a physical characterization of biological information and communication system model of ecosystems

J. theor. Biol. (1984) 110, 619-635

A Physical Characterization of Biological Information and Communication System Model of Ecosystems

MASAHIRO TANAKA

Department of Physics, Osaka Medical College, Takatsuki, Osaka 569, Japan

(Received 20 July 1983, and in final form 9 May 1984)

What is information for living organisms? An answer to this question is given on a physical basis and a contrast between genetic information and sensory information is stressed with a relation to information theory. A simple model of an environment of living organisms is investigated on the basis of communication system model proposed by the author and a cost of information transmission is taken into consideration through capacity- cost theory. It is shown that channel capacity of information theory can be interpreted as an environmental index measuring a mildness of the environment, and furthermore that a large diversity of genetic messages needs a large capacity of the environment. In addition, a definition of life in terms of information is proposed and a unified view on life processes is suggested.

1. Introduction

Significance of information for living organisms seems to be widely accepted. However, neither the functional nor the physical entity of information in biological processes has ever been revealed in any systematic manner. It is obvious that information is inherent in living organisms, because the genetic in format ion- -an intrinsic characteristic of only living organisms--has con- currently appeared with the emergence of life. Living organisms also use the sensory informat ion- -an attribute not only of living organisms but also of machines in the sense of cybernetics.

It can be said that cybernetics has stressed the control by means of information, though Wiener juxtaposed control and communication, in other words, it has studied nervous systems in a broad sense which process information and control the behaviors of the output devices through the information. On the other hand, information theory had studied transmission and reproduct ion of information. Actually, information theory is a theory concerning the reliability of signal transmission and reproduction, in which the error probabilities of the message reproduction in a communication system are evaluated and the condition of the reliability is described

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6 2 0 M. T A N A K A

in terms of the source entropy and the channel capacity. However, it had been difficult to get the channel capacities analytically or numerically except few specially symmetric cases, until Arimoto developed a powerful algorithm numerically calculating the capacity of arbitrary discrete memoryless channel (Arimoto, 1972; Blahut, 1972; Jimbo & Kunisawa, 1979).

Since the early days of information theory, several authors have talked about a metaphor regarding living organisms as messages. Goldman has suggested an idea that the activity of the chromosome is considered as a set of signals. He wrote, "Now the life history of the organism is regulated by the interactions of the various parts of the organism which is living in an environment subject to a considerable amount of random fluctuation". (Goldman, 1948). Wiener has given a metaphor considering living organisms as messages (Wiener, 1954). Jacobson has described evolutionary process elegantly in informational language (Jacobson, 1955). Rosie has also suggested an analogy between noisy communication systems and mutating cellular reproductive systems (Rosie, 1973). In my previous paper, a communication system model of biological population and the environment has been proposed, in this attempt a parallelism between information theory and genetics has been discussed and genetic information has been regarded as the message in information theory being transmitted from generation to generation in a biological population (Tanaka, 1980).

In this paper, this model is further developed especially including the following discussions. In section 2, a physical characterization of biological information is proposed and a contrast between genetic information and sensory information is suggested. In section 3, a parallelism between living organisms with the environments and communication systems is discussed in further detail. In section 4, capacity-cost theory is introduced, which is an extension of classical information theory to the cases in which each message has a cost of transmission. Also a possible biological interpretation of the theory is given. In section 5, a simple model of ecological environment is presented and it is shown that the channel capacity could be an environmental index. Finally, in section 6, an informational characterization of life is proposed and a parallelism between Maxwell's demon and DNA is suggested.

2. Physical Characterization of Biological Information

Information is indispensable to and inherent in living organisms as mentioned above. Concerning the definition of information, however, there seems to exist many ambiguities. Hence it is necessary to be careful of applying the concept of information to living organisms. Ordinary meaning,

I N F O R M A T I O N A N D O R G A N I S M S 621

"communicated knowledge" is anthropocentric and inadequate to biological use, of course. This anthropocentricism probably due to the verbal information which is peculiar to mankind almost exclusively, seems to be the main cause of the ambiguities. We shall try to eliminate them as far as possible in such a manner as to describe n physical terms and restrict our discussions only to biological information. In the next place, information theoretical meaning, "something which reduces uncertainty" is not anthropocentric but it is oversimplified by dismissing the semantic values. Rather biological meaning, "input to receptor" is insufficient as well, because it specifies neither the physical entity nor the effects of its action.

