Statistical-physics approaches to RNA molecules, families and networks
Simona Cocco, Andrea De Martino, Andrea Pagnani, Martin Weigt
备注：
19 pages, 6 figures, to appear in "Spin Glass Theory and Far Beyond - Replica Symmetry Breaking after 40 years" (edited by P Charbonneau, E Marinari, G Parisi, F Ricci Tersenghi, G Sicuro and F Zamponi)
This contribution focuses on the fascinating RNA molecule, its sequence-dependent folding driven by base-pairing interactions, the interplay between these interactions and natural evolution, and its multiple regulatory roles. The four of us have dug into these topics using the tools and the spirit of the statistical physics of disordered systems, and in particular the concept of a disordered (energy/fitness) landscape. After an introduction to RNA molecules and the perspectives they open not only in evolutionary and synthetic biology but also in medicine, we will introduce the important notions of energy and fitness landscapes for these molecules. In Section III we will review some models and algorithms for RNA sequence-to-secondary-structure mapping. Section IV discusses how the secondary-structure energy landscape can be derived from unzipping data. Section V deals with the inference of RNA structure from evolutionary sequence data sampled in different organisms. This will shift the focus from the `sequence-to-structure' mapping described in Section III to a `sequence-to-function' landscape that can be inferred from laboratory evolutionary data on DNA aptamers. Finally, in Section VI, we shall discuss the rich theoretical picture linking networks of interacting RNA molecules to the organization of robust, systemic regulatory programs. Along this path, we will therefore explore phenomena across multiple scales in space, number of molecules and time, showing how the biological complexity of the RNA world can be captured by the unifying concepts of statistical physics.
Stability of the branching structure of an RNA molecule is an important condition for its function. In this letter we show that the melting thermodynamics of RNA molecules is very sensitive to their branching geometry for the case of a molecule whose groundstate has the branching geometry of a Cayley Tree and whose pairing interactions are described by the Go model. Whereas RNA molecules with a linear geometry melt via a conventional continuous phase transition with classical exponents, molecules with a Cayley Tree geometry are found to have a free energy that seems smooth, at least within our…
Using a combination of theory and computer simulations, we study the translocation of an RNA molecule, pulled through a solid-state nanopore by an optical tweezer, as a method to determine its secondary structure. The resolution with which the elements of the secondary structure can be determined is limited by thermal fluctuations. We present a detailed study of these thermal fluctuations, including the frequency spectrum, and show that these rule out single-nucleotide resolution under the experimental conditions which we simulated. Two possible ways to improve this resolution are strong stret…
The mechanical unfolding of a simple RNA hairpin and of a 236--bases portion of the Tetrahymena thermophila ribozyme is studied by means of an Ising--like model. Phase diagrams and free energy landscapes are computed exactly and suggest a simple two--state behaviour for the hairpin and the presence of intermediate states for the ribozyme. Nonequilibrium simulations give the possible unfolding pathways for the ribozyme, and the dominant pathway corresponds to the experimentally observed one.
