FFungal Automata
Andrew Adamatzky , Eric Goles , Genaro J. Mart´ınez , Michail-AntisthenisTsompanas , Martin Tegelaar , and Han A. B. Wosten Unconventional Computing Laboratory, UWE, Bristol, UK Faculty of Engineering and Science, University of Adolfo Ib´a˜nez, Santiago, Chile High School of Computer Science, National Polytechnic Institute, Mexico Microbiology Department, University of Utrecht, Utrecht, The Netherlands
Abstract
We study a cellular automaton (CA) model of information dynamics on a single hypha of a fungalmycelium. Such a filament is divided in compartments (here also called cells) by septa. Thesesepta 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 closedby organelles called Woronin bodies. Septal closure is increased when the septa become olderand 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 everycompartment follows one of the elementary cellular automata (ECA) rules if its pores are openand either remains in state ‘0’ (first species of fungal automata) or its previous state (secondspecies of fungal automata) if its pores are closed. The Woronin bodies closing the pores are alsogoverned by ECA rules. We analyse a structure of the composition space of cell-state transitionand pore-state transitions rules, complexity of fungal automata with just few Woronin bodies, andexemplify several important local events in the automaton dynamics.
Keywords: fungi, ascomycete, Woronin body, cellular automata
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] andcan form microscopic fruit bodies or fruit bodies weighting up to half a ton [16]. The undergroundmycelium network can be seen as a distributed communication and information processing systemlinking together trees, fungi and bacteria [10]. Mechanisms and dynamics of information processing inmycelium networks form an unexplored field with just a handful of papers published related to spaceexploration by mycelium [20, 19], patterns of electrical activity of fungi [37, 34, 1] and potential useof 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 plugthe 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-hyphalheterogeneity [8, 7, 40, 40, 41].Woronin bodies can be located in different hyphal positions (Fig. 1a). When first formed, Woroninbodies 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 a r X i v : . [ n li n . C G ] M a r eptum associated Woronin body ~50umseptum leashinCytoplasmicWoronin 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 seesepta and associated Woronin bodies. (b) A scheme representing states of Woronin bodies: ‘0’ open,‘1’ closed.
Aspergillus nidulans , Aspergillus fumigatus , Magnaporthe grisea , Fusarium oxysporum , Zymoseptoriatritici ) where they are anchored with a leashin tether and largely immobile until they are translocatedto the septal pore due to cytoplasmic flow or ATP depletion [33, 40, 39, 30, 29, 43, 45, 23, 6]. Woroninbodies that are not anchored at the cellular cortex or the septum, are located in the cytoplasm andare highly mobile (
Aspergillus fumigatus , Aspergillus nidulans , Zymoseptoria tritici ) [5, 30, 40]. Septalpore occlusion can be induced by bulk cytoplasmic flow [40] or developmental [9] and environmentalcues, 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 septalpores [41].Aiming to lay a foundation of an emerging paradigm of fungal intelligence — distributed sensingand information processing in living mycelium networks — we decided to develop a formal model ofmycelium 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 thecomposition 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 isstudied in Sect. 4. Section 5 exemplifies local events, which could be useful for computation with fungalautomata, happening in fungal automata with sparsely but regularly positioned cells with Woroninbodies. The paper concludes with Sect. 6. 