Mar 16, 2001 - 3 Particle Artificial Chemistry. 31. 3.1 Dynamics .... a self-reproducing process and the concentration of all the elements of the cycle ..... Page 17 ...
Artificial Chemistry: Computational Studies on the Emergence of Self-Reproducing Units ?M9)2=3X?J2=$N%b%G%k:N< Institute of Physics, Graduate School of Arts and Sciences The University of Tokyo 3-8-1 Komaba Meguro-ku Tokyo, 153-8902, Japan March 16, 2001
For all living things in this world.
Contents 1 Introduction 1.1 Primitive Organisms . . . . . . . . . . . . . . . 1.1.1 Fossil Records . . . . . . . . . . . . . . . 1.1.2 Archaebacteria . . . . . . . . . . . . . . 1.2 Synthesizing Living Organisms . . . . . . . . . 1.2.1 Chemical Evolution . . . . . . . . . . . 1.2.2 Proto-Cells . . . . . . . . . . . . . . . . 1.3 Theoretical Approaches . . . . . . . . . . . . . 1.3.1 Models of Self-Replication . . . . . . . . 1.3.2 Evolvability; Replication v.s. Mutation 1.3.3 Metabolism . . . . . . . . . . . . . . . . 1.3.4 Models for Cell Organization . . . . . . 1.3.5 Aim of This Paper . . . . . . . . . . . .
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2 A Toy Model of Self-Maintaining Structure 2.1 Model . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Chemical Reactions . . . . . . . . . . 2.1.2 Repulsive Interaction . . . . . . . . . . 2.1.3 Computing Transition Probabilities . 2.1.4 Stability Analysis . . . . . . . . . . . . 2.1.5 Cluster Formation . . . . . . . . . . . 2.2 Results . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Self-Organization . . . . . . . . . . . . 2.2.2 Self-Maintenance . . . . . . . . . . . . 2.2.3 Fission of an Initially Large Cell . . . 2.2.4 Growth and Fission . . . . . . . . . . 2.3 Summary . . . . . . . . . . . . . . . . . . . .
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15 15 15 16 18 19 19 21 21 23 27 27 29
3 Particle Artificial Chemistry 3.1 Dynamics of Amphiphilic Molecules . . . . 3.1.1 Hydrophobic Interactions . . . . . . 3.1.2 Computing Forces . . . . . . . . . . 3.2 Results1 . . . . . . . . . . . . . . . . . . . . 3.2.1 Organization of Micelles and Vesicles 3.3 Chemical Reactions . . . . . . . . . . . . . . 3.3.1 Catalytic Effects . . . . . . . . . . . 3.3.2 Particle Bath . . . . . . . . . . . . . 3.3.3 Computing Procedures . . . . . . . .
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31 31 31 32 34 34 34 37 38 38
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4 Lattice Artificial Chemistry 4.1 Model . . . . . . . . . . . . . . . . . . . . . 4.1.1 Repulsive Interaction . . . . . . . . . 4.1.2 Chemical Reactions . . . . . . . . . 4.1.3 Reaction Kinetics . . . . . . . . . . . 4.2 Results . . . . . . . . . . . . . . . . . . . . . 4.2.1 Division of Self-Maintaining Cells . . 4.2.2 Variations of Cells . . . . . . . . . . 4.2.3 Emergence of Proto-Cell Structures 4.3 Summary . . . . . . . . . . . . . . . . . . .
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5 Discussion and Future Works 5.1 Self-Reproduction of Autopoietic Structures 5.2 Emergence of Proto-Cell Structures . . . . . 5.3 Inheritance and Evolution . . . . . . . . . . 5.4 Future Studies . . . . . . . . . . . . . . . .
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59 59 60 60 61
3.5
Results2 . . . . . . . . . . . . . . 3.4.1 Self-Maintenance of a Cell 3.4.2 Fission of a Cell . . . . . Summary . . . . . . . . . . . . .
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Publications • N. Ono and T. Ikegami. Self-maintenance and self-reproduction in an abstract cell model. J. theor. Biol., 206:243–253, 2000. (Chapter 2) • N. Ono and T. Ikegami. Computational approaches to self-reproducing proto-cell structures. (in preparation). (Chapter 3) • N. Ono and T. Ikegami. Model of self-replicating cell capable of selfmaintenance. In D. Floreano, J. D. Nicoud, and F. Mondada, editors, Proceedings of the 5th European Conference on Artificial Life (ECAL’99), pages 399–406, Lausanne, Switzerland, 1999. Springer. (Chapter4) • N. Ono and T. Ikegami. A computational model for emergence of primitive self-maintaining cells. (submitted to Journal of theoretical Biology). (Chapter 4)
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Chapter 1
Introduction In these decades, the studies on molecular biology have revealed the elemental processes of living organisms; what they consist of and how the components works. In summery, a living cell is composed of mainly four types of molecules; deoxyribonucleic acid (DNA), ribonucleic acid (RNA), protein and lipid. DNA and RNA is composed of a chain of nucleotide; a compound of sugar and nucleic acid base. There are four types of acid base, adenine (A), thymine (T), guanine (G), and cytosine (C); in RNA, thymine is replaces by uracil (U). These bases bond with specific partner, A pairs with T, G with C, respectively. According to this pairing, DNA forms the double helix which pairs each other complementary, and they can replicate themselves using each strand as a template. However the replication of DNA requires specific enzymes i.e. catalysts of proteins. A protein is a polymer of thousands of amino acids. There are 20 species of amino acids. Their order in a protein is determined according to the sequence of nucleotide of a RNA. These RNA are produced as a copy of the sequence of DNA with the help of the enzymes. Most of chemical reactions in the organisms are catalyzed by specific enzymes. There are other proteins which composes the structural frameworks of organisms. Lipids are catalyzed by enzymes and compose the membrane which envelope the whole cell system. Although those reactions in the levels of molecular mechanics are understood, none have succeeded in creating a “living” organism artificially yet. What is the difference between livings and non-livings; what a property does make it so unique, the most interesting phenomenon on earth? This question is deeply involved in the another question how the first living organism emerged.
1.1 1.1.1
Primitive Organisms Fossil Records
The oldest fossils of living organisms have been found the stratum of over 3 billion-years-old in Western Australia [54] and Sough Africa [3]. Most of these microfossils surprisingly look like the existing species of blue-green algae and other bacteria. However, no scientist believes that any organism as complex as an alga can spring into existence out of the soup of inorganic, small molecules. There must have been preceded some kind of proto-cellular evolution. Though there are a few indirect evidences on the existence of livings in the stratum of 3
3.8 billion-year-old from Green-Land, no fossil record of the earliest living cell are found.
1.1.2
Archaebacteria
The recent investigation on “Archaebacteria”; the groups of micro-organisms which have branched from other creatures at the earliest stage of evolution presents us clues to the forms of the primitive organisms. Some of their features seems the trace when they had been lived in the primitive earth condition [28], for example, most species of Archaebacteria live where there is no oxygen , some of them can live in a very hot environment [4] etc. In fact, it is difficult to distinguish the features which evolved lately from that have been conserved since their oldest days because they are not stoned fossils but living organisms which have evolved for billions of years in their own way.
1.2 1.2.1
Synthesizing Living Organisms Chemical Evolution
Miller’s Experiment There are attempts to play back the environment of the primordial earth to synthesize the primitive living organisms artificially. These studies are originated by Oparin [50]. The studies on the environment of the primordial earth revealed that the primitive atmosphere on the earth lacked free oxygens. It was probably composed of nitrogen, carbon dioxide as that of the Mars. In such anaerobic atmosphere, the organic molecules are prevented from oxidization and can exist much longer than today. Without the ozone layer, ultraviolet light from the sun would provide more energy to stimulate chemical reactions to synthesize complex molecules. In 1953, Stanley Miller performed an experience that pass an electric discharge through the anaerobic atmosphere in a test chamber which contains water, methane, and ammonia [45, 46]. In this and the following similar experiments, a variety of simple organic molecules including amino acids, sugars, nucleic acids are synthesized from inorganic chemicals. some of these obtained molecules are units of organic compounds which are essential for nowadays existing living organisms. There thus could be the another possible source of primitive organic molecules in the space. Amino acids and some other organic molecules are found in interplanetary dust particles or in the tails of comets. They could have been carried on the earth with meteorites which had fallen on the earth frequently at that time. Though, these facts may explain how the building blocks of the living organisms provided to the primordial earth, not yet how started the reactions as living organisms. One problem is, that the yields of these reactions are very low to sustain chemical reactions. Some concentrating process e.g. heating and drying a solution would be required. Some researchers proposed that the primitive chemical evolution could take place on the surface of minerals. When molecules are bounded on some surface, it increase the probability with which two molecules meet and react together. Cairns-Smith proposed that the earliest organic reaction is bounded on the surface of self-reproducing crystal of clay 4
[6] W¨ achtersh¨auser pointed out pyrite (FeS2 ) provides a charged surface which bind the organic molecules efficiently [62, 63]. It is widely supposed that the starting point of chemical evolution was a recursive catalytic interaction called “autocatalysis”. Autocatalysis is defined as a series of reaction which catalyzed by the products itself. By definition, it is a self-reproducing process and the concentration of all the elements of the cycle in the system increase until the resources are exhausted. However, there is a gap between the abiotical and biotical synthesis of organic molecules. The sources supposed here can only provides the molecules which are elemental units of organic compounds; amino acids, sugars and nucleic acids. Molecules in living organisms are composed of thousands of these elemental units. A molecule in the living organisms appears not as a random polymer. To process appropriate reactions of molecules, a polymer of hundreds of molecules would have to be synthesized in a specific order. If complex molecules are required in the all stage of the autocatalytic cycle, how can they synthesized from the primitive chemicals abiotically at first? RNA World RNA is less stable molecule which does not form the double helix. Though in the present living cells only DNA is used as a gene, RNA is also capable to duplicate information by template based replication DNA does. Further, a certain stand of RNA is folded into specific conformation and catalyze particular chemical reaction [8]. It was also reported that RNA can catalyze the replication of RNA itself [23]. Given appropriate monomers (i.e. four types of nucleotides) and a particular enzyme which is used in a virus, a certain sequence of RNA can replicate itself in a test tube. In vitro Darwinian evolution of self-replicating RNA is observed under selective conditions [52]. The hypothesis that proposes an autocatalytic cycle composed of RNA (today called as “RNA world”) could precede in the first cell was maintained many scientists, though the transition from this RNA world to the contemporary system of replication still remain unsolved. Protein World Though proteins can be more efficient catalyst than RNA and they support various reactions more widely, it seems difficult for them to carry and duplicate information as a strands of RNA does. However, in 1996, a helical peptide which can catalyze its own replication was created artificially [34]. This peptide is a polymer consist of certain thirty-two amino acids. It assembles its sub-chain (polymers of fifteen and seventeen amino acids) and connects them using itself as a template. A symbiotic self-replication of two different peptide chain was also reported [35]. Though these reactions are too simple and too artificial to discuss the origin of self-reproducing molecules, it suggests that an autocatalytic cycle consists of amino acids cab also produces a hereditable variations. One advantage of the peptide self-replicating system is that it is easier to synthesize amino acids than nucleotides which is composed of a sugar, a nucleic acid base and a phosphoric acid. It is still questioned which molecule played the leading part of the earliest self-reproducing reactions; DNA (less probable), RNA, proteins, (or others). 5
One hybrid hypothesis proposes the coexistence and cooperation of RNA and protein. At the earliest stage, there could be an autocatalytic cycle composed of both RNA and peptides and their compounds where all of them catalyze each other. Afterward, they could differentiate their role into genetic informations and functional enzymes according to their chemical properties. One of the circumstantial evidence for this hypothesis is that, there are catalysts which is compound of RNA and proteins. When a different type of molecule (called “co-factor”) is attached to a polymer, it increases the variation of the polymer’s conformation, and also, the range of specificity and activity of catalysis [65]. These co-enzymes are found globally in the present cells and some of them play essential roles in the organisms. For example, a ribosome; the translator from the information on RNA to the sequence of amino acids in a protein, is one of the most essential molecule which is found in all living cells. It is supposed that it has originates in the earliest stage of life.
