Biomech Model Mechanobiol DOI 10.1007/s10237-015-0661-5

ORIGINAL PAPER

Influence of disordered packing pattern on elastic modulus of single-stranded DNA film on substrate W. L. Meng · N. H. Zhang · H. S. Tang · Z. Q. Tan

Received: 2 December 2014 / Accepted: 13 February 2015 © Springer-Verlag Berlin Heidelberg 2015

Abstract Determining mechanical properties of singlestranded DNA film grafted on gold surface is critical for analysis and design of DNA-microcantilever biosensors. However, it remains an open issue to quantify the relations among the disordered packing patterns of DNA chains, the mechanical properties of DNA film and the resultant biodetection signals. In this paper, first, the bending experiment of microcantilever is carried out to provide the basic data for a refined multi-scale model of microcantilever deflection induced by ssDNA immobilization. In the model, the complicated interactions in DNA film (consisting of DNA, water molecules and salt ions) are simplified as effective interactions among coarse-grained soft cylinders, which can reveal the varieties of DNA structure in the circumstances of different lengths and salt concentrations; Ohshima’s distribution of net charge density is employed to incorporate compositional variations of salt ions along the thickness direction

into the Strey’s mesoscopic empirical potential on molecular interactions in DNA solutions, and the related model parameters for ssDNA film on substrate are obtained from the curve fitting with our microcantilever bending experiment. Second, the effect of nanoscopic distribution of DNA chains on elastic modulus of ssDNA film is studied by a thought experiment of uniaxial compression, and the disordered patterns of DNA chains are generated by Monte Carlo method. Simulation results point out that nanoscale ssDNA film shows size effect, gradient and diversity in elastic modulus and can achieve maximum stiffness by preferring a disordered and energetically favorable packing pattern collectively induced by electrostatic force, hydration force and configurational entropy. Keywords DNA film · Microcantilever biosensor · Elastic moduli · Multi-scale modeling · Monte Carlo method

1 Introduction Electronic supplementary material The online version of this article (doi:10.1007/s10237-015-0661-5) contains supplementary material, which is available to authorized users. W. L. Meng · N. H. Zhang (B)· H. S. Tang · Z. Q. Tan Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, China e-mail: [email protected] W. L. Meng (B) School of Engineering, Brown University, Providence, RI 02912, USA e-mail: [email protected] N. H. Zhang Department of Mechanics, College of Sciences, Shanghai University, Shanghai 200444, China

Mechanical nature of many fundamental biological processes justifies the development of nanomechanical tools to sense and actuate on biological systems (Tamayo et al. 2013). DNA-microcantilever system, one of micromechanical biosensors (Zhang and Shan 2011; Zhang et al. 2011b), can exhibit extremely low mechanical compliances translating biomolecular recognition events into measurable displacements. It is highlighted by recent experimental studies that biosensor signals will change at different experimental conditions, such as DNA concentration (Fritz et al. 2000; McKendry et al. 2002; Zhang et al. 2006), length and sequence, grafting and hybridization density (Wu et al. 2001; Stachowiak et al. 2006; Hansen and Thundat 2005), salt concentrations (McKendry et al. 2002; Wu et al. 2001; Stachowiak

