Compartmentalization is key to organizing cellular biochemistry. Biomolecular condensate formation underlies the compartmentalization of many cellular functions6. Considerable progress has been made towards understanding the biophysical properties of condensates in bulk. However, how these condensates interact with other cellular components such as polymers, membranes and chromatin remains unclear. Transcriptional hubs represent an example of compartments in the nucleus. These hubs involve the coalescence of transcription factors, biochemical regulators of transcription, and DNA11. The physical nature of these transcriptional hubs is under debate, though recent studies have proposed that transcriptional hubs can be understood as examples of biomolecular condensates12. In theory, the interactions between condensates composed of transcriptional machinery and the DNA polymer could deform DNA, potentially bridging distal regulatory elements, a critical step in gene regulation. However, we still lack a physical picture of how transcriptional regulators interact with each other and with the surface of the DNA polymer.

To investigate how transcription factors physically organize DNA, we attached linearized λ-phage DNA to a coverslip via biotin–streptavidin linkers (Fig. 1a). We used total internal reflection fluorescence microscopy to image the interactions between DNA and forkhead box protein A1 (FoxA1), a pioneer transcription factor that regulates tissue differentiation across a range of organisms13 (Fig. 1b). Upon addition of 10 nM FoxA1–mCherry (FoxA1) to the flow chamber in the presence of DNA, FoxA1 formed protein condensates that decorated the strand (Fig. 1c). In the absence of DNA, FoxA1 did not nucleate condensates in solution at concentrations ranging from 10 to 500 nM (Extended Data Fig. 1a). The requirement for DNA in condensate formation at low concentrations suggests that DNA mediates the condensation of a thin layer of FoxA1 on DNA.

Fig. 1: FoxA1 forms DNA–FoxA1 condensates in a tension-dependent manner.
figure 1

a, Schematic of single λ-phage DNA molecule assay. PLL, poly-l-lysine. b, Structure of FoxA1, consisting of a structured DNA-binding domain flanked by mostly disordered N and C termini. The DNA-binding domain has a sequence-specific binding region (blue) and two non-sequence-specific binding regions (green). c, Representative time-averaged projections of FoxA1 and DNA. The extent of FoxA1-mediated DNA condensation depends on the end-to-end distance of the strand. Note that the total amount of DNA is the same in each example. The DNA was imaged using 10 nM SYTOX Green. Scale bar, 2 μm. d, Schematic displaying three main quantities used to characterize DNA–FoxA1 condensation: L, the DNA’s end-to-end distance; cross-correlation of DNA and FoxA1 intensities; and DNA envelope width, a measure of transverse DNA fluctuations. e, Cross-correlation of FoxA1 and DNA signals shows that FoxA1 condenses DNA below a critical end-to-end distance. The grey dots represent individual strands, n = 107. The data are binned every 2 μm (black, mean ± s.d. for both correlations and strand lengths). f, DNA envelope width measurements (Methods) reveal that FoxA1–DNA condensation buffers DNA tension (blue and black dots correspond to control and DNA + FoxA1 conditions, n = 45 and n = 50 respectively). The data are binned every 2 μm (mean ± s.d. for both the envelope widths and strand lengths). The dashed black line represents the theoretical diffraction limit. g, Representative images of FoxA1 zipping two independent DNA strands over time. Scale bar, 2 μm.

Source data

In our assay, DNA molecules displayed a broad distribution of end-to-end distances (L), determined by the DNA–coverslip attachment points (Fig. 1c,d). This end-to-end distance tunes the tension of the DNA14. For DNA strands with end-to-end distances greater than approximately 10 μm, FoxA1 generated protein condensates on DNA (Fig. 1c). However, FoxA1 condensation did not influence the DNA molecule (Fig. 1c, leftmost pair of images). Strikingly, for DNA molecules with end-to-end distances below 10 μm, FoxA1 pulled the DNA into highly enriched condensates of FoxA1 and DNA (Fig. 1c and Extended Data Fig. 1b–e) with a density of roughly 750 molecules μm3 (Methods and Extended Data Fig. 2a–d). To quantify FoxA1-mediated DNA condensation, we measured the cross-correlation of FoxA1–DNA intensities as a function of end-to-end distance (Methods, Fig. 1d,e and Extended Data Fig. 3a). Consistent with the ability of FoxA1 to form FoxA1–DNA condensates at low tensions, the cross-correlation decayed from one to zero with increasing end-to-end distance (Fig. 1e). Thus, FoxA1 mediates the formation of a protein–DNA-rich phase in a tension-dependent manner.

The observation that FoxA1 drives DNA condensation suggests that it can overcome the DNA molecule’s entropic tension set by the end-to-end distance14. Incorporating DNA into the condensates increases the tension on the strand, thereby reducing the transverse DNA fluctuations of the non-condensed DNA. To quantify this, we measured the DNA envelope width of the non-condensed DNA fluctuations (Methods and Extended Data Fig. 3b). In buffer, the DNA envelope width decreased as a function of end-to-end distance, consistent with the corresponding increase of DNA strand tension for increasing end-to-end distance14 (Fig. 1f). However, in the presence of FoxA1, the DNA envelope width remained constant for all end-to-end distances as FoxA1 pulled DNA into one or more condensates. The magnitude of the DNA envelope width was lower in the presence of FoxA1 than in buffer conditions for all end-to-end distances (Fig. 1f). Taken together, this suggests that FoxA1–DNA condensates generate forces that can overcome the entropic tension of the non-condensed DNA and buffer its tension.

