Simple model for the DNA denaturation transition

Simulations of DNA denaturation dynamics under constrained conditions

We study the dynamics of double-stranded DNA (dsDNA) denaturation using Brownian dynamics simulations. We use a coarse-grained single nucleotide model for dsDNA which considers the helix structure. We compare the melting dynamics for free DNA of length 300 base pairs with that of a DNA of the same length but fixed from one end-mimicking DNA tethered to a substrate. We find that free DNA melts at faster rate because the entropic gain associated with denaturation is larger. Additionally, we insert the DNA in nanochannels of different widths to study the influence of the confinement on the melting dynamics. Our results suggest that there is no significant difference in the critical temperature or rate of melting when the channel width⩾Rg/2, whereR_gis the radius of gyration of DNA. Instead, at channel width ofR_g/4 we only see partial denaturation at the free DNA melting temperature. Surprisingly, this trend is reversed at higher temperature, and we find that at 110 °C tight confinement results in faster melting. This is due to the fact that confinement promotes segregation of the single-stranded segment, thereby acting as an effective entropic force aiding denaturation.

Melting of confined DNA: static and dynamic properties

We study dsDNA (double strand DNA) melting in detail within varying strip-like confinement in a two-dimensional lattice model. The interplay between reduced configurational entropy and attractive base-pairing energy results in a non-monotonic melting profile of DNA. Structural transitions associated with confined DNA melting reveal a stretched or extended state for very strong confinement. By using the exact enumeration method, we investigate the emergence of a local denatured zone e.g. bubbles during DNA melting. The survival time of a single bubble within varying strip width is studied from the Fokker-Planck formalism by considering the bubble size as a reaction co-ordinate. We show that a simple lattice model can capture the sequence heterogeneity effect on DNA melting and bubble dynamics within the strip. Different time scales of bubble zipping for different DNA sequences are found, which may have potential applications in denaturation mapping.

Efimov-DNA phase diagram: Three stranded DNA on a cubic lattice

We define a generalized model for three-stranded DNA consisting of two chains of one type and a third chain of a different type. The DNA strands are modeled by random walks on the three-dimensional cubic lattice with different interactions between two chains of the same type and two chains of different types. This model may be thought of as a classical analog of the quantum three-body problem. In the quantum situation, it is known that three identical quantum particles will form a triplet with an infinite tower of bound states at the point where any pair of particles would have zero binding energy. The phase diagram is mapped out, and the different phase transitions are examined using finite-size scaling. We look particularly at the scaling of the DNA model at the equivalent Efimov point for chains up to 10 000 steps in length. We find clear evidence of several bound states in the finite-size scaling. We compare these states with the expected Efimov behavior.

Elasticity of a DNA chain dotted with bubbles under force

The flexibility and the extension along the direction of the force are shown to be related to the bubble number fluctuation and the average number of bubbles, respectively, when the strands of the DNA are subjected to a force along the same direction, here called a stretching force. The force-temperature phase diagram shows the existence of a tricritical point, where the first-order force-induced zipping transition becomes continuous. On the other hand, when the forces are being applied in opposite directions, here called an unzipping force, the transition remains first order, with the possibility of vanishing of the low-temperature reentrant phase for a semiflexible DNA. Moreover, we found that the bulk elasticity changes only if an external force penetrates the bound phase and affects the bubble states.

Softening of DNA near melting as disappearance of an emergent property

Near the melting transition the bending elastic constant κ, an emergent property of double-stranded DNA (dsDNA), is shown not to follow the rodlike scaling for small-length N. The reduction in κ with temperature is determined by the denatured bubbles for a continuous transition, e.g., when the two strands are Gaussian, but by the broken bonds near the open end in a Y-like configuration for a first-order transition as for strands with excluded volume interactions. In the latter case, a lever rule is operational, implying a phase coexistence although dsDNA is known to be a single phase.

Role of hubs in the synergistic spread of behavior

The spread of behavior in a society has two major features: the synergy of multiple spreaders and the dominance of hubs. While strong synergy is known to induce mixed-order transitions (MOTs) at percolation, the effects of hubs on the phenomena are yet to be clarified. By analytically solving the generalized epidemic process on random scale-free networks with the power-law degree distribution pk∼k−α, we clarify how the dominance of hubs in social networks affects the conditions for MOTs. Our results show that, for α<4, an abundance of hubs drive MOTs, even if a synergistic spreading event requires an arbitrarily large number of adjacent spreaders. In particular, for 2<α<3, we find that a global cascade is possible even when only synergistic spreading events are allowed. These transition properties are substantially different from those of cooperative contagions, which are another class of synergistic cascading processes exhibiting MOTs.