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et al. 2006), humidity (Mertens et al. 2008), surface charge of gold layer (Godin et al. 2010), surface preparation technology (Arroyo-Hernandez et al. 2009), temperature variation (Yue et al. 2004) and the choice of substrate materials (Zhang et al. 2008). For DNA-microcantilever biosensors, identifying the relationship between the sensitive and specific signals and variations of environmental factors calls the demand of quantitative models (Zhang et al. 2011b; Yi and Duan 2009). In the multi-scale modeling processes, the core is to characterize the thermal/electrical/mechanical properties of DNA films including immobilized DNA chains, bounded water molecules and free salt ions on substrates. However, due to inherent varieties in structure and stiffness of DNA film and the limitation of the existing experiment methods, it brings great challenges to identify the local microstructure and constituent variations of nanoscale DNA films on substrates and their collective behaviors, especially in the liquid. Starting from macroscopic viewpoints, some quantitative models have been presented to describe the behaviors of DNA-microcantilever system (Begley et al. 2005; Huang et al. 2010; Liu et al. 2003; Zhang et al. 2007). However, these macroscopic continuum models might lose some microscale details and have limited benefits for clarifying key problems. With the help of Frank constants representing splay, twist and bending deformation of nematic liquid crystals, Strey et al. (1997) presented a mesoscopic empirical potential in which a DNA cylinder model was adopted to simplify the complicated interactions in DNA solution (without substrate). Although the interaction energy between DNA chains and substrate was not considered in the Strey’s potential, Hagan et al. (2002) introduced this potential into DNAmicrocantilever systems and presented favorable deflection predictions at given experimental conditions. For doublestranded DNA (dsDNA) film on substrate, Zhang et al. employed an alternative two-variable method (Zhang and Xing 2006; Zhang and Chen 2008; Zhang and Shan 2008; Zhang et al. 2011a) to characterize deformation field of multilayered structure and provided new empirical parameters (Zhang et al. 2010a, b). Weak interactions in single-stranded DNA (ssDNA) film due to counterion osmotic effects were also considered to model the ssDNA-microcantilever system (Zhang and Shan 2011). However, some microscale details, such as compositional variations of free salt ions along thickness direction and disordered patterns of DNA chains on gold layer, are usually neglected in these mentioned mesoscopic models. On the one hand, different kinds of ions, whatever free or fixed, are prone to migrate and reorganized to maintain a proper electrical balance and thus result in an inhomogeneous structure (Ohshima and Makino 1996; Zhulina and Borisov 1997; Jiang et al. 2007). The microstructure of salt solutions on substrate has been studied by the assumption of Boltzmann dis-

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tribution (Ohshima and Makino 1996; Zhulina and Borisov 1997), Monte Carlo method or density functional theory (Jiang et al. 2007), especially for those with several nanometer thicknesses. In fact, Pincus brush (Netz and Andelman 2003; Weir and Parnell 2011) of charged polymers has also been well studied by the scaling theory, in which the inhomogeneous interface property was widely accepted (de Gennes and Pincus 1983). For DNA film, this kind of inhomogeneous microstructure is more complicated because it is affected by hydration effect (Mertens et al. 2008) and steric hindrance effect (McKendry et al. 2002). Illuminated by the study about inhomogeneity (Sushko et al. 2008) of different ions and polymers calculated from DFT simulation, the compositional variation of salt ions was considered in a multi-scale dsDNAmicrocantilever model (Zhang et al. 2013). On the other hand, the packing patterns of DNA chains play an important role on mechanical properties (Pellegrino et al. 2007). The assumption of hexagonal packing pattern is often employed in the previous study of DNA film on gold (Dai et al. 2009). However, this sort of homogenous assumption is quite different from the disordered patterns observed by AFM (Casero et al. 2003; Erts et al. 2003). Although it is considered that the homogenous pattern is suitable for extremely high- density self-assembly packing cases (Yao et al. 2007) or typical nanografted patterns (Case et al. 2003), recent experiments reveal that even highly self-assembled packing DNA monolayer is not uniform (Kosaka et al. 2013) and the packing order become less defined upon hydration compared with its dry state (Milano and Bernhard 1999). Thermodynamically, motion of thiolated ssDNA molecules (Liu et al. 2002) emerges to be a significant factor. As a result, we need a sound and reasonable model to include the nonuniform and disordered microstructure at short-length scales (Hagan et al. 2002). Some stochastic simulations indicate a strong dependence of microcantilever deflection on disordered patterns of surface adsorbed probe molecules (Hagan et al. 2002; Zhang et al. 2010a, b; Zhao et al. 2012). How about the influence of disordered packing pattern on the mechanical properties of DNA film? Tang et al. (2014) introduced a Gaussian packing pattern into Strey’s empirical potential and provided an analytical prediction for elastic modulus of dsDNA film with an ordered hexagonal packing pattern; however, the analytical method is invalid for arbitrary packing patterns. Other recent theoretical works (Medalion et al. 2014) also pointed out interesting crosshybridization behaviors of complementary strands to grafted ssDNAs, which is not considered in the following analysis. The existing models for ssDNA film might not be valid for disordered packing patterns based on the two abovementioned aspects, so a comprehensive model incorporating microscale structural properties is needed. More specifically, first, this study measures the difference in bending deflections before and after ssDNA immobilization by com-

