Simulation of bio-like systems and processes using movable cellular automata Volodymyr Zhikharevich 1, Kateryna Hazdiuk2 and Serhiy Ostapov3 Yuriy Fedkovych Chernivtsi National University, Kotsyubynsky str., 2, Chernivtsi, 58012 Ukraine 1 vzhikhar81@gmail.com 2 kateryna.gazdyik@gmail.com 3 sergey.ostapov@gmail.com Abstract. In this work, the method of movable cellular automata is applied to the simulation of different biological systems and processes. A significant ad- vantage of this method is the possibility of transition from a static grid to the concept of neighbours. Four examples are considered, namely the worm-like movement, the amoeba-like locomotion, the self-replication process and the biological engine. An object of simulation in all cases is divided into small parts, which are called cellular automata, that interact with each other by the rules of interactions. The basic principles of these interactions are the same, al- though the observable examples are diverse. According to the concept of neighbours, rules of cellular automata interactions were found for the con- structed models. As a result, a computer models for each example were ob- tained. The models are qualitative, not quantitative, and allow you to demon- strate the fundamental possibility of movable cellular automata for such simula- tion. Keywords: Movable cellular automata, Bio-like systems, Computer simulation, Worm-like movement, Amoeba-like locomotion, Self-replication, Biological engine. 1 Introduction Nowadays computer modelling is one of the most powerful tools of scientific research and engineering applications. It provides an opportunity not only to reproduce the results of real experiments with a fairly high degree of confidence but also to reduce financial costs and eliminate of dangerous precedents. Moreover, in some cases, an alternative to computer models does not exist at all. Simulation biological processes and systems is one of the most relevant and most promising areas of research. In contradiction to examples of physical phenomena, the mechanisms of many bioprocesses are unknown at this time. Thus, one of the tasks that are solved through simulation is the search for answers to questions that are still open. These issues, in particular, include: the path of self-organization and the evolu- tion of living matter (Origin of Life); mechanisms of synergistic cooperation in the colonies of unicellular organisms and their organization in the form of multicellular organisms, accompanied by cell differentiation (acquisition of specific functions by individual cell types within a single organism); self-organization of various signal flow in the neural subsystem of organisms and the formation of the phenomenon of consciousness. The last problem is particularly relevant in the field of artificial intelli- gence. It is obvious that all the achievements in this direction have nothing to do with the real intelligence of the human brain today, since their implementations are not based on the processes that take place in it. At the same time, the identification of key mechanisms leading to the self-organization and the evolution of consciousness will give impetus to the next scientific, technological and humanitarian revolution. In this case, it will be possible to talk not about artificial intelligence, but about the computer model of real intelligence. The tools that are used to build models of biological objects and their dynamics are quite diverse – from describing investigating processes in terms of the theory of dif- ferential equations and then finding their numerical solution, to building imitational complex models that reflect the individual components of the corresponding proc- esses or systems. Simulation by the method of movable cellular automata (MCA) can be attributed precisely to the simulation approach that has long been used to imple- ment a wide range of artificial biosimilar structures. In general, this direction of simu- lation is name “Artificial Life” (Artificial Life), its purpose is to develop general ap- proaches for constructing artificial models of living organisms. At the same time, the use of MCA as a simulating tool attracts with the possibility of creating simple mod- els of rather complex systems. The first researcher on “Artificial Life” was John von Neumann, who studied the possibilities of implementing self-replicating structures. The ability of self- reproduction (or self-replication) is a unique phenomenon that characterizes all living organisms. Neumann showed and described [1] the possibility of constructing a dis- crete deterministic self-replicating automaton, the certain analogue of the Turing ma- chine. It was capable of copying the contents of a control program and fragments of the automaton structure. However, within the framework of the Neumann and similar models, no processes of self-organization and self-complication (evolution) are observed. This is due to the fact that in such structures there is no implementation of the mechanisms that occur in real living organisms. It is well