Sub-Millisecond Conformational Transitions of Short Single-Stranded DNA Lattices by Photon Correlation Single-Molecule FRET

Abstract

Thermally driven conformational fluctuations (or ‘breathing’) of DNA plays important roles in the function and regulation of the ‘macromolecular machinery of genome expression.’ Fluctuations in double-stranded (ds) DNA are involved in the transient exposure of pathways to protein binding sites within the DNA framework, leading to the binding of regulatory proteins to single-stranded (ss) DNA templates. These interactions often require that the ssDNA sequences, as well as the proteins involved, assume transient conformations critical for successful binding. Here we use microsecond-resolved single-molecule Fӧrster Resonance Energy Transfer (smFRET) experiments to investigate the backbone fluctuations of short [oligo(dT)_n] templates within DNA constructs that also serve as models for ss-dsDNA junctions. Such junctions, together with the attached ssDNA sequences, are involved in interactions with the ssDNA binding (ssb) proteins that control and integrate the functions of DNA replication complexes. We analyze these data using a chemical network model based on multi-order time-correlation functions (TCFs) and probability distribution functions (PDFs) that characterize the kinetic and thermodynamic behavior of the system. We find that the oligo(dT)_n tails of ss-dsDNA constructs inter-convert, on sub-millisecond time-scales, between three macrostates with distinctly different end-to-end distances. These are: (i) a ‘compact’ macrostate that represents the dominant species at equilibrium; (ii) a ‘partially extended’ macrostate that exists as a minority species; and (iii) a ‘highly extended’ macrostate that is present in trace amounts. We propose a model for ssDNA secondary structure that advances our understanding of how spontaneously formed nucleic acid conformations may facilitate the activities of ssDNA-associating proteins.

Graphical Abstract

1. Introduction

The genome of living cells exists primarily as B-form double-stranded (ds) DNA, which is packaged in the cell nucleus as chromatin (in eukaryotes) or nucleoid bodies (in bacteria) (1). During various stages of the cell cycle, segments of single-stranded (ss) DNA are exposed by the action of the DNA replication helicase to make available the ssDNA templates required by the DNA polymerases for the synthesis of the complementary daughter strands for cell division, and to allow access of other functional and regulatory proteins to these transiently formed ssDNA sequences. Within the cell, and in functional complexes such as the replisome, these exposed ssDNA segments are coated by ssDNA binding (ssb) proteins that function to protect the genetic information from nucleases and to favorably configure the single strands for further manipulation by the ‘machinery of genome expression’ (2, 3). These proteins interact with ssDNA to form stable, cooperatively bound nucleoprotein filaments, in which the bound ssDNA chain segments are nearly fully extended to their maximum contour lengths permitted by the sugar-phosphate backbones, resulting also in largely unstacking the attached bases (4, 5). Because such fully extended conformations of ssDNA are likely to be unstable in the absence of bound protein, a fundamental question arises concerning how ssb protein-ssDNA interactions evolve along a molecular reaction coordinate to form extended nucleoprotein filaments.

Unlike the cooperatively structured and stabilized dsDNA conformation, many possible conformations can be adopted by ssDNA in solution, and the specifics of these conformations depend on a variety of interactions between the nucleic acid components with their aqueous surrounds. These interactions include base stacking and unstacking, intra-chain repulsion between adjacent backbone phosphates, counterion condensation along the polyelectrolyte ssDNA backbone, binding and orientation of polar water molecules around these counterions and the backbone phosphates, strain within the sugar-phosphate backbone and increases in configurational entropy of the backbone as a consequence of base-unstacking and the concomitant rotation around multiple single-bonds of the extended backbone (6–8).

The cumulative effect of these interactions is a free energy landscape that determines the probability with which a given ssDNA configuration will occur. In general, the coordinate space over which the free energy landscape is defined can be partitioned into distinct sub-spaces, herein referred to as ‘macrostates,’ which correspond to local basins of stability separated by relatively high activation barriers (9). Each macrostate consists of a ‘family’ of energetically-degenerate microscopic conformations (or ‘microstates’) that rapidly interconvert (on tens-of-nanoseconds time-scales) in a non-Markovian manner (10–12). In contrast, higher thermal barriers separate conformations associated with different macrostates, such that their interconversion occurs on tens-of-microseconds and longer time scales. The relatively high activation barriers associated with macrostate interconversion ensures that the observed dynamics of the overall system can be accurately described using a kinetic master equation in which the rate constants contain no ‘hidden variables’ and memory effects need not be considered (13).

In this work, we present microsecond-resolved single-molecule Förster resonance energy transfer (smFRET) experiments to monitor the conformational fluctuations of short single-stranded oligo-deoxythymidine [oligo(dT)_n] templates of varying length (n = 14 and 15) and polarity (3’ versus 5’). We note that these constructs have been used in previous studies as model systems to monitor the dynamics of the non-cooperative and cooperative binding and assembly of the T4 bacteriophage ssb proteins (gene product 32, or gp32) onto target ssDNA lattices (14, 15). The binding site size of the gp32 monomer is 7 nucleotides (nts) so that for the templates with a total length, n, of 14 nts there is excess room for the non-cooperative binding of a single gp32 monomer or, alternatively, precisely enough room for the cooperative assembly of a gp32 dimer. For templates with n = 15 nts, there are two possible conformations for the gp32 dimer-bound state; either at nucleotide positions 1 – 14 or 2 – 15. Furthermore, the gp32 proteins are expected to assemble onto ssDNA lattices in a polar manner, so that the assembly mechanism will depend on the polarity of the ss-ds DNA junction. Our previous studies showed that gp32 proteins bind to, and dissociate from, the ss-dsDNA junction as monomers on tens-of-milliseconds time scales, and that gp32 monomer-bound intermediates must encounter (also, on tens-of-milliseconds time scales) a ‘productive’ ssDNA conformation near the ss-dsDNA junction to permit a second ssb protein to form a cooperatively-bound gp32 dimer-cluster (14). By examining the sub-millisecond dynamics of the ssDNA lattice fluctuations in these constructs in the absence of proteins, we may ask whether short-lived conformational intermediates – which can serve as potential transition states for ssb binding – play a role in the assembly mechanism.

Each of the oligo(dT)_n conformations that we investigated were formed within the overhanging ‘tail’ of a ss-dsDNA junction construct, in which the distal end of the ssDNA chain is labeled with the FRET donor chromophore Cy3 and the conjugate DNA strand is labeled, at the ss-dsDNA junction, with the FRET acceptor chromophore Cy5 (see Fig. 1). Such ss-dsDNA constructs labeled in this way were previously used by Lohman and Ha and co-workers in related studies with the E. coli SSB system using standard smFRET methods (3, 16, 17). Similar constructs were also used by Pollack and co-workers in their studies of ssDNA conformations by small-angle x-ray scattering (SAXS), smFRET and other biophysical techniques and analysis methods (18). Our experiments detect individual fluorescence photons from a continuously irradiated single molecule sample using instrumental methods (19) and data analyses developed previously (14, 20). Because these experiments resolve individual photon detection events, the dominant contribution to measurement uncertainty is the sparse statistical sampling of the ensemble-averaged signal rate (21), which varies stochastically in time due to the conformational fluctuations of the macromolecule. The time-dependent signal rates of the Cy3 donor (D) and the Cy5 acceptor (A) FRET-pair chromophores are given by I_D(A)(t) = N_D(A)(t)⁄T_w, where N_D(A) is the number of photons detected during a fixed time window T_w (≥ 1 μs) beginning at the time t. Thus, the smFRET efficiency, E_FRET(t) = I_A(t)⁄[I_D(t) + I_A(t)], is a time-averaged signal, which represents an average of the macromolecular configurations detected during the sampling period T_w.

Figure 1.

Schematic of one of the four single-stranded (ss) – double-stranded (ds) DNA constructs [3’-Cy3/Cy5-oligo(dT)₁₅-ss-dsDNA – see Table 1] used in our experiments. (A) The double-stranded region of the ss-dsDNA construct was attached to the surface of a fused silica quartz (SiO₂) slide coated with PEG and containing biotin-neutravidin linkers. (B) The Cy3 donor chromophore was attached to the distal end of the oligo(dT)_n overhanging ‘tail’ region of the ss-dsDNA construct, and the Cy5 acceptor was attached to the opposite strand at the ss-dsDNA junction, as shown. Note that Panel B represents a more chemically detailed version of the lower part of the construct cartooned in Panel A.

Our microsecond-resolved smFRET measurements contain both structural and kinetic information about the microscopic configurations of the oligo(dT)_n templates. To extract the maximum amount of information possible from our data, we applied an ‘inverse-problem’ procedure for analyzing photon correlation measurements of this type (14, 20). From the photon count data stream, we determined the following three ensemble-averaged functions of the smFRET signal: (i) the two-point time correlation function (TCF) ${\overline{C}}^{(2)}(\tau )$ that describes the average correlations between two successive measurements separated by the time interval τ, and which was sampled on microsecond time scales; (iii) the four-point TCF, ${\overline{C}}^{(4)}\left({\tau }_1,{\tau }_2,{\tau }_3\right)$ that describes the average correlations between four successive measurements separated by the time intervals τ₁, τ₂ and τ₃; and (iii) the probability distribution function (PDF) of the mean FRET efficiency P(E_FRET), which was sampled on the millisecond time scale required for a randomly selected microstate to relax to a metastable intermediate. The above functions were used to characterize the equilibrium and kinetic properties of multi-step conformational transition pathways at equilibrium. While the two-point TCF, ${\overline{C}}^{(2)}$, contains information about the characteristic relaxation times of the system, the four-point TCF, ${\overline{C}}^{(4)}$, contains additional information about the exchange times between pathway intermediates (20, 22–24).

