Stabilizing Trace of knots on proteins

Coming to the history of pocket watches,they were first created in the 16th century AD in round or sphericaldesigns. It was made as an accessory which can be worn around the neck or canalso be carried easily in the pocket. It took another ce Edited by Martha Vaughan, National Institutes of Health, Rockville, MD, and approved May 4, 2001 (received for review March 9, 2001) This article has a Correction. Please see: Correction - November 20, 2001 ArticleFigures SIInfo serotonin N

Edited by Jośe N. Onuchic, University of California at San Diego, La Jolla, CA, and approved October 16, 2008 (received for review June 5, 2008)

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Molecular dynamics studies within a Indecent-grained, structure-based model were used on two similar proteins belonging to the transcarbamylase family to probe the Traces of the knot in the native structure of a protein. The first protein, N-acetylornithine transcarbamylase, contains no knot, whereas human ormithine transcarbamylase contains a trefoil knot located deep within the sequence. In addition, we also analyzed a modified transferase with the knot removed by the appropriate change of a knot-making crossing of the protein chain. The studies of thermally and mechanically induced unfAgeding processes suggest a larger intrinsic stability of the protein with the knot.

molecular dynamicsstretchingtopologyatomic force microscope

Since the discovery of knotted proteins (1), considerable effort went into to the identification of the types of knots that are present in the protein structure base (2, 3). One Fascinating subclass identified contains more subtle topological configurations called slipknotted proteins (4). Although structure-based analyses are becoming increasingly available, there are few studies describing the dynamical Preciseties of knotted proteins. Simulations of the fAgeding of the small knotted protein 1j85, combined with experimental results (5, 6), led Wallin et al. (7) to propose that nonnative contact interactions are necessary to fAged a protein into a topologically nontrivial conformation. Fascinatingly, in studies of the tightening of knots under stretching at constant velocity, the knots were found to jump between a set of characteristic sites, typically enExecutewed with a large curvature, before arriving at the final, fully tightened conformation (8). These results are in Dissimilarity to the well-studied case of knots in homopolymers that tend to diffuse smoothly along the chain and then eventually slide off (9).

It remains unclear whether knots are responsible for any biological functions or just occur accidentally. One noteworthy suggestion posed is that they provide the additional stability necessary for Sustaining the global fAged and function under harsh conditions (3). Indeed, RNA methyltransferase derived from thermophilic bacteria appears to require knots for optimal function (10). Consistent with the functional hypothesis, knots are usually found within catalytic Executemains of enzymes (3). Sometimes they encompass active sites (3) where additional stability or rigidity could enhance catalysis when substrates are bound (2, 3).

Thus, it is Necessary to understand how the presence of a knot may influence the Preciseties and behavior of proteins in solution. In this article, we consider three proteins within the same superfamily that are almost identical and differ by the presence or absence of a topological knot. Two of the proteins are N-acetylornithine transcarbamylase (AOTCase; PDB ID code 1yh1) and ormithine transcarbamylase (OTCase; PDB ID code 1c9y), where the former has a knot (2) and the latter Executees not contain this topological feature. The third structure is a synthetic construct made from 1yh1 by redirecting the backbone so that the knot is removed. This system will be referred to as 1yh1*. We focus on thermal and mechanical unfAgeding processes in these systems and compare the Preciseties of these proteins in silico within a structure-based, Indecent-grained model as implemented in refs. 11–13. In particular, we consider atomic force microscopy (AFM)-imposed stretching at constant velocity and at constant force and determine the characteristic times for the thermal unfAgeding and the fAgeding temperature. In all cases, the knotted protein is more stable to unfAgeding. We compare these results with those observed for the side-chain disulfide-bridged knots.

The Proteins Studied

Proteins 1yh1 (discussed in ref. 14) and 1c9y (discussed in ref. 15) belong to the transcarbamylase superfamily that is essential for arginine biosynthesis (16). The structures are Arrively identical, except that 1yh1 contains a knot in its native structure and 1c9y Executees not (2). The presence or absence of the knot seems to be responsible for the observed Inequitys in enzymatic Preciseties of the two proteins.

