On dynamics of excitation in F-actin: automaton model

TL;DR

This paper models F-actin filaments as automata with nodes in three states, analyzing their dynamic behavior, limit cycles, and information storage capacity to understand excitation propagation.

Contribution

It introduces a novel automaton model of F-actin, exploring excitation rules, dynamic patterns, and information storage mechanisms in filamentous actin.

Findings

Distribution of transient periods analyzed

Limit cycle formation mechanisms proposed

Density of information storage evaluated

Abstract

We represent a filamentous actin molecule as a graph of finite-state machines (F-actin automaton). Each node in the graph takes three states --- resting, excited, refractory. All nodes update their states simultaneously and by the same rule, in discrete time steps. Two rules are considered: threshold rule --- a resting node is excited if it has at least one excited neighbour and narrow excitation interval rule --- a resting node is excited if it has exactly one excited neighbour. We analyse distributions of transient periods and lengths of limit cycles in evolution of F-actin automaton, propose mechanisms for formation of limit cycles and evaluate density of information storage in F-actin automata.

Tables

Table 1. Table 1: Characterisation of excitation dynamics in actin automata 𝒜 𝒜 \mathcal{A} . (abc) Dependence of the dynamic on ratios of stimulated nodes. (a)  𝒜 0 subscript 𝒜 0 {\mathcal{A}}_{0} , initially excited ratio ρ 𝜌 \rho of nodes. (b)  𝒜 1 subscript 𝒜 1 {\mathcal{A}}_{1} , initially excited ratio ρ 𝜌 \rho of nodes. (c)  𝒜 1 subscript 𝒜 1 {\mathcal{A}}_{1} , initial ratio ρ 𝜌 \rho of nodes assigned excited and refractory states at random. Data is collected in ten experiments for each value of ρ 𝜌 \rho . For ten ratios ρ 𝜌 \rho of initially stimulated nodes, ρ = 0.1 , 0.2 , … , 0.9 𝜌 0.1 0.2 … 0.9 \rho=0.1,0.2,\ldots,0.9 we calculated lengths of transient period p 𝑝 p , lengths of cycles c 𝑐 c , numbers e 𝑒 e of excited nodes in a cycle, and their standard deviations σ ​ ( p ) 𝜎 𝑝 \sigma(p) , σ ​ ( c ) 𝜎 𝑐 \sigma(c) , σ ​ ( e ) 𝜎 𝑒 \sigma(e) . Values are rounded to integer. (d) Characteristics averaged all over stimulation ratios for each rule and stimulation scenario.

$\rho$	$p$	$c$	$e$	$\sigma \left(p\right)$	$\sigma \left(c\right)$	$\sigma \left(e\right)$
10	1613	6	535	1820	1	55
20	1432	5	562	988	0	52
30	1984	7	626	1275	8	108
40	2536	14	598	3064	15	15
50	1583	13	786	610	8	206
60	2614	14	719	3322	14	191
70	2052	9	805	2236	4	207
80	1311	5	705	521	1	180
90	2850	16	651	2835	15	157

Table 2. (a) 𝒜 0 subscript 𝒜 0 {\mathcal{A}}_{0} , ( + − ) (+-) -start

$\rho$	$p$	$c$	$e$	$\sigma \left(p\right)$	$\sigma \left(c\right)$	$\sigma \left(e\right)$
10	1613	6	535	1820	1	55
20	1432	5	562	988	0	52
30	1984	7	626	1275	8	108
40	2536	14	598	3064	15	15
50	1583	13	786	610	8	206
60	2614	14	719	3322	14	191
70	2052	9	805	2236	4	207
80	1311	5	705	521	1	180
90	2850	16	651	2835	15	157

Table 3. (b) 𝒜 1 subscript 𝒜 1 {\mathcal{A}}_{1} , ( + ) (+) -start

$\rho$	$p$	$c$	$e$	$\sigma \left(p\right)$	$\sigma \left(c\right)$	$\sigma \left(e\right)$
0.1	1154	13	570	1251	12	39
0.2	1388	13	553	961	12	50
0.3	893	11	575	362	10	8
0.4	920	24	590	487	11	51
0.5	996	16	575	832	13	11
0.6	746	16	594	238	12	24
0.7	891	16	625	455	11	77
0.8	1354	16	639	408	12	109
0.9	1729	15	577	1368	13	13

