ALBERT

Notes: Abstract Freese’s Hypothesis states that a single specific alteration in the sequence of nucleotides of an information-bearing DNA molecule results in a specific mutational effect. Within the framework of the DNA-protein coding problem developed elsewhere, and assuming the quasi-ergodicity of the general coding process, it is shown that Freese’s Hypothesis allows us to derive expressions for the length of the smallest mutable DNA molecule and to obtain a bound for the maximal number of allelic molecules of fixed length. To illustrate these ideas, calculations are carried out on appropriate data from bacternophage and man, and the results are shown to differ by a factor of 10 (modulo the rather crude approximations used). It is further shown that, if ρ(N) and ϱ(N) are respectively the number of information-bearing words of lengthN in a given code and the number of words of lengthN, then the number lim ρ(N)/ϱ(N) depends sensitively on the parameter ∈ which specifiesN→∞ the given code. The implications of this result for the spontaneous aggregation of a sufficient number of information-bearing words to characterize an organism are discussed.

Notes: Abstract As a “base line” of memorization performance, the behavior of a “perfect learner” is considered. He is characterized by a perfect memory and by the ability to choose the best search procedure in problems where the correct response from a given repertoire is to be found to each of several stimuli under the condition of “right” and “wroing” promptings by the experimenter. Expected learning curves are derived for the case of disjoint response repertoires associated with the stimuli under cyclic and random presentation of the stimuli and for the case of a single response repertoire (a one-to-one matching problem) under cyclic presentation.

Notes: Abstract Detailed equations are given for the computation of aortic distensibility in the intact living human patient from measurements of systolic and diastolic arterial pressures, heart rate and cardiac output. From these equations, the aortic characteristics of a large series of normal men of different ages were computed. Comparing these results with measurements on excised aortas, a more pronounced trend toward increasing aortic stiffness with increasing age is evident in living than in dead aortas. Nor-epinephrine and exercise apparently cause the living aortas to constrict but to become more distensible. The same change occurs after 30 minutes of high spinal anesthesia. The ganglionic blocking agents hexamethonium, pentamethonium, and tetraethylammonium usually cause the living aorta to become stiffer, presumably due to dilatation. The aortas of patients with pulmonary disease usually react in different fashion to exercise or tetraethylammonium. The increased aortic distensibility that occurs with the hypertension induced by nor-epinephrine or exercise acts as a compensatory mechanism by decreasing systolic pressure.

Notes: Abstract Dimensional analysis is discussed from the viewpoint of its basic group properties and shown to be an algebraic Abelian group that is useful for analysis of physical measurements. The application of the method to various types of equations and the formulation of previously unclassified dimensions are discussed. Functional dimensional analysis is applied to the problems of cell size and biomass proliferation; future applications are also noted. A number of dimensionless terms have been formulated for cellular physiochemical phenomena. They apparently represent the first systematic study of biological dimensionless numbers recorded in the literature. A dimensionless proliferation law is suggested. A brief analysis of the physical dimensionality associated with information measures is carried out. Entropy and “information” are shown to be completely different in their dimensional meaning; other informational measures of possible interest in biology are proposed. The dimensional coding and computor analysis of biomathematical equations is suggested.

Notes: Abstract Expenditure of energy under several simultaneous forms (mechanical, chemical, etc.) is associated with all muscular activity. The energy is directly related to what is commonly called exertion or effort. This paper defines “muscular effort” quantitatively in terms of some of the elements of the dynamics of the human (and animal) body. It postulates that in all likelihood the individual will, consciously or otherwise, determine his motion (or his posture, if at rest) in such a manner as to reduce his total muscular effort to a minimum consistent with imposed conditions, or “constraints”. The principle, formulated in mathematical terms, is sufficient to ascribe to the moments at all body joints—a matter generally of free choice on the part of the individual—their most likely magnitudes. It therefore renders the equations of human (and animal) motion determinate within this context. The paper also describes briefly an iteration method for the solution of these equations, once they have been made determinate. A simple illustrative application of the principle is included.

Notes: Abstract It is pointed out that two fundamentally different views of primary genetic processes occur in the literature which are frequently confused. The first is a true communication-theoretic view, which regards the genetic apparatus as containing a real information-source and a transducer which converts that information to useful form. The second view is generally expressed as a template scheme based on the Watson-Crick model; it is shown that in this model there is actually no such thing as genetic information in a communication-theoretic sense. Both views are then discussed on the basis of microphysical principles developed in previous work of the author (Bull. Math. Biophysics,22, 227–255, 1960) in an attempt to find which approach is in closer accord with the biological facts. It is shown that, if the communication-theoretic view is correct, then the information-bearing object must act as a “catalyst,” but it is pointed out that the type of catalysis involved must be of a fundamentally different nature than that occurring in familiar enzyme-catalyzed reactions. On the basis of general considerations of irreversible changes in microphysical measuring systems, it is shown that any type of template must suffer a gradual and irreversible denaturation, which seems to make it unlikely that a template could play a primary role in fundamental genetic processes.

Notes: Abstract The theory developed in previous papers and based on distribution curves of definite form is generalized to any form of unimodel distributions. The time course of the change from one behavior to another is discussed and a general theorem about the time course is established.

Notes: Abstract It is observed that a dynamical continuity equation for biomass distribution yields the asymptotic steady-state exponential dependencen=A exp( $$ - m/\bar m$$ ) exhibited by certain fishery data, wherem is the biomass of an individual,n is the number of individuals per unit biomass interval, andA, $$\bar m$$ are positive constants. This dynamical approach to biomass distribution is an alternative to the global maximization principle proposed recently by Lurié and Wagensberg.

Notes: Abstract Linear birth and death processes are used to derive simple expressions for sequential extinction times and gene fixation probabilities in asexual populations.

Notes: Abstract We are here concerned with the functionf which assigns to each pointP of an object the numberf(P) which is the shortest distance fromP to the border. This function appears in various guises in diverse biological studies. The functionf(P) is itself a measure of shape—or more precisely, an infinite set of measures, one for each point (and hence, in view of its geometric definition, usually in a form inconvenient for use). Thus in this paper we sought a reasonable representative of this infinite set of measures, namely themean of the numbersf(P) asP ranges over all points of the entity. Computability studies are developed for various classes of shapes. For example, (1) the mean for a lamina bounded by a polygon circumscribable about a circle of radiusr isr/3; (2) the mean for a domain bounded by a polyhedron circumscribable about a sphere of radiusr isr/4. The transition from pointwise to piecewisef(P), especially in the non-convex case, requires working with inequalities.

