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A mathematical model for adaptive vein formation during exploratorymigration of Physarum polycephalum: routing while scoutingTo cite this article: Daniel Schenz et al 2017 J. Phys. D: Appl. Phys. 50 434001

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1. Introduction

The formation of transport networks is one of the most con-spicuous features of life. They efficiently distribute nutrients and signals in widespread body plans, enabling organisms to span sizable volumes while keeping costs comparatively low. In simple situations where the body shape is fixed, a unique optimal solution for the highest efficiency of the network can

be found. On the other hand, if the organism’s boundaries are changing in an environment that is unknown and not percep-tible to the organism, the task of constructing optimal network layouts becomes less straightforward.

This kind of challenge is regularly faced by the unicellular slime mould Physarum polycephalum, which expands over surfaces in the search for food and suitable environmental conditions. It is known that Physarum can utilise external

Journal of Physics D: Applied Physics

A mathematical model for adaptive vein formation during exploratory migration of Physarum polycephalum: routing while scouting

Daniel Schenz1 , Yasuaki Shima2, Shigeru Kuroda3 , Toshiyuki Nakagaki3 and Kei-Ichi Ueda2

1 Graduate School of Life Science, Hokkaido University, Sapporo, Hokkaido Prefecture 060-0808, Japan2 Graduate School of Science and Engineering, University of Toyama, 3190 Gofuku, Toyama, Toyama Prefecture 930-8555, Japan3 Research Institute for Electronic Science, Hokkaido University, Sapporo, Hokkaido Prefecture 060-0808, Japan

E-mail: kueda@sci.u-toyama.ac.jp

Received 16 February 2017, revised 22 August 2017Accepted for publication 29 August 2017Published 28 September 2017

AbstractExploring free space (scouting) efficiently is a non-trivial task for organisms of limited perception, such as the amoeboid Physarum polycephalum. However, the strategy behind its exploratory behaviour has not yet been characterised. In this organism, as the extension of the frontal part into free space is directly supported by the transport of body mass from behind, the formation of transport channels (routing) plays the main role in that strategy. Here, we study the organism’s exploration by letting it expand through a corridor of constant width. When turning at a corner of the corridor, the organism constructed a main transport vein tracing a centre-in-centre line. We argue that this is efficient for mass transport due to its short length, and check this intuition with a new algorithm that can predict the main vein’s position from the frontal tip’s progression. We then present a numerical model that incorporates reaction-diffusion dynamics for the behaviour of the organism’s growth front and current reinforcement dynamics for the formation of the vein network in its wake, as well as interactions between the two. The accuracy of the model is tested against the behaviour of the real organism and the importance of the interaction between growth tip dynamics and vein network development is analysed by studying variants of the model. We conclude by offering a biological interpretation of the well-known current reinforcement rule in the context of the natural exploratory behaviour of Physarum polycephalum.

Keywords: Physarum polycephalum, transport network, algorithm, cell migration

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D Schenz et al

A mathematical model for adaptive vein formation during exploratory migration of Physarum polycephalum: routing while scouting

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spatial memory to navigate complex environments to keep track of parts thereof that the organism has explored already but evacuated again [1]. However, this leaves unanswered how the organism explores a completely unknown environment in the first place.

Furthermore, in Physarum, body expansion and vein net-work formation are concurrent processes, where the latter has to constantly adapt to developments in the former. On the other hand, as the vein network in this organism is also responsible for transporting body mass from the rear of the organism to the extension front, the transport network is an integral element of the exploratory behaviour itself.

The migration behaviour of Physarum basically consists of the following processes: (i) leading edge formation at the peripheral region which drives unidirectional migration, (ii) protoplasm streaming generated by actomyosin contraction and (iii) plasmodial vein network formation occurring in the inner part of the body. In process (iii), in particular, we can observe the veins becoming thicker while handling high mass flow, and thinner for low flow rates. Appropriate mass redis-tribution is accomplished by this process. In previous studies of mass flow and vein network formation, only processes (ii) and (iii) have been focused on [2–11], while process (i) has been neglected. In [3–7], Nakagaki et al have shown that the organism has the capacity to create a minimum distance con-nection between two food sources. Mathematical modelling studies suggest that a key mechanism of vein development is based on current reinforcement [7].

For process (i), it has been found that a leading edge can be generated by specific chemical concentration patterns [12–15]. In fact, mathematical models based on a reaction-diffusion system have reproduced typical behaviours of the migration dynamics [16], making such systems a good can-didate for the mechanism underlying leading edge formation.

Combining these processes into one model promises not only to yield a mechanistic description of the natural explora-tory behaviour of Physarum but might also shed light on other biological, physical, and social phenomena where (nutrient, mass, or information) flow, network construction, and growth are inter-connected. Examples for these are river basin forma-tion, root growth, and road network evolution, to name just a few. This is in contrast to purely phenomenological models (for example [17, 18]) the results of which may resemble the real organism’s behaviour to some extent, but which do not allow gaining insights into the biological mechanism under-lying the behaviour or their generalisation to other organisms.

The aim of this study, therefore, is to construct a math-ematical model for the exploratory behaviour of Physarum, in order to understand the mechanisms it is based on and attempt its biological interpretation.

We start by observing the free-roaming behaviour of the organism, letting it explore corridors of constant width but dif-ferent shapes. This will restrict the developing vein network to only one or two parallel veins, making the interdependence of growth front behaviour and vein network formation more accessible to experimental manipulation and modelling.

These observations allow us to characterise a striking and typical feature of the organism’s behaviour during explora-tory behaviour. However, because we are interested in

understanding how the organism’s physiology determines it’s phenomenological behaviour, this characterisation is fol-lowed up with a simple draft model that translates our intuitive understanding of what lies at the root of the observed behav-iour into an algorithm. The accuracy of this algorithm is then checked against another series of experiments.

Having identified a key charactersitic of the organism’s exploratory behaviour and verified our intuitive understanding of its mechanism, we are then in a position to develop a model rooted in Physarum’s physiology, taking into considera-tion both above-mentioned processes: (i) reaction-diffusion dynamics and (ii) the current reinforcement rule, as well as other relevant processes that have been described for this organism. Finally, this physiological model is evaluated against experimental evidence and the afore-mentioned phe-nomenological model.

2. Materials and methods

2.1. Cultivation

Physarum polycephalum was raised in the dark on 1%(w/v) agar and fed oat flakes (Quaker Oats Co.) at regular intervals. Before the experiment, a part of the organism was transplanted onto a new agar plate and the organism allowed to extend [16].

