m EEGEMSM‘I’E A?‘PRQAC§~E m magma- QEFFgaemach Thesis. §or €59 Dogma at: M. S. MECMGAN STATE UNEVERSHY Fhiiip S. U‘Iinski 1967 AN EEGENS‘EATE A?’FROACE‘€ '50 CELLEJ‘LAfi QEFFERENUA‘FEON Thesis, fore- flm Degree of M. 5. MECEHGM STATE UNEE’ERS-ETY E’hiiip S. Ulinski 1967 .1 fatals A 5.5.16ij CT AN EIGLNSTATE AEEROACH TO CELLULAR DIFFERENTIATION by Philip 5. Ulinski The problem of the Thesis is to study the relationship be- tween the morphological type of a given cell and its biochemical system. This problem is shown to be a particular example of a more general set of problems typical of biology. The literature leading up to B. C. Goodwin's theory of cellular tensoral organi- zation is reviewed and criticized. This theory attempts to trans- fer physical concepts to the study of biological systems. It is argued that this transference is not warranted because physics and biology characteristically use different modes of analysis. It is assumed that a given cell can be in one of a finite number of discrete states and there is an isomorphism between mor- ph0105ical states and cellular biochemical states. These assump- tions imply that the probability density function for the proteins and the messenger ribonucleic acids of a cell satisfies a linear partial differential equation. This equation is solved for a sim- ple example. It is shown that in addition to states resulting from the activitation and inactivation of genes, cellular states may result frow the total organiZution of the cell. The assumptions are related to experimental data by discus- sing the ontogeny of the cerebral cortex and the embryology of the neural crest. The possibility of cellular organization states is related to current concepts of gene action. Susrestions are made g) 1.) for making the theory more directly testable. AN EIGENsTATE APPROACK Tb CLLLULAR DIthnhhTIATIUN By Philip S. Ulinski A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of Master of Science Department of BiOphysics 1967 ACKRCJLEJSEMLNTS I wish to thank Dr. 3. L. Kilner for aim congtunt encourige— ment and for his fnith in my often inurticulute inc ieculisr ideas. Dr. Karl Kornacker of the Department of Biology, Kasagchu- setts Institute of Technology, has read twrt; of the innuscriyt ;fld graciously took tine to<fiscuss my results with he. The preisreticn uf this Thesis was supported by Air Force Jumoridae Research Laboratories Grsnt AFl§(625)-507o and by Air Force Office of scientific neseirch Grunt AF-AFOéR-lOzj-ob. both ndninietered through he Division of Engineering Hesenrch, Michi- gan State University. For the conyletion of this Theis, I one most to my wife, Lee Anna, whose pitience end understanding made my work much less oner- 0115. ii TABLE OF CONTENTS CHAPTLR , Pt. 5‘6 I. INTRODUCTION: THE PROBLEM OF CCNCdSSCLNCE ........... l A. The Problem of Concrescence ...................... l l. The origin of the problem ................... l 2. The general nature of the problem ........... 4 3. The Specific problem of the Thesis .......... 7 B. Introductory comments ............................ 9 II. TACTICS nND sTRnTfiGY IN A Thuth CF CELLULAR TEMEOnAL ORGANIZATION ......................................... 11 A. The Mechanics and Thermodynamics of Denogruphic Systems .......................................... 12 1. Kinetic equations ........................... 12 2. The Volterra equations ...................... l} 5. Physical una105s ............................ 15 B. The StatistiCul Mechanics of Biolo;ical associa- tions ............................................ lo 1. The microcanonical ensemble ................. lo 2. The canonical ensemble ...................... a; C. Theories of Eyigenetic Systems ................... 27 l. Waddington and the cybernetics of develOpnent 27 2. A statistical mechanics of epigenetic systems El iii iv CHAPTER Pa 3e II. De TUCtiCS Jud Stratefiyooo...............oo.o........ 1+5 1. Tdctics eeeeeeeeeeeeoeeeeeeeeoeeeeeeeoooeeeee 1+5 20 Strateggy eeoeeoeeeoeeeeeeeeeeeeeeeeeeeeeeoeee 1+0 III. A TEEOKETICAL APIKUACH TC CLLLULAR JTATLS ............ 50 A. Assertions ....................................... so 1. The yroolem ................................. 58 2. Assertions .................................. 55 B. Definitions and the Eigenvalue Equation .......... 59 .l. (P Space, biochemical states and morphologi— cal states .................................. 59 2. Biochenical experiments ..................... ol 3. Expectation values and variances ............ 53 #. Stability ................................... 67 5. Embryologicnl experiments ................... 65 6. Two Preliminary results ..................... b9 7. The eigenvalue equation ..................... 72 8. Transition states ........................... 73 9. The relation of (RN to a):- linear oyerators on G)....................................... 74 10. Summary of argument ......................... 75 C. A Choice of the Morlhologicsl Operator L(p,r)0p... 75 1. Definition and proterties of L .............. 75 2. he Liouville equation ...................... 83 D. Deterniinttion 0f the Function Uk °eeoeoeeeeeeeeeee CL} HV' :1, ~\ VIIAIJJ. JIJH III. D. E. Ex lo Tillie derexldtllt ENJI't eeeeeeoe eeeeee do [liLlc indeiel‘ldellt 3;..I‘t 000000000000 ‘.‘.1'_JlQ lo Deter-Aflin'ittion Of D eeeeeeeeeoeeee k 4. Evaluation of exgectotion values . 3. switching states uni organization F. The Relationship oetween horphological a icnl Cellular btlteo ................... IV. THE NnTUKE AND LOCUS CF CELLULAR JTABILITY . Port 1: assertions A. The Relationsnir betaeen Morphologicul i iCi.Ll CCllUldr dt‘iteb eeeeeeeeeeeeeeeeeee l. histogenesis of the cerebral cort 2. CheniCul ontogeny of the cerebral 3. Other exuncles .................. [+0 QUilifiCé.tiorLE eeeeeeeeeeeeeeeeee B. Cellular Stubility ind Discreteneo ...... Ce T116 1. The embryology of the neural cres 2. Transition rules for develOping s Locus Part 2: Cellular States Of CCllUlLir D‘tibilit‘y 000000000 1. Ideas of cellular control ....... 20 cellUlLir Cigcnfitites 000000000000 D. Testabilitbr .0....OOOOOOOOOOOOOOOOOOOOOOO LITEKATURE CITED nd Bio' ex 0.. corte 1:1- 101 100 1o6 lll 11o 121+ 11+} 10. ll. [‘1 L15? 0F FIGUK'S Twe eriqenetic ltndscnpe .............................. A single cellular control loop ........................ The biocnemicul systen of n cell represented as an array of control loops ................................ Two control 10ops illustrating strong coupling ........ The results of the measurement of the concentration of the jth species of protein in a biochemical experiment Results of the measurenent of the cell type of a single cell in an embryological experiment ................... The eigenfunction u as a surface in p,r,u -spnce ..... k k Characteristic curves in p,r,uk-spece ................. Schematic diagram of the interactions between metabolic pathways important in cerebral ontogenecis ............ Transition diagram for prOsgective fates of neurul crest line cells ...................................... A model for cellular differentintion in n system with tu’o g‘ene lOCi eeeeeeeeeeeooeeeeeeeeeeeeeeeeeeooeeoeeeee vi 1+} CH PU 7O 68 115 126 137 APPENDIX LIST OF APTENDICEb THE EQUIVALENCE BETSEnN LINEAR DIFFEAEETIAL OPERIL‘J..U.L<’5 AI‘JD INLAI‘HICES OOOOOCCCOOOCOOIO00.0.0000... CHAPTER I. INTRODUCTION: THE PROBLEM OF CONCRESCENCES "All science is philOSOphical, and the only philosophies capable of validation are those of scientists." - G. G. Simpson A. The Problem of Concrescence l. The origin of the problem -- A liver cell, a cat, a tulip, a virus, and a mushroom are all examples of integrated biological systems. The purpose of this Thesis is to find some way of hand- ling, both conceptually and experimentally, a problem which arises in studying integrated biological systems. This problem will be called the problem of concrescence and it becomes important in the following way. The initial step in most scientifically conducted inquiries is to conceptually divide the system of interest into its constituent units. The prOper method of inquiry, Descartes instructs us, is to "reduce involved and obscure propositions step by step to simpler ones, and then attempt to ascend by the same steps from the intuition of all those that are entirely simple to the cognition of all the others". Bi010gists have, to date, been predominantly en- gaged in applying the first part of the algorithm to living systems, delineating and studying the "simplest units" of animals and plants. Exactly what these units are depends on which aspect of living sys- tems interests a given biologist: the geneticist has designated, 1 at various times, the gene, the "cistron", the "recon", etc. as the simplest unit of the genome; the neurolo;ist divides his system in— to neurons, fiber tracts, and nuclei; the anatomist takes cells to be the elemental units of all organisms; the biochemist gives pri- macy to proteins, sugars, coenzymes, etc.; the population bioloEist considers systems of gene pools and of animal and plant pOpulations. The second part of the algorithm, the ascension to the cogni- tion of all the other entities, those that are not entirely simple, is potentially much more difficult; but it is this undertaking which principally interests the biolOgist, for no living thing is a sim- ple unit and all living things are composed of interacting simple units. This idea of "organism" is, of course, not at all new. The philoSOphy of Organicisml developed in the first decades of this century championed exactly this idea, arguing that the total under- standing of bi010gical systems depends on studying simultaneously the action of all their components. Most groups of experimental biolOgists have long been aware of the problem. Very few, if any, phenotypic traits are determined by a single gene. Rather, pheno- types are usually determined by a series of multiple alleles sub- ject to the influence of various modifier and suppressor genes. To some extent, every trait is determined by the entire genome. The importance of interactions between the components of develOping systems is reflected in Roux's idea of Entwicklungsmechanik and was 1 The doctrine of the Organismic school is less well known than that of the Mechanists and the Vitalists. BibliOgraphies and ele— mentary statements of the argument are to be found in the books by Bertalanffy (1933,1952). A more detailed presentation is the one by Goldstein (1963). Discussions particularly relevant to this Thesis are by Weiss (1961,1963). A historical discussion is by Needham (1931). 5 emphasized by the discovery of embryonic inductionsa. The inter- connexity of the nervous system forces one to assume that all but the simplest physiological and behavioral processes result from the operation of many, interacting anatomical or functional units. Most neurologists seem to appreciate the need for what is called, following C. S. Peirce, a "calculus of relations", but -— indica- ting a sober appreciation of the difficulty of the problem - tend to place the search for such a calculus in the same category as the medieval serach for the Holy Grail. In the past ten years, biochem- ists have elaborated many examples of how interactions between meta- bolic processes result in their control. But, they still face the problem of understanding how all metabolic processes affect each 0- ther $3 3112. Although the importance of interactions is generally apprecia- ted and a descriptive analysis of many examples has been effected, the complexity of most systems has frustrated attempts to quantita— tively study integrated biolOgical systems. The impressive excep- tion is in the study of populations where an entire science has crystallized around the quantitative analyses of ecological and ge- netic pOpulations. The problem is difficult just because it is con- fusing to think about more than a few things at the same time. Most peOple in Western societies, for example, find it almost impossible to keep their family tree straight past the second cousin level. The task of keeping track of one's kinfolk is, however, unrepresen- tatively simple because it is a purely relational one and a static one: the relationship Father-Son is constant once it has been ex- 2 A clear and recent review of the various interactions Operative in embryogenesis is by Ebert (1965). L,. tablished and has no quantitative aspects. A developing embryo, however, is formed by virtue of a concatentation of changing inter- actions which have quantitative as well as relational aSpects. For instance, the eye lens of an adult salamander is only distantly re- lated, spatially and functionally, to the heart of the animal. At one point in development, however, the heart anlage comes into con- tact with the lens anlage and, in some way, contributes to its sub- sequent differentiation (Jacobson, 1966). Presumably, some induc- tor substance passes between these two components; if this is so, the differentiation of the lens would depend on how much inductor is transferred from the heart anlage to the lens anlage. Living systems are a reticulum of constantly changing correlations between subsystems. The point is, that the result of traditional inquiries into biological systems is an assemblage of subsystems and the problem of finding out what the relation is between the original integrated biological system and the conceptual or manufactured subsystems. We would like some way of specifying exactly what that relationship is. In particular, we would like that way of thinking to allow us to design experiments exploring the results of systematic variations of the subsystems upon the behavior of the integrated system. We will call the problem of finding such relations the problem 2£_222- crescence. To begin with, we introduce a vocabulary designed to discuss this kind of relationship. 2. The general nature of the problem -- The notion of con- crescence is borrowed from the work of A. N. Whitehead (1925, 1929). For Whitehead, the universe is a process merged from complex and 5 interdependent processes, called actual entities, much as the sea is an amalgam of water drOps. In the process of becoming an actual entity the potential unity of many entities —- actual and non-actual -— acquires the real unity of one actual entity. The actual entity is the real concrescence of many potentials. An electron, for ex- ample, is a process which the physicist may represent, say, as a plane wave. An atom is also a process, represented by a different, more complicated kind of wave. A hydrogen atom is an actual entity which is the real concrescence of its component electrons and pro- tons, each of which is an actual entity in its own right. The pro- cess of becoming a hydrogen atom is a series of eliminations of al- ternate possibilities: protonic, neutronic, and electronic proces- ses have the potentialities of becoming sulfur atoms, hydronium ions, carbon atoms, etc. The initial analysis of an actual entity reveals it to be a concrescence of prehensions which have originated in its process of becoming. A prehension consists of three factors: (a) the "subject" which is prehending, namely, the actual entity in which that prehen- sion is a concrete element, (b) the "datum" which is prehended, and (c) the "subjective form" which is how that subject prehends that datum. There are two kinds of prehensions: (a) positive prehen- sions are termed feelings, (b) negative prehensions are said to "eliminate from feeling". An electron and a proton and a neutron at the vertices of an infinite equilateral triangle, for example, are not a hydrogen atom because they do not prehend each other. A hyd- rogen atom results from a progessive concrescence of prehensions, the subject —— say, a proton -— incorporating various data such as an electron and a neutron into the real actual entity of the atom 6 and eliminating from feeling, considering as ineperative in the prOgressive concrescence of prehensions, data such as other hydro- gen atoms and the walls of the container. A ESEEE (plural nexus) is a set of actual entities in the uni- ty of the relatedness constituted by their prehensions of each ot- her. A society is a nexus of actual entities which are ordered among themselves in such a way that the nexus is self-sustaining A bot- tle of hydrogen gas, for instance, is a nexus but is not a society. A mouse is a nexus which is also a society. It is clear that the universe consists of a hierarchy of nexus ordered such that a gi- ven nexus consists of actual entities, each of which is a concres- cence of a subordinate nexus, and such that the specified nexus is a datum in the concrescences of a superordinate nexus. For practi- cal purposes, it is convenient to consider only some small portion of this hierarchy. For the metaphysician, Whitehead's thought is tremendously useful because it lays bare in full generality the structures of all possible processes. For the scientist, Whitehead's thought is a mask for ignorance because it does not specify the details of processes interesting to the scientist. Instead, the scientist must supply these details -— subject to the constraint that, inso- far as Whitehead is correct and the scientist does his work ade-— quately, the scientific description of a specific process must e- merge as a particularization of Whitehead's description of the gen- eral process. The scientist describes the processes of a specific nexus. In the example of the hydrogen atom, this description can be effected (in the "Heisenberg representation") by representing the electronic and the protonic pfirurftvs by separate Hamiltonian 7 Operators, H and, rougly speaking, representing the elec’ Hprot’ prehensions by an interaction Hamiltonian, H The atomic pro- int' cess is then represented by the Hamiltonian elec + Hprot Hint' Whitehead's cosmology can be especially useful to the scientist when he attempts to compare the specific descriptions of two dis- tantly related processes. 