known, that biological objects are a combination of a large number of regulatory processes interconnected among them and organized in the form of extensive networks of various transformations of molecular complexes. The overwhelming majority of stimuli, that is, deviations from organic natural dynamics (if they are not destructive for the organism), are perceived by the corresponding re- ceptor complexes that initiate certain chains of transformations and lead to the forma- tion of an appropriate reaction. Thus, all regulatory processes have the nature of feed- back. According to such principles, for example, the regeneration of damaged parts of organisms, the normalization of the state of the internal environment (homeostasis), the management of the organism's behaviour, its development, etc., occurs. The net- work of regulatory processes is determined by the organism’s genotype. Moreover, the genotype is also subject to self-regulation (for example, the process of DNA re- pair). But, at the same time, the biosystems are characterized by variability that asso- ciated with genotype mutations. These mutations are not renewed by regulatory mechanisms, because they harmoniously interact with them. During mutations proc- esses, the genotype may become more complex, that leads to a complication of the phenotype of the organism and expands the network of regulatory processes. And, against the backdrop of Darwinian natural selection, this leads to the evolution of biological objects. The basis of the above-mentioned dynamics of biological structures is also a proc- ess of self-organization, which means the absence of a single control centre. In con- trast to the centralized management approach, each element of self-organizing sys- tems acts by itself, interacting with only a small number of neighbouring elements. So, it turns out is enough to streamline chaotic structures. Based on the above, the method of mobile cellular automata is chosen as a simulation method. Or rather, the asynchronous stochastic MCA method with the symbol alphabet, which corresponds to the set of states of the MSA. This method is a further development of deterministic synchronous automata [2], which are used to simulate physico-mechanical processes in solid deformable bodies. The aim of the work is not only to build a model in terms of cellular automata, but also to search for cellular automaton interactions that lead to the dynamics of a system imitating bio-like processes and structures. That, in turn, is an important both theo- retical and practical task. 2 Simulation of worm-like movement It is quite promising to use a multi-agent approach implemented on the basis of the MCA for simulation the processes of locomotion. To do this, the biosystem is decom- posed into separate constituent elements and each element is represented by an MCA with a corresponding behaviour depending on the type of element and the state of the environment, that is, on the state of the neighbouring MCA. Similar artificial organ- ism is also called an “animat” (from Artificial Animal). The advantages of using MCA in creating animats are: the possibility of a fairly flexible formation of an arbi- trary morphology of an artificial organism and its organs; an imitation of the proc- esses of differentiation, division and death of cells; a wide range of functional proper- ties of cells, what form the corresponding tissues (muscular, nervous, skin, etc.). These approaches are used by many researchers. For example, in [3], the authors describe the process of morphology formation of elementary animats by using the idea of encoding the forms of artificial organisms as an analogue of the genotype. The change in the content of the genotype is displayed as a corresponding change in the shape of the organisms. It is possible to use the operators of genetic algorithms (muta- tions, crossing, selection) to looking for various biosimilar forms. In addition, in models of animats, imitation of the spreading of signals along the diffusion nervous system is carried out, and muscle contraction take place. The whole body of the ani- mat is a collection of cells and a kind of excitable medium in which nerve impulses can propagate in the form of waves. Under the influence of these waves, the distance between cells is reduced or increased in order to imitate the contraction of the muscles of an artificial organism. In fig. 1.a) shows an example of a worm-like organism con- structed in such a way. The examples of other forms of animats (jellyfish, fish, turtles, hydras, etc.) are also given in [3]. Nevertheless, many researchers pay attention precisely on the worms. Nematode Caenorhabditis elegans was chosen as an object for computer modelling in the international project OpenWorm [4]. The aim of the project is to build a dynamic computer model of C.Elegans at the cellular level. It was expected that, knowing the complete information about the structure of the organism, including the structure of the connectome, it would be possible to create an adequate artificial analogue of a real organism