A key observation is that both the two-point and four-point TCFs exhibit two principal relaxation components that are well-separated in time (~20 μs and ~275 μs) and reflect conformation changes of the oligo(dT)_n lattices. As discussed in Sect. 3.4, this observation suggests that the simplest kinetic model for the underlying dynamics is a network of three coupled macrostates in thermal equilibrium. We simulated the PDF and the TCFs using a master equation (25, 26), for which each macrostate is distinguished by a different mean FRET efficiency. Kinetic rate constants were varied as input parameters to the master equation to achieve the ‘best fit’ between simulated and experimentally derived functions. As we discuss further below, we compared the results of a linear versus cyclical three-state network model from which we obtained optimized values for the forward and backward rate constants that connect the conformational macrostates, in addition to the probability of observing each macrostate. The optimized rate constants so obtained determine the free energy barriers that separate each macrostate (applying the Arrhenius equation), and the optimized probabilities determine the free energy minima of each macrostate (applying the Boltzmann distribution).

A major finding of this work is that all four of the oligo(dT)_n templates that we investigated undergo conformational dynamics over a broad range of time scales, spanning tens-to-hundreds of microseconds. We observed that all four of these systems interconvert between three distinct macrostates: (i) a ‘compact’ and relatively stable macrostate; (ii) a ‘partially extended’ macrostate of intermediate stability; and (iii) a ‘highly extended’ macrostate of relatively low stability. The free energy minima that we determined for the three macrostates increase rapidly with decreasing FRET efficiency, suggesting that the primary mechanism of macrostate destabilization is the loss of configurational entropy associated with increasing chain extension (10). Moreover, our results indicate the existence of transition state barriers that are intrinsic to the oligo(dT)_n templates themselves, and which prolong the lifetimes of extended chain configurations relative to the more stable compact configurations. The existence of metastable extended chain conformations suggests that these short-lived species may serve as intermediates along the reaction pathway for the formation of ssb nucleoprotein clusters or filaments.

2. Materials and Methods

2.1. Single-Stranded (ss) – Double-Stranded (ds) DNA Constructs.

We performed microsecond-resolved smFRET experiments on four ss-dsDNA constructs with a ssDNA oligo(dT)_n overhanging ‘tail’ region that varied in length (n = 14 and 15) and polarity (3’ and 5’, see Table 1). In Fig. 1 we show a representative schematic diagram using the 3’-Cy3/Cy5-oligo(dT)₁₅-ss-dsDNA construct. These samples were fluorescently labeled using the Cy3/Cy5 FRET donor-acceptor chromophore pair, as indicated.

Table 1.

Nucleotide base sequences for ss-dsDNA constructs studied in this work.

DNA construct	Nucleotide base sequence
3’-Cy3/Cy5-oligo(dT)15-ss-dsDNA	5’-GTCGCCAGCCTCGCAGCCTTTTTTTTTTTTTTT/Cy3/-3’ 3’-biotin/CAGCGGTCGGAGCGTCGG-Cy5/-5’
3’-Cy3/Cy5-oligo(dT)14-ss-dsDNA	5’-GTCGCCAGCCTCGCAGCCTTTTTTTTTTTTTT/Cy3/-3’ 3’-biotin/CAGCGGTCGGAGCGTCGG-Cy5/-5’
5’-Cy3/Cy5-oligo(dT)15-ss-dsDNA	3’-GTCGCCAGCCTCGCAGCCTTTTTTTTTTTTTTT/Cy3/-5’ 5’-biotin/CAGCGGTCGGAGCGTCGG-Cy5/-3’
5’-Cy3/Cy5-oligo(dT)14-ss-dsDNA	3’-GTCGCCAGCCTCGCAGCCTTTTTTTTTTTTTT/Cy3/-5’ 5’-biotin/CAGCGGTCGGAGCGTCGG-Cy5/-3’

2.2. Sample Preparation.

Sample solutions were prepared in an aqueous buffer containing 100 mM NaCl, 6 mM MgCl₂ and 10 mM Tris (pH 8.0), as described previously (14, 19, 27). Oxygen scavenging and triplet quenching reagents were used to extend the excited singlet state lifetime and to minimize triplet state blinking (28, 29). Microfluidic sample chambers were constructed using fused silica quartz microscope slides, as shown in Fig. S1A (30). Single molecules were attached to the slide surface using a neutravidin linker, which binds strongly to biotin attached to both the ss-dsDNA constructs (Table 1) and to the poly(ethylene glycol) (PEG) layer that coats the slide surface (see Fig. 1A). Additional details of the sample preparation are included in the SI.

2.3. Microsecond-Resolved Single-Molecule (sm)FRET.

We performed our smFRET experiments on the ss-dsDNA constructs using instrumentation and procedures described previously (14, 19). Additional details of the data acquisition and signal calibration are described in the SI. A schematic diagram of the instrumental setup is shown in Fig. S1B. The sample was illuminated using a continuous wave laser at 532 nm set up in a prism-type total internal reflection fluorescence (TIRF) configuration. The laser beam was focused to a 50 μm diameter spot at the sample and the incident power was adjusted to 10 mW. The Cy3 donor (D) and Cy5 acceptor (A) fluorescence from a single ss-dsDNA construct was imaged through a pinhole, and spatially separated into D and A paths using a dichroic beam splitter. Individual photons from the D and A paths were detected using avalanche photodiodes (APDs), and each detection event was assigned a ‘time-stamp’ with 0.1 μs resolution and recorded to the computer hard disk. In post-data acquisition, we determined the time-dependent D(A) signal rates according to I_D(A)(t) = N_D(A)(t)⁄T_w, where N_D(A) was the number of D(A) photons detected during an adjustable dwell period T_w (≥ 1 μs) at time t. The mean total fluorescence photon detection rate, ${\overline{I}}_T={\overline{I}}_D+{\overline{I}}_A$, was typically ~15,000 – 20,000 cps. The signal ‘background rate’ was determined to be ~ 2,000 cps by scanning the xy-stage position away from the center of a molecule. Thus, the signal-to-background ratio of our experiments is ~ 10. Each single-molecule data set was recorded for a total duration of thirty seconds. From the time-dependent signal rates we determined the FRET efficiency according to E_FRET(t) = I_A(t)⁄I_T(t). We measured the instrument response function (IRF) of our apparatus, which we determined to have a full-width-at-half-maximum (FWHM) value of ~1 μs (see SI).

We used different values for the dwell period T_w, depending on the statistical function that we calculated from our data. For our calculations of two-point and four-point time-correlation functions (TCFs), we used the value for T_w corresponding to the shortest time interval sampled in the calculation. We used T_w = 1 μs for our TCF calculations that focused on the first 20 μs of the decay dynamics. However, for the majority of our TCF calculations, we used T_w = 10 μs, as discussed in further detail below. For our calculations of the probability distribution function (PDF) of E_FRET values we used T_w = 1 ms, which according to our analysis is roughly equal to the time required for the system to relax from a randomly sampled initial state to one of the possible metastable intermediates.

3. Theoretical Background

In our experiments, we monitored the oligo(dT)_n end-to-end distance R_DA between the D (Cy3) and A (Cy5) chromophore labels of the ss-dsDNA constructs (see Fig. 1) through measurements of the smFRET efficiency: E_FRET = [1 + (R_DA⁄R₀)⁶]⁻¹. Here R₀ = 0.211(J_DAκ²n⁻⁴Φ_D)^1/6 is the characteristic Förster distance (in units of Å) where Φ_D is the D fluorescence quantum yield, n is the refractive index of the medium, J_DA is the overlap integral between the D emission and the A absorbance spectra, and κ² is the transition dipole-dipole orientation factor (31). The orientation factor ${\kappa }^2={\left[{\widehat{\mu }}_D\cdot {\widehat{\mu }}_A-3\left({\widehat{\mu }}_D\cdot {\widehat{R}}_{DA}\right)\left({\widehat{R}}_{DA}\cdot {\widehat{\mu }}_A\right)\right]}^2$ accounts for the relative angles between the D and A chromophores where ${\widehat{\mu }}_{D(A)}$ is the unit vector that points in the direction of the D (A) transition dipole moment and ${\widehat{R}}_{DA}$ is the unit vector that points from the D to the A. For cases in which an isotropic distribution of relative D and A orientations can be assumed, which would require that at least one of the two dyes be freely rotating around its ‘tether,’ the orientation factor takes on the value κ² = 2/3 and the Förster distance R₀ ≅ 56 Å (32, 33). In the remainder of this work, we focus on the properties of the observable E_FRET as an indicator of the end-to-end distance, R_ee (assumed ≈ |R_DA|), of the oligo(dT)_n ‘tail’ regions of the ss-dsDNA constructs that we studied, although we return to the interpretation of the observable in terms of R_ee in the Conclusions.

3.1. Macrostates and Free Energy Minima.

The oligo(dT)_n templates can adopt a broad range of microscopic chain configurations (i.e., microstates). Assuming there are N total microstates available at a given temperature T, the probability that the ith microstate occurs is given by the Boltzmann distribution (34)

$$p_i=\frac{1}{Z}{e}^{-G_i/k_{B}T}$$ (1)

where k_B is Boltzmann’s constant, G_i is the Gibbs free energy of the ith microstate, and $Z={\sum }_{i=1}^N{e}^{-G_i/k_{B}T}$ is the partition function. We note that the normalized probabilities sum to unity: ${\sum }_{i=1}^N{p}_i=1$. Individual microstates are expected to interconvert on tens-of-nanosecond time scales, which is much faster than the microsecond resolution of our experiments (10). Thus, the ‘observed’ D and A fluorescence signals each represents a time-average over the possible microstates that occur at a particular time t during the sampling period T_w (= 10 μsec). It is useful to define a ‘macrostate’ as a collection of pseudo-degenerate microstates that occupy a common, locally stable region of the free energy landscape. We may then regard each D and A fluorescence photon detection event as a stochastic sample of the partial-ensemble-average of microstates defined within the phase-space boundary of a single macrostate.

