Topological descriptions of protein folding.

How knotted proteins fold has remained controversial since the identification of deeply knotted proteins nearly two decades ago. Both computational and experimental approaches have been used to investigate protein knot formation. Motivated by the computer simulations of Bölinger et al. [Bölinger D, et al. (2010) PLoS Comput Biol 6:e1000731] for the folding of the [Formula: see text]-knotted α-haloacid dehalogenase (DehI) protein, we introduce a topological description of knot folding that could describe pathways for the formation of all currently known protein knot types and predicts knot types that might be identified in the future. We analyze fingerprint data from crystal structures of protein knots as evidence that particular protein knots may fold according to specific pathways from our theory. Our results confirm Taylor's twisted hairpin theory of knot folding for the [Formula: see text]-knotted proteins and the [Formula: see text]-knotted ketol-acid reductoisomerases and present alternative folding mechanisms for the [Formula: see text]-knotted phytochromes and the [Formula: see text]- and [Formula: see text]-knotted proteins.

When protein knots were first identified, they were believed to be the result of randomly occurring misfolded conformations. However, the discovery of proteins containing deeply embedded knots forced a reevaluation of this belief (1–3). Systematic reviews of the ever-growing Protein Data Bank (PDB) and the development of specialized servers for detecting protein knots have led to the identification of hundreds of knotted proteins, and it is now generally accepted that a small but significant fraction of proteins contains knots (4–7). However, exactly how and why such knots form are still unknown.

As of now, only five distinct knot types have been found in proteins in the PDB. We illustrate these knots as closed curves in Fig. 1, although the proteins containing them are actually open chains. From a mathematical perspective, a knotted open chain is equivalent to an unknotted open chain, since an open chain can unknot via a continuous deformation. However, from a biophysical perspective, the energy necessary to undo a deeply embedded protein knot is prohibitively large, effectively trapping the knot in the open chain. Thus, it is common practice to close knotted proteins by bringing the ends of the knotted chain together to obtain a loop, to which a knot type can be assigned. Different methods of closing the chain can result in different knot types, and various approaches have been used to resolve this problem (1, 8–17).

Fig. 1.

As of now, these are the only knots that have been identified in proteins.

The standard notation to represent knots (18) uses a large numeral to denote the minimum number of crossings among all projections of the knot and a subscript to identify the particular knot with that number of crossings as in Fig. 1. To distinguish the two enantiomorphs of a chiral knot, we use the algorithm described by Mislow and coworkers (19, 20) to assign to one form and − to the other.

According to a recent survey, 23 families of knotted proteins have been identified (21). Of these families, 19 contain knots, only the ketol-acid reductoisomerases (KARIs) and the phytochromes contain knots, only the ubiquitin C-terminal hydrolases (UCHs) contain knots, and only the -haloacid dehalogenase (DehI) contains knots. It is unclear why the and knots have been found, but their mirror forms have not. It could be a matter of time before a protein is found to contain , , or any other knot (22).

The study of protein knotting has been approached using experimentation, simulation, and theoretical descriptions (refs. 21 and 22 have recent reviews). Inspired by the theory of knot folding put forth by Taylor (23) () together with the simulations of Bölinger et al. (24) for the folding of DehI (), this paper presents a theoretical description of protein knot folding, which could be applicable to any knotted protein. Because our theory builds on recent experimental and computational results about knot folding, it provides a step forward in current thinking about knot folding.

1. Taylor’s Twisted Hairpin Theory

Taylor (23) introduced a theory of protein knot folding where the protein assumes the form of a “twisted hairpin.” Then, one terminus threads through the “eye” of the hairpin to create a knot. We illustrate this in Fig. 2, with dotted segments added to make the chains into closed loops. Note that knotted protein conformations are more complex and contain more crossings than the projections drawn in Fig. 2. We use these simplified drawings to focus on the knotting mechanism.

Fig. 2.

The twisted hairpin folding mechanism proposed by Taylor (23).

We will refer to this mechanism of knot folding as a twisted hairpin pathway. Knots obtained in this way are in the family of twist knots. According to Taylor’s theory, all protein knots identified in the future must also be members of this family (23). For example, the , , and knots (Fig. 3) have not been found in any solved protein structures, although they have a projection with the same number of crossings as the and knots, which have been found. Since these three knots are not twist knots, Taylor’s theory of knot folding would explain their absence among solved protein structures.

Fig. 3.

Neither these knots nor their enantiomorphs have been found in any protein.

Taylor (23) argues that, in knot folding, loop penetration is the rate-limiting event and that knot formation depends primarily on the number of times that loop penetration occurs, then on the number of residues that must be threaded through the loop, and lastly, on the number of crossings in the resulting knot (23). This hierarchy suggests that proteins with deep knots should be less prevalent than those with shallow knots independent of the number of crossings. However, as can be seen on the database KnotProt (4, 5), proteins containing a deep knot are vastly more common than those containing a shallow knot. Thus, in contrast with Taylor’s hierarchy, we would expect the number of crossings in a given knot to be higher in the hierarchy than the number of residues that must be threaded through the loop.

