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Fungal Automata
Andrew Adamatzky1, Eric Goles1,2, Genaro J. Mart´ınez1,3, Michail-Antisthenis
Tsompanas1, Martin Tegelaar4, and Han A. B. Wosten4
1Unconventional Computing Laboratory, UWE, Bristol, UK
2Faculty of Engineering and Science, University of Adolfo Ib´a˜nez, Santiago, Chile
3High School of Computer Science, National Polytechnic Institute, Mexico
4Microbiology Department, University of Utrecht, Utrecht, The Netherlands
Abstract
We study a cellular automaton (CA) model of information dynamics on a single hypha of a fungal
mycelium. Such a filament is divided in compartments (here also called cells) by septa. These
septa are invaginations of the cell wall and their pores allow for flow of cytoplasm between com-
partments and hyphae. The septal pores of the fungal phylum of the Ascomycota can be closed
by organelles called Woronin bodies. Septal closure is increased when the septa become older
and when exposed to stress conditions. Thus, Woronin bodies act as informational flow valves.
The one dimensional fungal automata is a binary state ternary neighbourhood CA, where every
compartment follows one of the elementary cellular automata (ECA) rules if its pores are open
and either remains in state ‘0’ (first species of fungal automata) or its previous state (second
species of fungal automata) if its pores are closed. The Woronin bodies closing the pores are also
governed by ECA rules. We analyse a structure of the composition space of cell-state transition
and pore-state transitions rules, complexity of fungal automata with just few Woronin bodies, and
exemplify several important local events in the automaton dynamics.
Keywords: fungi, ascomycete, Woronin body, cellular automata
1 Introduction
The fungal kingdom represents organisms colonising all ecological niches [11] where they play a key role
[18, 15, 35, 12]. Fungi can consist of a single cell, can form enormous underground networks [38] and
can form microscopic fruit bodies or fruit bodies weighting up to half a ton [16]. The underground
mycelium network can be seen as a distributed communication and information processing system
linking together trees, fungi and bacteria [10]. Mechanisms and dynamics of information processing in
mycelium networks form an unexplored field with just a handful of papers published related to space
exploration by mycelium [20, 19], patterns of electrical activity of fungi [37, 34, 1] and potential use
of fungi as living electronic and computing devices [2, 3, 4].
Filamentous fungi grow by means of hyphae that grow at their tip and that branch sub-apically.
Hyphae may be coenocytic or divided in compartments by septa. Filamentous fungi in the phylum
Ascomycota have porous septa that allow for cytoplasmic streaming [31, 24]. Woronin bodies plug
the pores of these septa after hyphal wounding to prevent excessive bleeding of cytoplasm [44, 13, 22,
42, 39, 29]. In addition, they plug septa of intact growing hyphae to maintain intra- and inter-hyphal
heterogeneity [8, 7, 40, 40, 41].
Woronin bodies can be located in different hyphal positions (Fig. 1a). When first formed, Woronin
bodies are generally localised to the apex [30, 43, 5]. Subsequently, Woronin bodies are either trans-
ported to the cell cortex (Neurospora crassa,Sordaria fimicola) or to the septum (Aspergillus oryzae,
1
arXiv:2003.08168v1 [nlin.CG] 18 Mar 2020
Septum associated
Woronin body
~50um
septum leashin
Cytoplasmic
Woronin body
(a)
`
`
`
`
`
`
1 1 0 0 1 0 0 1 0
(b)
Figure 1: (a) A biological scheme of a fragment of a fungal hypha of an ascomycete, where we can see
septa and associated Woronin bodies. (b) A scheme representing states of Woronin bodies: ‘0’ open,
‘1’ closed.
Aspergillus nidulans,Aspergillus fumigatus,Magnaporthe grisea,Fusarium oxysporum,Zymoseptoria
tritici) where they are anchored with a leashin tether and largely immobile until they are translocated
to the septal pore due to cytoplasmic flow or ATP depletion [33, 40, 39, 30, 29, 43, 45, 23, 6]. Woronin
bodies that are not anchored at the cellular cortex or the septum, are located in the cytoplasm and
are highly mobile (Aspergillus fumigatus,Aspergillus nidulans,Zymoseptoria tritici) [5, 30, 40]. Septal
pore occlusion can be induced by bulk cytoplasmic flow [40] or developmental [9] and environmental
cues, like puncturing of the cell wall, high temperature, carbon and nitrogen starvation, high osmolar-
ity and low pH. Interestingly, high environmental pH reduces the proportion of occluded apical septal
pores [41].
