Chapter 1
For the logic of living system as a complex system
・ Preface
One of the most important purposes for biophysics is to extract some universal structure out of living phenomena and make clear the logic standing over the structure.
※ For example, thermodynamics and the theory of evolution are known as some of its successful examples.
We need to know processes in a living body deeperly to further understand living phenomena.
This tide itself was a main motivation for the birth of molecular biology.
But, at present ( about 2001 ), the viewpoint of molecular biology for life is insufficient to catch the logic of life.
In some sense, its viewpoint for life is very mechanical, it's suitable 'to compose an elaborate molecular machine' with the theory of evolution.
Roughly saying, stances of molecular biology are as follows.
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( 1 ) making clear roles of respective components in micro level (≒ drawing genes bearing specific roles)
( 2 ) next, drawing grammar rules to determine combination relations among respective components ( for example, searching for rules to activate and suppress some gene through other genes )
( 3 ) respective components are combined on rules to know how functions appear ( observing how respective genes in genomes are combined for the expressions of functions )
( 4 ) checking how efficient elements, patterns in function are chosen through evolution out of enormous cases in combination
For example, the cell-biology at present ( about 2001 ) is taking a way to think that the concentration of signal-molecule is varied through the appearance of gene, the gene also is generated beyond the threshold of the concentration.
Namely, it's still the main stream to find out a gene with specific role ( namely, '1 to 1 - correspondence' type method ).
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But, after the completion of the Human Genome Project, the course has been steered in the direction to look for functions of living being which are originated in combinations of such genetic expressions.
In other words, the progress of sequential generation is interpreted as a combination of logical operations such as 「If ( the concentration of signal-molecule is bigger than some threshold ), then ( some gene is at on or off )」.
As the concentration of signal-molecule is varied according to on/off of the gene, processes of generation are expressed with combinations of tremendous logical formulae.
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We take ways different from complexities in the combination theory.
Namely, as there are problems which can not be caught only by dividing into if-then type logical processes, we take a standpoint to first understand 'the whole' for understanding 'parts' .
Here, we review in-sufficiencies of molecular biology to understand life more detailedly.
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( a ) Considering a problem on if-then type logical rules, as only molecular reactions are generated in each cell, consequently, errors on logical operation can appear.
Meanwhile, living phenomena including cellular phenomena, molecular machines make progress under very large fluctuation.
※ For example, the number of signal-molecules N ＝ 1000 for one generation, taking the fluctuation √N into account, errors on the appearance of gene occur at the rate of 1/√N beyond some threshold.
Then, the variation on the expression of gene through the concentration of signal-molecule is equal to dividing the state of some cell into two fields for respective states of chemical composition.
In a word, considering many patterns of genetic expression is equal to dividing some state into many fields.
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But, taking a fact that some genetic expression itself is fed back to the state of gene into consideration, the number of border-lines for dividing rapidly becomes enormous.
And, if each division has a fluctuation of about 3 % , those divisions turn to be chaotic.
So that, the probability of the progress of generation according to 'program' gets to be about 0 .
※ Namely, dividing or classifying itself becomes a general catastrophe or a wild space in general topology.
So, the fed-back-ground tends to be a chaos or a random noise.
The above question is one of the universal problems in natural phenomena.
An other viewpoint for the stability of system is required to solve this problem.
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Actually, a group of cells may be stabilized through interactions among cells.
For example, in the case of equivalence (cellular-) group, if parts of equivalent cells are influenced by a fluctuation, the interaction among cells prevents surrounding cells from branching in the direction toward such influenced cells.
And, the community-effect is known, a synergetic effect makes embryonic cells determine the direction of differentiation in this effect.
Accordingly, mechanisms of stabilization through interactions must be investigated.
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( b ) Roles of molecules ( genes ) depend on various conditions such as time, position on the process of generation.
And, if a molecule as a cause ( origin ) is removed or specific genes are knocked out under some situation, other molecules or genes act in place of those removed carriers.
So, it's not desirable to determine roles of respective elements at the first step.
It's needed as a viewpoint that roles of respective elements are dynamically determined and varied through interactions over the whole system.
And, we have to consider of the cause of living system including the network of genes, the metabolic network and the transmission system of signals.
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( c ) Furthermore, the living system seems to have an universality which can't be reduced to mere combinations of individual genes with individual molecules.
※ For example, stability, irreversibility on the developmental process.
As in the thermodynamics, the stability in macro scale and the irreversibility are two different aspects of the same thing.
Actually, a specific directivity to lead from an ES ( Embryonic Stem ) cell to a determined cell through a stem cell can be found out in a cellular system.
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Possibilities of differentiation successively go to be reduced in an usual developmental process.
Of course, ultimately, we must be able to explain the irreversibility of life and/or death of an individual.
Clearly, the combinational method is not enough to this subject.
We think that it's effective to study dynamic complementary (≒ mutually compensating) relations between a whole and its part for making clear the universal properties of living system.
Outlines for that are shown in the following.
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Chapter 1 For the logic of living system as a complex system
・ Constructive Methods
By the way, which holds on life, A 「Complexity in the theory of combination」 or B 「Dynamic relation between part and whole」？
Probably, a living being has both features.
Therefore, to understand the essence of life, we should question which is more effective for that, A or B .
However, as living beings at present are historical, they are very intricate.
Accordingly, even if we study them in a frontal way, we can't decide whether they are originated from accidents on ways of evolution or necessary characters of living system.
So that, we have to face to 'synthetically created living systems' as a possible way.
