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# Modeling and design by hierarchical natural moves

Contributed by Michael Levitt, December 5, 2011 (sent for review November 16, 2011)

## Abstract

We develop a unique algorithm implemented in the program MOSAICS (Methodologies for Optimization and Sampling in Computational Studies) that is capable of nanoscale modeling without compromising the resolution of interest. This is achieved by modeling with customizable hierarchical degrees of freedom, thereby circumventing major limitations of conventional molecular modeling. With the emergence of RNA-based nanotechnology, large RNAs in all-atom representation are used here to benchmark our algorithm. Our method locates all favorable structural states of a model RNA of significant complexity while improving sampling accuracy and increasing speed many fold over existing all-atom RNA modeling methods. We also modeled the effects of sequence mutations on the structural building blocks of tRNA-based nanotechnology. With its flexibility in choosing arbitrary degrees of freedom as well as in allowing different all-atom energy functions, MOSAICS is an ideal tool to model and design biomolecules of the nanoscale.

Computational modeling is an important aspect of biology and nanotechnology. *In silico* design and manipulation of nanostructures often precedes experimental validation (1, 2), while effective computational structure prediction of biomolecules paves the way for biomolecular design (3). Most molecular modeling packages are either general but inefficient in modeling large molecular systems or designed to be effective for modeling only specific types of systems (e.g., coarse-grained modeling of DNA as an elastic rod) and are therefore not general purpose. These limitations are mainly a consequence of two major obstacles to computational modeling: (*i*) the high dimensionality of the systems studied and (*ii*) the complexity of the potential energy surface guiding the simulation. When attempting to model nanoscale systems at all-atom resolution, the combination of these two factors often leads to intractable complexity.

A wide range of methods has been proposed to remove obstacles presented by high dimensionality (4⇓–6) and complex energy surfaces (7⇓⇓⇓⇓–12), but none of these studies has considered both limitations in the same context. The difficulty of such a unified approach is that limitations arising from both obstacles are related in that reducing dimensionality often results in a more complex energy surface. For example, conformational sampling using dihedral angles, which is a common solution to the high dimensionality problem of macromolecular assemblies, often results in a more complex energy surface with energy barriers that could have been easily avoided in Cartesian space. The algorithm [implemented in MOSAICS (13)] proposed here overcomes the problem of high dimensionality without increasing the complexity of the underlying energy surface. This is achieved by sampling with hierarchical variables: Degrees of freedom that introduce large conformational changes are combined with degrees of freedom that allow for local rearrangement. The former help overcome obstacles raised by large dimensionality while the latter soften the resultant energy surface. Such degrees of freedom may break the molecular chain and even spoil stereochemistry. To prevent this, we have coupled hierarchical modeling with a stochastic chain-closure algorithm (6), permitting extensive *in silico* manipulation of large molecular structures at all-atom resolution without compromising chain connectivity or stereochemistry.

Our algorithm is general and therefore can be applied to diverse types of molecular structures. However, to benchmark our technique, we turn our attention to molecular junctions, a molecular topology pervasive in biology (14) and electronics (15⇓–17). Given the recent emergence of RNA-based nanotechnology (1), we focus on junctions found in RNA and compare our technique with other RNA computational modeling approaches. Naturally occurring RNA molecules are important in biology as they carry out a variety of roles in gene regulation while synthetically designed RNA nanostructures have increasingly been promoted as possible candidates for drug delivery (1). RNA molecules often fold in a hierarchical fashion, with the sequence determining stable base-pairings (secondary structure) that are preserved when these base-paired regions rearrange into tertiary or quaternary forms (18); RNA nanostructures are usually designed based on this premise and through the use of specific RNA structural motifs (2). Hence the hierarchical structural elements we used (see below) to define degrees of freedom for modeling RNA are natural physical descriptors of RNA (19). We emphasize that our method is general as the hierarchical organization of RNA assemblies is only one example of a more general phenomenon manifested in nanotechnology, where nanostructures are built from basic blocks that are themselves built hierarchically [e.g., nucleic acids (20, 21) or oxide nanostructures (22)].

