Reliable Protein Folding on Complex Energy Landscapes: The Free Energy Reaction Path

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Original Entry: Peter Foster, AP 225, Fall 2011

Figure 1, taken from [1].
Figure 2, taken from [1].
Figure 3, taken from [1].

General Information

Authors: Gregg Lois, Jerzy Blawzdziewicz, and Corey S. O’Hern

Publication: Lois and et al.. Reliable Protein Folding on Complex Energy Landscapes: The Free Energy Reaction Path. Biophysical journal (2008)


This paper deals with a computational approach to the protein folding problem. Exactly how a protein folds into its three dimensional native configuration is an ongoing unsolved problem that people have worked on for quite some time. Early on, it was shown that protein's cannot do a random search in order to find their native conformation because this would take an amount of time that's longer than the average human lifetime (Levinthal's Paradox). Instead, the protein's conformational change is thought to be driven by changing to conformations that lower the protein's free energy. Many people think that the energy landscape is like a funnel, with the lowest energy state being the native conformation.

This paper considers how proteins fold when the temperature is lowered from higher to lower temperatures at a set rate. If this rate is higher than the folding rate of the protein, it is possible that the protein can become trapped in a metastable conformation. It is possible to escape these misfolded metastable states. There is a certain rate (for the exact expression check out eqn 7 in the paper) that the proteins will escape from the metastable state. However it the rate at which one lowers temperature (defined simply as r) is greater than the rate that proteins can escape from metastable states, then the protein will be trapped in the metastable state, leading to misfolding in the final protein.

A Brownian Dynamics simulation was used to simulate protein folding in two and three dimensions. Brownian Dynamics simulations are a coarse grained approach that divides up the protein into hard spheres with an added potential between some of the spheres. For their protein model they choose a chain of 13 spheres. Four of these spheres (labeled as green in the picture) have an attractive Lennard-Jones potential. An addition potential is added between each sphere and the two adjacent spheres in order to keep the linear chain intact. There is also a repulsive Lennard-Jones potential between white-white pairs and green-white pairs. Figure 1 shows the free energy landscape for this protein, along with the conformations for the native state (lower left corner) and two populated metastable states. The vertical axis is the radius of gyration and the horizontal axis is the end to end distance. Even such a simple model as this leads to the result that metastable states are possible.

Figure 2 uses data from the same simulations and is a plot of the end to end distance vs the energy divided by Ec (Ec is the well depth for the attraction between the green spheres) for different values of c (c is the magnitude of Ec divided by the temperature). When c is small (small attractive well depth) the chain acts like a random coil polymer (Fig 2 left panel). However, as the well depth gets lower and lower (higher c) the protein folds more and more until the phase spaces collapses into the lower left hand corner. When they allow the value of c to change with time (c=rt) an interesting result arrises. The resulting data is plotted in Figure 3 for threedifferent rates defined in terms of rf, the rate of folding. As one can see, if the rate is slow enough (r = 0.5rf or r=5rf), the protein folds into conformation t5 which is the native conformation (shown in Fig 1). For the fastest rate (r=50rf) the protein can fold into either conformation t3 or t4, corresponding to the metastable states in figure 1. Thus, is one lowers the temperature slow enough, the protein will fold into its native conformation regardless of whether or not other metastable conformations exist!


The main result of this paper that proteins can find their native conformation regardless of whether metastable states exist is an interesting result. Many people like to use a model for protein folding where there exists a single minima in free energy, the native conformation and this paper could relax this condition a bit. I think that this is a step in the right direction. The vast majority of proteins fold into their native conformations on their own and with such an incredible diversity of proteins it would be a bit strange if none of them had folding pathways with metastable states.

The connection between this paper and surfactants has to do with the interaction between the spheres making up the protein. The green spheres are like the hydrophobic amino acids of proteins. These tend to become buried within the center of the protein with more hydrophilic amino acids between them and the surrounding aqueous environment. Even such a simply model of treating the hydrophobic interaction as a kind of Lennard-Jones potential reproduces the burying of hydrophobic residues in this system.


[1] Lois and et al.. Reliable Protein Folding on Complex Energy Landscapes: The Free Energy Reaction Path. Biophysical journal (2008)