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Modular genetic regulatory networks increase organization during pattern formation
Accepted Manuscript Title: Modular Genetic Regulatory Networks Increase Organization During Pattern Formation Author: Hamid Mohamadlou Gregory J. Podgorski Nicholas S. Flann PII: DOI: Reference:
S0303-2647(16)30037-5 http://dx.doi.org/doi:10.1016/j.biosystems.2016.04.004 BIO 3652
To appear in:
BioSystems
Received date: Accepted date:
15-2-2016 4-4-2016
Please cite this article as: Hamid Mohamadlou, Gregory J. Podgorski, Nicholas S. Flann, Modular Genetic Regulatory Networks Increase Organization During Pattern Formation, (2016), http://dx.doi.org/10.1016/j.biosystems.2016.04.004 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Modular Genetic Regulatory Networks Increase Organization During Pattern Formation Hamid Mohamadloua , Gregory J. Podgorskic,d , Nicholas S. Flanna,b a
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Department of Computer Science, Utah State University b Institute for Systems Biology, Seattle c Department of Biology, Utah State University d Center for Integrated BioSystems, Utah State University
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Abstract
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Studies have shown that genetic regulatory networks (GRNs) consist of modules that are densely connected subnetworks that function quasiautonomously. Modules may be recognized motifs that comprise of two or three genes with particular regulatory functions and connectivity or be purely structural and identified through connection density. It is unclear what evolutionary and developmental advantages modular structure and in particular motifs provide that have led to this enrichment. This study seeks to understand how modules within developmental GRNs influence the complexity of multicellular patterns that emerge from the dynamics of the regulatory networks. We apply an algorithmic complexity to measure the organization of the patterns. A computational study was performed by creating Boolean intracellular networks within a simulated epithelial field of embryonic cells, where each cell contains the same network and communicates with adjacent cells using contact-mediated signaling. Intracellular networks with random connectivity were compared to those with modular connectivity and with motifs. Results show that modularity effects network dynamics and pattern organization significantly. In particular: (1) Modular connectivity alone increases complexity in network dynamics and patterns; (2) Bistable switch motifs simplify both the pattern and network dynamics; (3) All other motifs with feedback loops increase multicellular pattern complexity while simplifyEmail addresses: [email protected] ( Hamid Mohamadlou), [email protected] ( Gregory J. Podgorski), [email protected] ( Nicholas S. Flann)
Preprint submitted to BioSystems
May 5, 2016
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ing the network dynamics. (4) Negative feedback loops affect the dynamics complexity more significantly than positive feedback loops.
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Multicellular organisms contain a large variety of cellular patterns. For instance, Figure 1 illustrates a Drosophila melanogaster embryo in which muscle and nervous system structures interconnect through sensory and activation signaling. These patterns are formed during development and are a consequence of genetic regulatory networks (GRNs) that operate within cells and that respond to communication between cells [1] [2], [3]. GRNs are networks of interacting genes where the expression or non-expression of genes determines the expression state of other genes. The dynamics of GRNs determine the gene expression profile for each cell leading to spatial patterns of cellular differentiation. This process is repeated to implement an organism’s body plan, and subsequent morphology [4]. GRNs contain subnetworks of genes referred to as modules. Smaller modules, usually consisting two or three genes with specific functioning, are known as motifs [5] [6]. Motifs are detected at a higher frequency than would be expected in random networks. Computational biologists have hypothesized that motifs play a determinative role in cell function [7] [8]. However, their influence on pattern formation during development is poorly understood. This work presents a computational study aimed at understanding how the presence of motifs within intracellular networks changes the GRN dynamics and the emergent multicellular patterns. In particular, this study measures the complexity of the organization of the dynamics and patterns. As can be seen in Figure 1, patterns can involve complex arrangements of specialized regions and interconnections that develop later, or earlier patterns such as simple segmentation [9] or mosaic arrangements [10] [11]. To quantify the level of organization we employ Kolmogorov complexity, also known as algorithmic complexity [12]. Such methods measure the information contained within an object, such as a cellular pattern, by considering the size of the algorithm needed to generate the object, hence the term algorithmic complexity. The smaller the algorithm, the
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1. Introduction
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Keywords: Network motifs, Kolmogorov complexity, Pattern formation, Genetic regulatory networks
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simpler the pattern. The calculations of Kolmogorov complexity is detailed in the methods section following. To understand how GRNs regulate biological events, scientists have developed mathematical and computational models to generate predictions and explain experimental observations. Among these modeling approaches is to represent a GRN as a Boolean network in which the activity of a gene is either on or off, determined by a set of logical functions over the activity of other genes [13]. It is this modeling framework that we apply in this study.
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To evaluate the influence of the modular structure of networks and in particular motifs on network dynamics and patterns, we design GRNs that are embedded into cells arranged in a 2D grid, simulating an epithelium. Such abstractions of the epithelium have been employed successfully in many developmental systems, such as the cellularized Drosophila embryo [9], and the sensory epithelia of the developing vertebrate retina [14] and the inner ear [10]. Each cell contains the same Boolean network, referred to as an intracellular network. We explore the impact on network dynamics and multicellular patterns of modules with random connectivity and regulatory functions, and when the modules are explicitly recognized motifs.
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Figure 1: Ventral view of stage 16 Drosophila melanogaster embryo immunostained for tropomyosin (green; a protein expressed in muscle), Pax 3/7 (blue; a regulatory protein expressed in central nervous system nuclei and ectoderm), and HRP (red; neurons). All nuclei shown in gray (DAPI). Courtesy of Julieta Mara Acevedo and Lucas Leclere, Marine Biological Laboratory, Woods Hole, www.mbl.edu/ dev.biologists.org/
2. Modular structure and motifs within gene regulatory networks Networks are structurally modular if they contain highly connected clusters of genes that are linked by sparser connections than those within the modules. Figure 2(a-b) shows a small network with a modular structure 3
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compared to randomly connected networks. Evidence suggests that modular network structure increases the functional modularity of gene regulatory networks by performing relatively independent tasks [15], [16] such as decision making, signal processing, and communication. The modular organization of biological structure is supported by experimental studies from pathogen structure, gene networks, and protein-protein interaction networks [17]. For example, Kim et al. [15] considered the connected subset of protein networks in protein-protein interaction data for budding yeast. Their analysis suggests that the yeast protein network is significantly modular, and it contains various motifs.
