Probing the effect of PEG-DNA interactions and buffer viscosity on tethered DNA in shear flow

Abstract

DNA flow-stretching is a widely employed, powerful technique for investigating the mechanisms of DNA-binding proteins involved in compacting and organizing chromosomal DNA. We combine single-molecule DNA flow-stretching experiments with Brownian dynamics simulations to study the effect of the crowding agent polyethylene glycol (PEG) in these experiments. PEG interacts with DNA by an excluded volume effect, resulting in compaction of single, free DNA molecules in PEG solutions. In addition, PEG increases the viscosity of the buffer solution. By stretching surface-tethered bacteriophage lambda DNA in a flow cell and tracking the positions of a quantum dot labeled at the free DNA end using total internal reflection fluorescence (TIRF) microscopy, we find that higher PEG concentrations result in increased end-to-end length of flow-stretched DNA and decreased fluctuations of the free DNA end. To better understand our experimental results, we perform Brownian dynamics simulations of a bead-spring chain model of flow-stretched DNA in a viscous buffer that models the excluded volume effect of PEG by an effective attractive interaction between DNA segments. We find quantitative agreement between our model and the experimental results for suitable PEG-DNA interaction parameters.

1 Introduction

Deoxyribonucleic acid (DNA) is a highly charged, semi-flexible polymer that stores genetic information in all living organisms. Importantly, the length of DNA is much longer than cell or nucleus sizes. For example, in each human cell, the total length of DNA confined within a nucleus of about 10 μm diameter [1] is approximately two meters [1,2]. Likewise, bacterial chromosomes are typically ~1,000 times longer than cell sizes [3–5] The large ratio of DNA to cell dimensions requires highly dynamic and organized DNA packaging. A variety of DNA-binding proteins employ various strategies to turn long linear DNA into a dense structure to package within the cellular volume. For instance, DNA supercoiling, bending (or wrapping), and loop formation by bridging different segments of DNA are well-characterized DNA compaction mechanisms [5,6]. Additionally, some DNA-binding proteins are capable of structuring DNAs by actively extruding DNA loops in an adenosine triphosphate (ATP)-dependent manner [7–11]. It has also been shown that liquid-liquid phase separation can assist in DNA compaction [12,13]. In addition to DNA packaging, DNA compaction and decompaction play crucial roles in various aspects of cellular events, such as gene regulation by changing chromatin accessibility and protection against chemical or biochemical stress [14,15].

The conformational dynamics of polymers and biopolymers in flow are of great experimental and theoretical interest. The stretching of single, tethered DNA molecules by a uniform (nonshearing) flow was observed by fluorescence microscopy in Perkins et al. [16]. The dynamics of single, free (untethered) DNA molecules in steady shear flow was observed by fluorescence microscopy in Smith et al. [17]. The dynamics of single DNA molecules tethered to a surface in shear flow was observed experimentally by fluorescence microscopy and studied by Brownian dynamics simulations of bead-spring chains [18–20]. The study in Doyle et al. [18] revealed the intriguing phenomenon of large temporal fluctuations in the chain extension due to a continual recirculating motion of the chain at moderate flow strengths, referred to as cyclic dynamics (see also Lueth and Shaqfeh [20] and Supporting Information, S1 Movie). The rheological and optical behavior of bead-rod chains in steady, linear flows was studied by Brownian dynamics simulations in Doyle et al. [21]. The effect of attractive surfaces on the stretching of confined tethered polymers under shear flow was studied by Brownian dynamics simulations in Ibáñez-García et al. [22] and reviewed in Refs. [23–25]. In the fluorescence microscopy experiments summarized above, the DNA was uniformly labeled with dye molecules, which alters the persistence length and other mechanical properties of the DNA being studied [23,24]. This problem was overcome by attaching a fluorescent quantum dot (with a diameter of a few tens nanometers) to the free end of the shear-stretched DNA and tracking the position of the quantum dot by fluorescence microscopy instead of using a DNA intercalating dye, allowing for the study of the DNA extension and the magnitude of fluctuations of the DNA as a function of the shear rate [26]. This new approach was refined by labeling DNA molecules with multiple fluorescent quantum dots at specific sites along the DNA contour and tracking their positions in time by fluorescence microscopy, revealing the dynamics of and correlations between mesoscopic subsegments of the DNA [27]. The single-molecule flow-stretching assay [28,29] has proved to be a powerful method for probing mechanisms of DNA-binding proteins and their implications in DNA [30–36] (Fig 1A) (and the related DNA curtain assay [37–39]). Another benefit of the single-molecule DNA flow-stretching assay is its high sensitivity. For example, by using this technique, we have recently demonstrated that attaching a small amino acid tag to DNA-binding proteins can alter their functional properties [40].

Fig 1. Schematics of our experimental setup and model.

(A) Biotinylated quantum dot (QD)-labeled λ-DNAs are tethered to a surface-passivated microfluidic flow cell via neutravidin-biotin interactions. As syringe plunger is withdrawn, buffer flows from the inlet tube side to the syringe through the microfluidic flow cell. The hydrodynamic drag force leads to DNA stretching. Polyethylene glycol (PEG) on the surface minimizes unwanted nonspecific DNA and QD binding to the surface. Air spring helps maintain the flow rate. (B) Coordinate system used in our model. λ-DNA (blue) is tethered close to the surface at one end (green dot) and labeled by a quantum dot (QD, red) at the other end. The surface of the flow cell is in the xy-plane at z = 0. The DNA is subject to a flow field whose speed in x-direction increases linearly with the distance z from the surface (green).

Cellular environments of all living organisms are densely crowded with macromolecules including proteins, metabolites, and small solutes [41]. For instance, the concentrations of cytoplasmic macromolecules were measured to be up to 450 g/L [42,43]. Such a crowded medium resembles a thick molecular soup where the average distances between macromolecules are significantly smaller than their sizes [44]. Although proteins are typically studied in vitro in diluted aqueous buffers, efforts have been made to conduct experiments in crowded conditions [31,45,46]. In this study, we investigate the effect of the commonly used crowding agent polyethylene glycol (PEG) on flow-stretched DNA. PEG affects biomolecular solutions through excluded volume effects [47,48] or co-condensation [49,50] and its physical properties are well-documented [41,51,52]. In particular, free DNA molecules in solution collapse upon the addition of PEG because the thermodynamically unfavorable contact between DNA and PEG decreases the available free space for coil DNA, leading to an effective, PEG-induced attractive force between DNA segments [53–55]. This effect is similar to the hydrophobic interaction of biomolecules in an aqueous solution, in which nonpolar (hydrophobic) residues seek to minimize the surface area exposed to water, resulting in an effective, nonspecific attractive force between nonpolar residues (see [56] for a recent review on the hydrophobic effect). In addition to crowding, PEG also increases the solvent viscosity [41], which leads to an increased drag on flow-stretched DNA. Therefore, the influence of PEG on solvent viscosity must be taken into account in single-molecule DNA flow-stretching assays in the presence of PEG.

We examine the dynamics of single bacteriophage λ-DNA molecules in a microfluidic flow cell for different concentrations of PEG by labeling the free DNA end with a fluorescent quantum dot (QD) and dynamically tracking the motion of the QD in real time using total internal reflection fluorescence (TIRF) microscopy. The single-molecule DNA flow-stretching experiments show that supplementing the imaging buffer with PEG increases DNA stretching under laminar flow and reduces fluctuations of the untethered free DNA ends. The observed dynamics results from the dual role of PEG, generating both an effective attraction between DNA segments and increasing the buffer viscosity. To better understand our experimental results, we perform Brownian dynamics simulations of a bead-spring chain model of flow-stretched DNA in a viscous buffer that incorporates the PEG-induced effective attraction between DNA segments. We find quantitative agreement between our model and the experimental results for the chain extension and the strength of the fluctuations of the free DNA end for suitable PEG-DNA interaction parameters, providing proof of principle of understanding the dynamics of DNA interacting with an agent in a crowded environment.

2 Experimental materials and methods

2.1 Viscosity measurements for buffers containing PEG

Buffer viscosities were measured using a HAAKE MARS rheometer (Thermo Scientific, Waltham, MA) with a parallel plate-plate geometry, where the radii of the rotor and the lower plates were 17.5 mm and 18.0 mm, respectively. The axial gap between the plates was 1 mm. For each PEG concentration, viscosities were measured at 80 different shear rates between 0.01 (s^-1) and 1000 (s^-1); however, for the geometry we used in our measurements, the data obtained were not reliable at low shear rates (between 0.01–0.1 s^-1). The measurements were repeated four or five times, and values averaged over these measurements were used. All experiments were performed at 23°C.

The composition of the imaging buffer used in the viscosity measurements was 10 mM Tris, pH 8.0, 150 mM NaCl, and 10 mM MgCl₂. Experiments were performed without PEG and with 3%, 5%, or 10% of PEG supplemented to the buffer. The Pearson correlation coefficient was calculated using Excel software.

2.2 DNA substrate preparation

The complementary single-strand 5’ overhangs of λ-phage DNA were utilized to tag one DNA end with biotin and the other with digoxigenin as described previously [32]. Biotin allows us to tether the DNA to the microfluidic flow cell surface via neutravidin-biotin interactions. The digoxigenin molecule was used to label the free DNA end with an anti-digoxigenin antibody-conjugated quantum dot 605 (Invitrogen, Waltham, MA). A biotinylated oligo was annealed to one of the complementary single-stranded overhangs and ligated, followed by annealing and ligation of a digoxigenin-oligo. Unreacted excess short oligos were removed by electrophoresis, and the DNA substrates in EB buffer (10 mM Tris, pH 8.5) were obtained by ethanol precipitation.

2.3 Microfluidic flow cell preparation

To minimize nonspecific DNA binding, cover glass surfaces were passivated by (3-aminopropyl)triethoxysilane (Millipore Sigma A3648, St. Louis, MO) followed by a mixture of PEG (MPEG-SVA-5000-1g) and its biotinylated version (Biotin-PEG-SVA-5000–100 mg) (Laysan Bio, Arab, AL) as described previously [30–32,40]. A microfluidic flow cell was constructed by applying double-sided tapes (Grace Bio-Labs, Bend, OR) between a PEGylated cover glass and a quartz plate (Technical Glass Product, Paineville, OH). The parallel-attached double-sided tapes form a channel of defined height and width (see Section 2.4). The PE60 inlet (13 cm in length) tube attached to one end of the channel was dipped into a tube containing a buffer. The outlet tube attached to the other end of the channel was connected to the syringe on a syringe pump (Harvard Apparatus, Holliston, MA) through an air spring (Fig 1A).

2.4 Single-molecule DNA flow-stretching assay

The height of the channel of a microfluidic flow cell, $\begin{array}{c}\hfill h=0.12\mathrm{m}\mathrm{m}\end{array}$ (z-direction in Fig 1B), was determined by the thickness of the tape. The channel width, $\begin{array}{c}\hfill w=1.8\mathrm{m}\mathrm{m}\end{array}$ (y-direction in Fig 1B, was the same as in our previous studies [30,31,40]. One end of 48.5-kb λ-phage DNA [57] was tethered to the sample chamber via neutravidin-biotin interactions, and the other end was tagged with a fluorescence quantum dot. The tethered DNAs were stretched by applying laminar flow at 50 μL/min generated by a syringe pump.