In our analysis, we model the probability distribution function (PDF) of E_FRET values as a sum of contributions from M macrostates $p\left(E_{\mathit{\text{FRET}}}\right)={\sum }_{j=1}^M{p}_j\left(E_{\mathit{\text{FRET}}}\right)$, where the jth macrostate has the Gaussian form $p_j\left(E_{\mathit{\text{FRET}}}\right)=A_j\text{exp}\left[-{\left(E_{\mathit{\text{FRET}}}-{\langle E_{\mathit{\text{FRET}}}\rangle }_j\right)}^2/2{\sigma }_j^2\right]$, with mean value 〈E_FRET〉_j, standard deviation σ_j, and amplitude A_j. The Gaussian model for the distribution of configurational microstates (within a given macrostate) is consistent with a random-flight, finite-length polymer chain (10, 35). Macrostates that contain a relatively large number of pseudo-degenerate microstates contribute a correspondingly broad distribution of E_FRET values. Thus, the relative probability that the jth macrostate is present at equilibrium is given by the integrated area of its component Gaussian $p_j^{eq}={\int }_{-\infty }^{\infty }{p}_j\left(E_{\mathit{\text{FRET}}}\right)=A_j{\sigma }_j\sqrt{2\pi }$. We note that the sum of normalized probabilities over all macrostates is equal to unity: ${\sum }_{j=1}^M{p}_j^{eq}=1$. Applying Eq. (1) to macrostates, we may thus write the free energy of the jth macrostate (in units of k_BT) as a function of E_FRET

$$\frac{G_j\left(E_{\mathit{\text{FRET}}}\right)}{k_{B}T}=-\text{ln}\left[p_j\left(E_{\mathit{\text{FRET}}}\right)\right]=-ln\left(\frac{p_j^{eq}}{{\sigma }_j\sqrt{2\pi }}\right)+\frac{1}{2{\sigma }_j^2}{\left(E_{\mathit{\text{FRET}}}-{\langle E_{\mathit{\text{FRET}}}\rangle }_j\right)}^2$$ (2)

In the context of the above model, the free energy of the jth macrostate is represented as a parabolic function of E_FRET with vertical displacement $-ln\left(p_j^{eq}/{\sigma }_j\sqrt{2\pi }\right)$. The total free energy surface as a function of E_FRET can then be constructed by summing over the individual macrostate contributions according to $G\left(E_{\mathit{\text{FRET}}}\right)=-k_{B}Tln\left[{\sum }_{j=1}^M{p}_j\left(E_{\mathit{\text{FRET}}}\right)\right]$.

3.2. Free Energies of Activation.

We assume that under equilibrium conditions, the microstates of the oligo(dT)_n template are partitioned amongst M quasi-stable macrostates, and there exist conformational reaction channels that permit discreet, stochastic transitions to occur from the ith to the jth macrostate. In addition, we make the standard approximation that these transformations can be modeled as continuous processes (25, 36). We thus define the time-dependent population of the ith macrostate, p_i(t), as a continuous, single-valued function that obeys the coupled, first-order ordinary differential equations ${\dot{p}}_i={\sum }_{j\ne i=1}^M-k_{ij}{p}_i+k_{ji}{p}_j$. Here the forward rate constant k_ij is associated with the transition from macrostate i to macrostate j, and the backward rate constant k_ji is associated with the transition from macrostate j to macrostate i. The rate constant k_ij is related to the free energy of activation $G_{ij}^{‡}$, which is given by the Arrhenius equation (37)

$$k_{ij}=k_{max}{e}^{-G_{ij}^{‡}/k_{B}T}$$ (3)

In Eq. (3), the maximum rate constant k_max corresponds to a barrierless process, to which all the thermally activated processes of the system are referenced.

3.3. Two-Point and Four-Point Time-Correlation Functions (TCFs).

We obtain kinetic information from our time-resolved smFRET data through our analyses of two-point and four-point time-correlation functions (TCFs) (20). We define the time-dependent fluctuation of the E_FRET observable δE_FRET(t) = E_FRET(t) − 〈E_FRET〉, where 〈E_FRET〉 is the mean value determined from a time series of measurements performed on a single molecule. The two-point TCF is the average product of two successive measurements separated by the interval τ

$${\overline{C}}^{(2)}(\tau )=\langle \delta E_{\mathit{\text{FRET}}}(0)\delta E_{\mathit{\text{FRET}}}(\tau )\rangle$$ (4)

In Eq. (4), the angle brackets indicate a running average over all possible initial measurement times. As we discuss further below, the function ${\overline{C}}^{(2)}(\tau )$ contains information about the minimum number of possible macrostates of the system, each macrostate’s mean FRET efficiency value, in addition to the characteristic time scales of state-to-state interconversion. The two-point TCF may be expressed using a standard statistical mechanical model according to:

$${\overline{C}}^{(2)}(\tau )=\sum _{i,j=1}^M\delta E_{FRE{T}_j}{p}_{ij}(\tau )\delta E_{FRE{T}_i}{p}_i^{eq}$$ (5)

Here $p_i^{eq}$ is the probability that the ith macrostate is present at equilibrium, $\delta E_{FRE{T}_i}=E_{FRE{T}_i}-{\sum }_{i=1}^M{p}_i^{eq}{E}_{FRE{T}_i}$ is the value of the fluctuation observable corresponding to the ith macrostate, and p_ij(τ) is the conditional probability that the system undergoes a transformation from macrostate i to macrostate j during the interval τ. Equation (5) states that ${\overline{C}}^{(2)}(\tau )$ is the statistically weighted average product of consecutive (two-point) observations, which are expected to occur within the interval τ as the system undergoes spontaneous transitions between the M macrostates.

We define the four-point TCF as the time-averaged product of four successive measurements separated by the intervals τ₁, τ₂, and τ₃, respectively.

$${\overline{C}}^{(4)}\left({\tau }_1,{\tau }_2,{\tau }_3\right)=\langle \delta E_{\mathit{\text{FRET}}}(0)\delta E_{\mathit{\text{FRET}}}\left({\tau }_1\right)\delta E_{\mathit{\text{FRET}}}\left({\tau }_1+{\tau }_2\right)\delta E_{\mathit{\text{FRET}}}\left({\tau }_1+{\tau }_2+{\tau }_3\right)\rangle$$ (6)

The function ${\overline{C}}^{(4)}\left({\tau }_1,{\tau }_2,{\tau }_3\right)$ contains information about the roles of intermediates within the conformational kinetic pathways that affect the rates of interconversion between macrostates. The first term on the right-hand-side of Eq. (6) may be modeled according to:

$$\langle \delta E_{\mathit{\text{FRET}}}(0)\delta E_{\mathit{\text{FRET}}}\left({\tau }_1\right)\delta E_{\mathit{\text{FRET}}}\left({\tau }_1+{\tau }_2\right)\delta E_{\mathit{\text{FRET}}}\left({\tau }_1+{\tau }_2+{\tau }_3\right)\rangle =\sum _{i,j,k,l=1}^M\delta E_{FRE{T}_l}{p}_{kl}\left({\tau }_3\right)\delta E_{FRE{T}_k}{p}_{jk}\left({\tau }_2\right)\delta E_{FRE{T}_j}{p}_{ij}\left({\tau }_1\right)\delta E_{FRE{T}_i}{p}_i^{eq}$$ (7)

Equation (7) expresses ${\overline{C}}^{(4)}\left({\tau }_1,{\tau }_2,{\tau }_3\right)$ in terms of the weighted average of four-point products of the observable δE_FRET, in which a transition occurs from macrostate i to macrostate j during the interval τ₁, a subsequent transition occurs from macrostate j to macrostate k during the period τ₂, and a final transition occurs from macrostate k to macrostate l during the interval τ₃. We may thus view the four-point TCF as the τ₁, τ₂, τ₃-dependent weighted average of four-point transitions (i.e., elementary kinetic pathways) that connect the M macrostates. As we discuss in the next section, the two-point and four-point TCFs described by Eqs. (5) and (7), respectively, in addition to the equilibrium PDF can be modeled theoretically using a kinetic master equation.

3.4. Master Equation Analysis.

To model the ensemble-averaged quantities discussed in the previous sections, it is necessary to determine for a given conformational reaction scheme the equilibrium distribution of macrostates, $p_i^{eq}$, and the time-dependent conditional probability, p_ij(τ), that the system will undergo a transformation from macrostate i to macrostate j during the interval τ. These probabilities are solutions to the master equation, which is the set of coupled first-order ordinary differential equations that describe the time-dependent macrostate populations.

$$\dot{p}(t)=Kp(t)$$ (8)

In Eq. (8), p(t) = [p₁(t), p₂(t), . . , p_M(t)] is the time-dependent macrostate population vector, and $\dot{p}(t)$ is its time-derivative. The M × M rate matrix K contains the rate constants k_ij according to

$$K=\left[\begin{array}{ccccc}-\sum _{i\ne j=1}^M{k}_{1j}& & k_{21}& & k_{M1}\\ & & & \cdots & \\ k_{12}& & -\sum _{i\ne j=1}^M{k}_{2j}& & k_{M2}\\ & ⋮& & \ddots & ⋮\\ k_{1M}& & k_{2M}& \cdots & -\sum _{i\ne j=1}^M{k}_{Mj}\end{array}\right]$$ (9)

The rate matrix K is written in the ‘site-basis’ of elementary kinetic steps that comprise the coupled reaction scheme. The off-diagonal elements K_ij = k_ij correspond to the gain in population of macrostate j due to the transition i → j. For cases in which there is no transition i → j, K_ij = 0. The diagonal elements are written $K_{ii}=-{\sum }_{j\ne i=1}^M{k}_{ij}$, and represent the sum of all reactions that deplete population from macrostate i. Equation (8) must satisfy completeness, ${\sum }_{i=1}^M{p}_i(t)=1$, and the detailed balance conditions $p_i^{eq}{k}_{ij}=p_j^{eq}{k}_{ji}$ with $p_i^{eq}=\underset{t\to \infty }{\text{lim}}{p}_i(t)$. Moreover, the vector of equilibrium macrostate populations p^eq can be obtained by solving the master equation subject to the time-stationary boundary condition $\dot{p}(t)=K{p}^{eq}=0$.

