Actin Networks Voltage Circuits
Stefano Siccardi, Andrew Adamatzky Jack Tuszyński, Florian Huber, Jörg Schnauß
AActin Networks Voltage Circuits
Stefano Siccardi and Andrew Adamatzky
Unconventional Computing Laboratory, Department of Computer Science, University of the West of England, Bristol, UK
Jack Tuszy´nski
Department of Oncology, University of Alberta, Edmonton,AB T6G 1Z2, Canada; DIMEAS, Politecnico di Torino,Corso Duca degli Abruzzi 24, 10129, TO, Turin, Italy
Florian Huber
Netherlands eScience Center, Science Park 140, 1098 XG Amsterdam, The Netherlands
J¨org Schnauß
Soft Matter Physics Division, Peter Debye Institute for Soft Matter Physics,Faculty of Physics and Earth Science, Leipzig University,Germany & Fraunhofer Institute for Cell Therapy and Immunology (IZI), DNA Nanodevices Group, Leipzig, Germany (Dated: December 3, 2019)Starting with an experimentally observed networks of actin bundles, we model their network struc-ture in terms of edges and nodes. We then compute and discuss the main electrical parameters,considering the bundles as electrical wires. A set of equations describing the network is solved withseveral initial conditions. Input voltages, that can be considered as information bits, are applied ina set of points and output voltages are computed in another set of positions. We consider both anidealized situation, where point-like electrodes can be inserted in any points of the bundles and amore realistic one, where electrodes lay on a surface and have typical dimensions available in theindustry. We find that in both cases such a system can implement the main logical gates and afinite state machine.
I. INTRODUCTION
Actin filaments (AFs) and tubulin microtubules (MTs)represent the key components of cytoskeleton net-works [13]. They have been experimentally demonstratedand modelled as ionic wave conducting biowires [18, 19,23–25, 28, 31], and predicted to support conformationalsolitons [8, 9, 12], as well as orientational transitionsof dipole moments [4, 5, 30]. These propagating local-izations could carry information and transform it wheninteracting with one another. Hence, these cytoskele-ton networks could be used as nano-scale computing de-vices [3]. This idea dates back to an early concept of sub-cellular computing on cytoskeleton networks [10, 11, 21]and was later developed further in the context of in-formation processing on actin-tubulin networks of neu-ron dendrites [20]. When immersed in an ion-rich liq-uid environment, AFs can be viewed as wires that canconduct electrical currents [16, 31]. Previously we havedemonstrated computationally that these electrical cur-rents can be used to implement Boolean gates [29]. Asingle AF hence can be conceived as a computing devicein computational experiments. Its practical implemen-tation under laboratory conditions, however, would bevery challenging and likely beyond current technologicalpossibilities. For this reason, we decided to adapt ourcomputing schemes to more realistic scenarios of bundlesof AFs instead of single AF units [26, 27]. We developeda model of an actin droplet computer, where informationis represented by travelling spikes of excitation and log- ical operations are implemented at the junctions of AFbundles [1, 2, 15]. The model developed treats the actinnetwork as a continuum with propagating abstract ex-citation waves – modelled with FitzHugh-Nagumo equa-tions. The model might be phenomenologically correct,but is not able to sufficiently describe the physics of thewaves in the AF networks. Therefore, we here propose amodel more solidly rooted in the underlying physics. Weconsider the AF networks to be made of wires and theirbundles to be connected at node locations. Each bundlehas its own set of electrical parameters and facilitates themovement of ions along its length. In the following wepresent the model in full detail and discuss the resultsderived from the model.
