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The Dynamic Properties of DNA-Programmable Nanoparticle Crystallization Qiuyan Yu, Xuena Zhang, Yi Hu, Zhihao Zhang, and Rong Wang ACS Nano, Just Accepted Manuscript • DOI: 10.1021/acsnano.6b02067 • Publication Date (Web): 13 Jul 2016 Downloaded from http://pubs.acs.org on July 18, 2016

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The Dynamic Properties of DNA-Programmable Nanoparticle Crystallization Qiuyan Yu, Xuena Zhang, Yi Hu, Zhihao Zhang, Rong Wang*

Key Laboratory of High Performance Polymer Material and Technology of Ministry of Education, State Key Laboratory of Coordination Chemistry and Collaborative Innovation Center of Chemistry for Life Sciences, Department of Polymer Science and Engineering, School of Chemistry and Chemical Engineering, Nanjing University, Nanjing 210023, China

ABSTRACT: The dynamics of DNA hybridization is very important in DNA-programmable nanoparticle crystallization. Here coarse-grained molecular dynamics (MD) is utilized to explore the structural and dynamic properties of DNA hybridizations for self-complementary DNA-directed nanoparticle self-assembly system. The hexagonal close-packed (HCP) and close-packed face-centered cubic (FCC) ordered structures are identified for the systems of different grafted DNA chains per nanoparticle, which are in good agreement with the experimental results. Most importantly, the dynamic crystallization processes of DNA hybridizations are elucidated by virtue of the mean square displacement, the percentage of hybridizations and the lifetime of DNA bonds. The lifetime can be modeled by the DNA dehybridization, which has the exponential form. The lifetime of DNA bonds closely depends on the temperature. The suitable temperature for the DNA-NPs crystallization is obtained in the work. Moreover, too large volume fraction hinders self-assembly process due to the steric effect. This work provides some essential information for future nanomaterials design. KEYWORDS: DNA, nanoparticle, crystallization, dynamic, structure 1 ACS Paragon Plus Environment

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Because the strength, length and nature of nucleic acid bonds (Watson-Crick base-pairing interaction) between particles can be adjusted systematically by varying the nucleobase sequence, length as well as the number of DNA strands attached to nanoparticles, DNA has been the ideal ligand to direct the nanoparticle interaction.1, 2 This DNA-functionalized nanoparticle (DNA-NP) allows the thermodynamically reversible and controlled assembly of inorganic nanoparticle into supramolecular structures.1 It can be widely severed as a programmable atom equivalent (PAE) which consists of a nanoparticle core densely functionalized by oligonucleotides with specific sequence and length, through which DNA mediates interactions between nanoparticles to program macroscopic materials with novel physical and chemical properties3, 4 and highly ordered structures.5-9 The rigid nanoparticle core and the oligonucleotide density impose a radial orientation of the DNA and valency to the nanoparticles. Unlike atoms, having a fixed series of physical and chemical properties and bonding possibilities induced by the inherent electronic, these properties and bonding behaviors of PAEs can be adjusted by controlling their structures in a wide range of parameters. Anisotropic spherical organic,10 inorganic,11, 12 polymeric7 nanoparticles have been regarded as PAE cores. Even RNA13 and proteins14 can achieve assemblies by the same design rules.15, 16 Actually, it's the DNA bond that programs nanoparticle interactions and drives their assemblies into ordered crystalline structure in the above studies.13 Nonetheless, it's concerned that how the DNA bond induces crystallization thermodynamically and kinetically. Although theoretical models15,

17

have been

proposed, they are too simplified to settle crucial problems of the DNA chain conformational analysis or dynamics. In order to have a better knowledge of factors that influence DNA base-pairing interaction and the crystallization kinetic process, it's essential to find scale-accurate coarse-grained model which can capture the stiffness and size of different parts of DNA chains as precisely as possible. The flanking bead model based on Molecular Dynamics18 is very helpful to research the DNA-based assembly of several superlattices containing body-centered cubic (BCC), CsCl, AlB2, 2 ACS Paragon Plus Environment

