We investigate the competition between two quasispecies residing on two disparate neutral networks. Under the assumption that the two neutral networks have different topologies and fitness levels, it is the mutation rate that determines which quasispecies will eventually be driven to extinction. For small mutation rates, we find that the quasispecies residing on the neutral network with the lower replication rate will disappear. For higher mutation rates, however, the faster replicating sequences may be outcompeted by the slower replicating ones in case the connection density on the second neutral network is sufficiently high. Our analytical results are in excellent agreement with flow-reactor simulations of replicating RNA sequences.
Loops are abundant in native RNA structures and proliferate close to the unfolding transition. By including a statistical weight ~ l^{-c} for loops of length l in the recursion relation for the partition function, we show that the calculated heat capacity depends sensitively on the presence and value of the exponent c, even of short t-RNA. For homo-RNA we analytically calculate the critical temperature and critical exponents which exhibit a non-universal dependence on c.
Background: RNA exhibits a variety of structural configurations. Here we consider a structure to be tantamount to the noncrossing Watson-Crick and \pairGU-base pairings (secondary structure) and additional cross-serial base pairs. These interactions are called pseudoknots and are observed across the whole spectrum of RNA functionalities. In the context of studying natural RNA structures, searching for new ribozymes and designing artificial RNA, it is of interest to find RNA sequences folding into a specific structure and to analyze their induced neutral networks. Since the established inverse …
We construct a minimalist model of RNA secondary-structure formation and use it to study the mapping from sequence to structure. There are strong, qualitative differences between two-letter and four or six-letter alphabets. With only two kinds of bases, there are many alternate folding configurations, yielding thermodynamically stable ground-states only for a small set of structures of high designability, i.e., total number of associated sequences. In contrast, sequences made from four bases, as found in nature, or six bases have far fewer competing folding configurations, resulting in a much …
The statistical mechanics of heteropolymer structure formation is studied in the context of RNA secondary structures. A designed RNA sequence biased energetically towards a particular native structure (a hairpin) is used to study the transition between the native and molten phase of the RNA as a function of temperature. The transition is driven by a competition between the energy gained from the polymer's overlap with the native structure and the entropic gain of forming random contacts. A simplified Go-like model is proposed and solved exactly. The predicted critical behavior is verified via …
We show that the folding rates (k_F) of RNA are determined by N, the number of nucleotides. By assuming that the distribution of free energy barriers separating the folded and the unfolded states is Gaussian, which follows from central limit theorem arguments and polymer physics concepts, we show that k_F ~ k_0 exp(-alpha N^0.5). Remarkably, the theory fits the experimental rates spanning over seven orders of magnitude with k_0 ~ 1.0 (microsec)^{-1}. An immediate consequence of our finding is that the speed limit of RNA folding is about one microsecond just as it is in the folding of globular …
Background: The rate at which fitness declines as an organism's genome accumulates random mutations is an important variable in several evolutionary theories. At an intuitive level, it might seem natural that random mutations should tend to interact synergistically, such that the rate of mean fitness decline accelerates as the number of random mutations is increased. However, in a number of recent studies, a prevalence of antagonistic epistasis (the tendency of multiple mutations to have a mitigating rather than reinforcing effect) has been observed. Results: We studied in silico the net amoun…
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…
Atomically detailed simulations of RNA folding have proven very challenging in view of the difficulties of developing realistic force fields and the intrinsic computational complexity of sampling rare conformational transitions. To tackle both these issues, we extend to RNA an enhanced path sampling method previously successfully applied to proteins. In this scheme, the information about the RNA's native structure is harnessed by a soft history-dependent biasing force, which is added to the atomistic force field, thus promoting the generation of productive folding trajectories. Here, we report…
We formulate the RNA folding problem as an $N\times N$ matrix field theory. This matrix formalism allows us to give a systematic classification of the terms in the partition function according to their topological character. The theory is set up in such a way that the limit $N\to \infty$ yields the so-called secondary structure (Hartree theory). Tertiary structure and pseudo-knots are obtained by calculating the $1/N^2$ corrections to the partition function. We propose a generalization of the Hartree recursion relation to generate the tertiary structure.