Notes: Abstract A mathematical model for traveling bands of motile and chemotactic bacteria in the presence of cell growth and death is examined. It is found that asymptotic traveling wave solutions exist in the absence of chemotaxis, due to the balance of growth, death and random motility. Thus random motility confers the ecological advantage of population propagation through migration into nutrient-rich regions. The presence of chemotaxis amplifies this advantage by moving more cells into higher nutrient concentration regions, resulting in larger and faster bands. Therefore there seem to be two types of traveling bands that can be attained by chemotactic bacteria in the presence of growth and death: (1) these growth/death/motility bands; and (2) pure chemotactic ‘Keller-Segel'-type bands. Comparison to experimental observations by Chapman in 1973 indicate that the latter seem to be formed. The relationship between these two types of solution is at present uncertain. The growth/death/motility bands may have relevance on longer time or distance scales characteristic of microbial ecological systems.

Notes: Abstract Based on the principle of minimum power, a mathematical model of the pathological functional state of the oxygen transport system is presented. The model is used to determine the optimal functional parameters of the oxygen transport system in hyperthyroidism, anemia and hypertension. Theoretical results are compared with clinical data.

Notes: Abstract A modified SIRS model is developed as a framework for the study of epizootiological dynamics in an insect-pathogen system. Linearized stability analysis reveals that the system with one immune and one susceptible host class can exhibit stable, periodic or unstable behavior depending on model parameters. In general, high pathogenicity, short pathogen propagule lifespan and high host reproductive rate are stabilizing influences. Pathogen transmissibility and propagule production/host do not influence local stability. The effect of seasonal host reproduction is studied because most insect hosts are seasonal in temperate climates. The basic stability dependence on model parameters holds except as modified by the length of the reproduction interval. The results of this study are compared with the recent work of Anderson and May.

Notes: Abstract The random predator-prey type interactions of the population species in a random varying environment have been investigated. The Fokker-Planck equation for the transition probability, corresponding to the stochastic differential equations established from Lotka-Volterra equations by the introduction of randomness and variability, has been integrated in the form of a path integral. The transition probabilities for extinction or survival of one or several species have been approximately evaluated and investigated.

Notes: Abstract If two microbial populations compete for a single resource in a homogeneous environment with time invariant inputs they cannot coexist indefinitely if the resource competed for is not renewed by biological activity within the system. Mathematical studies have shown that in a predator-prey system, where the resource (prey) is self-renewing, the two competitors (predators) can coexist in a limit cycle. This suggests that if the resource competed for is renewed by biological activity within the system coexistence can occur in any microbial system provided that it exhibits the same features as, but without being, a predator-prey one. A food chain involving commensalism, competition and amensalism is presented here. Two subcases are considered. It is only when maintenance effects are taken into account that coexistence, in limit cycles, can occur for this system. Limit cycle solutions for the system are demonstrated with the help of computer simulations. Some necessary conditions for coexistence are presented, as are some speculations regarding the possible physical explanations of the results.

Notes: Abstract The branching structure of the mammalian arterial tree has been known to be close to that of an optimal conduit system of the minimum work model characterized as the branch system of constant wall shear rate. The physiological mechanism producing such construction was considered to be based on the local response of arterial caliber induced by the wall shear stress (shear rate × blood viscosity) and thereby maintaining this stress constant, which was previously observed at the canine common carotid artery shunted to the external jugular vein. The stress levels at various parts of the arterial system estimated from available data fell within ±50% of the mean (15 dyn/cm2), which was consistent with the value predicted from the model. Theoretical analyses on the cost function of the model indicated that the suspected variation of shear rate levels in the arterial tree due to the anomalous changes in blood viscosity which might bring about 3- to 4-fold differences between the minimum and maximum shear rates would cause less than 10% increase in the total energy cost. It was concluded that a local adaptive response to wall shear stress is the mechanism which effectively optimizes the design of the arterial tree.

Notes: Abstract The structure of the genetic code is related to a Gray code, which is a plausible theoretical model for an amino acid code. The proposed model implies that the most important factor in shaping the code was the effects of mistakes in translation, not effects of mutations. Another possible implication is that the preservation of stiffness and flexibility at appropriate places in a protein chain is as important in protein structure as the appropriate placement of hydrophilic (external) and hydrophobic (internal) residues. Other results are a simple conceptualization of the relationships among the 20 amino acids and their relations to their codons. The detailed relationships are summarized in the following ‘similarity alphabet’: ala, thr, gly, pro, ser; asp, asn, glu, gln, lys; his, arg, trp, tyr, phe; leu, met, ile, val, cys; (ATGPS DNEQK HRWYF LMIVC in the one-letter code). This alphabet falls into four groups of amino acids: small, external, large, internal. The approximate relation of the groups to their codons is expressed as: the first base of a codon controls size—a purine means a small amino acid, a pyrimidine means large; the middle base controls cloisterednes—purine means external, pyrimidine means internal. These relationships express the minimum change principle upon which the code appears to be founded.

Notes: Abstract A theoretical study of the Brusselator model with non-uniform distribution of component A and a concentration-dependent diffusion coefficient has been performed. Numerical simulation reveals that a variable diffusion coefficient alters the bifurcation pattern and the stability properties of the steady-state as well as periodic solutions. A simple approximate method, based on one-point collocation, has been proposed to analyze the bifurcation phenomena for the case of fixed boundary conditions and low system size.

Notes: Abstract In this paper the effects of the occurrence of cut trees in the topological analysis of branching patterns have been studied. It is assumed that branches are removed at random from the trees. We prove that, for both the segmental and terminal growth models, the probability distributions of the cut trees are identical to those of complete trees.

Notes: Abstract The goal of this work is an examination of capillary exchange models as mathematical operators. The concentration function relations for the Krogh cylinder of a single capillary, basic to many organ models, are studied via the theory of operators on the Lebesgue normed spacesL p[0,∞], (1〈-p〈-∞). A discussion is included of theL p -normsvis-à-vis the coefficient of variation currently used in finding capillary parameters and evaluating parameter searches. The capillary model determines two operators on the space of locally integrable functions: O K (relating extravascular concentration to intravascular) and K a, k (relating intravascular concentration to input), wherek is the ratio of permeabilitysurface area (PS) to extravascular volume, and α is the ratio of PS to flow. These operators are shown to induce contractive (‖O K ‖ p 〈-1, ‖K a, k ‖ p 〈-1), isotone, linear operators onL p . The uniform convergence relation $$K_{a,k} = \mathop {\lim _{(p)} }\limits_{N \to \infty } \left( {\sum\limits_{n = 0}^N {P_n (a)O_k^n } } \right)$$ (as operators onL p) is derived, whereP n (a) is the Poisson probabilitye −a a n /n!. For the important special cases ofp=∞, 1, 2 the norms are found (‖Ok‖=‖Ka,k‖p=1). Consideration is also given to the norms and operators when the functions involved are limited to a finite interval of time.