Anomalous critical slowdown at a first order phase transition in single polymer chains

Using Brownian dynamics, we study the dynamical behavior of a polymer grafted onto an adhesive surface close to the mechanically induced adsorption-stretching transition. Even though the transition is first order (in the infinite chain length limit, the stretching degree of the chain jumps discontinuously), the characteristic relaxation time is found to grow according to a power law as the transition point is approached. We present a dynamic effective interface model which reproduces these observations and provides an excellent quantitative description of the simulation data. The generic nature of the theoretical model suggests that the unconventional mixing of features that are characteristic for first-order transitions (a jump in an order parameter) and features that are characteristic of critical points (an anomalous slowdown) may be a common phenomenon in force-driven phase transitions of macromolecules.

Sufficient minimal model for DNA denaturation: Integration of harmonic scalar elasticity and bond energies

We study DNA denaturation by integrating elasticity - as described by the Gaussian network model - with bond binding energies, distinguishing between different base pairs and stacking energies. We use exact calculation, within the model, of the Helmholtz free-energy of any partial denaturation state, which implies that the entropy of all formed "bubbles" ("loops") is accounted for. Considering base pair bond removal single events, the bond designated for opening is chosen by minimizing the free-energy difference for the process, over all remaining base pair bonds. Despite of its great simplicity, for several known DNA sequences our results are in accord with available theoretical and experimental studies. Moreover, we report free-energy profiles along the denaturation pathway, which allow to detect stable or meta-stable partial denaturation states, composed of bubble, as local free-energy minima separated by barriers. Our approach allows to study very long DNA strands with commonly available computational power, as we demonstrate for a few random sequences in the range 200-800 base-pairs. For the latter, we also elucidate the self-averaging property of the system. Implications for the well known breathing dynamics of DNA are elucidated.

Mixed-order phase transition of the contact process near multiple junctions

We have studied the phase transition of the contact process near a multiple junction of M semi-infinite chains by Monte Carlo simulations. As opposed to the continuous transitions of the translationally invariant (M=2) and semi-infinite (M=1) system, the local order parameter is found to be discontinuous for M>2. Furthermore, the temporal correlation length diverges algebraically as the critical point is approached, but with different exponents on the two sides of the transition. In the active phase, the estimate is compatible with the bulk value, while in the inactive phase it exceeds the bulk value and increases with M. The unusual local critical behavior is explained by a scaling theory with an irrelevant variable, which becomes dangerous in the inactive phase. Quenched spatial disorder is found to make the transition continuous in agreement with earlier renormalization group results.

Critical phenomena on k-booklets

We define a "k-booklet" to be a set of k semi-infinite planes with -∞<x<∞ and y≥0, glued together at the edges (the "spine") y=0. On such booklets we study three critical phenomena: self-avoiding random walks, the Ising model, and percolation. For k=2, a booklet is equivalent to a single infinite lattice, and for k=1 to a semi-infinite lattice. In both these cases the systems show standard critical phenomena. This is not so for k≥3. Self-avoiding walks starting at y=0 show a first-order transition at a shifted critical point, with no power-behaved scaling laws. The Ising model and percolation show hybrid transitions, i.e., the scaling laws of the standard models coexist with discontinuities of the order parameter at y≈0, and the critical points are not shifted. In the case of the Ising model, ergodicity is already broken at T=T_{c}, and not only for T<T_{c} as in the standard geometry. In all three models, correlations (as measured by walk and cluster shapes) are highly anisotropic for small y.

Coil-helix transition of polypeptide at water-lipid interface

We present the exact solution of a microscopic statistical mechanical model for the transformation of a long polypeptide between an unstructured coil conformation and an α-helix conformation. The polypeptide is assumed to be adsorbed to the interface between a polar and a non-polar environment such as realized by water and the lipid bilayer of a membrane. The interfacial coil-helix transformation is the first stage in the folding process of helical membrane proteins. Depending on the values of model parameters, the conformation changes as a crossover, a discontinuous transition, or a continuous transition with helicity in the role of order parameter. Our model is constructed as a system of statistically interacting quasiparticles that are activated from the helix pseudo-vacuum. The particles represent links between adjacent residues in coil conformation that form a self-avoiding random walk in two dimensions. Explicit results are presented for helicity, entropy, heat capacity, and the average numbers and sizes of sboth coil and helix segments.

