Information given at this resource https://predictioncenter.org/ is close to impossible to digest (as with everything in this field), so if anyone could tell me what is the accuracy we can predict tertiary protein structure now - I would be grateful.
Also would love to hear your thoughts on 'why a cell can make exactly the same protein structure thousands of time using known to us physical laws, but we have to guess it using machine learning'? Why is it difficult?
what is the accuracy we can predict tertiary protein structure
That depends on the protein. If the primary sequence closely matches the sequence of a protein for which the structure is already resolved, then template-based methods to model 3D structure can be used (a.k.a homology modeling). These methods tend to be accurate, as assessed by template-modeling score, though crystal structure confirmation is only available for a minority of models (1%, per this 2010 paper).
For proteins without structurally-resolved homologs, ab initio folding is often used, which relies on molecular mechanics assessment of iterative peptide chain folding to find structures that minimize the Gibbs free energy. Popular software for protein molecular mechanical modeling include CHARMM and AMBER. Ab initio methods are computationally intensive and are more difficult to validate.
'why a cell can make exactly the same protein structure thousands of time using known to us physical laws, but we have to guess it using machine learning'? Why is it difficult?
It's hard to know all the cellular factors present when a particular protein is synthesized and how those factors affect the folding of the protein. What is the temperature and pH proximal to the ribosome? Are chaperone proteins involved? Is the lowest-energy structure the true structure, or does the native structure fall in a local, stable minimum with a functional potential selected by evolution? A good discussion of that last point can be found on Quora.
The predictioncenter.org runs an open contest every two years - Critical Assessment of Structure Prediction (CASP). CASP 14 is underway now.
The best programs for ab initio folding are highly augmented molecular dynamics with machine learning and a bunch of predictive algorithms aggregated to create a structure. Look at David Baker's Rosetta software. More recently Google's DeepMind beat out Rosetta and a gaggle of others.
predictioncenter.org/casp13/zscores_final.cgi
Deepmind is a Deep Belief driven network https://deepmind.com/blog/article/AlphaFold-Using-AI-for-scientific-discovery
We trained a neural network to predict a distribution of distances between every pair of residues in a protein (visualised in Figure 2). These probabilities were then combined into a score that estimates how accurate a proposed protein structure is. We also trained a separate neural network that uses all distances in aggregate to estimate how close the proposed structure is to the right answer.
The problem of protein folding is known as Levinthal's paradox: a typical protein of a few hundreds amino-acids can fold into an astronomically big number of configurations, all of them consistent with the laws of physics (e.g., having the same or nearly the same energy). Yet the actual protein in a cell (nearly) always folds in the same structure and all the known proteins fold in about a thousand well-defined structures (moreover, sometimes proteins with little sequence similarity adopt exactly the same structure).
For those who want to know more about the physics behind, a rather readable introduction is Huang's Lectures on statistical physics of protein folding, whereas the other answers in this thread have already given a rather good review of the methods used in practice - note that these are not necessarily relying on the machine learning, as the OP suggests, although the ML has been used for this purpose for a few decades already, e.g., see this book.
