Statistical physics and mesoscopic modeling to interpret tethered particle motion experiments
SStatistical physics and mesoscopic modeling to interpret tethered particle motionexperiments
Manoel Manghi, ∗ Nicolas Destainville, † and Anna¨el Brunet ‡ Laboratoire de Physique Th´eorique (IRSAMC), Universit´e de Toulouse, CNRS, UPS, France Department of Molecular Medicine, Institute of Basic Medical Sciences,Faculty of Medicine, University of Oslo, 0317 Oslo, Norway (Dated: September 5, 2019)Tethered particle motion experiments are versatile single-molecule techniques enabling one toaddress in vitro the molecular properties of DNA and its interactions with various partners involvedin genetic regulations. These techniques provide raw data such as the tracked particle amplitude ofmovement, from which relevant information about DNA conformations or states must be recovered.Solving this inverse problem appeals to specific theoretical tools that have been designed in thetwo last decades, together with the data pre-processing procedures that ought to be implementedto avoid biases inherent to these experimental techniques. These statistical tools and models arereviewed in this paper.
I. INTRODUCTION
The main advantage of single-molecule techniques over traditional bulk experiments is the possibility to disentanglesample heterogeneity and to gain insight into subpopulation properties. The tethered particle motion (TPM[104])single-molecule technique has been developed in the early 1990’s [1, 2] to detect and quantify conformational changesof biopolymers induced by their interaction with other molecular partners or changes in their environment [3–5]. Itconsists in tracking the Brownian motion of a nano-particle (tens to hundreds of nanometers in diameter) attached toa glass surface by a biopolymer such as a DNA molecule and measuring the particle amplitude of movement and itschanges when experimental conditions are modified. TPM experiments do not require expensive experimental set-ups,in particular because the particle is tracked by an optical microscope. This explains why numbers of experimentalgroups adopt this technique to investigate the effects of agents (e.g., enzymes, drugs, ions, or more generally p H, ionicstrength or temperature) acting on the characteristics of the tethering polymer, such as its persistence length, itsconformation, or its denaturation properties. A variant of TPM is Tethered fluorophore motion (TFM) [6, 7]. It usesthe same principles as TPM but employs a fluorophore (and sometimes two [8]) in place of the particle. TFM canthus be combined with fluorescence techniques such as F¨orster resonance energy transfer. However, TFM is limitedin observation time because of fluorophore photobleaching [7].Optical and magnetic tweezers [9, 10] are another class of powerful tools to investigate the elastic properties ofDNA molecules. Optical tweezers [11, 12] rely on a focused laser beam to provide an attractive force on the orderof the pico-Newton (pN) to manipulate micrometric particles. Magnetic tweezers [13, 14] consist of two permanentmagnets producing a horizontal magnetic field at the location of a magnetic particle. In both cases, as in TPM, theparticle is attached to a biopolymer tether, itself grafted to the glass surface. Magnetic torque tweezers (MTT) are anextension of conventional magnetic tweezers where a cylindrical magnet creates a vertical magnetic field and permitsto apply both forces and torques. At zero turn the particle is at its rotational equilibrium position and the tetheredDNA is torsionally relaxed. After applying turns the DNA molecule is twisted, which gives access to the torsionalelastic properties of DNA and also its non-linear response when twist is converted into writhe through the creationof superhelical DNA regions [15].As compared to optical or magnetic tweezers, no external force is applied to the particle in TPM [4], if not theweak repulsion exerted by the glass surface on the polymer and the bead, in the tens of fN range [16]. Studying thebiopolymer in quasi-force-free conditions enables one to tackle its equilibrium properties, as well as reaction ratesbetween two (or more) states such a assembly/disassembly rates of DNA constructs or binding/unbinding rates ofenzymes on a tensionless DNA [17]. These different techniques are thus complementary.Recent improvements of TPM rely on multiplexing, hundreds of DNA-bead complexes being positioned in a con-trolled manner by soft nano-lithography and monitored in parallel. The ensuing technique is called high-throughput ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] a r X i v : . [ phy s i c s . b i o - ph ] S e p TPM (htTPM) [18]. By drastically reducing acquisition time as compared to anterior setups where the moleculeswhere observed one by one, multiplexing gives access to highly refined statistics allowing one to distinguish betweenclosely related conformations [19–23]. Dealing with these refined statistics justifies the development of improvedstatistical tools and modeling to interpret them, as detailed in this Review.From a biological perspective, single-molecule techniques have enabled many research groups to decipher key mech-anisms at play in cells, starting with the pioneering paper by Schafer and coworkers in 1991, where the progressiveshortening of the tether gave access to the processivity of immobilized RNA polymerases [1]. The estimated value,even though rather rough at this time, was of a dozen of bps/s (see also Ref. [2]), in satisfying agreement with mea-surements in solution. Later, using a similar strategy, the RuvAB-directed branch migration of individual Hollidayjunctions [24] was measured, also on the order of 10 to 20 bps/s, as well as its dependence on the construct sequence.Several other examples of biological applications can be found in previous review articles [5, 25] or will be discussedin the present Review. The interpretation of these experiments not only depends on accurate measurements but alsoon adequate and reliable physical models able to account for sometimes very weak and subtle effects. It allows one toidentify the relative roles of the intrinsic parameters of the system. The bead size, the tether length, the surface stateof the substrate and the solvent are all likely to play pivotal roles in this context, as well as the acquisition rate ofthe camera used to track the particle. Their different contributions must be precisely quantified, as discussed below.This Review article can be seen as a companion article to the one by Jean-Fran¸cois Allemand, Catherine Tardin andLaurence Salom´e in the same issue [23]. It gives additional details about physical, mathematical and algorithmicissues related to TPM and the ways to tackle them. Even though they are not the main focus of this Review, opticaland magnetic tweezers or AFM often come as complementary tools to study single-DNA molecules under force and/ortorque. They also need theoretical and algorithmic tools for the interpretation and modeling of experiments. Thelast decade has witnessed their rapid development, and the reader can for example refer to Refs. [26–33] for furtherdetail. When the connection with TPM experiments is meaningful, we shall discuss some works dealing with forceand torque experiments in the present Review.
II. MODELING SINGLE DNA-MOLECULE EXPERIMENTS AND THEIR DYNAMICS
Several coarse-grained models have been developed in the past decades to model a single DNA molecule. Thesemodels are either numerical and/or analytical, the simplest one being the Gaussian chain, for which the end-to-enddistance is R ee ≡ (cid:112) (cid:104) R (cid:105) (cid:39) ( bL ) / , a valid expression[105] as soon as the DNA contour length L is much largerthan the Kuhn length b = 2 (cid:96) p . The DNA persistence length is (cid:96) p = κ/ ( k B T ) where κ is the bending modulus and isapproximately 50 nm for double-stranded DNA in physiological conditions [19]. The associated numerical model is thebead-spring model which is easy to implement. It consists in modeling the DNA by N beads, whose diameter is equalto the Kuhn length, connected by springs, in Brownian dynamics or Monte Carlo simulations. Although this model iscentral to understand polymer properties at large scales [34], it is not adapted to single-DNA molecule experiments,which are interested in DNA lengths from few hundreds to few thousand base-pairs, on the order of the Kuhn length.Moreover the bead has a radius much larger than the dsDNA radius (cid:39) R ee (cid:39) ( v/b ) / L / (for L (cid:29) b ) where the excluded volume v depends on the salt concentration.The most adapted model is the worm-like (or semi-flexible) chain model because it reveals the mechanical andstatistical properties that are probed in single-molecule experiments, without describing the DNA structure in detail.This model considers the chain as a homogeneous stiff rod, the bending energy of which leads to a short-rangedtangent-tangent correlation, (cid:104) t ( s ) · t (0) (cid:105) = e − s/(cid:96) p where s is the curvilinear index along the chain and t ( s ) thenormalised tangent vector. The end-to-end distance is then given by the Kratky-Porod result [35] R (cid:39) (cid:96) p (cid:18) L(cid:96) p − e − L/(cid:96) p (cid:19) (1)which yields the two good limits of the rigid rod, R (cid:39) L , when L (cid:28) (cid:96) p and the Gaussian chain, R (cid:39) L(cid:96) p , when L (cid:29) (cid:96) p . The associated discretised numerical model is a bead-spring model with a large spring stiffness to enforcethe chain connectivity and an additional bending energy E b = κ (cid:80) N − i =1 (1 − cos θ i ) where κ = (cid:96) p k B T is the bendingmodulus and θ i the angle between two consecutive links. In this discrete worm-like chain model, the successive beadsare free to rotate with respect to each other (which is equivalent to set the torsional modulus C to 0). Note that inthe torque experiments, torsion must also be taken into account: the twist angle φ i is defined between two consecutivebase-pairs (or by defining a material frame for each bead) and the associated elastic energy of a torsional spring writes,in the limit of large C (valid for DNA), E t = C (cid:80) N − i =1 ( φ i − φ ) , where φ = 0 .