2.2. Experiments

A 20 ± 2 mg piece of the extending growth front was cut and lifted onto a 2% agar bounded by plastic film. These bounds created lanes with constant 5 mm width and 35 mm2 area, although the shapes varied for different experiments. The shapes used were a U-turn (15 replicates) and a snake shape (8 replicates). For the U-turn, 4 replicates were placed hori-zontally, 5 replicates vertically with the opening of the ‘U’ pointing downwards, and the remaining 6 replicates vertically with the opening of the ‘U’ pointing upwards. For the snake shape all 8 replicates were placed horizontally.

The organism was allowed to extend throughout the arena at 25 °C in the dark. At the same time, the extending plasmodia were illuminated from below (for the vertically mounted sam-ples: behind) using an infrared light source (λ ∼ 950 nm) that was passed through a light diffuser plate. The light passed through the organism and was captured by a CCD camera mounted above (in front), thus creating optical density images of the organism, with an image being stored every 3 min [6, 16].

2.3. Data analysis

2.3.1. Average position of the main vein. To understand the typical exploratory behaviour of Physarum, that is, the typi-cal position of its veins when growing in a corridor contain-ing a corner, we evaluated images of the organism’s extension through the U-shaped corridor at a stage just after having completely explored the arena. To do this, we measured the distance of the centre of the major vein(s) to the inner bound-ary at regular intervals of 2 mm along the corridor (figure 1(a)). Then, for each measurement site along the corridor, the respective results across the different samples were averaged

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to find the average vein position at that site. These averages were then connected with straight lines to estimate the average main vein’s course.

In order to see whether gravity, or rather, the spatial orien-tation of the organism influences its exploration behaviour, we tested the average vein positions for the different orientations using one-way analysis of variance. At the 95%-significance level, no difference could be found at any point. Therefore, all 15 samples were considered as belonging to the same distribu-tion for further analyses.

The same method to find an average vein trajectory was used for the results of the model. The shapes used for the model runs had the same proportions as the arenas used for the experiments, and measurement sites were established at regular intervals along the corridor. The distance between the measurement sites was 40% of the corridor’s width, which is equivalent to every 2 mm for a corridor of 5 mm width.

2.3.2. Statistics. We compared the average positions of the main vein in the model and the experiment by one-way analy-sis of variance pairwise for each measurement position, while

employing the Bonferroni correction to account for the mea-surements being part of series. Confidence levels were set at 95%, 98% and 99.9% for significant, highly significant, and very highly significant differences, respectively.

2.3.3. Prediction of the vein position. In the following, the ability to predict the position of the real organism’s main vein from the growth history of its extension front became important. This prediction of the vein position was based on the minimal distance between points on the growth front in two pictures taken at fixed time intervals, ∆T . Therefore, an algorithm to extract the position of the growth front was developed: the optical density images were binarised using an appropriate threshold to differentiate between the organism’s growth front and its main body; where the largest set of newly occupied pixels between two frames was then defined as the position of the growth front for the later frame.

To predict the position of the vein in a specified frame, the minimal distance connection of the vein position in the frame one time interval afterwards to the growth front in the frame under consideration was determined. This was done for every frame from the last to the first (figure 2).

We varied the time interval ∆T between two frames used for the prediction: that is, as we had saved images every three minutes, we used subsets of the time series to adapt the time difference between frames. For example, to achieve a time dif-ference of 9 min, we used only every third image. This proce-dure, however, resulted in the existence of different phases in parallel, because there are multiple ways of choosing subsets. To illustrate this ambiguity, and because no single phase can be considered to be ‘correct’ above the others, we plotted all ‘parallel’ predictions for a given time interval. For example, if every third image is used in a subset, there are three ways of choosing a subset and, therefore, three possibly different pre-dictions. All three predictions are plotted in the same figure.

2.3.4. Calculation of distances. To estimate the quality of the vein prediction algorithm, the distance from the major vein to the predicted vein was calculated as the average of the mini-mum Euclidian distances between each point of the real major vein and any point of the predicted vein. The distance of the real major vein from the shortest possible path was calculated in a similar way. These two distances were compared by a Mann–Whitney U test at the significance of p = 0.002. Here, for practicality, we define a ‘major vein’ as the part of the real vein that contains the thickest non-branching vein behind the frontal fan-like part, resulting in a singular, unbroken connec-tion from the start to the end point of the arena, as shown in figure 3(d).

3. Experimental results

3.1. Adaptive vein formation in a cornered space during exploratory migration

Figure 1 shows the typical way Physarum polycephalum con-structs its veins when growing in a corridor containing two 90 turns. The organism, instead of keeping to the centre line

Figure 1. U-turn experiment. While extending over a U-shaped arena, Physarum polycephalum constructs its main vein(s) near the centre, but cuts the corners. The organism started from the bottom left and grew towards the bottom right (as indicated by arrows). (a) Mean and standard deviation of major vein position. The positions of the main vein(s) were measured at regular intervals (2 mm) and averaged (x marks). Error bars are standard deviations. The dotted line is the centre line of the arena, the broken line represents the theoretically shortest possible path for the segment shown. (b) Relationship between the history of the growth front and the final major vein position. The black lines superimposed on the image of plasmodium indicate growth fronts which are traced after every hour of growth.

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of the corridor or establishing the vein along the line of the shortest possible path, cuts the corners and then returns to the centre line. The result is a centre-in-centre line at each corner. This is a very striking behaviour that is reproduced well throughout replicates and different arena shapes. As it is a clear characteristic of Physarum exploration, a mechanistic model needs be able to reproduce this behaviour.

Even though the resulting vein is constructed along a clearly distinct line from the shortest possible path, it is only 6% longer than the latter over the whole arch (containing two 90 turns). To put this into perspective, the path strictly along the centre line is 18% longer than the shortest possible path. How does the organism achieve such a high degree of effi-ciency constructing its vein network while still exploring the whole arena? What is the mechanism it employs?

3.2. Proposal for a vein prediction algorithm

During exploratory migration in the above experiments, the time scales of the domain growth and main vein formation are comparable, so the leading edge behaviour cannot be negli-gible for the vein formation process. As shown in figure 1(b), the thick vein position seems to connect the leading edge posi-tions at different points in time with the shortest path. This is plausible given that Tero et al have studied the geometry of

Physarum vein networks connecting food sources and showed that the vein with the minimum total length survives [3, 7].

