Actin droplet machine
AActin droplet machine
Andrew Adamatzky , Florian Huber , and J¨org Schnauß Unconventional Computing Laboratory, Department of ComputerScience, University of the West of England, Bristol, UK Netherlands eScience Center, Science Park 140, 1098 XG Amsterdam,The Netherlands Soft Matter Physics Division, Peter Debye Institute for Soft MatterPhysics, Faculty of Physics and Earth Science, Leipzig University,Germany & Fraunhofer Institute for Cell Therapy and Immunology (IZI),DNA Nanodevices Group, Leipzig, Germany
Abstract
The actin droplet machine is a computer model of a three-dimensional network ofactin bundles developed in a droplet of a physiological solution, which implementsmappings of sets of binary strings. The actin bundle network is conductive to travel-ling excitations, i.e. impulses. The machine is interfaced with an arbitrary selectedset of k electrodes through which stimuli, binary strings of length k represented byimpulses generated on the electrodes, are applied and responses are recorded. Theresponses are recorded in a form of impulses and then converted to binary strings.The machine’s state is a binary string of length k : if there is an impulse recordedon the i th electrode, there is a ‘1’ in the i th position of the string, and ‘0’ otherwise.We present a design of the machine and analyse its state transition graphs. Weenvisage that actin droplet machines could form an elementary processor of futuremassive parallel computers made from biopolymers. Keywords: actin network, computing, waves, logical gates, finite state machine,automata
Actin is a protein presented in forms of monomeric, globular actin (G-actin) and fila-mentous actin (F-actin) [34, 18, 38]. G-actin polymerises into filamentous actin forminga double helical structure [35, 19, 13]. The filaments can be further arranged into bun-dles by various different mechanisms such as crowding effects, cross-linking or counterioncondensation [46, 45, 7, 29, 30, 37, 36, 14, 15]. The bundles are conductive to travellinglocalisations — defects, ionic waves, solitons [41, 42, 43, 40, 22, 44, 25, 27, 26, 17]. By in-terpreting presence or absence of a travelling localisation at a given site of the network at agiven time step, we can implement a logical function. This approach was comprehensivelydeveloped and successfully tested on chemical systems in the framework of collision-basedcomputing [1, 16, 8, 39, 49, 2]. 1 a r X i v : . [ c s . ET ] M a y ur approach — computing with excitation waves propagating on overall ‘density’of the conductive material — has previously been presented by us in [5]. As conductivematerial we looked at networks of actin bundles which were arranged by crowding effectswithout the need of additional accessory proteins [29, 30]. We demonstrated how to dis-cover logical gates on a two-dimensional slice of the actin bundle network by representingBoolean inputs and outputs as spikes of the network activity.In the present paper we develop a novel concept and computer modelling implemen-tation of the actin network machine, which implements a mapping F : { , } k → { , } k ,where k is a number of electrodes, and ‘1’ signifies a presence of an impulse on the elec-trode and ‘0’ the absence. At a higher level, the machine acts as a finite state machine,at the lower level a structure of the mapping F is determined by interactions of impulsespropagating on the three-dimensional network of actin bundles.We also offer an alternative to a numerical integration used in [5]: an automaton modelof a three-dimensional actin network. There is a substantial body of evidence confirmingthat automaton models are sufficient and appropriate discrete tools for modelling dynam-ics of spatially extended non-linear excitable media [21, 10, 47], propagation [20], actionpotential [48, 6], electrical pulses in the heart [28, 9, 33]. A major advantage of automatais that they require less computational resources than typical numerical integration ap-proaches.The paper is structured as follows. Our modelling approach is described in detailin Sect. 2. This includes a representation of a three-dimensional actin bundle network(Subsect. 2.1), a structure of an automaton model to simulate propagation of impulses onthe actin bundle network (Subsect. 2.2), and an interface with the actin network (Sub-sect. 2.3). In Sect. 3 we analyse dependencies of a number of Boolean gates implementedin the network on an excitation threshold and refractory period. Thus, we justify theselection of these parameters for the construction of the actin machine. The actin dropletmachine is designed and analysed in Sect. 4. Section 5 discusses the results in a contextof cytoskeleton computing and outlines directions for future research. The overall approach is the following: we simulate the actin bundle network using three-dimensional arrays of finite-state machines, cellular automata. We select several domainsof the network and assign them as inputs and outputs. We represent Boolean logicvalues with spikes of electrical activity, which are schematically represented as a virtualexperiment in Fig. 1. We stimulate the network with all possible configurations of inputstrings and record spikes on the outputs. Based on the mapping of configurations of inputspikes to output spikes, we reconstruct logical functions implemented by the network. Inour design of the actin droplet machine we consider outputs recorded on all electrodes ata given time step as a binary string and then represent the actin droplet machine as afinite-state machine whose states are binary strings of a given length.