Notes: Abstract One particular kind of structure offers possible explanations, for long-term memory, efficient consolidation of stored information from the environment, clustering of data strings and multimodal functioning. It is a possible model for pieces of neural structure and its use offers a uniform method for both studying and constructing an extensive class of mechanisms.

Notes: Abstract Levenshtein dissimilarity measures are used to compare sequences in application areas including coding theory, computer science and macromolecular biology. In general, they measure sequence dissimilarity by the length of a shortest weighted sequence of insertions, deletions and substitutions required, to transform one sequence into another. Those Levenshtein dissimilarity measures based on insertions and deletions are analyzed by a model involving valuations on a partially ordered set. The model reveals structural relationships among poset, valuation and dissimilarity measure. As a consequence, certain Levenshtein dissimilarity measures are shown to be metrics characterized by betweenness properties and computable in terms of well-known measures of sequence similarity.

Notes: Abstract It is pointed out that the methane flux measured experimentally for certain ponds and swamps is quantitatively consistent with a commensal dependence of Methanobacteria on O2-chemotactic motile aerobic bacteria. The Methano species is thereby shielded from oxygen and provided with carbon dioxide for the anaerobic production of methane.

Notes: Abstract A sufficient condition is given for stochastic boundedness persistence of a top predator in generalized Lotka-Volterra-type stochastic food web models in arbitrary bounded regions of state space. The main result indicates that persistence in the corresponding deterministic system is preserved in the stochastic system if the intensities of the random fluctuations are not too large.

Notes: Abstract One of Bobisud's models for the evolution of cannibalism is reanalyzed by applying the method of finding evolutionarily stable strategies (or ESS's). It is demonstrated that ‘no cannibalism’ never will be an ESS if the initial rate of cannibalism is too large. It is further demonstrated that individual selection may even result in the evolution of cannibalism during food abundance. Some empirical case studies are briefly discussed in relation to this model.

Notes: Abstract A computer algorithm is presented which equiprobably generates any member of the set of all directed trees withk labeled terminal nodes and unlabeled interior nodes. The algorithm requires roughlyk 2 /2 storage locations. The one-time initialization requiresO(k 2 ) time, while generating each tree requiresO(k) time.

Notes: Abstract The comparison of several sequences is central to many problems of molecular biology. Finding consensus patterns that define genetic control regions or that determine structural or functional themes are examples of these problems. Previously proposed methods, such as dynamic programming, are not adequate for solving problems of realistic size. This paper gives a new and practical solution for finding unknown patterns that occur imperfectly above a preset frequency. Algorithms for finding the patterns are given as well as estimates of statistical significance.

Notes: Abstract A new development is introduced here in the use of dynamic programming in finding pattern similarities in genetic sequences, as was first done by Needleman and Wunsch (1969). A condition of pattern similarity is defined and an algorithm is given which scans any set of similarities and screens out those which fail to meet the condition. When the set to be scanned contains every pair of segments, one from each of two given sequences of lengthsm andn (i.e. every possible location for a pattern similarity), then it completes the scan in a number of computational steps proportional tom·n, leaving those pairs of segments which satisfy the similarity condition. The algorithm is based on the concept of match density, as suggested by Goad and Kanehisa (1982).

Notes: Abstract This paper concerns sequences of letters in which certain “distinguished” words are of interest. Such sequences arise as data in numerous fields including genetics and neuroscience. A probability distribution is given for the number of occurrences of a chosen word in a randomized sequence of letters. Such words are considered “favored” if they occur more than expected at random. Favored words have been discovered in nerve impulse trains and may reflect a neural coding scheme.

Notes: Abstract As the volume of protein sequence data grows, rapid methods for searching the protein sequence database become of primary importance. Rigorous comparison of sequences is obtained with the well-known dynamic programming algorithms. However, these algorithms are not rapid enough to use for routinely searching the entire database. In this paper we discuss some methods that can be used for rapid searches.

Notes: Abstract We give algorithms for computing the extent of similarity between two or three sequences of letters. The similarity measures we consider include a penalty for inserting gaps within the sequence in order to enhance similarity. The magnitude of the penalty for gaps is assumed to be independent of their size in order to accommodate certain biological applications. Our algorithm for three sequence comparisons, which is based on solving a system of recursive equations, improves upon the efficiency of existing methods. Although the system of recursive equations utilized by the algorithm is quite complicated as it stands, it has none the less been simplified by appeal to combinatorial considerations.

Notes: Abstract Well-known dynamic programming algorithms exist for comparing two finite sequences inO(N 2) time and storage, whereN is the common sequence length. Extensions to the comparison ofM finite sequences requireO((2N) M) time and storage, making such algorithms difficult even forM=3. A simple generalization of the sequences makes it possible to obtain some results about the geometry of sequence alignments. These ideas suggest heuristic approaches to problems of comparing several sequences. IfM sequences are known to be related by a binary tree, they can be aligned inO(MN 2) time andO(N 2+NM) storage.

Notes: Abstract Protein sequences of the Dayhoff databank of 1984 have been analyzed to evaluate the occurrences of the 400 dipeptides and 8000 tripeptides. Expected values and standard deviations for the di- and tripeptides were determined by Monte Carlo and binomial approximation. A condensed format containing this information, labeled a uniqueness diagram, is presented and made available in the form of a microfiche.

Notes: Abstract In two-state sliding filament models of muscle contraction a partial differential equation must be solved to find the cross-bridge distribution functionn(x, t). In this paper the analytical form of this function is obtained by integration along the characteristic line and special cases are presented in which the explicit expression forn(x, t) can be completely determined. These analytical solutions provide a direct mathematical connection between the microscopic contraction parameters contained in the kinetic theories and macroscopic muscle dynamics and are thus used to investigate what parameters influence the transient contractile tension in typical experimental conditions. The results of this investigation are consistent with relevant aspects of muscle physiology.

