ChapterPDF Available

Abstract and Figures

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.
Content may be subject to copyright.
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.
References
1. Barrat, A., Barthelemy, M., Vespignani, A.: Modeling the evolution of weighted networks.
Physical Review E 70, 066149 (2004)
2. Barrat, A., Barthelemy, M., Vespignani, A.: The effects of spatial constraints on the evolution
of weighted complex networks. Journal of Statistical Mechanics p. P05003 (2005)
3. Bebber, D., Hynes, J., Darrah, P., Boddy, L., Fricker, M.: Biological solutions to transport
network design. Proceedings of the Royal Society B 274, 2307–2315 (2007)
4. Bebber, D., Tlalka, M., Hynes, J., Darrah, P., Ashford, A., Watkinson, S., Boddy, L., Fricker,
M.: Imaging complex nutrient dynamics in mycelial networks. In: G. Gadd, S. Watkinson,
P. Dyer (eds.) Fungi in the Environment, vol. 25, pp. 3–21. Cambridge University Press,
Cambridge (2007)
5. Boddy, L., Jones, T.: Mycelial responses in heterogeneous environments: parallels with
macroorganisms. In: G. Gadd, S. Watkinson, P. Dyer (eds.) Fungi in the Environment, vol. 25,
pp. 112–158. Cambridge University Press, Cambridge (2007)
6. Boddy, L., Wells, J.M., Culshaw, C., Donnelly, D.P.: Fractal analysis in studies of mycelium
in soil. Geoderma 88, 301–328 (1999)
7. Bretherton, S., Tordoff, G.M., Jones, T.H., Boddy, L.: Compensatory growth of Phanerochaete
velutina mycelial systems grazed by Folsomia candida (collembola). FEMS Microbiology
Ecology 58, 33–40 (2006)
8. British Railways Board: The development of the major railway trunk routes (1965)
9. British Transport Commission: The reshaping of british railways - part 1: report (1963)
68 M.D. Fricker et al.
10. Buhl, J., Gautrais, J., Reeves, N., Sole, R.V., Valverde, S., Kuntz, P., Theraulaz, G.: Topological
patterns in street networks of self-organized urban settlements. European Physical Journal B
49, 513–522 (2006)
11. Buhl, J., Gautrais, J., Sole, R.V., Kuntz, P., Valverde, S., Deneubourg, J.L., Theraulaz, G.:
Efficiency and robustness in ant networks of galleries. European Physical Journal B 42, 123–
129 (2004)
12. Cairney, J.W.G.: Basidiomycete mycelia in forest soils: dimensions, dynamics and roles in
nutrient distribution. Mycological Research 109, 7–20 (2005)
13. Cardillo, A., Scellato, S., Latora, V., Porta, S.: Structural properties of planar graphs of urban
street patterns. Physical Review E 73, 066107 (2006)
14. Dorigo, M., Di Caro, G., Gambardella, L.M.: Ant algorithms for discrete optimization. Artifi-
cial Life 5, 137–172 (1999)
15. Elliott, M.L., Watkinson, S.C.: The effect of alpha-aminoisobutyric-acid on wood decay and
wood spoilage fungi. International Biodeterioration 25, 355–371 (1989)
16. Ferguson, B.A., Dreisbach, T.A., Parks, C.G., Filip, G.M., Schmitt, C.L.: Coarse-scale popula-
tion structure of pathogenic Armillaria species in a mixed-conifer forest in the blue mountains
of northeast oregon. Canadian Journal of Forest Research 33, 612–623 (2003)
17. Freeman, L.C.: Set of measures of centrality based on betweenness. Sociometry 40, 35–41
(1977)