The flow speed in the flow direction (x-direction in Fig 1B) at height $z$ from the surface of the flow cell is given by a parabola [58], i.e.,

$$u_x\left(z\right)=6u_{x,avg}\frac{hz-z^2}{h^2},$$ (1)

where $u_{x,avg}={\int }_0^{h}dz{u}_x\left(z\right)/\phantom{{\int }_0^{h}dz{u}_x\left(z\right)h}\mathrm{\backslash nulldelimiterspace}h$ is the average flow speed and $u_x\left(0\right)=u_x\left(h\right)=0$. In our experiments, we used a flow rate of $Q=0.833\mu \mathrm{L}/\mathrm{s}$ in a flow cell of cross-sectional area $\begin{array}{c}\hfill w·h=0.216\mathrm{m}{\mathrm{m}}^2\end{array}$ corresponding to an average flow speed of $\begin{array}{c}\hfill u_{x,avg}=Q/\phantom{Q\left(w·h\right)}\mathrm{\backslash nulldelimiterspace}\left(w·h\right)=3.856\mathrm{m}\mathrm{m}/\mathrm{s}\end{array}$.

2.5 Single-molecule DNA flow-stretching and data analysis

A small (~4%) percentage of the PEGylated surface of the microfluidic flow cell contained biotin molecules. Addition of 0.25 mg/mL neutravidin followed by quantum dot-labeled λ-DNA resulted in tethering of the DNA onto the surface at the biotin-tagged ends. Unlabeled quantum dots and untethered DNAs were washed away by flowing imaging buffer (10 mM Tris, pH 8.0, 150 mM NaCl, and 10 mM MgCl₂). When there was no buffer flow for at least two minutes, the movie acquisition was initiated. The average quantum dot position corresponds to the DNA tether point in the absence of flow. Subsequently, we stretched the DNAs by turning on the imaging buffer flow (flow rate $\begin{array}{c}\hfill Q=0.833\mu \mathrm{L}/\mathrm{s}\end{array}$) (Supporting Information, S2 Movie). The imaging buffer without polyethylene glycol (PEG) was switched to the buffer supplemented with a given amount (3% and 5%) of PEG while the flow rate remained constant. All experiments were performed on the IX-83 total internal reflection fluorescence (TIRF) microscope (Evident Scientific, Olympus, Waltham, MA) equipped with a 532 nm laser (Coherent, Santa Clara, CA). Micro-manager software [59] was employed to record images, and regions-of-interest (ROI) of DNAs were set using FIJI software [60].

The diameter of the quantum dots we used is around 15 nm, which is much smaller than the diffraction limit. (The anti-digoxigenin antibody quantum dot we used in our study is comparable to the quantum dot used in reference [27], who found a radius of 13.7 ± 0.4 nm for their quantum dot using dynamic light scattering.) We note that even the electron-multiplying charge-coupled device (EMCCD) images of much smaller fluorescent probes, such as organic dyes (e.g., Cy3) or fluorescent proteins, appear as diffraction-limited spots (~200 nm). Despite the limitation in resolution set by the diffraction limit, one can determine the positions of the fluorescent probes with much higher accuracy as long as the image background is low and the probe emits a large enough number of photons. For example, if the point spread function (PSF) of a fluorescent dye is 150 nm, and if 10,000 photos are collected during the camera exposure time, the position of the dye can be determined with an accuracy of about $\begin{array}{c}\hfill 150\mathrm{n}\mathrm{m}/\phantom{150\mathrm{n}\mathrm{m}\sqrt{10000}}\mathrm{\backslash nulldelimiterspace}\sqrt{10000}=1.5\mathrm{n}\mathrm{m}\end{array}$ [61]. This single-molecule imaging technique was termed fluorescence imaging with one-nanometer accuracy (FIONA) [62]. The quantum dots we used in our study are significantly brighter than organic dyes or fluorescent proteins, enabling position determination with even higher accuracy. However, the localization accuracy is compromised by the fluctuations of the quantum dot position under flow, and the determined position corresponds to the average position of the quantum dot within the EMCCD exposure time (100 msec). We used custom-written MATLAB software based on one-dimensional Gaussian fitting along the DNA length to determine the quantum dot positions. The MATLAB codes are available in our previous publication [30].

2.6 Averages over ensembles of single DNA molecules

Because of the heterogeneous nature of DNA stretching and fluctuations, the procedure described in Section 2.5 was repeated for 49 individual DNA molecules for each of the cases where the buffer without PEG was switched to buffers containing 3% and 5% PEG, respectively. This resulted in data for N = 98 DNA molecules for the buffer without PEG and N = 49 DNA molecules for buffers containing 3% and 5% PEG, respectively. Measured values for the mean extensions 〈x_i〉 and standard deviations Δx_i for DNA molecules $i=1,\dots ,N$ were averaged according to

$$\overline{X}=\frac{1}{N}\sum _{i=1}^N⟨x_i⟩,$$ (2)

$$\overline{\Delta X}=\frac{1}{N}\sum _{i=1}^N\Delta x_i,$$ (3)

where the overbar symbol indicates averages over the ensemble of N molecules. The changes of $\overline{X}$ and $\overline{\Delta X}$ upon adding to the buffer without PEG a given amount (3% or 5%) of PEG were statistically evaluated through the nonparametric Mann-Whitney test (Wilcoxon rank-sum test) using Prism software (GraphPad, San Diego, CA). The raw data are provided in Supplementary S1 Data.

3. DNA model and computational methods

3.1 Bead-spring model for DNA in a flow cell

To better understand the effect of the PEG-induced effective attraction between DNA segments and the buffer viscosity in our DNA flow-stretching experiments, we performed Brownian dynamics simulations of a bead-spring chain with parameters corresponding to a nearly inextensible worm-like chain such as DNA. The experimental setup (Fig 1A) is modeled as shown in Fig 1B. A linear DNA chain is modeled by a bead-spring model consisting of $N+1$ beads centered at positions ${\frac{\mathbf{\backslash buildrel}\mathbf{\backslash lower}\mathbf{3pt}\mathbf{\text{(scriptscriptstylerightharpoonup)}}}{\mathbf{r}}}_0,\dots ,{\frac{\mathbf{\backslash buildrel}\mathbf{\backslash lower}\mathbf{3pt}\mathbf{\text{(scriptscriptstylerightharpoonup)}}}{\mathbf{r}}}_N$ connected by $N$ stiff harmonic springs with equilibrium length a and spring constant $k_s$ (the bead radius does not enter our analysis) (Fig 2). As in our experiments, the chain is attached at one end close to the surface of the flow cell (bead 0 at position ${\frac{\mathbf{\backslash buildrel}\mathbf{\backslash lower}\mathbf{3pt}\mathbf{\text{(scriptscriptstylerightharpoonup)}}}{\mathbf{r}}}_0$) while the other end is moving freely (bead $N$ at position ${\frac{\mathbf{\backslash buildrel}\mathbf{\backslash lower}\mathbf{3pt}\mathbf{\text{(scriptscriptstylerightharpoonup)}}}{\mathbf{r}}}_N$). In our Brownian dynamics simulations (detailed below), we use chains with $N=50$ beads, corresponding to $\begin{array}{c}\hfill a=L/\phantom{LN}\mathrm{\backslash nulldelimiterspace}N=330\mathrm{n}\mathrm{m}\end{array}$ for the contour length $\begin{array}{c}\hfill L=16.49\mu \mathrm{m}\end{array}$ of the λ-DNA used in our experiments, and we assume that the chain is attached at ${\frac{\mathbf{\backslash buildrel}\mathbf{\backslash lower}\mathbf{3pt}\mathbf{\text{(scriptscriptstylerightharpoonup)}}}{\mathbf{r}}}_0=\left(x_0,y_0,z_0\right)=\left(0,0,a\right)$. The spring constant $k_s$ is chosen larger than all other force constants to account for the fact that biopolymers like DNA are nearly inextensible [63]. As a result, stretching modes relax fast compared to other dynamic processes, so that the simulated chain essentially behaves like an inextensible polymer chain. On the other hand, $k_s$ cannot be chosen too large as larger $k_s$ require smaller discretization time steps $\Delta t$ to ensure numerical stability of the Brownian dynamics simulations, thus limiting the maximal simulation time $t_{max}$. As a compromise, we chose the value $\begin{array}{c}\hfill k_s=1000k_{B}T/\phantom{k_{B}T{a}^2}\mathrm{\backslash nulldelimiterspace}{a}^2\end{array}$ where $T$ is the temperature and $k_B$ is the Boltzmann constant.

Fig 2. Bead-spring model for linear DNA.

$N+1$ beads are connected by springs with spring constant $k_s$ and equilibrium length $a$. The chain is tethered at ${\frac{\mathbf{\backslash buildrel}\mathbf{\backslash lower}\mathbf{3pt}\mathbf{\text{(scriptscriptstylerightharpoonup)}}}{\mathbf{r}}}_0$ close to the surface of the flow cell and ${\frac{\mathbf{\backslash buildrel}\mathbf{\backslash lower}\mathbf{3pt}\mathbf{\text{(scriptscriptstylerightharpoonup)}}}{\mathbf{r}}}_N$ is the position vector of the free end. (see Fig 1B).

Following previous studies [18–27], our bead-spring model of flow-stretched DNA does not include the electrostatic repulsion between the negatively charged phosphate groups of the DNA. For the high salt concentrations used in our experiments (150 mM NaCl), this can be justified by the fact that the DNA charges are screened by counterions in solution. More precisely, previous studies of the probability of knotting of λ-phage DNA at equilibrium (without flow) showed, by comparing experiments with simulations of a discrete semiflexible DNA model (a chain of cylindrical segments connected by semiflexible joints), that the electrostatic repulsion between DNA segments can be mapped on an effective diameter of the non-overlapping cylindrical segments used in the simulation [64–66]. Since flow-stretched DNA is dominated by extended DNA conformations, the self-avoiding interaction between DNA segments modeled by a finite diameter of the cylindrical segments is expected to be less relevant than for DNA at equilibrium, and is usually neglected [18–27]. However, for low salt concentrations, the electrostatic repulsion between the charged DNA phosphate groups and the dynamics of counterions becomes relevant, and the DNA should be described as a polyelectrolyte (see [67] for a review).

The non-hydrodynamic forces on the beads of the model chain include (i) the spring forces, (ii) the external force by the surface to which the chain is tethered, and (iii) the forces generated by the PEG-induced attractive interaction between DNA segments. The interaction of the model chain with the viscous solvent is described by (iv) a drag force and (v) a Brownian force due to random collisions of the solvent with the beads. The forces (i) – (v) are detailed below.

