Experimental visually-guided investigation of sub-structures in three-dimensional Turing-like patterns
EExperimental visually-guided investigation of sub-structuresin three-dimensional Turing-like patterns
Martin Skrodzki a,b , Ulrich Reitebuch c , and Eric Zimmermann c a ICERM, Brown University, Providence, RI, USA b RIKEN iTHEMS, Wako, Saitama, Japan c Institut f¨ur Mathematik und Informatik, Freie Universit¨at Berlin, Berlin, Germany
ARTICLE HISTORY
Compiled July 1, 2020
ABSTRACT
In his 1952 paper “The chemical basis of morphogenesis”, Alan M. Turing presenteda model for the formation of skin patterns. While it took several decades, the modelhas been validated by finding corresponding natural phenomena, e.g. in the skinpattern formation of zebrafish. More surprising, seemingly unrelated pattern for-mations can also be studied via the model, like e.g. the formation of plant patchesaround termite hills. In 1984, David A. Young proposed a discretization of Turing’smodel, reducing it to an activator/inhibitor process on a discrete domain. From thismodel, the concept of three-dimensional Turing-like patterns was derived.In this paper, we consider this generalization to pattern-formation in three-dimensional space. We are particularly interested in classifying the different arisingsub-structures of the patterns. By providing examples for the different structures, weprove a conjecture regarding these structures within the setup of three-dimensionalTuring-like pattern. Furthermore, we investigate–guided by visual experiments—how these sub-structures are distributed in the parameter space of the discretemodel. We found two-fold versions of zero- and one-dimensional sub-structures aswell as two-dimensional sub-structures and use our experimental findings to for-mulate several conjectures for three-dimensional Turing-like patterns and higher-dimensional cases.
KEYWORDS
Turing patterns; Cellular Automata; Parameter Space; Visual Experiments
AMS CLASSIFICATION
1. Introduction
Why do tigers and zebras have stripes while other animals like cows or leopards arespotted? This question is one of the aspects of the broader task to understand morpho-genesis . Composed of the two Greek words morph´e (shape) and g´enesis (creation), theterm describes the biological process of an organism developing its shape. One of theearliest authors to investigate this question was d’Arcy Wentworth Thompson, whodevoted his 1917 treatise “On Growth and Form” to it, see Thompson (1917). Anothernotable contribution was made by Alan M. Turing in his article “The chemical basis
CONTACT M. Skrodzki. Email: [email protected] a r X i v : . [ q - b i o . N C ] J un f morphogenesis”, issued in the philosophical transactions of the royal society of Lon-don, see Turing (1952). While Thompson’s approach to morphogenesis is largely basedon different growth rates of the animal, Turing focuses—as suggested by the title—ona chemical mechanism giving rise to different skin color patterns. He states in the ab-stract of his article “that a system of chemical substances, called morphogens, reactingtogether and diffusing through a tissue, is adequate to account for the main phenom-ena of morphogenesis,” (Turing 1952, p. 37). We will give a detailed presentation ofTuring’s model in Section 3.1.Indeed, though much later, the patterns predicted by Turing have been found inbiological settings and physical systems. While they indeed describe certain animalskin patterns, they surprisingly also arise in larger biological phenomena, like theformation of termite hills. Section 2.1 lists several of these applications.With the growing availability of computers, discretizations of models become in-creasingly important. Regarding the concept of Turing patterns, a most notably con-tribution was made by David A. Young in his 1984 paper “A local activator-inhibitormodel of vertebrate skin patterns,” see Young (1984). His model is not only discrete,but also reduces Turing’s setup to two simple morphogens with clear functionalities:one activator morphogen that causes cells to be differentiated (colored) and one in-hibitor morphogen that prevents the differentiation of cells. The structures obtainedfrom Young’s discretizations of Turing’s work are consequentially referred to as Turing-like patterns. While Young does never state it in his paper, he basically gave the de-scription of a two-dimensional cellular automaton to produce the patterns. We presentYoung’s approach in detail in Section 3.2 and give some general introduction to cellularautomata in Section 2.2.The formulation of Turing-like patterns by Young allows for a very efficient evalu-ation of different parameters and setups. While Turing has “obtained [his results] ina few hours by a manual computation” (Turing 1952, p. 59), modern computers cancreate respective imagery within seconds. This makes these patterns interesting for dif-ferent applications. For instance, a series of articles has been devoted to generalizationsof two-dimensional patterns to three-dimensional structures, see Skrodzki, Reitebuch,and Polthier (2016); Skrodzki and Polthier (2017, 2018). The second paper consideredthe emergence of Turing-like patterns from a corresponding three-dimensional cellularautomaton, see Section 3.3 for a detailed description. The resulting three-dimensionalpatterns in particular pose a visualization challenge: It is no longer feasible to rendereach cell as a colored solid as the outermost cells would obscure the view to the insideof the pattern. To illustrate the obtained findings, only the borders between differen-tiated and undifferentiated cells were shown, see Figure 5. The authors state at theend of their paper: “. . . with varying parameters, the [two-dimensional] pattern per-forms a phase transition from disconnected points (‘zero-dimensional’) to connectedstripes (‘one-dimensional’). A similar behavior is exhibited by three-dimensional pat-terns. . . . One would expect that in three-dimensional patterns also ‘two-dimensional’connections occur,” (Skrodzki and Polthier 2017, p. 417). However, they were unableto present a corresponding set of parameters to create such patterns.This article is dedicated to closing this gap. After discussing the works mentionedabove in greater detail, we present: • a classification of lower-dimensional sub-structures in Turing-like patterns, • an experimental investigation of these sub-structures in the three-dimensionalcase, • examples for all possible sub-structures, i.e. a proof to the conjecture raised2y Skrodzki and Polthier (2017), • and several conjectures on the sub-structures and parameter spaces for both thethree-dimensional case and higher-dimensional cases.Special emphasis is put on understanding and visualizing not only the Turing-likepatterns, but also their underlying parameter space. As we do not have a thoroughmathematical description available to formally distinguish different sub-structures inTuring-like patterns, see Section 4, our approach is experimental and visually guided.This is possible, as the different sub-structures are easily distinguishable by a humanobserver. Therefore, we can discretize the parameter space and classify each element byvisual inspection. As the number of elements is very large, we perform this classificationtask within a citizen science project, see Section 4.1. This approach provides us with alabeled partitioning of the parameter space. By visual and statistical analysis of theseresults, we are able to formulate conjectures for both the continuous setting as well ashigher-dimensional analogs.
