Molecular Dynamics vs energy minimization in protein folding I am a biologist . I "modelled" the behavior of a fully streched dipeptide using MD. In another exploration, I minimized the energy of the streched dipeptide. I used the same forcefield and implicit water for both models.
The 2 final dipeptides have different configurations, and their energy (which I guess is the potential energy) are different. As the sampling different molecular configurations for a dipeptide is not an issue (I guess), would a long MD converge to energy-minimized structure (which had lower energy)? Or is the result of MD the real "fluctuating but folded" structure? (assuming that a dipeptide would fold)
Stated in other words, do MD simulations (enough steps+ perfect forcefields + no cutt offs) lead to minimum free energy? What would be the role of energy conservation ? 
 A: At a finite (i.e. greater than zero) temperature, the structure will not minimise the potential energy due to thermal fluctuations. Instead the structure will attempt to minimise the free energy which balances the potential energy with the entropy. This behaviour naturally arises in molecular dynamics simulations as the instantaneous forces act to minimise the potential energy while the average forces (due to the thermal fluctuations) attempt to maximise the entropy.
What happens during a short simulation?
Depending on the conformation that you start the molecule in, it may become trapped in a local free energy minimum (what's known as a metastable state) rather than the global minimum (the thermodynamically stable state). It can take a long time to escape these metastable states and transition to a more stable configuration. In many cases (such as with protein folding) it would take prohibitively long, even with the aid of supercomputers.
What would happen during an infinitely long simulation?
The structure will ultimately sample the entire configuration space such that the probability, at any given moment, of the configuration being $x$ is proportional to the Boltzmann factor $\exp(-G(x)/k_BT)$ where $G(x)$ is the free energy of configuration $x$, $k_B$ Boltzmann's constant, and $T$ the temperature. Note, therefore, that the structure you are most likely to observe is that which minimises the free energy.
Are there ways of obtaining the minimum free energy structures without running really long simulations?
Yes. There is an entire field of research devoted to 'rare event' acceleration and phase space exploration. If you can identify a small number of coordinates (known as collective variables) that roughly capture the conformation of your molecule (for example, it may be a dihedral angle connecting your peptides) then you can either generate a free energy map with respect to your collective variables and identify the  free energy minima. Or you can simply add an auxiliary force to drive the system to explore the entire phase space. The most common technique to achieve this is metadynamics (I can recommend some papers if you'd like).
A much simpler method is parallel tempering which involves simulating your structure at different temperatures and then performing swaps based on whichever configuration is at the lowest energy. The idea is that the simulations at very high temperatures will be able to escape the free energy minima more easily.
A: MD will converge to a minimum, i.e. a local one, but not necessarily to the global one. If you get stuck in a local minimum, MD will likely not recover, except if you add energy of some kind (like heat). Optimization algorithms are designed to escape such local minima in order to find the global minimum.