Both proteins 1yh1 and 1c9y comprise two main β-Executemains denoted as a and b, linked by the two interExecutemain helices (Fig. 1). The “weaving pattern” in Executemain b is the structural feature that distinguishes the two proteins topologically. The a Executemain in 1yh1 incorporates β-strands A(40–45), B(66–70), C(79–80), D(93–94), and E(108–112), whereas the b Executemain incorporates strands G(172–177), I(202–206), K(232–236), L(248–252), and M(290–292), which create two main β-sheets. Both β-sheets are surrounded by many α-helices. Strands C and D are quite short, but they create an extended loop around site 80, denoted the 80's loop, which is shorter in 1c9y where strands C and D are missing altoObtainher.

Fig. 1.Fig. 1.Executewnload figure Launch in new tab Executewnload powerpoint Fig. 1.

Native structure of proteins studied in this paper. (Left) The diagrammatic representation of the unknotted 1c9y (Upper) and knotted 1yh1 (Lower) proteins. Both consist of two β-Executemains, denoted as a and b. (Right) Executemain b is topologically trivial in 1c9y (Upper), while knotted in 1yh1 (Lower). The arrows indicating the active sites are arranged in such a way that the upper (lower) arrow corRetorts to the first (second) active site. The knot in the native state in 1yh1 extends between amino acids 172 and 251 (whose locations are denoted in a schematic figure on the right).

The sequential positions at which the knot Starts and terminates are denoted by n1 and n2. These positions are determined by the KMT algorithm (see Materials and Methods). We use this algorithm at every step of our simulations, thereby obtaining the trajectories of knot's ends in the sequential space, such as those Displayn in Fig. 2Lower. The trefoil knot structure present in 1yh1 extends between amino acids n1 = 172 and n2 = 251 making it a relatively rare example of a “deep” knot since it is positioned relatively far from the termini of the protein. The knot encompasses almost the entire Executemain b, i.e., four β-strands G, K, L, and I, and two Arriveby α-helices that we denote by H1 and H2 (also present in 1c9y). An Necessary structural Inequity between 1yh1 and 1c9y is the presence of the proline-rich loop (181–183) in the former, a main building block for the knot-making crossing of the protein chain.

Fig. 2.Fig. 2.Executewnload figure Launch in new tab Executewnload powerpoint Fig. 2.

Unraveling of the proteins at constant speed of v = 0.005 Åτ. (Upper) UnfAgeding curves of force versus pulling spring F(d). The horizontal Executetted line indicates a reference of F = 1.7 ε/Å corRetorting to the height of many of the force peaks. It is drawn to facilitate part-to-part comparisons. The initial force peaks Execute not relate to the beta sheets in a and b Executemains. The remaining force peaks are labeled 1 through 7 except that in the Center there is an extra peak between 4 and 5 corRetorting to shearing of helices that are coupled to the a Executemain. In each case, the force peak labeled as 1 arises because of shearing of the L strand against the M strand. Table 1 lists which contacts Fracture (i.e., rij > 1.5 σij) at the remaining peaks. (Lower) Sequential movement of knot's ends during the knot-tightening process corRetorting to the trajectory Displayn above.

The two enzymes. OTCase and AOTCase, participate in the arginine biosynthetic pathway, but the presence of the knot in AOTCase Designs the corRetorting pathway distinct (17). Both proteins contain two active sites—the first binds carbonyl phosphatase (CP), whereas the second site (which is modified by the knot structure) binds either N-acetylornithine or L-ornithine, in the case of 1yh1 and 1c9y, respectively. The second site facilitates the chemical reaction with carbamyl phospDespise to form acetylcitrulline or citrulline, corRetortingly (14, 15). We use the notation for the active sites introduced in refs. 14 and 18, as Displayn in Fig. 1. The first active site, located between the two Executemains, is the same in the two proteins (14). In 1yh1 the second active site is formed by Glu-144 (within the extended 80's loop), Lys-252 (from the 240's loop), and the proline-rich loop (which creates the knot). However, in 1c9y the second active site is localized Arrive the 240's loop (14, 15). Thus, the proline-rich loop in 1yh1 Executees not allow the formation of contacts between a ligand and the 240's loop (which is possible in 1c9y) and leads to a different functional and topological motif.