Explosive percolation is continuous, but with unusual finite size behavior

We study four Achlioptas-type processes with "explosive" percolation transitions. All transitions are clearly continuous, but their finite size scaling functions are not entirely holomorphic. The distributions of the order parameter, i.e., the relative size s(max)/N of the largest cluster, are double humped. But-in contrast to first-order phase transitions-the distance between the two peaks decreases with system size N as N(-η) with η>0. We find different positive values of β (defined via (s(max)/N)∼(p-p(c))β for infinite systems) for each model, showing that they are all in different universality classes. In contrast, the exponent Θ (defined such that observables are homogeneous functions of (p-p(c))N(Θ)) is close to-or even equal to-1/2 for all models.

Aggregation of amphiphilic polymers in the presence of adhesive small colloidal particles

The interaction of amphiphilic polymers with small colloids, capable to reversibly stick onto the chains, is studied. Adhesive small colloids in solution are able to dynamically bind two polymer segments. This association leads to topological changes in the polymer network configurations, such as looping and cross-linking, although the reversible adhesion permits the colloid to slide along the chain backbone. Previous analyses only consider static topologies in the chain network. We show that the sliding degree of freedom ensures the dominance of small loops, over other structures, giving rise to a new perspective in the analysis of the problem. The results are applied to the analysis of the equilibrium between colloidal particles and star polymers, as well as to block copolymer micelles. The results are relevant for the reversible adsorption of silica particles onto hydrophilic polymers, used in the process of formation of mesoporous materials of the type SBA or MCM, cross-linked cyclodextrin molecules threading on the polymers and forming the structures known as polyrotaxanes. Adhesion of colloids on the corona of the latter induce micellization and growth of larger micelles as the number of colloids increase, in agreement with experimental data.

Analysis of a DNA simulation model through hairpin melting experiments

We compare the predictions of a two-bead Brownian dynamics simulation model to melting experiments of DNA hairpins with complementary AT or GC stems and noninteracting loops in buffer A. This system emphasizes the role of stacking and hydrogen bonding energies, which are characteristics of DNA, rather than backbone bending, stiffness, and excluded volume interactions, which are generic characteristics of semiflexible polymers. By comparing high throughput data on the open-close transition of various DNA hairpins to the corresponding simulation data, we (1) establish a suitable metric to compare the simulations to experiments, (2) find a conversion between the simulation and experimental temperatures, and (3) point out several limitations of the model, including the lack of G-quartets and cross stacking effects. Our approach and experimental data can be used to validate similar coarse-grained simulation models.

Prediction of RNA multiloop and pseudoknot conformations from a lattice-based, coarse-grain tertiary structure model

We present a semiquantitative lattice model of RNA folding, which is able to reproduce complex folded structures such as multiloops and pseudoknots without relying on the frequently employed ad hoc generalization of the Jacobson-Stockmayer loop entropy. We derive the model parameters from the Turner description of simple secondary structural elements and pay particular attention to the unification of mismatch and coaxial stacking parameters as well as of border and nonlocal loop parameters, resulting in a reduced, unified parameter set for simple loops of arbitrary type and size. For elementary structures, the predictive power of the model is comparable to the standard secondary structure approaches, from which its parameters are derived. For complex structures, our approach offers a systematic treatment of generic effects of chain connectivity as well as of excluded volume or attractive interactions between and within all elements of the secondary structure. We reproduce the native structures of tRNA multiloops and of viral frameshift signal pseudoknots.

Different pulling modes in DNA overstretching: a theoretical analysis

We study the thermally driven denaturation of a double-stranded polymer in the presence of a stretching force via Monte-Carlo simulations. When one strand only is stretched, the denaturation transition is first order, while when both strands are stretched, melting is second order. By revisiting the Poland-Scheraga model for DNA melting, we show that at room temperature, the most likely scenario is that DNA melts as it overstretches. Our results are in general agreement with the most recent experiments and suggest how varying temperature and stretching mode may help settle the question whether S-DNA exists or not.

Denaturation patterns in heterogeneous DNA

The thermodynamical properties of heterogeneous DNA sequences are computed by path-integral techniques applied to a nonlinear model Hamiltonian. The base pairs relative displacements are interpreted as time-dependent paths whose amplitudes are consistent with the model potential for the hydrogen bonds between complementary strands. The portion of configuration space contributing to the partition function is determined, at any temperature, by selecting the ensemble of paths which fulfill the second law of thermodynamics. For a short DNA fragment, the denaturation is signaled by a succession of peaks in the specific-heat plots while the entropy grows continuously versus T. Thus, the opening of the double strand with bubble formation appears as a smooth crossover due to base pair fluctuation effects which are accounted for by the path-integral method. The multistep transition is driven by the adenine-thymine- (AT) rich regions of the DNA fragment. The base pairs path ensemble shows an enhanced degree of cooperativity at about the same temperatures for which the specific-heat peaks occur. These findings establish a link between microscopic and macroscopic signatures of the transition. The fractions of mean base pair stretchings are computed by varying the AT base pairs content and taking some threshold values for the occurrence of the molecule denaturation.