The observation that FoxA1 can mediate DNA condensation suggests that it could bridge distant DNA strands. To investigate this possibility, we examined DNA molecules that were bound to the same streptavidin molecule at one end (Fig. 1g and Extended Data Fig. 3c). In the absence of FoxA1, these DNA molecules form a V-shaped morphology and fluctuate independently of one another. Upon addition of FoxA1, however, we observed that the two strands zipped together, generating a Y-shaped morphology as the condensation of FoxA1 increased over time (Fig. 1g and Extended Data Fig. 3c). Taken together, these data demonstrate that FoxA1 can physically bridge DNA strands in both cis and trans configurations.

Two mechanisms can be postulated to explain FoxA1-mediated DNA condensation in our experiments: (1) direct crosslinking via the multiple DNA-binding activities of FoxA115 or (2) weak protein–protein interactions driven by disordered regions of FoxA1. FoxA1 consists of a winged helix–turn–helix DNA-binding domain and two amino- and carboxy-terminus domains that are mostly disordered15. The DNA-binding domain contains a sequence-specific binding region composed of three alpha helices and a non-sequence-specific binding region composed of two wings. Two point mutations known to affect sequence-specific DNA binding (NH-FoxA115) had virtually no influence on DNA condensation activity (Fig. 2a). Although the presence of two point mutations known to affect non-sequence-specific DNA binding (RR-FoxA115) partially inhibited FoxA1 localization to the strand (Fig. 2b), this mutant still condensed DNA. In this case, condensation occurred on a time scale of minutes rather than seconds (as in WT-FoxA1), which can be explained by the delay in condensing sufficient RR-FoxA1 to the strand. These data suggest that non-sequence-specific binding drives the localization of FoxA1 to DNA but does not mediate DNA condensation through crosslinking. Furthermore, the sequence-specific binding domain of FoxA1 is dispensable for its localization to DNA in vitro. To probe whether FoxA1 protein–protein interactions through disordered domains mediate DNA condensation, we truncated both the N and C termini of FoxA1. Although ΔN-FoxA1 retained DNA condensation activity (Fig. 2c), truncating the disordered C terminus of FoxA1 largely inhibited DNA condensation activity (Fig. 2d). Additionally, we found that, at high FoxA1 concentrations in bulk (50 μM), 3% polyethylene glycol (PEG) (relative molecular mass 30,000—30K) nucleated highly enriched spherical FoxA1 condensates (Extended Data Fig. 4a), further suggesting the existence of weak FoxA1–FoxA1 interactions. Thus, non-sequence-specific binding drives FoxA1 localization to DNA, and the disordered C terminus of FoxA1 promotes DNA condensation.

Fig. 2: Mutant analysis reveals that the C terminus of FoxA1 drives DNA condensation.
figure 2

DNA envelope width measurements for FoxA1 mutants. The data are binned every 2 μm and the mean ± s.d. (for both the envelope width and the strand length) are shown in black for each mutant and in blue for the control (n = 45). a, Sequence-specific DNA-binding mutant NH-FoxA1 condenses DNA (n = 30). DBD, DNA-binding domain. b, Non-sequence-specific DNA-binding mutant RR-FoxA1 condenses DNA (n = 28). c, N-terminal truncation of FoxA1, ΔN-FoxA1, condenses DNA (n = 13). d, C-terminal truncation of FoxA1, ΔC-FoxA1, inhibits DNA condensation (n = 44). In all conditions, the protein concentration was 10 nM. See Supplementary Fig. 1 for representative protein–DNA images of the FoxA1 mutants.

Source data

Our results support the hypothesis that FoxA1 condenses onto DNA to generate a protein–DNA-rich condensate via weak protein–protein interactions that exerts a pulling force on the non-condensed strand (Thermodynamic description of DNA–protein condensation in Supplementary Information). To explore the thermodynamics of condensation, we developed a theoretical description based on a semiflexible polymer partially condensing into a liquid-like condensate. Here, the semiflexible polymer is DNA and the condensation is mediated by the transcription factor. The free energy of this process contains volume, \(\left( {\upsilon \frac{4}{3}\uppi R^3} \right)\), and surface contributions, (γR2), as well as a term representing the free energy of the non-condensed DNA (Fig. 3a), where υ is the condensation free energy per volume, R is the condensate radius and γ is the surface tension of the condensate. We assume that the DNA is fully collapsed inside the condensate and thus its volume is proportional to the condensed DNA contour length, V = αLd, where 1/α describes the packing density given as DNA length per condensate volume. The free energy of the polymer, \(F_{\mathrm{p}}(L,L_{\mathrm{p}}) = {\int}_0^L f (L,L_{\mathrm{p}}){\mathrm{d}}l,\) can be obtained from the force–extension curve of the polymer f(L, Lp), where Lp is the contour length of the non-condensed polymer. Using Lp = Lc − Ld, where Lc is the contour length of λ-phage DNA (16.5 μm), the free energy is as follows:

