A theoretical model suggests means to control condensation through biochemical reactions
The formation of biomolecular condensates depends on many parameters. If we consider an ideal, simplified case in which a single biomolecular species undergoes phase separation, this process is driven primarily by the species concentration and by the temperature of the system, as illustrated by well-understood phase diagrams22. However, it is less clear how temporal changes of the concentration of introduced by chemical reactions that deactivate or activate will affect the formation and the kinetics of phase separated condensates, despite recent theoretical work in this area 23,24
To investigate this problem, we built a mean field model of molecular condensation from the Cahn-Hilliard equation for phase separation in 2D, with the addition of mass action chemical kinetics that introduce temporal changes in the amount of monomers of phase separating species. We consider a case in which elementary chemical reactions convert the monomers between their active form , which phase separates, and their inactive form , which cannot: an inhibitor molecule mediates the deactivation of monomers, while an activator mediates their reactivation by removing inhibitor, a process that generates waste W:
The full derivation of the mean field equations, along with a description of their computational integration, is in Section 4.1 of the Supplementary Information (SI) file. First, we note that the stationary effect of including inhibitor species in the system is similar to a change in temperature, as evidenced by phase diagrams obtained under different dissociation constants (SI Fig. S62). Next, we perform computational simulations exploring the system’s kinetic response to the introduction of inhibitor and activator.
Figure 1A illustrates the behavior of a system in which droplets are allowed to phase separate first, with later sequential addition of inhibitor and activator at different ratios (.25x, .5x and 1x the level of monomer). These computationally generated images qualitatively confirm that addition of inhibitor causes droplet dissolution, and addition of activator promotes their regrowth. From these images, we gathered quantitative insight on the evolution of droplet size over time (normalized relative to the forward binding rate and the monomer concentration).
After addition of inhibitor, the half time for dissolution of individual droplets scales with a power law of the droplet radius (Fig. 1B), meaning that larger droplets take longer to dissolve. In contrast, the droplet ensemble dissolves faster when the total droplet surface is larger (Fig. 1C), i.e. when the monomers are allowed to phase separate for a longer period of time prior to adding inhibitor. Consistent with expectation, the average droplet size decreases faster when the inhibitor to monomer ratio is larger (Fig. 1D). This process is however affected by the inhibitor diffusion rate (), which manifests as a change in the temporal scaling law when the diffusion constant is between 1 and 10 (Fig. 1E). This indicates that inhibition may be surface-driven (slower diffusion), or volume driven (faster diffusion). Overall, increasing either the inhibitor concentration or its diffusion coefficient promotes faster droplet dissolution, though apparently with different scalings.
The addition of activator (in stoichiometric amount to the inhibitor) promotes droplet regrowth. However, Fig. 1F shows that the speed of regrowth depends on both the initial droplet size and on the activator to monomer ratio. If we focus on the average droplet area, we note a rapid increase at 0.25 and 0.5x ratios; in contrast, a slow increase is noticeable at a 1x ratio. This non-monotonic behavior emerges due to the presence of three processes driving growth: monomer activation, droplet nucleation, and droplet coarsening. When a large number of monomers are rapidly activated (Fig. 1A, post addition of 1x activator) nucleation of new, small droplets dominates over the coarsening of existing ones, resulting in slower increase of their mean size. As one would anticipate, the diffusion rate of the activator does not significantly influence droplet growth (Fig. 1G).
This simple theoretical model establishes a baseline expectation for the temporal behavior of droplets and for the degree of controllability of their average size. To test these expectations, we next develop an experimental platform to build dynamic biomolecular condensates using DNA nanotechnology.
The formation of DNA condensate droplets can be controlled via strand displacement
We used synthetic DNA motifs and strand displacement as a platform to probe the effects of chemical reactions on the condensation of droplets. We adopted multiarm DNA motifs, or DNA nanostars, that interact via complementary palindromic ‘sticky-end’ (SE) domains present on the end of each arm10,19,20. We started our investigation with three-arm motifs (valency equal to three) characterized by Sato et al.20, each arm having a four-base sticky-end, shown in Fig. 2A. After assembly by thermal annealing the nanostars can be considered monomers that spontaneously yield DNA-rich condensed droplets, which in our case are expected to remain liquid at room temperature (27°C) while coarsening and coalescing into larger droplets20. Nanostars were annealed at 5 µM concentration unless otherwise noted. To monitor the condensate droplets via fluorescence microscopy, we modified one of the strands of the nanostars to include a fluorescent dye (Cy3).