Supercoil formation in DNA denaturation

We generalize the Poland-Scheraga model to the case of a circular DNA, taking into account the twisting of the two strains around each other. Guided by recent single-molecule experiments on DNA strands, we assume that the torsional stress induced by denaturation enforces the formation of supercoils whose writhe absorbs the linking number expelled by the loops. Our model predicts that when the entropy parameter of a loop satisfies c<or=2, denaturation transition does not take place. On the other hand, for c>2, a first-order denaturation transition is consistent with our model and may take place in the actual system, as in the case with no supercoils. These results are in contrast with other treatments of circular DNA melting where denaturation is assumed to be accompanied by an increase in twist rather than writhe on the bound segments.

How adsorption influences DNA denaturation

The thermally induced denaturation of DNA in the presence of an attractive solid surface is studied. The two strands of DNA are modeled via two coupled flexible chains without volume interactions. If the two strands are adsorbed on the surface, the denaturation phase transition disappears. Instead, there is a smooth crossover to a weakly naturated state. Our second conclusion is that even when the interstrand attraction alone is too weak for creating a naturated state at the given temperature and also when the surface-strand attraction alone is too weak for creating an adsorbed state, the combined effect of the two attractions can lead to a naturated and adsorbed state.

Dynamics of DNA melting

The dynamics of loops at the DNA denaturation transition is studied. A scaling argument is used to evaluate the asymptotic behavior of the autocorrelation function of the state of complementary bases (either open or closed). The long-time asymptotic behavior of the autocorrelation function is expressed in terms of the entropy exponent, c, of a loop. The validity of the scaling argument is tested using a microscopic model of an isolated loop and a toy model of interacting loops. This suggests a method for measuring the entropy exponent using single-molecule experiments such as fluorescence correlation spectroscopy.

Efficient chain moves for Monte Carlo simulations of a wormlike DNA model: excluded volume, supercoils, site juxtapositions, knots, and comparisons with random-flight and lattice models

We develop two classes of Monte Carlo moves for efficient sampling of wormlike DNA chains that can have significant degrees of supercoiling, a conformational feature that is key to many aspects of biological function including replication, transcription, and recombination. One class of moves entails reversing the coordinates of a segment of the chain along one, two, or three axes of an appropriately chosen local frame of reference. These transformations may be viewed as a generalization, to the continuum, of the Madras-Orlitsky-Shepp algorithm for cubic lattices. Another class of moves, termed T+/-2, allows for interconversions between chains with different lengths by adding or subtracting two beads (monomer units) to or from the chain. Length-changing moves are generally useful for conformational sampling with a given site juxtaposition, as has been shown in previous lattice studies. Here, the continuum T+/-2 moves are designed to enhance their acceptance rate in supercoiled conformations. We apply these moves to a wormlike model in which excluded volume is accounted for by a bond-bond repulsion term. The computed autocorrelation functions for the relaxation of bond length, bond angle, writhe, and branch number indicate that the new moves lead to significantly more efficient sampling than conventional bead displacements and crankshaft rotations. A close correspondence is found in the equilibrium ensemble between the map of writhe computed for pair of chain segments and the map of site juxtapositions or self-contacts. To evaluate the more coarse-grained freely jointed chain (random-flight) and cubic lattice models that are commonly used in DNA investigations, twisting (torsional) potentials are introduced into these models. Conformational properties for a given superhelical density sigma may then be sampled by computing the writhe and using White's formula to relate the degree of twisting to writhe and sigma. Extensive comparisons of contact patterns and knot probabilities of the more coarse-grained models with the wormlike model show that the behaviors of the random-flight model are similar to that of DNA molecules in a solution environment with high ionic strengths, whereas the behaviors of the cubic lattice model with excluded volume are akin to that of DNA molecules under low ionic strengths.

Simplified Langevin approach to the Peyrard-Bishop-Dauxois model of DNA

A simple Langevin approach is used to study stationary properties of the Peyrard-Bishop-Dauxois model for DNA, allowing known properties to be recovered in an easy way. Results are shown for the denaturation transition in homogeneous samples, for which some implications, so far overlooked, of an analogy with equilibrium wetting transitions are highlighted. This analogy implies that the order parameter, asymptotically, exhibits a second-order transition even if it may be very abrupt for nonzero values of the stiffness parameter. Not surprisingly, we also find that, for heterogeneous DNA, within this model the largest bubbles in the premelting stage appear in adenine-thymine-rich regions, while we suggest the possibility of some sort of not strictly local effects owing to the merging of bubbles.

