Physarum polycephalum Percolation as a

week ending 17 AUGUST 2012

PHYSICAL REVIEW LETTERS

PRL 109, 078103 (2012)

Physarum polycephalum Percolation as a Paradigm for Topological Phase Transitions in Transportation Networks Adrian Fessel,1,2 Christina Oettmeier,1,2 Erik Bernitt,1,2 Nils C. Gauthier,2 and Hans-Gu¨nther Do¨bereiner1,2,* Institut fu¨r Biophysik, Universita¨t Bremen, Bremen, Germany Mechanobiology Institute, National University of Singapore, Singapore (Received 17 March 2012; published 16 August 2012) 1

2

We study the formation of transportation networks of the true slime mold Physarum polycephalum after fragmentation by shear. Small fragments, called microplasmodia, fuse to form macroplasmodia in a percolation transition. At this topological phase transition, one single giant component forms, connecting most of the previously isolated microplasmodia. Employing the configuration model of graph theory for small link degree, we have found analytically an exact solution for the phase transition. It is generally applicable to percolation as seen, e.g., in vascular networks. DOI: 10.1103/PhysRevLett.109.078103

PACS numbers: 87.18.Hf, 64.60.ah, 64.60.aq, 87.10.Ca

The theory of networks unifies a remarkable number of fields across virtually all disciplines [1,2]. In particular, evolving spatial networks [3,4] encompass transportation networks like railways and roads, power grids, and the World Wide Web, as well as the vascular system [5]. The true slime mold Physarum polycephalum forms a tubular network [6–9] as a foraging strategy [7,10] in order to maximize searchable area and minimize transport distances of nutrients and biochemical signals. P. polycephalum networks have been shown to solve mazes [10], possess apparent learning capabilites [11], and may lead to amorphous biological robots [12]. However, their network structure has largely been characterized in the late stages of development [6,9,13–15]. The early morphogenesis is not known well. Generally, we are interested in the initial development of a connected structure in a transportation network. Recently, adaptive network design has been studied in P. polycephalum and characterized with respect to cost, efficiency, and robustness in relation to general transportation networks [6]. This has been done with a fully developed, yet dynamically adapting network. But how might transportation networks form in the first place? We will show that global network connection in P. polycephalum occurs in a percolation transition [16] where topological constraints impose universal conditions. We find our general framework to be applicable to a model of percolation in vascular networks [17]. Thus, P. polycephalum percolation may serve as a paradigm for topological phase transitions in transportation networks. P. polycephalum is a unicellular, but multinucleate, amoeboid slime mold. While the plasmodia in their natural habitat can reach sizes of several hundred square centimeters, the microplasmodia utilized in this experiment measured only 200 to 500 m in diameter. These small, diploid amoebae, physiologically similar to big plasmodia, were created by cultivation in liquid medium in shaking flasks. Because of shear forces, big cellular structures are 0031-9007=12=109(7)=078103(4)

torn apart and give rise to the formation of small, quasispherical objects. These microplasmodia are very well suited to act as starting forms from which to grow networks, since they are easy to cultivate, very reproducible, and homogenous in both behavior and metabolism [18]. Experiments were performed at constant temperature and humidity on agar plates [19]. In Fig. 1, we show a typical sequence of network topology. Over time, plated microplasmodia fuse to form extended islands. These grow further and eventually connect to percolate the whole network in a single giant component. This process usually takes a few hours to occur. After one day, the microplasmodia have developed into a fully connected macroplasmodium. Apart from sheetlike growth regions at the network rim, it is mostly made up of tubular structures, which oscillate in their radii with a typical period of 1–2 minutes to drive cytosol along their paths. This is the wellknown shuttle streaming of P. polycephalum. In fact, microplasmodia already show these oscillations with similar periods [20]. However, within the scope of this work, we are not interested in these fast time scales, nor in the details of microplasmodium fusion. We rather want to concentrate on the dynamics of the overall network formation, which take place on the time scale of hours. Nevertheless, it is apparent from Fig. 1 and its corresponding movie [21], that the formation of a large networkspanning cluster happens quite suddenly. In fact, we proceed to show that it can be described within the framework of the well-known percolation transition. To this end, we characterize slime mold dynamics by extracting the topological structure of the network. Briefly, we create a binary image which is then skeletonized, as shown in the overlay of Fig. 1 and explained more in [19]. The skeleton represents a two-dimensional graph with nodes and links. Detailed inspection shows that we can describe the structure by nodes with a small number of links, see Fig. 2. Indeed, link degrees k larger or equal to five very rarely occur. Nodes with four links are only significant after

