WSEAS



A Principle of Prolation in Biology

CHARLES A. LONG

Department of Biology, University of Wisconsin, Stevens Point, Wisconsin 54481 U.S.A.

Abstract:--Prolation is a new principle in physiology, ontogeny, morphology, and evolution based on evolution of a negatively allometric thickness of an ellipsoidal organism relative to increased (without limit) area and length, when volume or weight does not vary.  The ellipsoid’s surface/volume may differ from classic surface/volume of spheres and metabolism/body weight comparing similar shapes of organisms and a variable mass. 

Key Words:  Prolate ellipsoids, Surface Area, Macroevolution, Dimorphism

1 Introduction

Even in pure mathematics but especially in biology [1], the relations of length, area and volume in ellipsoids are poorly understood. There have been practical attempts to estimate surface area by use of elliptic functions. For many organisms, the body mass (( volume or weight of matter) in vertebrates is related to an organism’s metabolism directly [2-5], and even for bird eggs [6]. Since any dimension, such as the radius of thickness b when squared is proportional to the body surface area, and b3 is proportional to weight or volume, then the ratio of surface-to-volume is about kV2/3 where k is a constant, and metabolism V0.7, with a similar exponent. Brown and Lasiewski [7] hypothesized the slender body in weasels has greater surface area than bodies of comparable volume such as ground squirrels. Since heat loss is a problem in cold climates, the compensation in fitness is assumed to be efficient predation in burrows of small mammals. Biologists have investigated form and function by the surface to volume ratio, inferring that any organ or organism is roughly spherical. By finding a constant for shape (= k) and an exponent m for slope of the log weight versus the log surface (= log S), approximations for “ovoid” forms are obtained [8-10]). There are properties of ellipsoids that reveal some undescribed geometric effects on biological development and evolution. We must understand why proportions of the body are important. The process of slenderization in spheroid bodies can be called “prolation”. If volume is constant, then the well-known volume equation can be solved for the dimensions. Prolation explains several problems.

1. The incremental changes of diameter (( b) and length (( a) are inversely related as components of length (2a) and radius (b) (see Fig. 1).

2. Lengthening proceeds, both in rapidity and extent. It results from even small reductions in diameter. Similar curves were noted for hydrostats of constant volume [11-12]).

3. The lengthening increases at faster and greater rates depending upon reductions of diameter.

4. In prolation, surface area increases without limit.

5. During prolation, divergence occurs between changing isometric forms that have different starting points (different initial diameters or mass). Divergence may enhance dimorphism.

6. The inner structures or materials are brought nearer the surface by prolation.

2 Problem Formulation

Since volume is constant, the spheroid elongates as the transverse diameter of the body narrows. For functions f (x) for length and f (y) for radius, their products determine area or volume. The formula of the ellipsoid is,

x2 / a 2 + y 2 / b 2 + z 2 / c2 = 1 (1)

In ellipsoids, a is the major semi-axis along either x direction, and b is the minor semi-axis along the y axis. The value c is on the z axis, and equal to b. In an ellipsoid a > b > 0. There are two foci around which any point might circle in a long elliptical path from a back to a, or, if within a sphere, the point revolves around the center (Fig. 1). The area of a sphere is determined by rotating a circle x 2 + y 2 = r 2 around the x axis, so that any point x, y lies on the arc for d x and d y, then d S is the product of 2 ( y and d s, which is the slant hypotenuse (d s = (x 2 + y 2)1/2).

S = [pic]2 ( (sin ( + 1) r 2d ( = 4 ( r 2 [when r = a = b = c > 0]. (2)

When V has a radius r , a sphere can be generated by revolving the circle x 2 + y 2 = r 2 , then y = f(x) = (r 2 - x 2 )1/2. V = [pic]( (r2 – x2) d x = 4/3 ( r 3. V = 4/3 ( a b c [when a ( b = c = r]; a = V/4/3((b2). (3)

When b = c > 0, the ellipsoid’s surface is a revolution around 2 a. By using parametric equations the surface area for the ellipsoid [13] is

Sellipsoid = 2( b 2 + 2 ( b / (a2 – b2) ½ [(a2 - b2) E (( + b2 ()] where E is the elliptic interval as used in studies of ellipses, and ( is eccentricity.