As a template for our actin droplet machine we used an actual three-dimensional actinbundle network produced in laboratory experiments with purified proteins (Fig. 2). Theunderlying experimental method was shown to reliably produce regularly spaced bundlenetworks from homogeneous filament solutions inside small isolated droplets in the absence2 ctin network E E E E E Figure 1: A scheme of a virtual experiment. The actin bundle network is shown as a three-dimensional Delaunay triangulation. Electrodes are shown by thick lines and labelled E to E . Exemplary trains of spikes are shown near the electrodes.3 Figure 1: A scheme of a virtual experiment. The actin bundle network is shown as a three-dimensional Delaunay triangulation. Electrodes are shown by thick lines and labelled E to E . Exemplary trains of spikes are shown near the electrodes.of molecular motor-driven processes or other accessory proteins [15]. These structureseffectively form very stable and long-living three-dimensional networks, which can bereadily imaged with confocal microscopy resulting in stacks of optical two-dimensionalslices (Fig. 2). Dimensions of the network are the following: size along x coordinate is225 µ m (width), along y coordinate is 222 µ m (height), along x coordinate is 112 µ m(depth), voxel width is 0.22 µ m, height 0.22 µ m and depth 4 µ m.Original image: A z = ( a ijz ) ≤ i,j ≤ n, ≤ z ≤ m , a ijz ∈ { r ijz , g ijz , b ijz } , where n = 1024, m = 30, r ijz , g ijz , b ijz are RGB values of the element at ijz , 1 ≤ r ijz , g ijz , b ijz ≤
255 wasconverted to a conductive matrix C = ( c ijz ) ≤ i,j ≤ n, ≤ z ≤ m as follows: c ijz = 1 if r ijz > g ijz >
19 and b ijz >
19. The conductive matrices are shown in Fig. 3. The 3D conductivematrix is compressed along z -axis to reduce consumption of computational resources,scenario of the non-compressed matrix will be considered in future papers. To model activity of an actin bundle network we represent it as an automaton A = (cid:104) C , Q , r, h, θ, δ (cid:105) . C ⊂ Z is a set of voxels, or a conductive matrix C defined in Sect. 2.1.Each voxel p ∈ C takes states from the set Q = { (cid:63), • , ◦} , excited ( (cid:63) ), refractory ( • ),resting ( ◦ ) and is complemented by a counter h p to handle the temporal decay of therefractory state. Following discrete time steps, each voxel p updates its state dependingon its current state and the states of its neighbourhood u ( p ) = { q ∈ C : d ( p, q ) ≤ r } ,where d ( p, q ) is an Euclidean distance between voxels p and q ; r ∈ N is a neighbourhoodradius. θ ∈ N is an excitation threshold and δ ∈ N is refractory delay. All voxels update3igure 2: Exemplary z -slices of a three-dimensional actin bundle network reconstructedas described in [15]. 4igure 3: Exemplary z -slices of ‘conductive’ geometries C selected from the three-dimensional actin bundle network shown in Fig. 2, which were reconstructed as describedin [15]. 5 i j z E . e i j z E .their states in parallel and by the same rule: p t +1 (cid:63), if ( p t = ◦ ) and ( σ ( p ) t > θ ) • , if ( p t = ◦ ) or (( p t = • ) and ( h tp > ◦ , otherwise h t +1 p = δ, if ( p t +1 = • ) and ( p t = (cid:63) ) h tp − , if ( p t +1 = • ) and ( h tp > , otherwise . Every resting ( ◦ ) voxel of C excites ( (cid:63) ) at the moment t + 1 if a number of its excitedneighbours at the moment t , σ ( p ) t = |{ q ∈ u ( p ) : q t = (cid:63) }| , exceeds a threshold θ . Anexcited voxel p t = (cid:63) takes the refractory state • at the next time step t + 1 and at thesame moment a counter of refractory state h p is set to the refractory delay δ . The counteris decremented, h t +1 p = h tp − h p becomes zero the voxel p returns to the resting state ◦ . For all results shown in thismanuscript, the neighbourhood radius was set to r = 3. Choices of θ and δ are consideredin Sect. 3. To stimulate the network and to record activity of the network we assigned several domainsof C