However, knot-folding rates are not only a function of the number of crossings and the depth of a knot. In particular, chaperones can speed up the kinetics of knot folding as has been observed for the -knotted proteins YibK, YbeA (25–27), VirC2, and DndE (28) and for the -knotted UCHs (29, 30). Furthermore, the work of Wallin et al. (31) shows that nonnative interactions increase the probability of correct knot folding for the deeply -knotted YibK protein, and simulations of Covino et al. (32) confirm this for the shallow trefoil knot in the MJ0366 protein.

In addition, the work of Chwastyk and Cieplak (33) shows that ribosomes may play a significant role in knot folding. In particular, they argue that the deep knot in the YibK protein is a result of on-ribosome folding. Recent computer simulations by Dabrowski-Tumanski et al. (34) also indicate that ribosomes play an active role in the folding of the protein with PDB ID code 5JIR, which contains the deepest knot that has been identified in a protein. In particular, according to their simulations, one end of the nascent chain comes out of the ribosome and forms a twisted loop, which attaches to the ribosome around the exit tunnel. While this loop is held in place, the ribosome pushes a piece of the protein through the exit tunnel so that it is surrounded by the first loop, creating a slipknot. Finally, the rest of the chain is threaded through the exit tunnel to form a knot.

While Taylor’s twisted hairpin theory is useful to describe knot folding independent of any particular protein, there is computational and experimental evidence that this may not be the only pathway to protein knotting. In particular, it has been shown that encapsulation in a chaperonin can facilitate multiple folding pathways (28, 29). Even without chaperones (25, 35–39), knotted proteins can have complex energy landscapes that include knotted intermediates and parallel pathways. This is supported by simulations indicating that some trefoil-knotted proteins fold via multiple pathways (31, 40–42), including a newly described pathway where each terminus threads through a separate loop (42). Also, computational studies of the folding of the knot in UCHs (29) and the folding of the knot in DehI (24, 43) produced pathways that involved knotted intermediates. Such intermediates would not occur if the knots folded via a twisted hairpin pathway, because the chain remains unknotted until threading occurs at the last step.

For all of the above reasons, even if a twisted hairpin pathway is the primary folding mechanism for knotted proteins, it is worth considering alternative pathways that permit partially folded knotted intermediates. In the next section, we describe loop flipping as a knot-folding mechanism. Then, in , we introduce our theory of knot folding.

2. Knot Folding via Loop Flipping

While Taylor’s theory of knot folding assumes that a terminus threads through the loop of a twisted hairpin, the same conformation would be produced if the loop of the hairpin was to flip over the terminus. In fact, the mobility of the loop may confer thermodynamical advantages, making it easier for knotting to occur by a loop-flipping motion rather than by threading. Furthermore, experimental results on the thermodynamic and kinetic properties of a -knotted protein similar to the MTase protein (44), a -knotted UCH protein (37), and the -knotted DehI protein (43) have all been consistent with loop flipping as a knotting mechanism. Loop flipping (also known as a “mousetrap-like” or “jump-rope-like” motion) is increasingly being observed in structure-based simulations of knot folding. For example, simulations show that some -knotted proteins (41, 42, 45) as well as the -knotted UCH proteins (29, 46) have at least one folding pathway involving a large loop flipping over a terminus.

Furthermore, using molecular dynamics simulations with a coarse-grained Gō model of the folding of DehI, Bölinger et al. (24) found two pathways to the knot, which each involved a large loop flipping over a mostly folded smaller loop. They then used crystallographic B-factor data from the DehI protein to verify that the relevant pieces of the protein are flexible enough to permit the loop flipping required by this pathway. While their simulations did not take into account nonnative interactions, they assert based on the work of Wallin et al. (31) that nonnative interactions should, in fact, increase the rate of knot folding via their pathways.

In Fig. 4, we illustrate the steps of the simulation of Bölinger et al. (24). In Steps 1 and 2, the polypeptide forms a large green loop and a smaller red loop, which are aligned. At this point, the folding mechanism splits into two parallel pathways. In Step 3a, the red loop twists one more time, and the green loop flips over both the red loop and the blue end, causing the blue end to thread through the green loop. Then, the blue end threads through the red loop to obtain Step 4. In Step 3b, the red loop twists one more time, and the blue end threads through it. From here, the green loop flips over both the red loop and the blue end, causing the blue end to thread through the green loop to again obtain Step 4. In both pathways, loop flipping enables the efficient threading of the terminus through the two loops. Note that these pathways are distinct, because the intermediate in Step 3a is a knot, while that of Step 3b is the unknot ().

Fig. 4.

The loop-flipping mechanism identified in structure-based simulations by Bölinger et al. (24). In Steps 1 and 2, green and red loops are formed. In Steps 3 and 4, the red loop adds a second twist, after which the larger green loop flips over the red loop and the blue end threads through the red and green loops in either of the orders illustrated in Steps 3a and 3b.

According to Bölinger et al. (24), the loop-flipping motion is facilitated by the existence of glycine and proline in the flexible regions of the protein. However, loop flipping is the rate-limiting step independent of how far the C terminus has threaded through the smaller loop. Thus, the depth of the knot does not slow the process down. This is in contrast to Taylor’s twisted hairpin theory, which assumes that a deep knot will fold less efficiently than a shallow one (23).