Non-coding RNA sequences play a great role in controlling a number of cellular functions, thus raising the need to understand their complex conformational dynamics in quantitative detail. In this perspective, we first show that single molecule pulling experiments when combined with with theory and simulations can be used to quantitatively explore the folding landscape of nucleic acid hairpins, and riboswitches with tertiary interactions. Applications to riboswitches, which are non-coding RNA elements that control gene expression by undergoing dynamical conformational changes in response to bin…
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 …
We present a thermodynamically robust coarse-grained model to simulate folding of RNA in monovalent salt solutions. The model includes stacking, hydrogen bond and electrostatic interactions as fundamental components in describing the stability of RNA structures. The stacking interactions are parametrized using a set of nucleotide-specific parameters, which were calibrated against the thermodynamic measurements for single-base stacks and base-pair stacks. All hydrogen bonds are assumed to have the same strength, regardless of their context in the RNA structure. The ionic buffer is modeled impli…
RNA molecules are essential cellular machines performing a wide variety of functions for which a specific three-dimensional structure is required. Over the last several years, experimental determination of RNA structures through X-ray crystallography and NMR seems to have reached a plateau in the number of structures resolved each year, but as more and more RNA sequences are being discovered, need for structure prediction tools to complement experimental data is strong. Theoretical approaches to RNA folding have been developed since the late nineties when the first algorithms for secondary str…
We investigated frictional effects on the folding rates of a human telomerase hairpin (hTR HP) and H-type pseudoknot from the Beet Western Yellow Virus (BWYV PK) using simulations of the Three Interaction Site (TIS) model for RNA. The heat capacity from TIS model simulations, calculated using temperature replica exchange simulations, reproduces nearly quantitatively the available experimental data for the hTR HP. The corresponding results for BWYV PK serve as predictions. We calculated the folding rates ($k_\mathrm{F}$) from more than 100 folding trajectories for each value of the solvent visc…
We propose a new deterministic methodology to predict RNA sequence and protein folding. Is stem enough for structure prediction? The main idea is to consider all possible stem formation in the given sequence. With the stem loop energy and the strength of stem, we explore how to deterministically utilize stem information for RNA sequence and protein folding structure prediction. We use graph notation, where all possible stems are represented as vertices, and co-existence as edges. This full Stem-graph presents all possible folding structure, and we pick sub-graph(s) which give the best matching…
We introduce a method for predicting RNA folding pathways, with an application to the most important RNA tetraloops. The method is based on the idea that ensembles of three-dimensional fragments extracted from high-resolution crystal structures are heterogeneous enough to describe metastable as well as intermediate states. These ensembles are first validated by performing a quantitative comparison against available solution NMR data of a set of RNA tetranucleotides. Notably, the agreement is better with respect to the one obtained by comparing NMR with extensive all-atom molecular dynamics sim…
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 computer-aided folding of biomolecules, particularly RNAs, is one of the most difficult challenges in computational structural biology. RNA tetraloops are fundamental RNA motifs playing key roles in RNA folding and RNA-RNA and RNA-protein interactions. Although state-of-the-art Molecular Dynamics (MD) force fields correctly describe the native state of these tetraloops as a stable free-energy basin on the microsecond time scale, enhanced sampling techniques reveal that the native state is not the global free energy minimum, suggesting yet unidentified significant imbalances in the force fi…
Accurate knowledge of RNA hybridization is essential for understanding RNA structure and function. Here we mechanically unzip and rezip a 2-kbp RNA hairpin and derive the 10 nearest-neighbor base pair (NNBP) RNA free energies in sodium and magnesium with 0.1 kcal/mol precision using optical tweezers. Notably, force-distance curves (FDCs) exhibit strong irreversible effects with hysteresis and several intermediates, precluding the extraction of the NNBP energies with currently available methods. The combination of a suitable RNA synthesis with a tailored pulling protocol allowed us to obtain th…
