Putting it very simplistic: it is all about (physical) chemistry. Molecule X binds to a receptor, which changes its form, which triggers another reaction, etc.

It’s a huge chain reaction with some probability distribution (depending on binding strength, temperature, etc.)

Structure determines function. The structure of the Golgi apparatus determines its function, like the capacity for retrograde and anterograde vesicle transport. Built within the super structure are a myriad of sensors and regulatory pathways. https://www.frontiersin.org/articles/10.3389/fpls.2020.609516/full

There are also overlapping pathways and direct connections between many of the eukaryotic cell organelles, resulting in “organelle-organelle communication”, allowing the entire cell to function and each organelle to “know” what some others are doing. https://cellandbioscience.biomedcentral.com/articles/10.1186/s13578-019-0289-8

So a protein is a very big molecule. It’s so big that parts of it are attracted to other parts and cling together, think of a balloon you’ve rubbed on your hair and stuck to a wall.

These self-stickiness gives proteins a folded shape. There’s still some charge, but since the protein is folded a specific way, most other molecules don’t fit to bind to those charged areas.

However, most proteins have at least one other protein that does fit. So when a protein “meets its match” the two can bind together. When they do that, their charges mingle and change and the proteins change shape.

This act, proteins changing shape as a result of things binding to them, is how your cell “does” stuff.

So in your Golgi apparatus example, the proteins don’t know what they’re doing at all. But they have a specific shape and charge and so attract and bind specific other proteins. When they do so, they change shape and that results in the golgi doing something.

That's a little like asking how does a taco truck know that its a taco truck. The truck itself doesn't know what it does, but inside it has the ingredients for tacos, a taco chef, and all the signage on the outside of the truck indicating its a taco truck.

The same works in cells. The golgi apparatus is essentially just a collection of membranes but its the proteins within the membranes that give it the function. And if you took out all those proteins and swapped in endoplasmic reticulum proteins, then it'd be the ER (just like if you swapped out taco ingredients for ice cream in your food truck). The proteins within organelles gather together through inter-molecular interactions. There are multiple proteins whose role it is to gather all the golgi components and keep them together (NSF and p97 for example).

Something else to consider is how do we define organelles? While in textbooks the organelles look quite distinct from one another, they actually exist on a spectrum. For example, it was thought each cell had a single golgi apparatus housed within the cell body. Recent research has shown that there are actually other golgi-like organelles ("golgi outposts") positioned far from the cell body (like in axons of neurons) that perform some functions of the golgi but not others.

Another example would be lysosomes (a round, acidic compartment used in autophagy). The lysosomal associated membrane protein 1 (LAMP1) was considered the marker for lysosomes for a long time. However, there have been multiple studies showing that there vesicles with LAMP1 on them that are not acidified which calls into question their identity. Another marker that researchers use is a protein called spinster which, in muscle cells, localizes to a tubular network rather than vesicles. Are these networks lysosomes? Are non-acidified LAMP1 compartments lysosomes? These questions are still being debated.

Going back to the original question, how does an organelle define its function? By the proteins associated with it. And these proteins assemble using a large assortment of chaperones that collect the protein components, bring them to their destination, and assist in forming complexes. And while we talk about organelles as distinct parts that have distinct functions, they exist on a spectrum.

Assembly of the golgi: www.ncbi.nlm.nih.gov/pmc/articles/PMC5710388

Golgi outposts: www.cell.com/trends/cell-biology/fulltext/S0962-8924(20)30145-8

Heterogeneity of LAMP1 vesicles: https://pubmed.ncbi.nlm.nih.gov/29940787/, www.ncbi.nlm.nih.gov/pmc/articles/PMC6123004/

Tubular lysosomes: https://pubmed.ncbi.nlm.nih.gov/35646899/

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