18. Fricker, M., Bebber, D., Boddy, L.: Mycelial networks: structure and dynamics. In: L. Boddy,
J. Frankland, P. van West (eds.) Ecology of Saprotrophic Basidiomycetes, vol. 28, pp. 3–18.
Academic Press, Amsterdam (2008)
19. Fricker, M., Boddy, L., Bebber, D.: Network organisation of mycelial fungi. In: R. Howard,
N. Gow (eds.) The Mycota, vol. VIII, pp. 309–330. Springer-Verlag, Berlin (2007)
20. Fricker, M., Lee, J., Bebber, D., Tlalka, M., Hynes, J., Darrah, P., Watkinson, S., Boddy, L.:
Imaging complex nutrient dynamics in mycelial networks. Journal of Microscopy 231, 299–
316 (2008)
21. Fricker, M., Lee, J., Boddy, L., Bebber, D.: The interplay between structure and function in
fungal networks. Topologica 1, 004 (2008)
22. Fricker, M.D., Tlalka, M., Bebber, D., Takagi, S., Watkinson, S.C., Darrah, P.R.: Fourier-
based spatial mapping of oscillatory phenomena in fungi. Fungal Genetics and Biology 44,
1077–1084 (2007)
23. Gastner, M.T., Newman, M.E.J.: Shape and efficiency in spatial distribution networks. Journal
of Statistical Mechanics p. P01015 (2006)
24. Glass, N.L., Jacobson, D.J., Shiu, P.K.T.: The genetics of hyphal fusion and vegetative incom-
patibility in filamentous ascomycete fungi. Annual Review of Genetics 34, 165–186 (2000)
25. Glass, N.L., Rasmussen, C., Roca, M.G., Read, N.D.: Hyphal homing, fusion and mycelial
interconnectedness. Trends in Microbiology 12, 135–141 (2004)
26. Haggett, P., Chorley, R.: Network Analysis in Geography. Arnold, London (1969)
27. Harold, S., Tordoff, G.M., Jones, T.H., Boddy, L.: Mycelial responses of Hypholoma fasci-
culare to collembola grazing: effect of inoculum age, nutrient status and resource quality.
Mycological Research 109, 927–935 (2005)
28. Helbing, D.: Traffic and related self-driven many-particle systems. Reviews of Modern
Physics 73, 1067–1141 (2001)
29. Hitchcock, D., Glasbey, C.A., Ritz, K.: Image analysis of space-filling by networks: applica-
tion to a fungal mycelium. Biotechnology Techniques 10, 205–210 (1996)
30. Kampichler, C., Rolschewski, J., Donnelly, D.P., Boddy, L.: Collembolan grazing affects the
growth strategy of the cord-forming fungus Hypholoma fasciculare. Soil Biology and Bio-
chemistry 36, 591–599 (2004)
31. Kim, K.W., Roon, R.J.: Transport and metabolic effects of alpha-aminoisobutyric-acid in Sac-
charomyces cerevisiae. Biochimica et Biophysica Acta 719, 356–362 (1982)
32. Kobayashi, R., Tero, A., Nakagaki, T.: Mathematical model for rhythmic protoplasmic move-
ment in the true slime mold. Journal of Mathematical Biology 53, 273–286 (2006)
4 Adaptive Biological Networks 69
33. Lamour, A., Termorshuizen, A.J., Volker, D., Jeger, M.J.: Network formation by rhizomorphs
of Armillaria lutea in natural soil: their description and ecological significance. FEMS Micro-
biology Ecology 62, 222–232 (2007)
34. Latora, V., Marchiori, M.: Efficient behavior of small-world networks. Physical Review Let-
ters 87, 198701 (2001)
35. Latora, V., Marchiori, M.: Economic small-world behavior in weighted networks. European
Physical Journal B 32, 249–263 (2003)
36. Latora, V., Marchiori, M.: A measure of centrality based on network efficiency. New Journal
of Physics 9, 188 (2007)
