Rene Markovič

Network science of biological systems at different scales: A review

Network science is today established as a backbone for description of structure and function of various physical, chemical, biological, technological, and social systems. Here we review recent advances in the study of complex biological systems that were inspired and enabled by methods of network science. First, we present research highlights ranging from determination of the molecular interaction network within a cell to studies of architectural and functional properties of brain networks and biological transportation networks. Second, we focus on synergies between network science and data analysis, which enable us to determine functional connectivity patterns in multicellular systems. Until now, this intermediate scale of biological organization received the least attention from the network perspective. As an example, we review the methodology for the extraction of functional beta cell networks in pancreatic islets of Langerhans by means of advanced imaging techniques. Third, we concentrate on the emerging field of multilayer networks and review the first endeavors and novel perspectives offered by this framework in exploring biological complexity. We conclude by outlining challenges and directions for future research that encompass utilization of the multilayer network formalism in exploring intercellular communication patterns in tissues, and we advocate for network science being one of the key pillars for assessing physiological function of complex biological systems-from organelles to organs-in health and disease.

Progressive glucose stimulation of islet beta cells reveals a transition from segregated to integrated modular functional connectivity patterns

Collective beta cell activity in islets of Langerhans is critical for the supply of insulin within an organism. Even though individual beta cells are intrinsically heterogeneous, the presence of intercellular coupling mechanisms ensures coordinated activity and a well-regulated exocytosis of insulin. In order to get a detailed insight into the functional organization of the syncytium, we applied advanced analytical tools from the realm of complex network theory to uncover the functional connectivity pattern among cells composing the intact islet. The procedure is based on the determination of correlations between long temporal traces obtained from confocal functional multicellular calcium imaging of beta cells stimulated in a stepwise manner with a range of physiological glucose concentrations. Our results revealed that the extracted connectivity networks are sparse for low glucose concentrations, whereas for higher stimulatory levels they become more densely connected. Most importantly, for all ranges of glucose concentration beta cells within the islets form locally clustered functional sub-compartments, thereby indicating that their collective activity profiles exhibit a modular nature. Moreover, we show that the observed non-linear functional relationship between different network metrics and glucose concentration represents a well-balanced setup that parallels physiological insulin release.

The relationship between node degree and dissipation rate in networks of diffusively coupled oscillators and its significance for pancreatic beta cells.

Self-sustained oscillatory dynamics is a motion along a stable limit cycle in the phase space, and it arises in a wide variety of mechanical, electrical, and biological systems. Typically, oscillations are due to a balance between energy dissipation and generation. Their stability depends on the properties of the attractor, in particular, its dissipative characteristics, which in turn determine the flexibility of a given dynamical system. In a network of oscillators, the coupling additionally contributes to the dissipation, and hence affects the robustness of the oscillatory solution. Here, we therefore investigate how a heterogeneous network structure affects the dissipation rate of individual oscillators. First, we show that in a network of diffusively coupled oscillators, the dissipation is a linearly decreasing function of the node degree, and we demonstrate this numerically by calculating the average divergence of coupled Hopf oscillators. Subsequently, we use recordings of intracellular calcium dynamics in pancreatic beta cells in mouse acute tissue slices and the corresponding functional connectivity networks for an experimental verification of the presented theory. We use methods of nonlinear time series analysis to reconstruct the phase space and calculate the sum of Lyapunov exponents. Our analysis reveals a clear tendency of cells with a higher degree, that is, more interconnected cells, having more negative values of divergence, thus confirming our theoretical predictions. We discuss these findings in the context of energetic aspects of signaling in beta cells and potential risks for pathological changes in the tissue.