Recent single-molecule pulling experiments have shown how it is possible to manipulate RNA molecules using optical tweezers force microscopy. We investigate a minimal model for the experimental setup which includes a RNA molecule connected to two polymers (handles) and a bead, trapped in the optical potential, attached to one of the handles. Initially, we focus on small single-domain RNA molecules which unfold in a cooperative way. The model qualitatively reproduces the experimental results and allow us to investigate the influence of the bead and handles on the unfolding reaction. A main ingr…
Single-stranded (ss) RNA viruses self-assemble spontaneously in solutions that contain the viral RNA genome molecules and the viral capsid proteins. The self-assembly of empty capsids can be understood on the basis of free energy minimization of rather simple models. However, during the self-assembly of complete viral particles in the cytoplasm of an infected cell, the viral genome molecules must be selected from a large pool of very similar host messenger RNA molecules. It is known that the assembly process takes the form of preferential heterogeneous nucleation of capsid proteins on viral RN…
Extracting the intrinsic kinetic information of biological molecule from its single-molecule kinetic data is of considerable biophysical interest. In this work, we theoretically investigate the feasibility of inferring single RNA's intrinsic kinetic parameters from the time series obtained by forced folding/unfolding experiment done in the light tweezer, where the molecule is flanked by long double-stranded DNA/RNA handles and tethered between two big beads. We first construct a coarse-grain physical model of the experimental system. The model has captured the major physical factors: the Brown…
We use mesoscopic non-equilibrium thermodynamics theory to describe RNA unfolding under tension. The theory introduces reaction coordinates, characterizing a continuum of states for each bond in the molecule. The unfolding considered is so slow that one can assume local equilibrium in the space of the reaction coordinates. In the quasi-stationary limit of high sequential barriers, our theory yields the master equation of a recently proposed sequential-step model. Non-linear switching kinetics is found between open and closed states. Our theory unifies the thermodynamic and kinetic descriptions…
By exerting mechanical force it is possible to unfold/refold RNA molecules one at a time. In a small range of forces, an RNA molecule can hop between the folded and the unfolded state with force-dependent kinetic rates. Here, we introduce a mesoscopic model to analyze the hopping kinetics of RNA hairpins in an optical tweezers setup. The model includes different elements of the experimental setup (beads, handles and RNA sequence) and limitations of the instrument (time lag of the force-feedback mechanism and finite bandwidth of data acquisition). We investigated the influence of the instrument…
Single-stranded (ss) RNA viruses self-assemble spontaneously in solutions that contain the viral RNA genome molecules and viral capsid proteins. The self-assembly of empty capsids can be understood on the basis of free energy minimization. However, during the self-assembly of complete viral particles in the cytoplasm of an infected cell, the viral genome molecules must be selected from a large pool of very similar host messenger RNA molecules and it is not known whether this also can be understood by free energy minimization. We address this question using a simple mathematical model recently …
RNA folding is a kinetic process governed by the competition of a large number of structures stabilized by the transient formation of base pairs that may induce complex folding pathways and the formation of misfolded structures. Despite of its importance in modern biophysics, the current understanding of RNA folding kinetics is limited by the complex interplay between the weak base-pair interactions that stabilize the native structure and the disordering effect of thermal forces. The possibility of mechanically pulling individual molecules offers a new perspective to understand the folding of …
Nowadays different experimental techniques, such as single molecule or relaxation experiments, can provide dynamic properties of biomolecular systems, but the amount of detail obtainable with these methods is often limited in terms of time or spatial resolution. Here we use state-of-the-art computational techniques, namely atomistic molecular dynamics and Markov state models, to provide insight into the rapid dynamics of short RNA oligonucleotides, in order to elucidate the kinetics of stacking interactions. Analysis of multiple microsecond-long simulations indicates that the main relaxation m…
A hierarchical model for the growth of planar arch structures for RNA secondary structures is presented, and shown to be equivalent to a tree-growth model. Both models can be solved analytically, giving access to scaling functions for large molecules, and corrections to scaling, checked by numerical simulations of up to 6500 bases. The equivalence of both models should be helpful in understanding more general tree-growth processes.
We study the statistical mechanics of RNA secondary structures designed to have an attraction between two different types of structures as a model system for heteropolymer aggregation. The competition between the branching entropy of the secondary structure and the energy gained by pairing drives the RNA to undergo a `temperature independent' second order phase transition from a molten to an aggregated phase'. The aggregated phase thus obtained has a macroscopically large number of contacts between different RNAs. The partition function scaling exponent for this phase is θ~ 1/2 and the crossov…
Many non-coding RNAs are known to play a role in the cell directly linked to their structure. Structure prediction based on the sole sequence is however a challenging task. On the other hand, thanks to the low cost of sequencing technologies, a very large number of homologous sequences are becoming available for many RNA families. In the protein community, it has emerged in the last decade the idea of exploiting the covariance of mutations within a family to predict the protein structure using the direct-coupling-analysis (DCA) method. The application of DCA to RNA systems has been limited so …
Ab initio RNA secondary structure predictions have long dismissed helices interior to loops, so-called pseudoknots, despite their structural importance. Here, we report that many pseudoknots can be predicted through long time scales RNA folding simulations, which follow the stochastic closing and opening of individual RNA helices. The numerical efficacy of these stochastic simulations relies on an O(n^2) clustering algorithm which computes time averages over a continously updated set of n reference structures. Applying this exact stochastic clustering approach, we typically obtain a 5- to 100-…
The ensemble of RNA secondary structures of uniform sequences is studied analytically. We calculate the partition function for very long sequences and discuss how the cross-over length, beyond which asymptotic scaling laws apply, depends on thermodynamic parameters. For realistic choices of parameters this length can be much longer than natural RNA molecules. This has to be taken into account when applying asymptotic theory to interpret experiments or numerical results.
