Orig Life Evol Biosph (2014) 44:125–141 DOI 10.1007/s11084-014-9372-7 THEORETICAL MODELING
Simulations of Living Cell Origins Using a Cellular Automata Model Takeshi Ishida
Received: 24 April 2014 / Accepted: 19 September 2014 / Published online: 6 December 2014 # Springer Science+Business Media Dordrecht 2014
Abstract Understanding the generalized mechanisms of cell self-assembly is fundamental for applications in various fields, such as mass producing molecular machines in nanotechnology. Thus, the details of real cellular reaction networks and the necessary conditions for selforganized cells must be elucidated. We constructed a 2-dimensional cellular automata model to investigate the emergence of biological cell formation, which incorporated a looped membrane and a membrane-bound information system (akin to a genetic code and gene expression system). In particular, with an artificial reaction system coupled with a thermal system, the simultaneous formation of a looped membrane and an inner reaction process resulted in a more stable structure. These double structures inspired the primitive biological cell formation process from chemical evolution stage. With a model to simulate cellular self-organization in a 2-dimensional cellular automata model, 3 phenomena could be realized: (1) an inner reaction system developed as an information carrier precursor (akin to DNA); (2) a cell border emerged (akin to a cell membrane); and (3) these cell structures could divide into 2. This double-structured cell was considered to be a primary biological cell. The outer loop evolved toward a lipid bilayer membrane, and inner polymeric particles evolved toward precursor information carriers (evolved toward DNA). This model did not completely clarify all the necessary and sufficient conditions for biological cell self-organization. Further, our virtual cells remained unstable and fragile. However, the “garbage bag model” of Dyson proposed that the first living cells were deficient; thus, it would be reasonable that the earliest cells were more unstable and fragile than the simplest current unicellular organisms. Keywords Cellular automata model . Emergence of life . Synthetic life . Prebiotic chemistry
Introduction Overview Understanding the general mechanisms of cell self-assembly phenomena is considered to be fundamental for applications in various fields, such as mass production of molecular machines
T. Ishida (*) National Fisheries University, Japan, 2-7–1, Nagatahonmachi, Shimonoseki, Yamaguchi 759-6595, Japan e-mail: [email protected]
126
T. Ishida
in nanotechnology and artificial synthetic biology. Thus, not only the details of real cellular reaction networks but also the necessary conditions for self-organized cells must be elucidated. There have been numerous hypotheses and experiments to describe the formation processes of living cells, including genetic codes derived from simple chemicals and molecules like amino acids that existed on primitive earth. In the earliest studies, these included artificial syntheses of amino acids by Miller (Miller and Orgel 1974) and the coacervate hypothesis of Oparin (Oparin and Synge 1957). Several subsequent studies showed that molecules like amino acids that constitute life were abundant on the early earth. Furthermore, we now understand that membrane-like capsules, such as colloids or films of fatty acids, can readily form under the conditions of self-organization phenomena. However, the conditions required to construct a perpetuating cell system that includes both metabolic and self-replicating capabilities are unknown. This probably includes mechanisms to catalyze the synthesis of proteins from simple amino acids and DNA from nucleic acids. To determine these processes, some hypotheses were recently proposed. The replicationearly theory (Pross 2003, 2004) proposed that reproduction systems, such as those with RNA, were produced first. In particular, the “RNA world hypothesis” (Gilbert 1986; Gesteland et al. 1999) that argued that RNA was the first replication system is well known. However, it is difficult to naturally generate RNA, so there are problems with the “RNA world hypothesis.” In contrast, one paper argued that peptide nucleic acids (PNAs) (Böhler et al. 1995), which are simpler than RNA, were the first replication system. In comparison, the metabolism-early theory (Dyson 1985; Kauffman 1993) proposed that a replicator emerged from a simple chemical network through chemical evolution. However, it is unknown how a catalyst system can be formed from simple chemical reactions. Recently, Froese et al. (2014) demonstrated an integrated theory that combined these 2 theories. These theories are categorized as “chemical evolution hypotheses” (Holm 1992) that argue that the