Motivated by these observations, we propose an algorithm that predicts the position of an organism’s main vein from the expansion of its domain, that is, from the movement of the frontal tip, alone (see section 2.3.3 and figure 2). In particular, the algorithm consists of choosing the shortest connections between two adequately spaced positions of the organism’s growth front and joining them up in a coherent connection from the first to the last. This naturally results in a connec-tion following the centre line in straight corridors and also cutting corners, if there are any. That is, the prediction of the vein position follows the same efficient ‘centre-in-centre’ line observed in the real organism.

3.3. Verification of the algorithm

In order to test the predictive power of the proposed vein pre-diction algorithm, we applied it to different (subsets of) series of images of the expanding plasmodia on a snake-shaped lane. Figure 3(a) shows an example of the migrating plasmodia and the predicted vein positions. It can be seen that the predic-tions (on the right) closely trace the real major vein of the organism (on the left). The prediction algorithm also man-ages to accurately capture the branching pattern of the vein network directly behind the growth front, with many parallel veins coalescing into a few major veins, which finally merge into a single vein. It is also worth mentioning that the data presented here are based on time differences of 9 min, or on subsets of the full time series using only every third picture. Therefore, three possible predictions were possible, which here we show superimposed. All three predictions yielded almost identical results. We observed the convergence of dif-ferent subset phases in all samples (n = 8) but one, which we will discuss in the last section of this paper (section 5.2).

The algorithm was able to predict the position of the final main vein accurately. This finds expression in the very low average distance of the predicted vein from the actual major vein (figure 3(c)). By comparison, the distance between the real vein and the shortest possible path (illustrated in figure 3(d)) is, on average, more than twice as large.

4. Mathematical model of adaptive vein formation coupled with leading edge dynamics

Above, we reconstructed the dynamics of the vein network from the behaviour of the growth front, thus demonstrating that these are not independent processes, but, at least the one depends on the other. In addition, its accuracy hints at it capturing some central aspect of the organism’s behaviour. However, this reconstruction is purely phenomenological and it is unclear whether its assumptions are physiologically mean-ingful. Thus, we now derive a more mechanistic model whose assumptions are matched by already established processes in Physarum. This will allow us to arrive at some degree of biological understanding of the mechanism underlying vein

Figure 2. Illustration of the procedure used to obtain a prediction of the position of the main vein from the history of the expansion of the organism’s growth front. The organism extends from the (top) left to the (bottom) right. The wave front is defined as the largest set of newly occupied pixels in each frame. To predict the position of the vein at (m − 1)∆T , the minimal distance connection of the vein position at m∆T to the growth front at (m − 1)∆T was determined. This was done for every frame from the last (M∆T ) to the first.

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network development and its interdependence with growth front expansion.

In detail, we consider the following processes: (i) Chemical reactions taking place at the leading edge control the migra-tion activity and also generate a pressure gradient of the cell’s sol. Due to this pressure gradient, the flux of the sol there increases. (ii) As discussed in [7], the vein thickness increases as the flux increases in a process called the ‘current rein-forcement rule’. This rule has been shown to produce results very similar to finding shortest distances in practice, but is also affected by mass distribution inhomogeneity. (iii) Since the chemical substances are transported through the vein, the migration activity increases too. Therefore, not only does the formation of the vein network depend on the behaviour of the growth front, but, in turn, the formation of the leading edge is driven by the transport of body mass from the rear of the organism to the front, through the vein network (see figure 4), as well.

4.1. Modelling

4.1.1. Fluid dynamics in the vein network. Tero et al have proposed a model to reproduce the time evolution of the vein formation [7]. In the model, the flux flowing from the source node ( Iout) and flux flowing into the sink node ( Iin) are taken as constant values and the constraint Iout = Iin should hold for all time. That is, the total mass in the tube is conserved. However, in our model setting, the flux flowing into and flowing out from the tube dynamically change and the total mass of the

sol is not necessarily conserved. Thus, we employ the model based on a network model for incompressible fluid dynam-ics in a one-dimensional, distensible tube derived in [19]. The

Figure 3. Application of the vein prediction algorithm to the migrating plasmodium on the snake-shaped lane. (a) Snapshots of a migrating plasmodium (left panels) and the predicted vein positions (black lines). The frame interval for the prediction is three frames (∆T = 9 min), the resulting three parallel predictions are superimposed. The times between the snapshots presented here are 63, 90, 63, and 51 min. (b) An example of partially incorrect prediction. One of the three predictions diverges from the actual major vein (middle and bottom panel on the right), but traces a secondary vein in the real organism (small black arrows in middle panel on the left). The times between the snapshots presented here are 51 and 99 min. (c) Boxplots of the distance from the major real veins (RV) to the predicted vein (PV) or the shortest path (ST). Differences in the distance from the real vein to the predicted vein (∆T = 18 min) and the distance from the real vein to the shortest path were statistically significant (Mann–Whitney U test, p = 0.002). (d) Examples of a trace of the major real vein (black line in the upper panel) and the shortest path in the lane (black dashed line in the lower panel).

Figure 4. Schematic figure of the model setting. (a) (i) A pressure gradient is generated by the chemical reaction taking place at the leading edge. (ii) Vein formation due to sol streaming. (iii) Transport of the endoplasmic sol to the leading edge. (b) Schematic figures of the profiles of u, v, w and Γ.

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change of the flux between two nodes in an idealised network is due to the pressure difference along the edge connecting the nodes, and the flux, in turn, results in a change in pressure at each node:

dqij

dt=

1Lij

( pi − pj)− Rijqij,

dpj

dt=

!

i

1Cij

qij,

(1)

where t is time, i, j = 1, . . . , N are node labels. The flux of the sol between nodes i and j in the tube is denoted by qij(t) and the pressure of the sol at the jth node is pj(t). The vari-ables Lij, Rij and Cij give the fluid inductance, resistance and capacitance, respectively, and are defined in [19] by Lij = ∆zij/(ald2

ij), Rij = ∆zij/(ard4ij), Cij = acd3

ij∆zij, where al = 4π/9ρ, ar = 8π/81ν, ac = 2π(1 − σ2)/Eθ. ∆zij and dij(t) indicate the length and radius, respectively, of the tube connecting the ith and jth nodes. ρ is the fluid density, ν is the ordinary coefficient of viscosity, E is Young’s modulus for the tube wall, θ is the thickness of the boundary wall and σ is the Poisson ratio for the wall. We assume distensibility of the tubes because for rigid walls the sum of all flows into and out of the system needs to vanish exactly, as is the case in the model used by Tero et al [7]. Here, we consider the more real-istic scenario of some difference between inflow and deposi-tion at the leading edge (see section 4.1.3) being absorbed by extending tube walls.