Secondary structure formation of nucleic acids strongly depends on salt concentration and temperature. We develop a theory for RNA folding that correctly accounts for sequence effects, the entropic contributions associated with loop formation, and salt effects. Using an iterative expression for the partition function that neglects pseudoknots, we calculate folding free energies and minimum free energy configurations based on the experimentally derived base pairing free energies. The configurational entropy of loop formation is modeled by the asymptotic expression -c ln m, where m is the length…
We propose a scenario for the prebiotic co-evolution of RNA and of fast folding proteins with large entropy gaps as observed today. We show from very general principles that the folding and unfolding of the proteins synthesized by RNA can function as a heat pump. Rock surfaces can facilitate the folding of amino acid chains having polar and hydrophobic residues, with an accompanying heat loss to the surrounding rock. These chains then absorb heat from the soup as they unfold. This opens the way to the enhancement of RNA replication rates, by the enzymatic action of folded proteins present in g…
We review the basic concepts and tools for mechanically unzipping RNA hairpins using force spectroscopy. By pulling apart the ends of an RNA molecule using optical tweezers, it is possible to measure the folding free energy at varying experimental conditions. Energy measurements permit us to characterize the thermodynamics of RNA hybridization (base pairing and stacking), the dynamics of the formation of native and kinetic (intermediates and misfolded) molecular states, and interactions with metallic ions. This paper introduces basic concepts and reviews recent developments related to RNA forc…
The sequence-dependent folding landscapes of nucleic acid hairpins reflect much of the complexity of biomolecular folding. Folding trajectories, generated using single molecule force clamp experiments by attaching semiflexible polymers to the ends of hairpins have been used to infer their folding landscapes. Using simulations and theory, we study the effect of the dynamics of the attached handles on the handle-free RNA free energy profile $F^o_{eq}(z_m)$, where $z_m$ is the molecular extension of the hairpin. Accurate measurements of $F^o_{eq}(z_m)$ requires stiff polymers with small $L/l_p$, …
Molecular chaperones are ATP-consuming biological machines, which facilitate the folding of proteins and RNA molecules that are kinetically trapped in misfolded states for long times. Unassisted folding occurs by the kinetic partitioning mechanism according to which folding to the native state, with low probability as well as misfolding to one of the many metastable states, with high probability, occur rapidly on similar time scales. GroEL is an all-purpose stochastic machine that assists misfolded substrate proteins (SPs) to fold. The RNA chaperones (CYT-19) help the folding of ribozymes that…
The functions of RNA pseudoknots (PKs), which are minimal tertiary structural motifs and an integral part of several ribozymes and ribonucleoprotein complexes, are determined by their structure, stability and dynamics. Therefore, it is important to elucidate the general principles governing their thermodynamics/folding mechanisms. Here, we combine experiments and simulations to examine the folding/unfolding pathways of the VPK pseudoknot, a variant of the Mouse Mammary Tumor Virus (MMTV) PK involved in ribosomal frameshifting. Fluorescent nucleotide analogs (2-aminopurine and pyrrolocytidine) …
As a consequence of the rugged landscape of RNA molecules their folding is described by the kinetic partitioning mechanism according to which only a small fraction ($ϕ_F$) reaches the folded state while the remaining fraction of molecules is kinetically trapped in misfolded intermediates. The transition from the misfolded states to the native state can far exceed biologically relevant time. Thus, RNA folding in vivo is often aided by protein cofactors, called RNA chaperones, that can rescue RNAs from a multitude of misfolded structures. We consider two models, based on chemical kinetics and ch…
Motivation: Predicting the secondary structure of an RNA sequence is useful in many applications. Existing algorithms (based on dynamic programming) suffer from a major limitation: their runtimes scale cubically with the RNA length, and this slowness limits their use in genome-wide applications. Results: We present a novel alternative $O(n^3)$-time dynamic programming algorithm for RNA folding that is amenable to heuristics that make it run in $O(n)$ time and $O(n)$ space, while producing a high-quality approximation to the optimal solution. Inspired by incremental parsing for context-free gra…