• (i) Spring forces: The total elastic (el) force on a bead $n$ within the chain is given by

$$\begin{array}{c}\hfill {\frac{\mathbf{\backslash buildrel}\mathbf{\backslash lower}\mathbf{3pt}\mathbf{\text{(scriptscriptstylerightharpoonup)}}}{\mathbf{F}}}_n^{el}=-k_s\left(\left|{\frac{\mathbf{\backslash buildrel}\mathbf{\backslash lower}\mathbf{3pt}\mathbf{\text{(scriptscriptstylerightharpoonup)}}}{\mathbf{r}}}_{n+1}-{\frac{\mathbf{\backslash buildrel}\mathbf{\backslash lower}\mathbf{3pt}\mathbf{\text{(scriptscriptstylerightharpoonup)}}}{\mathbf{r}}}_n\right|-a\right){\widehat{𝐫}}_{n,n+1}-k_s\left(\left|{\frac{\mathbf{\backslash buildrel}\mathbf{\backslash lower}\mathbf{3pt}\mathbf{\text{(scriptscriptstylerightharpoonup)}}}{\mathbf{r}}}_n-{\frac{\mathbf{\backslash buildrel}\mathbf{\backslash lower}\mathbf{3pt}\mathbf{\text{(scriptscriptstylerightharpoonup)}}}{\mathbf{r}}}_{n-1}\right|-a\right){\widehat{𝐫}}_{n,n-1},0<n<N,\end{array}$$ (4)

where $k_s$ is the spring constant and ${\widehat{𝐫}}_{n,n+1}$, ${\widehat{𝐫}}_{n,n-1}$ are unit vectors from bead $n+1$ and $n-1$ to bead $n$, respectively. The elastic force on bead $N$ at the free end of the chain is given by ${\frac{\mathbf{\backslash buildrel}\mathbf{\backslash lower}\mathbf{3pt}\mathbf{\text{(scriptscriptstylerightharpoonup)}}}{\mathbf{F}}}_N^{el}=-k_s\left(\left|{\frac{\mathbf{\backslash buildrel}\mathbf{\backslash lower}\mathbf{3pt}\mathbf{\text{(scriptscriptstylerightharpoonup)}}}{\mathbf{r}}}_N-{\frac{\mathbf{\backslash buildrel}\mathbf{\backslash lower}\mathbf{3pt}\mathbf{\text{(scriptscriptstylerightharpoonup)}}}{\mathbf{r}}}_{N-1}\right|-a\right){\widehat{𝐫}}_{N,N-1}$. As discussed above, $\begin{array}{c}\hfill a=330\mathrm{n}\mathrm{m}\end{array}$ for the λ-DNA used in our experiments modeled by a chain with 50 beads, and we used the value $\begin{array}{c}\hfill k_s=1000k_{B}T/\phantom{k_{B}T{a}^2}\mathrm{\backslash nulldelimiterspace}{a}^2\end{array}$ in our simulations.

• (ii)Surface interaction: For the repulsive interaction between the polymer chain and the surface (s) we assume a truncated soft-wall potential of the form

$$\begin{array}{c}\hfill {\frac{\mathbf{\backslash buildrel}\mathbf{\backslash lower}\mathbf{3pt}\mathbf{\text{scriptscriptstylerightharpoonup)}}}{\mathbf{F}}}_n^s=6\frac{\epsilon }{a}\left[{\left(\frac{a}{z_n}\right)}^7-{\left(\frac{a}{z_n}\right)}^4\right]\Theta (a-z_n,\widehat{𝐳},\end{array}$$ (5)

where $z_n$ is the distance of bead n from the surface (Fig 1B), and $\Theta \left(a-z_n\right)=1$ for $0\le z_n\le a$ and $\Theta \left(a-z_n\right)=0$ for $z_n>a$. In our Brownian dynamics simulations, we set $\epsilon =k_{B}T$. The force ${\frac{\mathbf{\backslash buildrel}\mathbf{\backslash lower}\mathbf{3pt}\mathbf{\text{(scriptscriptstylerightharpoonup)}}}{\mathbf{F}}}_n^s=-d{U}^s\left(z_n\right)/\phantom{d{U}^s\left(z_n\right)d{z}_n}\mathrm{\backslash nulldelimiterspace}d{z}_n\widehat{𝐳}$ on bead $n$ resulting from Eq. (5) is given by

$$\begin{array}{c}\hfill {\frac{\mathbf{\backslash buildrel}\mathbf{\backslash lower}\mathbf{3pt}\mathbf{\text{scriptscriptstylerightharpoonup)}}}{\mathbf{F}}}_n^s=6\frac{\epsilon }{a}\left[{\left(\frac{a}{z_n}\right)}^7-{\left(\frac{a}{z_n}\right)}^4\right]\Theta (a-z_n,\widehat{𝐳},\end{array}$$ (6)

where $\widehat{𝐳}$ is a unit vector perpendicular to the surface pointing into the flow cell. ${\frac{\mathbf{\backslash buildrel}\mathbf{\backslash lower}\mathbf{3pt}\mathbf{\text{(scriptscriptstylerightharpoonup)}}}{\mathbf{F}}}_n^s$ is repulsive for $0\le z_n<a$, equal to zero for $z_n\ge a$, and continuous at $z_n=a$. According to the size of our flow cell and the length of the λ-DNA used in our experiments, the DNA only interacts with the surface at $z=0$ to which it is tethered (Fig 1B).

• (iii) PEG-induced effective attraction between DNA segments: A theoretical free-energy potential (potential of mean force, PMF) $\Delta G_{PEG}\left(r\right)$ for the PEG-induced effective interaction between DNA segments separated by a distance r in solution is not available. (This is different from direct DNA-ligand interactions in which the ligand is explicitly included in the model and interacts with the DNA by repulsive and attractive (e.g., Debye-Hückel) terms; see, e.g., references [68,69].) In references [53,54] the free energy of mixing $\Delta G_{mix}$ of DNA and PEG in solution at given concentrations ${\varphi }_{DNA}$ and ${\varphi }_{PEG}$ was obtained using the Flory-Huggins model [70], but this approach did not provide an expression for $\Delta G_{PEG}\left(r\right)$. However, as outlined in the Introduction, the physical mechanism for the PEG-induced attractive force between DNA segments is similar to the hydrophobic force between nonpolar components in aqueous solution, as in both cases the respective constituents (DNA segments/ nonpolar components) tend to cluster together to avoid thermodynamically unfavorable contacts with other molecules (PEG/ water molecules), resulting in an effective, attractive force between them. In this work, we will use this analogy to obtain a phenomenological expression for $\Delta G_{PEG}\left(r\right)$. Such indirect, effective interactions are notoriously difficult to model because both entropy and enthalpy terms between all components in the system contribute. As a result, even for the fundamentally important hydrophobic force between nonpolar components in aqueous solution (reviewed in [56]), there is no generally accepted, distance-dependent energy potential (PMF) between the nonpolar components (compare text above equation (1) in reference [71]). In early experiments by Israelachvili and Pashley, the attractive force between hydrophobic (hp) surfaces in aqueous solution was found to decay exponentially with the distance, corresponding to a free-energy potential (PMF) of the form $\Delta G_{hp}\left(d\right)=-A\exp \left(-d/\phantom{d{D}_{hp}}\mathrm{\backslash nulldelimiterspace}{D}_{hp}\right)$ where d is the distance between the surfaces, D_hp is the hydrophobic decay length, and A is a positive amplitude [72,73]. Later experiments showed that the hydrophobic force is more complex than initially thought (see [74–76] for reviews). Nevertheless, the simple free-energy potential $\Delta G_{hp}\left(d\right)$ quoted above was used to describe the hydrophobic interaction between self-assembled surfactant bilayers at interfaces [71] (see equation (1) therein) and between nanoparticles [77] (see equation (1) therein).

Guided by the analogy between the PEG-induced attractive interaction between DNA segments and the hydrophobic interaction between nonpolar components in aqueous solution, in this work we assume a pairwise, attractive interaction between beads $n$ and $n\prime$ of our bead-spring model of DNA governed by a potential energy of the same form as $\Delta G_{hp}\left(d\right)$ quoted above, i.e.,

$$\Delta G_{PEG}\left(r_{nn\prime }\right)=-\alpha \left[\mathrm{P}\mathrm{E}\mathrm{G}\right]\exp \left(-\frac{r_{nn\prime }}{D}\right)$$ (7)

where $r_{nn\prime }=\left|{\frac{\mathbf{\backslash buildrel}\mathbf{\backslash lower}\mathbf{3pt}\mathbf{\text{(scriptscriptstylerightharpoonup)}}}{\mathbf{r}}}_n-{\frac{\mathbf{\backslash buildrel}\mathbf{\backslash lower}\mathbf{3pt}\mathbf{\text{(scriptscriptstylerightharpoonup)}}}{\mathbf{r}}}_{n\prime }\right|$ is the distance between the beads located at ${\frac{\mathbf{\backslash buildrel}\mathbf{\backslash lower}\mathbf{3pt}\mathbf{\text{(scriptscriptstylerightharpoonup)}}}{\mathbf{r}}}_n$ and ${\frac{\mathbf{\backslash buildrel}\mathbf{\backslash lower}\mathbf{3pt}\mathbf{\text{(scriptscriptstylerightharpoonup)}}}{\mathbf{r}}}_{n\prime }$, respectively, α is a positive constant, [PEG] is the PEG concentration in percent (%), and D is the characteristic decay length of the PEG-induced attractive force between DNA segments. The phenomenological energy potential in Eq. (7) for the PEG-induced, effective interaction between the DNA segments of our coarse-grained DNA model incorporates the assumptions that the effective force between DNA segments (corresponding to the beads in our model) is proportional to [PEG] and decays exponentially with the bead separation. The remaining parameters in Eq. (7), α and D, were determined by comparison with our DNA flow stretching experiments (Section 4.3). In a model with atomistic resolution, the decay length $D$ is expected to be related to the radius of gyration (size) $R_g$ of the PEG polymers, which in our experiment is a few nanometers [78,79]; however, in our coarse-grained DNA model using a bond length of $a=330\mathrm{n}\mathrm{m}$ (Fig 2) the decay length $D$ should be comparable to the bond length $a$ to obtain a notable PEG-induced change in the mean DNA length $⟨x_e⟩$. We performed Brownian dynamics simulations for two different values of the decay length, namely $D=a$ and $D=a/\phantom{a2}\mathrm{\backslash nulldelimiterspace}2$, to show that the constant $\alpha$ in Eq. (7) is indeed independent of $\left[\mathrm{P}\mathrm{E}\mathrm{G}\right]$ and the buffer viscosity $\eta$, regardless of the value of $D$ (more precisely, $\alpha$ depends on $D$ but for given $D$ is independent of [PEG] and $\eta$). Using Eq. (7) the net force ${\frac{\mathbf{\backslash buildrel}\mathbf{\backslash lower}\mathbf{3pt}\mathbf{\text{(scriptscriptstylerightharpoonup)}}}{\mathbf{F}}}_n^{PEG}$ on bead $n$ due to the PEG-induced attraction between nearby beads is given by

$$\begin{array}{c}\hfill {\frac{\mathbf{\backslash buildrel}\mathbf{\backslash lower}\mathbf{3pt}\mathbf{\text{scriptscriptstylerightharpoonup)}}}{\mathbf{F}}}_n^{PEG}=-\frac{\alpha }{D}\left[\mathrm{P}\mathrm{E}\mathrm{G}\right]\sum _{n^{\prime }\ne n}\exp (-\frac{r_{n{n}^{\prime }}}{D},{\widehat{𝐫}}_{n{n}^{\prime }},\end{array}$$ (8)

where the sum includes all beads $n\prime$ different from $n$ and ${\widehat{𝐫}}_{nn\prime }$ is the unit vector directed from bead $n\prime$ to $n$. For the elongated conformations occurring in our simulations of flow-stretched DNA, only beads $n\prime$ close to bead $n$ along the contour of the chain contribute significantly to the net force in Eq. (8). To reduce the computational cost in our simulations, we therefore restricted the sum in Eq. (8) to beads $n\prime$ with $1\le \left|n-n\prime \right|\le 5$; including more than 5 nearest-neighboring beads did not change our results within the statistical error of our simulations (we tested this by simulations including 8 and 10 nearest neighbors).

