Design of a flexible component gathering algorithm for converting cell-based models to graph representations for use in evolutionary search
© Budnikova et al.; licensee BioMed Central Ltd. 2014
Received: 31 August 2013
Accepted: 30 April 2014
Published: 10 June 2014
The ability of science to produce experimental data has outpaced the ability to effectively visualize and integrate the data into a conceptual framework that can further higher order understanding. Multidimensional and shape-based observational data of regenerative biology presents a particularly daunting challenge in this regard. Large amounts of data are available in regenerative biology, but little progress has been made in understanding how organisms such as planaria robustly achieve and maintain body form. An example of this kind of data can be found in a new repository (PlanformDB) that encodes descriptions of planaria experiments and morphological outcomes using a graph formalism.
We are developing a model discovery framework that uses a cell-based modeling platform combined with evolutionary search to automatically search for and identify plausible mechanisms for the biological behavior described in PlanformDB. To automate the evolutionary search we developed a way to compare the output of the modeling platform to the morphological descriptions stored in PlanformDB. We used a flexible connected component algorithm to create a graph representation of the virtual worm from the robust, cell-based simulation data. These graphs can then be validated and compared with target data from PlanformDB using the well-known graph-edit distance calculation, which provides a quantitative metric of similarity between graphs. The graph edit distance calculation was integrated into a fitness function that was able to guide automated searches for unbiased models of planarian regeneration. We present a cell-based model of planarian that can regenerate anatomical regions following bisection of the organism, and show that the automated model discovery framework is capable of searching for and finding models of planarian regeneration that match experimental data stored in PlanformDB.
The work presented here, including our algorithm for converting cell-based models into graphs for comparison with data stored in an external data repository, has made feasible the automated development, training, and validation of computational models using morphology-based data. This work is part of an ongoing project to automate the search process, which will greatly expand our ability to identify, consider, and test biological mechanisms in the field of regenerative biology.
High-throughput technologies have led to an accumulation of large amounts of data that can be used to advance scientific inquiry given the appropriate tools. However, our inability to effectively visualize or conceptualize these data, particularly multidimensional data, is one of the factors preventing its integration into the scientific process. One of the promising means of using these data is to develop, train, and validate computational models, preferably those with interactive visual interfaces. Advances in computational modeling platforms are beginning to allow simulation of biological systems from the single cell biochemical level to more abstract multicellular environments, such as representative tissues, organs, or even organisms. These emerging computational tools are poised to put the power of bioinformatics and data interpretation back into the wet-bench biologists hands by automatically incorporating data from the aforementioned datasets with tools for visualization, experimentation, and data analysis.
Many high-throughput technologies collect large amounts of measurement data that are conducive to being stored in databases. For example, a database can easily house multi-scale gene expression data obtained from a single cell to a whole organism while also documenting the source and experimental methods associated with the data. Such repositories are well suited for data consisting of lists of gene and protein abundance, for example. However, new ontologies and formalisms are required for collecting and describing certain kinds of higher-order data. For instance, the outcome of experiments involving shape or morphology can be challenging to describe accurately, particularly in a way that others can search for or interpret computationally. This problem has been particularly challenging in areas of development and regeneration where a description of the organ, appendage, or organism is one of the key reported observations.
The planarian worm is a model organism in regenerative biology that perfectly illustrates the problem of storing shape-based experimental results in a formal database. These free-living flatworms have exceptional regenerative properties that have fascinated biologists for centuries . They are able to regenerate aged, damaged, or lost tissues with the help of a large adult stem cell population . Despite being complex organisms possessing bilateral symmetry, musculature, intestine, and a central nervous system including a true brain [3, 4], fragments smaller than 1/200th of the adult size can remodel and regenerate an intact worm . This astonishing regenerative ability has stimulated an effort to understand its underlying mechanisms , producing an extensive number of experiments based on amputations , drug-induced phenotypes [7, 8], and RNAi gene-knockdowns [9–13]. However, despite these important efforts, we still lack a comprehensive model that can explain more than one or two aspects of planarian regeneration .