Critical and Supercritical Spatiotemporal Calcium Dynamics in Beta Cells

A coordinated functioning of beta cells within pancreatic islets is mediated by oscillatory membrane depolarization and subsequent changes in cytoplasmic calcium concentration. While gap junctions allow for intraislet information exchange, beta cells within islets form complex syncytia that are intrinsically nonlinear and highly heterogeneous. To study spatiotemporal calcium dynamics within these syncytia, we make use of computational modeling and confocal high-speed functional multicellular imaging. We show that model predictions are in good agreement with experimental data, especially if a high degree of heterogeneity in the intercellular coupling term is assumed. In particular, during the first few minutes after stimulation, the probability distribution of calcium wave sizes is characterized by a power law, thus indicating critical behavior. After this period, the dynamics changes qualitatively such that the number of global intercellular calcium events increases to the point where the behavior becomes supercritical. To better mimic normal in vivo conditions, we compare the described behavior during supraphysiological non-oscillatory stimulation with the behavior during exposure to a slightly lower and oscillatory glucose challenge. In the case of this protocol, we observe only critical behavior in both experiment and model. Our results indicate that the loss of oscillatory changes, along with the rise in plasma glucose observed in diabetes, could be associated with a switch to supercritical calcium dynamics and loss of beta cell functionality.

Planar cell polarity genes frizzled4 and frizzled6 exert patterning influence on arterial vessel morphogenesis

Quantitative analysis of the vascular network anatomy is critical for the understanding of the vasculature structure and function. In this study, we have combined microcomputed tomography (microCT) and computational analysis to provide quantitative three-dimensional geometrical and topological characterization of the normal kidney vasculature, and to investigate how 2 core genes of the Wnt/planar cell polarity, Frizzled4 and Frizzled6, affect vascular network morphogenesis. Experiments were performed on frizzled4 (Fzd4-/-) and frizzled6 (Fzd6-/-) deleted mice and littermate controls (WT) perfused with a contrast medium after euthanasia and exsanguination. The kidneys were scanned with a high-resolution (16 μm) microCT imaging system, followed by 3D reconstruction of the arterial vasculature. Computational treatment includes decomposition of 3D networks based on Diameter-Defined Strahler Order (DDSO). We have calculated quantitative (i) Global scale parameters, such as the volume of the vasculature and its fractal dimension (ii) Structural parameters depending on the DDSO hierarchical levels such as hierarchical ordering, diameter, length and branching angles of the vessel segments, and (iii) Functional parameters such as estimated resistance to blood flow alongside the vascular tree and average density of terminal arterioles. In normal kidneys, fractal dimension was 2.07±0.11 (n = 7), and was significantly lower in Fzd4-/- (1.71±0.04; n = 4), and Fzd6-/- (1.54±0.09; n = 3) kidneys. The DDSO number was 5 in WT and Fzd4-/-, and only 4 in Fzd6-/-. Scaling characteristics such as diameter and length of vessel segments were altered in mutants, whereas bifurcation angles were not different from WT. Fzd4 and Fzd6 deletion increased vessel resistance, calculated using the Hagen-Poiseuille equation, for each DDSO, and decreased the density and the homogeneity of the distal vessel segments. Our results show that our methodology is suitable for 3D quantitative characterization of vascular networks, and that Fzd4 and Fzd6 genes have a deep patterning effect on arterial vessel morphogenesis that may determine its functional efficiency.

The analysis of intracellular and intercellular calcium signaling in human anterior lens capsule epithelial cells with regard to different types and stages of the cataract

In this work we investigated how modifications of the Ca2+ homeostasis in anterior lens epithelial cells (LECs) are associated with different types of cataract (cortical or nuclear) and how the progression of the cataract (mild or moderate) affects the Ca2+ signaling. We systematically analyzed different aspects of intra- and inter-cellular Ca2+ signaling in the human LECs, which are attached to surgically isolated lens capsule (LC), obtained during cataract surgery. We monitored the temporal and spatial changes in intracellular Ca2+ concentration after stimulation with acetylcholine by means of Fura-2 fluorescence captured with an inverted microscope. In our analysis we compared the features of Ca2+ signals in individual cells, synchronized activations, spatio-temporal grouping and the nature of intercellular communication between LECs. The latter was assessed by using the methodologies of the complex network theory. Our results point out that at the level of individual cells there are no significant differences when comparing the features of the signals with regard either to the type or the stage of the cataract. On the other hand, noticeable differences are observed at the multicellular level, despite inter-capsule variability. LCs associated with more developed cataracts were found to exhibit a slower collective response to stimulation, a less pronounced spatio-temporal clustering of LECs with similar signaling characteristics. The reconstructed intercellular networks were found to be sparser and more segregated than in LCs associated with mild cataracts. Moreover, we show that spontaneously active LECs often operate in localized groups with quite well aligned Ca2+ activity. The presence of spontaneous activity was also found to affect the stimulated Ca2+ responses of individual cells. Our findings indicate that the cataract progression entails the impairment of intercellular signaling thereby suggesting the functional importance of altered Ca2+ signaling of LECs in cataractogenesis.