$$\begin{array}{rcl}F\left( {L,L_{\mathrm{d}}} \right) & = & - \upsilon \alpha L_{\mathrm{d}} + \gamma 4\uppi \left( {\frac{{3\alpha }}{{4\uppi }}} \right)^{\frac{2}{3}}L_{\mathrm{d}}^{\frac{2}{3}} \\ && + \kappa \left( {\frac{{(L_{\mathrm{c}} - L_{\mathrm{d}})^2}}{{4\left( {L_{\mathrm{c}} - L_{\mathrm{d}} - L} \right)}} - \frac{L}{4} + \frac{{L^2}}{{2\left( {L_{\mathrm{c}} - L_{\mathrm{d}}} \right)}} - \frac{{\left( {L_{\mathrm{c}} - L_{\mathrm{d}}} \right)}}{4}} \right)\end{array}$$

where \(\kappa = \frac{{k_{\mathrm{B}}T}}{P}\), kB is the Boltzmann constant, T is the temperature and P is the persistence length of DNA (Thermodynamic description of DNA–protein condensation in Supplementary Information). For fixed L, the minimum of F(L, Ld) determines the preferred size of the condensate. This free energy predicts upon variation of L a stochastic first-order phase transition for the formation of protein–DNA condensates (Fig. 3b). The distribution of condensate sizes is then given by \(P\left( {L_{\mathrm{d}}} \right)\approx {\mathrm{e}}^{ - \beta F(L,L_{\mathrm{d}})}\) for fixed L (Fig. 3c). This accounts for a sharp transition of DNA condensation controlled by the end-to-end distance and thus the tension of the DNA molecule. The first-order nature of this behaviour implies regimes of hysteresis and bistability. Our theory also predicts that the condensation forces exerted on the non-condensed DNA are kept roughly constant.

Fig. 3: Thermodynamic description of a liquid phase condensing onto a semiflexible polymer explains FoxA1-mediated DNA condensation.
figure 3

a, Schematic representing DNA–FoxA1 condensation (orange). DNA can be in a condensed state (black) or a non-condensed state (green). DNA condensation depends on γ, υ and α. b, Free energy profiles as a function of Ld for different L reveal a first-order phase transition for protein–DNA condensation (orange and blue correspond to favourable and unfavourable condensation, respectively). c, Boltzmann distributions corresponding to the free energy profiles in b. d, Condensate volume increases linearly with Ld. The orange curve represents a linear fit to individual strands (n = 47). For df, individual strands are represented as grey dots and binned mean ± s.e.m. is in black. e, The length of condensed DNA as a function of L (n = 63) reveals a sharp transition. The orange curve represents the optimal theoretical fit. The grey dashed line corresponds to the limit of maximum condensation where Ld is 16.5 μm minus L. f, Condensation forces that DNA–protein condensates exert on non-condensed DNA are buffered (n = 62). The orange curve is the theoretical prediction. The grey dashed line represents the force when Ld = 0. g, Pcond reveals a sharp transition at a critical end-to-end distance. Pcond is computed from binned local correlation data (n = 181 condensates). The end-to-end distance error bars are the s.d. and the Pcond error bars are the 95% confidence intervals from a beta distribution.

Source data

To test this theory, we first measured DNA condensate volumes and found that they increase linearly with Ld, with α = 0.04 ± 0.01 μm2 (Fig. 3d, Extended Data Fig. 4d and Methods). This confirms that DNA is in a collapsed conformation inside the condensates. Next, we simultaneously fitted the predictions to Ld and the probability of nucleating a DNA condensate (Pcond) as a function of end-to-end distance (Methods). We calculated Ld (Fig. 3e and Extended Data Figs. 4e and 5) and Pcond (Fig. 3g and Extended Data Fig. 4f) using the Boltzmann probability distributions (Fig. 3c) from the free energy. Our fits agree quantitatively with the data and show that Ld decreases with L until a critical end-to-end distance beyond which DNA condensates do not form. Below this critical length, we observed that the force exerted by the condensate is buffered at 0.21 pN (0.18–0.30 pN confidence interval), consistent with the theory (Fig. 3f). To complement our force measurements, we performed optical-tweezer measurements of FoxA1-mediated DNA condensation. Incubating a single λ-phage DNA molecule at either L = 6 or 8 μm in the presence of 150 nM FoxA1 generated forces of the order of 0.4–0.6 pN, consistent with the force measurements using fluorescence microscopy (Methods and Extended Data Figs. 6 and 7). Finally, Pcond exhibits a sharp transition at L = 10.5 μm (9.4–10.9 μm confidence interval), in agreement with a stochastic first-order phase transition (Fig. 3g). We also observed a sudden force jump during the onset of condensate formation (as measured by the individual temporal force trajectories in the optical-tweezer experiments), consistent with a first-order phase transition (Extended Data Figs. 6c and 7). Close to the transition point FoxA1-mediated DNA condensation displayed bistability. This bistability was observed in strands that contained multiple FoxA1 condensates, but where only some of them condensed DNA (Extended Data Fig. 8a). Our fits allowed us to extract the physical parameters associated with condensate formation, namely υ = 2.6 pN μm2 (2.3–5.2 pN μm2 confidence interval) and γ = 0.04 pN μm−1 (0.04–0.28 pN μm−1 confidence interval) (Methods). These parameters are consistent with previous measurements for in vitro and in vivo condensates16,17.

