extfieldsofstudy: Chemistry Medicine Physics

text:
The effect of topology on the structure and free energy landscape of DNA kissing complexes.

We use a recently developed coarse-grained model for DNA to study kissing complexes formed by hybridization of complementary hairpin loops. The binding of the loops is topologically constrained because their linking number must remain constant. By studying systems with linking numbers -1, 0 or 1 we show that the average number of interstrand base pairs is larger when the topology is more favourable for the right-handed wrapping of strands around each other. The thermodynamic stability of the kissing complex also decreases when the linking number changes from -1 to 0 to 1. The structures of the kissing complexes typically involve two intermolecular helices that coaxially stack with the hairpin stems at a parallel four-way junction.

I. INTRODUCTION
Not only is DNA the genetic information carrier of life, but, given the degree of control achieved in the chemistry of DNA 1 -molecule synthesis is fast, reliable and relatively cheapthese information-rich building blocks can be exploited to reliably self-assemble two-and three-dimensional structures [2][3][4] and to build functional nanodevices. 5 Hairpins probably represent the simplest structure that DNA can form besides the standard double helix. These are secondary structure motifs formed by single-stranded DNA molecules that have complementary regions that selfhybridize. The intramolecular double helix formed from the self-complementary sections is known as stem or neck, while the section that connects two of the stem ends is called a loop.
In contrast to RNA, which for the most part is singlestranded in vivo, so that hairpins are a common structural element, 6 DNA in vivo is mostly in its duplex form. Nevertheless, there are occasions when it is single-stranded, and examples have been identified where DNA hairpins play a biological role, 7 including in replication, transcription and recombination. 8,9 However, hairpin formation can sometimes be an undesired process, and has been implicated in certain diseases. 10,11 DNA hairpins also play an important role in DNA nanotechnology [12][13][14][15][16][17][18] and can be used as the "fuel" to provide the energy to power autonomous DNA motors, although they can also be an unwanted secondary structural motif in DNA designed to be unstructured. 19 In this paper we study "kissing" complexes that can form when the loops of two DNA hairpins are complementary and partially hybridize. In particular, we focus on the interplay between topology and the shape and stability of these complexes. For example, when two hairpin loops hybridize, the a) Present address: Department of Chemistry, University of California, Berkeley, California 94720, USA b) Author for correspondence right-handed wrapping of the DNA strands in the intermolecular double helix must be compensated by a region where the loops wrap around each other in the opposite sense. Thus, even when the two loops are fully complementary, topological effects will restrict the number of bonds that can be formed.
Kissing loop interactions are also an important RNA tertiary structure motif 20 and play key biological roles in processes such as the regulatory action of antisense RNAs and the dimerization of viral genomic RNA. 21 They have therefore been much better characterized for RNA, both in terms of their structure [22][23][24] and mechanical properties, 25,26 than for DNA. This structural knowledge has even been exploited in structural RNA nanotechnology, where kissing loop interactions have also been used as a means to join RNA components with a well-defined geometry. [27][28][29] Although these RNA systems provide an interesting comparison, the kissing loop interactions typically involve shorter sequences of complementary bases than for the DNA systems we consider here, and so topological effects are less significant. Interestingly, the NMR solution structure of a DNA kissing complex has been obtained for sequences analogous to that of a previously characterized RNA kissing complex. 30 Although there are differences in the details of the two structures, they are generally very similar.
The topological effects associated with kissing loop interactions have been exploited in DNA nanotechnology, particularly to allow the design of autonomous motors. 13,14,31,32 One way of driving a DNA nanodevice through a cycle is through the use of complementary single-stranded DNA strands as "fuel", where the first strand is designed to partially hybridize with the device to induce a conformational change, and the second then reverses this change by displacing it to form a "waste" duplex. The first example of such a device was DNA nanotweezers, where the strands induced the device to open and close. 33 However, one problem with such a device is the two strands have to be added sequentially, since, if both are present at the same time, they will preferentially directly hybridize with each other rather than with the device.
One way to circumvent this problem is through the use of fuel strands that can form hairpins, 13,14 since the topological restriction on binding between the hairpin loops will effectively prevent the duplexes from being formed, even though the duplexes are more stable. Given that the two strands are complementary, these hairpins naturally form kissing complexes. While the hairpins are unable to open each other's stems by displacement, the motor can be designed to be a catalyst for the hybridization. By having a single-stranded DNA section that both is partially complementary to one of the hairpins and has a free end, the motor can open the hairpin by displacement, unconstrained by topological effects. A second, and similar, solution is to prepare the fuel strands complexed to partially complementary protective strands that bind to either end of the fuels but leave a loop region in the middle unhybridized. 31,32 A variety of autonomous motors have been designed based on these principles. [15][16][17][18] Here, we investigate the system of fully complementary 40 base DNA hairpins studied by Bois et al. 13 using computer simulations of a nucleotide-level coarse-grained model of DNA. 19,34 This recently introduced model provides an excellent description of the structural, thermodynamic and mechanical properties of both single-stranded and duplex DNA, and has now made it feasible to study the free energy landscapes of such DNA nanotechnology systems in detail, as previously illustrated for DNA nanotweezers. 19 In particular, we focus on the effects of topology on the free energy landscape for the binding of the hairpin loops, and how the structure of the resulting kissing complex reflects these topological constraints. To further illustrate the role played by the topology, we also consider kissing complex formation in systems of linked hairpins.

A. Model
We use the coarse-grained DNA model developed by Ouldridge et. al.. 19,34 In this model, a DNA strand is described as a polymer of nucleotides that interact via excluded volume repulsion and anisotropic attractive potentials that mimic the Watson-Crick base-pairing, stacking, cross stacking and coaxial stacking. The model has been parameterized to reproduce the structural and thermodynamic properties of singlestranded and double-stranded DNA molecules at the high salt concentrations that are typically used in DNA nanotechnology applications. Since this model is described in detail in Ref. 34 we shall repeat here only the fundamental ingredients.
Each nucleotide is represented as a rigid body with three interaction sites, all on the same axis. Although the interaction sites are collinear, we stress that a nucleotide does not possess cylindrical symmetry, since the potential also depends on a vector perpendicular to the nucleotide axis to capture the effects of the orientation of a base on the interactions. The potential energy V can be written as: The first sum runs over all pairs of nucleotides that are adjacent along a strand (neighbours in our terminology) and the second sum runs over all other pairs. The interaction between neighbours consists of a backbone term that is designed to represent the connectivity of a DNA strand, a stacking term that is designed to mimic stacking interaction between nucleotides, and an excluded volume part that prevents nucleotides from overlapping. The interaction between non-neighbouring pairs consists of four different terms: (i) a hydrogen-bonding term that mimics directional Watson-Crick base-pairing; (ii) a cross-stacking term that accounts for stacking interactions between nucleotides that are second neighbours on different strands; (iii) a coaxial stacking term that is designed to capture stacking interactions between nonneighbouring bases; and (iv) an excluded volume term. The full forms of each of these terms is reported in Ref. 34, with the exception of the coaxial stacking term, which is described in Ref. 35. Its parametrization will be described in detail elsewhere. 36 Features of the model that are particularly important for the current study are the relative flexibility of single-stranded DNA and its ability to describe the thermodynamics of hairpin formation accurately, as well as hybridization in general. We are also confident of the general robustness of the model, based on the wide range of DNA systems on which we 19,34,35 and others 37 have tested the model. These so far include DNA nanotweezers, 33 "burnt bridges" 38 and twofooted 39 DNA walkers, as well as processes such as DNA displacement, 40 overstretching 41 and cruciform formation 42 and the formation of liquid crystalline phases. 43 However, we should also note that the model does introduce a significant level of coarse graining and neglects several features of the DNA structure and interactions. Firstly, all four nucleotides have the same structure and interaction properties, except for the hydrogen-bonding term, for which interactions are only allowed between Watson-Crick complementary bases. Although this approximation of course precludes the study of much of the sequence dependence of properties and behaviour, it is not a problem when, as here, we are interested in the generic behaviour of a system. Secondly, the double helix in our model is symmetrical rather than having different sizes for the minor and major grooves. Again this is unlikely to be an issue, unless we are interested in the DNA structure at a quite fine level of detail. Finally, the interactions have been fitted for a single, fairly high, salt concentration (namely, 0.5M), where the Debye screening length is short. This is the regime relevant to most DNA nanotechnology experiments.