Computer simulation study of probe-target hybridization in model DNA microarrays: effect of probe surface density and target concentration

We use lattice Monte Carlo simulations to study the thermodynamics of hybridization of single-stranded "target" genes in solution with complementary "probe" DNA molecules immobilized on a microarray surface. The target molecules in our system contain 48 segments and the probes tethered on a hard surface contain 8-24 segments. The segments on the probe and target are distinct, with each segment representing a sequence of nucleotides that interacts exclusively with its unique complementary target segment with a single hybridization energy; all other interactions are zero. We examine how surface density (number of probes per unit surface area) and concentration of target molecules affect the extent of hybridization. For short probe lengths, as the surface density increases, the probability of binding long stretches of target segments increases at low surface density, reaches a maximum at an intermediate surface density, and then decreases at high surface density. Furthermore, as the surface density increases, the target is less likely to bind completely to one probe; instead, it binds simultaneously to multiple probes. At short probe lengths, as the target concentration increases, the fraction of targets binding completely to the probes (specificity) decreases. At long probe lengths, varying the target concentration does not affect the specificity. At all target concentrations as the probe length increases, the fraction of target molecules bound to the probes by at least one segment (sensitivity) increases while the fraction of target molecules completely bound to the probes (specificity) decreases. This work provides general guidelines to maximizing microarray sensitivity and specificity. Our results suggest that the sensitivity and specificity can be maximized by using probes 130-180 nucleotides long at a surface density in the range of 7 x 10(-5)- 3 x 10(-4) probe molecules per nm(2).

Multifractal statistics of the local order parameter at random critical points: application to wetting transitions with disorder

Disordered systems present multifractal properties at criticality. In particular, as discovered by Ludwig [A.W.W. Ludwig, Nucl. Phys. B 330, 639 (1990)] in the case of a diluted two-dimensional Potts model, the moments rho(q) (r) of the local order parameter rho(r) scale with a set x(q) of nontrivial exponents x(q) not = qx(1). We reexamine these ideas to incorporate more recent findings: (i) whenever a multifractal measure w(r) normalized over space sum(r) w(r) = 1 occurs in a random system, it is crucial to distinguish between the typical values and the disorder-averaged values of the generalized moments Y(q) = sum(r) w(q) (r), since they may scale with different generalized dimensions D(q) and D(q), and (ii), as discovered by Wiseman and Domany [S. Wiseman and E. Domany, Phys. Rev. E 52, 3469 (1995)], the presence of an infinite correlation length induces a lack of self-averaging at critical points for thermodynamic observables, in particular for the order parameter. After this general discussion, valid for any random critical point, we apply these ideas to random polymer models that can be studied numerically for large sizes and good statistics over the samples. We study the bidimensional wetting or the Poland-Scheraga DNA model with loop exponents c = 1.5 (marginal disorder) and c = 1.75 (relevant disorder). Finally, we argue that the presence of finite Griffiths-ordered clusters at criticality determines the asymptotic value x(q-->infinity) = d and the minimal value alpha(min) = D(q-->infinity) = d - x(1) of the typical multifractal spectrum f(alpha).

Unified description of poly- and oligonucleotide DNA melting: nearest-neighbor, Poland-Sheraga, and lattice models

We show that a simple lattice model can provide a unified description of the thermal denaturation of DNA oligomers and polymers. The model quantitatively reproduces experimental melting curves and reduces in limiting cases to the nearest-neighbor model and a suitable modified Poland-Sheraga model. Our results support the interpretation of the cooperativity parameter sigma for bubble opening in terms of an interfacial (forking) free energy which also affects chain opening from the ends. The lattice model treats long-ranged excluded volume interactions between all parts of the molecule explicitly, provides access to an ensemble of three dimensional structures (and hence the response to external mechanical forces), and includes fluctuations in situations without pre-determined secondary structure.

Loop dynamics in DNA denaturation

The dynamics of a loop in DNA molecules at the denaturation transition is studied by scaling arguments and numerical simulations. The autocorrelation function of the state of complementary bases (either closed or open) is calculated. The long-time decay of the autocorrelation function is expressed in terms of the loop exponent c both for homopolymers and heteropolymers. This suggests an experimental method for measuring the exponent c using florescence correlation spectroscopy.

