Entropy-driven collective interactions in DNA brushes on a biochip

Abstract

Cell-free gene expression in localized DNA brushes on a biochip has been shown to depend on gene density and orientation, suggesting that brushes form compartments with partitioned conditions. At high density, the interplay of DNA entropic elasticity, electrostatics, and excluded volume interactions leads to collective conformations that affect the function of DNA-associated proteins. Hence, measuring the collective interactions in dense DNA, free of proteins, is essential for understanding crowded cellular environments and for the design of cell-free synthetic biochips. Here, we assembled dense DNA polymer brushes on a biochip along a density gradient and directly measured the collective extension of DNA using evanescent fluorescence. DNA of 1 kbp in a brush undergoes major conformational changes, from a relaxed random coil to a stretched configuration, following a universal function of density to ionic strength ratio with scaling exponent of 1/3. DNA extends because of the swelling force induced by the osmotic pressure of ions, which are trapped in the brush to maintain local charge neutrality, in competition with the restoring force of DNA entropic elasticity. The measurements reveal in DNA crossover between regimes of osmotic, salted, mushroom, and quasineutral brush. It is surprising to note that, at physiological ionic strength, DNA density does not induce collective stretch despite significant chain overlap, which implies that excluded volume interactions in DNA are weak.

Double-helix DNA polymers exhibit relaxed random-walk configurations at lengths beyond the persistence scale l_p = 50 nm, occupying volume to maximize their entropy. Unfolding DNA entropic degrees of freedom to full contour-length stretch requires large forces of 500 k_BT/l_p (50 pN) using a force-extension apparatus (1, 2). However, the transition of DNA into an ordered stretched state can also result from an entropy increase in a coupled system when chains experience significant overlap. Such is the case of polymer brushes where individual polymers stretch to minimize the free energy of the brush (3). For charged polymers, such as DNA, the collective extension also increases the mixing entropy of the ions that are trapped within the brush to maintain local charge neutrality (4–7).

In the past two decades, DNA brushes have become useful in a range of applications such as next-generation sequencing (8, 9), hybridization arrays (10–14), protein biosynthesis compartments (15–18), and coated particle assembly (19, 20). The utility of the diverse DNA-based reactions carried out in such brushes requires an in-depth understanding of their basic materials properties. To date, the focus has been on short DNA brushes (∼100 bp) (21, 22) having negligible polymer degrees of freedom. The compression of a few kilobase-pair DNA brushes on beads, as deduced from optical trapping force measurement and Brownian motion analysis (23, 24), was shown to behave as a power of −1/3 with ionic strength. For flexible polyelectrolytes, brush-height scaling with ionic strength and contour length has been observed by force measurements using synthetic diblock copolymers composed of hundreds of monomers (25). However, the scaling properties and phase behavior of DNA polymer brushes with varying density, ionic strength, and length have remained a challenge due to difficulty in controlling DNA grafting density.

In this paper we describe the following: (i) the method of brush assembly on continuous density gradients combined with direct height measurement. (ii) Observation of major conformational changes for a brush of 968 bp, exhibiting the universal scaling of polyelectrolyte brush height as a power of 1/3 with density-to-salt ratio (4). (iii) Theoretical considerations of the brush with no added salt, where the osmotic pressure of the confined counterions swells the brush to maximal extension (osmotic brush). The addition of salt reduces the osmotic pressure, leading to brush compression and scaling behavior (salted brush), from which the effective fraction of charged DNA phosphates = 5–10% is deduced. (iv) The crossover to regimes of high salt at low density (mushroom) and high density (quasineutral brush), where only minor extension with density is observed, from which we infer that excluded volume interactions of DNA are weak.