1.2.2
Proto-Cells
Lipid Bilayer Membranes The next question is how the first cell was organized . In general, membranes which envelope existing living cells consist of lipid bilayers. They globally share the basic structures (though, it is reported that the membranes of Archaebacteria are composed of distinct lipids. their structure is different from that of Eubacteria and Eukaryotes. The membrane of Eubacteria and Eukaryotes are composed of lipids which are composed of glycerol and fatty acids bonded by ester bonds on the other hand, that of Archaebacteria is composed of glycerol and polyisoprenoid bonded by ether bonds). A popular membrane molecule is a linear, long molecule which has two chemically different ends (see Fig. 1.1a). One end has hydrophilic bases which are polarized and likes to contact with water. The other end are hydrophobic molecules which are not polarized and tend to avoid the water molecules. Actually, a lipid bilayer can be organized spontaneously in the proper environment. It needs no special device so that it can be demonstrated in vitro. When amphiphilic molecules are mixed in water, they automatically form clusters together to conceal their hydrophobic ends from water. A structure of the clusters depends on the shape of the amphiphilic molecules. When its hydrophobic tail is thinner than the hydrophilic head, it likes to form microspheres called “micelle” (Fig. 1.1b). When its shape is columnar, the cluster tends to form a sheat of bilayer (Fig. 1.1c) which is the basic structure of the cell membrane. A membrane usually curls up and forms a spherical vesicle spontaneously to avoid the ends of the sheat to contact with water. Recently, it was reported that lipid compounds can be formed abiotically in hydrothermal systems [42]. The long chain fatty acids obtained in these reactions could be the components of the membranes of proto-cells. Lipid World Luisi and others presented an experiment that demonstrate spontaneous growth and self-reproduction of vesicles [1, 64]. They have focused on the catalytic effect of these micelles/vesicles on the hydrolysis of anhydride of fatty acids. Since 6
(b)
(a) CH3 CH3 +
N CH2
CH3
CH2 O O P O
CH2 O O C
CH2
O CH2
O
O C
(c)
Figure 1.1: (a) The structure of a representative membrane molecule: phosphatidylcholine. There are hydrophilic bases on the head (shown in gray) and the tail is consisted of long hydrocarbon chains which are hydrophobic. (b) Organization of a micelle. Mixed in water (represented by small gray particles), amphiphilic molecules form clusters enveloping their hydrophobic tails by hydrophilic heads. If they are wedge-shaped (i.e. their tails are thinner than thiner heads), they form a spherical structure. (b) Organization of a bilayer membrane. If their shapes are columnar, they like to form bilayers instead of micelle.
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anhydride of fatty acid is water insolvable, the hydrolysis takes place at the interface of hydrophobic and hydrophilic environment which can be provided by the surface of micelles or vesicles. When the anhydride of oleic acid is added as a neat oil to the suspension of oleic acid vesicle, the vesicles catalyze the hydrolysis and increasing of the surfactant molecules. As a consequence, vesicles grow both in size and number. This process is able to start without the vesicles in the initial solution. The hydrolysis starts with a low rate at first, and after a certain amount of fatty acids are catalyzed, vesicles form spontaneously, then the rate of hydrolysis increases autocatalytically. Inspired these results, Lancet and others proposed the hypothesis of “Lipid World” [55]. They suggest that self-reproducing micelles which catalyze the production of amphiphilic molecules from the precursor could precede more complex self-reproducing system (e.g. RNA world). They also discussed about the hereditary reproduction of these micelles [56]. Though there are a few major classes of membrane molecules found in the cell, thousands of variant of amphiphilic molecules can be defined with respect to their detail structures. If molecules in a micelle catalyses specific lipid, the daughter micelles which inherit the component molecules will inherit the specificity, too. They presented a simulation of a catalytic network of lipids which compose self-reproducing micelles, and displayed that a mutually catalytic cycle evolves through the compositional heredity. Marisomes In reality, there is no evidence to know what kinds of molecule composed membranes of the first cell. In the earliest stage of cell evolution, membrane molecules could be more simple (and not so adopted) than that of present cells so that it could be synthesized more easily, and the membranes could pass various molecules. For example, Yanagawa and others reported the formation of spherical shell structures consisted of amino acids named “marisome” [66]. They simulated some conditions of a warm or hot sea in the early earth environment which enriched with amino acids and metal ions as catalysts. The polypeptides synthesized in this experiments formed spherical shell structures. It is also shown that the shell can accumulate amino acids in the inside. Because the shell is so coarse that they allow small molecules of amino acids to pass through. When the amino acids are polymerized in the shell, they can not pass the shell any longer. This fact suggests that a shell could support the chemical reactions inside with the accumulation of substance.
1.3
Theoretical Approaches
While the experimental studies attempt to reveal the present creature in chemical and physical properties, there are other approaches to realize the universal rules which underlies the history of life. Though, we know only a unique type of life which is composed of carbon compounds on the earth, none cannot tell whether there exist other forms of life. How can we tell whether it is alive or not, when we find some thing on a planet out of the soler system? Many sciencefiction writers have imagined living creatures of alien spaces which is composed 8
of different material from that of ours. From the microscopic viewpoints, we do not find any essential difference between livings and non-livings in the levels of molecules and atoms. The difference is supposed to appear when we consider the organization of higher-order behaviors of organic molecules. We have to realize the features of life as a dynamical profiles rather than fixed, given ones. One cardinal aspect to realize the behaviors of life is “evolvability” [25]. It is undoubtedly the unique property of life that it subject to the Darwinian evolution which brings life great diversity. The question is, what the essential ability needed in evolution is, and how such an evolvable individual emerges from unordered initial conditions.
1.3.1
Models of Self-Replication
First of all, to evolve through the natural selection, living creatures must reproduce themselves recursively. It has been pointed out that there are two different classes of self-reproduction in the respect to whether the variation of their copy is limited or unlimited [41, 58]. For example, a crystal of a mineral can “reproduce” itself. They can incorporate molecules from the environment and reproduce their particular lattice patterns. However, its structure depends on the specific properties of the component molecule, so that a unique (or, at most, a few) structures is realizable. Another example is presented by reproducing patterns in a reaction-diffusion system. The Grey-Scott model: a partial differential system which simulates the chemical reaction of autocatalyst demonstrates the reproduction of spots in a two-dimensional reaction space [51, 36]. Though the patterns are able to reproduce recursively, there is no hereditable variations in their structure. On the other hand, when a living creature bears offsprings, there are individual difference between parents and children, between brothers and sisters. Even when a creature produces its clone by virgin generation, a mutation can be take place. Their diversity derived the genetic information, that is, the sequence of nucleotide in their DNA so the combinatorial variations are virtually unlimited. It is obvious that the latter self-reproducers has the potential for evolution and the former does not1 . Neumann’s Self-Replicating Cellular Automaton Most of recent theoretical studies related to self-replication can be traced back the paper of von Neumann [61]. He proposed the model of self-replicating system based on a discrete dynamics which is known as cellular automata (CA) today. A cellular automata models a dynamical structure by a pattern of symbols arranged on a two dimensional lattice. At each step, the symbols changes according to deterministic rules given as a function of the configuration of neighboring symbols. He pointed out that when a replicator produces its copy, it must know the heretical information to pass to the next generation, and the information must 1 There seems to remain a little confusion in the terminology of ”self-reproduction” and ”self-replication”. We use in this paper, the term ”self-reproduction” is used for general reproduction of some structures, patterns, and etc., while the term ”self-replication” is limited to the reproduction which is followed by the duplication of genetic informations and implies their diversity is unlimited.
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be separated from the replicating device itself. He demonstrated this idea in the model of “universal constructor” which controlled by “coding tape”. There is informations on the tape which tell the constructor how to move and the constructor reads the tape to put new symbols as it direct. Given an appropriate tape as a blueprint, the constructor can write down arbitrary structures. As a natural consequence, if the constructor is fed a tape which code the constructor itself, it will reproduce the same constructor. For a complete self-replicating structure, the another structure which reads the tape and duplicate it on the other site literally is required. He showed that the compounds of these two structure and a tape which code the two structures will reproduce itself recursively. There is arbitrariness in the additional tapes and structures with the extent that does not interfere its replication so its variation is unlimited. It is note worthing that when he present this concept, neither the double helix structure of DNA nor its role as a code of proteins have been revealed yet. Self-Reproducing CA Neumann’s design for self-replicating structure needed a huge complex pattern composed of thousands of symbols to implement a complete self-replicating structure. If such complexity is the minimum requirements for self-replication, an spontaneous emergence of such a complex structure seems improbable. There were various attempts to design more simple replicating (or, at least, reproducing) structure which is likely able to spring up from an random initial condition. Langton proposed a simpler rule of CA which demonstrates the recursive self-reproduction of a simple loop pattern [33]. Reggia has present more simple replicating patterns which can emerge from even a random initial configuration of symbols [10]. Though these reproducers does not have the potential to evolve, Sayama presented an extension of Langton’s CA model which present the evolvable self-replicating loop patterns [53]. In this model, a various size of loop patters can replicate themselves and a mutation could take place when two loop patters collide in the lattice space. It is demonstrated that the smallest loop which reproduce the fastest are selected and dominate the system at last. The reproducing patterns of simpler CA usually looses the universality, namely, in these models, only a few patterns are able to reproduce, while, the Neumann’s model can implement infinite variation of self-replicating structure. In general, the diversity of a self-replicator is derived from the combinatorial variety of the sequence of elemental symbols. It is supposed that primitive life had started from self-reproducers which is simple enough to emerge from inorganic chemical reactions but its variations are limited, and developed into self-replicators which has genetic information of unlimited variation afterward though it is still an outstanding problem how such a transition can take place.
1.3.2
Evolvability; Replication v.s. Mutation
When a self-replicating system which is evolvable through mutations and natural selection emerges there arises other difficulties. It is supposed that the earliest replication of genetic information is primitive and defective so that it causes many errors on the replication. Eigen have shown that there is an error threshold in such a defective self-replicating system, namely, to hand the information down 10
to its posterity, the error rate of replication must inferior to a certain threshold, otherwise, the number of correct copies decays through generation [15]. This crisis become more severe as the information to be replicated become larger. However, to catalyze the replication properly, the catalyst should be a rather large molecule to keep the required particular conformation. Hypercycle To avoid this paradoxical situation, he proposed a model called “hypercycle” [16]. In this model, the molecules are assumed not to perform whole function of replication by itself, but to catalyze a part of the replication in collaboration. It allows smaller molecules to compose a self-replicating system so that the affect of replication error is relatively small. Fontana and Buss presented an abstract model of replicators which catalyze replication each other to demonstrate that a hypercycle can be organized from an random set of molecules [17, 18]. A hypercycle has another weakness against mutations. When a new molecule is introduced into a hypercycle due to a random mutation, there is a danger that the new molecules destroy the cycle. If the new molecule can short-circuit the cycle, it can take over the rest of cycle. The more serious danger is a parasite. If the new molecule can be replicated faster than other molecules but it does not catalyze the replication of the others, the parasite will finally wipe out the all replicators in the cycle. Though it is difficult to simulate the evolution of a hypercycle experimentally, computational models provide a way to investigate its evolution. For example, the evolution of a network of hypercyclic replicators which is tolerant of external noise is presented by Ikegami and Hashimoto [26]. Dittrich and Banzhaf presented another reaction system implemented by a binary string system which demonstrates autonomous evolution of replicators [12]. Another solution to evade the problems which arise from mutation is the separation of the rection space [5, 58, 11]. It is assumed that; (1) Self-replicating molecules are separated into subsets (e.g. cells enclosed membranes) which do not interchange molecules. (2) The subset itself reproduces. The natural selection in the levels of these subsets allows symbiotic sets of replicators to survive.