3. Specific problem of the Thesis -- In this Thesis, we will restrict our attention to the relation between the nexus of chemi- cal molecules and the special set of societies which are integra- ted biological systems and, more particularly, are individual cells. The classical biologists consider a cell to be a unit of structure and describe it in terms of its shape, its size, its spatial rela- tion to other cells, its extracellular matrix, its staining prop- erties, its embryolOgical potentialities and prospective fates, etc. The biochemist, typically, does not deal with the cell as a unit, but homogenizes it to produce a variety of components which react with each other by certain types of chemical processes. The cell that the classical biologist studies and the abstracted cel- lular elements that the biochemist studies are not the same kind of thing. The cell is a society and the chemical processes are a nexus. In Whitehead’s terms, it seems clear that an intact, 1i- ving, individual cell may be regarded as the concrescence of all its biochemical processes. The goal of this Thesis will be to solve the problem of this concrescence. That is, we seek to spe- 8 cify the relation between the biochemical state of a cell and its morph0103ical state. We will consider the problem solved if a rule is found Jthh will allow us to specify what the morphological type of a given cell is if we are given the current state of all of its biochemical pro- cesses. Such a rule would allow one to predict the results of ex- periments in which the biochemical components of a living cell were systematically varied. This is exactly the kind of experiments per— formed by embryolOgists. A typical embryolog experiment might be conducted as follows: First, specify a certain cell in embryo A and determine its fate in the resulting adult. Second, in embryo B, alter the environment of the corresponding cell and determine its fate in the resulting adult. One way to do this is to trans- fer the cell to a foreign environment in a third embryo, C. Often, one will discover that two homologous cells will have different mor- ph0105ical characteristics in adult A and in adult C. The interpre- tation of such an experiment is that the environment of the cell af- fects its biochemical processes in some way and that this disturbance shows up in a deviation from its normal morphological type. Unfor- tunately, it is not usually possible to predict the results of such an experiment unless it has been done before. Some general guide- lines do exist for such predictions, but it is not easy to genera— lize them into an overall picture of deve10pmental processes couch- ed in terms of biochemical processes. A Specification of the rela- tionship between biochemical and morphological cellular states would provide such a generalization. B. Introductory Comments The discussion in this Thesis requires the following prelimi- nary comments. First, although a specification of the relation be- tween biochemical and morphological states of individual cells will be proffered, it is probably an entirely unsatisfactory one. What is really being attempted is a formulation of the problem and some of its possible avenues of attack which is eXplicit and heuristic enough to provide a foundation for subsequent work. Secondly, a fairly detailed knowledge of the biochemistry of the cells being studied will be assumed as "given". This is, of course, a poor assumption. A critique of how useful this assumption is will be found in Chapter IV. Thirdly, the purpose of the work is to devel- Op a ggneral method for solving the problem. In any particular case, the method presented would have to be fitted to the nature of the system under study. This would require a detailed knowledge of the system and would be a fairly involved mathematical problem. Thus, the solution of the problem for an actual biological system must be regarded as a project in itself. The example worked out in Chapter III is designed to illustrate how the calculations are to be done. Fourthly, the discussions of this Chapter and of Chap- ter II boarder the domain of speculative philosOphy. However, they are deemed to be a necessary preface to the critical development of the theory of Chapter III. The primary goal of this theory is a theoretical scheme which the eXperimental biologist can use to de- sign new experiments and to systematize the results of old ones. Fifthly: "The true method of discovery", whitehead writes, ”is like the flight of an aerOplane. It starts from the ground of par- lO ticular observations; it makes a flight in the thin air of imagi- native geneialization; and it again lands for renewed observation rendered acute by rational interpretation." For the sake of ef- ficiency, it seems best to relieve the people who elaborate theo- retical generalizations from the reaping of particular observations. This freedom must not be interpreted as a license for the theore- tician to permanently reside in SOme domain of ethereal verities, but imposes the responsibility of assurring that the aerOplane will have somewhere to land. In an attempt to discharge this responsi- bility, the theory of Chapter III has been reworded in empirical terms in Chapter IV. CHAPTER II. TACTICS AND STRATEGY IN A THEORY OF CLLLULAR TLMPOR- AL ORGANIZATION "We shall probably fare better if we constantly recall that the physical object before us is an undivided sys- tem, that the divisions we make therein are_more or less arbitrary importations, psychological rather than physical, and as such, are likely to introduce compli— cations into the expression of natural laws Operating upon the system as a whole." Our problem of studying the relationship between the morpho— logical type of a cell and its biochemical system has remained al- most entirely unscarred by theoretical assaults. But Brian Goodwin (l963,l964a,l96#b) has made an impressive attack on the related problem of the relationship between the components of the biochem- ical systems of cells and the time behavior of cells. By way of establishing the tactics with which to approach our problem, we will use this Chapter to review Goodwin's theory, requisitioning those of its aspects which bear upon our probleu and noting in what ways it can be altered to satisfy our needs. The Goodwin theory is also of interest in that it illustrates a strategy which has pervaded theoretical biology: Most students who have attempted ll 12 to formulate the relationships exhibited by integrated biological systems have almost tacitly mide the initial assumption that they are subject to the same kinds of laws Operative in the systems stud- ied by the physicist, so that theoretical biology should parallel theoretical physics, the laws of the former being analogous to the laws of the latter. To allow a judgement of the utility of this strategy, we will digress slightly by discussing the theories -— which are the grandparents of Goodwin's theory -— of animal and plant pOpulations elaborated by A. J. Lotka and Vito Volterra. A. The Mechanics and Thermodynamics of Demographic Systems 1. Kinetic equations —- Attempts to describe the drowth or evolution of pOpulations of species by general mathematical laws date back to Malthus' essay, but one of the earliest workers to produce exhaustive theoretical and empirical studies of the beha— vior of populations was A. J. Lotkaa. By analOgy to chemical ki- netics, Lotka considers the evolution of populations as transfer- ences of masses and energies between the components of a system, each species of organism comprising a component and the number of individuals of a species constituting the "mass" of the component. Interactions between species are represented as stoichiometric re- lationships, and the time rate of change of each component is des- cribed by a first order differential equation so that a set of El- netic equations 1 (SN ( ) Ezr = Fr(N1.--o.Nn): r = 1:2.--'~n' 3 Lotka's work is summarized in his Elements of Mathematical Bio- logy (1956) originally published as Elements oi Physical Biology (192M). 13 where Nr is the mass of the rip species, describes an ecolOgical system of n species. Of particular interest are systems in which a balance between prey and predator exists so that dNr/dt = O for all r. This situation is defined as a steady state. Steady states are classified as stable or unstable depending on whether the Nr's remain at or oscillate about some mean values qr, or whether they will gradually decrease to zero or increase to some upper limit. The task of the demographer is to find the form of the functions Fr(Nl’°'°’Nn) and to detail the conditions which will cause the sys- tem to be in a stable or unstable steady state. Lotka considers a variety of empirical examples of demagraphic systems, but for our purposes it is more instructive to consider Volterra's descussion of the demographic problem. 2. The Volterra eguations -- In two papers (1931,1937), Vol- terra considered demographic systems of the Lotka type from a fair— ly mathematical viewpoint. He takes the kinetic equations (1) to be of the form ‘2’ iN—r=e.l aNN. dt r figr sr s r The coefficients Er describe the growth (or death) of the rth ape- cies when it is in isolation; the coefficients l/pr -— if one again thinks of a pOpulation of individuals as a mass -— describe the mass of the rip component which is transformed in unit time into another component, i.e. l/Qr individuals of the r313 species are equivalent to 1/95 individuals of the 5.122 species; the constants asr describe 14 the interaction between the rth and the s32 species,and must be antisymmetric, i.e. asr = - ars' Ne will call the Equations (2) Volterra equations. If we introuuce the notation Nr = er/dt, the Volterra equations can also be written as second order system: 2 n (3) £3 dX ax dX r 2r— erg + dts dtr . dt 5 _ d The conditions that the system (2) have stable steady states are well known in the theory of differentiatl equations. The solu- tions to these equations will be of the form 31,; A2. Ant alt + G e + 000 +G e +G e + 000 N = G r2 rn 2rl r r1 where the parameters ‘Ar are the roots of the characteristic equa- tion iel’A “21/51 aL[11/91 D = 1.312432 ez')‘ axe/32 = O E i i aln/(gn aZn/(gn "' E'n " A The nature of these roots determines the stability of the states. IFor example, if all the AT'S are pure real and positive, the sys- 15 tem will persist near to the stable state values ql,...,qr,...,qn; if they are pure imaginary and positive, they will oscillate about the stable state values; if they are negative, the state is unstab- le. Of particular interest is a result which Volterra calls the law of the conservation of fluctuations and which guarantees that if a system has n even, has stable steady states, and if the Nr's are bounded between two positive limits, then at least several of the Nr's will exhibit undamped oscillations about their mean values. This means that in systems of sufficient complexity (as biological systems undoubtedly are) we may be justified in treating the sys- tem as ergodic and replacing time averages by averages over the ranges of the variables Nr' 3. Physical ana103_ -- In whitehead's terms, the kinetic e- quations describe feelings in the demographic universe. On the face of it, the idea of an actual entity resulting from a concres- cence of these feelings seems poorly conceived, for the masses Nr seem to be the only attributes which could be applied to such an entity, and they are the attributes of the feelings. However, both Lotka and Volterra suggest that such a concrescence is justifiable. Lotka, arguing by analOgy to the kinetic equations of chemical ki- netics, suggests that the demOgraphic system considered as an en- tity should evolve in accordance with an analog of the second law of thermodynamics. His argument hinges on the construction of a quadratic form. If we define the vector N = N(Nl,...,Nn) in an Ad n-dimensional space and designate the transpose of N by N, a Quadratic form is a scalar function, E , defined byi = NAN where 16 A is an nxn constant matrix. After a suitable orthogonal transfor— mation, it can be shown that the condition for the stability of the system is that §§ be a minimum at the stable state values qr' This is analogous to the classical mechanics where it can be shown that the potential enerey is a quadratic form. Thus, as in mechanics and thermodynamics, important ideas about demOgraphic systems can be phra- sed in terms of a minimum principle, so there might exist functions § analogous to the functions known as thermodynamic potentials, in terms of which the behavior of the system can be concisely epitomized, after the man- ner of thermodynamics. If such a plan could be success- fully carried out, the result would be a species of quasi- dynamics of evolving systems, in which certain parameters P played a role analogous to forces, without being in any sense identical with forces (or even with generalized for- ces); certain other conjugate parameters p would play a role analogous to displacements, and certain functions § would resemble in their relations to certain events in the system, the energy functions (free energy, thermodynamic potentials) of thermodynamics. 3/ Although Volterra sometimes follows Lotka in considering the kinetic equations of a demOgrsphic system as analOgous to chemical kinetic equations, he usually considers them as the equations of motion of the system. In this case, the attributes of the actual entity are parameters analOgous to those of classical mechanics. Thus, Volterra writes n n E : dX . L = rd? and M Br Er Xr C I‘ = 1 I‘ = l where C is a constant, and shows that L + M = constant. L is cal- led the demOgraphic kinetic energy and M is the demOgraphic poten- l, A. J. Lotka (1956), p 321. l7 tial energy, so tr at we have a conservation of demo graphic energy. Similarly, demo graphic work is given by Also, Volterra was able to derive equations (3) by considering the first variation of the function n dX C13— >(3ra-Frlnd wr+yaén 15:;- :srdtrxs +P r=l r where P is the demographic potential n n n E 9 6r X + % i E c X X , r r , rs r s r = l r = l e = l whence come the Euler-Lagrange equations an an 0 air a. me- These reduce to the kinetic equations (3). Now, if we put pr =CVQXr, we have the Hamiltonian 18 and the canonical equations of motion 9.2 .. 32.12 512‘. _ - 5.3.3 I‘ I‘ It is clear, then, that given sufficient ingenuity, one could ea- sily construct a "demographic mechanics" analogous to classical mechanics. B. The Statistical Mechanics of Biological Associations 1. The microcanonical ensemble -- Lotka's suggestion is that there may exist for demOgraphic systems analogs to the thermodyna- mic state parameters of chemical systems. Volterra suggests consi- dering the kinetic equations of the demographic system as equations of motion. The obvious next step is to develOp an analog to the program of statistical mechanics and to relate the equations of mo- tion to the parameters of state, a task which has been accepted by E. H . Kerner (1957. 1959, l96h). (a) To obtain a system of equations which satisfies the Lion- ville equation, Kerner introduces the change of variables vr = 1n Nr/qr° Since -— at steady state -— equations (2) imply the e- quations the equations of motion become (A) 6r ;r = E :asr qs (e S - 1) ° Now, as in the statistical mechanics, we think of these equations as guiding the movements of a point in an n-dimensional phase space. If we consider a Gibbsian ensemble of such phase Spaces, we can de- fine a density of points in phase space, (>(vl,...,vn). We also I ‘ . . I l I define V = (vl,...,vn). The conservation of den81ty in phase Space gives us n a? . a - 9P at T+divPV= Zlvravr+fi=oo I‘: From the equations of motion (1+), we have atr/avr = 0; but we note in passing that this is a consequence of the particular form of E- quations (2), and will not hold for the general kinetic equations. In this case, the, we have the Liouville equation n 3? . &P '37- -+ E::;vf :;;¥ = O . r: As is done in classical statistical mechanics, Kerner restricts the discussion to the equilibrium case where dP/é t = O. (b) The starting point of statistical mechanics is some state- ment about the way energy is distributed between the possible state of the system. Thus, Kerner must (1) introduce an analog to total energy, and (2) make some statement about its distribution. He 21 chooses to discuss first the micrccanonical ensemble in which the energy of the system is held constant, but he could equally well have elected to treat first tne canonical ensemble in which the total energy varies but the temperature is fixed. v (1) By multiplying Equations (4) by qr(e r - 1), setting ?} = @} qr’ and summing over all r, we obtain n 0 VI‘ V O -l = O E : 7} r ( ) ' r = l and we have the integral of motion n vr G = E 7} (e -vr) = constant. r=l Since, if we set Ysr = Oér/Qs Fr : - Y , G satisfies the ca- I‘S nonical equations 2" a r Ysr '1??; G ’ 8:]. 40 II Kerner takes G as the Hamiltonian of his svstem. Again, we note that the success of this choice depends on the particular form of Equations (2); G may not exist for some systems. (2) It is customary in statistical mechanics to postulate that there is, 3 priori, an equal probability of finding each of the possible states of an isolated system to have a specific total energy, G. Then, it is shown that if the number of systems in the 22 ensemble is sufficiently large, the probability distribution ap- proaches the frequency of finding the system in the "most probable" state of the system. This is eXpressed mathematically by writing the density function as a delta function f=fo(G-GO) where F; is a constantx Kerner adOpts this proceedure. 2. The canonical ensemble -- (a) Consider, now, the canoni- Cal ensemble formed by studying only V of the n Species in a sys- tem and regarding the remaining n — V species as a "heat bath" with energy being transferred into and out of the system of V species. In studying the canonical ensemble, it is customary to take 1 P” = exp —G—-(“P- G”) to be the density function for the Species;'v is a parameter inde- pendent of the variables vr, and the meaning of G will be made clear later; G is the total energy of the V Species. Since we must have V F» d1}: 1 where d7; is the incremental volume in phase space, and the inte- gration is over all of phase space, we have the phase integral Z = exp(-"P/9) = exp (-C'I/G) dz, . T A slice the :1“ C‘te 1 v‘ 23 This relationship is used in the evaluation of ensenble averages. Thus, the ensemble average F for some function F = F(vl,...,vn) is MI I: fi,Fd'Jj,= Fexp(‘P-G,/e)dl,, 12 :: cup/9 SF €XP('Gy/e) (171.1: F eXp(-Gu/6)dTy e- /6 5 F exp(-Gy/6)d7; exp(-Gy/6)4ix}. Since the actual evaluation of such averages is a formidable excer- cise in integration, ”e will merely catalog the results. 1 I - (l) The ensemole average of N is N = q . r r r (2) The demographic analog to kinetic-energy is the function T = v C)G/c) v . Kerner shows that T :6 . r r r r Since this result holds for all r, this is the ana- log of the equipartition theorem for physical sys- tems: the average value of the demagraphic Kine— tic energy for each of the U Species is the same, and one thinks of the kinetic energy being distri- buted equally between the different Species. (3) To see the signifigance of E}, consider the ensem- ble average of the functionTi(Nr/qr - l)d. It Can be shown that ”Tim/qr - l)2 = 9 . 1‘ Thus, 6: (Tr/qr)(Nr - qr)2, SO that, within a (b) that, we 3."- ‘I Q 21+ constant factor, €5is the variance of the system about its mean value qr. Since temperature plays the analogous role in physical systems,€3 is the demOgraphic tempera- ture. (b) If 5 is the "tenperature" and G is the "total energy", what, we might ask, are the analoas of the other themodynamic st- ate parameters? These are all easily calculated from the formulae of statistical thermodynamics: (l) The free energy is'W = - % ln Z (2) Heat capacity is C = (Ba/()9 (3) EntrOpy is S = 5—:73/9 (4) Work due to environmental influences if Fi = -<5G/Acxi for some external variable Cki. Under certain conditions, we can derive the demographic ideal gas law F T = kha- (c) However, ensemble averages of certain parameters rela- 'ting to the oscillations of the variables Nr are of more value. We introduce the notation xr = Tr/e = @rqr/e . (1) First, we define T_/T as the fraction of time, over a lLong interval, that the values of Nr are below the steady state ‘Value qr' To calculate T_/T, Kerner (1959) uses the function a CT] 'iluati V. n (_J (w ,) vuu‘zzi v ,_.‘ Y; ‘ .- U' ‘45 " d V ,. I V» fit “'86?- H N T... .(‘EEJ HEPA,‘ "\or t ‘ .. ”q 25 Then 00 T /T = “PW/e h(V ) e-G/e dV 0 - r r ~00 Evaluation of this integral gives 0.5 \< T_/‘1‘ = mar, x-1)41 where I(J§, x - l) is the incomplete gamma function. When evaluated numerically, this result indicates that for 9 small, i.e. the sys- tem is near steady state, T_/T = T+/T, (T*/T = l - T_/T). But for 69 large, T_/T b T+/T. Thus, in pOpulations which show large fluc- tuations, the number of individuals of any species will usually be below the steady state level. (2) In a similar way” one can calculate the mean amplitude .A+ of oscillations above the steady state value as X -X X e A+ : (T+7T)x! ° (3) Also of interest is the mean frequency with which the ixbpulation numbers take on a given value. It is possible to car- 12y out a variety of calculations of this sort, but the simplest is e. . .. . .3 . lie determination of the ratio of the mean frequency tnat the sys- tlemxtakes on steady state values (i.e. Nr/qr = l) to the mean fre- ‘allency that it takes on some other value, 7’: Nr/qr° This is found ‘t<3 be ’;\ '2 L (I to (7?) = exr(.)?e-7)xr rel for the rig speCies. we note that CJrel(l) = l and cu}el(fi>)<:1 if'q,£ l, which means that Nr takes on its steady state value more frequently than any other value. (d) Kerner offers two empirical tests of his theory. First, he (1957) notes that the probability of finding the pOpulation of r between vr and vr + dvr is given by —Gr/6 P dv = e dv / Z . r r r r Then, by setting nr Nr/qr he finds °'~ Tr":L '0‘Trnr n e r dnr P(n ) dn : i r r -dT t"Tr 1‘ fear) 51nd observes that this form of distribution has been used to des- <:ribe actual pOpulations by R. A. Fisher and his group. Second, ICerner (1959) makes use of data on the fluctuations in a pOpula- tlion of Labrador foxes over a period of 91 years. From these da- ‘t£1, he is able to calculate parameters such as T_/T, A+, and A_ ikbr the system. He then solves his theoretical expression for each C>f’these parameters for x and finds that within reasonable limits the empirically calculated values of x are in accord. Also, Ker- ner finds good agreement between theoretical and empirical Core-10?) 