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Cr3Si and more complex Cs6C60.9,

19, 20

We can obtain relatively stable structural

information and dynamic details of the assembly process by tracking the position and velocity of NP in different time and monitor bond-breaking or reforming of DNA hybridizations. In previous work, several factors, such as NP size and shape, DNA length and sequence, have been taken into consideration to independently adjust each of the relevant crystallographic parameters, including particle size, periodicity, inter-particle distance.15, 21 However, they mainly focus on lattice structures, ignoring dynamic properties of crystallization, which plays an important role in the process of lattice formation. In this paper, therefore, we largely focus on the kinetic process of DNA-NP self-assembly and crystallization to study DNA hybridization properties by coarse-grained MD simulation (Figure 1).

Herein, effects of temperature, the number

of DNA chains attached to per NP and volume fraction of NPs on properties of DNA hybridizations are systematically investigated to solve the existing problems.

Figure 1. Schematic illustration of the coarse-grained model for A-A self-assembly system. Top: a model of ssDNA chain. ns and nl are the number of coarse-grained space beads and linker beads, respectively. Bottom: An example of two ssDNA-NPs hybridization. Each particle can bind to every other particle with equal affinity by linker-linker pairing interaction in the single-component system.

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RESULTS AND DISCUSSION Knorowski and Li et al.18, 20 theoretically validated the experimental phase diagram of binary mixtures of DNA-mediated NPs15 based on the coarse-grained model. In addition to obtain some complicated superlattice structures by tuning temperature, the number of DNA chains, the stoichiometric ratio of two DNA-NP types and building blocks, they also investigate the DNA hybridization kinetics, the number of hybridizations and DNA behavior during the assembly process,18-20 which is widely aroused extensive concern previously by many researchers.15,

22, 23

However, the

attention to self-complementary single-component system (called A-A system here), in which each particle can bind to every other particle with equal affinity, is absent and deserved to be studied. Herein, two different kinds of close-packed crystal structures are observed: hexagonal close-packed (HCP) and face-centered-cubic (FCC) structures where each particle has 12 nearest neighbors, for different linker loadings (from n = 20 to 80 DNA chains per nanoparticle, where n represents the number of DNA chains attached to a nanoparticle) by simulating the single-component self-assembly (Figure 2). Compared with the binary system, the single-component system increases the probability of DNA linker-linker hybridization interactions, thus, allowing for closer packing and a less-significant thermodynamic difference between the HCP and FCC crystallographic arrangements.24 The HCP and FCC are arranged in different packing ways: the former is AB periodic layered type whose first layer overlaps the third one (Figure 2b), and the latter is ABC periodic layered type whose first layer overlaps the fourth one (Figure 2d), where A, B, C show the different periodic layer.


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(a)

(b)

(c)

(d)

Figure 2. Crystal structures obtained from MD simulation under the condition T = 1.4, Lx × Ly × Lz = 60 × 60 × 60, η =1.15. (a, c) Hexagonal close-packed (HCP) nanoparticle lattices and Face-centered cubic (FCC) nanoparticle lattices; (b, d) The packing type of HCP (AB periodic layered) and FCC (ABC periodic layered).

The number of grafted DNA chains per NP. The pair radial distribution function g(r) for n = 10~80 is shown in Figure 3a, giving the probability of finding a particle (or molecule) at distance r away from a reference particle, relative to that for the ideal gas distribution. DNA-NPs in the low-loading regime (i.e. n = 25~40) own hexagonal close-packed (HCP) or random hexagonal close-packed (rHCP) structure. When n = 45~55, peak positions of g(r) have a slow shift, attributed to small FCC domains. When n = 60~80, an infinite number of sharp peaks appear, whose separations and heights are 5 ACS Paragon Plus Environment