Notes: Abstract Experiments on random binary, ternary, etc. (P=2, 3,…, 10) switching nets are reported. Behavioral cycle lengths are examined as functions of output variety,P, input connectance,K, and net size,N. Overall, output variety appears an influential, well-behaved net property. Strong, but well-behaved interactions appear among net variables. In high connectance nets, median cycle length grows approx. asP N/2. Other factors constant, one-connected nets show the shortest cycles, and connectance effects appear to converge asymptotically aroundN. Data for cycle length as a function of net size suggest a concavity not compatible with the Kauffman “square root law” (Kauffman, 1969). Evidence of a positive relationship between cycle length and run-in length is found in two-input nets; weaker evidence is obtained that in higher connectance nets this relationship becomes negative in sign. The “modular complexity” ofP〉2 nets is examined briefly.

Notes: Abstract The influence of topographical situation on the spread of infection is studied. The investigation is based on a multigroup model. The population under consideration is thought to be divided into subpopulations living in regions that are separated from each other by natural barriers (mountains). Infection is carried from one region to another by migrating infectives. Migration is possible only along the river system so that the structure of the epidemiological network is that of a symmetric tree. The results allow comparison of the velocity of propagation of the epidemic for different geographical situations and allow quantification of the “channel-effect”, according to which mountainous regions are channels rather than barriers to the spread of an epidemic.

Notes: Abstract Some progress has been made on the problem of the interaction of respiratory gases with whole blood. A working mathematical model for the O2−CO2 interaction phenomena has been developed from mathematical studies of the data. The Edsall-Wyman (1958) model for CO2 absorption is improved upon in this paper by consolidating it with the O2 absorption model developed in paper I of this set (Bernard, S. R.,Bull. Math. Biophysics,22, 391–415, 1960). This improved model assumed the effect of O2 on CO2 absorption is mediated through the electrical charge possessed by the hemoglobin molecule,i.e., O2 molecules bound to hemoglobin displace protons from the hemoglobin thereby increasing the negative charge on the hemoglobin and at the same time increasing the acidity of the solution. The model is tested against the data.

Notes: Abstract On the basis of previously proposed mathematical models of social behavior, the present note investigates the possibility of the control of behavior remaining permanently in the hands of one class, if this class possesses sufficient means for influencing mass behavior. The conclusion is reached that, with the assumptions made, if the behavior imposed by the controlling class leads to sufficiently strong dissatisfaction, the control will pass to another class, no matter how strong the controlling power of the first.

Notes: Abstract Traffic in one direction on a multilane highway is considered, and a general expression for the number of cars which pass a car travelling at a given velocity, as well as the number of cars which the given car passes, is derived for the case when the speeds of different cars are distributed in some arbitrary manner. Closed expressions are derived and discussed for a rectangular distribution. Each passing by another car or of another car is considered as a distracting stimulus which affects the reaction times of the driver. Using previously derived expressions for the safe speed as a function of reaction times, expressions for the safe average speed are derived, in terms of the volume of traffic and of the spread of the distribution of speeds.

Notes: Abstract Emphasis upon the importance of homeostatic feedback has drawn attention away from the complexity of biological processes. A study of glucose metabolism indicates the importance of open-cycle as well as closed-cycle mechanisms. Besides the glucose-dependent mechanism of insulin secretion, many open-cycle processes involving the liver, adrenal glands and kidneys, play important roles in the variation of blood glucose. In addition, glucose utilization by the tissues is essentially open-cycle in nature.

Notes: Abstract In certain situations like the aftermath of a revolution when discontent rises amongst certain groups of the population, it is frequently observed that the discontented groups are firmly convinced that their point of view is shared by the majority of the population. Yet future events prove that this is far from being the case. This effect is partly attributable to “wishful thinking,” partly to a purely social mechanism. The wishful-thinking effect may be considered as a case of psycho-physical discrimination in which a bias is introduced proportional to the degree of satisfaction anticipated from a given situation. H. D. Landahl's well-known equations can be applied to this case. The social factor is based on the circumstance that an individual associates by preference with such other individuals as have similar opinions. This results in an actual error of estimation of the relative minority or majority because of different frequencies of contact with individuals of the two opposing groups. Both factors may be combined into one equation.

Notes: Abstract Pipeline techniques have been successfully applied to speeding up processing in both general- and special-purpose digital computers. Application of these techniques to nonrecursive (FIR) filters has been suggested and is quite straightforward. Application to recursive (IIR) filters has not previously been shown. In this paper, the technique for applying pipeline techniques to recursive filters is shown and the advantages and disadvantages of the technique are discussed. Using these techniques, recursive digital filters operating at hitherto impossibly high rates can be designed.

Notes: Abstract The long-standing problem of reconstructing a function from its samples is considered again. Assuming a sequence of oversampled values, a set of appropriate idealized reconstruction filters can be defined, which do not suffer from instability or slow convergence. The realization — a cascade of a nonrecursive digital filter, D/A-converter, and a fixed/analog smoothing filter — demands the design of the digital filter for the increase of the sampling rate. The design of this nonrecursive filter is the purpose of this paper. Approximations in the frequency as well as in the time domain are presented.

Notes: Abstract In this paper we formulate power systems as nonlinear nearly Hamiltonian systems. Using the invariance principle for ordinary differential equations, necessary and sufficient conditions for asymptotic stability are established and a new method of estimating the domain of attraction of the stable equilibrium point is developed. The present results constitute a novel approach to stability analysis and involve the following three steps: a. Given a system with dissipation, the stability of its equilibrium is ascertained by determining the stability of the associated conservative system. b. Attractivity of the stable equilibrium of the entire system (with dissipation) is determined from the system topology. c. An estimate of the domain of attraction of the asymptotically stable equilibrium is obtained by making use of results obtained in (a) and (b). The stability criterion developed in this paper sheds new light on the mechanism of instability in power systems and it provides analytical verification to the concept of the potential-energy boundary surface (PEBS). The PEBS is a hypersurface which makes up a part of the boundary of the domain of attraction of the stable equilibrium in a power system. The existence and properties of the PEBS have thus far been deduced primarily via simulations and heuristic methods.

Notes: Abstract In conventional television systems, picture scanning in vertical and temporal directions is usually very defective with regard to the sampling theorem. In this paper some deficiencies such as aliasing, line-structure distortion, line flickering, and large-area flickering are investigated with regard to their dependence on the inter-lacedpicture-scanning process. The three-dimensional reconstruction filtering of the sampled picture is especially analyzed with respect of the viewer's perception. Furthermore, it will be shown that in connection with a new concept of picture scanning published earlier [1], [2], a flat-field reproduction without any 25/30-Hz flicker can be achieved by vertical filtering only. This is true even though the final reproduction by the monitor is performed with interlace. The vertical filtering can then be optimized in the sense of maximum picture sharpness and resolution with negligible ringing as well. Practical results are given in this paper.