27 values. C. Theories of Epigenetic Systems 1. Weddington and the cybernetics of development -- The idea of using the tools developed to study demographic systems to elu- cidate embryolOgical problems is due to C. H. Waddington. As an example of Waddinton's ideas, we will consider the first essay in his book The Strategy 2; the Genes (1957). As the starting point of this essay, Waddington takes an observation about deveIOpmental systems: The major empirical fact about the develOpment of ani- mals - g fact which has no theoretical inevitability, but which is so obstrusive that only the crudest obser- vation is necessary to establish it —— is that the end products which it brings into existence usually vary dis— continuously. The tissues of an animal are in most cases quite sharply distinct from one another; skin, nerve, muscle, lung, kidney, etc., with of course many subtypes in the bodies of highly evolved and complex creatures, but with very few kinds of cells which could be considered as providing a range of intermediates con— necting two of the major varieties. Similarly, each organ has its well defined and characteristic morpholo- sy- 2/ Waddington sees one of the central problems in developmental bio- logy to be that of accounting for these macrosc0pic discontinui- ‘ties in terms of genes and "gene products" which are present only in.small concentrations in the organism and which vary continuous- ly. This is a different statement of our general problem of the relationship between feelings and actual entity, the processes of gene action being feelings and morpholOgical discontinuities being attributes of the actual entity which results from a concrescence —_ 5 C. H. Waddington (1957), p 13. of the genetic processes. COS Wadaington's approach to a solution is to regard the substan- Of the genetic-metabolic system as pOpulations which can be described by kinetic equations similar to those of Lotka and Vol- terra; he notes The If we regard the system as closed ... and if the sup- plies of raw materials are taken as constant, the e- quations which result are of the same type as those which arise in the study Of the growth of two pOpula- tions of animals which compete with one another for a limited food supply. é/ generalization to Open systems was made by H. Kacser in his epilogue to Waddington's The Strategy 2; the Genes. Kacser notes that, in general, the kinetic equations of an Open system show that the composition of the final steady state of the system is dependent upon the nature and the quantity of catlysts (i.e. en- zymes) present, that these steady state values are independent of the initial concentrations of the components Of the system, and that the flux of materials into the system enters as a factor into the pen For how steady state values. He then uses these properties of an 0- system to suggest explanations for SOme biological phenomena. example, the independence of initial conditions might explain in regulatory eggs a half of an embryo produces and apparent- ly normal adult. Or, pleiotrOpic effects in which a single gene Inay lead to a variety of phenotypic characteristics might be ex- plained by differences in enzyme concentration or state. In all likelihood, these explanations are too simple to be adequate, but the point of interest is that it is possible to use kinetic equa- tions to study Open systems. k. Ibid., p 21. ievelopin; J the notion t tate thingy, two niex er; 29 Wadcington realizes that in using the kinetic ap;roach to study develOping systems, a likely method is to consider them as guiding the motion of points in a multi-dimensional phase space. To facili- tate thinking about develOpmental problems in these terms, he intro- duces a surface, which he calls the epigenetic landscape, defined in this phase space. His description of an epigenetic landscape for two independent variables is as follows (Figure 1): Consider a more or less flat, or rather undulating, sur- face, which is tilted so that points representing later states are lower than those representing earlier ones. Then if something, such as a ball, were placed on the surface it would run down towards some final end state at the bottom edge. There are, Of course, not enough dimensions available along the botton edge to Specify all the components in these end states, but we can, very diagrammatically, mark along it one position to correspond, say, to the eye, and another to the brain, a third to the spinal cord, and so on for each type of tissue or organ. Similarly, along the tOp edge we can suppose that the points represent different cytOplasm- ic states in the various parts of the egg. 2/ In a rough way, one can conceive of the position Of the ball repre- senting the develOpmental state Of, say, a cell. The depth of the trough the ball finds itself in is a measure of the competencies of the cell at that time -— the deeper the trough, the more dif- ficult it being to deter the cell from its prospective fate. The concepts which Waddington eXpounds are particularly in- structive because they bring to bear upon our problem of the geno- type tO phenotype relationship all Of the ideas and methods we have discussed both in this Chapter and in Chapter I. As we have already noted, Waddinton suggests that one of the central problems in embryology is that of the concrescence of genetic processes to those of morphogenesis. He also emphasizes the importance of sta- Ibid., p 29. 30 Figure l. The epigenetic landscape. See text for description. Drawn after Wadwington (1957). Undifferentiated Cell . \ \ //////Mm\\n ///‘ / 31 bility and the discontinuity of the morpholOJical aspects of orga- nisms in this iroolem. he Observes that genetic processes can be described by kinetic equations in the same way that Lotka and Vol- terra used them to describe the evolution of pOpulations of orga- nisms. The idea of the epigenetic landscape is similar to Lotka's potential functiong, and Waddington, apparantly independently of Kerner, used the kinetic equations to move a point in a phase space, albeit Waddington expressed himself qualitatively. A significant difference between Waddington and the demographers is that Wadding- ton's discussion stems only from biological considerations, ig- noring the use use of physical analogs. 2. A statistical mechanics of epigenetic systems -- (a) The task of following up Waddington's suggestion and attempting to es— tablish a relationship between a set of genetic-metabolic equations and parameters which can be used to study an intact developing cell was accepted by Waddington's student, Brian Goodwin. This under- taking necessitated, as a preliminary, some attempt to resolve three problems: (1) The Lotka-Volterra-Kerner theory discusses pOpulations of organisms, and to use it as the basis for an epi- genetic theory one must first argue that the demographic and epi- genetic systems have the same formal characteristics, (2) the epi- genetic system is imbedded in the genetic system, in the physio- logical system, in the evolutionary system, etc. Before proceed- ing, one must decide in what way this hierarchy of nexus is to be included or excluded from the discussion of the epigenetic system, (3) the Volterra equations describe interactions between individ— uals. To transfer the Volterra-Kerner theory to a consideration g 8 Cf. J. Needham (1936). CI 3 be} til a... PE. I.I-. l» 32 of epigenetic systems, one must first derive a set of kinetic equations which describe the genetic-metabolic system. We will examine Goodwin's answer to each of these problems. (1) Waddington's writings tend to emphasize the similarities between embryological and evolutionary processes, and his sugges- tion that pOpulations of organisms be considered as analogous to the chemical components of an organism carries this theme just one step further. Phylogeny and ontogeny are similar in that they study the rise and fall of something (pcpulations or components) with time; they are both looking at develOping systems, in the broadest sense of the term. The Lotka and Volterra theories are potentially capable of describing evolving systems of p0pulations, but their principal application is to systems of animal and plant populations which are in equilibrium or, more precisely, in a steady state. The Kerner theory, however, does not intrinsically have the potential of describing develOping systems and can deal only with steady state systems. This is, of course, because Ker- ner models his theory after the classical statistical mechanics. Mathematically, this condition of reversibility is introduced by setting dR/ét = O in the Liouville equation or by making the er- godic hypothesis which allows the replacement of time averages by ensemble averages. An ergodic system is not necessarily ab- solutely invariant with time, but may vary about some mean value; all that the hypothesis requires is the variation be periodic with at most a period of very long duration. Insofar as Kerner is in- terested in studying the periodic fluctuations of pOpulations a- bout a steady state value, the modus Operandi of classical sta- 33 tistical mechanics is ideally suited to the problem. Thus, since Goodwin aptlies the Kerner theory to the study of cellular problems, his theory must be viewed as a failure for the purposes of studying embryolOgical systems. But, such an equilib— rium theory should apply to an important class of cellular steady state activities, the most notable of which are the clock rhythyms characteristic of many biolOgical systems. Goodwin is well aware of this limitation of his theory and applies his results primarily to problems in the study of biolOgical clocks, realizing that ap- plications to embryological problems would require the elaboration of a theory analogous to non-equilibrium statistical mechanics. (2) There is nothing new in the observation that living or- ganisms present to the biologist a hierarchy of nexfis. Comte (18- 58), for instance, noted this and suggested that a classification of these levels can be either "biotaxic" or "bicstatic" depending on whether it is in terms of dynamic and functional or of struc- tural characteristics. Waddington (1957) chooses a classification in terms of "time scales", the history of the ancestors of an or- ganism constituting the evolutionary or longest time scale, the deve10pment of the individual making up the embryological or medium time scale, and the constant activities of the organism forming the shortest or physiolOgical time scale. Goodwin also constructs a classification in terms of time scales, but uses the relaxation time of a given hierarchical level as a defining criterion. He distinguishes a metabolic system of cells comprising the diffusion and interaction processes and the enzymatic transformations of small molecules (i.e. not macromolecules) and having a relaxation 34 . . -l -2 . . . . time of lo —lO seconds, an epigenetic system comprising the biosynthesis, diffusion, and interaction of macromolecules and 2 A . . having a relaxation time of 10 -10 seconds, and a genetic system which has a very long relaxation time and is not relevant to his discussion. Haddington realizes that an accurate theory of biological sys- tems demands the simultaneous consideration of all levels of or- ganization in a mathematical theory of living systems would induce a complexity prohibiting the comprehension of the theory by every- one save, perhaps, a Laplacian calculator ratiocinatrix, and the primary task of the theoretician becomes one of performing some einklamnerung or bracketing off of all the organizational levels except the one in which he is most interested. Goonwin meets this problem by making a distinction between parameters and variables: if the two systems have very different relaxation times (say one is 100 times larger than the other), then rela- tive to the time required for significant changes to occur in the "slower" system (larger relaxation time), the variables of the "faster" one (shorter relaxation time) can be regarded as being always in a steady state. Therefore only these steady state quantities will enter into the dynamic equations describing the slower system, and a very considerable economy of motional equations can be achieved. On the other hand, the variables of the "slow" system will enter into the equations of mo- tion of the "fast" ne as parameters, not as variables. These parameters have a slow rate of change, and the faster system will gradually move in time in response to these slow changes; but for the purpose of studying the short-term dynamics of the fast system, the slowly changing quantities which define the notion of the slow system can be regarded as environmental parameters. 2/ (3) As the fundamental "unit" of the epigenetic system, Good- win takes a control loop, pictured in Figure 2. Lj represents a certain gene locus or perhaps an Operon; R is a ribosome which *— 9 B. C. Goodwin (1965), p 10. 11H 35 uses the messenger RNA, rj, of the th Species of protein, pj; C is a cellular locus where the jth species of protein is utilized to produce the metabolite Mj from some precursors. The 100p shown in Figure 2 has a feedback mechanism so that when the concentra- tion of Mj exceeds some Value Sj’ the production of the enzyme pj is halted by some mechanism of.enzyme repression. Presumably the entire genetic-metabolic system of the cell could be represented by an array of such lOOpS with varying degrees of interaction be- tween loops (Figure 5). For the purposes of develOping an analytic theory of the epi- ggenetic system, the array of control 100ps in Figure 3 must be re- ;placed by a set of kinetic equations which will serve the same :pupose fulfilled by the Volterra equations in the Kerner theory. ESince these control equations are undoubtedly greatly oversimpli- .fied, we will only sketch Goodwin's derivation. The principal eassumption made is that the gene-repressor interaction in enzyme Ifiepression and induction follows the same mechanism that enzyme- sxlbstrate interactions follow. To derive the control equations for the control 100p in Fig- ulme 2, Goodwin assumes, to begin with, that each protein is syn- thesized at a rate proportional to the concentration of its mRNA Eirui that it is degraded at a constant rate. Thus, “alere 0% and @j are constants and where pj designates the concen- tration of the 1th Species of protein and rj the concentration of 56 “"9 F18 JJ Figure 2. A single cellular control 100p. Lj is a certain locus; R3. is a ribosome; Cfi is the cellular locus where the jth protein is used) pj is the jth Species of protein and r3. is the corresponding mRNA; M3. is the metabolite produced by the protein. Drawn after Goodwin (1963). 38 Figure 3. The biochemical system of a cell represented as an array of control loops. See Figure 2 for explanation of notation. Drawn after Goodwin (1963). M2 Mn ‘—> 00.00... ‘—_> Cl. M’) hn /\ pn ......... R n r n 00000... L P0 :3 39 the j£h species of mRNA. To obtain a control equation for rj, Good.vin a ssumes that the precursors (nucleotides) of the mRNA are present in the cell in the constant average amount L Kj] , and that the precursors and the re- pressor (nucleohistone?) for the jth locus compete for the available gene template, Tj' Then, we have the stoichiometric equations T. + K."‘T.K. and R. T.““I‘.R. a J‘— J J J + J‘— J J’ the equilibrium constants [13.12.] [T1 -] K. = -__J-_J-j—l and L. 2 w , T. R. T. . 3 [3H J 3 J J and the conservation relationship ‘where [Tj]o is the total amount of template and R3 is the concen- tration of the repressor of the jth template. Thus, L [3133]“? _ [K [TjAj] = l + LJle] + Iqow, if we assume that the concentration of repressor is prOportion- éil.to the "excess" of metabolite [R3] ‘ 051m ‘ SJ] ’ 40 O; a constant, and assume that rj is degraded at a constant rate bj' then we have 9.11.- Twine-Io _ ,3, dt " 1 L. . K. .IV'.-S. ° + 3[ 3I + JUJDJ J] J for the rate of synthesis of rj may be assumed prOportional to the concentration of "activated" template, [TSAd]. Setting aj = L. K. T. B.=l+L.[K.] andm=K.O'.' J[J][J]O‘J JJ’ 33’ a. l (6) dr. dt ‘3. m.M.-S. ' 3+ at.) J J The distinction between parameter and variable discussed in £Section 2.a.(2). is used to remove Mj from Equation (6). Suppose izhat Mj has a kinetic equation of the form dM. —- = C.p.-8., <3j and s. constants. Since Mj is a parameter in the epigenetic syustem (i.e. it is a variable in the metabolic system), we may l?e111ace it by its steady state value, cjpj/Sj° Thus, Equation ( 6 ) becomes (7) dr. 31 01“ “A. + k.p. j 41 A. = B. - m.S. and k. = m.c. s.. E nations ( ) ind (7) are 3 J J J J J/ J q 5 a the control equations for the control loop shown in Figure 2. This where method of removing Mj from Equation (6) is clearly just a first ap- proximation; a more detailed study would necessitate obtaining Mj as a function or the pj's and rj's so that the control equations ‘would be more complex. (b) Starting from control equations of the type just derived, (Eoodwin deve10ps a theory of the epigenetic system parallel to the I§erner theory. Because we have already discussed this deve10pment :in.Section B, we will only (1) discuss a detail in which the Good- vvin.theory differs from the Kerner theory, and (2) discuss the re- Eaults and predictions of the Goodwin theory. (1) In Section B.l.(b) we saw that a simple set of transfor- znations serves to put the Volterra equations into a form which will salways lead to a Hamiltonian function G. However, the control equa- txions for different arrays of control 100ps can be expected to vary wcidely in form, and it may not always be possible to find a G func- tjuon. In the case of Equations (5) and (7) there is no difficulty. LVer let 5. and Fj designate the steady state values of p3. and r3. and j J definedb {5. =r. -?. andl ". = A. k. .) Euations () y 3 J J + pJ ( J + JpJ /QJ. q a 5 arid. (7) become :3e1: Qj = A + k.p.. Then if we introduce the variables 9j and fij 22. _ b 1 _ 1 dt j 1 +63 A 22-- « dtJ ‘ OSrj' 42 From these equations we obtain dG(f>‘j,?.) = o<3r3d§§j - ba 1 +131 — 1 dfij = o 00 c» and G(p},?j) = JP ' dG(pj,?j) = constant r (Aj/Qj—l) a. ’92. = ——%——l + bj[f3j - ln(l+pj)] = constant. This integral is easily exPressed in terms of the original variab- les pj and rj. But, in the case of the control loops shown in Fig- ure 4, there is a serious difficulty. Goodwin shows that the con- trol equations can be put in the form £121 - b Y1 - 1 dt ' l Yl + pl d—Qa - b Y2 - 1 A r 22 _ _; A A dtl “ Q1 “1103 r1 * k12 (*2 r2 ) A Y 22 _ _.2. « A . . . . . . - A A A A However, it is possible to integrate the differential 06(p1’p2'rl’r“) L 43 .Figure 4. Two control lOOps illustrating strong coupling. See .Figure 2 for explanation of the notation. Drawn after Goodwin (1953). c- 4 P2 R2 Ml V V M.2 { r1 re \ 41+ only in the special case when KlKIZCXZ/Ql : r-kalCXl/QZ, for then 2 the terms rl A A r2 A A Q-Ikll darldra + 2-1-2- kZlderdrl becomes a perfect differential. In general, it will be possible to find a Hamiltonian function for any set of control equations (Ker- ner, l96fi), but this example shows that it may not be possible to actually integrate the differential form. This is an important shortcoming, for the integrated form is needed to calculate all ensemble averages. (2) Goodwin follows Kerner's methods and calculates the var- ious thermodynamic functions, but replaces the adjective "demogra- phic" by the word "talandic" meaning "oscillatory" so that his analyses are in terms of talandic energy, talandic temperature, talandic entrOpy, etc. As in the Kerner theory the actual use of these state functions is that the total energy is needed to calcu- late ensemble averages and that the talandic temperature is used to measure the variance of fluctuations about steady state values. The equilibrium assumption is reflected in the equipartition of talandic energy between the components of the system. Kerner's results on the functions A+/A, T+/T, OJ, etc. are found also to apply in the case of epigenetic systems. Goodwin extends his study one step further than the Kerner theory by studying the statistical prOperties of oscillating sys— tems in some detail. In particular, he discusses the consequences of strong coupling between control 100ps; that is, situations in 45 which macromolecules from one control loop directly affect other control loops. This type of occurrence is illustrated by the loops in Figure 4 and is the opposite of weak coupling which occurs when control lOOpS interact throujh the metabolic system. Strongly coupled control loops can exhibit the phenomenon of entrainment in which one loop may recruit other lOOpS to its particular frequency and amplitude. As his choice of an adjective indicates, most of Goodwin's results are aimed at studies of the oscillating systems exemplified by bi0105ical clock phenomena. Thus, he suggests ex- periments in which the character of oscillations are changed fol- lowing pulses of amino acis or RNA's. However, as Goodwin points out, these results on periodically varying systems will probably be of little use in the study of deve10pmental problems. D. Tactics and Strategy The five workers we have discussed constitute something of a movement throughout which the tendency has been to develOp theories and concepts which parallel those of physics. The general program of the movement has been to effect a description of a system in terms of kinetic equations and to use these equations to formulate a statistical description of the system considered macroscoPically. If we use this approach on our problem of the genotype to phenotype relationship, we must decide (1) to what extent it is advantageous to continue the tactics of their statistical description, and (2) to what extent it is appropriate to follow their strategy of paral- leling physics. 