2. Related Work
To set the stage for our work, we briefly present related research from four differentareas. First, we consider how the morphogenetic patterns by Turing actually arise inbiological contexts. Second, we consider basic literature on cellular automata and howthese mechanisms in turn inspire research in biology. Third, as one of the main fo-cuses of this article is on visualization and illustration, we consider how Turing(-like)patterns have given rise to different visual artworks. Finally, as our experiment is con-cerned with the parameter space of three-dimensional Turing-like patterns, we presentprevious results on the analysis of parameter spaces of two-dimensional patterns aswell as on three-dimensional patterns.
Turing patterns in biology and related applications
While Turing predicted his morphogenetic patterns in the 1950s, it was only severaldecades later that these patterns were actually discovered in animal skins. This longprocess gave Turing patterns a bad standing in the community, which was allevi-ated with more and more examples for self-regulated, reaction-diffusion based patternformation in nature, see Kondo and Miura (2010). A particular, well-studied exam-ple in this context is the zebrafish, see Figure 1. Studies have connected genes (Asaiet al. (1999)), chemical properties (Wertheim and Roose (2019)), and cell-network in-teractions (Nakamasu et al. (2009)) to the formation of Turing patterns within thezebrafish. Recently, researchers revealed how mechanical stresses can lead to tissueanisotropies and thus morphogen gradients in the formation of Turing patterns, caus-ing the characteristic stripes of a common zebrafish, see Hiscock and Megason (2015).Other works also incorporate different patterns to combine e.g. stripe and dot forma-tion to show how more complex structures can emerge from Turing’s simple basics,see for instance Scoones and Hiscock (2020).While Turing patterns thus describe actual skin coloring of certain animals in na-ture, they have further, surprising applications in the realm of biology. The reaction-diffusion theory started by Turing forms the mathematical foundation of the notionof scale-dependent feedback (SDF). These mechanisms play an important role in thespatial self-organization of ecosystems. Recent works provide evidence that vegetation3 igure 1.
Four fully-grown (56 months) female zebrafish, with different alleles of the leopard gene, exhibitingstriped and dappled patterns respectively. Image taken from Asai et al. (1999). patterns generated by ants, termites, and other subterranean animals are inherentlycaused by such feedback procedures, see Pringle and Tarnita (2017). The authorsalso provide a list of references which cover the formation of gaps, labyrinths, stripes,spots, and rings in different ecosystems, such as tropical deserts and savannas as well assubarctic peatlands, subtropical swamps, montane forests, Pacific reefs, and Atlanticmussel beds. On a way smaller scale, diffusion-activation patterns help to understandprocesses in the visual cortex. In this context, they can be used for instance to modelgeometric visual hallucinations, see Bressloff et al. (2002).
Cellular automata
As hinted at in Section 1 and as will be explained in detail in Section 3.2, the contin-uous formulation of Turing was discretized in the form of a cellular automaton (CA)by Young (1984). First steps towards CA have been made by Stanislav Ulam and Johnvon Neumann in the late 1940s (cf. Wolfram 2002, p. 876). However, the mechanism ofCA only became popular with John H. Conway’s
Game of Life in the 1970s, reportedon in the article by Gardener (1970). After several articles on cellular automata inthe 1980s, Stephen Wolfram published his widely received treatise “A new kind ofscience”, see Wolfram (2002). Despite criticism on several parts of the book as beingwrong or unscientific, it serves as a concise introduction to the theory and applicationsof CA.Generally, a CA is given by: • a discrete domain of cells, • a finite neighborhood relation between the cells, • a set of states each cell can be in, • and a transition function which attributes a new state to each cell, based on thecell states in its neighborhood.In the case of Turing-like patterns as discretized by Young, there are only two differentstates. Namely, cells can be either differentiated or not. The neighborhood relation isbased on a Moore neighborhood, also referred to as the 1-ring under the Chebyshevdistance. Therefore, in a two-dimensional square grid, it consists of the eight cellsaround a center cell that share at least one corner with the center cell. Accordingly,the three-dimensional Moore neighborhood includes 26 cells. The transition functionwill be described in detail in Section 3.2.Interestingly, CA are themselves the object of study in the search for a basic under-standing of life. An article by Chan (2018) explores the emergence of life-like structuresfrom CA. Relevant to the work in this paper, the author also provides descriptions of4he parameter spaces for the described systems. Several different behaviors are distin-guished both in a two-dimensional (Chan 2018, Fig. 9) and a three-dimensional (Chan2018, Fig. 10) parameter space. The latter is related to our investigations in Section 4.In a series of articles, Beer (2004, 2014, 2015, 2020) uses elements from the Gameof Life both as model and study object. The author applies the notion of autopoiesis to the comparably compact setup of the cellular automaton to investigate its conse-quences, implications, and shortcomings. While these works tackle large-scale questionsin the emergence of life and cognition, the work of Tonello and Siebert (2019) considersthe small-scale question of inhibition-models in the context of cell-to-cell communica-tion. Similar to the activator-inhibitor approach presented in Section 3.2, their modelassumes differentiated and undifferentiated cells, characterized by delta and notch .Similar to the work of Beer, they also consider the stability of patterns under pertur-bations. We will have to keep these basins of stability in mind when interpreting thepartitioning of the parameter space in our application.
