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Mycelial fungi and acellular slime molds grow as self-organized networks that explore new territory to search for resources, whilst maintaining an effective internal transport system in the face of continuous attack or random damage. These networks adapt during development by selective reinforcement of major transport routes and recycling of the intervening redundant material to support further extension. In the case of fungi, the predicted transport efficiency of the weighted network is better than evenly weighted networks with the same topology, or standard reference networks. Experimentally, nutrient movement can be mapped using radio-tracers and scintillation imaging, and shows more complex transport dynamics, with synchronized oscillations and switching between different pre-existing routes. The significance of such dynamics to the interplay between transport control and topology is not yet known. In a similar manner, the resilience of the network can be tested in silico and experimentally using grazing invertebrates. Both approaches suggest that the same structures that confer good transport efficiency also show good resilience, with the persistence of a centrally connected core. The acellular slime mold, Physarum polycephalum also forms efficient networks between food sources, with a good balance between total cost, transit distance and fault tolerance. In this case, network formation can be captured by a mathematical model driven by non-linear positive reinforcement of tubes with high flux, and decay of tubes with low flux. We argue that organization of these simple planar networks has been honed by evolution, and they may exemplify potential solutions to real-world compromises between search strategy, transport efficiency, resilience and cost in other domains.
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Chapter 4
Adaptive Biological Networks
Mark D. Fricker, Lynne Boddy, Toshiyuki Nakagaki, and Daniel P. Bebber
Abstract Mycelial fungi and acellular slime molds grow as self-organized networks
that explore new territory to search for resources, whilst maintaining an effective
internal transport system in the face of continuous attack or random damage. These
networks adapt during development by selective reinforcement of major transport
routes and recycling of the intervening redundant material to support further exten-
sion. In the case of fungi, the predicted transport efficiency of the weighted net-
work is better than evenly weighted networks with the same topology, or standard
reference networks. Experimentally, nutrient movement can be mapped using radio-
tracers and scintillation imaging, and shows more complex transport dynamics, with
synchronized oscillations and switching between different pre-existing routes. The
significance of such dynamics to the interplay between transport control and topol-
ogy is not yet known. In a similar manner, the resilience of the network can be
tested in silico and experimentally using grazing invertebrates. Both approaches
suggest that the same structures that confer good transport efficiency also show
good resilience, with the persistence of a centrally connected core. The acellular
slime mold, Physarum polycephalum also forms efficient networks between food
sources, with a good balance between total cost, transit distance and fault tolerance.
In this case, network formation can be captured by a mathematical model driven by
non-linear positive reinforcement of tubes with high flux, and decay of tubes with
low flux. We argue that organization of these simple planar networks has been honed
by evolution, and they may exemplify potential solutions to real-world compromises
between search strategy, transport efficiency, resilience and cost in other domains.
4.1 Introduction
Networks are common within biological systems and have been characterized in
a range of different contexts that include metabolism, protein–protein interaction,
neural circuits and ecological food webs. Despite the recent progress in biological
M.D. Fricker (B)
Department of Plant Sciences, University of Oxford, Oxford, OX1 3RB, UK
e-mail: mark.fricker@plants.ox.ac.uk
T. Gross, H. Sayama (eds.), Adaptive Networks, Understanding Complex Systems,
DOI 10.1007/978-3-642-01284-6 4, C
NECSI Cambridge/Massachusetts 2009
51
52 M.D. Fricker et al.
network analysis, one area that has received relatively little attention is the charac-
terization of organisms whose entire growth form is as a network. In particular, both
plasmodial slime molds (myxomycetes) and mycelial fungi form elaborate intercon-
nected networks that are highly responsive to local environmental conditions. Unlike
the other biological networks described, the network formed by these organisms is
not part of the organism, it is the organism. These networks develop as the organism
forages for new resources in a patchy environment and must both transport nutri-
ents between spatially separated source and sink regions, and also maintain their
integrity in the face of predation or random damage [4, 5]. The challenges that these
conflicting demands place on the network organization have strong parallels with
those faced in the design of anthropogenic infrastructure networks. The balance
the biological systems have achieved between cost, efficiency and resilience may
represent a good compromise to such a combinatorial optimization problem, and
may yield useful insights into the design of delocalized, robust infrastructure net-
works. This presumes that solutions adopted by biological networks will exemplify
useful generic theoretical principles, such as persistence, robustness, error-handling
or appropriate redundancy, as they have been honed by evolution. The expectation
is that the process of Darwinian natural selection based on variation, competition
and survival has explored a significant range of possible network organizations and
the resulting systems are likely to be well-adapted to survive and reproduce under
particular biotic and abiotic conditions to solve certain ecological problems. A range
of network architectures, development and dynamics can be found within the fungi
and myxomycetes, suggesting a comparative approach may be instructive. How-
ever, the constraints imposed by the components used to construct the network (i.e.
branching tubes) may have a profound effect on the possible network organization
and dynamics, so it is possible that any result can only be generalized to a very
limited set of real-world problems.
In this Chapter we focus on recent work describing the structure and function of
foraging woodland fungi [3, 33], to illustrate how these essentially planar, weighted
spatial networks resolve the conflicting demands of exploration, exploitation, trans-
port and resilience [3]. We provide a brief introduction to network development in
Sect. 4.2 then describe predicted transport of such networks in Sect. 4.3 and how
it compares with experimentally measured nutrient movement in Sect. 4.4. We fur-
ther comment on the experimentally observed oscillations and pulsatile transport in
Sect. 4.5. Section 4.6 covers both predicted and experimentally determined network
robustness. In Sect. 4.7, we compare the results from mycelial fungi with network
development in Physarum, as a second exemplar of an adaptive biological network,
before speculating on the universal features of such biological networks in Sect. 4.8.
