Toward a Computational Model of Actin Filament Networks

Andrew Schumann

University of Information Technology and Management in Rzeszow, Sucharskiego 2, Rzeszow, Poland

Keywords:

Actin, Swarm Computing, Slime Mould Computing, Networks.

Abstract:

Actin is one of the most important proteins responsible for a reaction of cells to external stimuli (stresses).

There are monomeric actin or G-actin and polimeric actin or F-actin. Monomers of G-actin are connected

into double helical ﬁlaments of F-actin by the processes of nucleation, polymerization, and depolymerization.

Filaments are of 7-8 nm in diameter. They are of several microns in length. Furthermore, ﬁlaments can be

organized as complex networks of different forms: unstable bunches (parallel unbranched ﬁlaments), trees

(branched ﬁlaments), stable bunches (cross-linked ﬁlaments). Actin ﬁlament networks can be considered a

natural computational model of cells to perform different responses to different external stimuli. So, in this

model we have inputs as different stresses and outputs as formations and destructions of ﬁlaments, on the

one hand, and as assemblies and disassemblies of actin ﬁlament networks, on the other hand. Hence, under

different external conditions we observe dynamic changes in the length of actin ﬁlaments and in the outlook

of ﬁlament networks. As we see, the main difference of actin ﬁlament networks from others including neural

networks is that the topology of actin ﬁlament networks changes in responses to dynamics of external stimuli.

For instance, a neural network is a sorted triple (N,V, w), where N is the set of neurons/processors, V is a set

of connections among neurons/processors, and w is a weight for each connection. In the case of actin ﬁlament

networks we deal with a variability of ﬁlaments/processors. Some new ﬁlaments/processors can appear in one

conditions and they can disappear in other conditions. The same situation when the computational substratum

changes during the time of computations is faced in the so-called swarm computing, e.g. in slime mould

computing. In this paper we propose a swarm computing on the medium of actin ﬁlament networks.

1 INTRODUCTION

In machine learning there have been used some bio-

logically inspired networks such as artiﬁcial neural

networks, where we have a ﬁxed number of proces-

sors involved into computations. However, elemen-

tary computational units understood as processors in

real biological networks are never ﬁxed. They are be-

ing built and rebuilt permanently in responses to dif-

ferent stresses as external conditions. One of the best

examples of these units in biological networks is pre-

sented by actin ﬁlaments. Their networks are most

important in remodelling cell conﬁgurations and in

the cell motility. The point is that actin ﬁlament net-

works are engaged in changing the cell shape, for ex-

ample in the division of one cell into two daughter

cells and in the protrusion of parts of the cells, e.g. in

the cell deformation by means of growing pseudopo-

dia during phagocytosis. Meanwhile, the actin ﬁla-

ments are being assembled and disassembled during

the time. As a consequence, we face a permanent as-

sembly and disassembly of actin ﬁlament networks.

These changes in networking are cell responses to

different stimuli. For instance, actin ﬁlament net-

works by the own reconstruction can transmit inter-

nal stresses and govern the spatial organization of the

cytoskeleton. So, these networks can provide signal

transduction pathways and make a mechanical equi-

librium of the cell and its environment.

The actin ﬁlament networks react to external

stresses. These stresses are inputs for the networks.

There are three main types of the actin ﬁlament net-

works: unstable bunches (parallel unbranched ﬁla-

ments), trees (branched ﬁlaments), stable bunches

(cross-linked ﬁlaments), see: (Furukawa et al., 1993;

Steinmetz et al., 1997). Outputs are different for

different network types. For unstable bunches and

trees, the outputs are represented by chemotaxis, cell

spreading, nerve growth-cone movement, etc. For sta-

ble bunches, the outputs have the form of mechanic

stress transduction such as (i) a tensional stress in

the distortion to the network; (ii) a curvature stress in

the deformation of the network; (iii) an orientational

stress in the deformation of the network.

290

Schumann, A.

Toward a Computational Model of Actin Filament Networks.