62 rad is the equilibrium DNA twistin physiological conditions).Depending on the experimental setup, possible boundary conditions and/or interactions between the DNA andexternal objects might be considered. In TPM, one DNA end is tethered to a substrate, which is usually modeledas a freely rotative joint. The other DNA end is attached to a spherical particle. The DNA-particle link is alsotreated as freely rotative joint, except in torsion experiments. The glass coverslip is treated as a hard wall boundarycondition limiting the motion of the DNA and the particle to the upper half plane. Although easy to implementnumerically, the hard wall condition and the large particle size modify the equilibrium statistics of the DNA in TPMexperiments. The amplitude of movement of the particle is defined as σ = (cid:113) (cid:104) r (cid:107) (cid:105) where r (cid:107) is the two-dimensionalparticle position parallel to the coverslip. Note that without loss of generality, we have set here (cid:104) r (cid:107) ( s ) (cid:105) = 0. Theamplitude of movement σ can be estimated analytically only in the limit of flexible DNA ( L (cid:29) (cid:96) p ) [16, 17, 36], andsome interpolation formulas have been proposed in the semi-flexible regime [37].Concerning the dynamical properties of the DNA, the relaxation time (see next Section) is also modified by theexperimental setup. In particular, the no-slip boundary condition enforced by the presence of the coverslip slows downthe DNA–particle dynamics. Hence in the numerical simulations, the hydrodynamics interactions induced by the wallare encoded using Fax´en’s law prescribing how the diffusion coefficient of both the DNA molecule and the particle isreduced close to the wall, in order to satisfy the no-slip condition for the solvent velocity field at the wall [17]. RO ρϕ x yz FIG. 1: A TPM numerical model: the DNA molecule is modeled as a polymer chain made of N connected beads (variouscolours), anchored to the coverglass (in blue) at one extremity and to the tracked particle (in red) at the other end. The 2Dposition r (cid:107) of the particle center is represented by the polar coordinates ρ and ϕ in the ( xOy ) plane. III. DEALING WITH STATISTICAL AND SYSTEMATIC ERRORS
We first introduce the different experimental times of interest, here and in the sequel. The camera acquisitionperiod is denoted by T ac . It will play an important role when dealing with the blurring effect below. It typicallyranges from few ms for fast acquisition devices [38] to few tens of ms at video rate. The camera exposure time is T ex ≤ T ac , whilst in general T ex = T ac . It must be shorter than the characteristic physical times of interest in theexperiment in order to have access to all relevant events. As for the trajectory duration, it must be long as possible inorder to reduce statistical uncertainties. Since the tethered particle is not subject to photobleaching, the trajectorycan be recorded for several minutes in TPM, which is a great advantage in terms signal-to-noise ratio as compared tosingle-fluorophore techniques.One of the objectives here is to measure the useful correlation (or relaxation) time τ of the 2D tethered-particleposition r (cid:107) . It sets the typical time needed for the particle to explore its configuration space. It is defined throughthe auto-correlation function C ( t ) = (cid:104) r (cid:107) ( s + t ) · r (cid:107) ( s ) (cid:105) − (cid:104) r (cid:107) ( s ) (cid:105) (2)where the average (cid:104) . . . (cid:105) is taken along the trajectory. For sake of simplicity, we assume that the time correlationfunction has the form C ( t ) = C (0) e − t/τ m . This expression is exact only in the case of a quadratic confining potential.In the case of a more complicated confining potential induced by the polymer tether, it is only an approximationbecause the auto-correlation function is a sum of decreasing exponentials and τ m is then associated with the slowestdiffusion mode, which dominates at long times [35]. In practice, we shall see below that the measured correlation time τ m is slightly larger than the real one τ because of blurring effets that we aim at quantifying in the present section. A. Cleaning data from spurious points
A first data preprocessing step is essential to minimize the contributions of unwanted biases and experimentalvariability. First of all, it is essential to deal with the measurement heterogeneity induced by undesirable artefactsdue, e.g., to ill-assembled objects. For all single molecule approaches involving anchored DNA molecules, the substratesurface state is crucial [4]. This becomes even more critical when studying dynamically acting proteins or moleculesbinding on the DNA, and especially for AFM [39] or tweezer [40] manipulations. Such approaches require that theDNA molecules binds to substrate surface without modifying physiological functions and properties, and that onlydesired anchoring is realized, without any sort of spurious secondary attachments [17].When advanced experimental protocols are not sufficient to prevent those artefacts, statistical tools offer an alter-native way to do so during the data preprocessing step. In TPM experiments, malformed DNA-particle complexes,e.g. with two grafted DNA molecules instead of an expected unique one would interfere with main population ofinterest and hamper the authentic experimental noise. Applying a selecting filter based on the asymmetry factorof the 2D trajectories allows one to select only well-defined tethered DNA/particle complexes as mentioned in thearticle by Allemand et al. in this issue [23]. An asymmetry factor (or aspect ratio) larger than 1.35 is assumed to beassociated with two DNA tethers cross-linked to the same tracked particle [19].However, this criterion may appear to be insufficient to deal with spurious non-specific binding of some particlesto the coverslip. One must get rid of trajectories laying in the far tails of the amplitude of movement distribution.Hence, if a single population is expected, for example when extracting a bending angle or the persistence length (seeSection IV C below), trajectories with amplitude of movement outside the interval (mean ± . outliers . The correlation time for each 2D TPM-trajectories, τ , is determined as described above. The resulting data for the global DNA population is expected tobehave as a single population for a given DNA state. Based on that, and as above for amplitude of movements, atrajectory is declared as an outlier if its τ lays in the far tails of this distribution and is then sorted out. In practice,sorting out points deviating by more than 1 or 2 standard deviations from the average value is also a reasonablecriterion [20]. Based on the same principle for filtering experimental data, Schickinger et al. [8] exploit the averagedwell-time. It is determined in each specific state, unbound and bound DNA, for each particle. Comparing boundvs. unbound dwell times reveals multiple data point clusters and provided a criterion of selection. Considerablediscrepancy with the main populations leads to the discrimination of outliers.More generally, all single-molecule approaches come with their own specific protocol-induced defects, which needto be taken into account in order to clean data from outliers before confronting them to statistical analysis andtheoretical modeling. This is even more true when using high-throughput approaches that provide high samplinglevels and allow to define precisely the main conformation populations. B. Subtracting instrumental drift