Biologically, quality of information must be significant as well as quantity of information, though few authors have discussed it (Tansky, 1976; Volken- stein, 1979). Weaver distinguished three levels of communication problems (Shannon & Weaver, 1949): the technical problem (level A), the semantic problem (level B), and the effectiveness problem (level C).

While information theory has dealt with mostly the technical problems of level A, some extensions of the theory have been developed probably as a first step towards the semantic problems of level B. They are rate-distortion theory concerning source coding and capacity-cost theory concerning channel coding. They seem to provide a way to evaluate objectively the significance of information as the linguistic meaning is being evaluated through multidimensional scaling in psychology. However, in order to apply these theories to biology without ambiguity, it must be necessary to put living systems within the framework of information theory together with a proper characterization of biological information. The former has been treated in the communication system model presented in the previous paper (Tanaka, 1980) and here we shall discuss the latter theme.

Occasionally it is said that matter, energy, and information are the most fundamental .concepts in science. Matter may be partitioned into inorganic matter and organisms. An organism is a network of organismic elements which convert input energy to output depending on the received information (see Fig. l). Energy governs directly the motion of either inorganic matter or organisms through physical law (equations of motion), while information indirectly controls the behaviors of organisms probably through the boundary conditions which are independent of equations of motion i.e. the physical law (Rothstein, 1979). Now let us consider this point of view in more detail.

First, at the stage of the reception of information, a frog, for example, recognized food only by size and movement, so he starves to death surroun- ded by food if it is not moving (Lettvin et al., 1959). He extracts only "significant" data from the surrounding through his eyes, in other words,

622 M. T A N A K A

t Energy Informotion ' I Orgonismicelemen, I

t Energy

FIG. I. An organism is a network of organismic elements which convert input energy from an energy source to the output . Transistors, neurons , and operons are typical examples of the organismic elements.

whole data on food are compressed into the data on only size and movement. Organisms accept not complete but only significant data selectively. This data compression is the central theme of rate-distortion theory mentioned above and it is a way to speed up the rate of information transmission and to simplify data processing. Receptor organs, cells, or molecules may be regarded as source encoders with fidelity criterion which select and transmit only the significant messages. Data compression may also be a necessary means to build a model of the world in a finite brain of a frog, a tiny part of the world. Thus extractedness is an essential characteristic of information and the extraction is entirely dependent on the organism receiving the information. Information is subjective in this sense, hence negentropy does not necessarily convert to information though the reverse is possible.

Another characteristic of information at the stage of reception is smallness of the amount of its energy compared with the energy it controls. For example, in the above case, the energy received by frog's eyes is much smaller than that of his motion catching the food. Reception of information is a measurement, hence if a large amount of energy is taken out through the measurement, the object under measurement changes its state, which in turn implies the received information to be nonsense. Smallness of the energy as well as the extractedness may be the reason why copying information is easier than copying matter.

At the stage of information transmission, disequilibrium and metastability are the characteristics of information. Brillouin has pointed out that Max- well's demon would never see the molecules without the departure from equilibrium provided by the torch (Brillouin, 1962). Tribus and Mclrvine have also pointed out that thermodynamic information is conceptually the same as the degree of departure from equilibrium (Tribus & Mclrvine, 1971). A system in equilibrium with its surrounding can not transfer any

INFORMATION AND ORGANISMS 623

energy substantially to the surrounding, in other words, the system can not act with certainty, on the surrounding. Hence the carriers of information must not be in equilibrium, however, they must be stable enough to be transmitted to the user. The metastability of the energy partly guarantees the noise immunity at the stage of transmission. An important aspect at thi~ stage is the cost of information, which can be taken into consideration through capacity-cost theory (McEliece, 1977).

At the last stage of use of information, specificity and triggerability are the important features of information. For example, nuptial coloration of a male stickleback releases an aggressive behavior of another male stickleback. This means that biological information triggers the specific motion of an organism and the specificity is entirely dependent on the organism which uses the information. Ethological releaser is a typical example of sensory information and the key stimulus is the copressed data of the releaser.

