Implications of Molecular Topology for Nanoscale Mechanical

Article Cite This: J. Phys. Chem. B XXXX, XXX, XXX−XXX

pubs.acs.org/JPCB

Implications of Molecular Topology for Nanoscale Mechanical Unfolding Narges Nikoofard† and Alireza Mashaghi*,‡ †

Institute of Nanoscience and Nanotechnology, University of Kashan, Kashan 51167-87317, Iran Leiden Academic Centre for Drug Research, Faculty of Science, Leiden University, Leiden 2333 CC, The Netherlands

‡

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S Supporting Information *

ABSTRACT: Biopolymer unfolding events are ubiquitous in biology and mechanical unfolding is an established approach to study the structure and function of biomolecules, yet whether and how mechanical unfolding processes depend on native state topology remain unexplored. Here, we investigate how the number of unfolding pathways via mechanical methods depends on the circuit topology of a folded chain, which categorizes the arrangement of intrachain contacts into parallel, crossing, and series. Three unfolding strategies, namely, threading through a pore, pulling from the ends, and pulling by threading, are compared. Considering that some contacts may be unbreakable within the relevant forces, we also study the dependence of the unfolding efficiency on the chain topology. Our analysis reveals that the number of pathways and the efficiency of unfolding are critically determined by topology in a manner that depends on the employed mechanical approach, a significant result for interpretation of the unfolding experiments.

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INTRODUCTION Polymer unfolding occurs in many critical biological processes, such as translocation, degradation, and even folding of proteins and nucleic acids. Translocation of proteins through cellular nanochannels is typically coupled to a partial or complete unfolding of the protein chains.1,2 Disaggregation of protein aggregates may also be achieved by translocation-mediated unfolding.3 Furthermore, translocation can be coupled to degradation of the translocated polymer segment. This process, called digestion, is carried out by exopeptidases that process proteins4 or exonucleases that carry out degradation of unwanted nucleic acids.5 Also, folding of a protein to its final native state often involves partial unfolding and subsequent refolding.6 Unfolding can be mediated by molecular chaperones7 or processive motor-associated pores8 or can be achieved through a flow-mediated pulling process.9 Other than these fundamental aspects, understanding protein unfolding and disaggregation is important for biotechnological applications and pharmaceutical industries.10 Furthermore, advances in nanotechnology have enabled mechanical unfolding of single molecules using nanopores, optical tweezers, and atomic force microscopes.11,12 These advances have made unfolding a popular method for investigating the structure and function of different bipolymers.13−15 Despite significant progress in analyzing unfolding processes,16−19 we have limited mechanistic insights into the unfolding of biopolymers as it happens in nature or in engineered settings. A fundamental open question is whether and how inherent properties of a folded chain and properties of © XXXX American Chemical Society

the environment affect the unfolding processes. In particular, it is not clear whether the initial topology of a folded or misfolded protein or nucleic acid determines their unfolding and how that depends on the unfolding machinery that mediates molecular unfolding. Experimentally, there are three mechanical methods for unfolding a chain: (1) pulling the chain from its ends, (2) threading the chain through a nanopore, and (3) pulling by threading the chain through a nanopore. The third method is a combination of the first two methods. In this method, one end of the chain is tethered and the other end is pulled through a nanopore. Figure 1 provides a schematic representation of the three methods. These approaches are commonly used in biological organisms as well as in engineered systems.3,9,20 Here, we study the effect of chain topology on the mechanical unfolding of a chain. Unfolding involves disruption of intrachain contacts, which can be due to direct intrachain interactions or can be mediated by a linking molecule (e.g., CTCF-mediated genome folding). We use the circuit topology approach to describe the folded configuration of the polymer chains. Circuit topology is a generic method for parameterizing the arrangement of intrachain contacts of a folded linear chain, e.g., a protein, an RNA, or a DNA molecule (genome).21−28 In this study, we identify the number of pathways for unfolding different chain topologies using the three mechanical unfolding methods. We also investigate the unfolding efficiency of the three methods, Received: September 27, 2018