as electrodes. We calculated a potential p tx at an electrode location c ∈ C as p c = | z : d ( c, z ) < r e and z t = + | , where d ( c, z ) is an Euclidean distance between sites x and z in 3D space. We have chosen an electrode radius of r e = 4 voxels and conductedtwo families of experiments with two configurations of electrodes.In the first family of experiments E we studied frequencies of two-input-one-outputBoolean functions implementable in the network. We used ten electrodes, their coor-dinates are listed in Tab. 2.3 and a configuration is shown in Fig. 4(a). Electrodes E representing input x and E representing input y are the input electrodes, all others areoutput electrodes representing outputs z , . . . , z . Results are presented in Sect. 3. Inthe second family of experiments E we used six electrodes (Tab. 2.3 and Fig. 4(b)). Allelectrodes were considered as inputs during stimulation and outputs during recording ofthe network activity. 6 a) (b) Figure 4: Configurations of electrodes in the three-dimensional network of actin bundlesused in (a) E and (b) E . Depth of the network is shown by level of grey. Sizes of theelectrodes are shown in perspective.Exemplary snapshots of excitation dynamics on the network are shown in Fig. 5.Domains corresponding to the two electrodes e and e (Tab. 2.3 and Fig. 4(a)) havebeen excited (Fig. 5(a)). The excitation wave fronts propagates away from e and e (Fig. 5(b)). The fronts traverse the whole breadth of the network (Fig. 5(c)). Due to thepresence of circular conductive paths in the network, the repetitive patterns of activityemerge (Fig. 5(d)). Videos of the experiments can be found in http://doi.org/10.5281/zenodo.2649293 . To map dynamics of the network onto sets of gates, we undertook the following trials ofstimulation1. fixed refractory delay δ = 20 and excitation threshold θ = 4 , , . . . , θ = 7, and refractory delay δ = 10 , , , . . . , , ρ, θ ) we counted numbers of gates or , and , xor , not-and , and-not and select . We found that in overall a total number of gates ν ( θ ) realisedby the network decreases with increase of θ (Fig. 6(a)). The function ν ( θ ) is non-linearand could be adequately described by a five degree polynomial. The function reaches itsmaximal value at θ = 7 (Fig. 6(a)). or gates are most commonly realised at θ = 11, and gates at θ = 6 and xor gates at θ = 5 as well as θ = 7 (Fig. 6(b). A number of and-not gates implemented by the network reaches its highest value at θ = 6 then drops sharplyafter θ (Fig. 6(c)). not-and gates are more common at θ = 5 , , ,
11, while select ( x )has its peak at θ = 7 and select ( y ) at θ = 8 , θ = 7 decreases with the7 a) t = 13 (b) t = 50(c) t = 200 (d) t = 500 Figure 5: Snapshots of excitation dynamics on the network. The excitation wave frontis red and the refractory tail is magenta. The excitation threshold is θ = 7 and therefractory delay is δ = 20. 8 θ (a) ν θ (b) ν θ (c) ν ρ
10 15 20 25 30 (d) ν ρ
10 15 20 25 30 (e)
Figure 6: An average number ν of gates realisable on each of the electrodes e , . . . , e depends on threshold θ of excitation when the refractory delay δ is fixed to 20 (abc) andon refractory delays δ when the threshold θ is fixed to 7 (def). (a) Number of gates ν versusthreshold θ , δ = 20. (b) Number of or (black circle), and (orange solid triangle) and xor (red blank triangle) gates, δ = 20. (c) Number of not-and (yellow blank triangle), and-not (magenta solid triangle), select ( x ) (cyan blank rhombus), select ( y ) (lightblue disc), δ = 20. (d) Number of gates ν versus delay δ , θ = 7. (e) Number of or (blackcircle), and (orange solid triangle) and xor (red blank triangle) gates, θ = 7.9igure 7: All spikes recorded at each electrode for input binary strings from 1 to 63. Therepresentation is implemented as follows. We stimulate the M with strings from { , } and represent a spike detected at time t by a black pixel at position t along horizontalaxis. A plot of each electrode e i represents a binary matrix S = ( s zt ), where 1 ≤ z ≤ ≤ t ≤ s zt = 1 if the input configuration was z and a spike was detected atmoment t , and s zt = 1 otherwise.increase of δ . Oscillations of ν ( δ ) are visible at 15 ≤ δ ≤