Then, in the viewpoint of 「the theory of combination」, as existing lives have been extracted from very ingenious combinations, we must imitate existing living beings as faithfully as possible to create a living system.
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Meanwhile, in the viewpoint of 「the complex system」, as life has some universal properties, if respective conditions are properly prepared, such universal characters spontaneously appear.
By the way, what are those universal characters ？
Then, it's useless to adhere to shapes and chemical compositions of life on the earth.
Conversely, we have to consider universal properties which are independent of evolutions and accidents on the earth and stand on every planet.
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Roughly saying, living being has following features.
・ A living being has the inside and the outside.
・ The inside has an autonomy to maintain its state at some extent.
・ And, the metabolic activity for sustaining state has complexity at some extent.
・ However, life has possibility of multiplication.
・ Offsprings which are not completely same as the original but are very similar to that can be replicated.
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・ The living system has parts to bear informations for replication.
・ In the case of multi-cellular organism, branched cells respectively bear different roles, have proper developmental processes and such properties are inherited to next generations.
・ Diversities in shape, function are promoted through evolution.
What are minimum conditions to bring out such characters namely universal properties ?
If such minimum conditions can be caught, we will be able to compose a living system by preparing those conditions.
This way antagonizes the claim from the side of the theory of combination, this is called 'constructive biology' .
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This method considers only minimum processes to understand universality and necessity in living phenomena rather than adding numerous details according to this reality.
For example, a 'pouch' of sufficiently many chemical components is supposed, its boundary ( membrane ) can penetrate only a part of components.
The 'pouch' grows with taking chemical components from the outside for the inside and makes reaction to that in the inside.
And, the pouch is divided beyond some critical point in size.
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Fundamental properties in living system are explored (≒ groped) under minimum settings as above.
We investigate constructive conditions to make a class of phenomena corresponding to those fundamental properties appear.
Namely, the constructive method is a method to re-interpret a real world as a class obtained through such ways.
In the sense, this can be regarded as a 'very physical' way to interpret life ( or a 'very bio-logical' way to approach to a physical world ).
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Such a research can be carried out within the range of '① pure thought experiment' .
But, we have never been able to sufficiently get to the intuition for dynamic mutual circulating system between the whole and its parts.
Accordingly, it's needed to compensate the intuition hard to get with logical thinking through '② computer-simulation' .
Furthermore, when the subject is the system with very various degrees of freedom such as living being, even computer-simulation is in-sufficient to approach to that.
Namely, we may overlook significances of processes ignored as minor details.
So, the third way namely 'construction in a laboratory' also is required.
Actually, these three ways and factored processes are jointly used to explore 'biologies in possible worlds' .
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Here, we should emphasize to search for fundamental logic and universality underlying of living system through creating rather than composing a living system 'like a magic' .
Next, taking inter-acting dynamical systems into consideration, it's introduced as a viewpoint that the flexibility of system itself is settled into differentiations in species.
However, some theoretical unsolved points are included in getting the flexibility of system as a premise.
※ The 'flexible handle' (≒ Casson-handle) with non-integer valued dimension between 4 and 5 ( non-integer valued correlation dimension ) is considered as a model.
And, as a hyper-cube which is often referred in network-topology is homeomorphic to a B＾4 ( 4-dimensional ball ) as a bubble, such a 'flexibility' of 4-dimensional topology can be conjugated as the back-ground to support the flexibility of system.
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・ About construction of replicating system
The construction of proto-type of cellular system is known as one of the typical constructive methods.
※ Multiplicating systems from systems of chemical reactions including various components such as DNA-polymerase have been already constructed.
Researches of the origin of life have had a bias toward exploring good replicated machines until now.
On the other hand, recently ( about the end of the 1990's 〜 the beginning of the 2000's ), we are investigating how recursive productions appear out of various and inexact multiplications and factors to bear informations are generated.
And then, if a 'minimal ideal cellular system' is composed by enclosing such a reaction system into a membrane in multiplication, we can check the validity of constructive method.
※ Such experiments are being continued by Yomo ( 四方 ) and others ( about the beginning of the 2000's ).
By the way, what are theoretical problems in the construction of replicating system ?
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For example, in the case of a pouch of chemical reactions, the proto-type of differentiation or developmental process appears without distinctly assigning a role to each gene.
Of course, this does not mean that genetic informations are in-necessary to generate such processes.
However, it's important that differentiating process results from interacting systems without 'programming' other respective reactions by genes.
But, as it seems that genes control other factors, it can be regarded as a fact that memories with genes become more stable.
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By the way, how should we understand roles of such genes ?
How were specific components so as to control other factors consequently branched out of many chemical components ?
The separation of roles ( or factors ) into a side to produce genetic informations and another side bearing metabolism is equal to a transition from an in-differentiated state like a chicken-and-egg problem to a state of divided roles.
Then, what are properties which are possessed by the side of genes controlling other factors ?
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Taking usual DNA s and proteins into consideration,
① differences in quantity
② differences in stability of molecule, in other words, time-scale for reaction
③ differences in number of kinds of factors functionable as catalysts
are presented.
In general, considering a network of chemical reactions, the difference in reactivity depends on chemical components and the difference in time scale also depends on kinds of chemical components.
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So, the difference in quantity appears according those conditions.
In this case, does it hold that a minority in number controls other factors for a long time-scale of generative-decomposing reaction ?
In the model to describe variations on the concentration of chemical component regarded as a 'quantity' with differential equations, we have never known how the above logic is described at present ( in 2001 ).
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It's possible that the initial condition of slowly varying quantity namely a variable is set up so as to finally have a role as a controlling parameter.
