Crowding-induced Cooperativity in DNA Surface Hybridization

High density DNA brush is not only used to model cellular crowding, but also has a wide application in DNA-functionalized materials. Experiments have shown complicated cooperative hybridization/melting phenomena in these systems, raising the question that how molecular crowding influences DNA hybridization. In this work, a theoretical modeling including all possible inter and intramolecular interactions, as well as molecular details for different species, is proposed. We find that molecular crowding can lead to two distinct cooperative behaviours: negatively cooperative hybridization marked by a broader transition width, and positively cooperative hybridization with a sharper transition, well reconciling the experimental findings. Moreover, a phase transition as a result of positive cooperativity is also found. Our study provides new insights in crowding and compartmentation in cell, and has the potential value in controlling surface morphologies of DNA functionalized nano-particles.

D NA hybridization/dehybridization (dsDNA ? / 2ssDNA, ds: double-stranded, ss: single-stranded) is an important biological process for genetic functions of cell. In recent years, there are increasing interests in using dense DNA brush to model cellular crowding 1 and compartmentation [2][3][4] . DNA functionalized surfaces have also attracted intensive attentions in areas of bimolecular detection, gene therapy and nanomaterial. For example, DNA microarrays 5 are widely used for DNA sequencing; DNA functionalized nanoparticles, or Spherical Nucleic Acids (SNA) 6 , are designed as vectors for gene delivery 7,8 or as basic units for programmable colloid self-assembly 9,10 . The DNA brush system can create genetic molecular density approximating cellular value (10 7 bp/mm 3 ) or higher 4 . Such a crowded environment leads to distinctive cooperative behaviors in DNA hybridization. Experiments [11][12][13] showed that the width of hybridization/melting transition for planar DNA brush is broader than that in solution. This implies that preceding hybridized DNA impede succeeding hybridizing process, a typical negatively cooperative behaviour 14 . However, a 'sharp melting' transition was also observed in surfacesurface hybridization between DNA-coated nanoparticles, raising the discussion of existence of positively cooperative hybridization, where individual hybridizing events are mutually facilitated 15 . This problem is further complicated in a microcantilevers experiment, where Wu et al. 16 found that under certain conditions surface tension goes down during the process of DNA hybridization.
Another issue deeply related to the hybridization cooperativity is the possible phase separation in dense DNA brush. Since dsDNA and ssDNA are very different in length, size, charge and conformation, it is interesting to ask whether hybridization can cause phase separation in such a crowded condition. Similar phase behaviors have been observed in other systems 17,18 . Researches in this area not only can provide new ideas in explanation of the emergence of compartments in cell 1 , but also enables us to design surface morphologies through specific DNA-DNA hybridization. Recently, DNA patchy particles [19][20][21] have shown such potentials.
There have already been several models which focus on either negative [22][23][24] or positive 15,25,26 cooperativity under different scales and based on distinct mechanisms. However, whether both positive and negative cooperativity can be explained within the same framework at the molecular level and how the molecular crowding influences the cooperativity are currently unaddressed. In our theory, molecular crowding is well described by explicitly taking into account the size, shape, charge and conformation of different species, and various subtle interactions among them are included. We find that molecular crowding can lead to both positively and negatively cooperative hybridization by completely different mechanisms. A first-order phase separation is also found as a result of positively cooperative hybridization, and the dsDNA-rich and ssDNA-rich coexisting phases are obtained.

