Folding Rate Optimization Promotes Frustrated Interactions in Entangled Protein Structures

Abstract

Many native structures of proteins accomodate complex topological motifs such as knots, lassos, and other geometrical entanglements. How proteins can fold quickly even in the presence of such topological obstacles is a debated question in structural biology. Recently, the hypothesis that energetic frustration might be a mechanism to avoid topological frustration has been put forward based on the empirical observation that loops involved in entanglements are stabilized by weak interactions between amino-acids at their extrema. To verify this idea, we use a toy lattice model for the folding of proteins into two almost identical structures, one entangled and one not. As expected, the folding time is longer when random sequences folds into the entangled structure. This holds also under an evolutionary pressure simulated by optimizing the folding time. It turns out that optmized protein sequences in the entangled structure are in fact characterized by frustrated interactions at the closures of entangled loops. This phenomenon is much less enhanced in the control case where the entanglement is not present. Our findings, which are in agreement with experimental observations, corroborate the idea that an evolutionary pressure shapes the folding funnel to avoid topological and kinetic traps.

Keywords: entanglement; protein folding; topological frustration.

Figure 1

(a) Entangled native state. (b) Non-entangled native state. The putative “twin” native states are two self-avoiding walks on the fcc lattice with $N=18$ sites. The two loops shown in red and blue are concatenated in the entangled structure, and not concatenated in the non-entangled structure. The amino acids at the ends of each loop form the contacts (dashed yellow and green lines) whose energy is studied in this work.

Figure 2

Average folding times into the entangled native state (circles) and into the twin native state without entanglement (crosses), as a function of the weaker of the contact energies involved at the ends of the two loops ($V_1$). (a) 15 independent random sequences; (b) fastest proteins after $G=1000$ generations of the evolutionary process for 31 independent replicas. Evolution leads to a dramatic drop in time scales (note the log scale for the random sequences), yet the entangled proteins, with respect to their non-entangled twins, still fold more slowly. Note also that the evolved entangled proteins have on average more unstable energetic closures of the loops.

Figure 3

Frequency of amino acid single potentials for (a) protein with entangled loops and (b) its twin without entanglement, collecting the statistics of proteins at the end of the evolutionary process. The three curves show the frequency regardless of the position along the chain, the frequency at the 4 sites closing the loops, and the frequency in the complementary set of sites.

Figure 4

Fraction of configurations in the native states as a function of the inverse temperature $\beta =1/k_{B}T$, for proteins with entanglement (left) and without entanglement (right). Each curve is for a given random sequence. The mean folding inverse temperature is found by averaging the points where the curves cross the $50\%$. The inverse temperature used in the folding simulations within the evolutionary process is fixed to this value.

Figure 5

Illustration of the types of moves used in the simulations: (a) end-flip (frequency $P=2/19$, the figure illustrates the random choice at the end with site N), (b) crankshaft or internal flip ($P=16/19$), (c) reptation ($P=1/19$). In reptation, to satisfy detailed balance, the frequency of attempted moves in the two directions satisfy $W^R=3W^F$ (“F” and “R” following the notation in the figure) because there are 12 possible points for the added end vs only 4 internal sites for the added corner. The internal flip is in one out of the 4 possible sites if the corner is of ${60}^{\circ }$ or ${90}^{\circ }$, while only two sites are allowed when the corner is ${120}^{\circ }$. All attempted moves are then validated with self-avoidance constraints and eventually accepted with a Metropolis algorithm.