For practical reasons, we replace Lij, Rij and Cij. In the real organism, at the time when a new vein is born, the radius of the vein, dij, should be close to 0. This implies Rij, Lij and C−1

ij tend to diverge at the initial stage of vein genera-tion. From (1), Rij, Lij → ∞ means that a large pressure gra-dient is required to generate sol flows and C−1

ij → ∞ means that dpj/dt is very sensitive to a change in qij. These situa-tions cannot be observed in the real organism. Thus, we add saturation terms in Lij, Rij and C−1

ij to prevent divergence: !Lij(Dij) = ∆zij/(al + alD

1/2ij ), !Rij(Dij) = ∆zij/(ar + arDij),

!Cij(Dij) = (ac + acD3/4ij )∆zij, where we define Dij(t) := d4

ij(t) for convenience and will refer to it as ‘conductivity’. Thus, we obtain

dqij

dt=

1!Lij(Dij)

( pi − pj)− !Rij(Dij)qij,

dpj

dt=

"

i

1!Cij(Dij)

qij.

(2)

It has been observed that tube thickness and flux amplify each other in a positive feedback. We employ the equation pro-posed in [7] to describe this feedback process:

dDij

dt= f (qij)− rDij, (3)

where f (q) := |q|µ/(1 + ad|q|µ) and r > 0. Owing to the decay term −rDij, the radius of the vein converges to 0 if qij ≡ 0. The function f is a monotonically increasing function

of |q|, so Dij increases as |qij| increases. The parameter µ con-trols the nonlinearity of the feedback.

4.1.2. Actomyosin contraction. Experimental studies have shown that actomyosin contraction is generated by chemi-cal oscillations of Ca2+ and adenosine triphosphate (ATP) [20]. Here we employ the models proposed in [21, 22] of the interdependence of the Ca2+ concentration and phosphory-lation probability at any position in the organism, which is based on the biochemical analysis of the time period of Ca2+ fluctuations:

τ1∂c∂t

= F(c,ϕ) + Dcc,

τ1∂ϕ

∂t= G(c,ϕ) + Dϕϕ t > 0,

(4)

where = ∂2/∂x2 + ∂2/∂y2, x = (x, y) ∈ R2, Ω ⊂ R2 , τ1, Dc and Dϕ are positive constants, c = c(x, t) is the con-centration of Ca2+ outside of vacuoles. ϕ = ϕ(x, t) is the probability of a myosin light-chain kinase being phosphory-lated. F(c,ϕ) = −kVnc(c,ϕ) + kL(Nc − c) and G(c,ϕ) = KQ(nc(c,ϕ))(1 − ϕ)− kEϕ. \Nc is the equilibrium cal-cium concentration; kV and kL are pumping and leaking rate of calcium vacuoles, respectively; kE is the dephosphory-lation rates of the myosin light chain kinase; An activa-tion function for kinase phosphorylation is described by KQ(nc) = kQ[K∗nc(c,ϕ)/(1 + K∗nc(c,ϕ))] where nc(c,ϕ) = [1 + 0.17(c − 7.5)](a0 + a1ϕ+ a2ϕ2 + a3ϕ3), kQ, K∗ are coefficients for a Hill-equation and a0, a1, a2 and a3 are fitting parameters for nc (see [22]). The first terms of (4) generate a limit cycle.

4.1.3. Leading edge. We assume that migration is driven by the chemical reaction occurring at the leading edge. The stiff-ness of the gel depends on the aggregation state of the acto-myosin filaments, and the gel state can be converted to the sol state in response to solation factors.

It has also been experimentally observed that struc-tures containing high concentrations of ATP [12, 13], Ca2+ [14, 23], and Fragmin [24, 25] are present at the leading edge. The chemical reactions that control the stiffness of the gel are then responsible for generating the leading edge. It has also been found that many other chemicals, including H+, Ca2+ and K+, are involved in the reaction network [12, 23, 26–28]. Details of all these chemical reactions are currently unclear and the chemical reaction network in the organism is com-plex. Therefore, we adopt a simple reaction term to describe the fundamental dynamics of the chemical reactions. We use the following reaction-diffusion system for which the reaction term is same as that used in [16]:

τ2∂u∂t

= Duu − uv2,

τ2∂v∂t

= Dvv + uv2 − α1v, x ∈ Ω, t > 0,

τ2dwdt

= h(v),

(5)

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where Du and Dv are diffusion coefficients for u and v, respec-tively, α1 is a positive constant, and v = v(x, t) and u = u(x, t) represent the concentration of the solation factors and that of the regulation factors, respectively. The thickness of the organism is w. Phenomenologically, the materials to construct the organism body are supplied at the leading edge. Thus, we assume that the thickness of the organism w increases where the leading edge, or the localised pattern of v, exists. Here we assume that h is a function of v and take h(v) = α2v, where α2 is a positive constant. The second term represents autocata-lytic chemical reactions between the solation factors and the regulation factors. The third term in the v-equation represents a decay process.

It is noted that, in the actual organism, the total amount of sol fluctuates over time due to the sol-gel transformation. In particular, when the organism expands its domain, the trans-formation from sol to gel occurs at the leading edge, and the amount of the sol decreases there. This means that, in order to continue migration, the transformation from gel to sol should occur somewhere in the body. However, in order to make the model simple, we assume that enough sol is supplied to the leading edge from the rear part and the effect of the transfor-mation from gel to sol is negligible. That is, the total length of the vein is not limited. Such an assumption is plausible in biological experiments when enough of the organism is ini-tially introduced.

4.1.4. Implementation and interactions. The models above fall into two broad categories: equation (2) is a discrete model for the dynamics at pre-defined nodes and edges of a planar network. On the other hand, equations (4) and (5) are descrip-tions of processes in continuous space. We therefore imple-ment the model as the extension of a continuous ‘organism’ over a discrete and pre-defined network, with the reactions taking place in the appropriate ‘layer’. However, the dynamics of vein formation, actomyosin contraction and leading edge formation interact with each other.

