Contents:
Finally, we question ourselves about the mechanisms put in place by nature to fold these peculiar proteins. As earlier pointed out, the most remarkable difference between the knots found in viral DNA strands and the ones in proteins is reprodu-cibility: always the same knot, always in the same place. By analogy with human-sized cords, this is the difference between a messy tangle of earphones wire and the precise, elegant knot on a tie or a pair of handmade shoes—the pivotal role missing in the first case being played by a skilled, well-intentioned human.
If, on the one hand, the presence of chaperones represents a considerable aid to the folding process, on the other hand it should be noted that, in many cases, knotted proteins do not require them to spontaneously attain the self-entangled conformation. Albeit inefficiently, though, knotted proteins are sufficiently emancipated. Unfolding and refolding experiments have largely supported this observation [ 8 , 38 — 47 ], thereby highlighting the fact that not only can a knotted topology be fully embedded in an otherwise unspecific amino acid sequence, rather it can also be attained by the sole means that this sequence commonly relies on—intra-protein and protein-solvent interactions.
Nonetheless, relatively small knotted proteins can snap into a tied native structure without helping hands. Recent work [ 48 — 51 ] has highlighted the potential impact and importance of the protein synthesis itself on knotting. In fact, simulations [ 48 — 50 ] have shown that the chain folding and the knot formation can contextually occur with the polymerisation the chain in the ribosome, a process dubbed cotranslational folding [ 52 ].
The chain, in fact, transolcates through a pore, so that the newly synthesised stretch of the sequence has a lower conformational entropy than it would have if the remainder of the protein were present.
This, as well as other factors, can favour the self-entanglement of the protein. Differences with respect to the folding pathways followed by unknotted proteins, however, are present. In general, folding can proceed through various routes, characterised by several milestones and intermediate states among which the molecule can interconvert before landing onto the native, biologically functional conformation. The existence of a 'topological bottleneck', on the contrary, forces self-entangled proteins to avoid all those intermediate steps which are incompatible with the topology of the native state and might prevent the achievement of the latter [ 44 ].
Backtracking is required in these cases, that is, a partial unfolding necessary to solve the undesired tangle and attempt anew to obtain the desired one. The formation of the native topology is thus identified in the most relevant rate-limiting step in the folding process of self-entangled proteins. This fact has been most clearly highlighted by the experiments carried out in the Jackson group by means of de novo folding [ 8 ]. When available, chaperones help knotted proteins to fold correctly, reaching the aforementioned factor in speedup.
Other self-entangled polypeptides, however, do not take advantage of this kind of aid, and have by default to manage knotting by themselves. This is especially the case of small knotted proteins, whose folding pathways have been thoroughly studied. Backtracking is avoided in these mainly through a fairly polarised free energy landscape, which resembles a funnel-shaped highway rather than a mountain pass [ 44 , 53 — 55 ]. Many relevant cases have shown folding mechanisms involving the formation of loops through which a terminus penetrates, thereby establishing the native topology and the native structure almost in a single step.
The piercing terminus can happen to be 'straight' or bent in a hairpin-like conformation; in this second case, the knotting event takes place first through the formation of a slipknot [ 7 , 30 , 37 , 56 — 61 ], which subsequently opens up into a regular knot. Larger proteins, as anticipated, can feature more complex folding pathways, involving intermediate steps [ 42 , 46 , 47 , 62 , 63 ] and inspiring novel schemes for the description of the knotting process [ 64 ]. The experimental characterisation of knotted proteins and other types of self-entangled polypeptides has achieved remarkable successes.
These have required a broad spectrum of techniques, such as mechanical stretching of proteins by means of optical tweezers and AFM, in vitro translation-transcription, recombinant and cell-free protein expression, SAXS, fluorescence etc. Several different tools had to be put in place to synthesise wild type and mutant knotted proteins, create new ones, determine and characterise their structure, investigate their response to mechanical stresses, and above everything pinpoint their topological state.
Important pieces of knowledge have been obtained by these means. However, it appears evident that the experimental tools alone are not sufficient. This is more and more true in every facet of science, and the investigation of self-entangled proteins makes no exception. The insight contributed by in silico studies—be that through accurate and realistic all-atom models or simplified, effective coarse-grained representations—is of paramount importance to comprehend the fundamental properties and mechanisms that underlie the formation of protein knots.
The flexibility given by this instrument in constructing ad hoc models endowed with specific features and studying their properties is an invaluable help in understanding how does a protein tie a knot, what biological role such an entanglement might play, and which inescapable features its sequence must have in order to entail the capability of doing all this. The objective of the following chapters is to present the reader a list of the most popular, effective, and efficient techniques that have been employed in this endeavour so far.
The theoretical basis underlying all computational methods and models is illustrated with the aim of being clear and informative rather than detailed and comprehensive, in order to provide an agile resource to refer to when in search of the appropriate tool to tackle a given problem involving self-entangled proteins. The vastness of this yet rather young field of research, combined with the rapidity with which it evolves, makes it difficult to imagine that this resource will stand the proof of time, the latest edge-cutting development surely being only a few months away from the time of this writing.
