Twist-bend coupling, twist waves, and the shape of DNA loops.

By combining analytical and numerical calculations, we investigate the minimal-energy shape of short DNA loops of approximately 100 base pairs (bp). We show that in these loops the excess twist density oscillates as a response to an imposed bending stress, as recently found in DNA minicircles and observed in nucleosomal DNA. These twist oscillations, here referred to as twist waves, are due to the coupling between twist and bending deformations, which in turn originates from the asymmetry between DNA major and minor grooves. We introduce a simple analytical variational shape that reproduces the exact loop energy up to the fourth significant digit and is in very good agreement with shapes obtained from coarse-grained simulations. We, finally, analyze the loop dynamics at room temperature, and show that the twist waves are robust against thermal fluctuations. They perform a normal diffusive motion, whose origin is briefly discussed.

Effect of Memory and Active Forces on Transition Path Time Distributions.

An analytical expression is derived for the transition path time distribution for a one-dimensional particle crossing of a parabolic barrier. Two cases are analyzed: (i) a non-Markovian process described by a generalized Langevin equation with a power-law memory kernel and (ii) a Markovian process with a noise violating the fluctuation-dissipation theorem, modeling the stochastic dynamics generated by active forces. In case i, we show that the anomalous dynamics strongly affect the short time behavior of the distributions, but this happens only for very rare events not influencing the overall statistics. At long times the decay is always exponential, in disagreement with a recent study suggesting a stretched exponential decay. In case ii, the active forces do not substantially modify the short time behavior of the distribution but do lead to an overall decrease of the average transition path time. These findings offer some novel insights, useful for the analysis of experiments of transition path times in (bio)molecular systems.

Transition path time distributions.

Biomolecular folding, at least in simple systems, can be described as a two state transition in a free energy landscape with two deep wells separated by a high barrier. Transition paths are the short part of the trajectories that cross the barrier. Average transition path times and, recently, their full probability distribution have been measured for several biomolecular systems, e.g., in the folding of nucleic acids or proteins. Motivated by these experiments, we have calculated the full transition path time distribution for a single stochastic particle crossing a parabolic barrier, including inertial terms which were neglected in previous studies. These terms influence the short time scale dynamics of a stochastic system and can be of experimental relevance in view of the short duration of transition paths. We derive the full transition path time distribution as well as the average transition path times and discuss the similarities and differences with the high friction limit.

Non-Markovian dynamics of reaction coordinate in polymer folding.

We develop a theoretical description of the critical zipping dynamics of a self-folding polymer. We use tension propagation theory and the formalism of the generalized Langevin equation applied to a polymer that contains two complementary parts which can bind to each other. At the critical temperature, the (un)zipping is unbiased and the two strands open and close as a zipper. The number of broken base pairs n(t) displays a subdiffusive motion characterized by a variance growing as 〈Δn2(t)〉 ∼ tα with α < 1 at long times. Our theory provides an estimate of both the asymptotic anomalous exponent α and of the subleading correction term, which are both in excellent agreement with numerical simulations. The results indicate that the tension propagation theory captures the relevant features of the dynamics and shed some new insights on related polymer problems characterized by anomalous dynamical behavior.

Heterodimer Autorepression Loop: A Robust and Flexible Pulse-Generating Genetic Module.

We investigate the dynamics of the heterodimer autorepression loop (HAL), a small genetic module in which a protein A acts as an autorepressor and binds to a second protein B to form an AB dimer. For suitable values of the rate constants, the HAL produces pulses of A alternating with pulses of B. By means of analytical and numerical calculations, we show that the duration of A pulses is extremely robust against variation of the rate constants while the duration of the B pulses can be flexibly adjusted. The HAL is thus a minimal genetic module generating robust pulses with a tunable duration, an interesting property for cellular signaling.

Dynamic range extension of hybridization sensors.

In hybridization based nucleic acid sensors the stringency of hybridization poses a challenge to design and experiment. For a given set of experimental parameters the affinity window of probe-target interaction is always limited and vice versa for a given probe set design, changes in experimental conditions can easily bring some measurements out of detection range. In this paper we introduce and apply a strategy to extend this dynamic range for affinity sensors, sensors which measure the amount of hybridized molecules after equilibrium is reached. The method relies on concepts of additivity of nucleic acids hybridization free energies and on equilibrium isotherms. It consists in combining the measurements from probes with different lengths, by appropriately rescaling the measured signals. We test the validity of the approach on experiments and show that by combining probes with hybridizing regions of length 21, 23 and 25 nucleotides we manage to extend the dynamic range of the intensity signals by a factor of 25. The presented concept is easy to extend, platform free and applies to any hybridization based affinity sensor.