Sellipsoid = 2(b2 +2(b (a) (sin-1 ()/( (4)

The (, which works out by the Pythagorean Theorem to (2 = (a2 - b2)/ a2, was also defined [13] as (12 ( (a2 - b2)/ a2 , (22 ( (b2 - c2)/ b2, k ( ((2 / b), and (1 = sin ((, k).

Vsphere = 4/3 ( r 2 < Vellipsoid = 4/3 ( a b c [when a > b = c]. Setting V constant,

dV = 0 = 2 ( b2 + [2 ( b (a b) (sin-1 () / (]

= 2 d b + dl (l) (sin-1 () / (. (5)

= l (2 d b b)/ d l ) (sin-1 () / ()

where sin-1 ( / ( ( 0.8 – 0.9. (6)

Since the diameter of the cross-section is 2 r, the spheroid lengthens by approximately four times the change in r. Appendix 1 shows the same relation, by differentiating the volume equation with respect to time. The decrease in y extends a at both ends. Such extension more than doubles ( a, and was never described. Furthermore, 2 a increases without limit as b decreases to its limit 0. For linear measures of the right triangle in Fig.1, where b varies from 5 to 0, where the volume is held constant, “a” begins at 5. The extension of the focal distance to F can be calculated by the Pythagorean theorem or solving for a from Equation (4). Therefore, a increases generally, and for each reduction a1 < a2 < a3 . . . < an .

The eccentricity of the spheroid is of little consequence when ( is small; i.e., the sphere resembles the spheroid if ( has no value. This raises the question, how does slenderizing increase area if b can drop to zero? In Fig.1, distance to the focal point is, determined by Pythagoras’ theorem, 0 to F, which is the square root of (a2 - b 2). Obviously, a is the hypotenuse, the distance from point b to F. As the ellipsoid prolates, F moves farther along x and the hypotenuse is extended to it, at each reduction of b. Thus, eccentricity, related to the extension, is important in determining surface area. An approximate way to avoid the complexity of ( is given below.

[pic]

Fig. 1 A prolate ellipsoid. 2 a is the major semi- axis along the x axis (a to – a), b is the radius to the highest point b, b = c, which is the radius along the z axis, F is one focus of the ellipsoid, and a hypotenuse of a right triangle extends from b to F ( a.

3 Problem Solution

3.1 Lineages

If two lineages of mammals (M-1, M-2) evolved by prolation in order to invade tunnels, and one was slightly larger to begin with (e.g., diameters 10 > 8), in order to maintain their metabolic needs the c mass (or V) is assumed constant. If the surface area and heat loss increased, other adaptations might compensate in survival. Although different in diameter, the volumes are quite different. A reduction of 0.8 causes comparable extensions of a, for the larger mammal the greater length. Both increased 56 %.

By use of a handbook formula (equation (4)), surface areas were determined. Where prolation continues until b approaches lim 0, and the ratio ( approaches one, the semi-axis a extends rapidly. With the V constant and the changes parallel by similar decreases of b, then the initial b value and the quantity of V produce different curves. In one curve beginning with five, when b ( 2.04, the length of 2 a ( 60 for the larger V, and 30.5 for the smaller.

If the spheres were initially twice as large (r = 10 or 8), then 2 a attains the surprisingly great length of 458 or 234, even with similar reductions to b ( 2.09. Further reductions approach b = lim 0, and at b = 0.69 the 2 a length is 4,199 or 2,150. The disparity between these two points is enormous, the larger is twice as great. That was true of the original volumes (Fig. 2).

The length ratio 10/8 ( 15.6/ 12.5, which ratio is that of the two new lengths of a obtained by reduction of b by 0.8. Surprisingly, the divergence between the new lengths is much greater than it was initially with 10 and 8. Thus, narrowing the body of a mammal causes the length to extend geometrically, but especially the larger one, and less markedly the smaller. Where metabolic costs are even, this partially may explain marked sexual dimorphism in length.