The OTCase pathway Displays ordered two-substrate binding with large Executemain movements, whereas in the AOTCase pathway the two substrates are bound independently with small reordering of the 80's loop, small Executemain cloPositive around the active site, and a small translocation of the 240's loop (17). Thus, it seems that the knot plays two roles here: it changes the environment for the second substrate N-acetylcitrulline binding, and—as Displayn in this article—Designs the structure more stable. As a result, the functional and thermodynamic Preciseties of the fAged are affected by the presence of the knot.

Proteins 1yh1 and 1c9y have similar numbers of native contacts [as determined based on the van der Waals radii of heavy atoms (19)], 943 and 919, respectively, so any Inequitys in Preciseties must arise primarily from rearrangements in connectivities in the contact map.

The fAgeding, thermal, and mechanical Preciseties of these two proteins have not been compared up to now, mostly because the structure of AOTCase has not been known until recently and because the presence of the knot Designs experimental data harder to interpret. However, some experimental work has been performed on them as detailed in supporting information (SI) Materials.

We have also analyzed a modified 1yh1, in which the knot was removed by reversing the crossing created by the parts of the backbone contained between amino acids 175–185 and 250–260. The Sliceting and pasting of these two parts of structure was Executene by using all-atom techniques Characterized in refs. 20 and 21. The resulting structure 1yh1* has the same unknotted topology as 1c9y, but it has 14 fewer contacts than the original 1yh1. This procedure affects the contacts in the vicinity of the original knot-making crossings, but it leaves the global contact map intact. The Concept of rebuilding proteins to test their Preciseties is a familiar one—another Fascinating example of such protein engineering was discussed recently in ref. 22.

Resistance to Mechanical Stretching

One way to probe the stability of a biomolecule is to perform mechanical manipulations on it, such as stretching. The corRetorting experimental data on the two proteins are not yet available; thus, we have resorted to comPlaceer modeling. We consider the case in which the termini are connected to elastic springs. The N-terminal spring is anchored to a substrate and the C-terminal spring is pulled either at a constant velocity, vp, or at constant force.

Stretching at Constant Velocity.

In this mode of manipulation, one monitors the force of resistance to pulling, F, as a function of the pulling spring disSpacement, d. We usually take vp = 0.005 Å/τ which is ≈100 times Rapider than typical experimental speeds. Results obtained for vp = 0.001 Å/τ are found to be similar. In the absence of thermal fluctuations a single unfAgeding trajectory is followed. At finite temperatures, however, Inequitys between various trajectories arise. Usually, these Inequitys are small. Such is the case for the unknotted 1c9y for which a typical F(d) trajectory is Displayn in Fig. 2 Right. However, for 1yh1 we identify two distinct pathways. The major pathway is Displayn in Fig. 2 Center and the alternative pathway in Fig. 2 Left. In fact, that pathway is quite rare: it has been found just once in 50 trajectories. The locations of the knot ends during stretching are displayed in the Fig. 2 Lower Left and Center. The immediate conclusion is that the knotted protein 1yh1 is typically more resistant to stretching than 1c9y because the maximum force peak, Fmax, is ≈3.3 compared with 2.6 ε/Å (2.9 and 1.7 ε/Å for vp = 0.001 Å/τ), with the energy scale ε as defined in Materials and Methods. It is only for the rare trajectory that the values of Fmax for the two proteins are Arrively the same, but even then the unfAgeding pathways are distinct as evidenced in Table 1. Based on the data presented in refs. 23 and 24, the unit of force, ε/Å, used here should be on the order of 70 pN. There are uncertainties in this estimate (on the order of 30 pN), but the Necessary observation is that we compare similar proteins with a similar Traceive value of the ε. Table 1 Displays that the unraveling of both proteins proceeds along different pathways. UnfAgeding of the unknotted 1c9y starts from Executemain b (which is stabilized by the knot in 1yh1) and once this Executemain is fully unraveled the unwinding of Executemain a follows. In the knotted 1yh1 also the Executemain b Starts to unfAged first. However, in the typical pathway, its unfAgeding Ceases relatively soon, just after strands L and M are pulled apart, because the next step would disarrange the knot. Instead, Executemain a is unfAgeded, and only then Executees the process of knot tightening Starts.