Once data have been cleaned from outliers, the first systematic error to correct comes from the instrumental drift,due, inter alia , to thermal expansion of the observation setup [17]. Along a given trajectory, the DNA approximateanchoring point at each time t is determined by averaging the particle position over an interval of duration T av (typically 1 or 2 s, sometimes even more [41]) centered at t , and then subtracted from r (cid:107) . If T av is chosen to bemuch larger than the measured relaxation time τ m , this anchoring point is determined with a good accuracy. Thissets (cid:104) r (cid:107) (cid:105) (cid:39)
0, as desired.Note that subtraction of drift induces non-trivial systematic anti-correlations at short times t (cid:28) T av . Indeed, letus assume that the measured time correlation function in absence of drift also has the form C m ( t ) = C m (0) e − t/τ m .Then subtraction of drift modifies it to C m ( t ) C m (0) = (cid:18) τ m T av (cid:19) e − t/τ m − τ m T av (3)at first order in 2 τ m /T av (cid:28) τ m below. C. Correcting blurring effect
Another important source of systematic error in TPM comes from the finiteness of the camera exposure time T ex .An image in fact represents the optical signal averaged over a time interval of duration T ex . In Refs. [17, 37, 42, 43],the ensuing time-averaging (or blurring) effect in single-particle tracking experiments was investigated. It occurswhenever the trajectory of the tracked particle is confined in a bounded domain, not only in TPM but also whentracking plasma membrane constituants, for instance. Indeed, in the extreme case where the frame exposure timewould be much larger than the system auto-correlation time, itself inversely proportional to the domain area (seebelow), the measured particle position during this period would remain very close to the anchoring point, giving theerroneous impression that the amplitude of motion is much smaller than its actual value. However, we shall see thatwhen it is not too strong, this effect can efficiently be corrected.By using Eq. (3), the measured correlation function C m ( t ) is first fitted to obtain the measured correlation time τ m ,the only free parameter in this expression (the value of T av has been chosen from the beginning). It can be proventhat the real correlation time τ is given by τ (cid:39) τ m − T ex τ m ≥ T ex / σ ≡ (cid:113) (cid:104) r (cid:107) (cid:105) , also called “amplitude of movement”,measured on a sufficiently long interval in order to accurately sample configurations [38] (see Section V A). If σ m isits measured value, then the real one is recovered in its turn from σ (cid:39) σ m (cid:34) τT ex − (cid:18) τT ex (cid:19) (cid:16) − e − T ex τ (cid:17)(cid:35) − / . (5)For example, if T ex = 2 τ then σ m (cid:39) . σ [42]. In the case where T ex (cid:28) τ , we naturally get σ (cid:39) σ m . Note thatthe particle diffusion coefficient D depends on both σ and τ , through D = Const . σ /τ , where Const . depends on thedomain geometry [44]. The measured value D m can be substantially different from the actual one D , e.g., D m (cid:39) . D if T ex = 2 τ . It is necessary to correct both σ and τ thanks to the above formulae to get the correct value of D .Incorrectly dealing with this blurring phenomenon can have dramatic effects when varying the experimental tem-perature T . Indeed, the water viscosity η w ( T ) decreases rapidly with increasing T . This leads to a 4-fold fall of τ ∝ D − ∝ η w ( T ) /T when T grows from 15 to 70 ◦ C [22]. Even though larger than T ex at low T , τ likely becomescomparable to or smaller than τ at high T , then requiring correction of the blurring phenomenon. This issue haspreviously led to erroneous conclusions about denaturation profiles of DNA as observed by TPM [22, 45] (see alsoSection IV C 4 below). D. Estimating error bars
The last step is to determine average values and associated error bars. The most straightforward way is to estimatethe single average of an observable O over the distribution of the set of N data point as its mean µ O ≡ (cid:104)O(cid:105) . Statisticalfluctuations are estimated through the the variance σ O ≡ (cid:104)O (cid:105) − (cid:104)O(cid:105) , and the standard deviation σ O . Assumingstatistical independence of samples, the error bar on µ O is then σ O / √ N (68% confidence interval) or twice this value(95% confidence).More advanced methods can be used as the jackknife , used in Refs. [46–48], or the bootstrap , used in Refs. [19–22, 49], to estimate the error bar. Both are resampling methods. In the jackknife, one considers N resampled setsof data, each containing all but one of the original data points. The bootstrap uses M sets of data, each containing N data points obtained by random sampling, performed by Monte Carlo, of the original set of N points. Duringthe Monte Carlo sampling with replacement, the probability that a data point is selected is 1 /N . The number ofbootstrap samplings, M , should be chosen to be large enough so that the average bootstrap sampling is reproduciblewith sufficient accuracy. The jackknife approach leads to identical results each time it is run on the same set of data,which is not true for bootstrap. IV. SOLVING THE INVERSE PROBLEM “Solving the inverse problem” can be generically stated as calculating, from a set of experimental measurements, thecauses that produced them. The aim is to gain insight into the physical properties of a system by indirect measurementsor conversely to set up a predictive model that can reproduce observations. In the present context, coupling theory andexperiments offers appropriate tools to probe the intrinsic physical properties of the DNA macromolecule: for instance,DNA persistence length or its local defects, from the apparent end-to-end distance of the polymer as accessible fromTPM.