Influence of disordered packing pattern on elastic modulus

mercial cantilever arrays from Concentris GmbH. Second, an enriched multi-scale model is formulated for deflection of DNA-microcantilever. Typically, on one side, we use a soft cylinder model to coarse-grain the flexible structures of ssDNA film on substrates. On the other side, interactions between DNA cylinders are modeled by enriching Strey’s mesoscopic empirical potential (Strey et al. 1997) with the inclusion of Ohshima’s inhomogeneous distribution of net charge density (Ohshima and Makino 1996), and the related parameters in the present multi-scale model are determined by curve fitting to our experimental results. Third, we employ Monte Carlo method to mimic the scattered patterns of DNA cylinders. Three ensembles are compared, i.e., average spacing, Gaussian-perturbed random selection and Gaussianperturbed energy minimization ensemble. By using a thought experiment of uniaxial compression, the effect of packing patterns on elastic modulus of ssDNA film is investigated, and the simulation results are compared with the related AFM indentation and microcantilever bending experiments.

2 Methods 2.1 Bending experiment of DNA-microcantilever We use commercially available cantilever arrays (CLA-750010-08, Concentris GmbH, Switzerland) to measure bending deflections of microcantilevers due to ssDNA immobilizations. The arrays have eight identical microcantilevers in parallel, each of which contains Si, Cr and Au layers from bottom to top as a substrate for immobilization of ssDNA molecules. The related mechanical and geometrical parameters of each layer are listed in Table 1, in which E Au , E Cr and E Si are individual elastic moduli, h Au , h Cr , h Si are individual thicknesses, l and b are length and width, respectively. Measure-

Table 1 Parameters of microcantilevers E (GPa)

h (nm)

Au

81

20

Cr

116

3

Si

160

1000

Table 2 Reagent information

l (µm)

b (µm)

750

100

ments are taken on the commercial Cantisens sensor platform (CSR-801, Concentris GmbH, Switzerland) equipped with a 5 µL measurement cell, an automated liquid-handling system and an integrated temperature controller. Real-time cantilever deflections at the free ends are monitored and recorded during measurement. All DNA samples are synthesized in Sangon Biotech Co., Ltd. The sequences of probe ssDNA molecules are listed in Table 2, in which N is fragment length of a DNA molecule. Note that a thiol modification with a 5 HS(CH2 )6 linker enables covalent binding for DNA molecules on gold-coated cantilever surfaces. To start with, we insert a microcantilever array into the measurement cell and clean the cell by continuous injection of 0.1 M sodium buffer at pH = 7.0. Second, after the cell is filled in, we monitor real-time cantilever tip deflection and wait for 5 min until all the cantilevers are stabilized from fluctuation and turbulence induced by injected solution. Third, we introduce 0.5 µM probe ssDNA solution into the cell for immobilization. The solution is carefully conducted with a tiny constant flow velocity (0.42 µL/s) to reduce fluctuated noisy signals. The difference of steadystate microcantilever deflections before and after the injection of ssDNA solutions are recorded to obtain the deflection change resulting from DNA immobilization.

2.2 Deflection prediction for microcantilever by a multi-scale model As shown in Fig. 1, the DNA-microcantilever consists of a three-layered substrate and a ssDNA film. The interface between the DNA film and the Au layer is taken as the coordinate axis x. Following our previous methods for dsDNAmicrocantilever (Zhang et al. 2010, 2013), Zhang’s twovariable method (Zhang and Xing 2006; Zhang and Chen 2008; Zhang and Shan 2008; Zhang et al. 2011a) is used to precisely characterize the strain field of laminated substrate; as for the quantification of free energy of DNA film, the Strey’s potential (Strey et al. 1997) will be extended to ssDNA solution on substrate. The cylinder model is often used in dsDNA solutions (Strey et al. 1997; Hagan et al. 2002; Zhang and Shan 2008; Zhang et al. 2010); some researchers also showed its applicability in ssDNA film with strong interactions at high grafting density (Hagan et al. 2002; Mertens