4.2 Network Development in Mycelial Fungi
Filamentous fungi grow by apical extension of slender hyphae (Fig. 4.1a) that then
branch sub-apically to form a fractal, tree-like mycelium. In ascomycetes and basid-
iomycetes, tangential hyphal fusions or anastomoses occur as the colony develops to
4 Adaptive Biological Networks 53
F
l
a
E
D
ABC
50 μm
500 μm
50 mm
10 mm
25 cm
t=0h t = 60h t = 120h
G
t = 9d t = 25d t = 39d
25 mm
Fig. 4.1 Development of mycelial networks. (a) Bright-field image of hyphae of Phanerochaete
velutina growing across agar. (b) Mycelial system of P. velutina grown from 4 cm3beech wood
inocula on a 24 ×24 cm tray of compressed, non-sterile soil after 39 d. (c)a75×75 cm portion
of an extensive network of the saprotrophic basidiomycete Megacollybia platyphylla interconnect-
ing dead wood resources in Wytham Wood, Oxfordshire, UK. (d) Time lapse imaging of cord-
formation through hyphal aggregation in a growing colony of P. velutina on compressed sand/soil.
(e) Scanning electron micrograph showing the aggregation of hyphae in a cord. (f) Schematic
representation of a cord illustrating the length (l) and area (a) measures used to weight each link.
(g) Time lapse imaging of cord-regression and thinning-out of the network in a growing colony of
P. velutina. Modified from [21]
form an interconnected mycelial network [24, 25, 48, 49]. In the larger, more persis-
tent saprotrophic and ectomycorrhizal basidiomycetes that grow out into soil from
colonized food sources, the network architecture develops further with the formation
of specialized high-conductivity organs, termed cords or rhizomorphs [12]. These
form visible networks interconnecting food resources on a scale of centimetres in
laboratory microcosms (Fig. 4.1b) to meters in undisturbed woodland (Fig. 4.1c),
through parallel aggregation of many individual hyphae (Fig. 4.1d, e), [18]. Indeed
mycelial fungi form the most extensive biological networks so far characterized
[16, 33, 52–54], popularly known as the Wood Wide Web [50, 53].
54 M.D. Fricker et al.
The network topology is defined by classifying junctions (branch-points and
anastomoses) as nodes, and the cords between nodes as links. In general, during
foraging the number of nodes, number of links and the total material in the network,
increase through time. However, the local scale network evolution is also charac-
terized by selective loss of connections and thinning out of the fine mycelium and
weaker cords (Fig. 4.1g). This behaviour is also apparent in the box-count mass
fractal dimension of these networks, which shows a decrease as the networks thin
out [6]. Thus, fungal networks progress from a radial branching tree to a weakly
connected lattice-like network behind the growing margin, through a process of
fusion and reinforcement to form loops, and selective removal and recycling of
excess redundant material [3]. This shift can be quantified by the meshedness or
alpha coefficient [11, 26], that gives the number of closed loops or cycles present as
a fraction of the maximum possible for a planar network with the same number of
nodes. The alpha coefficient measured over the whole colony increased over time
from near zero, as expected for a branching tree, to 0.11 ±0.04 in control systems,
and to 0.20±0.05 in systems with an additional wood block resource [3]. The values
of the alpha index for Phanerochaete velutina were similar to those for networks of
tunnels in ant galleries [11], Physarum polycephalum (unpublished observations)
and street networks in cities [10, 13], suggesting that addition of around 20% of the
maximum number of cross-links into a planar network may be sufficient to achieve
desirable network properties in a range of different scenarios.
Other topological network measures have not proved to be very informative
as they are heavily constrained by the developmental processes of branching and
fusion, and crowding effects restricting the maximum number of connections pos-
sible in a planar network [2, 3, 3, 19, 29, 33]. Thus, the possible degree (k) of each
node is limited to 1 for tips, 3 for branch points or fusion, or occasionally 4 for
initially overlapping cords that then fuse. Likewise, the mean clustering coefficient,
C [70], is of limited relevance for fungal networks, as their growth habit effectively
precludes formation of triads. The frequency distribution of node strength shows
more diversity than node degree alone, and follows an approximately log-normal
distribution for P. velutina networks [3]. However, we have not found evidence
for power law relationships that have attracted so much attention in other network
analyses.
4.3 Predicted Transport Characteristics of the Mycelial Network
One approach to investigate the transport capacity of the network is to assume that
nutrient fluxes will follow the shortest path between pairs of nodes, calculated from
the predicted resistance of each link where longer, thinner cords have greater resis-
tance to flow. The changes in thickness of the cords during growth and network
re-modeling can be captured by image analysis of the reflected intensity of each
cord, with appropriate calibration, to give each link a weight that depends on its
length (l) and cross-sectional area (a). Each cord is modeled as a cylinder packed
4 Adaptive Biological Networks 55
with identical hyphae (Fig. 4.1f), rather than a single tube that increases in diam-
eter, although the internal structure of cords can be much more complex [62]. An
overall measure of transport is the average network efficiency (E), defined as the
mean of the reciprocal of shortest path lengths for transport through the network
[34, 35].