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characteristics of the FCC lattice structure. For n < 20, it has only one shape peak (r ≈ 22 in unit of σ) at short distances, superimposed on a steady decay to a constant value at longer distances. For short distances, g(r) is zero, due to the strong repulsive forces. At longer distances, g(r) tends to the ideal gas value, indicating that there is no long-range order and it is a typical liquid for n < 20.25 As n increases, a series of sharp and wide peaks appear at larger distances, with the first peak position r0 (r0 ≈ 21.22) (Figure S1) as the average distance between nearest-neighbor center-to-center nanoparticles. Theoretically, if the three linker beads of DNA chains achieve partial or complete hybridizations, r0 ≈ 2 R'−(1 ~ 3)σ (R' ≈ 11.4~11.9, R' is radius of a DNA tethered nanoparticle), then r0 ≈ 19.8~22.8 with σ as the diameter of a linker bead. Our simulation results agree with the theoretical prediction. To better determine the lattice structure, we also give the structure factor S(q) of nanoparticles. Figure 3b and 3c present S(q) for n = 60 and 35, respectively. From the figure, we can clearly see FCC (n = 60) and HCP (n = 35) structures. For FCC, q1 : q2 : q3 : q4 : q5 : q6 : q7 : q8 is equal to

3 : 4 : 8 : 11 : 12 : 16 : 19 : 20 ,

corresponding to crystallographic planes (111), (200), (220), (311), (222), (400), (331), (420). For HCP, q1 : q2 : q3 : q4 : q5 : q6 : q7 : q8 is equal to 1:1.061:1.132 :1.458 :1.732 :1.879 : 2 : 2.031 , corresponding to (010), (002), (011), (012),

(110), (013), (020), (112). Our simulation result agrees with the theoretical peaks very well (Figure 3b, 3c). We can distinguish between FCC crystal and other structures by comparing the relative positions of order peaks qx / q1 = 1, 4 / 3, 8 / 3, 11/ 3 , corresponding to FCC crystallographic planes (200), (220), (311) etc.24, 26 The number integral over g(r) for the first peak square corresponds to coordination numbers ro

( N coor = 2 ∗ ∫ 4πr 2 ρg ( r ) dr , ρ is the average number density) of NP. For n > 20, 0

coordination numbers Ncoor = 12, once again determines HCP and FCC crystal structures (Figure 3d).


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11.5

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11 121/2 1/2 16 191/2 1/2 20 1/2 24 271/2 1/2 1/2 32 35 1/2

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n = 15

19.6 9.8 0.0 0

5

10

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r

(d) Figure 3. Radial distribution functions g(r), static structure factor S(q) and integral of g(r). (a) g(r) for n = 10~80. (b, c) S(q) for n = 60 and n = 35 respectively. The structure factor is defined as S (q ) =

1 N

∑e

− iq ( R j − Rk )

, where q is the wave vector, N is the

jk


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number of NPs, Rj, Rk ( j, k = 1, 2, 3, ... N) are the nanoparticle positions.. (d) Integral of g(r) for n = 15, 35, 60, representing coordination numbers Ncoor = 12 for n = 35, 60.

It's difficult to form ordered structure when n < 20 and it forms HCP structure easily in relative low DNA loading regime, while FCC structure in higher regime. In fact, the radius ratio of a nanoparticle and a DNA bead is 6 : 1. If we set the volume of a DNA bead as 1, we obtain the overall particle density ρ (including the DNA beads and the nanoparticles) easily according to the model (Figure 1):

ρ=

N [63 + (ns + nl )n] L3

(1)

In the Methods section, we give the relevant simulation conditions, such as the DNA beads and the nanoparticles (nl = 3, ns = 8, N = 32). Therefore, the overall particle density ρ increases linearly for fixed box size. Maybe this increasing density would be the main factor inducing the transition from disorder to HCP, or even to FCC. Here we will introduce the kinetic process analysis in the following work by p(H) and f(H). p(H) is the percentage of complete hybridizations of the total DNA chains, where complete hybridizations are defined if every linker bead of a DNA strand forms a hydrogen bond, namely, within the distance of σ far away from its complementary linker bead. f(H) is the fraction of hybridizations that live up to with time ( i.e. all pairs of the hybridized sticky ends are recorded at time t = 0 and then are tracked with the simulation time). f(H) closely decreases exponentially with simulation time step and the decrease rate of f(H) mirrors the rate of DNA dehybridization. It conforms the following expression: f ( H ) = f ( H ) 0 + A * exp ( −t / τ 1 )