Because, in this case, as the 'multiplication' is requested as a character, the type of cell at some initial condition must be same as that of the next generation at another initial condition in order that only offsprings with same cell-type as that of the former generation will be left.
Respective initial conditions receive restrictions due to such repetitions.
Furthermore, some group of specific variables is precisely limited in its variation because of those restrictions, meanwhile other variables can be chaotic in its variation.
※ In other words, the set of initial conditions is regarded as a generative system namely 'core' .
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So that, the former group effectively play a role as a parameter for controlling to the latter.
Namely, then, as the progress of the latter is regulated by the former, the former is used as an information to determine the initial condition (→ synchronization or hemi-synchronization & control of chaos).
Besides, we must consider about the limits to way to think with dynamical systems on such continuous quantities.
Actually, numbers of respective chemical molecules included in each cell are not so much.
Each number is about below 1000 , it's far from Avogadro's number.
※ On the other hand, researches taking fluctuation into account also were carried out, it has been demonstrated that the process of differentiation is 'robust' to fluctuations in some range.
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But, when each molecular number takes a discrete value ( for example, 0, 1, 2, ・・・ ), properties which are not catchable from a viewpoint of 「continuous quantity ＋ fluctuation」appear.
For example, when one chemical component synergetically increases as a catalyst for another, in general, composite velocities of respective components are not equal each other.
Then, according to developmental equations of dynamical system, only faster components in velocity for multiplication get to be more than other components.
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Meanwhile, each component needs one molecule in a minimum in order that a 'cell' of these both components continues to divide itself.
Then, the dis-agreement between the evolution of dynamical system and that of multiplicating system is caused, so that the remarkable difference between a state with one molecule and a state without molecules appears.
Actually, in such a model, a component at slower composite speed chooses a very rare initial condition to suppress errors in replication.
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In a word, characters originated from differences of components at slower speed in the number of discrete molecules ( 0, 1, 2, ・・・ ) can not be captured by continuous dynamical systems.
And, these differences give remarkable influences on behaviors of system.
So, a structure that a small number of molecules controls other many molecules appears as a pattern, this pattern can be regarded as the appearance of information.
Of course, as this means that variations in minor part give remarkable influences on the whole system, it has a significance even for 'evolvability' .
Furthermore, this agrees with an experimental result to show that the number of genes must be small for evolution.
※ For example, the above mismatch ( between the evolution of dynamical system and that of multiplicating system ) is equal to a dis-agreement between dimensions to be able to cause topo-logical defects.
And, 'the formation of irregular nebula ( a little mass brings a 'disturbance' to the majority of the system in mass )' in cosmology is similar to the above pattern.
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Chapter 1 ,
1.2 Interactive dynamical systems 〜 ( 1 )
To interpret the living system as a kind of dynamical interactive system, we considered about interactions among some multiplicating units with inside (≒ endo-) degrees of freedom.
In this case, the dynamics of internal degrees of freedom is influenced by interactions among units, meanwhile interactions among units also are influenced by variations on internal states.
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Such reciprocal feedback between inside dynamics and interactions among units , and relations of dynamical characters in macro level with those in micro level are being considered in dynamical systems with many degrees of freedom.
In the sense, we may follow a stance as 'dynamical system with many degrees of freedom ＋ fluctuation' .
The exploration for biology on such dynamical systems is originated from morphogenesis (≒ topo-genesis) by A.Turing.
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However, we can't ignore properties for multiplication to consider about living phenomena including developments, evolutions.
Namely, when each unit is settled in a specific stable state through developments on dynamical system, the stable state is not necessarily a fast state in multiplication.
Accordingly, we must take dynamical systems to gather with characters for multiplication into consideration for the above aim.
Here, the multiplication of cell as unit is started with the di-vision of one cell.
And, a 'society' is formed of respective cells generated through repeated divisions.
※ In an other viewpoint, this means that the limit of those divisions corresponds to a Cantor set, the 'society' of those fractions corresponds to a lattice or a semi ( hemi )- lattice re-composed of subsets.
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Next, we show an ideal of cellular system.
Then, the concentration of chemical component j in each cell i is expressed with xj ( t )i as a variable.
This system is composed of only following three processes.
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① The network of bio-chemical reactions is simplified as an internal dynamics.
② Taking chemical matters from outside culture media into cells and its scattering ( diffusion ) are regarded as interactions.
③ Cell divisions to satisfy conditions for multiplication are caused by progresses of chemical reaction.
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Then, the internal dynamics is an independent system, has some degrees of freedom and is not simplified so much.
Namely, this system is robust to fluctuations, autonomously grows and can maintain its structure regardless of heating.
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Following simplifications were prepared to compose the above model.
( 1 ) Spatial structures and organs in a cell are ignored, merely, a soup of some chemical components is assumed.
( 2 ) Respective states are expressed with only quantities of respective chemical components without considering variations of respective molecules on internal structure.
Namely, the time-scale of variation in a molecule is sufficiently shorter than the variation in the dynamics of cell.
Or, in this case, variations at long scale are ignored to consider of developmental processes.
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( 3 ) To describe variations of chemical components, a method to roughly catch called rate-equation on chemical reaction is used.
And, fluctuations are introduced through methods like Langevin equation, the discreteness of the number of molecules is taken into account.
The same as that, methods to roughly catch such as diffusion or active transposition are used to describe interactions.
( 4 ) Owing to the accumulation of chemical matters in a cell, when the size of the cell gets at some critical point, the cell is divided into equal parts.
It's not possible to determine whether an important feature of cellular system is obtained or not at this step.
