The short answer is that cells come from other cells. Read on to find out how they do this.

One way is through cell division: an existing cell doubles its DNA and then divides in half. That DNA then makes the new cell into an exact copy of the old cell. Single-celled bacteria, single-celled algae and single-celled animals quite often just divide in half to make more cells. (An example of a single-celled animal is an amoeba and an example of single-celled algae is a diatom. You can look at pictures of amoeba and diatoms on Google by typing the words "pictures of amoebas and diatoms".

Certain cells in your body are formed this way (from existing cells that double their DNA and divide in half). You are constantly shedding dead skin cells and they are replaced by a layer of cells just at the base of your skin that are constantly dividing.

You make millions and millions of new blood cells each day from cells inside your bones (bone marrow). If you've heard of stem cells, these are cells that divide to make new cells but the difference is that these new cells can become any type of cell in your body that you need. You can see how this would be really helpful if you needed to repair a damaged organ or a bad heart, for example. This is unusual, however. We have very few natural stem cells in our bodies. But biologists are trying to find ways to use stem cells to help our bodies heal themselves. An example that looks really promising is a potential cure for diabetes. People with diabetes cannot produce insulin, and so have to take regular injections. Researchers think that by harvesting stem cells from brain tissue, transforming those stem cells in the lab into cells that can produce insulin, and then injecting the cells into the kidney of diabetes patients, the cells will survive and reproduce so that the patients can make their own insulin.
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The US Federal Government has made stem cell research very difficult, however, and so other countries are doing a lot of the pioneering research now. Case in point: Don Ho, who lives in Hawaii, recently had a stem cell procedure done in Thailand to repair his heart.

A zygote is a brand new cell that divides rapidly into a ball of new cells (an embryo) which then produces stem cells. These stem cells then transformed into all the different cells types you have in your body today: muscle cells, nerve cells, blood cells, skin cells, kidney cells, lung cells, brain cells, etc.

The ultimate question of biology, then, is how did the first cell come about? The study of evolutionary biology seeks to answer that question (along with many others), although it's hard to say if we'll ever really figure out exactly how it happened.

There are two answers to this question: proximate (i.e., short-term) and ultimate (i.e., long-term). The proximate origin of cells is, of course, other cells. Most cells reproduce asexually through cell division, which is called binary fission in prokaryotes and mitosis in eukaryotes (with the exception of sex cells or gametes that form through meiosis). However, some cells, notably bacteria and other prokaryotes are able to borrow parts of each others DNA or RNA in a process more akin to sexual reproduction.

Prokaryotes are more primitive cells that do not have a nucleus to enclose their genome and do not have membrane-bound organelles, whereas eukaryotes contain their genome (DNA orRNA) within a nucleus and have membrane-bound organelles.

The ultimate origin of cells is a much more difficult question and involves the actual evolution of cells from the elements and molecules that occurred on Earth hundreds of millions of years ago. Cells are essentially just self-replicating bundles of ribonucleic acids (RNA or DNA) that are enclosed in a semi-permeable membrane. There is quite a bit of experimental evidence to show that certain types of molecules called phospholipids automatically assemble into spherical membranes when exposed to an aqueous environment (thephospholipids have a water-attracting hydrophilic end that points outwards and a water-repellant hydrophobic end that points inwards). Likewise, other types of molecules called nucleic acids are capable of making copies of themselves by polymerizing nucleotides. At some point in time, phospholipids membranes enclosed nucleic acids, forming a primitive cell.

The phospholipid membrane must have been permeable to the nucleotides that nucleic acids are composed of, or the cell must have had some means to assemble nucleotides from basic molecules or elements that could penetrate the membrane. The nucleic acids then began to duplicate themselves by polymerizing nucleotides. Then, the phospholipid membrane must have increased in size until it was no longer stable and split into two smaller membranes (binary fission), each containing replicate nucleic acids.

From here, different cells adapted or evolved to perform specialized functions by producing or incorporating new molecules into their body. At some point, the nucleic acids began to code for the production of amino acids and their polymerization into proteins, such as enzymes. These proteins then became involved in carrying out important biochemical reactions.

For instance, some of these enzymes may have assisted in transporting necessary elements or molecules through the membrane (e.g., via endocytosis) to assist the replication of the nucleic acids or the production of new proteins. Others may have been responsible for isolating and excreting waste materials from the cell (e.g., via exocytosis). Others may have assisted in translating nucleic acids into proteins, packaging and folding the proteins so that they function correctly, or breaking down and recycling used or defective proteins. Mutations in the nucleotide sequence of nucleic acids would cause changes in these proteins that are produced, and these changes, if adaptive, would eventually lead to the evolution of new cell types.

Finally, some prokaryotic cells were engulfed but not digested by other cells (probably amoeboid-like eukaryotes), were incorporated into the larger cells body, and then co-opted to carry out novel functions. For instance mitochondrion and chloroplasts are organelles that have their own double phospholipid membranes and separate genomes from their parent cells. Some scientists believe that mitochondria and chloroplasts were independent prokaryotic cells many millions of years ago, but at some point were incorporated into eukaryotes and used by their new host cells for oxidative phosphoryltation and photosynthesis, respectively. This is called the Theory of Endosymbiosis.