37. Lilly, W.W., Higgins, S.M., Wallweber, G.J.: Uptake and translocation of 2-aminoisobutyric
acid by Schizophyllum commune. Experimental Mycology 14, 169–177 (1990)
38. Nakagaki, T.: Smart behavior of true slime mold in a labyrinth. Research in Microbiology
152, 767–770 (2001)
39. Nakagaki, T., Guy, R.D.: Intelligent behaviors of amoeboid movement based on complex
dynamics of soft matter. Soft Matter 4, 57–67 (2008)
40. Nakagaki, T., Iima, M., Ueda, T., Nishiura, Y., Saigusa, T., Tero, A., Kobayashi, R., Showalter,
K.: Minimum-risk path finding by an adaptive amoebal network. Physical Review Letters 99,
068104 (2007)
41. Nakagaki, T., Kobayashi, R., Nishiura, Y., Ueda, T.: Obtaining multiple separate food sources:
behavioural intelligence in the Physarum plasmodium. Proceedings of the Royal Society of
London Series B 271, 2305–2310 (2004)
42. Nakagaki, T., Saigusa, T., Tero, A., Kobayashi, R.: Effects of amount of food on path selection
in the transport network of an amoeboid organism. In: Proceedings of the International Sym-
posium on Topological Aspects of Critical Systems and Networks. World Scientific (2007)
43. Nakagaki, T., Yamada, H., Hara, M.: Smart network solutions in an amoeboid organism. Bio-
physical Chemistry 107, 1–5 (2004)
44. Nakagaki, T., Yamada, H., Toth, A.: Maze-solving by an amoeboid organism. Nature 407,
470–470 (2000)
45. Nakagaki, T., Yamada, H., Toth, A.: Path finding by tube morphogenesis in an amoeboid
organism. Biophysical Chemistry 92, 47–52 (2001)
46. Ogilvie-Villa, S., Debusk, R.M., Debusk, A.G.: Characterization of 2-aminoisobutyric acid
transport in Neurospora crassa – a general amino-acid permease-specific substrate. Journal of
Bacteriology 147, 944–948 (1981)
47. Olsson, S., Gray, S.N.: Patterns and dynamics of 32P-phosphate and labelled
2-aminoisobutyric acid (14C-AIB) translocation in intact basidiomycete mycelia. FEMS
Microbiology Ecology 26, 109–120 (1998)
48. Rayner, A., Griffith, G., Ainsworth, A.: Mycelial interconnectedness. In: N. Gow, G. Gadd
(eds.) The Growing Fungus, pp. 21–40. Chapman and Hall, London (1994)
49. Rayner, A., Watkins, Z., Beeching, J.: Self-integration - an emerging concept from the fungal
mycelium. In: N. Gow, G. Robson, G. Gadd (eds.) The Fungal Colony, pp. 1–24. Cambridge
University Press, Cambridge (1999)
50. Read, D.: Mycorrhizal fungi – the ties that bind. Nature 388, 517–518 (1997)
51. Salathe, M., May, R.M., Bonhoeffer, S.: The evolution of network topology by selective
removal. Journal of the Royal Society Interface 2, 533–536 (2005)
52. Simard, S.W., Durall, D.M.: Mycorrhizal networks: a review of their extent, function, and
importance. Canadian Journal of Botany 82, 1140–1165 (2004)
53. Simard, S.W., Perry, D.A., Jones, M.D., Myrold, D.D., Durall, D.M., Molina, R.: Net transfer
of carbon between ectomycorrhizal tree species in the field. Nature 388, 579–582 (1997)
54. Smith, M.L., Bruhn, J.N., Anderson, J.B.: The fungus Armillaria bulbosa is among the largest
and oldest living organisms. Nature 356, 428–431 (1992)
55. Strogatz, S.H.: From kuramoto to crawford: exploring the onset of synchronization in popula-
tions of coupled oscillators. Physica D 143, 1–20 (2000)