To control the growth and dissolution of droplets, we designed DNA species with the capacity to react with the nanostars, specifically to switch the state of their sticky-ends between inactive and active. To enable the inactivation of nanostars, we designed a strand displacement reaction controlling the availability of one of the sticky-ends: we reasoned that a reduction of the valency from three to two would be sufficient to dissolve the DNA-dense droplets. To make it possible to deactivate one sticky-end, we modified it to include a 7 nucleotide (nt) ‘toehold’ domain that is designed to have no secondary structure and to remain single-stranded (Fig. 2A and 2B). The toehold enables strand invasion by a DNA inhibitor molecule, henceforth dubbed ‘invader’, designed to be complementary to the toehold, the sticky-end sequence, and 6 nt of the nanostar arm. While an invader can weakly interact with any unpaired sticky-end and arm on the nanostars, it cannot invade base-paired sticky-ends that do not have a toehold.
To allow for nanostar activation, the invader molecules are in turn designed to include their own toehold domain (6 nt), so that a complementary ‘anti-invader’ (AI) strand can displace invader bound to a nanostar. This displacement reaction releases the sticky-end of the nanostar, which recovers its original valency and its capacity to yield condensate droplets. Invader and anti-invader serve the purpose of activating and deactivating nanostar units via chemical reactions that are equivalent to those described in our theoretical model.
After forming condensate droplets with our toeholded DNA nanostars, we sequentially added invader and anti-invader at equimolar levels relative to the nanostar concentration, and characterized the droplets kinetics during each reaction. Before starting invasion and anti-invasion, we annealed the DNA nanostars at a concentration of 5 µM and allowed the samples to grow at 27°C for 30 mins, measuring a coefficient of variation of 0.12 for droplet diameter, 0.23 for area, and 0.005 for number across three experimental replicates. These statistics set a baseline for the significance of differences observed across kinetic droplet experiments that have a significant dependence on initial conditions, as illustrated in our theoretical example. Example fluorescence microscopy images of condensates before invasion, after invasion and after anti-invasion are shown in Fig. 2C. The microscopy images were used to gather statistics of the condensate droplet diameter as a function of time, shown in the histograms in Fig. 2C (our methods for image processing are described in SI Section 2.5). The addition of invader causes condensates to disappear within two minutes (addition of a scramble sequence has no effect on the droplets, Fig S6), indicating that the dissolution time-scale is driven by the rapid forward rate of strand invasion. In contrast, droplet regrowth after the addition of anti-invader is a slow process driven by nucleation, coarsening, and coalescence of droplets.
The concentration of invader and anti-invader determine the kinetics of droplet dissolution and formation.
The kinetics of droplet dissolution and formation should depend on the level of inhibiting (invader) and activating (anti-invader) molecules relative to the level of monomer (nanostar), as anticipated by our simple model (Fig. 1D). To quantify this dependence we measured the average droplet area over time after sequential addition of invader and anti-invader (AI). The area was normalized relative to its initial value prior to initiating invasion and anti-invasion (Fig. 2D). Like the model predicts, the time it takes to dissolve droplets decreases monotonically with the invader to monomer ratio, with 1X invader resulting in droplet dissolution within two minutes, and 0.5X invader dissolving droplets in 15 minutes. However, a 0.25X level of invader only reduces the average droplet size without causing their complete dissolution. In contrast, the regrowth kinetics after addition of AI do not monotonically scale with the amount of AI, as anticipated by our model (Fig. 1F). A 0.5X amount of AI (equimolar to 0.5X invader added) yields a significantly faster regrowth of droplets. The 0.25X and 1X experiments produce a slower regrowth, albeit presumably for different reasons: at 0.25X, slow coarsening is likely the driving process, while at 1X the rapid activation of a large number of monomers makes nucleation of small droplets the dominant process (which yields an overall smaller average size). The addition of excess AI either does not significantly change (2X relative to nanostar level) or suppresses (4X) droplet regrowth (Fig. 2E). This can be explained by the fact that the AI sequence includes the 4-base palindromic domain of the nanostar sticky-ends, making it possible for the AI to associate with nanostars and prevent their separation, an effect that becomes predominant when AI is present in excess.