1.3.3
Metabolism
When the evolution of self-replicators is demonstrated in an artificial model, they often evolve into the smallest structures which replicate most rapidly, instead of developing a complex ones. If the replication rate depends only the size of genetic informations, it is obvious that a smaller replicator which is able to propagate faster will be dominant. This situation may be compared to the evolution of viruses that lives with its host. They needs to care about neither materials nor energy because they do not metabolize them by themselves but just exploit that of the host. However, in general, a living organism has to struggle for finite resource to survive. This material constraint should be taken into account to understand the evolution of metabolic systems which provide building blocks of self-replicators. Kauffman discussed that an autocatalytic metabolic cycle could be found in the certain size of reaction network of chemicals [29]. There remains a ques11
tion; under what conditions such a chemical cycles is able to continue? There were some approaches to investigate the emergence and evolution of a primitive metabolic system from dynamical aspects. Dyson presented a toy model which argue about the stability of a self-maintaining state. He pointed out that an active state in which molecules keep reacting each other may emerge from an unorganized state through a stochastic transition [13]. Bagley and Farmer presented a model which demonstrates the spontaneous emergence of autocatalytic metabolism [2]. Mitsuzawa and Watanabe further discussed about metabolism of polymers analyzing the contribution of the flows of resource and waste molecules toward the synthesis of polymers [47]. Self-Maintenance and Autopoiesis A living organism is an open system which takes energy from environment to sustain itself stably against the thermal equilibrium. A living system is not only a non-equilibrium open system but a self-maintaining system which synthesizes and the all components and sustain the whole structure by itself. In other words, it is a subsystem which determines itself. For example, a cell metabolizes membrane molecules to sustain its membranes which determine its boundary by itself. When the chemical reaction within cell dies, the whole cell structures will disintegrate finally. In this respect, self-maintenance of a living organism is distinct from that of a mere negative feedback system (such as a refrigerator). On the other hand, the boundaries separate the subsystem from the environment. In contrast, a dissipative structure such as a typhoon does not have clear boundaries. Maturana and Varela called this feature to create its borders by itself as “autopoiesis” [60, 37, 38]. It is worth noting that this is the very property which we demanded for the group selection of autocatalyst to avoid parasites. G´ anti proposed a model named “chemoton” [21, 22] to understand the minimal properties of life. A chemoton is composed of three autocatalytic cycle; metabolic, genetic, and membrane subsystems. The metabolic subsystem which is composed of autocatalytic cycle reproduces itself. This subsystem also produces monomers which are used in the genetic subsystems, and unit molecules of membrane. The genetic subsystem polymerizes the monomers to replicate the template which carries genetic information. The third subsystem is membrane. It encloses the whole systems and separate it from environment spatially as a unit of chemical reactions. The membrane incorporates unit molecules automatically to grow the surface. Assuming that the membrane divide itself into daughter chemotons when the system grows enough, it presents enough properties required for a primitive cell.
1.3.4
Models for Cell Organization
Through the argument above, the organization of cell membranes plays a significant role in the emergence and evolution of primitive life. First, it creates a group of autocatalytic molecules; a unit of evolution to encourage co-evolution of these replicators. Second, it defines a subsystem which would be a nonequilibrium, open system. The difference of concentration of substrates across the membrane is indispensable to sustain the metabolism. Third, a membrane 12
is not only a boundary but a catalyst which is able to interfere the chemical reactions. Last, but most essential, it defines an individual autopoietically. There were various attempts to model the formation of micelles and vesicles. Because the organization of these structure is meso-scopic, super-molecular dynamics, computational studies of theoretical models which simulate are useful approach to investigate their behavior, the conditions and parameters, the process of evolution. Molecular Dynamics Models One possible approach is to model membrane molecules precisely in the levels of individual atoms and compute the electric interaction between each atoms to simulate their dynamics. However , in practical, such an approach is computationally overbearing. The system should consist of a hundred of membrane molecules (long chain fatty acids which consist of dozen atoms each) at least, and of course more water molecules around them to simulate even a small piece of membrane. Particle Models Edwards, Peng and Reggia presented a simple computational model to simulate the self-assembly process of lipid [14]. The model simulates the dynamics of abstract particles in the two dimensional space. A lipid molecule is simulated as an amphiphilic particle composed of a hydrophilic head and a hydrophobic tail. They defined inter-particle interaction between particles; repulsion between head-tail and attraction between head-head and tail-tail and so on. A particle is driven by these interactive forces from all other particles and a random force representing thermal effects. Started from randomized initial distribution of hundreds of lipid particles, the system reaches a stable state and micelle-like structures are formed spontaneously. they also presented the organization of reversed micelles introducing abstract water drops and new interactions between the lipid particles and the water drops. Lattice Models Mayer and Rasmussen [39, 40] introduced a computational model of organization of micelles in the discrete system named Lattice Polymer Automata. this model is based on lattice-gas model; an extension of CA which simulates the dynamics of a fluid. a polymer is represented by a chain of monomers which are placed and move on a two-dimensional triangular lattice. Coveney and others also developed a lattice-gas model to simulate the dynamics of amphiphilic systems (micro-emulsion, self-reproducing micelle). Kier and Cheng proposed another CA model of a micelle formation [30]. Computational Autopoiesis Varela and others proposed an abstract computational model which present the autopoietic behavior of a proto-cell [60] (the original model was re-implemented by McMullin [43]). This model is based on a two-dimensional CA. Chemicals are represented by particles which take place in a discrete two dimensional lattice and moves in random walks. There are three different particle types which 13
engage in distinct reactions, those are, catalyst, substrate and link particle (i.e. membrane molecule). A catalyst produce a link particle from substrates. A link particle can bond with another link particle to form membranes, when they are on the adjacent site. The membranes does not permit other particles to pass through except substrate. A link particle naturally disintegrate in to substrate at a constant rate. Breyer and others proposed another extension of this model [5] introducing self-reproduction of catalysts.
1.3.5
Aim of This Paper
One of the advantage of these computational studies of abstract models is, it can simulate an evolution of a very long time scale which is difficult to simulate experimentally. Computer simulations of abstract models provides a way to bridge the experiments on the organization of proto-cell structures and the theoretical studies on the evolution of primitive organisms [48, 49] In the following chapters, we introduce computational approaches which attempt to realize the conceptual models of self-maintenance and self-reproduction discussed above to simulate their behaviors actually. Our purpose here is to understand how such a system emerges through the pre-cellular evolution and analyse the condition which allows it.
14
Chapter 2
A Toy Model of Self-Maintaining Structure 2.1
Model
First, we introduce the simplest model to realize the essence of self-maintaining structure. We consider an reaction cycle of autocatalytic molecules which metabolize themselves under the supply of resource materials. We assumed that the metabolic system has two stable state; a “dead” state where few reacting molecules exist and an “alive” state where the autocatalysts sustain their reproduction stably. It is also assumed that the metabolic system produces membrane molecules which organize spatial structures automatically.
2.1.1
Chemical Reactions
Figure 2.1 illustrates the reaction network studied in this model. There are five types of particles; A, E, M, X and Y. An autocatalytic particle (A) becomes an active enzyme particle (E). This reaction is catalyzed by another particle A at the rate PE . An enzyme particles E catalyses the reaction which produces a particle A from a resource particle (X) at the rate PA . It also catalyses the production of a membrane particle (M) from a particle X at the rate PM . Though a particle A can be spontaneously produced from a particle X, the rate Pa is much slower than that of catalyzed reaction. The rate of spontaneous production of a particle M is assumed to be so small that can be neglected. These particles spontaneously disintegrate into waste particle (Y) at a common rate PY . We assume that there is an external supply of resource particle which exchange a particle Y into a particle X at a constant rate PX . This represents an external source of energy. Those reactions are described by the following stoichiometric equations: P
(2.1)
E+X
PA
−→ E + A
(2.2)
E+X
PM
(2.3)
Pa
(2.4)
E A+E A + A −→
−→ E + M −→ A
X 15
P
(2.5)
PX
(2.6)
Y Y {A, E, X, M } −→
−→ X
Y
X chemical reaction
A catalysis
E M resource supply
Y Figure 2.1: Reaction paths. An autocatalytic particle (A) catalyses the production of an enzyme particle (E) from another particle A. A particle E catalyses the production of a particle A or a membrane particle (M) from a resource particle (X). All particles decay into a waste particle (Y) at a constant rate. There is an external source which changes a particle Y into a particle X.
2.1.2
Repulsive Interaction
We introduce spatial kinetics of particles. The above chemical reactions happen in a one-dimensional reaction space which is coarse-grained into discrete reaction sites. There are always 100 particles in each reaction site. A partile moves on the sites stochastically according to two spatial processes; diffusion and repulsion processes. When a particle on a i-th site moves to the adjacent j-th site, we choose a counter-particle from the destination site randomly and moved it to the i-th site to keep the total number of particles on the site constant. In the diffusion process, each particle moves to its neighboring sites with a constant probability (Fig. 2.2a). The repulsion process is caused by the hydrophobic interaction between particles. We assume there are hydrophilic particles (A, E and X) and hydrophobic ones (M) which repel each other in the space. In this process, a particle moves to neighboring sites at a certain rate due to repulsion between particles on the same (Fig. 2.2b) or neighboring sites (Fig. 2.2c). Therefore, a particle M on the i-th site moves to the (i ± 1)th site at the rate which is in proportion to the number of these three types of particles on the same site and the counter-neighboring (i ∓ 1)th site. Contrary, the three types of particles move to the neighboring (i ± 1)th site at the rate which is in proportion to the number of particle M on the i-th and the (i ∓ 1)th sites. We study two cases below. In the first case, all particle share the common mobility rates (i.e. diffusion and repulsion rates). In the second case, we assume that the mobility rates of autocatalytic particles is smaller that those of other particles. 16
(a)
(b)
(c)
Figure 2.2: Schematic illustrations of diffusion and repulsion processes. (a) Diffusion. A particle moves into neighboring sites with a constant probability. (b) Repulsion in the same site. A hydrophobic (represented by a white square) repels hydrophilic particles (a gray circle) in the same site with the probability PR0 removing it into the adjacent site, and vice versa. The direction is chosen at random. (c) Repulsion between the neighboring sites. Repulsion also takes place between particles in the adjacent two sites with the probability PR1 . The repelled particle moves one step aside. In any case, a counter-particle which is randomly chosen from the destination site replaces the removed particle to keep the total number of particles on a site constant.
17
2.1.3
Computing Transition Probabilities
A state of the system at the time t is represented by a set of vector, n(i, t) = nA (i, t), nE (i, t), nM (i, t), nX (i, t), nY (i, t)
(2.7)
where nα (i, t) denotes the number of particle α in the i-th site respectively. The transition probabilities of chemical reactions are calculated by the following formula. i PA→E (t) i PX→A (t) i PX→M (t)
= PE nA (i, t)
(2.8)
= PA nE (i, t) + Pa = PM nE (i, t)
(2.9) (2.10)
i P{A,E,M,X}→Y (t)
≡ PY
(2.11)
PYi →X (t)
≡ PX
(2.12)
i Here Pα→β (t) denotes the transition probability from particle α to β on the i-th site at the time t respectively. The values of the reaction rate below are fixed through the present simulations: PE = 0.5 × 10−6, Pa = 2 × 10−6, PX = 100 × 10−6 and PY = 100 × 10−6. The coefficients of catalytic effects for the production of particle A and M (PA and PM ) are used as controlling parameters. The mobility probabilities of the particles are given as below, i→i±1 PX (t) = i→i±1 (t) = P{A,E} i→i±1 PM (t) =
PD + PR0 nM (i, t) + PR1 nM (i ∓ 1, t) (2.13) 1 PD + PR0 nM (i, t) + PR1 nM (i ∓ 1, t) (2.14) m A,E,X A,E,X X X PD + PR0 nα (i, t) + PR1 nα (i ∓ 1, t) (2.15) α
PYi→i±1 (t)
≡
α
PD
(2.16)
Here Pαi→i±1 (t) denotes the mobility probability of particle α from the i-th site to the (i ± 1)th site respectively. The rates are PD = 7 × 10−3 , PR0 = 5 × 10−3 and PR1 = PR0 × 0.5. There are another mobility parameter m which denotes the mass of particles is different in the section 2.2.4, the mobility difference of particles is not considered (m = 1) but in the section 2.2.4, we assume that particle A and E are heavier than other particles so that they move slowly (m = 5). Using the variables and parameters above, execute the below algorithm on one-dimensional distributed particle system. There are 100 sites each contains always 100 particles. The boundary of reaction space is periodic. In each iteration, the chemical and spatial state of particles are updated by following procedures. 1. Calculate the transition probabilities for every particle. Then, change the chemical state of the particles according to the probabilities. 2. Particles diffuse on sites by the following steps. (a) Choose a site in a random order. 18
(b) Calculate the probabilities of diffusion and repulsion of the particles on the site. (c) Let particles move to the adjacent site according to the evaluated probabilities. In order to conserve the number of the particles on the site, when a particle on the i-th site moves to the adjacent j-th site, a randomly chosen particle on the j-th site will in turn occupy the i th site. (d) Repeat these steps untill all site is precessed. These procedures complete a unit simulation iteration.