Table 4. (c) 𝒜 1 subscript 𝒜 1 {\mathcal{A}}_{1} , ( + − ) (+-) -start

$\rho$	$p$	$c$	$e$	$\sigma \left(p\right)$	$\sigma \left(c\right)$	$\sigma \left(e\right)$
0.1	893	118	496	934	355	172
0.2	2328	5	583	1525	0	6
0.3	1953	13	599	1791	12	16
0.4	1957	8	591	2400	8	5
0.5	1785	14	636	2567	12	130
0.6	976	6	706	345	3	179
0.7	1342	11	709	464	13	175
0.8	1170	13	625	620	14	109
0.9	2599	5	593	2452	1	14

	${𝒜}_0$		${𝒜}_1$
	$(+)$-start	$(+-)$-start	$(+)$-start	$(+-)$-start
$p$	840	1997	1118	1667
$c$	1	10	15	21
$e$	0	665	588	615

Full text

actin; excitation; automata; dynamics; computation

I Introduction

Actin is a protein presented in all eukaryotic cells in the forms of globular actin (G-actin) and filamentous actin (F-actin) straub1943actin ; korn1982actin ; szent2004early . G-actin, polymerises in double helix of filamentous actin (Fig. 1a), during polymerisation G-actin units slightly change their shapes and thus become F-actin units oda2009nature . The actin filaments form a skeleton of single cells, where they play key roles in motility and shape changing — together myosin — and signal transduction – together with tubulin microtubules cooper2000cell . Actin filaments networks are key components of neural synapses cingolani2008actin . The actin networks is a substrate of cell-level learning hameroff1988coherence ; rasmussen1990computational ; ludin1993neuronal ; conrad1996cross ; tuszynski1998dielectric ; priel2006dendritic ; debanne2004information ; priel2010neural ; jaeken2007new and information processing fifkova1982cytoplasmic ; kim1999role ; dillon2005actin ; cingolani2008actin . Actin filaments process information in synapses and cells, they compute in a hardwired sense, as specialised processors. If we did manage to uncover exact mechanisms of information transmission and processing in the actin filaments and establish an interface with actin filaments we would be able to make a large-scale massive-parallel nano-computing devices. In  adamatzky2015actin we proposed a model of actin filaments as two chains of one-dimensional binary-state semi-totalistic automaton arrays. We discovered automaton state transitions rules that support travelling localisations, compact clusters of non-resting states. These travelling localisations are analogous to ionic waves proposed in actin filaments tuszynski2004ionic . We speculated that a computation in actin filaments could be implemented when localisations (defects, conformation changes, ionic clouds, solitons), which represent data, collide with each other and change their velocity vectors or states. Parameters of the localisations before a collision are interpreted as values of input variables. Parameters of the localisation after the collision are values of output variables. We implemented a range of computing schemes in several families of actin filament models, from quantum automata to lattice with Morse potential siccardi2015actin ; siccardi2016boolean ; siccardi2016quantum ; siccardi2016logical ; siccardi2017models . These models considered a unit (F-actin) of an actin filament as a single, discrete, entity which can take just two or three states, and carriers of information occupied one-two actin units. These were models of rather coarse-grained computation siccardi2015actin ; siccardi2016boolean ; siccardi2016quantum ; siccardi2016logical ; siccardi2017models . To take the paradigm of computation via interaction travelling localisations at the sub-molecular level we must understand how information, presented by a perturbation of some part of an F-actin unit from its resting state, propagates in the F-actin unit. The paper is structured as follows. We define a model of F-actin automata in Sect. II. In Sect. III we study excitation dynamics of automata with a threshold excitation rule, and in Sect. IV with a rule of narrow excitation interval. Implications of our findings for designs of actin-based information storage devices are discussed in Sect. VII.

II Model

We use a structure of F-actin molecule produced using X-ray fibre diffraction intensities obtained from well oriented sols of rabbit skeletal muscles oda2009nature . The structure was calculated with resolution 3.3Å in radial direction and 5.6Å along the axis (Fig. 1b) oda2009nature . The molecular structure was converted to a non-directed graph $\mathcal{A}$, where every node represents an atom and an edge corresponds to a bond between the atoms. The graph $\mathcal{A}$ has 2961 nodes, 3025 edges. Minimum degree is 1, maximum is 4, average is 2.044 (with standard deviation 0.8) and median degree 2. There are 883 nodes with degree 1, 1009 nodes with degree 2, 1066 nodes with degree 3 and two nodes with degree 4. The graph $\mathcal{A}$ has a diameter (longest shortest path) 1130 nodes, and a mean distance (mean shortest path between any two nodes) 376, and a median distance 338.