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DOI: 10.1021/acs.jpcb.8b09454 J. Phys. Chem. B XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry B

binding energies and the characteristic times change considerably with the type of the contacts. Here, we aim to study the problem of unfolding a chain generically; i.e., for any type of contacts, we are interested in knowing the effect of the arrangement of intermolecular contacts and existence of much stronger contacts. However, once the type of the contacts is defined, one can easily incorporate the relevant physiochemical parameters in the model (examples are given in the Supporting Information). In the circuit topology of folded linear chains, three different topologies are possible for a 2-contact chain: series (S), cross (X), and parallel (P) (Figure 2). Consider the case that one

Figure 2. According to circuit topology of linear chains, any two contacts can be arranged in three different manners. These arrangements are called series (S), cross (X), and parallel (P).21

Figure 1. Schematic representation of the three mechanical unfolding methods: pulling, threading, and pulling by threading. As a representative example, the number of pathways and efficiency of unfolding are shown for a 3-contact chain with a specific topology.

aims to mechanically unfold a 2-contact chain. In the pulling method, when the contacts are in series or cross relations relative to each other, any of the two contacts can be opened independently. Thus, there exist two unfolding pathways for series and cross topologies. However, for a 2-contact chain with parallel contact arrangement, there exists only one pathway. The contact nested inside the other contact cannot be opened unless the upper contact is opened first. In the threading method, only the contact in the nanopore can be opened at each time. Therefore, there exists only one pathway for the three topologies in the threading method. For a polymer with N contacts (or 2N contact sites), there are (2N − 1)!! different possible topologies. The topology of a chain is measured as the number of binary contacts that are in series, cross, and parallel relations relative to each other (Ns, Nc, and Np, respectively). For each topology, the number of pathways for unfolding the chain depends on the unfolding method. In the threading method, the chain is unfolded from the end inside the nanopore, consecutively. In the pulling method, the chain can be unfolded from any contact along the chain, under the condition that this contact is not nested inside an intact parallel contact. In other words, if two contacts are in parallel topology relation with each other, such that contact A is inside contact B, contact A does not feel the pulling force until contact B is disrupted. In pulling by the threading method, the relative magnitude of two lengths determines whether pulling by threading primarily acts like pulling or threading. If the length of the released chain behind the nanopore L is smaller than the distance between the nanopore and the tethered end of chain d, the pulling component would be dominant. However, if L > d, this method would be similar to simple threading. In pulling and threading experiments, one often passes the entropic regime (for polypeptides and nucleic acids) before the native contacts are broken. Aggregations are often made of tight contacts (relative to thermal energy) that break at high forces. As a result, the chain becomes nearly completely

in the case when one of the contacts is nonbreakable. This scenario is relevant to situations where a contact is so tight (has a high binding energy) that it cannot be disrupted with the applied force.29 Another example of a nonbreakable bond can be seen when exonucleases are unable to unfold a contact; this is because the RNA region right before the contact has a particular sequence.30

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MODEL AND SIMULATION METHOD We are interested in single-molecule unfolding techniques; thus, a single isolated chain is considered in the model. The aim is to understand how the internal arrangement of the contacts affects the unfolding process of the chain. One advantage of this knowledge would be the development of new methods for discovering the internal arrangement of a folded polymer using the data obtained by unfolding that polymer. Previous studies on the unfolding experiments focus on contact energies, whereas here, we take contact arrangements into account as well. Thus, our work is an essential complement to previous studies. To this end, a chain is taken to have an arbitrary contour length but a specified number of contacts. It should be noted that other parameters such as the distance between the contact sites are not included in the model, for the sake of simplicity. To quantify the arrangement of the contacts, the circuit topology of folded linear chains is used. The circuit topology is a general method for the classification of intermolecular contacts. Prior to using this method, it is necessary to choose the type of contacts that is relevant to the polymer of interest and the process to be studied. Hybridized regions would be relevant, when one aims to study the structure of RNA molecules. For proteins, the contact sites may be differently defined according to β strands, disulfide bonds, or neighboring regions with large interaction energies.21 The B


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Figure 3. Number of pathways for unfolding a 5-contact chain using the pulling, the pulling by threading, and the threading methods. In the pulling by threading method, the distance between the nanopore and the wall, d, also matters.