Notes: Abstract In this paper a new type of “velocity-selecting/rejecting filter” which passes or stops a particular event in a seismic signal is proposed. The velocity-selecting filter is based on a time-space band-pass filter with sharp passband for a particular direction, and similarly, the velocity-rejecting filter is based on a time-space band-stop filter. A technique for designing such filters, in terms of an infinite-impulse-response (IIR) filter, is presented, in which a rotated version of separable filter is used. Finally, numerical examples are included to illustrate the design theory.

Notes: Abstract A special type of factorization for operators defined on partially ordered Hilbert resolution spaces is considered. The main result includes, as a particular case, the classical Schur-Coleski triangular factorization. Connections with stochastic optimization and image-processing problems are established.

Notes: Abstract In a recent paper on nonlocal expansions necessary and sufficient conditions are given under whichf −1 has a generalized power series expansion, whenf is an invertible locally Lipschitz map between certain general subsets of a complex Banach space. Here we establish the validity of a conceptually interesting algorithm for obtaining the expansion. Basically, we show that a certain contraction mapping iteration generates iteratesℐ 1,ℐ 2,... such that eachℐ k yields all of the terms of the generalized power series expansion off −1 up to order (k + 1), assuming merely that the expansion off −1 exists. An earlier different result along related lines is mentioned.

Notes: Abstract The representation of functions in a basis function expansionz(t)= ∑k=1/∞=,a k〉 x k (t) is straightforward when the basis functionsx k (t) are orthogonal. There has been very little work up to this time in determining how to use nonorthogonal bases in signal representation. On the other hand, applications in data compression and signal synthesis often require using specific tailor-made bases. Presented here is a method for constructing very general nonorthogonal bases. Orthogonality has often been used to show that a basis spans the set of functions of interest and to calculate the coefficients of the representation. In this paper, both of these fundamental aspects are addressed for nonorthogonal bases. A new basis {y k (t)} is obtained by performing a linear transformation on a known existing basis {x k (t)}. This transformation is constructed such that the coefficients of signal representation on the new basis are readily found. Then, a useful and sufficient condition is placed upon the new basis such that representations converge. The fundamental methods are applied to the standard examples of signal representation. The complex sinusoids, the Rademacher functions, the orthogonal polynomials, and the decaying exponentials are used as the original basis {x k (t)} from which a new basis {y k (t)} is generated. Two examples are given to illustrate general applications: one in signal synthesis and one in signal analysis.

Notes: Abstract This report describes a method by which the concepts of the lead vector and the lead field may be extended to include equivalent cardiac singularities above the first order. The theoretical treatment results in generalization of the well-known lead vector principle into a lead tensor apparatus. The specific electrocardiographic contribution of annth order multipole is found to be the product of annth rank contravariant (“heart”) tensor by a covariant (“lead”) tensor of the same rank, and with identical indices. Methods are also described whereby lead tensor components may be cast into a form which relates them directly to the more usual, spherical harmonic notation of multipolar current generators. Some attention is devoted to the possibility of applying lead tensors to biomedical problems, particularly to the prospect of quantitatively evaluating the significant multipolar components of the human heart.

Notes: Abstract A simple model is presented for the formation of functional groups in a random neural net. They show the following characteristics: 1. They can maintain autonomous activity which might serve as temporary memory traces. 2. Early in the process of formation they become resistant to contraction. 3. Later they become resistant to expansion. 4. Nearby groups inhibit one another. 5. Two groups may contain some cells in common.

Notes: Abstract Quantum biology is the quantum mechanical study of electrons in molecules of biological interest. This requires the solution of problems involving many electrons. Approximation methods are therefore necessary and are discussed. The present study, concerned with the mobile electrons in riboflavin (FMN) and its radicals (FMN−, FMNH and FMNH2 +), is based on the approximation method, developed by B. Pullman and A. Pullman. The solution of the eigenvalue problems so obtained gives the energy levels of the mobile electron systems involved. The corresponding eigenvectors yield the mobile electronic charges of the atoms of riboflavin radicals which have contributed mobile electrons. Important differences of the net charge distributions of these radicals are emphasized. The longest wave length of light absorption is calculated from the obtained energy levels and agrees, within the accuracy of the method, with corresponding experimental results. From the appropriate calculated results, electronic assignments are obtained for the experimental transitions involved.

Notes: Abstract A postulated scheme for sequential enzyme synthesis coupled with linear chain reaction is proposed. Mathematical calculations are carried out for the transient-state formation of enzyme coupled with enzyme-substrate reactions. Examples are presented to demonstrate the relations between various operational parameters during enzyme synthesis as a function of time. Significance of the phenomenon of sequential synthesis is discussed in relation to abnormal growth and cellular injury.

Notes: Abstract By introducing a plausible model for the initiation of axonal impulses the output is obtained as a function of the input incoming impulses. If the temporal aspects of the excitatory process resulting from the afferent impulses are sufficiently rapid one obtains the discontinuous or microscopic model of McCulloch-Pitts. If these are sufficiently slow a continuous model, such as Rashevsky’s one or two factor theory, is a natural model. But the linear relation between the strength of excitation of one axon and excitatory factor of the next will not in general hold. However, under conditions which are not too restrictive the linear relation with threshold can be considered as satisfactory approximation over a fairly wide range of values.

Notes: Abstract With a view to future applications in relational biology, the notion of relations between sets is introduced and several theorems are demonstrated.

Notes: Abstract A mathematical theory is developed to explain the observed enhancement of O2 transport through solutions by hemoglobin. At high partial pressures of O2, ordinary diffusion through the solvent accounts for all transport of O2, but at low partial pressures the transport may be increased manyfold by the presence of hemoglobin. This phenomenon is explained and its possible role in living organisms is discussed. The theory also indicates a new method of determining dissociation curves from diffusion experiments.

Notes: Abstract A solution in quadratures is found for a special class of differential equations suggested by studies in the growth of competing populations.

Notes: Abstract The tremendous complexity of even the simplest living unit makes a correct theoretical guess as to its mechanism very difficult. It is therefore suggested that, following the example of the physical sciences, a number of purely abstract cases in molecular biology be studied mathematically at first. Subsequent comparison of the different conclusions of such an abstract theory with available data would enable us to decide which of the conceivable situations are likely to occur in reality. As a first step toward such a study the problem of the minimal size of a living unit is studied. Usually the minimal size is considered to be determined by information-theoretical requirements. It is shown that the minimal size may have a very different origin. It may be determined by the possibility that too small a system, even though performing all necessary biological functions, may not be viable unless it is a member of a large group of other similar systems. This approach is developed both from a metric and from a relational point of view. Some relational characteristics of systems of reactions which possess the elementary metabolic properties of organisms are studied.