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Figure 2: This figure illustrates networks with modular structures and example of variety of motifs. (a) A network with modular structure where intra-modular connectivity is higher than inter-module connectivity. (b) A randomly connected network. (c) A positive feedback loop (a double inhibitory loop with two positive autoregulatory loops). (d) A positive feedback loop (a double excitatory loop with two positive autoregulatory loops).(e) A negative feedback loop [18] with two positive autoregulatory loops. (f) Coupled positivepositive feedback loops. (g) Coupled positive-negative feedback loops. (h) The type-1 coherent feed-forward loop [19].
We study two kinds of modularity in this work: structural modularity when the modules within the intracellular network are connected randomly and functional modularity when the subnetworks are recognized motifs. Motifs frequently occur and consist of few interacting genes [8] and were first noted in Escherichia coli, where they were detected at a higher frequency than would be expected in random networks. Since then multiple motifs have been identified in bacteria and yeast [20], the immune system [21], and Drosophila [15]. This finding suggests that motifs are building blocks of transcription networks and that they may have evolved to achieve specific regulatory behaviors in cellular transcription networks [18]. Regulatory motifs have been found in regulatory networks that perform two distinct functions: 4
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3. Multicellular model and implementation
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(1) Developmental networks that guide differentiation and cell fate determination by transducing signals into irreversible cell-fate decisions [22] [23]; and (2) Sensory networks that respond to signals such as stresses and nutrients rapidly and make reversible decisions [5]. The motifs that are associated with developmental networks are commonly comprised of feedback loops. Positive-feedback loops are most common and are made up of two transcription factors that regulate each other. There are two kinds of positive-feedback loops, a double excitatory loop (Figure 2(b)) and a double-inhibitory loop (Figure 2(a)). The regulatory dynamics of these gene pairs coupled by positive feedback loops often results in two or more steady states and is referred to as multistability [18]. Positive feedback loops amplify signals and elongate the time required to reach a steady state, referred to as a network attractor [20]. This slowed response can be helpful when a cell makes significant decisions such as irreversible cell specification and apoptosis. Unlike positive feedback loops, negative feedback loops (Figure 2(c)) often enhance attractor stability [8]. They also function as noise filters and make cells more robust to signal noises. Additionally, positive and negative feedback loops are coupled into structures containing two feedback loops, such as positive-positive, positive-negative and negative-negative feedback loops (Figure 2(d,e)). Coupled feedback loops perform functions that single feedback loops cannot. In particular, Kim et al. [18] found that a positive-positive feedback loop enhances signal amplification and bistability, and a positive-negative feedback loop increases reliable decision-making by modulating signal responses and effectively dealing with noise. Feed-forward loops (FFL) are another family of motifs that do not contain feedback and are associated with sensory networks. FFL are found in a variety of organisms such as Saccharomyces cerevisiae, Bacillus subtilis, Caenorhabditis Elegans and humans [19]. FFLs consists of three genes (Figure 2(f)). The first regulatory gene controls the second and the third genes. The third gene is also regulated by the second gene. Logical gates such AND or OR could be applied to the three regulatory interactions in the FFL. The best known FFL which frequently occurs in (E. coli and yeast), is the coherent type-1 FFL [24] with all AND gates.
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The size of gene networks for multicellular pattern formation drove the decision to use Boolean networks as the framework for this computational 5
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study. A Boolean network is a simplified model of a genetic regulatory network. In this application, each gene is represented as a network node that takes binary values (1 for expressed and 0 for not expressed). The state of a gene (0 or 1) is determined by its Boolean function defined as the expressions over AND, OR, NOT on the inputs from other genes. These inputs are represented as directed edges in the network graph. Boolean networks provide a qualitative description of gene states and their interactions, first introduced by Kauffman [13], [25]. (b)
(a) Epithelium field
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Figure 3: (a) Structural modularity. Cell-cell signaling is implemented through specific modules referred to as signaling modules (depicted in green). In this example, cell-cell signaling is orthogonal such that two adjacent cells signal directionally [north-south and east-west]. These directions can be thought of as corresponding to the anterior-posterior and dorsal-ventral embryonic axis. (b) Motif insertion. Example motif insertion into a random GRN. Dashed arrows represent random outgoing signals from the motif. Outgoing and incoming signals from and to the random GRN are randomly connected to genes within the random network.
For the first part of this study, to evaluate the effect of modularity in network dynamics and pattern complexity, structurally modular networks were created in the critical or chaotic domain, see Figure 3(a). For the latter part of this study, motifs were inserted in randomly created networks, see Figure 3(b). For instance, the insertion of a double positive loop is produced by adding two autoregulatory genes that regulate each other and also regulate two random genes from the rest of the network. This work extends our previous study [3] that investigated the role of different dynamical regimes (ordered, critical or chaotic) on the complexity 6
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of multicellular pattern formation. The simulation model was unchanged along with the epithelial model as a lattice of cells, with each cell holding an intracellular Boolean network. Cell-cell signaling was implemented in the model as an edge connecting the state of one gene in a cell to an input of a Boolean function of one or more of its neighbors. Such genes are called communicating genes. The number of communicating genes is referred to as the signaling bandwidth. Signaling bandwidth was set to half of the total number of genes in a cell as our previous study showed that this configuration established the most efficient cell-cell signaling. Experiments were run for two different intercellular connection configurations, orthogonal and symmetric signaling (see Figure 4). When connected symmetrically, each cell receives signals from its four neighbors, inputs them to a logical OR function creating a single input to the intracellular network. This architecture biologically implements contact-mediated signalling [3] [26]. When connected orthogonally, each cell connects with its four neighbors, but the signals are independent inputs to the intracellular GRN [27]. Orthogonal signals correspond to intercellular communication along the anteriorposterior and dorsal-ventral embryonic axes essential to generate complex patterns illustrated in Figure 1. Networks in two complexity classes were considered: critical and chaotic. To produce networks of different complexity domains, the number of inputs to each Boolean function (node in the network) is set according to s = 2kp(1−p) where s is the sensitivity of the network to perturbations in gene values, p is the probability of the output of each Boolean function being 1, and k is the count of inputs to each Boolean function [28]. When s = 1 a single bit change is on average propagated to one other node, and the network is in the critical domain. In a chaotic network, s > 1 and perturbations tend to grow. In this work, p was fixed at 0.5 and k was changed to create networks of different domains: k = 2 for critical and k = 3 for chaotic. The state of each cellular GRN is initialized randomly by setting the state of each gene to 0 or 1. Randomly generated logic functions are assigned to networks as the transition rules used to determine the state of genes [3]. The state of the system during simulation is clocked synchronously until a steady or cyclic state (in up to 300 updates) is reached for all individual cells. When the state of genes change in a repetitive cycle or reach a fixed state, then the cell is in attractor state [13]. Since cells are connected through directional signaling, neighboring cells may converge to different attractors, forming a regular pattern, where cells in the same attractor have differentiated into