## Results and Discussion

### Benchmarking with an RNA Four-Way Junction.

We show that accurate sampling of a large RNA structure with all-atom representation and secondary structure constraints is impossible with currently available approaches. Specifically, they fail to generate diverse and canonical ensembles (Fig. 1). Regular Monte Carlo, molecular dynamics, and natural move Monte Carlo (4, 6) are generally confined to local structural basins close to the initial model (Fig. 1*A*). Fragment assembly, which is designed to generate a diverse set of conformations and commonly used for RNA (3, 23) and protein modeling (24), does not generate a canonically weighted ensemble of conformations because the preference for certain structures may depend on the particular fragment libraries used. As a result, thermodynamic observables cannot be calculated from such ensembles. Two state-of-the-art implementations of fragment assembly, Macromolecular Conformations by SYMbolic programming or MC-Sym (23) and Rosetta (3), produce differently biased distance distributions (Fig. 1 *B* and *C*). Both distributions feature unphysical cusps and additionally for MC-Sym, symmetry is not preserved. By contrast, our method, hierarchical natural move Monte Carlo as implemented in MOSAICS (13), has been designed to efficiently model large molecules and produce the canonical distribution of observables, regardless of starting structure (Fig. 1*D*).

Our hierarchical natural move Monte Carlo first generates a complex move of the molecular system by ignoring molecular connectivity, such that the system is likened to a collection of independent nucleotides, which are sequentially moved based on the user-defined degrees of freedom (a collection of independent sets of moves, *L*_{i}). This complex move may result in spoiling stereochemistry and chain breakage between any neighboring nucleotides, but these are corrected by our all-atom chain-closure algorithm (6) that adjusts a small number of dependent variables (*L*_{d}) before the actual energy evaluation and acceptance/rejection of the complex Monte Carlo move along *L*_{i} take place. (If chain closure is unsuccessful, the proposed complex move is immediately rejected.) Therefore the degrees of freedom used are the combination of both *L*_{i} and *L*_{d}.

For nucleic acids, an obvious degree of freedom is one that describes the position, orientation, and internal flexibility of each nucleotide (set *L*_{1}; Fig. 2*A* and ref. 6). Although such a set of natural moves uses a small number of essential degrees of freedom to reduce dimensionality [by an order of magnitude compared to Cartesian coordinates (6)], high dimensionality and energy surface complexity still present major challenges for large macromolecules. It may seem possible to avoid the dimensionality problem by introducing new degrees of freedom (*L*_{2}) that move larger parts of the structure. In the case of RNA, these degrees of freedom could describe the relative orientation and absolute position of all its rigid base-paired helices (Fig. 2*A*). Unfortunately, sampling helices as rigid bodies alone is inadequate: Tight packing of two helices may be prevented by the artificially high energy barriers that arise from the lack of flexibility within these rigid bodies.

To overcome this limitation we embed the smaller, local moves as described by *L*_{1} into the more global sampling by *L*_{2} moves; i.e., our complex independent move *L*_{i} is a combination of moves from *L*_{1} and *L*_{2}. In this way each RNA helix is no longer completely rigid and nucleotides within each helix are allowed to move. Sampling along collective degrees of freedom (*L*_{2}) solves the dimensionality problem, while the increased complexity of the energy surface is circumvented by introducing internal flexibility within rigid bodies (see *Materials and Methods*).

Analogously for protein complexes, global moves of protein domains (say *L*_{3}) can be incorporated with moving secondary structure elements in each domain independently (*L*_{2}). Further flexibility can be added with residue-level degrees of freedom (*L*_{1}).

### Improving Sampling Efficiency by Adding Hierarchical Degrees of Freedom.

As an illustration of the effects of modeling with different embedded degrees of freedom, consider a biomolecular assembly with many levels of hierarchical complexity like the large symmetric RNA system with multiple junctions illustrated in Fig. 2*A*. We can define seven arbitrary independent sets of degrees of freedom, *L*_{1} to *L*_{7}. We use the distance between the ends of neighboring full arms that consist of four helices each as our physical observable. By symmetry, all distance distributions are expected to converge to the same distribution; the overlap between distributions is a quantitative indicator of reaching convergence with unbiased sampling (see *Materials and Methods*). Fig. 2*B* shows that successive incorporation of degrees of freedom that introduce additional collective global rearrangements systematically reduces the number of Monte Carlo iterations needed to reach convergence. Specifically, sampling with sets *L*_{1} to *L*_{7} leads to convergence using at least an order of magnitude less computational effort than sampling with only sets *L*_{1} to *L*_{4}. The effects of introducing additional degrees of freedom appear to gradually saturate, as judged by the similarity between sampling with *L*_{1} to *L*_{6} and *L*_{1} to *L*_{7}. This suggests that additional sets are not required.