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SIMULATION RESULTS In this section, the simulation results are explained in detail. Our findings can be a basis for a new data analysis approach for mechanical unfolding experiments. Several examples about the relevance of these findings to experiments are discussed in the next section. Number of Pathways. The number of possible pathways for unfolding a 5-contact chain is shown as colormaps in Figure 3 for the three methods. In these plots, the number of parallel and series relations (Np and Ns) are shown on the axes and the colors represent the pathway counts. Considering that the total number of relations between all possible contact pairs in a 5contact chain is equal to 10, the number of cross relations is not an independent parameter (Ns + Np + Nc = 10). The results are shown for different values of the distance between the nanopore and the tethered end, d, for the pulling by threading method. All points are the result of averaging over different chains with the same topology numbers. The upper left panel in Figure 3 shows the number of pathways for the pulling method. The number of pathways depends only on the number of parallel relations (as is directly concluded from the simulation method). The number of pathways increases from 1 to 120 when the number of parallel relations decreases from 10 to 0. This is because in a chain with no parallel topology any of the contacts along the chain can be opened during the pulling method independently. Thus, there are 5! = 120 possible ways for unfolding a 5-contact chain with no parallel relations. However, when there are parallel contacts, a contact can only be opened if the upper contact is disrupted before. Thus, for a chain with all contacts being in parallel to each other, there is only one way to unfold the chain. The lower right panel in Figure 3 shows the number of pathways for the threading method. In the threading method, the contacts open consecutively from the chain end at the nanopore. Thus, there is only one pathway for unfolding of all topologies. The upper middle panel in Figure 3 corresponds to the pulling by threading method, for d = 8. The number of pathways has decreased for the topologies with the most series relations (points inside the oval). This is a result of the large

stretched before breaking the contacts. Therefore, entropic effects are ignored in our simulations, at least to a good approximation. The other important parameter in the unfolding process is heterogeneity of the contact energies. The unfolding efficiency is defined as the amount of length released normalized by the total length released after disruption of all contacts. If all of the contacts are weak, all of them will break during the unfolding process for the three methods and the unfolding efficiency would be 100%. However, if one of the contacts is too tight to be broken, then the efficiency of unfolding will be less than 100% and would differ for the three methods. (See Figure 1 for a schematic representation.) If ΔL is taken to be equal to the length released during unfolding, in the presence of a tight contact, the unfolding efficiency would be equal to ΔL . Here, ΔL tot

ΔLtot is the length released during unfolding, in the absence of any tight contacts. (For more details of the circuit topology and the simulation method, see Supporting Information.) Importantly, as we demonstrate below, the unfolding efficiency depends on the unfolding method. The location of the nonbreakable contact is another determining factor. We define the location of the contacts using the position of their contact sites on the polymer contour from one specific end of the polymer. For each contact, the contact site that is closer to the specified end defines the location of the contact. In the following, we take a 5-contact chain as a representative example of a chain of complex topology. Thus, in the rest of the paper, all of the analyses are conducted for a 5-contact chain. Simulations for chains with various numbers of contacts are also performed (see Supporting Information); the results indicate that the reported trends do not depend on the number of contacts. Our study is primarily focused on the contact topology of the chain, and as such, without loss of generality, we assume equally spaced contact sites on the chain. We note that this assumption particularly impacts the number of pathways for the pulling by the threading method as well as the unfolding efficiency. C