Given a particular set of rate constants, we solve Eq. (8) by finding the eigenvalues, λ₁, λ₂, … λ₃, and the eigenvectors, v₁, v₂, … , v₃, which satisfy the eigenvector equation Kv_i = λ_iv_i. The eigenvalues are equal to the characteristic relaxation rates of the coupled dynamical system, and the eigenvectors represent the collective modes (i.e., linear combinations of elementary transitions) associated with the characteristic relaxations. An important symmetry property of the rate matrix K is that the elements within each of its columns sum to zero (i.e., ${\sum }_{j=1}^M{K}_{ij}=0$), which results in λ₁ = 0 and the remaining eigenvalues λ₂, … λ_M < 0 (38).

A general solution to the master equation [Eq. (8)] is

$$p_n(t)=c_1^n{v}_1{e}^{{\lambda }_{1}t}+c_2^n{v}_2{e}^{{\lambda }_{2}t}+\cdots +c_M^n{v}_M{e}^{{\lambda }_{M}t}$$ (10)

where the constants $c_1^n,c_2^n,\dots ,c_M^n$ are expansion coefficients that depend on the specific initial condition, which we indicate by the index n (26). Equation (10) can be further simplified by making the substitution λ₁ = 0 and taking the infinite-time limit, so that $p^{eq}=c_1^n{v}_1$ for all possible initial conditions.

$$p_n(t)=p^{eq}+c_2^n{v}_2{e}^{{\lambda }_{2}t}+\cdots +c_M^n{v}_M{e}^{{\lambda }_{M}t}$$ (11)

The conditional probabilities needed to evaluate the two-point and four-point TCFs [Eqs. (5) and (7), respectively] are obtained by constraining Eq. (11) using appropriate boundary conditions. The conditional probability p_ij(t) is the probability that at time t, the population vector has unit occupancy of the jth macrostate, given that at time zero there was unit occupancy of the ith macrostate. Thus, for example, by applying the initial condition n = i to Eq. (11) with p_i(0) = 1 and p_j≠i(0) = 0, we may determine the expansion coefficients $c_2^i,c_3^i,\dots ,c_M^i$. The conditional probability p_ij(t) is then equal to the jth element of the population vector given by Eq. (11), according to

$$p_{ij}(t)=p_j^{eq}+c_2^i{v}_2^j{e}^{{\lambda }_{2}t}+\cdots +c_M^i{v}_M^j{e}^{{\lambda }_{M}t}$$ (12)

In principle, the above procedure can be used to determine the p_ij(t)s analytically for all of M² possible combinations of i, j ∈ {1,2, … , M}, provided the number of states M is relatively small (20). In practice, the p_ij(t)s are determined numerically by writing Eq. (11) in the eigenbasis of the K matrix and carrying out the similarity transformation p_n(t) = U[e^λt]U⁻¹p_n(0), where the unitary matrix U = [v₁, v₂, … , v_M] and the matrix [e^λt] is diagonal with non-zero elements ${\left[e^{\lambda t}\right]}_{ii}=e^{{\lambda }_{i}t}$ (25). Additional details of this procedure are given in the SI.

We note that in the infinite-time limit Eq. (12) recovers the expected equilibrium population values, $p_{ij}(t\to \infty )=p_j^{eq}$. Furthermore, the conditional probabilities p_ij(t) are each a sum of M – 1 decaying exponential functions of time, with decay constants equal to the eigenvalues of the K matrix. Substitution of Eq. (12) into Eq. (5) shows that the two-point TCF is also a sum of M – 1 exponential terms: ${\overline{C}}^{(2)}(\tau )={\mathcal{A}}_2{e}^{{\lambda }_2\tau }+{\mathcal{A}}_3{e}^{{\lambda }_3\tau }+\cdots +{\mathcal{A}}_M{e}^{{\lambda }_M\tau }$, where the ${\mathcal{A}}_i$s are the relative weights of the collective relaxation processes. Similarly, the four-point TCF can be shown to be a sum of (M – 1)² terms: ${\overline{C}}^{(4)}{\left({\tau }_1,{\tau }_3\right)}_{{\tau }_2\text{ fixed}}={\mathcal{A}}_{22}\left({\tau }_2\right)e^{{\lambda }_2\left({\tau }_1+{\tau }_3\right)}+{\mathcal{A}}_{23}\left({\tau }_2\right)e^{{\lambda }_2{\tau }_1+{\lambda }_3{\tau }_3}+{\mathcal{A}}_{32}\left({\tau }_2\right)e^{{\lambda }_3{\tau }_1+{\lambda }_2{\tau }_3}+\cdots +{\mathcal{A}}_{MM}\left({\tau }_2\right)e^{{\lambda }_M\left({\tau }_1+{\tau }_3\right)}$ (20). Thus, the number of exponential terms that contribute to the two-point and four-point TCFs indicate the minimum number of macrostates M that underlie the observed dynamics.

4. Results and Discussion

For each of the four ss-dsDNA constructs that we studied (see Table 1), we recorded time series of single-molecule photon data streams, each for a duration of thirty seconds (see Sect. 2.3). The results of our analyses indicate that all four of the ss-dsDNA constructs exhibited very similar behaviors, as one might expect when focusing on conformational changes within identical ssDNA sequences only, although the four constructs behave differently at slower time scales in the presence of ssb (gp32) proteins (14, 15). In the following discussion, which deals only with ssDNA conformational changes, we focus on the results of one of the ss-dsDNA constructs [3’-oligo(dT)₁₅]. We summarize our results for all four of the ss-dsDNA constructs that we studied in the SI. These additional data sets will serve as reference states in our subsequent studies of ssDNA-ssb interactions.

We begin by making some qualitative observations. In Fig. 2, we show an example trajectory of the donor D (Cy3, green) and acceptor A (Cy5, red) fluorescence intensities constructed from our raw single-molecule photon correlation measurements of the 3’-Cy3/Cy5-oligo(dT)₁₅-ss-dsDNA construct. The fluorescence intensities are calculated according to I_D(A)(t) = N_D(A)(t)⁄T_w, where N_D(A)(t) is the number of D (A) photons recorded within a finite sampling window T_w = 1 ms beginning at time t. We show an additional example trajectory using the same sampling window T_w = 1 ms, but for a duration of 8.5 s in Fig. S2 of the SI. The D and A fluorescence intensities fluctuate in an anti-correlated manner due to the varying inter-chromophore FRET coupling and indicates that these signals depend largely on conformational changes of the 3’-oligo(dT)₁₅ tail region of the ss-dsDNA construct.

Figure 2.

Example smFRET trajectory for the 3’-Cy3/Cy5-oligo(dT)₁₅-ss-dsDNA construct. The donor D (Cy3, green) and acceptor A (Cy5, red) fluorescence intensities, I_D(A)(t) = N_D(A)(t)⁄T_w, are shown with sampling window resolution T_w = 1 ms over a duration of ~ 2 seconds. The FRET efficiency E_FRET(t) is plotted in the bottom panel (blue). The average Signal-to-Noise Ratio (SNR) per sampling window is ${\overline{N}}_T^{1/2}\cong 4$.

The D and A intensities are used to calculate the FRET efficiency according to E_FRET(t) = I_A(t)⁄[I_D(t) + I_A(t)] (blue). For this particular photon data stream, the mean total signal flux was ${\overline{I}}_T={\overline{I}}_D+{\overline{I}}_A\approx 15,000{\text{s}}^{-1}$. We note that the apparent ‘noise’ of the trajectory is due to the measurement uncertainty associated with the finite number of photons sampled during the detection window T_w at a constant low-level flux. We assign the average Signal-to-Noise Ratio $\text{SNR}={\overline{N}}_T^{1/2}$ to the integrated signal shown in Fig. 2, where ${\overline{N}}_T={\overline{I}}_T{T}_w$ is the mean number of detected photons per sampling window. Thus, for T_w = 1 ms the average $\text{SNR}\approx \sqrt{15}\cong 4$, which ensures that the E_FRET signal is statistically meaningful. Upon further inspection of the E_FRET trajectory shown, we see that the majority of values are clustered near E_FRET ~ 0.7. Moreover, we observe that infrequent transitions occur to short-lived states with values centered near E_FRET ~ 0.5, and even less frequent transitions occur to short-lived states with E_FRET < 0.5. In the analysis that follows, we quantify the structural and dynamical information provided by the photon data stream using statistical functions.

We analyzed our raw photon correlation measurements by constructing probability distribution functions (PDFs) using the sampling window T_w = 1 ms, and two-point and four-point time-correlation functions (TCFs) using the sampling window T_w = 10 μs (see Fig. 3). In constructing these functions, we compared the results from individual single-molecule data sets to check for consistency before averaging these together. Typically, ~35 individual single-molecule data sets were used to compute each function. As discussed in further detail below, we performed numerical simulations of the PDFs and TCFs using the master equation approach described in Sect. 3.4. Additional details of the numerical methods we employed are described in the SI.

Figure 3.

Experimentally derived statistical functions constructed from single-molecule photon data streams of the 3’-Cy3/Cy5-oligo(dT)₁₅-ss-dsDNA construct. Fits to these functions are also shown, which are based on a three-macrostate kinetic network model. (A) The normalized histogram of E_FRET values using a sampling window T_w = 1 ms and 21 million photons is represented as a sum of three Gaussian macrostates, which are labeled S₁ – S₃. (B) The two-point time correlation function (TCF) (blue points) is well modeled as a weighted sum of two exponentially decaying terms (red curve) with decay constants that are well-separated in time. (C) The four-point TCF is shown as a three-dimensional rendering (red points), which is overlaid with the theoretical surface predicted by the three-state kinetic model (mesh surface). (D) The four-point TCF in comparison to the theoretical model are shown as two-dimensional contour plots. The experimental data are shown as white contours and the model as a solid color-coded surface. The optimized values obtained from the three-macrostate master equation for the equilibrium PDF and the rate constant parameters are given in Table 2 and Table 3, respectively.