II. THE MODEL
A detailed description of key models that we will use asfoundation in this study, can be found in [31], which wasaiming at a description of AFs. Let us highlight the as-sumptions on which the model was build. Each monomerin the filament has 11 negative excess charges. The dou-ble helical structure of the filament provides regions ofuneven charge distribution such that pockets of higherand lower charge density exist. There is a well-defineddistance, the so-called Bjerrum length λ B , beyond whichthermal fluctuations are stronger than the electrostaticattraction or repulsion between charges in solution. It isinversely proportional to temperature and directly pro- a r X i v : . [ c s . ET ] N ov portional to the ions’ valence z: λ B = ze π(cid:15)(cid:15) k B T , (1)where e is the electrical charge, (cid:15) the permittivity of thevacuum, (cid:15) the dielectric constant of the solution with AFsimmersed in (estimated similar to (cid:15) water ≈ k B Boltz-mann’s constant and T the absolute temperature. If δ isthe mean distance between charges, counterion conden-sation is expected when λ B /δ >
1. Considering the tem-perature is T=293 K and the ions are monovalent, [31]finds λ B = 7 . × − m and [16] λ B = 13 . × − mfor Ca at T=310 K. Considering actin filaments , δ isestimated to be 0 .
25 nm because assuming an averageof 370 monomers per µ m there are c. 4 e /nm. Eachmonomer behaves like an electrical circuit with induc-tive, capacitive, and resistive components. The model isbased on the transmission line analogy.The capacitance is computed considering the chargescontained in the space between two concentric cylin-ders, the inner with radius half the width of a monomer( r actin = 2 . r actin + λ B ;both cylinders are one monomer high (5.4 nm). Thus, C = 2 π(cid:15)lln ( r actin + λ B r actin ) , (2)where l ≈ . Q n = C ( V n − bV n ) . (3)The inductance is computed as: L = µN π ( r actin + λ B ) l , (4)where µ is the magnetic permeability of water and N isthe number of turns of the coil, that is the number ofwindings of the distribution of ions around the filament.It is approximated by counting how many ions can belined up along the length of a monomer as N = l/r h ,and it is supposed that the size of a typical ion is r h ≈ . × − m.The resistance is estimated considering the current be-tween the two concentric cylinders, obtaining: R = ρ ln (( r actin + λ B ) /r actin )2 πl , (5)where resistivity ρ is approximately given by: ρ = 1Λ K + c K + + Λ Na + c Na + . (6)Here, c K + and c Na + are the concentrations of sodiumand potassium ions, which were considered in previouspapers to be 0.15 M and 0.02 M, respectively; Λ K + ≈ . m ) − M − and Λ Na + ≈ . m ) − M − dependonly on the type of salts. With this formula R is com-puted and R is taken as ˜1 / R . Here, R accounts forviscosity.Figure 1 illustrates the circuit schema, where an actinmonomer unit in a filament is delimited by the dottedlines. FIG. 1. A circuit diagram for the n -th unit of an actin fila-ment. From [31]. The main equation for filaments is the following, de-rived from [31] (see Fig. 1 for the meaning of R etc.). LC d dt ( V n − bV n ) == V n +1 + V n − − V n − R C ddt ( V n − bV n ) − R C { ddt ( V n − bV n ) − ddt ( V n +1 − bV n +1 ) − ddt ( V n − − bV n − ) } (7)In [29] we used this equation to compute the evolutionof some tens of monomers in a filament. III. EXTENSION TO BUNDLE NETWORKS
In order to extend the model to bundle networks, wemust compute the suitable electrical parameters. We willconsider two possibilities:1. The filament density in the bundle is so low thateach filament stands at a distance greater thantwice λ B from all the others. In this situation, weassume that filaments do not interact and that eachone behaves as if it would not be in the bundle.2. The inner-bundle density is high enough that areascloser than λ B to the filaments intersect. In thissituation, we will conservatively assume that theinfluences of the filaments’ ions cancel out.In case 1, we can either consider the parameters for afilament and solely multiply results by the number offilaments in the bundle or compute C , L and R using theformulas for elements connected in parallel. In case 2, weonly use the bundle radius instead of the filament one inthe above formulas.Considering a Bjerrum length λ B = 7 . × − m[31], results for high density bundles at different bundlewidths are displayed in table I. Results for low densitybundles made of varying filament numbers are shown intable II. Width 200 nm 450 nm 700 nm C in pF 33.8 10 −