Recently, the Levin lab has developed a new tool (Planform) to aid in the assimilation of these data using a graph-based formalism to describe anatomy and morphology along with a new ontology for describing experimental manipulations and observations [15, 16]. The flexible and extensible graph notation allows worm regions and organs to be described as nodes connected by linkages with associated angles and length parameters. Based on this approach, the Planform Database (PlanformDB) was designed and curated to include a complete description of the many planarian experiments and outcomes that exist throughout the literature. Such a resource does not only make it possible for scientists to search and compare worm morphologies, but it also provides an extractable resource for bioinformatics applications.
We are currently combining Planform, agent-based modeling, and an evolutionary search engine to develop an automated system for searching and validating computational models of regeneration. Agent-based modeling holds promise for studying the emergent behavior and complex interactions between signaling networks involved in directing regeneration, when multi-scale or multi-cellular systems are supported. To this end, we are using a modeling platform (CellSim) where the central agents are autonomous cells containing many of the biological primitives necessary for simulating living systems . The current version of this software contains a number of useful features to support this endeavor, including a 3-D interface for visualization and tools for performing experimental manipulations within the client-server architecture. The process of developing, testing, and validating a complex model by hand can be a daunting task, particularly when many individual experimental outcomes are combined. To simplify this process, we have incorporated an evolutionary search engine that can automate this process using a genetic algorithm driven by appropriate fitness metrics that are informed by the Planform Database (PlanformDB). Our ultimate goal is for this integrated system to identify computational models that can account for many, if not all, of the available experimental outcomes related to planarian regeneration. We believe that this general approach holds the promise to spur biological discovery, develop novel insights into long-standing problems and biases, and elucidate previously unobserved biological behaviors.
This paper presents a novel agent-based planarian model capable of simulating basic biological behavior. The model is suitable for automated and varied experimental manipulations akin to those traditionally performed by wet-bench biologists and represented in the PlanformDB. This model includes a reaction network that responds to manipulations by initiating appropriate head and tail regeneration. Importantly, we describe an algorithm that allows translation of multicellular simulation output into a formal graph representation equivalent to that described by Lobo and colleagues [15, 16]. This real-time translation is central to the automation of model discovery as it enables use of a fitness metric based upon a graph-edit distance calculation, which quantitatively compares simulation output and target morphologies stored in the PlanformDB. The combination of the model, translation algorithm, and fitness metric provide the basis for future automated model discovery in regeneration biology.
Modeling a classic planaria regeneration experiment
Our representation of this work included a simple architecture of 420 planar cells arranged as a rectangular abstraction of an intact worm (Figure 1b). The number of cells was chosen empirically to provide a robust system that could be reasonably manipulated by one or more simultaneous or sequential cuts. The implementation of our morphogen-based model was represented in Cellsim using a series of metabolic and transcription reactions (Figure 2(b)) where every cell was controlled autonomously by this same network. In response to a simulated cut, a Regeneration signal activates a Regeneration pathway, which simultaneously promotes head and tail development responses. The head and tail development pathways are constitutively repressed by the presence of a morphogen (i.e. Head gradient and Tail gradient) emanating from existing head or tail cells in the simulation and spread to neighboring cells through gap junctions. The morphogen gradients will disappear or be diminished following a transverse cut when the source (head or tail) is physically removed, causing one developmental pathway to be favored over the other. Furthermore, the head and tail resources repress each other to ensure a unique cell state is ultimately achieved.
At the start of a simulation, the head, trunk, and tail regions were defined by introducing one of three cell-state resources (head, hCell; trunk, iCell; tail, tCell) into each cell. Simulations were then run for approximately 200 steps to allow the network to reach homeostasis and provide sufficient time to develop long-range morphogen gradients. As shown in panel 1 of Figure 1b, worms consisted of head (blue) and tail (purple) regions separated by a trunk (orange). Next, a transverse cut was simulated by injecting a resource, Lysis, into a cross-section of cells located at or near the mid-line of the worm. The presence of Lysis results in a localized cell death response that results in separation of the initial worm into two worm fragments lacking either a head or tail (panel 2). Nearby cells respond to the cut by inducing a localized Regeneration signal, which in turn activates a cell’s Regeneration pathway. At this point, simulations consisting of two worm fragments were advanced another 200 steps prior to evaluating their emergent outcomes. The regulatory network parameters were optimized by hand for the network (Figure 2(b)), which resulted in proper regeneration of head and tail regions as shown in panel 2 of Figure 1b. For simplicity, these simulations did not include cell growth, division, or rearrangements, but these properties will be introduced in future studies.