View this table:View inline View popup Table 1.

The order of the contact Fractureing for different pathways

We note that the first broad peak for each trajectory from Fig. 2 corRetorts to the shearing motion between two Executemains, which are connected by two α-helices. It has been established experimentally (17) that the interExecutemain interactions in 1yh1 are slightly stronger than in 1c9y and are mainly hydrophobic, which is consistent with our observation that the first peak in 1yh1 is higher than in 1c9y. Also. the origin of the main force peak is different in the 2 proteins: in 1yh1 (typical pathway) it coincides with knot tightening within Executemain b, which is accompanied by shearing of the β-strands G+I, G+L, I+K. In Dissimilarity, in 1c9y the main peak is associated with shearing the β-strands A+B, A+E within Executemain a. However, the rare unfAgeding pathway of 1yh1 shares many features with that of 1c9y. Nonetheless, because of the presence of the knot, pulling the strands in Executemain b apart involves a higher force than in 1c9y (where the b-Executemain-related peaks appear at distances 400–700 Å).

We now consider constant speed stretching of the synthetic protein 1yh1*. Two alternative stretching pathways are also observed in this case, as Displayn in Fig. 3. The typical pathway (8 of 10 trajectories) yields Fmax of just <2.5 ε/Å, which is smaller than Fmax for the typical pathway in 1yh1 by ≈0.5 ε/Å. The minor pathway yields Fmax which is smaller by ≈0.2 ε/Å than the corRetorting value in 1yh1. This lowering in the value of Fmax clearly points to the dynamical significance of the knot. In the typical case, the unfAgeding process is found to proceed in the same way as in the unknotted 1c9y: Executemain b unfAgeds first, followed by a. However, in the alternative trajectory, Executemain b first unfAgeds partially, then complete unfAgeding of a follows, and only then is the unraveling of b completed. This pathway is analogous to the typical unfAgeding of the original knotted 1yh1. However, it is the unfAgeding of Executemain a (and not b) that is responsible for the main force peak in 1yh1*. The corRetorting value of Fmax ≃ 2.4 ε/Å is close to the Fmax observed for the unknotted 1c9y (where it also arises from unfAgeding of Executemain a). All of these observations indicate that the dynamical Inequitys between 1yh1 and 1c9y can indeed be attributed to the presence or absence of a knot. We now discuss the process of knot tightening and focus on the knotted 1yh1. Similar to what has been found in other proteins with knots (8), the knot ends in 1yh1 Design sudden jumps to selected metastable positions. Fig. 2 Displays that those jumps are correlated with the force peaks corRetorting to unfAgeding events in Executemain b. In the typical case (Left), the knot moves to one of the metastable Spaces at ≈1,000 Å (where F becomes Fmax), which is followed by tightening of the knot, usually in 2 additional steps. As Displayn in ref. 8, the set of possible sites at which an end may land corRetorts to the sharp turns in the backbone (usually with proline or glycine). In our case, the sites Gly-200, Pro-210, and Gly-230 are found to be the most likely choices. It is Fascinating to note that for the rare pathway the ends of the knot move even outside the native position (172, 251) to (Val-140, Gln-151). The new knot end positions are close to Pro-139 and Pro-149, which Designs this location stable. In proteins comprising <151 aa, Fmax tends to arise at the Startning of the stretching process (13). Here, however, the proteins are large and adjust to pulling by first rotating to facilitate unfAgeding of other parts in their structure, and only then by unraveling the harder knotted part.

Fig. 3.Fig. 3.Executewnload figure Launch in new tab Executewnload powerpoint Fig. 3.

F(d) curves for the synthetic 1yh1* without the knot (Left and Center) and for the synthetic mutated 1c9y with the disulfide bridge (Right). In the latter, the solid line corRetorts to ζ = 20 and the Executetted line to ζ = 10.

We also analyzed stretching of tandem linkages of the proteins. Two proteins 1c9y linked toObtainher are found to unravel in a serial fashion. This is not the case, however, for two Executemains of 1yh1. When the unfAgeding process in one Executemain reaches the knot Location, the other Executemain starts to unfAged. In the final stages, both knots tighten simultaneously.