But, as results of a system of stem cells shown by Furusawa, actually, proto-types of development, differentiation appear under above settings.
In the following, the logic derived from such researches is briefly explained.
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1.2〜( 2 ) Interactive dynamical systems
In the developmental process, as a cell containing sets of same genes is divided into some parts, each divided cell has a chemical composition different from others.
Such divided cells respectively bear different roles in a living body.
The morphogenesis of living being makes progress
together with the placememnt of these divided cells .
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By the way, is such a differentiation - developmental process very cleverly designed - controlled ？
Or, is it an universal phenomenon as seen on above settings ？
Here, we paid attention to a network of reactions whose concentration of chemical component oscillates as the dynamics of one cell, and we investigated multiplications started with one cell.
Then, a group of cells is developed in order of following steps together with multiplications.
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Step 1 ; All oscillations of chemical components in respective cells are in synchronization, then respective cells are simultaneously divided.
Step 2 ; Oscillating phases are grouped into some clusters ; the synchronization of respective cells disappears, those cells are grouped into many aggregations oscillating at different phases.
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Step 3 ; differentiation ;
respective cells start to have chemical compositions different from others.
Namely, respective cells start to use (≒ be conjugated with) different domains of the phase space.
So that, respective cells bear different roles from others.
Step 4 ; recurrence ;
characters of divided respective groups are inherited to cells as offsprings.
Namely, a cell differentiated from some type of cell has a same type as the original cell.
Step 5 ; hierarchy ;
as those cells further go to be sub-divided, the differentiation of cell-kinds becomes hierarchical.
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In the step 2 , respective cells go through a same path in the phase space of chemical components.
For example, comparing the orbit of x i ( t ) m with that of x i ( t ) l , we find out that respective cells oscillate at different phase on same path.
In a word, if oscillations of all cells are in synchronization, those cells simultaneously scramble for a nutritive source.
So that, the multiplication gets to be hard for those cells.
※ This is a kind of resonance to lead to catastrophes.
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The temporal habitat segregation of cells is realized through clusterization to avoid such an event namely to properly share the nutritive source.
Such clusterizations are originated from specific unstabilities in reactive dynamics.
And, those dynamic variations are settled into some stable states through interactions.
So that, the difference of cell-states in micro level is amplified into the difference in macro level, the discrete pattern of types is formed.
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In the step 3 , following orbits of x i ( t ) m , x i ( t ) l , we find out that respective cells go through different paths in a phase space.
Namely, in this case, not only differences in phase but also those in amplitude, chemical composition appear.
According to these differences, groups of cells are categorized into groups which actively perform metabolic reactions and are quickly split & other groups which don't actively perform reactions so much and are dormant.
※ Then, former numbers of cells are less than latter numbers.
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This means that a gap between the rich and the poor appears over the scrambling for finite source.
Namely, in this step 3 , double-codings of phases and amplitudes are generated.
Then, differences in amplitude are grouped into discrete types.
The step 4 means the appearance of recurrence. Namely, the initial condition of chemical component after dissociation is inherited into the next generation.
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Actually, drawing a return map of the average on concentrations of chemical component in parental cell as the horizontal axis and the average on those in child cell as the vertical axis, the remarkable gap between parent and child is recognized in the beginning of dissociation.
But, those plots go to be gathered near the diagonal line through the repetition of dissociation.
Namely, the memory of cell state appears.
※ Such the memory of cell-type and hierarchical differentiations from a stem cell are explained by Furusawa in detail.
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・ About cellular memory
When recursive types are formed, what type of respective cells are memorized ?
In dynamical system, it's regarded that there are several attractors in which dynamics is settled, respective types are memorized as attractors.
So, are respective cellular types stored in respective attractors with internal-cellular dynamics ?
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According to numerical calculations of some models, respective types are regulated by dynamics in respective cells to some extent.
On the other hand, respective types are stabilized by interactions with surrounding cells.
Namely, cellular types are formed by interactions among cells and its internal dynamics rather than attractors.
Here, we should pay attention that 'differentiation' and 'division (≒ dissociation)' don't happen at the same time.
The number of cells is changed due to division, so that interactions among cells also are varied.
Therefore, differentiation seems to have close relation to division.
But, direct causalities between both processes have never been known ( about 2001 ).
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And, 'memories' of differentiated states are not necessarily generated through variations on the appearance of gene.
Differentiated states are memorized as variations of states including cytoplasms.
So, we should regard that when variations of those states are most 'robustly' memorized, variations on the appearance of gene also are caused.
Accordingly, inversely, states of cytoplasm are changed by giving stimuli on cells from the outside, those 'variations' can be memories.
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In the case of neural network, a static memory structure is often regarded as a strong connection among synapses.
Meanwhile, in the case of brain, it seems that memories are organized by dynamics related to inputs from the outside.
Furthermore, in the case of unicellular organism, variations on chemical signals are memorized as variations of cellular state.
Accordingly, we should understand a memory as a pattern of dynamics.
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For example, it's one of the fundamental problems how a structure like a 'memory' ー such a state that writes-in and reads-out is almost stably carried out and recorded contents are kept for long time ー is generated out of dynamically varying states.
If we can re-consider various memory-types from short term to long term to categorize characters of dynamical system into some patterns, we may be able to understand the 'learning process' from an other viewpoint.
Because states of dynamical system are changed by the hysteresis of influences from the outside.
It's significant to review responses through the system for signal-transmission over cells and variations of cellular types according to those responses.
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Chapter 1, 1-3 Stabiloty
Following four types of stability in developmental process are known at least.