Given that droplets rapidly dissolve and reform upon addition of 0.5X invader and AI, we asked whether their addition in multiple, sequential cycles would yield similar results. Figure 2F shows six separate cycles of invasion, and anti-invasion: invader and AI added in each cycle bind and form a fully double stranded ‘waste’ species that accumulates over time but is expected to be inert. Incorporation of inert waste in the condensates may be the reason why cycles 5 and 6 show a more pronounced regrowth. Overall, these results show that toehold-mediated strand displacement, a programmable biochemical reaction, can be used to direct self-assembly of large DNA condensates isothermally and reversibly, by activating and deactivating the motifs participating in the condensation process.
Valency reduction is a high-gain mechanism to control condensate kinetics
The observation that only 0.5X invader results in complete droplet dissolution prompted us to examine more carefully the influence of concentration of active DNA nanostars on their capacity to phase separate. Because high nanostar concentrations were adopted in previous work (typically 5 µM), we tested whether condensation is at all possible at lower nanostar concentrations, with an expectation that no condensation would occur below 2.5 µM (the level of active nanostars remaining after addition of 0.5X invader in our experiments). In contrast, we found that droplet condensates form at concentrations as low as 0.25 µM. Figure 3A (orange) reports the normalized total droplet area 30 minutes after annealing for separate experiments with nanostars at 0.25 µM, 1 µM, 2.5 µM, and 5 µM, and Fig. 3B reports normalized average area and Fig. 3C reports normalized droplet number (areas are normalized relative to the 5µM measurement). These results indicate that in our invasion experiments, dissolution of droplets is not due to the fact that the concentration of active nanostars in solution is brought below their critical threshold for condensation.
To further investigate this, we monitored the formation of droplets when 5µM DNA nanostars are annealed in the presence of invader (.25X to .5X). The normalized total droplet area 30 minutes after anneal is shown in Fig. 3A (dark red) versus the concentration of active nanostar, calculated as the difference between their total concentration and the concentration of invader. Average area and number are in Fig. 3B and Fig. 3C (dark red). These graphs make it possible to compare two distinct series of experiments that have the same amount of active monomers, but one includes invader and the other does not. In the presence of invader, samples include smaller and more numerous droplets, and condensation is completely suppressed by reducing the valency of only 1/2 the nanostar population. In contrast, in the absence of invader condensation may be suppressed with a reduction of more than 1/20 the nominal nanostar concentration (5µM).
Overall, this comparison indicates that using invader to reduce nanostar valency yields more than 2x “gain” toward suppressing condensation, as qualitatively suggested by the slope of the graphs in Fig. 3A. This behavior can be regarded as ultrasensitive, in that condensate formation is triggered under a small increment of the concentration of active monomers exceeding half the total. The fact that only half of the nanostar population requires deactivation for droplet dissolution prompts us to suggest a model in which inactive nanostars (valency two) interact with active ones, sequestering them and reducing the critical concentration for phase separation (Fig. 3E). In other words, a single invasion event affects multiple nanostars resulting in the observed “gain”.
This behavior can be captured with a theoretical model by performing linear stability analysis around the stationary state under different condition concentrations of inactivated nanostar and net attraction to the activated nanostar (SI Section 4.1). This stability analysis yields the phase diagram in Fig. 3D and indicates that there are conditions where the presence of inactivated nanostars alone is sufficient to disrupt the formation of condensates. It does this by raising the free energy of the droplet state relative to the disperse state. The theory predicts this behavior even in the absence of any chemical reactions, suggesting a thermodynamic origin of this phenomenon. Theoretically, this suggests that if the inactivated nanostars are not strongly excluded from the droplet, they will diffuse into the droplet, thus reducing the relative energetic benefit of the droplet existing by interrupting the multivalent droplet interactions, leading to droplet dissolution.
Droplet size can be controlled by modifying invasion and anti-invasion domains
We next tested how changes in the forward rate of invasion affect the dissolution of droplets. As is well-known from methods in DNA nanotechnology, the speed of strand displacement (forward rate) depends on the length of the toehold domain, so we designed nanostar variants (and corresponding invaders) with a toehold of 5 and 3 nucleotides, and one variant without a toehold (Fig. 4A). The speed of invasion should scale exponentially with the reduction of toehold length25, but invaders (1X) with shorter toeholds yield dissolution that is only marginally slower (Fig. 4B). A drastic shift in the behavior of the droplets occurs when the toehold domain is eliminated: in this case, droplets do not dissolve but their average size remains constant over 6 hours/360 mins post addition of invaders; also the number of droplets shows limited change over time (SI Section 3.6). We expect the invader weakly binds and unbinds to the free (unpaired) sticky-ends of the nanostars, that may be in solution or on the surface of the condensate droplets, thereby slowing down coarsening and fusion events (the nanostar arm-invader interaction domain is the same as a nanostar-nanostar arm interaction). Without a toehold, the invader may be considered a “decoy” molecule that stabilizes condensates.