• (iv)Viscous drag: The drag force between the viscous solvent and the polymer chain is represented by Stoke’s drag force on the beads of the model chain. The drag force on the bead $n$ is given by

$${\frac{\mathbf{\backslash buildrel}\mathbf{\backslash lower}\mathbf{3pt}\mathbf{\text{(scriptscriptstylerightharpoonup)}}}{\mathbf{F}}}_n^{drag}=-\zeta \left(\frac{d{\frac{\mathbf{\backslash buildrel}\mathbf{\backslash lower}\mathbf{3pt}\mathbf{\text{(scriptscriptstylerightharpoonup)}}}{\mathbf{r}}}_n}{dt}-\frac{\mathbf{\backslash buildrel}\mathbf{\backslash lower}\mathbf{3pt}\mathbf{\text{(scriptscriptstylerightharpoonup)}}}{\mathbf{u}}\left({\frac{\mathbf{\backslash buildrel}\mathbf{\backslash lower}\mathbf{3pt}\mathbf{\text{(scriptscriptstylerightharpoonup)}}}{\mathbf{r}}}_n\right)\right),$$ (9)

where $\zeta$ is the drag coefficient, $d{\frac{\mathbf{\backslash buildrel}\mathbf{\backslash lower}\mathbf{3pt}\mathbf{\text{(scriptscriptstylerightharpoonup)}}}{\mathbf{r}}}_n/\phantom{d{\frac{\mathbf{\backslash buildrel}\mathbf{\backslash lower}\mathbf{3pt}\mathbf{\text{(scriptscriptstylerightharpoonup)}}}{\mathbf{r}}}_{n}dt}\mathrm{\backslash nulldelimiterspace}dt$ is the velocity vector of bead n, and $\frac{\mathbf{\backslash buildrel}\mathbf{\backslash lower}\mathbf{3pt}\mathbf{\text{(scriptscriptstylerightharpoonup)}}}{\mathbf{u}}\left({\frac{\mathbf{\backslash buildrel}\mathbf{\backslash lower}\mathbf{3pt}\mathbf{\text{(scriptscriptstylerightharpoonup)}}}{\mathbf{r}}}_n\right)$ is the unperturbed flow velocity vector of the solvent at position ${\frac{\mathbf{\backslash buildrel}\mathbf{\backslash lower}\mathbf{3pt}\mathbf{\text{(scriptscriptstylerightharpoonup)}}}{\mathbf{r}}}_n$ of bead n. Assuming laminar shear flow in x-direction (Fig 1B), the velocity vector of the solvent at position $\frac{\mathbf{\backslash buildrel}\mathbf{\backslash lower}\mathbf{3pt}\mathbf{\text{(scriptscriptstylerightharpoonup)}}}{\mathbf{r}}=\left(x,y,z\right)$ in the flow cell has the form

$$\frac{\mathbf{\backslash buildrel}\mathbf{\backslash lower}\mathbf{3pt}\mathbf{\text{(scriptscriptstylerightharpoonup)}}}{\mathbf{u}}\left(\frac{\mathbf{\backslash buildrel}\mathbf{\backslash lower}\mathbf{3pt}\mathbf{\text{(scriptscriptstylerightharpoonup)}}}{\mathbf{r}}\right)=u_x\left(z\right)\widehat{𝐱}$$ (10)

where $\widehat{𝐱}$ is a unit vector in x-direction and the velocity profile $u_x\left(z\right)$ is given by Eq. (1). The shear rate of the fluid is given by

$$\dot{\gamma }\left(z\right)=\frac{d{u}_x}{dz}=\frac{6u_{x,avg}}{h}\left(1-2\frac{z}{h}\right)=192.9\left(1-2\frac{z}{h}\right){\mathrm{s}}^{-1}$$ (11)

where we used $\begin{array}{c}\hfill u_{x,avg}=3.856\mathrm{m}\mathrm{m}/\mathrm{s}\end{array}$ and $\begin{array}{c}\hfill h=0.12\mathrm{n}\mathrm{m}\end{array}$ for our flow cell (see text below Eq. (1)). In our DNA flow stretching experiments the ratio $⟨z_e⟩/\phantom{⟨z_e⟩h}\mathrm{\backslash nulldelimiterspace}h$, where $⟨z_e⟩$ is the average height of the free end of the DNA chain above the surface (Fig 1B), is always smaller than 0.002; thus, the term $2z/\phantom{zh}\mathrm{\backslash nulldelimiterspace}h$ in Eq. (11) is only a small correction to the leading term. In our Brownian dynamics simulations, it was therefore justified to neglect this correction, and we assumed a constant shear rate

$$\dot{\gamma }\simeq \dot{\gamma }\left(0\right)=192.9{\mathrm{s}}^{-1}.$$ (12)

Accordingly, in Eq. (9) we used the approximation

$$\frac{\mathbf{\backslash buildrel}\mathbf{\backslash lower}\mathbf{3pt}\mathbf{\text{(scriptscriptstylerightharpoonup)}}}{\mathbf{u}}\left({\frac{\mathbf{\backslash buildrel}\mathbf{\backslash lower}\mathbf{3pt}\mathbf{\text{(scriptscriptstylerightharpoonup)}}}{\mathbf{r}}}_n\right)\simeq \dot{\gamma }{z}_n\widehat{𝐱}$$ (13)

with the constant shear rate $\dot{\gamma }$ in Eq. (12).

3.2 Langevin equation and Brownian dynamics simulations

The total force on bead $n$ of the model chain (Fig 2) includes the forces ${\frac{\mathbf{\backslash buildrel}\mathbf{\backslash lower}\mathbf{3pt}\mathbf{\text{(scriptscriptstylerightharpoonup)}}}{\mathbf{F}}}_n^{el}$, ${\frac{\mathbf{\backslash buildrel}\mathbf{\backslash lower}\mathbf{3pt}\mathbf{\text{(scriptscriptstylerightharpoonup)}}}{\mathbf{F}}}_n^s$, ${\frac{\mathbf{\backslash buildrel}\mathbf{\backslash lower}\mathbf{3pt}\mathbf{\text{(scriptscriptstylerightharpoonup)}}}{\mathbf{F}}}_n^{PEG}$, ${\frac{\mathbf{\backslash buildrel}\mathbf{\backslash lower}\mathbf{3pt}\mathbf{\text{(scriptscriptstylerightharpoonup)}}}{\mathbf{F}}}_n^{drag}$ described in (i) – (iv) above, and (v) a Brownian (thermal) force ${\frac{\mathbf{\backslash buildrel}\mathbf{\backslash lower}\mathbf{3pt}\mathbf{\text{(scriptscriptstylerightharpoonup)}}}{\mathbf{F}}}_n^B$ due to random collisions of the solvent particles with the bead. According to Newton’s second law of motion and neglecting inertia, the net force ${\frac{\mathbf{\backslash buildrel}\mathbf{\backslash lower}\mathbf{3pt}\mathbf{\text{(scriptscriptstylerightharpoonup)}}}{\mathbf{F}}}_n={\frac{\mathbf{\backslash buildrel}\mathbf{\backslash lower}\mathbf{3pt}\mathbf{\text{(scriptscriptstylerightharpoonup)}}}{\mathbf{F}}}_n^{el}+{\frac{\mathbf{\backslash buildrel}\mathbf{\backslash lower}\mathbf{3pt}\mathbf{\text{(scriptscriptstylerightharpoonup)}}}{\mathbf{F}}}_n^s+{\frac{\mathbf{\backslash buildrel}\mathbf{\backslash lower}\mathbf{3pt}\mathbf{\text{(scriptscriptstylerightharpoonup)}}}{\mathbf{F}}}_n^{PEG}+{\frac{\mathbf{\backslash buildrel}\mathbf{\backslash lower}\mathbf{3pt}\mathbf{\text{(scriptscriptstylerightharpoonup)}}}{\mathbf{F}}}_n^{drag}+{\frac{\mathbf{\backslash buildrel}\mathbf{\backslash lower}\mathbf{3pt}\mathbf{\text{(scriptscriptstylerightharpoonup)}}}{\mathbf{F}}}_n^B$ on any bead $n$ is zero, resulting in the Langevin equation for bead $n$ [23]

$$\zeta \frac{d{\frac{\mathbf{\backslash buildrel}\mathbf{\backslash lower}\mathbf{3pt}\mathbf{\text{(scriptscriptstylerightharpoonup)}}}{\mathbf{r}}}_n}{dt}={\frac{\mathbf{\backslash buildrel}\mathbf{\backslash lower}\mathbf{3pt}\mathbf{\text{(scriptscriptstylerightharpoonup)}}}{\mathbf{F}}}_n^{el}+{\frac{\mathbf{\backslash buildrel}\mathbf{\backslash lower}\mathbf{3pt}\mathbf{\text{(scriptscriptstylerightharpoonup)}}}{\mathbf{F}}}_n^s+{\frac{\mathbf{\backslash buildrel}\mathbf{\backslash lower}\mathbf{3pt}\mathbf{\text{(scriptscriptstylerightharpoonup)}}}{\mathbf{F}}}_n^{PEG}+\zeta {\frac{\mathbf{\backslash buildrel}\mathbf{\backslash lower}\mathbf{3pt}\mathbf{\text{(scriptscriptstylerightharpoonup)}}}{\mathbf{u}}}^{\infty }\left({\frac{\mathbf{\backslash buildrel}\mathbf{\backslash lower}\mathbf{3pt}\mathbf{\text{(scriptscriptstylerightharpoonup)}}}{\mathbf{r}}}_n\right)+{\frac{\mathbf{\backslash buildrel}\mathbf{\backslash lower}\mathbf{3pt}\mathbf{\text{(scriptscriptstylerightharpoonup)}}}{\mathbf{F}}}_n^B.$$ (14)

The Brownian force ${\frac{\mathbf{\backslash buildrel}\mathbf{\backslash lower}\mathbf{3pt}\mathbf{\text{(scriptscriptstylerightharpoonup)}}}{\mathbf{F}}}_n^B$ is a stochastic (random) force, taken from a Gaussian distribution in our simulations. For the dynamics to satisfy the fluctuation-dissipation theorem [23], the expectation values of the Brownian force with components $\nu =x,y,z$ are given by

$$⟨F_{n,\nu }^B\left(t\right)⟩=0$$ (15)

$$⟨F_{n,\nu }^B\left(t\right)F_{n\prime ,\nu \prime }^B\left(t\prime \right)⟩=2k_{B}T\zeta {\delta }_{nn\prime }{\delta }_{\nu \nu \prime }\delta \left(t-t\prime \right)$$ (16)

with the drag coefficient $\zeta$ in Eq. (9); ${\delta }_{nn\prime }$ is the Kronecker delta, $\delta \left(t-t\prime \right)$ is the Dirac delta function, $T$ is the temperature, and $k_B$ is the Boltzmann constant.