Among these interactions, in this study, we focus on the following. (I) The actomyosin contraction wave generates a pressure gradient of the sol in the vein. Thus, we assume that the dynamics of the pressure pj depends on the Ca2+ concen-tration c. (II) Since the solation factors produced by the chem-ical reactions taking place at the leading edge decrease the stiffness of the vein, we assume that the dynamics of pj also depends on the concentration of the solation factors v. (III) Although a fine and dense grid of veins pre-exist in this imple-mentation of the model (see above), they are only allowed to expand their diameter to assume non-zero size when the vein is in the region occupied by the organism. Therefore, we assume that the equation of the conductivity Dij depends on the organism’s thickness w. (IV) The chemical reactions occurring at the leading edge are activated by the chemical substances supplied through the vein. Since the sol is trans-ported from the rear to the leading edge and is released in the frontal area, we assume that the equation of v depends on the supply rate of sol at the leading edge. The supply rate of sol is phenomenologically described by the radius of the veins at the leading edge.

Interactions (I) and (II). It has been shown in [21, 27] that periodic actin-myosin contraction occurs according to the time periodic oscillation of [Ca2+], or c; the contraction occurs when c is small. We incorporate this effect simply into (2) with a linear coupling. In addition, since the stiffness of the gel at the leading edge decreases as the concentration of the solation factor increases, we assume that !Cij depends on v, changing !Cij to !Cij(v, Dij) := "Cij(Dij)(1 + avv). There-fore, the governing equation of the pressure is given by

dpj

dt=

!

i

1"Cij(vij, Dij)

qij − κ(cj − c), (6)

where cj := c(xj), c is the basal concentration of Ca2+, vij := [v(xi) + v(xj)]/2, and xi ∈ R2 is the position of ith node. Since the relaxation and contraction of the actomyosin oscil-lation occur when [Ca2+] is large and small, respectively, we take κ positive [27].

Interaction (III). Since the vein structure is made of actin filaments present in the organism, the growth term f should depend on the thickness of the organism, w. A function Γ is given to determine the region occupied by the organism as follows:

Γ(w) :=!

1 w > wth

0 otherwise, (7)

where wth is a threshold thickness (figure 4). Since the vein can grow only in the region occupied by the organism, we multiply the growth term f by Γ:

dDij

dt= Γ(wij) f (qij)− rDij, (8)

where wij := [w(xi) + w(xj)]/2. It is noted that Dij → 0 as t → 0 when Γ ≡ 0.

Interaction (IV). It has been observed in [29] that a pro-trusion of the cell surface is initiated by the exocytosis of vesicles in the sol at the leading edge. We assume that the contents of the vesicles control the activity of the reaction-diffusion systems described in (5), that is, the vesicles con-tain the activator v. (For a justification of this assumption, refer to section 4.3.2.) Thus, because vesicles are trans-ported to the boundary of the organism through the vein system, the concentration of v changes according to the time average of the supply rate. As this increases as the num-ber and the radius of the veins increase, we simply describe this contribution of v as a linear function of Dij . Thus, we change (5) to

∂u∂t

= (Duu − uv2)/τ2,

∂v∂t

= (Dvv + uv2 − α1v)/τ2 + kB(w)!D(x),

∂w∂t

= h(v)/τ2,

(9)

where B indicates the position of the leading edge which is defined by

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B(w) =!

1 w ∈ [wext, win]

0 otherwise. (10)

The function !D at node point x = xi is defined by !D(xi) :=

"j Dij , where j is the number of a node connected to

node i. In general, the value of !D(x) for x = xi (i = 1, . . . , N) should be determined by using a smoothing method or aver-aging. However, in the numerical scheme explained later, we only use the values !D at x = xi when calculating numerical solutions because the discrete model and the continuous models use common node points. Therefore, we omit the defi-nition of !D for x = xi .

As a result, we obtain the following equations.Vein network:

dqij

dt=

1!Lij(Dij)

( pi − pj)− !Rij(Dij)qij,

dpj

dt=

"

i

1#Cij(vij, Dij)

qij − κ(cj − c),

dDij

dt= Γ(wij) f (qij)− rDij.

(11)

Actomyosin contraction:

τ1∂c∂t

= F(c,ϕ) + Dcc,

τ1∂ϕ

∂t= G(c,ϕ) + Dϕϕ.

(12)

Leading edge:

∂u∂t

= (Duu − uv2)/τ2,

∂v∂t

= (Dvv + uv2 − α1v)/τ2 + kB(w)!D(x),

∂w∂t

= h(v)/τ2.

(13)

4.2. Numerical results

4.2.1. Numerical scheme and settings. We use the finite ele-ment method to solve (12) and (13). A spatial mesh was gener-ated by using the software FreeFEM++ [30]. The values of pj, c, ϕ, u, v and w are defined at the node points and Dij , qij are given at the edges. Neumann boundary conditions are used for (12) and (13).

The initial data for pj, qij, Dij are

pj(0) = 0 ( j = 1, · · · , N),qij(0) = 0 (i, j = 1, · · · , N),

Dij(0) =!

0.1 if xi ∈ Ω0 or xj ∈ Ω0(i, j = 1, · · · , N),0 otherwise,

where Ω0 = x ∈ R2 | x ∈ [39, 40]× [0, 5] indicates the region in which the organism is initially placed. The initial data for the variables in (11)–(13) are

c(x, 0) =

!1.0 x ∈ Ω0,0.0 x ∈ Ω\Ω0,

ϕ(x, 0) = 0 x ∈ Ω,u(x, 0) = 3 x ∈ Ω,

v(x, 0) =

!1.0 x ∈ Ω0,0.0 x ∈ Ω\Ω0,

w(x, 0) = 0 x ∈ Ω.

In the actual organism, an actomyosin contraction wave is generated and the wave propagation enables the endoplasm to be transported to the leading edge [31]. In order to generate wave propagation, we take τ1 to be a function of space. In particular, we assume that τ1 in Ω0 is smaller:

τ1(x) =!τ1/3, x ∈ Ω0,τ1, otherwise. (14)

With this setting, time periodic contraction wave propagation from Ω0 to the leading edge is observed. In addition, due to the interaction term κ(cj − c) in (11), a back and forth sol flow called ‘shuttle streaming’ between Ω0 and the leading edge occurs.