B. Simulation Methods
Throughout this work, we use Monte Carlo simulations employing the Virtual Move algorithm (VMMC) introduced by Whitelam and coworkers. 44,45 The latter is a modification to the standard (Metropolis) Monte Carlo algorithm specifically designed to promote the collective diffusion of highly interacting clusters that would otherwise be suppressed. In our model DNA strands are effectively clusters of interacting nucleotides and that VMMC significantly speeds up sampling, particularly when strand diffusion is important, which is the case when studying hybridization processes.
Because of the presence of large free-energy barriers, we have used umbrella sampling 46 to accurately sample transitions between different states. In practice, this is accomplished by adding an additional term to the system Hamiltonian designed to flatten the free energy profile along a particular reaction coordinate, and then subsequently unbiasing the results. 47 In the present case, the natural choice for the reaction coordinate is the number of base pairs between the two strands. This choice requires a definition of a base pairing, and we define a pair of nucleotides as base paired if the hydrogen bonding interaction term between them is at least 0.093 times the well depth. Of course this choice is somewhat arbitrary, but changing the threshold does not significantly alter the results.

C. DNA Sequences
We have studied the same 40-base nucleotide sequences as in Ref. 13. The two DNA strands are fully complementary, and so can form a duplex as well as hairpins with a stem of 10 base pairs and a loop of 20 bases. The sequences of the two strands are, in 5 ′ to 3 ′ direction: gcgttgctgc-attttactcttctcccctcg-gcagcaacgc and gcgttgctgc-cgaggggagaagagtaaaat-gcagcaacgc where the hyphens separate stem and loop regions. All the results we present are at room temperature, taken as T = 296.15 K (23 • C). This compares to a hairpin melting temperature of around 350 K. Hence, at the temperature we consider, the probability of spontaneous hairpin opening is effectively zero in our simulations.

D. DNA Topology
A commonly used property to characterise systems with respect to their topology is the linking number, Lk, a number that describes how two closed curves are linked in threedimensional space. Given the projection of two closed curves onto any plane, a crossing is taken to be positive (negative) if the upper curve can be superimposed onto the lower by a counterclockwise (clockwise) rotation (see Fig. 1(a)). According to this definition, each crossing in the right-handed helix formed by dsDNA is negative as the strands in the helix are antiparallel. 48 The linking number Lk is then defined as where the index i runs over the crossings with c i = +1 for a positive crossing and c i = −1 for a negative crossing. In more intuitive terms, Lk is the number of times that each curve wraps around the other. For a detailed discussion of the linking number in the context of nucleic acids we refer the reader to Ref. 49.
Although we are considering a system of two unclosed DNA strands, because the rate of hairpin opening is negligible in our simulations at the temperature we consider, the hairpin loops can be effectively considered as closed loops, and so can exhibit topologically different states ( Fig. 1(b)). The most experimentally relevant state is that with Lk = 0, as the likelihood that two hairpin loops would interlink during their formation process is very low. Since linking number is conserved, in this state any negative crossings of the two hairpin loops due to hybridization of the complimentary loops must be compensated by positive crossings, i.e. sections where the loops wind round each other in the opposite sense. This topological effect will frustrate the hybridization process. the topologically unlinked (Lk = 0) kissing complex. Note that in (a) it almost looks as if the stem is longer than 10 base pairs, because the stacking tends to propagate beyond the end of the stem at this temperature. For (b) the chosen structure has 14 interstrand base pairs. To its right is a topological sketch of the configuration illustrating that the zero linking number is achieved by balancing positive and negative crossings. In panels (c)-(e) we show example structures for partially formed complexes with a total of 2, 6 and 10 base pairs formed between the loops, respectively.

A. Topologically unlinked complex; Lk = 0
Firstly, we consider two hairpins that are topologically unlinked ( Fig. 1(b)(i)) and are free to bind through the complementary loop regions. One of the unbound hairpins is illustrated in Fig. 2(a). The room temperature free energy profile for bonding is reported in Fig. 3. The jump associated with the  Fig. 1(c)) is also plotted (circles). In the inset, the full profiles are plotted, showing the large (> 30 kBT ) free energy difference between the most stable states of the kissing complex and the control system. The free energy profiles have been taken to have the same value when the number of base pairs is 1, assuming the same value for the association barrier. formation of the first bond is due to the loss of translational entropy associated with hybridization, and is dependent on concentration as well as temperature. Our data were collected with two strands in a volume of 4944 nm 3 , i.e. 0.336 mM.
For the hybridization of a duplex in our model, we have previously shown that, after the barrier for forming the first base pair, there is then a linear decrease in the free energy as the number of base pairs increases (aside from a possible small rise at the end due to the fraying of the ends of the duplex). 34 The behaviour seen for the binding of the two hairpin loops is significantly different from this scenario. After an initial roughly linear decrease (up to about 3-4 base pairs), the line begins to exhibit some positive curvature reaching a minimum at 14 base pairs (roughly one and a half helical turns) before rising steeply. This curvature is a result of the topological constraint that as the two hairpin loops wind around each other to form a duplex the linking number must remain constant, and this constraint is increasingly felt as the number of base pair increases.
To compare the thermodynamics to a system where topological constraints do not play a role we introduce a control system where the hairpin loops have been opened by breaking the backbones between bases 10 and 11 of the first strand and 30 and 31 of the second strand, as shown in Fig. 1(c). The free-energy profile of the control shows the expected linear decrease in free energy for hybridization in the absence of topological effects. It is also worth noting that this system gains 30k B T more in free energy from hybridization than the hairpins do from forming the kissing complex. This value is roughly the amount of free energy stored in the metastable kissing complex (the most stable state for this system is the full duplex, since the two sequences are fully complementary). In DNA nanotechnology systems where such hairpins are used as fuels, this would be the amount of free energy that is potentially available to do work.
It is also interesting to look at the structure of the kissing complex as hybridization progresses. Fig. 2 shows example structures with different numbers of base pairs, and in Fig. 4 the probability that a given base is bound as a function of its position along the loop is depicted for different numbers of total base pairs. For kissing complexes with a few base pairs there is not a strong thermodynamic preference for binding at a particular position in the loops, hence the distribution in Fig. 4 is roughly uniform. The exceptions are the first and last bases in the loops for which base-pairing is disfavoured, presumably because the more crowded environment that would result makes binding at these positions entropically less favourable.
By contrast, for kissing complexes with the most favourable number of base pairs (N tot = 14), there is a very clear pattern for bonding. The six bases closest to each stem are invariably base-paired, while the four central bases have virtually no probability of binding. Therefore, the structure of the ensemble of configurations with 14 base pairs are all very similar to that in Fig. 4(b). We should also note that this structure is very different from the typical schematics of kissing complexes that tend to assume a single hybridized region, normally between the central regions of the loops.
The reason for this well defined pattern of bonding becomes clear when we examine the structure in more detail. It is simply the best way to maximise the base pairing whilst satisfying the topological constraints. In particular, the base pairing adjacent to each stem continues the two helices formed by the stems (i.e. there is coaxial stacking) and there is a parallel four-way junction 50 at the coaxial stacking site associated with the exchange of strands between the two helices. Importantly, as the junction is parallel, the strand exchange leads to a crossing with positive sign that helps to counterbalance the negative crossings associated with each region of base pairing between the loops (Fig. 2(a)). This positive crossing comes with little free energy cost because it does not lead to any loss of base pairing. By contrast, the second positive crossing near to the centre of each hairpin loop is associated with a reversal of the direction of wrapping of the two chains around each other and is responsible for the inability of the two loops to hybridize further without significant free energy cost.
The base pairing probability distributions for kissing complexes with 6 and 10 base pairs in Fig. 4 illustrate how this tendency to base pair at the extremes of the loops becomes more pronounced as the number of base pairs is increased and the system becomes more topologically constrained. However, the ensembles of such structures are still much more diverse than for the fully formed kissing complex. Both the examples in Fig. 2(d) and (e) only have a single hybridized region, with only the latter being adjacent to one of the hairpin stems.
The positive crossings of DNA strands associated with parallel four-way junctions have previously been used to offset the negative crossings associated with hybridization in so-called "paranemic crossover" motifs. 51,52 In this motif hybridization occurs between bubbles (a series of unpaired bases) in two duplexes leading to parallel four way junctions at either end of the newly hybridized section (rather than at just one end as for the hairpin loops). These junctions can exactly offset half a turn in each of the helices that result, and so can lead to complete hybridization between topologically closed species. These paranemic crossover motifs have been proposed as an alternative to "sticky" single-stranded ends as a means for binding together different molecules in DNA nanotechnology, 51,53 and have been used in making DNA triangles 54 and octahedra. 55 Such paranemic crossovers have also been shown to form between negatively supercoiled homologous duplexes because their zero linking number helps to alleviate the supercoiling. 56 Interestingly, a very similar structure to that depicted in Fig. 2(b) has been identified for the inhibitory complex between the antisense RNA CopA and its target messenger RNA CopT 57-59 and has also been suggested for other such complexes. 60 CopA and CopT have small hairpin loops that associate to form an initial kissing complex, which then progresses to form an "extended" kissing complex, where some of the base pairs in the hairpin stems are lost in favour of two intermolecular helices that coaxially stack with the rest of the hairpin stems at a parallel four-way junction like that in Fig. 2(a). This progression is dependent on the presence of bulges in the hairpin stems 59 that presumably aid the transformation by destabilizing the stems. Intriguingly, the number of base pairs in the two intermolecular helices is thought to be 15 with six bases in each of the loops connecting the ends of these helices, 57,58 which is extremely similar to the detailed structure of our kissing complexes.
Our results for the topologically unlinked system can be compared to the experimental results of Refs. 13 and 14 on the stability of these DNA kissing complexes. Ref. 13 reported a high yeld (nearly 100%) of kissing complexes at a single strand concentration of 8 µM in a buffer of relatively high salt concentration, while Ref. 14 reported that kissing complexes were not stable for their 21-base loop hairpins but at a significantly lower strand concentration (0.1 µM) and at low salt. The relative probability Φ of the system being in a bound state (one or more interstrand base pairs) compared to an unbound state interstrand base pairs) can be inferred from the data in Fig. 3. We find Φ ≃ 3100. Assuming high dilution, this ratio can be extrapolated to a different simulation volume v ′ simply by dividing Φ by the ratio v ′ /v, where v is the original simulation volume. The relative probabilities of bound and unbound states can than be related to the bulk yields f ∞ as described in Ref. 61. Extrapolating our results to the conditions of Ref. 13 we get Φ = 73.7 and a bulk yield f ∞ = 0.89, which is consistent with the experimental result that the kissing complexes were significantly more stable than the unbound state. By contrast, extrapolating our results to the concentration used in Ref. 14 we get Φ = 0.92 and f ∞ = 0.37, indicating that the kissing complexes are less stable than the unbound states, which is again consistent with the experimental findings, especially when we take into account that our model is fitted to a much higher salt concentration than that used in Ref. 14 and thus is expected to give an overestimate for the bulk yield in this case since the electrostatic penalty for bringing two strands close together would be larger at the experimental salt concentration. It should also be pointed out that the sequences we study are quite asymmetrical in GC content, but that our model does not account for sequence-dependent effects. It is thus possible, depending on temperature and salt concentration, that the resulting bonding pattern is actually a single helix between the GC-rich regions of the loops. We next consider topologically linked hairpins. Although they are less experimentally relevant than the unlinked case, they nicely further illustrate the effect of topology on hybridization. First we consider hairpins with a linking number of -1 ( Fig. 1(b)(ii)). In this case the linkage has the same sense as the wrapping in duplex DNA and we thus expect hybridization of the two hairpin loops to be easier than for the unlinked case. The typical structure of the resulting kissing complex and the free energy profile for hybridization are shown in Fig. 5. There is a much lower entropic cost for initial binding as compared to the topologically unlinked system, because the two strands are already constrained to be close to each other due to the linkage. The free energy profile also exhibits a much closer to linear decrease with the number of base pairs formed than for the unlinked case, and has a minimum at 17 base pairs.
The structure of the resulting kissing complex is quite similar to that for the unlinked case in that it also has a parallel four-way junction at the point where the two stems end. The reason for this structure is again that the junction provides a positive crossing of the strands without any base pairs being lost. The typical structure assumed by the kissing complex is effectively two parallel helices and a small (2-4 base pairs) unbound region where the strands bend back on themselves. The topological sketch in Fig. 5(a) shows how the two positive crossings (at the four-way junction and at the end of the free energy profile associated with the formation with the kissing complex, compared to that for the control system in Fig. 1(c). To aid this comparison, the two free energy profiles were set to have the same value at 1 base pair.
helices) and the linking number of −1 allow the system to have four negative crossings, which is topologically sufficient to form two double helical turns. Thus, that there is still a free energy cost associated with the formation of the last base pairs is due not so much to topological constraints but to geometric constraints arising because the backbones have to bend around to bridge the two helices. One interesting feature of the structure shown in Fig. 5(a) is that the stems of both hairpins are only nine base pairs in length because the hairpin loops have displaced one base pair from each stem. This then raises the question of whether the four-way junction could migrate further and lead to the opening of both hairpins. We note that there are a number of features hindering the junction diffusion. Firstly, junction migration is easiest when the junction adopts an "open" configuration where there is no stacking across the junction, 50,62 rather than the parallel stacked configuration typical of the kissing complex. Secondly, the junction migration is resisted by the topology. If the total number of base pairs is to remain constant during migration then the number of base pairs in the duplex regions of the hybridized hairpin loops must increase. However, as the linking number must also remain constant, this also means that the unfavourable left-handed wrapping of the unhybridized sections of the loops must increase. Our simulations corroborate this picture. We observed that the position of the crossover between the end of the two stems is rather stable, although it occasionally did move one or two base pairs down.