1. Tactics -- The Goodwin theory serves the important pur- A6 pose of makinr clear what is inadeauate as a general theory of cellular deve10pment. The central result of the theory is that en- vironmental stimuli result in changes in the talandic temperature -— or variance of the epigenetic variables p3. and r3. about their steady state values -— so that the epigenetic "system can exist in many different talandic energy states without any change occuring in the steady state values of the microscopic variables"lo. But, if we think of cellular states as reflecting the enzymes present in the cell, development must consist of changes of the steady state values of the pj's. Thus, "it is necessary to have a model which is irreversible in the sense that ... the steady state quan- tities p3. and qj undergo permanent change"ll. "The difficulty is to produce a model which switches under an environmental stimulus (a temporary parametric alteration) and then stabilizes itself by other changes of internal activities so that even when the stimulus is removed the altered state persists"12. Our tactics, then, will be to use the crux of the Goodwin method and represent the genetic- metabolic system of the cell by control equations and the state of the cell by points in a multidimensiOnel phase space, but to attempt a more general theory able to describe irreversible and quantized processes. 2. Strategy -- The tendency to use physics as a model for theoretical biology is clear in Lotka's use of "stoichiometric" equations to study the "transformations of masses and energies" between demes. It is to be seen in Volterra's law of the conser- vation of demoSraphic energy and his pains to show that his ”equa- 10 ll 12 Ibid., p 13}. Ibid., p 151. Goodwin uses q instead of our r. Ibid., p 151. Italics in original. l+7 tions of motion" can be derived from a variational principle. It is to be seen in Kerner's even greater pains to derive the Volterra equations from a least-action principle, in his use of the Gibbsian statistical mechanics, and in his decision to derive the various thermodynamic functions. It is important to realize that none of these workers is directly applying physics to their problems. One can no longer doubt that the laws of chemistry and physics are sa- tisfied in living systems, so that it behooves the biologist to pay attention to entrOpy and force and molecule and a sizeable frac- tion of the physicist's armamentarium of concepts. But, each of the workers we have discussed realizes that it is inapprOpriate —- in general -— to use these concepts to discuss integrated bio- logical systems. It is meaningless, say, to speak of the phhysical kinetic energy of a population of animals when studying pOpulation genetics. The tendency is one to proceed in analogy to physics and the outcome is an analOg to temperature, an analog to entropy, and an analog to Hamilton's equations rather than a novel use of the physical quantities 2££.§£' An example of an argument offered in support of this analo- gizing occurs in the first chapter of Goodwin's book. First, Good- win points out that biologists are faced with a form of what we have called the problem of concrescence: From the properties of the "elementary particles" of cells, such as the cistron, the zymon, the replicon, etc., must emerge those characteristics which are the recognized attributes of living cells. 1’‘ {This is analogous to the problem of statistical mechanics, but un- like the case in thermodynamics where a quantiative set of macro- 13 Ibid., p 1. #8 scopic relationships in the form of phenomenolOgical thermodynamics exist, All that there is in biolOgy is a set of concepts such as organization, adaptation, regulation, competence, homeostasis, etc., which must carry an enormous burden of more or less intuitive understanding and experience about the essential principles of biological structure and function. Although some of these concepts have been analyzed into more exact notions which could lead to quantitative definitions satisfying to some extent their intuitively-perceived content, there is certain- ly no set of relations which order them into phenomeno— logical laws of cellular biology. lfi/ Goodwin then asserts that phenomenolOgical laws of cellular systems are not apt to be found: The singular absence of precisely-formulated laws of cellular organization suggests that there simply are no obvious general quantities for measuring cell be- havior which are presented to our senses in the manner that heat, pressure, and volume are in the study of physical phenomena. lé/ Thus, Goodwin elects to introduce analOgs of the known "macrOSCOpic parameters" from thermodynamics: The present theory ... sets out to derive some general lacrOSCOpiC or "thermodynam 0" functions which arise from certain dynamic characteristics of molecular con- trol mechanisms in living cells. The programme is, then, to use the present knowledge of the molecular organization of cells, so brilliantly eXposed by mo— lecular biologists, as the microstructure for a sta- tistical theory from which the general behavioural consequences of this organization can be deduced in terms of functions which bear a complete formal ana- logy with the classical thermodynamic quantities of temperature, free energy, work etc. lé/ ,Again, it must be emphasized that the posited relationship between physical thermodynamics and talandic thermodynamics is one of simi— larity or analogy rather than one of identity. l3? Ibid., p 2. 15 Ibid., p 3. 16 Ibid., p 3. 1+9 If we were to adopt this strategy, we would generalize the Goodwin theory to obtain the talandic analog of quantum mechanics or non-equilibrium statistical mechanics. However, we shall ar- gue that this strategy is not valid and should be abandoned. In- stead, we suggest that the problems of the mechanical—thermOdynah- ic relationship and of the genotype-phenotype relationship are re- lated in that they are elements of the sane class of proolems -— those of concrescences —— and that there is no formal similarity between the systems studied in physics and those studied in holis— tic biOIOgy. Our reasons are that (a) the difference in the modes of analysis of the physicist and of the biologist suggest that con- cepts cannot easily be transferred between the disciplines, and (b) assumming that the subjects of the two sciences are similar and that similar concepts exist in the two sciences has led to no use- ful results. (a) (1) To see how the analyses of the physicist and the bio- lOgist are different we must realize that all of the physicist's concepts and results are derivative from his eXperience as a human being in what, for lack of a better term, we shall call eXperential Space, the "Space in which" he perceives things and moves around in and endures in. The mathematical description of experential Space is that of a Euclidean 5-space and a Newtonian inertial reference system. For some purposes, experential space is too naive of a concept and it is necessary to alter our conception of space and time to one of space-time. This change is tolerable because this new idea still has eXperential Space as a "classical" limit and, Inore importantly, even in relativistic situations events re still SO constrained oy the most basic properties of experential space. For example, it is not possible to escape from space, the result of any movement is that the moved object is still in experentiul space. Also, an object can be made to move about in eXperential space in a continuous motion from one point to another. Even the host ab— stract results in physics must conform to such basic prOperties of experential Space. Mathematically, the point is that all the things that can be done to a physical object must be expressable in terms of a group of continuous transformations of Objects in experential space. There is no force of logic behind this "must", it is just the intuitively established standard of our intellectual tradition. The mathenatician remains unfettered by considerations of ex- perential space. In fact, mathematics is difficult just because it requires one to abandon his "real” eXperential space for a considera— tion of the more abstract idea of a set of elements which nay exhi- bit peculiar characteristics when judged by the standard of eXperen- tial space. This mathematical license tends to obscure the primacy of eXperential space in physics, for the physicist often uses a va- riety of mathematical spaces as tools to study events that occur in experential space. The Lagrangian and Hamiltonian formulations of mechanics represent events that happen in experential space as points in configuration or phase Space. Quantum mechanics represents "states" of objects in eXperential space as vectors in Hilbert space. But, the physicist never studies events "in phase Space" or "in hilbert space"; instead, he uses these spaces to emphasize the characteris- tics of events in experential space, and they are useful only as long as he can translate his findings to the coordinates of experen— 51 tial space-time. An apparent exception to this is the affine pres- sure-volume-temperature space of thermodynamics which imposes the odd restriction that state transformations can be made only along isobars or isotherms instead of along the shortest route from point to point.17 Perhaps, as Bridgman (1961) points out, it is because of this degree of abstraction that it has seemed necessary to "re- duce" thermodynamics to mechanics and, thus, to events in experen- tial space. By virtue of such reductions, all of the concepts of physics become utensils for thinking about events in experential space: concepts such as "force", "energy", and "angular momentum" being clearly related to experential space, the concept of electri- city and magnetism such as "charge" being less immediately stated in terms of experential Space, and the parameters of thermodynamics being related to experential space only through the SOphisticated transformation of statistical mechanics. In sum, physics may be defined as the study of events from the point of view of eXperential space. (2) However, this is not the only point of view from which to view natural events; entirely different orientations are often use- ful and intelligible. As a simple example, suppose one had a pile of building materials - boards, bricks, nails, etc. These compo- nents could be used to build one of several objects; say, a house, ' barn, and a row boat. It would, of course, be possible to des- cribe the construction of each item from a physicist's viewpoint as a series of transformations in experential space. .Each step in construction would be described as a change in space-time, particu- lar pmoblems being formulated in terms of "forces" and "stresses" 17 or. L. Brillouin (1961+). 52 and "strains". But, it is also pos;ible to describe the building of any object from alternative points of view. One might define a sequence of states such as that of being unassembled materials, that of being a house, that of being a boat, etc. Now, if we wish to study the transformations between such states, this viewpoint can give unique insights. It is clear, for instance, that a direct transformation from the state of being a hous to the state of being a barn could probibly be carried out by a suitable remodeling pro— cess. But, it is unlikely that a direct transformation between the state of being a house and the state of being a boat could be mana- ged; the radical differences in the macrosc0pic organizations of the two objects would demand an intermediate transformation to the unas- sembled state. All of these transformations possible or impossible must conform to tne physical laws hf transformations in experential space. On the other hand, it does not seem worth the effort to try to eXplain the impossibility of the house-to-boat transformation in the physicist's language: it is not clear, for example, that the entrOpy of a boat is more or less than that of a house so that it seems to be a misplaced effort to dream up an analogy to the Se- cond Law to describe the "evolution" of such objects. This example is important to us because much of the biologist's work is done from a point of view which emphasizes organization rather than processes viewed from the point of view of experential space. In particular, the ideas of "gene action" and "cellular state" and "induction” do ant really gain much by being interpreted in terms of eXperential :apace. Biology might be characterized (but not definej) as the study of a certain class of objects from the point of view of or- 53 ganization. A more realistic example of this point of viex is discussed in Section B of Chapter IV. (3) This reCOgnition of the possibility of viewing natural events from several points of view —- what Jhitehead calls dif- ferent modes of analysis —— puts us in tne position to reilize that an insistance on making physical fiUQlO5iGJ night tend to aC- tually vitiate th power of the tools the Volterra-herner-Goodwin movement has develOped. The central theme of these theories is the representation of a demographic or an epigenetic system by a point in a ”phase” Space. These spaces, however, are only distant- ly related to the pnysicist's phase space in uhich the pesitions of objects in eXperential space and their momenta are plotte;, for the points in these spaces are in no way related to events repre- sented frOm the point of view of experential space. Rather, they represent the results of interactions between organisms or between control 100ps, and are better described by a representation from the point of view of the organization of a pOpulation of organisms or of a cell. Most of the physicist's concepts are ideas about events in experential space and it is inappIOpriate to assume that they are meaningful when used 'n conjunction with the viewpoint of organization expressed by the "phase” spaces of Kerner and Goodwin. The danger in this transference of concepts between points of View is that the prevalence of "physical analogs” may obscur parameters and concepts which faithfully describe events from the organiza- tional point of view. Since both the Kerner and the Goodwin theor— ies have achieved some success in describing biolOgical events it is necessary to tentatively accept their ”phase” spaces as fritw— Eh ully re resenting tiolo;ic l i,1ena,'»;.'n;-n.:gx. jut since these ”Lhase” spaces manifest a yoint of view radically different from the point of view of experential stace represented by yhfibe srpces it is ne- cessary to assune, until lrcven otterwise, that the systems being analyzed by the yhysicist and by the biOlOJiSE are not formally similar so that ”physical analogs" should not be exrected to have aiytling to do with bioloaical systems. (b) At this taint, we must raise tto auestions. First, hgd the use of thysical analo;s received any entiricel Justification? Secondly, is Goodwin Justified in using ghysical anlegS as a last resort beCause the comflexity of biOlOgiCQl systems has mashed all phenomenological relationships? he will consider Goodwin's point first. (1) Immediately after suggesting the use of rhysical analoss, Goodwin laroceeds to a discussion of cybernetics and negative feed- back as dpflied to cells. This is ironic, for ”negative feedback” is the prime example of an intuitive, basically biolo;ical concept which has been rendered quantitatively precise. The notion of neg- ative feedback is pretty much equiValent to N. B. Cannon's concegt of hOmeostasis in that each homeostatic system is bound to exhi- bit negative feedback and each instance of n gative feedback in an organism is apt tc be associated with a hemeostntic system. Further- more, Rosenblueth, hiener, and Bigelow (1943) have suggested that negative feedback is a grecise may of describin; ”purposiveness”, the Lrototype of metaphysical entities. The concept of negative feedback is also an example of an idea oriented towards the organi- zational vie point. It says sonething about the way in which the 55 organizqtional comfonents (as oggosed to the material co gonents) of a system interact; although this interaction can usually Le des- cribed as events or configurations in experential space, the impor— tant consideration is only that a set of cosyonents actually do in- teract; how the components are arranged Spatially is pretty inch besides the point. Because the idea of negative feedoach is geared to the organizational viewpoint, it is not likely to be used and has not been used in any of the branches of :hysics is e fundamen- tal concept. The utility of negative feedback in studying biologi- cal systems does not prove thut a set of thenomenclOQical rel;tion- ships will soon be found for cellular systems, out since it, like the phenomenological relationships of thermodynamics, allows some definite statements ubout the behavior of the system of interest, the fact that theoretical biology and theoretical physics night not look alike no longer seems to be a serious shortc0uing. (2) he return now to our first question. In this regard, it must be pointed out that most of the yhysiCdl analon derived by Kerner and Goodwin are actually never used and do not contribute to the derivation of any testable results. The only yhysical ana— logs which are used are the total energy, G, and the temperature, 63. Physical total energy is defined directly in terms of the coordinates of experentisl space. The function G is chosen as the talandic total energy because it satisfies the Hemiltoninn efiuatichs; on this basis, the grobability distribution in phase spice is de— fined. If we retnin our scepticism towards the transferrence of Physical concept to biolOgiCdl groblems, this becones a rather i3 hoc choice of a probability distribution. Regardless, it is doubt- f 50 ful how much physical content is actually seine introduced into n 3 . . , , . , _ _ .A 15 ‘ . a . aOOQWln s theory, Ior a Single calculation shows that the {rous- bility that the variable r; falls between rj and r.j + dr. is given u by just the Gauss'an distribution (cf. Chapter III, Section D.l.(a)). We haVe already noted hat the talanuic teggerature is more or less simply related to the variances of the variables p4 and rj. The d other Paraneters (T+/T etc.) which are actually used are measures of the Various oscillatory characteristics of the systeu and are not yhySiCal analogs. Thus, all of the useful results of the Ker- ner and Goodwin theories could be obtained, devoid of ghysiCalis— tic trappings, by using the usual statistical methods to study the oscillations of the variables Ij and rj. There is no engiriCal justification for the transference of Physical concepts to the study of biolo;ical systems. Je have argued that because the physicist and the biolOgist tend to look at natural events from two different viewyoinns, the yhysicist abstracting out of nature ideas which relate directly to experential space while the Oi0103i5t has found it useful to ab— stract out of nature the functional relationships manifest in the svstem he is studying, it should not generally oe possiole to trans- fer particular concepts from one science to tne other. We have pointed out that, at least in one instance, a bioloiical cohcept has been put into a form which allows definite statements aoout the behevior of intact systems. And, we have noted that the im- portant results obtained by Kerner and Goodain are not actually degendent on the use of yhysical analOgs. he conclude, therefore, 18 \ See B. C. Goodwin (1&03}, L) C‘ t. K) I . 57 that the strategy of paralleling Physics is an unnecessary one which is likely to ooscur already Opaque proolems uy introducing inapproyriate formulae and unneeded entities. A gore aggrogrinte strategy is to begin by doubting the applicability of all physi- cal concepts to integrated biolOgical systems until such concepts have been shown to be actually neceSSury to understand living sys- tems and to taKe as central to a theoretical biology those ideas and parameters which DiOlOSiStS have found helpful in thinking a- bout living systens. No doubt the two nodes of analysis will ul— timately find seme concepts mutually cowpatible, but by using this strategy we will feel confident that these connon concepts reflect some basic similarity in the two types of systens rather than mir- roring the scientist's human tendency to over generalize well un- derstood ideas. CHAPTER III. A THLCRETICAL APPROACh TO CELLULAR STATES "The aim is not to ape physics, but to mine the uni— versal mathematical quarries." -- E. H. Kerner A. Assertions l. The problem -- Our problem is to discover the relation- ship of the genetic—metabolic system of a single cell to the mor- pholo;ical type of the cell. We will assume that the genetic-meta— bolic system of the cell is adequately represented by a set of 2n kinetic equations in terms of the variables p. and rj. If we in— troduce the variable A which measures the morphological type of the cell, the problem will be solved if we can express A as a func- tion of the variables p and r (we abbreviate the set of n pj's by p and the set of n rj's by r). In this Thesis, we will restrict the discussion to cells which persist in a cellular state and do not pass from state to state. 