N

Base sequence 5 –3

10

HS(CH2 )6 -AGCTAGCTAG

20

HS(CH2 )6 -AGCTAGCTAGCTAGCTAGCT

30

HS(CH2 )6 -AGCTAGCTAGCTAGCTAGCTAGCTAGCTAG

40

HS(CH2 )6 -AGCTAGCTAGCTAGCTAGCTAGCTAGCTAGCTAGCTAGCT

50

HS(CH2 )6 -AGCTAGCTAGCTAGCTAGCTAGCTAGCTAGCTAGCTAGCTAGCTAGCTAG

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two parallel DNA cylinders is written as (Zhang et al. 2013) Wb = We + Wh + Wc ,

(5)

in which We , Wh and Wc are, respectively, electrostatic energy, hydration energy and configurational entropy, and We (z, d) = a0 W¯ e (z, λD )

Fig. 1 Schematic shows a DNA-microcantilever and its coordinate system

et al. 2008; Ortega et al. 2014). However, the rigid cylinder model may lose some microscale details on substrate, so here we will use a soft cylinder model to reflect the effect of salt concentration and fragment length on flexible structure of ssDNA film (i.e., grafting density). In order to relax the assumption at extreme dense grafting density (Zhang et al. 2013; Tinland et al. 1997), the relation between the initial interaxial distance d0 and the gyration radius Rg of ssDNA was modified as follows d0 = k0 + 2Rg , Rg = (k1 Nlpss /3)1/2 ,

(1)

in which k0 = 2.64 nm and k1 = 0.072 nm/nt (Pellegrino et al. 2007), lpss is the persistence length of ssDNA and lpss = (1.481 + 0.03797/I ) nm (Ambia-Garrido et al. 2010), in which I is buffer salt concentration. The grafting density in the ordered hexagonal packing pattern goes to √ η = 2/( 3d02 ).

(2)

The thickness of ssDNA film could be approximately taken as follows (Halperin et al. 2005) h DNA = N a,

(3)

where a = 0.22 nm is nucleotide length directly obtained from STM experiment on gold substrate (Zhao et al. 2012). For DNA film on substrate, the inhomogeneous distribution of free salt ions could be captured by Poisson–Boltzmann equation. Different from the previous form of distribution function for dsDNA solution on substrate (Zhang et al. 2013), based on Ohshima’s distribution of net charge (including ion charges and fixed DNA charges) density (Ohshima and Makino 1996; Zhang et al. 2013), here we will introduce an alternative and more simple form as follows ρnet (z) ∝ − cosh(z/λD ), 0 < z < H.

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W¯ e (z, λD ) = cosh2 (z/λD ), exp(−d/λH ) Wh = b0 √ , d/λH  2 1 ∂(We + Wh ) −1/4 4 ∂ (We + Wh ) − Wc = c0 kB T kc , ∂d 2 d ∂d √ where λD = 0.308 nm/ I is Debye screening length, I is buffer salt concentration, λH = 0.29 nm is correlation length of water (Strey et al. 1997), a0 , b0 and c0 are undetermined parameters for DNA interactions. And kc = kB T lp is the bending stiffness of single-molecule ssDNA, kB the Boltzmann constant, T the temperature and lp the persistence length of ssDNA. By using the perpendicular chain assumption in the liquid (Hagan et al. 2002; Zhang et al. 2010a, b; Zhang et al. 2013), the interaxial distance, d, between parallel DNA cylinders after microcantilever bending is given as d = (1 + ε0 + κz)d0 , in which ε0 is the normal strain at the axis, κ is the curvature of the neutral axis. The total free energy of DNA-microcantilever WT consists  of the free energy of ssDNA film, WB = VB Wb dVB , and the  mechanical energy of the substrate, WM = VM Wm dVM , in which Wm is strain energy density, and VB and VM are volumes of DNA film and nonbiolayers, respectively. By the principle of minimum energy, δWT = δWB + δWM = 0, the tip deflection of the microcantilever could be provided as (Zhang and Shan 2008; Zhang et al. 2010a, b) w = κl 2 /2, in which

(6)

 B2 B1 − 2C2 C1  , A = W (ε0 ,κz)=(0,0) , 1 B 2 4 A2 C2 − B2 ∂ WB  ∂ WB  B1 = ,  (ε0 ,κz)=(0,0) , C1 = h DNA ∂ε0 ∂(κz) (ε0 ,κz)=(0,0)   bl  E Si h Si 3h 2Au + 6h Au h Cr + 3h 2Cr + h 2Si A2 = 24   κ=

