The geometrization of biology

The twentieth century has witnessed a geometrization of physics, that is, a reduction of the basic concepts of physics to geometric concepts. The topological approach to biology, recently proposed and to some extent developed by the author, is a small step in the direction of geometrization of biology, but is unable to achieve the main purpose of such a geometrization of biology, namely, the reduction to geometric concepts of such purely biological concepts as ingestion, digestion, assimilation, etc. To achieve this purpose we must find geometric structures or spaces, in which different geometric properties stand to each other in the same formal logical relation, as the different concepts of biology stand to each other. If this were possible, then a set of geometric theorems could be “translated” by an appropriate “glossary” into a set of biological laws. While not offering a solution to this problem, the present paper illustrates the possibility of such an approach on several examples. Certain new types of topological spaces are introduced, which are used for illustration purposes only. It is shown, however, how from a theorem about such spaces a verifiable biological prediction could be made, if these spaces were to be taken seriously. A possible application to biology of E. Artin's theory of braids is outlined.     

A note on topological biology

In line with a recent suggestion by the author (Bull. Math. Biophysics,20, 267–73, September, 1958) that not only does the organism as a whole map on the primordial, but that each organ can also be thus mapped, it is shown that the previously introduced abstract spaces, which represent an organism, contain subspaces which map continuously on the space of the primordial. Several theorems about those subspaces are proven.     

A note on the mathematical biology of automobile driving

It is pointed out that the three different stimuli for a corrective turn, namely the distance from the edge of the lane, the rate of approach to the edge, and the angle between the direction of the car and the direction of the lane (Bull. Math. Biophysics,28, 645–654, 1966,29, 181–186, 1967) may act all three simultaneously. It is found that in that case the tracking curve of the car is stable below a critical speed and becomes unstable above it.     

Topological biology: A note on Rashevsky's transformation T.

In the bio-topological transformation between graphs denoted by (T ⁽¹⁾ X) N. Rashevsky (Bull. Math. Biophysics,18, 173–88, 1956) considers the number of fundamental sets which (a) have only one specialized point as source (and no other sources), (b) have no points in common (are “disjoined”); he proves that this number is an invariant of the transformation. In this note we show that Rashevsky's Theorem can be extended as follows:The number of fundamental sets of the first category is an invariant of the transformation. We must, however, count the subsidiary points of the transformed graph as specialized points. We recall that fundamental sets of the first category are those whose sources consist of specialized points only (Trucco,Bull. Math. Biophysics,18, 65–85, 1956). But in this modified version of the Theorem the fundamental sets may have more than one source and need not be disjoined.     

A note on a variational problem in biology and sociology

The author's suggestion that the behavior of an individual or of a society may be described not by maximizing the satisfaction function but by maximizing the integral of it over a period of time (Rashevsky,Mathematical Biology of Social Behavior, rev. ed., p. 233, Chicago: The University of Chicago Press) has been considered applicable to other biological functions in a recent paper by R. Rosen (Bull. Math. Biophysics,24, 279–290, 1962). It is pointed out that the possibility of introducing such variational principles into biology or sociology requires that the corresponding functions depend explicitly and in a nonlinear manner on the rate of change of some of the variables. An example of a satisfaction function is given that describes the behavior of an individual who derives more satisfaction from the process of approaching a goal than from the actual goal itself.     

A note on Rashevsky's theorem about point-bases in topological biology

A graphG may have more than one point-baseB _G. In a primordial graphP of Rashevsky's (1954) TransformationT, some of the pointbases may consist of nonspecialized points only, and some other pointbases may contains specialized points. In this case, Rashevsky's Theorem (1955a) on point-bases may not hold. The Theorem is certainly true ifall point-bases ofP consists of nonspecialized points. A rigorous proof is given. Some results are derived for the more general case, when point-bases include both kinds of points. A general expression for the point-base ratio of the transformed graphP(T) is obtained. It is shown that with some biologically plausible assumptions Rashevsky's interpretation of the point-base ratio and his conclusions are still true. A few simple Theorems on point-bases of graphs are included in this work.     

Language and zoo biology training outcomes: A cautionary note

The Jersey Wildlife Preservation Trust (JWPT)/University of Kent Diploma in Endangered Species Management (DESMAN) was initiated in 1985 to provide an academically recognized qualification to trainees attending the JWPT's International Training Centre (ITC). Since its inception, 59 of 118 trainees enrolled were awarded the diploma. Examination and practical work performance of 141 trainees from 1994 to 1998 showed a significant variance in grades. DESMAN participants were by far the highest achievers with high mean examination grades and high practical work grades. More than one third (38%) of trainees, all of whom were DESMAN participants, obtained above median grades in practical work and the examination. Another third of all trainees attained below median grades in both examination and practical work. Given that most course participants (93%) had a strong academic background in biology or the veterinary sciences, low course performance could have been related to the inability of some trainees to fully understand the language used. Before the start of a course, trainees are requested to produce evidence of English language proficiency, at least level 7 of the International English Language Testing System (IELTS). When individual trainee language skills were tested in Jersey, through a self‐assessment exercise (named English Language Proficiency or ELP), a significant variation in language skills was found. These ELP scores positively correlated with the trainees' mean and median examination performance. Thus, entrance into the DESMAN seems to be affected by trainee's fluency in English. Since most trainees are graduates or have some form of tertiary education, the perceived influence of language proficiency on the attainment of a university‐accredited qualification such as the DESMAN needs to be taken most seriously. Zoo Biol 18:71–76, 1999. © 1999 Wiley‐Liss, Inc.     

Molecular biology and the kinetics of cell growth. II. A note on mass-volume relations

Using a previously derived theory of cell mass and volume growth, we show that a cell mass doubling criterion for cell division leads to a progressive deviation of cell volume from an arbitrary initial value to a value dependent only on the metabolic and osmotic parameters of the cell and its environment. This research was supported in part by USPHS Grant HE14177-01.     

A note on the horopter

By assuming the fixity (but not the symmetry) of corresponding points on the two retinae, it is possible to derive the equation of any horopter when one is known. In particular when, as experiment shows, one horopter is linear, then all horopters must be conics. These have the form given by Ogle, but whereas Ogle leaves one parameter undetermined at each fixation, on our assumption the only arbitrary parameter is determined by the position of the linear horopter.     

A note on the reafference principle

Two alternative circuits to realize the reafference principle are considered and quantitatively analyzed. Both are combinations of a conventional control loop with negative feedback and linear transfer functions, as well as of an efference copy branch. The feedback control loop compensates for passive movements of the body or of its parts, and generates active movements, whenever the set point differs from zero. The efference copy branch should eliminate sensory messages to higher brain centres during active body movements and, thus, mediate perceptual stability. In one of the combinations, discussed also briefly by Mittelstaedt (1971), the efference copy branch interacts with and thus modifies the properties of the feedback loop. The performance of this circuit is very sensitive to variations of the system parameters and, therefore, requires precise adjustment. When the transfer function of the efference copy branch exactly matches that of the feedback loop and its gain amounts to 1, this circuit performs as a control loop with an integrator in the negative feedback branch: there is no steady state control error. However, for certain parameter combination the circuit becomes unstable. In the alternative circuit proposed here, the efference copy branch does not interfere with the feedback loop. It is robust against parameter variations. The transient properties of both circuits are described under simplified assumptions regarding the linear transfer functions in the different branches.