2.1.4
Stability Analysis
Before considering the effects of membrane, we analyze the behaviors of this metabolic system under the conditions where it stays spatially homogeneous. When we set PM = 0 and start the simulation from homogeneous initial configuration, this system stays in a homogeneous state, within the parameter regions considered here and always falls into a stable state, so that the behaviors of the system can be represented by three variables, the mean number of particle X, A and E (the mean number of particle Y falls into a stable fixed value (Y¯ ≡ 100/(1 + PX /PY )). Concerning these conditions we study mean-field type equations below, in order to investigate the stability of the fixed points of the system. ¯ + PW )R ¯ ¯˙ = PR Y¯ − (PA E R ˙¯ = P Y¯ + P E¯ R ¯ − (PE A¯ + PW )A¯ A A a ¯ ¯˙ = P A¯2 − P E E E
W
(2.17) (2.18) (2.19)
Numerically solving the equation and analyzing the stability of the solutions, we obtain the bifurcation diagram Fig. 2.3. Below the critical value PA1 , the reproduction rate of particle A is too low to sustain metabolism, so that this system has only one stable state (named “dead” state) where few particles A exist. On this value, a saddle-node bifurcation generates two stable states and one unstable state. We named the upper branch as “alive” state because in which the metabolic reaction sustains. Above the second critical point PA2 , the upper steady state becomes a unique equilibrium state through the inverse saddle-node bifurcation.
2.1.5
Cluster Formation
Particle M tends to form clusters due to the repulsion with hydrophilic particles. In the case of one-dimensional space, the cluster becomes like a wall which occupies a few sites and expel other particles except Y. In the case of two-dimensional space, a potential membrane will form chains or stripes. It is relatively easy to trap catalysts in the one-dimensional case, as the two-sided walls are sufficient to trap catalytic particles inside. In the present model, a membrane cannot be sustained just by having some catalyst particles. Rather, it needs an active and dynamic state of the chemical network. In other words, 19
Number of A particles
A 35 30 25 20 15 10 5
PA1 5
PA2 10
15
20
25
PA 10-6
Production rate of A
Figure 2.3: The bifurcation diagram of the expected average number of particle ¯ against their reproduction rate (PA ). Solid lines denote stable and a A (A) ¯ respectively. On the value dashed line denotes unstable fixed point branch of A, PA = PA1 and PA2 , there occur saddle-node bifurcations. In the region below PA1 , A¯ has only a lower fixed point branch. Between PA1 and PA2 , there are two stable and one unstable fixed point branches. Beyond PA2 , A¯ has only a upper fixed point branch.
20
a mechanical aspect of self-reproduction, such as catalyst trapping will be investigated from a spatially extended reaction-diffusion system. Simulations in the two-dimensional case will be discussed later.
2.2
Results
We have conducted a series of three experiments. The first one is from a homogeneous initial configuration. The second one is from a cell-like initial structure, and the last one is an experiment with giving different diffusion and repulsion parameters allocated to particles.
2.2.1
Self-Organization
A final stage derived from a homogeneous initial configuration (Fig. 2.4a) will be categorized into three regions with respect to the concentration of particle A. When there is sufficient amount of particle A, they produce particle E which catalyses themselves and particle M Therefore, the high density state of particle A indicates an ongoing reaction cycle. The particles M build periodic membranes automatically (Fig. 2.4b). 100
100 X A E M
X A E M
80 Number of particles
Number of particles
80
60
40
20
60
40
20
0
0 0
20
40 60 Reaction sites
80
100
0
(a)
20
40 60 Reaction sites
80
100
(b)
Figure 2.4: Snapshots of the dynamics in the spatially extended reaction system. The X-axis denotes the reaction sites and the Y-axis denotes the number of particles on each site. Each line shows the distribution of particles (Dotted line: resource (X), Thick dashed line: autocatalyst (A), Dashed line: enzyme (E), Thick line: membrane (M), waste particles (Y) are not displayed). (a) The homogeneous initial state. The number of particles are n(i, 0) ≡ (20, 0, 0, 30, 50). (b) A snapshot after 500,000 iteration. When particle A produces enough particles E, they reproduce particles A and M, and particle M forms periodic clusters. (PA = 9 × 10−6 , PM = 7 × 10−6 ). Figure 2.5 shows the phase diagram against parameters PA and PM . Where they produce no particle M (PM = 0), it corresponds to the diagram shown in Fig 2.3. The boundary between regions 0 and I is well estimated by the mean-field approximation. However, the boundary between the regions I and II are overestimated. This is because the simulation is based on stochastic dynamics, the 21
-6
PM 10
Production Rate of M
15
10
0
5
0
- - - - - - - - - - = - = - = - = - = - = - = - = - = - - 5
- = - = - = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
I
= = = = = = = = = = = = = = = = = = = +
+ + + + = = = = = = = = = = + + + + + +
10
+ + + + + + + + + + + + + + + + + + + + 15
II
-6
PA 10
Production Rate of A Figure 2.5: The phase diagram with respect to the final density of particle A from a homogeneous initial configuration. The X- and Y-axes represent the production rate of particle A and of M (i.e. PA and PM ) respectively. Each mark represents the run of 1,000,000 iteration from the homogenous initial configuration shown in Fig. 2.4a. Region 0: A mark (−) denotes the region where the dead state is a unique stable state. The density of particle A falls low even though any large amount of particle A is given initially. Region I: A mark (=) denotes the bistable region where the final state depends on the initial density. Region II: A mark (+) denotes the region where the alive state is the unique stable state. The density of particle A rises even evolving from any low initial density.
22
system can leap into the upper steady state from the lower due to the fluctuation of the number of particles, which is not taken into consideration in this mean-field type equation. In region 0, particle A can not sustain their own replication. Because the reproduction rate is set too low, they disappear due to the spontaneous decay process. In region I, a final steady state is either dead or alive state according to the initial density of particle A. There is a threshold with respect to an initial density of particle A. In region II, a final state is always alive state regardless of the initial density of particle A.
2.2.2
Self-Maintenance
We are especially interested in region I, where the final state depends on the initial configuration of autocatalytic particles. We study the time evolution of an inhomogeneous initial configuration with the parameters in region I. First, to discuss the evolution of the inhomogeneous initial configuration briefly, consider a configuration of particles where autocatalytic particles localized in a small domain, and particle M are neither given initially nor produced by particle E (i.e. PM = 0). The fate of this state depends on the initial amount of particle A, or, the size of the domain. If the amount is sufficiently large, they gradually spread over the whole reaction space (Fig. 2.6a,b). Although there is a uncertainty due to the stochastic process, particle A decays away when the initial amounts is smaller than a certain threshold, (Fig. 2.6c,d). If the metabolic system can produce membrane particles, the time evolution becomes very different ones. The initial configuration of this experiment is shown in Fig. 2.7a. Between the clusters of partile M, there is a domain containing the rich amount of particle A. We simply call this configuration as “cell” and the adjacent M particles as “membrane” of the cell for the present. A rough diagram is computed from the final states evolving from this initial cell-like configuration, as in Fig. 2.8. It is found that region I is sensitive to the cell initial condition. Region I in Fig. 2.5 is now categorized into three subregions according to their temporal behavior. In region Ia, the supply of particle M catalyzed by particle E is too low to sustain the initial membranes. As the membranes degenerate, particle A diffuses away quickly. The more particles escape from the cell, the less likely that the membrane can be re-organized. Thus, the initial cell disappears finally (Fig. 2.7b). This behavior is compared to the case where the initial membranes fail to enclose the inside catalyst, described by McMullin & Varela (1997), although their model assumes a two-dimensional cell pattern. In region Ib, the production and degeneration rate of particles are balanced to sustain the cellular structure. The size of cell is kept constant (Fig. 2.7c). In region Ic, When the reproduction rate of A is rather high, and those particles increase their density outside the cell due to the autocatalytic cycle. This region differs from the regions Ia and Ib, with new membranes being generated outside the cell (Fig. 2.7d). We then have new cells. the above process is repeated to successively create new cells. Finally, the whole space is filled with the cells of a typical size. We call this process which observed in region Ic as “open reproduction”, because, in this process, the enclosed particles of the daughter cell are not only transferred directly from the mother cell. The permeated particle is mixed with 23
100
100 X A E M
60
40
20
60
40
20
0
0 0
20
40
60
80
100
0
20
40
60
Reaction sites
Reaction sites
(a)
(b)
100
80
100
100 X A E M
X A E M
80 Number of particles
80 Number of particles
X A E M
80 Number of particles
Number of particles
80
60
40
20
60
40
20
0
0 0
20
40
60
80
100
0
20
40
60
Reaction sites
Reaction sites
(c)
(d)
80
100
Figure 2.6: Evolutions of an inhomogeneous initial state without membranes. (a0) The initial configuration without membranes but contains a large amount of A particles. The parameters are chosen from the region Ic (PA = 8 × 10−6 , PM = 0). (a1) A state after 50,000 iteration. A particles increase their number and spread to fill the space finally. (b0) The smaller initial configuration. (b1) A state after 10,000 iteration. A particles diffuse away before they begin to replicate themselves stably. The parameters are as same as that of (a) (PA = 8 × 10−6 , PM = 0).
24
100
100 X A E M
X A E M
80 Number of particles
Number of particles
80
60
40
20
60
40
20
0
0 0
20
40 60 Reaction sites
80
100
0
20
(a)
80
100
(b)
100
100 X A E M
X A E M
80 Number of particles
80 Number of particles
40 60 Reaction sites
60
40
20
60
40
20
0
0 0
20
40 60 Reaction sites
80
100
0
(c)
20
40 60 Reaction sites
80
100
(d)
Figure 2.7: Snapshots of the dynamics of cells. (a) The initial configuration. There is a region containing A particles (as same size as that of Fig. 2.7b) between the “membranes”(i.e. clusters of M). (b)-(d) Snapshots after 100,000 iteration. (b) When PA and PM are set low, the production of M is too small so that the membranes can not keep particles inside the cell (PA = 5 × 10−6 , PM = 3 × 10−6 ). By loosing A particles by diffusion and decomposition, this cell is going to disappear. (c) When PA and PM are balanced, a cell can sustain itself stably (PA = 7 × 10−6 , PM = 9 × 10−6 ). (d) When PM is low or PA is high, A particles can increase in the outside of the membranes. They construct new membranes and new cells. (PA = 11 × 10−6 , PM = 5 × 10−6 ).
25
-6
PM 10
Ia
Production Rate of M
15
10
0
5
0
5
- = - = - = - = - = - = - = - = = = = = = = = = = = = = - + - +
= = = = = =
= = = = = =
Ib = = = = = = = = = = + + +
= = = = = = + + + + + + +
= = = + = + = + = + = + = + + + + + + + + + + + Ic + + + + + + + + + + + + + + + + + + 10
II
15
-6
PA 10
Production Rate of A
Figure 2.8: The phase diagram with respect to the time evolution from a cellular configuration. There are three sub-classifications in region I. Each mark represents a run of 1,000,000 iteration starting from the initial cell state (Fig. 2.7a). Region Ia: A mark (−) denotes where cells cannot sustain membranes. Region Ib: A mark (=) denotes where cells can maintain themselves in a stable manner. Region Ic: A mark (+) denotes where cells can reproduce.