76 10 − − L in pH 1668 8378 20227 R in M Ω 0.173 0.077 0.049TABLE I. C , L and R for high density bundles.Filaments 1 25 50 75 C in pF 102.6 10 − − − − L in pH 1.92 7.66 10 − − − R in M Ω 5.7 0.23 0.11 0.08TABLE II. C , L and R for low density bundles. In the following we will define equations for nodes.Equation (7) applies to elements inside the bundle, sowe will use equation (8) instead, where M is the numberof edges linked to a node, and the suffix n k ranges in theset of elements of those edges that are directly connectedto the node.The term F n represents an input voltage, which is sup-posed to be non-zero only for some values of n . d dt ( V n − bV n ) = LC ×× { (cid:80) Mk =1 V n k − M × V n + F n − R C ddt ( V n − bV n ) − R C { M × ddt ( V n − bV n ) − M (cid:88) k =1 ddt ( V n k − bV n k ) } (8)These equations can represent any type of element inthe network. When M = 2, they coincide with (7) andrepresent internal elements of a bundle. When M = 1,they refer to a free terminal element of a bundle that isnot connected to anything else. We note that in [29] weused a slightly different equation for this case, namely wealways kept M = 2. The present form is more consistentwith the model and its generalization. Other values of M represent generic nodes. IV. THE NETWORK
We used a stack of low-dimensional images of the three-dimensional actin network, produced in experiments onthe formation of regularly spaced bundle networks from homogeneous filament solutions [15]. The network waschosen because it resulted from a protocol that reliablyproduces regularly spaced networks due to self-assemblyeffects [6, 15] and thus could be used in prototyping ofcytoskeleton computers. From the stack of images weextracted a network description, in terms of edges andnodes, and used it as a substrate to compute the electricalbehavior. The extracted structure takes into account themain bundles in each image, with their intersections, andan estimate of bundles that can connect nodes in twoadjacent images. It is not an accurate portrait of all thebundles, but it captures the main characteristics of thenetwork.As an illustration, in Fig. 2 the green lines drawn onthe original image represent the computed edges.
FIG. 2. Two-dimensional centers and edges of a Z-slice.
In a realistic experiment, one would have to set a sup-port with electrodes in contact with the network. In atypical configuration, we will consider a grid of 5 × µm and center-to-center distances are 30 µm . Weconsidered two situations: (1) the network is grown indroplets sitting on the glass surface which holds the elec-trodes. This is actually very close to the experimentalsetup described in [14, 15]. And (2) the array of elec-trodes is set inside the network, i.e. in the middle ofthe actin droplet along its vertical axis. This might bethe case if the networks were grown around the electrodelayer or if it were placed in the network later. Figure 3is an illustration of the network grown on top of theelectrode-containing glass.The general features of the network are listed inTab. III. FIG. 3. The grid of electrodes as it appears when the networkformed on top of the supporting glass.Parameter ValueNumber of nodes in the main connected graph 2968Number of edges in the main connected graph 7583Max number of nodes linked to a node 13Average number of nodes linked to a node 5.07Standard deviation of nodes linked to a node 2.14Average radius of edges in pixels 8.48Max radius of edge in pixels 20Min radius of edge in pixels 3Standard deviation of radii of edges in pixels 2.62Average edge radius, one pixel is 244.14 nm 2.07 µ mAverage length of edges in pixels 70.11Max length of edge in pixels 465.40Min length of edge in pixels 4.12Standard deviation of lengths of edges in pixels 41.54Average edge length, one pixel is 244.14 nm 17.12 µ mTABLE III. Parameters of the actin network used in the mod-elling. V. PRELIMINARY RESULTS
We used the simplest possible form for the input func-tions F n , that is constant functions:for 0 < t < t F n ≡ n ∈ N n ∈ N − n ∈ N − , (9)where N , N and N − are three sets of indices and t the duration of the input stimuli, which can be equal toor less than the whole experiment time.Numerical integration has been performed for openand closed bundles consisting of some tens of elementsusing various stimuli and electrical values. Examples can be found in Fig. 4 (high density open bundle) and Fig. 5(low density closed bundle). V H.D. bundle 450nm thick, total 32 elements v0v16v31
FIG. 4. Evolution of some elements of an open high density(H.D.) bundle 32 elements long, with input = 1 and -1. V L.D. bundle of 50 filaments, total 32 elements v0v16v31
FIG. 5. Evolution of some elements of a closed low density(L.D.) bundle 32 elements long, with input = 1 and -1.