(2)

where f(H)0 is the fraction of hybridizations in equilibrium and A represents pre-exponential factors. τ1 reflects the lifetime of DNA bonds (in unit of (mσ 2 / ε )1 / 2 ) . Figure 4 shows p(H) as a function of simulation time step for different loading DNA chains. For n < 20, great thermal fluctuation of the simulation system occurs due to very limited DNA chain interaction, which makes NP fluctuate in the simulation box. On the


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one hand, average percentage of hybridizations (less than 30%) is so small that hybridization strength is not enough to serve as an appropriate stabilizing interaction between the DNA-NPs and guide them into crystallization. On the other hand, too few hybridized bead pairs (Figure 5) mean that there are not enough hydrogen bonds to counteract the thermal motion of the DNA-NPs and the system remains liquid.20 Martinez et al.27 has also explained colloids coated with a small number of DNA strands cannot crystallize in dilute solution by Monte Carlo simulation.

0.5 0.4 0.3 p(H)

0.2 0.44 0.40

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0.1

n = 10 n = 20 n = 30 n = 50 n = 60

0.36 0.32 0.28

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t Figure 4. p(H) as a function of time step for n = 10~60 at T = 1.4, η = 1.15, Lx × Ly × Lz = 60 × 60× 60. The inset is the average percentage of hybridization after equilibrium.

In addition, because f(H) realizes equilibrium comparatively as n increases from 10 to 80 and the rate of dehybridization is unaffected by DNA concentration,28, 29 increasing the number of DNA strands can raise the rate of DNA hybridization and the lifetime τ1 distributes at 1.2×103 ~ 2.6×103 in unit of (mσ 2 / ε )1 / 2 (Figure S2). It is slightly larger for n = 10 than for larger DNA loadings because it can easily reform DNA bonds after


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breaking, which also can explain the great thermal fluctuation of p(H). For 20 ≤ n ≤ 50, the effect of thermal fluctuation diminishes, promoting the process of DNA hybridization. Therefore, f(H) and τ1 have little relation to DNA loadings. However, the difference of f(H) and τ for 60 ≤ n ≤ 80 from n < 60 suggests dense DNA shells hinder NPs violent motion and allow them reorganization in the limited space, where DNA bond interaction is neither too strong nor too weak as a suitable stable interaction between DNA-NPs and induce them assemble into perfect FCC crystals. As the number of DNA strands increases from 80 to 100, the surface of NP is full of DNA strands and the coverage reaches nearly saturated. The dense DNA shells restrict the reorganization processes and prevent NPs from arranging themselves into large and perfect domains. Macfarlane et al. has observes the reorganization of nanoparticle superlattices and obtained the same conclusions by using time-resolved synchrotron small-angle X-ray scattering in experiment.30 Although the stable is found as the nanoparticle surface has been saturated, the total number of hybridized bead pairs (Figure 5) still linearly increases as the DNA chain number increases. Therefore, more DNA linkers result in more hybridizations and promote their dynamic process, in accord with the design role of Mirkin's group.15

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Simulation results Linear fit results

400

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Figure 5. The total number of hybridized bead pairs n(H) in equilibrium for n = 10~100 11 ACS Paragon Plus Environment

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at T = 1.4, η = 1.15, Lx × Ly × Lz = 60 × 60× 60. The inset table displays the relative parameters of linear fit results.