origin of life occurred on early earth, such as in deep-sea hydrothermal vents. The “Panspermia hypothesis” (Melosh 1988) is another theory that proposed that the origin of life occurred in space, which was recently recognized as Astrobiology (Gargaud 2011). Based on these theoretical considerations that underlie recent molecular biological understanding and technological progress, the area of synthetic biology has become increasingly popular, which attempts to construct a minimal cell from nonliving substances in the laboratory. Rasmussen et al. (2004) placed PNAs in small triglyceride vesicles and constructed a self-replication system under a repeated heating and cooling environment. Craig Venter (Daniel G. Gibson et al. 2010) succeeded in replicating artificial bacteria that carried an artificial genome after removing the original one (Gibson et al. 2010a, b). In addition, Pier Luigi Luisi’s group is attempting to construct a minimal cell “protocell” using various chemical substances added to vesicles (Pier 2002). Computer Model of Artificial Life Study In addition to these experimental trials, computer simulations are being used. E-CELL (Tomita et al. 1999) is a pioneering study of a cell model. E-Cell is the first study to simulate an entire metabolic network within a cell. Covert (Karr et al. 2012) is currently producing a computer model of Mycoplasma genitalium, which considers every gene. Their model considered all biological processes, including DNA replication, RNA transcription and regulation, protein synthesis, metabolism, and cell division. However, there is no current model that can reproduce the shape of a cell and reproduce the dynamics of cell division with a computer based on first principle. In addition these models are not constructed for simulation of emergence of first cell. Froese et al. (2014) attempted to
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simulate the emergence of the functions of first life, but these simulations did not include the emergence of a cell membrane or a heredity code. The study of self-reproduction phenomena using mathematics began when von Neumann theoretically proved this possibility half a century ago. Froese et al. (2014) showed a method by which a machine could reproduce itself based on the cell state and transition rules for 2dimensional square cells. However, it is difficult to apply this model to a biochemically complex reaction model. In the field of the artificial life study, many computer models are reported. However, all most of these artificial life models are too simple model to replicate real biochemistry phenomenon exactly. Actual chemical reaction system is far more complex than these artificial reaction. As shown in Fig. 1, the spatio-temporal scale of life-emergent phenomena occurs within a range that is dominated by statistical mechanics along with the Brownian motion in the world, where various proteins move and interact. Furthermore, this scale is a few orders of magnitude greater than the scale for molecular dynamics. A model to simulate the real biochemical world on this scale is comparatively undeveloped because continuum dynamics cannot be used, and molecular dynamics requires far too many calculations to be applicable. As noted above, the processes involved in assembling early living cells from simple materials are unknown, although both computer modeling and experimental research have been performed. Under the present computer performance, it is difficult to simulate whole intracellular reaction and emergence process from a molecular level. Therefore, we considered the coarse scope model, which applied the cellular automata (CA) model to simulate characteristically such that the huge molecules such as the protein moved randomly, and it is not a model to directly calculate the molecules dynamics. Although the present model is not able to express macromolecular behavior (aspect of dynamic and static electricity) directly, but uses the simple model that considered macromolecule to be one particle, it is believed that various discussion of the life emergence could be possible. Outline of a Cellular Automata Model Regarding an approach for a CA model, von Neumann studied the mathematics of selfreproduction phenomena with a CA model comprising cell-states and transition rules for 2-
Large
Phenomenon Macro System
Organ formation Tissue formation
Micro -scopic System
Cell formation Cell organelle
Simulation Numerical solution of continuous mechanics equations(FEM, FDM, etc.)
Simulation of the cell formation and the selfreproduction phenomenon with cellular automata system
Application - Formation of artificial organ - Self-organization of huge system
- Artificial synthetic of biology - Self-organization of micro machines etc.