While the toehold length affects the forward kinetics of invasion, the overall complementarity of invader-nanostar arm sequences determines the stability of the complex they form. The effects of this design parameter on the behavior of droplets is reported in Fig. 4D, that compares invader variants including 9, 6, 3 (nominal case shown in Fig. 2), and 0 nt complementary to the arm sequence. At 0.5X invader level, all the variants dissolve droplets within 15 minutes, indicating that invasion depth is not a major driver of droplet dissolution kinetics. In contrast, there is a significant difference in the anti-invasion behavior when comparing the variants with 0, 3 nt depth and those with 6, 9 nt depth (0.5X AI fully complementary to invader). Shorter variants yield droplets that regrow slowly, and do not exceed 30% their original average area. This is likely due to interactions between the nanostar and the AI, which can yield kinetically trapped complexes including nanostar, invader, and AI, which are equivalent to inactive nanostars.
The speed of droplet dissolution depends on the size of DNA nanostars.
Our theoretical model predicts that dissolution speed depends on the diffusion rate of the inhibitor molecule (Fig. 1D). To test whether we could change the diffusion of invader into the condensate, and thus droplet dissolution speed, we varied the length of nanostar arms between 8 and 24 base pairs (bp) in Fig. 1A. Because we considered arm lengths well below the persistence length of double stranded DNA (50 bp), it is reasonable to expect that longer arms yield droplets that are less densely packed and more permeable to invaders.
We first verified that all variants form droplets in the absence of a toehold, with 8 bp arm nanostars yielding noticeably smaller droplets than the 24 bp arm variant post anneal, as we found in recent work26. However, the 24 bp arm design did not yield droplets in the presence of a 7 nt toehold (similarly, 32 bp arm nanostars did not condense in the presence of any toehold; SI Fig. S34). For this reason, in all variants we adopted a 3 nt toehold, which in the nominal 16 bp arm design promoted droplet dissolution within 15 minutes at 1X invader (Fig. 4B).
At 1X invader level, droplets generated by all variants dissolve within 15 minutes: Fig. 5B reports normalized droplet average area versus number, parameterized with respect to time (time points are associated with markers). Following the arrow on each trace, the droplet number sharply decreases by 80–90% the initial level within 2 minutes, with few large droplets persisting for the 16 and 24 base design. The 8 bp variant dissolves significantly faster.
At 0.5X invader level, droplets in the 8 bp arm variant take about 20 minutes to dissolve, with droplet number and size gradually decreasing, as shown in Fig. 5C. Droplets in the 16 and 24 bp arm variants decrease in number, but their average area increases relative to the initial conditions (by 40% and 20% respectively). Over time, large droplets persist in the 16 bp arm design; while we observe minor change in their number, droplet average area fluctuates significantly, likely because few large droplets are captured per field of view (example images in Fig. 5D). Over time droplets persist also in the 24 bp arm variant, but their average size is around half the initial, with more significant fluctuations in their number. Notably, after 24 hours, droplets from 16 and 24 bp variants grow well above their initial average. Additional images and data (diameter histograms, average and total droplet area, and droplet number) are in SI Section 3.7.
These experiments do not indicate a clear correlation between nanostar size and invasion speed, suggesting that diffusion rate of invader may not significantly differ in the size range we considered. However, our theoretical model predicts that the time it takes a droplet to disappear depends on its radius as R3/2, and by making the nanostars smaller, we reduce the radii of the droplets 26. However, we increase the concentration of invadable sticky-ends inside droplets (which also decreases the time taken for a droplet to dissolve by rescaling the time). The competition of these effects together suggest that the time taken for droplets of smaller nanostars to dissolve would be faster, as is indeed observed in experiment.