Turing(-like) patterns in illustration and arts
Because of their simple formulation as well as their relation to naturally occurringpatterns, Turing patterns have lent themselves to quite some extent to illustration andarts. By rendering a cell based on the variation around it, McCabe (2010) arrives at thenotion of multi-scale Turing patterns. By imposing symmetry conditions, the authorcreates a juxtaposition of both mathematically complex and organic looking imagery.In order to speed up the generation of these patterns, Schwehm (2016) introduces aGPU-based algorithm. The reached performance allows for real-time interaction withthe system, as was done in a corresponding art installation. Furthermore, the usage ofcolor is explored within the renderings. Turning not to GPUs, but to the Photoshopsoftware, Werth (2015) creates Turing patterns by iteratively blurring and sharpeningan initial white noise image. The author uses these patterns as the starting point inthe design of abstract acrylic paintings.While these works employ Turing patterns, the approach of Greenfield (2016) usesYoung’s discretization scheme. The author experiments with non-uniform distribu-tions for cells being initially differentiated and also provides examples for non-circularareas of influence for both the activator and inhibitor. Final results are achieved byoverlaying several different images. This cursory overview on Turing(-like) patterns inthe context of illustration and arts proves their versatility as well as their wide usagewithin this field. The work of Greenfield (2016) served as a motivation for Skrodzkiand Polthier (2017) which in turn motivates the research presented in this article.
Parameter spaces and three-dimensional Turing patterns
Several works have been devoted to understanding the parameter space of two-dimensional Turing(-like) patterns. In the context of a predator-prey model, Weide Ro-drigues, Mistro, and D´ıaz Rodrigues (2020) consider the spatial distribution of preyand find several qualitative transitions when changing their model parameters. Namely,they transition from spots of undifferentiated cells, via differentiated stripes and un-differentiated stripes to spots of differentiated cells (Weide Rodrigues, Mistro, andD´ıaz Rodrigues 2020, Fig. 5). Taking a step further, considering the variation of twoparameters results in two-dimensional maps indicating a wide range of behaviors, fromall-undifferentiated cells via the states listed above to all-differentiated cells (cf. Ishida5020, Fig. 8,9). Similar results can be found in the investigation of dynamic Isingmodels, see Merle, Messio, and Mozziconacci (2019). Note that these investigations ofthe parameter space alter one or two parameters simultaneously. We divert from this,by identifying a set of three parameters and by investigating an entire section of thethree-dimensional parameter space.Furthermore, we are interested in the parameter space of three-dimensional ratherthan two-dimensional Turing-like patterns. In the context of diffusion-reaction in mi-croemulsion, researchers assert that “actual [Turing] patterns can never be truly pla-nar. . . at a molecular level,” (B´ans´agi, Vanag, and Epstein 2011, p. 1309). The au-thors consider the Belousov-Zhabotinsky reaction within a cylindrical domain andprovide experimental evidence for zero-dimensional “spot” patterns, one-dimensional“labyrinthine” patterns, and two-dimensional “lamellar” patterns. While they provideexperiments and numerical simulations for each of these cases, it remains unclear howmuch the shape of the domain effects the pattern formation. Furthermore, it is unclearwhat parameter changes cause transitions from one pattern to another or whether theeffect scales to larger domains.When going to a cubical domain with periodic boundary conditions, similar pat-terns as listed above can be observed, see Lepp¨anen et al. (2002). The authors statethat “starting from completely random initial conditions, it is not likely for the systemto converge into a purely lamellar state,” (Lepp¨anen et al. 2002, p. 38). Indeed, theauthors obtain two-dimensional patterns by manually choosing the initially differen-tiated cell sets. Again, it remains unclear how stable the respective patterns are, i.e.how slight parameter changes effect the pattern formation and how the patterns scalewith the domain size. These shortcomings of previous work motivate us to investigatethe complete parameter space of three-dimensional Turing-like patterns.
3. Turing patterns and their Turing-like discretizations
In this section, we will present the underlying theoretical models that form the basisof our experiments. First, we will describe Turing’s chemical model of morphogenesis.Second, we present the discretization into an activator-inhibitor model by Young.Finally, we show how Young’s model can be transferred into a three-dimensional setup.For this, we briefly discuss the resulting parameter space as well as challenges to thevisualization of the corresponding patterns.
The chemical model of Turing
As stated in Section 1, the morphogenetic model of Turing is based on a reaction-diffusion system of chemical substances. While his model allows arbitrary numbers ofmorphogenetic substances, throughout most of his article, Turing assumes two mor-phogens, X and Y , which are produced at a constant rate. Furthermore, the followinginteractions between the two morphogens are established: Y is destroyed dependingon the concentration of Y , X is converted into Y at a rate depending on X , X isadditionally produced depending on the concentration of X , and X is destroyed at arate depending on Y .More specifically, Turing assumes a ring of N ∈ N cells, with the two morphogens X and Y present in cell r = 1 , . . . , N via concentrations X r and Y r respectively. Further-more, he assumes cell-to-cell diffusion of X by µ and of Y by ν . Finally, the last as-sumption is that the concentrations of X and Y are increasing: X at the rate f ( X, Y )6 igure 2. A two-dimensional result, obtained by Turing from his morphogen system (Turing 1952, Fig. 2).Here, a one-morphogen system is used, the line indicates a marker of unit length. and Y at the rate g ( X, Y ). Given this setup, the behavior of the system may bedescribed by the 2 N differential equationsd X r d t = f ( X r , Y r ) + µ ( X r +1 − X r + X r − )d Y r d t = g ( X r , Y r ) + ν ( Y r +1 − Y r + Y r − ) r = 1 , . . . , N. In the case that the functions f ( X, Y ) and g ( X, Y ) can be approximated linearly,Turing provides general solutions of the above set of differential equations (cf. Turing1952, Sec. 6). Furthermore, he extends this discrete model from a set of N cells to acontinuous ring of tissue (cf. Turing 1952, Sec. 7,8).To give a two-dimensional example, Turing turns to a homogeneous one-morphogensystem. To this system, he applies random disturbances without diffusion for a pe-riod of time and then diffusion without disturbance. Namely, he chooses numbers u r,s initially at random to be ± f ( x, y ) = (cid:88) r,s u r,s exp (cid:18) − (cid:0) ( x − hr ) + ( y − hs ) (cid:1)(cid:19) , where h ≈ .