### Application 1: Modeling RNA Nanostructure Flexibility.

While we made use of synthetic four-way RNA junctions as model systems to benchmark our technique, four-way junctions are common in naturally occurring biological nucleic acids. For instance, during homologous recombination, four-way DNA Holliday junctions are formed and have preferential stacking and orientations (14). The flexibility of RNA junctions has strong influence on its dynamics and fold and affects the RNA’s functionality and capability to bind to different ligands (25). Therefore understanding the behavior of nucleic acid junctions has important biological implications and our method facilitates extensive *in silico* study of the physics of such systems.

Beyond naturally occurring systems, nucleic acid junctions are also important in bionanotechnology. There has been extensive research on DNA and RNA nanostructures because of their potential in a variety of fields like drug delivery (1), material science [e.g., DNA nanotubes (26)], biomolecular computing [e.g., algorithmic self-assembly (27)], and nanoelectronics (1, 27). Nanostructures consist of basic motifs of different junction types to design appropriate bends and kinks that are assembled into large and complex structures. The flexibility of these motifs typically depends on sequences of the single stranded regions connecting the helices and/or the presence of tertiary contacts that help stabilize the RNA/DNA junctions to particular conformations. From a practical standpoint related to experimental synthesis, it is important to understand the effects of specific sequences on the flexibility of the motif as the yield and stability of large nanostructures is affected by the flexibility of its smaller building blocks. For example, it was shown that a tRNA-square (four tRNA monomers connected via kissing loops to form a square; see Fig. 3) was less stable when its individual monomers carried sequence mutations that negated the tertiary interaction likely crucial in maintaining the right-angle motif within the monomer (28). This series of mutations also resulted in a decreased yield of the tRNA-square, instead facilitating formation of smaller dimers or trimers (28).

Using hierarchical move-sets and a simple base-pairing “potential” (see *Materials and Methods*) we show that single mutations of the tertiary contact do not strongly affect the angular distribution of the junction within each tRNA monomer (Fig. 3*A*). However, when both mutations U15G and C16G are made in concert, the tRNA monomer has significantly higher flexibility (see Fig. 3*A*), which could result in an increased ability to form complexes of different order, thereby decreasing tRNA-square yield. Further, based on thermal melting experiments, it was found that the same mutations affect the tRNA-square stability after it forms (28). Our simulations of the tRNA-square suggest that double mutations on each monomer result in more distorted squares than in the absence of any mutations (Fig. 3*B*). These distortions could lead to strains on the kissing loops that connect individual monomers, thereby decreasing the stability of the mutated tRNA-square relative to the native. Distortions in the tRNA-square could then propagate and result in lower yield and/or distortions of larger assemblies (such as a one-dimensional ladder).

### Application 2: Interpreting Limited Experimental Data with Modeling Results.

Another application of our algorithm is to run simulations with appropriate selection of degrees of freedom to aid the interpretation of experiments. As an example, we made use of unique move sets to identify the relative orientation of two-stacked helices within an RNA four-way junction (Fig. 4), specifically an RNA four-way junction derived from the hairpin ribozyme with its helix D stacked on A and its helix C on B. These two sets of stacked helices can orient in parallel or antiparallel, as determined by the direction of continuous RNA strands (pointing in the same or opposite directions, respectively) (14). The interhelical angle *θ* is 0 for the extreme parallel structure and 180° for the antiparallel equivalent. For 0 < *θ* < 180°, the junction can take either a left- or right-handed form (Fig. 4*A*). Using distance measurements between the ends of helices determined by fluorescence resonance energy transfer (FRET), experimentalists identified this junction as preferring the antiparallel conformation in the presence of Mg^{2+}, but the handedness of the antiparallel form could not be determined from this experimental FRET data (29). In the same set of experiments, it was found that switching of the four-way junction from its antiparallel form to its corresponding parallel form does not take place in the absence of helix unstacking (i.e., rotation of stacked helix BC relative to AD is not sufficient).