In isolation, the average efficiency is not useful without some frame of reference.
It is not straightforward to generate suitable reference models against which to test
the extent that differential cord weighting improves the performance of the network.
Indeed elucidation of such biologically-inspired algorithms is a key goal of current
research. At present there are no suitable algorithms available to generate weighted
planar networks with defined properties. In other areas of network theory, compar-
isons are typically made with a reference network produced by random rewiring of
the links. However, this does not make sense biologically. Likewise, randomly reas-
signing the weights to different links does not give an intuitively satisfying model
to test performance, as it also has no biological basis. We currently use a two stage
procedure to evaluate the performance of the fungal networks [3]. In the first step,
nodes within the Euclidean fungal network (Fig. 4.2b), were used to construct model
networks using well defined neighborhood graphs, including the minimum spanning
tree (MST, Fig. 4.2c) as a lower bound giving a low cost, but extremely vulnerable
network, the relative neighborhood graph (RNG, Fig. 4.2d), Gabriel graph (GAB,
Fig. 4.2e) and the Delaunay triangulation (DT, Fig. 4.2f), giving an upper bound
for a well-connected, robust, but rather expensive network [11, 13, 23, 43]. In the
second step of analysis, the effect of including a fixed amount of material in the
network, equivalent to the total material in the real network (Fig. 4.2a), was exam-
ined. Thus, each link in the “uniform” fungal and model networks was allocated a
constant weight, such that the total construction cost was the same. Effectively we
asked what the consequences for transport would be if the fungus had chosen to
allocate the same amount of resource evenly over the existing or model networks,
to determine the functional efficiency of the network. This also allowed comparison
with the real, differentially weighted network (Fig. 4.2a) as the network measures
were in comparable units.
Visual inspection of the resultant networks suggested that the topology of the
fungal network had some similarity to the RNG, in terms of the density of cross-
linking outside of the inoculum itself (Fig. 4.2d). Quantitatively, the RNGs had an
alpha coefficient of 0.12, slightly lower than the alpha coefficient of the fungal
networks. It was also apparent that regression of some links triggered substantial re-
arrangements in the layout of the model networks, particularly for the MST, which
showed dramatic alteration in the connections between neighboring nodes over time
(Fig. 4.2c) as the biological network developed.
Perhaps unsurprisingly, the real weighted networks had much shorter physio-
logical paths, especially in the central region, than their corresponding uniform
networks [3]. More surprisingly, the weighted fungal network outperformed both
the uniform DT and the uniform MST when the predicted transport from just the
inoculum to all other nodes was considered (Fig. 4.3). Although very well con-
nected, the DT performed poorly, as distributing material across the large number
56 M.D. Fricker et al.
Fig. 4.2 Comparison of weighted fungal networks and neighborhood graphs. P. velutina was
grown from a wood block inoculum over compressed soil in the presence of an additional wood-
block resource, and the weighted network digitized at 9, 18, 25 and 31 d. (A) The weighted fungal
network, in which line thickness and intensity indicate the relative cross-sectional area of each
cord. (B) A simplified version of the network that retains nodes arising from branching or fusion,
but not nodes simply required to trace the outline of each cord correctly. The amount of material
present in the network is distributed evenly across all links to give a uniform network. The nodes
present in the simplified graph were then connected according to well-defined rules to give: the
minimum spanning tree (C); the relative neighborhood graph (D); the Gabriel graph (E); or the
fully connected Delaunay triangulation (F). Modified from [21]
4 Adaptive Biological Networks 57
20000 30000
convex hull area (mm2)
functional efficiency (mm)
40000
0.0005
0.0010
0.0015
0.0020
0.0025
0.0030
50000
Fig. 4.3 Comparison of transport efficiency between weighted fungal and uniform model net-
works. The functional efficiency of the fungal network was predicted from the sum of the inverse
of the shortest paths from the inoculum to every node as the colony increased in area. The weighted
fungal network (,) has the highest functional efficiency, in comparison to uniform networks
constructed with the same topology (,·−··−), or connected using a Delaunay Triangulation
(,···), or Minimum Spanning Tree (,−−). Redrawn from [3]
of links present gave each one low cross-sectional area and consequent high resis-
tance. Conversely, the MST performed better than the DT as it was populated with
few, but extremely thick, links. The uniform fungal networks were similar in perfor-
mance to the MST, although they clearly have a different architecture, but the real
weighted fungal network showed the best predicted transport behavior (Fig. 4.3).
By normalizing to the DT, the local efficiency (Eloc) of the real network, uniform
network and MST were calculated as 4.4±0.11, 2.22 ±0.07 and 2.08 ±0.12,
respectively [3]. Thus, differential weighting of links in the real network gave a >4
fold improvement in local efficiency in comparison to a fully connected uniform
network constructed with the same total cost. The ability of fungal networks to mod-
ify link strengths in a dynamic way is, therefore, crucial to achieve high transport
capacity.
Subtle shifts in the predicted transport performance of the network as it grows
can be identified by which links carry the greatest number of shortest paths and
therefore have a high shortest-path betweenness centrality (SPBC) [17, 36]. The
relative importance of particular links between the inoculum and added resource,
as judged by their SPBC, fluctuate in the early stage of growth with several cords
competing before one thickens up sufficiently to achieve dominance [21]. Equally,
one of the disadvantages of using shortest path analysis is that comparable par-
allel pathways that are only marginally longer do not feature prominently in the
analysis, but might be expected to participate in transport in a real system. A key
area for future development will be to evaluate comparable parallel flow centrality
measures.