While the simulation of Bölinger et al. (24) shows that loop flipping is implicated in the folding of a deep knot in DehI, loop flipping has also been identified as a folding pathway for a shallow knot in the protein MJ0366 (41). More recently, Chwastyk and Cieplak (42) have shown that MJ0366 has multiple folding pathways, including newly described two-loop mechanisms, and some of the pathways in the one-loop and two-loop mechanisms involve loop flipping. Given these examples of loop flipping as a folding mechanism for both deep and shallow knots, we assert that loop flipping should be considered as a possible folding mechanism for any protein knot whether deep or shallow.

3. Our Proposed Theory of Knot Folding

Motivated by the steps described in the simulation of Bölinger et al. (24) for the folding of DehI, we developed the following general theory of knot folding, which includes the pathways described by Bölinger et al. (24) and Taylor’s twisted hairpin theory as special cases. Like the twisted hairpin theory, our theory is not obtained via a computer simulation and is not focused on any particular protein or family of proteins. In fact, we will show in that all known protein knots can be obtained by applying our steps. Thus, while we do not claim, as Taylor (23) did, that our theory is the only knot-folding mechanism, we believe that our theory is a possibility that should be considered for any knotted protein.

The Steps of Our Theory.

An unknotted open chain is colored as in Fig. 4.

A small red loop and a large green loop each containing zero, one, or two twists form and come close together.

The blue end approaches the two loops, causing the black arc to pass either behind or in front of the red arc.

One of the following occurs.

The green loop flips over the red loop and threads the blue end. Then, the loops align, and the blue end threads through the red loop.

The blue end threads through the red loop. Then, the green loop flips over both the blue end and the red loop so that the loops are aligned and the green loop is threaded.

Fig. 5 illustrates how the knot could be folded using these steps. In Step 1, red and green loops are formed and brought close together. In Step 2, the blue end approaches the two loops, causing the black arc to pass in front of the red arc. In Step 3a, the green loop flips over the red loop and threads the blue end, after which the red loop aligns with the green loop and the blue end threads through the red loop. Alternatively, in Step 3b, the blue end threads through the red loop, after which the green loop flips over both the blue end and the red loop so that the two loops are aligned and the green loop is now threaded.

Fig. 5.

An example of how a knot could be folded with the above steps.

The steps of our theory are closely related to the steps in the simulation of Bölinger et al. (24) for the knotting of DehI. The primary difference between our theory and the simulation of Bölinger et al. (24) is that we allow zero, one, or two twists in each of the loops, while they mandate two twists in the red loop and one twist in the green loop. The other differences are that we do not specify which direction the loops should twist in or whether the black arc should go behind or in front of the red arc.

Because of the parallels between our theory and the simulation of Bölinger et al. (24), we adopt the same assumptions. In particular, following Bölinger et al. (24), we assume that nonnative interactions and chaperones are not required for our steps to occur, although such interactions are likely to speed up the folding rate. Also, for our steps and those of Bölinger et al. (24), the blue terminus is the only one that is required to move during the knot folding. Thus, in a cotranslational model, where the red terminus is attached to the ribosome and the blue terminus remains free, knot folding could still occur with our steps.

In fact, Sorokina and Mushegian (47, 48) have argued that, for many proteins, knot formation is significantly facilitated when the protein is formed on the ribosome. This is consistent with the simulation of Chwastyk and Cieplak (33), which shows that the probability of knot formation for the protein YibK is increased substantially when the protein is formed on the ribosome. More recently, Dabrowski-Tumanski et al. (34) obtained similar results for the deep -knotted protein with PDB ID code 5JIR. Because of the role of the ribosome in promoting knotting in all of these studies, we expect a cotranslational model to promote knotting for our theory as well. In particular, for knots that are deeply embedded on the blue end or on both ends, the loop-flipping mechanism described by our steps might be difficult to achieve. In this case, the ribosome could facilitate the mechanism by acting as a scaffolding during the steps. For example, in our Step 1, the red loop and the green loop could exit from the ribosome and then be held in place while the blue end exits the ribosome in Step 2 and threads through the red loop in Step 3b. Afterward, the ribosome would release the green loop and hold the red loop and the blue end close together while the green loop flips over both to obtain the conformation in Step 4.

Our requirement in Step 1 that there are no more than two twists in each loop is related to an observation of Taylor (23) that the more twists in a hairpin, the farther the termini may be from the loop, making threading less likely. By an analogous argument, the more twists there are in either or both of the loops in our theory, the farther the blue terminus may be from the loops, making loop flipping over the terminus less likely (although this distance might be diminished if the protein is encapsulated in a chaperonin). In addition, according to Banavar and Maritan (49), proteins should be considered as tubes of nonzero thickness, and according to Taylor (23), this means that knots with more twisting require longer chains. Since a given protein has a fixed length, its thickness will favor a knotting mechanism requiring as few twists as possible in each loop. For all of these reasons, whenever multiple pathways produce the same knot, we assume that those with fewer twists in each loop will be more likely to describe successful knotting. We will refer to this principle henceforth as the Minimal Twisting Principle.