26
the outside particles when create a new membrane to enclose the new catalytic particles. It make a difference with respect to the heredity of a cell, because the molecules in the original cell are not inherited strictly through this selfreproduction due to the contamination. The internal content is updated for the new cells, which is why we also call this reproduction open reproduction. However, “closed reproduction” which inherits the molecules correctly is also made possible by introducing mobility differences. We will present this alternate reproduction process in the next section.
2.2.3
Fission of an Initially Large Cell
In general, a stable size of cell is determined both by the reaction and mobility parameters. For example, an initially large cell (Fig. 2.9a) will break into two by creating a new membrane in the middle of the original cell (Fig. 2.9b), because the larger cell can produce surplus particle M. 100
100 X A E M
X A E M
80 Number of particles
Number of particles
80
60
40
20
60
40
20
0
0 0
20
40 60 Reaction sites
80
100
0
(a)
20
40 60 Reaction sites
80
100
(b)
Figure 2.9: Division of an initially large cell. (a) The initial large cell configuration. (b) After 100,000 iteration, the large cell produces surplus M particles which create a new cluster in the middle of the original cell (around the 60 th site). (PA = 7 × 10−6 , PM = 9 × 10−6 , m = 1).
2.2.4
Growth and Fission
We found another dynamic aspect of cell reproduction. In the next experiment, we give different mobility rates to particles, assuming that particles A and E have lower diffusion and repulsion rates than M and X. According to this mobility difference, particles outside the cell (i.e. X) are repelled more frequently than particles inside (i.e. A and E). It results in the movement of the membranes outwards. When the cell reaches a certain critical size, it divides itself spontaneously as it shown in the previous section. This process can take place recursively (Fig. 2.10). Compared with the previous open reproduction dynamics, we call this “closed reproduction” dynamics, because the internal particles are kept from outside through the reproduction process. 27
400000
Iteration
300000
200000
100000
0 0
50
100
150
Reaction sites
Figure 2.10: The motion of membranes in the recursive cell division. The X-axis denotes the reaction sites and the Y-axis denotes the time axis. The distribution of M particles is only displayed in the figure. Starting from a cell configuration which is the same configuration shown as Fig. 2.7a, membranes move outwards because the mobility of particles inside the cell is smaller than that of outside the cell. When the cell reaches a critical size, new membranes are created in the middle of the mother cell. This division takes place recursively (PA = 7 × 10−6 , PM = 5 × 10−6 ).
28
2.3
Summary
We presented an abstract model of self-maintaining structures consisted of autocatalysts and self-organizing particles. which also demonstrates the spontaneous growth and division. This is an autopoietic structure in that the autocatalytic cycle inside the cellular structure supplies the membrane particles which compose the clusters trapping the metabolic system itself. It makes a difference between this structure and reproducing patterns observed in reaction-diffusion system e.g. GrayScott model [51, 36]. Though the reproducing spots in the Grey-Scott model are at a distance, molecules can diffuse between the spots continuously. While the autocatalyst in the cell are separated by the membranes which suppress the diffusion of the molecules. This allows the cell to be regarded as a distinct structure. We demonstrated two different reproduction of cellular structure: open and closed reproduction. In the former process, autocatalysts reproduce and spread outside the membranes. Although the diffusion of particles are restricted in the one-dimensional space in this model, the decline of the density of autocatalyst due to the diffusion is much faster and their propagation is more difficult in a space of higher-dimensions. We will discuss the dependence of metabolic system on the membrane in a two-dimensional space, lately.
29
Chapter 3
Particle Artificial Chemistry In the following chapters, we will introduce computational models which presents self-organization of a proto-cell structure in a two-dimensional space. Since, none knows the accurate information of the earliest membrane molecule, our purpose here is not to simulate the membrane of existing cell in detail, but to approach to what is the minimal requirement for the first cell membrane. Therefore, we will not simulate the interaction in the level of individual atoms nor molecules but consider an abstract model which simulates a meso-scopic behavior of molecules. Chemicals are represented by abstract particles to save the computation. Within this framework, we simulate the organization of membrane and its role in the interaction with metabolic reaction within cell.
3.1
Dynamics of Amphiphilic Molecules
This model simulates the repulsive interaction of abstract particles in the twodimensional space and metabolism of them. Particles are driven by random forces which representing thermal effects and by interactive forces between particles. To model the organization of membrane, we will introduce the repulsive interaction of chemicals. Chemical reaction of particle is simulated by a change of particle type. Throughout the reactions, the total number of particles is conserved. To model a primitive metabolic system, we consider an autocatalytic cycle which metabolizes itself and also produces membrane molecules as co-products. We assume that there exists a particle bath which brings energy flow into the system: supplies resource and removes waste particles.
3.1.1
Hydrophobic Interactions
In an aqueous environment, hydrophobic molecules tend to stick together as if they are attracted by “hydrophobic force” each other. In reality, the attraction is derived from the repulsion between water and hydrophobic molecules (more precisely, it is explained by the effect of free energy which depends on the arrangement of water molecules). Hydrophobic molecules cluster to avoid 31
contacting with water molecules to minimize the potential. Therefore, we focused on the repulsive interaction between particles to model the dynamics of membrane molecules. This formalization allows us to compute their interaction in a simple, uniformed way. We introduce some classes of particle; “hydrophobic” and “hydrophilic” (including water) particles and different repulsion forces between them. Repulsion between hydrophobic and water particles is assumed stronger than those among the particles of the same class. There is the third class of “neutral” particles which interact with other particles relatively weakly. The repulsion force between hydrophobic and hydrophilic particles (f1 ) is the strongest. Force f2 is the repulsion between particles of the same group. Force f3 is the weak repulsive force between particles of neutral class and all particle. Within this framework, an amphiphilic molecule is modeled by a compound of hydrophilic and hydrophobic particles that are bonded by an elastic spring. Figure 3.1 illustrates a simplest conformation of an amphiphilic molecule consists of two particles; a hydrophilic (head) and a hydrophobic (tail) particle.
H
f2 W
Rb
f1
fb T
Figure 3.1: Interaction between particles. An amphiphilic molecule is represented by a compound of two particles. The head (shown by a gray circle) particle is hydrophilic, and the tail (white box) is hydrophobic. They are connected by a liner spring of coefficient kb whose natural length is Rb .
3.1.2
Computing Forces
Because it is not our purpose to demonstrate the molecular dynamics in detail, but to simulate the meso-scopic behavior of membranes and metabolic system, we approximated the repulsive and bonding force potential by simple functions of the distance between two particles (see Fig 3.2) instead of using more realistic functions (e.g. the Lennard-Jones potential [57]) in order to save computation. The repulsion (Hfl ) and bonding (Hfb ) potential are given as follows,
(
2
r −r+ kl ( 2R l 0
Hfl (r)
=
Hfb (r)
= kb (−r +
r2 ). 2Rb 32
Rl 2 )
(r ≤ Rl ) (r > Rl )
(3.1) (3.2)
Potential (f)
0.25 0.2
f1
0.15 0.1 0.05
f2 f3 0.02
0.06
0.1
0.14
Distance (r) Figure 3.2: Each repulsion potential is depicted against a distance r. The repulsion between hydrophobic and hydrophilic is the strongest force f1 .
The coefficient kl and the length Rl are the magnitude and the range of the correspond force fl which depends on the type of each particle, respectively1 . The coefficient kb and the length Rb are the spring constant and the natural length of the bond between particles respectively2 . The total potential field for the i-th particle is computed by the following summation, Ui (xi (t))
=
X
Hfl (|xi (t) − xj (t)|) + B(i, j)Hfb (|xi (t) − xj (t)|) (3.3)
j6=i
where xj (t) denotes the position of particle j at the time t. The first term gives the total repulsion effects from all other particles. The coefficient in the second term B(i, j) is a characteristic function which is equal to 1 if the particle i and j are bonded, otherwise 0. The total force on particle i is given by below, Fi (t) = ∇Ui (xi (t)) − γvi + rand(t).
(3.4)
The first term, the gradient of the potential Umi gives the total interactive force on particle of type mi from other neighboring particles. The second term is the friction, where γ > 0 denotes the friction coefficient which is given common constant for all particles and vi is the velocity of particle i. The third term rand(t) represents thermal effects. All particle except water particle share a common unit mass (M = 1). A water particle has the lighter mass than other particles (MW = 0.5). Euler’s simple calculation method is used to determine the movement of particles for each time step (dt = 0.003). 1 The values of repulsion are; k = 3, R = 0.075, k = 2, R = 0.05, k = 1, R = 0.025, 1 1 2 2 3 3 where the reaction space shown in Fig. 3.3 is a square of 1 unit length on a side. 2 The values of bonding force are k = 10, R = 0.025. b b
33
3.2 3.2.1
Results1 Organization of Micelles and Vesicles
This model presents the formation of micelles and bilayer membranes successfully. Figure 3.3 shows the representative snapshots of the simulations which present the formation of those structures. The initial configuration of particles on a space is a random distribution, and their initial velocities are null vectors3 . Structures of clusters are controlled by the interaction of particles. The next two simulations are both started from a random initial configuration which is shown in Fig. 3.3a. In the case where the repulsion between two head particle is stronger than that of tails, the amphiphilic molecules like to form micelles. To simulate organization of wedge-shaped molecules, different values of repulsion are used in this simulation 4 . Figure. 3.3b shows the result of this simulation. While, columner amphiphilic particles form bilayer membrane as is shown Fig. 3.3c. Figures 3.3d, e show the double magnified pictures, respectively to present the details of the structure. Phase transition from micelle to membrane is difficult to observe clearly because of the stochastic behavior of the model. The structure also depends on the density of particles. For example, multi-layer structures or reversed micelles are formed where there are a large amount of amphiphilic or hydrophobic particles and a few number of water particles (figures are not shown). A membrane often forms closed vesicles because the both ends avoid to contact with hydrophilic particles. Figure 3.4 present the time series of spontaneous organization of vesicles 5 .
3.3
Chemical Reactions
We introduce chemical reactions among particles. To model a minimal metabolic system of proto-cell, we define six different types of particles A, H, T, W, X and Y. Chemical reaction is represented by stochastic transitions of particle types and bonding/decomposition of particles. Figure 3.5 shows the paths of chemical reactions among the particle types. We assume that there are four reaction paths, X + X −→ A-A
(3.5)
X + X −→ H-T
(3.6)
A-A −→ Y + Y H-T −→ Y + Y
(3.7) (3.8)
We attribute several chemical properties to each particle. A particle X represents basic material of metabolism, When the distance of two particle X is smaller than 2Rb , there is a chance that a bond is created between them 3 There are 1200 particles in the system; 1000 particles W and 100 pairs of particle H and T which are bonded together. The boundary conditions are periodic. The friction coefficient γ = 4. The thermal effects is given by a Gaussian noise with standard deviation σ = 0.06. 4 The repulsion between H is given by f and that between T is given by f , respectively, 1 3 instead of f2 . 5 The initial configuration is random distribution of 1200 particles including 150 pairs of membrane particles.
34
(a)
(b)
(c)
(d)
(e)
Figure 3.3: Organization of micelles and membranes. A red box and a purple circle represent the hydrophobic head and the hydrophilic tail of an amphiphilic molecule respectively. A blue dot represents a water particle. (a) The initial configuration (t=0). After 60000 time steps (t=180), particles gather together to form micelles (b) or vesicles (c) according to the shapes of particles. Pictures (d) and (e) show the magnifications of the upper pictures, respectively.
35
(a)
(b)
(c)
(d)
(e)
(f)
Figure 3.4: Organization of a vesicle: time series snapshots are depicted (t=0 (a), 180 (b), 360 (c), 540 (d), 720 (e), 900 (f)). A closed vesicle of amphiphilic particles is automatically formed from a random initial configuration.