Temperature. The rearrangement of DNA bonds and the thermally active hybridization are the important requirements to realize the equilibrium and long-range ordered assembly of DNA-directed systems.5,

19, 31

The so-called thermally active

hybridization means that linkers should easily attach to and attach from their complementary counterparts because of thermal fluctuations, thus allowing to escape from any kinetically trapped structure easily.19 It can be obviously achieved by thermal annealing that induces partial melting or dissociation of DNA duplexes chains, allowing particles to optimize their location in order to maximize the number of DNA hybridizations.5 By very slow cooling through the melting temperature over several days, DNA-modified nanoparticle solutions can give thermodynamic products and achieve Wulff equilibrium crystal structure.31 In addition, DNA bonds between the complementary sticky ends will also form, break and reform discontinuously at a constant temperature T because of the thermal fluctuation of simulation system itself. Herein, both the above controlled temperature styles are used to discuss the process of the crystallization: either starting from Tinitial = 1.5 then cooling slowly to Tfinal = 1.3 (linearly decreasing temperature) or at a constant temperature T. The dynamic process of bond-breaking and reforming is mirrored by p(H) and f(H). The crystalline process is determined by mean square displacement (MSD) of NPs as a function of the simulation time step. In Figure S3, the temperature of the system first remains constant T = 1.5 for 5×105 simulation steps to create a random conformation. As the temperature linearly decreases to T = 1.4, the MSD of NPs reaches a steady state at t = 2.0×107, indicating fully crystallized. This annealing process helps us determine the melting temperature (Tm) and optimal annealing temperature at which DNA bonds between particles readily break and reform, leading to reorganization within the particles.30 However, it takes only 7×106 time steps to realize fully crystallization at constant temperature T = 1.4 where the system only depends on thermal fluctuation of simulation system itself 12 ACS Paragon Plus Environment

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(Figure 6). The point is that in the steady state of the p(H) only the average number of the hybridization is constant, bond-breaking and reforming between the sticky ends are dynamic, even when thermodynamic equilibrium has been achieved, in good agreement with the binary experimental system.20

30

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25

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0.4 0.3 6

7×10 1x107

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Figure 6. Mean square displacement (MSD) of NPs and the percentage of hybridizations p(H) at constant temperature T = 1.4 for n = 60, η =1.15, and Lx× Ly× Lz = 60 × 60× 60.

The temperature is a crucial factor to control DNA-NP self-assembly and crystallization. Figure 7 gives p(H), f(H) and as a function of simulation time step under different temperatures, showing both p(H) and f(H) strongly depend on temperature. As T increases, the average decreases and f(H) decrease more and more quickly, indicating faster and faster rate of dehybridizations. This also accounts for the fact that the p(H) of the above annealing process increases as T decreases (Figure S3). When T is high enough (T > 1.45), DNA bonds break easily and the lifetime of DNA bonds τ1 is so short (less than 1×103 (mσ 2 / ε )1 / 2 ), leading to too high rate of dehybridizations. f(H) reaches equilibrium quickly and becomes lower and lower as T increases (Figure 7), which make the self-assembly occur hardly.


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Moreover, too few hybridized complementary bead pairs result in not enough attractive interactions to counteract the thermal motion of the system.20 Under this condition, the system melts or remains liquid. Under relatively low temperature condition, the rate of bond-forming is larger than the rate of bond-breaking, so p(H) increases quickly and hybridized DNA will be predominant. The lifetime τ1 of DNA bonds is larger than 1×104 (mσ 2 / ε )1 / 2 , allowing that too many interconnecting duplexes form and the attraction between DNA-NPs becomes so strong that they break and reform difficultly. The system is kinetically trapped either in a metastable state or an amorphous state. From Figure 7(b), we can see that it becomes very difficult for the system to reach equilibrium when T ≤ 1.2, which limits the reorganization process of the assembled DNA-NPs into a thermodynamically favored structure. Therefore, the suitable temperature for the DNA-NP crystallization is between 1.2 and 1.45, and the lifetime τ1 ranges from 1×103 to 1×104 (mσ 2 / ε )1 / 2 .

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1.0 T = 1.1 T = 1.2 T = 1.3 T = 1.4 T = 1.5 T = 1.6

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(c) Figure 7. (a) p(H) and (b) f(H) as a function of time steps under different temperatures, (c) The lifetime τ1 of DNA bonds as a function of temperature T for n = 60, η = 1.15, Lx× Ly× Lz = 60 × 60× 60. The inset of (c) is the average percentage of hybridization after equilibrium.