We study dynamics of excitation in the actin graph $\mathcal{A}$ using the following models. Each node $s$ of $\mathcal{A}$ takes three states: resting ($\circ$), excited ($\oplus$) and refractory ($\ominus$). Each node $s$ has a neighbourhood $u(s)$ which is a set of nodes connected to the node $s$ by edges in $\mathcal{A}$. The nodes update their states simultaneously in discrete time by the same rule. Each step of simulated discrete time corresponds to one attosecond of real time.

A resting node $s^{t}=\circ$ excites depending on a number $\sigma_{s}^{t}$ of its excited neighbours in neighbourhood $u(s)$: $\sigma_{s}^{t}=\sum_{w\in u(s)}\{w^{t}=\oplus\}$ We consider two excitation rules. In rule ${\mathcal{A}}_{0}$ a resting node excites if it has at least one excited neighbour: $\sigma_{s}^{t}>0$. In rule ${\mathcal{A}}_{1}$ a resting node excites if it has exactly one excited neighbour: $\sigma_{s}^{t}=1$ (we do not consider rules where $\sigma_{s}^{t}>1$ because excitation there extincts quickly). Transitions from excited state to refractory state and from refractory state to resting state are unconditional, i.e. these transitions take place independently on neighbourhood state. The rules can be written as follows

${\mathcal{A}}_{0}$

${\mathcal{A}}_{1}$

$s^{t+1}=\begin{cases}\oplus,\text{ if }\sigma_{s}^{t}>0\\ \ominus,\text{ if }s^{t}=\oplus\\ \circ,\text{ otherwise }\end{cases}$

$s^{t+1}=\begin{cases}\oplus,\text{ if }\sigma_{s}^{t}=1\\ \ominus,\text{ if }s^{t}=\oplus\\ \circ,\text{ otherwise }\end{cases}$

At the beginning of each computational experiment the F-actin automaton $\mathcal{A}$ is in a global resting state, every node is assigned state $\circ$. An excitation dynamic is initiated by assigning non-resting states $\oplus$ or $\ominus$ or both to a portion of randomly selected nodes. Three stimulation scenarios are considered:

• •

Single node stimulation: a single node is selected at random and this node is assigned excited state $\oplus$

• •

$(+)$-stimulation: a specified ratio of nodes is selected at random and the selected nodes are assigned excited state $\oplus$

• •

$(+-)$-stimulation: a specified ratio of nodes is selected at random and the selected nodes are assigned either excited state $\oplus$ or refractory state $\ominus$ at random.

The automaton $\mathcal{A}$ is deterministic, therefore from any initial configuration the automaton evolves into in a limit cycle in its state space (where its configuration is repeated after a finite number of steps) or an a absorbing state (this is limit cycle length one). For the rules selected there is only one absorbing state — all nodes are in the resting state. A limit cycle is comprised of configurations where compact patterns of excitation travel along closed paths. A transient period is an interval of automaton evolution from initial configuration to entering a limit cycle or an absorbing state.

For modelling we used C and Processing, for visualisation and analyses we used R, iGraph, Chimera.

III Dynamics of ${\mathcal{A}}_{0}$

III.1 Single node stimulation

The excitation propagates as a localised pattern (Fig. 2a-f). A number of nodes excited at every single step of time varies between one and five (Fig. 3a). Sometimes an excitation pattern splits into two localisations which travel along their independent pathways. The automaton ${\mathcal{A}}_{0}$ always evolves into the absorbing state where all nodes are resting. This is because travelling localisation either cancel each other when collide or reach cul-de-sacs of their pathways. A distribution of transition periods is shown in Fig. 3b. The mean transition period is 840 time steps, median 847, minimum 2 and maximum 1131. Only 29 nodes, when stimulated lead to excitation development with a transition period between 2 and 15 steps. Stimulation of all others 2932 trigger excitation dynamics for at least 568 steps. The longest transition period is observed when the localised excitation runs along a longest shortest path where initially stimulated node is a source. The path of the longest excitation is visualised in Fig. 4a; the path matches the backbone of the actin unit.