The effect of sequence correlation on bubble statistics in double-stranded DNA

DNA exists stably in the double-stranded structure at physiological temperatures, but base pairs are observed to unbind locally, giving way to bubbles (i.e., locally denatured states) due to thermal fluctuation. In this study, we consider the effect of sequence on the bubble statistics. On the basis of the Edwards equation description [W. Sung and J.-H. Jeons, Phys. Rev. E 69, 031902 (2004) ], we develop a stochastic model incorporating the sequence randomness as a dichotomic noise, where the bubble and its size are identified as a returning random walk and its first passage time, respectively. By simulating the model Langevin equation, we obtain the bubble size distribution and show how it is affected by the sequence correlation. We find that the bubble size distribution of DNA with finite sequence correlation deviates from the Poland-Scheraga-type distribution. In particular, the formation of large bubbles is dramatically enhanced as sequence correlation length gets longer.

Computer simulation study of molecular recognition in model DNA microarrays

DNA microarrays have been widely adopted by the scientific community for a variety of applications. To improve the performance of microarrays there is a need for a fundamental understanding of the interplay between the various factors that affect microarray sensitivity and specificity. We use lattice Monte Carlo simulations to study the thermodynamics and kinetics of hybridization of single-stranded target genes in solution with complementary probe DNA molecules immobilized on a microarray surface. The target molecules in our system contain 48 segments and the probes tethered on a hard surface contain 8-24 segments. The segments on the probe and target are distinct and each segment represents a sequence of nucleotides ( approximately 11 nucleotides). Each probe segment interacts exclusively with its unique complementary target segment with a single hybridization energy; all other interactions are zero. We examine how the probe length, temperature, or hybridization energy, and the stretch along the target that the probe segments complement, affect the extent of hybridization. For systems containing single probe and single target molecules, we observe that as the probe length increases, the probability of binding all probe segments to the target decreases, implying that the specificity decreases. We observe that probes 12-16 segments ( approximately 132-176 nucleotides) long gave the highest specificity and sensitivity. This agrees with the experimental results obtained by another research group, who found an optimal probe length of 150 nucleotides. As the hybridization energy increases, the longer probes are able to bind all their segments to the target, thus improving their specificity. The hybridization kinetics reveals that the segments at the ends of the probe are most likely to start the hybridization. The segments toward the center of the probe remain bound to the target for a longer time than the segments at the ends of the probe.

Single-molecule experiments in biological physics: methods and applications

I review single-molecule experiments (SMEs) in biological physics. Recent technological developments have provided the tools to design and build scientific instruments of high enough sensitivity and precision to manipulate and visualize individual molecules and measure microscopic forces. Using SMEs it is possible to manipulate molecules one at a time and measure distributions describing molecular properties, characterize the kinetics of biomolecular reactions and detect molecular intermediates. SMEs provide additional information about thermodynamics and kinetics of biomolecular processes. This complements information obtained in traditional bulk assays. In SMEs it is also possible to measure small energies and detect large Brownian deviations in biomolecular reactions, thereby offering new methods and systems to scrutinize the basic foundations of statistical mechanics. This review is written at a very introductory level, emphasizing the importance of SMEs to scientists interested in knowing the common playground of ideas and the interdisciplinary topics accessible by these techniques. The review discusses SMEs from an experimental perspective, first exposing the most common experimental methodologies and later presenting various molecular systems where such techniques have been applied. I briefly discuss experimental techniques such as atomic-force microscopy (AFM), laser optical tweezers (LOTs), magnetic tweezers (MTs), biomembrane force probes (BFPs) and single-molecule fluorescence (SMF). I then present several applications of SME to the study of nucleic acids (DNA, RNA and DNA condensation) and proteins (protein-protein interactions, protein folding and molecular motors). Finally, I discuss applications of SMEs to the study of the nonequilibrium thermodynamics of small systems and the experimental verification of fluctuation theorems. I conclude with a discussion of open questions and future perspectives.

A semiflexible chain model of local denaturation in double-stranded DNA

Double-stranded DNA (dsDNA) is known to form a locally denatured structure ("bubble") below its denaturation temperature (T(c)). We have investigated the physical behavior of the bubbles using a model of dsDNA as two wormlike chains containing interacting complementary base pairs. The model incorporates two biologically relevant features, i.e., semiflexibility of the strand and overdamping nature of aqueous background. Computer simulations using the Langevin equation are performed to examine the size distribution and dynamics of bubbles. The results show that the entropy associated with semiflexibility of DNA sensitively affects the size distribution and lifetime of bubble. In particular, the lifetime grows with bubble size m as m(2.7) at temperature close to T(c), which is consistent with our analysis based on a stochastic model of bubble size dynamics.

