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Can photosynthesis take place in a plant even if the plant is kept in ice cold water? I have googled this question, but so far have not found a satisfactory answer.

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Before answering the question, I assume that the plant is somehow getting enough O2 to survive.

There can be a number of variations in the answer depending upon the more particular situation you give. You can have a look here to see a list of factors affecting rate of photosynthesis in plants. I put graphs of factors important for this question:

light intensityCO2 concentrationtemperature

If we talk about only cold climate, then your answer is given here as it says:

Plants of tropical or subtropical origin may incur irreversible damage to photosynthesis by temperatures below about 10°C, whereas plants from cooler climates may photosynthesize and develop normally at temperatures down to around 0°C.

So, I assume now that the plant you are talking about is adapted to cold climate i.e. you are not bringing a plant from Africa to North Pole. But, as you would have seen in the list of factors, you must have noticed that photosynthesis also depends on a lot more factors.

Another crucial factor under water is CO2 concentration. The plant needs enough amount of CO2 to carry on photosynthesis. In an ideal water body, if a plant can survive, then algae/bacteria can thrive, which would together produce lots of CO2 (by respiration) which would dissolve in water. For effect of temperature on CO2 solubility in water, see this link. It states:

solution of CO2 in water gives carbonic acid

As the forward reaction is exothermic, a decrease in temperature causes the system to move in the forward direction.

We don't need to worry about how plant would use H2CO3: plants have carbonic anhydrase for it. See this article:

Carbonic anhydrase, an enzyme which catalyzes the reversible hydration of CO2, is a major protein component of most photosynthetic microorganisms and higher plant tissues.

So, ideally, we can say that Yes, photosynthesis can occur in ice cold water if water is not frozen, has enough purity to let sufficient amount of light pass through it, and the plant has reached almost at the surface of water body to get enough carbon dioxide both from the atmosphere as well as the dissolved CO2 in the water body (which was produced by respiration of bacteria, aquatic animals and plant itself).

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Biochemically, it appears it can, as some Algae can. The question is then whether there are any real plants who do it.

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    $\begingroup$ Algae are "real" plants. However, diatoms are not really algae. That is an obsolete classification. Perhaps you can edit your answer to clarify this. $\endgroup$ – WYSIWYG Apr 20 '16 at 8:55

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