Results
Quantification of cooperativity and two distinctive cooperative behaviours. Cooperative hybridization arises because different grafted DNA molecules begin to interact with each others. If DNA molecular density is low, hybridization is non-cooperative and the hybridization curve obeys the classical Langmuir isotherm 22,27 h 1{h~c tar : e {bDG' , with the hybridization fraction h, the target ssDNA concentration c tar and the surface hybridization free energy DG9. This isotherm fails when the interactions between grafted DNA emerge (See Figure 1). In order to study the cooperative hybridization in crowding condition, a general expression of DNA hybridization isotherm that incorporates complex intermolecular interactions is needed. This task is accomplished by using the molecular theory (see Supplementary section 1 and 2). The result is rather simple: with the standard hybridization free energy in DNA solution DG 0 . Equation (1) differs from classical Langmuir isotherm mainly in the excess hybridization free energy DG ex (h), which satisfied DG ex 5 U ds 2 U ss , with U ds and U ss the potentials of mean force (PMF) of dsDNA and ssDNA molecules staying in DNA layer, respectively. The PMF reflects the crowdedness of the layer, which includes isotropic excluded volume and orientational interactions between DNA molecules, ionic osmotic pressure, electrostatic interaction, hydration repulsion among dsDNA and entropic force arising from DNA conformational deformation (see Supplementary section 3). The purpose of this work is to investigate cooperativity of DNA hybridization in dense DNA layer. A common measurement of hybridization cooperativity is the full width at half-maximum (FWHM) of its hybridization or melting curve. For positive cooperativity, the transition will be sharper, leading to a narrower FWHM. Oppositely, negative cooperativity retards the transition, resulting in a broader FWHM, as demonstrated in Figure 2 at the melting point T m (h 5 0.5). For DNA brush system, it can be written as where W 0 is a reference value representing the melting width in DNA solution 28 (no intermolecular interaction). Hence, D is a normalized melting width. D 5 1, D , 1 and D . 1 can be used to represent non-cooperativity, positive cooperativity and negative cooperativity respectively. According to equation (1), D can be expressed as The interpretation of equation (3) is simple. For non-cooperative situation, DNA hybridizations or dehybridizations happen independently, hence DG ex is a constant and D 5 1. Cooperativity happens when DG ex varies with h. More specifically, if increasing h lowers the excess hybridization free energy, it is positive cooperativity, meaning that hybridization or dehybridization events in DNA layer are mutual-facilitated. In this situation we can easily get D , 1. Inversely, if DG ex (h) is an increasing function of h, it is negative cooperativity, indicating that hybridization or dissociation events are mutually impeded. In this case, we will have D . 1.
In fact, D can be also written into a more general form where Dm is the exchange chemical potential 23 of single DNA molecule from coil to helix state Dm 5 m ds 2 m ss . Equation (4) facilitates us to measure the cooperativity by Dm instead of plotting the melting curve. The comparison between Dm 2 h curve and melting curve is given in Figure 2(b). Moreover, a negative gradient of Dm is an indicator of phase transition. Therefore, equation (4) also enables us to identify phase separation by testing whether D , 0. Equation (3) and (4) also imply that hybridization cooperativity is insensitive to the specific DNA sequences and other factors which are unchanged during hybridization process. The insensitivity to sequence is verified by melting curves for four more DNA sequences with different G-C contents as shown in Supplementary Figure S1.
Phase diagram of cooperativity. In Figure 3,  Figure 3, where B is very close to the D 5 1 curve. PMF of dsDNA and ssDNA are also given in Figure 4(a)-(c), since DG ex 5 U ds 2 U ss . We can expect that when h increase, the layer becomes more and more crowded. So both U ds and U ss go up along with h. However, the increase of U ds at high ionic strength(1 M) obviously slows down with a modest h, indicating that some part of repulsions felt by dsDNA are relaxed during the hybridization process. This is the reason for the decrease of corresponding DG ex (h). To find the causation of this crowding relaxation, it is necessary to evaluate different contributions to the crowdedness separately.
In our model, molecular crowding is explicitly depicted by isotropic excluded volume and orientational interactions between DNA molecules, electrostatic interaction, ionic osmotic pressure and hydration repulsion among dsDNA in the layer. In particular, the orientational interaction between dsDNA can be viewed as attractive since it can reduce the free energy by aligning dsDNA in the same direction. The advantage of our model in such complex system, is that we can decompose the molecular crowdedness based on different mechanisms. Figure 4(d)-(f) show contributions to PMF of dsDNA molecule from different interactions under different ionic strengths. At high ion concentration(1 M), electrostatic interaction is rather weak due to the strong electrostatic screening. Therefore, isotropic excluded volume repulsion and anisotropic orientational attractions are the most two important interactions to determine the collective behavior of DNA molecules. Both of them increase with the formation of more dsDNA. However, orientational attraction increases faster, which leads to the decrease of DG ex (h) and the positive cooperativity. As the ion strength decreasing, electrostatic repulsions get stronger (Fig. 4(e)) and finally play the dominating role (Fig 4(f)), resulting in negative cooperativity. Therefore, by varying the ion strength from high to low, we show how the electrostatic interaction competes with orientational attraction, and leads to two completely different cooperative hybridization behaviours. It should also be mentioned that, for all three cases, when h is close to 1, hydration repulsions rise quickly, making a faster increase of DG ex (h) near h 5 1.