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The Journal of Physical Chemistry B end-to-end distance of a chain with all series relations. In the pulling by threading method, when the chain end-to-end distance becomes larger than d, the method is similar to threading. As was stated, for the threading method, the contacts can only be opened consecutively from the nanopore side. The upper right and the lower left plots of Figure 3 correspond to the pulling by threading method, for d = 6 and 7. It is seen that as the distance between the nanopore and the wall decreases, the number of pathways for more topologies reduces from that for the pulling method. The boxes distinguish the points that are mostly deviated from the plot of the pulling method. The lower middle panel in Figure 3 shows the number of pathways for the pulling by threading method, for d = 5. According to the required condition that d should be larger than the end-to-end distance for all topologies, this is the smallest value for d that we could examine. It is seen that the number of pathways has decreased again toward the threading method relative to the previous plots. If we reduce the distance more despite the required condition, we would asymptotically reach the threading method. Looking more closely at the lower middle panel in Figure 3, one can see that for Np = 0 the number of pathways is a nonmonotonic function of Ns (the points inside the box). Besides, the number of pathways for the topology with Ns = Np = 0 suddenly changes from 120 to 1 with decreasing the value of d (comparing with the previous plot for d = 6). To understand this nonmonotonic behavior, the number of pathways is plotted as a function of Ns, for Np = 0 and different values of d (Figure 4a). It is observed that the nonmonotonic behavior is observed only for d = 5. The nonmonotonicity is especially seen at Ns = 4 and 0. In Figure 4b, the number of pathways for different configurations of the chain is plotted separately (not averaged). The colors in this plot show the initial length of the chain (end-to-end distance of the chain before unfolding). It is seen that a sudden increase of the initial length of the chain at Ns = 0 causes a sudden decrease in the number of pathways. The nonmonotonicity at Ns = 4 also results from the two configurations with a large initial length. In contrast, all of the configurations with Ns = 3 and 5 have a small initial length. We also note that in this plot the maximum initial length of the configurations is 5. Thus, the nonmonotonic behavior is seen only for d = 5. Overall, we conclude that the initial length of the chain is a determining factor for the number of pathways when the pulling by threading method is applied. Unfolding Efficiency. Contacts in (folded and misfolded) proteins and nucleic acids have a wide range of contact energies, and some of the contacts may be so tight that they cannot be disrupted with a given pulling force. In this case, the unfolding efficiency gives a measure of the length ratio that can be released through the unfolding process. In Figure 5, we plotted the unfolding efficiency (colormap) as a function of Ns and Np for different unfolding methods and various positions of a nonbreakable contact. The results are averaged over different chains with the same topology numbers. It should be noted that for some topologies the unfolding efficiency depends on the unfolding pathway when the pulling by threading method is applied. In this case, the result is averaged over different pathways in Figure 5. Interestingly, for some other topologies, despite different set of broken contacts

Figure 4. Number of pathways for the pulling by threading method, for Np = 0. (a) Results are averaged over different configurations of the chain that have the same topology numbers. (b) Results are shown without averaging, for d = 5. The initial length of the chain is represented with color.

in different unfolding pathways, the final length of the chain and the unfolding efficiency are similar. In Figure 5, the first row (Nnbrk = 5) shows the results for the case when the nonbreakable contact is positioned at the farthest distance from the chain end (or the nanopore). It is seen that the unfolding efficiency is the same for the three methods. A similar trend is seen in the three plots: the efficiency increases with the sum of the parallel and the series numbers. In other words, the unfolding efficiency is generally constant on the isoclines of Ns + Np. Constant values of Ns + Np refer to constant values for the number of cross topologies. The maximum value for Ns + Np refers to the smallest value for Nc. This general behavior is perturbed only on the line with Ns = 0 for small values of Np (the points within the oval). The general behavior can be justified by considering a 2-contact chain with cross topology and a nonbreakable contact. In this chain, the cross topology releases no length upon disrupting the one breakable contact. Here, one sees a similar behavior for a 5-contact chain that increasing the number of cross topologies (direction of the arrow) decreases the unfolding efficiency. The second row (Nnbrk = 4) in Figure 5 shows that the efficiency for the three methods reduces by moving the location of the nonbreakable contact toward the chain end (or the nanopore). The reduction for the three methods is stronger on the line with Ns = 0. This change is also seen on line Ns = 1 for small values of Np. (Either mentioned points are shown with ovals.) This is because Ns = 0 refers to more nonlocal contacts that attach more distant parts of the chain. When the nonbreakable contact connects more distant parts, it prevents the release of a larger length of the chain in the unfolding process. It is also seen that the decrease in efficiency D


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Figure 5. Efficiency of unfolding a 5-contact chain with a nonbreakable contact, using the pulling, the threading, and the pulling by threading methods. For the pulling by threading method, d = 5 is assumed in all plots. The rows correspond to different positions of the nonbreakable contact.