RNAs are negatively charged molecules residing in macromolecular crowding cellular environments. Macromolecular confinement can influence the ion effects in RNA folding. In this work, using the recently developed tightly bound ion model for ion fluctuation and correlation, we investigate the confinement effect on the ion-mediated RNA structural collapse for a simple model system. We found that, for both Na$^+$ and Mg$^{2+}$, ion efficiencies in mediating structural collapse/folding are significantly enhanced by the structural confinement. Such an enhancement in the ion efficiency is attributed…
We investigate the possibility of extending the notion of temperature in a stochastic model for the RNA/protein folding driven out of equilibrium. We simulate the dynamics of a small RNA hairpin subject to an external pulling force, which is time-dependent. First, we consider a fluctuation-dissipation relation (FDR) whereby we verify that various effective temperatures can be obtained for different observables, only when the slowest intrinsic relaxation timescale of the system regulates the dynamics of the system. Then, we introduce a different nonequilibrium temperature, which is defined from…
We review the effects of macromolecular crowding on the folding of RNA by considering the simplest scenario when excluded volume interactions between crowding particles and RNA dominate. Using human telomerase enzyme as an example, we discuss how crowding can alter the equilibrium between pseudoknot and hairpin states of the same RNA molecule - a key aspect of crowder-RNA interactions. We summarize data showing that the crowding effect is significant only if the size of the spherical crowding particle is smaller than the radius of gyration of the RNA in the absence of crowding particles. The i…
We report the folding thermodynamics of ccUUCGgg and ccGAGAgg RNA tetraloops using atomistic molecular dynamics simulations. We obtain a previously unreported estimation of the folding free energy using parallel tempering in combination with well-tempered metadynamics. A key ingredient is the use of a recently developed metric distance, eRMSD, as a biased collective variable. We find that the native fold of both tetraloops is not the global free energy minimum using the Amber\c{hi}OL3 force field. The estimated folding free energies are 30.2kJ/mol for UUCG and 7.5 kJ/mol for GAGA, in striking …
We investigate the relationship between the average fitness decay due to single mutations and the strength of epistatic interactions in genetic sequences. We observe that epistatic interactions between mutations are correlated to the average fitness decay, both in RNA secondary structure prediction as well as in digital organisms replicating in silico. This correlation implies that during adaptation, epistasis and average mutational effect cannot be optimized independently. In experiments with RNA sequences evolving on a neutral network, the selective pressure to decrease the mutational load t…
The folding of biological macromolecules is a fundamental process of which we lack a full comprehension. Mostly studied in proteins and RNA, single-stranded DNA (ssDNA) also folds, at physiological salt conditions, by forming non-specific secondary structures that are difficult to characterize with biophysical techniques. Here we present a helix-coil model for secondary structure formation, where ssDNA bases are organized in two different types of domains (compact and free). The model contains two parameters: the energy gain per base in a compact domain, $ε$, and the cooperativity related to t…
RNA-RNA binding is an important phenomenon observed for many classes of non-coding RNAs and plays a crucial role in a number of regulatory processes. Recently several MFE folding algorithms for predicting the joint structure of two interacting RNA molecules have been proposed. Here joint structure means that in a diagram representation the intramolecular bonds of each partner are pseudoknot-free, that the intermolecular binding pairs are noncrossing, and that there is no so-called ``zig-zag'' configuration. This paper presents the combinatorics of RNA interaction structures including their gen…
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 notion…
Although both RNA and proteins have densely packed native structures, chain organizations of these two biopolymers are fundamentally different. Motivated by the recent discoveries in chromatin folding that interphase chromosomes have territorial organization with signatures pointing to metastability, we analyzed the biomolecular structures deposited in the Protein Data Bank and found that the intrachain contact probabilities, $P(s)$ as a function of the arc length $s$, decay in power-law $\sim s^{-γ}$ over the intermediate range of $s$, $10\lesssim s\lesssim 110$. We found that the contact pro…
