Atash what is the woody looking material in the photo of Tom digging?
It's not woody; it's just potato-"vines" that have been bleached down to their cellulose fibers.
I have more to say about the freeze resistance of the tubers, but will wait til I have more time. Thanks for sharing Atash. Funny how two people can write about the same subject, yet the imagery is so strikingly different.
Take it easy on me, Chief, I'm not a botanist and have very little understanding of biochemistry.

What I do know is about the functioning of anti-frost mechanisms on a very gross scale of things. You've seen my garden full of oddly exotic plants. How is it for example that a Banana--namely Musa basjoo--can survive in the decidedly non-tropical climate of Seattle?
What tends to kill plant tissues in cold weather is the formation of ice crystals in their cells, that destroy delicate cellular substructures.
The solution to frost is generally to keep those ice crystals from forming. The mechanisms include
* higher starch and sugar concentrations, that lower the freezing point of the cellular fluid
* specific proteins that inhibit the ice crystals from finding a nucleus to form around
* removal of water from the cells. This is how it is possible for tropical plant seeds to survive freezing! But non-seed tissues of coldhardy plants can generally do it to a certain degree--that's one of several reasons why coldhardy plants often wilt in cold weather (the other being that their roots can't supply water anyway when the ground is frozen...).
Interestingly, the ability to regulate water in plant tissues explains why the most cold-resistant Eucalyptus grown here in the Northwest are NOT the ones from the highest, coldest elevations in Australia and Tasmania, but oddly, from wetlands.
Eucalyptus neglecta is one of the most cold-resistant Eucalyptus grown here--it essentially never freezes back, even though its broad leaves which are prone to collecting snow that can damage the branches from too much weight, are poorly-adapted to snowy climates. Its native climate isn't snowy--it comes from only a few thousand feet elevation, a bit frosty but not prone to heavy snow. It can tolerate cold because it regulates the amount of moisture in its tissues, which it has to do, because its habitat is the banks of a single river.
On the other side of the coin, most cacti can tolerate frost in habitat, but not in maritime climates, where winter rains saturate their tissues and make them vulnerable to frost (the exceptions are mostly those species with so much hardiness to spare that it doesn't matter too much, and even they need good drainage).
The easiest frost-resistance mechanism to develop is probably to have higher concentration of starch and sugar.
One more factor partially related to frost-tolerance:
the temperature range at which biochemical processes are optimized can make the difference between life and death. I do not know a lot about the underlying biochemistry, but I do know that some plants grow best at relatively high temperatures (corn and amaranth, that do C4 metabolism), and some plants are adapted to grow at low temperatures. There are something like 3 or 4 bands of temperature ranges, that different plants tend to fall into.
One thing that can happen, is if a plant's active temperature range is too high compared to the ambient temperature, even if temperatures are above freezing, it can die from lack of enough activity to fend off pathogens active at much lower temperatures.
There are undoubtedly more mechanisms I am not aware of.