What does it mean by a "restraint" in the case of a protein chain?

Question

• Discrete restraint-based protein modeling and the Cα-trace problem

We present a novel de novo method to generate protein models from sparse, discretized restraints on the conformation of the main chain and side chain atoms. We focus on Cα-trace generation, the problem of constructing an accurate and complete model from approximate knowledge of the positions of the Cα atoms and, in some cases, the side chain centroids. Spatial restraints on the Cα atoms and side chain centroids are supplemented by constraints on main chain geometry, ϕ/ξ angles, rotameric side chain conformations, and inter-atomic separations derived from analyses of known protein structures. A novel conformational search algorithm, combining features of tree-search and genetic algorithms, generates models consistent with these restraints by propensity-weighted dihedral angle sampling. Models with ideal geometry, good ϕ/ξ angles, and no inter-atomic overlaps are produced with 0.8 Å main chain and, with side chain centroid restraints, 1.0 Å all-atom root-mean-square deviation (RMSD) from the crystal structure over a diverse set of target proteins. The mean model derived from 50 independently generated models is closer to the crystal structure than any individual model, with 0.5 Å main chain RMSD under only Cα restraints and 0.7 Å all-atom RMSD under both Cα and centroid restraints. The method is insensitive to randomly distributed errors of up to 4 Å in the Cα restraints. The conformational search algorithm is efficient, with the computational costs increase linearly with protein size. Issues relating to decoy set generation, experimental structure determination, the efficiency of conformational sampling, and homology modeling are discussed.

Note: The emphases are mine.

Can you tell me what they meant by "restraint" and "spatial restraint"?

Please give me an intuitive example so that I can understand it clearly.

Answer 1

They go on to describe exactly what they mean by a "restraint" later in the paper in the section "Materials and Methods::Cα and side chain centroid restraints".

A Cα atom at position p satisfies a spherical restraint centered at O with radius r when $$\Vert p-O\Vert < r$$

All this formula says is that they are restricting the position $p$ (which is initially unknown) of some atom to be no further than $r$ distance away from some point $O$ that they choose. They do this for all or most of the atoms in the hopes that the combination of restraints (e.g. carbon1 is no more than 2 angstrom away from point O1, atom2 is no more than 2.5 angstrom away from O2, ...) will be enough information to construct a reasonable 3D structure.

Similar to Cα restraints, side chain centroid restraints are enforced by spherical restraints on the mean position of the side chain atoms (here we take the set of side chain atoms to be all atoms outwards from Cβ, excluding the Cα as seen in other work). A centroid restraint of radius r at position O is satisfied when $$\Bigg\Vert\frac{\sum_{i\in\text{side chains}}p_i}{\Vert\text{side chains}\Vert}-O\Bigg\Vert<r$$

The notation is a bit sloppy for the side chain restraint (not clear what the norm of "side chains" is), but the meaning is the same as for the atom restraints: they determine the positions of these points (side chain centroids) by establishing a set of bounds (restraints) that each position has to fit into.

They mention in the paper that how to go from a set of restraints to a 3D geometry is a challenging problem and there isn't a general algorithm that can reliably/efficiently do this. In this context they make a distinction between a "restraint" which is a bound or inequality on some value, vs a "constraint", which explicitly enforces some particular value.