( 1 ) As respective numbers of molecules are not so much, its molecular fluctuations are magnified.
But, states of respective cells are settled into some stable types ( stability in micro level ).
( 2 ) Stability as a character of group of cells ;
Even if the disturbance in macro level such as the removal and the death of some cells is caused, properties of group including the distribution of cells with different types are recovered at once.
And, when respective cells are placed as a spatial pattern such as a lattice, the pattern is robust against disturbance ( stability in macro level ).
( 3 ) Stability of developmental process ( as a process along the temporal axis ) ー what type of cell appears, what kind of group is formed at which step ( stability in path ).
( 4 ) The developmental process and the individual in the next generation are similar to those in the previous generation ( recurrence over generations ).
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A real living being has above stabilities to some extent.
Furthermore, these stabilities can't be explained with only usual viewpoints such as the appearance of genes beyond the threshold of concentration of signal.
Seemingly, the approach on unstable dynamics is not suitable for considering stability.
But, actually, characters of respective cellular types, cellular genealogies and its distributions are stable.
For example, we added noises to rate-equations of chemical reaction to investigate fluctuations in molecular level.
Then, even if the remarkable large fluctuation had been prepared, types of generated cells, chemical compositions of respective types and proportions in the number of cellular types were invariable.
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We think that such stabilities are general properties of dynamical system with multiplying elements.
According to explanations by Furusawa, in the control of differentiation ratio for some model of differentiated stem cells, responses can be generated in the directions so as to cancel perturbations from the outside.
In this sense, the above control is same as the Le Chatelier-Braun principle in thermodynamics.
In the case of thermodynamics, this principle means the stability of system.
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Meanwhile, the control of differentiation ratio in a system of stem cells means the stability on a group of cells.
And, even in thermodynamics, unstabilities from molecular chaos are observed in micro level, this is similar to 'stability realized through unstability' in cellular system.
However, in the case of cellular system, as characters of respective cellular types depend on properties of the whole group of cells, the relation between the dynamics of chemical reaction in cell-level and the group of cells is complementary.
Such an inseparable relation of micro level from macro level can be regarded as a difference between cellular system and thermodynamics.
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By the way, stability has close relation with irreversibility.
For example, the relation of irreversibility with stability when the thermodynamical potential becomes minimum.
Meanwhile, in the cellular biology, we can find out specific directivities to lead from ES ( Embryonic Stem ) cells to determined cells in differentiation through stem cells.
Namely, as a developmental process makes progress, possibilities to differentiate some state into various cells are successively reduced.
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Furthermore, we found out following cell types.
( i ) The 'embryonic cellular class' has no recursiveness, its variations are continued through dissociations.
( i i ) The stem cell class ; this class is divided into two cases, namely one is a case to produce same types through dissociations, another is a case to change some type into different types through dissociations.
The transition depends on internal states and interactions among cells, it seems from outside (≒ objectively) that such choices are probabilistically carried out.
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( i i i ) The class of determined cells ; these classes basically become same types over individuals after dissociation.
Then, there are some grades for determining.
Namely, there are some types from a case in which the recursive character is invariable under remarkably changing environmental interactions to a case in which some cell types is transformed into another type through de-differentiation under changing environmental situation, those steps depend upon which is more dominant, internal state or interaction among cells.
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Multiplying a single cell as a start, the differentiation makes progress in order of ( i ) → ( ii ) → ( iii ) .
As types of respective cells are formed from balances of characters of internal dynamics with interactions among cells,
grades ( degrees ) to determine differentiations of cell have correlations with internal states of recursive type namely the dependence of respective cells on attractors after regulating interactions.
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As stated previously, removing some types of cells, determined cells can be switched into other types through 'de-differentiation' .
Even in real living beings, irreversible directions can be turned in cases different from usual developmental processes such as re-generative phenomena.
Then, first, we define only usual developmental processes without special cases including such large perturbations (≒ disturbances) from the outside as ideal developmental processes.
So that, we can clearly catch the irreversibility.
Next, if we compare those ideal processes with processes including outside perturbations, we may be able to consider conditions to break down irreversibility such as re-generation, somatic clone.
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For example, thermodynamics has a conception of entropy.
The order of adiabatically realized states is determined with this quantity.
The same as that, is there an index to express irreversible differentiative processes of cellular system ?
As explained by Furusawa, directivities to reduce diversities of chemical composition, unstabilities of cellular dynamics have been found out even in existing developmental models.
In other words, those can be regarded as reductions of dimensions of subspace to represent states of cell.
Then, diversities on the appearance of gene and grades for its variations can be regarded as indices of irreversibility.
And, we can consider possibilities to make differentiations turn back through outside operations with taking variations of these indices into account.
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・ The recursive characters of cell-group & the separation of generative series
To consider recursive characters over generations, we must understand the mechanism that a part of cells such as generative series is separated from other parts and the next generation with same shape as some generation is generated over developmental process of multicellular organism.
In this case, a multicellular organism has some special class of cells called generative cell, even if those classes are determined in developmental process, those classes taken out at some condition become un-differentiated cells, furthermore developmental processes can be re-produced from those taken-out un-differentiated cells.
Why are only specific classes of cells transferred into the next generation without using many various cells ?
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Of course, since the isolation of generative series by Weismann, this mechanism is understood as a mechanism to prevent Lamarckian mechanisms from evolution.
Because variations of somatic cell (≒ mutations) are not inherited into the next generation.
Here, two questions arise.
One is, why are Lamarckian mechanisms avoided in evolution ?