These results showed that we could develop a simple model of planarian regeneration using a long-range morphogen gradient that could faithfully respond to at least simple manipulations. However, it was clear that hand-design and tuning a single model to represent the many experimental outcomes described in the literature would be a daunting task without computational automation. This challenge could be alleviated using an automated method of model creation and evaluation, such as performed by genetic algorithms . These algorithms are based upon the principles of evolution where individuals in a population are generationally- modified through random genetic mutations and crossovers during reproduction. Individuals chosen to contribute to the offspring of the subsequent generation are selected, in part, based upon a fitness metric, which quantitatively defines how well each individual matches the characteristics of the target. This evolutionary search technique continues in an automated fashion until an individual matching the desired target (fitness value of 1.0) is generated.
Graph formalism provides a convenient means of storing morphologies and comparing worms
The challenge of automating searches to identify possible planarian regeneration mechanisms was made more tractable by the database and formalism developed to describe wet-bench experiments and outcomes [15, 16]. Within PlanformDB, worm morphologies are described using a graph-based formalism as part of a more general ontology for describing regeneration experiments. Briefly, a graph defines anatomic regions and organs as nodes where their size, spatial orientation, and connections are defined by parameters and linkages between adjoined nodes. For example, a simple description of the regions within a normal planarian consists of three connected nodes (head, trunk, and tail) as shown in Figure 1c. Although the formalism supports more complex descriptions of worms including organs, those aspects of worm anatomy were not considered in the current work. A particular experiment may include a description of the observed starting, intermediate, and ending morphologies of worms along with the physical or chemical manipulations performed in the laboratory. The database currently describes most, if not all, of the published planarian regeneration experiments for use by this and other projects.
In this study, we extended and adapted an existing genetic algorithm (CSGA) to fit our needs to model and evaluate planarian regeneration. One of the key adaptations was providing the CSGA access to the PlanformDB to facilitate simulation and fitness evaluation. However, the challenge of comparing our agent-based simulation output to a graph-based representation presented a significant challenge. In order to facilitate these comparisons, we chose to convert simulation results into a graph representation for a number of reasons, including increased flexibility as the CSGA could be extended to support additional modeling platforms as long as their output could also be translated into this graph formalism. More importantly, many methods currently exist for operating on, transforming, and comparing graphs which can be included as part of the fitness evaluation step of an automated evolutionary search [19–21]. Included in this repertoire are a number of algorithms suited for measuring similarity between two graphs . Of these, the graph edit distance algorithm is the most flexible and powerful and was chosen here as it deals with structural errors and any type of graph node and edge labels [23, 24].
The graph edit distance is defined as the minimum number of distortions required for transforming one graph into another. These distortions are referred to as graph edit operations, where each edit has a defined cost associated with it . A particular sequence of edit operations is called an edit path, and the total cost of the edit path is the graph edit distance. Graphs that are similar to each other typically have small edit distances, whereas dissimilar graphs have large edit distances. The cost of each type of graph edit operation varies and is dependent upon the perceived severity of the operation. For example, the deletion of a node from a graph is generally viewed as having a higher cost than a node parameter change. Thus, the graph edit distance can be used as a quantitative similarity measure to compare and order individuals within a population, and thus serve as a metric within a fitness evaluation to guide the evolutionary search process.
Design of a connected component analysis algorithm to convert cell simulation output into graph representations
Once each cell in the snapshot is assigned to a specific region, the algorithm determines the number of neighboring regions using the border cells found during the recursive process and establishes links between nodes where regions are considered linked if their border cells are adjacent to each other. Other necessary parameters for a complete graph representation include the distance between the connected regions’ centers (length of link), orientation with respect to each other (angle of the link relative to the x-axis), and the border between the two regions (location along the link where the two regions meet). The center of a region is calculated by averaging the spatial centers of every cell within a particular region. The Euclidian distance between these points of neighboring regions is used to define the length and orientation of each link. Finally, the graph component parameter defining the borders of regions in each direction is calculated from the location of the most distantly located cell in a specific direction. The number of parameters for a region depends on the number of links it has with other regions. Figures 1(b to c) show examples of simulation morphologies that have been converted to a graph formalism using this algorithm.