C. Topologically disfavoured complex; Lk = +1
Finally, we consider topologically linked hairpins with a linking number of +1 ( Fig. 1(b)(iii)). In this case, the crossings associated with the linkage are of the opposite sign to that for a duplex, and so hinder base pairing between the loops. The effects of this topological frustration are clear from the free energy profile in Fig. 6. Now, the most stable kissing complex has only 8 base pairs between the hairpin loops, and is only a few k B T more stable than the unhybridized state. Indeed, it is likely that for a slightly shorter loops the topological frustration would be sufficient to totally inhibit binding.
The effects of topology are underlined by the comparison with the topologically favoured configuration with linking number Lk = −1, for which a further 25k B T drop in free energy is obtained on forming the kissing complex. Visual inspection of the structure in Fig 6(a) indicates a much more distorted structure compared to the previous cases. In this configuration, there is only a single negative crossing (roughly enough for one half turn of the double helix), but three positive crossings associated with the strands wrapping in the wrong sense around each other.

D. Role of the backbone excluded volume
Here, we consider how the results for kissing complexes depend on our parametrization of the excluded volume interaction between backbone sites. We do this firstly because this interaction term will play a key role in determining how easy it is for the DNA chains to wrap around each other in the wrong sense, and hence how many base pairs can be formed between hairpin loops. But, secondly, in our original parametrization of the model many of the properties to which we fitted are relatively insensitive to this interaction, and so it is possible that this parameter does not have its optimal value. For example, in the duplex state, in the high-salt concentration to which our DNA model is fitted, the backbone sites are too far away from each other for their mutual excluded volume to significantly affect duplex properties. Other properties such as the single-stranded persistence length played a greater role in its parametrization. The shape of the interaction between backbone sites, modelled as a soft repulsion, is also a significant approximation especially when two strands are close together.
We have therefore repeated the calculation of the free energy profile for the complex with Lk = 0, but where we have changed the amount of repulsion between backbones by increasing the effective radius σ bb of the coarse-grained backbone site by up to 30%. As shown in Fig. 7, as the repulsion is increased, the average number of base pairs in the loop is diminished, and the free energy gain for association is significantly lowered. Of course, this change induces a large change the yield of kissing complexes at this temperature. Since our model's predictions for yields using the original value of σ bb are reasonably in line with the experimental studies reported in Refs. 13 and 14, the original parameterization appears to be robust. Moreover, that the detailed pattern of base pairing is consistent with known structures for RNA kissing complexes [22][23][24] further corroborates this conclusion.
We also note that at the largest value studied for the range of the repulsion the typical structure of the complex has a single intermolecular helix. Since one could regard the increase in the range of the repulsion as a very crude way to extrapolate the predictions of the model to a lower salt concentration than the one at which it was parameterized, it is possible that under those conditions binding between the hairpins' loops involves a single intermolecular helix between the GC-rich regions.