Our theory and experiments show that two key parameters govern protein–DNA co-condensation, namely υ and γ. We reasoned that different DNA-binding proteins may exhibit a range of behaviours depending on these parameters. First, we investigated the sequence-specific DNA-binding region mutant (NH-FoxA1), which also condensed DNA but to a lesser extent (Fig. 2a). Quantitatively, we found that the surface tension of condensates formed with this mutant was roughly unchanged compared with WT-FoxA1, γ = 0.065 pN μm−1 (0.05–0.07 pN μm−1 confidence interval), but the free energy per volume of condensation was reduced, consistent with reduced DNA binding, υ = 1.05 pN μm2 (0.9–1.1 pN μm2 confidence interval) (Extended Data Fig. 9 and Fig. 4a). This was also reflected in a decrease in the extent of DNA packing, with α = 0.09 ± 0.02 μm2 (Extended Data Fig. 9a). We also observed that NH-FoxA1-mediated condensates generated a force of 0.17 pN (0.16–0.19 pN confidence interval), lower than that for WT-FoxA1. In addition, NH-FoxA1 displayed bistable protein–DNA condensation activity in the neighbourhood of the transition point (Extended Data Fig. 8b). Next, we examined the interactions of a different transcription factor, TATA-binding protein (TBP), with DNA. We found that TBP also formed small condensates on DNA, but did not condense DNA even at the lowest imposed DNA tensions (Fig. 4b). Instead, TBP performed a diffusive motion along the DNA strand (Extended Data Fig. 10c), suggesting that protein–DNA condensation is not thermodynamically favoured. Another transcription factor, Gal4–VP16, formed condensates on DNA and condensed DNA in a tension-dependent manner consistent with FoxA1 (Extended Data Fig. 10e). Finally, we analysed somatic linker histone H1, a protein that is structurally similar to FoxA1. However, in contrast to FoxA1, one of the known functions of H1 is to compact chromatin18, so we expected H1 to strongly condense DNA. Consistent with this, we found that H1 displayed a stronger DNA condensation activity compared with FoxA1, condensing DNA for all measured end-to-end distances (Fig. 4c). Interestingly, the Xenopus embryonic linker histone B4 condensed DNA in a tension-dependent manner but not to the same extent as H1 (Extended Data Fig. 10f). Thus, we propose that the competition between condensation free energy per volume of the protein–DNA phase and surface tension regulates a spectrum of DNA condensation activities, which may be tuned by the structure of transcription factors.

Fig. 4: Universality of protein–DNA co-condensation.
figure 4

ac, Condensation probability quantification for sequence-specific DNA-binding mutant NH-FoxA1 (a), TBP (b) and somatic linker histone H1 (c). Pcond is computed from local correlation data with n = 361 condensates for NH-FoxA1 (a), n = 247 condensates for TBP (b) and n = 101 for H1 (c). The error bars for the end-to-end distance are s.d. and the Pcond error bars are the 95% confidence intervals from a beta distribution. We found that NH-FoxA1 condensed DNA less strongly than WT-FoxA1, TBP could not condense DNA for any end-to-end distance and H1 condensed DNA for all measured end-to-end distances. d, Biomolecular condensates generate condensation forces that could serve to recruit transcriptional regulators, and potentially remodel chromatin at physiologically relevant force scales to properly regulate transcription. See Supplementary Fig. 2 for representative protein–DNA images of NH-FoxA1, TBP and H1.

Source data

Here, we show that FoxA1 can condense DNA under tension to form a protein–DNA-rich phase that nucleates through a force-dependent first-order transition for forces below a critical value. This critical force, which is of the order of 0.2–0.6 pN for FoxA1, is set by co-condensation forces that the protein–DNA phase exerts on the non-condensed DNA. These forces are similar in magnitude to those recently measured for DNA loop extrusion, of the order of 0.2–1 pN (refs. 19,20), and those estimated in intact nuclei from nuclear condensate fusion21. Thus, we speculate that these weak forces we find in vitro may be of relevance to the mechanics of chromatin organization, though future studies are necessary to show this. Taken together, our work suggests that co-condensation forces may act as an additional mechanism to remodel chromatin in addition to molecular motors that extrude loops and complexes that remove or displace nucleosomes (Fig. 4d).

Transcription-factor-mediated protein–DNA condensation represents a possible mechanism by which transcription factors coordinate enhancer–promoter contacts in transcriptional hubs12. In this context, protein–DNA condensates could act as scaffolds, pulling co-factors into the droplet (Fig. 4d). Our theoretical description reveals that these protein–DNA condensates are formed via a first-order phase transition, suggesting that they can be assembled and disassembled rapidly by changing mechanical conditions. Near the transition point, assembly and disassembly of these in vitro protein–DNA condensates becomes highly stochastic, reminiscent of the rapid dynamics associated with the initiation and cessation of transcriptional bursts observed in vivo22.

