bioRxiv preprint doi: https://doi.org/10.1101/2020.06.01.127407; this version posted June 1, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY 4.0 International license. Biological network growth in complex environments { a computational framework Torsten Johann Paul1, Philip Kollmannsberger1 1 Center for Computational and Theoretical Biology, University of W¨urzburg, Campus Hubland Nord 32, W¨urzburg,Germany * [email protected] Abstract Spatial biological networks are abundant on all scales of life, from single cells to ecosystems, and perform various important functions including signal transmission and nutrient transport. These biological functions depend on the architecture of the network, which emerges as the result of a dynamic, feedback-driven developmental process. While cell behavior during growth can be genetically encoded, the resulting network structure depends on spatial constraints and tissue architecture. Since network growth is often difficult to observe experimentally, computer simulations can help to understand how local cell behavior determines the resulting network architecture. We present here a computational framework based on directional statistics to model network formation in space and time under arbitrary spatial constraints. Growth is described as a biased correlated random walk where direction and branching depend on the local environmental conditions and constraints, which are presented as 3D multilayer images. To demonstrate the application of our tool, we perform growth simulations of the osteocyte lacuno-canalicular system in bone and of the zebrafish sensory nervous system. Our generic framework might help to better understand how network patterns depend on spatial constraints, or to identify the biological cause of deviations from healthy network function. Author summary We present a novel modeling approach and computational implementation to better under- stand the development of spatial biological networks under the influence of external signals. Our tool allows to study the relationship between local biological growth parameters and the emerging macroscopic network function using simulations. This computational approach can generate plausible network graphs that take local feedback into account and provide a basis for comparative studies using graph-based methods. Introduction 1 Complex biological networks such as the neural connectome are striking examples of large- 2 scale functional structures arising from a locally controlled growth process [1,2]. The resulting 3 network architecture is not only genetically determined, but depends on biological and physical 4 interactions with the microenvironment during the growth process [3, 4]. In this context, 5 evolution has shaped diverse spatial networks on all length scales, from the cytoskeletal 6 network in cells [5], to multicellular networks such as the vascular system [6] or the osteocyte 7 1/19 bioRxiv preprint doi: https://doi.org/10.1101/2020.06.01.127407; this version posted June 1, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY 4.0 International license. lacuno-canalicular network [7], to macroscopic networks of slime molds [8], mycelia [9] and 8 plants [10]. Sophisticated imaging techniques together with large-scale automated analysis 9 provide increasingly detailed views of the architecture of biological networks, revealing e.g. 10 how neurons are wired together in the brain [11]. After extracting the topological connectivity 11 from such image data, quantitative methods from the physics of complex networks can 12 be applied to compare different types of networks and to uncover common organizational 13 principles [12{15]. To understand the functional role of networks in the context of evolution, 14 however, it is not sufficient to characterize static network structure [1]: we need to be able 15 to infer the dynamics of the underlying biological growth process that defines this structure. 16 This poses a major challenge, since the local dynamics of the growth process is in most 17 cases not experimentally accessible, except for simple model systems. Here, computer 18 simulations provide a solution: they allow to connect the observed patterns to the underlying 19 process by performing in-silico experiments. Different hypotheses can be tested by systemat- 20 ically varying the local growth rules in the simulation and analyzing the resulting network 21 patterns. Such a computational approach also helps to overcome an important limitation of 22 network physics with respect to spatial networks: the properties of complex networks are 23 often calculated with respect to canonical random graphs such as the Erd¨os-Renyi [16] or 24 Watts-Strogatz model [17]. These models are defined by the topological structure in an 25 adjacency matrix, but usually neglect spatial constraints [18,19]. A generic model of network 26 growth under spatial constraints would thus address two important problems: enabling a 27 meaningful quantification of spatial network patterns, and linking the observed patterns to 28 the underlying biological growth process. 29 The first computer simulation of spatial network development, published in 1967 [20], 30 already included the role of active and repulsive cues. Since then, several more advanced 31 approaches to model spatial network growth have been developed. One type of model 32 implements network growth by formation of edges between pre-existing nodes in space [21,22] 33 and was e.g. used to develop large-scale models of branching neurons [23] or leaf vascular 34 networks [24]. Another type of approach is based on growth processes emanating from 35 predefined seed points with branching and merging and has been used to model neural 36 network development [25,26], fibrous materials [27] or the development of branching organ 37 structures [28]. Most of these existing models are targeted towards a specific domain such 38 as neural networks and synapse formation, or include only simplified interactions with the 39 environment. The current state of modeling spatial network growth was recently summarized 40 in [1], arguing that there is "a crucial lack of theoretical models". 41 The aim of this work is to develop a generic framework for biological network development 42 in space and time under arbitrary spatial constraints. We propose a probabilistic agent-based 43 model to describe individual growth processes as biased, correlated random motion with rules 44 for branching, bifurcation, merging and termination. We apply mathematical concepts from 45 directional statistics to describe the influence of external cues on the direction of individual 46 growth processes without restricting the model by making too specific assumptions about 47 these cues. Structural and geometric factors are obtained directly from real image data 48 rather than formulated explicitly. The model is implemented as a computational framework 49 that allows to perform simulations for diverse types of biological networks on all scales, and 50 to monitor the evolving spatial structure and connectivity. 51 Materials and Methods 52 Growth Process 53 The elementary process of biological network formation is the outgrowth of new edges in 54 space from existing nodes, such as cells. Each individual growth agent, for example a 55 neuronal growth cone, randomly explores space and senses soluble signals as well as physical 56 2/19 bioRxiv preprint doi: https://doi.org/10.1101/2020.06.01.127407; this version posted June 1, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY 4.0 International license. cues and obstacles. This process can be formulated as a biased correlated random walk [29]: 57 the movement of edges is biased by attractive or repulsive signals and structural cues, and 58 correlated due to directional persistence of edge growth. Edges turn into trees if branching 59 and/or bifurcations occur. Finally, if edges can join other edges and form new nodes, a 60 connected network emerges. The elements of this discretized, agent-based formalization of 61 spatial network development are summarized in Figure 1. In the following, we describe our 62 mathematical approach for modeling the growth process. 63 ! ! P (r(t ),v(t ),t ) ! ! 2 i+n i+n i+n P (r(t ),v(t ),t ) ! ! 2 i i i P r(t ),v(t ),t 1 ( i i i ) Points on Edges Bifurcate/Split Branch Node ! ! Terminate P1 (r(ti+n ),v(ti+n ),ti+n ) Merge Active Cone Fig 1. Discrete local events during spatial network development. Two growth agents (green and blue, e.g. growth cones, filopodia, cells) start at time ti and randomly explore their environment in a biased correlated random walk. The environment is represented as a Fig. 1: Discretizeddiscrete local events grid during (grey spatial lines), network while development. the network T growswo growth in continuous agents (green space. and blue, Blue e.g. and growth green cones, dots filopodia, cells) start at time ti and randomlycorrespond explore their to subsequentenvironment positions in a biased of correlated the growth random cones. walk. Cones The environment can branch is offrepresented new daughter as a discrete grid (grey lines), while the network grows in continuous space. Blue and green dots correspond to subsequent positions of the growth cones. Cones can branch off new daughter cones,cones,bifurcate/split bifurcate/splitin two innew twocones, newmerge cones, with mergeexisting withedges, existing or terminate. edges, Each or terminate. of these events These occurs with a probability p and leads to formationevents of newleads nodesto in the formation network, of shown new as nodes red dots. in the Over network, time, the shown individual as processes red dots. form Over a connected time, the network of nodes and edges in space.