For the numerical integration of Eq. (14) we use an Euler-Maruyama iteration scheme by discretizing time in small steps $\Delta t$ [80]. We write Eq. (14) in terms of reduced (unitless) variables by expressing lengths in units of $a$, energies in units of $k_{B}T$, forces in units of $k_{B}T/\phantom{k_{B}Ta}\mathrm{\backslash nulldelimiterspace}a$, and times in terms of the unit time

$${\tau }_0=\zeta \frac{a^2}{k_{B}T}.$$ (17)

${\tau }_0$ is the time for a single, free bead with drag coefficient $\zeta$ to diffuse a distance $a$ in a solvent of temperature $T$. Discretizing time in steps $\Delta t$ the delta function in Eq. (16) becomes $\delta \left(t-t\prime \right)=1/\phantom{1\Delta t}\mathrm{\backslash nulldelimiterspace}\Delta t$ for $t=t\prime$ and zero otherwise. Each time step $i\to i+1$ advances the position ${\frac{\mathbf{\backslash buildrel}\mathbf{\backslash lower}\mathbf{3pt}\mathbf{\text{(scriptscriptstylerightharpoonup)}}}{\mathbf{r}}}_n\left(i\right)$ of bead $n$ at time $t\left(i\right)=i\Delta t$ according to

$${\stackrel{\rightharpoonup }{𝐫}}_n(i+1)={\stackrel{\rightharpoonup }{𝐫}}_n(i)+({\stackrel{\rightharpoonup }{𝐅}}_n^{el}+{\stackrel{\rightharpoonup }{𝐅}}_n^s+{\stackrel{\rightharpoonup }{𝐅}}_n^{PEG}+Pe·z·\widehat{𝐱})\Delta t+{\stackrel{\rightharpoonup }{𝐟}}_n^B,\left(\text{unitless}\right)$$ (18)

where $Pe$ is the Péclet number (see next paragraph below). According to Eqs. (15) and (16), the stochastic impulse ${\frac{\mathbf{\backslash buildrel}\mathbf{\backslash lower}\mathbf{3pt}\mathbf{\text{(scriptscriptstylerightharpoonup)}}}{\mathbf{f}}}_n^B={\frac{\mathbf{\backslash buildrel}\mathbf{\backslash lower}\mathbf{3pt}\mathbf{\text{(scriptscriptstylerightharpoonup)}}}{\mathbf{F}}}_n^B\Delta t$ is determined by the correlations

$$⟨f_{n,\nu }^B⟩=0,\left(\text{unitless}\right)$$ (19)

$$⟨f_{n,\nu }^B(i)f_{n\prime ,\nu \prime }^B(i\prime )⟩=2{\delta }_{nn\prime }{\delta }_{\nu \nu \prime }{\delta }_{ii\prime }\Delta t.\left(\text{unitless}\right)$$ (20)

To simplify notation in Eqs. (18)–(20) we use the same symbols for the reduced (unitless) variables defined above as for their dimensionful counterparts, as explicitly indicated; otherwise, we will use a tilde symbol to indicate reduced variables (see next paragraph below). Equation (20) implies that the stochastic impulse is independent of all beads $n$, spatial directions $v=x,y,z$, and time steps $i\to i+1$. The strength of the Brownian random force is characterized by the variance $⟨{\left[f_{n,\nu }^B\left(i\right)\right]}^2⟩=2\Delta t$ (unitless variables). In our numerical implementation, we used unitless time steps $\Delta t={10}^{-4}$ in Eq. (18), and values $f_{n,\nu }^B\left(i\right)$ were drawn from a Gaussian distribution with zero mean and variance $2\Delta t$ independently for each $n$, $v$, and $i$.

The Fortran code for the Brownian dynamics simulations (Langevin equation) used for Figs 3, 6, and 7 is available at the GitHub repository https://github.com/andreas-hanke/PEG-DNA-shear-flow/.

Fig 3. Determination of the longest relaxation time of the polymer chain, ${\tau }_R$, in our simulations in the absence of PEG ($\alpha =0$ in Eq. (8)).

${\tau }_R$ is determined by (a) the autocorrelation function $C\left(t\prime \right)=⟨\frac{\mathit{\backslash buildrel}\mathit{\backslash lower}\mathbf{3pt}\text{(scriptscriptstylerightharpoonup)}}{\mathit{\rho }}\left(t\right)·\frac{\mathit{\backslash buildrel}\mathit{\backslash lower}\mathbf{3pt}\text{(scriptscriptstylerightharpoonup)}}{\mathit{\rho }}\left(t+t\prime \right)⟩$ at equilibrium and (b) the relaxation of $⟨\frac{\mathit{\backslash buildrel}\mathit{\backslash lower}\mathbf{3pt}\text{(scriptscriptstylerightharpoonup)}}{\mathit{\rho }}\left(t\right)⟩$ to equilibrium, where $\frac{\mathit{\backslash buildrel}\mathit{\backslash lower}\mathbf{3pt}\text{(scriptscriptstylerightharpoonup)}}{\mathit{\rho }}=\left(x,y\right)$ is the projection of the chain end on the $xy$ – plane (Fig 1(b)). $C\left(t\prime \right)$ and $⟨\frac{\mathbf{\backslash buildrel}\mathbf{\backslash lower}\mathbf{3pt}\mathbf{\text{(scriptscriptstylerightharpoonup)}}}{\mathbf{\rho }}\left(t\right)⟩$ were fitted by least squares to single exponentials in the domain $200\le t\prime /\phantom{t\prime {\tau }_0}\mathrm{\backslash nulldelimiterspace}{\tau }_0,t/\phantom{t{\tau }_0\le 2000}\mathrm{\backslash nulldelimiterspace}{\tau }_0\le 2000$ (vertical blue lines), yielding ${\tilde{\tau }}_R^{corr}={\tau }_R^{corr}/\phantom{{\tau }_R^{corr}{\tau }_0}\mathrm{\backslash nulldelimiterspace}{\tau }_0=327$ and ${\tilde{\tau }}_R^{relax}={\tau }_R^{relax}/\phantom{{\tau }_R^{relax}{\tau }_0}\mathrm{\backslash nulldelimiterspace}{\tau }_0=339$, respectively.

Fig 6. Fractional extension of flow-stretched DNA.

Comparison of the fractional extension ξobtained in the simulations (sim, $\xi =⟨x⟩/\phantom{⟨x⟩L}\mathrm{\backslash nulldelimiterspace}L$) and the experiments (exp, $\xi =\overline{X}/\phantom{\overline{X}L}\mathrm{\backslash nulldelimiterspace}L$, see Eq. (2) and Figs 4, 5). (A) Simulated ξ versus Wi (bottom axis) and $\eta =Wi/\phantom{Wic\prime }\mathrm{\backslash nulldelimiterspace}c\prime$ (top axis) [where $c\prime =20.48\mathrm{c}{\mathrm{P}}^{-1}$, Eq. (28)] in the absence of a PEG-induced attractive interaction between DNA segments ($\alpha =0$ in Eq. (8)). η-values for buffers with 0%, 3%, and 5% PEG used in the experiments (Table 1) are indicated by green vertical lines labeled ${\eta }_0,{\eta }_1,{\eta }_2$, respectively. (B) Magnification of the η-region $0.7\mathrm{c}\mathrm{P}\le \eta \le 2.6\mathrm{c}\mathrm{P}$ highlighted by the blue box in (A). Experimental ξ-values for 0%, 3%, and 5% PEG are indicated by green circles with error bars (SEM, Table 1). The black dots show simulated ξ-values for ${\eta }_0,{\eta }_1,{\eta }_2$ in the absence of a PEG-induced attractive interaction between DNA segments. Blue and magenta symbols are simulated ξ-values for 3% PEG (η₁) and 5% PEG (η₂) using ($D=a$, $\alpha =0.345k_{B}T$) and ($D=a/\phantom{a2}\mathrm{\backslash nulldelimiterspace}2$, $\alpha =0.615k_{B}T$) in Eq. (8), respectively.

Fig 7. Fluctuations of the free DNA end of flow-stretched DNA.

Comparison of the standard deviation (fluctuations) of the DNA end (Fig 1B) along the x-direction obtained in the simulations (sim, $\varphi =\Delta x/\phantom{\Delta x{R}_e}\mathrm{\backslash nulldelimiterspace}{R}_e$) and the experiments [exp, $\varphi =\overline{\Delta X}/\phantom{\overline{\Delta X}{R}_e}\mathrm{\backslash nulldelimiterspace}{R}_e$, see Eq. (3) and Figs 4, 5] where $R_e$ is the root mean square end-to-end distance of the chain. The meaning of the symbols is the same as in Fig 6. (A) Simulated $\varphi$ vs Wi (bottom axis) and η (top axis) in the absence of a PEG-induced attractive interaction between DNA segments. (B) Magnification of the η-region $0.7\mathrm{c}\mathrm{P}\le \eta \le 2.6\mathrm{c}\mathrm{P}$ highlighted by the blue box in (A). Experimental $\varphi$ -values for 0%, 3%, and 5% PEG are indicated by green circles with error bars (SD, Table 1). The black dots show simulated $\varphi$ -values for ${\eta }_0,{\eta }_1,{\eta }_2$ in the absence of a PEG-induced attractive interaction between DNA segments. Blue and magenta symbols are simulated $\varphi$ -values for 3% PEG (η₁) and 5% PEG (η₂) using the same values for α, D as in Fig 6 (i.e., without adjustable parameters).

3.3 Relaxation time and Weissenberg number

In Eq. (18) arises the unitless Péclet number

$$Pe=\dot{\gamma }{\tau }_0=\dot{\gamma }\zeta \frac{a^2}{k_{B}T}$$ (21)

corresponding to the ratio of the time ${\tau }_0$ in Eq. (17) for a single bead to freely diffuse a distance $a$ to the time scale $1/\phantom{1\dot{\gamma }}\mathrm{\backslash nulldelimiterspace}\dot{\gamma }$ set by the shear rate $\dot{\gamma }$ of the flow in Eq. (12). For a long polymer chain rather than a single bead, a more relevant parameter is the unitless Weissenberg number

$$W_i=\dot{\gamma }{\tau }_R=Pe·{\tilde{\tau }}_R,$$ (22)

where ${\tau }_R$ is the longest relaxation time of the chain and ${\tilde{\tau }}_R={\tau }_R/\phantom{{\tau }_R{\tau }_0}\mathrm{\backslash nulldelimiterspace}{\tau }_0$; in Eq. (22) and below, we indicate reduced (unitless) variables by a tilde symbol to distinguish them from their dimensionful counterparts. In our simulations, we determined ${\tilde{\tau }}_R$ in two different ways (Fig 3A and 3B): (A) By the exponential decay of the autocorrelation function $C\left(t\prime \right)=⟨\frac{\mathit{\backslash buildrel}\mathit{\backslash lower}\mathbf{3pt}\text{(scriptscriptstylerightharpoonup)}}{\mathit{\rho }}\left(t\right)·\frac{\mathit{\backslash buildrel}\mathit{\backslash lower}\mathbf{3pt}\text{(scriptscriptstylerightharpoonup)}}{\mathit{\rho }}\left(t+t\prime \right)⟩$ at equilibrium (i.e., without flow and at thermal equilibrium), where $\frac{\mathbf{\backslash buildrel}\mathbf{\backslash lower}\mathbf{3pt}\mathbf{\text{(scriptscriptstylerightharpoonup)}}}{\mathbf{\rho }}=\left(x,y\right)$ is the two-dimensional projection of the three-dimensional end-to-end vector $\frac{\mathbf{\backslash buildrel}\mathbf{\backslash lower}\mathbf{3pt}\mathbf{\text{(scriptscriptstylerightharpoonup)}}}{\mathbf{r}}$ on the $xy$ – plane (Fig 1B). Fitting $C\left(t\prime \right)$ to a single exponential $\tilde{A}\exp \left(-\tilde{t}\prime /\phantom{\tilde{t}\prime {\tilde{\tau }}_R^{corr}}\mathrm{\backslash nulldelimiterspace}{\tilde{\tau }}_R^{corr}\right)$, where ${\tilde{\tau }}_R^{corr}$ and $\tilde{A}$ are unitless fit parameters and $\tilde{t}=t/\phantom{t{\tau }_0}\mathrm{\backslash nulldelimiterspace}{\tau }_0$, yields ${\tilde{\tau }}_R^{corr}={\tau }_R^{corr}/\phantom{{\tau }_R^{corr}{\tau }_0}\mathrm{\backslash nulldelimiterspace}{\tau }_0=327$ (with $\tilde{A}=30.57$) (Fig 3A) (we used the autocorrelation function of $\frac{\mathbf{\backslash buildrel}\mathbf{\backslash lower}\mathbf{3pt}\mathbf{\text{(scriptscriptstylerightharpoonup)}}}{\mathbf{\rho }}$ instead of $\frac{\mathbf{\backslash buildrel}\mathbf{\backslash lower}\mathbf{3pt}\mathbf{\text{(scriptscriptstylerightharpoonup)}}}{\mathbf{r}}$ because $⟨\frac{\mathit{\backslash buildrel}\mathit{\backslash lower}\mathbf{3pt}\text{(scriptscriptstylerightharpoonup)}}{\mathit{\rho }}⟩=0$ by symmetry, allowing for a fit of $C\left(t\prime \right)$ to a single exponential with only two free parameters). (B) By the exponential decay of the mean value $⟨\frac{\mathit{\backslash buildrel}\mathit{\backslash lower}\mathbf{3pt}\text{(scriptscriptstylerightharpoonup)}}{\mathit{\rho }}\left(t\right)⟩$ to equilibrium starting from a non-equilibrium state at $t=0$; in our simulations, the initial non-equilibrium state of the chain consisted in a fully extended conformation along the x-direction at $t=0$, allowing the chain to relax to equilibrium for $t>0$ (in the absence of flow). Fitting $⟨\frac{\mathit{\backslash buildrel}\mathit{\backslash lower}\mathbf{3pt}\text{(scriptscriptstylerightharpoonup)}}{\mathit{\rho }}\left(t\right)⟩$ to a single exponential $\tilde{B}\exp \left(-\tilde{t}\prime /\phantom{\tilde{t}\prime {\tilde{\tau }}_R^{relax}}\mathrm{\backslash nulldelimiterspace}{\tilde{\tau }}_R^{relax}\right)$, where ${\tilde{\tau }}_R^{relax}$ and $\tilde{B}$ are unitless fit parameters, yields ${\tilde{\tau }}_R^{relax}={\tau }_R^{relax}/\phantom{{\tau }_R^{relax}{\tau }_0}\mathrm{\backslash nulldelimiterspace}{\tau }_0=339$ (with $\tilde{B}=30.57$) (Fig 3B). According to Onsager’s regression hypothesis (or, more generally, the fluctuation-dissipation theorem) [23], we expect ${\tau }_R^{corr}={\tau }_R^{relax}$, which holds for our simulation results with an error of about 3%. In what follows, we use the average of the values obtained by the two methods, i.e.,