IV. CONCLUSIONS
Our simulations of the systems of kissing hairpins considered experimentally by Bois et al. 13 using a recently introduced coarse-grained DNA model clearly illustrate the effects of topology (due to the constraint that the linking number must remain constant) on the free energy landscape for the formation of a kissing complex. For the unlinked case, this topological frustrations leads to 30% of the bases being unpaired, and the binding free energy of the kissing complex is significantly smaller than for a fully formed duplex with equal strand length (it is equivalent to the binding of a duplex with about 7 or 8 bases). The free energy landscapes of the linked hairpins dramatically illustrate how manipulation of the linking number can increase or decrease this topological frustration. Compared to the unlinked system, the number of base pairs in the most stable kissing complex increases by 3 in the topologically more favourable case (L k = −1) , but decreases by 6 in the topologically less favourable case (Lk = 1). Even though the two sequences are fully complementary, the topological frustration also prevents the stems being opened by a displacement reaction involving the propagation of the intermolecular helices formed between the loops. It is this inhibition of duplex hybridization that underlies the use of hairpins as fuel for autonomous DNA nanodevices.
The structure of the kissing complex is also of particular interest, as this information is not straightforward to extract experimentally. We found that the kissing complex has a somewhat unusual structure. In particular, rather than having a single hybridized region between the loops, as might have been anticipated, the kissing complex involves two intermolecular helices that coaxially stack with the hairpin stems, and involves a parallel four-way junction. This structure is favoured because there is a positive crossing of the strands at the junction that helps to offset the negative crossings associated with hybridization, but without any loss of base pairing. By contrast, the positive crossing nearer the centre of the loops is associated with unhybridized bases. For similar topological reasons, parallel four-way junctions have also been observed for paranemic motifs in which bulges in two separate duplexes cross hybridize. Furthermore, a structure very similar to that reported here has also been identified for an "extended" kissing complex between messenger and antisense RNA that also involves parallel helices and a four-way junction. [57][58][59] As with any study with a coarse-grained model, one needs to consider how robust the results are and whether they might reflect any weaknesses of the model. We explicitly checked this for the repulsion between backbone sites in Section III D and found that although the results could change significantly when varying this parameter, our current value appears to be physically most reasonable. In particular, the thermodynam-ics in our model is consistent with the experimental stability of kissing complexes for 20-base hairpin loops in Refs. 13 and 14 (after taking into account differences in DNA and salt concentration). Furthermore, that similar structures are seen for systems with similar topological constraints suggests that our findings are physically robust.
One of the approximations in the model that we should particularly consider is the "average base" approximation, namely that that the bases in our model have identical interaction properties, except that hydrogen bonding can only occur between Watson-Crick base pairs. Although the G-C content of the hairpin loops is close to half, 7 of those G-C base pairs occur in one half of the loop. The consequences of this specific sequence might be to make the kissing complex asymmetrical with a longer intermolecular helix associated with the G-C rich half, or even to lead to the total loss of the second more weakly bound helical section. In this regard, it is interesting to note that the catalyst strand used by Bois et. al. to open the kissing complex binds to that half of one of the hairpins that is more weakly bound in the kissing complex 13 .
Although different topological states are only strictly welldefined for closed-loop molecules, as we have shown here, topological effects can be significant for linear DNA due to long-lived secondary structure that leads to the formation of internal loops. These topological constraints can inhibit hybridization and prevent the system reaching the lowest freeenergy state. DNA nanotechnology takes advantage of these effects when using DNA hairpins as fuel for autonomous motors, but they could also potentially be an obstacle to the successful self-assembly of DNA nanostructures. For example if a strand hybridizes at its two ends to parts of a second long strand, an internal loop results that will be potentially restricted in its binding by topological effects, unless one of the already hybridized ends unbinds, either due to melting or displacement. Therefore, the longer the strands involved in a structure, the more likely that topological constraints will have a significant effect on the ability of the system to self assemble. This argument suggest that for DNA origamis, 63 the shortness of the staple strands (typically having two or three binding domains) probably has the effect of reducing the potential for topological effects to hinder self-assembly. Furthermore, the excess of staple strands means that a topologicallyconstrained bound strand can be displaced by one of the equivalent staple strands from the reservoir in solution.

V. ACKNOWLEDGMENTS
We would like to thank the Engineering and Physical Sciences Research Council for financial support.


s2fieldsofstudy:
Chemistry
Materials Science

text:
Deposition of Nanofilms inside a Polymer Template:Formation of Metal Nanotubes

Given sufficient synthetic control, electroless plating is a suitable method for the fabrication of precisely defined nanomaterials. This study focuses on requirements which have to be met for the homogeneous and conformal electroless metal deposition within ion track etched polymer templates, allowing the fabrication of high aspect ratio metal nanotubes. Namely, the effects of (i) the seed density of the substrate, (ii) the surface-selectivity of the plating reaction and (iii) the deposition speed and the grain size of the resulting deposit are examined, and strategies to modify these characteristics in order to improve nanotube formation are illustrated. With optimized procedures, Ag, Au and Pt nanotubes were obtained. As an application example, metal nanotube membranes were utilized as flow-through microreactors, catalyzing the reduction of 4-nitrophenol with sodium borohydride. By partial exchange of Ag precursor structures with Pt, roughened Ag-Pt nanotubes with significantly improved activity were obtained. DOI: 10.1380/ejssnt.2012.578]

I. INTRODUCTION
Electroless plating displays a versatile route towards the deposition of polycrystalline metal films on substrates of variable composition and shape [1][2][3][4][5]. It is an industrially relevant reaction class which utilizes the surfaceselective reduction of a dissolved metal precursor by a chemical agent. Selectivity is achieved by choosing a metastable redox pair which does not react in the bulk solution, but only on catalytically active surfaces. As the obtained metal films are active in the plating reactions, the deposition continuously carries on by autocatalysis. Key advantages of electroless plating are the possibility to cover complex shaped [4] as well as non-conductive substrates [5] and to realize very homogeneous films with excellent properties [1,2].
Typically, this technique is used for the surface refinement of work pieces, e.g. to create electrically conducting films [1,6] or to enhance wear and corrosion resistance [1,2]. In these applications, besides other important issues such as bath stability and film adhesion, high deposition rates are demanded to accelerate production [3].
The electroless synthesis of metal NTs is usually based on the deposition on a template with channel-shaped pores, such as membranes consisting of ion track etched polymers [9][10][11][12][13][14][15] or anodized alumina [16]. This method combination enables extensive control over the product parameters (e.g., metal type, NT diameter, NT shape, wall thickness, NT density). However, the limited mass transport to the central channel regions causes reagent depletion within the templates, leading to varying plating rates which impede homogeneous metal deposition alongside the pore walls [11,17,18]. Therefore, contrary to the aforementioned case, slow reaction rates are desired as they facilitate NT formation.
In this study, we demonstrate how different reaction characteristics in electroless plating (seed density, nucleation behavior, deposition rate, particle size) can prevent NT formation and present synthetic strategies to overcome these problems. As an application example, arrays of metal NTs were used as catalysts for the reduction of 4-nitrophenol by sodium borohydride [10,11,16,19].

A. General, chemicals
Glassware was cleaned with aqua regia prior to use. All solutions were freshly prepared.

C. Catalysis
The reaction solution was prepared by mixing a solution containing 4-nitrophenol with a freshly prepared sodium borohydride solution. The final solution contained 8.4×10 −4 M 4-nitrophenol and a large excess of sodium borohydride (60 mM) to ensure pseudo-first order kinetics. For flow-through catalysis, 3 ml of the solution was pressed through a NT membrane using the setup described in section II.B.

D. Analysis
Scanning electron miscroscopy (SEM); JSM-7401F (JEOL), 5-10 kV acceleration voltage: The polymer was dissolved with dichloromethane and the metal nanostructures were collected on Si wafer pieces sputter-coated with Au. Transmission electron microscopy (TEM); CM20 microscope (FEI, Eindhoven, The Netherlands), 200 kV acceleration voltage, LaB 6 cathode: The NT containing templates were embedded in Araldit 502 c ⃝ (polymerization at 60 • C for 16 h) and examined as ultrathin sections (70 nm thickness, Ultracut E ultramicrotome (Reichert-Jung) with a diamond knife (DKK)). In combination with TEM, energy dispersive X-ray spectroscopy (EDS) was performed (Oxford Model 6767 EDS system (England)). UV-Vis spectroscopy (ATi Unicam UV/vis spectrometer UV4): The UV-Vis spectra were recorded using a high scan rate of 240 nm min −1 to evade problems caused by the formation of hydrogen bubbles. The observed range was 250-500 nm.

III. RESULTS AND DISCUSSIONS
The electroless fabrication of NTs in ion track etched polymer templates can be summarized as a sequence of redox reactions aiming to achieve conformal metal film growth on the whole template surface, especially inside the channels. Figure 1 contains a scheme of the involved synthetic steps and illustrates common problems which obstruct the formation of homogeneous NTs. First, a polymer foil (e.g. polycarbonate, polyethylene terephthalate or polyimide [20]) is irradiated with swift heavy ions. Due to their large momentum, the ions penetrate the foil on linear tracks, leaving damaged material along their paths. Second, the ion tracks are selectively etched out, leading to the evolution of channels. The shape and diameter of these channels can be adjusted by variation of the etching time and the etchant composition [20]. Third, metal nanoparticles (NPs) are attached to the polymer surface. As polymer substrates do not catalyze electroless plating reactions [5], this procedure is required to initiate metal deposition on the substrate surface. In the synthesis of NTs, this is mostly achieved by consecutively dipping the substrates in solutions of Sn(II) and Ag(I) [21]. During the first step (sensitization), Sn(II) ions form surface complexes with the polymer, which reduce the Ag(I) ions in the second step (activation), causing the precipitation of Ag NPs. Finally, the templates are transferred to an electroless plating bath, leading to the formation of metal NTs.
To obtain metal NTs, the activation as well as the plating reactions have to meet certain requirements. Without sufficient activation (ideally, a homogeneous and dense distribution of small, active seeds), electroless plating does not yield regular NTs [21] (see A) in Fig. 1). Several characteristics of the plating reaction can impede NT formation. Ideally, the metal film is slowly deposited only on the template surface, consists of particles of similar size (for narrow tubes, small particles are required) and is very homogeneous. If the metal reduction is not surfaceselective and homogeneous nucleation occurs as a sidereaction, uncontrolled particle formation occurs in the bulk solution. The particles precipitate on the template surface and reduce the life-time of the plating bath [22] (see B) in Fig. 1). High deposition rates lead to reagent depletion in the inner template regions and cause wall thickness inhomogeneities [17] (see C) in Fig. 1). In the worst case, the pore openings are clogged before NTs can form inside the template [11]. If the plated film consists of particles which are of similar size as the desired product feature (e.g. NT wall thickness), the morphology cannot be well reproduced (see D) in Fig. 1).
In the following, the outlined obstacles in the electroless synthesis of metal NTs as well as their overcoming will be discussed on the basis of experimental examples.