Brush Assembly and Height Measurement

Our measurements used the assembly of DNA polymer brushes along continuous density gradients on a photochemical biochip, combined with a direct measurement of DNA extension using total internal reflection fluorescence (TIRF) of fluorophores attached to the ends of the polymers (26) (Fig. 1A). The assembly of brushes from solution was irreversible, stable in time, and reproducible. Brush density increased with bulk DNA concentration, and was limited by the interactions between DNA polymers, rather than by shortage of surface binding sites (Fig. S1) (26). Based on a previous calibration using radioactive labeling of DNA (15), the maximal density for 1-kbp DNA is estimated to be ∼2,500 polymers/μm². Assuming that the polymers are stretched to their contour length, this brush density amounts to a local concentration of ∼10⁷ bp/µm³, which is comparable to dense DNA in live cells.

Fig. 1.

Measurement of 1-kbp DNA brush density and height. (A) Scheme of DNA brush along a density gradient. DNA ends labeled with a fluorophores (blue) excited by TIRF, and by standard epiFL. (B) (Upper) The epiFL image of a DNA density gradient. (Lower) DNA density profile in arbitrary fluorescence units along the x axis averaged along the symmetric y axis at varying NaCl concentration (color code bar in ionic strength). Data taken from within the gradient (white rectangle). Scale bar is 10 μm. (C) (Upper) TIRF image of a DNA density gradient. (Lower) TIRF profiles along gradient taken as a function of salt (NaCl) concentration. The TIRF profiles increase with salt-reflecting compression of height.

A 968-bp DNA brush was assembled on a rectangular pattern 90 × 45 μm², with the density continuously increasing by sevenfold along the x axis, symmetrically along the y axis (Fig. 1B). To extract the brush height h(x) along the DNA density gradient, TIRF, and standard height-independent epifluorescence (epiFL) images were taken consecutively. Averaging was done over the y axis to produce intensity profiles along the x axis (Fig. 1 B and C). The epiFL profile was proportional to local DNA density epiFL α σ(x), whereas the TIRF profile decayed exponentially with the mean height of the labeled DNA ends, TIRF ∝ σ(x)e^−h(x)/ξ, and estimated ξ ∼ 120 nm as the decay length of the evanescent excitation. The mean brush height as a function of density (in arbitrary fluorescence units) at each point along the gradient was then directly obtained by normalizing the TIRF signal with the epiFL signal (Supporting Information).

Collective Conformational Changes

To study the effect of ionic strength and density on brush height we varied monovalent salt (NaCl) concentration from no added salt to 150 mM. For the DNA density gradient profile (Fig. 1B), the TIRF profile gradually increased with salt, indicating compression of brush height (Fig. 1C); the fluorescence efficiency with salt was calibrated (Fig. S2). The resulting mean height profiles are plotted in Fig. 2A. Height deviations, because of the distribution of polymer ends, are estimated to be ∼10% and are neglected for simplicity (Fig. S3). It was noticeable that, at low ionic strength, c_s < 0.6 mM, the brush was maximally extended, nearly independent of density. The maximal height is estimated 300–330 nm within the experimental uncertainty.

Fig. 2.

DNA brush height and data collapse. (A) Brush height in units of evanescent decay length ξ ∼ 120 nm as a function of 968-bp DNA surface density and NaCl ionic strength, as deduced from the TIRF and epiFL profiles (color code bar in ionic strength). (B) Brush height as a function of ratio of density to ionic strength, σ/c_s. Data includes NaCl height for density σ < 460 (in arbitrary fluorescence units, estimated 2,000 μm⁻²). Solid line represents scaling of height ∝ (σ/c_s)^1/3. (C) Height as a function of density for varying MgCl₂ salt concentration. (D) Brush height as a function of ratio of density to ionic strength normalized by counterion valency, σ/c_sz². Data includes z = 1 NaCl height (gray circle) and z = 2 MgCl₂ (colored circle) for density σ < 460. Solid line represents scaling of h ∝ (σ/c_sz²)^1/3.

At high ionic strength, c_s = 150 mM, DNA attained a relaxed conformation at a height 100–120 nm that was also independent of density, corresponding to relaxed chains with minimal collective response, despite overlap between chains. Midrange, 0.6 < c_s < 150 mM, the brush stretched out gradually with increased density and reduced ionic strength. The height (Fig. 2A) displayed no hysteresis when increasing the salt concentration from pure water or vice versa (Fig. S4); identical results were obtained when replacing NaCl salt with Tris buffer (Fig. S5).