In the actual organism, pattern transitions of the vein structure are observed. Firstly, a large number of small veins appear around the leading edge. Secondly, due to the posi-tive feedback between the flux and the vein thickness growth, that is, the current reinforcement rule, a small number of the veins survive and form a tree-like structure. Finally, at the rear part of the organism, a few thick veins are left. It has been found that taking µ in an appropriate regime is crucial to reproducing these behaviours [7]. Based on the analysis in [32], we take µ = 1.3

The time scales of actomyosin contraction and migration are determined by τ1 and τ2. The ratio between τ1 and τ2 can be estimated based on experimental results. It takes about 10 h for the organism to reach the end point of the domain and the period of the actomyosin contraction is about 1 min. Thus, we determine the ratio between τ1 and τ2 such that the actomyosin contraction occurs about 600 times during the time it takes for the leading edge to reach the end point. The absolute values of these parameters determine the time scale relative to the vein network dynamics (equation (11)) and were chosen to obtain vein networks most closely resembling experimental results. (Refer section 4.3.3 for further analysis of this point).

The other parameters are set to al = 5.0, ac = 0.03, ar = 13.5, al = 10.0, ac = 0.1, ar = 2.5, ad = 0.01, av = 0.01, R = 1.0, κ = 1.0, r = 0.15, c = 7.0, a0 = 0.349 353, a1 = −0.045 4567, a2 = 1.159 05, a3 = 1.823 858, kV = 0.12, kL = 0.004, kQ = 1.0, kE = 0.1, K∗ = 1.75, Nc = 25.0, Dϕ = 0.012,Dc = 0.2, τ1 = 0.0072, Du = 0.2, Dv = 0.02, α1 = 0.01, α2 = 0.1, τ2 = 20, wth = 1.0, wext = 0.1, win = 0.9, and k = 0.01.

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4.2.2. Snake-shaped and U-shaped arena. The domain shape, size and spatial grids for a numerical simulation are displayed in figure 5(a). Running the simulation for each arena shape 8 times and using randomly generated grids, the grid number of the snake-shaped arena was between 2420 and 3422, and for the U-shaped arena between 1933 and 2331.

Figure 5(b) shows the time evolution of v and the main vein positions. A localised pattern of v propagates along the lane, and the leading edge moves forward according to the prop-agation of the localised pattern. A tree-like pattern of veins appears around the leading edge, and the main vein is formed at the rear. As shown in figure 5(c), the region occupied by the organism (the region with w > 1.0) increases and the vein network is formed in this region. These results qualitatively reproduce the behaviours of the actual organism.

4.3. Evaluation of the model

In section 3.1, we characterised the way Physarum constructs its main vein while exploring (see figure 1). We will now examine whether the model can recreate this behaviour.

4.3.1. Centre-in-centre vein trajectory. As shown in fig-ure 6(a), the numerical results of the model capture the charac-teristic centre-in-centre vein trajectory observed in Physarum during exploratory migration. That is, the main vein does not follow the global shortest path (broken line in figure 6(a)), but

instead keeps a significant distance from the boundary of the arena at the middle part between the first and second corner. In fact, not only is the shape of the trajectory qualitatively similar, but it is significantly different from the vein trajectory of the experiment in only one small area where the variance of the model’s vein is exceptionally small.

4.3.2. Variation of the interaction strength k. It is instructive to examine the performance of the algorithm under a different condition of the model. If, for instance, no interaction between the leading edge and the tube network is assumed, for exam-ple because the vesicles being transported to the leading edge in the sol do not contain the activator v as assumed in equa-tion (9), in other words, if in (9) k is set to 0, the vein trajectory assumes a shape that is much closer to the shortest possible path (see figure 6(b)). From a statistical view point, the model under these circumstances constructs the vein in significantly different positions over a large portion of its overall course.

4.3.3. Variation of the migration speed (τ2). Next, we inves-tigate the effect of the migration speed of the leading edge on the vein network formation. One of the possible param-eters that can control the migration speed is τ2 in (13), the time scale of the leading edge dynamics. We find that, as τ2 decreases, while keeping the ratio between τ1 and τ2 constant, the migration speed increases (figure 7(a)). For example, at τ2 = 2 the migration speed of the growth front is so large that

v

(b) (c)

w

t = 1040

t = 1280

t = 800

t = 1040

t = 1280

5

5

5

5

(a)

t = 800(d)

t = 429 min

t = 543 min

t = 726 min

Figure 5. Numerical scheme and settings. (a) The meshes for the numerical simulation in the snake-shaped and U-shaped arena. (b) Images taken from one exemplary sample of the organism’s exploratory behaviour for comparison. (c) Time evolution of v. (a) Black and gray dots indicate the positions of the leading edge (B(xi, ·) = 1) (i = 1, . . . , N), and the nodes satisfying vi > 1.5, where vi = v(xi, ·), respectively. Thick lines indicate the main vein with Dij > 10. (d) Time evolution of w. Gray dots indicate the region occupied by the organism (w > 1.0). Thick lines as in (c).

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it becomes increasingly disconnected from the closest main vein (see figure 7(b)). We observe centre-in-centre trajecto-ries when τ2 is large (τ2 = 20), but not for small τ2 (τ2 = 2) (figure 7(c)).

5. Discussion

This study links two well studied mechanisms in the behav-iour of Physarum polycephalum: growth front expansion dynamics and vein network formation.These two phenomena have been described separately by mathematical models that allowed their prediction in computer simulations. However, in nature, Physarum employs both mechanisms simultaneously to forage for food and amicable environmental conditions. As a consequence, the growth front’s expansion depends on the vein network, while the latter is developing and is being optimised in a constantly changing domain. Also, the expan-sion front acts as the ‘sensory horizon’ of the organism, deter-mining the area within which the vein network is to be formed.

5.1. Vein network lay-out of the expanding organism: the centre-in-centre trajectory

First, we observed how free-roaming Physarum constructs its vein network when allowed to explore and expand through a corridor with a 90° bend, and described a striking charac-teristic: Physarum did not construct its main vein along the centre line of its body, nor reconstruct the vein to achieve the globally shortest path. Instead, the organism generally fol-lowed the centre line, cut the corners, and then returned to the centre line, thus forming a ‘centre-in-centre’ trajectory. This vein network lay-out remained largely unchanged as long as exploration was ongoing, apart from minor side veins gradu-ally disappearing (for an example, refer to figure 3).

However, the lay-out thus realised achieved almost ideal efficiency, being, on average, only 6 percent longer than the globally shortest possible path. The value of ‘6 percent’ is somewhat arbitrary, though, as other values could be given depending on how much of the terminal stretches are included. But if compared to some kind of upper bound, in this case the ideal centre line throughout the corridor, the efficiency of the chosen vein lay-out can be appreciated.