Though we have discussed the characteristics of sensory information so far, the same characteristics are applicable to genetic information as well, for it is considered as a set of signals to a developing cell in the scheme of the operon theory. Indeed, there exists an interesting duality between sensory information and genetic information. Table 1 shows the duality. Genetic

TABLE 1

Duality between genetic information and sensory information

Genetic information Sensory information

Source Population Receptor Channel Environment Nervous system User Population Effector Message generation Internal generation External generation Adaptation Evolution Learning Feedback Positive feedback Negative feedback Origination Speciation Creation

information is the internal information by which an organism constructs its hardware. Living organisms can evolve through positive feedback of the internal information. Sensory information is the external information which an organism extracts from the surroundings. Organisms can learn through negative feedback of the external information. Wiener has pointed out an analogy between evolution and learning (Wiener, 1961), indeed both of them are adaptations in hardware and in software respectively which decrease uncertainty to get higher efficiency.

624 M. TANAKA

Now, let us propose a physical definition of biological information: Information is a small metastable energy which is out of equilibrium and selectively extracted. It triggers a specific motion in an organism. In other words, genetic information is the intracellular information, which triggers a series of the specific reactions of ontogenesis selected by nature. Sensory information is the extracellular information, which triggers the specific behaviors chosen by the organism. Although the physical entity of information is energy, the action cannot be measured in ergs or cals because it is not direct but indirect action on a system. In typical cases, however high the amount of the energy may be, the information has the same effect so long as it is below the threshold or it is above the threshold. The effect of the action of information is usually measured by entropy decrease which measures how definitely it specifies the uncertain behaviors of the system.

It can be said that information theoretical meaning, "something which reduces uncertainty" abstracts the aspect that information determines the behaviors of an organism. After all, information is the specific energy to turn the "switches" in an organism and controls the organism indirectly, whereas energy controls matter directly. Now, old proverb, "scientia est potentia" is interpreted as "information controls energy", i.e., information is an upper hierarchy of energy which controls energies.

Applying the above definition, some rather ambiguous concepts are clearly characterized. For example, a system is a set of parts interacting by information; a hardware is a system of which behaviors are undetermined unless information is applied; software is information which determines the behaviors of a hardware; and so on.

3. Communication System Model

Among the authors who suggested the metaphor discussed in the introduction, Jacobson described a model, which may be depicted as in Fig. 2, though he did not show the figure. In this scheme, self-reproduction is synonymous with positive feedback, because the input is identical to the reproductive output. Figure 2 also shows that Darwinism is not mere

Speciotion Ii

Lethal output i I Env'r°nment I I '

Reproductive output

FIG. 2. Jacobson's model may be depicted as in this figure. A genetic message generated through a speciation goes round the positive feedback loop.


Reproduced Survived J J P°pulation I 1 message i Environment = J JJ message ~ [I Population I

FIG. 3. Communication system model (Tanaka, 1980). This is essentially equivalent to Jacobson's model, because the two populations in this figure are identical to each other, hence the two signal lines form the same feedback loop as in Fig. 2.

tautology but is implying that organisms just going to live are the descendants of the previous survivals. If the loop in Fig. 2 is opened, it coincides with our communication system model shown in Fig. 3 (Tanaka, 1980). In Table 2, some important technical terms in genetics with their analogues in information theory are shown.

TABLE 2

Genetics--information theory dictionary

Information theory Genetics

Source Population Channel Environment User Population Message Gene action Code Species Code word Individual Noise Mutagen Error Mutation Source entropy Genetic diversity Channel capacity Transmission capacity Message generation Reproduction Transmission Inheritance Use Ontogenesis

In this model, a genetic message of an individual in a population is interpreted as a code word of an error correcting code (species). A genetic message is generated in a source (population) through reproduction and transmitted over the channel (environment). These messages suffer prob- abilistic disturbances, i.e., mutations and selections in the noisy channel of environment and only survived messages are reproduced in the population. The population is also interpreted as the user of the genetic information, in which the survival machines (code words) are constructed.

Using this model, the author has dealt with the cost of genetic information incompletely reproduced and got the suggestion that there seems to exist an optimum region of incompleteness (distortion) in order to keep the

626 M. TANAKA

mutant surviving. In this paper, we shall study the source statistics which maximizes the transmission rate of the channel in the case that every message has a cost of transmission.

4. C a p a c i t y - c o s t T h e o r y

In this section, a brief summary of capacity-cost theory will be given (Blahut, 1972; McEliece, 1977). Capacity-cost theory is an extension of the ordinary information theory, in which each input message j has a cost ej of transmission over the channel and a constraint is imposed by an inequality such that the average cost must not be larger than a given value E

E pjej <<- E (1) J

where pj is the probability of message j. A typical example of the cost is electric power of signal and the constraint is imposed on the average power.