Plotting the brush height as a function of the density to ionic strength ratio, σ/c_s, for density σ < 460(FL) (∼2,000 μm⁻²) the data collapse onto a single curve (Fig. 2B). The function h(σ/c_s) is sigmoid-like, where up to σ/c_s ∼ 10 (FL/mM), DNA attains relaxed conformations at a constant height. At σ/c_s ∼ 10 (FL/mM) there is a smooth crossover to a scaling regime with h ∝ (σ/c_s)^1/3 up to σ/c_s ∼ 100 (FL/mM), where the brush is maximally stretched and scaling breaks down. Similar data collapse was previously observed for long flexible polyelectrolytes with 70–80% maximal extension at the osmotic regime (25).

Compression with increased ionic strength was also observed with divalent salt MgCl₂ (without NaCl), but attaining reduced maximal extension (Fig. 2C and Fig. S6). It is notable that the MgCl₂ and NaCl height curves differ for the same ionic strength (Fig. 2 A and C). Expressed in normalized variables, σ/c_sz², with counterion valency z we find (Fig. 2D) that the data for MgCl₂ collapse onto the same universal curve as NaCl, but saturates at a reduced brush extension. At high ionic strength the minimal brush height is similar for MgCl₂ and NaCl, with no measureable effects of collapse or bridging in the presence of the divalent ion.

We next investigated the height dependence on the DNA contour length in the range of 0.3–2.5 kbp, each at its maximal density (Fig. S1) (26). We measured the height at low salt (0.1 mM NaCl) and at high salt (250mM NaCl, Fig. 3), where the height is nearly independent of the density (Fig. 2A). Without salt, the DNA was maximally stretched and we obtained a linear increase of the DNA height with the contour length in the range 0.5 < h/ξ < 3, above which the TIRF signal-to-noise ratio decreased. At high salt (250 mM NaCl) the brush was maximally compressed, exhibiting sublinear dependence of the height with the contour length.

Fig. 3.

DNA brush height at varying contour length. Brush height h in units of evanescent decay length ξ as a function of DNA contour length in base pairs at low salt (circles, 0.1 mM NaCl), and high salt (triangles; 250 mM NaCl). Fits correspond (dashed line) to h = N_al_a and wormlike chain model (solid line; see Mushroom Regime).

Theory of DNA Brush

DNA carries two negatively charged phosphates per base pair, each bound to a counterion. In a salt-free solution a fraction f of the z-valent counterions are released, reflecting a balance of the counterion entropy gain with the electrostatic attraction. This results in Manning condensation (27) with charge fraction of one per Bjerrum length along the DNA, f_M = a/2zl_B = 1/4 (for monovalent counterions); the Bjerrum length is l_B = e²/ɛk_BT ∼ 0.7 nm in water. The charge unit is e and water permittivity ɛ. Here, we limit the discussion to z = 1,2, and ignore bridging effects expected at higher valency (28).

When viewed from a distance, the brush is a charged surface with counterions dispersed in solution. The mean distance λ between the counterions and the surface is obtained by balancing the entropic gain of ionization and the electrostatic attraction. The electrostatic energy is estimated as for a plate capacitor with charge density Q, inter-plate distance λ and capacitance C = ɛ/λ; the entropy gain is that of an ideal gas with concentration Q/λe: k_BTQ/e ∼Q²/2C. The brush charge density is, Q = 2Nfeσ, for N base-pair-long DNA. We obtain the Gouy–Chapman length λ = (l_BfσN)⁻¹∼10 nm for σ∼10³ μm⁻², N = 10³ bp, and f∼0.1. Thus, λ ≪ h and, in a salt-free solution, the released counterions remain trapped within the brush to maintain its neutrality with an effective concentration of c₀ = 2Nσf/h∼1 mM. Within the brush, the electrostatic interactions are screened over the Debye length, κ⁻¹ = (8πc₀l_B)^−1/2∼10 nm, which is smaller than the interchain distance ∼30 nm. Hence, DNA in a brush can be considered as a neutral polymer slab coexisting with an ideal counterion gas (4).