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Volume fraction. The concentration of NPs is controllable in experiment but difficult in simulation, so we introduce volume fraction (η) to tune self-assembly process of DNA-NPs by the aid of the size of simulation box, based on Formula (7). As the end-to-end distance (R' = 11.4~11.9) is hardly influenced by the simulation box, the volume fraction decreases with the increase of simulation box. Figure S4 shows the dynamic hybridization process for different volume fractions. The p(H) reaches a steady platform quickly, f(H) realizes equilibrium in 2.0×106 time steps and the lifetime of DNA bonds ranges from 1.3×103 to 2.2×103 in unit of (mσ 2 / ε )1 / 2 which is hardly related to the volume fraction. As η increases from 0.67 to 1.15 (Lx = Ly = Lz = 73~60), the linearly increases more quickly than for η = 1.20~2.00 (Lx = Ly = Lz = 59.8~50) and have a maximum value at η = 1.15 (Figure 8). This is because DNA-NPs have enough space for free motion at low η, where NPs realize reorganization by bond-breaking and bond-reforming, while there exists the hindering effect in limited space for larger η. When η = 1.15, the space is optimized for reorganization process, thus achieving full crystallization and obtaining the perfect closed-packing FCC superlattice.

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0.27

Simulation results Linear fit for η = 0.67~1.15 Linear fit for η = 1.20~2.00

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CONCLUSIONS We focus on the dynamic process of DNA bonds for A-A self-complementary system by using Molecular Dynamics (MD) simulation based on the flanking bead model of Knorowski et al.18 The number of DNA chains per nanoparticle and temperature are the main factors to control the interaction strength between the nanoparticle building blocks. The low DNA loadings can promote the formation of HCP lattice to maximize the number of DNA hybridizations, but higher loadings makes for close-packed FCC structure (60 ≤ n ≤ 80). Moreover, very dense DNA shells (n ≥ 90) restrict the reorganization processes and prevent NPs from arranging themselves into large and perfect domains. The thermally active hybridization induces the sticky ends to attach to and attach from their complementary counterparts, allowing to escape from any kinetically trapped structure easily. As the temperature increases, p(H) decreases. The rate of dehybridizations speeds up more and more quickly and the rate of hybridizations is affected slightly. The lifetime τ1 of DNA bonds is closely related to the temperature, but there is little dependency on the number of DNA chains per nanoparticle and the volume fraction. When T ≤ 1.2, the interaction between complementary bead pairs is too strong to induce the crystallization and τ1 ≥ 1×104 (τ1 in unit of (mσ 2 / ε )1 / 2 ); but when T ≥ 1.45, the interaction is too weak to induce the crystallization and τ1 ≤ 1×103. When 1.2 ≤ T ≤ 1.45, 1×103 ≤ τ1 ≤ 1×104 and it is an appropriate temperature interval, for the nanoparticle crystallization because it owns neither too strong nor too weak DNA interaction. Our results are very helpful to control the properties of crystalline with potential applications in plasmonic metamaterials,12 energy conversion,3 medical diagnosis,32, 33 electrochemical DNA biosensor.4 More work is having in hand to better understand the kinetic process and factors of DNA-NP crystallization.

METHODS Figure 1 is the model of A-A self-assembly system where one particle can bind to 17 ACS Paragon Plus Environment

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any other particle by using linkers with self-complementary sticky ends used to simulate directional hydrogen bond interaction between complementary base pairs (A-T, C-G). Each nanoparticle is modeled as a spherical core of radius R = 3σ where σ ≈ 2.0nm and single-strand DNA (ssDNA) chains (with the chain number n) are distributed randomly on the NP surface. A ssDNA is modeled as spacer beads (ns) and linker beads (nl) and each bead with diameter of σ represents about 4-7 bases.18, 19 Each linker bead has an additional structure with the central bead (CT) to achieve hybridization and the flanking beads (FL) to protect CT from binding to more than one complementary bead. The diameter of each CT and FL bead is 0.6σ.18 The point is that it's difficult to simulate exact values of parameters of DNA chains and NPs, so we can only obtain some approximate parameters in the condition that the coarse-grained model can capture some relevant physical processes. Although flexible DNA chains are a crucial requirement to realize crystallization,10, 34, 35