Anomalous dynamics of DNA hairpin folding.

By means of computer simulations of a coarse-grained DNA model we show that the DNA hairpin zippering dynamics is anomalous; i.e., the characteristic time τ scales nonlinearly with N, the hairpin length, τ ∼ N(α) with α>1. This is in sharp contrast to the prediction of the zipper model for which τ ∼ N. We show that the anomalous dynamics originates from an increase in the friction during zippering due to the tension built in the closing strands. From a simple polymer model we get α = 1+ν ≈ 1.59 with ν being the Flory exponent, a result which is in agreement with the simulations. We discuss transition path times data where such effects should be detected.

Generation of oscillating gene regulatory network motifs.

Using an improved version of an evolutionary algorithm originally proposed by François and Hakim [Proc. Natl. Acad. Sci. USA 101, 580 (2004)], we generated small gene regulatory networks in which the concentration of a target protein oscillates in time. These networks may serve as candidates for oscillatory modules to be found in larger regulatory networks and protein interaction networks. The algorithm was run for 10(5) times to produce a large set of oscillating modules, which were systematically classified and analyzed. The robustness of the oscillations against variations of the kinetic rates was also determined, to filter out the least robust cases. Furthermore, we show that the set of evolved networks can serve as a database of models whose behavior can be compared to experimentally observed oscillations. The algorithm found three smallest (core) oscillators in which nonlinearities and number of components are minimal. Two of those are two-gene modules: the mixed feedback loop, already discussed in the literature, and an autorepressed gene coupled with a heterodimer. The third one is a single gene module which is competitively regulated by a monomer and a dimer. The evolutionary algorithm also generated larger oscillating networks, which are in part extensions of the three core modules and in part genuinely new modules. The latter includes oscillators which do not rely on feedback induced by transcription factors, but are purely of post-transcriptional type. Analysis of post-transcriptional mechanisms of oscillation may provide useful information for circadian clock research, as recent experiments showed that circadian rhythms are maintained even in the absence of transcription.

Unwinding relaxation dynamics of polymers.

The relaxation dynamics of a polymer wound around a fixed obstacle constitutes a fundamental instance of polymer with twist and torque, and it is also of relevance for DNA denaturation dynamics. We investigate it by simulations and Langevin equation analysis. The latter predicts a relaxation time scaling as a power of the polymer length times a logarithmic correction related to the equilibrium fluctuations of the winding angle. The numerical data support this result and show that at short times the winding angle decreases as a power law. This is also in agreement with the Langevin equation provided a winding-dependent friction is used, suggesting that such reduced description of the system captures the basic features of the problem.

Probing hybridization parameters from microarray experiments: nearest-neighbor model and beyond.

In this article, it is shown how optimized and dedicated microarray experiments can be used to study the thermodynamics of DNA hybridization for a large number of different conformations in a highly parallel fashion. In particular, free energy penalties for mismatches are obtained in two independent ways and are shown to be correlated with values from melting experiments in solution reported in the literature. The additivity principle, which is at the basis of the nearest-neighbor model, and according to which the penalty for two isolated mismatches is equal to the sum of the independent penalties, is thoroughly tested. Additivity is shown to break down for a mismatch distance below 5 nt. The behavior of mismatches in the vicinity of the helix edges, and the behavior of tandem mismatches are also investigated. Finally, some thermodynamic outlying sequences are observed and highlighted. These sequences contain combinations of GA mismatches. The analysis of the microarray data reported in this article provides new insights on the DNA hybridization parameters and can help to increase the accuracy of hybridization-based technologies.

Fractional Brownian motion and the critical dynamics of zipping polymers.