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 f and the Woronin bodies adjustment rule g : M = (cid:104) N , u, Q , f, g (cid:105) .Each cell x i has a unique index i ∈ N . Its state is updated from Q = { , } in discrete time depending ofits current state x ti , the states of its left x ti − and right neighbours x ti +1 and the state of cell x ’s Woroninbody w . Woronin bodies take states from Q : w t = 1 means Woronin bodies (Fig. 1) in cell x blocksthe pores and the cell has no communication with its neighbours, and w t = 0 means that Woroninbodies in cell x do not block the pores. Woronin bodies update their states g ( · ), w t +1 = g ( u ( x ) t ),depending on the state of neighbourhood u ( x ) t . Cells x update their states by function f ( · ) if theirWoronin bodies do not block the pores.Two species of mycelium automata are considered M , where each cell updates its state as following: x t +1 = (cid:40) w t = 1 f ( u ( x ) t ) otherwise2 a) M , ρ f = 133 , ρ g = 116 (b) M , ρ f = 133 , ρ g = 116 (c) M , ρ f = 73 , ρ g = 128(d) M , ρ f = 73 , ρ g = 128 (e) M , ρ f = 61 , ρ g = 132 (f) M , ρ f = 61 , ρ g = 132(g) M , ρ f = 57 , ρ g = 98 (h) M , ρ f = 57 , ρ g = 98 (i) M , ρ f = 26 , ρ g = 84(j) M , ρ f = 26 , ρ g = 84 (k) M , ρ f = 125 , ρ g = 105 (l) M , ρ f = 125 , ρ g = 105 Figure 2: Examples of space-time dynamics of M . The automata are 10 cells each. Initial configu-ration is random with probability of a cell x to be in state ‘1’, x = 1, equals 0.01. Each automatonevolved for 10 iterations. Binary values of ECA rules f and g are 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. 3nd M where each cell updates its state as following: x t +1 = (cid:40) x t if w t = 1 f ( u ( x ) t ) otherwisewhere w t = g ( u ( x ) t ).State ‘1’ in the cells of array x symbolises metabolites, signals exchanged between cells. Wherepores in a cell are open the cell updates its state by ECA rule f : { , } → { , } .In automaton M , when Woronin bodies block the pores in a cell, the cell does not update itsstate and remains in the state ‘0’ and left and right neighbours of the cells can not detect any ‘cargo’in this cell. In automaton, M , where Woronin bodies block the pores in a cell, the cell does notupdate its state and remains in its current state. In real living mycelium glucose and possibly othermetabolites [7] can still cross the septum even when septa are closed by Woronin bodies, but we canignore this fact in present abstract model.Both species are biologically plausible and, thus, will be studied in parallel. The rules for closingand opening Woronin bodies are also ECA rules g : { , } → { , } . If g ( u ( x ) t ) = 0 this means thatpores are open, if g ( u ( x ) t ) = 1 Woronin bodies block the pores. Examples of space-time configurationsof both species of M are shown in Fig. 2. f ◦ g Predecessor sets
Let F = { h : { , } → { , }} 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 ∈ F we can construct a set P ( h ) = { f ◦ g ∈ F × F | f ◦ g → h } . The sets P ( h ) for each h ∈ F are available online .A size of P ( h ) for each h is shown in Fig. 3c. The functions with largest size of P ( h ) are rule 0 inautomaton M and rule 51 (only neighbourhood configurations (010, 011, 110, 111 are mapped into1) in M .Size σ of P ( h ) vs a number γ of functions h having set P ( h ) of size σ is shown for automata M and M in Table 1a.With regards to Wolfram classification [47], sizes of P ( h ) for rules from Class III vary from 9 to729 in M (Tab. 1b). Rule 126 would be the most difficult to obtain in M by composition two ECArules chosen at random, it has only 9 ‘predecessor’ f ◦ g pairs. Rule 18 would be the easiest, for ClassIII rules, to be obtained, it has 729 predecessors, in both M (Tab. 1b) and M (Tab. 1d). In M ,one rule, rule 41, from the class IV has 243 f ◦ g predecessors, and all other rules in that class have81 (Tab. 1c). From Class IV rule 54 has the largest number of predecessors in M , and thus can beconsidered as most common (Tab. 1d). Diagonals
In automaton M for any f ∈ F f ◦ f = 0. Assume f : { , } → f produces state ‘0’. If f : { , } → f produce state ‘0’.For automaton M a structure of diagonal mapping f ◦ f → h , where f, h ∈ F is shown inTab. 2. The set of the diagonal outputs f ◦ f consists 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 ( x t ) = [ u ( x ) t = (111)] ∨ [ u ( x ) t = (111)] and B ( x t ) = [ u ( x ) t = (011)] ∨ [ u ( x ) t = (010)]. Then x t = 1if C ( x ) t ∨ C ( x ) t ∧ B ( x t ). https://figshare.com/s/b7750ee3fe6df7cbe228 a) (b) κ ( ρ f ) ρ f (c) Figure 3: Mapping F × F → F for automaton M (a) and M (b) is visualised as an array of pixels, P = ( p ) ≤ ρ f ≤ , ≤ ρ f ≤ . An entry at the intersection of any ρ f and ρ g is a coloured as follows: redif p ρ f ρ g = p ρ g ρ f , blue if ρ g = p ρ g ρ f , green if ρ f = p ρ g ρ f . (c) Sizes of P ( h ) sets for M , circle, and M ,solid discs, are shown for every function h apart of rule 0 ( M ) and rule 51 ( M ).5 a) Rules per | P ( h ) | σ γ (b) M : Class III rules Rule σ