25 (Fig. 6(d)). The three highestvalues of ν ( δ ) are achieved at δ = 10 ,
17 and 20. Let us look now at the dependence of thenumbers of or , and and xor gates of the refractory delay δ in Fig. 6(e). The number of or gates increases with δ increasing from 10 to 15, but then drops substantially at δ = 18to reach its maximum at δ = 19. Numbers of gates and and xor behave similarly toeach other. They both have a pronounced peak at δ = 20 (Fig. 6(e)). Thus, to maximisea number of logical gates produced and their diversity we selected θ = 7 and δ = 20 forour construction of the actin droplet machine. An actin droplet machine is defined as a tuple M = (cid:104)A , k, E , S , F (cid:105) , where A is an actinnetwork automaton, defined in Sect. 2.2, k is a number of electrodes, E is a configurationof electrodes, S = { , } k , F is a state-transition function F : S → S that implements amapping between sets of all possible configurations of binary strings of length k . In theexperiments reported here k = 6.In our experiments we have chosen six electrodes, their locations are shown in Fig. 4(b)and exact coordinates in Tab, 2.3. Thus, F : { , } → { , } and the machine M has64 states. We represent the inputs and the machine states in decimal encoding. Spikesdetected in response to every input from { , } are shown in Fig. 7.Global transition graphs of M for selected inputs are shown in Fig. 8. Nodes of thegraphs are states of M , edges show transitions between the states. These directed graphsare defined as follows. There is an edge from node a to node b if there is such 1 ≤ t ≤ M t = a and M t +1 = b .Let us now define a weighted global transition graph G = (cid:104) Q , E , w (cid:105) , where Q is a set ofnodes (isomorphic to the { , } ), and E is a set of edges, and weighting function w : E → [0 ,
1] assigning a number of a unit interval to each edge. Let a, b ∈ Q and e ( a, b ) ∈ E thena normalised weight is calculated as w ( e ( a, b )) = (cid:80) i ∈ Q ,t ∈ T χ ( s t = a and s t +1 = b ) (cid:80) d ∈ Q ,t ∈ T (cid:80) Q ,t ∈ T χ ( s t = a and s t +1 = d ) , with χ a) I = 5 (b) I = 15(c) I = 31 (d) I = 63 Figure 8: State transitions of machine M for selected inputs I . A node is a decimalencoding of the M state ( e t . . . e t ). 11
24 56 891617 18 192024 283236 48 5256 21 49310 123334 40 5025 37 3853 4144 (a) (b)
Figure 9: (a) Global graph of M state transitions. Edge weights are visualised by colours:from lowest weight in orange to highest weight in blue. (b) Pruned global graph of M : onlytransitions with maximum weight for any given predecessors are shown, each node/statehas at most one outgoing edge. 12 Figure 10: Graph of g at t = 41.takes value ‘1’ when the conditions are true and ‘0’ otherwise. In words, w ( e ( a, b )) is anumber of transitions from a to b observed in the evolution of M for all possible inputsfrom Q during time interval T normalised by a total number of transition from a to allother nodes. The graph G is visualised in Fig. 9(a). Nodes which have predecessors are1–6, 8–10, 12, 16–21, 24, 25, 28, 32–34, 36–38, 40, 41, 44, 48–50, 52, 53, 56. Nodeswithout predecessors are 7, 11, 13–15, 22, 23, 26, 27, 29–31, 35, 39, 42, 43, 45–47, 51, 54,55, 57–63.Let us convert G to an acyclic non-weighted graph of more likely transitions G ∗ (cid:104) Q , E ∗ (cid:105) ,where e ( a, b ) ∈ E ∗ if w ( e ( a, b )) = max { w ( e ( a, c )) | e ( a, c ) ∈ E } . That is for each nodewe select an outgoing edge with maximum weight. The graph is a tree, see Fig. 9(b).Most states apart of 1, 2, 4, 8, 16, 20, 32 are Garden-of-Eden configurations, whichhave no predecessors. Indegrees ν () of not-Garden-of-Eden nodes are ν (20) = 