We have demonstrated that protein–DNA co-condensation is associated with a difference in chemical potential between the condensed and non-condensed DNA. This difference in chemical potential is transduced by the condensate to perform mechanical work on the non-condensed DNA strand. Capillary forces represent another example of forces that involve liquid–surface interactions1,2,23. With both co-condensation and capillary forces, attractive interactions give rise to the transduction of free energy into work. Such forces may also be relevant beyond chromatin in other biological contexts, including membranes and the cytoskeleton.

Protein–DNA co-condensation not only provides mechanisms to facilitate enhancer–promoter contacts, but could also play a more general role in DNA compaction and maintenance of bulk chromatin rigidity in processes such as mitotic chromatid compaction24 and the formation of chromatin compartments8,25,26. Owing to the tension-dependent nature of protein–DNA co-condensation, our work suggests that these forces could play a key, and, as yet, underappreciated role in genome organization and transcriptional initiation. It is appealing to imagine that transcriptional outputs respond not only to concentrations of transcription factors in the nucleus, but also to mechanical cues from chromatin.


Cloning and protein purification

FoxA1–mCherry was introduced into a bacterial expression vector with an N-terminal His6 tag using Gateway cloning. Unlabelled FoxA1 was cloned and purified the same way. This vector was transformed into T7 Express cells (enhanced BL21 derivative, NEB C2566I), grown to optical density (OD) ≈ 0.4–0.8, whereupon we added 1 mM isopropyl-β-d-thiogalactoside and expressed His6–FoxA1–mCherry for 3–4 h at 37 °C. We thawed frozen pellets in binding buffer that contained 20 mM Tris-HCl (pH = 7.9), 500 mM NaCl, 20 mM imidazole and 1 mM MgCl2, supplemented with protease inhibitors and Benzonase. The redissolved pellets were lysed and clarified via centrifugation. Discarding the supernatant, we resuspended the pellets in binding buffer + 6 M urea, spun, collected the supernatant and poured it over an immobilized metal ion affinity chromatography column, eluting the protein with binding buffer + 6 M urea + 250 mM imidazole. We dialysed overnight into storage buffer, 20 mM HEPES (pH = 6.5), 100 mM KCl, 1 mM MgCl2, 3 mM dithiothreitol (DTT) and 5 M urea. Multiple dialysis rounds reduced the concentration of urea. Finally, the protein was dialysed into storage buffer + 2 M urea, spin-concentrated to 4–5 mg ml−1 (~50 μM), and then snap-frozen in nitrogen and stored at −80 °C. NH-FoxA1–mCherry and RR-FoxA1–mCherry were obtained following ref. 15 using a Q5 site-directed mutagenesis kit. The truncation constructs were generated using restriction digestion–ligation approaches coupled with PCR. We used Alexa-488-labelled somatic linker histone H1 purified from calf thymus (H-13188, Thermo Fisher). To purify mCherry–B4, the gene (GenScript) was cloned into a bacterial expression vector with N-terminal His6 and mCherry tags, transformed into T7 Express cells, grown to OD ≈ 0.7, supplemented with 0.8 mM isopropyl-β-d-thiogalactoside and expressed at 37 °C for 4 h. Resuspending the pellets in lysis buffer, 1×PBS with 500 mM NaCl, 1 mM DTT plus protease inhibitors and Benzonase, we then lysed the cells, collected the supernatant, ran the supernatant over an immobilized metal ion affinity chromatography column and eluted the protein with lysis buffer + 250 mM imidazole. The protein was dialysed into 1×PBS + 500 mM NaCl overnight, spin-concentrated, snap-frozen and stored at −80 °C. We purified labelled versions of TBP and Gal4–VP16 using similar purification strategies. Both vectors—His6–MBP–eGFP–zTBP and His6–Gal4–GFP–VP16—were transformed into T7 Express cells and grown to OD ≈ 0.6, whereupon we added 0.2 mM isopropyl-β-d-thiogalactoside, and expressed overnight at 18 °C. We lysed the cells into buffer containing 50 mM Tris-HCl (pH = 8.0), 1 M NaCl, 10% glycerol, 1 mM DTT and 1 mM MgCl2 supplemented with protease inhibitors. For subsequent steps, 10 μM ZnSO4 was added to buffers for the Gal4–VP16 purification. After lysis, we added NP40 to 0.1% and clarified via centrifugation. We performed a polyethyleneimine precipitation to precipitate DNA and then an ammonium sulfate precipitation to recover the protein, resuspending the precipitated proteins in buffer containing 50 mM Tris-HCl (pH = 8.0), 1 M NaCl, 10% glycerol, 1 mM DTT, 0.1% NP40 and 20 mM imidazole and clarified the soluble fraction via centrifugation. We poured the lysate over an immobilized metal ion affinity chromatography column and eluted the protein using 2×PBS, 250 mM imidazole, 10% glycerol and 1 mM DTT. We pooled protein fractions and dialysed TBP overnight into 20 mM HEPES (pH = 7.7), 150 mM KCl, 10% glycerol and 1 mM DTT and Gal4–VP16 into 20 mM HEPES (pH = 7.7), 100 mM KCl, 50 mM sucrose, 0.1 mM CaCl2, 1 mM MgCl2, 1 mM DTT and 10 μM ZnSO4. We then spin-concentrated the proteins, snap-froze them using liquid nitrogen and stored at −80 °C.