Capacity-cost function C(E) is an extension of channel capacity in the case of the constrained channel, it has the meaning as the maximum amount of information that can be transmitted reliably over the channel under the constraint.

Capacity-cost function C(E) is defined as follows:

C(E) = m a x I(p, Q) (2) P~Pe

I(p, Q)= ~ ~, pjQjR log~ Qjk J k PiQ~k (3)

J

• j

If the cost constraint (equation (1)) is removed, C(E) reduces to the ordinary channel capacity.

Capacity-cast function is a monotonic non-decreasing function of E, because if E ' > E then PE,> P~. Moreover, C(E) is a convex upward function of E, hence it is strictly increasing, implying that the maximum value must be achieved at the boundary of PE:

Y~ pjej = E. (5) J

Capacity-cost function can be expressed parametrically in terms of a parameter s e [0, oo] which is the Lagrange multiplier for the constraint


(equation (5)),

C(Es)=sEs+max[V~.~.kpjQjklOg QJ-------Z-k s'}'.pjet] (6) P~P J J

where p* achieves the maximum. Maximization problems under inequality constraints are typical examples

of nonlinear programming and the following specialization of the Kuhn- Tucker theorem is known (Blahut, 1972; Kuhn & Tucker, 1951).

A vector p achieves capacity with average cost E , if and only if there exists a number V such that

Qjk log Qjk set = V, Pt > 0 (7) k E PjQtk

J

L Qtk log Qjk se t <-- V, pj = 0. (8) k • PjQtk

J

We can assume that min ej = 0 without loss of generality, hence Emin = 0 and C(E) exists for all E >-0. Let us define C~x = max C(E)then

E

Cmax = max l(p, Q) (9)

where the maximum is taken over all p with no bound on the average cost. Cm~x is equal to the channel capacity. I f we define

Em~x = min {E: I(p, Q) = Cm~x} (10)

then clearly c(E) = Cmax for all E -> Emax. The minimum Cmin = C(E~in) of the capacity-cost function is achieved by the source which uses only the cheapest messages having emi,.

Now, a typical C(E) curve might look as shown in Fig. 4. The channel coding theorem and its converse state that the messages from a source can be transmitted and reproduced reliably over the channel with average cost E if and only if the source entropy is not larger than H * shown in Fig. 4.

A possible biological interpretation of capacity-cost theory may be as follows. A populat ion (source) in an environment (channel) transmits genetic messages over the environment through reproduction, however, physical, chemical, or biological resources (costs) of the environment are

628 M. TANAKA

~mox . . . . . . . . . . . . . . . . . . . . . . . .

H:~ . . . . . . . . . t t

C m i n - _ _

Emin E Emax

FIG. 4. Typical C(E) curve. Only the sources which have smaller entropy than H* can reliably transmit the messages over the channel if the average cost is bounded under a value E.

restricted in any case, hence the actual consumption of the resources (average cost) must be bounded within a limit E. This constraint represents the energy aspects of the system. Even under this constraint, all the messages can not pass through the noisy environment, i.e. they can not reproduce themselves over the disturbing environment, unless the diversity (entropy) is not larger than the specified value (C(E)). In other words, in a milder environment (larger E and C(E)), reliable transmission and reproduction may be possible with full diversity, however, in a harder environment (small E and C(E)), some messages having higher costs are supposed to be excluded in order to satisfy the constraint together with reducing the diversity. Thus, capacity-cost function can possibly be an index measuring a mildness of the environment.

The amount of information I(p, Q) of equation (3) being transmitted over the environment depends on the statistical structure of the population and actual biological populations are supposed to maximize the amount of the genetic information being transmitted over the environment. This maximization over the populations under the constraint determines the probability distribution of the genetic messages in the population. Biological evolution is interpreted as this maximization process by changing the statistical structure of the population. In this paper, only one constraint inequality is imposed, however, any number of constraints might possibly be imposed. Then the set of the costs assigned to a message might form a meaning of the message as being tried through multidimensional scaling in psychology.


Furthermore, the channel or the environment can be divided into subchan- nels corresponding to the various stages in the life cycle.

5. A S i m p l e M o d e l o f an E n v i r o n m e n t

According to the communication system model, an environment is interpreted as a noisy channel, which causes mutations and selects output of the messages either as reproductive or as lethal. These environmental features may be described by the following channel model.