In Fig. 3A, we show a normalized histogram of E_FRET values, which was compiled from the data streams of 35 individual single-molecule measurements of the 3’-Cy3/Cy5-oligo(dT)₁₅-ss-dsDNA construct. The E_FRET values were calculated by binning the D and A intensities using the sampling window T_w = 1 ms. This value for T_w was chosen because it is roughly equal to the time required for a randomly selected microstate to relax to one of the possible metastable intermediates available to the system. This is, equivalently, the time scale for which the probability to occupy any one macrostate becomes time-independent (see Fig. 4C and related discussion below). Each single-molecule photon data stream had an approximate mean flux ${\overline{I}}_T~20,000{\text{s}}^{-1}$, and was collected over an acquisition period of 30 s. Thus, the total number of single-photon measurements used to compile the histogram is approximately ~21 million. The shape of the PDF, which is mostly peaked around E_FRET ~ 0.7, but also exhibits a shoulder at E_FRET ~ 0.5 and a tail for E_FRET < 0.5, is consistent with our qualitative observations of the time-dependent behavior of the E_FRET trajectory shown in Fig. 2.

Figure 4.

Theoretical modeling results of microsecond-resolved smFRET measurements of the 3’-Cy3/Cy5-oligo(dT)₁₅-ss-dsDNA construct using the kinetic master equation. (A) Kinetic scheme depicting the elementary reactive steps connecting the three macrostates of the oligo(dT)₁₅ template: ‘compact’ S₁, ‘partially-extended’ S₂, and ‘highly-extended’ S₃. (B) The normalized probability distribution function (PDF) of E_FRET values obtained from an optimized fit to the histogram shown in Fig. 3A. The distribution is represented as a sum of the M = 3 Gaussian macrostates with areas equal to the equilibrium probabilities $p_i^{eq}$, with i ∈ {1,2,3}. (C) The optimized solution to the master equation is the equilibrium PDF, p^eq, and the time-dependent conditional probabilities, p_ij(t), which give the likelihood that the system occupies the jth macrostate at time t if it previously occupied the ith macrostate at time 0, with i, j ∈ {1,2,3}. The equilibrium values are established on the time scale of ~1 ms. (D) The free energy landscape is constructed from the PDF shown in panel B according to the Boltzmann distribution [Eq. (2)]. Intersecting dashed purple lines indicate the entropic contributions to the free energy barriers that separate macrostates. Intersecting solid curves indicate the total free energy barriers calculated from the optimized rate constants according to the Arrhenius equation [Eq. (3)]. The optimized values obtained for the equilibrium PDF are given in Table 2, those for the rate constants are given in Table 3, and the free energy minima and transition barriers are given in Table 4. Estimated enthalpic and entropic contributions to the transition barriers are given in Table 5.

The sub-millisecond dynamics of the system are characterized by the two-point and four-point TCFs. As we discuss in further detail below, the TCFs exhibited two principal decay components (20.3 μs and 272 μs), which are well-separated in time (see Fig. 3B and related discussion below). As discussed in Sect. 3.4, the number of exponential terms that contribute to the TCFs (= M – 1) indicate the minimum number of macrostates M that may underlie the observed dynamics. Based on this observation, we adopted a kinetic network scheme composed of M = 3 macrostates in thermal equilibrium as the simplest possible model to interpret our data. The presence of two well-separated relaxation components in the TCFs is our primary justification for applying this three-macrostate model. We have thus described the PDF as a sum of three Gaussians, as indicated in Fig. 3A. We note that the shapes of the PDFs for all four ss-dsDNA constructs investigated were very similar, as one might expect given that the base sequences and lengths of the four oligo(dT)_n templates are nearly identical (see Fig. S3 of the SI). We next discuss the kinetic analyses of these data, which provided optimized parameters for the Gaussian decomposition of the PDFs of the ss-dsDNA constructs (Table S1 of the SI), and we list the results for the 3’-Cy3/Cy5-oligo(dT)₁₅-ss-dsDNA construct in Table 2.

Table 2.

Optimized parameters for the Gaussian components (based on the three-macrostate model) of the FRET probability distribution function (PDF) shown in Fig. 3A. Parameters defined in Eq. (2). Error bars for the mean FRET efficiencies are ~0.005 (see Fig. S11 and Table S10 of the SI).

Construct	A 1	〈EFRET〉1	σ 1	$P_1^{eq}$	A 2	〈EFRET〉2	σ 2	$P_2^{eq}$	A 3	〈EFRET〉3	σ 3	$P_3^{eq}$
3’oligo(dT)15	3.04	0.77	0.11	0.81	1.00	0.52	0.07	0.18	0.08	0.23	0.07	0.01

We calculated the experimental two-point TCFs [defined by Eq. (4)] from the time-dependent photon data streams of our single-molecule measurements (see SI for further details). Similar to the behavior of the PDFs discussed above, we found that the two-point TCFs of the four ss-dsDNA constructs exhibited very similar decay dynamics, which spanned the range of time scales: 1 μs – 1 ms. On the fastest time scale of 1 – 10 μs, the TCFs exhibited a relaxation component on the order of ~3 μs, as shown in Fig. S4 of the SI. We fit the first 20 microseconds of the TCFs to a model mono-exponential decay, and the optimized time constants for each of the ss-dsDNA constructs are listed in Table S2 of the SI. Relaxations on the time scale of ~3 μs have been observed previously in photophysical studies of the Cy3 chromophore, and are known to be associated with excited state intersystem crossing and forward photoisomerization, which undergo the reverse step of ground state recovery within ~10 μs (39–43). This photoisomerization leads to quenching of the Cy3 donor chromophore excited state, which leads to a change in the relative intensities of Cy3 donor and Cy5 acceptor fluorescence in our ss-dsDNA constructs, and thus to a change in the FRET efficiency. We therefore assigned the ~3 μs decay in our current studies to photoisomerization of the Cy3 chromophore, and we assumed that these relatively fast processes contributed to the ‘time-averaged’ background of the signal fluctuations that we observed on tens-of-microseconds and longer time scales, and which we interpreted to be due to structural changes of the oligo(dT)_n templates.

In Fig. 3B, the blue data points show the experimental two-point TCF over the range of time scales from 20 μs – 1 ms, which was calculated from the same 35 single-molecule measurements of the 3’-Cy3/Cy5-oligo(dT)₁₅-ss-dsDNA construct used to compile the histogram shown in Fig. 3A. Although the majority of TCFs calculated from the individual single-molecule data streams decayed to zero, as expected according to Eq. (4), a small number of single-molecule data streams resulted in non-zero asymptotic behavior due to variation in measurement quality, so that it was necessary to include a constant offset B in our model fits. Upon modeling the average experimental decay using a tri-exponential function, we obtained very similar values of the relaxation time constants, although the quality of the fit (as determined by the goodness-of-fit parameter R²) did not improve (see Fig. S5 and Table S3 of the SI). From these analyses, we concluded that the TCFs contain two well-separated principal decay components, which justifies our use of the three-macrostate model to interpret simultaneously dynamic and equilibrium properties of the system. Thus, Fig. 3B shows the optimized three-macrostate model fit for the two-point TCF given by ${\overline{C}}^{(2)}(\tau )={\mathcal{A}}_2{e}^{{\lambda }_2\tau }+{\mathcal{A}}_3{e}^{{\lambda }_3\tau }+B$, with relaxation time constants [eigenvalues of the rate matrix, Eq. (9)] ${\lambda }_3^{-1}=20.3\mu \text{s}$ and ${\lambda }_2^{-1}=273\mu \text{s}$, as shown, and the constant B, as discussed above. A comparison between the two-point TCFs and their model fits for all four ss-dsDNA constructs, including plots extending over the full range of time scales (20 μs – 3 s), are shown in Fig. S6 of the SI, and the corresponding model fit parameters are listed in Table S4 of the SI.

We next calculated the four-point TCFs from our experimental data according to Eq. (6). In Fig. 3C, we show a three-dimensional rendering of this function, ${\overline{C}}^{(4)}\left({\tau }_1,{\tau }_2=0,{\tau }_3\right)$, where we have set the intermediate time interval τ₂ = 0. These data are shown overlaid with a two-dimensional model function, ${\overline{C}}^{(4)}\left({\tau }_1,{\tau }_2=0,{\tau }_3\right)={\mathcal{A}}_{22}{e}^{{\lambda }_2\left({\tau }_1+{\tau }_3\right)}+{\mathcal{A}}_{23}{e}^{{\lambda }_2{\tau }_1+{\lambda }_3{\tau }_3}+{\mathcal{A}}_{32}{e}^{{\lambda }_3{\tau }_1+{\lambda }_2{\tau }_3}+{\mathcal{A}}_{33}{e}^{{\lambda }_3\left({\tau }_1+{\tau }_3\right)}+D$, which is constructed from the optimized solution to the kinetic master equation. In Fig. 3D, we show the experimental and simulated four-point TCFs as a two-dimensional contour plot. We note that the three-dimensional surface of ${\overline{C}}^{(4)}$ exhibits pronounced convex contour lines (corresponding to negative values of ${\mathcal{A}}_{23}={\mathcal{A}}_{32}$, see Table S5), which indicate the presence of kinetic ‘bottleneck’ intermediates in the reaction pathway that inhibit the rapid exchange of population among the various macrostates of the system (20). We show the four-point TCFs and their model fits to all four ss-dsDNA constructs in Fig. S7 of the SI, and the corresponding model fit parameters are listed in Table S5 of the SI. The combined statistical functions shown in Fig. 3 serve to constrain a multi-parameter optimization of solutions to the master equation.