In our RNA simulations (see *Materials and Methods*) that conserved the stacking of helix D on A and B on C, starting from left- and right-handed conformations, we found that the two types of handedness can be easily distinguished (Fig. 4*B*): With helices kept stacked, the left-handed form can rotate between parallel and antiparallel types, while the right-handed form is confined to mostly antiparallel conformations. It appears that steric and connectivity constraints imposed by the junction prevent the right-handed antiparallel conformation from rotating to its corresponding parallel type. Therefore, conversion from the right-handed antiparallel conformation to the right-handed parallel conformation requires more than a simple rotation of stacked helices, and helix unstacking may be necessary. Our results indicate that a right-handed antiparallel conformation is most consistent with the FRET data. Subsequent gel electrophoresis experiments showed that this four-way junction indeed prefers the right-handed antiparallel form (30).

## Conclusion

We are currently applying our algorithm in the field of structural biology to model biological complexes with limited/low-resolution experimental information (such as cryoelectron microscopy and small-angle X-ray scattering density maps). A good ensemble of structures is required for such modeling, and our algorithm allows us to generate this ensemble even for large molecular systems. Moreover, due to the generality in our implementation, we can model structures with any scoring or energy function derived either from experimental input or by a physics-based or knowledge-based potential. Hence our hierarchical natural move Monte Carlo technique can be used in structure prediction of RNA and protein or else to assess the quality of molecular modeling force fields by comparing observables derived from *in silico* ensembles to experimentally determined ones. We are currently combining hierarchical natural move Monte Carlo with continuous sequence space sampling (treating the fractional presence of each type of nucleotide at each sequence position as an additional independent variable) so as to allow simultaneous exploration of sequence-structure space. We hope these approaches pave the way for large scale design of nanostructures (1, 2, 31) or molecules that mimic biological processes (3).

## Materials and Methods

Descriptions about modeling the tRNA nanostructure and four-way junction derived from the hairpin ribozyme are provided in *SI Text*.

### Sampling with Embedded Hierarchical Moves.

The typical objective of sampling is to obtain the expectation value and distribution of system specific observables over a phase space (Ω) of interest that is described by a probability distribution function, (function *f* maps elements from phase space Ω to real numbers ). Then, for any observable, , the expectation value is defined as [1]

where *L* spans Ω. In the case of the canonical ensemble, the probability distribution function *f* is given by the Boltzmann distribution, *f*(*L*) = exp(-*βE*(*L*))/*Q*, where the function represents the energy of the system, *β* = 1/*kT* (*k* is the Boltzmann constant and *T* the temperature) and *Q* is the partition function defined as *Q* = ∫_{Ω}*dL* exp(-*βE*(*L*)). Therefore, [2]

A common method for obtaining is to average over all states visited by a Markov chain Monte Carlo procedure sampling from probability distribution *f*. In a brute force approach, such sampling is performed along a particular set of independent degrees of freedom, *L*_{f}, covering phase space, Ω_{f}, (i.e., *L*_{f} spans Ω_{f}) with Ω_{f}⊂Ω (Ω_{f} is a proper subspace of Ω). In many practical applications, however, we are interested only in a proper subspace, Ω_{s}⊂Ω_{f} spanned by a small number of generalized degrees of freedom, *L*_{s} (*L*_{s} spans Ω_{s}). For example, consider a long polymer chain modeled by an energy function. If the observable of interest is the end-to-end distance of the polymer, a convenient choice for Ω_{s} is the space spanned by the dihedral angles about single bonds (torsion angle space with constant bond lengths or angles). On the other hand, Ω_{f}, would be the space spanned by the Cartesian degrees of freedom, which do not constrain bond lengths or angles. Because displacements from ideal bond angles and lengths are governed by stiff harmonic forces, and (the expectation values of *α* inferred from Markov chain Monte Carlo in spaces Ω_{f} and Ω_{s}, respectively) should both provide sufficiently good approximations to . However, the number of iterations required to reach convergence to give reliable expectation values and distributions of observables may vary significantly. In practical implementations, the large number of iterations required to sample the larger conformational space Ω_{f} makes the evaluation of Eq. **1** impractical.

