Not to my knowledge. DNA is inherently a double-stranded molecule. There are 4 bases all with a sugar and a phosphate (called nucleotide). The bases are on the inside of the DNA and the sugar and phosphate interact with other sugars and phosphates to make up a backbone that is held together by PHOSPHODIESTER bonds.

The four bases are Cytosine, Thymine, Adenine and Guanine and their linear sequence combines in an infinite number of combinations. It is one half of the DNA DOUBLE HELIX. One strand of DNA comes into contact with the other strand by hydrogen bonds, a sort of electrical attraction between partially negative atoms on the base of one side with the partially positive atoms on the other. Both sides have positive and negative charges. A single such pairing would not hold the molecule together well, but several million such bonds are quite effective.

The two bases Cytosine and Thymine are pyrimidines, and Adenine and Guanine are purines. Cytosine binds to Guanine and Thymine binds to Adenine. This also has the advantage that little effort is required to pull the two halves apart for replication, when the DNA is copied, and for transcription, when the DNA message is read. The message of DNA is the information from which the cell and its components are built.

DNA is only read as a double helix, so a triple helix would be a disadvantage. Also, all the enzymes (cellular proteins that act as 'machines') need the DNA to be in a double helix to recognize their substrate (landing spot) to do their work.

A triple helix is possible for a VERY SHORT amount of time (transient) during translation, but the other strand would be RNA not DNA, and the base pairing that actually HOLDS the helix together would not be there.

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Interesting question. I have personally never heard of any organism having triple helix DNA. That would mean there would be three separate strands of DNA to form this helix. Based on the way a double-stranded DNA helix is formed, a triple-stranded helix would not be possible.

Let me explain, in your cells (and all organisms' cells) there is lots of double-stranded DNA. When it is time for the cell to divide and make a copy of itself it has to duplicate all of the DNA inside of its nucleus. Here is the way DNA is copied: First, the double stranded helix is separated into two separate single strands of DNA. Second, each single strand is read and a mirror image of each strand is created. Third, the mirror image that was created binds to the original single strand that was read and a new double helix is formed. Because this happens to both single strands of the original double-stranded helix, you end up with two new double-stranded helices when you started with just one.

To help you understand the structure of DNA and how the two single strands fit together to form the helix, here is an analogy I thought up (hope it helps):
Imagine that each strand of DNA is composed of four different structures, such as: square pegs, round pegs, square holes and round holes. Obviously, square pegs match with square holes and round pegs match with round holes. When DNA is in its double-stranded configuration, the two strands have to fit together as you would imagine the pegs fitting into their appropriate shaped holes. So if you had one strand that had a sequence like: round peg-square hole-square hole-square peg-round hole. Then the strand of DNA that would fit together with this strand would be the exact opposite:round hole-square peg-square peg-square hole-round peg.

See how these two strands if lined up end to end, would fit together because they are the opposite of each other? One these two strands are fit together it is not possible for another strand to fit in because all of the pegs and holes are occupied by the perfect fit with the other strand.

When DNA coils up to form a double-stranded helix it is a very tight coil and it requires little energy to stay in this shape. Any other configuration would require more energy and is therefore not as desirable (cells, just like humans like to spend the least energy they can on any activity).

When it's time to make a copy of the DNA, a special molecule, called an cutting enzyme, comes along and separates the two strands of DNA (pulls the pegs out of the holes, in my example). After the two strands are separated, another special molecule (another enzyme) comes along and, beginning at one end of one of the strand, the enzyme "reads" the strand one piece at a time (to find out whether each piece is a round hole or a square peg or a round peg or a square hole), then it places the appropriate match on a growing new strand of DNA. So if this special enzyme reads the existing single strand of DNA has a square peg at one position, then it gets a square hole and puts it on the new, growing strand of DNA that will bind with the one it is reading.

This process continues along until the entire strand of DNA has been read and a new matching, but opposite strand has been created right next to it. Immediately after the new strand is made it binds to the strand that was read to form a new double-stranded helix that is an exact copy of the original double-stranded helix.Can you see how a triple helix would be impossible with this mode of DNA duplication and strand binding?

Do you know when and how the double-stranded helix configuration of DNA was discovered?

To answer your question: There is no organism known to me that have a tipple DNA strand.

It is, however, possible to bind short oligonucleotides (very short DNA pieces that are synthetically made from nucleotides ( = phosphate sugar and purine or pyrimidine base )) to a specific DNA site. This is extensively studied for gene regulation.

Let me try to explain this. Gene expression is the transfer of the information encoded within the genome into proteins via an intermediate message, mRNA. Each step involves an amplification step: multiple copies of mRNA are transcribed from the DNA and a large number of protein molecules are translated from each mRNA. Each step is highly controlled by the action of proteins. These proteins form specific protein-DNA complexes which are necessary so that the transcription or translation can happen.

It would be nice if we could control gene expression externally for instance to correct some malfunction in the cell or as a means of killing foreign cells (like bacteria,viruses etc). For example, inhibition of specific proteins (usually enzymes) by drugs is already used by the pharmaceutical industry. Here control is taken place after the proteins are translated.

But in addition to inhibit particular gene products, it may be more efficient to interfere with earlier stages of its production like at the level of the gene itself. It is possible to design drugs that bind to DNA. The complex formed between the Drug and the DNA is then unable to be transcribed. (This is pretty cool but we are only at the beginning). Another possibility which is very specific is using oligonucleotides that recognize DNA duplexes (the DNA helix) and by hydrogen bonding (H--H bonds between the bases) bind to the DNA in its major groove and form a DNA triple helix. These triple helical structures will play an important role in the control of gene expression in the future because the are highly specific. This means that if you know the DNA sequence which you want to block you can synthesize an oligonucleotide than binds only to that region and nowhere else. This triple helix can not be transcribed and therefor the "bad" protein will not be synthesized.