Sequence Determines Degree of Knottedness in a Coarse-Grained Protein Model

Knots are abundant in globular homopolymers but rare in globular proteins. To shed new light on this long-standing conundrum, we study the influence of sequence on the formation of knots in proteins under native conditions within the framework of the hydrophobic-polar lattice protein model. By employing large-scale Wang-Landau simulations combined with suitable Monte Carlo trial moves we show that even though knots are still abundant on average, sequence introduces large variability in the degree of self-entanglements. Moreover, we are able to design sequences which are either almost always or almost never knotted. Our findings serve as proof of concept that the introduction of just one additional degree of freedom per monomer (in our case sequence) facilitates evolution towards a protein universe in which knots are rare.

Figure 1

Left panel: Unknotting probabilities of 100 random HP sequences (gray dots) and selected, designed sequences (black dots with error bars) with $N=500$ monomers under native conditions; for the latter, representative snapshots of native conformations are shown too. The designed sequences are: (upper part, from left to right) sequence 1 (see Fig. 2); ${PH(PHPHPHHPHPHHPPHP)}_{31}HP$; self-avoiding walk ($[HP{]}_{250}$); $(P_2{H}_2{)}_{12}(P_4{H}_4{)}_8(P_2{H}_2{)}_{12}(P_4{H}_4{)}_8(P_2{H}_2{)}_{6}PH$ + same sequence in reverse order; (lower part, from left to right) homopolymer ($H_{500}$); $(H_{10}{P}_{10}{)}_{25}$; sequence 2 (see Fig. 2). Error bars of unknotting probabilities for the individual sequences have been estimated by a jackknife analysis (because of similarity, they are only shown for the designed sequences). Note that the distribution of points on the $x$ axis is arbitrary. The mean unknotting probability of the 100 random sequences is 0.460(5) (thin horizontal line). Right panel: Frequency distribution of unknotting probabilities of the 100 random HP sequences.

Figure 2

Snapshots of representative native structures. From left to right: Designed HP sequence featuring almost no knots (${[HHPP]}_{125}$); random HP sequence with a modest degree of knottedness; homopolymer (an exact ground-state structure); highly knotted designed HP sequence ($[P_{10}{(HP)}_7{H}_{10}{(HP)}_8{]}_5[{(PH)}_8{H}_{10}{(PH)}_7{P}_{10}{]}_5$). Upper structures: Monomer type coloring with hydrophobic monomers and polar monomers shown in red and blue, respectively. Lower structures: Monomer index coloring, with colors gradually changing from blue (monomers at the beginning of the sequence), over white (monomers in the center of the sequence), to red (monomers at the end of the sequence). $p_{\text{unknot}}$ denotes the unknotting probability of the corresponding sequence.

Figure 3

Average probabilities of finding unknots (upper curves) and trefoils (lower curves), respectively, in homopolymers (100% $H$) and random heteropolymers (50% $H$, 50% $P$) with $N=500$ as a function of the root-mean-squared radius of gyration, $\sqrt{⟨R_g^2⟩}$. The red symbols denote corresponding probabilities under native (ground-state-like) conditions. Error bars have been calculated by averaging over independent runs; they do not exceed symbol size and are, thus, not shown.