While confining sampling to Ω_{s} circumvents most challenges posed by high dimensionality, it may make the energy surface more difficult to explore: The energy surface in phase space Ω_{s} might be more rugged than that in Ω_{f}. As a result it is likely that the probability of moving from one state to another in subspace Ω_{s} is much smaller than that in subspace Ω_{f}. For instance, in the previous polymer example, the difficulty in attaining practical Markov chain Monte Carlo acceptance rates when sampling dihedral angles is mainly due to the rough energy surface present in torsion angle space. However, augmenting Ω_{s} (our subspace of interest) with sufficiently small additional phase space volume elements from (the subspace of Ω_{f} that is orthogonal to Ω_{s}, where refers to the orthogonal complement of Ω_{s}) may eliminate the sampling limitations arising from the rough energy surface in phase space Ω_{s} and so allow energy barriers to be overcome more easily. An example of such a phenomenon is the softening of the torsion energy surface by adding bond angle degrees of freedom (6).

In general, additional degrees of freedom needed to smooth the rough energy surface of Ω_{s} can be chosen to exploit the hierarchical nature of biological/inorganic structure. Efficient sampling can be achieved with collective moves that change conformations at a higher hierarchical level while still including moves at lower levels. Thus by embedding moves of different levels, dimensionality is reduced without significantly changing the ruggedness of the energy surface. To do this, embedded subspaces of increasing sizes are defined, from Ω_{s} to Ω_{f} such that Ω_{s} = Ω_{n}⊂Ω_{n-1}…⊂Ω_{1} = Ω_{f}, with each subspace representing a desired level of the hierarchy.

Fig. 5 illustrates an example with three embedded subspaces (*n* = 3): Ω_{1}, Ω_{2}, and Ω_{3}. The large symmetric RNA junction as discussed in the main text (Fig. 2) is a physical analog of such a scenario. In the full Ω_{f} = Ω_{1} space (analogous to moving in the volume of the whole cube), each nucleotide is flexible and free to move independently (Fig. 5*A*). Two examples of smaller nested subspaces are moving helices as rigid bodies independently (Ω_{2}, analogous to moving in the titled plane *abcd*) or moving each full arm consisting of four helices simultaneously (Ω_{3}, analogous to moving along the diagonal line *cb*).

These integrals are illustrated geometrically and mathematically in Fig. 5*B*. The first integration is along the line *cb*, whereas the second integration extends over the plane *abcd*. As a result, the final integration is over a volume of space (red mesh) much smaller than that of the full cube. Sampling within these embedded subspaces greatly improves efficiency: Less computational time is spent exploring conformations in phase space with negligible probability of occurrence. Using a smaller but more important region for integration solves the problem of high dimensionality.

Furthermore, augmenting spaces Ω_{3} with Ω_{2} and Ω_{1} ensures that additional physically reasonable solutions (beyond just rigid body motions) are explored and we avoid the rough energy surface problem arising due to confining sampling to Ω_{s} = Ω_{3}.

While integrating along Ω_{3} (to obtain ) gives a good initial approximation to (that is evaluated from a full integral over Ω_{1}), a more accurate approximation is obtained by extending the domain of integration into Ω_{2} and subsequently into Ω_{1} around the local neighborhood of Ω_{3}. This integral requires a more confined phase space exploration (red mesh opposed to full cube or space Ω_{1}), thereby significantly reducing the number of iterations needed to give converged averages of observables. All these lead to the main objective of our algorithm: is not only obtained more efficiently than , it is also a more accurate estimate of .