DOI: 10.5220/0005828902900297

In Proceedings of the 9th International Joint Conference on Biomedical Engineering Systems and Technologies (BIOSTEC 2016) - Volume 4: BIOSIGNALS, pages 290-297

ISBN: 978-989-758-170-0

Due to actin ﬁlaments, the amoeboid cell motil-

ity is possible. In robototechnics, one species of the

Mycetozoan group of the Amoebozoa, which is char-

acterized by this kind of motility, is best studied. It

is Physarum polycephalum. In the project Physarum

Chip Project: Growing Computers From Slime Mould

(Adamatzky et al., 2012) supported by FP7 we have

designed an unconventional computer on plasmodia

of Physarum polycephalum. Each plasmodium has

both an external stationary ectoplasm and an inter-

nal liquid endoplasm that moves to pseudopodia. The

amoeboid motility is classiﬁed by the following three

stages: (i) the stage of growing pseudopodia; (ii) the

stage of attaching the pseudopodium to a substrate;

(iii) the stage of dragging up the rest of the cell (Fack-

ler and Grosse, 2008; Pollard and Borisy, 2003). Such

a movement requires an oscillatory mode of contrac-

tility system where an actin ﬁlament network is be-

ing assembled and disassembled for an equilibrium

between ectoplasm and endoplasm (Furuhashi et al.,

1998). Hence, the Physarum polycephalum plasmod-

ium motility that is so intelligent and can be pro-

grammed by using chemotaxis so that we can design

computers on their media is based on some actin ﬁla-

ment network properties.

Actin ﬁlament networks are a universal mecha-

nism in the reception and further transmission of ex-

ternal stimuli/stresses in any biological organism. By

chemicals it is possible to govern an actin polymerisa-

tion and depolymerisation, i.e. the ﬁlament assembly

and disassembly. For instance, on the one hand, cy-

tochalasin B, the cell-permeable mycotoxin, strongly

inhibits the formation of actin ﬁlament networks. On

the other hand, the mycotoxin of Amanita phalloides

strongly activates the aggregation of all cell G-actin

into ﬁlaments which become Ph-actin and cannot be

depolymerised and thus this taxin avoids any dynam-

ics of cytoskeleton.

Hence, theoretically we can assume that it is pos-

sible to synthesize and then govern/program an ar-

tiﬁcial actin ﬁlament network. This network is fun-

damental in any cell reaction to stimuli. As a result,

this artiﬁcial network could be considered a biological

computer/chip as such. Its computational processes

would be primary for organic intelligence in princi-

ple. Some computational features of actin ﬁlament

network are as follows:

• We deal with a system (N

, w

), where (i) N

a non-well-founded set of processors called “ﬁl-

aments”; this set is non-well-founded, because it

is impossible to divide N

into atoms or even just

into excluded subsets n

which form a partition

of N

; in other words, processors are be-

ing redesigned permanently and they can appear

and disappear and ever change own features; (ii)

is a set of tuples {(i

, j

): i

, j

∈ N

} whose

elements are connections between ﬁlament i

and

ﬁlament j

at time step t; hence, the set V

is non-

well-founded, too, as its cardinality can change

during the time t; (iii) w

is a function from V

∗

R, where

∗

R is a set of hyperreal numbers

such that w

((

∗

j)), where

∗

i = i

, . . . and

∗

j = j

. . . , for short w

∗

, is called the

weight of the connection between ﬁlament

∗

i and

ﬁlament

∗

j at each time step t = 0, 1, 2, 3, . . . ; no-

tice that a ﬁlament

∗

i can be hidden (not present)

at actual time.

• Each ﬁlament behaves as an artiﬁcial organism

(e.g. as an artiﬁcial cell): (i) it can grow up; (ii)

its behaviour can be attracted/directed by chemo-

taxis. In cells there are ever ﬁlaments, some of

them ‘die’ and some others ‘born’.

• Each unstable actin ﬁlament network behaves as

an artiﬁcial swarm: (i) it can grow up; (ii) its be-

haviour can be attracted/directed by chemotaxis

(Van Haastert and Devreotes, 2004); (iii) there can

be a fusion of two swarms (two actin ﬁlament net-

works) into one swarm (one network); (iv) there

can be a splitting of one swarm (one network) into

two swarms (two networks). In cells there are ever

some actin ﬁlament networks.

• Each cross-linked actin ﬁlament network behaves

as a metaswarm: it reacts to mechanical external

stimuli and it can be partly reorganized.