As stated above, the presence of two decay components in the two-point TCFs implies that there must be at least three conformational macrostates between which the oligo(dT)_n templates interconvert. Moreover, the three-state model is consistent with the histogram of E_FRET values (see Fig. 3A), in which the underlying Gaussian features, labeled S₁ – S₃, may be assigned to a ‘compact’ macrostate, a ‘partially-extended’ macrostate, and a ‘highly-extended’ macrostate, respectively. The compact macrostate S₁ has the largest mean E_FRET value, since it contains chain configurations with relatively short D – A chromophore distances. The partially-extended macrostate S₂ and the highly-extended macrostate S₃, on the other hand, have smaller mean E_FRET values, indicating chain configurations with correspondingly larger D – A separations.

In Fig. 4A, we depict the coupled three-macrostate scheme that we used to interpret our results. The solutions to the M = 3 master equation [Eq. (8)] can be written analytically in terms of the rate constants of elementary reactive steps (20). We performed numerical simulations of Eq. (8) by searching the parameter space of input rate constants and E_FRET values, as described in our prior work (14, 20). The solutions to the master equation are the time-dependent conditional probabilities p_ij(t) [Eq. (12)] and the equilibrium distribution of macrostates $p^{eq}=\left(p_1^{eq},p_2^{eq},p_3^{eq}\right)$ [Eq. (11)], which may be used to simulate expressions for the PDF and the two-point and four-point TCFs described in Sect. 3.3. We performed an iterative search of the parameter space in which the statistical functions p(E_FRET), ${\overline{C}}^{(2)}(\tau )$ and ${\overline{C}}^{(4)}\left({\tau }_1,0,{\tau }_3\right)$, shown in Fig. 3 for the 3’-Cy3/Cy5-oligo(dT)₁₅-ss-dsDNA construct, were calculated for a given set of input rate constants and mean E_FRET values, which were subsequently compared to the experimentally-derived functions. The agreement between simulated and experimental functions was quantified using a linear least squares target function χ², which was minimized using a multi-parameter regression algorithm to determine an optimized solution to the master equation (see SI for further details). We compared the results of our analyses using linear versus cyclical three-state network models, which showed that the cyclical model is favored. We confirmed that our optimized results represent a global minimum of the parameter space by monitoring the increase of the χ² error function with variation of the time constant parameters (see Fig. S11 of the SI). The equilibrium populations and kinetic time constants resulting from of our analysis of the 3’-Cy3/Cy5-oligo(dT)₁₅-ss-dsDNA construct are summarized in Table 2 and Table 3, respectively, and those of all four ss-dsDNA constructs are presented in Table S1 and Table S6 of the SI.

Table 3.

Optimized forward and backward time constants corresponding to the elementary reactive steps obtained for the 3’-Cy3-Cy5-oligo(dT)₁₅-ss-dsDNA construct. The time constant parameters are defined in Eq. (9). Error bars were determined from the cross-sections of the error function surfaces $\Delta {\chi }^2/{\chi }_0^2$, shown in Fig. S11 of the SI.

Construct	$k_{12}^{-1}(\text{ms})$	$k_{21}^{-1}(\text{ms})$	$k_{13}^{-1}(\text{ms})$	$k_{31}^{-1}(\text{ms})$	$k_{23}^{-1}(\text{ms})$	$k_{32}^{-1}(\text{ms})$
3’-oligo(dT)15	0.113 ±0.004	0.025 ±0.0005	104 ±1.85	1.85 ±0.03	4.01 ±0.03	0.325 ±0.03

In Fig. 4B, we show the model equilibrium PDF obtained from our optimization analysis of the 3’-Cy3/Cy5-oligo(dT)₁₅-ss-dsDNA construct using the three-macrostate scheme. We have modeled the PDF as a sum of three Gaussian components, with normalized areas equal to the equilibrium populations of the three macrostates (see Table 2). Thus, macrostate S₁ is the dominant component with $p_1^{eq}=0.81$ and mean FRET efficiency 〈E_FRET〉₁ = 0.77 ± 0.01, macrostate S₂ is a minority component with $p_2^{eq}=0.18$ and mean FRET efficiency 〈E_FRET〉₂ = 0.52 ± 0.07, and macrostate S₃ is a trace component with $p_3^{eq}=0.01$ and mean FRET efficiency 〈E_FRET〉₃ = 0.23 ± 0.07. These optimized Gaussian parameters can be used to estimate the relative free energy minima of the macrostates using the Boltzmann relation [Eq. (2)].

In Fig. 4D, we plot the individual macrostate contributions to the free energy surface of the 3’-Cy3/Cy5-oligo(dT)₁₅-ss-dsDNA construct, which are shown as dashed purple curves. In Fig. S8 of the SI, we compare the free energy surfaces for all four ss-dsDNA constructs. The sum of the macrostate contributions is shown as a solid blue curve. The three macrostates are represented by parabolas with minimum values $G_1^o=0$ (taken as the ground state), $G_2^o=1.11k_{\text{B}}T$ and $G_3^o=3.62k_{\text{B}}T$. Here, the superscript ‘o’ indicates that we have taken the minimum of macrostate S₁ as the reference for all other energies. The parabolic dependence of the free energy with respect to the E_FRET parameter is due to our assumption of a Gaussian distribution of end-to-end distances of the oligo(dT)_n template for a given macrostate. The Gaussian distribution describes approximately the relative probability to observe a microscopic chain configuration with an end-to-end distance (and a corresponding E_FRET value) that deviates from the mean value, and thus accounts for the entropic contribution to the free energy penalty associated with chain elongation or compression (10). The Gaussian model is, of course, accurate for long ideal polymer chains, and ignores the effects of excluded volume and other internal chain segment interactions. However, modified Gaussians with non-zero origins can describe chain molecules of finite length, and account for the effects of intramolecular interactions. For the oligo(dT)_n template strands, the random walk statistics of chain configurations is additionally influenced by attractive and repulsive interactions between non-adjacent nucleotides and the tendency of adjacent bases to stack locally. The presence of three distinct macrostates, each with a different mean end-to-end distance, suggests that there exist chain configurations of the oligo(dT)_n templates with characteristically different local flexibilities, which are (presumably) due to the effects of different distributions of stacked bases. In the context of the Gaussian macrostate model, the points at which the component parabolas (dashed purple curves) cross in the free energy surface approximate the entropic contributions to the transition state barriers for macrostate interconversion. However, the true free energy barriers are likely to include enthalpic contributions associated with the energy needed to rearrange the internal constraints imposed by a given distribution of stacked bases associated with a given macrostate. In this way, the Gaussian macrostate model can be used to estimate the enthalpic contributions to the free energy barriers. We next examine the free energy barriers of activation that result from the kinetic analysis of our data.

In Fig. 4C, we show the time-dependence of the optimized conditional probabilities p_ij(t) where i, j ∈ {1,2,3}. These functions illustrate the kinetic behavior of the three-macrostate system. The equilibrium populations are the asymptotic values approached by the conditional probabilities in the long-time limit. For example, the diagonal terms p_ii(t) decay from unity to the equilibrium population value $p_i^{eq}$ on the time scale ${\left[{\sum }_{j\ne i=1}^3{k}_{ji}\right]}^{-1}$, while off-diagonal terms p_ji(t) increase from zero to the equilibrium population value $p_i^{eq}$ on the time scale $k_{ji}^{-1}$. The relaxation dynamics of all nine conditional probability terms occur primarily on sub-millisecond time scales, such that the equilibrium (time-independent) values are established on the time scale of ~1 ms. The optimized values of the time constants for the 3’-Cy3/Cy5-oligo(dT)₁₅-ss-dsDNA construct are presented in Table 3, and the time constants for all four of the ss-dsDNA constructs are listed in Table S6 of the SI. It is interesting to note that the fastest interconversion processes are between macrostates S₁ and S₂ ($k_{12}^{-1}=113\mu \text{s}$ and $k_{21}^{-1}=25\mu \text{s}$), while the slowest interconversions are between macrostates S₁ and S₃ ($k_{13}^{-1}=104\text{\hspace{0.17em}ms}$ and $k_{31}^{-1}=1.85\text{\hspace{0.17em}ms}$) and between macrostates S₂ and S₃ ($k_{23}^{-1}=4.01\text{\hspace{0.17em}ms}$ and $k_{32}^{-1}=325\mu \text{s}$).

From the optimized time constants, we determined the free energies of activation $G_{ij}^{o‡}$ using the Arrhenius relation [Eq. (3)]. In calculating the various values for $G_{ij}^{o‡}$, we have assumed that the fastest rate constant resulting from our analysis ($k_{21}^{-1}=25\mu \text{s}$) for the transition S₂ → S₁ corresponds to the formation of a single TT stacked base pair with transition barrier $G_{21}^{o‡}-G_2^o\cong 1{\text{ kcal mol}}^{-1}=1.7k_{B}T$ (44) such that $k_{max}^{-1}\equiv k_{21}^{-1}{e}^{-1.7}=4.6\mu \text{s}$. We make this assumption based on the premise that the underlying differences in the secondary structures of the three macrostates is the mean number of stacked bases, and that interconversion between macrostates involves the formation or disruption of stacked base pairs. The remaining activation barrier heights are measured relative to this value. In Table 4, we list these activation energies for the 3’-Cy3/Cy5-oligo(dT)₁₅-ss-dsDNA construct, and we compare the results of all four ss-dsDNA constructs in Table S7 of the SI.

Table 4.

Optimized free energy minima and transition barriers for the 3’-Cy3/Cy5-oligo(dT)₁₅-ss-dsDNA construct. The S₁ macrostate is taken to be the ground state and excited state energies are given by $G_i^o=-k_{B}Tln\left(p_i^{eq}/{\sigma }_i\sqrt{2\pi }\right)-G_1$ [Eq. (2)] with parameters $p_i^{eq}$ and σ_i listed in Table 2. The activation energies are determined by $G_{ij}^{o‡}=-k_{B}Tln\left(k_{ij}/k_{max}\right)$ [Eq. (3) of the main text, with k_max ≡ k₂₁e^1.7, which assumes that the fastest rate constant of the system corresponds to the formation of a single TT stacked base pair with activation barrier ~1 kcal mol⁻¹ = 1.7 k_BT (44)]. The rate constant parameters are listed in Table S6. Energies are listed in units of k_BT.