7. After applying diffusion and perturbation as indicated above, he obtainsa figure, where a point ( x, y ) is colored black if f ( x, y ) is positive, see Figure 2 (cf.Turing 1952, Sec. 9).Interpreting his figure, Turing states: “This is the least interesting of the cases. Itis possible, however, that it might account for ‘dappled’ colour patterns[. . . ]” (Turing1952, p. 66). Later on, he continues to interpret his results within the context of actualbiological processes. Namely, he reasons: “If dappled patterns are to be explained inthis way they must be laid down in a latent form when the foetus is only a fewinches long. Later the distances would be greater than the morphogens could travel bydiffusion.” (Turing 1952, p. 66–67). We have seen in Section 2.1 that Turing’s modelactually proved to be correct for several developments of skin patterns in the animalkingdom. See Woolley, Baker, and Maini (2017) for a more detailed discussion onboth the theoretical background and the computation of Turing patterns on arbitrary7 igure 3. Left: Distribution field around a discriminated cell, given by the continuous diffusion of activa-tor M and inhibitor M according to radius R around the cell. Right: Reinterpreting the impact of the twomorphogens around a discriminated cell by discretizing their summed contribution to an area of positive ornegative influence, depending on the radius R around the cell. (Young 1984, Fig. 1) surfaces. However, Turing himself recognized that the used equations can only beapproximately computed by hand, in a tedious process. Thus, subsequent work wasdevoted to providing simpler computational models for Turing’s patterns. Young’s activator-inhibitor discretization
Several years after Turing presented his model, it was still not tested experimen-tally, but evidence suggested even more localized inter-cellular interaction than pre-dicted by the model. Based on previous models for patterns in the visual cortex of thebrain (Swindale (1980)), Young presented his own two-morphogen model, see Young(1984). He assumes two types of cells: differentiated (pigmented) cells (DCs) and un-differentiated cells (UCs). Initially, the cell type is uniformly distributed in the domain.Then, each DC produces an activator morphogen M which differentiates nearby UCsand an inhibitor morphogen M which causes nearby DCs to become undifferentiated.Both morphogens diffuse through the domain, with the inhibitor morphogen havinglonger range. The UCs are passive in this model and do not produce any morphogen.The diffusion of the morphogens is modeled asd M i d t = ∇ · D · ∇ M i − KM i + Q, where M i = M i ( r, t ), i = 1 , r and time t , while the other terms describe diffusion, chemicaltransformation, and production of the respective morphogen, see Young (1984).With time passing, each DC produces the two morphogens at a constant rate andboth diffuse according to the above equation. This causes a distribution field aroundeach DC, see Figure 3 (left). Combining the two morphogens, attributing a positivevalue to the activator and a negative value to the inhibitor, the field has a positiveas well as a negative region, depending on the distance R to the considered DC, seeFigure 3 (right). Here, Young considers a field of constant positive circular region ofradius R , surrounded by an annulus of constant negative influence of inner radius R and outer radius R .From this model, Turing-like patterns are obtained by starting with a rectangulargrid of cells. We will model a cell at position ( x, y ) in the grid by its state s t ( x, y ) at8 igure 4. A Turing-like pattern, created by Young on 25 ×
100 grid cells, with R = 2 . R = 6 . w = 1 . w as indicated below the patterns (Young 1984, Fig. 2). time t , given by s t ( x, y ) := (cid:40) x, y ) is a DC , x, y ) is a UC . Initially, DCs are randomly chosen in the grid, i.e. s ( x, y ) = 1 for some cells chosenby some random process. Now, to determine the state of a cell ( x, y ) in the next timestep t + 1, we compute s t +1 ( x, y ) = (cid:80) ( x (cid:48) ,y (cid:48) ) ∈ B R ( x,y ) ω t, ( x,y ) ( x (cid:48) , y (cid:48) ) > ,s t ( x, y ) (cid:80) ( x (cid:48) ,y (cid:48) ) ∈ B R ( x,y ) ω t, ( x,y ) ( x (cid:48) , y (cid:48) ) = 0 , (cid:80) ( x (cid:48) ,y (cid:48) ) ∈ B R ( x,y ) ω t, ( x,y ) ( x (cid:48) , y (cid:48) ) < , (1)where B R ( x, y ) is the ball of radius R around ( x, y ) and ω t, ( x,y ) ( x (cid:48) , y (cid:48) ) = x − x (cid:48) ) + ( y − y (cid:48) ) > R ,w · s t ( x (cid:48) , y (cid:48) ) ( x − x (cid:48) ) + ( y − y (cid:48) ) < R ,w · s t ( x (cid:48) , y (cid:48) ) otherwise . (2)That is, for any grid cell ( x, y ), all DCs within the circular region of radius R are takeninto account. Those that lie within the smaller circular region of radius R contributeweight w while those in the annulus between R and R contribute weight w . If thesum of these weights is positive, the cell becomes differentiated; if the sum is zero, thecell does not change its state; if the sum is negative, the cell becomes undifferentiated.Young presents a corresponding experiment on 25 ×
100 grid cells, with R = 2 . R = 6 . w = 1 .