Mismatches and bubbles in DNA

Single mismatches in the DNA double helix form nucleation sites for bubbles. Although the overall melting temperature of the duplex is affected to different degrees depending on the probe length, the statistical weights of the bubble states around the defect are always strongly affected. Here we show experimentally that a single mismatch has indeed a dramatic effect on the distribution of intermediate (bubble) states in the melting transition of DNA oligomers. For probe lengths in the range 20-40 bases, the mismatch transforms a transition with many intermediates into a nearly two-state transition. One surprising consequence is the existence of a regime where the sensitivity of a mismatch detection assay based on monitoring intermediate states would increase with probe length. Our results provide experimental constraints on how mismatches should be implemented in models of DNA melting, such as the widely used thermodynamic nearest neighbor model, to which we compare our data.

Adhesion-induced DNA naturation

DNA adsorption and naturation is modeled via two interacting flexible homopolymers coupled to a solid surface. DNA denatures if the entropy gain for unbinding the two strands overcomes the loss of binding energy. When adsorbed to a surface, the entropy gain is smaller than in the bulk, leading to a stronger binding and, upon neglecting self-avoidance, absence of a denatured phase. Now consider conditions where the binding potentials are too weak for naturation, and the surface potential too weak to adsorb single strands. In a variational approach it is shown that their combined action may lead to a naturated adsorbed phase. Conditions for the absence of naturation and adsorption are derived too. The phase diagram is constructed qualitatively.

Numerical study of a disordered model for DNA denaturation transition

We numerically study a disordered version of the model for DNA denaturation transition consisting of two interacting self-avoiding walks in three dimensions, which undergoes a first order transition in the homogeneous case. The two possible values epsilonAT and epsilonGC of the interactions between base pairs are taken as quenched random variables distributed with equal probability along the chain. We measure quantities averaged over disorder such as the energy density, the specific heat, and the probability distribution of the loop lengths. When applying the scaling laws used in the homogeneous case we find that the transition seems to be smoother in the presence of disorder, in agreement with general theoretical arguments, although we cannot rule out the possibility of a first order transition.

Unzipping DNA from the condensed globule state—Effects of unraveling

We have studied theoretically the unzipping of a double-stranded DNA from a condensed globule state by an external force. At constant force, we found that the double-stranded DNA unzips an at critical force Fc and the number of unzipped monomers M goes as M approximately (Fc - F)-3, for both the homogeneous and heterogeneous double-stranded DNA sequence. This is different from the case of unzipping from an extended coil state in which the number of unzipped monomers M goes as M approximately (Fc - F)-chi, where the exponent chi is either 1 or 2 depending on whether the double-stranded DNA sequence is homogeneous or heterogeneous, respectively. In the case of unzipping at constant extension, we found that for a double-stranded DNA with a very large number N of base pairs, the force remains almost constant as a function of the extension, before the unraveling transition, at which the force drops abruptly to zero. Right at the unraveling transition, the number of base pairs remaining in the condensed globule state is still very large and goes as N(3/4), in agreement with theoretical predictions of the unraveling transition of polymers stretched by an external force.

Unbinding of mutually avoiding random walks and two-dimensional quantum gravity

We analyze the unbinding transition for a two-dimensional lattice polymer in which the constituent strands are mutually avoiding random walks. At low temperatures the strands are bound and form a single self-avoiding walk. We show that unbinding in this model is a strong first order transition. The entropic exponents associated with denaturated loops and end-segment distributions show sharp differences at the transition point and in the high temperature phase. Their values can be deduced from some exact arguments relying on a conformal mapping of copolymer networks into a fluctuating geometry, i.e., in the presence of quantum gravity. An excellent agreement between analytical and numerical estimates is observed for all cases analyzed.

Statistical mechanics of base stacking and pairing in DNA melting

We propose a statistical mechanics model for DNA melting in which base stacking and pairing are explicitly introduced as distinct degrees of freedom. Unlike previous approaches, this model describes thermal denaturation of DNA secondary structure in the whole experimentally accessible temperature range. Base pairing is described through a zipper model, base stacking through an Ising model. We present experimental data on the unstacking transition, obtained exploiting the observation that at moderately low pH this transition is moved down to experimentally accessible temperatures. These measurements confirm that the Ising model approach is indeed a good description of base stacking. On the other hand, comparison with the experiments points to the limitations of the simple zipper model description of base pairing.

Bubble nucleation and cooperativity in DNA melting

Bubbles in DNA are related to fundamental processes such as duplication and transcription. Using a new ensemble technique to trap intermediate states, we present direct measurements of the average length of the denaturation bubble and the statistical weights of the bubble states in the temperature-driven melting of DNA oligomers. For a bubble flanked by double-stranded regions, we find a nucleation size of approximately 20 bases, and a broad distribution of bubble sizes. However, for bubbles opening at the ends of the molecule there is no nucleation threshold. The measured statistical weights of different conformations agree with the predictions of the thermodynamic models in the case of unzipping from the ends; however, internal bubble states are not completely described by the models. The measurements further show that, due to end effects, the melting transition becomes a two-state process only in the limit of a molecule length L approximately 1 bp.