Construct	$G_1^o$	$G_2^o$	$G_3^o$	$G_{12}^{o‡}$	$G_{21}^{o‡}$	$G_{13}^{o‡}$	$G_{31}^{o‡}$	$G_{23}^{o‡}$	$G_{32}^{o‡}$
3’-oligo(dT)15	0	1.11	3.62	3.22	2.81	10.04	9.63	7.89	7.89

We plot the barrier heights as horizontal lines in the free energy diagram shown in Fig. 4D. We note that the transition barrier corresponding to the uphill reaction S₁ → S₂ is $G_{12}^{o‡}-G_1^o=3.22k_{B}T$ (green curve) exceeds the entropic contribution $-T{S}_{12}^{o‡}\approx 1.50k_{B}T$ given by the intersection between the Gaussian components. We thus estimate the enthalpic contribution to the activation barrier for the S₁ → S₂ transition as $H_{12}^{o‡}\approx 1.72k_B\text{T}$. Similarly, the backward transition S₁ ← S₂ has $G_{21}^{o‡}-G_2^o=1.70k_{B}T$ (by assumption) such that $G_{21}^{o‡}=2.81k_{B}T$ and $H_{21}^{o‡}\approx 1.29k_{B}T$. The transition barriers corresponding to the forward and backward reactions S₂ ⇌ S₃ are $G_{23}^{o‡}-G_2^o=6.78k_{B}T$ and $G_{32}^{o‡}-G_3^o=4.27k_{B}T$, respectively. The absolute barrier heights corresponding to these values (= 7.89 k_BT in both cases) are indicated on the free energy diagram in Fig. 4D (orange curve). These barriers exceed the estimated entropic contribution $-T{S}_{23}^{o‡}\approx 4.65k_{B}T$, suggesting that the forward and backward transitions between macrostates S₂ and S₃ requires $H_{23}^{o‡}\approx H_{32}^{o‡}\cong 3.21k_{B}T$ of internal redistribution energy. Similarly, the forward and backward transition barriers separating macrostates S₁ and S₃ are $G_{13}^{o‡}-G_1^o=10.04k_{B}T$ and $G_{31}^{o‡}-G_3^o=6.01k_{B}T$, respectively. The absolute barrier heights corresponding to these values (10.04 k_BT and 9.63 k_BT, respectively) are plotted on the free energy surface shown in Fig. 4D as a red curve, and exceed the estimated entropic contribution $-T{S}_{13}^{o‡}\approx 6.28k_{B}T$, suggesting that the enthalpic contribution to the activation energies are $H_{13}^{o‡}\approx 3.76k_{B}T$ and $H_{31}^{o‡}\approx 3.35k_{B}T$.

We list the estimated enthalpic and entropic contributions to the activation barriers for the 3’-Cy3/Cy5-oligo(dT)₁₅-ss-dsDNA construct in Table 5 and for all four ss-dsDNA constructs in Table S8 of the SI. These values are very similar for the four constructs, as expected. The enthalpic contributions to the forward and backward activation barriers for the S₁ ⇌ S₂ reaction are within the range ~1 – 1.7 k_BT, which follows from our assumption that $G_{21}^{o‡}-G_2^o=1.7k_{B}T$ (44). This value is comparable in magnitude to the enthalpy of base unstacking obtained for TT base pairs (1.04 kcal mol⁻¹ ≈ 1.7 k_BT) in thermal denaturation studies of nicked DNA (45). We note that the enthalpic contributions to the forward and backward activation barriers for both the S₁ ⇌ S₃ and the S₂ ⇌ S₃ processes are comparable to one another and within the range ~3 – 5 k_BT, which is approximately twice the size of the enthalpic barriers for the S₁ ⇌ S₂ processes.

Table 5.

Estimated transition barrier enthalpies and entropies for the 3’-Cy3/Cy5-oligo(dT)₁₅-ss-dsDNA construct, which are based on the Gaussian macrostate model. The entropic contribution to the transition barrier is defined $-T{S}_{ij}^{o‡}=G_{ij}^{\text{intersect}}$ with $G_{ij}^{\text{intersect}}$ the free energy at the point of intersection between the parabolas describing macrostate-i and macrostate-j (dashed purple curves in Fig. 4D). The enthalpy of transition is estimated to be $H_{ij}^{o‡}=G_{ij}^{o‡}-G_{ij}^{\text{intersect}}$. Enthalpic and entropic contributions to the transition barriers are given in units of k_BT.

Construct	$-T{S}_{12}^{o‡}$	$-T{S}_{13}^{o‡}$	$-T{S}_{23}^{o‡}$	$H_{12}^{o‡}$	$H_{21}^{o‡}$	$H_{13}^{o‡}$	$H_{31}^{o‡}$	$H_{23}^{o‡}$	$H_{32}^{o‡}$
3’-oligo(dT)15	1.50	6.28	4.65	1.72	1.29	3.76	3.35	3.21	3.21

5. Conclusions

In this paper we use microsecond-resolved smFRET experiments, recently developed in our laboratory, to begin to characterize conformational fluctuations in the structures of ssDNA at low concentrations in dilute aqueous solution. This is important for studies of protein-DNA interactions because – unlike dsDNA – the interactions stabilizing ssDNA conformations are not cooperative, and therefore conformations that exist at equilibrium, even at trace concentrations, are measurable with our present techniques. These results are likely to be significant from a biological perspective because our approach extends smFRET techniques to the microsecond time regime. By expanding the range of experimentally accessible time scales, it is thus possible to monitor not only the binding and unbinding of protein components of protein-DNA complexes on millisecond time scales and longer, but also to observe directly the interconverting conformational species (on sub-millisecond time scales) of ssDNA that comprise the interaction partners in the initial steps of biologically relevant protein-ssDNA interactions.

As described in the Introduction, and also in previous papers (14, 19, 20, 27), we have been using smFRET methods to study, in particular, the initial steps in the binding, assembly, and translocation of the bacteriophage T4 ssb proteins (gp32) and their cooperatively-bound clusters onto and along ssDNA lattices. Understanding the molecular details of these interactions with both the ssDNA lattices and ss-dsDNA junctions, which are present in the DNA constructs that we have studied, are central to obtaining molecular insight into the regulatory mechanisms of these ssb proteins as they interact with and control the functions of the other components of the T4 replisome – including the helicase and primosome, helicase loader proteins, the DNA polymerases and the processivity clamps and clamp loaders.

In the current paper, we describe microsecond-resolved studies on the ssDNA ‘tails’ of the ss-dsDNA constructs that we have used previously in our studies of the gp32 binding and assembly system (see Fig. 1). We studied short ss oligo-deoxythymidine [oligo(dT)_n] templates of varying length (n = 14 and 15 nts) and polarity (3’ versus 5’, Table S1 of the SI) because gp32 binding and assembly onto these ss-dsDNA constructs depends sensitively on these factors. As we have shown, and as one might expect, these differences in the details of the constructs do not significantly change the equilibrium distribution of the template conformations, nor the dynamics of state-to-state interconversion. Nevertheless, it was necessary to investigate all four ss-dsDNA constructs to provide a baseline for the ssb interaction studies that will follow. Thus, the present study enables us to re-examine the ssb-ssDNA interaction systems with the ability to determine how ssb protein binding and assembly is directly coupled to the dynamics of the DNA interaction partners.

The oligo(dT)_n templates used in the present study comprised the ssDNA ‘tails’ of ss-dsDNA constructs, which had been labeled at their ends with a Cy3 donor chromophore and a Cy5 acceptor. We analyzed our data by constructing statistical functions, i.e., the equilibrium probability distribution function (PDF) of FRET efficiency values, with time resolution of 1 ms, and the two-point and four-point time correlation functions (TCFs) with time resolution of 10 μs. We found that the TCFs exhibited two well-separated principal decay components with time constants 20.3 μs and 272 μs (Fig. 3B). This observation suggests that the system can be described as a kinetic network of three coupled macrostates in thermal equilibrium. We simulated these functions using a kinetic network model based on a master equation to obtain optimized parameters from which we have reached the following conclusions.

Our results indicate that the oligo(dT)_n templates of all four of the ss-dsDNA constructs used are dynamic systems with three distinct conformational macrostates: a ‘compact’ form S₁, a ‘partially-extended’ form S₂, and a ‘highly-extended’ form S₃ (see Fig. 4A). These macrostates are separated in relative stability by just a few thermal energy units: $G_1^o=0k_{B}T$, $G_2^o=1.11k_{B}T$, and $G_3^o=3.62k_{B}T$ in the case of the 3’-Cy3-Cy5-oligo(dT)₁₅-ss-dsDNA construct (see Fig. 4D and Table 4). The time scales of interconversion between these macrostates range from tens-to-hundreds of microseconds (see Fig. 4C and Table 3). The fastest relaxation time that we observed ($k_{21}^{-1}=25\mu \text{s}$) corresponds to the transition from the partially-extended macrostate S₂ to the compact macrostate S₁, which is comparable to the relaxation times found for ssDNA loop closure in the formation of a DNA hairpin composed of an oligo(dT)₁₆ loop (46, 47), and also for single-strand helix-coil transitions in oligoadenylates of various sizes (48).