Another is, is there the general mechanism as a developmental dynamics to inherit only a part of cells such as generative cells into offsprings ?
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The former is a question of evolvability.
As the path into the next generation gets narrower, the variation at each step also is easily amplified.
So that, the possibility of evolution also increases.
And, as interactions in Lamarckian mechanism are strong, when the environment of the individual is remarkably varied, its strength can be disadvantageous on the contrary.
Because, the Lamarckian mechanisms are based on an 'expectation' that an adaptability in some generation is effective to the next generation too, but if the environment around the individual remarkably is changed at the next generation, such an 'expectation' becomes ineffective.
Accordingly, the formation of next generation through narrow path such as generative cell contains a fundamental theme from viewpoints of the theory of evolution, the genetics and the dynamical system too.
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The latter is whether a group of cells forms a stable recursive process so as to inherit only a part of cells into the next generation through developmental process or not.
Actually, according to a model taking the glueing among cells into consideration, it's found out that only a part of types is issued after differentiation, those issued types become ancestors to the next generation.
However, it's not clear what class of cell-group the 'separation of generative series from others' is necessarily caused in.
And, it has never been clearly understood how the separation of generative series from others as 'a narrow path' to produce the next generation is correlated with recursive properties of cell-group ( about 2001 ).
※ By the way, we should refer to explanations by Ueda to know researches with myxomycete ( mold ) related to such the 'origin of individual' .
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Chapter 1 , 1-4. Evolution 〜 ( 1 )
A conception standing for a cell as an unit for multiplication can be applied to an individual ( of those cells ).
Then, consequences derived from 1 〜 3 mean that two individuals with same gene at same initial state can be states with different phenotypes respectively through interactions.
By the way, what does this mean for evolution ?
Namely, we must consider phenomena in long time-scale in which genetic informations are varied through multiplications, selections of respective individuals.
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First, except for differentiation of phenotype, we consider about this question on 'orthodox' ways to think of evolution.
Following three approaches correspond to those 'orthodox' ways.
( a ) A state of a living being is regulated with the genetic type and the phenotype, there are controls of genetic type on phenotype, but there are no actions in its reverse direction ( from phenotype to genotype ).
This is a central dogma in the molecular biology.
Furthermore, variations of genotype are inherited into the next generation, those of phenotype are not inherited.
Possibilities of heredity except for those through genes are often considered, but those are ignored here.
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( b ) The multiplying velocity namely the adaptive degree of the individual is determined by the phenotype and the environment.
In this case, interactions with other individuals also are included in the environment.
By the way, as the nutritive source is finite, the number of individuals does not increase infinitely.
Accordingly, the selection such as competition is applied to the multiplying process for controlling the number of individuals.
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( c ) In the 'usual' group genetics, the phenotype is one-sidedly determined according to the genotype under stationary environment.
Namely, then, the phenotype can be regarded as a single-valued function between the environment and the genotype.
If ( c ) holds, the adaptive degree as a function of phenotype gets to be a function of genotype ( with the environment as an outside parameter ).
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Then, evolution can be interpreted as a process in which adaptive genes are selected from others.
According to the 'usual' way, the combination of ( a ) with ( c ) can be regarded as a process to select 'adaptive' genes from others through mutation.
But, taking models of dynamical system into consideration, ( c ) does not necessarily hold.
Namely, even if two individuals begin to develop at same initial state in fluctuation and equations to express developmental dynamics of those two individuals are same, those two individuals can have different phenotypes through interactions.
Actually, such an example that individuals with same gene have different phenotypes is known as 'low penetrance' .
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Furthermore, according to reports by Kashiwagi, multiplying some colon bacillus at high density in a liquid culture medium, it's divided into a high group and a low group in activation of enzyme.
Then, further multiplying only high group in activation, the group also is divided into a high group and a low group in activation.
In a word, it can be regarded that the differentiation of phenotype is caused through interactions among individuals with same gene rather than mutations.
Accordingly, as 「phenotype ＝ single-valued function ( genotype ; environment )」is not necessary, we consider about evolution in the case without this premise.
In dynamical approach, genotype and phenotype are respectively expressed as follows.
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Genes are inherited into the next generation without varying so much, characters of other phenotypes are controlled by those genes.
In this case, parameters in dynamical system correspond to genes.
Namely, the dynamic variation of individual is expressed as a dynamical system for a variable ｛ x m ｝, the gene corresponds to a controlling parameter ｛ a l ｝of the equation.
Then, a parameter to give genotype affects on a variable to give phenotype, meanwhile it's difficult to directly realize its reverse pattern.
So that, ( a ) is satisfied.
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In this case, in multiplication, those parameters are slightly, at random, varied due to mutations, meanwhile variables to show phenotype are largerly varied ( or, these variables are re-set so as to be at proper initial condition ).
Actually, supposing a same model as the case of cellular differentiation, its reaction co-efficient is regarded as the parameter.
And, if the number of individuals increases to some critical level, a competition occurs to prevent the whole number from explosively increasing.
For example, members inhibited from growing are removed, or members are removed in probability.
So that, ( b ) is satisfied.
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Under the above setting,
some experiments were carried out with the dynamics of developmental process and some interactions.
Of course, the differentiation of phenotype depends on the number of individuals and the choice of developmental equation.
But, the competition over finite nutritive source gets to be severe because of the increase of the number of individuals, besides various interactions and internal dynamics are added to the developmental process, so that the process can be differentiated into discrete types.
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Then, we investigate what happens by applying mutations on genotype and natural selections to the developmental process when such a differentiation is caused.
Obtained results are as follows.