Protein formation Brownian motion Nano System
Small
Chemical reaction
Molecular dynamics Quantum chemical calculation
Mass productive machine systems
molecular
Fig. 1 Positioning of our simulation system within the physical hierarchical structure. We used a cellular automata model to simulate the scale of the real biochemical world, in which continuum dynamics cannot be used and molecular dynamics require far too many computations to be applicable
128
T. Ishida
dimensional square cells (von Neumann 1966). However, the size of the self-reproduction machine proposed by von Neumann was huge, and it is difficult to perfectly implement this self-reproduction machine using a computer until recently. Langton (1989) developed a system comprising a simple shaped machine that could reproduce by itself by abandoning the completeness of the self-reproduction machine of Neumann. However, in the case of this simple shape, the transition rules are complex and it can only reproduce the specific simple shape. Subsequently, studies were conducted to derive transition rules based on a genetic algorithm, although it was difficult to derive any generalized rules. In addition, there are few self-reproduction models that incorporate actual 3-dimensional continuous spaces. On the other hand, early studies with the “Game of Life” by Conway and Wolfram (Wolfram 1984) involved theoretical research with 1-dimensional CA models that have been applied to various fields involving natural phenomena. Furthermore, a Lattice Gas Automaton (LGA) model was developed (Fisch et al. 1986; d’Humieres et al. 1986), which was a kind of CA model. With this model, it is possible to discretely solve the fluid dynamics problem. The results with the LGA model are theoretically equivalent to solutions for the Navier–Stokes equation. A Lattice Boltzmann Method (LMB) (McNamara and Zanetti 1988) was expanded from the LGA model. This method was applied to discrete numerical simulations of physicochemical phenomena of multiphase flow and thermal fluid flow. In addition to these applications, research on “ultradiscretization” was used to deduce CA transition rules from nonlinear equations (Tokihiro et al. 1996). As noted above, a CA model can be used to incorporate a temporal–spatial structure and to model the self-organized phenomena, which occur with biochemical phenomena. Thus, the CA system is a useful method to investigate life-emerging processes. In our previous study (Ishida 2010), we developed a model to simulate cellular selfreproduction with a 2-dimensional cellular automata model. We demonstrated that 3functions could be realized by the transition between 2 adjacent cells. (1) Formation of a border similar to a cell membrane. (2) Self-replication while maintaining carrier-containing information (information carrier). (3) Division of the cell membrane while maintaining the total structure of the cell. In that study, we demonstrated the self-reproducing capability of a shape that was similar to that of a real living cell. Subsequently, we constructed a hybrid cellular automaton model (Ishida 2011). Our model exhibited self-reproduction of a cell-like shape based on some state transition rules. To reduce the number of these transition rules, we considered both the state transition rules and the concentration diffusion in the Gray Scott (GS) model (Gray and Scott 1984), in which the selfreproduction phenomenon emerges when using certain parameters. In this hybrid model, information carriers trigger the self-reproduction phenomenon of the GS model, and a cell membrane of the CA is formed by a part of the specific concentration of the GS model. These studies suggested that it was possible to construct the emergent processes of complex phenomena like those of a living cell, using combinations of self-organizing phenomena (Turing pattern, BZ reaction, Gray Scott model, etc.) and a CA model. Research Objective In this paper, we constructed a 2-dimensional cellular automata model to investigate the processes underlying the emergence of biological cell formation, which incorporated a looped membrane and a membrane-bound information system (e.g., a genetic code and gene
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expression system). In particular, under a random artificial reaction system coupled with a thermal system, we observed that the simultaneous formation of a looped membrane and an inner reaction process resulted in a comparatively stable structure. Besides, catalysis does not engage in these formations. It is believed that these double structures imply primitive biological cell formation processes from the chemical evolution stage. To determine these cell shape emerging processes, we constructed a model for simulating cellular self-organization using a 2dimensional cellular automata model. We demonstrated that the following 3 phenomena could be realized. (1) Emergence of a cell border (cell membrane); (2) an inner reaction system developed, which was an information carrier precursor (akin to DNA); and (3) these cell structures could divide into 2. This double-structured cell is considered to be a prototype of a primary biological cell. We considered that the outer loop evolved toward a lipid bilayer membrane and the inner polymeric particles evolved toward precursor information carriers (and could evolve toward DNA). However, this study did not completely clarify all the necessary and sufficient conditions for the self-organizing processes of a biological cell. On the other hand, such a model provides very useful information on discussing a process of the life emergence. Furthermore, our virtual cells remained unstable and fragile. However, as Dyson proposed in the garbage bag model (Dyson 1999), which argued that the first living cell was deficient, we considered that it would be reasonable and appropriate to assume that the earliest cell was more unstable and fragile compared with the simplest current unicellular organisms.
Model Specifications for the CA Model In this study we used the original cellular automata model of a virtual particle system. The features of our model are indicated below. –
– – –
A 2-dimensional hexagonal grid model was used (Fig. 2). Although square grids are typically used for 2-dimensional cellular automata, we used a hexagonal grid model for 2 reasons. (1) With a square grid, the state of an automaton in the next step is determined based on the state of the grid itself and the states of the 8 adjacent sites. This increases the number of transition rules and consequently, their complexity. With a hexagonal grid, the state of the next step is determined by the state of the grid itself and the states of the 6 adjacent sites. This reduces the number of transition rules. (2) Isotropy in the horizontal/vertical and diagonal directions is maintained with a hexagonal grid but not with a square grid. We considered 30×30 grids space and the periodic boundary condition which the border grids continued to the border of the other side. We assumed some number of virtual particles located on each grid. To treat different kinds of particles, we assumed a hierarchical model. Each layer considers one kind of particle. It is difficult to consider various kinds of particles in one layer, as the layer of one field is very narrow to express various reactions. Therefore, a multilayer model is used, which allows for the existence of several particles in the same position. This hierarchical model is a pseudo-3-dimensional model, which is more flexible than a 2-dimensional one. Figure 2 outlines our hierarchical layer model.