A. Analytical approaches and their limits
The exact equilibrium distribution, p ( r (cid:107) ), of the amplitude of movement of the TPM particle can only be computedin the rigid and flexible limits [16, 17]. Using the mirror reflection argument, the probability distribution in theGaussian (flexible) regime and in the limit where the particle radius is large, R (cid:29) (cid:112) L(cid:96) p , is given by [17]: p G ( | r (cid:107) | ) ≈ (cid:115) πL(cid:96) p | r (cid:107) | (cid:113) r (cid:107) + R exp (cid:20) − L(cid:96) p (cid:16) r (cid:107) + 2 R − R (cid:113) r (cid:107) + R (cid:17)(cid:21) . (6)The intermediate semi-flexible regime, L (cid:39) (cid:96) p , of interest in TPM experiments, is well described by the worm-likechain model. For instance, in the experiments described in Ref. [17], DNA lengths vary between 400 and 2080 bp,which corresponds to 2 < L/(cid:96) p <
14. This model can be tackled analytically [50, 51] but the boundary conditionsmust be handled with care and the final step should be solved numerically, which does not provide any analyticalformula for the probability distribution with fitting parameters. Moreover, it should be kept in mind that real chainsare self-avoiding and that the presence of the labelling particle renders the problem even more intricate. For instance,the effect of the excluded volume of the particle is to widen and shift to large | r (cid:107) | the Gaussian distributions. Itthen becomes analytically intractable for finite chains, but it can be tackled numerically by Monte Carlo or Browniandynamics simulations (see Section IV B).The over-damped dynamics of the DNA in the flexible regime is controlled by the Rouse time τ (cid:107) = N R G /D where R G is the radius of gyration and D = k B T / (6 πη(cid:96) p ) is the diffusion coefficient of a monomer sphere of radius (cid:96) p in a liquid of viscosity η [17, 52]. However, when the TPM particle is grafted at one end, the relaxation timeand the diffusion coefficient measured by tracking the particle dynamics are features of the dynamics of the wholeDNA–particle complex. Using the Langevin equation, it is shown that the diffusion coefficient D c is given by: D c = D part D DNA D part + D DNA (7)where D part = k B T / (6 πηR ). Thus the particle does not slow down the complex provided that D part (cid:29) D DNA .Knowing D part , the value of D DNA is inferred from the measurement of D c . Note however that these analyticalconsiderations do not allow us to compare with TPM experiments due to (i) the fact that the chain is in the semiflexibleregime; and (ii) the neglect of hydrodynamics interactions in this approach. B. Numerical simulations
Several types of numerical simulations have been developed to circumvent the above limitations. To test both thedynamics and the equilibrium properties in the TPM setup geometry, the adequate numerical methods are Browniandynamics simulations and kinetic Monte Carlo simulations.In Brownian dynamics simulations, the evolution of each sphere position r i ( t ) is governed by an iterative Langevinequation (discrete time step δt and discrete time variable n = t/δt )˜ r i ( n + 1) = ˜ r i ( n ) − ˜ D ∇ ˜ r i ˜ U ( n ) + (cid:113) D ˜ ξ i ( n ) , (8)where the rescaled random displacement has variance unity (cid:104) ˜ ξ i ( n ) · ˜ ξ j ( m ) (cid:105) = 3 δ ij δ nm . The rescaled bare diffusioncoefficient ˜ D = D δt/a is the diffusion constant in an unbounded space in units of the particle radius a and time step δt . For sufficient numerical accuracy the usual choice is ˜ D = 10 − –10 − . The dimensionless potential ˜ U = U/ ( k B T )is the sum of stretching and bending potentials described in Section II. The excluded volume interaction is modeledby a repulsive, truncated Lennard-Jones potential ˜ U LJ = (cid:80) i
0, we use the reflection boundary condition: if a sphere intersects the substrate, its height z i is replaced by itsmirror image z refl i = 2 a − z i ( z refl N = 2( R − a ) − z N for the particle). e x p s i m L/R
FIG. 2:
Experimental ( τ exp (cid:107) , solid symbols) and numerical ( τ sim (cid:107) , using z -dependent diffusion coefficients, open symbols) relaxation timesfor different DNA lengths, L , and particle radius, R , in linear-log coordinates. Inset: Ratio τ exp (cid:107) /τ sim (cid:107) versus L/R with the same symbolsas above, in log-log coordinates. The dashed line shows the best linear regression, with slope − . In Monte Carlo simulations, at each step (MCStep) δt , a bead is chosen uniformly at random among the N + 1possible ones (monomer spheres and labelling particle). Then a random move δ r is attempted for this bead, uniformlyin a ball of center 0 and radius R b . One shows that in this case (cid:104) δ r (cid:105) = 3 R b /
5. This quantity must be equal to6 D δt , where D is the diffusion coefficient of the spherical bead, depending on its diameter, which sets R b = √ D δt .Interactions between adjacent beads are treated via the potential U , whereas interactions between non-adjacent beadsare of hard core nature, like surface-bead interactions: whenever a move would lead to the penetration of a bead intoan other one or the surface, it is rejected. The physical time is incremented of δt following each Monte Carlo Sweep(MCS, equal to a sequence of N + 1 MCSteps). A simulation snapshot is shown in Fig. 1 at equilibrium. Note thatthe choice of excluded volume interaction in Monte Carlo simulations saves computational time as compared to thecalculation of truncated Lennard-Jones potentials used in Brownian Dynamics.The advantage of these simulations is to give access to dynamical properties. From this viewpoint, BrownianDynamics and kinetic Monter Carlo simulations are equivalent in the small δt limit. As an example, experimentaland numerical values of the relaxation times τ (extracted from C ( t ) defined in Eq. (2)) are shown in Figure 2 [17].Experimental and numerical values are found in good agreement, with ratios of experimental to numerical valuesvarying from 0.5 to 2 (shown in the inset).If one is only interested in equilibrium properties of the conformation of the DNA–particle complex, i.e. theprobability distribution p ( | r (cid:107) | ) or its standard deviation σ for various apparent DNA contour lengths L , a fasternumerical method is to compute DNA–particle conformations by exact Monte Carlo sampling. This method hasbeen developed by Segall and collaborators [16, 37] and used in subsequent works [7, 19]. It consists in generatinglabeled DNA as a random walk of N steps with a bending energy E b (defined in Section II) by step. The angles θ i between successive links are randomly chosen through a probability distribution in agreement with the Boltzmannweight at equilibrium ∝ exp[ − E b / ( k B T )]. The starting point is the bead tethered to the substrate, and at eachstep, self-intersecting trajectories (resp. trajectories intersecting the substrate) are discarded to take into accountintra-chain excluded volume interactions (resp. repulsive interactions with the substrate). Then statistical averagesare computed. This numerical method made possible comparisons with approximate analytical expressions, and thefinding of very accurate interpolation functions for σ ( L ) [37]. Moreover, it has been used in Ref. [19] to obtain agraphical reference for σ in function of the dsDNA persistence length (cid:96) p and therefore to study quantitatively thevariation of (cid:96) p with the salt concentration in the solution (see Section IV C 3). C. Examples
Weak magnitude changes in DNA conformation can be difficult to detect by TPM. Coupling multiplexed single-molecule experiments, statistical physics and mesoscopic modeling allows one to detect and investigate narrow changesin the amplitude of movement σ ( t ). Local modifications along the DNA molecule, such as kink, protein binding, loopformation, or more global effects due to a change in the surrounding environment, such as viscosity, p H, ionic strengthor temperature, will impact the physical properties of the DNA molecule. Changes in the distribution of DNAmolecule conformations induce a transition in the apparent contour length, a variable directly accessible throughsingle-molecules techniques. We now review representative examples.
1. Intrinsic curvature angle in TPM
Local bending of the DNA double helix axis can be induced by either the binding of proteins [4, 40, 53] (Figure 1,right) or by specific sub-sequences. Specific sub-sequences, such as short A-tracts composed of a succession of adenineson the same strand, can locally change the bio-molecule mechanical properties, which is measured through small localbends. Different single-molecule techniques can give access to quantitative measurement of local bending, includingAFM, fluorescence spectroscopy, tweezers and TPM [40, 54].Joint theory and simulation establish the adapted formalism to explore the effect of a local bend in TPM experi-ments. Using the worm-like chain model to describe a local bending deformation, modeled as a kink of angle θ (cid:54) = 0located at distance (cid:96) from one end, the end-to-end distance R ee is given by a modified version of Eq. (1) [19], theso-called kinked worm-like chain model: R = 2 (cid:96) p (cid:20)(cid:18) L(cid:96) p − e − (cid:96)/(cid:96) p + e − ( L − (cid:96) ) /(cid:96) p (cid:19) + cos( θ ) (cid:16) − e − (cid:96)/(cid:96) p − e − ( L − (cid:96) ) /(cid:96) p + e − L/(cid:96) p (cid:17)(cid:21) (9)From a numerical perspective, simulated TPM is adapted to model locally bent DNA by incorporating the preferredangle θ between three successive beads into the bead chain. Only the inserted sub-sequence is expected to inducean intrinsic curvature, the remainder of the DNA sequence is assumed to be a random one, without any intrinsiccurvature. For 575-bp-long DNA molecule, an angle of θ = π would induce a decrease of the apparent DNA contourlength of ≈
30% (absolute reduction of ≈ ≈
15% (absolute reduction of ≈ /
2. DNA looping in TPM
DNA looping is a common phenomenon, useful to gene regulation in both prokaryotes and eukaryotes. Probingprotein binding and the induced loop formation ([25] and references therein) helps to resolve more precisely thegeometry of the DNA-protein complexes involved in many biological processes. TPM measurements can be used tounravel the structure of the loop. Moreover, by knowing the length of the loop and the positions of protein-bindingsites, the change in apparent end-to-end distance measured by TPM can be used to infer geometrical or conformationalalteration of the looped protein-DNA structure. Using statistical mechanics models based upon elastic interactionsin small DNA molecules, Biton et al. [55] and Johnson et al. [41] probed the interactions of DNA molecules withLac repressor proteins. They performed TPM measurements to extract the distribution and changes in the apparentlength of tethered DNA in function of the operators. Due to the symmetry of the two identical dimeric arms of theLac repressor, different operators can bind to each arm with different possible orientations, yielding distinct looptypes, either parallel or anti-parallel.Monte Carlo simulations were performed [55], where the DNA molecule was modeled as a necklace of rigid beadsseparated by rigid cylinders, which takes into account the fluctuations of the Lac repressor-mediated DNA loopedsegment. Then, the looping probabilities for a specific DNA loop configuration were calculated from the simulations.This information gave access to the probability of a loop of a particular length and a given topology. Comparingcomputational results to TPM experimental results enabled the authors to identify the looped state topology. Inaddition, simulations were also used to estimate the dissociation constants associated with the binding of the Lacrepressor to each of the operators. This numerical study [55] suggests different looping probabilities for anti-paralleland parallel loop types formed along a 1632 bp-long DNA molecule, to which a particle of radius 160 nm is attached.The looping probabilities for the anti-parallel loop between binding site centers located at base-pairs 444 and 1044(resp. 1344) are 4-fold (resp. 6-fold) higher than those for the parallel loop. It also suggests that the anti-parallelloop is entropically favored.