Notes: Abstract In Yeh, Martinek and de Beaumont (Bull. Math. Biophysics,20, 203–16, 1958), a method is presented for determining successively better central multipole representations of the current generators in a homogeneous conducting sphere by measuring surface potentials at a successively increasing number of points. This paper generalizes the method such that the multipoles may be located at any chosen point in the conductor. The spherical harmonic expansion is advantageously used and the “interior sphere theorem” of Ludford, Martinek and Yeh (Proc. Cambridge Philos. Soc.,51, 389–93, 1955) makes possible disturbance potential expressions in closed forms. A method for approximate determination of the eccentricity is also presented. In the theory of electrocardiography, the eccentric multipoles can more accurately represent the heart as a current generator with fewer surface potential measurements than the central multipoles.

Notes: Abstract Given three primary events, occurrences or nonoccurrences of two response classes and nonoccurrence of both response classes, with their respective probabilities to be known, and assuming the first two events are disjoint, this paper deduces the error equation of the incorrect response and the latency equation of the correct response under the condition of correlated reward situation.

Notes: Abstract Schemes are presented for induced synthesis of the dissociable enzyme in which repeated use of the template is made. The role of the inducer is to release the repression. A mathematical analysis is carried out and expressions are obtained to describe the kinetics of enzyme formation. A practical case (penicillinase synthesis) is compared with theoretically derived equations by using an analogue computer to simulate an enzyme forming system. A good correlation between theoretical and experimental data is obtained.

Notes: Abstract The temperature regulation in homothermic animals is an example of a negative feedback system. It shows sudden changes of parameter values, and therefore suggests the use of catastrophe theory for its description. This paper reports on work done on the human temperature regulation mechanism to shows that the five-dimensional dual butterfly catastrophe model is sufficient for its description. Nearly all experiments reported in the literature overlook the dynamic multiparametric nature of the process. Use of catastrophe theory, on the other hand, shows that one cannot find a model with fewer than five parameters for such a system. From work by Benzinger and Kitzinger, the exact shape of part of the corresponding bifurcation set is determined.

Notes: Abstract This paper is an analytical study on the pulse wave velocities in the aorta. In conformity to a physiologic state of loading, the distensibility of the vessel wall has been accounted for. The wall material is treated by using the theory of large elastic deformations. The orthotropicity of wall tissues and the effect of the surrounding tissues have been incorporated in the analysis which is based on the use of the strain energy function suggested by Vaishnavet al. Numerical values of the wave velocities of the canine middle descending thoracic aorta are computed by using the derived analytical expressions.

Notes: Abstract The theory of a symmetrical 3-barrier, 4-site, single-filing ionic channel is developed. The model goes beyond earlier models by including additional sites, as well as barriers which need not be symmetrical in the applied field, and contains the earlier models as special cases. It is itself a special case of the most general 4-site model, which has 5 barriers. By considering the barriers at the mouth and middle of the channel to be sufficiently larger than the barriers separating the sites in each channel half, these barriers can be neglected; thus this case reduces to a 3-barrier model where the sites in each channel half can then be assumed to be in equilibrium with each other. The alternative 3-barrier, 4-site case, where the barrier between the sites is considered to be larger than that at the mouth of the channel, is considered elsewhere. Pure cation permeation is considered and only single-salt properties of the system are analyzed, namely occupancy, conductance, flux ratio exponent and current-voltage relation. The concentration dependences of these properties are computed and interrelated and, where possible, also given in analytical form. The mathematical relations are obtained for a channel which is symmetrical around its middle, which is the appropriate assumption for the gramicidin channel. However, the barriers themselves are allowed to be asymmetric with respect to the potential dependence, which has been found to be essential for gramicidin. Mathematically, a straight-forward matrix formulation is used; but a general theoretical method is presented for reducing a complex model (with more than 2 sites) to a simpler cases when equilibrium exists across one or several barriers, as is often the cases. This method is a prototype which makes analytical solutions of complex barrier models possible in many cases.

Notes: Abstract A chain-like arrangement of four urns (a catenary system) into which different color balls (white, corresponding to radio atoms, and black, corresponding to stable atoms) are being transferred is used to simulate the transport of atoms down the GI tract of man and animals. Into the first urn (stomach) are placedw o white balls andr black balls while in the 2nd (small intestines) and 3rd (large intestines) urn, onlyr blacks are put in, with no whites. A sample of sizer is transferred from the 1st, 2nd and 3rd urns to the 2nd, 3rd and 4th (infinite universe) urns. From the random variable difference equations the first and second moments for the distribution of the number of radio atoms present in each urn are obtained. The variance of the contents of radioatoms in the excretion urn is

Notes: Abstract A power series solution is presented which describes the steady-state concentration profiles for substrate and product molecules in immobilized enzyme systems. Diffusional effects and product inhibition are incorporated into this model. The kinetic consequences of diffusion limitation and product inhibition for immobilized enzymes are discussed and are compared to kinetic behavior characteristic of other types of effects, such as substrate inhibition and substrate activation.

Notes: Abstract The use of spheroids as a tumor model has become commonplace since it was discovered that many cell lines can form spheroids when grown on a surface to which the cells cannot attach. This culture system complicates experiments which depend on oxygen supply because the oxygen concentration in the vicinity of a stationary spheroid has not been well defined. We present in this paper solutions to the oxygen diffusion equation for simple geometries: a spheroid in an infinite stationary medium and in a finite spherical stationary medium. Comparison of these solutions provides an estimate of the oxygen supply to a spheroid in a Petri dish. We show that typical spheroids can be expected to cause a substantial depletion of the oxygen in the nearby medium. Any disturbance of the medium or the spheroids will temporarily increase the oxygen supply. We provide a method for estimating the rate of return to equilibrium in the finite cases. These results indicate that the oxygen supply to stationary spheroids can be altered temporarily by small movements or changes in temperature which cause convection currents, or permanently by changes in the depth of the medium.

Notes: Abstract Alternative sufficient conditions are derived that guarantee the stability of spatially heterogenous steady-state distributions of motile aerobic bacterial populations attracted chemotactically by oxygen, motile anaerobic populations repelled by oxygen, and the oxygen concentration itself through a stationary aqueous medium. In particular, it follows from the latter criteria that the heterogeneous steady-state distributions for cylindrical regions with arbitrary cross-sections, uniform depth and mixed Dirichlet-Neumann boundary conditions on the oxygen concentration (appropriate to certain still-water bodies in nature) are stable with respect to arbitrary perturbations in the bacteria cell and the oxygen distributions.

