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Natural products are chemicals produced by living organisms.
Sometimes we use this term to refer to complex and diverse molecules that have difficult-to-synthesize chemistries, but they don't have to be like that. For example, DMT is a hallucinogenic natural product synthesized by many plants and some animals, and it doesn't have a complicated structure.
Around two thirds of approved drugs originate from natural products #1, so it may seem that natural products are enriched in bioactive molecules. There are a couple of possible reasons for this apparent enrichment:
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For decades or even centuries, people have used organisms (and plants in particular) as living chemical libraries from which to draw inspiration for bioactive compounds. Currently, we have a wealth of artificial molecules that can be synthesized from scratch with combinatorial chemistry, but in the past we lacked such technology. One strategy to find new molecules and structures was to make extractions from living organisms, and so we ended up finding bioactive compounds in living organisms. You find what you look for.
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Some protein motifs and domains are highly conserved across different (eukaryotic) organisms. It may not make sense that a plant evolves antidepressant or anticancer compounds, but perhaps the interactions or the signalling pathways in which these molecules participate are closely related to some interactions or signalling pathways in mammals. Therefore, it may be reasonable to expect the chemical space of natural products to be enriched in bioactive compounds with respect to all chemical space (and in particular with respect to artificial combinatorial chemical libraries).
Therefore, natural products may indeed be enriched in bioactive molecules, but this enrichment may be blown out of proportion due to our historical sampling bias.
1 Conversation with Andreas Bender