2. Assertions -- As the basis for a solution we will use two assertions about the nature of cellular states. These state- ments are assertions in the sense that eXperimental data can be used to support them, but will not be presented until Chapter IV. The reader who is uneasy about this is invited to read Chapter IV before proceeding in this Chapter. First, we assert that a given 58 59 cell can be in one of a finite number of discrete morphological types and that the cell is a stable system (in a sense defined be- low) with respect to its morphological type. Secondly, we assert that there is a one-to-one correspondence or isomorphism between the morphOIOgical type of a cell and the state of its genetic—meta— bolic system. In view of the first assertion, this means that a cell can only be in one of a finite number of discrete genetic— netabolic states. B. Definitions and the Eigenvalue Equation 1. G>space, biochemical states, and morpholoEiCal states -- lNe designate the aggregate of all cellular systems as S. From now CHI, the genetic-metabolic system of the cell will also be called tale biochemical system of the cell. The biochemical system of a ceJJ.is.represented by a set of 2n kinetic equations a: . dr. . J = Rj(p'r) and E = Pj(p,r), J = 1.2,eee’no Pj is the concentration of the ith species of protein; rj is the CCJncentration of the corresponding mRNA. For the purposes of de- VEfiLOping the general theory of cellular states, we will allow the functions Rj(p,r) and Pj(p,r) to be any well behaved functions of P 61nd r. However, we assume that Rj(p,r) and Pj(p,r) do not de- Pe11d explicitly on small molecules such as ions, cofactors, hor- mcules, vitamins, phOSphorylating agents, etc. In almost every JJTteresting case it will be necessary to take account of these sflmill molecules; this can be done without changing the general 60 theory by expressing some of the constants in Rj(p,r) and Pj(p,r) as functions of p,r, and other variables; thus, Obtaining more complicated kinetic equations. A more practical description of S is suggested in Section D of Chapter IV. We define a vector space, 6), with the basis vectors a.j and bj’ j = l,2,...,n, such that each axis of 0 represents the con- centration of a species of protein or mRNA. A vector in 6>is gi- ven by n T ( a r'b ) = ..+ .- 21:33 33’ j = l and the time rate of change of T is :5 n dT “ = fl“. '3. = $2.". P76). 3;- = Q 2(1)an +133) {3385+ 33 ° 1 a: j:l We temporaily define the biochemical state of S as the Zn- tuple (p,r); that is, a point in O). This definition is implicit in Goodwin's "phase" space of epigenetic systems. The morphological state of a cell is a real positive number Xk such that each different morphological cell type correSponds to a different Ak' (It is easy to modify the theory to accomodate the possibility of several cell types corresponding to the same xk). In accord with our first assertion, we stipulate that there be N morphological states and N xk's; that is, k = l,2,...,N. Our second assertion maintains that there be a correspondence between the biochemical and morphological states of a cell. To obtain 61 such a correspondence, we introduce a function L(p,r) defined on O’which maps biochemical states (Zn-tuples) into the set of all Ak's; we will specify the form of L(p,r) in Section C. Our as- sertion requires an isomorphism between biochemical states and Ak's, but there are an infinity of 2n-tuples in G’and only a fi- nite number of Ak's. To obviate this difficulty, we will rede- fine our concept of biochemical state. 2. Biochemical experiments -- Experimentally, we can at least conceive of measuring the variables p and r for a given cell. Biochemical experiments consist of measuring the p and the r under normal and experimentally altered conditions. In either case, the result of any experiment is a 2n-tuple of values, (p,r), which can be represented as a point in 0). Suppose that it be possible to follow the variables p and r for a period of time in a cell of a given morphological type. The results of this eXperiment will be a configuration of points in (P. Suppose further that we pick one of the p or the r, say, pj, and prepare a plot of the density of points in 6>as a function of pj. If this plot be normalized, we can consider it as a probability density. The results of such an experiment might resemble the plot in Figure 5. To generalize this idea to the case of all 2n variables, we introduce a single 19 valued, continuous function Uk(p,r,t) defined on 0); the index k means that the cell was in the hth morphological state at the time of the experiment. Denote the density of experimental points in G’by Ui(p,r,t)Uk(p,r,t) where Ui(p,r,t) is the complex conjugate of Uk(p,r,t). After normalization, Ui(p,r,t)Uk(p,r,t)dpdr gives the probability that the results of a single measurement of pj is 19 Cf. Atkinson (1965), note 8. 62 Figure 5. The results of the measurement of the concentration of the th Species of protein in a biochemical experiment. Probability that p3 is between I I rnd {j + dpj fit A 0 ma ...: .5 Concentration cf the jth protein species 63 between p3 and pj + dp‘j and r3 is between rj and rj + drj for all j. In general, we must expect that an actual distribution of points and not a single point will result from a series of measure- ments. If we were to explicitly study the influence of small mole- cules on the p and r, variations in the activities of small mole- cules would produce variations in the experimentally obtained val- ues of p and r. But, even if we ignore these effects, the compo- nents of p and r are likely to oscillate about some mean values, so that measurements will result in values of p and r anywhere be- tween certain limits. The necessity of reCOgnizing such oscillations is another reason for rejecting the idea of a biochemical state be- ing represented by a point in 0). To meet this contingency, we now discard our first concept of biochemical state and define the biochemical state of S as a den- sity function in (P. We can fulfill our second assertion by con— sidering N such functions. The function Uk(p,r,t) serves to define such a density function, so we define the N biochemical states of S as the N functions Uk(p,r,t), k = l,2,...,N. Thus, we consider the function(a1) L(p,r) as mapping the set of functions Uk(p,r,t) into the set of the 4Akds. We will assume that all N functions Uk(p,r,t) are different. The distribution of points in GDare, in general, not disjoint. 3. Expectation values and variances -- In a bioloSical sys- tem, p. must have some upper bound; for example, if the average total protein concentration for a cell is p0, it will always be the case that pj‘(po. By a suitable choice of units for the p 64 and the r we can have O\and for all t Ui(l,l,t)Uk(l,l,t) = 0. Suppose that S be in its hth state; the expectation value of any function, A(p.r), defined on U’is given by20 l l Ak(p,r) = A(p,r)Ui(p,r,t)Uk(p,r,t) dpdr . O 0 Note that Ak(p,r) does not take into account the probability that S is in the hth state. Setting d7“: dpdr = dpl...dpndrl...drn and suppressing the limits of integration as understood, this can be written as Ak(p,r) = /[Ui(p,r,t)A(p,r)Uk(p,r,t) d7'. If S has N possible morphological states, we have a similar ex- pression for the expectation value of A(p,r) when S is in each state. Since these states are assumed disjoint, we may inquire what the expectation value is for a series of determinations of A(p,r) performed while S is in different stable states. We de- note the probability that S is in the hth state when a measure- ment is being made by cgck. Then, the expectation value of A(p,r), taking into account the probability that S is in different states, is 20 This is the usual definition of expectation value. For example, see Frazer (l95b). N e E ckck Ak(p,r) k = l A(p'r) = N I I i ,ckck k = 1 We require, of course, that chick = 1, so that N (1) A(p,r) = E ciUi(p,r,t)A(p,r)ckUk(p,r,t) dT . k = 1 A more compact form of (1) can be obtained by introducing the N x N matrix A(p,r)Ma = A(p,r) o e e e 0 O l O O 0 O l the column matrix 66 and the row matrix . i t t 1 see kkeee CNUN o In terms of this notation, (1) becomes A(p,r) = JU*A(p,r)MaU d7 . This form of the expectation value will be used throughout this work. As'a special case, note that if A(p,r) = pj or A(p,r) = rj, then .- U".Ud 3 [p3 T U! I (2) U‘r.U d . far a! l The variance of all measurements of A(p,r) in all N states is defined as u A p N V I A b! V N (3) (AA)2 -11 [U‘Afiav - JU'AMaU j Where A = A(p,r) for brevity. An equivalent way of writing (3) is 67 (4) (1310‘2 = JU’ [AMa-A12Ud7‘. Equation (3) is easily derived from Equation (h); these definitions and prOperties of variance are anaIOgous to those usually used in statistics (cf. Frazer (1958)). 4. Stability -- Let A = A(p,r) be some function of p and r. S is defined to be stable with respect £9 A(p,r) if and only if (£§.A)2 = O. This suggests that if a series of measurements of A are made at times t1,t2,...,th, then S should be considered stable with respect to (wrt) A if the time average deviation of the mea- surements from some value A of A is zero. This definition oses $1.. two technical difficulties. (a) The definition of expectation value introduced in B.?. is an average in (P-space, and it is not obvious that this kind of average is equivalent to a time average resulting from an actual ex- periment. The connection between the two kinds of averages is a- chieved by interpreting, for example, ‘ ckck Zlcick as the fraction of time spent in the kth state over a sufficiently long period of time. The difficulty comes in deciding how long is "sufficiently long". Physicists meet the difficulty be resorting 68 to time averages extended over infinite periods of time, for which the correspondence can be more or less established (ergodic hypo- thesis). In our case, the most realistic approach is to leave the matter entirely up to the judgement of the experimentalist and al- low him to decide how long he has to continue to make measurements to faithfully reflect the nature of the system under study. (b) The word "stability” has a precise meaning in mathematics; but, in general, it is not clear how the definition given here is related to the mathematical definition. However, if we set A(p,r) = p, the condition (.CxA)2 = 0 suggests that p is arbitrarily close to B for almost all values of t. Then, the solution 3 of the sys- tem of equations p = P(p,r) and 5 = R(p,r) will be stable in the sense of Poincare (for example, see Magiros (1966)) with p’as a point of stability. Also, if A(p,r) is a quadratic form of the p and the r, then (3,?) will be a stable point of the equations if A(p,r) has a minimum at (3,?). Thus, it should be understood that the "stability" used in this Thesis is related to but not ne- cessarily identical with mathematical stability. It is very possible that S may be stable wrt A and be unstable wrt some other function E = B(p,r), i.e. (13 B)2 i 0. An individ— ual cell is expected to be stable wrt morphological type but be un- stable wrt some of the p and the r most of the time. 5. EmbryOIOgical egperiments -— From an embryological point of view, an experiment consists of "measuring” the morpholOgical state of a cell under normal and experimentally altered conditions. Our first assertion was that cells are stable with respect to mor- phological type. Thus, the anticipated results of an embryologi- 69 cal experiment are of the form of Figure 6. In the experiment de- picted, the morphological state of a cell was measured, and the pa- rameter elk (the cell was of the REE type) has been equated to the expectation value of the measurement. Since L(p,r) serves to re— late the variables of G’to the values [A of the morphOIOgical k variable 1*, we choose to write zxk = EXETF3. L(p,r) is a func- tion defined on G’so that the expectation value and the variance of L(p,r) are defined as in the previous Section. The Figure and the first assertion indicate that llL(p,r) = o for the kt state (and, in fact, for all of the morphOIOgical states). Together, Figures 5 and 6 show that S is stable with respect to the function L(p,r) or the variable >\, but is not stable with respect to the variables p and r. These observations are the starting point of tne theory to be develOped here. In preparation for this develop- ment we will need the familiar Schwartz inequality and a certain integral. 6. Two preliminary results —- (a) In our notation, the Schwartz inequality 21 is A. (5) jU‘U JlAMau)'(AMaU) bl [§(AM8U)‘U] with equality holding if and only if A aU = CU for some scalar M e - _ ¢ C. In particular, we may take C - jU AMaU . (b) We will now show that the integral Irwin = [Maw 21 This is a standard result in analysis; e.g., see Schmeidler (1965). 70 Figure 6. Results of the measurement of the cell type of a single cell in an embryological experiment. ,X is the morpholOgical varia— ble. Probability that the Value of X is between A and )+ d.) 71 t can be written in the form J (AMaU) (AMaU) . To do this, con- sider the column matrix W = U + d(AMaU) , d is any scalar and the corresponding row matrix fl we = U‘ + d‘(AMaU) . By inspecting the integrand of the integral [W‘AMaW = [[Ue + d‘(A_MaU).J AMa [U + d (AllaUfl and recalling that A is just a compact way of writing the pos- Ma sible values of a biological measurement” we see that the integ- ral itself must be a real quantity. Written explicitly, this is JN‘AMaVJ .—. [H.AMaU + d IU'AMaUHMaU) + d*I(A,‘iaU)‘(1th) + dd’,[(AMaU).AMa(AMaU) ' But, in order for this to be real, we must have (Im d)fU‘AMaA_MaU + (Im at) [(AMaU). (AMa U) = c . 72 Thus, we have our result . 2 * , [U Abiau = Jm'biau) (Al/lab) ' 7. The eigenvalue equation -- we can now derive the funda- mental equation of the present theory. Recall that Z§L(p,r) = O; that is J. IU" 211 - [Iva U] k;Ma k - k Ja k for we are limiting ourselves to the measurement of a single cell in the hth state so that cick = l and cfick, = O for all k' f k. Using the results of Section B.6.b. and the fact that .{UEUK ll ...: a. «JUQUK Icf the probability distribution U'U. The ad- Vantage Of this method is that L(p,r) will be reflecting an aspect apprOpriate to a biolo iCal system, for it will be a parameter ap- plicable to any system thich undergoes changes. The risk involved in this case is that this parameter may be so general in nature that it can contribute no interesting or useful infornation about biological systems as Opposed to systems in general. The choice between these two approaches is, perhaps, really dependent on in- dividual intuition, but the only legitimate means of arbitrating the matter is to compare tne success of the experimental predictions of each approach. (b) Since we are proceeding in an arbitrary nanner at this point, there is nothing to prevent picking L(p,r) to be a differ- ential Operator defined on G). ‘The advantage of this choice is that, since we have assumed S to have stable states with respect to L(p,r) and have argued that there is an equivalence between the matrices defined on (RN and differential Operators defined on G), we can write immediately L(p,r) = L(p,r)Op = L(p,r)Ma where the eoual signs mean contextual equivalence, and conclude that L(p,r) Uk(p,r,t) : AkUk(p,r,t) . 76 Then, the ka's will be tne expectation Values of L(p,r) in tLe N stable states of S; the functions Uk(p,r,t) determining the kro- bability distributions will be the solutions of a differential e— quation. The total time derivative of the function U(p,r,t) is iiven an a “3% = ot+ :— jzl 3 by dr. dt ° C 24% Q/ C (.4. + *U (3’ *3 Uaing the kinetic equations of S and multiplying by i = 4-1 : (8) i'—= i au R.(p,r) + 3'3 5‘3 0-D: rtC: QIQ/ ”C3 + P m” Gale, Pot: Pj(p,r) . We now Eostulate that the function L(p,r) is the Liouville opera— tor n a 3 L(p,r) = L(p,r)Op = i Zijan) a pj + Pj(P.r) fij. J = Then, Equation (8) can be written I'd—t- = 1 dU '(BU+LU 3t where we have set L : L(p,r) for brevity. The Liouville Operator can oe written in a different form by use of standard vector notation: Recall that a gosition vector in (y denoted by-f has the time rate of change 79 n -,~ d? E —~ -~ - a? = . Rj(p,r) dj + Pj(p,r) bj e 3:1 U is a function defined on G’with gradient n <;U = E iLQ '3. + a P3 3 j=1 Q/ G gt 0/ a m In terms of this notation, we hive A .3 L = i Qo(V ) so that L U = i ZeU. (c) In Section B.l., we took the function L to describe the morphological state of S. The result of an embryological exyeriment is an expectation value of L, and picking a form for L is the same as stipulating what is "really" being measured in an embryolo;icnl experiment. In our case, if S is in its K32 state, the expecti- tion value of L is )\ T A 6 = U‘ L U d = i " n. U d . k k k k m k T d U o O n A 0 a c {Q is the veloc1ty of a pOint in G); <7U'1s a vector Wlth magnitude equal to the greatest rate of change of the function U and pointing in the direction of the greatest rate of change of the function U 60 and pointing in the direction of the greatest change of that func- tion; Uk determines the probability distribution in. t. That is, we want d O Anticipating the result of Section D.l., U‘(p,r,t)U(p,r,t) : u(p,r) x u(p,r) where u = u(p,r) is a time independent function of o and A r, and it is sufficient to demand that e e :3? (l0) ti°VP + at a 0 where f): n‘u. The identity div (pi) = flit») + 3.6m) allows this condition to oe written as 3p . e _ “‘_7—’~‘_, 5t + le (Fri) P '3— : le(F~o¢), and the condition that the integral (9) be tine invariant reduces to 6"; = O. . - In general,‘V-Q i O. For example, if p = ab + br + c, r : up .5 + er + 1‘, then 3-4 = a + e. Thus, to obtain an sober-table density function we introduce the function M = M(p,r), called the last mul- IN 62 tielier of 5, defined by the condition,Vn(M%) : O. hultiglying e- quation (10) through by h and setting D : Hf), we obtain a ‘ at=00 U _\ ..A MQ oVF-t- A —3 ..‘b A _5 Using the identity div (MP-Q) = PV'OM) + MQ-VP, gives 51) at A d .3 + div (Deg) —pVo(m) = o . If we use D as the corrected firebability density, we will have div (D3) = - a D/at and dD/dt = c. M can be determined from its defining condition 3- -‘-> . This condition implies that dt QHVM : - dt M‘V'e. Since M does U A uh .g-VM , and exp [-Jfis, cit] . The parameter t must be removed from M by solving the kinetic equa- not explicitly contain t, dM/dt 2:74: : - dtfio-‘z, or M tions of the system in terms of p and r. This proceedure is illus— trated in the Example, below. In some instances, t : t(p,r) will e be a complex function; then we will use D = (M‘M)’uiuk as the cor— rected probability density. (e) To be biologically meaningful, the ‘kkfs must be real constants. .L -b A .1 x #y. _ ' 4:. . - M4. 7* wk, vU _ 1h: *WUKUK) UK; ,mk] h H ....a. II P. [6'(U‘U I'A) “It A ‘3 “A- t] k k“ - LkUkGT he“) - Uknmovuk II P“ u I ,4 ¥-"\ :2 WC: gut 4 c We Here, 3 is the surface of the unit sphere in G’space, 3 is the nor- mal to S, and ds is an incremental area on 5. We have imbosed the boundary condition Uk = 0 along 5. since ‘AKLiS equal to its con- plex conjugate, it is a real quantity. .3 2. The Liouville equation -— The equation BID/a t + (.37de = 0 leads to an equation for the time deve10pment of U For sin- k' plicity, assumefii§ = 0 so that M = 1. Then C1 a 9“ t 9...: O = U‘[Ot + fi'VU] + U :3 + Q'VU'] . SetUzReU+iI;nU,U‘=ReU-iImU. Then 0=2ReU[gtReU + :WRe U] + ZIInU[-5a—t Ian+:.€IxiLT]. at. In renerxl, Re U and Im U are independent and non—zero, so that we must have c) ~34 fiReU + Q°VRCU:O and a A _s :gz'lu b + $0<7Im U = C . A .3 Thus, 3 U/d t + QoVTU = 0. Finally, we have the Liouville equation for U: Q, C‘. 0/ d- D. Determination of the Functions Uk(p,r,t) The time dependent behavior of the Uk(p,r,t)'s is governed by the Liouville equation; the time independent behavior is de- termined by the eigenvalue eiuution. To deteruine the functions Uk we assuhe that the variable: are gartially separable so that Uk(p,r,t) = 2}L(t)uk(p,r) for some functions of time 1}k(t) and some functions of p and r, uk(p,r). 1. Time detendent part -- Using U and the Liou— k: kuk ville equation we find that l}k(t) nuet satisfy the ordinary dif— ferential equation - d <11) 1a“ 1km + vkmkk = o . 