with similar behaviour. The Open-Worm project has been actively developed since 2011, but the goals have not been already achieved. On our opinion, it is not necessary to strive to realize the absolute equivalent of the real analogue, as the participants of the OpenWorm project try to do. Building a com- plex detailed model requires a lot of time and computational resources and, that most importantly, it can interfere with the search for regulatory mechanisms that maintain the wholeness and durability of an artificial organism, as well as ensure the natural harmony of its components. In addition, the development of a universal method for modelling bio-like systems, which could be applied to different objects, is quite rele- vant. Regarding to this theory, the MCA method, what, in fact, was used by the authors of [3], is very promising, in our opinion. In addition, this method provides the possi- bility of a variety flexible modifications, partially described earlier. In fig. 1.c). an example of a model of the worm-like organism proposed by us is showed. This model is also based on the MCA method. To implement the model, was chosen a scheme of neighbourhood. According the scheme each cell has four neighbours for a two-dimensional model and six in the case of a three-dimensional one. In this work we consider two-dimensional model, and in this case the neighbours are divided into two groups what simulate longitudinal and transverse muscles. Fig. 1. Examples of models of worm-like organisms: a) the model given in [3]; b) the model from [4]; c) the model proposed by us. When the transverse muscles are reduced, the corresponding body fragments should be increased in length and compressed, and with the reduction of longitudinal vice versa - decrease in length and expand. The signal for muscle contraction is the state of the corresponding "nerve ending" of the neural subsystem, which is associated with the corresponding MCA. In fig. 2 shows examples of the mutual arrangement of MCA in interaction. Fig. 2.a) reflects the state of the MCA in cases of whether ab- sence of muscle contraction signals for the selected cell, or attempts to reduce both muscles (longitudinal and transverse). Fig. 2.b) reflects the state of the MCA in the case of a signal of reduction of longitudinal muscle. And Fig. 2.c) reflects the state of the MCA in the case of a signal of reduction a transverse one. Fig. 2. Examples of the mutual arrangement of MCA The essence of MCA interactions is to establish the distance between neighbours depending on the state of the corresponding efferent neuron, what also locate in the structure of the automaton. Moreover, according to the properties of incompressibil- ity, the volume of the elements of the worm body is unchanged. The model provides for the possibility of the presence in the MCA structure, in addition to the efferent neuron, also of an afferent (receptor) one what responds to a specific state (form) of the surroundings. Connections between the neural parts of the MCA can be not only within the nearest neighbours, but also between distant ones. (fig. 3). Moreover, these connections can dynamically change, drifting from one cell to another, imitating the growth of dendrites and axons of real nerve cells. The signal flows in the nervous subsystem of an artificial organism lead to a corresponding contraction of the muscles and form a variety dynamic, examples of which are shown in Fig. 1.c). Fig. 3. Simple schematic representation of the neural subsystem To describe the mathematical model of the neural network, we define the transfor- mation function f: X  Y, which forms one of three output values (0 is the absence of a signal, 1 - reduce the longitudinal muscle, 2 - reduce the transverse muscle). It de- pends from the value of the amount of input signals multiplied by weight coefficients. Each input has its own synaptic weight what gives input the effect needing for the adder function of the processing element: 0, 0   K xi wi  2 K / 3,  i 1  (1) y j  0, 2 K / 3  i 1 xi wi  4 K / 3, K  2, 4 K / 3  i 1 xi wi  2 K , K where n – number of MCA in the modelled object; K – the maximum number of con- nections for neurons; xi (xi = 0, 1, 2) – input signals; wi  [0 ... 1] – weighted coeffi- cients, what are the measure of the force of the input bonds and simulate the various synaptic forces of the biological neurons. The weights of the substantial input are increased and, conversely, the weights of the non-essential input are reduced, which determines the intensity of the input signal. It should be noted that the values of the weight coefficients and the order of the condi- tional operators in formula (1) may change and they define a characteristic of the individual MCA. 3 Simulation of amoeba-like locomotion Another interesting microorganism for research is the unicellular amoeba protea (Amoeba Proteus). It is moving in space by changing the shape of its body. At the same time certain protrusions, so-called pseudopodia ("false-feet") and posterior uro- pods, are formed. Amoeba recognizes various