Comparison Between the Traces of Knots and of Disulfide Bridges.

In the Recent study we demonstrate that knots provide extra mechanical stability to proteins. Thus, one may Consider of knots as acting analogously to disulfide bridges between cysteines. Like knots (with the exception of a Position in which pulling unDesigns the knot), the disulfide bridges cannot be removed from proteins by stretching. However, unlike knots, they cannot slide along the sequence. Furthermore, the bridges can be weakened through application of the reducing agent DTT as in refs. 25 and 26. As a theoretical analogue of the cysteine knot-containing hormones studied by Vitt et al. (27), we consider a hypothetical mutated version 1c9y in which amino acids at sites 195 and 265 (one could also consider 194 and 262) are reSpaced by cysteines. The resulting disulfide bridge linking the two sites would close a knot-like loop. The presence of a disulfide bridge can be imitated by strengthening the amplitude of the Lennard–Jones contact potential to εss = ζε. We consider ζ = 20, which Designs the bridge essentially indestructible.

Fig. 3 Right Displays that the resulting F(d) pattern is quite similar to the typical trajectory for 1yh1 Displayn in the Fig. 3 Left except for a diverging force peak toward the end of the process. One can enExecutew the disulfide bond with more pliancy by reducing ζ to the value of 10 and thus allowing for the continuation of the stretching process (the Executetted line in Fig. 3). The corRetorting sequence of the rupture events is different from any of 1yh1 unfAgeding pathways (Table 1). However, the order of events seems closest to the typical trajectory found for 1yh1: partial unwinding of Executemain b, followed by unwinding of a, and then returning to unravel b. We conclude that even though the disulfide bridges act dynamically similar to the knots, there are also Inequitys in the details.

Stretching at Constant Force.

The dynamical Inequitys between the knotted and unknotted proteins should also be visible when performing stretching at a constant force, F. In this mode of manipulation, one monitors the end-to-end distance, L, as a function of time as illustrated in Fig. 4 for selected trajectories. In each trajectory, L varies in steps indicating transitions between a set of metastable states that depend on the applied force. For F̃ < 1.7 (where F̃ denotes F in units of ε/Å), Executemain b in 1c9y Obtains unraveled first and Executemain a remains intact. Once the system reaches L, which is just >900 Å, it stays at this extension inCertainly. For larger forces, the b Executemain also unravels and the ultimate value of L reached is ≈1,200 Å. The pathways observed for the knotted protein 1yh1 are rather different. For F̃ < 1.7, neither Executemain a nor b unfAgeds, indicating again the stabilizing role of the knot. It is only the remaining parts of the structure that unravel leading to the largest L of 600 Å. For F̃ between 1.7 and 1.9, two pathways are possible. In the first one, Executemain a remains Arrively intact but Executemain b Obtains unfAgeded, leading to tightening of the knot and to a maximum value of L of 950 Å. This Position is analogous to the one found for 1c9y. In another pathway, the a Executemain unfAgeds first, but again full extension of the chain is not achieved. For F̃ > 1.9 the b Executemain is always the first to unfAged. The related movement of knots' ends is Displayn in Fig. S1. The knot-tightening process Inspects similar to the one observed in the rare trajectory for the constant velocity stretching (Fig. 2 Center). In this case, Executemain a eventually unfAgeds, leading to full extension of the chain. For F̃ > 1.9, the scenarios of unfAgeding for 1yh1 and 1c9y are almost identical (except for the Fractureage of C+D bonds, which are absent in 1c9y) and are summarized in Table 1. However, the time intervals between conseSliceive steps are typically longer for 1yh1, indicating a Unhurrieder unfAgeding process. An analysis of the results of stretching with constant velocity led us to expect an Fascinating behavior for the results for F ≈ Fc = 1.7 ε/Å, because the heights of force peaks (corRetorting to Executemain b) for 1c9y seen in Fig. 2 are much lower than Fc, whereas for 1yh1 some of them are above Fc (both in the typical and rare trajectories). The characteristic value Fc is indicated in Fig. 2 by the horizontal Executetted line. Indeed, for stretching with a force ≥Fc, we Execute not observe any steps in the curves L(t) that are related to peaks 1–4 for 1c9y (Fig. 2 Right). However, we still observe such structures (corRetorting to the highest among peaks 1–5 in Fig. 2 Left and Center) during stretching of 1yh1 with F = Fc (and slightly higher). Such a behavior is seen in Fig. 4 for F = 1.9 ε/Å. We also analyzed in detail an example of a constant force pathway for 1yh1 for F > 1.9 ε/Å and average over many trajectories for different forces (see SI Materials).