Figure 6. Unfolding efficiency (shown in Figure 5) averaged over different positions of the nonbreakable contact. Dependence on series and parallel topology relations, Ns and Np, is shown.

correspond to the smaller values of Ns. This is because small values of Ns refer to more nonlocal contacts and more distant parts attached. Also, the trend for the pulling by threading method changes from that for the pulling method clearly. The efficiency decreases much more than that in the pulling method, for larger values of Ns. This results from the more extended size of the topologies with large Ns values. In the pulling by threading method, the more extended the size of the chain, the more the threading component dominates the unfolding process (rather than the pulling component). The last row (Nnbrk = 1) in Figure 5 corresponds to the nonbreakable contact located near the chain end (or the nanopore). In the pulling method, large values of Np and small values of Ns (or, equivalently, large values of Np and Nc) result

for the threading method is stronger than that for the other methods. The third row in Figure 5 corresponds to Nnbrk = 3. In previous rows, we had a general behavior that depended on Ns + Np, which was not obeyed around small values of Ns. Now, this discrepancy from the general behavior is seen for all points with Ns = 0, 1 (separated by a line), clearly. This description is true for the pulling and the pulling by threading methods. These two methods behave generally in the same manner. Also, a nonmonotonic behavior is observed on the line with Ns + Np = 10 for all of the three methods. The fourth row in Figure 5 corresponds to Nnbrk = 2. The previous general behavior has completely disappeared in the plots. A new trend is seen in which the smallest efficiencies E


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The Journal of Physical Chemistry B in smaller efficiencies. On the other hand, in the pulling method, large values of Ns and small values of Np lead to higher efficiencies. For the pulling by threading method, the efficiency is highest around the middle values of Ns (the points separated by the two lines). This is because small values of Ns correspond to more nonlocal contacts and large values of Ns result in more extended structures. In Figure 6, the unfolding efficiencies are averaged over different positions of the nonbreakable contact. It is seen that the efficiency for the pulling method increases mostly with increasing Ns (more local contacts). It also weakly increases with increasing Np. Again, this can be justified by looking at the unfolding efficiency of a 2-contact chain, which becomes maximum for the parallel topology. Indeed, a parallel topology connects more distant parts; consequently, it releases a larger length of the chain upon disruption. For the threading method, the unfolding efficiency is highest for large values of Ns + Np or small values of Nc. The pulling by threading method differs from the pulling method for large values of Ns. This is because the chains with a large number of series topology have more extended structures. Therefore, the pulling by threading method for unfolding these chains is primarily dominated by the threading component. Another way of approaching this problem is to identify the number of contacts that are parallel to the nonbreakable contact (rather than the total number of parallel relations). There is a formulation of the circuit topology framework in which one can distinguish contacts that are parallel or antiparallel to a given contact.21 Contact A is parallel (P) to contact B if the contour length of the chain between the two contact sites of A belongs to that of B. On the other hand, if the contour length between the two contact sites of B belongs to that of A, then A is antiparallel (P−1) to B. To avoid confusion, we use P for referring to topological relations that are ignorant of orientation and P* for topological relations that respect orientation. Thus, P = P−1, but P* and P*−1 are not the same. In this approach, two configurations with nonidentical Np and Ns but an identical fraction of P* contacts with respect to a given contact are classified into one category. Thus, here, the total number of Np and Ns or the location of the nonbreakable bond is not important, but the significant parameter is the number of contacts that are P* to the nonbreakable contact (excluding those that are P*−1 to). Figure 7 shows dependence of the unfolding efficiency on Np*. As one expects, the efficiency in the three methods decreases with increasing Np*. One may ask if the unfolding efficiency also depends on the position of the nonbreakable contact. The unfolding efficiency as a function of Np* and the position of the nonbreakable contact is shown as a colormap in Figure 8. Nnbrk refers to the position of the nonbreakable contact when the contacts are numbered from the chain end (or the nanopore). Only Nnbrk + Np* < 5 is meaningful. There is no significant dependence on the position of the nonbreakable contact for the pulling methods. However, for the threading method, this dependence is highest among the three methods, as one expects. The dependence on Nnbrk becomes maximum for small values of Np*.