microscopic organisms that serve it as food. It goes away from bright light, mechanical irritation, and elevated concentra- tions of substances dissolved in water. In addition to these unconditioned reflexes, some researchers have documented the formation of conditioned reflexes in amoebas. Conditioned reflexes of a unicellular organism with a missing nervous subsystem! Many scientists, which watch about amoebas, ask themselves: how it is possible? It is also interesting to note that the size of the amoeba genome is almost 200 times the size of the human genome. This paradox indicates that there is not always a direct connection between the size of the genome and the complexity of the organism. Let us focus on the problem of modelling the movement of the amoeba, which oc- curs, as already noted, due to the growth of the membrane in the form of so-called pseudopodia. At the same time some other parts of the membrane are reduced. Changes in the membrane are due to reversible transformations "sol" ↔ "gel". These processes are guided by the cytoskeleton, whose structure is constantly changing (Fig. 4. a)). The result of complex membrane-cytoskeletal interactions is the harmonious dynamics of the movement of the microorganism. All existing models of amoeba do not have taken into account the cytoskeletal sub- system of the body, but nevertheless quite realistically reflect the processes of move- ment, including in the direction of nutrients, or a leaving from lighting (phototaxis) or from chemical stimuli (chemotaxis). One of the implementation options described in [5]. In this work, the amoeba body is a set of related spatial elements, where a gradi- ent of a certain continuous parameter is formed. The internal environment of the unicellular organism (cytoplasm) moves along the gradient. This movement, leads to the growth of pseudopodia and contraction of the amoeba body in the opposite direc- tion (Fig. 4.b)). To build a model of the described microorganism, it is also possible to present it in the form of some multicellular analogue, and therefore use the MCA approach. This, in particular, will provide an opportunity to simulate membrane-cytoskeletal interac- tions. The MCA are able to change their state by defining interaction functions with neighbours as weakly or rigidly interconnected, when modelling the internal and membrane environment of a microorganism. That means forming the corresponding properties of the cellular environment, which form as "sol" or "gel". Also, the model includes self-assembly and self-destruction of cytoskeleton fragments. Self-assembly usually occurs in centrioles. Different types of filaments (filaments or microtubules) of the cytoskeleton can be distinguished, which have different effects on the transfor- mation in the membrane (actin and myosin filaments). In cases of interacting with the appropriate types of the end of filament thread, a transition from the gel state to the sol state occurs and the membrane reduced, in the other positions the reverse process occurs and the membrane grows in the form of a pseudopodia (Fig. 4.c)). Fig. 4. Examples of models: a) an image of a real amoeba with a cytoskeleton; b) the model of amoeba-like organism described in [5]; c) the model proposed by us. We used a dynamic index array, allowing not only to identify any MCA in the sys- tem, but also containing the indices of its nearest neighbours. As parameters, we choose: the type of MCA, which determines its properties; coordinates (for simplicity and clarity let us dwell on a two-dimensional model); indices of the nearest neighbours. The simulation used a hexagonal neighbourhood pattern (M = 6), provid- ing for weak and rigid connectivity of pairs. In addition, to reduce the number of neighbours, we add zero values to the index of neighbours. The method of finding for the nearest neighbours under the conditions of their fixed value is described in [5]. In this paper, the criterion for finding neighbouring cells is the minimum distance be- tween them, and the search is carried out by introducing an additional index array. Neglect of the inertial properties of the amoeba body allows one to abandon the con- tent of the MCA velocity vector components in the index array (Fig. 5). Fig. 5. The fragment of the MCA set. The main essence of the algorithm is iterative modification of the contents of the index array. For this, an asynchronous approach is implemented, in which one of the N MCAs and one of the M of its nearest neighbours are selected in an equiprobable way. The choice of the asynchronous approach when developing the MCA algorithm is due to the fact that it allows to avoid collisions, that is, to satisfy the criterion of correctness (there will not be a single attempt to change the state of the same cell more than once at the same time t). The cellular automaton model admits six possible states of movable cellular auto- mata, which have a different purpose and functionality and are described using the alphabet A = {a1; a2; …; a6}. For the difference are used circles of radii ri, i = 1, 2, 3 (r1