58 M.D. Fricker et al.
4.4 Comparison Between Predicted Transport
and Experimental Transport
In parallel to the theoretical network analysis, we have developed methods to image
nutrient movement directly in these microcosms by mapping the distribution of
the amino-acid analogue, 14C-amino isobutyrate (14C-AIB), using photon-counting
scintillation imaging (PCSI), [22, 63–66]. 14C-AIB accumulates in the free amino
acid pool and is not metabolized in a range of woodland fungi so far examined, as
judged by the lack of incorporation of 14C in other metabolites or released as 14CO2
[15, 31, 37, 46, 47, 69]. This allows it to be used as a proxy for nitrogen translocation
[69] and provides an opportunity to compare the predictions made by the theoretical
network analysis to the actual pattern of nutrient movement in the same microcosms
[20].
Networks were allowed to develop in microcosms for 45 d (Fig. 4.4a) and
the weighted network digitised (Fig. 4.4b) and analysed to give the link evolution
(Fig. 4.4c) and the SPBC (Fig. 4.4d). 14C-AIB was added at the end of the growth
period and imaged using photon-counting scintillation imaging (PCSI) to map nutri-
ent movement (Fig. 4.4e). The topological network was then superimposed on the
14C-AIB image to determine the amount of AIB present in each link from the inte-
grated 14C-AIB intensity (Fig. 4.4f). Ideally we would like to calculate the total
flux through each link rather than just the integrated amount using knowledge of
the amount of 14C-AIB appearing further downstream. However, this is challeng-
ing as it requires assumptions about the flow pathway to reallocate the AIB signal
correctly. Nevertheless, as a first approximation we have compared 14C-AIB maps
with various network parameters such as final link weight (Fig. 4.4g), link evolution
(Fig. 4.4h), based on linear regression of the change in link weight with time, and
SPBC (Fig. 4.4i). A number of different populations of links were identified. The
most prominent were a cluster with high 14C-AIB but low SPBC, corresponding to
the tips where the 14C-AIB accumulated. For the other links there was some degree
of correlation between the AIB distribution and the network parameter. Equally, the
AIB pattern did not always match expectations. For example, there was no obvious
reason from the weighted network image why there should be substantial accumu-
lation on the right-hand side of the colony, or little apparent transport to the added
resource or beyond (Fig. 4.4e) based on the final link weight (Fig. 4.4b, final panel),
link evolution (Fig. 4.4c) or SPBC (Fig. 4.4d). There are clearly additional features
governing the control of nutrient distribution that cannot be captured by simple pre-
dictions of flow, based solely on network measures or shortest path calculations.
4.5 Oscillations and Pulsatile Transport
In addition to the evolution of the longer term trends described above, a strong
pulsatile component was also associated with 14C-AIB transport [22, 64–66]. To
characterize this oscillatory behavior, we have analyzed the image-series in the
frequency domain and mapped the frequency, phase or magnitude, on a pixel-by-
pixel basis as the hue in pseudo-color coded images [22, 64–66]. In single juvenile
4 Adaptive Biological Networks 59
Fig. 4.4 Bright-field images of P. velutina growing from a beech wood block inoculum to a set
of additional resource wood blocks over compressed soil were obtained at 3 d intervals for 45
d(a). Branch points and anastomoses were manually coded as nodes connected by links, and the
cord diameter estimated by image analysis, to give a weighted network (b) in which thick cords are
represented in red and thin cords in blue, through a rainbow spectrum. Various network parameters
were calculated including link evolution (c), based on linear regression of the change in link weight
with time and color-coded by gradient of the regression equation, and shortest-path betweenness
centrality, measured as the number of shortest paths passing through each link (d). To compare
the predicted transport properties of the network with actual transport, 14C-AIB movement was
mapped by photon-counting scintillation imaging (PCSI) at the end of the time-series (e)andthe
amount of AIB present in each link extracted using the digitized network (f). The distribution of
AIB was then compared with link cross-sectional area at the last time point (g), link evolution (h)
or link betweenness centrality (i). Redrawn from [20]
mycelial systems with no additional resource, the mycelium beneath the inoculum
and that growing over the screen formed distinct oscillatory domains with the same
frequency, but almost 180 degrees out-of-phase with each other [22, 64]. When two
colonies were allowed to grow and fuse, the oscillations synchronized between the