Selective control of monomer valency via individually addressable toeholds results in distinct dissolution behaviors
The valency of DNA nanostar motifs is determined by the number of arms, a parameter that is known to affect the phase diagram of the system 10,19,20. Here we demonstrate that, like three-arm nanostars considered so far, nanostars with higher valency can be controlled by strand invasion reactions that target specific toehold domains. Further, we investigate the kinetics of droplet dissolution as valency is progressively reduced by invader binding.
Nanostars with 4 and 6 arms, presenting distinct sequences in each arm but identical palindromic 4 nt sticky-ends, were engineered to include a variable number of 7 nt toeholds, each characterized by a distinct sequence (schematics in Fig. 6A and C). We then designed a set of invaders, each complementary to a single specific toehold and arm, so that individual arms on a nanostar can be selectively deactivated. For these experiments, we used stoichiometric amounts (1X) of each invader with respect to the nanostar level, to deactivate all the corresponding sticky ends and fully reduce the motif valency.
For the 4 arm motif, we found that the invasion of a single sticky-end is sufficient to dissolve droplets within 15–20 minutes, causing a 90% drop in the number of droplets detected within 2 minutes. The invasion of two sticky-ends accelerates dissolution, which is complete in under 5 minutes (Fig. 6A and B). Removal of invaders by their specific anti-invader results in droplet regrowth (SI Section 3.9).
For the 6 arm motif, invasion of a single sticky-end is insufficient to completely dissolve droplets, however their number is reduced by 70–80% and their average size by 60% over the course of two hours of observation. Invasion of two sticky-ends reduces the droplet number by more than 70% in five minutes, however the droplet average size decreases very slowly, reaching 30% the initial size over two hours of observation. Invasion of 3 adjacent sticky-ends is necessary to completely dissolve the droplets, with over 90% of droplets dissolved in the first 5 minutes, and the remaining larger droplets requiring about two hours to be eliminated. In contrast, invading three staggered sticky-ends yields a droplet dissolution behavior comparable to the invasion of two adjacent sticky-ends. (After 24 hours, the staggered three arm design shows very large droplets, see SI Fig. S59). These results suggest that orientational order of the nanostar arms is important for condensation. If the remaining active arms cannot easily connect in a tessellating pattern, formation of condensates becomes more difficult.
Programmable dynamic control of dissolution and formation of distinct coexisting condensates
A major advantage of DNA nanotechnology is the possibility to build motifs and assemblies that are structurally identical but operate as orthogonal, non-interacting species. This principle can be extended to the design of motifs yielding condensates20,27, and to the strand displacement reactions that control them. To illustrate this advantage, we designed two orthogonal 3 arm DNA nanostars with different sticky-ends, and labeled them with distinct fluorophores (Nanostar 1, Cy3; and Nanostar 2, FAM; shown in Fig. 7). As DNA backbones are negatively charged, DNA nanostars do not spontaneously interact in the absence of base-pairing. To minimize the interactions between palindromic SEs of the two designs, nanostars were designed to have 6 nt sticky-ends. (Notably, palindromic 4 nt sticky-ends containing only ‘A’s and ‘T’s do not yield condensates, see also SI Fig S60). The use of 6 nt sticky-ends however is expected to raise the temperature at which the condensates transition from a liquid to a gel state as noted in the literature20,28. This is confirmed by the fact that even though invasion yields droplet dissolution at room temperature within a few minutes, we were unable to regrow these condensates even after 3 hours of anti-invader addition (SI Fig S61). By raising the temperature to 34°C, we were able to recover reversible phase transitions using invader and anti-invader.
The two nanostar types were annealed in the same pot, producing condensates of distinct colors, orange for Nanostar 1, labeled with Cy3, and green for Nanostar 2, labeled with FAM. The droplets remained demixed, and were not observed to fuse (Fig. 7, left). We then split the sample in two: we added sequentially the invader (1X) and the anti-invader (1X) designed for Nanostar 1 to the first aliquot, observing complete dissolution and then regrowth of the orange droplets, while the distribution of green droplets is qualitatively unaffected (Fig. 7, top row of images). Similarly, when invader and anti-invader for Nanostar 2 were sequentially added, we observed dissolution and regrowth of the green droplets, while the orange ones remained intact.
These results demonstrate the design of two orthogonal artificial DNA condensates that are dynamically controllable via tailored chemical reactions, and we expect they can be easily scaled to systems in which dozens of distinct condensates are individually addressable by sequence-specific DNA or RNA regulators.