0, and w as indicated below the patterns, see Figure 4. He doesnot report his boundary conditions, i.e. whether his grid is following a flat, a cylindrical,or a toric topology.In his paper, Young proceeds to present a second, elliptical kernel for his patterns.This anisotropic kernel—expectably—creates stripe patterns oriented along the longer9 igure 5. Two Turing-like patterns on a three-torus, discretized into 100 × × ρ = 0 . , R = 4 , R = 7. Right: ρ = 0 . , R = 8 , R = 10. (Skrodzki and Polthier 2017, Fig. 4) axis of the ellipse (see Young 1984, Fig. 3). Without having mentioned it, he reinter-preted Turning’s model in the context of cellular automata (CA). All images presentedso far where two-dimensional, but Young’s CA offers an easy generalization to arbi-trary dimensions. Stepping up one dimension
Turing’s model as described in Section 3.1 has been applied to three-dimensional set-tings in the medical (B´ans´agi, Vanag, and Epstein (2011)) and physical (Lepp¨anenet al. (2002)) realm. Analogously, the discretization of Young also gives rise to three-dimensional Turing-like patterns. The single necessary addition is a term ( z − z (cid:48) ) inthe right-hand side of Equation (2). To further simplify Young’s approach, it can bereduced to choosing initial DCs in the domain uniformly randomly with some probabil-ity ρ ∈ [0 ,
1] and furthermore fixing w = 1, w = −
1, see Skrodzki and Polthier (2017).Given a domain size, these simplifications reduce the parameter space of Turing-likepatterns to choosing a probability ρ ∈ [0 ,
1] as well as two radii R and R .In the two-dimensional patterns of Figures 2 and 4, DCs are colored black, whileUCs are rendered white. For the visualization of three-dimensional patterns, a dif-ferent approach is necessary in order to gain insights into the presented structures.Thus, the images from Skrodzki and Polthier (2017) show only the boundary betweenregions of DCs and UCs. Furthermore, these images are rendered on the domain ofa three-torus, i.e. two opposing sides of the cube are identified. Examples for three-dimensional Turing-like patterns on a domain of 100 × ×
100 cells in a three-torus,taken from Skrodzki and Polthier (2017), are given in Figure 5.The publication by Skrodzki and Polthier (2017) on three-dimensional Turing-likepatterns ends with the following paragraph: “Note how. . . with varying parameters,the pattern performs a phase transition from disconnected points (‘zero-dimensional’)to connected stripes (‘one-dimensional’). A similar behavior is exhibited by three-dimensional patterns. . . . One would expect that. . . also ‘two-dimensional’ connectionsoccur.” (Skrodzki and Polthier 2017, p. 418). The remainder of this paper is devoted to10
20 40 60 80 100 12000 . . . . x σ ( x ) Figure 6.
Sigmoid for initial cell activation proposed in Equation (3). The domain I = { , . . . , } is mappedonto probability values σ ( x ) ∈ [0 , systematically exploring the parameter space of three-dimensional Turing-like patternsand to investigate the variety of obtainable structures.
4. Partitioning the parameter space in 3D
In order to experimentally investigate the parameter space, we first fixed the size ofthe cubical three-torus to work on. To be able to fully classify the entire parameterspace in a somewhat interactive fashion, we chose a grid size of 70 × ×
70. Webase our parameter space on the discretizations of Young (1984) and the weightingchoices ω = 1, ω = − R ∈ N ofthe activator morphogen, radius R ∈ N of the inhibitor morphogen, and initial cellactivation probability ρ ∈ [0 , R and R in a comparably small domainwould obscure the patterns formed. Therefore, we reduced the choice for the tworadii to be from { , . . . , } ⊂ N . The value ρ determines the probability of initialactivity of each cell in our cubical grid. To discretize the probability, we chose adomain of I := { , . . . , n ρ } ⊂ N with n ρ = 120 in our experiment. As the probabili-ties ρ = 0 and ρ = 1 only provide completely undifferentiated or completely differen-tiated domains respectively, we chose the lowest probability to be investigated to beat exp( − min ρ ) with min ρ = 12, i.e. the lowest probability as exp( − ≈ · − .As Skrodzki and Polthier (2017) found their zero-dimensional behavior for extremelylarge values of ρ , we decided to discretize this dimension of the parameter space non-linearly. Therefore, the domain is mapped to a probability value via the followingsigmoid σ : I → [0 , ,x (cid:55)→ exp ( · min ρ · n − ρ · x − min ρ ) ( · min ρ · n − ρ · x − min ρ ) , (3)which provides good resolutions at very low and very high probabilities as well as adecent resolution of the behavior between these extremes, see Figure 6. In the following,we will assume that some x ∈ I was chosen and simply write ρ := σ ( x ) for brevity.The three parameters ρ , R , and R span a 121 × ×
40 grid where each includingtriple gives rise to a Turing-like pattern. Based on the works Lepp¨anen et al. (2002);11 a) DC spheres (cid:4) (b) DC pipes (cid:4) (c) areas (cid:4) (d) UC pipes (cid:4) (e) UC spheres (cid:4)
Figure 7.
Five of the seven possible Turing-like patterns in a three-torus, with the trivial cases of all DCs orall UCs not shown. Patterns include 0-dimensional, sphere-like structures ((a) and (e)), 1-dimensional, pipe-likestructures ((b) and (d)), and 2-dimensional, area-spanning structures (c).