$${\tilde{\tau }}_R=\frac{{\tau }_R}{{\tau }_0}\equiv \frac{1}{2}\left({\tilde{\tau }}_R^{corr}+{\tilde{\tau }}_R^{relax}\right)=333.$$ (23)

The simulations used to determine ${\tau }_R$ were done in the absence of any PEG-induced attractive interaction between DNA segments, i.e., $\alpha =0$ in Eqs. (7) and (8). Thus, the value ${\tau }_R=333{\tau }_0$ in Eq. (23) holds for the specific viscosity η₀ of the buffer without PEG (indicated by the index “0”). As will be argued below, for the salt concentrations in our experiments, ${\tau }_R$ is linearly proportional to the viscosity η, i.e., ${\tau }_R=c\eta$, where c is a constant (see Eq. (26) and text above it). In Section 4.3 we determine the constant c by comparing simulated and experimentally measured chain extensions for a buffer without PEG (see Eq. (28) and text above it). Once c is determined, we can perform our simulations for any viscosity η, including those viscosity values generated by adding 3% and 5% PEG to the buffer in our experiments. This is done by finding the Weissenberg number Wi for given η using Eq. (27), which gives the Péclet number Pe using Eqs. (22) and (23) ($Pe=Wi/\phantom{Wi333}\mathrm{\backslash nulldelimiterspace}333$), and using this value of Pe in Eq. (18) (see Section 4.3).

4 Results

4.1 PEG leads to higher buffer viscosities

To confirm that adding PEG to a buffer solution increases the viscosity, we first measured buffer viscosities while changing the shear rate using a rheometer with a parallel plate-plate configuration. In the rotational rheometer, one plate rotates relative to the other, inducing shear on the fluid between the two plates.

In our experiments, for the imaging buffer (see Section 2) without PEG, the viscosity $\eta$remained roughly constant (0.914 ± 0.02 cP) for shear rates $\dot{\gamma }$ in the range 60–500 s⁻¹ (Supporting Information, S1A and S1B Fig, Supplementary S1 Data), suggesting that the buffer behaves as a Newtonian fluid. Supplementing the buffer with 3%, 5%, or 10% concentrations of PEG, respectively, increases $\eta$ in this range of shear rates as follows: 1.675 ± 0.05 cP (3% PEG), 2.396 ± 0.03 cP (5% PEG), and 5.680 ± 0.04 cP (10% PEG) (Supporting Information, S1C Fig, Supplementary S1 Data), using the notation (mean value) ± (standard deviation, SD) (Pearson’s correlation coefficient: 0.976), and these PEG-containing buffers are again Newtonian fluids (see Table 1). The viscosity measurements directly demonstrate that PEG increases the viscosity of a buffer, which therefore needs to be taken into account in our single-molecule DNA flow-stretching assay.

Table 1. Measured buffer viscosity, DNA length, and fluctuations of the free DNA end, as a function of PEG concentration.

[PEG]	〈η〉 ± SD (cP)	$\overline{X}$ ± SD (SEM) (μm)	$\overline{\Delta X}$ ± SD (SEM) (μm)
0%	0.914 ± 0.02	11.897 ± 1.249 (0.126)	0.3966 ± 0.1095 (0.0111)
3%	1.675 ± 0.05	12.446 ± 1.545 (0.221)	0.3538 ± 0.1118 (0.0160)
5%	2.396 ± 0.03	12.554 ± 0.819 (0.117)	0.2652 ± 0.0601 (0.0086)
10%	5.680 ± 0.04	no data	no data

Measured buffer viscosity 〈η〉, average end-to-end DNA length $\overline{X}$ in x-direction (Eq. (2), and standard deviation $\overline{\Delta X}$ of the DNA end position in x-direction (Eq. (3), as a function of PEG concentration in our DNA flow-stretching experiments. Errors are given by the standard deviation (SD) followed by the standard error of the mean (SEM) in parentheses as indicated. SD and SEM were determined from N = 49 independent measurements for 3% and 5% [PEG] and N = 98 independent measurements for 0% [PEG] (Figs 4 and 5, and Section 2). 10% PEG led to nonspecific DNA binding to the flow cell, so that we were not able to obtain reproducible results for the DNA extension; therefore, no data for $\overline{X}$ and $\overline{\Delta X}$ are shown for 10% PEG.

4.2 Experimental DNA flow-stretching at different PEG concentrations

We experimentally investigated how the end-to-end length and the strength of the fluctuations of the free end of flow-stretched DNA depend on the PEG concentration. Comparisons for buffers with 5% and 3% [PEG] with 0% [PEG] are shown in Figs 4 and 5, respectively, and Table 1. (We also attempted to perform DNA flow-stretching experiments with 10% PEG; however, this elevated level of [PEG] led to nonspecific DNA binding to the flow cell surface, so that we were not able to obtain reproducible results for the DNA extension.)

Fig 4. Supplementing 5% PEG increases the end-to-end flow-stretched DNA length and reduces fluctuations.

(A) Representative graph of quantum dot (QD) position versus time for an individual DNA molecule. Data collection of QD positions x (red dots) vs time began in the absence of flow and continued while turning on flow in the x-direction, leading to DNA stretching. Data were collected for an imaging buffer without PEG and for a buffer supplemented with 5% PEG. For a given PEG concentration, the mean value 〈x〉 (horizontal blue lines) and the standard deviation (SD) Δx were determined from the corresponding set of QD positions. (B) Histograms of mean values 〈x〉 from independent measurements on 49 DNA molecules for buffers without PEG (upper panel) and 5% PEG (lower panel). (C) Same as (B) for SDs Δx. (D) Mean value $\overline{X}$ of mean values 〈x〉 for the ensemble of 49 DNA molecules corresponding to the histograms in (B) for buffers without PEG and 5% PEG, respectively (see Eq. (2)). The error bars show the SDs (larger error bars) and the standard errors of the mean (SEMs, smaller error bars) corresponding to the histograms in (B). ****: p < 0.0001 by Mann-hiteney test. (E) Same as (D) for the mean value $\overline{\Delta X}$ of SDs Δx corresponding to the histograms in (C) (see Eq. (3)). (F) Histogram of the relative increase in chain extensions $\mathbf{\delta }\mathbf{=}⟨x⟩(5\%)-⟨x⟩(0\%)$ obtained for the 49 individual DNA molecules upon supplementing the buffer with 5% PEG.

Fig 5. Supplementing the buffer with 3% PEG increases the end-to-end flow-stretched DNA length and reduces fluctuations.

(A) Histograms of mean values 〈x〉 from independent measurements on 49 DNA molecules for buffers without PEG (upper panel) and 3% PEG (lower panel). (B) Same as (A) for the standard deviations (SDs) Δx. (C) Mean value $\overline{X}$ of mean values 〈x〉 for the ensemble of 49 DNA molecules corresponding to the histograms in (A) for buffers without PEG and 3% PEG, respectively (see Eq. (2)). The error bars show the SDs (larger error bars) and the standard errors of the mean (SEMs, smaller error bars) corresponding to the histograms in (A). *: 0.01 < p < 0.05 by Mann-Whiteney test. (D) Same as (C) for the mean value $\overline{\Delta X}$ of SDs Δx corresponding to the histograms in (B) (see Eq. 3). **: 0.001 < p < 0.01 by Mann-Whiteney test. (E) Histogram of the relative increase in chain extensions $\mathbf{\delta }\mathbf{=}⟨x⟩(3\%)-⟨x⟩(0\%)$ obtained for the 49 individual DNA molecules upon supplementing the buffer with 3% PEG.

Linearized λ-phage DNA (48.5 kbp) has 12-nucleotide single-stranded overhangs at both ends. We utilized the overhangs to introduce biotin at one end of the DNA and digoxigenin at the other end (Section 2). One end of the DNA was tethered to the surface of the flow cell via biotin-neutravidin interaction, and the other end was labeled with an (anti-digoxigenin antibody-conjugated) fluorescent quantum dot (QD) (Fig 1A). Since these labels are specific (based on complementary oligo sequences), using the 12-nt overhangs of λ-phage DNA ensures a unique geometry of DNA tethering and QD labeling. Furthermore, the superior QD brightness (compared to fluorescence intensities from organic dyes or fluorescent proteins) enabled us to determine the average QD positions during the EMCCD exposure time (100 ms) with high accuracy (Section 2).

For a given DNA molecule and in the absence of flow, the average QD position in x-direction, 〈x〉, was approximately that of the DNA tether point. Applying flow in x-direction stretches the DNA (Supporting Information, S2 Movie), and the resulting increase of the average end-to-end DNA length 〈x〉 was obtained by comparing average QD positions before and after the DNA was stretched by flow (Fig 4A, red dots). We started with an imaging buffer without PEG followed by a buffer supplemented with 5% PEG (Fig 4A). We also calculated the standard deviation Δx (fluctuations) along the x-direction of the QD positions on the flow-stretched DNA. The average end-to-end length 〈x〉 and fluctuations Δx were obtained for both buffer conditions (no PEG and 5% PEG). Although multiple DNA molecules are observed in a single field of view, each of them behaves slightly differently in terms of stretching and fluctuations. Observing individual DNA molecules allows the detection of heterogeneous events. Thus, these measurements were performed for 49 different DNA molecules. For rigorous data reporting, three (5% PEG) and two (3% PEG) independent experiments for each condition were performed (Figs 4 and 5, respectively). These experiments were performed across multiple microfluidic flow cells and on different days. The averages of DNA length changes span between 0.46 ~ 0.86 μm for the 5% PEG experiments and between 0.59 ~ 0.62 μm for the 3% PEG experiments. Therefore, the Mann-Whitney test was employed to compare measured data rigorously despite the heterogeneities. Figs 4B and 4C show histograms of the resulting mean chain extensions 〈x_i〉 and standard deviations Δx_i, respectively, for no PEG and 5% PEG. The histograms in Fig 4B are broader than the fluctuations for a single DNA molecule in Fig 4A. This is because errors in determining DNA tether point positions, in addition to fluctuations of QD positions, affect the DNA end-to-end length measurements (see Section 2). Figs 4D and 4E show the averages $\overline{X}$ and $\overline{\Delta X}$, respectively, over these values (see Eqs. (2) and (3)), again for no PEG and 5% PEG. These data show that the addition of 5% PEG increases the average chain extension $\overline{X}$ (Fig 4D) and reduces the average fluctuations $\overline{\Delta X}$ (Fig 4E) compared to the buffer without PEG in a statistically significant way (p-value < 0.0001 obtained by Mann-Whitney test). Finally, Fig 4F shows a histogram of the relative increase in chain extensions, ${\delta }_i=⟨x_i⟩(5\%)-⟨x_i⟩(0\%)$ obtained for the individual DNA molecules upon supplementing the buffer with 5% PEG, resulting in an average $\overline{\delta }=\frac{1}{49}\sum _{i=1}^{49}{\delta }_i=0.68$ μm.