A. Ensuring sufficient seed density
To evaluate the influence of the activation on NT formation, electroless Ag plating from an ethylenediaminestabilized Ag(I)-tartrate bath [11] was chosen as this reaction leads to the formation of relatively large particles which easily allow to evaluate the nucleation density of the evolving metal films. The amount of seeds deposited on the polycarbonate template was varied by the number of activation cycles [21,23]. Besides a regularly pretreated polymer template [17] which was sensitized and activated once, templates with three and five consecutive seeding steps were used. All templates were subjected to electroless Ag plating under identical conditions (17 mM AgNO 3 , 100 mM ethylenediamine, 120 mM tartrate, pH 11.1, 24 h deposition at 8 • C). After metal deposition, the template was removed with dichloromethane, and the obtained nanostructures were analyzed with SEM (Fig. 2).
As can be seen, NTs were formed in all cases. However, due to the lower seed density, the walls of the NTs grown in triple (Fig. 2B) and particularly in single activated templates ( Fig. 2A) remained porous, and the re- sulting nanostructures were distinctly fragmented during the polymer dissolution. In the case of fivefold activation, many NTs were found spanning the whole template thickness of approx. 30 µm (Fig. 2C), demonstrating that high seed densities facilitate the formation of robust NTs with closed walls.

B. Suppression of homogeneous nucleation
To illustrate the effects of bath stability on the NT formation, two electroless Pt baths were prepared. The first reaction solution was adapted from the standard electroless plating bath for the synthesis of Au NTs [24] and contained 7.8 mM hexachloroplatinic acid as the metal precursor and 625 mM formaldehyde as the reducing agent. Besides heterogeneous deposition of Pt on the template surface, homogeneous nucleation occurred which was indicated by a slow change of the color of the plating solution color from initially yellow to dark brown. SEM analysis verified the development of a thin metal film on the template (Fig. 3A, left image). However, in addition to a thin film originating from heterogeneously deposited Pt, irregular particles were formed in the solution (Fig. 3A, right image) which fell on the template and mostly blocked the pore openings (Fig. 3A, left image). Therefore, further Pt deposition inside the template was inhibited, and no well-defined NTs could be obtained.
As the Pt (IV) source is only bound to relatively weak chloro ligands, one possibility to increase the stability of the plating solution is the introduction of suitable ligands which reduce the oxidizing strength of the metal precursor [1]. In the case of Pt, ethylenediamine can be used to continuously reduce the reactivity of hexachloroplatinate by substitution of an increasing number of chloro ligands, leading to the formation of complexes of the composition [PtCl 6−2x en x ] (2x−2) (x = 0-3) [13]. In combination with hydrazine as a reducing agent which is efficiently oxidized on Pt surfaces [25], a metastable plating solution was obtained (7.8 mM hexachloroplatinic acid, 18.7 mM ethylenediamine, 250 mM hydrazine). With homogeneous nucleation completely suppressed, free-standing Pt NTs were deposited inside the polycarbonate template (Fig. 3B).
As electroless reducing agents usually are more effective at high pH values [1,25], pH adjustment is also an important way to modify the reactivity of plating baths. For instance, in electroless Ag plating, bath stability was increased and the plating rate was reduced by addition of acid [11]. Another important strategy to suppress homogeneous nucleation is to add trace amounts of stabilizers which interfere with the electroless plating reaction, e.g. heavy metal ions, oxo anions or chalcogenide compounds [26].

C. Reduction of the deposition speed and the particle size
For the molding of a template, it is essential that the building blocks (in electroless nanotube synthesis: metal nanoparticles) are below the size of the morphological features to be reproduced. In addition, the deposition reaction must be slow enough to enable sufficient diffusion of the reagents to badly accessible regions of the template.
As ion track etched polymer membranes have a very complex surface structure, these issues are vital for the electroless synthesis of metal NTs. To illustrate the impact of these factors, electroless Au plating from different baths with pH 10.5 was performed. First, a standard Au plating solution [27] containing 7.8 mM disulfitoaurate, 125 mM sulfite and 625 mM formaldehyde was applied to a track-etched polycarbonate template (Fig. 4A). Due to the relatively high deposition rate, only a limited amount of Au was deposited within the template, as can be seen by the morphological transition of the nanostructures which formed in the pores. At the pore openings, solid structures are found, which after few micrometers get porous and finally fragment. The fact that not even fragments of well-defined tubular shape could be found is related to the relatively large grain size of this synthesis, which impedes NT formation in narrow pores [17,28].
To slow down electroless Au plating, additives such as 4-(dimethylamino)pyridine (DMAP) can be used which form adsorbates on the Au films and poison them to a certain degree, leading to reduced reaction rates [10]. In addition to the effect on the deposition kinetics, the grain size of the deposit can be reduced considerably [10]. Here, we used the stem compound pyridine and its derivative DMAP. The addition of 40 mM of each pyridine to Au plating baths led to a significant and simultaneous reduction of the reaction rate and the particle size. In case of pyridine, free-standing NTs with homogeneous walls and a diameter of approx. 200 nm were easily obtained (Fig. 4B). With DMAP, a more pronounced effect on the Au plating was observed, allowing the fabrication of extremely narrow NTs with diameters of 50 nm, consisting of particles of < 10 nm size. Although these filigree structures partly collapsed upon template removal (Fig. 4C, left image), TEM proved the formation of NTs and confirmed the efficient reproduction of even very small template features in the case of sufficiently reduced particle size (Fig. 4C, right image).
Probably, the stronger interference of DMAP with the Au plating reaction is caused by the mesomeric electron shift of the dimethylamino substituent. Compared to pyridine, the basicity of the endocyclic nitrogen is increased, which enhances bonding to the Au surface and suppression of surface reactions. This reasoning is in accord with results on the phase transfer and protection of Au NPs by different pyridines [29], of which only the strongly basic DMAP led to the formation of stable aqueous colloids.

D. Application of metal NTs as microreactors
Template-embedded metal NTs can be applied as highly efficient microreactors, heterogeneously catalyzing reactions in a solution which flows through them [10,11]. The reduction of 4-nitrophenol is frequently used as a model reaction to evaluate the performance of metal nanostructures [19]. With this reaction, it was shown that homogeneous metal deposition inside the ion track etched polymer templates is essential to achieve high activities [11].
Here, we demonstrate the addition of a second metal as another strategy to improve the performance of metal NT membranes in flow-through catalysis. First, Ag NTs were fabricated by electroless deposition from baths containing Ag(I)-ethylenediamine complexes as the oxidizing agent and tartrate as the reducing agent [11]. The product was characterized by TEM (Fig. 5A, left), showing tubes consisting of agglomerated particles of several 10 nm size. EDS proved that the nanostructures consist of Ag (Fig. 5A, right; the other elements relate to the template (C, O), the embedding resin (Si) and the TEM grid (Cu), respectively).
To introduce Pt as a second metal, one Ag NT membrane was purged with 3.9 mM hexachloroplatinic acid, leading to the cementation of the more noble Pt on the Ag nanostructures. After the exchange reaction, the membrane was cleaned with ammonia solution to remove AgCl precipitates. TEM revealed a change of the NT wall structure (Fig. 5B, left), which was rougher than in the case of the Ag NTs, consisted of smaller particles and showed some porosity. EDS confirmed the introduction of Pt, but significant amounts of Ag remained in the product ( Fig. 5B, right).
Finally, nitrophenol reduction was performed with the membranes containing the Ag-Pt and the Ag NTs. Reference experiments with catalyst-free membranes confirmed that nitrophenol conversion required the presence of metal NTs. Due to the large excess of the reducing agent sodium borohydride, the reaction follows a pseudo-first order rate law, and the apparent rate constant k app of catalysts can be determined by the conversion of the educt over time [19]. Usually, the change of the 4-nitrophenolate concentration-in the alkaline reaction solution, 4-nitrophenol is deprotonated-is followed by UV-Vis spectroscopy [19]. Assuming Beer-Lamberts law, the educt concentration and the absorbance are linearly correlated, and k app can be determined according to Eq. (1): The corresponding results for the NT membranes are shown in Fig. 6. It can be seen that after the passage through the catalytic membrane, the 4-nitrophenolatecaused absorbance at 400 nm is reduced, while a new peak evolves at a wavelength of 300 nm which corresponds to the product of the conversion, 4-aminophenolate. The calculated apparent rate constant of the Ag-Pt NT catalyst is 1.3×10 −2 s −1 , which exceeds the value of the Ag NT membrane (8.5×10 −3 s −1 ) by about 50%. As possible reasons for the considerable performance enhancement, the increased surface area of the roughened Ag-Pt NTs can be named aside the high intrinsic activity of Pt catalysts in the nitrophenol reduction [19].
FIG. 6: UV-Vis spectra of the reaction solution before and after passage through A) the Ag NT membrane B) the Ag-Pt NT membrane. During the reaction, the educt absorption peak at 400 nm fades, while a growing product absorption peak at 300 nm is found. The dots mark the nitrophenolate absorptions according to the apparent rate constants calculated with the pseudo-first order rate law.