1 , ν (32) =2 , ν (2) = 3 , ν (4) = 4 , ν (1) = 5 , ν (16) = 6 , ν (8) = 12. There is one fixed point, the state1, corresponding to the situation when a spike is recorded only on electrode e ; it has nosuccessors.By analysing G we can characterise a richness of M ’s responses to input stimuli. Wedefine a richness as a number of different states over all inputs, as shown in Tab. 3,and distribution in Fig. 11(a). A number of states produced increases from under fivefor beginning of M evolution and then reaches circa seven states in average. Oscillationsaround this value are seen in (Fig. 11(a)). Figure 11(b) shows a number of different nodes,generated in evolution of M , stimulated by a given input. There is below fifteen differentstates found in the evolution in responses to inputs 1 to 21 (21 corresponds to binary inputstring 010101); then a number of different nodes stay around 25. The diagram Fig. 11(c)shows how many inputs might lead to a given state/node of M . Some of the states/nodesare seen to be Garden-of-Eden configurations E (nodes without predecessors) and thuscould not be generated by stimulating M by sequences from Q − E .Assume T is a set of temporal moments when the machine responded at least to oneinput string with a non-zero state. Configurations at each transition t can be considered asoutputs representing the function g : 0 , → , . As we can see in Tab. 3, transitions at t = 41 and t = 53 correspond to the highest number of different binary strings ( e , . . . , e ).13 µ ( t ) P ( t )1 3 8, 9, 1,2 3 16, 32, 8,3 3 1, 16, 32,4 3 8, 1, 16,5 3 1, 8, 16,6 3 16, 8, 1,7 4 8, 1, 16, 4,8 4 1, 16, 8, 5,9 5 16, 1, 8, 4, 5,10 4 16, 1, 8, 4,11 5 8, 1, 16, 20, 4,12 4 1, 16, 8, 20,13 6 16, 8, 1, 17, 4, 20,14 8 8, 16, 17, 4, 20, 1, 32, 2,15 8 1, 16, 8, 4, 2, 10, 20, 32,16 6 16, 4, 8, 1, 10, 32,17 5 16, 1, 4, 8, 9,18 7 8, 16, 4, 1, 17, 10, 9,19 6 1, 8, 16, 17, 4, 10,20 8 16, 1, 8, 17, 4, 24, 10, 2,21 9 8, 16, 1, 17, 32, 24, 9, 4, 10,22 6 16, 1, 8, 32, 9, 4,23 7 8, 1, 16, 4, 32, 9, 17,24 6 1, 16, 17, 4, 32, 8,25 7 16, 1, 8, 4, 17, 32, 9,26 6 8, 16, 4, 12, 1, 17,27 6 1, 8, 16, 4, 17, 32,28 6 16, 8, 4, 1, 24, 32,29 7 8, 1, 4, 16, 12, 24, 32,30 7 16, 1, 8, 4, 17, 2, 32,31 9 8, 1, 24, 16, 12, 4, 2, 17, 32,32 7 1, 16, 8, 24, 17, 2, 40,33 9 16, 8, 1, 4, 40, 17, 24, 32, 2,34 7 8, 1, 16, 24, 40, 4, 32,35 6 1, 16, 8, 4, 24, 2,36 6 16, 8, 1, 17, 4, 32,37 7 8, 16, 17, 4, 1, 40, 2,38 7 1, 8, 16, 17, 4, 24, 2,39 7 16, 1, 8, 17, 9, 4, 2,40 7 8, 16, 4, 1, 24, 40, 2,41 10 1, 8, 16, 9, 17, 4, 18, 24, 40, 2,42 8 16, 1, 8, 4, 18, 33, 40, 24,43 9 8, 1, 16, 4, 24, 33, 18, 32, 34,44 9 1, 16, 8, 4, 17, 33, 24, 32, 40,45 7 16, 8, 4, 1, 12, 24, 34,46 7 8, 1, 16, 4, 24, 18, 34,47 5 1, 16, 8, 4, 33,48 5 16, 8, 1, 4, 17,49 8 8, 1, 16, 4, 20, 32, 24, 19,50 6 1, 16, 8, 4, 17, 32,51 8 16, 8, 1, 4, 17, 32, 41, 19,52 9 8, 16, 4, 1, 32, 33, 41, 2, 19,53 10 1, 8, 16, 4, 20, 10, 2, 41, 32, 19,54 9 16, 1, 8, 5, 17, 4, 2, 32, 19, Table 3: Fifty four state transitions of M over all possible inputs: t is a transition step, µ ( t ) is a number of different states appeared over all possible inputs, P ( t ) is a set of nodesappeared at t . 14 u m b e r o f d i ff e r e n t nod e s ν ( t ) t
10 20 30 40 50 (a) N od e s (b) I npu t s (c) Figure 11: Distributions characterising richness of M ’s responses. (a) Different states pertransitions over all inputs. Horizontal axis shows steps of M transitions. Vertical axis isa number of different states. (b) Nodes per input. Horizontal axis shows decimal valuesof input strings. Horizon axis shows a number of different states/nodes generates in theevolution of M . (c) Inputs per node.The graph corresponding to g (41) at t = 41 is shown in Fig. 10 and is not connected. Thesmall component consists of fixed point 40 (string ‘101000’) with two leafs 39 (‘100111’)and 38 (‘100110’). The largest component has a tree structure at large, with cycle 2(‘000010’) – 1 (‘000001’) as a