3. Effects of salt on elastic properties
The amplitude of movement σ not only depends on the physical properties of the DNA-particle system, but alsoon the physicochemical properties of the surrounding solution.The stiffness properties of nucleic acids molecules depend on the solution ionic strength, defined as I = (cid:80) i z i c i where z i and c i are respectively the valency and the concentration of ion i , that induces screening of the electrostaticrepulsion between the negatively charged phosphate groups along the sugar-phosphate backbones. Single-moleculetechniques permit to characterize changes of the polyelectrolyte mechanical properties induced by changes in ionicstrength as pioneered by Lambert et al. [56]. Studies based on high-throughput TPM investigated the dependence ofthe persistence length on salt concentration for monovalent (Li + , Na + , K + ) and divalent (Mg , Ca ) metallic valentions [19, 57]. In the first study [19], after correcting the blurring effect using Eq. (4), the end-to-end distance of thetethered 1201 and 2060 bp-long dsDNA was observed to decrease as a function of the ionic strength I (ranging from10 mM to 3 M), as expected. A TPM coarse-grained model, taking into account excluded volume interactions, wasused to extract (cid:96) p from measurements by resolving the inverse problem (described in Section IV B). It was observedthat (cid:96) p varies from 30 to 55 nm over the large range of ionic conditions, comparable to previous experimental results.Two main models were used to fit the data: the Odijk-Skolnick-Fixman model valid at large I , relying on a meanfield approach valid at low values of the electrostatic potential [58, 59]; the Odijk-Skolnick-Fixman-Manning modelwhere the above approach was corrected at low I accounting for the Manning condensation of a few counterionsthat decreases the effective charge along the DNA [60]. These models could not account quantitatively for the wholeexperimental data set obtained with Na + or Mg . The more recent Manning model [61] with internal electrostaticstretching force due to the repulsion of the charges along the polyelectrolyte well fitted the entire range of I for theNa + case only.The second experimental study [57] refined the physical understanding of the phenomenon by exploring the roleplayed by ion size and a larger range of I (from 0.5 mM to 6 M). The experimental protocol was also improvedbecause p H was observed to decrease significantly in phosphate buffer when ions were added. A 4-(2- hydroxyethyl)-1-piperazineethanesulfonic acid (HEPES) buffer was used instead and a slower decrease of the apparent length of thedNA molecule was now observed as comparer to the previous study. Then, the extracted persistence length of theDNA could be quantitatively described by more sophisticated theoretical approaches, the Netz-Orland [19, 62] fordivalent ions and Shen-Trizac [63] for monovalent ones. These theories, by including non-linear electrostatic effectsand the finite DNA radius, can account for the observed behavior of (cid:96) p over the whole I range. Interestingly, themetallic ion size does not influence the persistence length in contrast to alkyl ammonium monovalent ions at high I [57].A recent work also probed the salt dependence of the torsional stiffness of DNA by multiplexed MTT [64]. Fewdifferent Na + monovalent concentrations, 20, 100 and 500 mM, and a combination of 100 mM of Na + and 10 mM ofMg were tested. The extension-rotation and torque-rotation curves were collected. The effective torsional stiffness, C eff , was determined by fitting the linear torque-rotation regime for each of the ionic strength conditions over variousstretching forces. At high stretching forces ( f > C eff increaseswhen the ionic strength increases. Coupling MTT measurement and simulation of the twistable worm-like chainmodel, permits to examine in more detail the torsional persistence length of the DNA molecule [65]. The discrepancybetween experimental and numerical measurements underlay that bending and twisting are intrinsically coupled inDNA molecule because of the difference between the major and minor groves. The bending elasticity is not isotropicanymore. This effect can be implemented in the alternative twistable worm-like chain elastic model proposed byMarko and Siggia [15], where twist-bend coupling is fully taken into account. The systematic deviations of the twistresponse of dsDNA investigated by magnetic tweezers experiments with the numerical model reported in previousstudies could be explained by taking into account this direct coupling between twist and bend deformations.0
4. Effects of temperature – DNA denaturation
Due to base-pairing and stacking energies on the order of the thermal energy k B T , DNA flexibility is stronglydependent not only on the ionic strength, but also on the temperature T . It affects the cohesive interactions betweenthe DNA bases as well as the contribution of the chain configurational entropy in the free energy [66]. From abiological perspective, various species live in extreme environments and are subjected either to high temperatures orto large temperature fluctuations. This emphasizes the importance of knowing how DNA structure, properties andprotein-DNA interactions are affected by temperature. To this purpose temperature-controlled TPM studies wereperformed during the last decade [22, 45]. This technique allowed one to explore the temperature-dependence of theapparent DNA persistence length (cid:96) p .In the measured temperature ranges, from 23 to 52 ◦ C in Ref. [45] and from 15 to 75 ◦ C in Ref. [22], a correlationbetween the increase of T and the decrease of the apparent end-to-end distance of the DNA molecule was revealed.Driessen et al. [45] used a numerical procedure, by solving the Langevin equation, Eq. (8), to model the Brownianmotion of the tethered particle. The simulated results in function of the DNA persistence length were fitted witha quadratic function in order to extract the relation between (cid:96) p and the amplitude of movement σ . This empiricalequation was used to extrapolate the persistence length from the values of σ measured in TPM experiments. Asexpected, (cid:96) p is slightly dependent on the AT/GC base-pair composition of the DNA. More surprisingly, this studyrevealed that the intrinsic flexibility of dsDNA strongly and linearly depends on temperature in a range well belowthe DNA melting temperature, an effect much stronger than expected.Brunet et al. [22] investigated further the same question by coupling htTPM experiments and Monte Carlo simula-tions (Section IV B). The extracted values of (cid:96) p showed a slower decrease of the amplitude of movement as comparedto the previous study. Considering the changes in buffer viscosity with T , the authors put forward that the detec-tor time-averaging blurring effect (Section III C) needed to be cautiously corrected. The observed decrease of theapparent end-to-end distance of the tethered DNA well below the DNA melting temperature is mainly due to thiseffect. After carefully correcting the TPM measurements from Ref. [45], the variation of (cid:96) p = κ ( T ) / ( k B T ) with T issharper, in much better agreement with the expected dependency of the bending modulus κ with the temperature atphysiological salt conditions. Up to 60 ◦ C , the extracted values of (cid:96) p display a temperature dependency that can beassociated with an intact dsDNA molecule, without a significant fraction of denaturated base-pairs.Additional work focussed on the temperature dependence of the response of DNA to torsion [67]. This studycombined single-molecule magnetic tweezers measurements with all-atom molecular dynamics and coarse-grainedsimulations. DNA extension-rotation curves were measured over a temperature range of 24 to 42 ◦ C. Increasingtemperature systematically shifted the extension-rotation curves to a negative number of turns. In other words, thepoint where the DNA molecule is torsionally