Aiming to lay a foundation of an emerging paradigm of fungal intelligence — distributed sensing
and information processing in living mycelium networks — we decided to develop a formal model of
mycelium and investigate a role of Woronin bodies in potential information dynamics in the mycelium.
The paper is structured as follows. We introduce fungal automata in Sect. 2. Properties of the
composition of cell state transition and Woronin body state transition functions are studied in Sect. 3.
Complexity of space-time configuration of fungal automata, where just few cells have Woronin bodies is
studied in Sect. 4. Section 5 exemplifies local events, which could be useful for computation with fungal
automata, happening in fungal automata with sparsely but regularly positioned cells with Woronin
bodies. The paper concludes with Sect. 6.
2 Fungal automata M
A fungal automaton is a one-dimensional cellular automaton with binary cell states and ternary,
including central cell, cell neighbourhood, governed by two elementary cellular automata (ECA) rules,
namely the cell state transition rule fand the Woronin bodies adjustment rule g:M=hN, u, Q, f, gi.
Each cell xihas a unique index i∈N. Its state is updated from Q={0,1}in discrete time depending of
its current state xt
i, the states of its left xt
i−1and right neighbours xt
i+1 and the state of cell x’s Woronin
body w. Woronin bodies take states from Q:wt= 1 means Woronin bodies (Fig. 1) in cell xblocks
the pores and the cell has no communication with its neighbours, and wt= 0 means that Woronin
bodies in cell xdo not block the pores. Woronin bodies update their states g(·), wt+1 =g(u(x)t),
depending on the state of neighbourhood u(x)t. Cells xupdate their states by function f(·) if their
Woronin bodies do not block the pores.
Two species of mycelium automata are considered M1, where each cell updates its state as following:
xt+1 =(0 if wt= 1
f(u(x)t) otherwise
2
(a) M1, ρf= 133, ρg= 116 (b) M2, ρf= 133, ρg= 116 (c) M1, ρf= 73, ρg= 128
(d) M2, ρf= 73, ρg= 128 (e) M1, ρf= 61, ρg= 132 (f) M2, ρf= 61, ρg= 132
(g) M1, ρf= 57, ρg= 98 (h) M2, ρf= 57, ρg= 98 (i) M1, ρf= 26, ρg= 84
(j) M2, ρf= 26, ρg= 84 (k) M1, ρf= 125, ρg= 105 (l) M2, ρf= 125, ρg= 105
Figure 2: Examples of space-time dynamics of M. The automata are 103cells each. Initial configu-
ration is random with probability of a cell xto be in state ‘1’, x0= 1, equals 0.01. Each automaton
evolved for 103iterations. Binary values of ECA rules fand gare shown in sub-captions. Rule g
is applied to every iteration starting from 200th. Cells in state ‘0’ are white, in state ‘1’ are black,
cells with Woronin bodies blocking pores are red. Indexes of cells increase from the left to the right,
iterations are increasing from the to the bottom.
3
and M2where each cell updates its state as following:
xt+1 =(xtif wt= 1
f(u(x)t) otherwise
where wt=g(u(x)t).
State ‘1’ in the cells of array xsymbolises metabolites, signals exchanged between cells. Where
pores in a cell are open the cell updates its state by ECA rule f:{0,1}3→ {0,1}.
In automaton M1, when Woronin bodies block the pores in a cell, the cell does not update its
state and remains in the state ‘0’ and left and right neighbours of the cells can not detect any ‘cargo’
in this cell. In automaton, M2, where Woronin bodies block the pores in a cell, the cell does not
update its state and remains in its current state. In real living mycelium glucose and possibly other
metabolites [7] can still cross the septum even when septa are closed by Woronin bodies, but we can
ignore this fact in present abstract model.