root. Other nodes with most predecessors are 8 (‘001000’),16 (‘010000’), and 18 (‘010010’).From the transitions g (41) we can reconstruct Boolean functions realised at each ofsix electrodes (the functions are minimised and represented in a disjunctive normal form): e : f ( x , . . . , x ) = x · x · x · x + x · x · x · x · x + x · x · x · x · x + x · x · x · x · x + x · x · x · x · x + x · x · x · x · x + x · x · x · x · x · x + x · x · x · x · x · x e : f ( x , . . . , x ) = x · x · x · x + x · x · x · x · x + x · x · x · x · x + x · x · x · x · x + x · x · x · x · x + x · x · x · x · x + x · x · x · x · x · x + x · x · x · x · x · x e : f ( x , . . . , x ) = x · x · x · x + x · x · x · x · x + x · x · x · x · x + x · x · x · x · x + x · x · x · x · x + x · x · x · x · x + x · x · x · x · x · x + x · x · x · x · x · x e : f ( x , . . . , x ) = x · x · x · x + x · x · x · x · x + x · x · x · x · x + x · x · x · x · x + x · x · x · x · x + x · x · x · x · x + x · x · x · x · x · x + x · x · x · x · x · x e : f ( x , . . . , x ) = x · x · x · x + x · x · x · x · x + x · x · x · x · x + x · x · x · x · x + x · x · x · x · x + x · x · x · x · x + x · x · x · x · x · x + x · x · x · x · x · x e : f ( x , . . . , x ) = x · x · x · x + x · x · x · x · x + x · x · x · x · x + x · x · x · x · x + x · x · x · x · x + x · x · x · x · x + x · x · x · x · x · x + x · x · x · x · x · x Discussion
Early concepts of sub-cellular computing on cytoskeleton networks as microtubule au-tomata [12, 24, 11] and information processing in actin-tubulin networks [23] did notspecify what type of ‘computation’ or ‘information processing’ the cytoskeleton networkscould execute and how exactly they do this. We implemented several concrete implemen-tations of logical gates and functions on a single actin filament [32] and on an intersectionof several actin filaments [31] via collisions between solitons. We also used a reservoir-computing-like approach to discover functions on a single actin unit [3] and filament [4].Later, we realised that it might be unrealistic to expect someone to initiate and record atravelling localisations (solitons, impulses) on a single actin filament. Therefore, we de-veloped a numerical model of spikes propagating on a network of actin filament bundlesand demonstrated that such a network can implement Boolean gates [5].In present paper, we reconsidered the whole idea of the information processing onactin networks and designed an actin droplet machine. The machine is a model of athree-dimensional network, based on an experimental network developed in a droplet,which executes mapping F of a space of binary strings of length k on itself. The machineacts as a finite state machine, which behaviour at a low level is governed by localisationstravelling along the networks and interacting with each other. By focusing on a singleelement of a string, i.e. a single location of an electrode, we can reconstruct k functionswith k arguments, as we have exemplified at the end of the Sect. 4. Exact structureof each k -ary function is determined by F , which, in turn, is determined by the exactarchitecture of a three-dimensional actin network and a configuration of electrodes.Thus, potential future directions could be in detailed analysis of possible architecturesof actin networks developed in laboratory experiments and evaluation on how far an exactconfiguration of electrodes affects a structure of mapping F and corresponding distributionof functions implementable by the actin droplet machine. The ultimate goal would beto implement actin droplet machines in laboratory experiments and to cascade severalmachines into a multi-processors computing architecture. References [1] A. Adamatzky. Collision-based computing in biopolymers and their automata mod-els.
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Author contributions statements
A.A., F.H., J.S. undertook the research and wrote the manuscript.