relaxed changes linearly with temperature. Measurements show thatthe temperature-dependent helical change is not force-dependent for stretching forces < T ) = ( − . ± . ◦ /( ◦ C.kbp), in agreement with anterior studies. Usingthe oxDNA coarse-grained model, describing DNA as two inter-twined strings of rigid nucleotides, a 600 bp-longDNA molecule is simulated including Debye-H¨uckel screened electrostatic interactions, at I = 150 mM to matchexperimental conditions. Simulated temperatures ranged from 27 to 67 ◦ C. The mean twist angle for zero torquedecreased as when the temperature was raised. Linear fit over temperature yielded the slope ∆Tw( T ) = ( − . ± . ◦ /( ◦ C.kbp), smaller than the experimental value. Additionally, this question was addressed in all-atom simulationsof a 33 bp mixed DNA sequence with explicit water molecules and ions at five different temperatures ranging from7 to 47 ◦ C . The twist angle linearly decreases in the range, DNA-twist changes are equal to ∆Tw( T ) = ( − . ± . ◦ /( ◦ C.kbp), in close agreement with experimental values. The discrepancy in coarse-grained simulations suggestsan important role of structural local changes along the DNA molecule, only taken into account in the all-atomsimulation. Coupling experimental results with theoretical predictions, this work suggested that the temperature-dependent change in twist is predominantly due to partial and local loss of hydrogen bonds over the DNA backbonethat is not correctly considered in coarse-grained models.DNA structure modifications can be attained not only through temperature changes as discussed above, but alsoby applying torque and/or force with tweezers. At low longitudinally applied forces ( f < f above ≈ −
70 pN. Since its initial discovery in 1996 [11, 68] a debate has arisen as to whether this overstretchedstate is a new S-DNA form or more simply a denaturated state, i.e. a large denaturation bubble if the two endsare closed or peeled ssDNA if one end is open [69]. New experiments have then been done to probe (i) the impactof the experimental conditions of attachement on the coverslip and the bead [69], (ii) the effect of the NaCl salt1 (cid:64) pN (cid:68) z (cid:64) Μ m (cid:68) (a) (b) FIG. 3: (a)
Force-extension curve for a ssDNA. Data (symbols) are taken from H¨ugel et al. [74]. The black solid curvecorresponds to a fit using the discrete worm like chain interpolation with the non-linear bond elasticity, Eq. (A2) (the redone corresponds to discrete version of the Marko Siggia interpolation, Eq. (A1)). The parameters values are: L = 3 . µ m,˜ κ = 1 . a = 0 .
20 nm. (b)
Force-extension curve for a poly(dG-dC) dsDNA. Data (blue symbols) are taken from Rief et al. [75].Solid curves correspond to the discrete worm-like chain interpolation for B-DNA (red), S-DNA (green) and with non-linearextensibility for ssDNA (pink). The black curve corresponds to Eq. (A3), where linear stretching is included as shown by theblue curve for pure B-DNA. The red symbols correspond to the semi-analytical calculation using transfer matrix. Parametersvalues are: L B = 0 . µ m, ˜ κ B = 147, γ = 1 .
89, ˜ κ S = ˜ κ BS = 3 . E B = 1200 pN. Inset: Fraction of base-pairs ϕ S in the S statevs. force, and Ising correlation function 1 − (cid:104) σ i σ i +1 (cid:105) (dashed curve). Taken from Ref. [27]. concentration [70, 71], and (iii) the influence the base content [72, 73], on the overstretched states. It has been clearlyshown (see for instance Refs. [27, 69]) that the knowledge of the different formulas that fit these three possible statesare central in the interpretation of data. Although fitting the transitions itself needs complicated theories such asthe one presented in A leading to the fit shown in Figure 3, simpler formulas such as Eq. (A1) with fewer fittingparameters (the DNA length L , its persistence length (cid:96) p ) or Eq. (A2) for ssDNA stretching allows one to undoubtedlyrecognize the overstretched state.This issue is an example where analytical approaches cannot be replaced by numerical simulations since a goodnumerical model would necessitate both the structural details of the double helix and a dsDNA length L between 0.2and 4 µ m (i.e. 600 to 10000 bp). An attempt has been done using the oxDNA code [76] for a 100 bp dsDNA but theS-DNA state has not been observed. V. DYNAMICALLY DETECTING TWO (OR MORE) DISTINCT STATES
Detecting dynamical configurational changes of DNA molecules is challenging in many situations of interest ingenetic regulation. As already stressed in Section IV C 2, protein-induced DNA looping is a paradigmatic mechanismthat has drawn much attention during the last 25 years [25], because single-molecule techniques have enabled variousresearch groups to shed light not only on looping thermodynamics of different molecular systems but also on theirkinetics. As TPM minimizes mechanical constraints on DNA and proteins, it can give access to kinetics at themolecular scale, with high time-resolution. In 1995, lactose repressor-mediated loop formation and disassembly werekinetically monitored for the first time [3]. In this case, the total DNA molecule was L = 1150 bps long, the polystyreneparticle radius was R = 115 nm, the two DNA sites that are bound when the loop is formed were ∆ s (cid:39)
300 bpsaway, and the repressor concentration was 1 nM. The looping and disassembly lifetimes were found to be very longand both on the order of 100 s. These values were later refined [77]. Systematic exploration of the effect of thedistance ∆ s on looping time was performed in Ref. [8]. In 2006, the IS911 transpososome assembly was analyzed byfollowing a similar strategy [53]. During the last decade, the bridging activity of site-specific recombinases could alsobe studied by TPM or TFM by employing a construct where the synapse assembly also reduces the apparent lengthof the DNA molecule by forming a loop [5–7, 21, 78]. Further addition of sodium dodecyl sulfate (SDS) allows one toassess whether strand exchanged occurred or not within the synapse.Other two-state system kinetics have also been thoroughly studied by TPM. When protein binding provokes a DNA2bend, it is detected as an effective shortening [5, 53, 54, 78]. Nucleosome assembly in eukaryotes also leads to a shorterapparent length [79]. Furthermore, TPM can be used to probe the kinetics of (single) secondary bonds, which can,e.g., transiently form between the particle and the substrate [80].TPM is well adapted to follow dynamics of two-state systems when the dwell-times in both states are on the sameorder of magnitude, in other words when their free energies are comparable. In this section, we explain how recenttheoretical developments likely improve the indirect measurement accuracy of the transition rates between the two(or more) states with TPM. A. Thresholding and correlation time