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4. Methodology
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4.1. Information Complexity Set Complexity [29] is an information complexity metric derived from Kolmogorov’s original work on algorithmic complexity. In this seminal paper by Galis et al., the method was developed explicitly to measure the information contained in sets of biological data at different scales: the molecular, sequence and network. Recently, Set Complexity has been applied to study modularity in biological networks [30] and large-scale genetic interaction networks [31]. In this work, we extend the approach to the analysis of multicellular patterns and the network dynamics that create them. Kolmogorov complexity of an object is the size of the shortest program that can produce that object. The exact calculation is undecidable and, therefore, cannot be computed as defined. As an approximation, the compression size of an object is used. Kolmogorov analysis describes the similarity of two objects as the size of the shortest program that can translate one object to the other. This measure is approximated by the Normalized Compression Distance (NCD) [32], where the NCD of two objects is 0.0 if they are identical and 1.0 if they are random. Set Complexity combines the analysis of single objects and pairwise objects to arrive at a complexity measure of a set of objects, given in Equation 1. By employing NCD as a metric to evaluate the similarity between all pairs of objects in a set, set complexity discounts the influence of the pairs of objects that are randomly related or redundant. As long as any object can be encoded as a string, Set Complexity can compute the information content that resides in the set. Set Complexity of a set of n strings S = {s1 , . . . , sn } is defined:
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the same cell types. The state of all the genes as the networks along with multicellular patterns are recorded for analysis of information content. After running the randomly-generated GRNs, single and coupled feedback and feedforward loops are inserted into the randomly generated intracellular GRN (see Figure 3). The network is run again with the inserted motifs to identify the attractors and visualize the multicellular patterns that are formed. The Kolmogorov complexity of both the gene network dynamics and multicellular patterns is computed by an information theoretic measure called Set Complexity described below.
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X X 1 C(si ) NCD(si , sj )(1 − NCD(si , sj )) n(n − 1) s ∈S s 6=s i
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Where C(si ) is the compression size of string si . The term NCD(si , sj )(1− NCD(si , sj )) is maximized when NCD(si , sj ) = 0.5, which occurs when C(si + sj ) ' C(si )/2 − C(sj ), assuming C(si ) > C(sj ). The influence of similar and dissimilar strings is discounted by NCD(si , sj )(1 − NCD(si , sj )) because NCD(si , sj ) will be near 1.0 or 0.0. In this implementation, bZip is used as a compression algorithm. A one-to-one mapping is required to encode an object into a string so that no information is lost. The temporal dynamics are converted to a string by concatenating in column-order the state of the network at each time period. The spatial pattern is likewise converted by traversing the cells in row order and concatenating the type ID of each cell. To convert a GRN to a string, a more complex procedure is needed. Here, a unique ID is assigned to each logic gate and edge; then the whole epithelial network is traversed, and substrings created for each logic gate that includes its truth table and its input and output edge IDs. The string also includes a row order traversal of the connection matrix. See [33] for the exact algorithm for encoding Boolean networks. 4.2. Structural modularity score measurement The most common method used in the literature to score structural modularity is a method by Newman[34]. Newman’s method to compute the modularity score Q for a given network is as follows:
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where c is the total number of modules, eii is fraction of edges in module i, eij is the fraction of edges that connect module i to module j and ai is the fraction P of edges that connect module i to other modules and is as follows: ai = j eij In our experiments, the nodes in the modules are known since each module is generated with higher intra-connection than the sparse connections between the modules. 9
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5.1. Structural modularity To analyse how the structural modularity of a network influences the dynamics and pattern complexity, we partition a population of random networks by percentile of their modularity score distribution. Networks that lie in lower third are defined as non-modular, and networks that lie in upper third are defined as highly modular. Those networks considered non-modular exhibit the same low modularity score as random networks. Networks that lie in the middle third were not considered in this study. Figure 4 illustrates the average dynamics complexity and pattern complexity for 600 non-modular and 600 highly-modular networks. Results in Figure 4(a) show how highly modular networks produce higher dynamics and pattern complexity when cell-cell signaling is orthogonal. A statistical analysis confirms the distribution of the two classes of networks are significantly different Table 1 and Table 2.
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5. Results
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Figure 4: Non modular vs. modular. Average dynamics and pattern complexity for 60 non-modular (arrow tails) and 60 highly-modular (arrow heads) networks for orthogonal(orange) and symmetric signaling(dark red).
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Critical modular
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Table 1: Orthogonal signaling. Dynamics and pattern complexity change when non modular networks become highly modular networks.
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Table 2: Symmetric signaling. Dynamics and pattern complexity change when non modular networks become highly modular networks.
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The observation for symmetric signaling with a critical configuration for each module in Figure 4(b), suggest that the networks with modular structure can produce multicellular patterns with higher information content without an increase in the network dynamics complexity. Another interesting observation was that for symmetric signaling with chaotic configuration (Figure 4(b)), both network dynamics and pattern complexity decrease. As shown in the previous study [3] symmetric signaling produces only simple information patterns and dynamics because information transfer among adjacent cells combined using disjunction, which loses directional information. Now when the configuration of each module is chaotic, structural modularity of network 11
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5.2. Functional Modularity In this section, we explore the effect of insertion of the best-known motifs illustrated in Figure 2(b) into randomly generated intracellular GRNs in the critical domain. For each motif, 600 random networks were generated and then each network was modified by the insertion of that regulatory motif. To obtain the data for analysis, each network was executed from random initial conditions and the dynamics and resulting pattern recorded. Following this step, the network data is grouped into sets of 10 and the set complexity computed for the networks, the dynamics, and the pattern. Figure 5 shows the influence of insertion of a positive feedback loop (Figure 2(b)) into a random GRN. The vertical axis of both graphs is the pattern complexity; on the left shows the relationship with dynamics complexity, on the right shows the relationship with network complexity. Each point on the graph corresponds to a set of 10 runs described above. Insertion of a double positive feedback loop into random GRNs clearly modifies the behavior of the network. The addition of the motif increases pattern complexity significantly while slightly reducing the complexity of the dynamics (Figure 5 left). Little effect on network complexity is observed (Figure 5 right).