Effect of DNA molecular density. DNA molecular density, or surface coverage, which directly relates to molecular crowding, is another crucial factor to determine the cooperative hybridization. As disclosed by the phase diagrams, increasing the molecular density(crowdedness) has a non-monotonic effect on the hybridization cooperativity under high ion strength condition. From low molecular density (s # 0.02 nm 22 ) to moderate molecular density (s , 0.08 nm 22 ), the increasing crowdedness significantly enhances the orientational attraction as reflected by a fast increasing of orientational order parameter S or~1 2 v3 cos 2 h{1w given by Figure 5(a). This makes the hybridization cooperativity change from non-cooperativity to positive cooperativity. However, further increasing the molecular density from moderate molecular density to high molecular density (s . 0.1 nm 22 ) shifts the hybridization cooperativity from positive to negative. This is because under such highly crowded condition, DNA molecules get much closer to each other, short range hydration repulsion among dsDNAs begins to play the dominant role as shown in the Figure 5(b). Such a dramatic increment of hydration repulsion with h overwhelms that from orientational attraction, resulting in negative cooperativity. The threshold s 5 0.08 nm 2 for the strong    First order phase transition. From above analysis, we can conclude that both negatively and positively cooperative hybridization arise from molecular crowding. Negative cooperativity (D . 1) happens when crowding-strengthened repulsive interactions play the dominate roles, while positive cooperativity (D , 1) occurs when the entropy-favouring orientational attraction becomes more important. Moreover, a phase transition can happen in the region of positive cooperativity (D , 0). This kind of phase transition is a coupling between the helix-coil transition and nematic-isotropic transition 31 , which has been verified in polypeptide solutions 32,33 . Recently, short dsDNA molecules were also found to form domains in high density ssDNA solution 17 .
Generally speaking, there is no spontaneous isotropic-nematic phase transition in hard rod brush systems 34,35 . Here we show that the phase transition occurring in the DNA layer is a first-order one which can lead to a two-phases separation. In Figure 6, detailed structural informations are given for the two coexisting phases: dsDNA-rich state (a) and ssDNA-rich state (b). An interesting phenomenon is that ions are repelled from the DNA layer due to strong molecular crowding. For the charge neutrality of the system, anions are repelled much more than cations. Moreover, it's well known that for simple solid brush systems, only microscopic phase separation is possible due to the immobility of anchored chains 36 . In our system, however, the positions of dsDNA are mobile despite the immobility of anchored ssDNA as a result of the dynamic equilibrium of DNA hybridization/dehybridization. Calculated surface tensions near the transition point also show positive value(see Discussion Section). These suggest that the phase separation that can happen in our system is a macroscopic one, completely different from that of classical solid brush systems.

Discussion
In practical applications, surface curvature and chain length are two controllable factors. Experimental evidences have shown that accommodation of DNA on spherical nano-particles can be quite different from that of planar surface 37 . In view of the enormous applications of spherical nucleic acids, it is an important issue to understand how surface curvature influences molecular crowding and DNA cooperative hybridization.
In Figure 7, we show the cooperativity phase diagram of DNA hybridization on a spherical surface with radius of 10 nm. Compared with planar surface, intensity of cooperativity is remarkably reduced for both positive and negative cooperativity regions. This indicates that the molecular crowding is weakened since spherical surface has a larger spatial accessibility for DNA. As a result, phase separation is suppressed. However, the region of positive cooperativity is expanded to the area with high molecular density. Therefore, on the whole, high curvature favors positive cooperativity. In addition, high curved sphere has much smaller surface area compared with planar surface, which can effectively resist the surface heterogeneity 12 and avoid the kinetic trap 38 . All these factors could attribute to a smaller change of width of melting curve and a stronger binding affinity of DNA on nano-particle surface 7,39 .
To investigate how DNA length affects cooperativity, we also calculate the cooperativity phase diagrams for N 5 20, 40, 50 (see Supplementary Fig. S2) for both planar and spherical surfaces. We give the statistics on area fractions of positive cooperativity (D , 1) and phase transition (D , 0) in Figure 8(a). It can be found that increasing chain length can significantly enlarge the region of positive cooperativity and phase transition owing to enhanced orientational attraction. In Figure 8(b), we also plot surface tension as a function of h for different DNA lengths under the condition of s 5 0.07 nm 22 and ion strength of 1 M for the planar surface. We find that the surface tension increases monotonously in short length case. For the long DNA case, there is a maximum value followed by a drop when h approaching 1. This decrease of surface tension is a result of crowding relaxation due to a more ordered dsDNA alignment. Our finding is qualitatively consistent with experimental observation 16 and other theoretical calculations 40,41 .