With this principle in mind, we cannot allow arbitrarily many twists in the loops described by our theory. Since the red loop in Fig. 4 has two twists and we want our theory to encompass the results of the simulation of Bölinger et al. (24), our upper bound must be at least two. However, as we will see in , all known protein knots can be obtained by applying our steps with at most two twists in each loop. Hence, we use two twists as an upper bound. If new protein knots are identified that require more twists in one or both of the loops, this upper bound can be increased accordingly.

While our steps describe a general knotting mechanism, the particular parameters involved can vary as follows.

The number of twists in the green and red loops and the direction in which they twist

Whether the black arc crosses over or under the red arc

Whether the left or right side of each loop goes in front or behind the blue arc after it threads

The Parameters of Our Theory.

For example, in the final conformation of Fig. 5, the red loop and the green loop each have one twist but in opposite directions, the black arc crosses over the red arc, and the right sides of both loops are in front of the blue arc.

To symbolically represent different types of crossings, we introduce the following sign convention. If the slope of an overcrossing is positive, we designate the crossing by a sign, and if the slope of an overcrossing is negative, we designate the crossing by a − sign. For example, in the final conformation of Fig. 5, the crossing of the red loop is negative, the crossing of the green loop is positive, and the red–black crossing is positive. In some illustrations, it is hard to tell the sign of the red–black crossing. Thus, we remark that the red–black crossing is always positive if the black arc goes over the red arc and always negative if the red arc goes over the black arc. Note that our sign convention does not agree with standard practice in knot theory, which requires a uniform orientation on an entire knot before determining the sign of any crossing.

Fig. 6 shows projections of all of the knots that can be obtained with our steps together with the notation that we will use to represent them. We refer to these projections as the configurations of our theory. Observe that configurations encode the steps of the pathways used to obtain them and are useful for determining the knot types resulting from these pathways. However, these are simplified drawings of the final conformations obtained with our knotting mechanism. The actual conformations are much more complicated.

Fig. 6.

We illustrate all configurations, where and are 0, , or and a or − sign indicates whether the slope of an overcrossing is positive or negative.

We use the following notation for configurations. The letters and indicate whether the left or right side, respectively, of a loop goes in front of the blue arc. We always list an or for the red loop before we list it for the green loop. The first parameter inside of the parentheses indicates whether the red–black crossing at the bottom of the projection is positive or negative. The parameters and , which can be 0, , or , describe the number and slope of the vertical twists inside the boxes. As with and , we list the crossings of the red loop before we list the crossings of the green loop. For example, the knot in Fig. 5 has configuration , and the projection of the knot resulting from the simulation of Bölinger et al. (24) has configuration (see Fig. 12).

Fig. 12.

The knot fingerprint of the knot with configuration agrees with those of the -knotted DehIs.

Although the blue and red ends illustrated in Fig. 6 are very short, either or both ends could be much longer, yielding a deeper knot. A very deep knot, like the found in the protein with PDB ID code 5JIR, would correspond to a configuration where both ends are significantly longer. In this case, the protein would have three separate domains, with only the middle one knotted, and the external domains would remain unfolded until after the knotting mechanism begins.

4. Knots That Can Be Obtained with Our Theory

Table 1 lists the positive forms of all nontrivial knots that can be obtained with our theory together with the parameters of the configurations that are used to obtain them. This includes the positive forms of the knots , , , and , although has only been found in its negative form in a protein. lists the configurations for the unknot and all the nontrivial knots that can be obtained with our steps. displays the same information, but organized according to parameters rather than according to knot type. In particular, this includes the negative forms and , which have been found in proteins. Table 1 lists the positive forms of 10 additional knot types , , , , , , , , , and , which have not yet been identified in proteins. An explanation of how the tables were produced is given in , and detailed computations are provided in .

Table 1.

Right-handed (+) and achiral knots produced by our model and the configurations used to obtain them