70 M.D. Fricker et al.
56. Takamatsu, A., Tanaka, R., Yamada, H., Nakagaki, T., Fujii, T., Endo, I.: Spatiotemporal sym-
metry in rings of coupled biological oscillators of Physarum plasmodial slime mold. Physical
Review Letters 87, 078102 (2001)
57. Tero, A., Kobayashi, R., Nakagaki, T.: A coupled-oscillator model with a conservation law
for the rhythmic amoeboid movements of plasmodial slime molds. Physica D 205, 125–135
(2005)
58. Tero, A., Kobayashi, R., Nakagaki, T.: Physarum solver: A biologically inspired method of
road-network navigation. Physica A 363, 115–119 (2006)
59. Tero, A., Kobayashi, R., Nakagaki, T.: A mathematical model for adaptive transport network
in path finding by true slime mold. Journal of Theoretical Biology 244, 553–564 (2007)
60. Tero, A., Nakagaki, T., Toyabe, K., Yumili, K., Kobayashi, R.: A method inspired by physarum
for solving the steiner problem. International Journal for Unconventional Computing in press
(2009)
61. Tero, A., Yumiki, K., Kobayashi, R., Saigusa, T., Nakagaki, T.: Flow-network adaptation in
physarum amoebae. Theory in Biosciences 127, 89–94 (2008)
62. Thompson, W., Rayner, A.D.M.: Structure and development of mycelial cord systems of
Phanerochaete laevis in soil. Transactions of the British Mycological Society 78, 193–200
(1982)
63. Tlalka, M., Bebber, D., Darrah, P., Watkinson, S., Fricker, M.: Dynamic resource allocation
and foraging strategy in mycelial systems. Fungal Genetics and Biology 45, 1111–1121 (2008)
64. Tlalka, M., Bebber, D., Darrah, P.R., Watkinson, S.C., Fricker, M.D.: Emergence of self-
organised oscillatory domains in fungal mycelia. Fungal Genetics and Biology 44, 1085–1095
(2007)
65. Tlalka, M., Hensman, D., Darrah, P.R., Watkinson, S.C., Fricker, M.D.: Noncircadian oscilla-
tions in amino acid transport have complementary profiles in assimilatory and foraging hyphae
of Phanerochaete velutina. New Phytologist 158, 325–335 (2003)
66. Tlalka, M., Watkinson, S.C., Darrah, P.R., Fricker, M.D.: Continuous imaging of amino-acid
translocation in intact mycelia of Phanerochaete velutina reveals rapid, pulsatile fluxes. New
Phytologist 153, 173–184 (2002)
67. Tordoff, G.M., Boddy, L., Jones, T.H.: Grazing by Folsomia candida (collembola) differen-
tially affects mycelial morphology of the cord-forming basidiomycetes Hypholoma fascicu-
lare,Phanerochaete velutina and Resinicium bicolor. Mycological Research 110, 335–345
(2006)
68. Tordoff, G.M., Boddy, L., Jones, T.H.: Species-specific impacts of collembola grazing on fun-
gal foraging ecology. Soil Biology and Biochemistry 40, 434–442 (2008)
69. Watkinson, S.C.: Inhibition of growth and development of Serpula lacrimans by the non-
metabolized amino-acid analog alpha-aminoisobutyric-acid. FEMS Microbiology Letters 24,
247–250 (1984)
70. Watts, D.J., Strogatz, S.H.: Collective dynamics of ’small-world’ networks. Nature 393, 440–
442 (1998)
71. Wood, J., Tordoff, G.M., Jones, T.H., Boddy, L.: Reorganization of mycelial networks of
Phanerochaete velutina in response to new woody resources and collembola (Folsomia can-
dida) grazing. Mycological Research 110, 985–993 (2006)
... 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.
... Network-forming foragers, such as fungi and plasmodial slime molds, impress by sheer network size and continuous network reorganization [13,31,53,55,58]. As foragers transport resources throughout their network, reorganization of their network affects ecosystems [34,66,67,72,73]. ...
... As foragers transport resources throughout their network, reorganization of their network affects ecosystems [34,66,67,72,73]. Both plasmodial slime molds and foraging fungi are strikingly similar in their network reorganization dynamics [31,35,62] despite their different biological makeup: When foraging, body mass is recycled [25,29,65] as newly formed network fronts enter new territory [8,10,13,32,77], thereby changing network morphology [10,13,14,37]. However, how networks reorganize and what costs and functional requirements constrain their reorganization are still being determined. ...