55 1 Mt The solution to Equition (11) is I}k(t) = e . AR must have . ,. . , . —l the dimensions of (time) . Two important results follow directly from the form of Zflk(t). Thus, unless uk(p,r) is identically zero,dUk(p,r,t)/3 t 2 0 if and only if k = O. In this case, the eigenvulue equation oeCOhes just Luk = O, which is satisfied by uk(p,r) : constant. In pir— ticular, we may set uk(p,r) : eXp (constant), or uk(p,r) = Uk(p.r.t) = eX'p (G -W/G) where G is the tulandic total energy, 4’15 the talandic free ener- gy, and E91s the talsndic tenperature. Thus, the Goodwin theory is restricted to the special case in which k = O and is not equip- ped to discuss the N-l other cuses in which k A O. (b) The functions U form an orthonormal set. To verify this k property, we introduce a constant of normulizution, Nk' Consider the integral iAk,t -iAkt ' e _ t Nk'Nk Uk'Uk .. Nk'Nk e n k,e uK . There are tho cases: (1) if xk = X we can huve k. (I C‘\ H H P' E: EDb’ ‘t}qe Ergodic theorem (see Khincnin, 1949). This intedrul is eu- :5j_]_3r evaluuted: N N u v k' k in}: 1 -i(Ak'- AK” ) lim — e -i(Xk,- A hm k 'rlltis fCDIWn an orthonormal set. 2. Time independent tart -- ‘VC13.ue equation; written explicitly, n (12) i i Rj('pl’...,:pfl,rl,. j = l <)uk(pl,...,rn) N N U‘ U = (8 ° the functions U when normalize; ’ k' k k' k k'k’ k’ ’ uk(p,r) must satisfy the eigen- this is 3 uk(pl’....’rn) 0.,rn) + apj Pj(pl,oo o,I‘n)"—g = >k.uk(p],...,rn). brj 57 Equation (12) is u linear, first order, partial differentiul enun— tion in 2n Variables. The solutions of such un euuution is usuel- ly effected by reducing it to a systes of ordinary differential e- '3 . L. quations 3 . For the purposes of this section only, we use p and r to desi; nate just two variables out of the conplete set of an Vuriables of S (i.e. p = Pj' r = rj, for sone j). Thus, Equation (12) becones Auk(pir) ()uk(P’r) (13) R(p,r) <9 p + P(p,r) a r = - i,Akuk(p,r). We now introduce a three dimensiona space with coordinates given by p,r, and u : uk(p,r); and note that u is just a surface k k in this space (Figure 7). A general function in p,r,uk- Space is represented by C€(p,r,uk). For example, we might have 2 2 CP(PgI‘ouk) = O = (p-a)2 + (I‘-Q)2 + (u —a) — b , k which is the sphere in p,r,uk-space. At any given point, (p,r,uk), the normal to the surface CQ(p,r,uk) is given by (11+) ‘3 =‘W(p.r,uk) = 1% :1 . Q3;- '32 . dig-k :33, where we designate the unit vectors of p,r,u 23 _ . ...x A _\ k-space by el,e2,e3. This method is discussed in detail by Courant and Hilbert (1962), Chap. II of vol. 2. We will only outline the proceedure here. Our choice of boundary conditions guarantees that a unique solution to Equation (12) will always exist. CC ()0 5L 0 I Figure 7. The eigenfunction uK -' a surface in p,r,uk-space. -F_.—.—-_-J‘ .-- 89 Now consider the special case in which a C?(p,r,uk) is a so- lution to Equation (15). Later, we will specify a set of boundary conditions which determine a unique solution; but, for the modent, we must consider Ce(p,r,uk) a5 a family of surfaces in p,r,u -space, k each of wnich satisfies Equation (13) and a given boundary condi- tion. Since it is to be a solution of Equation (13), we have uk = %>(p,r,uk). Ne seek a set of ordinary differential equations which generate the family of surfaces uk(p,r). To find them, we use 47(p,r,uk) = uk(p,r) in equation (1h) and note that the nor- mal to any solution of Equation (13) must have the direction num- bers d uk/A p, ()uk/f§r, and 1. Since the third number is the same for all of the solutions, the normals to all of the solutions at any given point will be distributed about a point in a certain plane (Figure 5). This means that the planes tangent at a given point to all of the solutions of Equation (13) will form a pen— cil of planes; the axis of this pencil is called the Monge Efiii after the French mathematician Gaspard Monge. The characteristic curves of Equation (13) are defined to be a set of curves in p,r, uk-space such that each curve is tangent at every point to one of the components of the Monge axis at that point (Figure o). It is possible to show (Courant and Hilbert (1962)) that there is an equivalence between a set of characteristic curves and a given first order, linear, partial differential eq ation. Thus, we may take the differential equations of the characteristic curves as our desired ordinary differential equations. If we introduce the parameter s which measures the distance along a given characteris- tic curve, we see that the differential equations of the charac— Figure b. Characteristic curves in p,r,u “it 9O k'SpaCe e Tangent Planes / / bug \ \ /“ ' / -.. i... ii . ; ~----- ~- / \ r 11 / 7$!;/’ \ / /!/ A : O / ,. / -/ / / F / / . . fk”r- Characteristic Curves teristic curves are l . d d d . ( 5) 35 = B(p,r), 3% = P(p.r), 6%}: = -l Akuk(p.r) 0 Equations (15) are called the characteristic equations of Equation (13). We can single out one of the family of sets of Characteristic curves which generate solutions of Equation (13) by requiring that the characteristic curves of interest intersect a curvetr in p,r, uk-space. we consider_I‘to be described parametricslly in terms of a parameter 7]; and, for our purposes, we specify that I+is not itself a characteristic curve of the partial differential equation. If the solutions of the Equations (15) are given by p = p(s), r = r(s), and uk = uk(s), we specify that s = O alongir and determine p = p(s) and r = r(s) so that they satisfy the initial value con- ditions r = g(w?) and p = h(’7 ) wnen s = O. This gives -11Kks (16) p = pun") ), r ... 145,“? ), and 11km = . . Integration of Equation (13) is then accomplished by eliminating "W from the equations (16), solving the resulting system of eiua- tions for s, and substituting the value of s into the expression uk(p,r) = expi}i.)ks(p,ri]. uk(p,r) is a solution of Equation (15) which satisfies the desired boundary conditions along]? . For boun— dary conditions, we will always take 5 = O at the curve:F in p,r,u K -Space and require that uk = O at 1?, i.e. uk(O) : 0. We choose to define:r by the equations (17) p = cos W , r = sin’? in terms of the parameter 7 . That is, we require that the proba- bility of finding a cell with the concentration of its 33h protein or mRNA such that p3. = l or r3. = 1, be zero. This discussion is readily generalized to the case of an variables. E. An Example To illustrate the theory developed in this Chapter, we will study a simple example. Consider a system, S, described by the kinetic equations 92-. dt - ap + br + E (18) dr R-Cp+dr+e where a,b,c, and d are constants and.e«l. We could allow 6 to be any arbitrary constant, but assumming that it is small considerab- ly simplifies the calculations. No real biological system is apt to be describable by such a simple set of equations, but Ecuations A (18) are complex enough to illustrate a general principle which is of biological interest. p and r have the dimensions of molecules/cell, say; the con- stants have dimensions of reciprocal time units. Since this sys- 93 tem is unrealistic, no attempt will be made to assign values to the constants. Goodwin (1963) discusses in detail how this can be C1 one . The idea of the Examgle is as follows. The kinetic equations (16) are taken to describe the biochemical system of sOme cell of imaginary simplicity. The problezn is to predict the p05: ible mor- pholo ical states of the cell on the basis of these equations. Two steps are involved: (1) the deterninaticn of the probability density function Dk(p,r,t), and (2) the evaluation of the expec- tation values )k,5, and F. 1. Determination of D —- The probability density function is given by D =MU‘U U satisfies the eigenvalue equation LU k k k k k kak’ or more expliCitly (ap+br+€)%—p-uk+ (Cp+dr+€)'§—ruk :-Ii.Akuk -i)\kt The time dependent part of Uk is 19k(t) = e as above. The associated characteristic equations are = Cp + dr + 6 , -—k = -i/X kku (LID. U.‘ *1 d dfi = ap + br + 6 , It is easily ceternined by elementary methods that the solutions to these equations are --/\ e1 k8 (a + h C); (a + n c): (19) p " l I C e 1 L1 C e 5' 1+E(r1-b — . l - . fla-Lr. 2 l l 2 bC " .Isd c l (a + h C)” (a + r C)S l 2 ” ‘l E (x-c) 2 “l M where hl ind he Are given by l 2 . h = :- (d-a) + (u-d) + Abe 1 40 l 2 h = f— (d—a) - (m-d) + QbC 2 cc and Cl and 02 are constant of integration. Since the characteristic equations are linelr, the hl-tc‘ns and the hfi-terms are seyarately solutions. If th iroolem no wor- L ked through using just tne hl-terms, it is found thnt uk is irregu- lar at the origin. Thus, we set C1 = 0. CD is to be oetermined so thvt Equations (la) sitisfy the boundury conditions ill 2 CD is given, L. {urinetric form, by -(h -h ) ct = 2 1 sin.) _ 6(1 - b) 2 h bC - ad 1 If we .Eet ., E (d — b) d = R - (be - Lxdjfizg - 111) E3 = I‘ _ é (a - C) (be - ad7(h2 - hl) ‘ -(a + h c)s _ (d -b) _ (a - e) f _ a r-- be - ad ’ 6.— bc - ad ’ find x ‘ e ’ Equations (19) become dtx + r‘: sin7? and fix + 5 = cos‘? . Lhence ') (0L2+?2)x2 + (20(Y+ 295)): + (r2+5"-1)=o. If we neylect terms yuadratic in E.: 2_ 2 2€p(d-b) 2_ 2 2€r(a-c) 0k - p - bc - ad ’ P ' r - bc - ad Ep(d - b) 6r(a - c) 2 a f» our: "cc-ad ’ ¢S=bc-ad ’ 5— +8 -l~-l' Then, 1:. A -' .' w v ,. ' ' , »~. ' .. ' ‘_ ' ‘ 1r1b:., 1.1L _L.;. dimpn, L)", U“. 2: T .-'_ , fir; .. f. Chiltailv. -‘-;: 11‘ L‘. tpe curlj KfflCLCtg ble Jean .et 6 u.l to T. I g L;q;r: c;e- isn'— ‘A - 9- m : e“, - (J + k) e : c . l l- 'u: L LCI.-:AJ-116(A .’ {.4 w TULC Diff)-.. pi Y. Ru 1 ..A .L 1 ‘\2 Uk»lb;r v‘q VJ‘. iv. Q1: LL 1 "‘ t say be 2 con leg lunction of , .nn r fer Luge v lbs” 01 tle cop— ' , . 1. _ ‘ .* n e ‘ I“: —‘ . 73 V, ‘_ .‘ . A L r t;:, sc> = e .ILLJ' tn-e L) : (ra*xu)’ Lift}, . .ae l.lflt tvzr t LL : T , A- {L IL il- there a . 1 , - ' ‘ ‘ _.___._( f‘) a». + C(hf + n.) + 1>\LC(-‘i‘ - 1i ) (do) E = L " Ti + ii_".c) (l + n c) l C . I" W B untion (20) is v lid fcr all vtlLen or t ever, L gust be real ;unntity if tre GAICCti: r ure to be real (see oelcn). Thus, we :!-. I L. ‘< O and agree to regd ["(u - d)2 + 4befl for 80 that ha : l/2C Making these ussunrtions, we find that 2 e w (a + d) - 2 A. “In — d) + 40C . K L = a a J i Note tzmt if a : d = O and we take the lin (lb) becone e uiValent to tne nernonic oscillnt . . . 2 Under tleae conditions, we have In : l, T = (12 E = - Z‘Ak/«IEC . Tnus, the corrected grob.oility density tion for the harmonic oscillstor is 2).? (pg + rt) doc DK(1§,I‘) :. It is readily verifies tnet Dn for the LJTnCUiC maxina' it simvly increuses fren D.(0 U) = U to D (l l) = l for all , l K ’ K 9 values of the index K. Cn tLe other land, tLC oriiinal Case gives q- a v-6 (20:1. taint ; 0.“; ‘v’ ,‘z. U L' [(d -31).. 1 Ju. — d): + 4m, ]. it 6.90, 3.211” L-I‘ + rL) “l, QQCilJJLCT I fune- t'J 213:.431‘? 1'1 0 if: ”gnu-min u'» I U 3 Dk/A ; = C if (.1) E = 0, or (b) ATS/3;.) : t}. In tie first chute we iinu tint 3 DRE/3 1‘ = b. In tze secozd C...‘..e, we 11nd tut t A Dk/B p : C2 le:..d:s to an extregum :;.t 1" = O h - x pQV:_.’"d+Z. It is not difficult to show thut if tnis vilue of b is nitLin tne rende of'CDSEMSlW tien.the extremum is s Minimum. Thus, D~(p,r) IX incre ses fFOn DK(U,O) = c to j E 'E 63d)(d-b+n—e) A F 4 C H v I n _ a 2(bc - with a. m.-a_.. C if d > b and a > c. That is, in tnis systen one morpholoti- Cal state results from the gene beinq shut off exile tnere are in infinity of different morphologiCsl states possible with the gene active. he will ssy tnat txo cells are in different switcnin; states if they nave different morpholOgic 1 types (i.e. different will say that two cells are in different organization st tes if k's) and Love different configurations of innccive genes. Je they have different mor9n0105iCJl tyfes and have tne same configu- ration of inactive genes. If the 333 gene is inactive, 1"..11en":r-7‘j : O. For examyle, sugpose a cell contains three genes. It till then have a potency of being in six switcning states and an in- definite number of organization states. Une p ssibility 's: State Gene Configuration noryholohical state 1 r = O = O r O 1 r2 3) 1 2 r1 = 0 r2 ) 0 r3) 0 2 3 rl)O r2=0r3)0 3 103 stute Gene Configurution Hortholcgicel b 458 b, r- o r. > c r-.= o . J.) a j q. 5 r = O ri = 0 rs: C r l a J 2 t r O r“ O r” O t l > d > )> o r O r, - 7 1‘) & >.O rj:>C 7 States 1,:,5,4,5, und 0 are different switching ttxteo wnile states 0 and 7 are tne Sume switcniné state but different orgnniiatlon sta- tes. F. The Relationship between borehological end biochemical Cellular States The Problem of the Thesis is to discover tne relations 1; of the biochemical systen of a single cell to tne norphologiCil tyne of tne cell. The biochenicul state of u cell M15 defined in two ways: On the one hand, it was defined as a certain eigenvector Uk' In this case, tne morph0103ical state of S is the corresnon- ding eigenvalue, )\k° These definitions suffice to seecify the one can 31w:ys find A_, oy fornins concrescence because, given Uk { the scalar trodudt s-ULZLUk dT'. On the other hand, the oiocheni- cal state of 5 was defined as a goint in tne s,ace: tnese ]_ryers ire IHLJJEreu ixlsinsl0;3',.itr Brouurun1hs descri til». ()1‘ the Adult hUmlM cortex. Layer I, tle L0:t yerigner l, is one 1.3:nina sonw is and contuins only sclttered glisl cells. Tue QCCOLd ljx‘odngrn l yer, tne ll liu Aruguliris externu, is not listin uigi- fible in tle rat. Luver III is the l:-..::in_._ E:~gr:z..ii lie: which contlins «...—~— J. "€3e1m¢t1inint, Closely since; tyru i; 1 cells. Layer IV is tre li- “iltla franuliris interns CCflgiLtiflT of crowns: and lee —stuininc \ .A J L .J I3?21nules wrict resemble glial cells. The 1 din; 4:29lionwris, m.de ‘JI‘ of lmrge and loosely {sexed yfriliu 1 cells, is Lijer V. TLe in— r1€31ziost layer VI is tne i Linu multiforuis wnicn contlins {oljmor— i31¥;)is cells. By wennin; (which occurs on the ilst day xfter birth) tle cor— t‘53< has taken on adult crqr.oteristicu. ultlou;h t e cortex of t;e neingttc rcpt is roujilv; line til-u; of tne- -...u'lt, taro i.;tQI‘t«.1llt clif- 3E€¥rences can ce cone veu. First, tne loult ldJCF; ire not in es- 4 *;1~lg oistinguishe;, the lulifi; :yr did lis ipiewrin; to Live been llIVuoei of cells from the lsnina zonglis .nd tne l JiUl érnrullri: ‘4 Cf \ (P L l *1. rd L.— H \ interval neiny ind- unnotzole. lflue cells o lilve an in .wture :_.nrile t..r e. deconclw ,'t.o :zuz'litit.._; yer; A Are to oe seen bet een tie l2 ins multiforii; “wd tne ventricul r w ! 'wsll. .mre Cells of there lagers ure Iresunatlj ”i I‘Llu to .rls the cortex; tne lagers uieugtegr Curie; the 3r? uni %th Loetu.t.l days. Jed tnree vMJLe; in tne tr neititu tetyeen ;; IF“ (I) Sugita reco,1 zieontte:gnli tne _iult colteg. lflu: fir:t .'.m:6 l ate grow oirtl- tintil tie lute d.; sue is ch r;cterizeo :3 tie wrolirer ti 4 :f lieuroblu ts in; .“clf centrifug l mlsrdolog. Durinf t.e first rolnase tne Ittte or incre -xa:in tne tL ounces o;‘t-t.1 l ‘r i (i (h r) k H . , k. . C1 +- Ik.‘ C (~ , ;lnc levels or; to their tuult vtlue .t ;qut 4! o. s. bet.een HI 1 rarid 45 c; f) t; , tne Nissl eucltlnce li;e;rs sue the nut er in; sites of neuronll Processes- incre..t~e:;. Using; tne Eculen tecini 1.4.0, I lfeelle (lt3l) h.; correl;teu t»e tr nrfort tion of tne nucleoli fr;i ., densely Ltlinin, tjce ty Lcll of e rlg Neveloguwmtwl to e' to l “V .Cucl'te. tjqu tytic;l of tne ”Quit ;tines ,is. (c) In :dcition to an tumiC'l changer, t4; electrolyte in; r”xi:er t lguCG of tie cellui r :nfi tne eltr:Cellulzr tuneen j c .m- :irlg duriu- the critic l 'eriud. Flegner 3n; Flex er (1)5L-) found t37* t tne thTuCGllbldf water content necre we; fire; 95 to An 3‘1- 5‘1LC1 tieai irx:rer.;e:.'to . cc14;tlzu; v LLuc :;t 155 e., e .. lli? t 4; egtl‘ - “C3Jllul—i‘u~.ter i; incre.lJJr-euhi then necrewedlxg'b3\. censtunt vil— lie ' mm . w.%vr * u u v« n'LZ A L . s.,dq ms», ‘ \4ullr oll¢ Pullou. Almb, ocluln +/ Qth, hue oo.ccstrlcluu OI -- fi‘ . . . . v _ .‘ I | vfi - 3 ~.— .‘ ' . ' 4 I I " Doxillldl .1311). CillL‘I‘lCe lbuz- lfx t.;t‘. (BACK-g-‘pClltLJ. I L.?-..,‘,;,(‘,‘ 1.: e U- i1 "V L: .. ~/ d 3:, tLe extrHCcllulor sacium concentr tion incre.l;s m.rlegl J—exner‘ewmj Elexner, llflfij). Flexner'jnlter;reti t in; -,.¢ue to A ( M] L le: CitlpllpIMt t of a selective ,esle’oilit; _n. Loggt to it .1 I‘ I I“ 4‘ ‘ ' . ,_ " .‘ - l“ . ,,. ,Y , A , _ .- . ,, \.'. I C‘L"LL'J tC 14‘. LA; 'cJAA‘ \.t v‘j. eleCtrlQ- 'Lv‘uJ—VLt O ;L\O‘;RIIC1 ’ «v .1.-. .l ’L C iiscuccion in ‘lexncr (l:3C:)). *7 t c. :ir t CJJwITTlC.l.“JSVN’ H- J;;,cr et al. (l;}7) fotn) t+ llO tials apgeared in the brain of the fetal guinea ii; between #6 and 56 days. Flexner, Tyler, and Gallant (1950), however, were able to record cortical potentials on the Moth day but not on the LSEE day. Grain (1952) using neonate rate obtained spontaneous but ir- regular and intermittent electrical activity during the first week; during this period he could also elicit etrychnine potentials. By t:he tenth day, the regularity and rhythmicity of the yotentials “was adult in character. Kimel and Kavaler (1951) were first uble tcs induce muscular re5yorees to electrical stimulation of the mo— tcar areas of the guinea pig cortex between 42 and 46 days. Kava- .143r and Kinel (1952) found that the level of acetylcholinesterase cacrtivity begins before this at about 55 days. (d) In both the rat and tne guinea pig, the cerebral cortex {liaise aseeenw to acquire its adult churaoterietics during the second CDf' deve10pment. The histological arrangenent of the cortex has been attained by this time; during; the firit yhaee cells are moving .ir1t<3 their final position: and during the third ohase they are fi- nlifiihing their differentiation into adult neurone. Neuroblaete dif- f.I'entiate during; the second phase into immature neurons: they d‘3‘relop an adult configuration of axons and dendrites, they obtain Iqj~kial substance, and they undergo nuclemr and nucleolar changes Wilfi.ch give them adult character'stics. The electrolyte difference between the cell and its environment may be established at this tithe, and the newly formed neurons seem to begin fultioning during t}1e: second phase. In terms of cellular eigenstntes, a neuroblast dLKPing the first phase would be in a given eigenstate of the nor- Phoiogical operator, call it the 1:3}; eigenstate. That is, if it 111 were possible to actually measure the value of the morphOIOgical function L(p,r) for such a neuroblast, the eXpectation value for a series of measurements would be Ak' During the second phase, the neuroblast undergoes a transition to a different eigenstate, k', so that measurements of L(p,r) would yield Ah" as the expectation value. In general, we would eXpect tL&t the expectation Values of each of the cellular proteins would be different in the neuroblast 21nd in the neuron; if there is an isomorphism between morphOIOgical and biochemical states, the neuroblaets of the first phase should have an enzyme pattern different from the enzyme pattern of the neurons of the second phase. 