We consider two complementary polymer strands of length L attached by a common-end monomer. The two strands bind through complementary monomers and at low temperatures form a double-stranded conformation (zipping), while at high temperature they dissociate (unzipping). This is a simple model of DNA (or RNA) hairpin formation. Here we investigate the dynamics of the strands at the equilibrium critical temperature T=T(c) using Monte Carlo Rouse dynamics. We find that the dynamics is anomalous, with a characteristic time scaling as τ∼L(2.26(2)), exceeding the Rouse time ∼L(2.18). We investigate the probability distribution function, velocity autocorrelation function, survival probability, and boundary behavior of the underlying stochastic process. These quantities scale as expected from a fractional Brownian motion with a Hurst exponent H=0.44(1). We discuss similarities to and differences from unbiased polymer translocation.

Computing equilibrium concentrations for large heterodimerization networks.

We consider a chemical reaction network governed by mass action kinetics and composed of N different species which can reversibly form heterodimers. A fast iterative algorithm is introduced to compute te equilibrium concentrations of such networks. We show that the convergence is guaranteed by the Banach fixed point theorem. As a practical example of relevance for a quantitative analysis of microarray data, we consider a reaction network formed by N~10(6) mutually hybridizing different mRNA sequences. We show that, despite the large number of species involved, the convergence to equilibrium is very rapid for most species. The origin of slow convergence for some specific subnetworks is discussed. This provides some insights for improving the performance of the algorithm.

Nonequilibrium effects in DNA microarrays: a multiplatform study.

It has recently been shown that in some DNA microarrays the time needed to reach thermal equilibrium may largely exceed the typical experimental time, which is about 15 h in standard protocols (Hooyberghs et al. Phys. Rev. E2010, 81, 012901). In this paper we discuss how this breakdown of thermodynamic equilibrium could be detected in microarray experiments without resorting to real time hybridization data, which are difficult to implement in standard experimental conditions. The method is based on the analysis of the distribution of fluorescence intensities I from different spots for probes carrying base mismatches. In thermal equilibrium and at sufficiently low concentrations, log I is expected to be linearly related to the hybridization free energy ΔG with a slope equal to 1/RT(exp), where T(exp) is the experimental temperature and R is the gas constant. The breakdown of equilibrium results in the deviation from this law. A model for hybridization kinetics explaining the observed experimental behavior is discussed, the so-called 3-state model. It predicts that deviations from equilibrium yield a proportionality of log I to ΔG/RT(eff). Here, T(eff) is an "effective" temperature, higher than the experimental one. This behavior is indeed observed in some experiments on Agilent arrays [Hooyberghs et al. Phys. Rev. E2010, 81, 012901 and Hooyberghs et al. Nucleic Acids Res. 2009, 37, e53]. We analyze experimental data from two other microarray platforms and discuss, on the basis of the results, the attainment of equilibrium in these cases. Interestingly, the same 3-state model predicts a (dynamical) saturation of the signal at values below the expected one at equilibrium.

Unwinding dynamics of double-stranded polymers.

We consider the unwinding of two lattice polymer strands of length N that are initially wound around each other in a double-helical conformation and evolve through Rouse dynamics. The problem relates to quickly bringing a double-stranded polymer well above its melting temperature, i.e., the binding interactions between the strands are neglected, and the strands separate from each other as it is entropically favorable for them to do so. The strands unwind by rotating around each other until they separate. We find that the process proceeds from the ends inward; intermediate conformations can be characterized by a tightly wound inner part, from which loose strands are sticking out, with length l∼t(0.39). The total time needed for the two strands to unwind scales as a power of N as τ(u)∼N(2.57±0.03). We present a theoretical argument, which suggests that during this unwinding process, these loose strands are far out of equilibrium.

Elastic lattice polymers.

We study a model of "elastic" lattice polymer in which a fixed number of monomers m is hosted by a self-avoiding walk with fluctuating length l . We show that the stored length density ρm≡1-/m scales asymptotically for large m as ρm=ρ∞(1-θ/m+…) , where θ is the polymer entropic exponent, so that θ can be determined from the analysis of ρm. We perform simulations for elastic lattice polymer loops with various sizes and knots, in which we measure ρm. The resulting estimates support the hypothesis that the exponent θ is determined only by the number of prime knots and not by their type. However, if knots are present, we observe strong corrections to scaling, which help to understand how an entropic competition between knots is affected by the finite length of the chain.

Breakdown of thermodynamic equilibrium for DNA hybridization in microarrays.

Test experiments of hybridization in DNA microarrays show systematic deviations from the equilibrium isotherms. We argue that these deviations are due to the presence of a partially hybridized long-lived state, which we include in a kinetic model. Experiments confirm the model predictions for the intensity vs free-energy behavior. The existence of slow relaxation phenomena has important consequences for the specificity of microarrays as devices for the detection of a target sequence from a complex mixture of nucleic acids.