These numerical experiments demonstrate that in allthe cases considered, the solutions become constant aftera transient time. Moreover, when the inputs are blockedall the solution converge to the same constant value, sothat no currents can be detected.We therefore considered constant stimuli lasting for allthe experiment time and searched for constant solutions.Input bits are defined as a pair of points of the network,so that a +1 potential (in arbitrary units) is applied atone of them and -1 at the other to encode a value 1 of thebit; when no potential is applied, the bit value is zero.Analogously, we chose pairs of points and measuredthe difference of their potential to read an output bit. Asuitable threshold has been defined to distinguish the 1and 0 values.
VI. RESULTSA. Ideal electrodes
In this section we consider ideal electrodes that (1) canbe placed in any point on the network surface or inside itand (2) are so small that they would be in contact withone element of a single bundle only. Moreover we use aslightly idealized network of spherical shape.
1. Boolean gates
We randomly chose 8 sites in the network and consid-ered them as 4 pairs to represent 4 input bits.Then we applied in turn all the possible input statesfrom (0000) to (1111) and solved the system (8). It re-duces to a linear algebraic structure and simplifies find-ing the values of the potential in the nodes . Then wechecked, for all the sets of input states that correspondto a logical input, which output bits correspond to theexpected results for a gate.For instance, to find the not gates we considered inputstate sets ((0000),(0001)), ((1000),(1001)) etc.; then welooked for all the output bits that are 1 for the first stateand 0 for the second of one of the input sets.The same procedure was used to find or , and and xor gates. We used three values for the output threshold: 2,1, and 0.5.The results of three runs are shown in Tabs. IV, V, VIand VII revealing that once an input position is chosen, itis possible to find a suitable number of edges that behaveas output for the main gate types. Run Thresh. 2 Thresh. 1 Thresh. 0.51 8266 8944 84092 3688 4660 46823 5730 7043 7455TABLE IV. Number of possible not gatesRun Thresh. 2 Thresh. 1 Thresh. 0.51 4385 8191 124942 6360 8188 113363 5835 8260 11063TABLE V. Number of possible or gatesRun Thresh. 2 Thresh. 1 Thresh. 0.51 3600 3562 25772 4506 43842 31193 4954 5076 3726TABLE VI. Number of possible and gates Run Thresh. 2 Thresh. 1 Thresh. 0.51 1543 2155 37492 584 986 17993 1009 1499 3083TABLE VII. Number of possible xor gates
2. Time estimates
We also computed the time that the network wouldneed to converge to the constant solutions, taking intoaccount the time needed for an element to discharge. Asa first estimate, we used the value R C , that is the dis-charge time of a pure RC circuit. Using the parametersfor a single filament (or for low density bundles made ofindependent filaments), we got 2 . · − sec to travelthe 3843876 elements of the whole network. Parametersfor a high density network, adjusted for the estimatedwidth of each bundle, give a time of 2 . · − sec. Inboth cases, the velocity is of the order of 4.7 m/sec, twoorders of magnitude larger than the estimate found in[16] with a different model (pure RC), but in the rangeestimated in [31] using the present one. B. Realistic electrodes
In this section we consider electrodes that could beactually available, with their supporting glass. Moreover,we use the real network dimensions (the confocal imagesare 250 µm × µm and they are spaced 110 µm in depth).We considered both the case with the network beingon top of the glass holding the electrodes, and the casewhen the electrodes are placed inside the network, alongthe middle plane of the confocal image stack.