( i ) Individuals with same genotype are divided into two groups with different phenotype from another group.
However, at this step, offsprings belonging to respective groups are not same as groups in which parents are included in type.
( ii ) The difference in phenotype is settled into that in genotype.
For example, in this model, one group with some phenotype multiplies with using a specific reaction, another group uses a different reaction from that for multiplication.
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In this case, when the parameter is varied so as to raise the rate on the frequently used reaction, the group easily multiplies.
Meanwhile, when variations on a parameter in the reverse direction occur, another group easily multiplies.
Then, two groups different in phenotype correspond to both left-right systems on genotype respectively through mutations and selections.
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( iii ) So that, 1 to 1-correspondences between genotypes and phenotypes are recovered, those genotypes also are divided into different two groups.
As this step, geno/pheno-types of offsprings belonging to respective groups are same as those of parents.
The differentiation of species is completed in the sense.
Viewing only ( iii ) in this process, it seems that the same species is divided into two specieses with different genotypes through random mutations.
But, viewing ( i ), ( ii ), we should regard that the 'origin' of differentiation is the 'settlement' of phenotype under some situation into the difference in genotype for differentiation rather than the variation of gene.
Such a shift of sight brings the following new viewpoint on evolution.
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Chapter 1 For the logic of living system as a complex system
4. Evolution 〜 ( 2 )
A. The possibility of stable sympatric speciation
Specciations are roughly categorized into two cases.
One is a case that differentiations are caused even in a situation possible to mutually cross-breed at same place ( sympatric speciation ).
Meanwhile, in another case, as cross-breedings and interactions among individuals don't appear because of the difference in place and condition, differentiations are caused ( allopatric speciation ).
※ Examples of sympatric speciation have been found out in ecosystems of fishes in a lake, in a solitary island and in a tropical rain forest.
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On the other hand, it's pointed out that it's hard that sympatric speciation holds theoretically.
Because, if individuals with slightly different genes have similar phenotypes respectively, the competition in a same ecosystem happens, and it becomes difficult to co-exist.
Accordingly, in the theory of sympatric speciation until now, assuming that the difference between these two groups in adaptive degree is not big so much, besides we had to devise a model that even though these two groups are put at the same place, those cross-breedings don't occur.
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For example, the 'mating preference' was introduced as a process to satisfy this condition.
Namely, habitat segregations according to spaces of some phenotypes such as space for preference appear at a same location.
We can build a model to cause speciations on such a mechanism.
But, in the case, as these two groups substantially have no relations each other, if those are integrated through fluctuations, it won't be divided into two groups again.
Namely, very unnatural settings are required to compose the usual model of speciation, besides, the model is not robust to fluctuation ( unstable ).
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On the other hand, in our hypothesis, when two groups with same genotype have different phenotypes from another, those groups separately co-exist through interactions.
Therefore, even if one group disappears on the way of differentiating process, a state of two groups is recovered again.
Accordingly, the speciation makes a progress through interactions at the same position without being disturbed by fluctuations.
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By the way, in the case of cross-breeding in sexual reproduction, middle types between two groups are produced at some probability.
Surely, middle types are generated at about half probability without mating preference of cross-breeding.
※ Letting frequencies of both groups be p1 , p2＝1 − p1 respectively, those are produced at the probability of p1＾2 ＋ p2＾2 .
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But, even if middle types as 'hybrids' take any phenotypes, those become more disadvantageous in multiplication than groups under correspondences between genotypes and phenotypes.
Actually, according to calculations on this model, as the differentiation of genotypes makes a progress, the probability to make middle typed offsprings survive falls, finally middle-typed offsprings disappear.
Namely, 'the sterility of hybrid' as a definition of species is realized here.
Consequently, the speciation stably makes a progress even in sexual reproduction.
Namely, the 'mating preference' is developed so as to reduce the ratio to generate hybrids.
In other words, we think that the mating-preference itself results from 'the sterility of hybrid' for now.
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B. Deterministic Differentiation
In our theory, as long as mutations and selections exist, if the differentiation happens on the phenotype, the speciation also is necessarily caused.
Actually, according to results of this model, the differentiation of phenotype through interactions is a necessary and sufficient condition for genetic evolution.
In other words, if the phenotype is differentiated under strong interaction through the change of environment, the increase of the number of individuals, the speciation of genotype necessarily is caused.
This process is very different from usually supposed probabilitical processes.
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Accordingly, if specific conditions are satisfied, the speciation quickly makes progress and the differentiation on genotype is carried out in proportion to time.
For example, the hypothesis of intermittent balanced evolution asserts that an evolution always rapidly makes progress beyond a specific step as a critical point.
The speciation-process as stated here can be an effective example of such intermittent balances.
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In other words, the genetic evolution is promoted by the flexibility of phenotype in the developmental process and interactions.
At a glance, this looks like an introduction of Lamarckian evolution.
But, this holds within an 'orthodox' process in which uni-directional flows from genotypes to phenotypes are assumed.
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If the flexibility of developmental process has relation with evolution, new viewpoints are given to some problems, for example, the diversity of insect in species, why the penetrance of mutant is often lower than that of wild type.
But, as such an argument is easy to be an endless dispute, we are trying to compose an evolvable system in the level of laboratory to make clear whether the theory of evolution is truly valid or not.
As one of the concrete examples, viewing how the differentiation of colon bacillus in phenotype is varied into the specific genotype, we can demonstrate the validness of the theory of evolution in the time-scale of laboratory.
※ Kashiwagi and others are partially carrying out such inspections ( about 2001 ).
