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Mitochondria1 are found in virtually every eukaryotic2 cell. They are primarily a site of cellular respiration, that is, the breaking down of glucose to release energy, thus enabling the cell to grow, make proteins and lipids (fats), move and, most importantly, to make more cells.
What Do They Look Like?
Mitochondria can be a variety of shapes, but you usually see them drawn like a peanut. Well, a peanut with a funny squiggly blob thing inside it, actually. This represents a double membrane: a mitochondrion has a smooth outer membrane and an inner membrane which is very folded and wrinkly. The stuff inside both membranes is called 'the matrix' and the bit in between both membranes is called 'the intermembrane space'. (It would seem that certain biologists don't have much imagination when it comes to naming things). The folds of the inner membrane are called cristae.
How Do They Do What They Do?
First off, anyone wanting to know about cellular biology had better be familiar with the common three letter acronyms, also known as TLAs3:
DNA4 - deoxyribonucleic acid
This is basically the manager of the cell. It tells the rest of the cell what to do while it sits in the nucleus doing nothing.
RNA - Ribonucleic acid
This comes in three forms, mRNA, tRNA and ribosomal RNA (rRNA). mRNA is 'messenger' RNA. It is the DNA's secretary. It takes all its orders and makes sure that they get done. tRNA is 'transfer' RNA. This is the actual worker, as it takes the message the mRNA gives it and builds proteins.
IMP - Integral membrane proteins
If you think of a protein as a blob which can do stuff, then this is a blob which can do stuff which happens to be stuck right through (or just partially through) a membrane. Simple.
ADP - Adenosine DiPhosphate and ATP - Adenosine TriPhosphate
ADP is a flat battery. If you put energy into it (as the mitochondria do) it becomes ATP, a charged up battery. The cell can then transport it to where it is needed and use it, forming ADP again, which is cycled back to the mitochondria to be recharged.
The process of converting glucose into energy has several steps. This process is called cellular respiration.
Outside the mitochondrion, glucose is broken in half to form two molecules of pyruvate. This process also releases enough energy to make two ADP molecules into ATP molecules, and also some stuff called NADH. This is kind of like ATP.
Inside the matrix, this pyruvate is broken down even further into carbon dioxide (CO2) and water (H2O) through something called the Krebs Cycle. This releases enough energy for another two ATP molecules, and a lot more NADH.
The Inner Membrane
All this NADH floats around until it comes in contact with a special IMP on the inner membrane of the mitochondrion. Now, this NADH has a very excited electron, buzzing around it like an angry hornet. It gives this electron to the IMP and, through a series of complicated reactions, the IMP pumps Hydrogen Ions (H+, protons) across the membrane into the intermembrane space. The electron is then passed along to another IMP and the same thing happens. This repeats again, and then the electron, having lost much of its 'angryness', is palmed off on an oxygen molecule (along with some hydrogen ions) to make water.
Just like people don't like getting pushed around, hydrogen ions don't like getting pushed around either. In fact, hydrogen ions are pretty antisocial, in that they will always try to spread out until there is the maximum amount of space between them. Because of this, they keep trying to get back out of the intermembrane space (which is quite small, and is now packed with hydrogen ions), but they can't just go straight through the membrane, they have to go through a special IMP known as ATPsynthase. This is just like a turnstile, and in order to get back, the hydrogen ions must turn the turnstile, which, in turn, creates ATP from ADP.
Now, just to let you know just how much ADP is turned into 'charged up' ATP: step one makes two molecules, step two makes two molecules, and this final step creates about 32. You can see that the final step produces a lot more energy than the previous two steps.
Where Did It Come From?
Mitochondria contain their own DNA, RNA and ribosomes (things for making proteins) and are (almost) capable of replicating completely on their own. Their DNA is in a loop, like bacteria, and their ribosomes are smaller than normal, like bacteria. Also, mitochondria are able to self-replicate. They cannot replicate outside a living cell, but if the cell needs more energy, they can reproduce themselves. This has caused some to suggest that perhaps at one time mitochondria were bacteria. These bacteria found that by living inside another cell they got protected and provided with plenty of food. The cells they lived in found that the mitochondrial method of aerobic respiration (using oxygen, produces 34 ATP) was much more efficient than their own anaerobic respiration (without oxygen, produces two ATP), and so they lived together, happily ever after.
The fact that mitochondria contain their own DNA is also very useful as a tool for determining parentage, and in other genealogical exercises. As the sperm cell contains mitochondria only in its tail, which falls off when it enters the egg, mitochondrial DNA is only inherited from the mother. Using this we can trace the maternal line of a person very accurately (using ordinary nuclear DNA is much harder because it is all mixed up with each new generation).