Knot	RR(±,a,b)	RL(±,a,b)	LR(±,a,b)	LL(±,a,b)	No. of configurations
+3_1	(+,0,0),(−,1,−2), (−,−2,1),(−,2,2)	(+,0,−1),(−,−2,0), (−,2,1)	(+,−1,0),(−,0,−2), (−,1,2)	(+,−1,−1),(−,1,1)	12
4_1	(+,0,−1),(+,−1,0), (−,1,2),(−,2,1)	(+,−1,−1),(−,1,1), (−,2,0),(+,0,−2)	(+,−1,−1),(−,1,1), (+,−2,0),(−,0,2)	(−,0,1),(−,1,0), (+,−1,−2),(+,−2,−1)	16
+5_1	(+,−2,−2),(−,0,0)	(−,0,−1)	(−,−1,0)	(−,−1,−1)	5
+5_2	(+,0,−2),(+,−2,0), (−,0,2),(−,2,0)	(−,0,1),(+,−2,−1), (−,2,−1)	(−,1,0),(+,−1,−2), (−,−1,2)	(−,−1,1),(−,1,−1)	12
+6_1	(−,−1,2),(−,2,−1)	(−,−1,1),(−,2,−2)	(−,1,−1),(−,−2,2)	(−,1,−2),(−,−2,1)	8
+6_2	(−,0,−1),(−,−1,0)	(−,−1,−1),(−,0,−2)	(−,−1,−1),(−,−2,0)	(−,−1,−2),(−,−2,−1)	8
6_3	(+,1,−2),(+,−2,1)	(+,−2,0),(−,0,2)	(+,0,−2),(−,2,0)	(−,−1,2),(−,2,−1)	8
+7_2	(−,2,−2),(−,−2,2)	(−,−2,1)	(−,1,−2)		4
+7_5	(−,0,−2),(−,−2,0)	(−,−2,−1)	(−,−1,−2)		4
+7_6	(+,2,−2),(+,−2,2)	(+,−2,1)	(+,1,−2)		4
+7_7	(+,2,2)	(+,2,1)	(+,1,2)	(+,1,1)	4
+8_8		(−,−2,2)	(−,2,−2)		2
+8_14	(−,−1,−2),(−,−2,−1)	(−,−2,−2)	(−,−2,−2)		4
+9_23	(−,−2,−2)				1

For a table of all knots produced by our model and their configurations, see ..

Every knot in Table 1, except for , occurs with multiple configurations. Since each configuration represents a pair of pathways to a knot’s formation, this means that our steps produce many pathways to fold most of the knots. This makes biological sense, since different families of proteins with the same knot would not necessarily fold in the same way, and even one particular protein may have multiple knotting pathways.

To understand the relationship between a configuration and its mirror image, observe that all of the overcrossings and undercrossings are interchanged when a configuration is reflected in the plane of the paper. As a result, the mirror image of a configuration will interchange and and change the sign of each of the other parameters of the configuration. We summarize this in the following lemma.

Lemma 1.

Let and be integers. Then, the following relationships hold between configurations and their mirror forms (denoted by a minus sign in front of the configuration):

For example, we see from Table 1 that the knot is produced by 12 configurations:It now follows from Lemma 1 that the knot is produced by 12 configurations:This means that, in total, there are 24 configurations for the knot. By contrast, because the knot is achiral, the 16 configurations listed in Table 1 for the knot are the only ones that can produce it.

One of the key tools that we used to construct the tables is the following result, the proof of which is given in .

Theorem 1.

Let and be integers, and let denote or –. If one of the following configurations has knot type , then all of these configurations have knot type :Furthermore, all of the following configurations have knot type (the mirror image of ):Thus, for an achiral knot , if any configuration listed above has knot type , then all 16 configurations have knot type .

The following theorem, proved in , gives us information about the types of knots that can be produced by our theory (without restrictions on and ).

Theorem 2.

All knots obtained by our steps can be deformed to a conformation with projection that has only two local maxima.

This theorem does not imply that, if a protein becomes knotted via our steps, its final conformation has only two local maxima. In fact, due to physical and chemical properties, such as hydrophobic collapse, the final conformation of a knotted protein is quite complicated, containing many crossings and many local maxima. Saying that a knot can be deformed to have only two local maxima is saying something about its knot type rather than about its particular conformation.

To rephrase Theorem 2 in more mathematical language, all knots that can be produced with our steps are in the family of two-bridge knots. Such knots are a proper subset of the prime knots (i.e., those that cannot be split into two knotted arcs) (50). However, of the 84 prime knots with nine or fewer crossings, just 50 are two bridge (18), and only 14 of these can be obtained with our steps, where and are restricted to 0, , and .

5. Configurations That Are Consistent with Twisted Hairpin Pathways

While our theory of knot folding was motivated by the simulation of Bölinger et al. (24), it is consistent with Taylor’s twisted hairpin theory in the special case where the red loop does not play an essential role in the folding mechanism. In particular, we see in Fig. 7 that, if we start with the configuration , , , or and tighten the knot by pulling downward on the red end while fixing the blue end, then the red arc will slide down along the blue and black arcs so that the red loop disappears, leaving a single red–black crossing at the bottom of the picture. This means that, for these configurations, the knotting was entirely due to the threading of the green loop. In this sense, the configurations , , , and represent knotting mechanisms that are similar to a twisted hairpin pathway.

Fig. 7.

The red loop does not play an essential role in these pathways, since it can be eliminated by pulling down on the red end.

More generally, we say that a configuration is consistent with a twisted hairpin pathway if pulling down on the red end while fixing the blue end causes the red loop and all of the red twisting to disappear, leaving only a single red–black crossing at the bottom. Theorem 3 (proven in ) says that the only configurations with this property are those illustrated in Fig. 7 together with their mirror images.

Theorem 3.

The only configurations that are consistent with a twisted hairpin pathway are , , , , , , , and .

6. Knot Fingerprints of Configurations

King et al. (51) and Taylor (52) defined the fingerprint of a knotted protein to be the knot types of the protein and all of the partial structures obtained by clipping residues from each of the termini. Knot fingerprinting is useful, because it distinguishes different conformations of the same knot. For example, the knot has been identified in KARIs and phytochromes (4). However, according to KnotProt, if both termini of the KARIs are clipped, we obtain a knot, while no matter how much one or both termini of the phytochromes are clipped, we will not obtain a knot.