How optimal synchronization of oscillators depends on the network structure and the individual dynamical properties of the oscillators

The problem of making a network of dynamical systems synchronize onto a common evolution is the subject of much ongoing research in several scientific disciplines. It is nowadays a well-known fact that the synchronization processes are gradually influenced by the interaction topology between the dynamically interacting units. A complex coupling configuration can significantly affect the synchronization abilities of a networked system. However, the question arises what is the optimal network topology that provides enhancement of the synchronization features under given circumstances. In order to address this issue we make use of a network model in which we can smoothly tune the topology from a highly heterogeneous and efficient scale-free network to a homogeneous and less efficient network. The network is then populated with Poincare oscillators, a paradigmatic model for limit-cycle oscillations. This oscillator model exhibits a parameter that enables changes of the limit cycle attraction and is thus immediately related to flexibility/rigidity properties of the oscillator. Our results reveal that for weak attractions of the limit cycle, intermediate homogeneous topology ensures maximal synchronization, whereas highly heterogeneous scale-free topology ensures maximal synchronization for strong attractions of the limit cycle. We argue that the flexibility/rigidity of individual nodes of the networks defines the topology, where maximal global coherence is achieved.

Loosening the shackles of scientific disciplines with network science

a Institute of Physiology, Faculty of Medicine, University of Maribor, Taborska 8, SI-2000 Maribor, Slovenia b Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška cesta 160, SI-2000 Maribor, Slovenia c Faculty of Education, University of Maribor, Koroška cesta 160, SI-2000 Maribor, Slovenia d Faculty of Energy and Technology, University of Maribor, Hočevarjev Trg 1, SI-8270 Krško, Slovenia e Institute of Physiology and Pharmacology, Medical University of Vienna, Schwarzspanierstraße 17A, AU-1090 Vienna, Austria f CAMTP – Center for Applied Mathematics and Theoretical Physics, University of Maribor, Mladinska 3, SI-2000 Maribor, Slovenia g Complexity Science Hub, Josefstädterstraße 39, A-1080 Vienna, Austria

Chapter Eight - Defects in Planar Cell Polarity of Epithelium: What Can We Learn from Liquid Crystals?

Epithelial tissues are structured and highly organized monolayers of cells with many different tissue-specific functions. Ordering of epithelium cells in living tissues relies on spatially and temporally regulated cell behavior and is of vital importance for their functioning. The underlying mechanisms that govern the development of the tissue architecture and morphogenesis rely on planar cell polarity signaling pathways. Mutations and other disruptions of these pathways were found to cause developmental defects, leading to failures in lung branching or kidney development, for example, and are also involved in cancer cell migration. Here, we investigate how these defects affect the spatial arrangement and orientation of epithelium cells, giving special attention to tissue reorganization during development. For the characterization of the resulting polarized cytoarchitectures, we make use of methods developed in the field of liquid crystal (LC) research. In fact, epithelial tissues possess typical features of liquid crystalline systems albeit exhibiting a different local symmetry. Therefore, tools developed in the LC research community can be successfully applied for the description of the overall epithelial tissue topology and its orientational order. We additionally discuss and hypothesize the possibilities of using nanoparticles for structural defect stabilization and its application.

Applying network theory to fables: complexity in Slovene belles-lettres for different age groups

Words are the building blocks of human communication. They are arranged in sentences in a non-trivial and universal way, which implies the existence of fundamental organizational principles that have shaped language development. One of the fundamental examples is the Zipf’s law, which says that the frequency of word occurrence is roughly an inverse power-law function of its rank. In our article, we study the structure and complexity of texts in Slovene belles-lettres, with an emphasis on evaluating the differences in the texts for different age groups. We show that the co-occurrence connectivity of words forms a complex and heterogeneous network that is characterized by an efficient transfer of information. Moreover, we show