36
chemical reaction
2X catalysis
external flow
A-A H-T
W 2Y Figure 3.5: Reactions between particles. Two resource particles X can be bonded and changed into an autocatalytic molecule (A-A) or a membrane molecule (H-T). This reaction is catalyzed when there are molecules A-A in the neighborhood. Bonded particles spontaneously decompose into two waste particles Y.
and both particles change into particle A at the same time. There is another chance that two particles X create a bond and change into a particle H and T which represents hydrophilic (head) and hydrophobic (tail) part of membrane molecules, respectively. Two particles A which are bonded together represent an autocatalytic molecules which catalyze the both bonding reaction. A bond between particles spontaneously decompose at a constant rate, changing the particles into waste particle Y. A particle W represents water, which does not change its particle type by any chemical reaction.
3.3.1
Catalytic Effects
We assume that, bonding reaction between two particles X is increased when there are particles A in the neighborhood. The probabilities of the reactions 3.5 and 3.6 are given by the functions below, i kA (t)
=
BA +
A X
CA fc (|xj (t) − xi (t)|)
(3.9)
j
i kM (t)
=
BM +
A X
CM fc (|xj (t) − xi (t)|)
(3.10)
j
fc (r)
=
1− 0
r Rc
(r ≤ Rc ) (r > Rc )
(3.11) (3.12)
37
where BA and BM denotes the coefficient of the spontaneous reaction, CA and CM denotes the activity coefficient of catalyst, and Rc gives the truncation length of catalytic effects.
3.3.2
Particle Bath
In order to sustain the metabolism in this system, a source which supplies resource and remove waste particle is requested. We assume that the system can exchange particles with an external particle bath. The regions where particles are supplied and removed are defined as is shown in Fig. 3.6. To conserve the total number of particles, when a particle is supplied into or removed from the system, it is exchanged with water particles. Thus, when a particle is in this region, it changes its type with a given constant probability SW and SX as below, {X, Y} W
S
(3.13)
SX
(3.14)
W −→ W
−→ X.
Membrane and autocatalytic particles which are bonded together are not exchanged.
Figure 3.6: The region contacted with a particle bath. In the gray region at the corners, resource particles are supplied and waste particles are removed.
3.3.3
Computing Procedures
For each time step, chemical reactions are executed after the motion of all particles are calculated. Composition and decomposition of a bond between particles are simulated by a stochastic process, using following procedures. 1. For all pairs of particle X, (a) compose A-A, and (b) compose H-T with the probabilities given by 3.9 and 3.10 respectively. 38
2. For all pairs of particle A-A and H-T, decompose them with the constant probability kY . 3. For particle X, Y and particle W, if the particle is in the region presented in Fig. 3.6, exchange it with the probability SW and SX , respectively. The all reactions above take place exclusively 6 .
3.4 3.4.1
Results2 Self-Maintenance of a Cell
The simulations in this section starts from a cellular initial distribution. A cellular domain which is enveloped by a bilayer membrane and contains particle A. A representative snapshot of self-maintaining cell is shown in Fig. 3.7a 7 .
Figure 3.7: Self-maintaining cell structure. An yellow circle represents a particle A. A purple circle and a red box represent particle H and T, respectively. A blue dot represents a particle W. A green ring represents particle X. Particles Y are not displayed. This structure grows as assimilating resource from the environment. However, if the production rate of autocatalytic particles inside the cell is much faster than that of membrane particles, the pressure of particles insides the cell becomes higher as the particles A are produced. When the pressure becomes too high for the membrane to keep its tention, the cell bursts at last (Fig. 3.8)8 .
3.4.2
Fission of a Cell
When these production rates are comparable, the growth of the length of the membrane is possibly faster than that of cell size in two-dimensions, because the length of membrane of one-dimension grows in proportion to the amount 6 The values of reaction coefficients are, B −7 , k −7 , S A = BM = 10 Y = 10 W = 0.003 and SX = 0.0003, Rc = 2Rb . The catalytic coefficients CA and CM are used as controlling parameters. 7 The values of catalytic parameters are, C = 1 × 10−5 , C −5 . A M = 1 × 10 8 The values of catalytic parameters are, C = 3 × 10−5 , C −6 . = 3 × 10 A M
39
(a)
(b)
(c)
(d)
Figure 3.8: An example of cell burst process (t=30 (a), 900 (b), 1800 (c), 2700 (d)). When the production rate of inner particles exceed widely that of membrane particles, a cell grows fatter, and finally bursts due to the inside pressure.
40
of membrane particles while the cell size of two-dimensions is in proportion to the square root of the amount of inner particles. It is obvious the growth of three-dimensional cell can be argued as well. This results in the change of cell shapes. A cell becomes round due to the surface tention of membrane at first. The shape changes into oval as the membrane grows, further, it becomes dumbbell-shaped. Finally, recombination of the membrane takes place at the narrowest part spontaneously to divide a cell into two daughter cells. Figure 3.9 shows a representative snapshots of cell fission 9 .
(a)
(b)
(c)
(d)
Figure 3.9: An example of cell fission (t=900 (a), 1350 (b), 1800 (c), 2250 (d)). As the cell grows, it becomes dumbbell-shaped and brakes into daughter cells finally. Another type of cell fission is observed when the production rate of membrane is fast. In this case, membrane molecules which are incorporated into the inner layer of the membrane form an invaginate fragment before they transit to the outer layer. The growth of the fragment results in the formation of a new membrane separating the original cell. Figure 3.10 shows an example of the formation of a septum 10 . However, as is shown in Fig. 3.10f, a cell sometimes fails to fission. 9 The 10 The
values of catalytic parameters are, CA = 1 × 10−5 , CM = 1 × 10−5 . values of catalytic parameters are, CA = 1 × 10−5 , CM = 3 × 10−5 .
41
(a)
(b)
(c)
(d)
(e)
(f)
Figure 3.10: Another example of cell fission dynamics (t=900 (a), 1350 (b), 1800 (c), 2250 (d), 2700 (e), 3150 (f)). Here the second daughter cell fails to form a right boundary.
42
3.5
Summary
We presented an almost minimal model which simulates the organization of micelles, vesicles of bilayer membranes and other structures. Introducing metabolisms of molecules, we composed self-maintaining proto-cell structures. We also demonstrated that self-reproduction of a self-maintaining proto-cell structure takes place spontaneously. These two property; self-maintenance and self-reproduction will allow the structure to be an evolvable individual if it can carry genetic informations. Practically, the fission of existing living cell is carefully regulated. Otherwise, daughter cells often lacks essential organella. Specially, they must share a set of chromosomes evenly. Though it is supposed that such complex mechanisms for cell fission could not be available in the earliest stage of cellular evolution, the results presented in this chapter suggests the way of the spontaneous fission of a self-maintaining structure. When a cell grows, changes of cell shapes lead by the unbalance of growth between the volume of the cell and area of its surface. Koch discussed the division mechanisms of phospholipid vesicles by considering the property of mechanical energy [31, 32]. He pointed out that the membrane molecules synthesized by internal metabolic systems will be incorporated mostly into the inner layer of the cell membrane because the transition rate of membrane molecules between the layers is very slow. It causes an invagination of inner layer inside the cell which leads the cell fission finally. Our model demonstrates an analogous dynamic division mechanism. However, as is shown in Fig. 3.10f, a cell sometimes fails to fission. This is supposed to be caused by the instability of the conjunction of membrane. In this model, the motion of particles is restricted in the two-dimensional space. Though it can be compared to the hypothesis which proposes that the primitive chemical evolution could take place on the surface of minerals [62], it should be discussed how the membrane is organized in a three-dimensional space. We presume that the interactions we introduced here would be easily extendible into three-dimension and the organization of self-maintaining protocells and the fission of cells would be demonstrated in the same manner.
43
Chapter 4
Lattice Artificial Chemistry In the previous chapter, we introduced a model which simulate self-maintenance and self-reproduction of a proto-cell. The purpose of this chapter is to understand how such a structure as an evolutionary unit emerges from unorganized configurations through the pre-cellular evolution. We present a model to bridge two different levels of dynamics; molecular and cellular level. The time scale considered here is much longer than that of previous chapter. We thus introduce another variation of artificial cell model in order to save computation
4.1
Model
Dynamics of molecules is modeled with a discrete system of artificial chemistry. Chemicals are represented by abstract particles which react to each other and diffuse in a two-dimensional reaction space which is arrange as a triangular lattice. Note that, unlike popular CA models, any numbers of particles can be placed on each lattice site. A state of the system at the time t is represented by a set of vector, n(x, t)
= (n1 (x, t), . . . nm (x, t) . . . , nN (x, t))
(4.1)
where nm (x, t) denotes the number of the m-th type of particles in the site x. The reaction sites are arranged as a triangular lattice instead of a cubic one to avoid the effect of anisotropy of the lattice. However, the general formalism can be easily transformable to other arbitrary lattice topologies. The total number of particles is conserved throughout the simulation.
4.1.1
Repulsive Interaction
Particles are located on discrete lattice sites. They diffuse on the sites by random walk which is affected by interactive force between particles. To simulate the organization of membranes we introduce hydrophobic repulsion. The interactive potential of an m-th type of particle (Fm ) is given by a summation of all repulsion effects from the particles in the same and the adjacent six sites; Fm (x, t) =
X
|x′ −x|≤1
N X
m′ =1
45
nm′ (x, t)fmm′ (x′ − x)
(4.2)
where fmm′ (dr) gives the repulsion effects on an m-th type particle from an m′ -th type of particle. The movement of a particle is biased according to the gradient of the interactive potential. The probability with which a particle moves to the adjacent site is given by following formula, pm (x, x′ , t) =
kdif ge (Fm (x′ , t) − Fm (x, t))
(4.3)
where kdif gives the diffusion coefficient which is a common constant for all particles. The function of potential ge (∆E) is given by below, ge (−∆E)
=
∆E 1 − e−β∆E
(4.4)
where β denotes the Boltzmann constant which gives the inversed temperature of the system. We assume that the system is in contact with a heat bath so that the temperature is kept constant. The value of β is normalized and fixed to 1 throughout the all simulations below. Repulsion effect fmm′ (dr) between two particles depends on both their types and their configuration. We define three groups of particles; hydrophilic, hydrophobic, and neutral. Hydrophilic and hydrophobic particles repel each other strongly. While, a neutral particle interacts with other particles weakly. To compare the role of the organization of membranes, we consider two types of hydrophobic particles one forms a membrane and the another does not. An simple hydrophobic molecule is represented by a particle M1 (see Fig. 4.1a). Because it repels hydrophilic particles regardless of the orientation it does not form a membrane but a cluster like as oil in water. To model the organization of membrane-like structures, anisotropic interaction between particles must be taken into consideration. We introduce almost the simplest interaction of anisotropic particles which can demonstrate the organization of membranes. Figure 4.1b illustrates the repulsion of an membrane particle (M2 ). It has an axis in which it repels hydrophilic particles strongly, but it does not repel so strongly in the other directions. According to its symmetry, it can turn into three distinct orientation (M02 , M12 , M22 ). The values of repulsion potential are listed in Table 4.1.1. Rotation of an anisotropic particle is also simulated with stochastic dynamics. Probabilities of rotation also depend on the difference of the interactive potential as follows, pM h →M h′ (x, t) 2
2
= krot ge (FM2h (x, t) − FM h′ (x, t)) 2
(4.5)
where krot gives the rotation coefficient. As is shown in Table 4.1.1, membrane particles interact also among themselves. The repulsion is stronger when the orientations of membrane particles are different. It also depends on the configuration of the two membrane particles. According to these terms, membrane particles tend to arrange themselves in the same orientation. When hydrophobic particles are mixed with water (i.e. mixed with enough amount of hydrophilic particles), phase separation takes place automatically. Figure 4.2 shows representative snapshots of cluster formation both started from an homogenous initial distribution of particles 1 . Isotropic particles form 1 The numbers of each water, hydrophobic, and neutral particles on each site are 65,20,15, respectively.