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Fig. 2 Outline of the calculation model. A 2-dimensional multilayered hexagonal grid model was used in this study. We assumed four layers with four types of particles and a virtual temperature layer. Particle movement in each layer is controlled by transition rules
–
–
–
–
We assumed four types of particles. Particle 1: Construction factor Particle 2: Deconstruction factor Particle 3: Membrane component Particle 4: Energy factor Furthermore, each type of particle had 2 attributes. The first attribute indicated the moving/stopping conditions for the particle, and the other attribute indicated the connecting/nonconnecting conditions to an adjacent static particle. Therefore, each particle could assume 1 of 4 conditions: 1) moving particle; 2) static particle; 3) static particle connected to an adjacent particle; and 4) moving particle connected to an adjacent particle. Three transitions could occur among these particles: collision, reaction, and polymerization. These transition laws are described in the next section. Among these, laws were incorporated for heat generation and heat absorption; thus, we added one more layer for heat concentration. In this heat layer, it was assumed that heat was dissipated to the outside of the layer. At this stage of the investigation, our virtual particles are not considered to be equivalent to real chemical molecules, such as amino acids or nucleic acids. In addition, it is not a complete model of physical and chemical phenomena. However, this model is more realistic than those used in our previous studies cited above.
Simulations of Living Cell Origins
–
–
131
Three transitions could occur among these particles: collision, reaction, and polymerization. These transition laws are described in the next section. Among these, laws were incorporated for heat generation and heat absorption; thus, we added one more layer for heat concentration. In this heat layer, it was assumed that heat was dissipated to the outside of the layer. At this stage of the investigation, our virtual particles are not considered to be equivalent to real chemical molecules, such as amino acids or nucleic acids. In addition, it is not a complete model of physical and chemical phenomena. However, this model is more realistic than those used in our previous studies cited above.
Transition Rules for Our Virtual Particles Model We assumed 4 types of particles with directions indicating the particle movement direction as an attribute. A moving particle moves toward an adjacent grid during 1 time step. After moving, each particle randomly changes its direction. In this way, a Brownian-like particle motion is produced. As an initial condition, we randomly placed some particles in the grid space of each layer and randomly set the direction of each particle. The heat concentration layer was set at a virtual temperature, the central part of this layer was set at a higher temperature, and the other part was set to 0°. Furthermore, it was assumed that state 0 reflected no particle on the grid. It can be imaged that a medium, such as water, fills this space. Each grid state was renewed based on transition rules, and the state of the next step was determined by the state of the grid and the states of its 6 neighboring sites. The transition rules are given below. We have not yet determined a method that can be automatically used to derive transition rules according to a uniform law. Therefore, we constructed transition rules step-by-step with reference to the mechanisms involved in a reaction–diffusion system, like the Gray Scott model. These transition rules are indicated in Fig. 3. 1) Transition rule 1: Particle 1ðstaticÞ þ particle 4ðmovingÞ→Particle 1ðstaticÞ þ heat Transition condition: grid temperature of >0° Heat generation: temperature increases by 320° with reaction heat in the grid that occurred during this transition. 2) Transition rule 2: Particle 1ðstaticÞ þ particle 1ðmovingÞ→2 Particle 1ðstaticÞ Transition condition: grid temperature of >80° and grid temperature of 20° and grid temperature of 30°, a heat generation reaction occurred only in a high-temperature area. Under this condition, if polymerization moved to a low temperature area, the occurrence of a heat generation reaction was less likely compared with that in the standard case. Therefore, maintaining a high-temperature area was difficult and the number of extinct cases increased. Parameter change for transition rule 2: The condition for transition with grid temperature of >80° and a grid temperature of 20° and grid temperature of 40° and a grid temperature of 1/25, the movement of polymerization was faster and often jumped out of a cell membrane. In comparison, when this probability was lower, the movement of polymerization was slower and it was difficult for cell division to occur. In the case of Additional rule 3, the condition of “when the total number of particle 4 types in the grid space was 15° and a grid temperature of