Both species are biologically plausible and, thus, will be studied in parallel. The rules for closing
and opening Woronin bodies are also ECA rules g:{0,1}3→ {0,1}. If g(u(x)t) = 0 this means that
pores are open, if g(u(x)t) = 1 Woronin bodies block the pores. Examples of space-time configurations
of both species of Mare shown in Fig. 2.
3 Properties of composition f◦g
Predecessor sets
Let F={h:{0,1}3→ {0,1}} be a set of all ECA functions. Then for any composition f◦g,
where f, g ∈F, can be converted to a single function h∈F. For each h∈Fwe can construct a set
P(h) = {f◦g∈F×F|f◦g→h}. The sets P(h) for each h∈Fare available online1.
A size of P(h) for each his shown in Fig. 3c. The functions with largest size of P(h) are rule 0 in
automaton M1and rule 51 (only neighbourhood configurations (010, 011, 110, 111 are mapped into
1) in M2.
Size σof P(h) vs a number γof functions hhaving set P(h) of size σis shown for automata M1
and M2in Table 1a.
With regards to Wolfram classification [47], sizes of P(h) for rules from Class III vary from 9 to
729 in M1(Tab. 1b). Rule 126 would be the most difficult to obtain in M1by composition two ECA
rules chosen at random, it has only 9 ‘predecessor’ f◦gpairs. Rule 18 would be the easiest, for Class
III rules, to be obtained, it has 729 predecessors, in both M1(Tab. 1b) and M2(Tab. 1d). In M1,
one rule, rule 41, from the class IV has 243 f◦gpredecessors, and all other rules in that class have
81 (Tab. 1c). From Class IV rule 54 has the largest number of predecessors in M2, and thus can be
considered as most common (Tab. 1d).
Diagonals
In automaton M1for any f∈Ff◦f= 0. Assume f:{0,1}3→1 then Woronin bodies close the
pores and, thus, second application of fproduces state ‘0’. If f:{0,1}3→0 then Woronin bodes
does not close pores but yet second application of the fproduce state ‘0’.
For automaton M2a structure of diagonal mapping f◦f→h, where f, h ∈Fis shown in
Tab. 2. The set of the diagonal outputs f◦fconsists of 16 rules: (0, 1, 2, 3), (16, 17, 18, 19),
(32, 33, 34, 35), (48, 49, 40, 51). These set of rules can be reduced to the following rule. Let
C(xt) = [u(x)t= (111)] ∨[u(x)t= (111)] and B(xt) = [u(x)t= (011)] ∨[u(x)t= (010)]. Then xt= 1
if C(x)t∨C(x)t∧B(xt).
1https://figshare.com/s/b7750ee3fe6df7cbe228
4
(a) (b)
κ(ρf)
0
500
1000
1500
2000
ρf
0 20 40 60 80 100 120 140 160 180 200 220 240
(c)
Figure 3: Mapping F×F→Ffor automaton M1(a) and M2(b) is visualised as an array of pixels,
P= (p)0≤ρf≤255,0≤ρf≤255. An entry at the intersection of any ρfand ρgis a coloured as follows: red
if pρfρg=pρgρf, blue if ρg=pρgρf, green if ρf=pρgρf. (c) Sizes of P(h) sets for M1, circle, and M2,
solid discs, are shown for every function hapart of rule 0 (M1) and rule 51 (M2).
5
(a) Rules per
|P(h)|
σ γ
1 1
3 8
9 28
27 56
81 70
243 56
729 28
2187 8
6561 1
(b) M1: Class III rules
Rule σ
18 729
22, 146 243
30, 45, 60,
90, 105, 150
81
122 27
126 9
(c) M1: Class IV rules
Rule σ
41 243
54, 106, 110 81
(d) M2: Class III rules
Rule σ
18 729
22, 146 243
30, 45, 60 90,
105, 150
81
122 243
126 81
(e) M2: Class IV rules
Rule σ
41 243
54 729
106 81
110 27
Table 1: Characterisations of automaton mapping F×F→F. (a) Size σof P(h) vs a number γ
of functions hhaving set P(h) of size σ. T (b) Sizes of sets P(h) for rules from Wolfram class III.