B´ans´agi, Vanag, and Epstein (2011); Skrodzki and Polthier (2017), we expect to findseven different types of structures that can form in the three-dimensional domain. Twoof these are the trivial patterns of only differentiated or only undifferentiated cells.The remaining five structures are displayed in Figure 7. We can find 0-dimensionalstructures (more or less spheres, see Figures 7(a) and 7(e)), 1-dimensional structures(pipe-like, Figures 7(b) and 7(d)), and 2-dimensional structures (clear flat, planar areasvisible, Figure 7(c)). Observe, that except for the 2-dimensional case, we obtain twopossible scenarios for 0-dimensional and 1-dimensional structures, which come from thedistinction into interior and exterior cell types. That is, for instance the spheres couldbe made from DCs while being surrounded by UCs (Figure 7(a)) or they could containthe UCs while being surrounded by DCs (Figure 7(e)). An analogous distinction holdsfor the 1-dimensional patterns. The experimental part of the project now consists ofpartitioning the discretized parameter space into these seven cases, i.e. identifying foreach parameter triple ( ρ, R , R ) which type of Turing-like pattern arises. The experiment
To label all triples in the discretized parameter space, we set up a citizen scienceproject. It was run during the “Long Night of Sciences” in 2019. This is an annualpublic event in several cities in Germany taking place since 2001. Scientific institutionsopen their doors and present (hands-on) research exhibits to the public audience foryoung and old alike. At our booth, visitors were first introduced to three-dimensionalTuring-like pattern, via the results of Skrodzki and Polthier (2017). Furthermore, theywere acquainted with the different non-trivial three-dimensional patterns via imagescorresponding to those in Figure 7. Being thus prepared, the visitors could contributeto the exploration of the parameter space.In our software on display during the event, the visitors would first choose a value forthe initial differentiation probability ρ on the scale given by I = { , . . . , } . Makingthis choice reduces the three-dimensional to a two-dimensional parameter space. Thislower-dimensional parameter space was then shown to the visitors, see Figure 8. Fromthis display, a pair ( R , R ) is selected and the corresponding Turing-like pattern isgenerated iteratively following Equation (1). After the automaton has either converged(i.e. no changes occurred after a time step) or a maximal number of iterations orseconds has passed, the current state is rendered using the Marching Cubes Algorithm as described by Skrodzki and Polthier (2017). The visitors were then asked to eitherrate the resulting pattern according to the images in Figure 7 or (if the pattern clearly igure 8. Each choice of ρ reduces the parameter-space to two dimensions, where the x -axis represents R and the y -axis represents R with (1 ,
1) in the bottom left corner. Squares with a gray border indicate not yetclassified triples, those without a border are classified, the pink border indicates the currently selected triple.Colors indicate either the classified color of the triple or the type of the closest triple in the three-dimensionalparameter space, following the color coding introduced in Figure 7. Note that all pairs equal and above theblack diagonal represent scenarios with R ≥ R which result in all cells being differentiated. Both imagesdisplay the layer ρ = σ (24), where the left represents estimates during the progress and the right representsthe final stage of our experiments for this slice of the parameter space. Note that the bottom right corner withmajorly unvisited cells will generate “UC all” structures. had not converged yet) to continue the generation for several more iterations.While the citizen science approach harnesses the power of the masses in evaluatingas many triples as possible, it also bears problems. On the one hand, the visitors to theexhibition were not continuously monitored, thus they could deliberately mis-classifytriples of the parameter space. On the other hand, as some of the patterns take manyiterations to converge, visitors might unintentionally mis-classify a not yet fully con-verged pattern. After the initial classification during the public science event, we thushad to perform some cleaning operations to re-classify outliers in the parameter space.Furthermore, we classified as many remaining triples as necessary by hand to identifythe separating surfaces between connected volumes in the three-dimensional parameterspace, corresponding to the five different patterns as shown in Figure 7. The relevantpart of the three-dimensional parameter space still contains 121 · · / , Implementation details
We implemented the iterative procedure given by Equation (1) within the geometryvisualization framework
JavaView by Polthier et al. (2020). The framework is basedon
Java and provides several comfort functions and interaction tools (cf. Polthier et al.2002, Sec. 2). When starting the procedure for a given triple ( ρ, R , R ), a first initialchoice of differentiated cells is performed, uniformly random according to ρ . Then,iteratively, Equation (1) is evaluated for each cell. Therefore, the spherical kernel withradius R and the larger ball-cavity kernel with outer radius R and inner radius R are evaluated for each cell see Figure 9(a) for a two-dimensional illustration. Thisevaluation consists of counting the DCs within radius R of the center cell (shown inyellow) and weighting this number against the number of DCs that lie within radius R of the center cell, but farther away than R . The center cell will then be a DC in the13 a) (b) (c) (d) Figure 9.