Simulation snapshots are converted to well-ordered graphs using our conversion and graph-edit distance algorithms
This table presents the graph edit costs used for graph edit distance calculations in this manuscript
Change region type
Change region parameter
0.1 per unit changed
Change link distance
0.1 per unit moved
Change link angle
Change link angle > 90 penalty
A cell connectivity distance threshold effects region determination
A comparison of more complicated morphologies highlighted the need for a flexible distance threshold in the component gathering algorithm. Since the cells in our simulation have radii of 0.5 units, the Euclidian distance between two adjacent cells can be as low as one. However, using a very rigid measure for identifying neighboring cells and determining the borders of regions can have dramatic effects on the graph conversion. For example, consider the morphology of the second individual shown in Figure 5. In this individual thin lines of trunk cells dissect the head and tail regions into a number of potentially distinct heads and tails if the borders are considered rigidly. Comparison of this individual with the target results in a very high graph edit distance due to the cost associated with having multiple heads. However, in the context of a evolutionary search, this individual may be very close to producing the target morphology.
A flexible threshold parameter was introduced to reduce the rigidity of region definitions, which allowed neighboring regions separated by thin regions to be merged in the final graph representation. Increasing the threshold value reduces the stringency by increasing the search distance between cells for neighbors with the same state. Thus, in the example just discussed, increasing the threshold value allowed the multiple head regions to be lumped into a single head region. The graph edit distance of this worm is much lower resulting in a fitness value close to one.A second example highlighting the importance of this parameter to component gathering is presented at the bottom of Figure 5. This worm represents a classic experiment that involves bifurcating the head region into two fully-developed heads. These two heads are separated physically and should be classified as two-headed. A threshold parameter of less than three results in the desired graph conversion in our algorithm, whereas the larger value results in a worm with a single head.
These two examples show the necessity of a flexible parameter for determining local regions during a GA run. In the first case, a low threshold was shown to penalize a morphology that was very similar to the target, whereas a high threshold inappropriately favored a morphology containing a physical gap between head regions. An optimal threshold will depend upon the modeling platform and project, but in this work and from an evolutionary perspective a threshold of two was optimal.
Validation of component gathering and graph edit distance during evolutionary search
Our searches included populations of only 30 individuals, but much larger populations are feasible. In order to generate variability in the individuals, a set of mutation and crossover parameters were introduced and applied to each new generation of offspring. These operators and parameters were hand designed for this experiment, but parameter definition will eventually become automated. Searches were performed for individuals with proper regeneration of head or tail following removal of the anterior or posterior regions, respectively. During the evolutionary search, the GA pauses each simulation at a predefined step (e.g. 200 in this experiment) and requests a simulated experiment be performed (e.g. cutting off the head or tail). Each individual simulation is resumed and continues until the GA requests a snapshot to evaluate (e.g. step 400). At this point, the simulation snapshot is used to create a graph representation using our component gathering algorithm, which is then compared to the target individual from the PlanformDB to determine the individual’s fitness value. High scoring individuals were chosen for reproduction to generate the next generation of individuals which were independently mutated to increase variability in the population.The GA was successful in identifying individuals with fitness values very close to the target value of 1.0. Two such regulatory networks are shown in Figure 6. The network shown in Figure 6(b) is a representative evolved solution when selecting for a worm that properly regenerates head after removal of the anterior region of an intact worm. Similarly, the network in Figure 6(c) comes from a representative worm that was evolved to regenerate a tail following removal of the posterior end. There are noticeable similarities between the two solutions. In both solutions, a direct connection was made between the cut response to regeneration of the missing region, head or tail, respectively. Although each search found a solution to the experiment at hand, the solutions were limited in their flexibility to respond to other permutations. As expected, neither network was capable of solving the reverse problem as their was no selective pressure in the evolutionary search. Nonetheless, these results show that our evolutionary search process is capable of finding solutions using our connected component analysis to convert cell-based individuals into graphs, which are compared with the appropriate target extracted from the PlanformDB using the graph edit difference evaluation metric.