60 M.D. Fricker et al.
two connected inocula but still showed a phase shift with respect to the rest of the
colony [22]. Recently we have examined the phase relationships in more complex
systems in which arrays of colonies of both compatible and incompatible strains
were allowed to grow and fuse. A subset of inocula were labeled and rapid, long-
distance transport of 14C-AIB occurred between the connected compatible inocula
following fusion, with eventual distribution throughout the super-organism formed
AB
CD
EF
Fig. 4.5 Synchronized oscillations and phase domains in coupled networks. A 9 ×5arrayof
inocula from two incompatible isolates of Coniophora puteana (shown as red and green circles
in panel (a) was set up on a scintillation screen. Several inocula were labeled with 14C-AIB and
transport imaged using photon-counting scintillation imaging (PCSI) for 12 d. When compatible
growing colonies met, they fused and allowed rapid distribution of 14C-AIB throughout the newly
inter-connected system. Initially signals from the inoculum and growing mycelium of each colony
showed out-of-phase oscillations, which are shown in (b–d) as a difference in intensity following
subtraction of the long term trend during Fourier analysis. The phase of the oscillations determined
from the Fourier analysis on a pixel-by-pixel basis was color-coded (eand f), in which regions of
the same color are oscillating in phase. Before the network was fully connected there was con-
siderable variation in the phase relations across the system (e). Following fusion, three domains
of synchronized oscillations emerged, that differed in phase (f). Thus the interconnected inocula
were all synchronized with one phase (green), the central domain (purple) and the outer, growing
margin (blue)
4 Adaptive Biological Networks 61
(Fig. 4.5a–d). Furthermore, whilst oscillations in the individual colonies were not
coupled initially (Fig. 4.5e), they became synchronized following fusion to give
a network of linked cords and inocula with one phase, a central mycelial domain
within the new super-colony that was phase-shifted by a few hours, and a contiguous
foraging margin that was further phase-shifted by a few hours again (Fig. 4.5f).
At this stage we do not know what significance to attribute to these oscillating
phenomena.
4.6 Network Robustness
High transport capacity and low construction cost could have come at the expense
of other network properties, such as robustness to damage, as there is no a priori
reason why link weight allocation for one feature necessarily enhances another. This
is clearly seen in the improved global transport efficiency of the uniformly weighted
MST, even though the MST would be expected to be very vulnerable to disconnec-
tion during attack. Robustness to damage, e.g. by physical breakage or grazing by
invertebrates [5, 7, 27, 30, 67, 68, 71], is of major significance to long-lived mycelial
systems. Having a large number of alternate pathways is important in this context,
and the differential strengthening of links not only imparts high transport capacity
but also robustness to damage. This can be seen by examining the effects of breaking
links in models of the fungal networks in comparison to corresponding uniform
networks. We chose to look at link breakage rather than node removal, which is
commonly used in other networks, as the cord is the biologically relevant target
for attack. Links were broken in order, assuming that the probability of breakage
increased with length and decreased with the thickness of the link. That is long,
thin links were broken before short, thick ones. Robustness was quantified as the
proportion of the total material cost of the network that remained connected to the
inoculum. The fungal networks maintained a much greater system connected with
the inoculum than did the uniform fungal, DT or MST networks (Fig. 4.6), i.e. the
fungal networks were much more robust to damage.
This represents a minimum estimate of the real network resilience in nature, as
the network is also able to respond to local damage, by modification of adjacent link
strength, and to regrow and reconnect. Thus, for example, local mechanical damage
to a small region of the network promoted strengthening of distal circumferential
connections (Fig. 4.7a, c). Continuous grazing trimmed the network back to the
reinforced core, in support of the in silico predictions (Fig. 4.7b), but also promoted
an increase in tangential connections (Fig. 4.7d).
4.7 Simple Networks in the Plasmodial Slime Mould Physarum
Polycephalum
Whilst network analysis of mycelial fungi is in its infancy, considerable progress
has already been made in the analysis of simple networks in the plasmodial slime
62 M.D. Fricker et al.
connected fraction
broken link fraction
0.0
0.0 0.2 0.4 0.6 0.8 1.0
0.2
0.4
0.6
0.8
1.0
Fig. 4.6 Comparison of network resilience between weighted fungal and uniform model networks.
The amount of mycelium remaining connected to the inoculum was measured as an increasing
fraction of links were broken. When more than 0.3 of the total fraction of the link area was
broken, the weighted fungal networks (,) maintained a greater connected core than the uniform
fungal network (,·−··−), or networks connected using a Delaunay Triangulation (,···), or
Minimum Spanning Tree (,−−). Redrawn from [3]
mould Physarum polycephalum [32, 38–45, 57–61]. P. polycephalum is a large,
single-celled amoeboid organism that forages for food resources in a woodland
environment. During exploration, it spreads with a relatively contiguous foraging
margin to maximize the area searched. However, behind the margin, it resolves this
dense structure into a tubular network, interconnecting captured food resources and
acting as a supply network to support further exploration.
This natural capacity to construct a transport network can be exploited in experi-
mental microcosms in which food sources (FSs), typically oat flakes, are arranged in
specific geometric patterns [39]. As the plasmodium grows, it links each FS encoun-
tered in an efficient manner to form a network that includes both direct connections,
Steiner points and some additional cross-links that improve both transport efficiency
and resilience (Fig. 4.8) [41, 43]. In all cases, the network that is established by the
plasmodium has a relatively short total length of interconnecting tubes, but main-
tains close connections among all the food sources and exhibits a high tolerance to
accidental fragmentation.
Growth can also be constrained by physical barriers [44] or influenced by the
light regime [40], increasing the opportunity for experimental manipulation to
mimic real-world network problems. Thus, for example, P. polycephalum can find
the shortest path through a maze [40, 44, 45], or connect different arrays of FSs in an
efficient manner. For three or more FRs up to about 10, the system strikes a balance
between a low total length (TL) of the interconnected network whilst keeping a
short connection distance (CD) between any pair of FSs and a high degree of fault
tolerance (FT) against accidental disconnection of any tube [41, 43].