Let j be a channel input message and k the output. A discrete memoryless channel is characterized by the channel matrix Qjk , moreover, each input j has a cost ej of transmission, because every environmental resource is finite and every message might have different requirement for the resource. Con- sequently, actual consumption of the resource must be restricted within a bound, which imposes the constraint (equation (1)). Although the cost in the theory are assigned to the imputs, it is rather natural to assign cost e~, to the output k in our case, in which ej should be replaced by the conditional average ~.k Qjke'k

ej = ~ Q~ke'k (11) k

using this conversion, we can express the average cost at the output by that at the input:

~. qke'k = ~ ~, pjQjke'k = ~, pjej k k j j

(12) qk = ~ pjQjk.

J

To see the effect of transmission cost, let us consider a simplest case, i.e. a noiseless channel with three inputs. A noiseless channel is a channel which outputs the message just input, hence it is characterized by an identity matrix:

O = 1 .

0

Let the costs be e~, e2, and e3. Mutual information I(p, Q) reduces to the source entropy H ( p ) because Ojk = 8jk in this case.

l ( p , Q) = H ( p ) = -~ . pj log pj. (13) J

To execute the maximization (2), let us, at first, neglect the constraint pj-> 0 and get a tentative solution. The tentative solution will be proved to

6 3 0 M. T A N A K A

satisfy the Kuhn-Tucker condition (equations (7) and (8)). Now, capacity- cost function C(E) is given by

C(E) = max H(p) pePE

where the maximization is taken under the equality constraints

~ p j = 1 (14) J

and

pjej = E. (15) J

Let the Lagranre multipliers for equations (14) and (15) be A and s respectively. Then, from the conditions

a-~j[H(p)-A~pj-s~pjej]=O, ( j = 1,2,3)

we get the tentative solution p* parametrized by s,

e -se j

P* =~. e - '7 ' ( j = 1,2,3). (16)

J

This tentative solution proves to satisfy the Kuhn-Tucker condition (equation (7)), assuming V = log [~ e-S~j]:

J

Oj~ksej=l°g[PjOjk J ] Qjk log ~ e-'ej = V, ( j = 1,2, 3) (17)

J

Hence the tentative solution (equation (16)) gives the maximum points. It has the meaning as the most eitective probability distribution for the population to transmit the genetic messages over the environment.

A point (E, C(E)) of the C(E) curve is given by

E =Ep*e) (18) J

and

C(E) = H(p*), (19)

the actual consumption of the resource and Average cost E implies capacity-cost function the amount of the diversity possibly transmitted over the environment under the cost constraint.

I N F O R M A T I O N A N D O R G A N I S M S

Terminal points of the curve are given by

E m i n = min ej = el

C m i n = C ( E m i ~ ) = 0 , s =oo, Pl = 1, and

P2 = P3 = 0

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(20)

Ema x = min {E: I(p, Q) = Cmax}

=½(e, +e2 +es) (21)

C m a x = log 3, s = 0, P~ = P2 = P3 = ]"

In Fig. 5 and Fig. 7 some results of numerical calculations are plotted. These curves show that the optimum probabilities p*'s, hence E (equation (18)) and C(E) (equation (19)) depend both on the channel matrix Qjk and on the cost distribution (e~, e2, e3) of the environment, though they are determined by only Qjk in a mild environment (E -> E m a x ) . It is also observed that the most expensive message having the maximum cost e 3 is excluded from the population in rather milder environment. It probably represents the lethal messages in the limit of infinite cost. The only survival in a harder environment (smaller E) is the cheapest message in any case.

6. Conclusions

Biological information should be strictly distinguished from human information, because the latter is an extremely special case of a species having an extraordinary processability of biological information. A principal characteristic of biological information is to determine the behaviors of an organism. Usual definition that information is what reduces uncertainty reflects this aspect of biological information, not simply meaning the reduc- tion of uncertainty with the occurrence of an event. Another significant aspect of biological information is its value, which can be taken into account through the rate-distortion theory and capacity-cost theory. The physical entity of biological information is small specific energy triggering the specific behavior of an organism.

Living organisms use internal (genetic) information as well as external (sensory) information, whereas machines use only external information, which suggests a definition of life in terms of information. A living organism is a system which is built according to the internal information reproduced by the progenitor and adapts its behaviors to the environment through the external information. Note that this definition includes mules but excludes crystals. It also implies artificial organisms synthesized by genetic engineering.