Similar results were obtained from experiments comparing DNA length and fluctuations for buffers without PEG and with 3% PEG (Fig 5A-5E).

4.3 The dual role of PEG in DNA flow-stretching experiments

As discussed in Section 1, PEG plays a dual role in DNA flow-stretching experiments: On the one hand, PEG tends to reduce the DNA length by inducing an effective, attractive interaction between DNA segments; on the other hand, PEG increases the DNA length by increasing the buffer viscosity η. In general, the mean chain length 〈x〉 of flow-stretched DNA depends on both the viscosity η and the PEG-induced attractive interaction between DNA segments, which in turn depends on the PEG concentration of the buffer, [PEG]. In our theoretical treatment, we separate these two effects by considering 〈x〉 as a function of two independent variables η and [PEG]. This allows us to isolate the effect of the buffer viscosity on 〈x〉 by performing simulations for different values of η in the absence of any PEG-induced attractive interaction between DNA segments (experimentally, this corresponds to supplementing the buffer with a hypothetical agent that increases the viscosity η without interacting with the DNA in any other way). Conversely, in the experiments, the viscosity η is implicitly determined by [PEG], corresponding to a functional dependence ${\eta }_{\exp }=\eta \left(\left[\mathrm{P}\mathrm{E}\mathrm{G}\right]\right)$ (see Table 1, second column). Therefore, we also performed simulations for the specific combinations $\left(\eta \left(\left[\mathrm{P}\mathrm{E}\mathrm{G}\right]\right),\left[\mathrm{P}\mathrm{E}\mathrm{G}\right]\right)$ of the variables η and [PEG] that are realized in our experiments for 3% and 5% [PEG], respectively. This allows us to compare simulation results for which the PEG-induced attractive interaction between DNA segments is included or not included, respectively, at the corresponding values of the viscosity in the experiments.

In brief, the experimental results for flow-stretched DNA are compared with the simulations as follows. First, we simulate the fractional chain extension $\xi =⟨x⟩/\phantom{⟨x⟩L}\mathrm{\backslash nulldelimiterspace}L$, where L is the contour length of the DNA, as a function of η in the absence of any PEG-induced attractive interaction between DNA segments (corresponding to α = 0 in Eqs. (7) and (8)). This results in a simulated function ${\xi }_{\mathrm{s}\mathrm{i}\mathrm{m}}\left(\eta ,0\right)$ shown as the black solid lines and black open circles in Fig 6A and 6B. We find that the experimental values ${\xi }_{\exp }$ obtained for 3% and 5% [PEG] (green circles labeled η₁ and η₂ in Fig 6B, respectively) deviate from ${\xi }_{\mathrm{s}\mathrm{i}\mathrm{m}}\left(\eta ,0\right)$ for the corresponding values of η. However, turning on the PEG-induced attractive interaction between DNA segments in the simulations (corresponding to choosing suitable values of α and D in Eq. (8)) at the given viscosities η₁ and η₂ results in good agreement between the simulated and experimental results.

In what follows, we discuss the procedure sketched above in detail. As mentioned above, to distinguish between the counteracting effects of the buffer viscosity η and the PEG-induced attractive interaction between DNA segments, we assume that the mean DNA length 〈x〉 in flow direction (Fig 1B) is a function of $\eta$ and the PEG concentration, which we consider as independent variables in our theoretical treatment:

$$⟨x⟩=⟨x⟩\left(\eta ,\left[\mathrm{P}\mathrm{E}\mathrm{G}\right]\right).$$ (24)

That is, the dependence of 〈x〉 on the buffer viscosity is described by the first argument $\eta$, whereas the dependence of 〈x〉 on the PEG-induced attractive interaction between DNA segments is described by the second argument [PEG], where η and [PEG] are independent variables. For example, $⟨x⟩\left(\eta ,0\right)$ describes the dependence of 〈x〉 on the viscosity η in the absence of an attractive interaction between DNA segments. In our DNA flow-stretching experiments (indicated by the subscript “exp” below), the buffer viscosity $\eta$ is determined by the PEG concentration, i.e., ${\eta }_{\exp }=\eta \left(\left[\mathrm{P}\mathrm{E}\mathrm{G}\right]\right)$ (Table 1, second column); thus, the measured mean DNA length ${\overline{X}}_{\exp }$ for the ensemble of 49 DNA molecules (see Eq. (2)) is effectively a function of [PEG] only (Table 1, third column), which by Eq. (24) can be written as

$${\overline{X}}_{\exp }\left(\left[\mathrm{P}\mathrm{E}\mathrm{G}\right]\right)=\overline{X}\left(\eta \left(\left[\mathrm{P}\mathrm{E}\mathrm{G}\right]\right),\left[\mathrm{P}\mathrm{E}\mathrm{G}\right]\right).$$ (25)

Measurements for bacteriophage lambda DNA indicate that if the electrolyte concentration is high enough (~10 mM of NaCl and above), the relaxation time ${\tau }_R$ in Eq. (22) is proportional to the viscosity η of the solvent [23,81], i.e., ${\tau }_R=c\eta$, where c is a constant. The buffer used in our single-molecule flow-stretching assay meets this condition (10 mM Tris pH 8.0, 150 mM NaCl, 10 mM MgCl₂, 0.2 mg/mL bovine serum albumin, see Section 2). Assuming that the relaxation time of the PEG is much faster than ${\tau }_R$, so that ${\tau }_R$ is determined only by the viscosity, the Weissenberg number in Eq. (22) is given by

$$Wi=\dot{\gamma }{\tau }_R\equiv \dot{\gamma }c\eta .$$ (26)

Since the shear rate $\dot{\gamma }$ in our experiments is also constant ($\dot{\gamma }=192.9{\mathrm{s}}^{-1}$, see Eq. (12)), we obtain

$$Wi=c\prime \eta$$ (27)

where $c\prime :=c\dot{\gamma }$ is another constant. Equations (25)–(27) allow for a comparison of our experimental results for the mean DNA length, ${\overline{X}}_{\exp }\left(\left[\mathrm{P}\mathrm{E}\mathrm{G}\right]\right)$ in Eq. (25), with the Brownian dynamics simulations for a single chain as follows (Fig 6A and 6B). First, we simulated the fractional extension $\xi =⟨x⟩/\phantom{⟨x⟩L}\mathrm{\backslash nulldelimiterspace}L$ as a function of Wi in the absence of PEG, i.e., $\alpha =0$ in Eq. (8) (using the Péclet number $Pe=Wi/\phantom{Wi{\tilde{\tau }}_R}\mathrm{\backslash nulldelimiterspace}{\tilde{\tau }}_R$ in Eq. (18) with ${\tilde{\tau }}_R=333$ from Eq. (23)), where L is the contour length of the chain. This yields the simulated (sim) function ${\xi }_{\mathrm{s}\mathrm{i}\mathrm{m}}\left(Wi\right)$ (Fig 6A, labels on bottom axis). Using the condition ${\xi }_{\mathrm{s}\mathrm{i}\mathrm{m}}\left(W{i}_0\right)\equiv {\xi }_{\exp }\left({\eta }_0\right)$, where ${\xi }_{\text{exp}}\left({\eta }_0\right)=\frac{{\bar{X}}_{\text{exp}}(0\%)}{L}$ is the mean fractional extension measured for a flowing buffer without PEG and ${\eta }_0=0.914$ cP (first line in Table 1), yields the Weissenberg number $W{i}_0$ for the viscosity ${\eta }_0$, and thereby the constant $c\prime$ in Eq. (27):

$$c\prime =W{i}_0/\phantom{W{i}_0{\eta }_0}\mathrm{\backslash nulldelimiterspace}{\eta }_0=20.48\mathrm{c}{\mathrm{P}}^{-1}.$$ (28)

From ${\xi }_{\mathrm{s}\mathrm{i}\mathrm{m}}\left(Wi\right)$ and using Eqs. (27), (28), we obtain the simulated function ${\xi }_{\mathrm{s}\mathrm{i}\mathrm{m}}\left(\eta \right)=⟨x⟩\left(\eta ,0\right)/\phantom{⟨x⟩\left(\eta ,0\right)L}\mathrm{\backslash nulldelimiterspace}L$, i.e., the simulated fractional chain extension as a function of $\eta$ in the absence of PEG (see Eq. (24)) (Fig 6A, labels on top axis). Thus, in Fig 6B, the agreement of $\xi \left({\eta }_0\right)$ between the simulated value (leftmost black dot) and the measured value (green circle with error bars) results from the condition ${\xi }_{\mathrm{s}\mathrm{i}\mathrm{m}}\left(W{i}_0\right)={\xi }_{\exp }\left({\eta }_0\right)$ used to determine the constant $c\prime$ in Eq. (28).

Next, we considered a buffer with 3% PEG corresponding to the measured viscosity value ${\eta }_1=1.675\mathrm{c}\mathrm{P}$ (second line in Table 1). According to Eqs. (7) and (8), the PEG-induced attractive interaction between DNA segments is proportional to the PEG concentration and a constant α. The latter depends on the exponential decay length $D$ in Eqs. (7) and (8), but for given $D$ is independent of [PEG] and η. To find $\alpha$ for given $D$ (where $D=a$ and $D=a/\phantom{a2}\mathrm{\backslash nulldelimiterspace}2$ in our simulations), we used the condition

$${\xi }_{\text{sim}}\left({\eta }_1;\alpha \right)\equiv {\xi }_{\text{exp}}\left({\eta }_1\right)=\frac{{\bar{X}}_{\text{exp}}(3\%)}{L}$$ (29)

where ${\xi }_{\mathrm{s}\mathrm{i}\mathrm{m}}\left({\eta }_1;\alpha \right)$ is the simulated fractional chain extension for $\begin{array}{c}\hfill {\eta }_1=1.675\mathrm{c}\mathrm{P}\end{array}$ and given values for α and D in Eq. (8), and ${\xi }_{\exp }\left({\eta }_1\right)$ is the measured fractional chain extension for 3% PEG (see Table 1 and Eq. (25)). Equation (29) yields $\begin{array}{c}\hfill \alpha =0.345k_{B}T\end{array}$ for $D=a$ (Fig 6B, blue symbol for ${\eta }_1$) and $\alpha =0.615k_{B}T$ for $D=a/\phantom{a2}\mathrm{\backslash nulldelimiterspace}2$ (Fig 6B, magenta symbol for ${\eta }_1$). Thus, in Fig 6B, the agreement of $\xi \left({\eta }_1\right)$ between the simulated values (blue and magenta symbols) and the measured value (green circle with error bars) results from the condition Eq. (29) used to determine $\alpha$ for given $D$.