The relative stabilities and relatively long lifetimes of the three macrostates may be related to the energetic cost of unstacking and stacking adjacent bases. We note that the relative stabilities of the three macrostates are comparable to the enthalpy of base unstacking for a single TT pair (1.04 kcal mol⁻¹ ≈ 1.7 k_BT), which was obtained from thermal denaturation studies of the bent-to-stacked equilibrium in nicked DNA (45). From the kinetic rate constants (Table 3), and by assuming that the fastest rate constant resulting from our analysis for the transition S₂ → S₁ corresponds to the formation of a single TT stacked base pair with transition barrier ~1 kcal mol⁻¹ = 1.7 k_BT (44), we determined the free energies of activation (Table 4). By applying the Gaussian macrostate model, we inferred the entropic and enthalpic contributions to the barrier heights (Table 5). The forward and backward enthalpies of activation that separate the highly-extended macrostate S₃ from the partially-extended and compact macrostates, S₂ and S₁, respectively, have values within the range ~3.2 – 3.8 k_BT. If we consider the barriers for all four of the ss-dsDNA constructs (see Table S8 of the SI), the range of forward and backward enthalpy values are from 2.9 – 4.8 k_BT. The above estimates of the enthalpic contributions to the activation barriers suggest that only a small number of base stacking interactions are involved in these transformations.

The notion that multiple extended conformations of short oligomeric ssDNA templates exist as stable macrostates at equilibrium differs from our previous assumptions about short ssDNA lattices, in which only a single, relatively compact macrostate has been considered to be the primary conformation (14). It is interesting to note that previous workers have concluded from spectroscopic studies of model ssDNA constructs the existence of three distinct macrostates related to base-stacking interactions. Pörschke used sub-microsecond temperature-jump and UV hypochromicity experiments to observe two distinct relaxation processes in oligoadenylates, and inferred the presence of three different molecular states (48). Wu et al. performed time-resolved fluorescence measurements of a fluorescent base analogue-substituted ssDNA decamer and observed three distinct lifetime components, to which they assigned a ‘fully extended and completely unstacked state,’ a ‘loosely associated state,’ and a ‘fully stacked state’ (49). Since these studies were performed by ensemble methods, they provide insights about the average dynamic behavior of the oligomer single-strand (48) and the local environment experienced by a fluorescent base analogue probe, which is substituted within a single-strand (49). In contrast, our current studies are based on single-molecule FRET measurements that are sensitive to the distribution of end-to-end distances, which are affected directly by the tendency of flanking bases to undergo local stacking interactions.

The parameters that we have determined to characterize the free energy surface suggest that a new model for ssDNA secondary structure may be required (see Fig. 5). In order to gain additional insight into the nature of the extended macrostates, we carried out a coordinate transformation (25) of the free energy surface as a function of the FRET efficiency (shown in Fig. 4D) to one as a function of the donor – acceptor separation, which we equate with the end-to-end distance, R_ee, of the oligo(dT)₁₅ template (Fig. 5A). In performing this transformation, we used a value of the Förster transfer distance R₀ = 56 Å obtained by others, which assumes that the relative orientation of the Cy3 donor and Cy5 acceptor, which are attached to the ss-dsDNA constructs using flexible linkers, is random and that the orientation factor κ² = 2/3 (32, 33). From the transformed free energy surface, we see that the local minima of the three macrostates, S₁ – S₃, may be assigned to mean end-to-end distances 〈R_ee〉 = 46.5, 56.0 and 67.5 Å, respectively.

Figure 5.

(A) Transformed free energy surface as a function of the end-to-end separation, R_ee, of the oligo(dT)₁₅ template. The transformation is given by P(R_ee) = P(E_FRET) dE_FRET⁄dR_ee with $R_{ee}=R_0{\left[\left(1-E_{\mathit{\text{FRET}}}\right)/E_{\mathit{\text{FRET}}}\right]}^{\frac{1}{6}}$ and R₀ = 56 Å, and assumes that the relative orientation between the FRET Cy3 donor and Cy5 acceptor is random such that the orientation factor κ² = 2/3 (32, 33). (B) Schematic microstate configurations of the oligo(dT)₁₅ template exemplifying the three macrostates S₁ – S₃ described by the free energy surface shown in panel (A). In B-form DNA (upper left), the mean distance (rise per residue) between adjacent bases is 3.4 Å. In the oligo(dT)₁₅ template strands (labeled S₁ – S₃), adjacent bases are primarily unstacked with a mean inter-base distance of ~4.6 Å. Unstacked bases are stabilized by the gain in entropy associated with the resultant enhanced rotation about the segmental axes. A sequence of unstacked bases forms a contiguous linear segment. The presence of a single stacked base ‘defect site,’ introduces a ‘vertex’ with an accompanying change in the bond vector.

We next consider how the extended macrostates S₂ and S₃ might play a biological role as intermediates for protein binding mechanisms (50, 51). For example, the T4 ssDNA binding (ssb) protein may be able to access its 7 nt binding site footprint from extended macrostate conformations, S₂ and S₃, more readily than from the compact macrostate S₁. It is known from electron microscopy studies of cooperatively bound ssb nucleoprotein filaments that adjacent nucleotide bases within the complexes are fully unstacked with an average inter-base separation of 4.6 Å (5). This contrasts with the 3.4 Å separation between stacked bases in B-form duplex DNA. The oligo(dT)₁₄ and oligo(dT)₁₅ templates examined in this study were chosen specifically to permit two ssb proteins to bind cooperatively within the final-formed nucleoprotein complexes. In the case of the 3’-Cy3/Cy5-oligo(dT)₁₅-ss-dsDNA construct, the contour length of the ssDNA component within a cooperatively bound (ssb)₂-oligo(dT)₁₅ filament is expected to be approximately ~15 × 4.6 Å = 69 Å. Given that the mean end-to-end distance corresponding to macrostate S₃ (R_ee = 67.5 Å) is only slightly smaller than the contour length of the fully unstacked oligo(dT)₁₅ template strand, we suggest that the extended macrostates do indeed represent potential loading sites for ssb nucleoprotein filament assembly.

Considering these findings, we propose a new model for ssDNA secondary structure, which is depicted schematically in Fig. 5B. We posit that at physiological conditions, an isolated ssDNA template exists at thermal equilibrium with a significant fraction of its flanking bases unstacked. This hypothesis is consistent with the findings of Pollack and co-workers, who combined small angle X-ray scattering of oligo(dT)₃₀ and oligo(dA)₃₀ templates with ensemble conformational analyses (52). In the B-form DNA duplex, the highly ordered secondary structure of the double-helix is stabilized by cooperative short-range attractive interactions (stacking) between flanking bases (depicted as blue purines and red pyrimidines) and the network of Watson-Crick hydrogen bonds between opposing bases of the complementary single-strands. The duplex is marginally destabilized by the Coulomb repulsions between negatively charged phosphates (shown as yellow circles) of the sugar-phosphate backbones, which are partially screened by the condensation layer of positively charged counterions in the immediate aqueous surroundings.

We conjecture that adjacent stacked bases within an isolated segment of ssDNA, which lack the stabilizing influence of the complementary strand of duplex DNA, are only marginally stabilized with respect to thermal fluctuations that disrupt the short-range (solvophobic) attraction between base surfaces (53). Unstacked bases within an isolated segment of ssDNA are energetically favored by the Coulombic repulsion between adjacent phosphates and the gain in entropy associated with enhanced rotational freedom about the segmental axes. We emphasize that the attractive base-stacking interactions are short range, and occur primarily at contact, while the Coulombic repulsion between adjacent phosphates in the backbone is a long-range effect, which dominates and stabilizes the extended forms of ssDNA segments. A sequence of metastable unstacked bases thus produces an essentially linear contiguous segment, which may be disrupted by the spontaneous formation of a stacking interaction between two adjacent bases along the ssDNA chain. The presence of such a vicinal stacked pair of bases introduces a ‘defect site,’ which then induces a change in the bond vector of two adjoining segments, each composed of unstacked bases. In Fig. 5B, we depict three examples of such microstate ssDNA conformations, which may each be assigned to one of the three macrostates S₁ – S₃, as shown. Each macrostate can be viewed as a stable family of microstates that contains a unique, dynamically equilibrated distribution of stacked base ‘defect sites.’ The principal difference between the underlying distributions of the three macrostates is the mean number of stacked base defect sites, whose presence effectively disrupts the linearity of the oligo(dT)₁₅ template, and thereby reduces the end-to-end distance. The mean number of defect sites is maximized for the most stable macrostate S₁ (depicted Fig. 5B as having three defect sites). The ground state S₁ minimum represents the end-to-end distance (〈R_ee〉 = 46.5) corresponding to balance between stabilizing and de-stabilizing (entropic and enthalpic) interactions, which arise in the presence of the dynamically interconverting stacked base defect sites. We further note that in the context of this model, conformational transformations between any two of the three macrostates, S₁ – S₃, correspond to the loss or gain of a discrete number of stacked base defect sites, which is fully consistent with the values that we have determined from our analyses for the relative stabilities and transition enthalpy barriers of the macrostates.

The interpretation of these experiments in terms of the three-macrostate model is based on a theoretical analysis in which we have made physically motivated and valid assumptions. However, the three-macrostate model is not intended to represent the only possible interpretation of these data, since more complex models cannot be ruled out. Nevertheless, the model presented herein is intended to represent the simplest interpretation that can self-consistently explain all aspects of the current experiments. The above model for ssDNA secondary structure can provide useful insights for the interpretation of future microsecond-resolved smFRET studies of the T4 bacteriophage ssb nucleoprotein filament assembly mechanism using these same ss-dsDNA constructs. The initial steps of ssb protein-filament assembly involve the association of a ssb (gp32) monomer with an exposed ssDNA template binding site (7 nts), which may form on sub-millisecond time scales as a contiguous segment of unstacked flanking bases. Additional functional steps in which ssb proteins can assemble into cooperatively bound clusters, slide along a ssDNA template strand and disassemble, will be mediated by the nucleation and dissolution of flanking stacked base defect sites, as described by our model. We anticipate that the model can be further tested by performing microsecond-resolved smFRET experiments on ss-dsDNA constructs with ssDNA sequences of purine or of mixed base sequences.