(b) Sizes of sets P(h) for rules from Wolfram class IV.
f◦f f
0 0, 1, 2, 3, 16, 17, 18, 19, 32, 33, 34, 35, 48, 49, 50, 51
1 128, 129, 130, 131, 144, 145, 146, 147, 160, 161, 162, 163, 176, 177, 178, 179
2 64, 65, 66, 67, 80, 81, 82, 83, 96, 97, 98, 99, 112, 113, 114, 115
3 192, 193, 194, 195, 208, 209, 210, 211, 224, 225, 226, 227, 240, 241, 242, 243
16 8, 9, 10, 11, 24, 25, 26, 27, 40, 41, 42, 43, 56, 57, 58, 59
17 136, 137, 138, 139, 152, 153, 154, 155, 168, 169, 170, 171, 184, 185, 186, 187
18 72, 73, 74, 75, 88, 89, 90, 91, 104, 105, 106, 107, 120, 121, 122, 123
19 200, 201, 202, 203, 216, 217, 218, 219, 232, 233, 234, 235, 248, 249, 250, 251
32 4, 5, 6, 7, 20, 21, 22, 23, 36, 37, 38, 39, 52, 53, 54, 55
33 132, 133, 134, 135, 148, 149, 150, 151, 164, 165, 166, 167, 180, 181, 182, 183
34 68, 69, 70, 71, 84, 85, 86, 87, 100, 101, 102, 103, 116, 117, 118, 119
35 196, 197, 198, 199, 212, 213, 214, 215, 228, 229, 230, 231, 244, 245, 246, 247
48 12, 13, 14, 15, 28, 29, 30, 31, 44, 45, 46, 47, 60, 61, 62, 63
49 140, 141, 142, 143, 156, 157, 158, 159, 172, 173, 174, 175, 188, 189, 190, 191
50 76, 77, 78, 79, 92, 93, 94, 95, 108, 109, 110, 111, 124, 125, 126, 127
51 204, 205, 206, 207, 220, 221, 222, 223, 236, 237, 238, 239, 252, 253, 254, 255
Table 2: Diagonals of automaton M2.
6
Commutativity
In automaton M1, for any f, g ∈Ff◦g6=g◦fonly if f6=g. In automaton M2there are 32768
pairs of function which ◦is commutative, their distribution visualised in red in Fig. 3b.
Identities and zeros
In automaton M1there are no left or right identities, neither right zeros in hF,F,◦i. The only left
zero is the rule 0. In automaton M2there are no identities or zeros at all.
Associativity
In automaton M1there 456976 triples hf, g, hion which operation ◦is associative: (f◦g)◦h=
f◦(g◦h). The ratio of associative triples to the total number of triples is 0.027237892. There are
104976 associative triples in M2, a ratio of 0.006257057.
4 Tuning Complexity: Rule 110
To evaluate on how introduction of Woronin bodies could affect complexity of automaton evolution,
we undertook two series of experiments. In the first series we used fungal automaton where just one
cell has a Woronin body (Fig. 5). In the second series we employed fungal automaton where regularly
positioned cells (but not all cells of the array) have Woronin bodies.
State transition functions gof Woronin bodies were varied across the whole diapason but the state
transition function fof a cell was Rule 110, ρf= 110. We have chosen Rule 110 because the rule is
proven to be computationally universal [25, 14], P-complete [32], the rules belong to Wolfram class
IV renown for exhibiting complex and non-trivial interactions between travelling localisation [46], rich
families of gliders can be produce in collisions with other gliders [26, 27, 28].
We wanted to check how an introduction of Woronin bodies affect dynamics of most complex
space-time developed of Rule 110 automaton. Thus, we evolved the automata from all possible initial
configurations of 8 cells placed near the end of n= 1000 cells array of resting cells and allowing
to evolve for 950 iterations. Lempel–Ziv complexity (compressibility) LZ was evaluated via sizes of
space-time configurations saved as PNG files. This is sufficient because the ’deflation’ algorithm used
in PNG lossless compression [36, 21, 17] is a variation of the classical Lempel–Ziv 1977 algorithm [48].
Estimates of LZ complexity for each of 8-cell initial configurations are shown in Fig. 4a. The initial
configurations with highest estimated LZ complexity are 10110001 (decimal 177), 11010001 (209),
10000011 (131), 11111011 (253), see example of space-time dynamics in Fig. 4b.