The inflexible network solutions emphasize the importance of searching for solutions using rigorous fitness criteria and why an automated approach is necessary. Future experiments will target networks that are capable of handling both anterior and posterior ablations. Simulation snapshots contain a detailed description of the current state of the simulation, including a list of all cells, their location, shape, genomes, metabolic equations, environmental conditions, and the concentration of all resources. Thus, snapshots provide much richer information about the cells and individuals than the graph formalism and will provide opportunities to develop additional fitness metrics to complement graph edit distance in the future.
Discussion and conclusion
One of the challenges in the biological sciences is the development of new methods for data visualization and integration to provide informative and predictive insight into the scientific process. Computer models hold great promise in this area, but often involve significant human interaction and time. In this study, we laid the foundation for a system of automated model discovery and development that incorporates shape-based experimental data from a repository of documented experiments. Graphs are a powerful and convenient means of describing morphological data. Using comparison methods, such as the graph difference evaluation, one can easily search such a database for results that are similar or identical. We showed that the utility of the graph difference evaluation could be further extended as a fitness evaluation metric during evolutionary search. This method was combined with a cell-based modeling platform to model basic regeneration of the planarian flatworm. Agent-based models are particularly amenable to this approach as they are tractable to simulated experimental manipulations combined with fully emergent outcomes. The ability to automate these behaviors fits nicely into an automated discovery system that can be driven by a genetic algorithm search engine. Furthermore, simulators that include robust visualization capabilities make it very convenient for the scientist to evaluate or experiment on a set of search results.
Planarian worms provide an excellent model system for developing such an automated search process due to the plethora of experimental data in the literature, and now available in a curated database (PlanformDB). However, the principles inherent in this design are extensible to any system where shape is an integral component of the observable outcomes. That said, developing model discovery systems that automatically incorporate experimental data is a general and attainable goal that is not limited to systems dependent upon morphological data.The challenge of describing phenotypic outcomes based upon morphological characteristics is challenging for biological systems and cell-based computational models alike. We showed that converting cell-based simulation output into graphs can be achieved using a component gathering algorithm that identifies regions and their juxtaposition to each other and converts them into nodes joined by linkages. The resulting graphs can be stored to a database and easily reconstructed later and/or compared with other graphs using algorithms such as the graph edit distance. These comparisons and the resulting metric were incorporated into an evolutionary search where the genetic algorithm retrieved its target morphology from a data repository of experimental outcomes and used the graph edit distance as a fitness metric to drive development. Using a small population size of 30 individuals, our mutation and crossover frequencies were sufficient to generate a solution state in as few as 19 generations for the experiments shown in Figure 6(a).
The method for converting a simulation snapshot to a graph formulation works well, and the converted morphologies reflected shape and positions of body regions relative to each other in space. The graph-based fitness function thus accurately distinguishes the shapes of different morphologies, but does have difficulty predicting which morphology will more likely regenerate into the target morphology. To improve the effectiveness of our graph-based fitness function, future work will seek to automate the optimization of the graph edit costs to not only reflect the shapes of individuals, but also to favor morphologies that are more likely to regenerate into the target individual. The optimization of graph edit costs needs to be automated since tweaking graph edit costs allows one to achieve a more accurate graph comparison for some cases, but there is no perfect parameter assignment that covers all cases, and so it should be chosen depending on the needs and design of the experiment.
Future work will also develop additional morphological based fitness functions to act side by side with the graph edit distance fitness function. This will allow researchers to choose from among multiple morphological fitness operators, possibly combining the output of several evaluators instead of assigning a fitness value based on one fitness function evaluation.
Cell-based modeling platform
Cellsim is an agent-based modeling platform whose principle agents are cellular in nature. These cells and their behaviors are rooted in biology as the environment, interactions, metabolism, and signaling networks are designed based upon biological primitives. Cells are capable of proliferating, growing, dividing, dying, and regulating metabolic and genetic networks in response to changes in their local environment, including cell interactions and signaling. As a result, cells have emergent properties as they are autonomous, evolvable, exhibit inheritance, and are contingent upon their neighbors. Another important feature of this system is its ability to be automated and manipulated using a genetic algorithm search engine .