Notes: Abstract Closed positive feedback loops of catalytic reactions between macromolecules, or hypercycles, provide a kinetic mechanism whereby each Species serves to catalyze selfreproduction of its successor in the loop. Hypercycles of five members or more evolve into limit cycles characteristic of a biochemical clock. Computer study of the coupled non-linear differential equations which describe these systems shows that the periodT n of then-species limit cycle is given byT n=nτn, where τn is an elemental repeat period reflecting translational time invariance. Analytic solutions of the equations are developed so that the time evolution of elementaryn-hypercycles can be traced in dynamical detail. It is shown that the magnitude of τn is, to good approximation, a linear function ofn. For a givenn, τn is a very sensitive function of the relative concentration a given member of the loop has at the time its predecessor dominates the state of the hypercycle. These concentrations decrease with increasingn. Aroundn=15 they become so small that elementary hypercycles become unstable against disruptive concentration fluctuations. Species concentrations for more realistic hypercycles tend not to be as small, so that the present estimate of a maximum number of components is a lower bound.

Notes: Abstract The technique of the probability generating function is used to derive the stochastic differential equations for a nonlinear model based on Eigen and Schuster's theory of biomolecular selection and evolution. The stabilities of various steady states are analyzed by using the linear stability approximation. The instability of a small starting population is investigated numerically. The minimum starting populations required for steady-state survival are then estimated for a wide range of parameters.

Notes: Abstract Some morphological features of the human bronchial tree were simulated by computergenerated trees. The trees were ordered by the methods of Horsfield and Strahler. Delta, the difference between the Horsfield orders of the two branches at a bifurcation, was determined by pseudorandom numbers generated according to a distribution of probabilities defined on input. By trial and error a distribution was found which resulted in trees being generated with average Strahler order branching ratios of 2.82, similar to a real bronchial tree. Branching angles and length ratio could also be defined on input. By varying these input parameters it was found that the form of the tree was quite sensitive to them, and that by a suitable choice the intrasegmental part of the bronchial tree could be simulated. It is concluded that branching ratio, length ratio, mean branching angles and distribution of delta are controlled within tight limits in the bronchial tree, and this may support the concept of optimal design.

Notes: Abstract In this paper perturbation methods are used for the mathematical analysis of coupled relaxation oscillators. This study covers entrainment by an external periodic stimulus as well as mutual entrainment of coupled oscillators with different limit cycles. The oscillators are of a type one meets in the modeling of biological oscillators by chemical reactions and electronic circuits. Special attention is given to entrainment different from 1∶1. The results relate to phenomena occurring in physiological experiments, such as the periodic stimulation of neural and cardiac cells, and in the non-regular functioning of organs and organisms, such as the AV-block in the heart.

Notes: Abstract A new, more realistic model of the action of ionizing radiation on mammalian cells growingin vitro is presented. Although this model requires a large number of parameters, these are linked to biologically observable quantities rather than being abstract sensitivities, as had previously been the case. Three different stochastic processes are required: {X(t);t ∈ [0, τ]}, representing damage alterations during irradiation; {(X(t), S(t));t ∈ [τ, τ+T D]}, representing changes in both damageX(t) and cell cycle positionS(t) during the post-irradiation cell cycle; and {N x(t);t ∈ [0,T G]}, representing the subsequent colony growth process conditioned on the value ofX(τ+T D). The assumptions used to define these processes extend a previous model of short term DNA damage formation and repair (Nelson S. J. 1982,Radiat. Res. 92, 120–145) to include the influence of cell cycle progression on damage in the irradiated cell and the effect of permanent inherited damage on the daughter cells' colony growth pattern. Expressions corresponding to commonly measured radiation effects are derived from the model and compared with predictions from previous models. It is found that these previous models oversimplified the mechanism of radiation action because they did not adequately represent repair during irradiation, the influence of radiation-induced cycle delays and damage inheritance by any daughter cells. Suggestions are then made for ways in which the new model can be used to test the importance of these effects.

Notes: Abstract A stochastic approach is utilized to develop a model equation capable of describing the time course of germination in a sample of bacterial spores. The time required by a spore to complete the change characteristic of germination consists of an initial interval of no change followed immediately by the duration of the change itself. The experimental basis of the proposed model is the observation that each of these time intervals is distributed over a range of values in a spore sample. Mixed continuous and discrete probabilities are employed in arriving at an average single-spore germination curve which, to a different scale, describes the sample in time.

Notes: Abstract Mathematical methods for comparison of nucleic acid sequences are reviewed. There are two major methods of sequence comparison: dynamic programming and a method referred to here as the regions method. The problem types discussed are comparison of two sequences, location of long matching segments, efficient database searches and comparison of several sequences.

Notes: Abstract An algorithm for nucleic acid and protein sequence alignment is presented. It is a non-metric local similarity minimal-difference algorithm and in the current implementation, assembles the matching regions found into a pseudo-global format. Its strengths are its speed of execution and the especially convenient presentation of its output. The algorithm is intended for use in sequence melding and local (small-region) similarity searching. It is not designed to replace a metric Needleman-Wunsch-Sellers-type similarity algorithm. The program is written in FORTRAN and is designed to be easily transportable to a variety of computer systems.

Notes: Abstract This is a review of past and present attempts to predict the secondary structure of ribonucleic acids (RNAs) through mathematical and computer methods. Related areas covering classification, enumeration and graphical representations of structures are also covered. Various general prediction techniques are discussed, especially the use of thermodynamic criteria to construct an optimal structure. The emphasis in this approach is on the use of dynamic programming algorithms to minimize free energy. One such algorithm is introduced which comprises existing ones as special cases.

Notes: Abstract It is shown that the concepts of grammar complexity and syntactic structure provide a useful mathematical framework for the investigation of some current problems in protein structure. Grammar complexity gives a measure of the degree of aperiodicity of a sequence and also an optimization criterion for evaluating amino acid categorizations. Three systems of amino acid categorization are compared in relation to their value for describing molecular architecture.

Notes: Abstract A scheme for representing amino acids as vectors in the plane is presented and justified. The two dimensions of the plane are size and hydrophobicity. The vector representation is then applied to generate a consensus sequence for some sets of homologous proteins. A figure of merit for the degree of homology of a set of sequences results from the analysis. Some other applications of the scheme are considered also. This work grew from ideaseeds planted by Margaret Dayhoff's work in theAtlas of Protein Structure and Sequence. It is with gratitude that I dedicate this paper to her memory.