C(EI

l 0

e=(10 ,1 .7 ,30 ) / . ._ ._. - . - - - -

M. T A N A K A 632

0 I i I I k I I ~ i

I-0 2O E

(o)

1.0

p

0 IO 2 0

E

(b)

FIG. 5. C(E) curve of noiseless channel (a) and the optimum source distribution (b) ; e I = 1 - 0 , e 2 = 1 . 7 , e 3 = 3 . 0 .

Internal information plays the central role in life processes. Biological evolution is an adaptation to the environment by means of the internal information. Ontogenesis is the developing process of the internal information. According to our communication system model, lives of living organisms are transmission processes of the internal information. The genetic messages of a population of a species are reproduced as an error correcting code being transmitted over the noisy channel in excess of the channel capacity (Darwinism). A large diversity of genetic messages needs a large

I0

C(E)

I N F O R M A T I O N A N D O R G A N I S M S

0 I I I I I I I I I

IO 2-0 E

(o)

633

I O

p

0 i i I-0 2.0

E

(b)

FIG. 6. C(E) curve of noiseless channel (a) and the optimum source distribution (b) ; e~ = 1 . 0 , e 2 = 1- I , e~ = 3 . 0 .

transmission capacity of the environment (channel-coding theorem). Thus transmission capacity can be an environmental index measuring a mildness of the environment.

In the process of phylogenesis, a population of a species generates messages with higher negentropies. These messages, however, lose the negentropies as they age and eventually they die (somatic mutation theory). This process parallels that of Maxwell's demon. Theories of the demon show that he cannot violate the second law but increasing the negentropy

634 M . T A N A K A

tO

C(E)

e = ( 1 0 , 1 1 , 3 0 ) / - - - - - - - - - - - -

0 I T I [ Z t t T T

~-0 2-0 E

(o)

10

0 t t I '0 2 0

E

(b)

F I G . 7. C(E) curve of noiseless channel (a) and the opt imum source distribution ( b ) ; e I = l . 0 , e 2 = l - l , e 3 = 3 . 0 .

of the gas, the demon himself loses negentropy. Indeed, DNA, the carrier of the internal information, is a possible candidate for the demon. In the scheme of the operon theory, operators receive repressors in the same way as the demon receives the photons reflected from the gas molecules, and the gates of protein synthesis are opened or closed depending on the information. Consequently the DNAs increase the negentropy of the amino acids in the same manner as the demon increases .the negentropy of the gas. Teramoto pointed out, "Maxwell's demon must be a first target to study


the relation between the process of the fixation of information entropy and the second law of thermodynamics" (Teramoto, 1973). From this point of view, the parallelism can be pursued further.

External information is also used by living organisms to adapt their behaviors to the environment. Learning is a software adaptation by means of the externaLinformation in the subprocesses of the evolutionary process which itself is a hardware adaptation by means of the internal information.

The present theory is only a first step towards semantic information in biology (level B) and there remain a number of problems to be studied. In particular, information structure including the context-dependence and pragmatics dealing with the actions of information at the user (level C) are important.

The author wishes to thank Professor E. Teramoto of Kyoto Universi ty for his cont inuing support and the refinement of the manuscript . Helpful and stimulating discussions with Drs H. Nishio, N. Shigesada, I. Aoki and N. Kuroda together with their coworkers are gratefully acknowledged. The author also wishes to thank Professor S. Ar imoto of Osaka University for his kind suggestions and Professor R. Kawabe of Osaka Medical Col lege for his critical reading of the manuscript .

R E F E R E N C E S

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47, 1940. MCELIECE, R. J. (1977). The Theory of Information and Coding: A Mathematical Framework

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Nostrand-Reinhold. ROTHSTEIN, J. (1979). In: The Maximum Entropy Formalism. Levine, R. D. & Tribus, M.,

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II1.: University Illinois Press. TANAKA, M. (1980). J. theor. Biol. 85, 789. TANSKY, M. (1976). Prog. Theor. Biol. 4, 205. TERAMOTO, E. (1973). Energy and Entropy (in Japanese). Kyoto:Kagaku-Dojin. TRIaUS, M. & MCIRVINE, E. C. (1971). Sci. Amer. 225(3), 179. VOLKENSTEIN, M. V. (1979). J. theor. Biol. 80, 155. WIENER, N. (1954). The Human Use of Human Being. Doubleday: Anchor Books. WIENER, N. (1961). Cybernetics: or Control and Communication in the Animals and the

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a physical characterization of biological information and communication system model of ecosystems

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