Thus, in constructing the actin ﬁlament network

, w

) we have the following three levels of com-

putations: (i) an artiﬁcial organism (ﬁlament); (ii)

an artiﬁcial swarm (unstable bunch or tree); (iii) a

metaswarm (stable bunch). On all three levels we

face an instability of computation substratum. So, we

assume that an actin ﬁlament chip changes its con-

ﬁgurations during the computation processes. In de-

signing this chip we can appeal to swarm computing,

e.g. to slime mould computing. In the same measure

as the neural networks, the actin ﬁlament networks

can be used in pattern recognitions. So, due to the

actin ﬁlament networks any slime mould can recog-

nize its dynamic environment and occupy the pieces

of food/attractants.

In this paper we consider some basic features of

artiﬁcial actin ﬁlament networks. In Section 2 we

concentrate on computational properties of unstable

actin ﬁlament networks. In Section 3 on properties of

stable actin ﬁlament networks. All constructions of

this paper are pioneering.

Toward a Computational Model of Actin Filament Networks

291

2 ACTIN FILAMENTS AS

SWARMS

In any cell there is a huge number of actin monomers

or globular actin (the so-called G-actin) denoted by

. Monomers are involved into computations only

within actin ﬁlaments, minimally consisting of three

actin monomers. G-actin A

can assemble into double

helical ﬁlaments of 7-8 nm in diameter and of several

microns in length.

Filaments are (re)designed by the processes of nu-

cleation, polymerization, and depolymerization:

• Nucleation. For nucleation that allows monomers

to be assembled into ﬁlaments it is enough to

bound three G-actin monomers at ﬁrst. Then for

enlargement an actin ﬁlament must be polarised

with a barbed end that is plus at which monomer

addition is faster than at the pointed end that is

minus. So, actin ﬁlaments are morphologically

asymmetric with different kinetic characteristics

at two ends, and this fact helps to polymerize ﬁl-

amentous actin (the so-called F-actin) from the

head (plus end) to the tail (minus end), see please

(Holmes et al., 1990; J. Hu et al., 2007).

• Polymerization. It is an association of new ﬁl-

aments mediated by actin cross-linking proteins:

α-actinin, fascin, ﬁmbrin, and ﬁlamin. This as-

sembly is carried out on the basis of adding new

monomers at the barbed end. Meanwhile, at the

steady state, when a ﬁlament ﬁnishes to grow, the

net rate of disassembly matches the rate of assem-

bly at the plus end (Goldmann et al., 1998).

• Depolymerization. It is a dissociation of ﬁlaments

which takes place when the critical concentration

for actin polymerization is less than the dissocia-

tion constants at the two ﬁlament ends (Coluccio

and Tilney, 1983).

Thus, an actin ﬁlament consists of two ﬁlament

strands in the helical form. For each actin monomer

in both strands there are three possible states:

bound on the left, lA

; bound on the right, rA

; and

bound on both sides, bA

. Out of any ﬁlament, the

monomer A

is considered free, f A

, or turned off.

Monomers in the ﬁrst strand are distinguished by sA

•

where s ∈ {l, r, b}. Monomers in the second strand are

distinguished by sA

◦

, where s ∈ {l, r, b}.

Deﬁnitions 1. Polymerization is a process calculus

deﬁned as follows:

names ::= f A

|, lA

•

|rA

•

|bA

•

|lA

◦

|rA

◦

|bA

◦

;

&, bounding ::= if f A



&rA



, then

f A



→



i+1

and rA



→



if f A



&lA



, then

f A



→



i−1

and lA



→



if lA



&rA



, then



→



i+1

and rA



→



if bA



i+i

&lA



, then

if lA



→



j+1

, then bA



i+1

→



j+2

if rA



i+i

&lA



, then

if lA



→



j+1

, then rA



i+1

→



j+2

if rA



i+i

&bA



, then

if bA



→



j+1

, then rA



i+1

→



j+2

if f A



& f A



, then

f A



→



j+1

and f A



→



f A



→



and f A



→



i+1

where  ∈ {◦, •} and A →

B is a renaming of

monomer A by a new name B after the bounding;

P, processes ::= & | P

In this deﬁnition we have deﬁned ﬁlaments as

growing from the right side. So, a free monomer f A



can interact with another free monomer f A



and, as

a result, we can have a strand rA



j+1



or a strand



i+1



. A free monomer f A



can also interact with

a monomer lA



, and, as a result, we obtain a strand

. . . bA



j−1

. A free monomer f A



can also interact

with a monomer rA



, and, as a result, we obtain a

strand rA



j+1



. . . . A monomer rA



can associate

with a monomer lA



to give a strand . . . bA



i+1



. . .