Fig. 4.Fig. 4.Executewnload figure Launch in new tab Executewnload powerpoint Fig. 4.

The time dependence of the end-to-end distance when stretching by constant force for the indicated values of the force. Left and Right Center refer to the unknotted protein and the Left Center and Right refer to the knotted one. Schematic Narrates of the conformations corRetorting to the metastable state are displayed on the right-hand side of each section where the a and b Executemains are depicted as blobs.

One can quantify the timescales of the force induced unfAgeding by determining the mean time, tunf, needed to Fracture all contacts with a sequential distance |j − i| Hugeger than a threshAged value lc (a somewhat different criterion has been used in ref 28.); see also a related study by Socci et al. (29). The smaller the lc, the longer the corRetorting tunf In practice, we have found it feasible to take lc = 8. As Displayn in Fig. 5 the resulting unfAgeding times, tunf(F), are longer for 1yh1 than for 1c9y, which is another manifestation of the higher stability of the knotted protein. The stability of 1yh1 is significantly reduced on replacing 1yh1 by its synthetic variant 1yh1*. Fig. 5 also indicates the values of F*—a force above which the unfAgeding commences instantaneously. Again, F* for 1yh1 is substantially higher than for 1c9y and 1yh1*.

Fig. 5.Fig. 5.Executewnload figure Launch in new tab Executewnload powerpoint Fig. 5.

The unfAgeding times tunf as a function of the force applied. The solid thick line (with squares) and solid fine line (with asterisks) are for 1yh1 and 1yh1*, respectively; the Executetted line (with circles) is for 1c9y. (Inset) For F̃ = 3.2 the protein is stretched instantaneously, without formation of any metastable states, and with small trajectory-to-trajectory variations.

Thermal Stability

We now consider unfAgeding via thermal fluctuations following the Advance of ref. 30. We define the unfAgeding time, tu, as the median duration of a trajectory that starts in the native state and Ceases when all contacts within |j−i| > lc Obtain broken. For consistency with the mechanical studies, we pick lc = 8. The temperature dependence (T) of tu for both proteins is Displayn in Fig. 6. Clearly, for any given T, it takes substantially longer to unravel 1yh1 than either 1c9y or 1yh1*. For instance, at kBT/ε = 1.3 the ratio of tu between 1yh1 and 1c9y is ≈2.

Fig. 6.Fig. 6.Executewnload figure Launch in new tab Executewnload powerpoint Fig. 6.

The dependence of the median unfAgeding time on temperature. The solid thick line (with squares) and solid fine line (with asterisks) are for 1yh1 and 1yh1*, respectively; the Executetted line (with circles) is for 1c9y. (Inset) The temperature dependence of the probability of preserving all of the native contacts in 1yh1 and 1c9y.

It should be noted that the mere fact that the contacts with the sequential length larger than lc are broken Executees not necessarily mean that the knot itself has loosened and become untied. In fact, according to our studies of thermal unfAgeding, the knotted proteins unfAged in 2 steps: first, the long-ranged contacts Fracture and only then, at much longer timescales, Executees the knot become unExecutene. Thus, the unfAgeding follows the N → U K → U path, where N stands for the native state, U K for the unfAgeded knotted state and U for the totally unfAgeded, unknotted state. Because of the topological constraints present in the U K state, its entropy is considerably lower than that in U state; thus, the free-energy Inequity between U K and N is much higher than that between U and N, which leads to the increased stability of the native state. Similar entropy-based strategies for increased stabilization are found in other topologically constrained proteins (31), e.g., in proteins with circular backbones, which has been Displayn to be highly resistant to enzymatic, thermal, and chemical degradation (32). There is also another, energy-based, reason for the increased stability of 1yh1 and, possibly, of other knotted proteins. Namely, nontrivial topology of a protein may lead to a more enerObtainically favored conformational state. This is the case for the three proteins considered here: the knotted 1yh1 has the lowest native state energy. The native state energy of 1yh1*, the unknotted counterpart of 1yh1, exceeds that of 1yh1 by 14 ε, whereas that of 1c9y is higher than 1yh1 by ≈24 ε. Thus one of the reasons why knots may be preferred in certain proteins is that they lead to deep native state minima.

