Why do we need to find minimum energy in a protein chain? 
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*High–quality protein backbone reconstruction from alpha carbons using Gaussian mixture models
The above research paper is about a software tool for reconstructing a protein's main chain model only from its alpha-carbon backbone (aka, C-alpha trace).
This tool provides two options: conversion (1) with energy minimization, and (2) without energy minimization.

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*Energy Minimization
The above paper says that:

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The goal of energy Minimization is to find a set of coordinates representing the minimum energy conformation for the given structure.
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In molecular modeling we tend to are particularly curious about minimum points on the energy surface. Minimum energy arrangements of the atoms correspond to stable states of the system; any movement off from a minimum provides a configuration with a better energy.
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I haven't understood the quote from the second paper.
Why do we need to know the minimum energy conformation of a protein chain? I.e., why do we want to do energy minimization?
 A: One of the biggest reasons: degeneracy of states in an energy minima is much much lower than states with higher energy! Meaning you don't have to compute a huge number of random states which have higher energy, and can focus on constraining models (for e.g. toy-models a la Ken Dill/Hue Sun Chan) to figure out the few states with lower energy. And we observe this in nature.
Proteins have have an increased probability of being found here (related to the Botlzmann factor) due to a combination of favourable physico-chemical interactions and random fluctuations.
Any protein not in an energy minimum is being 'pulled' or 'pushed' toward some other minimum, by thermal motions, influence of solvent, pH, ligand-binding, or another reason. Non-minima are not stable precisely because the protein has energy to 'sample' its landscape and determine how it can rearrange in 3-D space to reach a point where there are no unfavourable interactions (examples of unfavourable interactions could be excessive hydrophobic groups facing solvent or steric clashes).
You want to do energy minimization because it is likely to be a 'real' conformational state of your protein. After all, everything we know about the world suggests systems tend to energy minima without being constantly perturbed. It is also how you have to start MD simulations (otherwise the computer gets mad when you have energetically unfavourable interactions) or compute possible structures using toy-models (H-P polymer chains).
Large potential energy wells are only a feature of very well-folded and stable proteins. If you consider intrinsically disordered proteins, their 'conformational landscape' is like the grand canyon rather than the Mariana's Trench, so they rapidly inter-convert between numerous minima, which all have similar energies and much smaller inter-conversion energy barriers.
