Yes, Molecular self-assembly is the process by which molecules adopt a defined structure without external intervention.
It can be either intramolecular (folding) or intermolecular.
Protein Folding is the process via which peptides, that can display several conformations in unfolded state, adopt a stable (lowest free energy) native fold. Thus, the native fold is determined by the primary structure (aminoacidic sequence) according to the Anfinsen's dogma.
Nicolò M. Villa, the native structure is the lowest free energy state accessible in reasonable timescales. From a theoretical point of view it is impossible to say if the native structure is indeed the lowest possible free energy state, since the protein does not have the possibility to sample its phase space completely.
You got an excellent question and a few partially correct but misleading answers.
There is no place not time to explain all the intricacies of the protein folding problem here which is mostly misleading and ill-posed problem. Let's suffice to state that if the answers of two previous responders were true, life on earth, or as a matter of fact anywhere would not exist. A table, or a brick are in the lowest free energy state (with a caveat of the individual environment) and they are dead as a stone. Anfinsen's principles were invalidated or severely limited a very long time ago. Proteins as well as all four basic biochemical macromolecules (proteins, nucleic acids, sugars and lipids) have similar self organization rules governed by Van Der Waals or London forces (hydrophobicity, electrostatics, dynamic stability etc). In order to function as carriers of life they have to combine two contradictory features: having some semi-stable basic form, and having enough mobility (read disorder) to perform the function. All of these classes of macromolecules are dynamically coupled with water based solvents. All of them have some characteristic proportion of atoms in predictable three-dimensional arrangement (existence of stable structure) and the other proportion in partially, or completely dynamic (disordered) states. The Anfinsen's results were challenged by himself when he extended the folding studies to variable conditions (heat and cold denaturation). But a real death of the naive folding paradigm was dealt by discovery of prions and amyloids, Paracelsus challenge, ambivalent sequences in proteins, statistical studies on proportion of folded, to partially folded, to unfolded proteins in genomes and many, many other studies. Actually there was a full edition of Science in 2008 devoted to the issue of structure and its uniqueness with calmodulin picture on the cover. I personally contributed to this business with several contributions but mostly by formulating a new paradigm that states that every protein has a characteristic proportion of stable to unstable elements needed to perform the evolutionary attained function. The results were published in PNAS 106(2009)10505. From a physics point of view proteins are called "self-organized criticality". Actually every biological macromolecule is. This means like all life we are semi stable (conditionally stable) "pattern" that survives in time and space for some time to disappear in the end to thermal death when real physical equilibration happens. So in essence proteins are self assembling, but only sometimes and in appropriate conditions. However, the same word (self-assembly) can be assigned to formation, and emergence of prions or amyloids formed by different proteins not in native forms that cause terrible diseases as those that are essential components of life. In complex eukaryotes a minority of the proteins are actually ever stably folded.
However, what seems to follow from quantum chemistry/physics is that proteins are unique molecules in terms of their flexibility and sensitivity to the env. A protein has a 3D network of the quantum/tunneling H-bonds, a network of the ‘aromatic’ fragments with their highly-sensitive pi-electrons, a network of trans metals/ions with their highly-responsive 3d, 4f, etc electrons - just to name a few active contributors. All that makes a protein, in particular being in a structured env, a unique substrate that is non-stop involved in an active
interaction with its local and remote ‘neighbors’ that includes electronic, vibrational, and rotational degrees of freedom that allow for non-stop transformative changes.
So, for a naive reader, like me, no folding and no ‘self’-assembling AT ALL. All said makes a protein the unique molecule indeed, deeply-involved in the non-stop transformations until the (local) conditions/constraints STOP these dynamics and make possible (depending on the local context) (de)polymerization, crystallization, melting, and basically any unlimited transformations.
Also what seems to be important is protein’s memory (based on H-bonds). The memory makes a protein a substrate learning the env and adapting its structure and responses properly - making the protein structure consistent/matching its local env. Thus, this is more a local env-driven highly-specific reorganization of the protein structure, and less so-called ‘self’-organization. In other words, such a protein every (milli/micro/nano second) changes its structure to match dynamic local env. This is why nature invented/made them highly sensitive to match the local variable constraints inside each part of an organism.
So a protein, being in an eye, designed for visual programming the memory, becomes translucent, and if being at the periphery of an organism becomes various sensitive skins, hairs, tactile surfaces, etc. This is possible only if a protein molecule has a control memory to be locally tuned/programmed by the memory of a local genome in each cell. A local genome having a two-way interface to the local env has a location-specific memory content and follows the local constraints. So the local eye’s genome uses the multi-functional protein substrate (P-substrate) to make an eye’s lens, retina, optic nerve, etc. Or to use it to make different hairs on various body parts. The local control likely originates from the local genome, not from ‘a’ (generic) genome with its remote ‘controlling genes’.
Yes. Protein folding is an example of self-assembly. What is understood poorly is the way the self-assembly get affected by other proteins, epigenetics, probably hormones, food, diseases, medical drugs, water quality, chemicals, pesticides, and other unwanted substances.
We tend to understand protein folding in terms of single proteins. The reality is different. Many entities/players are affecting the final shape of the protein. Those are either part of epigenetic networks or other cellular components. We have almost zero knowledge about these processes.
Think about it. Just the number of components, not atoms, present within a single cell is comparable with the total number of people living on the Earth!
Hence, the complexity of processes present in the post-ribosomal creation of proteins is hardly comprehensible.
We know that many proteins can hold different shapes according to epigenetics and the cellular environment. It is one of the reasons leading to diseases. Cellular set up & the environment is the key to the understanding of protein folding.
This means from the computational point of view to simulate not only protein but as well all other cellular components. The whole ecosystem. We are not there yet.
We can simulate only bits and pieces here or there. Those localized simulations are just the beginning of the way towards the whole picture. This research is hard, very hard.