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Figure 5: Effect of insertion of a double positive feedback loop on network, network dynamics and pattern complexity for GRNs in the critical domain. 272
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complexity for 600 network sets for each of the motifs. The influence of each motif is represented as an arrow. The start of the arrow is the set of random networks and is the same for all motifs. The arrowhead is the average complexity of the set of all networks with the indicated motif inserted. The insertion of a negative feedback loop, a double negative feedback loop or a double positive feedback loop all have the same qualitative effect of decreasing the complexity of network dynamics and increasing pattern complexity. However, a double positive loop increases pattern complexity much more than either of the negative feedback loops (Figure 6(a)). Insertion of feed-forward motifs decreases dynamics with almost no effect on the pattern complexity. Only the double negative motif leads to a reduction in pattern complexity. Statistical analysis confirms that insertion of these motifs makes a significant change in the average network dynamics and patterns complexity. ANOVA analysis presented in Table.3 and Table 4 shows that insertion of these motifs makes a significant change in the average network dynamics and patterns complexity.
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Figure 6: (a) Effects on network dynamics and pattern complexity of inserting regulatory motifs into random GRNs. Average dynamics and pattern complexity for 60 random sets of GRNs (arrow tails) and 60 sets of GRNs with the indicated inserted motifs (arrow heads). (b) Effects on network and pattern complexity of inserting regulatory motifs into random GRNs.
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Table 3: ANOVA analysis of dynamics complexity illustrates differences for various groups of motifs over networks with orthogonal and symmetric signaling
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Symmetric signaling Sum Average Variance 1560.25 20.80 15.96 1423.40 18.97 8.18 1828.02 24.37 7.64 1658.00 22.10 8.15 1466.11 19.54 4.17 1718.50 22.91 4.02
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MS 318.94 8.02
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6. Discussion
In this study, we explored how network dynamics and pattern complexity are affected by structural modularity in general and regulatory motifs in particular. The results show that networks with a modular structure, in general, tend to produce more complex multicellular patterns without a significant increase in gene expression dynamics. The results indicate that network motifs that contain feedback loops increase the information complexity of the multicellular patterns. Another important observation was negative feedback loops do not affect the dynamics complexity significantly as positive feedback loops do. We hypothesize that variation in dynamics complexity associated with adding different types of motifs to random GRNs originates from the time it takes for the GRN to reach a steady state and the proportion of single state versus cyclic attractors produced by the GRN. Since the motifs studied in this work act as multistable switches, they simplify the complex cyclic attractors to attractors with a few states. Gene expression in feedback loops
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Orthogonal signaling ANOVA Source of Variation SS Between Groups 17099.6 Within Groups 118226.6 Total 135326.3 Symmetric signaling ANOVA Source of Variation SS Between Groups 1594.73 Within Groups 3563.42 Total 5158.16
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Orthogonal signaling Sum Average Variance 3911.45 52.15 474.35 3017.06 40.22 284.05 2633.55 35.11 101.26 3322.99 44.30 181.91 3355.92 44.74 306.40 2560.45 34.13 249.67
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Table 4: ANOVA analysis of pattern complexity illustrates differences for various groups of motifs over networks with orthogonal and symmetric signaling.
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Symmetric signaling Sum Average Variance 592.51 7.90 3.55 456.57 6.08 1.97 805.59 10.74 5.08 687.03 9.16 2.66 530.80 7.07 2.35 692.85 9.23 3.22
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MS 211.53 3.14
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reach a steady state quickly, and so reduce the length and complexity of cyclic attractors. We hypothesize that this is why in all cases dynamics complexity decreases from that of the original random networks. The rate of dynamics complexity reduction associated with the addition of the motifs with negative feedback loops is significantly lower than for positive loops. Unlike positive feedback loops, negative feedback loops do not increase the time to reach steady states [18]. Therefore, they don’t affect the dynamics complexity noticeably. As results show, negative feedback loop motifs (such as single negative feedback loop and coupled positive-negative loops) have the lowest reduction in their dynamics complexity. All the motifs with feedback loops impact the pattern complexity. In fact, the only motifs in this study that has no effect on pattern complexity are feed-forward loops. This observation confirms the association of feedback loop motifs with developmental networks that mediate important cell fate decision. The increase in information complexity of the multicellular patterns shows more certain cell-fate decisions have been taken. Only double nega-
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Orthogonal signaling ANOVA Source of Variation SS Between Groups 2366.77 Within Groups 8466.32 Total 10833.10 Symmetric signaling ANOVA Source of Variation SS Between Groups 1057.66 Within Groups 1396.30 Total 2453.97
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Orthogonal signaling Sum Average Variance 1233.65 16.44 25.15 1026.61 13.68 22.74 1559.75 20.79 13.10 1457.84 19.43 19.91 1372.30 18.29 24.27 1250.56 16.67 9.22
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Acknowledgements
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tive motifs decrease the complexity of the patterns formed. We hypothesize that this is due to the bistable behavior of these motifs, where they rapidly converge to one state or another producing few distinct non-cyclic attractors. Overall, we find that all motifs with feedback decrease the complexity of the dynamics. By decreasing the complexity of the dynamics, these motifs could increase the robustness in pattern formation in the presence of noise as shown in [8]. Feedback motifs tend to increase pattern complexity while decreasing dynamics complexity. This implies that these motifs increase the“efficiency” of pattern formation where simpler dynamics leads to more complex patterns. Such an increase in efficiency may be one of the principle reasons that feedback motifs are enriched within developmental networks.
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Research reported in this publication was supported by the National Institute Of General Medical Sciences of the National Institutes of Health under Award Number P50GM076547, Thanks to Ilya Shmulevich for helpful discussions. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health.
