or they wouldn't clean water, they would just remove it.

Plants in the soil must be able to do this too.

How do they manage this?

This might help...

Plants have genes encoding a wide range of nutrient uptake transporters.
Each transporter has a characteristic binding affinity for a particular nutrient, and as long as the dissolved concentration of the nutrient is higher than the transporter binding affinity (binding constant), the transporter can bind the nutrient.
Then the uptake can be passive (flowing from high to low concentration), or active (using energy to pull nutrients in against a concentration gradient).

Different plants have different sets of uptake transporter genes, and may have differing intrinsic nutrient requirements to build tissues, and may change the expression of their set of transporters depending upon external conditions.
cheers, Doug Campbell (Ph.D., Plant Science).