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・ An acquisition of plasticity ?
〜 the extensibility of system
A scenario found out here to diversification can be regarded as a process that the plasticity of the dynamics of an original developmental process is consumed due to the settlement of that into genotype.
Then, if its environment is stationary, the possibility of differentiation gradually goes to be reduced with the course of time, and it faces the limit of diversification.
Accordingly, this theory is still in-sufficient to consider about sequential diversifications such as adaptive radiations.
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It's required to increase the dimension of phase space to represent phenotype for solving the problem.
For example, new chemical components are used through variation to obtain 'variables' for representing new phenotypes, separate two living systems are combined to obtain new degrees of freedom ( dimension between phases ) ー namely a kind of 'symbiosis' , etc.
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So that, if the number of individuals with the degree of freedom for new phenotype increases, its phenotype is further differentiated owing to strengthened interactions.
Furthermore, the differentiation is genetically settled, and the new species is established.
As the phenotype is expressed as a function of genotype at this step, stable correspondences between both types in balanced state hold.
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Besides, if the number of individuals further increases and the environment is varied, variables for new phenotypes different from those at previous step appear and those also are differentiated.
Such scenarios of evolution are expected as the extension of our theory.
And, interactions must be intensified to some extent to continue such positive feed-back processes.
It can be regarded that the difference ( phase transition ? ) between a diverse ecosystem such as tropical rain forest and an in-diverse ecosystem is brought through such conditions.
However, this 'opened evolutionary process' is still on the way of research ( about 2001 ).
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Chapter 1 For the logic of living system as a complex system
5. Dynamical Complementariness
Here, we have regarded the essence of life as the universal stable structures revealed from very random and complex dynamics rather than ingeniously tuned machines.
※ For example, the 'invariant set' to be the limit of Smale's horseshoe mappings also is a kind of such universal stable structures.
This 'invariant set' is the product (＝ intersection) of a Cantor set of horizontal components with a Cantor set of vertical components.
And, as a Rossler-attractor can be parameterized with a Cantor set, the correlation between two types of Rossler-attractor can be considered through this invariant set.
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In such a context, dynamic components with internal (≒ endo-)degrees of freedom spontaneously multiply without intricate mechanisms, and then living systems as universal structures appear with interacting with other factors.
Obtained views are as follows.
① In the cellular biology including developmental processes, the prototype of a process for which higher degree-controls seem to be needed is a universal property of dynamical system in some range.
Therefore, this property appears even in the rather loosely composed system.
② We can draw the logic of stability and irreversibility which is satisfied in multiplying systems through ① .
③ First, assuming a more complex and 'flexible' dynamics system such as DNA-informations, recursive types are formed from such a dynamics and those are settled into symbolic systems.
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Such a viewpoint is essential for the stability of development and evolvability.
The significance of 'flexibility ( or plasticity )' in living phenomena is originated from a question of 'loose or tight ?' suggested by Fumio.Ohsawa in the research of molecular machine.
※ This conception can be extended into the spontaneity and the appearance of individuality proper to life.
As the above, the complex systematic life science considers such a 'flexibility' as one of the main problems to catch complementary relations between parts and the whole.
Considering the life as a system, we find out that living phenomena are categorized into following antagonizing two groups.
〈 A 〉
・ Characters as deterministic machines
・ Digital symbols
・ Logical rules described with combinations of if-then trees
・ Individuality
〈 B 〉
・ Probabilistic ( stochastic ) characters
・ Continuous patterns
・ Analogue and dynamic behaviors
・ Wholeness
・・・
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A way to explain latters (〈B〉) with formers (〈A〉) is the main stream of biology at present ( in 2001 ).
Meanwhile, the research of complex system pays attention to the reciprocal gives and takes between respective parts and the whole and tries to draw paths from latters (〈B〉) to formers (〈A〉) , for example, the generations of rules from behaviors, mechanisms to embed characters proper to the whole into respective elements.
If we try to compose a whole from respective parts, we need rules on combinations of respective parts to form a whole.
※ For example, blueprints for development of combinations of on-off genetic switches in molecular biology.
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Meanwhile, the standpoint of complex system pays attention to 'why such blueprints are spontaneously generated and stably function rather than preparing blueprints artificially.
Accordingly, in this case, the viewpoints from the whole to parts are consequently required.
Then, the following scenario is drawn, that is, rules of on/off of genes in respective cells are generated through interactions with other cells in the context of wholeness, so that the stability on development is realized.
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And, we can consider how Turing-patterns resulted from physical and chemical phenomena are related to more symbolized tendencies including the appearance of gene through the above method.
Actually, Sawai and Sawada have studied about the influence of structures in macro level such as Turing-patterns on the appearance of gene.
Furthermore, the viewpoint of complementarity can be applied to not only the correspondences between genotypes and phenotypes in life but also cognitive processes of brain.
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The viewpoint pays attention to a standpoint that symbols and logic are formed out of the chaotic 'sea of images' rather than a standpoint that the logic expressed with symbols precedes everything.
There are relations between symbols and patterns or syntax and semantics as more general problems.
The standpoint of generative grammar by Chomsky pays attention to previous〈 A 〉, ignores semantic features.
Meanwhile, the cognitive semantics pays attention to〈 B 〉, tries to catch stable structures out of many behaviors related to body.
The standpoint of research of complex system agrees with that of cognitive semantics, that is, how symbols including syntax are formed out of embodied expressions.
※ Repeatedly saying, we should notice that (5, 2) or (2, 5)-type torus knot can be conjugated as a symbol to abstract a human body.
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