Preprint
Full-text available
Network-forming organisms, like fungi and slime molds, dynamically reorganize their networks during foraging. The resulting re-routing of resource flows within the organism's network can significantly impact local ecosystems. In current analysis limitations stem from a focus on single-time-point morphology, hindering understanding of continuous dynamics and underlying constraints. Here, we study ongoing network reorganization in the foraging slime mold \textit{Physarum~polycephalum}, identifying three distinct states with varying morphology and migration velocity. We estimate the energetic cost of each state and find a trade-off between building and transport costs within the morphological variability, facilitating different search strategies. Adaptation of state population to the environment suggests that diverse network morphologies support varied foraging strategies, though constrained by associated costs. Our findings provide insights for evaluating the impact of resource flow re-routing in changing ecosystems.
... Network-forming foragers, such as fungi and plasmodial slime molds, impress by sheer network size and continuous network reorganization [1][2][3][4][5]. As foragers transport resources throughout their network, reorganization of their network affects ecosystems [6][7][8][9][10]. ...
... As foragers transport resources throughout their network, reorganization of their network affects ecosystems [6][7][8][9][10]. Both plasmodial slime molds and foraging fungi are strikingly similar in their network reorganization dynamics [2,11,12] despite their different biological makeup: When foraging, body mass is recycled [13][14][15] as newly formed network fronts enter new territory [1,[16][17][18][19], thereby changing network morphology [1,17,20,21]. However, how networks reorganize and what costs and functional requirements constrain their reorganization are still being determined. ...
Article
Full-text available
Network-forming organisms, like fungi and slime molds, dynamically reorganize their networks during foraging. The resulting rerouting of resource flows within the organism's network can significantly impact local ecosystems. In current analysis limitations stem from a focus on single-time-point morphology, hindering understanding of continuous dynamics and underlying constraints. Here we study ongoing network reorganization in the foraging slime mold , identifying three distinct states with varying morphology and migration velocity. We estimate the energetic cost of each state and find a trade-off between building and transport costs within the morphological variability, facilitating different search strategies. Adaptation of state population to the environment suggests that diverse network morphologies support varied foraging strategies, though constrained by associated costs. Our findings provide insights for evaluating the impact of resource flow rerouting in changing ecosystems. Published by the American Physical Society 2024
... Biological systems such as vascular networks 19 , hyphal networks 20, 21 , neurons 19,22 , slime molds 23 , and bacterial colonies 24 , 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 of the environment [25][26][27][28] . One characteristic of these systems is their physical fractality, often quantified by the fractal dimension 19,[29][30][31] . ...
Article
Full-text available
The functional features of spatial networks depend upon a non-trivial relationship between the topological and physical structure. Here, we explore that relationship for spatial networks with radial symmetry and disordered fractal morphology. Under a geometric graphs approach, we quantify the effectiveness of the exchange of information in the system from center to perimeter and over the entire network structure. We mainly consider two paradigmatic models of disordered fractal formation, the Ballistic Aggregation and Diffusion-Limited Aggregation models, and complementary, the Viscek and Hexaflake fractals, and Kagome and Hexagonal lattices. First, we show that complex tree morphologies provide important advantages over regular configurations, such as an invariant structural cost for different fractal dimensions. Furthermore, although these systems are known to be scale-free in space, they have bounded degree distributions for different values of an euclidean connectivity parameter and, therefore, do not represent ordinary scale-free networks. Finally, compared to regular structures, fractal trees are fragile and overall inefficient as expected, however, we show that this efficiency can become similar to that of a robust hexagonal lattice, at a similar cost, by just considering a very short euclidean connectivity beyond first neighbors.
... Despite its apparent simplicity, its potentially reversible life cycle made the slime mold a rich model for developmental biologists with research ranged into three topics: differentiation and development, cell cycle, and motility [1]. More recently it also became a subject of interest for the self-organizing behavior of its tubular network and the use of these behaviors to solve complex problems [2]. This problem solving ability which sometimes makes it being deemed as "intelligent", shows that it must have a mechanism to move information along its tubes. ...