46
M 02
M1
0 3
2
3
2 1
M 12
1 2
1
1
3 0
3
M 22
1
3
2
0 2
(a) Case i
3
2 3
2
1
1 0
1
2 3
(b) Case ii
Figure 4.1: Repulsion potential field around a hydrophobic particle. Depth of red corresponds the degree of repulsion on a hydrophilic particle from the hydrophobic particle in the center site. Numbers on each sites correspond to the columns in Table 4.1.1 (a) Isotropic repulsion of a particle M1 (b) Anisotropic repulsion of particles M2 .
Table 4.1: The values of repulsion potential where ǫ = 5 × 10−5 . The columns correspond to the numbered sites present in Fig. 4.1. The index of orientation is modulo 3. (a) Isotropic particle (M1 ). (b) Anisotropic particle (M2 ).
{W,A,X,Y} ↔ {W,A,X,Y} {W,A} ↔ M1 {X,Y} ↔ M1 M1 ↔ M1 (a)
{W,A,X,Y} ↔ {W,A,X,Y} {W,A} ↔ Mi2 {X,Y} ↔ Mi2 Mi2 ↔ Mi2 Mi2 → Mi+1 2 Mi2 → Mi+2 2 (b)
47
0 3ǫ 60ǫ 3ǫ 18ǫ
configuration 1 2 0 0 20ǫ 18ǫ 0 0 0 0
0 3ǫ 60ǫ 3ǫ 6ǫ 60ǫ 60ǫ
configuration 1 2 0 0 42ǫ 6ǫ 0 0 0 0 6ǫ 6ǫ 6ǫ 42ǫ
3 0 18ǫ 0 0
3 0 6ǫ 0 0 42ǫ 6ǫ
spot-shaped clusters like as oil on the surface of a chicken soup (see Fig. 4.2a). While, clusters of anisotropic membrane particles form membrane-like structures (Fig. 4.2b).
(a)
(b)
Figure 4.2: Phase separation. Red and blue represent hydrophobic and hydrophilic region, respectively. (a) Clusters formed by hydrophobic particles become round-shaped due to the surface tention. (b) Membranes formed by anisotropic particles. Due to the anisotropic repulsion, their cluster becomes thin. The lattice size is 32 by 32 and the boundaries are periodic.
4.1.2
Chemical Reactions
We introduce chemical reactions to simulate metabolism of these particles. In order to present a minimal model of metabolic system of a proto-cell, we define five distinct chemical species; A, M, X, Y and W. Chemical reactions are simulated by the transitions among these particle types. Figure 4.3 illustrates the reaction paths considered in this model. A particle A represents “autocatalytic” particle which catalyze its own replication by consuming a “resource” particle X. It can also catalyze the production of “membrane” particle M by consuming a particle X. These particles decay into “waste” particles Y which are recycled into X by an external source. A particle W plays the role of “water”, and do not change into any other chemical. We assume that particles A and W are hydrophilic, a particle M is hydrophobic, particle X and Y are neutral, respectively.
4.1.3
Reaction Kinetics
We assume that particle A replicate itself using itself as a template as below, X + Atemplate + Acatalyst ↔ Acopy + Atemplate + Acatalyst .
(4.6)
Therefore, the reaction rate depends on the both concentration of template and catalyst particles. Since, they are both particles A, the effect of catalyst should be approximated by a function proportional to the square of the concentration of the autocatalyst below, k1 (x, t)
= k0 + CA n2A (x, t)
(4.7)
k2 (x, t)
= k0 + CM n2A (x, t)
(4.8)
where k1 and k2 represent the coefficients of the reaction X ↔ A and X ↔ M respectively, k0 denotes the coefficient of spontaneous reaction, CA and CM 48
X
chemical reaction
k1 catalysis
k2
Sx
A resource supply
k3
M k3
k3
W
Y
Figure 4.3: Reaction paths. Particle A produce A or M from X. All particles decay into Y. There is a source which supply X denote the activity coefficient of the catalysts, and nA (x, t) gives the number of particle A on the site x at the time t. The coefficient of spontaneous decay is given as a common constant (k3 ). In addition to these reactions, we assume there is an external source which change Y into X with a constant probability SX . The reaction coefficients between particle X and Y are given as below, X Y
k
3 −→
k3 +SX
−→
Y
(4.9)
X.
(4.10)
In general, a catalyst lowers the potential barrier of a chemical reaction so that it increases the reaction rate of the substrates, but it do not alter the rate constant of the reaction itself. Therefore, we assign chemical energy (Gm ) to each chemical species and calculate the transition probability from the difference of potential between two species. The potential of resource particle X is the highest and that of waste particle Y is the lowest 2 . The transition probability of chemical reaction is given by following formula, pm→m′ (x, t)
=
kl ge (Gm′ + Um′ (x, t) − Gm − Um (x, t))
(4.11)
where kl gives the corresponding reaction coefficient which depends on the number of catalyst on the site as it shown in Equ. 4.8. It is worth noting that the change of interactive potential (Um′ − Um ) of the particles is also taken into account here. According to this term, the environment of particles effects on the rate constant, for example, it becomes difficult to synthesize a hydrophobic particle in the hydrophilic environment. The coefficient β which represents 2 The
values of chemical energy are, GX = 12, GA = 6, GM = 2, GY = 0.
49
the Boltzmann constant (i.e. inversed temperature of the system) is fixed to 1 throughout the simulations below. The simulation is based on a Monte-Carlo method. At each time step, we calculate the transition probabilities for all lattice sites and all particle types as follows3 , 1. Compute the potential of particles 2. Compute the probabilities of particle moves (to adjacent site) and rotations. 3. Compute the effects of catalysts 4. Compute the probabilities of chemical reactions and resource supply 5. Update the state of particles according to these probabilities synchronously.
4.2 4.2.1
Results Division of Self-Maintaining Cells
First, we demonstrate the evolution of a single cell configuration shown in Fig. 4.4. The cell contains the enough number of particle A to sustain metabolism inside and it is enveloped by membrane of particle M.
Figure 4.4: Initial configuration. The depth of red, green and blue corresponds to the number of particle M, A, and W on the site, respectively. We have shown that this structure can maintain itself. The on-going processes are, particle A within the cell keeps replication and products particle M which is incorporated into the membrane. The generated membrane keeps inner particles from diffusing away. Further evolution of a cell is shown in Fig. 4.5. Because the resource and waste particles are not repelled by the membrane particles, particles X and Y can permeate through the membranes. This causes osmosis of resource particles. Namely, particle X is consumed within the cell, its density becomes lower inside the cell than in the environment. According to the gradient of the density, particle X diffuses into a cell from outside. A cell assimilates the external resource particles resulting in the growth of its size (Fig. 4.5b). When the cell reaches a certain size at which it has outgrown its stability, it begins to generate a new membrane inside because it produce 3 The values of other reaction coefficients are, k −6 , dif = 0.02, krot = 0.001, k0 = 1 × 10 CA = 7 × 10−7 , CM = 4 × 10−7 , and k3 = 1 × 10−4 , SX = 25.
50
more particle M which can be incorporated into the membrane (Fig. 4.5c,d). This finally divides the mother cell into daughter cells (Fig. 4.5e,f). These new cells repeat the process of growing and dividing recursively.
(a)
(b)
(c)
(d)
(e)
(f)
Figure 4.5: Snapshots of cell division (t=3000 (a), 6000 (b), 9000 (c), 12000 (d), 15000 (e), and 18000 (f)). A cell grows assimilating resource particles from the environment. When the size reaches to a certain threshold, a new membrane starts to grow inward. Finally, it divides the original cell into daughter cells. It is worth noting that the metabolic system inside the cell and the membranes are mutually dependent. For example, Fig. 4.6 shows that this structure decays if the membrane has a defect, though the composition of particles and reaction coefficients are identical to the previous simulation. Without enclosed by membrane, particle A diffuses fast, their density becomes too low to sustain their replication. Then, without the supply from the metabolism inside, the membranes also decay into Y and disappear finally.
4.2.2
Variations of Cells
Flexibility of membranes depends on the values of repulsion among membrane particles. It also affect the process of cell fission. Examples are presented in Fig. 4.7. A strong repulsion between particle M results in the formation of stiff membranes, and the shape of cells becomes more regular (Fig. 4.7a and 4.7b). On the other hand, Fig. 4.7c represents a cell with more flexible membranes, which form in the presence of low repulsion values for particle M. These cells divide themselves rather irregularly. The values of their repulsion potential are listed in table 4.2.2.
4.2.3
Emergence of Proto-Cell Structures
Though in the experiments above, a cellular structure is given as an initial configuration, it should be discussed how the first cell is organized, whether 51
(a)
(b)
(c)
(d)
Figure 4.6: A breakdown of a cell (t=2000, 4000, 6000, 8000). If autocatalytic particles are not fully enclosed by membrane, they diffuse away through the defect. Their density falls too low to sustain metabolism. The rest of membrane also decays at last.
(a)
(b)
(c)
Figure 4.7: A variations of cell shapes and their way of division (t=18000). According to the strength of repulsion potential around a particle M, the cell shapes and the way of division changes in the various manners.
52
Table 4.2: Variation of membranes. Each table corresponds to the pictures in Fig. 4.7.
{W,A,X,Y} ↔ {W,A,X,Y} {W,A} ↔ M2 {X,Y} ↔ M2 Mi2 ↔ Mi2 Mi2 → Mi+1 2 Mi2 → Mi+2 2 (a)
{W,A,X,Y} ↔ {W,A,X,Y} {W,A} ↔ M2 {X,Y} ↔ M2 Mi2 ↔ Mi2 Mi2 → Mi+1 2 Mi2 → Mi+2 2 (b)
{W,A,X,Y} ↔ {W,A,X,Y} {W,A} ↔ M2 {X,Y} ↔ M2 Mi2 ↔ Mi2 Mi2 → Mi+1 2 Mi2 → Mi+2 2 (c)
53
0 3ǫ 60ǫ 3ǫ 3ǫ 60ǫ 60ǫ
configuration 1 2 0 0 48ǫ 3ǫ 0 0 0 0 3ǫ 3ǫ 3ǫ 48ǫ
0 3ǫ 60ǫ 3ǫ 6ǫ 60ǫ 60ǫ
configuration 1 2 0 0 48ǫ 6ǫ 0 0 0 0 3ǫ 3ǫ 3ǫ 48ǫ
0 3ǫ 60ǫ 3ǫ 6ǫ 60ǫ 60ǫ
configuration 1 2 0 0 42ǫ 6ǫ 0 0 0 0 1ǫ 1ǫ 1ǫ 42ǫ
3 0 3ǫ 0 0 48ǫ 3ǫ
3 0 6ǫ 0 0 48ǫ 3ǫ
3 0 6ǫ 0 0 42ǫ 1ǫ
such a structure is able to be organized by chance. In this section, we study an evolution from an unorganized initial configuration. We compared two cases; (i) A metabolic system produces isotropic hydrophobic molecules which do not form membranes (see Fig. 4.2a). (ii) It produces anisotropic molecules which organize membranes automatically (see Fig. 4.2b). Simulations start from a homogeneous initial configuration 4 , where each site has the enough number of X and A particles to sustain replication. At first, the dominant evolution is the production of A and M from X, catalyzed by A. When a sufficient number of M have been produced, a phase separation between M and other particles takes place. In the isotropic case, M accumulates into clusters or spots, like oil in water. Because the spots Being maintained by the metabolic cycle, the spots present a stable periodic pattern which is similar to Turing patterns [59, 44, 36]. While, in the anisotropic case, membranes are formed according to the anisotropy potential. Sometimes they form a closed barrier which creates a domain separated from the rest of reaction space.
(a)
(b)
Figure 4.8: Evolution from a homogenous initial configuration (t=6000). (a) Isotropic case: they form periodic spots. (b) Anisotropic case: they compose membrane-like structures. The lattice size is 64 by 64.