+E Cr h Cr 3h 2Au + h 2Cr − 6h Au h Si + 3h 2Si   +E Ai h Au h 2Au + 3(h Cr + h Si )2 ,

(4)

Using Eq. (4) and Strey’s potential (Strey et al. 1997), the refined formula for interaction energy per unit length between

exp(−d/λD ) , √ d/λD

B2 =

bl [E Cr h Cr (−h Au + h Si ) + E Au h Au (h Cr + h Si ) 2 −E Si (h Au + h Cr )h Si ] ,

Influence of disordered packing pattern on elastic modulus

C2 =

bl (E Au h Au + E Cr h Cr + E Si h Si ). 2

2.3 Packing pattern simulation of DNA chain by Monte Carlo method Once the aforementioned parameters a0 , b0 and c0 in the present empirical formula (i.e., Eq. 5) are determined, the free energy of the ssDNA film on substrate could be derived. Note that the free energy varies with different patterns of DNA cylinders, which can be captured by the Monte Carlo method (Gupta et al. 2010). For a given grafting density, three different ensembles of immobilized DNA molecules are constructed in order to evaluate the influence of DNA packing patterns on mechanical properties of DNA film on substrate. The three ensembles are, respectively, average spacing, Gaussian-perturbed random selection and energy minimization ensembles (Zhao et al. 2012). All the three ensembles are required to satisfy periodic boundary conditions in simulation. In the first ensemble (i.e., average spacing), the DNA cylinders are self-assembled in a hexagonal close-packed uniform arrangement. Although it is easy to construct the ordered pattern, it in fact neglects some details found in the delicate experiments (Zhao et al. 2012). In the second ensemble (i.e., Gaussian-perturbed random selection), DNA cylinders are distributed on substrate by imposing random position coordinates on the 2D plane of a simulation cell. The free energy in the simulation cell is calculated by the sum of interaction energies for all possible DNA pairs within a cutoff interaxial distance. The Gaussian-perturbed random selection ensemble might be the simplest way to simulate a high degree of disorder (Case et al. 2003), but the final distribution patterns are determined only by statistical principles, unrelated to DNA interactions. It makes sense to involve free energy in the selection process of distribution pattern. In the third ensemble (i.e., Gaussian-perturbed energy minimization), Monte Carlo method is used. First, the DNA cylinders are immobilized on substrate with a random posi(0,l ) (0,l ) tion coordinate [Ux1 cell , U y1 cell ] as an initial distribution C1 like Gaussian-perturbed random does, in which U(0,lcell ) represents a random number in uniform distribution between zero and the side length of the cell. As for the choice of DNA number in a simulation cell, we show the detailed process in Online Resource 1. The free energy of the DNA film at this initial distribution W1 is calculated. Second, one DNA cylinder in the simulation cell is randomly selected and imposed on a displacement (−π/2,π/2) (−π/2,π/2) (0,d) (0,d) ), N r sin(U θ ] as a trial of [N r cos(U θ (0,d) distribution Ct , in which N r represents a random number in normal distribution with mean 0 and standard devi¯ The free energy of the DNA film at this trail ation d.

distribution Wt is calculated. Third, we define probability R(C1 , Ct ) = exp(−(Wt − W1 )/(kB T )) and compare it with (0,l ) a uniform random number P = U p cell . If R > P, the trail distribution is accepted as C2 = Ct with free energy W2 = Wt . Otherwise, we reject the trail and make C2 = C1 . The second and third steps are repeated N times, where N is a sufficiently large number to make sure to obtain an energetically favorable state. 2.4 Elastic analysis of DNA film by a thought experiment As for the mechanical properties of ssDNA film with three ensembles, we could apply the thought experiment method to study the elastic modulus once the free energy of the DNA film is obtained. Assume that the DNA film (without substrate) is strain-controlled and compressed along x direction. During the deformation, the change in free energy is given as   W¯ B = W¯ B ε=1+ε − W¯ B ε=1 . 0

(7)

On the other hand, in the viewpoint of continuum mechanics, the change in strain energy is written as WS =