The effects of mismatches on hybridization in DNA microarrays: determination of nearest neighbor parameters.

Quantifying interactions in DNA microarrays is of central importance for a better understanding of their functioning. Hybridization thermodynamics for nucleic acid strands in aqueous solution can be described by the so-called nearest neighbor model, which estimates the hybridization free energy of a given sequence as a sum of dinucleotide terms. Compared with its solution counterparts, hybridization in DNA microarrays may be hindered due to the presence of a solid surface and of a high density of DNA strands. We present here a study aimed at the determination of hybridization free energies in DNA microarrays. Experiments are performed on custom Agilent slides. The solution contains a single oligonucleotide. The microarray contains spots with a perfect matching (PM) complementary sequence and other spots with one or two mismatches (MM) : in total 1006 different probe spots, each replicated 15 times per microarray. The free energy parameters are directly fitted from microarray data. The experiments demonstrate a clear correlation between hybridization free energies in the microarray and in solution. The experiments are fully consistent with the Langmuir model at low intensities, but show a clear deviation at intermediate (non-saturating) intensities. These results provide new interesting insights for the quantification of molecular interactions in DNA microarrays.

On the relationship between perfect matches and mismatches in Affymetrix Genechips.

A relationship which links the fluorescence intensity histograms for perfect match (PM) and mismatch (MM) probes in Affymetrix Genechips is derived using inputs from physical-chemistry as the Langmuir and the Nearest Neighbor models. This relationship is in good agreement with experimental data from few dozens of chips belonging to about 10 different organisms. Principles of physical-chemistry impose some constant value for the average ratios of PM and MM intensities. Experimental data, however, show quite some variations in these parameters, although they follow the same inequalities as expected from hybridization free energies for oligonucleotides melting in solution. It is suggested that the anomalous experiment to experiment differences are due to 1) problems with chip design and 2) excessive fragmentation of the target in solution. The histogram analysis developed may be a useful preprocessing step to evaluate the global quality of the experimental data, prior to the calculation of the gene expression level.

Modeling background intensity in DNA microarrays.

DNA microarrays are devices that are able, in principle, to detect and quantify the presence of specific nucleic acid sequences in complex biological mixtures. The measurement consists in detecting fluorescence signals from several spots on the microarray surface onto which different probe sequences are grafted. One of the problems of the data analysis is that the signal contains a noisy background component due to nonspecific binding. We present a physical model for background estimation in Affymetrix Genechips. It combines two different approaches. The first is based on the sequence composition, specifically its sequence-dependent hybridization affinity. The second is based on the strong correlation of intensities from locations which are the physical neighbors of a specific spot on the chip. Both effects are incorporated in a background estimator which contains 24 free parameters, fixed by minimization on a training data set. In all data analyzed the sequence-specific parameters, obtained by minimization, are found to strongly correlate with empirically determined stacking free energies for RNA-DNA hybridization in solution. Moreover, there is an overall agreement with experimental background data and we show that the physics-based model that we propose performs on average better than purely statistical approaches for background calculations. The model thus provides an interesting alternative method for background subtraction schemes in Affymetrix Genechips.

Stability domains of actin genes and genomic evolution.

In eukaryotic genes, the protein coding sequence is split into several fragments, the exons, separated by noncoding DNA stretches, the introns. Prokaryotes do not have introns in their genomes. We report calculations of the stability domains of actin genes for various organisms in the animal, plant, and fungi kingdoms. Actin genes have been chosen because they have been highly conserved during evolution. In these genes, all introns were removed so as to mimic ancient genes at the time of the early eukaryotic development, i.e., before intron insertion. Common stability boundaries are found in evolutionarily distant organisms, which implies that these boundaries date from the early origin of eukaryotes. In general, the boundaries correspond with intron positions in the actins of vertebrates and other animals, but not much for plants and fungi. The sharpest boundary is found in a locus where fungi, algae, and animals have introns in positions separated by one nucleotide only, which identifies a hot spot for insertion. These results suggest that some introns may have been incorporated into the genomes through a thermodynamically driven mechanism, in agreement with previous observations on human genes. They also suggest a different mechanism for intron insertion in plants and animals.