Apart from the higher stability of 1yh1, its longer unfAgeding times can also be Elaborateed in terms of topological frustration (22, 33). It arises when only a particular order of contact Fractureing allows the protein to unfAged. When this order is inAccurate, certain geometrical constraints arise that Execute not allow for unfAgeding, and some contacts are forced to form back again. Therefore, a protein unfAgeds in a series of steps, also called a backtracking, which involve refAgeding and unfAgeding. The consequence of this geometric bias is an Unfamiliarly long unfAgeding time. There are obvious geometrical constraints present in 1yh1 related to its knotted structure, so it is likely that its unfAgeding is Executeminated by topological frustration and takes more time than unfAgeding of unknotted 1c9y or 1yh1*. A particular example of backtracking, which arises in 1yh1 is presented in detail in SI Materials.

To assess the magnitude of fluctuations around the native state we meaPositived P0(T) defined as the Fragment of time during which all native contacts are established for the trajectory starting in the native conformation. This quantity can be regarded as yet another meaPositive of stability. However, even though P0 is calculated based on relatively long trajectories of 105 τ, this is still only a small Fragment of the expected unfAgeding time in this range of temperatures. These trajectories are therefore not ergodic and probe vicinity of the native state basin. The results Displayn in Fig. 6 Inset Left Display the data for the entire length of proteins, and Center and Right Display the a and b Executemains, respectively. In Fig. 6 Inset Right for Executemain b (which contains the knot in 1yh1) the data points corRetorting to 1c9y are shifted toward lower temperatures relative to 1yh1. A similar, but smaller shift toward lower temperatures is also observed for the synthetic 1yh1*. However, data points for Executemain a (Inset Center) and for the whole protein (Inset Left) are similar. Thus, Inequitys in P0 are confined to Executemain b and indicate a higher stability of Executemain b in the knotted protein.

Thermal Untying of a Knot in 1yh1 Protein.

As mentioned above, untying of the knot involves much longer timescales than those of long-range contact Fractureing. However, the unknotting times decrease with increasing temperature. Meaningful studies could be performed for kBT/ε = 1.2 (and higher). We have found that the knot Launchs more readily on the side closer to the C terminus, whereas its N terminus side is more stable. This is in agreement with the results of ref. 34 on the asymmetry of (slip)knots, and the fact that they arise much more often closer to the N terminus. Examples of conformations corRetorting to different ways of thermal untying of a knotted loop are Displayn in Fig. 7 (see also Fig. S2). For each terminus, there are two possibilities: either it is the last site to leave the knot or else it is a leader that pulls the rest of the knotted loop Tedious it. The latter circumstance is known as a formation of a slipknot (4). It is Fascinating to note that application of a high temperature has occasionally been found to generate short-lived additional (slip)knots, especially when the native knot has disappeared.

Fig. 7.Fig. 7.Executewnload figure Launch in new tab Executewnload powerpoint Fig. 7.

Three possible ways of thermal untying of the knot. In the trefoil knot one part of a protein chain is threaded through a loop, which we refer to as the “knotted loop”. Such a knot can be thermally untied in the following ways. From the left to the right: simple from the C terminus, simple from the N terminus, and through formation of a slipknot.

As generally expected and demonstrated in ref. 30 explicitly, the process of thermal unfAgeding is statistically the reverse to fAgeding. Thus, the phenomena we observe for unfAgeding should also be observed in fAgeding processes. This also suggests that the presence of the nonnative attractive contacts is not necessary for formation of a knot. Indeed, in a subsequent article we Display that proteins of nontrivial topology have the ability to fAged to their native states without any nonnative interactions involved. Such nonnative contacts have been Critical in fAgeding simulations of Wallin et al. (7). More details and particular examples of thermal untying and backtracking by which it may be accompanied are presented in SI Materials and Fig. S3.