Two-dimensional illustrations of our implementation of the kernel evaluation. (a) Inner kernelof radius R (yellow) and outer kernel (annulus in the two-dimensional setup) with radii R and R (blue)including several DCs. (b) Box of size R × R as naive evaluation of the kernel. (c) Exact evaluation of thekernel by computing involved cell indices beforehand. (d) Evaluation of the difference when shifting the kernelkeeps most information constant and only causes evaluation on the edge of the kernel. next time step as prescribed by Equation (1).The calculation of one time-step of the cubical grid is time consuming. As rea-soned above, our cubical size choice was a justifiable trade-off between obtainablestructure size and calculation time. A naive implementation would consider a box ofsize R × R × R around the center cell and then compute for each cell whether itshould be considered in either kernel, see Figure 9(b) for a two-dimensional illustra-tion. By computing the kernels beforehand, it is possible to drop R − π R operations,causing for a significant speedup, see Figure 9(c) for a two-dimensional illustration.However, we can further enhance the computation by considering a spherical differ-ence area. That is, when we switch from one central cell to the next and query itsneighboring cells, we only consider those that got changed w.r.t. the kernel shift fromone cell to the other, see Figure 9(d) for a two-dimensional illustration. This reducesthe number of cells to be evaluated to 4 π ( R + R ).A last optimization was achieved via parallelization. We split the cubical domainin chunks up to the number of available threads and process them in parallel. Themachine used in the citizen science setup described above used 24 threads for parallelprocessing of the iterations. We also employ parallelization for the Marching Cubesisosurface construction for the visualization of the patterns. However, even with theseimprovements, timely convergence of the patterns is not guaranteed. To ensure the userhaving an interactive experience, we further integrated an iteration and a time limit, asindicated above, such that the computation will automatically stop after convergenceor after one of the given limits is reached. In the experimental setup, computation washalted after 20 iterations or seconds respectively, if no convergence was reached yet.Following the work of Schwehm (2016), the generation of the patterns as well as theirvisualization could be sped up further by utilizing GPU computation. The results
In this section, we are going to evaluate the experimentally found structures andhow they are distributed. According to the histogram in Figure 10, the dominatingstructures are “DC all” followed by “UC all”, which enclose all other structures inthe parameter space. This encapsulating behavior can be seen in the two-dimensionallayers of the parameter space shown in Figure 8 and in the left image of Figure 10,showing the entire discretized parameter space. Note that in Figure 10 and in theupcoming illustrations, we decided to neglect the “DC all” triples for visual reasons:14 C a ll D C s p h e r e s D C p i p e s a r e a s U C p i p e s U C s p h e r e s U C a ll . · ,
348 2 ,
218 607 408 98 63 , Figure 10.
Left: Plot of 3D space consisting of activator radius R , inhibitor radius R , and activation ( x as argument for ρ = σ ( x )). The parameter triples are colored by found structures. The shown hyperplane withincrease √ − indicates for what elements of the parameter space the volumes of inhibitor and activator kernelsare equal. Note that the color for all-DC triples is black, and we disabled their rendering for enhanced visibilityof the other structures, cf. Figure 11. Note that the x -axis in the left picture is not a linear depiction of theactivation probability, but the argument for σ ( x ), see Equation (3). Right: Histogram over the found structures.Note that the y -axis in the histogram follows a logarithmic scale. Both figures follow the color coding accordingto Figure 7. Rendering these would hide the distribution of the other structures within the pa-rameter space. Going back to the histogram in Figure 10, the remaining non-trivialstructures follow with decreasing amount, with “UC pipes” and ”UC spheres” havingthe rarest occurrence.One first trivial observation about the distribution of structures can be drawn fromthe behavior of the activator and inhibitor radii: If R ≥ R , all cells are differentiated(given that the random process guided by ρ > ⇒ πR = 43 πR − πR ⇒ R = R √ . Therefore, in each probability layer (i.e. for a fixed value ρ ) we can draw a separatingline with increase √ − . Respectively, we can separate the entire three-dimensionalparameter space by a separating hyperplane, see Figure 10(a). Above this separation,the triples are more likely to produce “DC all” structures and below it, they tendto form “UC all” structures. However, as we are dealing with a discretization—i.e.the volume of kernels is not πR i , i = 1 , igure 11. Left: Separating isosurfaces of volumetric distributed triples “UC all” (behind white), “DCspheres” (between white and beige), “DC pipes” (between beige and green), and “DC all” (remaining volumeon top). Right: The respective triple groups from the parameter set, giving rise to the separating isosurfaces,i.e. “DC all” (cid:4) , “DC pipes” (cid:4) , “DC spheres” (cid:4) , and “UC all” in white. separation, see Figure 10).Within our discretization of the parameter space, we find that those triples formvolumetric groups that give rise to the following four structures: “UC all”, “DC all”,“DC pipes”, and “DC spheres”. On the left in Figure 11, we draw separation areas (asisosurfaces) in white, beige, and dark green that separate the respective triple groups.The groups are shown on the right in Figure 11. That is, most of the cube is filled with“DC all” triples, filling it from the top. From the dark green up to the beige isosurface,we find a slender wedge consisting of “DC pipes”. The volume between the beigeisosurface and the highly non-planar white surface is filled with “DC spheres” whilethe rest of the parameter space consists of “UC all”. The other structures (“areas”,“UC pipes”, and “UC spheres”) do not obtain volumetric extent in our discretizationand with the chosen parameter bounds. They all lie close to the linear separation planeshown in Figure 10. These observations motivate two conjectures.
Conjecture 1.
The wedge, formed by the “DC pipes” triples, given as volume be-tween the beige and dark green isosurface, will grow larger and wider with increasingvalues for R and R . Conjecture 2.
The structures: “areas”, “UC pipes”, and “UC spheres” will gainvolumetric extent when further increasing the values for R and R .Figure 12 illustrates the top view, showing the occurrence and distributions of thosestructures lying close to the separating plane shown in Figure 10. Note how the blue“area” structures exhibit an almost linear behavior when increasing ρ while keeping R and R fixed. These blue spikes exhibit almost equidistant starting points.The “UC spheres” shown in yellow also exhibit a curve-like behavior, not as straightas the blue “area” structures, though. They start below the blue lines and graduallystep down towards the next blue line with growing values of ρ . The “UC pipes” shownin light green, however, behave different as they clearly extend over more than justcurves, but rather span the two-dimensional space between blue and yellow lines. Note16 e+001e+052e+053e+05 0e+00 1e+05 2e+05 3e+05 Volume S u r f a c e Volume A c t i v a t o r R ad i u s Figure 12.