Simulations of Living Cell Origins Using a Cellular Automata Model Takeshi Ishida
Received: 24 April 2014 / Accepted: 19 September 2014 / Published online: 6 December 2014 # Springer Science+Business Media Dordrecht 2014
Abstract Understanding the generalized mechanisms of cell self-assembly is fundamental for applications in various fields, such as mass producing molecular machines in nanotechnology. Thus, the details of real cellular reaction networks and the necessary conditions for selforganized cells must be elucidated. We constructed a 2-dimensional cellular automata model to investigate the emergence of biological cell formation, which incorporated a looped membrane and a membrane-bound information system (akin to a genetic code and gene expression system). In particular, with an artificial reaction system coupled with a thermal system, the simultaneous formation of a looped membrane and an inner reaction process resulted in a more stable structure. These double structures inspired the primitive biological cell formation process from chemical evolution stage. With a model to simulate cellular self-organization in a 2-dimensional cellular automata model, 3 phenomena could be realized: (1) an inner reaction system developed as an information carrier precursor (akin to DNA); (2) a cell border emerged (akin to a cell membrane); and (3) these cell structures could divide into 2. This double-structured cell was considered to be a primary biological cell. The outer loop evolved toward a lipid bilayer membrane, and inner polymeric particles evolved toward precursor information carriers (evolved toward DNA). This model did not completely clarify all the necessary and sufficient conditions for biological cell self-organization. Further, our virtual cells remained unstable and fragile. However, the “garbage bag model” of Dyson proposed that the first living cells were deficient; thus, it would be reasonable that the earliest cells were more unstable and fragile than the simplest current unicellular organisms. Keywords Cellular automata model . Emergence of life . Synthetic life . Prebiotic chemistry
Introduction Overview Understanding the general mechanisms of cell self-assembly phenomena is considered to be fundamental for applications in various fields, such as mass production of molecular machines
T. Ishida (*) National Fisheries University, Japan, 2-7–1, Nagatahonmachi, Shimonoseki, Yamaguchi 759-6595, Japan e-mail: [email protected]
126
T. Ishida
in nanotechnology and artificial synthetic biology. Thus, not only the details of real cellular reaction networks but also the necessary conditions for self-organized cells must be elucidated. There have been numerous hypotheses and experiments to describe the formation processes of living cells, including genetic codes derived from simple chemicals and molecules like amino acids that existed on primitive earth. In the earliest studies, these included artificial syntheses of amino acids by Miller (Miller and Orgel 1974) and the coacervate hypothesis of Oparin (Oparin and Synge 1957). Several subsequent studies showed that molecules like amino acids that constitute life were abundant on the early earth. Furthermore, we now understand that membrane-like capsules, such as colloids or films of fatty acids, can readily form under the conditions of self-organization phenomena. However, the conditions required to construct a perpetuating cell system that includes both metabolic and self-replicating capabilities are unknown. This probably includes mechanisms to catalyze the synthesis of proteins from simple amino acids and DNA from nucleic acids. To determine these processes, some hypotheses were recently proposed. The replicationearly theory (Pross 2003, 2004) proposed that reproduction systems, such as those with RNA, were produced first. In particular, the “RNA world hypothesis” (Gilbert 1986; Gesteland et al. 1999) that argued that RNA was the first replication system is well known. However, it is difficult to naturally generate RNA, so there are problems with the “RNA world hypothesis.” In contrast, one paper argued that peptide nucleic acids (PNAs) (Böhler et al. 1995), which are simpler than RNA, were the first replication system. In comparison, the metabolism-early theory (Dyson 1985; Kauffman 1993) proposed that a replicator emerged from a simple chemical network through chemical evolution. However, it is unknown how a catalyst system can be formed from simple chemical reactions. Recently, Froese et al. (2014) demonstrated an integrated theory that combined these 2 theories. These theories are categorized as “chemical evolution hypotheses” (Holm 1992) that argue that the origin of life occurred on early earth, such as in deep-sea hydrothermal vents. The “Panspermia hypothesis” (Melosh 1988) is another theory that proposed that the origin of life occurred in space, which was recently recognized as Astrobiology (Gargaud 2011). Based on these theoretical considerations that underlie recent molecular biological understanding and technological progress, the area of synthetic biology has become increasingly popular, which attempts to construct a minimal cell from nonliving substances in the laboratory. Rasmussen et al. (2004) placed PNAs in small triglyceride vesicles and constructed a self-replication system under a repeated heating and cooling environment. Craig Venter (Daniel G. Gibson et al. 2010) succeeded in replicating artificial bacteria that carried an artificial genome after removing the original one (Gibson et al. 2010a, b). In addition, Pier Luigi Luisi’s group is attempting to construct a minimal cell “protocell” using various chemical substances added to vesicles (Pier 2002). Computer Model of Artificial Life Study In addition to these experimental trials, computer simulations are being used. E-CELL (Tomita et al. 1999) is a pioneering study of a cell model. E-Cell is the first study to simulate an entire metabolic network within a cell. Covert (Karr et al. 2012) is currently producing a computer model of Mycoplasma genitalium, which considers every gene. Their model considered all biological processes, including DNA replication, RNA transcription and regulation, protein synthesis, metabolism, and cell division. However, there is no current model that can reproduce the shape of a cell and reproduce the dynamics of cell division with a computer based on first principle. In addition these models are not constructed for simulation of emergence of first cell. Froese et al. (2014) attempted to
Simulations of Living Cell Origins
127
simulate the emergence of the functions of first life, but these simulations did not include the emergence of a cell membrane or a heredity code. The study of self-reproduction phenomena using mathematics began when von Neumann theoretically proved this possibility half a century ago. Froese et al. (2014) showed a method by which a machine could reproduce itself based on the cell state and transition rules for 2dimensional square cells. However, it is difficult to apply this model to a biochemically complex reaction model. In the field of the artificial life study, many computer models are reported. However, all most of these artificial life models are too simple model to replicate real biochemistry phenomenon exactly. Actual chemical reaction system is far more complex than these artificial reaction. As shown in Fig. 1, the spatio-temporal scale of life-emergent phenomena occurs within a range that is dominated by statistical mechanics along with the Brownian motion in the world, where various proteins move and interact. Furthermore, this scale is a few orders of magnitude greater than the scale for molecular dynamics. A model to simulate the real biochemical world on this scale is comparatively undeveloped because continuum dynamics cannot be used, and molecular dynamics requires far too many calculations to be applicable. As noted above, the processes involved in assembling early living cells from simple materials are unknown, although both computer modeling and experimental research have been performed. Under the present computer performance, it is difficult to simulate whole intracellular reaction and emergence process from a molecular level. Therefore, we considered the coarse scope model, which applied the cellular automata (CA) model to simulate characteristically such that the huge molecules such as the protein moved randomly, and it is not a model to directly calculate the molecules dynamics. Although the present model is not able to express macromolecular behavior (aspect of dynamic and static electricity) directly, but uses the simple model that considered macromolecule to be one particle, it is believed that various discussion of the life emergence could be possible. Outline of a Cellular Automata Model Regarding an approach for a CA model, von Neumann studied the mathematics of selfreproduction phenomena with a CA model comprising cell-states and transition rules for 2-
Large
Phenomenon Macro System
Organ formation Tissue formation
Micro -scopic System
Cell formation Cell organelle
Simulation Numerical solution of continuous mechanics equations(FEM, FDM, etc.)
Simulation of the cell formation and the selfreproduction phenomenon with cellular automata system
Application - Formation of artificial organ - Self-organization of huge system
- Artificial synthetic of biology - Self-organization of micro machines etc.