IV. CONCLUSION
In this study, important characteristics of electroless plating reactions were investigated concerning the confor-mal deposition of homogeneous metal films on ion track etched polycarbonate substrates. It was shown how the activation and plating steps can be modified to ensure a high seed density, surface-selective metal deposition, sufficient mass transport and low grain sizes, facilitating the formation of metal NTs as a challenging product morphology.
Due to their general nature, the discussed strategies can be used to enhance electroless metal deposition on other substrates with intricate shape, especially with extended inner surfaces. However, due to the complex interaction of the reaction conditions, the deposition kinetics and the film structure as well as the strong dependence of these parameters on the applied metastable redox pair, individual optimization is required.
As an application example, arrays of templateembedded metal NTs were utilized as flow-through reactors, catalyzing the reduction of 4-nitrophenol. By employing Ag NTs as sacrificial templates for Pt deposition, a possible way was illustrated how the functional properties of electrolessly plated nanomaterials can be further improved.


extfieldsofstudy: Chemistry Medicine

text:
Förster Resonance Energy Transfer (FRET) between Heterogeneously Distributed Probes: Application to Lipid Nanodomains and Pores

The formation of membrane heterogeneities, e.g., lipid domains and pores, leads to a redistribution of donor (D) and acceptor (A) molecules according to their affinity to the structures formed and the remaining bilayer. If such changes sufficiently influence the Förster resonance energy transfer (FRET) efficiency, these changes can be further analyzed in terms of nanodomain/pore size. This paper is a continuation of previous work on this theme. In particular, it is demonstrated how FRET experiments should be planned and how data should be analyzed in order to achieve the best possible resolution. The limiting resolution of domains and pores are discussed simultaneously, in order to enable direct comparison. It appears that choice of suitable donor/acceptor pairs is the most crucial step in the design of experiments. For instance, it is recommended to use DA pairs, which exhibit an increased affinity to pores (i.e., partition coefficients KD,A > 10) for the determination of pore sizes with radii comparable to the Förster radius R0. On the other hand, donors and acceptors exhibiting a high affinity to different phases are better suited for the determination of domain sizes. The experimental setup where donors and acceptors are excluded from the domains/pores should be avoided.

Introduction
Liquid ordered L o domains (referred to as rafts) [1] and pores [2] represent heterogeneities usually encountered in lipid bilayers. The lipid domains are usually small (10-200 nm), highly dynamic and are enriched in the presence of cholesterol and sphingomyelin [3]. The accumulation of cholesterol and sphingomyelin into the L o domains could be ascribed to the existence of attractive forces between cholesterol and shingomyelin or between cholesterol and a saturated phosphatidylcholine [4]. The acyl chains are ordered in the domains, but molecules diffuse across the domains at a speed comparable to the remaining liquid-disordered L d phase. The domains adopt a circular shape in order to minimize the line tension originating from the height mismatch between the L o and L d phases and the steric interactions at the phase interface [5]. In cells, the domains are connected with several functional roles, such as signaling, membrane trafficking or viral infection.
Model lipid bilayers represent a very simplified system, as compared to the plasma membrane of a cell. First of all, the plasma membrane has a more complex composition. This will influence the nature of the phase boundaries and probably decrease the line tension, which in turn may lead to the formation of smaller domains. Actually, domains of micrometer size were hitherto never observed in cell membranes. Also, cell membranes exhibit an asymmetric composition and are not studied under equilibrium conditions, as is the case with model lipid bilayers. Therefore, the nature of the domains in the model bilayer and in the plasma membrane is probably different.
Since the size of L o domains is beyond the optical resolution of a conventional microscope [6,7], elaborate techniques, such as neutron scattering, stimulated emission depletion spectroscopy (STED) [8] or Förster resonance energy transfer (FRET) [9,10], are required for the size determination. However, solely the lastly named technique is able to determine domain/pore sizes smaller than 20 nm. Usually, the spectroscopic characterization of lipid domains is complicated by the fact that all lipid molecules partition to some extent into both the L d and L o phase. Nevertheless, as will be shown here, domain/pore sizes comparable to the Förster radius R 0 can be estimated by FRET, despite a limited partitioning of the probes into the L o phase. Here, the FRET analysis relies on a non-random distribution of donors (D:s) and acceptors (A:s) between domains/pores and the remaining bilayer surface. Actually, the best resolution can be expected if all D:s and/or A:s are excluded from the remaining bilayer surface. So far, this seems hypothetical, because few fluorophores exhibit the desirable property. For liquid-ordered (L o ) domains, most probes are excluded from the densely packed region of ordered acyl chains. Cholera toxin labeled by Alexa 488 exhibits an increased affinity with a partition coefficients (K) ranging from 6 to 10 [7,11], depending on the lipid composition, as well as dipalmitoylphosphatidylethanolamine labeled by NBD with K = 4.3 [12]. The situation can be more favorable in the case of pores, which are usually built by bigger (as compared to the size of a fluorescent dye) protein/peptide molecules. Attachment of a dye to such a big molecule should not influence its partitioning, and the efficiency of pore labeling should thus only depend on whether the peptide/protein can also be found outside the pore.
In a typical qualitative FRET experiment, the FRET efficiencies of a DA pair localized in a bilayer, which does and does not contain membrane heterogeneities, are compared. This should preferentially be done by comparison of time-resolved fluorescence (TRF) decays for the two situations, but measurement of steady-state (SS) intensities is in principle possible as well. In order to determine domain/pore sizes, the measured TRF-decays must be fitted by an appropriate model, where the domain/pore radius is one of the optimized parameters. In this paper, it is assumed that the domains/pores have circular/cylindrical shape, respectively.
The present paper is a continuation of our previous work [13] and aims at showing how FRET experiments should be performed and the data analyzed in order to achieve the best possible resolution in estimating nanodomain and pore sizes. On the contrary to the previous work [13], D:s and A:s are assumed to reside only at the lipid/water interface. Concerning pores, the donors are assumed to be distributed at the bottom/top part of the pores, while the acceptors are distributed over the entire pore surface (cf. Figure 1). Such an arrangement intends to mimic donors attached at a well defined position of a protein/peptide that forms toroidal pores, whereas the acceptor is a lipid analogue labeled in its headgroup. The present paper also reviews the Baumann-Fayer (B-F) model [14], which accounts for FRET between uniformly distributed probes in a homogenous bilayer. The B-F model establishes a basis for an extended modeling of heterogeneous distributions. In the simulation, the shape of the pores has been approximated by a cylinder. The following extreme situation is assumed in the picture: The pore is formed by several peptide molecules, which exclusively reside in the pores. A certain peptide fraction is labeled by a fluorescent donor (green balls) in the polar end part. This means that donors are localized in the pores at the lipid/water interface only (K D = ∞). Acceptors (red balls), on the other hand, have been distributed over the entire surface of the cylindrical pore (K A = ∞). In order to highlight the pore-structure in the illustration, the pores have been heavily stained by the fluorophores.