Preprint
Full-text available
Network dynamics is a subject of general interest, often hard to investigate because of its slow dynamics. \emph{Physarum Polycephalum}, a true slime mould is an unicellular organism taking the shape of a network with complex self-organizing behaviors. Here, we explore its network emergence and directionality by suddenly confining it in a ring chambers. By using a structure tensor based image analysis method specifically adapted to our configuration, we have been able to derive quantities which capture these two dynamics. We show that confinement induces a reorganization of the network with a 10410^{4}s time scale, making its directionality align with the ring chamber main axis and that network emerges with two different time scales, one being compatible with network reorganization. Furthermore, we also observe that network dynamics for both emergence and reorganization conserve local features. Interestingly, these observations hint simultaneously towards flows driven network morphogenesis with no particular coordination at the organism scale.
... 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
Full-text available
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.
... Although recycling probably involves autophagy and apoptotic-like mechanisms, it is not clear how changes are made in order to trigger autophagy and renewed growth [34]. Previous studies have demonstrated bidirectional translocation of materials across mycelial networks, both acropetal and basipetal, based on local demand within the mycelium [35][36][37][38][39][40][41][42]. However, the underlying mechanisms of the rhythmic expansion-reduction-re-expansion cycle observed in the present study may be different from those of relatively short-term (10-60 h) oscillatory material transfers observed in the previous studies [40,43,44], because the former cycle is considerably longer (2-4 weeks). ...
Article
Full-text available
Studies of fungal behavior are essential for a better understanding of fungal-driven ecological processes. Here, we evaluated the effects of timing of resource (bait) addition on the behavior of fungal mycelia when it remains in the inoculum and when it migrates from it towards a bait, using cord-forming basidiomycetes. Experiments allowed mycelium to grow from an inoculum wood across the surface of a soil microcosm, where it encountered a new wood bait 14 or 98 d after the start of growth. After the 42-d colonization of the bait, inoculum and bait were individually moved to a dish containing fresh soil to determine whether the mycelia were able to grow out. When the inoculum and bait of mycelia baited after 14 d were transferred to new soil, there was 100% regrowth from both inoculum and bait in Pholiota brunnescens and Phanerochaete velutina, indicating that no migration occurred. However, when mycelium was baited after 98 d, 3 and 4 out of 10 replicates of P. brunnescens and P. velutina, respectively, regrew only from bait and not from inoculum, indicating migration. These results suggest that prolonged periods without new resources alter the behavior of mycelium, probably due to the exhaustion of resources.
Article
Full-text available
The architectural features of cellular life and its ecologies at larger scales are built upon foundational networks of reactions between molecules that avoid a collapse to equilibrium. The search for life’s origins is, in some respects, a search for biotic network attributes in abiotic chemical systems. Radiation chemistry has long been employed to model prebiotic reaction networks, and here we report network-level analyses carried out on a compiled database of radiolysis reactions, acquired by the scientific community over decades of research. The resulting network shows robust connections between abundant geochemical reservoirs and the production of carboxylic acids, amino acids, and ribonucleotide precursors—the chemistry of which is predominantly dependent on radicals. Moreover, the network exhibits the following measurable attributes associated with biological systems: (1) the species connectivity histogram exhibits a heterogeneous (heavy-tailed) distribution, (2) overlapping families of closed-loop cycles, and (3) a hierarchical arrangement of chemical species with a bottom-heavy energy-size spectrum. The latter attribute is implicated with stability and entropy production in complex systems, notably in ecology where it is known as a trophic pyramid. Radiolysis is implicated as a driver of abiotic chemical organization and could provide insights about the complex and perhaps radical-dependent mechanisms associated with life’s origins.
Article
Full-text available
Saprotrophic woodland fungi form self-organised transport networks as they forage for resources across the forest floor. These networks adapt during development by selective reinforcement of major transport routes and recycling of the intervening redundant mycelium to support further extension. The predicted transport performance of the resulting weighted networks show improved efficiency in comparison to evenly weighted networks with the same topology, or standard reference networks. Experimental measurement of nutrient movement using radiotracers and scintillation imaging show that fluxes are more dynamic, with synchronised oscillations and switching between different pre-existing routes. The same structures that confer good transport efficiency also show good resilience to both simulated damage and experimental attack by grazing insects, with persistence of a centrally connected core. We argue that fungi grow as self-organised planar spatial networks, honed by evolution, which may exemplify potential solutions to real-world compromises between search strategy, transport efficiency, resilience and cost.