However, the construction lies and relies on a solid bedrock, and it is the intent and hope of the writers to present the readers with a sufficiently good guide of the old town so as to allow them to confidently explore the modern neighbourhoods of the city—and why not, motivate them to build a new block. Knots in proteins represent an example of physical knots , topological entanglements of linear objects, characterized by physical properties such as thickness, friction, or flexibility.
These properties distinguish such objects from the immaterial curves considered by the formal knot theory.
Nonetheless, the study of physical knots mutuates several concepts from knot theory, crucial to define and classify the topological states of proteins [ 65 — 67 ]. In the present section we report these theoretical concepts, costituting the necessary background for the study of entanglements in proteins. According to theory, knots can be rigorously defined only as a property of closed curves [ 1 — 4 ]. However, as mentioned before, objects such as proteins, whose geometry can be represented by an open curve, are found in stable, deeply entangled states.
The entangled configurations of open curves can be traced back to a well-defined knot if a closure operation is performed, namely if its two ends are artificially connected by extending the curve. The definition of the closure represents therefore a crucial step in the detection and classification of knotted polymers and proteins.
Recently, Turaev has proposed the mathematical definition of knotoids [ 68 ], which generalizes the concept of knots including open curves entanglements. As such, these topological objects are well-suited to characterize the topology of proteins, without requiring the definition of a closure [ 69 — 71 ].
A further approach to the classification of topology in open curves is adopted in [ 72 ], where protein structures are analized as virtual knots. Both these classification methods build on a statistical treatment of all possible planar projections of three-dimensional curves. While knotoids do not require closure by definition, in virtual knots a so-called virtual closure is performed on each planar projections, keeping trace of the possible topological ambiguities introduced while closing the curve.
Despite the existence of these novel, more general concepts, ordinary knot theory is used in most of the literature about entangled proteins, therefore we shall rather focus on the knot-classification than on knotoids or virtual knots. We first consider immaterial closed curves, introducing the knot theory insights required to classify their topological state, in section 3. Then, in section 3.
After that, in section 3. The definition and detection of entangled states other than knots is addressed in section 3. As mentioned before, knot theory defines the topological state of closed curves. By common experience we know that, while a knot on an open string can be undone by proper manipulation, this is impossible if the ends of the string are attached to form a loop see figures 1 a and b.
Any possible deformation in space of this closed loop preserves its topological state. This suggests that the knotted state of a closed curve can be operationally defined by means of spatial deformations.
Biochemistry. Dec 3;52(48) doi: /bi Epub Nov Computational and theoretical methods for protein folding. Compiani. Share. Email; Facebook; Twitter; Linked In; Reddit; CiteULike. View Table of Contents for Computational Methods for Protein Folding.
Figure 1. Pictorial representation of a knotted open string a , of a knotted closed string b , and of a knotted closed curve in red in the three dimensional space. To render the three-dimensionality of the curve it is embedded on the surface of a torus gray dotted mesh. Let us mathematically represent the knotted closed string of figure 1 b as a closed curve embedded in the three-dimensional euclidean space , as shown in figure 1 c. In mathematics, all the possible continuous deformations of X in space are called homotopies see e.
To define the topological state of X we need to restrict the class of transformations to those homotopies that prevent the curve from passing through itself, named isotopies. This is still not sufficient, since X has no thickness, any entanglement hosted by the curve can be continuously reduced to a single point, transforming the curve into an un-knotted loop, also called trivial knot. This leads to the definition of an ambient isotopy AI , which deforms X through the continuous transformation of its embedding space, in this case.
The action of an AI on X does not change its topological state, thus the AIs are the mathematical analogous of the string manipulation mentioned before. The topological state of a curve is defined through AIs as an equivalence class, named knot type. Two curves that can be transformed into each other by AIs belong to the same knot type, namely they are topologically equivalent. This definition is related to the concept of knot complement , namely , where S is a compact region embedding the curve and is a tubular region that indicates the neighboring space of X.
Indeed, K defines the knot type, as equivalent knots have homeomorphic complements [ 73 ]. For example, all the curves that can be transformed into a circle belong to the trivial knot type, and are said to be unknotted. There exist infinitely many possible knot types, the known ones being classified in catalogs [ 74 , 75 ]. From this general definition of knots, we shall restrict to those knot types that can describe a physical objects, which are named tame knots.
Tame knots, as demonstrated in [ 2 ], can always be represented by closed, non-intersecting and finite polygonal curves, called polygonal knots. An example of a knot type excluded from this definition is the so-called wild knot, shown in figure 2 , which features an infinitely recursive character and cannot represent a physical knot.
Figure 2. A wild knot, featuring a pattern tangle indicated by the dashed boxes that is repeated and rescaled infinite times, towards the limit where the tangle reduces to a point. The repeated tangle is trivial, but the whole curve is not see e. To represent the three dimensional character of the curve we have used the knot diagram notation, explained in section 3.