In addition to thermal noise, which is essential to promote conformational transitions in biopolymers, cellular environment is replete with a spectrum of athermal fluctuations that are produced from a plethora of active processes. To understand the effect of athermal noise on biological processes, we studied how a small oscillatory force affects the thermally induced folding and unfolding transition of an RNA hairpin, whose response to constant tension had been investigated extensively in both theory and experiments. Strikingly, our molecular simulations performed under overdamped condition sh…
Urea titration of RNA by urea is an effective approach to investigate the forces stabilizing this biologically important molecule. We used all atom molecular dynamics simulations using two urea force fields and two RNA constructs to elucidate in atomic detail the destabilization mechanism of folded RNA in aqueous urea solutions. Urea denatures RNA by forming multiple hydrogen bonds with the RNA bases and has little influence on the phosphodiester backbone. Most significantly we discovered that urea engages in stacking interactions with the bases. We also estimate, for the first time, m-value f…
Single molecule force spectroscopy methods can be used to generate folding trajectories of biopolymers from arbitrary regions of the folding landscape. We illustrate the complexity of the folding kinetics and generic aspects of the collapse of RNA and proteins upon force quench, using simulations of an RNA hairpin and theory based on the de Gennes model for homopolymer collapse. The folding time, $τ_F$, depends asymmetrically on $δf_S = f_S - f_m$ and $δf_Q = f_m - f_Q$ where $f_S$ ($f_Q$) is the stretch (quench) force, and $f_m$ is the transition mid-force of the RNA hairpin. In accord with e…
We study the folding of RNA secondary structures with quenched sequence randomness by means of the constrained annealing method. A thermodynamic phase transition is induced by including the conformational weight of loop structures. In addition to the expected melting at high temperature, a cold melting transition appears. Our results suggest that the cold denaturation of RNA found experimentally is, in fact, a continuous phase transition triggered by quenched sequence disorder. We calculate both hot and cold melting critical temperatures for the competing energy scenario between favorable and …
We describe a dynamic programming algorithm for predicting optimal RNA secondary structure, including pseudoknots. The algorithm has a worst case complexity of ${\cal O}(N^6)$ in time and ${\cal O}(N^4)$ in storage. The description of the algorithm is complex, which led us to adopt a useful graphical representation (Feynman diagrams) borrowed from quantum field theory. We present an implementation of the algorithm that generates the optimal minimum energy structure for a single RNA sequence, using standard RNA folding thermodynamic parameters augmented by a few parameters describing the thermo…
RNA secondary structures of increasing complexity are probed combining single molecule stretching experiments and stochastic unfolding/refolding simulations. We find that force-induced unfolding pathways cannot usually be interpretated by solely invoking successive openings of native helices. Indeed, typical force-extension responses of complex RNA molecules are largely shaped by stretching-induced, long-lived intermediates including non-native helices. This is first shown for a set of generic structural motifs found in larger RNA structures, and then for Escherichia coli's 1540-base long 16S …
The ongoing effort to detect and characterize physical entanglement in biopolymers has so far established that knots are present in many globular proteins and also abound in viral DNA packaged inside bacteriophages. RNA molecules, on the other hand, have not yet been systematically screened for the occurrence of physical knots. We have accordingly undertaken the systematic profiling of the ~6,000 RNA structures present in the protein data bank. The search identified no more than three deeply-knotted RNA molecules. These are ribosomal RNAs solved by cryo-em and consist of about 3,000 nucleotide…
The prediction of protein secondary and tertiary structures from the primary amino acid sequence is both an incredibly important and incredibly difficult problem. Accurate prediction of a protein's native structure can provide critical insights about its function, ultimately leading to breakthoughs in drug design and disease diagnosis. The field has a rich history, from the earliest folding experiments in the 1960's to the use of state-of-the-art algorithms today; this article reviews protein folding's history with an emphasis on how modern methods are tackling the protein folding problem. Ass…
