Metadynamics is usually used to construct 1D or 2D free-energy surfaces. There seem to be a few examples for 3D free-energy surfaces:
Paper 1: Exploring the Free Energy Landscape of Solutes Embedded in Lipid Bilayers
Paper 2: Exploring Complex Protein−Ligand Recognition Mechanisms with Coarse Metadynamics.
Has anyone ever seen a 4D or 5D free-energy surface constructed with metadynamics? If so, how did they manage to converge the simulation?
This paper from 2015 describes a modified metadynamics method (parallel-bias metadynamics) by which the authors demonstrate enhanced sampling over six collective variables for the tryptophan cage system1. They do not mention any attempts using plain metadynamics to get anywhere near that number of CVs, so I assume (from silence) that 3D is as far as plain metadynamics has went.
Even before worrying about convergence, I think you'd have a problem simply calculating the metadynamics bias. Doing so "by hand" (storing all visited points and calculating the gradient from each previous point at a new step) is very slow. But if you try to deposit the weights on a grid, the number of grid points needed increases exponentially with dimensionality. Storing 50^3 floats takes 50 KB of RAM; storing 50^6 floats takes 62.5 GB of RAM, which you won't have per processor on almost any cluster. The paper I linked above gets around this by marginalising the landscape onto each CV separately. Somehow the authors can reconstruct the whole landscape afterwards; I'm not sure exactly how they do it, and I'm keen to learn.
1 Sadly, they lacked six-dimensional paper to print their six-dimensional free energy landscape upon. More funding needed, surely.