In this section, we compare the knot fingerprints of configurations from our theory with those of proteins on KnotProt to see if the pathways described by these configurations could correspond to folding pathways for the proteins. In particular, for each knot, we use Theorem 3 to determine which configurations are consistent with a twisted hairpin pathway for that knot. Then, we compare the knotted subchains of these configurations with those on KnotProt. For any protein where these do not agree, we propose an alternative configuration.

We use the following rules for creating fingerprints. At each step, we clip an end roughly at the first place where the knot type of the subchain is distinct from what it was before the cut, indicating which end has been cut with a red or blue arrow. Since the green and red loops are closely aligned, if we clip the blue end so that it goes through the red loop but not the green loop, then an extension of the blue end is likely to again pass through both loops. Thus, our first cut of the blue end will always remove enough of the blue arc so that it no longer passes through either loop, and hence, it will occur near where the blue and black arcs meet.

Each time that we clip one or both ends, we join the ends together with a dotted arc to show the most likely knot in a subchain. These dotted arcs are not part of the structure, and therefore, we remove them before we do any additional clipping.

As explained in , wiggles can be added to any configuration to obtain repeated occurrences of a given knot. Thus, here, we focus only on comparing distinct knot types of subchains in configurations with those of proteins on KnotProt. More information on how fingerprints are determined by KnotProt and in the figures below is in and .

Fingerprints of -Knotted Proteins.

None of the -knotted proteins on KnotProt have subchains containing any other knots. Thus, any configuration for with no other nontrivial knots in its fingerprint will agree with these data.

By the Minimal Twisting Principle, the configuration would describe the most likely pathways for folding the knot, because it requires the least twisting of any configuration for the knot (Table 1). We show in that the fingerprint of has two occurrences of the separated by an unknot. This coincides with the fingerprints of some -knotted proteins (designated by on KnotProt). Since most -knotted proteins have only one knot in their fingerprint, we consider other configurations as well.

After , the configurations for with the least twisting are and . We see in Fig. 8 that the fingerprint for has only one occurrence of the knot. In particular, at the top and center, we illustrate . Then, we clip the blue end on the right and the red end on the left (as indicated by the colored arrows). In both cases, we get the unknot. If we clip either or both ends any farther, we also get the unknot. Thus, the pathways described by could correspond to folding pathways for -knotted proteins. We show in that the fingerprint of contains a knot, and hence, is unlikely to describe folding pathways for the -knotted proteins.

Fig. 8.

The fingerprint of a knot with configuration agrees with the fingerprints on KnotProt containing only one .

The configurations and are consistent with a twisted hairpin pathway (as shown by Theorem 3), and hence, the agreement of their fingerprints with KnotProt provides evidence that the -knotted proteins fold via a twisted hairpin pathway. By taking the mirror images of the configurations for , we obtain the configurations and , which could describe pathways for the folding of -knotted proteins.

We begin by considering the fingerprints of the -knotted KARIs. According to KnotProt, all subchains obtained by clipping either end alone are unknots, but removing a sufficient number of residues from both ends produces a knot. We show in Fig. 9 that the fingerprint of the configuration for the knot agrees with this. Since this configuration is consistent with a twisted hairpin pathway by Theorem 3, our theory supports such a folding pathway for the KARIs.

Fig. 9.

The knot fingerprint of a knot with configuration agrees with those of the -knotted KARIs.

However, is not the only configuration for the knot, which is consistent with a twisted hairpin pathway. In , we show that, among all of the configurations for that are consistent with a twisted hairpin pathway, the only one other than with a fingerprint that agrees with the KARIs is . However, the configuration requires that both loops contain twists, while requires only one loop to have twists. Because of the Minimal Twisting Principle, we propose that the KARIs are more likely to fold according to the twisted hairpin pathways described by .

Next, we consider the fingerprints for the -knotted phytochromes. According to KnotProt, no matter how much either or both ends of the phytochromes are clipped, the knot is the only nontrivial knot that can be obtained. In , we show that all of the configurations for that are consistent with a twisted hairpin pathway contain either a knot or a knot in their fingerprint. Thus, we suggest that the -knotted phytochromes fold via a configuration that is inconsistent with a twisted hairpin pathway.

Fig. 10 illustrates the fingerprint of the configuration for the knot. Clipping either or both ends enough to change the knot type results in the unknot. Thus, the fingerprint of agrees with those of the phytochromes on KnotProt. Since the knot is achiral, it follows that the fingerprint of its mirror form also agrees with those of the phytochromes on KnotProt.

Fig. 10.

The knot fingerprint of the knot with configuration is consistent with the -knotted phytochromes.

In fact, we show in that the fingerprints of all of the configurations for that are inconsistent with a twisted hairpin pathway contain no knots other than the . Thus, any of these configurations could describe the folding pathways of the phytochromes. However, the configurations and are the only ones with just one twist. Thus, because of the Minimal Twisting Principle, we believe that one of these configurations is most likely to describe pathways for the folding of the phytochromes.