The versatile genetic algorithm associated with Cellsim was expanded as part of this work and includes many parameters to customize the common elements of a genetic algorithm, such as number of crossovers, mutation rates, selection criteria, and population size.
Graph edit distance algorithm
where P(g1,g2) is the set of edit paths that transform graph g1 into g2, c is the edit cost function and e i denotes an edit operation. Generally speaking, determining the graph edit distance requires that we examine the set of paths that transform g1 to g2 and calculate the path costs for each. This is non-trivial but can be achieved in an optimally efficient manner using the A* best-first search algorithm .
The graph edit distance calculation has been adapted for comparing planaria graph representations. A list of edit operations and an example of the corresponding costs used in this paper are given in Table 1.
For simplicity, we are working with worms whose graph representations are devoid of organs for this analysis, but will include them in later experiments.
The minimal path cost between two graphs g1 and g2 is found using the A* search algorithm. The possible edit paths can be viewed as forming a tree, where the edges of the tree are individual edit operations and the nodes are graphs. The inner nodes of the tree correspond to partially edited graphs, and the leaf nodes represent complete edit paths, all of which terminate at the target graph. The search for the minimal edit path starts with one of the graphs, say g1, as the root of the tree, and defines the possible branches from this node to be all possible single edit operations that could be applied to the original graph g1. At each step of the A* search, the algorithm expands (explores) the branch of the tree that leads to a node on the search frontier that has the minimum estimated total path cost. The minimum estimated total path cost is derived from the sum of the cost of the edits required to reach the node being expanded from the root, plus an estimate of the total cost of the edits required to reach the goal state (graph g2) from the node being expanded. The algorithm terminates the first time it expands the goal state, as the path that it finds at this point is guaranteed to be optimal.
We would like to thank past and present employees of Crowley Davis Research for their hard work on the Cellsim platform and for making it available for these studies. This work was made available through an NSF-CDI grant(EF-1124665 and EF-1124651).
- Pallas PS: Spicilegia zoologica quibus novae imprimis et obscurae animalium species iconibus, descriptionibus atque commentariis illustrantur. Berolini, Prostant, Apud Gottl. 1774, 8 (4): 23-
- Aboobaker AA: Planarian stem cells: a simple paradigm for regeneration. Trends Cell Biol. 2011, 21 (5): 304-311. 10.1016/j.tcb.2011.01.005. [http://www.sciencedirect.com/science/article/pii/S0962892411000158]View ArticlePubMed
- Sarnat HB, Netsky MG: The brain of the planarian as the ancestor of the human brain. Can J Neurol Sci. 1985, 12 (4): 296-302. [http://view.ncbi.nlm.nih.gov/pubmed/4084864]PubMed
- Reddien PW, Alvarado AS: Fundamentals of planarian regeneration. Annu Rev Cell Dev Biol. 2004, 20: 725-757. 10.1146/annurev.cellbio.20.010403.095114. [http://www.annualreviews.org/doi/abs/10.1146/annurev.cellbio.20.010403.095114] [PMID15473858]View ArticlePubMed
- Morgan TH: Experimental studies of the regeneration of Planaria Maculata. Arch Entwickelungsmech Org. 1898, 7: 364-397. 10.1007/BF02161491.View Article
- Adell T, Cebrià F: Gradients in planarian regeneration and homeostasis. Cold Spring Harb Perspect Biol. 2010, 2: [http://dx.doi.org/10.1101/cshperspect.a000505]