We assumed that a cell in the position n−100 has a Woronin body which can be activated (Fig. 5),
i.e. start updating its state by rule f, after 100th iteration of the automaton evolution. We then
run 950 iteration of automaton evolution for each of 256 Woronin rules and estimated LZ complexity.
In experiments with M1we found that 128 rules, with even decimal representations, do not affect
space time dynamics of evolution and 128 rules, with even decimal representations, reduce complexity
of the space-time configuration. The key reasons for the complexity reduction (compare Fig. 4b and
c) are cancellation of three gliders at c. 300th iteration and simplification of the behaviour of glider
guns positioned at the tail of the propagating wave-front. In experiments with M2128 rules, with
even decimal representations, do not change the space-time configuration of the author. Other 128
rules reduce complexity and modify space-time configuration by re-arranging the structures of glider
guns and establishing one oscillators at the site surrounding position of the cell with Woronin body
(Fig. 4d).
In second series of experiments we regularly positioned cells with Woronin bodies along the 1D
array: every 50th cell has a Woronin body. Then we evolved fungal automata M1and M2from
exactly the same initial random configuration with density of ‘1’ equal to 0.3. Space-time configuration
of the automaton without Woronin bodies is shown in Fig. 6a. Exemplar of space-time configurations
7
Size, kB
20
30
40
50
60
70
Initial configuration, decimal encoding
0 50 100 150 200 250
(a)
(b) (c)
(d)
Figure 4: (a) Estimates of LZ complexity of space-time configurations of ECA Rule 110 without
Woronin bodies. (b) A space-time configuration of ECA Rule 110 evolving from initial configuration
10110001 (177), no Woronin bodies are activated. (c) A space-time configuration of M1Rule 110
evolving from initial configuration 10110001 (177), Woronin body is governed by rule 43; red lines
indicate time when the body was activated and position of the cell with the body. In (bcd), a pixel in
position (i, t) is black if xt
i= 1.
8
`
`
`
`
`
`
1 1 0 0 1 0 0 1 0
Figure 5: Only one cell has Woronin body.
of automata with Woronin bodies are shown in Fig. 6b–h. As seen in Fig. 7 both species of fungal
automata show similar dynamics of complexity along the Woronin transition functions ordered by their
decimal values. The automaton M1has average LZ complexity 82.2 (σ= 24.6) and the automaton
M278.4 (σ= 22.1). Woronin rules gwhich generate most LZ complex space-time configurations are
ρg= 133 in M)1(Fig. 6b) and ρg= 193 in M)2(Fig. 6e). The space-time dynamics of the automaton
is characterised by a substantial number of gliders guns and gliders (Fig. 6b). Functions being in the
middle of the descending hierarchy of LZ complexity produce space-time configurations with declined
number of travelling localisation and growing domains of homogeneous states (Fig. 6cg). Automata
with Woronin functions at the bottom of the complexity hierarchy quickly (i.e. after 200-300 iterations)
evolve towards stable, equilibrium states (Fig. 6dh).
5 Local events
Let us consider some local events happening in the fungal automata discussed in Sect. 4: every 50th
cell of an array has a Woronin body.
Retaining gliders. A glider can be stopped and converted into a station localisation by a cell
with Woronin body. As exemplified in Fig. 8a, the localisation travelling left was stopped from further
propagation by a cell with Woronin body yet the localisation did not annihilate but remained stationary.
Register memory. Different substrings of input string (initial configuration) might lead to differ-
ent equilibrium configurations achieved in the domains of the array separated by cells with Woronin
bodies. When there is just two types of equilibrium configurations they be seen as ‘bit up’ and ‘bit
down’ and therefore such fungal automaton can be used a memory register (Fig. 8b).
Reflectors. In many cases cells with Woronin bodies induce local domains of stationary, sometimes
time oscillations, inhomogeneities which might act as reflectors for travelling localisations. An example
is shown in Fig. 8c where several localisations are repeatedly bouncing between two cells with Woronin
bodies.