We illustrate the concept of thresholding [17, 81, 82] (Figure 4) in the case of DNA looping between two specificoperators and mediated by DNA-binding proteins [3, 8, 41, 53, 77]. The average dwell-times in the unlooped and loopedstates are respectively denoted by τ LF and τ LB . By definition, the transition rates between these two states are τ − and τ − . These quantities depend on the binding energy of the DNA-protein complex, on the protein concentration,and on the DNA elastic properties [77]. We assume that the slowly diffusing bead does not significantly alter loopingkinetics if it is sufficiently small, as discussed in Ref. [17] and in Section IV A.The most basic idea is to start from the fact that the amplitude of movement σ of the tethered particle is smallerin the looped state. Hence the plot of σ (or the variance σ ) in function of time will display an alternance of timeintervals where σ is small and large, as displayed in Figure 4. On must define a threshold, denoted here by σ c , belowwhich the molecule is considered to be looped, and above which it is unlooped. The value of σ c can be set by examiningthe bimodal distribution of amplitudes of movement [53, 77]. One can for example set it at half-amplitude betweenthe two maxima of the bimodal distribution. Since σ ≡ (cid:104) r (cid:107) (cid:105) , the average (cid:104) . . . (cid:105) must be estimated on a (sliding)time-window, the duration of which, also called the averaging time, is denoted here by T av . From a signal-processingperspective, this averaging scheme corresponds to window-filtering, i.e. convolution of r (cid:107) ( t ) with a window function.One might choose to switch to more elaborate exponential or Gaussian filters [77, 83, 84], without gaining a significantadvantage, however. t (s) ! r || ( n m ) FIG. 4: Thresholding: The plots of the variance σ (in nm ) versus time for a simulated TPM experiment of a DNA moleculeof length L = 798 bp, a particle of radius R = 20 nm and three averaging times T av = 3 (yellow), 30 (red) and 300 ms (black).The vertical lines indicate the transition events (looping or unlooping) that are forced in the simulation [17]. The horizontalsolid lines show the average values of σ in the looped (bottom) and unlooped (top) states. The dashed horizontal line showsthe threshold value σ c separating these two states for detection purposes. Detecting close-lying transitions, as in the right partof the figure, is the most critical issue of thresholding together with detection of false transition events (see text). Taken fromRef. [17]. One of the main difficulties comes from the fact that the probability distributions of σ for the looped and unloopedstates can overlap substantially when using a too short averaging time T av [85], as illustrated in Figure 4. However,only events occurring at a time-scale larger than T av can be detected. Efficient thresholding thus relies on a compromisebetween a large value of T av needed to estimate at best the amplitude of movement (and avoid at best false detections),and a short value needed to get the best time resolution (and minimize missed transition events) [17, 83]. In particular,3the measured values of the rate constants can depend significantly on the window size [85] because the rate is the inverseof the average dwell-time in a state, and measured dwell-times are bounded below by the window size. Consequently,the T av must be chosen consistently with the dwell-times, typically T av < τ LF , τ LB , even though some improvementscan substantially correct for missed events [83].A lower bound on T av comes from the correlation time τ introduced in Section III C, as discussed in Ref. [17]. Letus assume for simplicity that the two states have comparable correlation times τ . Their amplitudes of movement are σ < σ . We introduce the parameter λ = 1 − ( σ /σ ) . It was demonstrated [17] that the minimal averaging time T av needed to resolve them with good accuracy (i.e. with few false detections of transitions) is typically equal to τ /λ .All in all, optimal T av must satisfy τλ < T av < τ LF , τ LB . (10) B. Hidden Markov chains
Hidden Markov modeling (HMM) combined with a maximum-likelihood approach can be used to determine thenumerical values of model parameters such as the transition rates. In contrast to thresholding, no windowing isrequired, nor the prior selection of a threshold. HMM, initially developed by mathematicians [86], has nowadaysplenty of applications in various fields of science. It has been adapted to TPM in 2007 by Beausang et al. [85, 87].The underlying idea is that the system under consideration can be modeled by a Markov chain [88], the states ofwhich are not directly observed in the experiment, i.e., they are “hidden”.Here we again illustrate these ideas in the case of DNA looping, even though it can be generalized, e.g., to protein-binding. The hidden state, denoted by q ( t ), can be “looped” or “unlooped” DNA. We again look for the averagedwell-times τ LF and τ LB . The most basic idea [85] would be to consider that this two-state system is governed by atwo-state Markov chain with a 2 × r (cid:107) , itself governed by an over-damped Langevinequation in a harmonic potential. The system state as it appears in the Markov chain is now ( q ( t ) , r (cid:107) ( t )).Once an experimental time series ( q ( t ) , r (cid:107) ( t )) t has been recorder, the idea is then to calculate the likelihood that itis observed for a given pair of dwell-times ( τ LF , τ LB ). This can be done with the standard tools of probability theory.Then the dwell-time values maximizing this likelihood are considered to be the most probable ones. This procedurehas been tested on numerically generated trajectories for which the dwell times were exactly known. It was able tocorrectly recover these values, up to statistical error bars.However, in spite of its conceptual simplicity, the practical implementation of the method relies on some strongapproximations about the looping process, as stated by the authors themselves [85]. For example, looping is allowed ifand only if the particle excursion (cid:107) r (cid:107) ( t ) (cid:107) is smaller than a threshold ρ max . This is modeled by a crude Θ step-functionin the Markov chain. In addition, a simplification is “to ignore the unobserved height variable z [above the glasssubstrate plane], in effect treating the bead motion as diffusion in two dimensions.” This is again an issue whendeciding whether looping is possible or not for a given value of (cid:107) r (cid:107) ( t ) (cid:107) because z can be large even though (cid:107) r (cid:107) ( t ) (cid:107) is small. “A better analysis might treat z as another hidden (unobserved) variable.” Finally, as in the thresholdingapproach, the polymer is considered to be in quasi-equilibrium in both the looped and unlooped state, from whichtransition probabilities are inferred. This assumption is only valid if the free-energy wells around each state aresufficiently deep.A refinement of the HMM method relies on variational Bayesian inference, first used in the study of Lac repressor-mediated looping [41] that we have already mentioned in Section IV C 2. Bayesian inference is able to determine notonly the most likely model parameters but also the most likely number of hidden states, that was fixed a priori inthe above HMM method, by observing the number of peaks in the probability distribution of σ . The improved abilityof this method to resolve close-lying transitions (see Figure 4) has also been tested in this work. For instance, twostates separated by an amplitude of movement σ of 40 nm could be resolved by their technique at a mean lifetime ofabout 0.5 s. In contrast, lifetimes of more than 4 s were necessary for states to be resolvable by simple thresholding.The reason for this difference is that Bayesian inference does not require time-filtering as evoked above.When the free energies of the states are significantly different, one state is favoured with respect to the other. Forexample, let us assume that once the loop is formed, it is very stable because breaking it would require a free energy∆ F (cid:29) kT . Then loop opening events become rare and cannot be observed in practice. It becomes interesting toapply a force f (with the help of tweezers) in the pN range on the tethered particle because the associated potentialenergy difference between both states can compensate ∆ F and make them roughly equiprobable, thus allowing one4to observe more frequent opening events. Indeed, the dwell-times are known to depend exponentially on the appliedforce [89].Beyond looping, this approach has been successfully used in combination with HMM analysis to study single-nucleosome unwrapping [90]. Since DNA is wrapped in about two turns around a histone octamer, three distinctconformations can be observed: fully wrapped, about one turn unwrapped and fully unwrapped. Applying a 2.5 pNforce allows one to observe transitions between the two first states, and at a stretching force of 6 pN, the secondturn unwraps. Detailed information about nucleosome unwrapping such as bending angles of nucleosomal DNA ordwell-times could be extracted from these experiments. Zero-force dwell-times can then be extrapolated from thesemeasurements. This method can in principle be used to detect any transient DNA-protein complex formation anddwell-times. Similar approaches have been used to study DNA hairpins opening/closure [91] and DNA G-quadruplexunfolding/refolding [92]. VI. CONCLUSION
