Kinetic Signature of Fractal-like Filament Networks Formed by Orientational Linear Epitaxy

Wonmuk Hwang and Esma Eryilmaz

Phys. Rev. Lett. 113, 025502 – Published 8 July 2014

Abstract

We study a broad class of epitaxial assembly of filament networks on lattice surfaces. Over time, a scale-free behavior emerges with a 2.5–3 power-law exponent in filament length distribution. Partitioning between the power-law and exponential behaviors in a network can be used to find the stage and kinetic parameters of the assembly process. To analyze real-world networks, we develop a computer program that measures the network architecture in experimental images. Application to triaxial networks of collagen fibrils shows quantitative agreement with our model. Our unifying approach can be used for characterizing and controlling the network formation that is observed across biological and nonbiological systems.

Received 22 February 2014

DOI:https://doi.org/10.1103/PhysRevLett.113.025502

Authors & Affiliations

Wonmuk Hwang^*

Departments of Biomedical Engineering and Materials Science & Engineering, Texas A&M University, College Station, Texas 77845, USA and School of Computational Sciences, Korea Institute for Advanced Study, Seoul 130-722, Korea

Esma Eryilmaz

Department of Physics, Texas A&M University, College Station, Texas 77845, USA

^*hwm@tamu.edu

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Issue

Vol. 113, Iss. 2 — 11 July 2014

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Images

Figure 1
Four types of OLE networks that have been experimentally observed: (a) Triaxial [8, 9, 10, 11, 12, 13, 14, 15, 18, 19, 21, 27], (b) biaxial [5, 6], (c) rectangular [16, 17, 28], and (d) isotropic [22, 23, 24, 25, 26]. (a),(b) Triaxial and biaxial form on hexagonal lattices, notably on mica. (c) Rectangular forms on cubic lattices such as KCl. (d) Istotropic forms in the absence of orientational bias. See also Supplemental Material, videos [29].
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Figure 2
Length distribution of OLE networks. Measurements were made over 100–500 runs in each case. (a) Time evolution of $P_{L}$ for the triaxial network ( $g = 1$ ). For $t ≲ 1200$ (early nucleation regime), $P_{L}$ follows the MFT solution, Eq. (2). For example, the front of $P_{L}$ moves with speed $g = 1$ , so that at $t = 400$ , it is located at $\log (400) = 6.0$ . As filaments encounter ( $t ≳ 2000$ ), $P_{L}$ becomes “frozen,” starting from longer filaments and, subsequently, to shorter ones (dashed arrows). (b) Near-steady state profiles at $t = 4 \times 10^{5}$ . Dependence of $P_{L}$ on $g$ (dashed arrows) is analogous to the $t$ dependence in (a). Inset: log-normal plot showing exponential distribution of longer filaments. (c) Local slopes of the curves in (b). For the intermediate range of $L$ where the power law holds, $b = 2.5 - 3$ . The power law behavior becomes more prominent with greater $g$ as this is similar to a longer-time case. Similar behaviors are observed for the networks in Figs. 1 (Supplemental Material, Fig. S.2 [29]). (d) Log-log plot of $k$ vs $g$ , indicating $k \sim g^{- 1 / 3}$ across different networks. Numbers in legends are slopes of linear fits. The prefactor in this relation ( $k$ at $g = 1$ ) correlates with the propensity for encounter between filaments, which is the greatest for the isotropic network, followed by triaxial, and the smallest for biaxial and rectangular networks that behave very similarly.
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Figure 3
Analysis of a triaxial collagen network. (a) AFM scan. (b) Corresponding in silico network extracted via CAFE. Filaments are individually recognized and colored randomly. (c) Filament angle distribution. Angle $θ$ is measured relative to the horizontal direction in (a). The asymmetry in $P_{θ}$ is visible in (a). (d) Length distribution. Solid circle: From the network in (b). Dashed and solid lines: stochastic simulation with $g = 0.25$ and 0.75, respectively (each averaged over 50 runs). (e) An example simulation with $g = 0.50$ , which looks qualitatively similar to the AFM image in (a).
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