We discuss the physics of RNA as described by its secondary structure. We examine the static properties of a homogeneous RNA-model that includes pairing and base stacking energies as well as entropic costs for internal loops. For large enough costs the model exhibits a thermal denaturation transition which we analyze in terms of the radius of gyration. We point out an inconsistency in the standard approach to RNA secondary structure prediction for large molecules. Under an external force a second order phase transition between a globular and an extended phase takes place. A Harris-type criteri…
MicroRNAs (miRNAs) are small RNA molecules, about 22 nucleotide long, which post-transcriptionally regulate their target messenger RNAs (mRNAs). They accomplish key roles in gene regulatory networks, ranging from signaling pathways to tissue morphogenesis, and their aberrant behavior is often associated with the development of various diseases. Recently it has been shown that, in analogy with the better understood case of small RNAs in bacteria, the way miRNAs interact with their targets can be described in terms of a titration mechanism characterized by threshold effects, hypersensitivity of …
Elastic network models (ENMs) are valuable and efficient tools for characterizing the collective internal dynamics of proteins based on the knowledge of their native structures. The increasing evidence that the biological functionality of RNAs is often linked to their innate internal motions, poses the question of whether ENM approaches can be successfully extended to this class of biomolecules. This issue is tackled here by considering various families of elastic networks of increasing complexity applied to a representative set of RNAs. The fluctuations predicted by the alternative ENMs are s…
{\it Transcription} is the process whereby RNA molecules are polymerized by molecular machines, called RNA polymerase (RNAP), using the corresponding DNA as the template. Recent {\it in-vivo} experiments with single cells have established that transcription takes place in "bursts" or "pulses". In this letter we present a model that captures not only the mechano-chemistry of individual RNAPs and their steric interactions but also the switching of the gene between the ON and OFF states. This model accounts for the statistical properties of the transcriptional bursts. It also shows how the quanti…
We study the force-induced unfolding of random disordered RNA or single-stranded DNA polymers. The system undergoes a second order phase transition from a collapsed globular phase at low forces to an extensive necklace phase with a macroscopic end-to-end distance at high forces. At low temperatures, the sequence inhomogeneities modify the critical behaviour. We provide numerical evidence for the universality of the critical exponents which, by extrapolation of the scaling laws to zero force, contain useful information on the ground state (f=0) properties. This provides a good method for quanti…
We study the fraction $f$ of nucleotides involved in the formation of a cactus--like secondary structure of random heteropolymer RNA--like molecules. In the low--temperature limit we study this fraction as a function of the number $c$ of different nucleotide species. We show, that with changing $c$, the secondary structures of random RNAs undergo a morphological transition: $f(c)\to 1$ for $c \le c_{\rm cr}$ as the chain length $n$ goes to infinity, signaling the formation of a virtually "perfect" gapless secondary structure; while $f(c)<1$ for $c>c_{\rm cr}$, what means that a non-perfect str…
A new statistical method of alignment of two heteropolymers which can form hierarchical cloverleaf-like secondary structures is proposed. This offers a new constructive algorithm for quantitative determination of binding free energy of two noncoding RNAs with arbitrary primary sequences. The alignment of ncRNAs differs from the complete alignment of two RNA sequences: in ncRNA case we align only the sequences of nucleotides which constitute pairs between two different RNAs, while the secondary structure of each RNA comes into play only by the combinatorial factors affecting the entropc contrib…