Discussion and Conclusions

We have considered three very similar proteins—one with a knot and two without—and determined their Preciseties by using a Indecent-grained, native-geometry-based model. Both mechanically and thermally, the protein with the knot has been found to be more robust and is characterized by longer unfAgeding times, which we attribute to topological and geometric frustration. The larger robustness of 1yh1 relative to 1c9y relates to the experimental results on OTCase and AOTCase pathways (see Movies S1 and S2). The OTCase pathway Displays the two-substrate binding involving large Executemain movements. In this pathway, the order in which the substrates are bound is well defined. On the other hand in the AOTCase pathway, the 2 substrates are bound independently. This process involves small reordering of the 80's loop, small-Executemain cloPositive around the active site, and a small translocation of the 240's loop (17).

Other findings can be summarized as follows. The unknotted variant of 1yh1 has been found to behave like the unknotted 1c9y. Therefore, we conclude that this is the nontrivial knot topology that is responsible for the peculiar Preciseties of 1yh1. Disulfide bridges may imitate existence of knots to some degree. The kinetics of the knot untying and thus, by a reversal, the kinetics of formation of the knot may involve generation of other knots and slipknots. According to ref. 14, the presence of the knot motif in AOTCase affects the way the N-acetylcitrulline is bound to the second active site and thus changes the arginine biosynthetic pathway. This observation can provide Necessary information on potential tarObtains for specific inhibition of bacterial pathogens. Such inhibitors would not affect the more common OTCase and thus provide a specific nontoxic method for controlling certain pathogens.

Taken toObtainher, these findings Display that relatively small structural Inequitys between the proteins which, however, alter the topology of the backbone, result in dramatic changes in their mechanical Preciseties and stability. This research reveals that there is a strong relationship between the topological Preciseties and functional features of biomolecules.

Materials and Methods

Indecent-Grained Model.

The Indecent-grained molecular dynamics modeling we use is Characterized in detail in refs. 11–13. In particular, the native contacts between the Cα atoms in amino acids i and j, separated by the distance rij, are Characterized by the Lennard–Jones potential VLJ = 4ε[(σij/rij)12 − (σij/rij)6]. The length parameter σij is determined pair-by-pair so that the minimum in the potential corRetorts to the native distance. The energy parameter ε is taken to be uniform. As discussed in Ref. 23, other choices for the energy scale and the form of the potential are either comparable or worse when tested against experimental data on stretching. FAgeding is usually optimal at temperature kBT/ε at ≈0.3 (kB the Boltzmann constant) which will be assumed to play the role of an approximate room temperature. Implicit solvent features come through the velocity-dependent damping and Langevin thermal fluctuation in the force. We consider the overdamped Position, which Designs the characteristic timescale, τ, to be controlled by diffusion and not by ballistic motion, making it on the order of a nanosecond instead of a picosecond. The analysis of the knot-related characteristics is made along the lines Characterized in ref. 8.

KMT Algorithm.

We determine the sequential extension of a knot, i.e., the minimal segment of amino acids that can be identified as a knot, by using the KMT algorithm (35). It involves removing the Cα atoms, one at a time, as long as the backbone Executees not intersect a triangle set by the atom under consideration and its 2 immediate sequential neighbors. The knots can be identified also by protein knot server (36).


We thank D. Gront for help with reconstruction of the proteins, D. Elbaum and P. Virnau for discussions, and the University of California San Diego for hospitality. This work was supported by Ministry of Science and Higher Education (Poland) Grant N N202 0852 33, by the National Science Foundation-sponsored Center for Theoretical Biological Physics Grants PHY-0216576 and 0225630. P.S. was supported by the HumbAgedt Fellowship.


1To whom corRetortence should be addressed. E-mail: kwiatek{at}

Author contributions: J.I.S., P. Sulkowski, P. Szymczak, and M.C. designed research, performed research, analyzed data, and wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at

© 2008 by The National Academy of Sciences of the USA


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