In reality, many other factors may influence DNA hybridizing behaviours. For instance, unwanted DNA self-hybridization and  cross hybridization 30 would decrease the local order of DNA layer and undermine the effect of orientational attraction. Moreover, surface heterogeneity can widen the melting curve, enhancing the negative cooperativity 12 . On the other side, aggregation on the surface was reported during the preparation of both dsDNA and ssDNA monolayers, giving dense and spare domains [42][43][44] . This implies the existence of favourable interactions between same kind of DNA, which would promote positive cooperativity and phase separation. Depletion effect may also have the same effect 17 . We also note that molecular crowding conditions can change the water activity, thus affects the solvation of DNA and thermodynamic of DNA hybridization 48,49 . In addition, the rearrangement of water's hydrogen-bonds network and change of local dielectric properties in the crowding situation is also expected to modify the DNA hybridization cooperativity 1 .
In fact, a more sensitive system to investigate the DNA cooperative hybridization is the liquid DNA brush system, in which tethered DNA molecules have mobility on the surface. Experimental researches in this direction have just gained its momentum [45][46][47] . Since the adding of the mobility usually promotes phase separation as well as weakening cross-hybridization and surface heterogeneity, hybridization cooperativity can be more easily perceived. The detecting sensitivity can also be improved by using DNA tetrahedral nanostructures as demonstrated by Lin et al. 50 .

Conclusion
In this work, we systematically investigate the molecular crowding and its effect on DNA surface hybridization. We find molecular crowding can lead to two types of cooperativity due to the competition between various interactions. Generally speaking, DNA molecules feel isotropic excluded volume and orientational interaction, electrostatic repulsion, hydration repulsion and osmotic pressure from ions in the layer. It is found that the crowding-strengthened repulsions cause the negative cooperativity, while the entropyfavouring orientational attraction is the driving force for positive cooperativity. Under certain conditions, this positive cooperativity can induce a first order phase separation on surface. We discussed various factors that affect DNA surface hybridization, including DNA molecular density, ion strength, surface curvature and DNA chain length. Our discovery is not only important for practical applications, but also of great significance to understand complex crowding-induced biological phenomena in cell.

Methods
In the present work, we focus on the thermodynamics of DNA surface hybridization based on the assumption that two ssDNA hybridize into one perfect dsDNA(twostate model). In our model, unhybridized ssDNA molecules are tethered on planar or spherical surface, shown in Figure 1. Hybridization happens when a tethered ssDNA captures its complementary ssDNA in solution and turns into a rigid dsDNA. The coil state ssDNA, which has the persistence length of 2 nm 51 , is described by the wormlike chain model(see Supplementary Section 5). While helix state dsDNA is modeled as rigid rod, which can rotate freely on its anchored point. Due to counterion condensation effect, dsDNA is assumed to take 0.75 e 2 charge per nucleotide pair, while ssDNA takes 0.5 e 2 per nucleotide, according to recent experimental results 52 . Cations (Na 1 ) and anions (Cl 2 ) are explicitly included and are assumed to have a hydrated radius of 0.35 nm, while solvents only enter the theory implicitly through the dielectric constant e 5 78 53 .
We use a molecular theory 36,54 that explicitly considers the size, rigidity, conformation, charge, and inter-molecular interactions between all molecular species in the system. The theory is formulated by writing down the free energy of the system. In general terms, it can be expressed as where A is the surface area of the system; s is the molecular density of DNA; DG9 is the surface hybridization free energy; c tar is the target ssDNA concentration. The first term describes the associating entropy of DNA between two states (hybridized and unhybridized); S conf represents the configurational entropy of flexible ssDNA, while S orient represents the orientational entropy of dsDNA helix. F hc is the helix-coil excluded volume interaction 41,55 between DNA molecules, which consists of isotropic and orientation-dependent parts. F hyd is the short-range hydration repulsion between parallel dsDNA. S ion is the entropy of ions, and F free is its free volume modification due to the existence of crowded DNA. The last F elect represents the electrostatic energy of the system. Each of these terms is a function of distributions of the different molecular species, charge, the probabilities of the dsDNA and ssDNA conformations. We minimize F with respect to these functions to determine the equilibrium structure of the layer. In Supplementary Information, we present a detailed description of the molecular model, the free energy expression, the minimization procedure, etc.