### Sampling a Small Symmetric Four-Way Junction RNA.

A small symmetric four-way junction RNA (Fig. S1) was chosen to benchmark different sampling methods. Only secondary structure information was used in both fragment assembly approaches [MC-Sym (23) and Rosetta (3)]. A local version of MC-Sym was implemented, and because we were only interested in the distance distribution between ends of neighboring helices, small chain discontinuities in the RNA were not corrected either by chain closure or minimization. Rosetta 3.0 was used, with default parameters, and its built-in minimization phase using a high resolution RNA potential.

Natural move Monte Carlo and hierarchical natural move Monte Carlo sampling were both done using the MOSAICS (13) software package. The initial starting structure consisted of four perfect A-form helices (from the make-na server) stitched together using MC-Sym and a short (100 steps) all-atom “relaxation” procedure [minimization using the AMBER 99 force field (32) and the Generalized Born treatment of electrostatics (33) with an inverse Debye–Hückel length of 0.19 *Å*^{-1} as implemented in Nucleic Acid Builder (34)]. This approach efficiently restored chain connectivity and local stereochemistry, while not perturbing base-pair interactions significantly.

In both natural move Monte Carlo and hierarchical natural move Monte Carlo sampling, we are interested in obtaining distance distributions of a symmetric RNA. A hard sphere potential was used to increase the efficiency of exploring the conformational space using the all-atom representation. To prevent unnatural distortions of the A-form helices due to the hard sphere potential, base-pairs were only allowed to move as single units except in chain closure. The same starting structure was used to run 50 independent Markov chain Monte Carlo trajectories, and the last 1,000 conformations from each of these independently equilibrated runs were used to generate statistics for Fig. 1 and S2.

### Hierarchical Natural Move Monte Carlo Sampling of a Large Symmetric RNA.

The conformational sampling protocol for the large symmetric RNA was similar to the one for the symmetric four-way junction RNA. MC-Sym was used to assemble the starting structure from perfect A-form helices of eight and four base-pairs, respectively, and the chain connectivity was repaired with the “relaxation” procedure described above. Data for Fig. 2 and Fig. S3 were generated using the last 4,000 conformations of 50 independently equilibrated simulations sampled using a hard sphere potential.

### Measuring Interhelical Distances.

To facilitate the calculation of interhelical distances, we determined the origins and ends of all helices within the RNA. A perfect A-form RNA helix of the same length was superimposed to each base-paired region of the model (using C4′, C2, C4, and C6 atoms), and its helix origin and end were determined using the program X3DNA (4).

For the large symmetric RNA, the distance distributions between full arms were based on the average positions of the distal four-way junction (averaged over the positions of three helices of four base-pairs, and one helix of eight base-pairs).

### Symmetry Deviation Metric.

By symmetry, the four distributions of distances between neighboring full arms of the large symmetric RNA were expected to converge to identical distributions. Therefore, a metric to measure deviation from symmetry was defined as

where *y*_{i}(*d*) is a histogram distribution of *N*_{bins} distance bins for the *i*th set of distances. This metric quantitatively measures the degree of deviation from the expected symmetric conformational distribution. The horizontal dashed line in Fig. 2 indicates the limiting deviation achieved by sampling the large symmetric RNA with move-sets *L*_{1} to *L*_{7}.

## Acknowledgments

We thank the Jaeger Lab for generously sharing their tRNA nanostructure models used in our simulations, the Levitt Lab for useful discussions, and S. Doniach and J. A. Izaguirre for careful reading. Computations were done on Stanford’s Bio-*X*^{2} computers (National Science Foundation award CNS-0619926). A.Y.L.S. is funded by the Agency of Science, Technology, and Research, Singapore. This work was supported by National Institutes of Health Grant GM041455 and by a Human Frontier Science Program grant to M.L. M.L. is the Robert W. and Vivian K. Cahill Professor of Cancer Research.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. E-mail: peter.minary{at}stanford.edu.

Author contributions: A.Y.L.S., M.L., and P.M. designed research; A.Y.L.S. and P.M. performed research; A.Y.L.S. and P.M. contributed new reagents/analytic tools; A.Y.L.S., M.L., and P.M. analyzed data; and A.Y.L.S., M.L., and P.M. wrote the paper.

The authors declare no conflict of interest.

See Commentary on page 2691.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1119918109/-/DCSupplemental.

Freely available online through the PNAS open access option.

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