Also, real veins are almost never perfectly straight, but show numerous undulations. But this is true for both the actual vein position here and for situations where the organism approaches the globally minimal network length [4]. We therefore deem the underestimation of the path length to be negligible.

Figure 6. Average vein positions for the numerical results of the model. (a) Interaction strength k = 0.001. The model’s average vein position agrees well with the average position of the organism’s vein in experiments. Significant differences are very limited and only where the model’s vein exhibits very small variation. (b) Interaction strength k = 0 (no interaction). The model’s average vein differs much more and in more places from the experimental data. Moreover, it seems to follow the globally shortest path over wide stretches.

Figure 7. (a) Dependence of the migrating velocity on τ2, where τ1 is given such that the ratio τ2/τ1 = 20/0.0072. T ′ is the time when the leading edge first touches the line ℓ = (x, y) | x ∈ [25, 30], y = 10 shown in (b). (b) Distance of the main vein from the growth front. (Definitions as in figure 5.) (c) Typical vein network patterns of the proposed model.

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5.2. Vein prediction algorithm

As a first approach to the mechanism behind the efficient network construction we presented an algorithm that can ‘predict’ the position of the main vein if fed only with the consecutive positions of the growth front. The high accuracy of this algorithm is remarkable, because it strongly supports the notion that tip growth and vein growth are not indepen-dent, but interdependent. Therefore, one can be reconstructed from the other. Moreover, in order to do so, the information contained in the position of the growth front needs to be inter-preted correctly. In this case it is the time course, or more specifically, the shortest connection between the positions of the wave front at two points in time, which yields a good re-construction, and not, for example, the point of the lowest or highest curvature. It is reasonable to suggest that this principle is employed by the organism, as well.

Figure 3(b) shows an example of a partially failed pre-diction. In this superimposition of three parallel predictions, based on the three different subsets that can be chosen for a time interval ∆T = 9 min, two of these predictions trace the actual major vein of the organism, but one prediction devi-ates between the third and the fourth 90° bend. Interestingly, though, this deviation traces a secondary vein that the organism did in fact construct, but dismantled shortly thereafter.

Generally, a multiplicity of veins is encountered either immediately behind the growth front or where the growth front has expanded to a wide arch, e.g. at a corner where a new direction for expansion opens up. In both cases this is due to the geometric fact that if the angle of aspect of the growth front is large from a given distance from it, local path length minimisation will require more than one end point. But both in the organism as well as in the algorithm, as the growth front moves away and refocuses, the number of paths is reduced, as many of the now disappearing veins do not con-nect to the growth front in an efficient way anymore. However, sometimes (usually: two) parallel veins survive, at least for an extended time. This is true for the organism in a literal sense, but even in the algorithm this can occur when looking at the composite image of numerous parallel predictions based on different subsets of the same image series (as described above and shown in figure 3(b)).

We interpret this occurrence as examples for situations in which, in the real organism, two veins are virtually equally efficient for supplying body mass to the growth front at one point in time, so that current reinforcement (or, in the algo-rithm, local path minimisation) cannot decide on one par-ticular path. In these cases, the organism keeps both veins functional, a trait that might pay off at a later point in time, when the geometry of the environment is explored to a higher degree. Then, one of the multiple veins might be in a pref-erable position and will be used to a greater extent than the other, which will disappear as a consequence. Therefore, the appearance of a multiplicity of veins can be interpreted as an adaption to a fundamental problem of Physarum, that is, the lack of remote sensory abilities, and the need to entertain mul-tiple behavioural options to be able to react to ‘unforeseen’ environmental conditions.

5.3. Significance of the frame interval

Physarum polycephalum is, of course, blind and has no other remote-sensing organs. It is therefore unaware of its environ-ment’s general lay-out and only has information about the area already extended over. In this sense, its growth front is its sensory organ, feeding back information on whether growth in a given direction is desirable or even possible. This means that the decision where to construct its vein network cannot be made in anticipation of future developments, but an efficient, shortest path can, therefore, only be ‘short’ within the organ-ism’s boundaries.

This problem could be circumvented by delaying the vein construction as much as possible to integrate more and more information about the body’s shape and growth direction. But this raises a dilemma: the vein network is needed for efficient mass transport [33], and waiting until the whole environ-ment is explored is not an option. Therefore, the organism has to find a balance between immediate and delayed network construction.

This is mirrored by the question of the adequate choice of the time interval for the vein prediction algorithm. Using time intervals that are too small often fails to correctly identify the main vein. In these cases the algorithm returned predic-tions that retrieved the position of some secondary vein or a temporary main vein (see figure 3(b) for an example). On the other hand, if the time interval for the vein prediction is too large the accuracy of the prediction decreases (see figure 8). This is due to the fact that the growth reacts to environmental conditions (e.g. corners or obstacles) and in turn influences the position of the vein, and this information is ignored for the frames skipped when working with large time intervals. In practice, time intervals around 15 min yielded the smallest median distance of the predicted vein from the actual major vein. However, owing to the large variation among the sam-ples, the differences between the time intervals in figure 8 were not statistically significant, and we could not establish a single time interval ∆T (⩽30 min) yielding the best result for all studied samples. We interpret this as hinting at a variability in the duration of the determination process in the organism itself, possibly depending on expansion speed and environ-mental conditions.

5.4. Numerical simulation

The model proposed here contains some phenomenological formulations, but key characteristics of the organism, first and foremost the centre-in-centre trajectory of the main vein, and also the tree-like vein structure, the rough leading edge pro-file, or shuttle streaming were successfully reproduced.

The centre-in-centre trajectory seems to mainly be the con-sequence of the coupling of the growth front expansion to the vein formation process (equation (9)). This can be seen from the change of the trajectory when the explicit interaction term in interaction (IV) is disregarded (section 4.3.2, k set to 0, thus no direct feedback of the vein network’s state to the frontal tip dynamics) or when the vein development is dissociated from the frontal tip dynamics (section 4.3.3, τ1 and τ2 set very low,

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slowing down the time scale of the vein development rela-tive to the other processes). In these cases, the resulting vein assumes a shape much closer to the shortest possible vein (see figures 6(b) and 7(c)).

An extreme version of the ‘no-interaction case’ is the ‘Physarum Solver’ model introduced in [7], where frontal tip dynamics are not considered at all (or, it could be said, migra-tion speeds are infinite), and which indeed yields shortest possible paths under comparable conditions (U-shaped arena, 1086 nodes, µ = 1.3; see figure 9).