Modifiers. Cells with Woronin bodies can act as modifiers of states of gliders reflected from them
and of outcomes of collision between travelling localizations. In Fig. 8d we can see how a travelling
localisation is reflected from the vicinity of Woronin bodies three times: every time the state of the
localisation changes. On the third reflection the localisation becomes stationary. In the fragment
(Fig. 8e) of space-time configuration of automaton with Woronin bodies governed by ρg= 201 of the
fragment we can see how two localisations got into contact with each in the vicinity of the Woronin
body and an advanced structure is formed two breathing stationary localisations act as mirror, and
there are streams of travelling localisations between them. A multi-step chain reaction can be observed
in Fig. 8f: there are two stationary, breathing, localisations at the sites of the cells with Woronin bodies.
A glider is formed on the left stationary localisation. The glider travel to the right and collide into
right breather. In the result of the collision the breath undergoes structural transitions, emits a glider
travelling left and transforms itself into a pair of stationary breathers. Meantime the newly born glider
collided into left breather and changes its state.
6 Discussion
As a first step towards formalisation of fungal intelligence we introduced one-dimensional fungal au-
tomata operated by two local transition function: one, g, governs states of Woronin bodies (pores are
open or closed), another, f, governs cells states: ‘0’ and ‘1’. We provided a detailed analysis of the
9
(a) (b) M1,ρg= 133 (c) M1,ρg= 29 (d) M1,ρg= 49
(e) M2,ρg= 193 (f) M2,ρg= 5 (g) M2,ρg= 221 (h) M2,ρg= 174
Figure 6: (a) ECA Rule 110, no Woronin bodies. Space-time evolution of M∞(bcd) and M∈(e–h)
for Woronin rules shown in subcaption. LZ complexity of space-time configurations decreases from (b)
to (d) and from (e) to (h). Every 50th cell has a Woronin body.
10
Size, kB
40
60
80
100
120
140
Woronin rule ρg, decimal encoding
0 50 100 150 200 250
Figure 7: Estimations of LZ complexity of space-time, 500 cells by 500 iterations, configurations of
M1, discs, and M1, circles, for all Woronin functions g.
(a) M1,
ρg= 2
(b) M1,ρg= 15 (c) M1,
ρg= 21
(d) M1,
ρg= 31
(e) M1,ρg=
29
(f) M1,
ρg= 201
Figure 8: (a) Localisation travelling left was stopped by the Woronin body. (b) Analog of a memory
register. (c) Reflections of travelling localisations from cells with Woronin bodies. (d) Modification of
glider state in the vicitinity of Woronin bodies. (e) A fragment of configuration of automaton with
ρg= 29, left cell states, right Woronin bodies states. (f) Enlarged sub-fragment of the fragment (d)
where Wonorin body tunes the outcome of the collision. For both automata ρf= 110.
11
Figure 9: An example of 5-inputs-7-outputs collision in M2,ρf= 110, ρg= 40. Every 50th cell has a
Woronin body. Cells state transitions are shown on the left, Woronin bodies state transitions on the
right. A pixel in position (i, t) is black if xt
i= 1, left, or wt
i= 1, right.
12
magma hf, g, ◦i, results of which might be useful for future designs of computational and language
recognition structures with fungal automata. The magma as a whole does not satisfy any other prop-
erty but closure. Chances are high that there are subsets of the magma which might satisfy conditions
of other algebraic structures. A search for such subsets could be one of the topics of further studies.
Another topic could be an implementation of computational circuits in fungal automata. For
certain combination of fand gwe can find quite sophisticated families of stationary and travelling
localisations and many outcomes of the collisions and interactions between these localisations, an
illustration is shown in Fig. 9. Thus the target could be, for example, to construct a n-binary full
adder which is as compact in space and time as possible.
The theoretical results reported show that by controlling just a few cells with Woronin bodies it is
possible to drastically change dynamics of the automaton array. Third direction of future studies could
be in implemented information processing in a single hypha. In such a hypothetical experimental setup
input strings will be represented by arrays of illumination and outputs could be patterns of electrical
activity recorded from the mycelium hypha resting on an electrode array.
Acknowledgement
AA, MT, HABW have received funding from the European Union’s Horizon 2020 research and in-
novation programme FET OPEN “Challenging current thinking” under grant agreement No 858132.
EG residency in UWE has been supported by funding from the Leverhulme Trust under the Visiting
Research Professorship grant VP2-2018-001.
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