1. Boolean gates
In the case of the network on top of the glass, we ran-domly chose 8 electrodes and considered them as 4 pairsto represent 4 input bits.We applied in turn all the possible input states from(0000) to (1111) and solved the system (8). Then wecomputed the potential differences for all the pairs ofelectrodes that were not used as input and applied a suit-able threshold to distinguish 0 and 1 bits. The thresholdwe used was the median of the differences.As 10 electrodes out of the 18 connected to the networkwere not used as input, we had 45 potential output bits.We found that, considering all the possible input andoutput bits, we have 101 not gates, 113 or gates, 46 and gates and 13 xor gates.It must be noted that the same pair of output elec-trodes may have been counted many times in these fig-ures. For instance, the potential difference of electrodesbetween 46th and 32nd electrode (electrodes in row 4 col-umn 6 and in row 3 column 2), were considered a possible not gate for all the cases listed in table VIII. Input state Output value not on bit1100 1 41101 01010 1 41011 01000 1 41001 00110 1 40111 00100 1 40101 00111 1 20011 00101 1 20001 01011 1 10011 01001 1 10001 0TABLE VIII. Possible not with a single edge
In the case of the network with electrodes placed in theinterior, we randomly chose 12 electrodes and consideredthem as 6 couples to represent 6 input bits.We applied in turn all the possible input states from(000000) to (111111) and solved the system (8). Thenwe computed the potential differences for all the pairsof electrodes that were not used as input and applieda suitable threshold to distinguish 0 and 1 bits. Thethreshold we used was the median of the differences.As 15 electrodes out of the 27 connected to the networkwere not used as input, we had 105 potential output bits.We found that, considering all the possible input andoutput bits, we have 1885 not gates, 1279 or gates, 783 and gates and 467 xor gates. VII. FINITE STATE MACHINE
The actin network implements a mapping from { , } k to { , } k , where k is a number of input bits, representedby potential difference in pairs of electrodes, as describedabove. Thus, the network can be considered as an au-tomaton or a finite state machine, A k = (cid:104){ , } , C, k, f (cid:105) .The behaviour of the automaton is governed by the func-tion f : { , } k → { , } k , k ∈ Z + . The structure of themapping f is determined by exact configuration of elec-trodes C ∈ R and geometry of the AF bundle network.
1. Using two values of k = 4 and k = 6 . The machine A represents the actin network placedonto an array of electrodes. In this case, having at ourdisposal 45 potential output bits, the number of com-binations of 4 of them is 148995. We therefore limited the study at the output positions that assume a 1 valuemore than 6 and less than 11 times for the 16 input states.In this way we found 11 output bits and computed thestate transitions for the 330 machines that one can obtainchoosing 4 out of them, k = 4.The machine A represents the actin network wherethe array of electrodes is inside the network. In thiscase, having at our disposal 105 potential output bits,the number of combinations of 6 of them is quite large.We therefore limited the study at the output positionsthat assume a 1 value 32 times for the 64 input states.In this way we found again 11 output bits and computedthe state transitions for the 462 machines that one canobtain choosing 6 out of them, k = 6.We derived structures of functions f and f , govern-ing behavior of automata A and A , as follows. Thereis potentially an infinite number of electrode configu-rations from R . Therefore, we selected 330 and 462configurations C for machines A and A , respectively,and calculated the frequencies of connections of input tooutput states, obtaining two probabilistic state machines= (cid:104){ , } , p, k, f (cid:105) , where p : { , } k { , } → [0 , p assigns a probability to each mapping from { , } k to { , } . Thus, a state transition of A k is a directed weightgraph, where weight represents a probability of the tran-sition between states of A k corresponding to nodes ofthe graph. The weighted graph can be converted to anon-weighted directed graph by removing all edges withweight less than a given threshold θ . Let us performtrimming for several thresholds with 0.1 increment.The graph remains connected for θ till 0.1 (Fig. VII 1).The graph for A is characterising for having no unreach-able nodes and several absorbing states (Fig. VII 1)a)while the graph for A has a number of unreachablenodes (Garden-of-Eden states) and less, than A , ab-sorbing states (Fig. VII 1)b).The state transition graph of A becomes disconnectedfor θ = 0 . A remainsconnected (Fig. VII 1b).Another way of converting weighted, probabilistic,state transition graphs into non-weighted graphs is by se-lecting for each node x a successor y such that the weightof the arc ( xy ) is the highest among all arcs outgoing from x . These graphs G and G of most likely transitions areshown in Fig. 8. The graph G (Fig. 8a) has two dis-connected sub-graphs, 8 Garden-of-Eden states and twoabsorbing states corresponding to (1111) and (0000); thegraph has no cycles. The graph G has 5 disconnectedsub-graphs (Fig. 8a). Two of them have only absorbingstates, corresponding to (00000) and (101010), and no cy-cles. Three of the sub-graphs do not have an absorbingstate but have cycles: (111110) → (111111) → (111110),(001111) → (001111) → (100110) → (111100) → (001111) and (000011) → (110001) → (001001) → (000010) → (100001) → (001100) → (000011).
15 121413 6710 11 8 9 42 0513 (a) (b)
FIG. 6. State transitions graphs for (a) A and (b) A , trimming threshold is θ = 0 .
1. Nodes are labelled by digitalrepresentation of 4-bit (a) and 6-bit (b) states.
VIII. DISCUSSION
By using a physical model of ionic currents on non-linear transmission network we demonstrated how a com-putation of Boolean functions can be implemented onactin networks and what type of distribution of Booleangates can be obtained. In the model, we employed ageometry of the three-dimensional actin bundle networkderived from experimental laboratory data. Our resultsmight act as a feasibility study for future experimentallaboratory prototypes of cytoskeleton computing devices.We have also derived finite state machines realizable onthe actin networks. The importance of the machines istwo-folded. First, their state transition graphs might actas unique fingerprints of actin networks formed at thedifferent experimental or physiological conditions. Sec-ond, the structure of the machines could advance studiesin computational power of actin networks in the contextof formal language recognition.The following issues could be addressed in the future.We did not consider the currents generated by ions flow-ing in the liquid medium containing the network. These could produce some amount of noise. The model couldbe improved in this aspect. We did not take into accountthe fact that differences of electrical potential along thebundles could give rise to local patterns of ion concen-trations, and these, in turn could change the bundle re-sistance R . This could be a retrofit mechanism that wepropose to study in future work.It should also be noted that similar implementationsof the model involving MTs instead of AFs are possi-ble with minor modifications. Random arrangements ofMTs in buffer solutions have been analyzed experimen-tally regarding their conductive and capacitive proper-ties [7, 17, 22]. It was observed that MTs measurably in-crease the solutions conductance compared to free tubu-lin at lower ionic concentrations while the opposite is trueat high ionic concentrations. This effect can be explainedusing the Debye-Hueckel model as due to the formation ofa counter-ionic layer whose thickness is concentration andtemperature dependent according to the Debye lengthformula. At high ionic concentrations MTs act as low-resistance cables while at low ionic concentrations theircontribution to impedance is mainly capacitive. At the
15 12 1413710 116 48 0512 (a) (b)
FIG. 7. State transitions graphs for (a) A and (b) A , trimming threshold is θ = 0 .
2. Nodes are labelled by digitalrepresentation of 4-bit (a) and 6-bit (b) states, respectively. peak value, the intrinsic conductivity of MTs has beenfound to be two orders of magnitude greater than thatof the buffer solution. On the other hand, at low pro-tein concentration, free tubulin dimers decrease solutionsconductance, and it was modelled as being due to tubu-lin attracting ionic charges and lowering their mobility.Both tubulin and MTs were found to increase capaci- tance of buffer solutions, due to their formation of ionicdouble layers. Consequently, MT networks exhibit fasci-nating electrical properties, which change as a functionof ionic concentration and pH providing an opportunityto additionally control the functional characteristics ofthe networks assembled from MTs. We intend to de-rive a complete model based on MT networks in a futurestudy. [1] Andrew Adamatzky, Florian Huber, and J¨org Schnauß.Actin droplet machine. arXiv preprint arXiv:1905.07860 ,2019.[2] Andrew Adamatzky, Florian Huber, and J¨org Schnauß.Computing on actin bundles network.