Article
Full-text available
The coarse-scale population structure of pathogenic Armillaria (Fr.) Staude species was determined on ap- proximately 16 100 ha of relatively dry, mixed-conifer forest in the Blue Mountains of northeast Oregon. Sampling of recently dead or live, symptomatic conifers produced 112 isolates of Armillaria from six tree species. Armillaria spe- cies identifications done by using a polymerase chain reaction based diagnostic and diploid-diploid pairings produced identical results: 108 of the isolates were Armillaria ostoyae (Romagn.) Herink and four were North American Biologi- cal Species X (NABS X). Five genets of A. ostoyae and one of NABS X were identified through the use of somatic incompatibility pairings among the putatively diploid isolates. Armillaria ostoyae genet sizes were approximately 20, 95, 195, 260, and 965 ha; cumulative colonization of the study area was at least 9.5%. The maximum distance between isolates from the 965-ha A. ostoyae genet was approximately 3810 m, and use of three estimates of A. ostoyae spread rate in conifer forests resulted in age estimates for the genet ranging from 1900 to 8650 years. Results are discussed in relation to possible mechanisms that influenced the establishment, expansion, and expression of these genets; the ge- netic structure and stability of Armillaria; and the implications for disease management in this and similar forests. Résumé : La structure grossière des populations des espèces pathogènes d'Armillaria (Fr.) Staude a été établie sur une
Chapter
Fungi are of fundamental importance in the terrestrial environment. They have roles as decomposers, plant pathogens, symbionts, and in elemental cycles. Fungi are often dominant, and in soil can comprise the largest pool of biomass (including other microorganisms and invertebrates). They also play a role in maintenance of soil structure due to their filamentous growth habit and exopolymer production. Despite their important roles in the biosphere, fungi are frequently neglected within broader environmental and microbiological spheres. Additionally, mycological interests can be somewhat fragmented between traditional subject boundaries. This multi-disciplinary volume explores the roles and importance of fungi in the environment. Particular emphasis is given to major research advances made in recent years as a result of molecular and genomic approaches, and in cell imaging and biology. Drawing together microbiologists, mycologists, and environmental scientists, this work is a unique account of modern environmental mycology, and a pivotal contribution to the field.
Chapter
Fungi are amongst the simplest of eukaryotes. Their study has provided useful paradigms for processes that are fundamental to the way in which higher cells grow, divide, establish form and shape, and communicate with one another. The majority of work has been carried out on the budding yeast Saccharomyces cerevisiae, but in nature unicellular fungi are greatly outnumbered by filamentous forms for which our knowledge is much less well developed. This volume focuses on the analysis of the filamentous life style, particularly on the hyphae which constitute the fungal mycelial colony. It provides the most recent insights into the molecular genetics and physiological mechanisms underlying the elaboration of the branching mycelium and the interactions between individual fungal mycelia. As such it offers much to interest mycologists and, equally, those working in the fields of cell biology, developmental biology, physiology and biochemistry.
Article
Terrestrial fungi are commonly studied in the laboratory, growing on artificial media in which nutrients are typically homogeneously distributed and supplied in superabundance, the environment is sterile and microclimate (temperature, moisture, gaseous regime) usually relatively constant. This contrasts with the natural environment, in which: nutrients are often patchily and sparsely distributed or not readily available, because they are locked in recalcitrant material (e.g. lignin); many other organisms are encountered, including other fungi, bacteria and invertebrates; and microclimate is constantly changing, both temporally and spatially. This chapter explores the ways in which fungi cope with environmental heterogeneity. Similar situations are faced by macroorganisms and analogies are drawn. Emphasis is placed on basidiomycetes, not only because they have been studied in most detail, but because of their dominant role as decomposers and mutualistic symbionts (Boddy & Watkinson, 1995; Smith & Read, 1997) and because they are better adapted to respond to environmental heterogeneity over scales ranging from micrometres to many metres than are other fungi. Both saprotrophic and ectomycorrhizal Basidiomycota form extensive mycelial systems in woodland soil and litter, but it is the former that are the focus of this review. Saprotrophic, cord-forming Basidiomycota that ramify at the soil–litter interface, interconnecting disparate litter components, provide most examples. The key feature of these fungi that fits them for growth in environments where resources are heterogeneously distributed is that they are non-resource-unit restricted, i.e. they can grow out of one resource in search of others. © Cambridge University Press 2007 and Cambridge University Press, 2009.