Deﬁnitions 2. Depolymerization is a process calcu-

lus deﬁned as follows:

names ::= f A

|, lA

•

|rA

•

|bA

•

|lA

◦

|rA

◦

|bA

◦

;

&, unbounding ::= if bA



i+1

&bA



, then



i+1

→



and bA



→



if bA



i+1

&lA



, then



i+1

→



i+1

and lA



→

f A



if bA



&rA



j+1

, then



→



and rA



j+1

→

f A



j+1

if rA



i+i

&lA



, then



i+1

→

f A



i+1

and lA



→

f A



where  ∈ {◦, •} and A →

B is a renaming of

monomer A by a new name B after the unbounding;

P, processes ::= & | P

BIOSIGNALS 2016 - 9th International Conference on Bio-inspired Systems and Signal Processing

292

Hence, a monomer lA



can dissociate from a

strand . . . bA



i+1



or a strand rA



i+1



and to be-

come a free monomer f A



. A monomer rA



i+1

can dissociate from a strand rA



i+1



or a strand



i+1



. . . and to become a free monomer f A



i+1

. A

strand . . . bA



i+1



. . . can be divided into two strands

. . . lA



i+1

and rA



. . . .

Let [sA



], where s ∈ {l, r, b}, mean a value of sA



(i) [sA



] = 1 iff sA



is excited; and (ii) [sA



] = 0 iff



is not excited. A signal is transmitted in ﬁlaments

. . . bA



i+1



form the left to the right hand and the

signal transmission deﬁned as follows:

[bA

◦

] =







1, if either [bA

◦

i+1

] ∧ [bA

•

i+1

] = 1 or

[bA

◦

] ∧ [bA

•

] = 1;

0, otherwise.

[bA

•

] =







1, if either [bA

•

i+1

] ∧ [bA

◦

i+1

] = 1 or

[bA

•

] ∧ [bA

◦

] = 1;

0, otherwise.

[lA

◦

] =



1, if [bA

◦

i+1

] ∧ [bA

•

i+1

] = 1;

0, otherwise.

[lA

•

] =



1, if [bA

•

i+1

] ∧ [bA

◦

i+1

] = 1;

0, otherwise.

Each ﬁlament is instable because of the faster net

loss of G-actin at the pointed end than at the barbed

end and the faster net addition at the barbed end than

at the pointed end. If there is an equilibrium of two

rates of association and dissociation, it gives rise to a

treadmilling when there is a net ﬂow of actin subunits

through the ﬁlament (Svitkina and Borisy, 1999). The

rate of treadmilling may be altered by the inhibition

of disassembly at the pointed end, e.g., by mM phos-

phate (Coluccio and Tilney, 1983). Increasing the rate

of monomer dissociation at the pointed end, e.g., by

coﬁlin can accelerate treadmilling, also. Proteins that

control treadmilling of actin ﬁlaments and their adhe-

sions are classiﬁed as follows:

C, control ::= SeqP | CrossP | SevP | NucP,

where

• SeqP are sequestering proteins: (i) β-thymosins

which sequester G-actin to prevent spontaneous

nucleation; (ii) proﬁlin which interacts with actin

monomers to enhance nucleotide exchange (Car-

lier et al., 1994);

• CrossP are cross-linking proteins: (i) α-actinin

which cross-links the actin ﬁlaments; (ii) vin-

culin, talin, and zyxin which link the cortex to

the plasma membrane (Chhabra and Higgs, 2007;

Choi et al., 2008);

• SevP are severing proteins: (i) coﬁlin (actin de-

polymerization factor) which sever F-actin to gen-

erate more ﬁlament ends for assembly or dis-

assembly and enhances subunit dissociation; (ii)

gelsolin and the Arp2/3 complex (containing the

actin-related proteins, Arp2 and Arp3) which cap

ﬁlament ends to regulate addition or loss of actin

subunits, see: (Iwasa and Mullins, 2007);

• NucP are nucleating proteins: (i) formin, the

Arp2/3 complex which nucleate ﬁlament growth;

(ii) integrins which nucleate the formation of as-

semblies of structural and signaling proteins for

ﬁlament adhesions (Balaban et al., 2001).