σ dε0 ,

(8)

VB

in which σ is effective stress. No matter what kind of description methods we use, the energy stored in the DNA film should have the same value, i.e., W¯ B = WS . With the equality, one could obtain the effective stress. Under small deformation assumption, the elastic modulus is determined from the initial slope of stress– strain curve (Zhang et al. 2010a, b; Zhang et al. 2013), which is written as E(z) =

∂ 2 ( W¯ B /(lb)) . ∂ε02

(9)

3 Results and discussion 3.1 Deflections In our microcantilever bending experiment, the related material and geometrical parameters for nonbiolayers are given in Tables 1 and 2, and the other solution conditions are given as kB = 1.38 × 10−23 J/K, T = 298 K. By curve fitting with the experimental data of microcantilever deflection w versus DNA fragment length N, the empirical parameters in Eq. (5) could be determined as a0 = 2.56 × 10−11 J/m, b0 = 6.85 × 10−7 J/m, c0 = 0.18. Here, we focus on the static mode, so the steady-state values of tip deflections are

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Fig. 2 Curve fitting of our theoretical model with experiment on microcantilever deflection changes with DNA fragment length

Fig. 3 Elastic modulus versus DNA fragment length obtained from energy minimization, random selection and average spacing ensembles in Monte Carlo simulation, respectively. Exponential fitting curve is shown in dashed lines

used. As shown in Fig. 2, the model curve tracks the experimental data fairly well with correlation coefficient of 0.985, which indicates our theoretical model is suitable for ssDNAmicrocantilever bending systems. 3.2 Elastic modulus 3.2.1 Influence of packing pattern and size effect under different fragment length Figure 3 plots elastic modulus at I = 0.5 M versus DNA fragment length in the three ensembles, i.e., energy minimization, random selection and average spacing. All of the predicted results are 40 nt, when the elastic modulus maintain a relative stable value 15 MPa according to the fitting formula. 3.2.2 Influence of salt concentration and contribution of three microscopic interactions At low salt concentration (I < 0.1 M) in Fig. 4, elastic modulus is very sensitive to salt concentration, varying from 50 to 170 MPa. As for I > 0.1 M, the modulus enters into a saltinsensitive region, remaining at about 170 MPa. A similar phenomenon has been found in microcantilever deflection experiment done by Wu et al. (2001), in which there was a threshold value at 0.1 M between salt-sensitive and saltinsensitive region. As shown in Fig. 5, when salt concentration varies from 0.05 to 0.2 M, the contribution of electrostatic energy on elastic modulus of DNA film drops from 3.3 to 0.3 % due to electrostatic screening. Meanwhile, hydration energy exceeds electrostatic energy and configurational entropy and becomes dominant in DNA film. Note that more salt ions imply higher grafting density and less configurational entropy owing to physical steric crowding (McKendry et al. 2002). Thus,

Influence of disordered packing pattern on elastic modulus

Fig. 4 Elastic modulus for energy minimization ensemble versus salt concentration

Fig. 5 Contribution of electrostatic energy, hydration energy and configurational entropy on elastic modulus for energy minimization ensemble when I = 0.05, 0.1 and 0.2 M, respectively

hydration energy becomes larger and makes the other two interactions almost neglected. This result greatly supports Hagan’s assumption that the most probable cause of strong repulsive interactions is hydration forces in ssDNA molecules with strong interactions (Hagan et al. 2002). Mertens et al. (2008) and Domínguez et al. (2014) suggested that this kind of hydration-induced tension in the channels between the DNA molecules could lead to the development of a labelfree DNA biosensor that can detect single mutations in saltfree DNA films. However, for dsDNA film in salt solution, the situation is more complicated (Zhang et al. 2013). 3.2.3 Modulus gradient and its diversity along the thickness direction Given buffer salt concentration and DNA fragment length, Fig. 6 shows elastic modulus gradient along the thickness direction, which shows a distinctive mechanical property in biointerfaces (Seidi et al. 2011). At the very bottom of DNA film, the local modulus appears to be

Influence of disordered packing pattern on elastic modulus of single-stranded DNA film on substrate.

Determining mechanical properties of single-stranded DNA film grafted on gold surface is critical for analysis and design of DNA-microcantilever biose...
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