- Oviedo NJ, Morokuma J, Walentek P, Kema IP, Gu MB, Ahn JM, Hwang JS, Gojobori T, Levin M: Long-range neural and gap junction protein-mediated cues control polarity during planarian regeneration. Dev Biol. 2010, 339: 188-199. 10.1016/j.ydbio.2009.12.012. [http://www.sciencedirect.com/science/article/pii/S001216060901402X]View ArticlePubMed CentralPubMed
- Beane WS, Morokuma J, Adams DS, Levin M: A chemical genetics approach reveals H,K-ATPase-mediated membrane voltage is required for planarian head regeneration. Chem Biol. 2011, 18: 77-89. 10.1016/j.chembiol.2010.11.012. [http://www.sciencedirect.com/science/article/pii/S1074552110004473]View ArticlePubMed CentralPubMed
- Petersen CP, Reddien PW: Smed-βcatenin-1 is required for anteroposterior blastema polarity in planarian regeneration. Science. 2008, 319 (5861): 327-330. 10.1126/science.1149943. [http://www.sciencemag.org/content/319/5861/327.abstract]View ArticlePubMed
- Rink JC, Gurley KA, Elliott SA, Alvarado AS: Planarian Hh signaling regulates regeneration polarity and links Hh pathway evolution to cilia. Science. 2009, 326 (5958): 1406-1410. 10.1126/science.1178712. [http://www.sciencemag.org/content/326/5958/1406.abstract]View ArticlePubMed CentralPubMed
- Petersen CP, Reddien PW: Polarized notum activation at wounds inhibits wnt function to promote planarian head regeneration. Science. 2011, 332 (6031): 852-855. 10.1126/science.1202143. [http://www.sciencemag.org/content/332/6031/852.abstract]View ArticlePubMed CentralPubMed
- Umesono Y, Tasaki J, Nishimura Y, Hrouda M, Kawaguchi E, Yazawa S, Nishimura O, Hosoda K, Inoue T, Agata K: The molecular logic for planarian regeneration along the anterior-posterior axis. Nature. 2013, [http://dx.doi.org/10.1038/nature12359]
- Sikes JM, Newmark PA: Restoration of anterior regeneration in a planarian with limited regenerative ability. Nature. 2013, [http://dx.doi.org/10.1038/nature12403]
- Lobo D, Beane WS, Levin M: Modeling planarian regeneration: a primer for reverse-engineering the worm. PLoS Comput Biol. 2012, 8 (4): [http://dx.doi.org/10.1371/journal.pcbi.1002481]
- Lobo D, Malone TJ, Levin M: Planform: an application and database of graph-encoded planarian regenerative experiments. Bioinformatics. 2013, 29 (8): 1098-1100. 10.1093/bioinformatics/btt088.View ArticlePubMed CentralPubMed
- Lobo D, Malone TJ, Levin M: Towards a bioinformatics of patterning: a computational approach to understanding regulative morphogensis. Biol Open. 2013, 2 (2): 156-169. 10.1242/bio.20123400.View ArticlePubMed CentralPubMed
- Andersen T, Newman R, Otter T: Shape homeostasis in virtual embryos. Artif Life. 2009, 15 (2): 161-183. 10.1162/artl.2009.15.2.15201.View ArticlePubMed
- Grandstrand O: Agent Based Evolutionary Search. 2010, Berlin, Heidelberg: Springer-Verlag, chap. Agent Based Evolutionary Approach: An Introduction
- Lobo D, Vico V, Dassow J: Graph grammars with string-regulated rewriting. Theor Comput Sci. 2011, 412: 6101-6111. 10.1016/j.tcs.2011.07.004.View Article
- Bandelt H, Chepoi V: Metric graph tehory and geometry: a survey. Contemp Math. 2008, 453: 49-86.View Article
- Mason O, Verwoerd M: Graph theory and networks in biology. IET Syst Biol. 2007, 1: 89-119. 10.1049/iet-syb:20060038.View ArticlePubMed
- Conte D, Foggia P, Sansone C: Thirty years of graph matching in pattern recognition. Int J Pattern Recogn Artif Intell. 2004, 18: 265-298. 10.1142/S0218001404003228.View Article
- Neuhaus M, Bunke H: Bridging the Gap Between Graph Edit Distance and Kernel Machines. 2007, Singapore: World Scientific
- Sanfeliu A, Fu K: A distance measure between attributed relational graphs for pattern recognition. IEEE Trans Syst Man Cybern. 1983, 13: 353-362.View Article
- Rosenfeld A, Pfaltz J: Sequential operations in digital image processing. J ACM. 1966, 13: 471-494. 10.1145/321356.321357.View Article
- Hart P: A formal basis for the heuristic determination of minimum cost paths. IEEE Trans Syst Man Cybern. 1968, 4: 100-107.
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