Actin ﬁlaments can be organized as bunches in re-

sponse to the activation of signalling pathways by ex-

ternal stimuli. The basic processes in ﬁlament bunch-

ing are as follows:

• Anchoring. As response to external stimuli the

actin ﬁlaments anchor to membranes. Meanwhile,

ﬁlaments are attached at their plus ends so that

the ﬁlament elongation occurs at anchored ends

which can cause a membrane deformation in the

form of growing pseudopodia.

• Parallel orientation. Usually, actin ﬁlaments are

short, randomly oriented, and not bundled. Exter-

nal stimuli organize actin ﬁlaments in linear pat-

terns with orientation of ﬁlament heads towards

stimuli.

• Branching. The Arp2/3 complex anchors the

pointed end of the future daughter ﬁlament to the

existing mother ﬁlament. Then the daughter ﬁl-

ament grows up at its barbed end. As a result, a

branch appears. Continuing in the same way, a

tree is assembled. More often the Arp2/3 com-

plex nucleates the tree assembly close to the cell

membrane at the point of external stimulus.

• Cross-linking. The actin ﬁlaments are cross-

linked and bound to the cytoskeleton due to

the actin-bounding proteins: proﬁlin, the Arp2/3

complex, ﬁlamin, spectrin, and α-actinin. These

proteins transform unbundled actin ﬁlaments with

small adhesions into the bundled actin ﬁlaments

with larger adhesions. Notice that the actin cy-

toskeleton built up by cross-linkers is used then

for mechanotransduction and signal transmission.

• Adhesion. Integrins and myosin II are mainly re-

sponsible for adhesion and actin organization as

bunches. For adhesion stability talin activates in-

tegrins and links them to actin. Adhesions dis-

assemble at the back edge of the lamellipodium,

which is a region of active actin depolymerization.

Toward a Computational Model of Actin Filament Networks

293

Each actin ﬁlament is attracted by a stimulus so

that the ﬁlament grows up towards this stimulus. In

case we have many actin ﬁlaments in one bunch, they

behave as a swarm in front of stimuli attracting each

individual of that bunch. This grouping behaviour can

be represented as the following process calculus:

Deﬁnitions 3. Let P

bunch

= {n

, n

, . . . , n

} be initial

states of transitions for k ﬁlament bunches, A

bunch

, a

, . . . , a

} be a set of external stimuli (stresses)

localized at different places, V

bunch

= {r

, r

, . . . , r

}

be a set of ﬁlament trees. Then the bunch transition

system, T S

bunch

= (S

bunch

, E

bunch

, T

bunch

, I

bunch

), is de-

ﬁned in the following manner:

• σ : P

bunch

∪ A

bunch

→ S

bunch

assigning a state to

each original point of bunches as well as to each

external stimulus;

• τ : V

bunch

→ T

bunch

assigning a transition to each

ﬁlament tree attracted by one stimulus;

• ι : P

bunch

→ I

bunch

assigning an initial state to each

bunch localization.

Each event of the set of events E

bunch

is assigned to

bunch transitions in accordance with the following

process types:

• direction (the ﬁlament tree grows from one state /

localization / initial point to another state / stim-

ulus),

• fusion (the ﬁlament tree grows from different

states / localizations to the same one state / stim-

ulus),

• splitting (the ﬁlament tree grows from one state /

localization / initial point to different states / stim-

uli),

• repelling (the ﬁlament tree can dissociate).

The system T S

bunch

just deﬁned behaves like the

slime mould computer (Adamatzky et al., 2012) and

it can solve the same tasks: transporting net optimiza-

tion, pattern recognition, etc. Hence, in T S

bunch

can design reversible logic gates (Schumann, 2015).