However, it must be pointed out that recently centre-in-centre trajectories reproduced by the Physarum Solver model were published [34], although the arena used in that publica-tion was much more complex than the ones used here. As a consequence, the appearance of centre-in-centre trajectories depended on the configuration and interaction of neighbouring corners, on the local ‘randomness’ of the mesh, and on fluctua-tions in the initial vein thickness. Therefore, in order to discuss the differences between the results of the ‘Physarum Solver’ model and the model proposed here conclusively, we need to clarify the precise meaning of the term ‘centre-in-centre tra-jectory’. Here, we mean the return of the vein to the centre, unguided by the boundary, after each corner (given sufficient space to do so), even if two consecutive corners turn in the same direction. This is in contradiction to strict global optimisation and can be considered a consequence of local optimisation.

Interactions (I) and (II) are essential to reproduce the typ-ical vein patterns. As studied in [35], the flux of sol across the organism increases when the stiffness of the organism at the peripheral region (leading edge) locally decreases and the pressure of the sol at the leading edge becomes lower than in other regions. This is the meaning of the term avv in !Cij.

With the the actomyosin contraction, or interaction (I), as the driving force of the sol streaming, a large flux of the sol occurs across the whole body. Because of these, vein generation occurs and a large number of small veins appear at the leading edge. We also note that the growth rate of a vein is determined by the flux (see f in (3)). As interaction (III) determines the place in which veins can grow, veins emanate basically at the boundary of the organism. This constraint results in a tree-like structure around the leading edge, which is a characteristic pattern observed in the actual organism.

As shown in figure 5(a), the leading edge profile is usually not smooth but undulated. Since the sol is transported from the rear to the leading edge through the vein, the supply rate of the sol should depend on the position of the end point of the vein. In turn, since the vein can grow in the region occu-pied by the organism, veins extend along the direction of the domain growth, as governed by interaction (IV). Therefore, we can see a positive feedback between the direction of the domain growth of the organism and that of the vein’s exten-sion, which causes the instability and wavy appearance of the profile.

In the organism, a back and forth mass flow (shuttle streaming) is generated by the actomyosin contraction. In our model, this is reproduced by interaction (I), the effect of the periodic oscillation of the Ca2+ concentration on sol pres-sure. To generate ‘unidirectional’ mass transportation in such a regime, symmetry breaking of the mass transportation is required; that is, a global spatial asymmetric pressure gradient is needed. In our mathematical model, the spatial asymmetry is generated by the localised pattern of v and the term 1 + avv in (11).

Finally, the vein prediction algorithm is based on the assumption that vein network and growth front development are deeply interconnected, so that for images of the time evo-lution of the real organism’s shape, it is possible to derive the one from the other (see section 5.2). In this way, it can also serve as a test whether that interconnectedness has been

Figure 8. The average distance of the predicted vein from the real vein depends on the chosen time interval ∆T . The boxes stretch from the first quartile Q1 to the third quartile Q3 (Q3 − Q1 being the interquartile range I), the mark gives the median value Q2. The ‘error bars’ encompass all values within the interval [Q1 − 1.5I, Q3 + 1.5I]. Every value outside this range is displayed as an outlier (dot). The median is lowest at ∆T = 15 min, but the high variation does not allow to choose one time interval that is superior for all samples. However, after ∆T = 30 min, the distances steadily increase.

mesh

vein (Dij > 10)

A B

Figure 9. Result of a ‘Physarum Solver’ simulation under conditions comparable to our model: U-shaped arena, 1086 nodes, µ = 1.3. The function f in equation (3) with r = 1 is given by f (q) = |q|1.3. The veins satisfying Dij > 10 are displayed. Points A and B are a source and sink node, respectively. The vein follows a shortest path and does not exhibit a ‘centre-in-centre’ trajectory.

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preserved through modeling. And in fact, the algorithm does predict the vein of the numerical model correctly, showing that there, too, the vein network formation can be derived from growth front development.

5.5. Biological interpretation

The motivation of this study was to understand (an aspect of) biological transport networks, and specifically, how the struc-ture and temporal behaviour of this feature connects with the organism’s natural use of it. In our context this means that, in order to understand whether and how they interact to produce Physarum’s exploratory behaviour, we wanted to relate the well-known mechanisms of reaction diffusion dynamics in the growth tip and the current reinforcement dynamics in the vein network to the exploratory behaviour of Physarum polyceph-alum in a mathematical model.

In particular, the current reinforcement rule is well studied in static situations [7]. There, the food sources act as attractors for body mass and generate a directional body mass flow nec-essary for the mechanism to work, thus creating a highly effi-cient network connecting the food sources. But being spread out over a large area and then be presented with food sources placed at different positions in the area occupied is not a nat-ural setting. However, in nature the organism has an incentive to transport body mass efficiently to the growth front, that is, to expend as little energy as possible and to maximise locomo-tion speed [32]. Here it is the expansion of the growth front, in particular the sol-gel transformation, that acts as the body mass sink causing a directional flow. Therefore, we claim that the advantage the current reinforcement mechanism confers to Physarum mainly lies in it affording the organism very effi-cient locomotion.

In this context it would be interesting to see how this under-standing translates to other systems in which the expansion of the system’s boundaries are in step with the construction of its internal network. Can, for example, the lay-out of overland road networks be understood in terms of its contribution to the expansion of civilisation, or the growth of roots by the flow of nutrients necessary for its construction? There are, of course, differences, as civilisations have ‘remote-sensing capabili-ties’, and roots have no body other than the transport network (they are the transport network). Nevertheless, in both cases current reinforcement as well as some kind of leading edge dynamics play an important role [36, 37], and in both cases they interact with each other strongly [38, 39]. Therefore, we think that the phenomenology and the mechanism discussed in this paper are shared among a wide and significant group of natural networks, namely networks that form within an expanding domain.

Acknowledgments

This research was supported by Grant-in-Aid for Scientific Research on Innovative Area from MEXT 25111726 and 25103006 and Japan Society for the Promotion of Science

KAKENHI grant numbers 26310202, 26247015 and 25400199, as well as by the MEXT Scholarship Number 152503.

ORCID iDs

Daniel Schenz https://orcid.org/0000-0003-2930-641XShigeru Kuroda https://orcid.org/0000-0002-8079-6322Toshiyuki Nakagaki https://orcid.org/0000-0002-3446-4888Kei-Ichi Ueda https://orcid.org/0000-0003-3078-1162

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