Notes: Abstract An annotated bibliography of mathematical and computer analyses of protein and nucleic acid sequences is presented. The major subject areas represented are the determination of sequences, restriction mapping, similarity searching, sequence alignment, codon utilization, statistical analysis, information theoretic analysis, the construction of secondary and tertiary structure and DNA topology.

Notes: Abstract DISGEO is a new implementation of a distance geometry algorithm which has been specialized for the calculation of macromolecular conformation from distance measurements obtained by two-dimensional nuclear Overhauser enhancement spectroscopy. The improvements include (1) a decomposition of the complete embedding process into two successive, more tractable calculations by the use of “substructures”, (2) a compact data structure for storing incomplete distance information on a structure, (3) a more efficient shortest-path algorithm for computing the triangle inequality limits on all distances from this information, (4) a new algorithm for selecting random metric spaces from within these limits, (5) the use of chirality constraints to obtain good covalent geometry without the use ofad hoc weights or excessive optimization. The utility of the resultant program with nuclear magnetic resonance data is demonstrated by embedding complete spatial structures for the protein basic pancreatic trypsin inhibitor vs all 508 intramolecular, interresidue proton-proton contacts shorter than 4.0 Å that were present in its crystal structure. The crystal structure could be reproduced from this data set to within 1.3 Å minimum root mean square coordinate difference between the backbone atoms. We conclude that the information potentially available from nuclear magnetic resonance experiments in solution is sufficient to define the spatial structure of small proteins.

Notes: Abstract Recent evidence suggests that the cyclic nucleotides play a central role in the intracellular processing of neural signals. The dynamics of this system may be seen as a realization of the enzymatic neuron model. Enzymatic neurons are formal neurons which map binary afferent signals into patterns of excitation across an abstract membrane. The distribution of enzyme-like elements called excitases enables a set of local threshold functions to determine the firing activity of the neuron. This paper analyzes the basic properties of enzymatic neurons in a simple continuous-time framework, and shows how they may be presented as reaction-diffusion networks which model the cyclic nucleotide system. We present the results of computer simulations of this neuron and discuss its implications for selectional learning and its relation to conventional two-factor systems. One fundamental property of the reaction-diffusion neuron is its so-called “double-dynamics” property; examination of this property and its contribution to the computing power of the neuron provides some insight into the obscure relation between microscopic and macroscopic models of computation.

Notes: Abstract A model for enzymic catalysis is presented using the mathematical theories of differential geometry and Stieltjes integration. The Stieltjesintegrator is a complex-valued function of bounded variation which represents the curvature and torsion, hence the conformation, of the backbone of an enzyme molecule. Theintegrand is a complex-valued continuous function which describes the shape of the surface of a substrate molecule. We postulate that enzyme-substrate interactions correspond to evaluations of Stieltjes integrals, and that observables of enzymic catalysis correspond to projections. Results from the mathematical theory of the Stieltjes integral are discussed together with their biological interpretations. We contrast the difference between structural and functional proteins, and construct analogues of enzyme cofactors, modifications, and regulation. Various techniques of locating the active site on enzymes are also given. We construct a total variation metric, which is particularly useful for detecting similarities among proteins. An examination on the many different modes of convergence of mathematical functions representing biological molecules leads to a mathematical statement of the fundamental dogma of molecular biology, that ‘structure implies function’. Similar arguments also result in the converse statement ‘function dictates structure’, which is a basic premise of relational biology. Stepped-helical approximations of the backbone space curves of enzymes provide a concrete computational tool with which to calculate the Stieltjes integrals that model enzymic catalysis, by replacing the integral with a finite series. The duality between enzymes and substrates (that they aremeters ‘observing’ one another) is shown to be a consequence of the mathematical duality of Banach spaces. The Stieltjes integrals of enzyme-substrate interactions are hence shown to be bounded bilinear functionals. The mechanism of enzymic catalysis, the transformation from substrate to product, is also formulated in the Stieltjes integration context via the mathematical theory of adjoints. The paper closes with suggestions for generalizations, prospects for future studies, and a review of the correspondence between mathematical and biological concepts.

Notes: Abstract An earlier theory of cell differentiation and morphogenesis (Wassermann, 1972, 1973, 1978) is combined with the genetic control model of Davidson and Britten (e.g. 1979). The resulting new theory suggests how, bysystematic process algorithms, specifically enumerated combinations of batteries of structural genes can become switched on in particularly enumerated cells, via battery-specific enumerable regulator genes. The systematization is idealized. Up to a certain stage of development in each mitotically arising cell a unique cell-specific combination of structural genes called ‘marker genes’ is active. Marker genes are assumed to code for cell-specifying marker proteins (CSMPs) which permit cells carrying related markers to recognize each other, thus permitting specific cell sorting.Batteries of marker genes could ensure great developmental precision and can safeguard—via redundancies of CSMP types—against accidental loss or detrimental mutational modification of CSMPs or marker genes, respectively. This paper is much concerned with cell lineage in relation to ‘microdifferentiation’, where ‘microdifferentiation’ of a cell refers to a cell's active marker genes and its syntheses of CSMPs. A drastic distinction is made between ‘microdifferentiation’ and ‘gross’ differentiation of a cell, where the same ‘gross’ differentiation may be shared by a large number of cells that could each be uniquely ‘microdifferentiated’. Typical ‘gross’ differentiation could manifest itself in tissue specificity, whereas, up to certain stages of development, all cells of the same gross differentiation type (say tissue specificity) could each be uniquely ‘microdifferentiated’. The theory also assumes that at certain stages of the developmental process some (or in some organisms all) of the previously uniquely specified cells could give rise to small (or occasionally large) clones of equispecified cells, some of which might form clusters that represent complete ‘morphogenetic fields’ Tentative implementation mechanisms are proposed which suggest how the theory could operate in molecular biological terms. In particular, CSMPs could endow cell surface membranes with a highly specific protein network, and an associated equally specific cell surface coat. It is suggested how these highly specified cell surface coats and other systems could provide an ‘extracellular guidance network’ which could help to direct cells to attain energetically optimal locations relative to each other based on the matching of their surface specificities. In numerous experimental situations, where normally present optimal matching of cells is excluded, ‘alternative matching’ based on experiment-specific suboptimal matching could explain many data, notably in experimental development neurobiology (Wassermann, 1978).

Notes: Abstract Two equations describing one-dimensional food chains are known to possess soliton solutions. It is demonstrated that both equations are embraced within another equation, which arises in the theory of chains of enzymic reactions. We find an elliptic function solution to this equation. We obtain a one-soliton solution from it and re-derive the elliptic function solutions of the two ecological equations.