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S0303-2647(16)30037-5 http://dx.doi.org/doi:10.1016/j.biosystems.2016.04.004 BIO 3652
To appear in:
BioSystems
Received date: Accepted date:
15-2-2016 4-4-2016
Please cite this article as: Hamid Mohamadlou, Gregory J. Podgorski, Nicholas S. Flann, Modular Genetic Regulatory Networks Increase Organization During Pattern Formation, (2016), http://dx.doi.org/10.1016/j.biosystems.2016.04.004 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Modular Genetic Regulatory Networks Increase Organization During Pattern Formation Hamid Mohamadloua , Gregory J. Podgorskic,d , Nicholas S. Flanna,b a
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Department of Computer Science, Utah State University b Institute for Systems Biology, Seattle c Department of Biology, Utah State University d Center for Integrated BioSystems, Utah State University
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Abstract
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Studies have shown that genetic regulatory networks (GRNs) consist of modules that are densely connected subnetworks that function quasiautonomously. Modules may be recognized motifs that comprise of two or three genes with particular regulatory functions and connectivity or be purely structural and identified through connection density. It is unclear what evolutionary and developmental advantages modular structure and in particular motifs provide that have led to this enrichment. This study seeks to understand how modules within developmental GRNs influence the complexity of multicellular patterns that emerge from the dynamics of the regulatory networks. We apply an algorithmic complexity to measure the organization of the patterns. A computational study was performed by creating Boolean intracellular networks within a simulated epithelial field of embryonic cells, where each cell contains the same network and communicates with adjacent cells using contact-mediated signaling. Intracellular networks with random connectivity were compared to those with modular connectivity and with motifs. Results show that modularity effects network dynamics and pattern organization significantly. In particular: (1) Modular connectivity alone increases complexity in network dynamics and patterns; (2) Bistable switch motifs simplify both the pattern and network dynamics; (3) All other motifs with feedback loops increase multicellular pattern complexity while simplifyEmail addresses: [email protected] ( Hamid Mohamadlou), [email protected] ( Gregory J. Podgorski), [email protected] ( Nicholas S. Flann)
Preprint submitted to BioSystems
May 5, 2016
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ing the network dynamics. (4) Negative feedback loops affect the dynamics complexity more significantly than positive feedback loops.
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Multicellular organisms contain a large variety of cellular patterns. For instance, Figure 1 illustrates a Drosophila melanogaster embryo in which muscle and nervous system structures interconnect through sensory and activation signaling. These patterns are formed during development and are a consequence of genetic regulatory networks (GRNs) that operate within cells and that respond to communication between cells [1] [2], [3]. GRNs are networks of interacting genes where the expression or non-expression of genes determines the expression state of other genes. The dynamics of GRNs determine the gene expression profile for each cell leading to spatial patterns of cellular differentiation. This process is repeated to implement an organism’s body plan, and subsequent morphology [4]. GRNs contain subnetworks of genes referred to as modules. Smaller modules, usually consisting two or three genes with specific functioning, are known as motifs [5] [6]. Motifs are detected at a higher frequency than would be expected in random networks. Computational biologists have hypothesized that motifs play a determinative role in cell function [7] [8]. However, their influence on pattern formation during development is poorly understood. This work presents a computational study aimed at understanding how the presence of motifs within intracellular networks changes the GRN dynamics and the emergent multicellular patterns. In particular, this study measures the complexity of the organization of the dynamics and patterns. As can be seen in Figure 1, patterns can involve complex arrangements of specialized regions and interconnections that develop later, or earlier patterns such as simple segmentation [9] or mosaic arrangements [10] [11]. To quantify the level of organization we employ Kolmogorov complexity, also known as algorithmic complexity [12]. Such methods measure the information contained within an object, such as a cellular pattern, by considering the size of the algorithm needed to generate the object, hence the term algorithmic complexity. The smaller the algorithm, the
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Keywords: Network motifs, Kolmogorov complexity, Pattern formation, Genetic regulatory networks
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simpler the pattern. The calculations of Kolmogorov complexity is detailed in the methods section following. To understand how GRNs regulate biological events, scientists have developed mathematical and computational models to generate predictions and explain experimental observations. Among these modeling approaches is to represent a GRN as a Boolean network in which the activity of a gene is either on or off, determined by a set of logical functions over the activity of other genes [13]. It is this modeling framework that we apply in this study.
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To evaluate the influence of the modular structure of networks and in particular motifs on network dynamics and patterns, we design GRNs that are embedded into cells arranged in a 2D grid, simulating an epithelium. Such abstractions of the epithelium have been employed successfully in many developmental systems, such as the cellularized Drosophila embryo [9], and the sensory epithelia of the developing vertebrate retina [14] and the inner ear [10]. Each cell contains the same Boolean network, referred to as an intracellular network. We explore the impact on network dynamics and multicellular patterns of modules with random connectivity and regulatory functions, and when the modules are explicitly recognized motifs.
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Figure 1: Ventral view of stage 16 Drosophila melanogaster embryo immunostained for tropomyosin (green; a protein expressed in muscle), Pax 3/7 (blue; a regulatory protein expressed in central nervous system nuclei and ectoderm), and HRP (red; neurons). All nuclei shown in gray (DAPI). Courtesy of Julieta Mara Acevedo and Lucas Leclere, Marine Biological Laboratory, Woods Hole, www.mbl.edu/ dev.biologists.org/
2. Modular structure and motifs within gene regulatory networks Networks are structurally modular if they contain highly connected clusters of genes that are linked by sparser connections than those within the modules. Figure 2(a-b) shows a small network with a modular structure 3
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compared to randomly connected networks. Evidence suggests that modular network structure increases the functional modularity of gene regulatory networks by performing relatively independent tasks [15], [16] such as decision making, signal processing, and communication. The modular organization of biological structure is supported by experimental studies from pathogen structure, gene networks, and protein-protein interaction networks [17]. For example, Kim et al. [15] considered the connected subset of protein networks in protein-protein interaction data for budding yeast. Their analysis suggests that the yeast protein network is significantly modular, and it contains various motifs.