Osmotic Brush

The counterion entropy within the brush provides a driving force for swelling the volume, which comes at the expense of reducing the entropic elasticity of the DNA chains. For each persistence length the DNA has a single degree of freedom while there are n₀ = 2l_pf/a ∼30 counter-ions (for f ∼0.1) each with a free energy of order k_BT. This imbalance leads to a significant extension of the DNA due to the osmotic pressure of the counterions confined to the brush (osmotic regime).

At large extension h ∼ Na, the brush height can be evaluated based on the wormlike chain model, taking into account the DNA finite length (2). For a single molecule, the relation between DNA height and the applied stretching force, F, at large extension h/Na = 60–95% was measured, h/Na = 1 − (k_BT/4Fl_p)^1/2, with asymptotic full extension at large force, F ≫ k_BT/l_p (2). The osmotic pressure of the counterions Π_o/k_BT = Δc = c₀ applies a force on each DNA polymer, F = Π/σ = 2Nf/h ∼ 30k_BT/l_p, which is sufficient to stretch a single DNA molecule up to h/Na ∼ 90% of its contour length, in agreement with the observed maximal extension at low ionic strength (Fig. 2A). It is notable that the force per DNA, and hence brush height, is density independent in this regime (Fig. 2A; c_s < 0.6 mM).

In the presence of a minute bulk concentration of z-valent counterions, the counterions replace both condensed and uncondensed monovalent ions in the brush. Due to their high charge, fewer counterions are required to neutralize the brush. Furthermore, z-valent counterions condense more readily on the DNA leading to a reduction in the DNA charge, f ∝ z⁻¹ (27). Therefore, a minute concentration of MgCl₂, leads to a drop in the osmotic pressure and a discontinuous height reduction in the osmotic regime (Fig. 2C).

Salted Brush

At higher ionic strength c_s > c₀ the brush neutrality is maintained but Donnan equilibrium (detailed in Supporting Information) leads to a reduction in the ionic concentration imbalance, and Δc decreases to (4–6). The ionic osmotic pressure is then Π_s/k_BT = σ²N²f²/c_sh². For a small extension the DNA can be described as N_a freely joint Kuhn segments of size l_a = 2l_p. The DNA entropy loss is quadratic in the brush height , resembling a harmonic spring that resists stretching with a force (2). Balancing DNA entropic elasticity with counterion osmotic pressure we find the brush height in the salted regime

which is consistent with the observed scaling (Fig. 2B; 10 < σ/c_s < 100 FL/mM). It is notable that the brush-height data collapse for NaCl and MgCl₂ (Fig. 2D). Thus, the differences between mono- and divalent salt curves are quantitatively explained by considering their difference in condensation on DNA, f ∝ z⁻¹.

We fit the collapsed data using ξ = 120 nm, l_a = 50 − 100 nm, a = 0.34 nm, and maximal DNA density σ = 1,000–2,500 μm⁻². We deduce the charge fraction in the brush, f = 5–10%, which is somewhat smaller than the theoretical prediction, f_M = 1/4. It is notable that our measurement only counts the ions that contribute to the osmotic pressure in the brush. A small charge fraction was also observed for long polyelectrolytes where only 20% of the counterions were free (within the brush) instead of the theoretical prediction of 40% (25). Finally, DNA in dilute solution is expected to stiffen at low salt concentration (29, 30). We did not observe this effect. This is understood by observing that the brush holds a minimal concentration of ions ∼1 mM, which limits the electrostatic stiffening.

Mushroom Regime

At high salt and low density, the osmotic pressure vanishes as Π_s ∝ σ²/c_s and DNA in a brush assumes a density- and salt-independent configuration (Fig. 2B; σ/c_s < 10 FL/mM), suggesting that each chain is independent of its neighbors, and that there are no collective stretching effects (mushroom regime). Indeed, the sublinear dependence of height with contour length at high salt (Fig. 3; 250 mM) fits well with the end-to-end distance of a semiflexible polymer in solution (wormlike chain model), h ∝ R_WLC, with stiff rod behavior for short polymers R_WLC(N_a → 1) ≈ N_al_a and ideal chain behavior for long polymers . The interpolating formula is (31).