Figure 7. Unfolding efficiency versus Np* for the pulling, the threading, and the pulling by threading method. Np* is the number of contacts that are parallel (and not antiparallel) to the nonbreakable contact.

number of pathways depends on the number of parallel relations mostly when the pulling and the pulling by threading methods are applied. This is because a contact that is nested inside a parallel relation cannot be disrupted unless the upper contact is broken. This finding is expectedly important for interpreting the experimental data on the unfolding pathways.12,17,18 These works discuss the number of intermediates and steps in the unfolding process toward the complete structural loss. Most recent works confirm multistate equilibrium unfolding.18 This is in agreement with our findings that unfolding involves multiple steps that, in turn, depend on the chosen pathway. Besides, our work shows that topology is a major determinant of the number of available pathways for unfolding a chain. To the best of our knowledge, this point has not been explicitly addressed in previous studies. We also discussed the unfolding efficiency of a chain containing a nonbreakable contact and its dependency on the chain topology and the employed nanomechanical methods. When the nonbreakable contact is farthest from the chain end, the unfolding efficiency depends mostly on the number of cross relations for all of the three methods. In the opposite case, when the nonbreakable contact is closest to the chain end, the unfolding efficiency decreases with the number of parallel relations and increases with series ones when the pulling method is used. For the pulling by threading method, the efficiency is highest for middle values of the number of series relations. We found that the efficiency depends on the number of contacts that are nested inside a parallel relation with the nonbreakable contact. These results are important in determining the efficiency of some of the enzymes. For example, AAA+ proteases catalyze protein degradation by threading the proteins through a pore. Sometimes, the energy landscape is such that the protein unfolds through a short-lived intermediate. In this case, the enzyme needs to thread the protein fast, unless the protein refolds and the enzyme is unable to degrade protein.29 Besides, it is shown that a pathogenic viral RNA has specific secondary structures that resist degradation by host cell exonuclease.30 In general, if the resisting amino acid (or nucleotide) sequence of the protein (or nucleic acid) is known, it is possible to determine the efficiency of the enzyme in the degradation of the substrate using our results presented above. Our results showed that unfolding a chain using the pulling by threading approach is completely different from that using both pulling and threading methods when the tethered end of

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DISCUSSIONS In the above section, we studied the unfolding pathways of different fold topologies and their dependency on the employed nanoscale mechanical methods. We found that the F


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Figure 8. Unfolding efficiency for the three methods, pulling, threading, and pulling by threading with d = 5.

Here, it would be useful to discuss the relevance of some of our findings with mechanical unfolding experiments. In interpreting these experiments, the energy landscape framework is commonly used in the literature.33−39 Our results are complementary, as they help obtaining the internal arrangement of the contacts along the chain. This is mostly possible using new methods that are able to give unprecedented details of the unfolding process.34 Let us consider unfolding by pulling from the ends. There are two possible ways for performing the experiment. If the two ends of the chain are pulled at a constant speed, then the exerted force is recorded versus the chain extension (force−extension curve). The force required to unfold a contact depends on its bonding energy. Every time when the experiment is repeated, the chain takes one of the possible pathways, and the resulting force−extension curve is different. Different unfolding pathways of the chain are found by repeating the experiments many times to allow visiting all available pathways. Using the results in the previous section, one is able to obtain the internal arrangement of the contacts of different bonding energies relative to each other. Here, we discuss some cartoon examples on the interpretation of the force−extension curves. To this end, the contacts are marked according to their bonding energy found in the energy landscape. If the folded biopolymer has two series contacts with different energies, mostly the contact that has lower energy would unfold sooner. However, it is possible that the contact with higher energy could unfold sooner, rarely, at few pathways. See as a typical example Figure 2A,C of ref 35 that shows ascending forces for unfolding titin that has its folded domains in series with each other. The other example is unfolding of polyproteins with the unfolded domain in series, shown in Figure 3b of ref 36. If the folded polymer has two parallel contacts, one of the contacts would always unfold sooner, at all pathways, irrespective of their bonding energies. Notably, if a contact with higher energy always unfold sooner than some contacts with lower energy, then that contact is likely parallel to those contacts. See, e.g., Figure 1D of ref 34. If the two contacts are cross to each other, they would likely unfold at the same time because unfolding one contact in a cross relation does not change the chain extension. This means that two cross contacts are sometimes not recognized separately, in force−extension curves, irrespective of their bonding energies. Surely, these statements are applicable to unfolding by pulling experiments. In threading experiments, the series and the cross contacts always unfold according to their arrangement along the chain. This makes the results of unfolding by the nanopore depend on the chain orientation in the nanopore.37 It is worth noting that if the two contacts are parallel to each other, dependence on orientation would not be observed in the unfolding process.