Protein formation Brownian motion Nano System
Small
Chemical reaction
Molecular dynamics Quantum chemical calculation
Mass productive machine systems
molecular
Fig. 1 Positioning of our simulation system within the physical hierarchical structure. We used a cellular automata model to simulate the scale of the real biochemical world, in which continuum dynamics cannot be used and molecular dynamics require far too many computations to be applicable
128
T. Ishida
dimensional square cells (von Neumann 1966). However, the size of the self-reproduction machine proposed by von Neumann was huge, and it is difficult to perfectly implement this self-reproduction machine using a computer until recently. Langton (1989) developed a system comprising a simple shaped machine that could reproduce by itself by abandoning the completeness of the self-reproduction machine of Neumann. However, in the case of this simple shape, the transition rules are complex and it can only reproduce the specific simple shape. Subsequently, studies were conducted to derive transition rules based on a genetic algorithm, although it was difficult to derive any generalized rules. In addition, there are few self-reproduction models that incorporate actual 3-dimensional continuous spaces. On the other hand, early studies with the “Game of Life” by Conway and Wolfram (Wolfram 1984) involved theoretical research with 1-dimensional CA models that have been applied to various fields involving natural phenomena. Furthermore, a Lattice Gas Automaton (LGA) model was developed (Fisch et al. 1986; d’Humieres et al. 1986), which was a kind of CA model. With this model, it is possible to discretely solve the fluid dynamics problem. The results with the LGA model are theoretically equivalent to solutions for the Navier–Stokes equation. A Lattice Boltzmann Method (LMB) (McNamara and Zanetti 1988) was expanded from the LGA model. This method was applied to discrete numerical simulations of physicochemical phenomena of multiphase flow and thermal fluid flow. In addition to these applications, research on “ultradiscretization” was used to deduce CA transition rules from nonlinear equations (Tokihiro et al. 1996). As noted above, a CA model can be used to incorporate a temporal–spatial structure and to model the self-organized phenomena, which occur with biochemical phenomena. Thus, the CA system is a useful method to investigate life-emerging processes. In our previous study (Ishida 2010), we developed a model to simulate cellular selfreproduction with a 2-dimensional cellular automata model. We demonstrated that 3functions could be realized by the transition between 2 adjacent cells. (1) Formation of a border similar to a cell membrane. (2) Self-replication while maintaining carrier-containing information (information carrier). (3) Division of the cell membrane while maintaining the total structure of the cell. In that study, we demonstrated the self-reproducing capability of a shape that was similar to that of a real living cell. Subsequently, we constructed a hybrid cellular automaton model (Ishida 2011). Our model exhibited self-reproduction of a cell-like shape based on some state transition rules. To reduce the number of these transition rules, we considered both the state transition rules and the concentration diffusion in the Gray Scott (GS) model (Gray and Scott 1984), in which the selfreproduction phenomenon emerges when using certain parameters. In this hybrid model, information carriers trigger the self-reproduction phenomenon of the GS model, and a cell membrane of the CA is formed by a part of the specific concentration of the GS model. These studies suggested that it was possible to construct the emergent processes of complex phenomena like those of a living cell, using combinations of self-organizing phenomena (Turing pattern, BZ reaction, Gray Scott model, etc.) and a CA model. Research Objective In this paper, we constructed a 2-dimensional cellular automata model to investigate the processes underlying the emergence of biological cell formation, which incorporated a looped membrane and a membrane-bound information system (e.g., a genetic code and gene
Simulations of Living Cell Origins
129
expression system). In particular, under a random artificial reaction system coupled with a thermal system, we observed that the simultaneous formation of a looped membrane and an inner reaction process resulted in a comparatively stable structure. Besides, catalysis does not engage in these formations. It is believed that these double structures imply primitive biological cell formation processes from the chemical evolution stage. To determine these cell shape emerging processes, we constructed a model for simulating cellular self-organization using a 2dimensional cellular automata model. We demonstrated that the following 3 phenomena could be realized. (1) Emergence of a cell border (cell membrane); (2) an inner reaction system developed, which was an information carrier precursor (akin to DNA); and (3) these cell structures could divide into 2. This double-structured cell is considered to be a prototype of a primary biological cell. We considered that the outer loop evolved toward a lipid bilayer membrane and the inner polymeric particles evolved toward precursor information carriers (and could evolve toward DNA). However, this study did not completely clarify all the necessary and sufficient conditions for the self-organizing processes of a biological cell. On the other hand, such a model provides very useful information on discussing a process of the life emergence. Furthermore, our virtual cells remained unstable and fragile. However, as Dyson proposed in the garbage bag model (Dyson 1999), which argued that the first living cell was deficient, we considered that it would be reasonable and appropriate to assume that the earliest cell was more unstable and fragile compared with the simplest current unicellular organisms.
Model Specifications for the CA Model In this study we used the original cellular automata model of a virtual particle system. The features of our model are indicated below. –
– – –
A 2-dimensional hexagonal grid model was used (Fig. 2). Although square grids are typically used for 2-dimensional cellular automata, we used a hexagonal grid model for 2 reasons. (1) With a square grid, the state of an automaton in the next step is determined based on the state of the grid itself and the states of the 8 adjacent sites. This increases the number of transition rules and consequently, their complexity. With a hexagonal grid, the state of the next step is determined by the state of the grid itself and the states of the 6 adjacent sites. This reduces the number of transition rules. (2) Isotropy in the horizontal/vertical and diagonal directions is maintained with a hexagonal grid but not with a square grid. We considered 30×30 grids space and the periodic boundary condition which the border grids continued to the border of the other side. We assumed some number of virtual particles located on each grid. To treat different kinds of particles, we assumed a hierarchical model. Each layer considers one kind of particle. It is difficult to consider various kinds of particles in one layer, as the layer of one field is very narrow to express various reactions. Therefore, a multilayer model is used, which allows for the existence of several particles in the same position. This hierarchical model is a pseudo-3-dimensional model, which is more flexible than a 2-dimensional one. Figure 2 outlines our hierarchical layer model.