Article
Following uptake of 32P-orthophosphate and 14C-aminoisobutyric acid (14C-AIB) the patterns of distribution of the isotopes through intact basidiomycete mycelia were non-destructively mapped at regular intervals using a β-scanner. Analysis of the results suggests that translocation of 32P and 14C-AIB through mycelia of Pleurotus ostreatus and Schizophyllum commune occurred along a restricted number of clearly defined, but macroscopically invisible, routes through the mycelium. In contrast to this, 32P added to mycelia of Coprinus cinereus remained immobilised at the addition point. Simultaneous acropetal and basipetal translocation of 32P and 14C-AIB was observed in different regions of colonies of P. ostreatus and S. commune. Translocation of label around the periphery of colonies strongly suggested the existence of anastomoses around the colony margin. Both 32P and 14C-AIB were initially immobilised at the addition point, from which each was subsequently translocated to other parts of the mycelium. The observed translocation of nutrients could not be explained by simple diffusion alone. The velocity of translocation and the complexity of the translocation pattern of 32P were greatest in mycelia of P. ostreatus, a hardwood decomposer, followed by S. commune, a wood and litter decomposer and parasite. Translocation through mycelia of C. cinereus, a coprophilus saprophyte, was very slow. This study provides the first detailed description of nutrient translocation through intact, entire fungal mycelia over time.
Article
We studied the effect of the size of food sources (FSs) presented to the true slime mould Physarum polycephalum on the tubular networks formed by the organism to absorb nutrient. The amount of plasmodium gathering at an FS was shown to be proportional to both the concentration of nutrient and the surface area of the FS. We presented two FSs to test which connection the organism selected in response to varying amounts of food and derived a simple rule for connection persistence: the longer connection collapses earlier. A mathematical model for tube selection in response to amount of food was derived and predicted our experimental findings regarding the choice of connection. When three FSs were presented to the organism, the longer tubes were also the first to collapse, explained by the relative probability of disconnection. The size of the FS is thus a key parameter determining network formation.
Article
In most higher plants, symbiotic fungi are central to the process of nutrient capture from soil1. Evidence from fossils of the earliest land plants2, as well as molecular studies3, confirms that roots co-evolved with fungal partners to form structures known as mycorrhizas — literally, 'fungus-roots'. These are almost universally distributed through present-day terrestrial plant communities, yet most researchers (deterred, one suspects, from experimental analysis of mycorrhizal function in natural communities by the complexity of these systems) have instead used excised roots or pot-grown plants to examine the relationships between partners in the symbiosis. Unfortunately, reductionist approaches cannot answer larger questions about the effect of symbiosis on interactions between the individual plants that form natural ecosystems.
Article
To survive saprotrophic fungi must be able to capture organic resources discontinuously dispersed in space and time. Some basidiomycetes can only achieve this by production of sexual and asexual spores or sclerotia — categorized as ‘resource-unit-restricted’, whereas ‘non-resource-unit-restricted’ basidiomycetes can also spread between organic resources as mycelium. Mycelial distribution and foraging within organic resources and among relatively homogeneously and heterogeneously distributed resources is reviewed. ‘Non-resource-unit-restricted’ Basidiomycota have evolved different patterns of mycelial spread appropriate to discovery of resources of different sizes and distributions. They show remarkable patterns of reallocation of biomass and mineral nutrients on discovery and colonization of new resources. Network architecture is a significant factor in the acquisition and distribution of nutrients, and in survival when parts of the network are destroyed. The costs and benefits of different architectures to large mycelial networks are considered.