DNA functionalization, coverslip PEGylation and DNA microchannel preparation

To biotinylate DNA purified from λ-phage, we followed the protocol given in ref. 19. Each end of the biotinylated λ-phage DNA had two biotin molecules. To PEGylate the cover slips and prepare the DNA microchannels we followed the protocol given in ref. 19.

DNA and protein imaging

We fluorescently stained immobilized DNA strands with 10 nM SYTOX Green in Cirillo buffer (20 mM HEPES, pH = 7.8, 50 mM KCl, 2 or 3 mM DTT, 5% glycerol, 100 μg ml−1 BSA). For experiments with H1 and TBP, we imaged DNA using 25 nM SYTOX Orange. We used protein concentrations of 10 nM. We used a Nikon Eclipse microscope with a Nikon ×100/numerical aperture (NA) 1.49 oil SR Apo total internal reflection fluorescence microscope and an Andor iXon3 EMCCD camera using a frame-rate of 100–300 ms. A highly inclined and laminated optical sheet was established using a Nikon Ti-TIRF-E unit mounted on the microscope stand.

Optical-tweezer measurements

We performed optical-tweezer experiments using a C-Trap G2 system (LUMICKS) in a microfluidics flowcell (LUMICKS), providing separate laminar flow channels. For each experiment, we trapped two streptavidin-coated polystyrene beads (Spherotech SVP-40-5). Once trapped, we moved these beads to a channel containing biotinylated λ-phage DNA (LUMICKS) at a concentration of 0.5 μg ml−1, whereupon we used an automated ‘tether-finder’ routine to capture a single molecule between the two beads. Once a single λ-phage DNA molecule was attached to the two beads, we moved the trapped beads to a buffer-only channel (containing Cirillo buffer with 3 mM DTT). In the buffer-only channel, we fixed L at either 6 or 8 μm. We then moved the tethered DNA to a channel containing 150 nM FoxA1 in Cirillo buffer or another buffer-only channel (as a control) and tracked the force and imaged the FoxA1–mCherry fluorescence for 100 s.

Bulk phase-separation assays

We performed bulk phase-separation assays with FoxA1–mCherry, NH-FoxA1–mCherry and somatic linker histone H1. The storage buffer for FoxA1 and NH-FoxA1 was 20 mM HEPES (pH = 6.5), 100 mM KCl, 1 mM MgCl2, 3 mM DTT and 2 M urea. The storage buffer for H1 was 1×PBS. For FoxA1, we combined 6 μl of FoxA1 (at 50 μM) and 1 μl of 20% 30K PEG. For NH-FoxA1, we combined 9 μl and 1 μl of 20% 30K PEG. For H1, we combined 9 μl H1 and 1 μl 100 μM 32-base-pair single-stranded DNA. We prepared flow channels with double-sided tape on the cover slide and attached a PEGylated coverslip to the tape. We imaged the condensates using spinning disc microscopy and a ×60 objective.

FoxA1 molecule number estimation

To estimate the number of FoxA1–mCherry molecules per condensate, we quantified the intensity of single FoxA1–mCherry molecules bound non-specifically to the slide. Around each segmented spot of DNA-independent FoxA1 intensity, we cropped an area of 10 pixels × 10 pixels, performed a background subtraction and summed the remaining intensity in the cropped area. To determine the contribution of the background, the same method was applied to 10 pixel × 10 pixel areas void of FoxA1 signal intensity. The resulting distribution of these integrated signal intensities reveals consecutive peaks that are evenly spaced by an average intensity of about 400 a.u., allowing us to calculate the number of molecules. This approach should be interpreted as a lower-bound estimate of the number of FoxA1–mCherry molecules per condensate, as it neglects effects such as fluorescent quenching27.

Hydrodynamic stretching of DNA

DNA molecules bound at only one end to the slide were hydrodynamically stretched using a constant flow rate of 100 μl min−1 of 0.5 nM FoxA1–mCherry in Cirillo buffer with 10 nM SYTOX Orange. The flow rate was sustained for tens of seconds using a programmable syringe pump (Pro Sense NE-501).

Strand length calculation

To calculate the end-to-end distance, we generated time-averaged projections of FoxA1 and DNA and integrated these projections along the strand’s orthogonal axis. To find the profile’s ‘left’ edge, we computed the gradient of the signal and determined the position where the gradient went through a threshold (defined as 0.2). We then took all the points from the start of the signal to this position, performed a background subtraction, and fitted an exponential to these points. To ensure that we included the entire DNA signal, we defined the fitted threshold for both the left and the right edges as three-quarters of the value of the fitted exponential value at the point when the gradient had gone through the intensity threshold. Using this fitted threshold, we computed the position values for the left and the right sides, and computed the end-to-end distance as the difference between these two positions.