This Review has illustrated in many situations the powerful capabilities of TPM experiments coupled to theoreticaland/or numerical modeling to give access to quantities of interest in both biological and biophysical contexts. Wehave identified two levels of difficulties that must be overcome in order to infer the physical parameters of interestwith a good accuracy.First, raw data must be carefully processed thanks to well-established protocols in order to deal with both statisticaland systematic sources of errors. The existence of outliers seems inherent to single molecule experiments because itis both difficult to prepare samples with 100 % of identical molecules and to graft all molecules in ideal conditionslimiting unwanted non-specific interactions. This is in part solved thanks to nano-lithography techniques, but notentirely. After having discarded outliers, the most critical systematic effect to deal with is the blurring effect inherentto the finiteness of the detector exposure. We have proposed an efficient way to solve this issue through simpleinversion formulae, as summarized in Eqs. (4) and (5).Once the data have been corrected from these sources of error, solving the inverse problem enables one to inferthe DNA state or the physical parameters of interest. This concerns not only quantities measured at thermodynamicequilibrium (e.g., elastic parameters C and (cid:96) p or intrinsic bending angles), but also out-of-equilibrium properties ofgreat biological interest such as transition rates (e.g. binding/unbinding or looping/unlooping rates). In the lattercase, we have shown that hidden Markov chain approaches are promising even though they are more complex toimplement than simple thresholding. In all cases, the underlying quantitative model must rely on solid physicalgrounds, appealing to polymer and elasticity theory, and out-of-equilibrium statistical mechanics. We have listedseveral successes of such approaches in this Review.However, to our point of view, few issues remain to be solved in the future in order to provide a fully operationaltool to biophysicists and biologists. We have explained that intrinsic curvature modifies the amplitude of movementin a way that can be quantified with good accuracy. However, weak intrinsic curvature is spread all over the molecule,and it is not only localized at specific high-curvature loci. This “quenched” disorder likely modifies the amplitude ofmovement in a systematic but ill-controlled manner. This phenomenon should be quantified, for example by usingmore sophisticated mesoscopic models fully taking account such subtle effects [93].When a region of DNA is modified or affected in any way, for example through binding of a protein, it can bearnot only a different spontaneous curvature that one wants to quantify, but also a different bending modulus. Inparticular, DNA flexibility is a sequence-dependent quantity [94]. Disentangling sequence-dependent spontaneouscurvature and sequence-dependent elastic properties in single-molecule experiments is another issue that will requireextensive modeling work in the future.As far as dynamical properties are concerned, we have just stressed that methods relying on hidden Markov chainsare quite promising. However, to our knowledge, no systematic quantification of their capabilities, in the spirit ofEq. (10), has been performed so far. Probability theory together with numerical modeling should be able to givedefinitive and robust conclusions on the strengths and limits of these approaches. Filling this gap seems importantto us in order to eventually provide an easy-to-use tool to experimentalists. Acknowledgments
We warmly thank our colleagues Catherine Tardin and Laurence Salom´e for fruitful discussions and sound advicesduring the writing of this Review. We are also tributary to the Universit´e Toulouse III-Paul Sabatier and the CentreNational de la Recherche Scientifique (CNRS).5
Appendix A: Analytical modeling of stretching experiments
The classical model for semi-flexible polymer stretching has been developed by Marko and Siggia [95] using thecontinuous worm-like chain model by developing a formula that interpolates between the exact results in the limitsof low ( f (cid:28) k B T /(cid:96) p (cid:39) .
08 pN) and strong forces ( f (cid:29) k B T /(cid:96) p ): f = k B T(cid:96) p (cid:20) zL −
14 + 14(1 − z/L ) (cid:21) . (A1)This formula has been successfully used to fit the force-extension curves for f (cid:46)
65 pN for a λ -phage dsDNA (see thereview [96] and references therein). One technical difficulty in fitting these curves is to set the origin. This might becorrelated to the fact that experimentally the tethers are not stretched exactly in the direction perpendicular to thecoverslip [97].For larger forces, the DNA internal structure starts to come into play (see below). However this type of approachremains valid for very high stretching of ssDNA [75] provided that both the discrete nature of the chain is taken intoaccount [98] and non-linear stretching terms are included [74]. Fits of force-extension curves of ssDNA up to 1200 pN,as shown in Figure 3a, have been nicely fitted using the modified formula that includes both terms [27]: f = k B Ta (cid:34) zL (1 + U nl ( f )) (cid:18) − u (˜ κ )1 + u (˜ κ ) − √ κ (cid:19) + (cid:18) − z (1 + U nl ( f )) /L ] + 4˜ κ (cid:19) / − (cid:112) κ (cid:35) (A2)where ˜ κ = κ/ ( k B T ) is the bending modulus in units of k B T (cid:39) × − J (at room temperature), u (˜ κ ) = coth(˜ κ ) − / ˜ κ ,and U nl ( f ) = 1 . f − . f + 4 . f (where f is in units of 10 nN) [74]. The fit leads to the bendingmodulus value κ (ssDNA) = 1 . k B T and an effective monomer length a = 0 .
20 nm (see Figure 3a). Unexpectedlythis value of a is much smaller than the distance between two consecutive bases in ssDNA a ss (cid:39) . za B N = (cid:18) F ˜ E B − α B (cid:19) ϕ B + γ (cid:18) − α S (cid:19) ϕ S + (cid:104) σ i σ i +1 (cid:105) − (cid:18) α B ˜ κ B − ˜ κ BS ˜ κ BS + F/ α B + γ α S ˜ κ S − ˜ κ BS ˜ κ BS + γF/ α S (cid:19) (A3)where B and S refer to the DNA state, F = a B f / ( k B T ), ˜ κ BS is the bending modulus at the BS domainwall, γ = a S /a B , ˜ E B = a B E B / ( k B T ) where E B is the stretching modulus, α B = (˜ κ B F + F / / and α S = [˜ κ S γF +( γF ) / / . The fraction of base-pairs in the B (respectively S) state are ϕ B (respectively ϕ S = 1 − ϕ B ).Finally (cid:104) σ i σ i +1 (cid:105) is the two-point correlation function of the effective Ising model which is non-zero only close to thetransition [27]. An example of a fit of the force-extension curve for a poly(dG-dC) DNA is shown in Figure 3b. Theinset shows the variation of the fraction of base pairs in the S state and the Ising correlation function as a functionof the applied force. Eq. (A3) has also been used to fit the S-DNA to ssDNA transition for poly(dG-dC) where thelinear stretching term is replaced by the non-linear stretching one for ssDNA [27]. References [1] D.A. Schafer, J. Gelles, M.P. Sheetz, R. Landick, Transcription by single molecules of RNA polymerase observed by lightmicroscopy, Nature (1991) 352, 444-448. [2] H. Yin, R. Landick, J. Gelles, Tethered particle motion method for studying transcript elongation by a single RNApolymerase molecule. Biophys. J. (1994) 67, 2468-2478.[3] L. Finzi, J. Gelles, Measurement of lactose repressor-mediated loop formation and breakdown in single DNA molecules.Science (1995) 267, 378-380.[4] G. Zocchi, Analytical assays based on detecting conformational changes of single molecules, ChemPhysChem (2006) 7,555-560.[5] H.F. Fan, C.H. Ma, M. 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