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Ó 2012 American Physical Society



PRL 109, 078103 (2012)

FIG. 1 (color). Formation of an extended tubular network by fusion and growth of microplasmodia on agar. The time is given in hours and minutes (hh:mm). The networks’ skeletons are overlaid on top of bright field images. The largest network component is colored in red. Around 6.7 hours (06:40) the largest component grows to the size of the whole system and percolates most of the structure. The sequence is available as a movie [21].

percolation. Thus, before and during the transition, we can characterize the network by the fractions of nodes pk with zero, one, and three links (k ¼ 0, 1, 3). We omit nodes with two links, since morphologically there is no detectable difference between such a node and a link without an inserted node. In Fig. 3, we show the time dependence of the fraction of nodes pk for the particular, yet typical, sequence depicted in Fig. 1. Indeed, we confirm that link degrees with k < 4 are dominant up to and shortly beyond the transition, which we will find analytically occurs at p0 =p3 ¼ 2. From the corresponding time point on this ratio continues to fall with p0 decaying to zero and p3 steadily increasing. In fact, asymptotically, the network is dominated by three-legged vertices (k ¼ 3) [15], see Fig. 4, panel B. Before characterizing our data further, let us give a theoretical description of the percolation transition. We will do so using the configuration model of graph theory [1] where the configuration is determined by the fraction of p0

p1

p2

p3

p4

nodes pk with given link degree k. We are interested in the existence of a giant component covering a fraction S of the whole graph. The probability of a random node not to belong to this giant component is then given by u ¼ 1 S. There is a self-consistent relation [1] for u in the limit of large graph size X u ¼ pk uk (1) k

which states that the probability of a node not to belong to the giant component equals the probability of all its connected neighbors not to belong to it. In the following, we evaluate all expressions for thePspecial case k 3. Using the normalization condition k pk ¼ 1 and Cardanos’ method the physically relevant cubic root of Eq. (1) gives the fraction of the graph with a single giant component SðqÞ ¼ 1 uðqÞ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 1 q 27 ¼1þ ðq þ 1Þ cos arccos þ 3 3 2 ðq þ 1Þ3 3 (2)

FIG. 2 (color). Definition of link degrees. The probability of a node with link degree k is denoted by pk . Nodes with two links have no biological significance and are omitted. Nodes with a link degree higher than four rarely occur and can be neglected.

as a function of the ratio q ¼ pp03 . The function S is plotted in Fig. 4, panel A (solid line). There is a giant component, i.e., S Þ 0, for 0 < q < 2. Thus, the percolation transition occurs at q ¼ 2. Further, it was shown [22] that at percolation

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PRL 109, 078103 (2012)

A 0.6 8

0.8

6

0,6 S

0

pk

p /p

3

0.4

1

4

0.4

2

0.2

0.2

0

0

5

10

15

20

0 −2 10

0

t [hours]

X k

kðk 2Þpk ¼ 0:

(3)

0

10 p0/p3

1

10

2

10

3 2.5 2 1.5 1

This exact condition has quite an intuitive meaning in our situation. We find p1 ¼ 3p3 ; i.e., the number of nodes with one link equals 3 times the number of nodes with three links, as expected for a nearly fully connected network. Link degrees pk at the percolation transition are given by (4) ðp0 ; p1 ; p3 Þ ¼ 13; 12; 16 : The mean link degree near percolation X 1 p0 hki ¼ pk k ¼ 1 2 6 p3 k

B

FIG. 3 (color). Left axis: Distribution of link degrees pk for k < 5 as a function of time t (p0 red, p1 green, p3 blue, p4 yellow) corresponding to the sequence shown in Fig. 1. Right axis: The ratio p0 =p3 (black) drives the percolation transition at p0 =p3 ¼ 2 (dashed horizontal line). In this particular example, the network percolates at t ¼ 6:7 h (dashed vertical line).