Global cross-correlation analysis

We generated time-averaged projections from videos of both FoxA1 and DNA, and then summed the intensities in the orthogonal axis to the strand, generating line profiles. We then calculated the strand length and cropped both the FoxA1 and DNA line profiles from the edges of the strand. We then subtracted the mean value from these cropped line profiles, normalized the amplitudes of the signals by their Euclidean distances and computed the zero-lag cross-correlation coefficient of the normalized signals, which we defined as ‘correlation’:\(R(\tau = 0) = \mathop {\sum}\nolimits_{n = 1}^N {\overline {x_n} \overline {y_n} }\), where τ is the number of lags, N is the number of points in the normalized FoxA1 and DNA signals, \(\overline {x_n}\) is the nth entry of the normalized FoxA1 signal and \(\overline {y_n}\) is the nth entry of the normalized DNA signal. In general, correlation values range from −1 to 1, but in our experimental data the values range from roughly 0 to 1, where 1 represents the formation of DNA–FoxA1 condensates and 0 represents the formation of only FoxA1 condensates (no DNA condensation).

DNA envelope width calculation

To compute the DNA envelope width, we first generated time-averaged projections from videos of FoxA1 and DNA. We then selected segments of the strand that did not contain FoxA1—regions of non-condensed DNA. Using these segments, we extracted a line profile of the DNA signal orthogonal to the strand that gave the maximum width. We then subtracted the background of the DNA profile, normalized the signal’s amplitude using the Euclidean distance and fitted a Gaussian. We defined the DNA envelope width as \(\sqrt 2 \sigma\), which represents the square root of twice the s.d. of the fitted Gaussian. The theoretical diffraction limit is calculated using the Rayleigh criterion, a measure of the minimal resolvable distance between two point sources in close proximity for a given set of imaging conditions: \(d = \frac{{0.61\lambda }}{{\mathrm{NA}}}\), where λ represents the imaging wavelength. For our imaging set-up, d = 0.2 μm, which is approximately 2σ of the fluorescent source from the DNA. As the DNA envelope width is defined as \(\sqrt 2 \sigma\), our ‘diffraction limit’ as given by the dashed line in Fig. 1f is 0.14 μm.

Condensate volume analysis

To calculate condensate volumes, we generated time-averaged DNA–FoxA1 projections and then localized the peaks of the DNA condensates. Using the peak locations, we extracted background-subtracted one-dimensional profiles of the DNA condensates in the orthogonal axis to the strand—these profiles went through the peak location. We fitted Gaussians to these profiles without normalizing the amplitude. To define the radii of the condensates, we computed the gradients of the fitted Gaussians and defined the condensate ‘edges’ as when the absolute value of the gradient of the Gaussian function gradient went through a threshold value (defined as unity, determined by comparing with fluorescence). Assuming that condensates are spherical, we computed the condensate volume as \(V = \frac{4}{3}\uppi R^3\). To compute a condensate volume for strands with multiple condensates, we simply added up the volumes for each condensate.

Condensed DNA length analysis

To compute Ld, we generated time-averaged projections of DNA and FoxA1 signals, integrating the DNA signal in the orthogonal direction to the strand. We then defined condensed versus non-condensed DNA by Thresholddrop: the median value of the profile plus a tolerance. Intensity values below Thresholddrop were defined as pixels of non-condensed DNA, and intensity values above Thresholddrop were defined as pixels of condensed DNA. This assumption was also consistent with the measured FoxA1 signal, where FoxA1 signals clearly localized to regions of condensed DNA, as defined by Thresholddrop. The tolerance value was used to suppress artefactual fluctuations of the non-condensed DNA signal in the neighbourhood of the median. To optimize the tolerance value, we assume that Ld as a function of L is linear for lower values of L (<5 μm) with a y intercept equal to the contour length of the DNA molecule (16.5 μm), as this is consistent with our theoretical description. We plotted the y intercepts of the linear fits as a function of tolerance and found that tolerance = 500 gives a y intercept equal to 16.5 and generates DNA–FoxA1 condensates up to 10 μm, consistent with our data and analysis (Extended Data Fig. 5). To calculate the DNA length contained within the droplet, we integrated the intensities from pixels above Thresholddrop, divided this value by the sum of the total intensity of the profile and then multiplied this ratio by the contour length of λ-phage DNA, 16.5 μm. The non-condensed DNA length was calculated as simply the contour length minus Ld. We used the same tolerance = 500 for the NH-FoxA1 mutant analysis.

Force analysis

To calculate the force that the condensate exerts on the non-condensed DNA, we used the worm-like-chain model, which relates λ-phage DNA’s extension and force. Upon addition of FoxA1, the amount of non-condensed DNA reduces, and the extension changes as follows: \(E = \frac{L}{{L_{\mathrm{c}} - L_{\mathrm{d}}}}\). We then directly compute the force using the worm-like-chain model,

$$F = \kappa \left( {\frac{1}{4}(1 - E)^{ - 2} - \frac{1}{4} + E} \right).$$

Condensation probability analysis

To calculate the probability of the formation of a protein–DNA condensate as a function of end-to-end distance, we localized the peaks of the FoxA1 condensates from time-averaged projections of FoxA1 and DNA. We then extracted 0.9 μm × 0.5 μm windows centred around the localized FoxA1 peaks of both the FoxA1 and DNA signals—with the window’s long axis going with the strand and the short axis orthogonal to the strand. We then computed the zero-lag normalized cross-correlation coefficient as follows:

$$C_{\mathrm{loc}} = \frac{{{\sum} {\left( {f\left( {x,y} \right) - \mu _f} \right)\left( {g\left( {x,y} \right) - \mu _g} \right)} }}{{\sqrt {{\sum} {f(x,y)^2 - \mu _f^2} } \sqrt {{\sum} {g(x,y)^2 - \mu _g^2} } }}$$

where f(x, y) is the DNA, g(x, y) is FoxA1, μf is the mean of the DNA image and μg is the mean of the FoxA1 image. This generates values from −1 to 1. For FoxA1-mediated DNA condensation, the values for particular condensates are close to 1. When FoxA1 fails to condense DNA, owing to the morphology of the underlying DNA strand and the small number of pixels, we obtain values that range from −1 to roughly 0.5. To obtain a value for Pcond as a function of end-to-end distance, we selected a threshold of 0.75—Cloc values above the threshold are considered ‘condensed’ and values below ‘non-condensed’. We binned the Cloc data in 2 μm increments as a function of end-to-end distance, and calculated Pcond by taking the number of condensed condensates and dividing it by the total number of condensates within the bin. The confidence intervals for Pcond in each respective bin are computed by computing the 95% confidence interval of a beta distribution, which represents the probability distribution for a Bernoulli process that takes into account the total number of successes with respect to the total number of attempts.

Parameter fitting of the thermodynamic description and confidence intervals

To fit α, we used a linear fit of the condensate volumes for individual strands as a function of Ld. The confidence intervals are the 95% confidence intervals generated from directly fitting the points. To fit γ and υ, we minimized the error of the average \(\overline {L_{\mathrm{d}}} \left( L \right)\) and Pcond(L) with respect to the data to optimize the parameter values. We used the normalized Boltzmann distribution \(P\left( {L_{\mathrm{d}}} \right) = \frac{{{\mathrm{e}}^{ - \beta F(L,L_{\mathrm{d}})}}}{{\mathop {\smallint }\nolimits_0^{L_{\mathrm{c}} - L} {\mathrm{e}}^{ - \beta F(l)}\,{\mathrm{d}}l}}\) to calculate \(\overline {L_{\mathrm{d}}} = {\int}_0^{L_{\mathrm{c}} - L} {lP\left( l \right)\,{\mathrm{d}}l}\). To compute Pcond(L), we localized the position of the local maximum in the free energy, \(L_{\mathrm{d}}^{\mathrm{max}}\), for a given L and then computed the probability to ‘not’ nucleate a droplet from the Boltzmann distribution \({\int}_0^{L_{\mathrm{d}}^{\mathrm{max}}} P \left( l \right)\,{\mathrm{d}}l\), which gives \(P_{\mathrm{cond}} = 1 - {\int}_0^{L_{\mathrm{d}}^{\mathrm{max}}} P \left( l \right)\,{\mathrm{d}}l.\) To minimize the error, we binned the data in 2-μm-width bins. For each ‘binned’ mean for both condensed DNA and condensation probability, we computed the squared residual of the mean value with respect to the theoretical expression. For residuals calculated from \(\overline {L_{\mathrm{d}}} \left( L \right)\), we normalized each residual by the squared standard error of the mean, and then summed the normalized residuals to obtain the error. For residuals calculated from Pnuc(L), we normalized each residual by the variance of the beta distribution, \(P_{{{\rm{nuc}}\,{\rm{cond}}}}^{\sigma ^2} = \frac{{(1 + k)(1 - k + N)}}{{\left( {2N^2(3 + N)} \right)}}\) and then summed the normalized residuals to obtain the error. For the global error, we simply added the error from both deviations in \(\overline {L_{\mathrm{d}}} \left( L \right)\) and Pcond(L). We then iterated through a range of values for (γ, υ) and computed the total error associated with each set of parameter values, exponentiated the negative values of the total error matrix and computed the largest combined value to select the parameter values. To calculate the parameters’ confidence intervals, we obtained one-dimensional profiles of the integrated exponentiated total error for υ as a function of γ and γ as a function of υ. The peaks of these profiles represented the values that we selected for our best-fit parameters. We assumed that these profiles represented probability distributions for parameter selection, and then calculated the left and right bounds, where the area under the curve between these bounds represented 95% of the area. These left and right bounds represent the lower and upper values of our confidence intervals. To compute the 95% confidence interval for the force for each end-to-end distance value, we scanned through (γ, υ) parameter space and computed the value of Ld for each set of parameters. We then plotted these values against the probability that these parameter values were the ‘true’ values—simply the probability from the exponentiated error matrix. Integrating the points under the probability versus Ld curve and dividing this by the total area under this curve, we generated a probability distribution function from which we could compute the 95% confidence intervals for Ld. Because the force was constant, to compute the confidence intervals for the force we calculated the force using the worm-like-chain model with corresponding Ld values for an end-to-end distance that retained FoxA1-mediated DNA condensation. To compute the confidence intervals for Lcrit, we scanned through (γ, υ) parameter space and computed Lcrit for each set of parameters. We then plotted Lcrit values with the corresponding values from the probability that these parameter values were true (again, the exponentiated error matrix). Integrating the points under the probability versus Lcrit curve and dividing this by the total area under this curve, we generated a probability distribution function from which we could compute the 95% confidence intervals for Lcrit.

Reporting Summary

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