Observe that the knot fingerprints illustrated in Figs. 9 and 10 partition the -knotted proteins according to biological function. If these protein classes have different knotting pathways as indicated by their different configurations, it could suggest that the knot plays a different functional role in the phytochromes than it does in the KARIs.

The only proteins that are known to contain the knot are the UCHs. These proteins are shallowly knotted; however, as shown on KnotProt, clipping the C terminus produces a knot. There are no other knots in the fingerprints of the UCHs.

together with Theorem 3 show that the only configurations that are consistent with a twisted hairpin pathway for the knot are and . However, shows that, if we clip both ends of either of these configurations, we obtain a knot. Since the fingerprints of the UCHs do not contain a knot, the UCHs are unlikely to fold via a configuration that is consistent with a twisted hairpin pathway.

We see in Fig. 11 that the fingerprint of the configuration agrees with that of the UCHs. In particular, by clipping the blue end, both ends, or the red end, we get the unknot, while clipping the red end more gives us a knot. If we clip either end any farther, the unknot is produced. Additional support for this configuration comes from its intermediates shown in , which agree with those obtained for the UCHs in computer simulations by Zhao et al. (29).

Fig. 11.

The knot fingerprint of a knot with configuration agrees with those of the -knotted UCHs.

By the Minimal Twisting Principle, the folding pathways described by , which has only one twist, are more likely than those described by a configuration with multiple twists in one loop or a single twist in each loop. The only other configuration for the knot that has a single twist is . However, we show in that the fingerprint of contains a knot, which does not agree with the fingerprints for the UCHs on KnotProt. Thus, we assert that the configuration describes the most likely folding pathways for the UCHs.

The only proteins known to contain a knot are the DehIs. According to KnotProt, clipping either end of DehI a little yields an unknot, but clipping the N terminus a lot yields a knot, and clipping both ends a moderate amount yields the knot.

None of the configurations for listed in Table 1 are of the form described in Theorem 3, and hence, no configuration for is consistent with a twisted hairpin pathway. Thus, we begin with the configuration produced by the simulation of Bölinger et al. (24).

In Fig. 12, we see that clipping the blue end of produces the unknot. Clipping the red end a little also gives us an unknot, while clipping the red end a lot yields a knot. Clipping both the red and blue ends gives us the knot. Any additional clipping of either end enough to change the knot type yields the unknot. This fingerprint matches that of DehI on KnotProt, providing additional evidence that DehI may fold according to the pathways described by the simulations of Bölinger et al. (24).

To determine if another configuration could also describe the folding of the knot in DehI, we considered the simplified illustration of the crystal structure obtained by Wang et al. (43). shows that, with only very minor changes, this simplified crystal structure corresponds to the configuration .

In Fig. 13, we illustrate the fingerprint of . As shown, clipping the red end a little yields the unknot, and clipping it more substantially yields the knot. If we clip the blue end alone, we again get the unknot. However, if we clip both the red end and the blue end, we obtain the knot. Additional clipping of either end enough to change the knot type yields the unknot.

Fig. 13.

The knot fingerprint of the knot with configuration agrees with those of the -knotted DehIs.

Thus, both and have fingerprints that agree with DehI on KnotProt, and hence, either could describe the folding pathways of DehI. However, since requires less twisting and corresponds to the simplified crystal structure for DehI (43), the folding pathways described by are a reasonable alternative to the pathways described by .

7. Discussion

Motivated by the simulations of Bölinger et al. (24) for the knotting of DehI, we introduced a theory that could describe folding pathways for any knotted protein. We expressed our theory in terms of steps that are encoded by the configurations in Fig. 6. Since multiple configurations produce the same knot (as listed in Table 1), our theory shows that different families of proteins containing the same knot could fold in distinct ways. This would apply, for example, to the KARIs and the phytochromes, which both fold into a knot.

The differences between our theory and the twisted hairpin theory introduced by Taylor (23) are the number of folding pathways of a given knot, the number of loops, threading vs. loop flipping, and the possibility of knotted intermediates. According to Taylor’s theory, all knots occur as the result of a terminus threading through the single loop of a twisted hairpin as illustrated in Fig. 2. This means that the complexity of a knot is entirely the result of the twists in the hairpin, and there can be no knotted intermediates. By contrast, according to our theory, a loop-flipping move causes a terminus to be threaded through two loops that are closely aligned but have no more than two twists each. Our theory produces two parallel folding pathways, which can each lead to knotted intermediates. This is consistent with recent experimental and computational results (24, 29, 35, 39).

In , we compared fingerprints of knotted proteins obtained by KnotProt with fingerprints of particular configurations. Our results show that the fingerprints of the configurations and for the -knotted proteins (Fig. 8), and for the -knotted proteins, and for the -knotted KARIs (9) contain the same knots as the fingerprints for these proteins on KnotProt. Since these configurations are consistent with twisted hairpin pathways as shown in Theorem 3, our theory supports Taylor’s twisted hairpin theory in these cases.