III.2 $(+)$-stimulation

When we stimulate more than one node the automaton ${\mathcal{A}}_{0}$ exhibits several ‘epicentres’ of excitation, the patterns of excitation propagate away from their origins (Fig. 5), and populate the graph. This stage is manifested in increasing a number of excited states at each step of the evolution (Fig. 6a). Eventually, depending on distances between sources of excitation, the graph becomes filled with waves and localisations, e.g. in illustration Fig. 6a a peak is reached in 7-8 steps. Then patterns of excitation start colliding with each other. They annihilate in the results of the collisions. A number of excited nodes decreases over time (Fig. 6a). The graph returns to the totally resting state. The larger is the portion of initially excited nodes the quicker evolution halts in the resting state (Fig. 6b). The ‘quicker’ can be quantified by a polynomial function $p=4.7\cdot\rho^{-0.6}$, where $p$ is a length of transient period and $\rho$ is a ratio of initially excited nodes.

III.3 $(+-)$-stimulation

In a ’classical’ two-dimensional discrete excitable medium stimulation of the medium with an excited node neighbouring with a refractory node leads to formation of a spiral wave. Due to the spiral waves excitation can persist in a modelled medium indefinitely. F-actin automata follow this principle. When we stimulate nodes of ${\mathcal{A}}_{0}$ such that some of the nodes get excited states and some refractory states we evoke the excitation patterns. Average level of excitation over trials is proportional to a number of nodes stimulated (see row $e$ in Tab. 1a). The automaton enters a limit cycle (Fig. 7). The limit cycle’s length varies between 5 to 14 time steps (see row $c$ in Tab. 1a). Apparently, the automaton falls into longest limit cycles when nearly half of nodes are stimulated, however, due to high deviation of the results (see row $\sigma(c)$ in Tab. 1a), we would not state this as a fact. Lengths of transient periods, from stimulation to entering the limit cycle, is over a half of the number of nodes in $\mathcal{A}$.

IV Dynamics of ${\mathcal{A}}_{1}$

IV.1 Single node stimulation

When a single node is excited initially, the automaton ${\mathcal{A}}_{1}$ always evolves to a globally resting state. In sampling of seventy trials we found that average length of the transient period is 862 time steps (standard deviation 230) and median transition period is 869. The average transient period to the resting state is 22 steps longer then the one in the automaton ${\mathcal{A}}_{0}$.

IV.2 (+)-stimulation

In contrast to automaton ${\mathcal{A}}_{0}$, automaton ${\mathcal{A}}_{1}$ does not show a pronounced sensitivity to a ratio $\rho$ of initially excited nodes. Transition periods for all values of $\rho$ are grouped around 1112 (Tab. 1b). The automata always evolve to limit cycles. Cycle lengths are around 15 time steps with excitation level (number of excited nodes) of just below 600 nodes. The system shows a high degree of variability in lengths of transition periods and cycles, as manifested in large values of standard deviations $\sigma(p)$ and $\sigma(c)$. Level of excitation typically remains preserved.

IV.3 (+-)-stimulation

${\mathcal{A}}_{1}$ behaves similarly to the scenario of $(+)$-start: there are many travelling localisation, which collide and, mostly, annihilate each other. Few localisations survive by finding a cyclic path to travel: if no other localisation enters their path, the remaining localisations can cycle indefinitely. The surviving localisations are responsible for ${\mathcal{A}}_{1}$ falling into the limit cycle. Automaton starting with a mix of randomly excited and refractory states usually travels one-and-half times longer to its limit cycle than the same automaton starting only with randomly excited states (compare Tab. 1b and Tab. 1c).

V Stability of the dynamics

How does repeated stimulation affect excitation dynamics of ${\mathcal{A}}_{0}$ and ${\mathcal{A}}_{1}$? $(+)$-stimulation of ${\mathcal{A}}_{0}$ at any stage of its evolution raises level of excitation by amount equivalent to that of stimulated resting automaton (Fig. 8). Thus repeated stimulation prolongs return of the automaton to its resting state. In scenario of $(+-)$-stimulation ${\mathcal{A}}_{0}$ evolves to a limit cycle. Repeated $(+-)$-stimulation of the automaton while it is in the limit cycle causes the automaton to change its trajectory in a state space. This change is characterised by initially reduced level of excitation. Typically, excitation level drops by 100-150 nodes at the moment of stimulation. The level of excitation returns to its ‘pre-stimulation’ value in 400-500 time steps.

