G. Lois, J. Blawzdziewicz, and C. S. O'Hern, "Protein folding on rugged energy landscapes: Conformational diffusion on fractal networks", Phys. Rev. E 81 (2010) 051907

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Wiki Entry by Robin Kirkpatrick, AP 225, Fall 2011


Levinthal's paradox is the thought experiment that points out that it would take an infinite amount of time for a system to sample all of the possible protein conformations. However, proteins reach a stable conformation in a matter of miliseconds. From a thermodynamic perspective, one may ask "How does a protein reach a stable state quickly when a large number of metastable states exist?". The timescales of protein folding and probability of misfolding is largely dependent on the nature of the energy landscape. For instance, rough energy landscapes in which the local variation in energy is small compared to the total energy barrier. Proteins with rugged energy landscapes, in which the local variation is on the same magnitude as the total energy barrier, exhibit much different first passage properties than proteins with funneled landscapes. An example is shown below in Figure 1. The left figure represents a funneled energy landscape, where the local energy fluctuation is much less than the depth of the well. The figure on the right show an example of a rugged energy landscape, where there are several metastable states.


While funneled energy landscapes may describe quick a reliable protein folding, rough energy landscapes may describe a mechanism for misfolding, and the conformational dynamics of proteins in a metastable state. In this paper, the author's present a computational study on the dynamics of protein folding in two dimensions of a model protein with a rugged energy landscape. They demonstrate power law scaling, which is consistent with describing conformational diffusion as a fractal network.


The authors employ a 2D model to study the conformational dynamics of a heteropolymer. Their model was designed to incorporate many metastable energy minima, one unique native state, and large energy barriers separating local minima. Each monomer is equal sized with either hydrophobic or hydrophilic interactions. In Figure 2, the hydrophilic monomers are white, and the two types of hydrophobic monomers are green and blue. Green and red monomers interact via a Lennard-Jones potential, while all other monomer interactions are repulsive. An nonlinear elastic potential is included between adjacent monomers to retain the polymer constraint. The authors choose to simulate a 18-mer sequence of ggggwwwrrrrwwwgggg which has a total possible number of permutations of 10^5 distinct energy minima. The global minima of the system (ie. the native conformation) is given in Figure 2c. The authors vary the temperature relative to the interaction energies used.


To measure the folding time <math> \tau_f </math>, the heteropolymer was put into a random extended states, and the average time to folding was measured. The unfolding time was measured as the first time that the folded state reached an extended state with the absence of any red-green (ie attractive ) contacts. This was done for a variety of temperatures and is shown below in Figure 3. The unfolding time was measured in a similar fashion. It is immediately clear from this figure that folding will occur for T<0.8.


First Passage Networks

For each heterpolymer, a list of contacting green a red monomers is made which is associated with an energy basin. Rugged energy landscapes sample a large number of basins. As illustrated in Figure 2d, the pathways from the extended to native state is comprised of hitting several energy basins, and is termed a "first passage network". For T<0.8 (stable conformation), the authors simulated 10^6 first-passage networks. They map the conformation to an energy basin every q time steps to construct a series of nodes. They count the distinct number of basins (nodes) Ni, the number of transitions (Nt), and the number of bonds (Nb). There are 850 data points for each temperature. The number of transitions is also counted. In Figure 4, the number of transitions and number of bonds are plotted vs the number of nodes over a range of temperatures. Noting that the log-log plots are roughly linear, the observed linearity implies a power law scaling of Nb vs. Ni and Nt vs Ni.

<math>Nb \propto Ni^\Lambda</math>

<math>Nt \propto Ni^\Gamma</math>


The scaling exponents are plotted vs. temperature in Figure 5, along with the predicted scaling for a fractal network (described below).


At low temperature, <math>\Lambda</math> reaching an asymptote and is roughly invariant to temperature, while <math>\Gamma</math> increases substantially when temperature is decreasing, signifying temperature dependent sampling of the configuration space in a system with rugged energy landscapes. The authors note that proteins with funneled landscapes (confirmed using a Go model), the sampling is much more uniform over temperature.

Because the power law scaling and that the results are independent of q (results not shown), it can be inferred that the first passage network is self-similar and fractal.

The authors make an argument for the validity of the power law observation by noting that, if we define a chemical distance <math>/Delta c</math>, that, if the network is fractal

<math>/Delta c /propto t^k</math>

<math> N(/Delta c) = /Delta c ^{df}</math>

where N denotes the number of basins sampled in the chemical distance, and df is the chemical fractal dimension. It follows from these relations that

<math>\Gamma = 1/(kdf)</math>

which is consistent with the experimental observation. As expected, k decreases with temperature, implying that the kinetics of the search are reduced with reduction in temperature. The authors test this scaling and show the results in Figure 6. They show that, for a certain range of chemical distances, df varies linearly with temperature.



First of all, the authors appear to neglect the entropic drive of the system, and if they did include it, it is not immediately clear to the reader where it enters the simulation. The authors assert that they expect similar results for a 3D system. However, it is unclear to the reader whether or not this is immediately obvious. It is a well known fact that diffusion in 1D and 2D guarantee that after an infinite amount of time, the probability of returning to the prior position is 1, which does not hold for 3D first passage problems. In other words, the nature of the solution in 3D is significantly different compared to 2D and 1D, which is one reason that transport along surfaces of along a 1 or 2 dimensions (for instance RNA polymerase zipping along DNA) is so much more efficient. I would think that reordering of the polymer is a constrained problem (ie, not truly analogous to 3D diffusion), thus its not truly analogous to a 3D problem, but it is still unclear to the reader that their results are immediately applicable to 3D (ie, the scaling and fractal dimensions may be quite different).


Energy Landscape Protein Folding Polymer Physics Fractal Dimension Fractal Network Levinthal's Paradox Lennard-Jones Potential