4 Adaptive Biological Networks 63
B
CD
2 cm 2 cm
A
Fig. 4.7 Adaptive network resilience. Colonies of P. velutina were grown from 2 ×2×2 beech
wood blocks over compressed soil in the absence (a)orpresence(b) of grazing by Folsomia can-
dida.In(a) a localized region of physical damage (indicated by the arrow) stimulated a localized
increase in tangential connections (c). In (b) grazing continuously trimmed the finest hyphae, stim-
ulating more local sprouting, and accentuated growth both of the dominant radial cords and also
tangential connections (d). Redrawn from [19]
The degree of separation is defined as the number of food sources along the
shortest path between two food sources. The average separation (AS) is the degree
of separation averaged over all pairs of food sources, and decreases as food sources
are more closely coupled. To allow comparisons between different arrangements,
AS is normalized to the average separation for the minimum spanning tree [41].
The fault tolerance (FT) is the probability that the organism is not fragmented
into separate pieces if an accidental breakage occurs at a random point along the
tubes. Since the probability of disconnection of a tube is proportional to its length,
a longer tube has a higher risk of disconnection. The combined index, FT/TL, can
be regarded as a measure of the ratio of benefit to cost. By judicious positioning of
food sources, the geometry of the network can be compared to possible theoretical
solutions in terms of path length and fault tolerance, such as the minimal spanning
tree (MST), the Steiner minimal tree (SMT) and a Delaunay triangulation network
(DTN) [41]. Examples are given in Fig. 4.8 for the predicted network with 3 food
sources (Fig. 4.8d) and experimental results for 3 food sources (Fig. 4.8e–g), 6 and
7 food sources (Fig. 4.8h, i) with the associated analysis of path length and fault
64 M.D. Fricker et al.
Fig. 4.8 (continued)
4 Adaptive Biological Networks 65
tolerance (Fig. 4.8j), rings of 12 food sources (Fig. 4.8k, l) and grids of 64 food
sources (Fig. 4.8m, n).
In computational terms, it becomes progressively more challenging to find a
good solution to such a combinatorial optimization problem as the number of FSs
increases, particularly with the inclusion of Steiner points [61]. It is remarkable,
therefore, that solutions reached by P. polycephalum, whilst not necessarily optimal
individually, cluster around the predicted optimal solution in replicate experiments.
Furthermore, the solution is reached rapidly, based only on local information and
parallel analogue computing. If it were possible to capture the essence of such a
system in simple rules, it might have significant potential to guide de-centralized
network development in other domains [39, 42, 58–60].
4.8 Universal Features of Biological Networks?
Characterization of mycelial networks is still in its infancy. However, the network
approach provides a way of quantifying and analyzing complex fungal systems for
the first time, and also makes it possible to link measurements in microcosms in
the laboratory to observations of networks in the field. The simple models predict-
ing transport through such networks, so far based on shortest path considerations
through the weighted network, only capture part of the experimentally determined
transport behavior. We anticipate that models that include parallel-flow pathways
and evolution of the network should improve the match between simulation and
experiment, and will benefit from the recent advances in fast algorithms to calculate
the necessary metrics. The next conceptual advance will be to identify the rules that
allow the network iteratively to refine its structure and transport behavior to yield
the network architectures observed. It is conceivable that the identification of such
rules will allow development of generic “fungal colony optimization” algorithms
Fig. 4.8 (Opposite page) Self-organization of robust network architecture in Physarum poly-
cephalum. (a–c) Development of a network between three food sources, starting from a continuous
sheet of plasmodium on the surface of agar. Network structure at 0h (a), 6 h (b) and 36 h (c).
Scale bar = 1 cm. (d) Schematic illustration of the arrangement of food sources (black dots). The
orange, green and blue lines represent the network of minimum spanning tree (MST), Steiner’s
minimal tree (SMT) and Delaunay triangulation network (DTN), respectively. (e–g) Three typical
networks in ascending order of total length (TL) after 35 h. Scale bar = 1 cm. (h, i) Typical emergent
network structure with six (h) and seven (i) food sources (FS) and schematic representation of the
corresponding MST (orange), SMT (green)andDTN(blue). (j) Properties of the plasmodial net-
works, defined by average separation of food sources (AS), normalised to the value for the minimal
spanning tree, and benefit to cost ratio, defined as the fault tolerance over the total length (FT/TL).
Black symbols give the value for each specimen, and red, the value of the mean with associated
s.e.m. Orange, green and blue symbols give the values of MST, SMT and DTN respectively. The
organism maintains a short total length of tubes with close connections between food sources yet
high tolerance of accidental disconnection. (k–n) Network organization with ring (k, l) and grid
(m, n) arrangement of food sources. These systems also show robust network architecture, with
short path length but high fault tolerance. Redrawn from [43]
66 M.D. Fricker et al.
similar to those that have evolved from the study of ant colony foraging patterns
[14] or based on P. polycephalum [40, 41, 58–60].
Even at this stage, some common features of biological network formation seem
to emerge. Fungal networks are constructed by local iterative developmental pro-
cesses rather than predetermined blueprints or centralized control, with growth
involving over-production of links and nodes, followed by selective pruning of some
links and reinforcement of others. Such a process mimics the process of Darwinian
evolution in which natural selection removes less fit offspring. This “Darwinian
network model” may be applicable to other biological systems, including foraging
ant trails, P. polycephalum, axon development and angiogenesis, and may represent
a generalized model for growth of physical biological networks. Based on the ant
colony and P. polycephalum models, we might expect the generic ingredients in
such a model will involve a non-linear positive reinforcement term related to the
local flux and a linear decay term. Notably this model differs from other models of
weighted network evolution that incorporate differential strengthening of links, i.e.