but excessively long flexible DNA chains easily lead to short-range ordered, even

disordered assemblies due to destroying the directional interaction5. Therefore, only eight spacer beads (about 32 bases), enough flexible to form long-range ordered structures10 are used in our modeled ssDNA. Since only relatively weak DNA interactions between particles make for forming crystalline lattices of nanoparticles,34 we employed as few as three complementary linker beads (nl = 3), corresponding to approximate 12~20 base pair linkers10, 18 rather than 30 or more complementary linkers. Moreover, weak interactions result in reorganization of PAEs in the lattices to break any DNA bonds easily which force particles to trap in thermodynamically unfavorable states even when they are bonding to another, thus achieving particle reorganization into ordered superlattice.20, 24, 28 The system volume fraction is defined as18, 19 4 N × πR'3 3 η= L3

(3)

where N is the number of DNA-NPs, L is the linear size of the simulation box, R' is the average distance between the center of a nanoparticle and the end of the relaxed DNA 18 ACS Paragon Plus Environment

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chain. The harmonic spring potential is used to model the covalent bonds along the ssDNA chain with the spring constant ks = 330ε/σ2 and the native spring length r0 = 0.84σ.

Vbond (r ) =

1 k s ( r − r0 ) 2 2

(4)

Harmonic angel potential is used to model the stiffness of the ssDNA chain, and also aligns neighboring CT beads and its flanking beads (FL). Vangle (θ ) =

1 kθ (θ − θ 0 ) 2 2

(5)

where θ represents the harmonic angle between three consecutive bead (θ0 =180º) and the spring constant kθ = 100 kbT. The short-range repulsive interactions between any other pair of beads are modeled with Weeks-Chandler-Andersen (WCA) soft-core repulsive potential36 with the cut-off distance rc = 21/6σ and the unit of energy ε = 1.0kbT. σ

σ

U ( r ) = 4ε [( )12 − ( ) 6 ] ( r ≤ r c ) r r

(6)

The attractive interactions between central beads (CT) are modeled via Lennard-Jones (LJ) potential and the cut-off distance is rc = 2.5σ.

σ

σ

U LJ = 4ε ′[( )12 − ( )6 ] ( r ≤ r c ) r r

(7)

where ε' = 10ε =10kbT. The MD simulation is performed under the constant number of particles, volume and temperature (NVT) ensemble condition and the temperature is controlled by a dissipative particle dynamics (DPD) thermostat37 in a three-dimensional Lx × Ly × Lz lattice. 32 NPs coated with DNA chains are placed in a periodic simulation box by applying periodic boundary conditions, thus simulating a bulk system effectively. The total number of DNA-NPs (N) is selected to be compatible with the number of NPs, necessarily creating the given number of unit cells (2×2×2).20 For simplicity, all beads are assumed to have equal mass, m = 1. Each simulation runs 1.0×107 ~ 4.0×107 time steps and a time step is ∆t = 0.005 τ ( τ = (mσ 2 / ε )1/ 2 ).


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ASSOCIATED CONTENT Supporting information Additional figures of Radial distribution functions g(r), Static structure factors S(q) and dynamic process for DNA-NP crystallization under different conditions. These materials are available free of charge via the Internet at http://pubs.acs.org.

AUTHOR INFORMATION Corresponding author *E-mail: [email protected]; Notes The authors declare no competing financial interest.

ACKNOWLEDGMENTS We thank Prof. Jinglei Hu and Prof. Liangshun Zhang for useful discussion. This work was financially supported by the National Natural Science Foundation of China (grant nos. 21474051, 21074053 and 51133002), and Program for Changjiang Scholars and Innovative Research Team in University (PCSIRT). The numerical calculations in this paper have been done on the IBM Blade cluster system in the High Performance Computing Center (HPCC) of Nanjing University.

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For Table of Contents use only The Dynamic Properties of DNA-Programmable Nanoparticle Crystallization Qiuyan Yu, Xuena Zhang, Yi Hu, Zhihao Zhang, Rong Wang*


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