Finally, we considered a buffer with 5% PEG corresponding to the measured viscosity value $\begin{array}{c}\hfill {\eta }_2=2.396\mathrm{c}\mathrm{P}\end{array}$ (third line in Table 1). Since $\alpha$ for given $D$ was determined by Eq. (29) the simulated values $\xi \left({\eta }_2\right)$ for $D=a$ (Fig 6B, blue symbol for ${\eta }_2$) and $D=a/\phantom{a2}\mathrm{\backslash nulldelimiterspace}2$ (Fig 6B, magenta symbol for ${\eta }_2$) were obtained without any adjustable parameters. The agreement with the corresponding experimental results (green circle with error bars) provides evidence for the validity of our DNA model (Eqs. (14)–(16)) incorporating the PEG-induced attractive interaction between DNA segments by Eqs. (7) and (8). This agreement is obtained independently of the specific choice of the exponential decay length D in Eq. (8) (i.e., $D=a$ or $D=a/\phantom{a2}\mathrm{\backslash nulldelimiterspace}2$); this shows that the interaction energy in Eq. (7) is robust against a change of the parameter D in the sense that such a change can be compensated by a corresponding adjustment of the parameter α.

Fig 7 shows a comparison between the standard deviation Δx of the DNA end (Fig 1B) along the x-direction obtained in the simulations for a single chain (sim) and the average ${\overline{\Delta X}}_{\exp }$ in Eq. (3) for the ensemble of 49 DNA molecules in the experiments (Fig 4 and 5). The meaning of the symbols is the same as in Fig 6. According to polymer theory, the ratio $\varphi =\Delta x/\phantom{\Delta x{R}_e}\mathrm{\backslash nulldelimiterspace}{R}_e$, where

$$R_e=\sqrt{⟨{\left|{\frac{\mathbf{\backslash buildrel}\mathbf{\backslash lower}\mathbf{3pt}\mathbf{\text{(scriptscriptstylerightharpoonup)}}}{\mathbf{r}}}_N-{\frac{\mathbf{\backslash buildrel}\mathbf{\backslash lower}\mathbf{3pt}\mathbf{\text{(scriptscriptstylerightharpoonup)}}}{\mathbf{r}}}_0\right|}^2⟩}$$ (30)

is the root mean square end-to-end distance of the free chain at equilibrium (i.e., without the surface of the flow cell and without flow, see Fig 2), is a universal function of Wi [18–20]. Fig 7A shows the simulated ratio $\varphi =\Delta x/\phantom{\Delta x{R}_e}\mathrm{\backslash nulldelimiterspace}{R}_e$ vs Wi (bottom axis) and $\eta$ (top axis) in the absence of a PEG-induced attractive interaction between DNA segments. Here, $R_e=\sqrt{50}a$ for our model chain with $N=50$ segments of length $a$ (Fig 2). Fig 7B is a magnification of the η-region $0.7\mathrm{c}\mathrm{P}\le \eta \le 2.6\mathrm{c}\mathrm{P}$ highlighted by the blue box in Fig 7A. Experimental ϕ -values for 0%, 3%, and 5% PEG are indicated by green circles with error bars (standard deviation). These values were determined as follows. Experimental values for Δx are given in Table 1. To find $R_e$ for the λ-DNA used in our experiments, we used $R_e=\sqrt{N_K}{b}_K$ where $N_K$ is the number of Kuhn segments (K) and $b_K$ is the Kuhn length of the DNA. Using the DNA persistence length $\begin{array}{c}\hfill {\ell }_P=50\mathrm{n}\mathrm{m}\end{array}$ corresponding to the buffer in our experiments (see Section 2) and the contour length $\begin{array}{c}\hfill L=16.49\mu \mathrm{m}\end{array}$ of the λ-DNA gives $\begin{array}{c}\hfill b_K=2{\ell }_P=0.1\mu \mathrm{m}\end{array}$, thus $N_K=L/\phantom{L{b}_K}\mathrm{\backslash nulldelimiterspace}{b}_K=165$ and $\begin{array}{c}\hfill R_e=1.284\mu \mathrm{m}\end{array}$. The blue and magenta symbols in Fig 7B are simulated ϕ -values for 3% PEG (η₁) and 5% PEG (η₂) using the same values for $\alpha$, D as determined for Fig 6, i.e., without adjustable parameters. Both the experimental and the simulated results show that Δx (fluctuations of the DNA end position, see Fig 1B) decreases with increasing PEG concentration. The simulated results for $\varphi =\Delta x/\phantom{\Delta x{R}_e}\mathrm{\backslash nulldelimiterspace}{R}_e$ agree with the experiments within the standard error, providing further evidence for the validity of our DNA model incorporating the PEG-induced attractive interaction between DNA segments by Eqs. (7) and (8). However, for Δx (Fig 7) the influence of the attractive interaction between DNA segments is less pronounced than for the chain extension 〈x〉 (Fig 6).

5 Discussion

We studied the dynamics of singly-tethered flow-stretched λ-DNA molecules by supplementing different concentrations of PEG to one of the commonly used single-molecule buffer conditions (10 mM Tris, pH 8.0, 150 mM NaCl, and 10 mM MgCl₂). For this purpose, we labeled the free DNA end with a fluorescent quantum dot (QD) and dynamically tracked the motion of the QD in real time using fluorescence microscopy. Out of different crowding reagents commonly used in in vitro experiments, we were particularly interested in the effects of PEG due to its twofold properties: (1) While another commonly used crowding agent, sucrose, shows no evidence of DNA compaction [82], PEG can compact DNA [83] by inducing an excluded volume effect, generating an effective attraction between DNA segments [53–55]; (2) PEG increases the buffer viscosity, which can increase DNA stretching under flow. The high sensitivity of our single-molecule DNA flow-stretching assay [28,29] allowed us to resolve these counteracting effects of PEG on DNA experimentally. Furthermore, our Brownian dynamics simulations of a bead-spring chain model of flow-stretched DNA in a viscous buffer that incorporates the PEG-induced attractive interaction between DNA segments results in quantitative agreement with the experimental results for suitable PEG-DNA interaction parameters. When a PEG-induced attractive interaction between DNA segments was not considered ($\alpha =0$; Fig 6), deviations of ξ-values between experiments and simulations were prominent. This result underscores the importance of considering both counteracting effects of PEG. In summary, our approach provides proof of principle of understanding the dynamics of DNA interacting with an agent in a crowded environment. This is relevant for probing mechanisms of DNA-binding proteins and their implications in DNA [30–36] because the cellular environment in which these proteins act is crowded.

It would be interesting to extend this work in the following directions:

• (1) Study the balance of PEG-induced compaction and viscous stretching as a function of the DNA chain length: The current work considers 48.5-kb λ-phage DNA to compare experiments on λ-phage DNA with simulations. However, polymer dynamics, including compaction and flow response, are strongly dependent on chain length.

• (2) Probe correlations of fluctuations within the chain for flow-stretched DNA by labeling DNA molecules with multiple fluorescent quantum dots at specific sites along the DNA contour [27] and how these correlations change in the presence of PEG or DNA-binding ligands.

• (3) Extend higher-resolution DNA models to flow-stretched DNA: Coarse-grained DNA models, such as OxDNA [84] and reference [85], represent the three-dimensional structure of DNA explicitly, including DNA’s helical structure and sequence of nucleotides. It would be interesting to extend these models to probe DNA and DNA-ligand interactions under flow for short DNA sequences.

• (4) Even without PEG, the flow-stretched chain exhibits the intriguing phenomenon of large temporal fluctuations in the chain extension (backfolding and looping), at moderate flow strength, referred to as cyclic dynamics (see [18–20], and Supporting Information, Movie S1). It would be interesting to study this phenomenon for flow-stretched DNA in the presence of PEG or DNA-binding ligands.

• (5) Our approach, combining singly-tethered DNA stretching experiments with Brownian dynamics simulations, lays the groundwork for future DNA stretching research. For example, in single-molecule DNA-protein studies, highly negatively charged casein and bovine serum albumin (BSA) are commonly supplemented to a buffer to minimize nonspecific adsorption of proteins onto the microfluidic flow cell [86,87]. Building on previous reports of the effects of negatively charged proteins and colloids on DNA compaction [88,89], our approach is expected to provide a deeper understanding of the effects of those unbound charged reagents in solution to the flow-stretched DNA compaction by a DNA-binding protein. Results obtained by exploring different flow cell channel geometry and flow rates will ensure more accurate DNA flow-stretching data interpretation when crowding agents are involved.

Supporting information

S1 Fig. Measurements of buffer viscosity $\mathit{\eta }$ as a function of shear rate $\dot{\mathit{\gamma }}$ in the absence and presence of polyethylene glycol (PEG).

(A) Buffer viscosity $\eta$ in the absence of PEG. Shown are the results for $\eta$in units of centipoise (cP) for four measurements (gray) and the average of these measurements (black) ($\text{1}\text{cP}=\text{1mP}·\text{s}$). (B) For better visibility, shown is the average of $\eta$in (A) for shear rates $\dot{\gamma }$ in the range 20–1000 ${\text{s}}^{-\text{1}}$ as indicated by the red box in (A). Error bars: SD. (C) Buffer viscosity $\eta$ in the absence of PEG and with 3%, 5%, and 10% concentrations of PEG, respectively, for shear rates $\dot{\gamma }$ in the range 20–1000 ${\text{s}}^{-\text{1}}$ obtained as the average of five measurements for each concentration of PEG. Shear rates were averaged over the range 60–500 ${\text{s}}^{-\text{1}}$ for subsequent analysis in this study.

(TIF)

S1 Movie. Animation of a Brownian dynamics simulation of our model chain.

The movie was obtained for a model chain with N = 50 beads (Fig 2) by numerical iteration of the Langevin equation (Eq. 18). Shown is the projection of the bead positions on the xz-plane. Initially, the chain is aligned with the z-axis perpendicular to the surface in the xy-plane (Fig 1B). It is then stretched by shear flow in the x-direction characterized by the Péclet number Pe = 0.02 (Eq. 21) corresponding to the Weissenberg number Wi = 333 Pe = 6.66 (Eqs. 22 and 23). The time steps in Eq. 18 are $\Delta t={10}^{-4}$ in units of ${\tau }_0$ (Eq. 17). The animation contains 500 frames with a delay of ${10}^5\Delta t$ between frames (in units of ${\tau }_0$) corresponding to a total time of $5000{\tau }_0$. Note the cyclic dynamics reported in references 18 and 20.

(AVI)

S2 Movie. Flow-stretching of quantum dot-labeled bacteriophage lambda DNA.

The movie was recorded by an EMCCD camera (see main text, Section 2). The movie is 8 times faster than in real-time. The dimensions of the frame are 3.20 $\times$17.12 μm. In the beginning of the movie where there is no flow, the average quantum dot position corresponds to the DNA tether point location. Applying buffer stretches the surface-tethered quantum dot-labeled DNA in the direction of the buffer flow (from the bottom to the top). See Fig 1A.

(AVI)

S1 Data. Experimental raw data.

The experimental raw data for Figs 4, 5, and S1 are summarized in each tab.

(XLSX)

Data Availability

All relevant data are within the manuscript and its Supporting Information files (Supplementary S1 Data). In addition, the Fortran code for the Brownian dynamics simulations (Langevin equation) is available at the GitHub repository https://github.com/andreas-hanke/PEG-DNA-shear-flow/.

Funding Statement

This work was supported by the National Institutes of Health (https://www.nigms.nih.gov/) R35GM143093 (to H.K.).