“the busiest get busier” [1], rather than differential weakening and loss that is the
hallmark of evolution by natural selection. However, the model has parallels with
the selective link removal model recently proposed for unweighted networks [51].
In infrastructure networks where costs are associated with creation and maintenance
of links, where links differ in some measure of fitness, and where material can be
recycled, such a Darwinian model may be applicable. In practical terms such a pro-
cess may also be witnessed in the evolution of real infrastructure networks, such as
British railways following the Beeching reviews in the early ’60s [8, 9]. In these
reviews, the flux along various routes was measured and routes with too low a level
of traffic, mainly branch lines, were targeted for closure. At the same time, major
routes were strengthened to cope with the expected source-sink relationships for
both passenger and freight traffic. Interestingly, the reports focussed on efficiency
rather than any explicit consideration of resilience, which may explain the sensitivity
of the current UK rail network to disruption.
A second feature of interest emerging, particularly through consideration of the
P. polycephalum and fungal networks, is the extent that coupled flows may contain
global information. Networks involving physical flows obey continuity equations
and are therefore intrinsically coupled across the network. This automatically means
that increasing the flow in one part of the network will lead to reductions elsewhere,
even though the local conditions in the distal region remain the same. Thus each part
of the network is influenced by and can influence the whole network, but without any
global assessment of behavior. Useful properties of the network may emerge from
the interaction between the local update rules governing topology and flows without
the need for long-distance communication or calculation of aggregate properties of
the network. It is this coupling in the P. polycephalum model that allows the network
to resolve from a fine mesh into a quasi-optimal solution [40, 58–60]. Furthermore,
the computational overhead for such self-organized networks scales well with the
number of additional nodes.
The third general observation on these biological networks is the prevalence of
some form of oscillatory process. In P. polycephalum it is an actin-myosin contrac-
4 Adaptive Biological Networks 67
tion with a short (min) period whilst in the fungal networks it is manifest as a change
in the amount of radiolabeled nutrient with a longer (hr to day) period. In both cases
the oscillations can synchronize across large regions of the developing system, even
if the individual components are asynchronous initially [22, 56, 57]. That the oscilla-
tions manage to synchronize is not surprising [55], but the extent that the organisms
may be able to interpret and act upon the oscillations is not known. In other contexts,
such as supply chains or traffic flow, the existing strategy is to minimize oscilla-
tions to achieve maximum throughput [28]. This suggests that either the biological
systems lack the additional sensory and feedback systems to suppress oscillations,
or that maintaining an oscillatory system is an alternative means to achieve a sta-
ble long-term quasi-optimal solution, potentially with less control infrastructure.
In P. polycephalum, oscillations drive protoplasmic shuttle streaming and generate
flows considerably greater than the volume needed simply for extension growth at
the margin. It seems likely therefore that the additional energy demands of rhyth-
mic contraction represent the cost of this indirect information transfer. Nevertheless,
such a cost is minimal compared to the developmental and behavioral complexity
and metabolic cost of the more sophisticated neuron-based sensory systems used by
higher organisms.
Acknowledgements Research has been supported by BBSRC (43/P19284), EPSRC (GR/S63090/
01), NERC (GR3/12946 and NER/A/S/2002/882), EU Framework 6 (STREP No. 12999), Oxford
University Research Infrastructure Fund and the University Dunston Bequest. We thank A. Ashford,
K. Burton, P.R. Darrah, D.P. Donnelly, D. Eastwood, J. Efstathiou, J. Hynes, N. Johnson,
F. Reed-Tsochas, M. Tlalka, G.M. Tordoff, S.C. Watkinson and members of CABDyN for stimu-
lating discussions.
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... Prior measures of fungal network architecture have characterized population distributions of morphological features in fungal colonies. Such features include both local measures (such as number of tips, node degree at branch points, and branch length [10,11,12,13,14]) and global measures (such as fractal dimension [15], predicted global transport efficiency, and resilience [1,10,11,12]). However, most of the phenotypic plasticity and behaviour is likely to arise at an intermediate scale (i.e., a mesoscale) that reflects how smaller units (hyphae, branches, and cords) are organized locally to produce spatial domains with differing architecture and behaviour that collectively yield global behaviour and temporal changes in such behaviour. ...
... Prior measures of fungal network architecture have characterized population distributions of morphological features in fungal colonies. Such features include both local measures (such as number of tips, node degree at branch points, and branch length [10,11,12,13,14]) and global measures (such as fractal dimension [15], predicted global transport efficiency, and resilience [1,10,11,12]). However, most of the phenotypic plasticity and behaviour is likely to arise at an intermediate scale (i.e., a mesoscale) that reflects how smaller units (hyphae, branches, and cords) are organized locally to produce spatial domains with differing architecture and behaviour that collectively yield global behaviour and temporal changes in such behaviour. ...
... One can grow fungi, slime moulds, and hydractiniid hydroids in a laboratory, and it is consequently possible to expose them to a wide variety of experimental conditions and species interactions, in multiple replicates, to generate a rich collection of networks. Therefore, investigating such adaptive, selforganized networks -which are honed by evolution -provides a fascinating opportunity to uncover underlying principles of biological network organization, evaluate the relevance of network descriptors (which have been developed in related disciplines) to evolved network behaviour, and explore how much utility biologically-inspired algorithms have in other domains [11,24,26]. ...