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Figure 2: This figure illustrates networks with modular structures and example of variety of motifs. (a) A network with modular structure where intra-modular connectivity is higher than inter-module connectivity. (b) A randomly connected network. (c) A positive feedback loop (a double inhibitory loop with two positive autoregulatory loops). (d) A positive feedback loop (a double excitatory loop with two positive autoregulatory loops).(e) A negative feedback loop [18] with two positive autoregulatory loops. (f) Coupled positivepositive feedback loops. (g) Coupled positive-negative feedback loops. (h) The type-1 coherent feed-forward loop [19].
We study two kinds of modularity in this work: structural modularity when the modules within the intracellular network are connected randomly and functional modularity when the subnetworks are recognized motifs. Motifs frequently occur and consist of few interacting genes [8] and were first noted in Escherichia coli, where they were detected at a higher frequency than would be expected in random networks. Since then multiple motifs have been identified in bacteria and yeast [20], the immune system [21], and Drosophila [15]. This finding suggests that motifs are building blocks of transcription networks and that they may have evolved to achieve specific regulatory behaviors in cellular transcription networks [18]. Regulatory motifs have been found in regulatory networks that perform two distinct functions: 4
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3. Multicellular model and implementation
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(1) Developmental networks that guide differentiation and cell fate determination by transducing signals into irreversible cell-fate decisions [22] [23]; and (2) Sensory networks that respond to signals such as stresses and nutrients rapidly and make reversible decisions [5]. The motifs that are associated with developmental networks are commonly comprised of feedback loops. Positive-feedback loops are most common and are made up of two transcription factors that regulate each other. There are two kinds of positive-feedback loops, a double excitatory loop (Figure 2(b)) and a double-inhibitory loop (Figure 2(a)). The regulatory dynamics of these gene pairs coupled by positive feedback loops often results in two or more steady states and is referred to as multistability [18]. Positive feedback loops amplify signals and elongate the time required to reach a steady state, referred to as a network attractor [20]. This slowed response can be helpful when a cell makes significant decisions such as irreversible cell specification and apoptosis. Unlike positive feedback loops, negative feedback loops (Figure 2(c)) often enhance attractor stability [8]. They also function as noise filters and make cells more robust to signal noises. Additionally, positive and negative feedback loops are coupled into structures containing two feedback loops, such as positive-positive, positive-negative and negative-negative feedback loops (Figure 2(d,e)). Coupled feedback loops perform functions that single feedback loops cannot. In particular, Kim et al. [18] found that a positive-positive feedback loop enhances signal amplification and bistability, and a positive-negative feedback loop increases reliable decision-making by modulating signal responses and effectively dealing with noise. Feed-forward loops (FFL) are another family of motifs that do not contain feedback and are associated with sensory networks. FFL are found in a variety of organisms such as Saccharomyces cerevisiae, Bacillus subtilis, Caenorhabditis Elegans and humans [19]. FFLs consists of three genes (Figure 2(f)). The first regulatory gene controls the second and the third genes. The third gene is also regulated by the second gene. Logical gates such AND or OR could be applied to the three regulatory interactions in the FFL. The best known FFL which frequently occurs in (E. coli and yeast), is the coherent type-1 FFL [24] with all AND gates.
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The size of gene networks for multicellular pattern formation drove the decision to use Boolean networks as the framework for this computational 5
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study. A Boolean network is a simplified model of a genetic regulatory network. In this application, each gene is represented as a network node that takes binary values (1 for expressed and 0 for not expressed). The state of a gene (0 or 1) is determined by its Boolean function defined as the expressions over AND, OR, NOT on the inputs from other genes. These inputs are represented as directed edges in the network graph. Boolean networks provide a qualitative description of gene states and their interactions, first introduced by Kauffman [13], [25]. (b)
(a) Epithelium field
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Figure 3: (a) Structural modularity. Cell-cell signaling is implemented through specific modules referred to as signaling modules (depicted in green). In this example, cell-cell signaling is orthogonal such that two adjacent cells signal directionally [north-south and east-west]. These directions can be thought of as corresponding to the anterior-posterior and dorsal-ventral embryonic axis. (b) Motif insertion. Example motif insertion into a random GRN. Dashed arrows represent random outgoing signals from the motif. Outgoing and incoming signals from and to the random GRN are randomly connected to genes within the random network.
For the first part of this study, to evaluate the effect of modularity in network dynamics and pattern complexity, structurally modular networks were created in the critical or chaotic domain, see Figure 3(a). For the latter part of this study, motifs were inserted in randomly created networks, see Figure 3(b). For instance, the insertion of a double positive loop is produced by adding two autoregulatory genes that regulate each other and also regulate two random genes from the rest of the network. This work extends our previous study [3] that investigated the role of different dynamical regimes (ordered, critical or chaotic) on the complexity 6
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of multicellular pattern formation. The simulation model was unchanged along with the epithelial model as a lattice of cells, with each cell holding an intracellular Boolean network. Cell-cell signaling was implemented in the model as an edge connecting the state of one gene in a cell to an input of a Boolean function of one or more of its neighbors. Such genes are called communicating genes. The number of communicating genes is referred to as the signaling bandwidth. Signaling bandwidth was set to half of the total number of genes in a cell as our previous study showed that this configuration established the most efficient cell-cell signaling. Experiments were run for two different intercellular connection configurations, orthogonal and symmetric signaling (see Figure 4). When connected symmetrically, each cell receives signals from its four neighbors, inputs them to a logical OR function creating a single input to the intracellular network. This architecture biologically implements contact-mediated signalling [3] [26]. When connected orthogonally, each cell connects with its four neighbors, but the signals are independent inputs to the intracellular GRN [27]. Orthogonal signals correspond to intercellular communication along the anteriorposterior and dorsal-ventral embryonic axes essential to generate complex patterns illustrated in Figure 1. Networks in two complexity classes were considered: critical and chaotic. To produce networks of different complexity domains, the number of inputs to each Boolean function (node in the network) is set according to s = 2kp(1−p) where s is the sensitivity of the network to perturbations in gene values, p is the probability of the output of each Boolean function being 1, and k is the count of inputs to each Boolean function [28]. When s = 1 a single bit change is on average propagated to one other node, and the network is in the critical domain. In a chaotic network, s > 1 and perturbations tend to grow. In this work, p was fixed at 0.5 and k was changed to create networks of different domains: k = 2 for critical and k = 3 for chaotic. The state of each cellular GRN is initialized randomly by setting the state of each gene to 0 or 1. Randomly generated logic functions are assigned to networks as the transition rules used to determine the state of genes [3]. The state of the system during simulation is clocked synchronously until a steady or cyclic state (in up to 300 updates) is reached for all individual cells. When the state of genes change in a repetitive cycle or reach a fixed state, then the cell is in attractor state [13]. Since cells are connected through directional signaling, neighboring cells may converge to different attractors, forming a regular pattern, where cells in the same attractor have differentiated into