Limitations of FRET in the Detection of Nanodomains and Cylindrical Pores
It can be expected that the formation of domains/pores in the lipid bilayer disturbs a homogenous distribution of fluorescent probes. Under certain conditions, which are discussed throughout this paper, the appearance of these heterogeneities can be detected by Förster resonance energy transfer (FRET). This technique enables not only detection of membrane heterogeneities, but also a size determination of these structures. Sensitivity of FRET to the formation of membrane heterogeneities depends on how donors and acceptors are distributed between these structures and the remaining bilayer. In practice four situations are of interest: (1) The donors and acceptors are excluded from the domains/pores; (2) The donors and acceptors are preferentially localized within the domains/pores; (3) The donors prefer a localization within the domains/pores, while the acceptors are localized in the remaining bilayer; and (4) Acceptors are localized in the domains/pores and donors are excluded from them. Because the last situation gives similar FRET efficiencies as the third situation, it will not be discussed here. In general, the first two cases lead to increased FRET efficiencies (as compared to the uniform probe distribution), i.e., to a faster fluorescence decay of donors and to the decrease in the SS-intensity. On the contrary, the last two cases result in the decrease of FRET efficiency, i.e., in slower fluorescence decays and an increased SS-intensity. This happens due the formation of structures that segregate the donors from the acceptors. For discussing the limitations of FRET, it is convenient to introduce the ratio of steady-state (SS) intensities for the heterogeneous I D and uniform I D (uni) distribution of probes. The ratio is formed by I D /I D (uni) when formation of domains/pores leads to decreased FRET efficiency or by I D (uni)/I D , respectively, when the energy transfer rate (=ω) is increased due to the formation of domains/pores. The ratio is formed in the way that it is always ≥ 1. According to our experience, domains are for ratio < 1.05 (red) beyond the resolution of TR-FRET and SS-FRET. For the range 1.05 < ratio < 1.1 (orange), the domains are close to the detection limit of TR-FRET, whereas the range 1.1 < ratio < 1.2 (yellow) represent domains that can be resolved by TR-FRET, but remain close to the detection limit of SS-FRET [13]. Finally, for the ratios > 1.2 (green), the domains can be detected by TR-FRET and SS-FRET.
The affinity of probes to the domains/pores and the remaining bilayer is characterized by the partition coefficient K D,A = [probes inside]/[probes outside]). In the case of domains, the concentration is clearly calculated as the average number of probes in a domain, N in , divided by the domain area, πR 2 . Due to a large complexity of pores, it is not possible to find a general expression that would yield the entire pore surface for which donors and acceptors compete with other molecules. Therefore, it seems a good compromise to define [probes inside] in the simulation as N in /V pore , where V pore = πR pore 2 d is the average volume of a pore. However, due to direct proportionality of this value to πR pore 2 (since d is a constant) and due to resemblance of πR 2 to πR pore 2 , it is more convenient to define [probes inside] as N in /πR pore 2 . As will be shown further in the text, this enables direct comparison between FRET diagrams for the domains and the rafts, respectively. Situation 1: This situation is relevant when donors and acceptors are excluded from the domains/pores. Concerning the estimation of domain-sizes, this experimental setup appears attractive, because most probes exhibit an increased affinity to the liquid-disordered non-raft phase. However, as is illustrated in Figure 2a, FRET cannot resolve domains with radii comparable to the Förster radius R 0 , and domains occupying up to 40% of the entire bilayer area (tested region). The resolution improves with increasing domain size and fraction. The reason is that (i) domains larger than R 0 make FRET across the domains impossible and (ii) higher amount of domains concentrates the acceptors in the non-raft phase. Regarding pores and their structure, two main types exist [2,15]. Toroidal pores exhibit high curvature made up by peptides and lipids. For example, these include antibacterial peptides, such as magainin-2 [16,17]. On the other hand, barrel-stave pores are exclusively formed by peptides (e.g., alamethicin [18]), and the bilayer does not bend in the pores [19]. Pores may also form proteins, such as Bax [20,21], or a bacterial toxin α-hemolysin [22], but the pore structure is less known. Situation 1 seems thus convenient for barrel-stave pores, which do not offer any free area for the lipid-based probes, as well as for toroidal pores, where it is to expect that K D,A of cylindrically shaped probes would not exceed unity due to high pore-curvature. As is shown in Figure 3a, resolution of pore-sizes comparable to R 0 is as poor as in the case of domains and improves a little with increasing the pore radius and the area fraction. Naturally, resolution can be further improved by using higher acceptor concentrations (cf. Figure 3a,b with acceptor to lipid ratios 1:200 or 1:100, respectively).

(c)
In the Situation 2 donors and acceptors have an increased affinity to domains/pores. This experimental setup leads to a closer contact between donors and acceptors, even for small domains/pores covering a low area fraction. Therefore, this is better suited for the determination of domain/pore sizes that are comparable to R 0 . For instance, structures with R ≈ R 0 and occupying 10% of the entire bilayer area are resolvable by D/A pairs with K D,A = 10. Utilization of probes with K D,A = 5 will shrink the resolvable range of R and A r to still experimentally interesting values 2 < R/R 0 < 8 and 15 < A r < 40 (Figure 2a, Figure 3a-c and Figure 4a,b). It is evident from the figures shown that situation 2 is more convenient in the determination of pores rather than domain sizes for the same combination of K D,A values, i.e., for the same number of probes within domains/pores at the given R. This is also true regarding the availability of suitable fluorescent probes. Available probes exhibit only modest affinity to the L o domains with the highest known value of K = 6 for Cholera toxin labeled by Alexa 488 [7,11]. Most probes are excluded from L o domains, mainly due to more packed environment of L o phase, as compared to the L d phase [23].
Attachment of a dye to a peptide/protein that builds the pore appears as the best solution for preferential labeling of the pores. Theoretically, if all peptide/protein molecules formed the pores, 100% labeling efficiency of pores could be achieved. Such extreme cases have been simulated as well and are displayed in Figure 4a,b. The best resolution is practically achieved when K D,A reaches 100 (compare Figures 4a and 3b). Such a high partition coefficient results in a dramatic change of the decay (with respect to the uniform distribution of probes in the homogenous bilayer) and allows for detection of pore sizes as small as 0.25 R 0 and occupying a few percent of the bilayer area. The only region where the resolution is poor has been found for small domains/pores (R ≈ 0.5R 0 ), which occupy more than 30% of the bilayer area. If the pores were formed by six peptide/protein molecules and these were exclusively found in the pores, such conditions would correspond to a rather high peptide-to-lipid ratio of 1:30. Despite this limitation, situation 2 remains as the best choice in the determination of pore sizes.  In the Situation 3, it is assumed that donors have increased affinity to the domains/pores, whereas acceptors are excluded from them. This situation appears as a reasonable compromise between the availability of fluorescent probes and the resolution of FRET. Recently, we have shown [7] that a DA pair represented by CTxB labeled by Alexa 488 (K D = 6) and DiD (K A = 0.01-0.005) could determine domains with R = 8 nm and A r = 6% at a relatively high acceptor to lipid ratio 1:80. In order to detect domains comparable to R 0, the ratio 1:100 is recommended, but it can further be reduced to 1:200 if probes with K D = 10 are used (cf. Figure 2a). Since the pore interior must remain empty (cf. Figure 1, donors which reside in the pores cannot be as efficiently separated from the acceptors as in the case of domains. This is why situation 3 is less useful for the determination of pore sizes at the same combination of K D and K A (compare Figures 2a and 3a). In this case, K D > 100 or the acceptor-to-lipid ratio of 1:100 are recommended, respectively.
The fourth and last situation would arise when at least a member of the DA pair had the same affinity to the domains/rafts and the remaining bilayer surface. In the extreme case, probes are uniformly distributed as in the homogenous bilayer (K D = 1 and K A = 1), and consequently no domains can be detected. On the other hand, such unfavorable conditions should still enable the detection of pores, because FRET efficiency for probes uniformly distributed between the pores and the remaining bilayer must at certain R and A r start deviating from the uniform distribution of probes in the homogenous bilayer. The difference allows for the determination of pore sizes when R/R 0 > 4 and A r > 30. This limiting case would correspond to the protein/peptide-to-lipid ratio of 1: 578 if the pores were built by six protein/peptide molecules [6] and the peptide/protein worked as a donor/acceptor as well (see also above). A slightly better resolution is achieved for K D = 10 6 and K A = 1 (cf. Figure 4a). In this case, donors are efficiently concentrated into the pores, while acceptors are uniformly distributed between the pores and the remaining bilayer. This causes a partial separation of donors from the acceptors and decreased FRET efficiency. On the other hand, a seemingly identical situation characterized by K D = 1 and K A = 10 6 leads to a significant increase in FRET efficiency, enabling the determination of either small pores (R ≈ R 0 ) occupying only a few percent of the bilayer area, or larger pores (R > 2R 0 ) covering several tens of percent of the bilayer area (cf. Figure 4a). Formation of L o domains leads to an increased bilayer thickness and to a decreased lipid headgroup area. Risselada et al. [24] report, for instance, that the bilayer thickness increases from 4.0 nm to 4.6 nm and that the headgroup area decreases from 0.58 nm 2 to 0.4 nm 2 when going from the L d to the L o phase.
These two phenomena have an opposite effect on the FRET efficiency. A reasonable quantitative estimate can be done by means of the B-F model, which accounts for FRET occurring in a homogenous lipid bilayer. Increase of d of about 10% or 20% results in the increase of the steady-state ratio from 1 to 1.035 or 1.073, respectively. On the other hand, shrinking of the average headgroup area of about 6% (which corresponds to the domain area fraction of 20%) results in the decrease of the ratio from 1 to 0.965. Because both effects are small and compensate each other, the bilayer thickness and lipid headgroup area were assumed to be constant in the simulations.