18 72922, 146 24330, 45, 60,90, 105, 150 81122 27126 9 (c) M : Class IV rules Rule σ
41 24354, 106, 110 81 (d) M : Class III rules Rule σ
18 72922, 146 24330, 45, 60 90,105, 150 81122 243126 81 (e) M : Class IV rules Rule σ
41 24354 729106 81110 27Table 1: Characterisations of automaton mapping F × F → F . (a) Size σ of P ( h ) vs a number γ of functions h having 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 M .6 ommutativity In automaton M , for any f, g ∈ F f ◦ g (cid:54) = g ◦ f only if f (cid:54) = g . In automaton M there are 32768pairs of function which ◦ is commutative, their distribution visualised in red in Fig. 3b. Identities and zeros
In automaton M there are no left or right identities, neither right zeros in (cid:104) F , F , ◦(cid:105) . The only leftzero is the rule 0. In automaton M there are no identities or zeros at all. Associativity
In automaton M there 456976 triples (cid:104) f, g, h (cid:105) on 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 are104976 associative triples in M , a ratio of 0.006257057. 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 onecell has a Woronin body (Fig. 5). In the second series we employed fungal automaton where regularlypositioned cells (but not all cells of the array) have Woronin bodies.State transition functions g of Woronin bodies were varied across the whole diapason but the statetransition function f of a cell was Rule 110, ρ f = 110. We have chosen Rule 110 because the rule isproven to be computationally universal [25, 14], P-complete [32], the rules belong to Wolfram classIV renown for exhibiting complex and non-trivial interactions between travelling localisation [46], richfamilies 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 complexspace-time developed of Rule 110 automaton. Thus, we evolved the automata from all possible initialconfigurations of 8 cells placed near the end of n = 1000 cells array of resting cells and allowingto evolve for 950 iterations. Lempel–Ziv complexity (compressibility) LZ was evaluated via sizes ofspace-time configurations saved as PNG files. This is sufficient because the ’deflation’ algorithm usedin 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 initialconfigurations 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 thenrun 950 iteration of automaton evolution for each of 256 Woronin rules and estimated LZ complexity.In experiments with M we found that 128 rules, with even decimal representations, do not affectspace time dynamics of evolution and 128 rules, with even decimal representations, reduce complexityof the space-time configuration. The key reasons for the complexity reduction (compare Fig. 4b andc) are cancellation of three gliders at c. 300th iteration and simplification of the behaviour of gliderguns positioned at the tail of the propagating wave-front. In experiments with M
128 rules, witheven decimal representations, do not change the space-time configuration of the author. Other 128rules reduce complexity and modify space-time configuration by re-arranging the structures of gliderguns 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 1Darray: every 50th cell has a Woronin body. Then we evolved fungal automata M and M fromexactly the same initial random configuration with density of ‘1’ equal to 0.3. Space-time configurationof the automaton without Woronin bodies is shown in Fig. 6a. Exemplar of space-time configurations7 i ze , k B (a)(b) (c)(d) Figure 4: (a) Estimates of LZ complexity of space-time configurations of ECA Rule 110 withoutWoronin bodies. (b) A space-time configuration of ECA Rule 110 evolving from initial configuration10110001 (177), no Woronin bodies are activated. (c) A space-time configuration of M Rule 110evolving from initial configuration 10110001 (177), Woronin body is governed by rule 43; red linesindicate time when the body was activated and position of the cell with the body. In (bcd), a pixel inposition ( i, t ) is black if x ti = 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 fungalautomata show similar dynamics of complexity along the Woronin transition functions ordered by theirdecimal values. The automaton M has average LZ complexity 82.2 ( σ = 24 .