VI Implementation of memory

F-actin automata entering limit cycles could play a role of information storage in actin filaments. The minimal length of a limit cycle detected is 5 time steps. Thus aromatic rings could be a substrate responsible for some patterns of cycling excitation dynamics. Let an aromatic ring automaton be stimulated such that a node is assigned an excited state and one of its neighbours is assigned refractory state. The wave of excitation (comprised of one excited and one refractory states) propagates into the direction of its excited head (Fig. 10a). The excitation running along the aromatic ring can not be extinguished by stimulation of one resting node (Fig. 10bcde) or two resting nodes (Fig. 10fgh). This is because an excited node surrounded by two resting neighbours excites both resting neighbours. Thus excitation waves propagate along the ring in both direction. Therefore even if original excitation wave is cancelled by external stimulation then similar running wave will emerge. To extinguish the excitation in an aromatic ring we must externally excite all four resting nodes or force them into a refractory state.

The excited aromatic rings act as generators of excitation in F-actin automata. Let us consider an example. In Fig. 11 we see a histidine’s aromatic ring stimulated: one node is assigned excited state and its neighbour refractory state. The wave of excitation travels along the ring clockwise (Fig. 12abc). When excitation reaches a node linked to the rest of the graph the excitation propagates along the ‘bridge’ (Fig. 12d). The excitation then propagates further inside the graph (Fig. 12ef) splitting into two compact excitation patterns at the junction (Fig. 12gh). The overall pattern of excitation in ${\mathcal{A}}_{0}$ recorded at 90th step of evolution is shown in Fig. 13.

VII Discussion

Automaton model of F-actin unit is a fast prototyping tool for studying dynamics of excitation in actin filaments allowing for controlled propagation of localisations at atomic level. Two rules of excitation were analysed. First rule states that a resting node is excited if it has at least one excited neighbour (${\mathcal{A}}_{0}$): this is a classical threshold excitation rule. Second rule states that a resting node is excited if it has exactly one excited neighbour (${\mathcal{A}}_{1}$): this may be seen as a rule of non-linear excitation because only narrow band of local activity triggers excitation in the node. We did not consider other ranges of thresholds or excitation intervals, because they always lead to extinction of excitation at the very beginning of the evolution. Both rules support travelling patterns of excitation. Automata ${\mathcal{A}}_{0}$ show longer transient periods, smaller limit cycles and larger average levels of excitation than automata ${\mathcal{A}}_{1}$ (Tab. 1d). When a resting automaton ${\mathcal{A}}_{0}$ is stimulated by external excitation of some nodes the excitation patterns spread all over the automaton graph but then activity declines to a global resting state. Stimulation of actin automata with a mix of excited and refractory states leads to excitation dynamics with longer transient periods and formation of repeated patterns of excitation, analogous to oscillatory structures. The limit cycles are stable: an automaton subjected to repeated stimulation always slide back to its pre-stimulation activity level.

Due to substantial noise-tolerance of excitation waves propagating in aromatic rings, the rings could be seen as memory devices in a hypothetical actin computer. Assume excited aromatic ring represents one bit. To write a bit we excite one node and inhibit (force into a refractory state) one of its neighbours. To erase a bit we must excite or inhibit all resting nodes. An F-actin unit contains 40 rings (8 of histidine, 12 of phenylalanine, 4 of tryptophane, and 16 tyrosine), see configuration of the aromatic rings in Fig. 14. Thus an F-actin unit can store 40 bits. Maximum diameter of an actin filament is 8 nm moore1970three ; spudich1972regulation . An actin filament is composed of overlapping units of F-actin (Fig. 1a). Thus, diameter of a single unit is c. 4 nm. Persistent length of F-actin polymer is 17 $\mu$m gittes1993flexural , therefore we can assume that it is feasible to write $32\cdot 10^{4}$ bit on a double-strand actin filament. Given appropriate tools to read and write dynamics of excitation in predetermined parts of F-actin molecule we can assume the actin polymer offers us a memory density 64 Petabit per square inch ($6.452\cdot 10^{16}$ per square inch).