130
T. Ishida
Fig. 2 Outline of the calculation model. A 2-dimensional multilayered hexagonal grid model was used in this study. We assumed four layers with four types of particles and a virtual temperature layer. Particle movement in each layer is controlled by transition rules
–
–
–
–
We assumed four types of particles. Particle 1: Construction factor Particle 2: Deconstruction factor Particle 3: Membrane component Particle 4: Energy factor Furthermore, each type of particle had 2 attributes. The first attribute indicated the moving/stopping conditions for the particle, and the other attribute indicated the connecting/nonconnecting conditions to an adjacent static particle. Therefore, each particle could assume 1 of 4 conditions: 1) moving particle; 2) static particle; 3) static particle connected to an adjacent particle; and 4) moving particle connected to an adjacent particle. Three transitions could occur among these particles: collision, reaction, and polymerization. These transition laws are described in the next section. Among these, laws were incorporated for heat generation and heat absorption; thus, we added one more layer for heat concentration. In this heat layer, it was assumed that heat was dissipated to the outside of the layer. At this stage of the investigation, our virtual particles are not considered to be equivalent to real chemical molecules, such as amino acids or nucleic acids. In addition, it is not a complete model of physical and chemical phenomena. However, this model is more realistic than those used in our previous studies cited above.
Simulations of Living Cell Origins
–
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Three transitions could occur among these particles: collision, reaction, and polymerization. These transition laws are described in the next section. Among these, laws were incorporated for heat generation and heat absorption; thus, we added one more layer for heat concentration. In this heat layer, it was assumed that heat was dissipated to the outside of the layer. At this stage of the investigation, our virtual particles are not considered to be equivalent to real chemical molecules, such as amino acids or nucleic acids. In addition, it is not a complete model of physical and chemical phenomena. However, this model is more realistic than those used in our previous studies cited above.
Transition Rules for Our Virtual Particles Model We assumed 4 types of particles with directions indicating the particle movement direction as an attribute. A moving particle moves toward an adjacent grid during 1 time step. After moving, each particle randomly changes its direction. In this way, a Brownian-like particle motion is produced. As an initial condition, we randomly placed some particles in the grid space of each layer and randomly set the direction of each particle. The heat concentration layer was set at a virtual temperature, the central part of this layer was set at a higher temperature, and the other part was set to 0°. Furthermore, it was assumed that state 0 reflected no particle on the grid. It can be imaged that a medium, such as water, fills this space. Each grid state was renewed based on transition rules, and the state of the next step was determined by the state of the grid and the states of its 6 neighboring sites. The transition rules are given below. We have not yet determined a method that can be automatically used to derive transition rules according to a uniform law. Therefore, we constructed transition rules step-by-step with reference to the mechanisms involved in a reaction–diffusion system, like the Gray Scott model. These transition rules are indicated in Fig. 3. 1) Transition rule 1: Particle 1ðstaticÞ þ particle 4ðmovingÞ→Particle 1ðstaticÞ þ heat Transition condition: grid temperature of >0° Heat generation: temperature increases by 320° with reaction heat in the grid that occurred during this transition. 2) Transition rule 2: Particle 1ðstaticÞ þ particle 1ðmovingÞ→2 Particle 1ðstaticÞ Transition condition: grid temperature of >80° and grid temperature of 20° and grid temperature of 30°, a heat generation reaction occurred only in a high-temperature area. Under this condition, if polymerization moved to a low temperature area, the occurrence of a heat generation reaction was less likely compared with that in the standard case. Therefore, maintaining a high-temperature area was difficult and the number of extinct cases increased. Parameter change for transition rule 2: The condition for transition with grid temperature of >80° and a grid temperature of 20° and grid temperature of 40° and a grid temperature of 1/25, the movement of polymerization was faster and often jumped out of a cell membrane. In comparison, when this probability was lower, the movement of polymerization was slower and it was difficult for cell division to occur. In the case of Additional rule 3, the condition of “when the total number of particle 4 types in the grid space was 15° and a grid temperature of