The evolution after the organization of separated domains. different from isotropic case. A competition occurs between the different sides of the membrane due to the osmotic effects. When two domains share a common bounding membrane, one of the domains outperforms the other in the sense that one loses the internal metabolic reaction. The mechanism of this competition is simply osmosis caused by the gradient of X between the two sides. If the concentration of A is slightly higher on one side than the other, X particles are consumed more rapidly on this side. According to the gradient, more X particles will come to this side, and more are catalyzed into A particles. This positive feedback makes one side rich in A and the other poor. As a result, one domain wins whenever two domains share the same membrane. Once a few selected cells exist, a reproduction process will start. With the 4 The numbers of particles A, M, X, Y, and W on each site are 15, 15, 10, 10, 50, respectively.
54
(a)
(b)
(c)
(d)
Figure 4.9: Emergence of a proto-cell structure from a homogenous initial configuration (t=12000 (a), 18000 (b), 24000 (c), 30000 (d)). Due to osmosis between separated domains, only a few domains enclosed by membranes can survive.
55
assimilation of particles by osmosis, a cell grows in size. When the size of a cell reaches a certain threshold, new membrane partitions begin to form within it. At last, a cell is divided into several daughter cells. This process takes place recursively as long as X particles are supplied. The amount of A can be computed as a function of time. In the isotropic case, it gradually decreases to zero. In the anisotropic case, it first decreases to a certain level, then begins to increase as cells reproduce. In sum, the emergence of reproductive cell goes as follows: 1) Starting from a random initial state, membrane particles form entangled clusters. 2) When those clusters form a closed barrier, they can enclose metabolic reaction. 3) Competition occurs among domains and a few domains outperform others. 4) Reproduction starts from these selected domains, which we name “cells”. The process is clearly demonstrated in Fig. 4.10. In this experiment, the parameter SX (supply of resource X) is set too low to cause autonomous reaction (SX = 20). In the isotropic case, A and M particles gradually decay and finally disappear (Fig. 4.10a). In the anisotropic case, on the other hand, a few cells are selected to further grow and reproduce (Fig. 4.10b). At last, a cell is divided into several daughter cells . This process takes place recursively as long as X particles are supplied.
4.3
Summary
We demonstrated the model of self-maintaining proto-cells. It develops a symbiotic structure of metabolic system and membrane: the internal autocatalytic cycle supplies membrane molecules, and the membrane keeps the internal molecules from diffusion. This self-maintaining cell grows and fissions spontaneously. Moreover, we showed that a proto-cell emerges from an unorganized initial configuration and that the organization of a cell gives an advantage to internal metabolic system in respect to accumulation of resources. These results suggests that the evolution from molecular to cellular reproducers could take place automatically. When it is assumed that a variety of autocatalytic molecules can be contained inside the proto-cell, the relative amounts of those components will be inherited to their daughters. Such a componental genetic information could bridge limited variation of reproducers and an unlimited variation of replicators. Fission of cells in this model mostly depends on the instability of cell size. It can be compared to reproducing spots in the two-dimensional reaction-diffusion system [51] [36]. In the simulations presented above, cells do not leave away but stick together when they fission. It is because that, membrane formation is simulated by anisotropic clustering of hydrophobic particles instead of organization of bilayer membrane which consists of amphiphilic molecules. We presume that it does not make difference when we consider a proto-cell structure as a unit of evolution because the It is the selective permeability that leads the emergence of proto-cell. It is supposed that the earliest membranes might have been simpler and rougher than the phospholipid membranes of present cells, then, acquire the more adopted membranes and mechanism of transportation through the co-evolution of metabolic system and membranes. The evolution of function of the membranes e.g. speci56
(a)
(b)
(c)
(d)
(e)
(f)
Figure 4.10: Emergence of proto-cell under a low resource supply (t=15000 (a,b), 30000 (c,d), 45000 (e,f)). Autocatalytic particles cannot sustain metabolism without enclosure (c,e), while they reproduce with proto-cell structure (d,f). 57
ficity of permeability, active transportation signaling etc. must be considered in the future studies. It is worth noting that, in this model, the effects of repulsion potential are taken into account when the transition probabilities of chemical reactions are computed. According to this term, when there are membrane particles in the neighbor of a site, they lower the supposed potential of a new membrane particle, so that they let the synthesis of a membrane particle easy. It represents the autocatalytic function of proto-cells discussed by Segre and others [55]. When a living cell modeled theoretically, membranes were often disregarded and represented mere boundary conditions. However, it was pointed out that a membrane can catalyze chemical reactions directly and would play a significant role in the pre-cellular evolution. Our model provides a basic framework to investigate the roles of membrane simulating the interaction between metabolic systems and the spatial patterns. We have shown that the cell divides in a different manner according to variation in the interaction strength between M. It can be compared that the flexibility of phospholipid membranes depends on the component molecules. For example, lipids consisted of unsaturated fatty acids soften the membrane, while some cholesterols harden it [9].
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Chapter 5
Discussion and Future Works To demonstrate the models of self-maintaining proto-cell, we have designed a metabolic system composed of primitive autocatalysts (we here considering ribozymes or self-replicating peptides) which also produces molecules to organize membrane-like structures. We presented three different models of autopoietic self-maintaining structures in a one-dimensional and two-dimensional space, discrete and continuous. Our models are based on stochastic dynamics of artificial chemistry. Though the models are abstract and artificial, they provide a way to investigate the dynamics of meso-scopic and long-time scale evolution (e.g. from a pre-cellular stage to formation of multicellular organisms) which is difficult to simulate with in vitro experiments, or rigid theoretical models. Further, because the interactions of particles are all calculated locally, so that it could be readily implemented with parallel distributed computing system.
5.1
Self-Reproduction of Autopoietic Structures
As Maturana and Varela pointed out, it is the autonomous organization of boundaries that creates a living structure. A boundary of a proto-cell is considered as a structure which restricts the interaction between a subsystem and its environment— in other words, this restriction allows us to find a subsystem which is separated from the environment from the viewpoint of the dynamics of molecules. With this respect, self-organization of an autopoietic structure is distinct from mere cluster formation. The difference becomes clearer when some new molecules are introduced into the system because membranes could protect the inner metabolic system from invasion of molecules which may destroy it. While, a membrane is not only produced to maintain the cell boundary but related to various chemical reactions. For example, a lipid membrane catalyzes the synthesis of membrane molecules by itself [1, 64, 55]. And a membrane provides the difference of concentration of chemicals between inside and outside the cell to sustain metabolic reactions. Our third model introduced in section 4.1 may be generalized to analyze how membranes influence on metabolism in a direct fashion because we simulated both chemical reactions and spatial 59
interaction of molecules in the unified manner. In the section 4.2.2, we showed that stability of cell shapes and a process of cell fission depends on the property of membrane molecules. Our models also provide a way to understand the dynamics of membranes coupled with the metabolic system, this allows to bridge between molecular and cellular dynamics; the growth, reproduction and death of a cell. We showed that an autopoietic self-maintaining structure readily performs self-reproduction. It is observed in the all three models as a natural consequence of the growth dynamics of a cell (see section 2.2.4, 3.4.2, and 4.2.1). The fission of cell seems to be triggered by (1) instability of too large cell and (2) instability of cell shape due to the dynamics of membranes. Similar process of cell fission is expected to be observed in the models extended into a three-dimensional space. We presume that spontaneous reproduction of autopoietic systems would be observed more generally. Because the boundary of an autopoietic system is not defined as a given condition but as a dynamical structure, there would always remain a potential for the creation of a new boundary to produce a new structure.
5.2
Emergence of Proto-Cell Structures
A notable observation in the section 4.2.3 presents that proto-cell structures emerge automatically from an unorganized initial condition.Further, the result exemplified in Fig. 4.10 numerically proves that a proto-cell structure enclosed by membranes is indispensable to sustain metabolism in a severe environment. A metabolic system which produces membranes could evolve through natural selection and it would result in the emergence of cellular structures. This result presents an example of the organization of a unit of self-reproduction. This model can be further extended introducing heredity of metabolic systems to demonstrate the emergence of a new evolvable individual of higher-order; the transition from pre-cellular to cellular replication. We hope that investigation along this line provides a key to understand the other major transitions [41], for example, the establishment an individual of multicellular organisms. We have assumed that the production of membrane molecules depends on the catalyst of inner metabolic system. Though self-reproducing micelles/vesicles (e.g. lipid world [55]) could precede such a system, the symbiosis of membrane and metabolic system is necessary to compose a self-maintaining cell.
5.3
Inheritance and Evolution
In the chapter 2, we demonstrated two different ways of self-reproduction of a proto-cell structure; open and closed reproduction. Though open reproduction is not observed in the two-dimensional model within the parameter regions where we studied, such process may serve as a because it would provide wide range of variety when the reproducing molecules are contaminated. When a proto-cell divides, molecules inside the cell are distributed randomly between the daughter cells. Assuming that some distinct linage of selfreproducing molecules are included in the cell, their composition is also inherited by their daughter cells with some random fluctuations. Through this composi60
tional heredity, this cell structure can be a unit of Darwinian evolution. It can also inherit dynamical states of metabolic system, e.g. the attractor it stays, the phase of oscillation, etc. which would provide another way of inheritance [20]. It is worth noting that evolvability of a proto-cell structure is controlled by the membrane system. For example, a cell may come to smaller which contain fewer molecules in order to reproduce faster. Because a smaller cell contains fewer molecules within it, the fluctuations of their composition is amplified. Therefore, their evolution through compositional heredity will be accelerated. They can also evolve some mechanism to divide the necessary compositions in more accurate ways. For example, the genes of bacteria are attached to their membrane. It seems to allow to distribute each gene surely when the cell fissions [7]. It was pointed out that the diversification of a cell society depends on the intercellular communications [26, 27]. A cell membrane can decide which molecules permeate through it to control the diversification.
5.4
Future Studies
There remains many promising future works. Our next objective is to simulate the development of functions of membranes. First, a more complex metabolic system which produces diverse molecules should be considered in the future studies. The experiment shown in Fig. 4.7 presents that this model is able to simulate various shapes of cells. Other properties of membranes also can be controlled through the local interactions of each membrane molecules. For example, we have already mentioned of the diversification of cells which depends on the intercellular communications. Varying each values of repulsion between a membrane molecule and other molecules, a cell membrane can perform various selective permeability. Difference of adhesion among membrane can be introduced to model the formation of a multicellular organism. When cells form a cluster, their metabolic dynamics may differentiate spontaneously through the interaction among them [20]. Assuming that their membranes show different adhesion when they differentiated, various multicellular structure patterns are observed [19, 24]. The results shown in Fig. 4.9 and Fig. 4.10 present that the selective permeability of the membrane is the key property which causes the spontaneous emergence of cell structure from a homogeneous configuration. Though a membrane of a proto-cell is supposed not to contain complex devices, which control transportation of molecules as the membrane of present cells does, it can filter molecules simply with regard to their size because a primitive membrane could be rough enough to pass small molecules but not large complexes. It should be discussed the evolution of permeability of primitive membrane including the acquisition of active transportation capability. The organization of membranes which perform selective permeability can provide flows of particles in a non-equilibrium environment. Though it is difficult to formulate the “efficiency” of a metabolic system, it is supposed that the evolution of cells improves the method of assimilating nutrients, in a sense. Our model allows to analyze the flow of materials across a cell membrane to investigate how it develops. Effects of the organization of an autopoietic structure on 61
the metabolic system and its role in the evolution of cell should be discussed in the future studies.
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Acknowledgments I greatly appreciate the tuition of Dr. Takashi Ikegami throughout these five years. I fully owe this work for his supports. I’m thankful to Dr. Hashimoto, Dr. Yoshikawa, Dr. Harada, Dr. Sato, Dr. Nishimura, Ms. Matsumoto, and Dr. Igari, for their obligingness and favors. To Mr. Gentaro Morimoto, I would give lots of thanks for all the troubles he has taken. I would like to express my gratitude to Dr. Kunihiko Kanako and Dr. Shinichi Sasa for their helpful advice. Thanks so mush to the all colleagues of Kaneko, Ikegami and Sasa Laboratories.
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