We present a new, nucleotide-level model for RNA, oxRNA, based on the coarse-graining methodology recently developed for the oxDNA model of DNA. The model is designed to reproduce structural, mechanical and thermodynamic properties of RNA, and the coarse-graining level aims to retain the relevant physics for RNA hybridization and the structure of single- and double-stranded RNA. In order to explore its strengths and weaknesses, we test the model in a range of nanotechnological and biological settings. Applications explored include the folding thermodynamics of a pseudoknot, the formation of a …
A full understanding of RNA-mediated biology would require the knowledge of three-dimensional (3D) structures, structural flexibility and stability of RNAs. To predict RNA 3D structures and stability, we have previously proposed a three-bead coarse-grained predictive model with implicit salt/solvent potentials. In this study, we will further develop the model by improving the implicit-salt electrostatic potential and involving a sequence-dependent coaxial stacking potential to enable the model to simulate RNA 3D structure folding in divalent/monovalent ion solutions. As compared with the exper…
Using force as a probe to map the folding landscapes of RNA molecules has become a reality thanks to major advances in single molecule pulling experiments. Although the unfolding pathways under tension are complicated to predict studies in the context of proteins have shown that topology plays is the major determinant of the unfolding landscapes. By building on this finding we study the responses of RNA molecules to force by adapting Gaussian network model (GNM) that represents RNAs using a bead-spring network with isotropic interactions. Cross-correlation matrices of residue fluctuations, whi…
In this paper we study the evolution of the mutation rate for simple organisms in dynamic environments. A model with multiple fitness coding loci tracking a moving fitness peak is developed and an analytical expression for the optimal mutation rate is derived. Surprisingly it turns out that the optimal mutation rate per genome is approximately independent of genome length, something that also has been observed in nature. Simulations confirm the theoretical predictions. We also suggest an explanation for the difference in mutation frequency between RNA and DNA based organisms.
Experimental variables of optical tweezers instrumentation that affect RNA folding/unfolding kinetics were investigated. A model RNA hairpin, P5ab, was attached to two micron-sized beads through hybrid RNA/DNA handles; one bead was trapped by dual-beam lasers and the other was held by a micropipette. Several experimental variables were changed while measuring the unfolding/refolding kinetics, including handle lengths, trap stiffness, and modes of force applied to the molecule. In constant-force mode where the tension applied to the RNA was maintained through feedback control, the measured rate…
Many ncRNAs function through RNA-RNA interactions. Fast and reliable RNA structure prediction with consideration of RNA-RNA interaction is useful. Some existing tools are less accurate due to omitting the competing of intermolecular and intramolecular base pairs, or focus more on predicting the binding region rather than predicting the complete secondary structure of two interacting strands. Vienna RNAcofold, which reduces the problem into the classical single sequence folding by concatenating two strands, scales in cubic time against the combined sequence length, and is slow for long sequence…
RNA structure prediction is a challenging problem, especially with pseudoknots. Recently, there has been a shift from the classical minimum free energy-based methods (MFE) to partition function-based ones that assemble structures using base-pairing probabilities. Two examples of the latter group are the popular maximum expected accuracy (MEA) method and the ProbKnot method. ProbKnot is a fast heuristic that pairs nucleotides that are reciprocally most probable pairing partners, and unlike MEA, can also predict structures with pseudoknots. However, ProbKnot's full potential has been largely ove…
Riboswitches, RNA elements found in the untranslated region, regulate gene expression by binding to target metaboloites with exquisite specificity. Binding of metabolites to the conserved aptamer domain allosterically alters the conformation in the downstream expression platform. The fate of gene expression is determined by the changes in the downstream RNA sequence. As the metabolite-dependent cotranscriptional folding and unfolding dynamics of riboswitches is the key determinant of gene expression, it is important to investigate both the thermodynamics and kinetics of riboswitches both in th…