Preprint
Full-text available
We investigate the application of mesoscopic response functions (MRFs) to characterize a large set of networks of fungi and slime moulds grown under a wide variety of different experimental treatments, including inter-species competition and attack by fungivores. We construct ‘structural networks’ by estimating cord conductances (which yield edge weights) from the experimental data, and we construct ‘functional networks’ by calculating edge weights based on how much nutrient traffic is predicted to occur along each edge. Both types of networks have the same topology, and we compute MRFs for both families of networks to illustrate two different ways of constructing taxonomies to group the networks into clusters of related fungi and slime moulds. Although both network taxonomies generate intuitively sensible groupings of networks across species, treatments, and laboratories, we find that clustering using the functional-network measure appears to give groups with lower intra-group variation in species or treatments. We argue that MRFs provide a useful quantitative analysis of network behaviour that can (1) help summarize an expanding set of increasingly complex biological networks and (2) help extract information that captures subtle changes in intra-specific and inter-specific phenotypic traits that are integral to a mechanistic understanding of fungal behaviour and ecology. As an accompaniment to our paper, we also make a large data set of fungal networks available in the public domain.
... The graph representation is based on translation of the pixel skeleton to a planar, weighted, undirected graph (Fig. 7C), with nodes located at hyphal tips, branches, and anastomoses, and edges representing hyphae or cords, with weights based on the Euclidean length (L) and radius (r) of each cord, combined either as the cylindrical volume (V = πr 2 L), to represent the material cost of the cord, or the predicted conductance (G = r 2 /L), assuming cords are bundles of equally sized vessels rather than a single vessel with increasing radius (144,173,175,182,(199)(200)(201). The network cannot be resolved within the resource, so it is represented as a single node connected to all the edges incident on the resource boundary (Fig. 7C). ...
... Summary statistics, such as the number of tips, junctions, and edges; the total hyphal length, area, and volume; or the distribution of branching angles and internodal lengths can be readily extracted from the pixel skeleton or graph representation of the fungal network (116,144,173,175,(199)(200)(201)(202). Measures can also be referenced to 2-D space to give tip densities and fractal dimensions (173,175). ...
... Measures can also be referenced to 2-D space to give tip densities and fractal dimensions (173,175). Topological network measures, such as node degree, α-index or shortest path metrics, such as betweenness centrality (Fig. 7D), can also be readily calculated and show species-dependent developmental changes over time (144,173,(199)(200)(201)(202)(203). ...
... Biological systems, such as vascular networks [1], hyphal networks [2,3], neurons [1,4], slime molds [5], and bacterial colonies [6], display complex structures rich in branched or tree-like spatial features. The morphology of these systems seem to solve an adaptive exploration problem related to the maximization of the space that a connected structure can cover in order to retain or gain conditions for survival given limited amounts of matter, energy and information, and according to the demands and restrictions of the environment [7,8,9,10]. ...
Preprint
Full-text available
Biological systems with tree-like morphological features emerge as nature's solution to an adaptive spatial exploration problem. The morphological complexity of these systems is often described in terms of its fractality, however, the network topology plays a relevant role behind the system's biological function. Therefore, here we considered a structural analysis of bio-inspired spatial systems based on fractal and network approaches in order to identify the features that could make tree-like morphologies better at exploring space under limited matter, energy and information. We considered connected clusters of particles in two-dimensions: the Ballistic and Diffusion-Limited Aggregation stochastic fractals, the Viscek and Hexaflake deterministic fractals, and the Kagome and Hexagonal lattices. We characterized their structure in terms of the range (linear extension), coverage (plane-filling), cost (assembly connections), configurational complexity (local connectivity), and efficiency (network communication). We found that tree-like systems have a lower configurational complexity and an invariant structural cost for different fractal dimensions, however, they are also fragile and inefficient. Nevertheless, this efficiency can become similar to that of an hexagonal lattice, at a similar cost, by considering euclidean connectivity beyond first neighbors. These results provide relevant insights into the interplay between the morphological and network properties of complex spatial systems.
... Estimation of the relative cost of the network can be made by scaling observed network traits to those from a baseline network model calculated as the minimum spanning tree (MST) [36]. This baseline model has minimal connectivity that allows at least unidirectional transport from the centre of the network to the growing tips at the foraging margin, where usually most of the resources are required (see below). ...
Article
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Colonization of terrestrial environments by filamentous fungi relies on their ability to form networks that can forage for and connect resource patches. Despite the importance of these networks, ecologists rarely consider network features as functional traits because their measurement and interpretation are conceptually and methodologically difficult. To address these challenges, we have developed a pipeline to translate images of fungal mycelia, from both micro- and macro-scales, to weighted network graphs that capture ecologically relevant fungal behaviour. We focus on four properties that we hypothesize determine how fungi forage for resources, specifically: connectivity; relative construction cost; transport efficiency; and robustness against attack by fungivores. Constrained ordination and Pareto front analysis of these traits revealed that foraging strategies can be distinguished predominantly along a gradient of connectivity for micro- and macro-scale mycelial networks that is reminiscent of the qualitative ‘phalanx’ and ‘guerilla’ descriptors previously proposed in the literature. At one extreme are species with many inter-connections that increase the paths for multidirectional transport and robustness to damage, but with a high construction cost; at the other extreme are species with an opposite phenotype. Thus, we propose this approach represents a significant advance in quantifying ecological strategies for fungi using network information.
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