Ordered, Random, Monotonic and Non-Monotonic Digital Nanodot Gradients

Cell navigation is directed by inhomogeneous distributions of extracellular cues. It is well known that noise plays a key role in biology and is present in naturally occurring gradients at the micro- and nanoscale, yet it has not been studied with gradients in vitro. Here, we introduce novel algorithms to produce ordered and random gradients of discrete nanodots – called digital nanodot gradients (DNGs) – according to monotonic and non-monotonic density functions. The algorithms generate continuous DNGs, with dot spacing changing in two dimensions along the gradient direction according to arbitrary mathematical functions, with densities ranging from 0.02% to 44.44%. The random gradient algorithm compensates for random nanodot overlap, and the randomness and spatial homogeneity of the DNGs were confirmed with Ripley's K function. An array of 100 DNGs, each 400×400 µm2, comprising a total of 57 million 200×200 nm2 dots was designed and patterned into silicon using electron-beam lithography, then patterned as fluorescently labeled IgGs on glass using lift-off nanocontact printing. DNGs will facilitate the study of the effects of noise and randomness at the micro- and nanoscales on cell migration and growth.


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Gradients are fundamental to many phenomena of biology, from directing axonal 2! navigation during neural development to the differentiation of stem cells in response to ! 4 limitations, we previously developed digital nanodot gradients (DNGs), where the

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spacing between nanodots (200 nm in diameter) was changed in two dimensions to 28! produce a dynamic range exceeding 3 OM. These designs were implemented using a 29! low-cost, lift-off nanocontact printing method to pattern substrate-bound gradients of

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proteins and peptides. We employed these patterns for adhesion and migration studies 31! of C2C12 myoblasts on RGD peptide and netrin-1 gradients, respectively [14].

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Noise is ubiquitous in biology [15], and modern patterning technologies can be exploited

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to introduce defined amounts of noise and randomness into otherwise regular patterns.

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The effect of randomness was evaluated in ordered arrays and repetitive patterns with a 43! constant average density. In one study, disorder was introduced in arrays of 120 nm 44! dots spaced 300 nm apart by randomly displacing the dot by up to 50% of the spacing to 45! avoid overlap [16]. While this approach introduced a controlled amount of noise by 46! restricting the maximum displacement, it was not random since dots were each 47! contained within the original grid. Nonetheless, cellular adhesion and stem cell 48! differentiation of osteoblasts were markedly altered as a result of increasing disorder. In

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another study, whole proteome analysis of cells grown on the same disordered patterns 50! resulted in differential expression of certain proteins in the extracellular signal-regulated 51! kinase (ERK1/2) pathway [17]. Similarly, controlled amounts of topographical noise on 52! ! 5 nanogratings of 500 nm ridges and grooves has shown to effect PC12 neuronal cell 53! alignment to the gratings, focal adhesion maturation and directionality [18].

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Randomness and noise are also highly relevant to directed cell migration. The

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stochasticity of chemo-and haptotaxis has been well studied, and is apparent from the 57! random-walk like traces of migrating cells [19]. It is well understood that biological 58! gradients, which appear continuous, are in fact quantized since they are comprised of

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individual molecules adsorbed to a surface. The distribution of these molecules is not 60! deterministic, but stochastic at the nanoscale. The engagement of receptors from 61! migrating cells with these guidance cues has been modeled within a stochastic 62! framework [19,20]. Random

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The unit cell is largest at low densities and decreases at higher densities, matching the The probability that this point will not be covered by N nanodots simultaneously can then 306! be found (Eq. 5).

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To determine the total area of the box covered by nanodots (A cov ), the probability that a 311! point will be covered is integrated over the area of the box (Eq. 6).

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We compared the programmed density with the measured density of a monotonic 337! exponential curve with decay constant k (Eq. 9) using both the ordered and random 338! gradient generation approaches. Eq. 9 is normalized to span the dynamic range over the

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length of the gradient. The random gradients produced with this algorithm were found to 340! follow the programmed curve with high fidelity (R 2 = 0.99986), and accurately match the 341! measured density of ordered gradients produced with the same input function (Fig. 3).

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The measured density follows the programmed exponential curve with high fidelity for

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Deviations from πs 2 indicate regions of clustering or dispersion [31]. Using the

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coordinates of nanodots from the DNGs, K(s) was found to lie within a 95% confidence 365! interval obtained from 10 simulations of randomly distributed coordinates at the same

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density. This suggests the nanodots are spatially homogenous and randomly distributed 367! compared to ordered gradients, which lie outside the confidence interval (Fig. 4).

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We have shown that linear and exponential gradients can be produced using either the 379! ordered or random algorithms. These curves are monotonic, meaning they only ever 380! increase or decrease. Given the approaches outlined here, more complex gradients can 381! ! 20 be easily generated from any input density function, specifically non-monotonous curves.

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To study the ability of cells to recognize an average gradient in a non-monotonous 383! environment, we propose linear and exponential gradients that are superposed with

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The flexibility of the gradient algorithms and the fabrication method discussed below was 408! leveraged by producing an array of 100 distinct gradients within a 35 mm 2 area (Fig. 6,

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The hundred-gradient array was etched 100 nm deep into a Si wafer by e-beam 2! lithography (Fig. 7). The integrity of individual dots for ordered and random gradients

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The pattern overlay in Fig. 9 was characterized by thresholding the printed image with 34! boundaries of 31 and 255 in ImageJ and comparing it to the bitmap image (Fig. S3)