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4.1. Information Complexity Set Complexity [29] is an information complexity metric derived from Kolmogorov’s original work on algorithmic complexity. In this seminal paper by Galis et al., the method was developed explicitly to measure the information contained in sets of biological data at different scales: the molecular, sequence and network. Recently, Set Complexity has been applied to study modularity in biological networks [30] and large-scale genetic interaction networks [31]. In this work, we extend the approach to the analysis of multicellular patterns and the network dynamics that create them. Kolmogorov complexity of an object is the size of the shortest program that can produce that object. The exact calculation is undecidable and, therefore, cannot be computed as defined. As an approximation, the compression size of an object is used. Kolmogorov analysis describes the similarity of two objects as the size of the shortest program that can translate one object to the other. This measure is approximated by the Normalized Compression Distance (NCD) [32], where the NCD of two objects is 0.0 if they are identical and 1.0 if they are random. Set Complexity combines the analysis of single objects and pairwise objects to arrive at a complexity measure of a set of objects, given in Equation 1. By employing NCD as a metric to evaluate the similarity between all pairs of objects in a set, set complexity discounts the influence of the pairs of objects that are randomly related or redundant. As long as any object can be encoded as a string, Set Complexity can compute the information content that resides in the set. Set Complexity of a set of n strings S = {s1 , . . . , sn } is defined:
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the same cell types. The state of all the genes as the networks along with multicellular patterns are recorded for analysis of information content. After running the randomly-generated GRNs, single and coupled feedback and feedforward loops are inserted into the randomly generated intracellular GRN (see Figure 3). The network is run again with the inserted motifs to identify the attractors and visualize the multicellular patterns that are formed. The Kolmogorov complexity of both the gene network dynamics and multicellular patterns is computed by an information theoretic measure called Set Complexity described below.
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X X 1 C(si ) NCD(si , sj )(1 − NCD(si , sj )) n(n − 1) s ∈S s 6=s i
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Where C(si ) is the compression size of string si . The term NCD(si , sj )(1− NCD(si , sj )) is maximized when NCD(si , sj ) = 0.5, which occurs when C(si + sj ) ' C(si )/2 − C(sj ), assuming C(si ) > C(sj ). The influence of similar and dissimilar strings is discounted by NCD(si , sj )(1 − NCD(si , sj )) because NCD(si , sj ) will be near 1.0 or 0.0. In this implementation, bZip is used as a compression algorithm. A one-to-one mapping is required to encode an object into a string so that no information is lost. The temporal dynamics are converted to a string by concatenating in column-order the state of the network at each time period. The spatial pattern is likewise converted by traversing the cells in row order and concatenating the type ID of each cell. To convert a GRN to a string, a more complex procedure is needed. Here, a unique ID is assigned to each logic gate and edge; then the whole epithelial network is traversed, and substrings created for each logic gate that includes its truth table and its input and output edge IDs. The string also includes a row order traversal of the connection matrix. See [33] for the exact algorithm for encoding Boolean networks. 4.2. Structural modularity score measurement The most common method used in the literature to score structural modularity is a method by Newman[34]. Newman’s method to compute the modularity score Q for a given network is as follows:
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5.1. Structural modularity To analyse how the structural modularity of a network influences the dynamics and pattern complexity, we partition a population of random networks by percentile of their modularity score distribution. Networks that lie in lower third are defined as non-modular, and networks that lie in upper third are defined as highly modular. Those networks considered non-modular exhibit the same low modularity score as random networks. Networks that lie in the middle third were not considered in this study. Figure 4 illustrates the average dynamics complexity and pattern complexity for 600 non-modular and 600 highly-modular networks. Results in Figure 4(a) show how highly modular networks produce higher dynamics and pattern complexity when cell-cell signaling is orthogonal. A statistical analysis confirms the distribution of the two classes of networks are significantly different Table 1 and Table 2.
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Figure 4: Non modular vs. modular. Average dynamics and pattern complexity for 60 non-modular (arrow tails) and 60 highly-modular (arrow heads) networks for orthogonal(orange) and symmetric signaling(dark red).
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Critical modular
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90.059
112.25
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18.212
20.827
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273.23
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The observation for symmetric signaling with a critical configuration for each module in Figure 4(b), suggest that the networks with modular structure can produce multicellular patterns with higher information content without an increase in the network dynamics complexity. Another interesting observation was that for symmetric signaling with chaotic configuration (Figure 4(b)), both network dynamics and pattern complexity decrease. As shown in the previous study [3] symmetric signaling produces only simple information patterns and dynamics because information transfer among adjacent cells combined using disjunction, which loses directional information. Now when the configuration of each module is chaotic, structural modularity of network 11
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5.2. Functional Modularity In this section, we explore the effect of insertion of the best-known motifs illustrated in Figure 2(b) into randomly generated intracellular GRNs in the critical domain. For each motif, 600 random networks were generated and then each network was modified by the insertion of that regulatory motif. To obtain the data for analysis, each network was executed from random initial conditions and the dynamics and resulting pattern recorded. Following this step, the network data is grouped into sets of 10 and the set complexity computed for the networks, the dynamics, and the pattern. Figure 5 shows the influence of insertion of a positive feedback loop (Figure 2(b)) into a random GRN. The vertical axis of both graphs is the pattern complexity; on the left shows the relationship with dynamics complexity, on the right shows the relationship with network complexity. Each point on the graph corresponds to a set of 10 runs described above. Insertion of a double positive feedback loop into random GRNs clearly modifies the behavior of the network. The addition of the motif increases pattern complexity significantly while slightly reducing the complexity of the dynamics (Figure 5 left). Little effect on network complexity is observed (Figure 5 right).
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helps to produce a more simplified pattern and dynamics.
