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A broad range of membrane proteins display anomalous diffusion on the cell surface. Different methods provide evidence for obstructed subdiffusion and diffusion on a fractal space, but the underlying structure inducing anomalous diffusion has never been visualized because of experimental challenges. We addressed this problem by imaging the cortical actin at high resolution while simultaneously tracking individual membrane proteins in live mammalian cells. Our data confirm that actin introduces barriers leading to compartmentalization of the plasma membrane and that membrane proteins are transiently confined within actin fences. Furthermore, superresolution imaging shows that the cortical actin is organized into a self-similar meshwork. These results present a hierarchical nanoscale picture of the plasma membrane.
Using superresolution imaging of human embryonic kidney (HEK 293) cells, we simultaneously study the random motion of ion channels on the cell membrane and the dynamic actin meshwork. Actin acts as a transient barrier to protein motion, and it partitions the membrane into compartments of various sizes. Close to the cell membrane, the actin meshwork also forms a statistically self-similar fractal that exhibits similar structural properties at many sizes. This structure enables the compartmentalization of the membrane across multiple length scales.
The concept of fractals introduced by Mandelbrot was initially applied to the geometric description of irregular objects but quickly spread to the field of dynamics where it joined chaos. Initially described by Lorenz in the context of meteorology, chaos extended his influence from fluid dynamics to biology and biochemistry.
Analysis of time series of biological variables with techniques inspired by fractal and chaos theories are providing a more clear understanding of the relationship between scaling in space and time exhibited by cells and...
Through an elegant geometrical interpretation, the multi-fractal analysis quantifies the spatial and temporal irregularities of the structural and dynamical formation of complex networks. Despite its effectiveness in unweighted networks, the multi-fractal geometry of weighted complex networks, the role of interaction intensity, the influence of the embedding metric spaces and the design of reliable estimation algorithms remain open challenges. To address these challenges, we present a set of reliable multi-fractal estimation algorithms for quantifying the structural complexity and heterogeneity of weighted complex networks. Our methodology uncovers that (i) the weights of complex networks and their underlying metric spaces play a key role in dictating the existence of multi-fractal scaling and (ii) the multi-fractal scaling can be localized in both space and scales. In addition, this multi-fractal characterization framework enables the construction of a scaling-based similarity metric and the identification of community structure of human brain connectome. The detected communities are accurately aligned with the biological brain connectivity patterns. This characterization framework has no constraint on the target network and can thus be leveraged as a basis for both structural and dynamic analysis of networks in a wide spectrum of applications.
The uncovered self-similarity in complex networks connects to the important fractal and multi-fractal geometry domain where a family of objects are distinguished based on their self-repeating patterns and invariability under scale-length operations. Such objects are known as fractal objects. A mono-fractal object obeys a perfect self-repeating law at all scales. When embedded in Euclidean metric space and tiled by equally sized boxes at different scales, it becomes apparent that an important property of fractals is the power-law dependence between the mass distribution M(r) (e.g., the number of points in a box) and the scale factor r:
In Eq. (1), D is the fractal dimension and represents a real-valued number in contrast to the embedded space dimension which is always an integer. Fractal dimension is the major tool for describing the fractal geometry and the heterogeneity of irregular geometric objects that the dimension of its embedded space fails to capture. For instance, in Euclidean geometry, a straight line and a crooked line share the same geometrical dimension but have very distinct properties. Multi-fractals could be seen as an extension to fractals with increased complexity. They are invariant by translation although a distortion factor q needs to be considered to distinguish the details of different regions of the objects as a consequence of inhomogenous mass distribution. Intuitively, multi-fractals are not perfect self-repetitions but rich in localized variations of detailed geometric configurations. Consequently, a single fractal dimension is not sufficient to characterize the irregularity of the geometric shapes as the scaling factor measured across the object could be different. As a result, multi-fractal analysis (MFA, see Methods for details) is proposed to capture the localized and heterogenous self-similarity by learning a generalized fractal dimension D(q) under different distortion factors q.
To overcome these issues, we first analytically study the multi-fractal structure of the Sierpinski fractal network family to set up the theoretical ground for evaluation and comparative analysis of our proposed algorithms (See Supplementary Material Section 1). We find that the multi-fractality identified by SBw can be just the side effect of the limited size of the network considered. The analytical discussion of multi-fractality in Sierpinksi family \(\mathcalS\) provides the theoretical basis on which not only we can quantitatively reason about the existence of multi-fractality/fractality from an asymptotic perspective that numerical approaches will surely fail to offer, but also we can shed some light on the design of numerical algorithms for reliable estimation of multi-fractal spectrum of the complex networks.
Based on both our theoretical findings and numerical experimental results, we propose the finite box-covering algorithm for weighted network (FBCw) and the finite sandbox algorithm for weighted network (FSBw) with improved performance. We compare the accuracy of the estimates obtained by FBCw and FSBw with our analytical results of Sierpinski fractal network as well as with those obtained by BCANw and SBw. The comparison shows that the proposed algorithms are not only able to give reliable numerical estimates of fractality with insensitivity to the distribution of link weights, but also are capable of detecting the fractal scaling dependence when it holds within a finite range of scales (i.e., scale-localized).
More importantly, we apply the proposed algorithms to learn the multi-fractal structure of a set of real world weighted networks. We show the link weights play a definitive role in governing the existence of fractality in the network. The investigated weighted networks exhibit a phase transition from self-similar networks to small-world networks when converted to binary networks. Furthermore, we demonstrate that the fractal and multi-fractal scaling behaviors can be spatially localized and co-exist in the same network. Learning from our observations on the locality of the scaling behavior of real world weighted networks, we finally propose a network characterization framework based on the localized scaling feature space learned by the construction of scaling feature vectors for each node in the network. The proposed characterization is general and not limited to complex networks that are fractal or multi-fractal. It can be easily interfaced with subsequent analytical tools (e.g., machine learning algorithms) to unveil the intrinsic properties of the weighted complex networks. To illustrate the benefits of our methodology, we apply our algorithms to the network community detection in the human brain connectome. The identified communities are consistent with our biological knowledge.
The following discussion is organized in three parts. In the first part, we present the estimation error of previous numerical algorithms. In the second part of discussion, we compare the performance of BCANw, SBw with the proposed FBCw and FSBw. Finally, we present the multi-fractal analysis on a set of weighted real world complex network and propose a localized scaling based approach for the characterization of the weighted complex networks. We provide an illustrative application example in network community detection to show its effectiveness.
The link weights distribution of complex networks largely depend on the growth rule and weights allocation process. For instance, the distribution of link weights of Sierpinski family is shaped by the scaling factor s and growth rule b. Interestingly, for small scaling factor s, we prove G k approaches a monofractal that has no explicit dependence on weights distribution (See the proof in Supplementary Material Section 1). Yet this is valid only for a complex network that has infinite resolution in the sense that box/sandbox can grow by infinitely small steps (but not continuously) in a network of unbounded range of scales. In most of cases, this does not hold for complex networks and perfect fractals of limited size (e.g., Sierpinski network of limited size). Therefore, when it comes to numerical calculation of the limit in Eqs (13) and (17) using linear regression which is shared by both box-covering and sandbox methods, we are able to show that the box/sandbox should grow in a regulated way that is compatible with link weights distribution such that the stairway effect is minimized. 2ff7e9595c
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