6) and the automaton M σ = 22 . g which generate most LZ complex space-time configurations are ρ g = 133 in M ) (Fig. 6b) and ρ g = 193 in M ) (Fig. 6e). The space-time dynamics of the automatonis characterised by a substantial number of gliders guns and gliders (Fig. 6b). Functions being in themiddle of the descending hierarchy of LZ complexity produce space-time configurations with declinednumber of travelling localisation and growing domains of homogeneous states (Fig. 6cg). Automatawith Woronin functions at the bottom of the complexity hierarchy quickly (i.e. after 200-300 iterations)evolve towards stable, equilibrium states (Fig. 6dh). Let us consider some local events happening in the fungal automata discussed in Sect. 4: every 50thcell of an array has a Woronin body.
Retaining gliders.
A glider can be stopped and converted into a station localisation by a cellwith Woronin body. As exemplified in Fig. 8a, the localisation travelling left was stopped from furtherpropagation 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 Woroninbodies. When there is just two types of equilibrium configurations they be seen as ‘bit up’ and ‘bitdown’ 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, sometimestime oscillations, inhomogeneities which might act as reflectors for travelling localisations. An exampleis shown in Fig. 8c where several localisations are repeatedly bouncing between two cells with Woroninbodies.
Modifiers.
Cells with Woronin bodies can act as modifiers of states of gliders reflected from themand of outcomes of collision between travelling localizations. In Fig. 8d we can see how a travellinglocalisation is reflected from the vicinity of Woronin bodies three times: every time the state of thelocalisation 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 ρ g = 201 of thefragment we can see how two localisations got into contact with each in the vicinity of the Woroninbody and an advanced structure is formed two breathing stationary localisations act as mirror, andthere are streams of travelling localisations between them. A multi-step chain reaction can be observedin 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 intoright breather. In the result of the collision the breath undergoes structural transitions, emits a glidertravelling left and transforms itself into a pair of stationary breathers. Meantime the newly born glidercollided into left breather and changes its state. 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 areopen or closed), another, f , governs cells states: ‘0’ and ‘1’. We provided a detailed analysis of the9 a) (b) M , ρ g = 133 (c) M , ρ g = 29 (d) M , ρ g = 49(e) M , ρ g = 193 (f) M , ρ g = 5 (g) M , ρ g = 221 (h) M , ρ g = 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 i ze , k B ρ g , decimal encoding0 50 100 150 200 250 Figure 7: Estimations of LZ complexity of space-time, 500 cells by 500 iterations, configurations of M , discs, and M , circles, for all Woronin functions g . (a) M , ρ g = 2 (b) M , ρ g = 15 (c) M , ρ g = 21 (d) M , ρ g = 31 (e) M , ρ g =29 (f) M , ρ g = 201 Figure 8: (a) Localisation travelling left was stopped by the Woronin body. (b) Analog of a memoryregister. (c) Reflections of travelling localisations from cells with Woronin bodies. (d) Modification ofglider state in the vicitinity of Woronin bodies. (e) A fragment of configuration of automaton with ρ g = 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 ρ f = 110.11igure 9: An example of 5-inputs-7-outputs collision in M , ρ f = 110, ρ g = 40. Every 50th cell has aWoronin body. Cells state transitions are shown on the left, Woronin bodies state transitions on theright. A pixel in position ( i, t ) is black if x ti = 1, left, or w ti = 1, right.12agma (cid:104) f, g, ◦(cid:105) , results of which might be useful for future designs of computational and languagerecognition 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 conditionsof 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. Forcertain combination of f and g we can find quite sophisticated families of stationary and travellinglocalisations and many outcomes of the collisions and interactions between these localisations, anillustration is shown in Fig. 9. Thus the target could be, for example, to construct a n -binary fulladder 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 ispossible to drastically change dynamics of the automaton array. Third direction of future studies couldbe in implemented information processing in a single hypha. In such a hypothetical experimental setupinput strings will be represented by arrays of illumination and outputs could be patterns of electricalactivity 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 VisitingResearch Professorship grant VP2-2018-001.
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