How big could celled organisms grow? Is there a limit?

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I'm going to answer this for multi-cellular organisms, as all organisms are celled, so I'm not completely sure what you mean by the question.

There is certainly a limit on how large organisms could grow. Remember MRS NERG, the definition of life. Every living organism moves, respires, is sensitive, requires nutrition, excretes, reproduces and grows. Specifically, every organism will die if it cannot respire, ingest nutrition or excrete the byproducts of respiriation. 

These processes require some way of providing the organisms cells with the things it needs. For example, mammals take in oxygen for respiration via breathing through their lungs. This oxygen is then transported to the cells by the blood. They ingest nutrition through the digestive tract and the nutrients are also delivered to the cells via the blood. The byproducts of respiration and other cellular processes are removed from the mammal by the excretion of carbon dioxide during exhalation and by the kidneys in urine.

This isn't the same throughout the animal kingdom though. Insects don't 'breathe' through lungs but instead the oxygen is diffused through their bodies through pores in their exoskeleton. The larger the insect gets, the further the oxygen has to diffuse and the longer it takes to get there. As an insect gets larger, this becomes less and less efficient until it is no longer possible to use this as a method of oxygen transfer to sustain life. This is why you never see huge, human sized insects. In pre-historic times however, there has been more oxygen in the atmosphere and this meant that there have been dragonflies with wingspans over 50cm!

So, the same thing happens with the mammal respiratory system. There is only a certain size range in which lungs and blood circulation are efficient ways of moving oxygen aroung the body. Tiny animals the size of an ant could not really use lungs, a heart and blood as at such minute scales, the fluids involved would be much more viscous and hard to pump.

Similarly with mammals the size of a mountain (I.E. much bigger than a blue whale), the amount of effort required to pump the blood around the body would be enormous and simply not feasable for any life form. Not only that, but actually getting the sheer amount of nutrients required to grow that big would take an incredible digestive system or a very long period of time.

So yes, there are certainly limits to the size of an organism. This is simply by the fact that methods of getting oxygen only work to a point and the fact that no organism has evolved methods of getting quite that much nutrition and protein to grow to such sizes.

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