However, the knots in the fingerprints for the -knotted phytochromes, the -knotted UCHs, and the -knotted DehI do not correspond to those of configurations that are consistent with a twisted hairpin pathway. Rather, they agree with the configurations for the phytochromes (Fig. 10), for the UCHs (Fig. 11), and (Fig. 12) and (Fig. 13) for DehI. Thus, these configurations describe pathways that could correspond to the folding of these proteins. Furthermore, the configuration resembles the simplified crystal structure for DehI found by Wang et al. (43) and requires less twisting than . Thus, the pathways described by the configuration could be a good alternative to both the pathways described by the simulation of Bölinger et al. (24) and the twisted hairpin pathway proposed by Taylor (23).

While Taylor’s twisted hairpin theory of knot folding may be correct for most knotted proteins with three or four crossings, our results show that, for more complex protein knots, there may be other viable folding pathways. Furthermore, Taylor’s theory predicts that all future protein knots will be members of the twist knot family, whereas our theory predicts that some nontwist knots in the family of two-bridge knots might also eventually be found in proteins. Nonetheless, we predict that only the 14 knots listed in Table 1 (or their enantiomers) are likely to occur in proteins, although this list would be longer if we allowed three twists in each loop.

We see from Table 1 and that the , , , and knots (which have been found in proteins) have 12, 12, 16, and 12 configurations, respectively, while enantiomers of the knots , , , , , , , and (which have not been identified in proteins) have only 5, 4, 4, 4, 4, 2, 4, and 1 configurations, respectively. Since configurations describe folding pathways, this means that, according to our theory, there are fewer pathways to obtain the knots in the latter group than to obtain those in the former group. This may explain why none of the latter knots have been found thus far in any protein. This is not surprising for complex knots with seven or more crossings, but it offers a different hypothesis for why the knot has not been found.

The situation for the six-crossing knots is somewhat more subtle. The knot has eight configurations as does the achiral knot as well as each enantiomer of the knot. However, all eight of the configurations for the knot require at least one loop to contain two twists. Thus, by the Minimal Twisting Principle, we believe that, of the three six-crossing knots, the knot is the least likely to occur in a protein. By contrast, for each enantiomer of the knot, four of its eight configurations require no more than one twist in each loop. We compare this with the knot, where only two of its eight configurations require no more than one twist in each loop. Because the knot has been found in DehI, we predict that at least one of the enantiomers of the knot will eventually be identified in a protein. If this turns out to be the case, it would be the first nontwist knot found in a protein.

8. Materials and Methods

Method for Obtaining Table 1.

lists all possible configurations. To determine the knot types associated with these configurations, we started with a configuration and deformed it into a knot projection in the standard knot tables (18). Next, we applied Theorem 1 to that configuration to obtain other configurations that represent the same knot. We did this repeatedly until all 200 configurations listed in were identified. We then reorganized the information into Table 1 and , which group the information by knot type rather than by the parameters of the configuration. Note that the configurations for the negative forms of the chiral knots in can be deduced from the configurations for their positive forms in Table 1 by applying Lemma 1; also, Table 1 does not include the unknot, which is among the configurations listed in .

As an example, below we show how all configurations for the knot were obtained. We begin by deforming the configuration to the standard projection of in Fig. 14.

Fig. 14.

is the knot.

Next, we apply Theorem 1 to the configuration to conclude that the following configurations also produce the knot: , , , , , , and . Since is achiral, we can apply Lemma 1 to conclude that is also produced by the configurations , , , , , , , and . This gives us all 16 configurations for that are listed in Table 1.

Methods for Computing Knot Fingerprints.

KnotProt (4, 5) determines fingerprints by starting with a crystal structure from the PDB, which is positioned in the center of a large ball. The termini are then extended to the boundary of the ball in several hundred randomly chosen directions. In each case, the endpoints are joined together by an arc in the boundary of the ball. To identify the knot type of the closed loop, the Alexander polynomial is computed. If the knot is chiral, the HOMFLYPT is computed to identify the exact enantiomer. Of the several hundred knot types obtained in this way, the most frequently occurring one is then assigned to the protein. To determine the subknots in a protein, the endpoints are clipped to specified residues, and the same method is used.

In contrast with KnotProt, we determine the fingerprints of our configurations qualitatively rather than quantitatively. In particular, we do not do a probabilistic analysis of hundreds of ways to extend the ends of a configuration to the boundary of a ball to get a closed knot. Rather, we assert that it is possible to draw the configurations and subchains as we have in and join the ends with dotted arcs in such a way that the knots that we obtain are indeed the most probable ones.

For simplicity, we do not include wiggles when we draw configurations, although wiggles occur in protein conformations and are important, because they can result in slipknots in subchains (45, 51, 53). Also, wiggles can cause the same knot to appear multiple times in a fingerprint, separated briefly by an unknot. For example, according to KnotProt, the fingerprint for the protein 5m4sA is , meaning that the entire chain contains a knot, but there is also a knot in a subchain. As we see in , this fingerprint occurs for the configuration . In , we show that this same fingerprint can also occur with the configuration by adding a wiggle. All of the fingerprints on KnotProt with multiple occurrences of a given knot can be similarly obtained from those in by adding wiggles at appropriate places.