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Figure 5: Effect of insertion of a double positive feedback loop on network, network dynamics and pattern complexity for GRNs in the critical domain. 272
Figure 6 illustrates the average network, network dynamics and pattern 12
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complexity for 600 network sets for each of the motifs. The influence of each motif is represented as an arrow. The start of the arrow is the set of random networks and is the same for all motifs. The arrowhead is the average complexity of the set of all networks with the indicated motif inserted. The insertion of a negative feedback loop, a double negative feedback loop or a double positive feedback loop all have the same qualitative effect of decreasing the complexity of network dynamics and increasing pattern complexity. However, a double positive loop increases pattern complexity much more than either of the negative feedback loops (Figure 6(a)). Insertion of feed-forward motifs decreases dynamics with almost no effect on the pattern complexity. Only the double negative motif leads to a reduction in pattern complexity. Statistical analysis confirms that insertion of these motifs makes a significant change in the average network dynamics and patterns complexity. ANOVA analysis presented in Table.3 and Table 4 shows that insertion of these motifs makes a significant change in the average network dynamics and patterns complexity.
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Figure 6: (a) Effects on network dynamics and pattern complexity of inserting regulatory motifs into random GRNs. Average dynamics and pattern complexity for 60 random sets of GRNs (arrow tails) and 60 sets of GRNs with the indicated inserted motifs (arrow heads). (b) Effects on network and pattern complexity of inserting regulatory motifs into random GRNs.
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Table 3: ANOVA analysis of dynamics complexity illustrates differences for various groups of motifs over networks with orthogonal and symmetric signaling
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MS 3419.93 266.27
F 12.84
P-value 1.1E-11
F crit 2.234
F 39.74
P-value 9.5E-34
F crit 2.234
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df 5 444 449
Symmetric signaling Sum Average Variance 1560.25 20.80 15.96 1423.40 18.97 8.18 1828.02 24.37 7.64 1658.00 22.10 8.15 1466.11 19.54 4.17 1718.50 22.91 4.02
df 5 444 449
MS 318.94 8.02
290 291 292 293 294 295 296 297 298 299 300 301 302 303 304
6. Discussion
In this study, we explored how network dynamics and pattern complexity are affected by structural modularity in general and regulatory motifs in particular. The results show that networks with a modular structure, in general, tend to produce more complex multicellular patterns without a significant increase in gene expression dynamics. The results indicate that network motifs that contain feedback loops increase the information complexity of the multicellular patterns. Another important observation was negative feedback loops do not affect the dynamics complexity significantly as positive feedback loops do. We hypothesize that variation in dynamics complexity associated with adding different types of motifs to random GRNs originates from the time it takes for the GRN to reach a steady state and the proportion of single state versus cyclic attractors produced by the GRN. Since the motifs studied in this work act as multistable switches, they simplify the complex cyclic attractors to attractors with a few states. Gene expression in feedback loops
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Orthogonal signaling ANOVA Source of Variation SS Between Groups 17099.6 Within Groups 118226.6 Total 135326.3 Symmetric signaling ANOVA Source of Variation SS Between Groups 1594.73 Within Groups 3563.42 Total 5158.16
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Groups Without Motifs Double-negative Double-positive Negative-Feedback Coupled P-P Type-1 feed Forward
Orthogonal signaling Sum Average Variance 3911.45 52.15 474.35 3017.06 40.22 284.05 2633.55 35.11 101.26 3322.99 44.30 181.91 3355.92 44.74 306.40 2560.45 34.13 249.67
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Table 4: ANOVA analysis of pattern complexity illustrates differences for various groups of motifs over networks with orthogonal and symmetric signaling.
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MS 473.35 19.06
F 24.82
P-value 4.4E-22
F crit 2.234
F 67.26
P-value 3.0E-52
F crit 2.234
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df 5 444 449
Symmetric signaling Sum Average Variance 592.51 7.90 3.55 456.57 6.08 1.97 805.59 10.74 5.08 687.03 9.16 2.66 530.80 7.07 2.35 692.85 9.23 3.22
df 5 444 449
MS 211.53 3.14
306 307 308 309 310 311 312 313 314 315 316 317 318 319 320
reach a steady state quickly, and so reduce the length and complexity of cyclic attractors. We hypothesize that this is why in all cases dynamics complexity decreases from that of the original random networks. The rate of dynamics complexity reduction associated with the addition of the motifs with negative feedback loops is significantly lower than for positive loops. Unlike positive feedback loops, negative feedback loops do not increase the time to reach steady states [18]. Therefore, they don’t affect the dynamics complexity noticeably. As results show, negative feedback loop motifs (such as single negative feedback loop and coupled positive-negative loops) have the lowest reduction in their dynamics complexity. All the motifs with feedback loops impact the pattern complexity. In fact, the only motifs in this study that has no effect on pattern complexity are feed-forward loops. This observation confirms the association of feedback loop motifs with developmental networks that mediate important cell fate decision. The increase in information complexity of the multicellular patterns shows more certain cell-fate decisions have been taken. Only double nega-
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Orthogonal signaling ANOVA Source of Variation SS Between Groups 2366.77 Within Groups 8466.32 Total 10833.10 Symmetric signaling ANOVA Source of Variation SS Between Groups 1057.66 Within Groups 1396.30 Total 2453.97
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Groups Without Motifs Double-negative Double-positive Negative-Feedback Coupled P-P Type-1 feed Forward
Orthogonal signaling Sum Average Variance 1233.65 16.44 25.15 1026.61 13.68 22.74 1559.75 20.79 13.10 1457.84 19.43 19.91 1372.30 18.29 24.27 1250.56 16.67 9.22
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Acknowledgements
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tive motifs decrease the complexity of the patterns formed. We hypothesize that this is due to the bistable behavior of these motifs, where they rapidly converge to one state or another producing few distinct non-cyclic attractors. Overall, we find that all motifs with feedback decrease the complexity of the dynamics. By decreasing the complexity of the dynamics, these motifs could increase the robustness in pattern formation in the presence of noise as shown in [8]. Feedback motifs tend to increase pattern complexity while decreasing dynamics complexity. This implies that these motifs increase the“efficiency” of pattern formation where simpler dynamics leads to more complex patterns. Such an increase in efficiency may be one of the principle reasons that feedback motifs are enriched within developmental networks.
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Research reported in this publication was supported by the National Institute Of General Medical Sciences of the National Institutes of Health under Award Number P50GM076547, Thanks to Ilya Shmulevich for helpful discussions. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health.
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