the chain is close to the nanopore. Indeed, pulling by threading becomes different from pulling at large frequencies of the series relations. A chain with a larger number of series relations has a more extended structure; thus, the threading component dominates the unfolding process. As a result, both the number of pathways and the efficiency for the pulling by threading method decrease below those for the pulling method for large number of series relations. These findings are experimentally significant in the interpretation of the efficiency of AAA+ disaggregases that reactivate protein aggregates formed at cell exposure to severe conditions. The central disaggregation machinery of most bacteria is composed of ClpB disaggregase that partners with an aggregate-binding protein DnaK. ClpB and DnaK form a complex, in which ClpB threads the proteins and DnaK tethers one end of the chain. DnaK is positioned close to the opening mouth of ClpB. Thus, unfolding by the ClpB/DnaK complex is carried out using the pulling by threading method.3,20 Despite ClpB that needs to partner DnaK, ClpG is a stand-alone disaggregase,31 which unfolds the proteins using the threading method (Figure 9). According to

Figure 9. ClpB forms a complex with partner proteins to disaggregase proteins, whereas ClpG is a stand-alone disaggregase. They are hexamers encoded with similar sequences, and both work to extract single proteins from protein aggregates.

our findings, the pulling by threading method has a higher efficiency compared to that of the threading one. A comparison between ClpG and ClpB efficiencies for two different protein aggregates is given in Figure 3 of ref 31. Figure 3A,D of ref 31 shows higher and lower efficiencies for ClpB (pulling by threading method) and ClpG (threading method), respectively, when acting on different protein aggregates. As is obvious, the method of unfolding should be considered besides various parameters that affect the disaggregase efficiency (such as the affinity for the substrate32) to have an accurate comparison. G


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sufficiently lower than the unfolding energy unfold as soon as they feel the force. (2) Contacts with bond energies on the order of the unfolding energy would take some time to unfold, as a result of thermal fluctuations. These contacts mostly affect the unfolding rate; however, they can make the chain prefer to take some of the unfolding pathways more than other. (3) Contacts sufficiently stronger than the unfolding energy would not unfold at a reasonable time. Here, the latter contacts are called nonbreakable and determine the unfolding efficiency as explained (a more accurate description of the unfolding process is given for a specific case in the Supporting Information). Consequently, in our model, heterogeneity of the contact energies is considered through the assumption of two kinds of contacts: breakable and nonbreakable contacts. As explained above, the nonbreakable contacts are those that do not unfold at a reasonable time in the experiment or biological process. In this study, we considered unfolding of biological polymers using nanoscale mechanical methods. One may wonder about the relevance of the results to other unfolding mechanisms such as thermal,40 chemical,41 and other available unfolding methods.42,43 The mentioned nonmechanical methods resemble the pulling method rather than the threading method. Threading acts on the contacts one-by-one; however, thermal and chemical methods act on all of the contacts at once. There is one fundamental difference between the pulling method and the thermal or the chemical methods. In the pulling method, the drag force acts only on the outer contact and not on the inner contact in a parallel topology relation. This is in contrast to thermal and chemical methods wherein the unfolding agent (chemicals or thermal motions) acts on all of the contacts at the same time. In the mentioned nonmechanical methods, the contacts that are nested inside other contacts in parallel topology relations may unfold independently. As a result, the unfolding pathway of nonmechanical methods would become different from what is discussed here. This point should be considered in comparing the nanoscale mechanical methods with thermal and chemical methods.44 How would the dependency of the unfolding efficiency on topology change when thermal and chemical methods are employed? As stated above, in thermal and chemical methods, a nested contact inside a parallel relation can be unfolded independently, in contrast to the pulling method. However, unfolding of these nested contacts would not release any length of the polymer as long as their upper contacts are not unfolded. As a result, the unfolding efficiency of the thermal and the chemical methods would be similar to that of the pulling method.