Scientific Reports ,9(1):15887, 2019.[3] Andrew Adamatzky, Jack Tuszynski, Joerg Pieper,Dan V Nicolau, Rossalia Rinalndi, Georgios Sirakoulis,Victor Erokhin, Joerg Schnauss, and David M Smith.Towards cytoskeleton computers. A proposal. In AndrewAdamatzky, Selim Akl, and Georgios Sirakoulis, editors,
From parallel to emergent computing . CRC Group/Taylor& Francis, 2019.[4] JA Brown and JA Tuszy´nski. Dipole interactions in ax-onal microtubules as a mechanism of signal propagation.
Physical Review E , 56(5):5834, 1997.[5] Michal Cifra, Jir´ı Pokorn`y, Daniel Havelka, andO Kuˇcera. Electric field generated by axial longitudinalvibration modes of microtubule.
BioSystems , 100(2):122–131, 2010.[6] Martin Glaser, J¨org Schnauß, Teresa Tschirner, B U Se-bastian Schmidt, Maximilian Moebius-Winkler, Josef A. (a)
10 2 33 3494 5 636 7378 9 1035 115112 1330143815 391617 18192021 2232 232425 2643 27 2829 3431 5660 4041 4262 44454663474850 52 535455 575859 61 (b)
FIG. 8. Graphs representing most likely transitions G (a) and G (b) of A (a) and A (b).K¨as, and David M. Smith. Self-assembly of hierarchi-cally ordered structures in dna nanotube systems. NewJournal of Physics , 18(5):055001, 2016.[7] Jose Rafael Guzman-Sepulveda, Ruitao Wu, Aarat PKalra, Maral Aminpour, Jack A Tuszynski, and AristideDogariu. Tubulin polarizability in aqueous suspensions.
ACS Omega , 4(5):9144–9149, 2019.[8] Scott Hagan, Stuart R Hameroff, and Jack A Tuszy´nski.Quantum computation in brain microtubules: Deco-herence and biological feasibility.
Physical Review E ,65(6):061901, 2002.[9] Stuart Hameroff, Alex Nip, Mitchell Porter, and JackTuszynski. Conduction pathways in microtubules, bio-logical quantum computation, and consciousness.
Biosys-tems , 64(1-3):149–168, 2002.[10] Stuart Hameroff and Steen Rasmussen. Microtubule au-tomata: Sub-neural information processing in biologicalneural networks, 1990.[11] Stuart R Hameroff and Steen Rasmussen. Informationprocessing in microtubules: Biomolecular automata andnanocomputers. In
Molecular Electronics , pages 243–257.Springer, 1989.[12] Stuart R Hameroff and Richard C Watt. Information pro-cessing in microtubules.
Journal of Theoretical Biology , 98(4):549–561, 1982.[13] Florian Huber, J¨org Schnauß, Susanne Rnicke, PhilippRauch, Karla Mller, Claus Ftterer, and Josef A. K¨as.Emergent complexity of the cytoskeleton: from single fil-aments to tissue.
Advances in Physics , 62(1):1–112, 2013.[14] Florian Huber, Dan Strehle, and Josef K¨as. Counterion-induced formation of regular actin bundle networks.
SoftMatter , 8:931–936, 2012.[15] Florian Huber, Dan Strehle, J¨org Schnauß, and JosefK¨as. Formation of regularly spaced networks as a generalfeature of actin bundle condensation by entropic forces.