Let us remember that in these gates the number of in-

puts and outputs is the same. Let us show how we can

implement the CNOT gate (the 2-bit controlled-NOT

gate) into ﬁlament trees. In the CNOT gate, the four

possible input bit strings are 00, 01, 10, 11 and these

are mapped into 00, 01, 11, and 10 respectively (see

Table 1).

Table 1: The CNOT gate in the matrix form.

00 01 10 11

00 1 0 0 0

01 0 1 0 0

10 0 0 0 1

11 0 0 1 0

In the unexcited state ﬁlaments are chaotically ori-

ented. Let us consider an artiﬁcial plane cell with the

four zones on the cell surface: a, b, c, d, see Figure 1.

Assume that two external stimuli are transmitted from

both fronts: ab and cd. At the same time, suppose that

the protein pool of the cell activates the polymeriza-

tion and branching of ﬁlaments only in the four pos-

sible ways pictured in Figures 1, 2, 3, 4. Then the cell

can be regarded as the CNOT gate. In this gate, me-

chanical stimuli generate a ﬁlament tree building in

accordance with proteins which control treadmilling

of actin ﬁlaments and their adhesions.

Figure 1: The ﬁrst possible state of reacting to two stimuli

(one from ab and one from cd): mapping 00 into 00, i.e. the

string [ab] is mapped into the string [cd].

Figure 2: The second possible state of reacting to two stim-

uli: mapping 01 into 01.

In Figure 1, there are no mechanical stimuli and,

as a consequence, there is no transmission of signal

from ab to cd or from cd to ab, i.e. there are no

ﬁlament trees and, therefore, we have a transmission

from 00 to 00. In Figure 2, we have one mechanical

stimulus at the zone b (or at the zone d) that implies a

tree building at the whole right side of the cell and we

have a transmission of signal from 01 to 01. In Fig-

ure 3, we have two mechanical stimuli at the zones

a and b (or one mechanical stimulus at the zone c)

that result a tree building at the zones a, b, and c and

from that we have a transmission of signal from 11

to 10 (or from 10 to 11). In Figure 3, we deal with

BIOSIGNALS 2016 - 9th International Conference on Bio-inspired Systems and Signal Processing

294

Figure 3: The third possible state of reacting to two stimuli:

mapping 11 into 10.

Figure 4: The fourth possible state of reacting to two stim-

uli: mapping 10 into 11.

one mechanical stimulus at the zone a (or with two

mechanical stimuli at the zones c and d) that causes a

tree building at the zones a, c, and d and we observe

a transmission of signal from 10 to 11 (or from 11 to

10).

3 ACTIN FILAMENTS AS

METASWARM

Highly cross-linked ﬁlaments are used in cells for

transferring the mechanical stimulus resulted by me-

chanical forces applied to cell surfaces. These forces

generate elastic stress waves which rapidly propa-

gate through actin stress ﬁbers. Such external me-

chanical stresses can imply a diffusion of actin ﬁla-

ment networks if ﬁlaments were uncross-linked semi-

dilute (Kas et al., 1995). So, the rotation around

their axis cause colliding with other ﬁlaments for

uncross-linked ﬁlament networks, but in the case of

cytoskeleton the mechanical stimulus is being trans-

mitted by ﬁlament strands. As a consequence, in

response to the physical force the stress ﬁber dis-

placement is activated in the cytoskeleton (Hotulainen

and Lappalainen, 2006; K. Hu, 2007; Ridley et al.,

2003). Some forms of that displacement which trans-

mit the external stress: ﬁber inertia, ﬁber viscoelas-

ticity, and cytosolic damping. Hence, if highly cross-

linked actin ﬁlament networks are organized in paral-

lel arrays of ﬁlaments, they become stress ﬁbers with

the following kinds of deformation which strongly

inﬂuence on transmitting signals (ben Avraham and

Tirion, 1995; Guo and Wang, 2007; Higuchi et al.,

1995; Xu et al., 1998):

• Shear deformation. In a highly cross-linked

actin ﬁlament network both sides do not change

their lengths under shear, but the diagonals are

stretched and compressed respectively. This

stretch or compression causes a large tensional

stress in the actin ﬁlament network.

• Bending deformation. It is a curvature stress that

is a result of force that is generated by the local

extension of the material on the convex side and

compression of the material on the concave side

of the bend.