2. Chemical entOgeny of the cerebral cortex -- Louis and Josefa Flexner have approached the relationship between biochemi- cal and morphological states by studying the enzymes in the fron- tal cortex of the rat, pig, and guinea pig during the first and the second phases (Reviews: Sperry (1962), Flexner (1952,195561, 1955b,l950)). When the information obtained by the Flexnere is combined with the data of other workers, a fairly detailed picture 9f the biochemical differentiation of the cortex can be obtained. T0 provide a perspective for a review of these studies it will be helpful to outline the general nature of brain metabolism. (a) (Reviews: Richter (1955), Balaze and Richter (1961)). Br1min is a unique tissue in that it depends almost entirely for its ‘utrition on the metabolism of glucose, :1 peculiarity imposed by the "blood-brain" barrier which restricts the entrance of almost everything except glucose into the brain from the blood. In the braLin, part of the glucose is utilized for the synthesis of amino 112 acids (e.g. Roberts ot al. (1959)) and other compounds, part is degraded via the Warburg-Dickens pathway to provide NADPE2 for the metabolism of fatty acids and a source of five carbon sugars, but most of the glucose is used in the Embdon-Myorhoff scheme which degrades glucose to pyruvic acid. This compound has two principal fates: in the presence of oxygen, it is used in the citric acid cycle and the electron transport system of mitochondria to provide 38 moles of ATP per mole of glucose (aerobic glycolysis). In the absence of oxygen, it is converted to lactic acid and two moles of ATP (anerobic glycolysis). The relationship between these metabol- ic pathways is diagrammed in Figure 9. Adult and fetal brain tissue differ in their modes of carbo- hydrate metabolism. Fetal brain tissue almost exclusively uses pyruvio acid to form lactic acid. Since anerobic glycolysis does not require molecular oxygen, fetal brain is remarkably insensi- tive to anoxia (Fazeka et a1. (1914)). In adult brain, however, anorobic glycolysis prevails. Most tissues retain the compounds icedod for anerobic glycolysis so that they can function, albeit 103s efficiently, under anerobic conditions. Adult brain tissue 1. peculiar in having seemingly lost the capacity to metabolize glucose anerobically and is notoriously sensitive to anoxia. If thgre is a biochemical—morphological isomorphism, it should be P$asib1e to show that the components of the citric acid cycle and the electron transport chain become active during the second phase °f deve10pment when the cortical neurons are differentiating. In alCldition, some mechanism responsible for the adult sensitivity to ‘noxia should become operative. I I. E g.l1f§n.t§ Figure 9. ll} Schematic diagram of the interactions between metabolic pathways important in cerebral categenesis. Glucose -—%7-Gluceae - 6 - Phosphate ATP i ,/ l /////// 1,6 - Glucose Dipheaphate 1,6 - Fructose Diphosphate Wcrburg-Dickens l Dihydroxyacetone Phosphate Pathway and Glyceraldehydo Phosphate / 1 //’ 1,3 - DiphosRhOElyceric Acid ./ if / /// Pyruvic Acid L ctic v—" a NADPH2 I, //////’ / J// l Acid / / Acetyl CoA W lr Citric Acid Trigl:[erides Cycle Myelin \ ADP 5‘ Electron Trans- ‘——-? ATP ‘ t Chr' : per dln A H20 02 >‘ -~*->c02 11% (b) The data on the biochemistry of differentiation will be discussed under four headings: (l) metabolism of phosphocompounds, (2) Embden-Myerhoff enzymes, (3) citric acid cycle and electron transport components, and (h) lactic dehydrogenases. (l) Flexner and Flexner (1950b) found that the total phos- 13horous content per kilogram of the cortical cellular phase fluc- tiuates about a mean during the last half of gestation. The indi- VGidual phosphorous containing fractions vary, however: between the 1+(lgh and the 45th days, the concentration of phospholipids increas- css , presumably in conjunction with the synthesis of myelin. Pre- ‘rj.ous to this period, the content of pentose nucleic acids had been increasing, but during this period it decreases so that the total Phosphorous content is appreximotely oenstant (Flexner and Flexner (314951)). Flexner and Flexner (1948) found that the concentrations 01 adenylpyrophosphatase increases sharply at 42 days. These chan- 80. suggest an increase in protein synthesis during the critical poriod (Flexner et al. (1951)). (2) The degradation of glucose begins by two phosphoryla- 'tiixons and an isomerization to produce diphosphofructose which is EBIDIit into one molecule of dihydroxyacetone phosphate and one mole— <=Inle of glyceraldehyde phosphate by aldolase. Flexner et a1. (19— 5563) report that the activity of aldolase is approximately constant in the cortex of the guinea pig until about the 33rd day when it “gins to increase slightly up to the 35% day; then it increases to its adult value at term. Dihydroxyacetone phosphate is readily isomerized to glycer— aldehyde phosphate so that the degradation of one molecule of di- 115 phospho fructose results in the formation of two molecules of the aldehyde- Glyceraldehyde phosphate has two metabolic fates of in- terest: (a) it may be reduced by glycerol phosphate dehydrogenase to glycerol phosphat which combines with fatty acids to form the Llipids needed for myelination, and (b) it may be oxidized by gly- <:eraldehyde phosphate dehydrogenase to phosphoglyceric acid which iJS metabolized to pyruvic acid. Laetsch (1962) found that the rat fWDrebrain showed a 20% increase in glyceraldehyde phosphate dehy- cizrogenase activity between the #32 and 21st days post partum and an fivefold increase in glycerol phosphate dehydrogenase activity t:c> reach the adult level between the 21st and 335g days. (3) Succinic dehydrOgenase is one of the enzymes of the cit- 1?j.c acid cycle. Flexner and Flexner (1946) have shown that in the CWDzstex of the fetal pig the activity of succinic dehydrogenase is ‘3 isozymes could be identified, but in the adult all four com- IPCDrzents could be found. Bonavita, Ponte, and Amore (1964) found tllsit there is a progressive dominance of the H subunit during de- velopment in the neonate rat. Plageman et a1. (1960) have shown 'tlient the H subunit is inhibited by an excess of pyruvate while the ”4 ssubunit is not. In contrast to the cortex, the retina is highly resistant to anox’ia (Noell (1958)) and has a predominance of the b4 subunit (Bonavita, Ponte, and Amore (1963)). (c) The biochemical differentiation of the cortex proceeds cOncomitantly with morphogenesis. The enzymes of the Embden-Myer- hNfo patheway are present during all three phases, as is lactic de- llydrOgenase. The citric acid cycle and electron transport compo- !lents studied, however, increase during the second phase in step With differentiation of adult neurons. The concentration of lactic [III I‘ll It 118 dehydrogenase also increases during the second phase, but there is a relative increase of the H subunit. Thus, as development pro- ceeds the lactic dehydrogenase system will be less and less able to sustain the brain when anoxia and a build up of pyruVate occurs. The cytological differentiation of cortical neuroblasts, then, ac- companies their vauisition of the equipment needed for anerobic fizlycolysis and the development of the adult sensitivity to anoxia. 3. Other examples -- A correspondence between biochemical sn-d.morphologica1 development is exhibited by a number of other ssgrstems. For example, in the development of epidermal structures j_:i the chick embryo (Review: Bell (1966)) the determination of the Iarwospective epidermis seems correlated with the acquisition of three jmnununologically detectable proteins, and the final appearance of 1:11e2adult x—ray diffraction pattern of feather Gi-keratin is con- <3c>umdtant with the appearance of a protein with high cystine con- tGent. In the fetal lung, the form of the bronchial tree is laid Cicrwn by a group of rapidly dividing and relatively undifferentia- tle!d.cells. Histochemical data (Review: Sorokin (1966)) suggest t3kLat this process is accompanied by glycogen storage and reliance Ilpon anerobic glycolysis for metabolic needs. The subsequent per- j-C>d of differentiation and low mitotic rate is accompanied by an 1Increase of citric acid cycle enzymes, g1ucose-6—phosphate dehy— (ixrogenase, and the components of the cytochrome system. In both 1Heart and skeletal muscle, contractions first appear in the embryo EBhortly after the appearance of the second of the two contractile :proteins (Ogawa (19588. 1958b. 1961. 1962), Ebert (1952. 1953. 1955. 1956)). 119 Greenstein (195%) has argued that a similar biochemical-mor- phological isomorphism is important in the anaplastic transforma- tions diagnostic of cancerous growth. he attempts to summarize the changes in chemical patterns that accompany the morphological tran- sition from restrained growth and relative cellular specialization to unrestrained growth, loss of the tendency towards segregation ‘by'cellular type, and less of cellular differentiation in five gen- eral conclusionsa6: (a) Each normal tissue is characterized by the Exasession of an individual pattern of enzymatic activity which may swerve to distinguish it from all other tissues., (b) No enzyme pe- <:Iiliar to tumors has ever been found so that tumors must be assumed 't<3 have enzymes qualitatively the same as those of normal cells., (<3) The enzymes pattern of a tumor appears to be constant even- iiliough the environment of the tumor may change. The enzyme pattern C>f‘ a tumor will not change if the tumor be explanted to a culture Inedium or transplanted to a host of different genetic composition (31‘ just allowed to grow in a constant environment., (d) The range ‘>17 enzyme activity or concentrations of small molecules in cancer- CDLis cells is narrow relative to the great Variations shown by nor- Yneal cells. Cancerous cells, thus, tend to approximate a uniform Cell type distinct from normal cells., (e) Specific functional ac- tivities tend to become reduced or entirely lost in tumor cells. Ioiyer cells, for instance, are characterized by high arginase, cat- afllase, and cystine desulfurase activities; in hepatomas, all of these {fictivities are reduced or absent. E‘s . . . . . The literature of cancer biochemistry is extremely volumi— nous, and the present author is not competent to judge Greenstein's thesis. His peers seem, generally, to accept the interpretation; e.g. Reid (1965) and LeBreton and noulé (1961). 120 4. Qualifications -- The embryogenesis of the cerebral cor- tex and the examples of develOping and metaplastic systems cited above support the assertion of a biochemical-morphological isomor- phism. However, at least three kinds of considerations suggest that it is not an accurate generalization. (a) The cellular slime mold Dictyostelium discoideum demon- ,strates two morphological forms. During favorable conditions, the (Jrganism exists as unicellular myxoamoebae. But when starvation cconditions obtain, the myxoamoebae aggregate and differentiate in— ‘t<3 an organism consisting of a stalk and a sorocarp or spore Case. 1111 important aspect of this differentiation is the formation of the gfilyc05en and cellulose cell wall constituents from endogenous pro— ‘teein. Wright (1963,1966) has studied the metabolism of glutamic £1<:id in Dictyostelium. This compound is metabolized via its oxi- Cisztion by glutamic dehydrogenase to ck-ketoglutaric acid, a citric €1c:id cycle intermediate, to CO This sequence, along with others, 2. IDI‘ovides energy for the synthesis of cell wall carbohydrates. The C30ncentration of glutamic dehydrogenase is approximately constant tll‘iroughout differentiation. However, the rate of in vivo evolution <31? CO measured by C11+ tracer studies, increases sevenfold during 21 13 his period. This paradox is resolvable by taking into account the concen- thration of glutamic acid. The Michaelis-Menten equation is vmax [S] V=Km+ESJ 121 where v is the velocity of an enzyme reaction —— in this case, the glutamic dehydrOgenase reaction —— K is a const;nt, anx V n A reaction, and [S] is the con- is t;e >v-‘ theoretical maximal velocity of the centration of :uhstrate —— glutamic acid, here. V is :iven bir max 8 J Vmay = k3[:E] ; k3 a rate constant and [E] the enzyme concentration. ‘Ihis equation can he rewritten as k, [s] KmZtESJ+ 1 ° V : fIVbe reaction velocity can be increased in two ways. First, the Clancentration of E can be increased. But, Wright's data show that [.EJ is constant throughout differentiation. Secondly, the sun- =313rate concentration could increase, making v larger. Nright has Inseasured the ratio Km/[:S] and found that it decreases during dif- lfeerentiation at the right rate to account for the inerease of the I‘axte of CO2 evolution. Thus, in this instance, the substrate con- czeantration is limiting and the concentration of an enzyme, alone, \usas not an adequate incex of biochemical differentiation. (b) Rudnick and fiaelsch (1955) have studied the ontoSeny of gfllutamotransferase, an enzyme which catalyzes the formation of ifilutamohydroxamic acid from glutamine and hydroxylamine. In the Cthick embryo nervous system, the level of glutamotransferase ac- ttivity increases slowly frOi the 9th day of incubation to a geak (Eight weeks after hatching. Except for minor variations, the le- ‘Vel of activity in the optic lobes, the cerebellum, the diencepha— lon, the medulla, the cerebral hehispheres, and the spinal cord follow the same trend. In the retina, however, the level of clu- t'." C1 122 tamotransferase activity is low and shows no rise until the 1733 day. On that day it increases twentyfold. On the lSth day, the retina begins a critical period which results in the final differ- entiation of the rods and cones (Coulombre (1955)). Between the lbth and the 20th days, the level of acetylcholine increases sharp- .ly and the retina beCOmeS functional as judged by the appearance of tzhe pupillary constrictor reflex. But, it is not clear what role gglutamotransferase could play in these anatomical and physiologi- cal changes. (c) R. L. Friede (1959a) has studied the distribution of suc- c:j.nic dehydrOgenase in frozen slices of rat brain by histochemical zueeans throughout postnatal development. He reports a general in— <3I‘ease of succinic dehydroSenase activity (SDA) in the cerebral CKDzrtex from birth to weaning. There is a general caudal-cranial £§rwadient in the time at which SDA first deve10ps. In general, en- ZHYYne activity deve10ps in cell bodies before it does in the neuro- IliJl. Distinct lamina of SDA develOp in the layers of some corti- ‘3Ell.areas, and some areas show signs of temporary hyperactivity (nucleus coeruleus) which might reflect Special events such as "neilanin deposition. In other papers (1959b, 1959c, 1960), Friede 11615 extended his techniques to the adult guinea pig and construc- ‘tted an atlas depicting regions of high and low SDA throughout the Ineedulla, the midbrain, and the cortex. He finds that the distri- bution of SDA closely follows the general morphology of the brain. 'Tlrus, he (1961) was able to distinguish on the basis of SDA several histologically defined nuclei of the reticular system. His data also suggest that functionally related systems exhibit similar 12} values of SDA. For instance, he was able to associate gradations in various thalamic nuclei with parallel gradations in SDA in their cortical projections. In a general way he suggests a relation be- tween cellular function and intracellular SDA distribution: cells in regions of integrative or receptive function show high enzyme ac- tivity in the neurOpil (e.g. olfactory bulb) while cells in regions of effector or relay function show high SDA in the perikarya (e.g. nucleus of the mesencephalic root of the trigeminal nerve). Thus, in some instances, it may not be possible to assume an isomorphism between biochemical and morphological states in the way we have done in Chapter III. It may be necessary to explicitly take into account more biochemical parameters than we have done or might <:are to do. It may turn out for some systems that there simply is :10 correspondence. Or, it may turn out that the correspondence is Ilot apparent when looking at a single cell because it involves gra- Ciients of characteristics spread over many cells or it involves (Iharacteristics localized within the cell. Nevertheless, if these 61nd similar difficulties are kept in mind it seems legitimate to Eiccept the biochemical-morpholOgical isomorphism as a first approxi- nuation which is applicable to at least some interesting cases. I3. Cellular Stability and Discreteness The first assertion of Chapter III was that cellular morpho- 10gical states are discrete and are stable. A cell was said to be fstable with respect to a given variable if the variance of measure- ments of that variable is zero. The quote from C. H. fiaddington in 124 Chapter II (Page 27) is a statement of this assertion.. Like the second assertion, it is valid only within certain limits. fie will try to establish these limits by means of an example. 1. The embry010gy of the neural crest -- (a) As the neural tube of a vertebrate embryo forms, a certain amount of neural plate material is excluded from the tube itself and comes to rest on top of the developing CNS. This group of cells is known as the neural crest. The pertinent literature has been reviewed by Horstadius (1950); a recent corroboration of the classical work by radioactive tracers is by Weston (1963). References to the primary literature will be found in these sources. The histological descriptions used here are from Bloom and Fawcett (1962). The prOpensity of a given type of cell to become transformed :into another type of cell is described by its transition vector mG (see Chapter III Section B.8.), where the original cell starts in iihe mph morphologiCal state and changes to the kth morphological EState. If mG = O, the transition mn—>k:is not possible; if mG = 1., the transition mn+>k;will definitely occur. If cells are stab- 3~e with respect to morphological type m, mG should equal zero or ‘De very small for most values of k. To gain some idea of the na- 11ure of cellular stability, we will outline the potencies of a sin- ggle neural crest cell. Figure 10 represents the possible morpho- Ilogical transformations of a neural crest cell. Each circle rep- resents a morphological state; ® is the state with eigenvalue >\ k' Of course, these assignments of eigenvalues are entirely arbitrary. Arrows between circles represent possible transformations; if there is no arrow between two states, a transformation is not possible Figure 10. State 125 (Description of Figure) Description Undifferentiated ectoderm Neural plate (presumptive CNS) General neural crest Head crest Trunk crest Gill arch cartilage Cartilage of anterior trabeculae Dorsal spinal ganglia Sympathetic nervous system Schwann cells Meninges Dentin of teeth Corium of skin Connective tissue of dorsal fin Pigment cells (melanocytes) Mechanocytes Amoebocytes Generalized dedifferentiated cell 126 Figure 10. Transition diagram for prospective fates of neural crest line cells. Numbers represent cellular morph010gical states. Arrows represent possible transitions. 18 127 (i.e. it has never been observed). (b) The possible transformations are: V and V : neural crest cells form on the lateral margins of the neural plate The from the ectoderm covering of the embryo. Like the neural plate cells proper, the neural crest cells form only when they have been in contact with the roof of the archenteron which lies below the neural plate in the embryo. The medial portions of the archenteron roof induce central nervous system while the lateral portions in- duce neural crest. V3,“: The anterior neural crest has the potency to form car- tilage. The fate of these cells is determined quite suddenly at ‘the end of the yolk plug stage. Before this stage they are fairly ‘totipotent. Even after determination, however, anterior neural crest material can form cartilage only when it has been in contact with another tissue, such as pharynx. Cartilage cells are spher- oidal in shape and characteristically produce a matrix of glyCOpro- teins containing chondroitin sulfate. V4,6: The posterior part of the anterior neural crest forms the cartilage of the gill arches, the hyoid arch, and Meckel's car- tilage (lower portion of the lower jaw). It has the capability of forming trabeculae when transplanted rostrally. V4.7: The anterior part of the anterior neural crest forms the cartilaginous trabeculae of the chondrocranium which, after ossification, form the nasal portion of the skull. The major por- tion of the skull is of endomesodermal origin. This part of the anterior neural crest does not have the ability to form visceral arch cartilage. 