New Journal of Physics , 17(4):043029, 2015.[16] J.A. J. A.Tuszynski, M.V. Sataric, D.L. Sekulic, B.M.Sataric, and Zdravkovic S. Nonlinear calcium ion wavesalong actin filaments control active hairbundle motility. bioxiv.org , doi.org/10.1101/292292, 2018.[17] Aarat Pratyaksh Kalra, Piyush Kar, Jordane Preto,Vahid Rezania, Aristide Dogariu, John D Lewis, JackTuszynski, and Karthik Shankar. Behavior of α , β tubu-lin in dmso-containing electrolytes. Nanoscale Advances ,2019.[18] A Priel and JA Tuszy´nski. A nonlinear cable-like modelof amplified ionic wave propagation along microtubules.
EPL (Europhysics Letters) , 83(6):68004, 2008. [19] Avner Priel, Jack A Tuszynski, and Horacio F Cantiello.Ionic waves propagation along the dendritic cytoskeletonas a signaling mechanism. Advances in Molecular andCell Biology , 37:163–180, 2006.[20] Avner Priel, Jack A Tuszynski, and Horacion F Cantiello.The dendritic cytoskeleton as a computational device: anhypothesis. In
The emerging physics of consciousness ,pages 293–325. Springer, 2006.[21] Steen Rasmussen, Hasnain Karampurwala, RajeshVaidyanath, Klaus S Jensen, and Stuart Hameroff. Com-putational connectionism within neurons: A model of cy-toskeletal automata subserving neural networks.
PhysicaD: Nonlinear Phenomena , 42(1-3):428–449, 1990.[22] Iara B Santelices, Douglas E Friesen, Clayton Bell,Cameron M Hough, Jack Xiao, Aarat Kalra, Piyush Kar,Holly Freedman, Vahid Rezania, John D Lewis, et al. Re-sponse to alternating electric fields of tubulin dimers andmicrotubule ensembles in electrolytic solutions.
Scientificreports , 7(1):9594, 2017.[23] MV Satari´c, DI Ili´c, N Ralevi´c, and Jack Adam Tuszyn-ski. A nonlinear model of ionic wave propagation alongmicrotubules.
European biophysics journal , 38(5):637–647, 2009.[24] MV Satari´c and BM Satari´c. Ionic pulses along cytoskele-tal protophilaments. In
Journal of Physics: ConferenceSeries , volume 329, page 012009. IOP Publishing, 2011. [25] MV Satari´c, D Sekuli´c, and M ˇZivanov. Solitonic ioniccurrents along microtubules.
Journal of Computationaland Theoretical Nanoscience , 7(11):2281–2290, 2010.[26] Jrg Schnau, Tom Golde, Carsten Schuldt, B. U. Sebas-tian Schmidt, Martin Glaser, Dan Strehle, Tina Hndler,Claus Heussinger, and Josef A. Ks. Transition froma linear to a harmonic potential in collective dynam-ics of a multifilament actin bundle.
Phys. Rev. Lett. ,116(10):108102, 2016.[27] Jrg Schnau, Tina Hndler, and Josef A. Ks. Semiflex-ible biopolymers in bundled arrangements.
Polymers ,8(8):274, 2016.[28] Dalibor L Sekuli´c, Bogdan M Satari´c, Jack A Tuszynski,and Miljko V Satari´c. Nonlinear ionic pulses along mi-crotubules.
The European Physical Journal E , 34(5):49,2011.[29] Stefano Siccardi, Jack A Tuszynski, and AndrewAdamatzky. Boolean gates on actin filaments.
PhysicsLetters A , 380(1):88–97, 2016.[30] JA Tuszy´nski, S Hameroff, MV Satari´c, B Trpisova, andMLA Nip. Ferroelectric behavior in microtubule dipolelattices: implications for information processing, signal-ing and assembly/disassembly.
Journal of Theoretical Bi-ology , 174(4):371–380, 1995.[31] JA Tuszy´nski, S Portet, JM Dixon, C Luxford, andHF Cantiello. Ionic wave propagation along actin fila-ments.