• Orientation deformation. It is a mechanical stress,

when the orientation changes.

Thus, we can consider all the possible states in the

cell deformation as a form of signal transmission by

ﬁlaments. Each deformation state is caused by an ap-

propriate mechanical force. In turn, the mechanical

signal generated by the force is transmitted in differ-

ent ways according to a kind of the cell deformation.

So, the cell can be regarded as a reversible logic gate

which has the same number of inputs (mechanical

stresses) and outputs (stress transmissions by defor-

mation). The form of the cell deformation is a form

of signal transmission. Let us examine an artiﬁcial

plane cell pictured in Figure 5. Then let us concen-

trate on the four zones on the cell surface: a, b, c,

d. Each letter runs over the two values: 1 or 0. It

has the value 1 if a mechanical stress goes through

an appropriate zone. Notice that the stress transmis-

sion depends on the cytoskeleton structure and there

may be zones, where the mechanical stress cannot go

through at all. Let us suppose that the cell pictured

in Figure 5 can have only four kinds of deformation

in transmitting mechanical signals: from Figure 6 to

Figure 9. Then this cell can be regarded as the CNOT

gate.

In this gate we assume that a compression of zone

x means the value [x] = 1 and a stretching of zone x

means the value [x] = 0. Hence, in Figure 6, we deal

with the cell deformation where all four zones a, b, c,

d are stretched and, as a result, we have a transmission

from 00 to 00. In Figure 7, zones a and c are stretched

and zones b and d are compressed and, therefore, we

have a transmission from 01 to 01. It is a bending in

the right side of the cell from b to d and vice versa. In

Toward a Computational Model of Actin Filament Networks

295

Figure 5: The artiﬁcial cell considered the CNOT gate. Its

cytoskeleton is at the normal stage without external stresses.

We assume that after stresses the string [ab] is transformed

into the string [cd] and vice versa.

Figure 6: The ﬁrst possible state of the cell after one me-

chanical stress: mapping 00 into 00.

Figure 7: The second possible state of the cell after one

mechanical stress: mapping 01 into 01.

Figure 8: The third possible state of the cell after one me-

chanical stress: mapping 11 into 10.

Figure 9: The fourth possible state of the cell after one me-

chanical stress: mapping 10 into 11.

Figure 8, zones a, b, and c are compressed and zone

d is stretched and we have a transmission from 11 to

10 (or from 10 to 11). It is a bending from a, b, c to

d and vice versa. In Figure 9, zones a, c, and d are

compressed and zone b is stretched and, hence, we

have a transmission from 10 to 11 (or from 11 to 10).

It is a bending from a, c, d to b and vice versa.

Any deformation of cells is a transduction of me-

chanical stress to their neighboring cells. This allows

us to construct artiﬁcial networks uniting thousands

cells with a ﬁxed number of neighbors for each cell.

Let us suppose that for each cell with n inputs and

k outputs we have just n + k neighbor in our actin

ﬁlament network. For instance, in the CNOT gate

we deal with 2 inputs and 2 outputs, thus, we have

4 neighbors: for each zone from a to d pictured in

Figure 5 we have just one neighbor. So, the cell de-

formation of Figure 6 implies no mechanical stress

transduction for the four neighbors of the cell. The

cell deformation of Figure 7 is a mechanical stress

transduction for the two neighbors of the cell at zones

b and d. The cell deformation of Figure 8 is a me-

chanical stress transduction for the three neighbors of

the cell at zones a, b, and c. And, ﬁnally, the cell

deformation of Figure 9 is a mechanical stress trans-

duction for the three neighbors of the cell at zones a,

c, and d.

4 CONCLUSIONS

We have considered two types of actin ﬁlament net-

works as a computational substratum: branched ﬁla-

ments (Section 2) and cross-linked ﬁlaments (Section

3). In both cases we can design reversible logic gates.

In the second case we can design actin ﬁlament net-

works with thousands cells which by their deforma-

tions transmit mechanical stresses to their neighbors.

BIOSIGNALS 2016 - 9th International Conference on Bio-inspired Systems and Signal Processing

296

ACKNOWLEDGEMENTS

This research is supported by FP7-ICT-2011-8.

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