125 Vj,5: The posterior portion of the neural crest material is totipotent until the late yolk plug stage when it attains the ab- ility to autonomously differentiate into a number of cell types and looses the ability to form cartilage. V5‘8: Neural crest cells migrate down between the myotomes to form the dorsal ganglia of the spinal nerves. A given ganglion is formed in part by cells from the ipsolateral side and in part by cells which cross over from the contralateral side. Although the ganglia differentiate autonomously, the regularity of their Spacing is dependent on the Spacing of the myotomes. The neurons of the dorsal ganglia are typically T-shaped with a process which extends from the cell body and divides into an axon and a dendrite. They are myelinated and surrounded by small, flat, satellite cells. V5.9: The neural crest cells contribute to the formation of the sympathetic ganglia. Material from the ventral part of the spinal cord also contributes. The neurons of the sympathetic gang- lia are usually small, polymorphous multipolar cells. V5,10: Schwann cells are considered as resulting, at least in part, from neural crest cells. Schwann cells are long, flat cells which spiral about neurons in a large number of laminae and constitute the myelin sheath of the neuron. Myelin is a complex of lipids and mucopolysaccharides. V5,11: The majority of the pia and the arachnoid matters are derived from neural crest cells. The dura is of endomesodermal origin. The arachnoid and the pia are composed of intertwining collagenous bundles interspersed with fibroblasts, macrOphages, mast cells, and other connective tissue elements. The surfaces of 129 both meninges are covered Wlth a layer of squamous mesenchymal epi- thelium. V5'12: The dentin of the teeth is derived from neural crest cells. Dentin is a yellowish semitransparent substance, similar in structure to bone, which contains a network of tubules. Dentin contains both inorganic materials and a glyc0protein. V5‘13: The innermost layer of the skin, the corium, is deri- ved from neural crest material. This layer is a webwork of colla- genous and elastic fibers. V5.1“: In fishes, the connective tissue of the dorsal fin is of neural crest origin. V5.15: The melanocytes or pigment cells are derivative from the neural crest. These are long cells with irregular outgrowths; they can elaborate melanin. (c) The cell types 6 through 15 are found in adult vertebrates, and, in slip, they tend to maintain their identity as long as they are alive eventhough they may be constantly changing their shape or the character of their metabolism. It is this preservation of identity that we refer to as cellular stability. Cells of the neu- ral crest lineage demonstrate SOme remarkable diversities. Gill arch chondrocytes and dorsal ganglionic neurons represent extrene- 1y different morphological types. At the same time, there is a considerable overlap between some states: the meninges and the corium are basically similar in histology and both contain pigment cells. (d) When a cell in one of the states 6 through 15 is removed from its normal environment, as in tissue or cell culture, it can 130 undergo further transitions. Hhen animal cells are cultured in lumps of tissue or in a mass of dissociated cells they undergo transformations known as dedifferentiation. Willmer (1960) has classified dedifferentiated cells into three main groups in terms of their morphological characteristics, their lccomotory behavior, and their biochemical and physiological prOperties. Epitheliocytes characteristically grow in vitro as an intact sheet of cells. They tend to grow on the surfaces of culture flasks and move by a gli- ding movement of the sheet followed by extensions of the peripheral cells. They may produce carcinomas. They store mucosubstances in the cytOplasm and may form keratin. Mechanocytes grow as a reticu- lum or network of cells which ataches itself to the coverslip only at its extremities. The individual cells move with a definite po- larity. They may form sarcomas. They exude mucosubstances and may form collagen. Amoebocytes grow as isolated cells which move in an amoeboid fashion without a definite polarity. They may produce leu— cemias. They actively pinocytose and collect mucosubstances in their cytoplasm. More detailed descriptions of these classes of cells can be found in Willmer's book. These three classes seen to account for most dedifferentiated cells of all of the vertebrates and at least some of the invertebrates. Cells derived from neural crest material show three types of transitions in tissue culture: V Spinal gang1ion and sympathetic ganglion cells V8,k’v9,k‘ 10,18 retain their basic Characteristics in vitro. That is, they do not dedifferentiate. They may temporarily loose their Nissl substance or develop axonal swellings. Schwann cells seem to retain their Characteristics, also. V 6,16‘V7,16’vll,16’v12,lo‘vlj,lo’vlh,lb: Cnundrocytes and connective tissue cells become mechanocytes in tissue culture. V15’17: Melanocytes and other pigment cells become amoebo- cytes in tissue culture. It is not entirely certain whether or not dedifferentiation is reversible. There is a good deal of evidence suggesting that cells in tissue cultures may revert to their original types under ii_fii. appropriate conditions. On the other hand, isolated cells in cell culture seem to permanently loose their identities and become "gen- eralized dedifferentiated cells". (Review: Grobstein (1959)). (e) The "adult" cell states appear to be discrete in that they have different characteristics (shape, size, biochemistry, potencies, etc.) and in that they do not appear to become direct- ly transformed between themselves. A chondrocyte, honever, could conceivably become a mechanocyte and then become a corium cell. The adult states are stable in that they tend not to become transformed as long as they are in their normal environment. They are clearly not totally stable for some of them may dedifferentiate. Since this is quite arbitrary, the proper use of the first assertion should be as a hypothesis in need of verification. 2. Transition rules for developing systems —- The transition diagram (Figure 10) for neural crest cells can serve as the basis for a formal description of the behavior of developing systems. Verbal descriptions of such systems are, of course, well known, and in effecting a symbolic description we are contributing nothing new to the science. (A recent "axiomatization" of developing systems 132 in a slightly different context has been done by Apter (l9b6).) This is an important point because scientific mythology fosters the fallacy that a discussion is necessarilly rendered more accurate or otherwise more "scientific" by the introduction of symbols. The reason for formally describing embryological ”rules“ is that a pic- ture of cells in terms of stable states is not very useful without a way of calculating the transition vectors V An apprOpriate mk' description of the constraints placed by the cell on transitions may allow the development of a method for calculating these vectors. The hOpe is that the successful completion of this project would contribute something new to embryology. Figure 10 suggests four rules governing transitions between morphOIOgical states. (a) Transitions are dependent on the state in which the cell is at the start of the transition. In our nota- tion this is clear because each transition vector,V depends on mk‘ the index m. Thus, if we are given an undifferentiated ectoderm eell, we could not confidently predict that it will become a chon- drocyte until it makes the transitions described by V13 and V3“. This kind of behavior is to be contrasted with that of some physi- cal systems in which the development of the system depends only on the initial state of the system and not on intermediate states. (b) Transitions are dependent on the environment of the cell; cells remain in a stable state unless they are perturbed by an ex- ternal stimulus. The embryelogist calls such stimuli "inductions". The transition l-+>2, is known to depend upon such an induction; it has already been mentioned that the transition h~+>7 depends upon the inductive influence of mesodermal tissue. This rule is some- 133 thing of an overstatement, for although a great many transitions are known to depend on inductions, there is nothing to rule out the pos- sibility that some cells may spontaneously initiate transitions. (c) Most transitions are impossible. That is, mG = O for most values of m and k. In Figure 10 there are 18 states and, thus, 32# transitions. But only 30 of these are possible. This kind of behavior is to be contrasted with the behavior of some physical sys- terms in experential space for which all transitions are possible. It has already been noted (Chapter II, Section D.2.) that this dif- ference between integrated biological systems and physical systems suggests that they cannot be described by the same set of laws. (d) Most transitions are irreversible. That is, mG fl Vkm for most values of m and k. This rule does not apply to plants and it is not clear whether or not it applies to animals. However, it is generally clear that most cases of "dedifferentiation" actually involve transitions to new states. Thus, if mG)h 0, it is most likely that Vkm = 0. Part 2: Cellular States C. The'Locus of Cellular Stability The realization that cells show the kind of "stable state" be- havior we have just sketched leads one naturally to inquire what agents are responsible for maintaining a cell in its current stable state, guiding it to its next stable state, and determining which states it can change to. Historically, this question has been ap- proached by supposing that some "thing", some cellular subunit per- haps, serves to effect cellular stability. The results of Chapter 13h III, however, suggest that some kinds of cellular stability reflect the total cellular organization rather than the action of any cel- lular subentity. To set the stage for this concept, we will brief- ly outline the current dogma on the loci of cellular stability. 1. Ideas of cellular control -— (a) The early ideas about cellular control centered about the notion that a cell's phenotype was determined by the presence or absence of one of several kinds of "determinants" in the cell. Weissmann argued that the deter— minants were actually fragments of the chromosomes. During embryo- genesis, he suggested, the chromosomes undergo a fragmentation at each cleavage so that different blastcmeres get different chromo- some fragments. If the determinants are located on these fragments, each cell would get a different determinant and have a different phenotype. The somatic line cells in Ascaris actually undergo this kind of fragmentation. However, the karyotypes of almost every other species show that most somatic cells retain intact chromoso- mal complements, so that Weissmann's thesis is generally without support. A more viable application of the same idea is that cytoplas— mic substances act as determinants. The mollusc Dentalium or the oligochaete Tubifex, for example, have eggs with different kinds of cytOplasm. In Dentalium, a mass of clear cytoplasm localized in the lower half of the zygote ends up in just one of the blasto- meres in the four cell stage. This "polar lobe" can be shown to be necessary for the formation of the mesodermal components of the embryo. No other part of the egg can form mesoderm. Similar sit- uations occur in a variety of groups of animals (e.g. see Balinsky 135 (1960)). It is difficult to generalize the role of cytOplasmic determinants; but in most animals, the cytOplasmic organization of the egg seems to play an important part in guiding deve10pment up to about the gastrula stage. (b) After gastrulation, development is thought to become pro- gressively more controlled by the genes. If one rejects Neiasmann's thesis, it becomes necessary to explain how the expression of genes can be regulated to allow eye pigment genes to direct the produc- tion of eye pigments only in eye cells, to allow keratin genes to direct the synthesis of keratin only in skin cells, etc. The pic- ture which has been developed in the past twenty years is that dif- ferentiation is the result of a highly synchronized sequence of gene "activations" and "inactivations" with the result that each differentiated cell type has the genes appropriate to its pheno- type in the "active" state while the inapprOpriate genes are in the "inactive" state. In terms of Chapter III, different types of cells are in different switching states. (Review: Ursprung (1966)). On the molecular level, these events are currently interpreted as analogous to the repression of gene loci known to occur in the bac- teria (Jacob and Monod (1961)) although there is little direct evi- dence to warrant the generalization. A totipotent zygote has a full complement of "active", non-repressed genes. During differen— tiation various loci are repressed, accounting for the loss of po- tencies which accompanies development. There is some evidence that the repressor substance is a histone protein (Review: Bonner and T'so (1964)). A finer level of control is known to be effected by the inactivations of enzymes by substrates and small molecules 136 (Changeux (1965)). 2. Cellular eigenstates -- A model of this picture of cellu- lar differentiation has been presented by Simon (1965). A system of two loci is considered (Figure 11); G; locus and G; represents the inactive allele. R1 and R2 represent the concentrations of two repressor substances. The system is des- represents an active gene cribed by the stoichiometric equations + + x1 - G1 + R2 ;:;2. G1 x 1 + .352; - G + R {—Z“ G e 2 1 2 x 2 If we assume that G: + G; = 1 and that dGi/dt = 0, we find that + G-- _ x1R2 l ‘ - + X1 + Xle G+ _ x1 1 .- X- + X+R . 1 1 2 We may take the kinetic equations of the system to be dR + 32"]. = -klRl + VlGl 93-2 = k R + v G“ 137 Figure 11. A model for cellular differentiation in a system with two gene loci. GI and G+ are active gene loci; G- and 2 l G; are inactive gene loci. R1 and R2 are repressors. kl and k2 are constants. G _) R1 ——————) Inactive Inactive Q v N340 t 138 where v and v are constants. At steady state 1 2 v1x1 R1 = - + kl(X1 + leZ) (l) - R _ v2X2 2 ‘ - + ° 1:20:2 xanl) The system has three distinct states: :0 II 4 ...: k ....I :0 H O 5U \/ O n: u (D to I <1 m \ w a; n1 \/ C) For 33 the values of R1 and R2 are determined by solving Equations (1). Simon's claim is that states S1 and S are more generally im- 2 portant than state S}. If it be assumed that a gene must be active a minimum period of time for the synthesis of an mRNA molecule, he argues that the system will respond to external inducers by either retaining its current state (either S1 or $2) or "switching" to the other state, depending on the concentration and the nature of the inducer. However, his analysis does not seem to exclude the possibility of an external inducer causing a transition from $1 or S to S 2 3' To see the relation between Simon's model and the theory of Chapter III, assume that the units are arranged such that R1v1,v2. Although the formulae of the Example may not be applicable if v1 and v2 are too large, the general results about the nature of the morphologi- cal states is relevant to Simon's system. Simon's states 51 and S. correspond to the switching states of the Example in which p = O and r = 0; his state S- corresponds to one of the organization 5 states of the Example in which'p ) O and.r'> Ch However, bimon's analysis misses the N-B other states included in the results of the Example. This omis.ion arises because he does not include states for which de/dt i O and d8 /dt i C host of the tine but , 2 for which Rl : constant > O and R2 = constant > O. 1% The “oint is that in many cases it may be meaningless to as- sign a locus to cellular stability. The stability of cell type may be a result of the total organization of the cell rather than of the inactivation of plrt of the cell's genome. The traditional picture of gene action contains a non seouitor: The facts that the entire genome is retained in each cell of an organism and that these cells differ biochemically and morphologiCally do not necessarilly imply that genes are inactivated during differentiation. It is possible, in fact more likely, that two different types of cells have eXactly the sane configuration of active and inactive genes, their differences arising beCause they are in two distinct organi- zation states. It nust be emphasized that switching states and or— ganization states are not incompatible so that sone differences in cell type nay arise from genie inactivation while others result frou cell-environment interactions maintaining two cells with the same active benome in different morphological states. The possibil- ity of organization states follows from the structure of cellular biochemical systems, but this does not mean that they actually oc- cur in living cells. Only an experinental ap,roach can establish this e D. Testability It is obvious that the theory of cellular states developed in Chapter III cannot, in its present form, be readily used to design eXperiments. The only way to distinmuish snitching states and or- ganization states in a living cell, for example, would be to mea- sure the mRNA concentration of all Species of mRNA. This is not 141 technically feasible. Nevertheless, it should be possible to put yhe theory into a more practical form without making any modifica- tions in the arguments. This would involve the following steps. 1. The theory is applicable to any organism with any finite number of cellular types. However, it needlessly complicates cal— culations to deal with metazoa or metaphyta which may have hundreds of different cellular states. Equally interesting problems can be undertaken using organisms with two or three cellular states. ‘The slime molds, for instance, undergo a transition from an amoeboid state to a sporOphyte state. The relationship between this morpho- 105ical transformation and tne underlying biochemical transforma— tion is within the grasp of current biochemical methods (see Sec- tion A.3.a.). The protozoan Naegleria has two distinct cellular states, a flagellate state and an amoeboid state. Transitions be- tween these states can be effected through changes in the cell's environment (Willmer (1956a, 1956b)). 2. The exact relationships between gene loci and enzymes are known in only a very few instances, so that it is not possible to write authoritative kinetic equations in terms of the pj's and the rj's. Even if it were possible, a prohibitively large number of loci would have to be considered. It is more practical, therefore, to represent the biochemical system of a cell by its enzymes and substrates. Thus, the Goodwin control lOOpS of Figure 3 would be replaced by the metabolic 100ps of Figure 9. In many instances, these pathways are known in some detail. Also, some important pathways of, say, carbohydrate metabolism can be considered as independent of other pathways to a first approximation. Thus, rei- 142 listic and interesting problems could be stated in terms of ki— netic equations involving between 20 and #0 independent variables. 3. The most important extension of the theory itself which is necessary (to some extent) for practical applications is a me- thod of calculating the transition vectors V~ This would allow mk' predictions of the effects of controlled perturbations on cellular states. It is not obvious how to calculate these vectors. LITERATURE CITED. NOTE: Only the second half of the title of papers in the series "Biochemical and Physiological Differentiation During horphogene— sis" is given. Each paper in that series which is cited here is prefixed by a Roman nuaeral. 10. ll. 12. Apter, E., 1966, beernetics and Development, Pergamon, N. Y. Atkinson, D., 1965, BiolOgical feedback control at the mo— lecular level, Science, 150: 651-657. Balazs, R. and D. Richter, 1961, The regulation of Glucose metabolism in the brain, in Regional Neurochemistry, S. S. 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E., 1965, Endogenous substrate control in bio- che11011 differentiation, Bact. ReVs., 27: 272-201. wright, B. E., 1966, Multiple Causes 1nd controls in differ- entiation, Science, 153: CEO-637. APPEBDIX. ThE E,UIVALENCL BETWEEN LINEAR DIFFEKLLBIAL OPEdATOAo AND MATRICES The assertion that there is an equivalence between linear dif- ferential Operators and square matrices is nathemuticully identi- cal te asserting the equivalence of the Schrodinger and the Heisen- berg pictures of quantum mechanics, and it can be troven in a Vu- riety of Ways. ochredinger (1926) established the equivalence of the two systems by arguing fron the fact that the comnutition re- lationships between the variables of 1 necneniC1l system ur: the sane whether the vuriuoles are represented as ogerators on confid- wuce. Dirac (1926) es- uration synce or as matrices in hilbert s, tublished a unificstion cf the two pictures in his "trunsformition” theory by showing that the Schrgdinger wave functions correshond to similarity tr1nsfornaticns of the Heisenberg nltrices. however, von Neununn (1955) criticized Dirac's theory because it relied on the use of the "iuyroper” delta functions, and offered, instead, a more rigorous discussion of the equivalence. More recently, such functions have been p11ced on a better mathematiC1l foundation27. The demonstration given here is b1sed on von Neununn's disCuseion of the Diruc theory. It is not intended to be rigorous. Ne denote the elements of A(p,r)Mé by akk, k : 1,2,...,N, and consider the vectors Uk as functions of the discrete variable, k. 27 _ ..A ~ . -1 . 1-1 H .‘ The theory of such imprcter functions or "dlmtrlbuclunt is beyond the scoye of this work. A readable discussion is to be found in Streeter 1nd flightman (1964). 153 The eigenvalue equation in just a linear transfornetion in GKN which can be written as I Z akk'Uk' "kkUk ' k' The yreblem is how to define an analogous linear transformation on G). To do this, we must retlace the discrete variable k by the continuous variables r and r; thus, we also reyloce the cunnation *4 by the integral L...) <1?” I where d?” = dp'dr'. Similarly, we replace the matrix elements akk' by a function h(p,r,p',r'). Then, the desired transformation is ”U b(p,r,p',rl) Uk(p',r') dJ ‘—€>‘akkUk(p,r) . But, thie means that L(p,r)op must be an integral operator defined by n(p,r)OpUk(p,r,t) = h(p,r,p',r')Uk(p',r',t) djr. where h(p,r,p',r') is the kernel of the operator. This is a recu- liar circumstance, for ne have asserted that A(p,r)cfi should be u l. 154 differential onerator. To complete the discussion of the corres- pondence we must anzwer three questions: (a) Do there exist on a set of operators which are at once differential and integral 0p- erators?, (b) If such coerators do exist, how does one proceed to find a particular operator on.G?mtich,corresyonds to a given metrix in (RN?' (6) Conversely, given such on operator on Q , how does one orcceed to find a CGFPQJLOfldiHJ matrix in GKN? (a) To show the existence of such «fl eterntor, A(p,r)OU, it is sufficient to show tne existence of the corresgonding kernel. First, sulpoee that A(p,r)ou is just the identity onerator, i.e. .L (l) Uk(Pv1'vt) = h(Par9P'9r')Uk(Pier'vt) dT'0 Equation (1) is satisfied if we set h(p,r,p',r') :(;(p - p')S(r - r') where 6.(p - p') is the well known delta function. The Lroperties of these functions are well Known2 . In gar- ticulnr, for a function, f(x), of x: )3 n(x - x') f(x') dx' = {-1)In dnf(x)/dxn . n . . . , . wnere 6' (x - x') 18 the nth derivative of the delta function. Thus, for Any linear differential oterator, we can construct a ker- nel in terms of delta functions and the derivatives of delta func- tions. For exanple, if 28 For exnmgle, see Messiah (1964), vol 1, p 4bo et neq. . 2 2 uhnflcp ‘ 1+ 5 /ap . '7) tLen we take h(p,r,p',r") ==