Studies of cricket malpighian tubules revealed that potassium ions accumulated inside the tubule, moving against the potassium concentration gradient. how would you expect the movement of water to be influenced by the distribution of potassium ions?

There would be a net movement of water into the lumen of the tubules.                  

Explanation:

The Malpighian tubule system is a kind of osmoregulatory  as well as excretory system found in some insects like crickets. Potassium ions gets accumulated in the inner face of the tubule. Thus the water movement in the tubule will increase in the lumen of the tubules by the potassium ions distribution.

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