Searching for the Size of the Liquid-Ordered Domains
In our approach, the estimation of domain sizes is based on fitting the computed decay curve provided by MC simulations to the experimental one. Shape of the decay depends on the following parameters, which can be kept either free or fixed during the optimization routine: the donor fluorescence decay [F(t)]; the Förster radius, the bilayer thickness (d); the partition coefficients of donors (K D ) and acceptors (K A ); the reduced surface concentration of acceptors (C A ); the domain radius (R); as well as the area fraction occupied by the domains in the bilayer (A r ). In order to decrease the number of local minima, the number of optimized parameters is reduced as much as possible. For instance, the output parameter of the domain radius was rather insensitive to changes of the input parameters K D and K A, as well as to moderate changes of C A in DOPC (49%)/Chol (25%)/Sph (19%)/DOPG (5%)/G M1 (2%) bilayer (cf. the list of abbreviations) containing small amounts of Cholera toxin (CTxB) [7]. By using the correct input values of K D = 6 and K A = 0.01, the domain radius found was 8 nm, and the domains occupy 6% of the bilayer surface. Interestingly, we have observed that reduction of K D from six to three shifted the position of the chi-squared minimum to R = 8 nm and A r = 12%, while the increase of K D to 9 resulted in R = 7 nm and A r = 5 % (cf. Table 1). A variation of K A in the range ± 0.002 has not affected the shape of the decay, while a change of C A in the range ±10% had a negligible impact on both R and A r . A low sensitivity of R to K D , K A and C A may open new possibilities and advantages for the data analysis: due to experimental difficulties to measure partition coefficients at the point of the phase diagram where the domains are not visible by a microscope, K D and K A can be measured at closest points of the phase diagram where the bilayer is microscopically phase separated and used at the point of interest as well. Values of K D and K A do not necessarily need to be the same as at the original point, but due to the low impact of partition coefficients on R, values of K D and K A being lower or higher than the true ones are not expected to influence the R estimation substantially. Similarly, reduced surface concentration C A can be estimated for the non-separated homogenous lipid bilayer by means of the Baumann-Fayer model (see the section above) and used at the point of interest. By this approach, the number of optimized parameters could be reduced to only two, i.e., the domain radius (R) and the area fraction of domains (A r ) occupying a bilayer. Table 1. The table shows to which extent the input parameters of the partition coefficient for donors K D and acceptors K A and the acceptor surface concentration C A influence the optimized values of the domain radius, R, and the area fraction of the domains, A r . The real domain sizes are shown in the first row marked as "reference".

FRET Modeling of Uniformly Distributed Probes
Energy transfer/migration within an ensemble of fluorescent molecules is usually described within a two-particle approximation [14]. It assumes an independent pairwise dipole-dipole interaction between the excited donor and any acceptor. In DDEM, the excitation energy reversibly jumps from an excited to an unexcited donor. The two-particle approximation, however, excludes transfer steps via the previously unexcited molecule further to another unexcited donor. For describing FRET in a system where the excited energy can be transferred to more than one acceptor, it is convenient to introduce a function G(t), which describes the probability that the initially excited donor is still excited at a time t later. Then the decay F(t) of a donor surrounded by acceptors is given by [14]: where ∑ exp / represents the decay of a donor in the absence of FRET. By summing up over all pairwise interactions exp in a lattice, where each lattice site is occupied by an acceptor with the probability p, one obtains a general expression that accounts for energy transfer/migration between uniformly distributed donors and acceptors in systems with unspecified geometry. See [14,25] for the derivation.
Here, λ is a constant equal to 1 for FRET and to 2 for DDEM, respectively, ω is the energy transfer rate and ρ is the number density of acceptors. u(r) denotes a homogenous spatial distribution of donor-acceptor distances (r). The dynamic limit means that the reorienting rate of the interacting molecules is much faster than an energy transfer event. Hence, ω can be averaged before any transfer occurs and is therefore time-independent. For an ensemble of donors and acceptors, the dynamic limit is a reasonable approximation, provided that an excited donor may interact with many acceptors, which represent a wide range of orientations at the instant of the transfer.
In a lipid bilayer the excitation energy can be transferred either to a molecule localized in the same or the opposite leaflet ( Figure 5). These processes are referred to as intra-FRET [G intra (t)], and inter-FRET [G inter (t)], respectively. The total survival probability G tot (t) is given by the joint probability according to: ,inter/intra (3) when FRET/DDEM occurs in the same plane the distribution function u(r) = 2πr. The integration of Equation 2 yields the final expression for G intra (t) [14]: Similarly, one can obtain the analytical expression for the survival probability G inter [14]: In this expression, cos and . The distance between the bilayer sheets (d) approximately corresponds to the bilayer thickness, the angel (θ) is between the bilayer normal and the vector connecting the donor and the acceptor ( Figure 5) and is the mean-square average of the angular part of the dipole-dipole interaction.

Figure 5. (left)
Intra-and inter-energy transfer from donors (green spheres) to acceptors (red spheres) in a lipid bilayer. d denotes the bilayer thickness and θ r the angle between the bilayer normal and the vector connecting D/A. (right) A schematic top view of the lipid bilayer divided into a replicated box (in the middle) and nine neighboring replicas. The size of the replicated box is 20R 0 . Energy transfer rate of the acceptors that are found outside the cut-off radius R c = 10R 0 is calculated via the continuous approach.

Handling the Lack of Analytical Models for Heterogeneous Probe Distributions
Donors solubilized in a lipid bilayer that contains domains/pores are surrounded by unique density distributions of acceptors. This complicates the derivation of an analytical model that would account for FRET/DDEM processes. Despite this complication, several approaches relying on utilization of analytical equations have been developed and were described in a recent review [26]. So far, the most frequently used approach assumes an infinite phase separation [26,27], where FRET across the phase separation boundary is not considered. In this case, TRF-decays can be analyzed by means of the Bauman-Fayer (B-F) model (cf. Equations 4 and 5). This yields the acceptor surface concentrations C 2A and C 2B in phases A and B. These are related to the partition coefficient of the acceptors K A = C 2A /C 2B . Not surprisingly, K A values estimated by B-F model deviate from the true partition coefficient with decreasing domain size. Because the true partition coefficient also can be obtained by using other methods, the differently obtained K A values can be compared and the domain size can be estimated. The limitations following from the use of an analytical model can be circumvented by a numerical analysis of the decays.

Monte Carlo Simulations
A typical run of the MC simulation will be exemplified on the phase separated bilayer containing liquid-ordered domains. Small modifications are necessary in applications to a bilayer containing cylindrical pores. The following steps are usually carried out: (i) A certain random number of circular domains with uniform size is generated. The mean number of domains corresponds to the predefined area fraction the domains occupy in the bilayer; (ii) the donors and acceptors are generated to exhibit a certain probability of localization within and outside the domains. The distribution is described by an equilibrium constant K D,A (= [probes within L o domains]/[probes outside L o domains]); (iii) A donor is randomly excited and its excitation energy is transferred to an acceptor. The time for this event to occur depends on the overall energy transfer rate Ω i according to [28,29]: where α denotes a random number between 0 and 1. The total energy transfer rate is calculated as a sum of energy transfer rates from the excited donor i to all acceptors (first term of Equation 7). Acceptors that are beyond the cut-off distance 10R 0 are included via the continuum approach (second term of Equation 7) [28]: Here j (1 ≤ j ≤ N) refers to the number of acceptors (N) found within a cut-off distance R c . R ij is the distance between the i-th donor and j-th acceptor. Each generated configuration set is used 100 times in the calculation step before a new configuration set is generated. In order to assure for a good statistics, new configurations are usually generated 3000 times. Periodic boundary conditions with the size of the replicated box of 20R 0 × 20R 0 mimic an infinitely large membrane. The simulations correspond to energy transfer taking place under the dynamic limit condition and among isotropically oriented donors and acceptors. Therefore, kappa factor <κ ij 2 > = 2/3. The outcome of each simulation step is the time interval Δt i between the excitation and energy transfer event. By constructing a histogram made up by the number Δt i intervals belonging to each channel, G(t) function is obtained and the decay of donors undergoing FRET calculated (cf. Equation 1).

Conclusions
This paper demonstrates that the liquid-ordered nanodomains and pores with radii of a few nanometers and occupying several percent of the entire lipid bilayer can be determined by Förster resonance energy transfer combined with Monte Carlo simulations. The FRET resolution mainly depends on the affinity of donors and acceptors to the domains/pores. Therefore, to reach the best possible resolution choice of a suitable donor/acceptor pair is a most crucial step in the design of an experiment. When both donors and acceptors are preferentially localized in the pores appears to be the best experimental setup for the determination of pore sizes. This could for example be achieved by attaching a probe to a peptide/protein molecule, which usually forms the pore. The same experimental setup would also be convenient for the determination of domain sizes. But because probes with high affinity to the liquid-ordered domains are still missing, it is recommended to use a DA pair with high affinity to different phases instead. For instance, a pair with realistic partition coefficients of K D = 6 and K A = 0.01 resolves a broad range of domain sizes. Although majority of probes is excluded from the domains/rafts or is evenly distributed between these structures and the remaining bilayer, the case when the DA pair is excluded from the structures is not recommended.