Left: A top view of the left image in Figure 10, i.e. a projection along the R -axis. Each squarerepresents the coordinate triple with corresponding x and R value and the color of the largest R triple thatdoes not represent “DC all”. Right: Two scatterplots indicating the relationship between volume and surfacearea (top) as well as volume and activator radius (bottom) of the sub-structures. that with low R , R values (towards the bottom in the left Figure 12) we do not find“UC pipes” or “UC spheres”, which might however be caused by the chosen resolutionof our discretization.Consequently, we were able to find all structures displayed in Figure 7. Thereby, wepositively answer the corresponding conjecture, posed in Skrodzki and Polthier (2017).Our experiments lead us to the following conjecture. Conjecture 3.
A Turing-like pattern of dimension d exhibits sub-structures of alldimensions from 0 to d −
1. Furthermore, each structure of dimension 0 , . . . , d − d − ,
214 hand-classifiedsub-structures to train the predictor. Note that this data is not uniformly selected,but rather chosen along the border of larger patches as shown in Figure 8. For cross-correlation, we separate the data set into five folds, with the distribution of sub-structures as shown in Table 1. To train our predictor, we perform the followingoperation on all folds. For two neighboring sub-structure types, we use the trainingdata (all data points not in the fold) to find a certain volume-threshold to optimallydistinguish these. Thus, we obtain four volume-thresholds for each fold, indicating fiveintervals that are associated with the respective sub-structures. Then, we use thesethresholds to classify the testing data from the fold. Table 1 indicates the classificationerror made within each fold. Overall, the prediction of the sub-structure by the volumeadmits an average classification error of 8 . , ] ( , ] ( , ] ( , ] ( , ] ( , ] ( , ] ( , ] ( , ] ( , ] ( , ] ( , ] ( , ] ( , ] ( , ] ( , ] ( , ] ( , ] ( , ] ( , ) Percentage of active cells
Figure 13.
Histogram over 2,214 data points from the parameter space, selected from the border of two sub-structure areas. The bins on the x -axis represent the percentage of activated cells in the final state (excluding“UC all” and “DC all” at the far ends). The stacked bars are colored following the color scheme of Figure 7.Bars are printed up to 85 elements and cut from there, where the actual values are given on top. V o l u m e Figure 14.
Violin plot over the different types of sub-structures, indicating the volume that is taken by eachsub-structure. Note the box-plot overlay indicating the mean as well as the first and third quartile. Outliersare shown in red, most notably for the “areas”. Furthermore, “UC pipes” and “UC spheres” do not show aclear separation by their volume.
Table 1.
Distribution of the different sub-structures over the folds of the data set used for cross-correlation.The errors given indicate classification errors of the sub-structures solely based on the volume of the sub-structure. a very strong predictor, the results differ drastically over the different sub-structures.While “DC spheres” exhibit an extremely low classification error of 3 .
47% across allfolds and “DC pipes” are within the average with an error of 8 . .
86% of “UC spheres” beingwrongly classified, see Table 1. This confirms the impression from Figure 14 that “UCpipes” and “UC spheres” are not well separated solely by their volume.The scatterplots in Figure 12 suggest that the surface area of the sub-structuredoes not carry a lot of information regarding the type of the structure, but that theactivator radius, in conjunction with the volume could provide a better classifier. Onthe same folds as used in the training of the above predictor, we train a k -nearestneighbor predictor. The used neighborhood sizes were k ∈ { , . . . , } . As the twofeatures, volume and activator radius, exhibit very different sizes, we normalize thesetwo dimensions to exclude any artifacts. For our data set and the number of chosenfolds, we obtain a minimal classification error for k = 1, where the mean classificationerror over all folds is 7 .
5. Conclusion
In this paper, we have presented a visually-guided investigation of sub-structures inthree-dimensional Turing-like patterns. Based on a discretization, we gave a completepartition of the parameter space and found five different non-trivial sub-structures,see Figure 7. Thereby, we provided a positive answer to the conjecture of Skrodzkiand Polthier (2017), who proposed the existence of two-dimensional sub-structuresin three-dimensional Turing-like patterns. Based on our experiments, we furthermoreprovided statistical insight into the distribution of the different sub-structures withinthe parameter space. From these data, we presented several conjectures both regardingthree-dimensional and higher-dimensional Turing-like patterns.The experimental approach chosen in this paper has several weaknesses. First, thediscretization of the activation probability, as given in Equation (3), is not linear,but emphasizes very low and very high activation probabilities equally. Therefore, ourstatistical findings as reported in the histogram in Figure 13 are correspondingly dis-19orted. Second, the identification and distinction of the different sub-structure casesdepends solely on visual inspection, the current iteration number, and subjective in-terpretation. Aside from the clearly distinguishable “DC all” and “UC all” cases, thereis no automatism for classifying the sub-structures yet.Future research has to be directed at finding a thorough mathematical descriptionof the different sub-structures described in this work. We have shown that the volumeis a strong predictor to classify the “DC” sub-structures, but it does not work wellon the “UC” sub-structures. Furthermore, a larger portion of the parameter spacehas to be investigated to better understand the zero-, one-, and two-dimensional sub-structures that arise. The experiments presented in this paper can help narrow downa corresponding portion of the parameter space and thus help focus on the relevantparts.
Funding
This material is based upon work supported by the National Science Foundation un-der Grant No. DMS-1439786 and the Alfred P. Sloan Foundation award G-2019-11406while the author was in residence at the Institute for Computational and Experimen-tal Research in Mathematics in Providence, RI, during the Illustrating Mathematicsprogram.Furthermore, this research was supported by the DFG Collaborative Research CenterTRR 109, “Discretization in Geometry and Dynamics”, RIKEN iTHEMS, and theGerman National Academic Foundation.
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