# Protein folding on rugged energy landscapes: Conformational diffusion on fractal networks

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Gregg Lois, Jerzy Blawzdziewicz, and Corey S. O'Hern

Physical Review E 81 (2010) 051907

wiki entry by Emily Russell, Fall 2010

The article can be found here.

This paper tackles the difficult problem of how a protein navigates a wide selection of metastable states to fold into its lowest-energy native state. This is a question of great importance to biology, where the correct conformation of a protein is essential to its function. The authors find that there is no fixed pathway a model protein follows during folding, but rather there is an ensemble of paths leading to the native state. Furthermore, for a "rugged" energy landscape (that is, a landscape in which the activation barriers between local minima are comparable to the depth of the global minimum), the local kinetics of transitions between metastable states are less important than the fractal nature of the network of states visited and transitions between them in predicting the folding time.

## The Model

Fig. 1: Schematics of (a) funneled and (b) rugged energy landscapes. In (a), the depth of the energy minimum that drives folding $\delta E \ll \Delta E$, where $\delta E$ gives the root-mean-square energy fluctuations over the given range of the reaction coordinate. In (b), $\delta E \approx \Delta E$.

The paper begins by briefly discussing previous approaches to protein folding which operate in the framework of a "funneled" energy landscape, in which activation barriers between minima are relatively small, and the thermodynamics of the system strongly drives the protein to the stable state (see Fig. 1). While some proteins do experience such a landscape, are are also many which live in a rugged energy landscape, with large activation barriers. In these cases it is not immediately clear that simple thermodynamics will drive the protein to its native state in any reasonable timescale.

The authors develop a particular model, of an 18-mer protein composed of 12 "hydrophobic" monomers which have an attractive Lennard-Jones interaction with one another, and 6 "hydrophilic" monomers which have only repulsive interactions; this model thus mimics the hydrophobic interactions which are the primary drivers of biological protein folding. The model protein is calculated to have a single global energy minimum, the native state (see Fig. 2), and approximately $10^5$ local minima or metastable states. A Brownian dynamics model is used to follow the conformation pathway of the protein folding from an initial extended state to the native state. The path is then considered as a network, with the states of local minima that the protein samples as nodes, and transitions between those states as edges or bonds. About a million networks were computed.

Fig. 2: The heteropolymer model in its (a) extended, (b) metastable misfolded, and (c) native states.

It was found that the protein typically sampled between 100 and 10,000 energy minima before reaching the native state, showing that there is no single path to the native state. A power law relation was also found between $N_i$, the number of sampled energy minima (counting each minimum once even if it was sampled multiple times), and $N_b$, the number of sampled transitions (also counted only once even if sampled multiples times): $N_b \propto N_i^\Lambda$. The exponent depends only weakly on temperature (see Fig. 3).

Fig. 3 (original Fig. 5): The scaling exponents $\Gamma$ and $\Lambda$ and the prediction $1/\kappa df$ for $\Gamma$ from Eq. 4. Error bars for $\Gamma$ and $\Lambda$ are smaller than the symbol sizes.

A power-law relation was also found between $N_i$ and $N_t$, the total number of transitions made: $N_t \propto N_i^\Gamma$. The exponent here depends strongly on the temperature (Fig. 3.). The authors argue that the origin of this power law is the fractal nature of the network. They define a chemical distance $\Delta c$ as the shortest distance between two nodes. Then for random motion on a fractal network, $\Delta c$ is expressed as a power law in the time interval t, $\Delta c \propto t^\kappa$, while the number of basins sampled within that chemical distance is $N_i \propto \Delta c ^ {d_f}$, by definition of the fractal dimension $d_f$. Assuming $N_b \propto \tau_f$, the folding time, these two relations are rearranged to give a prediction for the exponent $\Gamma$ found above, $\Gamma = \frac{1}{\kappa d_f}$. Power law scalings of $\Delta c$ and $N(\Delta c)$ were indeed found over limited but relevant ranges of time and chemical distance, respectively, and the prediction for $\Gamma$ from these two power laws found to be very consistent with the $\Gamma$s measured directly (Fig. 3). The authors argue that this correspondence justifies a posteriori the assumption of the fractal nature of the network.

It is also pointed out that it is only the network of sampled minima which is here considered; the full network of all possible minima is also fractal in nature, but its properties are not relevant to the problem. The sampled network is significantly smaller than the full network, and an argument is made that the overall kinetic effects in a rugged landscape do not significantly affect the dynamics.

## Results and Discussion

Investigation of the properties of the network is a clever approach to this problem, and seems to provide some fruitful insights. The folding of protein polymer chains is of clear interest in biology, and an excellent arena for soft-matter physicists to offer new ways of thinking. The fractal nature of the network neatly recalls the fractal nature of a general polymer, albeit in a higher-dimensional space.

There are to my mind two weaknesses to this paper. Only a single model protein is considered, bringing into question the generality of the results. Furthermore, the analysis of the fractal network in terms of power laws of the chemical distance seems to be simply breaking down the power law relations of the numbers of minima and of bonds into two pieces; the authors have not argued strongly enough to convince me that this treatment of the "origin of power laws" is independent enough to justify the claim that they can "predict" the exponent $\Gamma$.

Nonetheless, it is an intriguing paper, and if followed up by comparisons to real biological protein folding, could provide an important new approach to this challenging problem.