−1

10

(5)

is found to be a linear function of the ratio q ¼ pp03 which is the system variable driving the phase transition. The fraction S can be considered as the order parameter. In Fig. 4, we show experimental trajectories of 48 P. polycephalum networks together with theoretical expectations [Eq. (2), see panel A, and Eq. (5), see panel B, inset]. Our data collapse rather well on the theoretical master curves. The skeleton of the slime mold visible in Fig. 1 and corresponding movie [21] is shown together with the trajectory Sðp0 =p3 Þ as an inset in an additional movie [21]. Note that for a finite network, there is always a largest connected component. Thus, the observed smearing out of the percolation transition in Sðp0 =p3 Þ is a finite size effect as compared to the theoretical curve which describes the limit of infinite network size. Nevertheless, the mean link degree near the transition is determined by such a strong topological constraint that all trajectories go through the same point with virtually identical slope. At percolation, we find a distribution of link degree pk [23] characterized in Table I. Indeed hp4 i is found to be an order of magnitude smaller than the other link degrees, validating our model assumption to set it to zero. Average link degrees add up to

0.5 0 −2 10

−1

10

0

10 p0/p3

1

10

2

10

FIG. 4. Panel A: Fraction SðqÞ of all nodes included in the largest connected network component as a function of the ratio q ¼ p0 =p3 in a semilogarithmic plot. The solid line depicts the theoretical value [Eq. (2)]. The percolation is visible as a sharp increase of S ¼ 1 u from zero at p0 =p3 ¼ 2. The gray dots correspond to 48 experimental trajectories, while the dashed line shows the median. The black markers correspond to one particular trajectory of a vascular network assembly of endothelial cells [17,21]. Panel B: Average link degree as a function of the ratio p0 =p3 for the data sets shown in panel A. Near the percolation transition at ðhki; p0 =p3 Þ ¼ ð1; 2Þ the data collapse onto a line with slope 1=6, see inset [Eq. (5)].

unity. Within fluctuations induced by p4 , experimental values agree quite well with theoretical expectations (4). With a more general theoretical calculation for the percolation with p4 Þ 0, we find a two-dimensional phase diagram in the new ratios ( pp34 , pp04 ) with the percolation occurring along the line pp34 ¼ 12 ðpp04 3Þ [24]. The strong topological constraint at the percolation transition led us to believe that one should find this behavior in all systems which form a network from disconnected pieces. Indeed, the development of the vascular network in embryos initiates in blood islands, which connect to form an extended network [5]. This process has been modeled in vitro with human microvascular endothelial cells (EC) on Matrigel [17]. The authors find a percolation transition as a function of cell density . We reanalyzed their supplemental movie. The experimental trajectories

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PRL 109, 078103 (2012)

TABLE I. Average values hpk i and standard deviations k of link degree distributions at percolation. k hpk i k

0

1

3

4

0.367 0.057

0.418 0.092

0.190 0.037

0.026 0.017

for S and hki are shown in Fig. 4 and corresponding movie [21]. Remarkably, the human cells exhibit the same universal behavior as the microplasmodia of the slime mold P. polycephalum. We conclude that it is not so much the cell density which drives the transition, but rather the ratio q ¼ pp03 . EC density at time of plating and the number N0 of initially unconnected nodes, i.e., isolated cells, set the initial node fraction p0 ¼ NA0 1 where A is the plating area. Nevertheless, we do find in our experiments with P. polycephalum that sparsely plated microplasmodia take a longer time to percolate than samples with a denser initial area coverage [23]. Within experimental resolution there is no dependence on glucose concentration as a nutrient. In summary, we showed that global percolation in transportation networks can be described by a single topological parameter characterizing local connectivity. Details, like the system size or specific material parameters, do not enter our description of this topological phase transition. The universality of percolation may be used as a general gauge in the analysis of transportation networks. Especially, this should be applicable to vasculogenic mimicry [25] and alternative strategies of vascularization [26] in tumors. Some malignant tissues derive their blood vessels not by angiogenesis, i.e., remodeling of existing vessels, but rather by denovo vascularization like embryos. Since topologically, percolation is independent from detailed mechanisms and even space dimensions, i.e., 2D versus 3D growth, it may serve as a reference point in space and time when comparing the dynamics of network formation in tumors of varying size and shape. Since restricting blood supply via hindering vessel percolation is paramount for suppressing tumor growth, this may foster development of antiangiogenic therapy. H.-G. D. thanks Klaudia Brix for arousing his interest in Physarum, as well as Anna Kaufmann and Siddharth Deshpande who have been involved in an early stage of the project. We are grateful to Anja Bammann and Alexander Seupt for technical assistance. Last but not least, we thank Wolfgang Marwan for his generous gift of Physarum strains and valuable advice.


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