This point can help in obtaining the chain arrangement using unfolding by threading experiments. The other possible way to unfold a chain using the pulling method is to exert a constant force on the polymer ends. Then, the chain extension is recorded by increasing the exerted force (extension−force curve). In these experiments, upon increasing the force, stronger contacts can be unfolded.38,39 However, once a strong contact unfolds, those weaker contacts that are nested parallel inside the just-unfolded strong contact would suddenly unfold. Typically, these weaker contacts would not be detectable in the extension−force curve. Thus, these curves are not useful for detecting weaker contacts in a chain. This is also true for the threading method if the applied voltage is increased to rupture a strong contact, and once the strong contact is unfolded, all subsequent weak contacts would suddenly unfold and would not be detectable. However, extension−force curves are comparable with our results on unfolding efficiency. At each middle strength of the force, only those contacts that are weaker than the unfolding energy are unfolded. The chain extension at this stage divided by the total chain extension (at ultimately stronger forces) gives the unfolding efficiency. Notably, the extension−force curves may also depend on the pathway (Figure 2 of refs 38, 39). However, in an ideal pulling experiment performed at very small loading rates (the rate of increasing the force per second), it is expected to obtain one single curve. Using Figure 7 of the previous section, it is deduced that the strong contact that stops the unfolding process at each stage is parallel relative to the other not-yet-unfolded contacts. This is another useful conclusion for determining the internal arrangement of contacts. A more detailed comparison of our results with available experimental reports on various biopolymers of different known topologies is possible but is beyond the scope of the present manuscript. Even, obtaining a map of the internal arrangement of different contacts in a chain using unfolding experiments seems feasible by developing software packages based on our results. There are a few points that need to be mentioned about the presented results. First, the effect of the chain orientation (the chain end that enters the pore) is ignored here for simplicity. This is done by averaging the results over different chains with the same topology numbers. Indeed, the phase space of all possible topologies of a chain is symmetric; i.e., each topology has a counterpart in the phase space, which is completely the same but reverse in orientation. Thus, averaging over orientation is included in our results. Second, bond energies are critically important here. Consider unfolding a chain by pulling with a constant force f. Competition between two energies determines unfolding of a contact: unfolding energy and bond energy of the contact. Most of the forces that stabilize biopolymers are short-ranged (except electrostatics), such as covalent or hydrogen bonds, dispersion−repulsion forces between large complementary patches of proteins, or hydrophobic interactions. Thus, roughly speaking, the unfolding energy is defined as the multiplication of the unfolding force and the distance where the binding energy becomes minimum. One may define the unfolding energy as the unfolding force multiplied by the total length released during the unfolding; however, this is not true for short-range binding energies. On the basis of competition between unfolding and binding energies, the contacts can be divided into three groups: (1) contacts with bond energies

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CONCLUSIONS In summary, the dependence of the number of pathways and efficiency of unfolding on the topology of the chain was studied when different mechanical methods were employed. It was shown that the number of available pathways depends on the number of intrachain contact pairs that are arranged in parallel when chains are unfolded via pulling or pulling by threading. We also found that the unfolding efficiency depends on the number of cross topology relations (or the number of parallel and series relations) when the nonbreakable contact is located farthest (or closest) to the chain ends. Besides, the analysis proved that the unfolding efficiency decreases with the number of contacts that are arranged in parallel with the nonbreakable contact (and not all parallel relations in the chain) for all studied cases. H


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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcb.8b09454.

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Simulation method and two examples of incorporating physiochemical parameters into the model (PDF)

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Narges Nikoofard: 0000-0001-8623-3144 Alireza Mashaghi: 0000-0002-2157-1211 Notes

The authors declare no competing financial interest.

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Implications of Molecular Topology for Nanoscale Mechanical

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