Conformation of local denaturation in double-stranded DNA

Double-stranded DNA (dsDNA) undergoes a denaturing transition above which the strands unbind completely. At temperatures (including the physiological temperature) below the transition the base pairs tend to unbind locally, giving way to loops, i.e., locally denatured states. In the flexible-chain model, the imaginary time Schrödinger equation describes the interstrand distance distribution of dsDNA with the time variable replaced by the sequence number. We transform the equation to the Fokker-Planck equation (FPE), which provides a convenient and powerful analytical method and, via the equivalent Langevin equation, a simulation scheme. The temperature-dependent potential that emerges in the FPE manifests how the DNA conformation changes dramatically near the transition temperature. We present several simulation plots along with analytical results illustrating the order parameter (concentration of bound base pairs), base pair distance correlation function, and loop size distribution at different temperatures.

Thermal denaturation of a helicoidal DNA model

We study the static and dynamical properties of DNA in the vicinity of its melting transition, i.e., the separation of the two strands upon heating. The investigation is based on a simple mechanical model which includes the helicoidal geometry of the molecule and allows an exact numerical evaluation of its thermodynamical properties. Dynamical simulations of long-enough molecular segments allow the study of the structure factors and of the properties of the denaturated regions. Simulations of finite chains display the hallmarks of a first order transition for sufficiently long-ranged stacking forces although a study of the model's "universality class" strongly suggests the presence of an "underlying" continuous transition.

Long-range interaction and heterogeneity yield a different kind of critical phenomenon

DNA denaturation, wetting in two dimensions, depinning of a flux line, and other problems are known to map onto a phase transition with effective long-range interaction. In a disordered system the latter yields a giant, nonuniversal, temperature-dependent critical index, and macroscopic fluctuations at a finite distance from the critical temperature. In the vicinity of the critical region the Gibbs distribution is invalid, and thermodynamics must be calculated from first principles. There are no fluctuations above the critical temperature.

Reparametrizing the loop entropy weights: effect on DNA melting curves

Recent advances in the understanding of the melting behavior of double-stranded DNA with statistical mechanics methods lead to improved estimates of the weight factors for the dissociation events of the chains, in particular for interior loop melting. So far, in the modeling of DNA melting, the entropy of denaturated loops has been estimated from the number of configurations of a closed self-avoiding walk. It is well understood now that a loop embedded in a chain is characterized by a loop closure exponent c which is higher than that of an isolated loop. Here we report an analysis of DNA melting curves for sequences of a broad range of lengths (from 10 to 10(6) base pairs) calculated with a program based on the algorithms underlying MELTSIM. Using the embedded loop exponent we find that the cooperativity parameter is one order of magnitude bigger than current estimates. We argue that in the melting region the double helix persistence length is greatly reduced compared to its room temperature value, so that the use of the embedded loop closure exponent for real DNA sequences is justified.

Statistical mechanics of RNA folding: a lattice approach

We propose a lattice model for the secondary structure of RNA based on a self-interacting two-tolerant trail. Self-avoidance and pseudoknots are taken into account. We investigate a simple version of the model in which the native state of RNA consists of just one hairpin. Using exact arguments and Monte Carlo simulations we determine the phase diagram for this case. We show that the denaturation transition is first order and can either occur directly or through an intermediate molten phase.

RNA denaturation: excluded volume, pseudoknots, and transition scenarios

A lattice model of RNA denaturation which fully accounts for the excluded volume effects among nucleotides is proposed. A numerical study shows that interactions forming pseudoknots must be included in order to get a sharp continuous transition. Otherwise a smooth crossover occurs from the swollen linear polymer behavior to highly ramified, almost compact conformations with secondary structures. In the latter scenario, which is appropriate when these structures are much more stable than pseudoknot links, probability distributions for the lengths of both loops and main branches obey scaling with nonclassical exponents.

Length and statistical weight of bubbles in DNA melting

We present measurements of the length of the bubble versus temperature in the melting transition of DNA oligomers. The sequences are clamped at the ends of GC pairs (strong binding), and have AT rich (weaker binding) middle regions of variable length B relative to the molecule size. We use a quenching technique to trap intermediate states. We find that the average relative bubble size <l> grows from zero for increasing temperature, but reaches a plateau at a length of order B, producing a discontinuous transition with finite bubble size. The statistical weight of the bubble states decreases as B is reduced, the transition becoming more cooperative.