High-Density Quasineutral Brush

For the highest density values in the experiment, σ > 460 (FL) (∼2,000 μm⁻²) the brush height is no longer a scaling function of σ/c_s. We plot the brush height for NaCl and MgCl₂ as a function of ionic strength at high density σ = 500 (FL) (Fig. 4A), and as a function of the density at σ > 460 (FL) for several ionic strength values (Fig. 4B). At low ionic strength, c_s < 1.5 mM, the brush is fully extended independently of salt (Fig. 4A) and density (Fig. 4B). Midrange, 1.5 < c_s < 100 mM we observe (Fig. 4 A and B) a scaling regime, h ∝ (σ/c_s)^1/3, similar to the scaling at σ < 460 (FL) (Fig. 2B). At high ionic strength, c_s > 100 mM, the height enters a salt-independent regime (Fig. 4A), but maintains the scaling with density h ∝ σ^1/3 (Fig. 4B). This is in contrast to the behavior at σ < 460 (FL), where at high salt the height is salt and density independent. Scaling with density but not with salt is suggestive of excluded volume interactions. However, the height extension at high salt is small Δh/h_min ∼0.1. The excluded volume between two Kuhn segments is (32) leading to a swelling pressure, . Balancing the excluded volume interactions with the DNA entropic elasticity leads to equilibrium height, h_n = N_al_a(σl_ad/3)^1/3 (quasineutral regime). Fitting the brush-height density slope at c_s = 150 mM with the parameters l_a = 50 − 100 nm and maximal DNA density σ = 1000 − 2,500 μm^-2 we obtain d = 1–4 nm, in agreement with DNA diameter of d = 2 nm.

Fig. 4.

Scaling at high DNA brush density. Brush height in units of evanescent decay length ξ as a function of (A) NaCl ionic strength at high density σ = 500[FL]; color coding as in Fig. 2. Solid line represents scaling of , and (B) DNA density σ, in double distilled water (ddw), and for c_s = 5, 150 mM. Solid lines represent scaling of h ∝ σ^1/3.

Summary

We summarize the data in a phase diagram for a 1-kbp DNA (Fig. 5) and at a maximal density of ∼2,500 chains/μm². We identified four DNA brush regimes: osmotic, salted, mushroom, and quasineutral, with crossover between regimes determined as described (Table S1). The mean field theory of polyelectrolyte brushes was derived for infinitely long polymers. It was surprising that the theory successfully provides a quantitative description for a DNA polymer brush having only six persistence lengths, suggesting that DNA behaves as a Gaussian chain even at these short contour lengths, except at high extension where the finite polymer contour length must be taken into account. We found that the main swelling force is the osmotic pressure (entropy) of the counterions, and that the excluded volume interactions are weak even at the highest densities.

Fig. 5.

Phase diagram, scaling regimes. A plot in the density and ionic strength plane of the four brush-height scaling regimes—osmotic, salted, quasineutral, and mushroom—as identified in the data. The crossovers between the regimes are derived in Table S1.

The density at which the chains begin to overlap is of the order of , yet at high salt, no significant stretching was observed even at 40-fold higher density. Indeed, the onset of extension due to excluded volume interactions is expected at the transition from the mushroom regime to the quasineutral brush, h_n(σ_e) = R, with for d∼2 nm (Fig. 5). The separation of scales, σ_e ≫ R⁻² occurs because DNA is a rigid polymer l_a/d∼10², and the monomers excluded volume is small compared to the total polymer volume allowing the polymers to overlap without colliding. This is in contrast to flexible polymers where excluded volume interactions swell the single polymer, R ∝ N^3/5, and the onset of extension occurs with chains overlap, σ_e ∝ R⁻². The weak excluded volume interactions, at high salt, are in agreement with the ideal chain behavior for DNA < 100 kbp in solution (33–35) but disagree with the observed self-avoiding scaling R ∝ N^3/5 in DNA (36, 37).