When the log10 values for b and 2 a were plotted obtaining three straight-line curves, the slopes were nearly identical (- m = 0.49-0.50). The equation log10 b = - m log10 (2 a) + log K can be rewritten as b = (2 a)- ½ (K), where K = y at x0.

[pic]

Fig. 2 The log10 exponent of the length of the ellipsoid 2 a plotted as 3 curves based on the prolation of spheres where b = 10, = 8, = 5.

3.2 Areas

Areas also increase during prolation without limit. If surface area is important for survival, prolation may be important. This alteration of form may occur without change in body mass (weight or V). Even with slight changes in girth, remarkable extension is possible. Surface areas of ellipsoids become as large as necessary. Considering an ellipse as the longitudinal (i.e., mid-sagittal) section of an ellipsoid with length 2 a and greatest width 2 b, geometrically the surface area of the ellipsoid > twice the area bounded by the longitudinal ellipse, i.e., both sides of it. Since the area of the ellipse equals ( a b, we have Sellipsoid > 2 ( a b. Because V = 4/3 ( a b2, it follows that V = 2/3 b (2 ( a b) < 2/3 b S and S = 3V/ 2 b. Upon solving for b in the volume formula, we have (3V/4 ( a)1/2.

Sellipsoid > 3V/ 2 b = (3 ( V)1/2 (a1/2) = c (a1/2) [c is a constant 3 ( V.]

S can increase without limit by extending a and reducing b to its limit. From another explanation, the complicated Sellipsoid formula also yields the area (Equation (3)). Because V = 4/3 ( a b2, it follows that b = (3V/ 4 ( a )1/2. When the volume is constant and the eccentricity ( = (a2 - b2)1/2 / a ,

Sellipsoid = 2 ( (3V/ 4 ( a) + 2 ( a [3V/ 4 ( a]1/2 [a sin-1(a2-3V/4(a)1/2/a/(a2-3V/4(a)1/2]. (7)

When a is large, (a2- 3V/ 4 ( a)1/2 ( a. Therefore, when a becomes large, and 1/ a approaches 0, then

Sellipsoid ( 0 + (3 ( V)1/2 a1/2 [a sin-1 a /a / a ] = a1/2 ( (3 ( V)1/2 / 2 [because sin-1 = ( /2] . (8)

By choosing a large enough, Sellipsoid has no limit, as long as c = ( (3 ( V)1/2 / 2 is a positive constant. Approximation works for large values of a (Fig. 3).

[pic]

Fig. 3 Plot of Sellipsoid and a based on arcsin formula, and approximation(dashed line) based on the constant c (a )1/2.

3.3 Comparisons with Cylinders

An elongated, ellipsoidal form may become, by its elongation, similar in form and related biological functions to a cylindrical form. Wainwright [12] described 3-dimensional organisms as spheres, leaf-like forms, and cylinders. He considered cylinders to include ellipsoids, even mobile and highly specialized running, jumping, and flying organisms. He said little about the areas of his “cylinders”. His approximation of biradial, ellipsoidal forms worked out for his purposes. Many structures and organisms that are elongate true cylinders add increments, as the tips of antlers, twigs on trees, or cylindrical threads. Extensions are usually created by increments adding to mass, not by keeping V constant.

There is a worm-like similarity, but cylinders always have larger volumes than ellipsoids of comparable dimension; the cylinder sheaths or encloses the ellipsoid having the same radius b, axis 2a, and radius c. Compare the surface areas of a cylinder including its two ends (area approximately 2(b2) with that of an ellipsoid of similar dimension, and compare both with a sphere, where b = c, but a = length/2. Then a varies inversely with the value of b to a greater positive number, and b = c < a > 0.

Ssphere = 2 ( b 2 + 2 ( b 2

Sellipsoid = 2 ( b 2 + 2 ( a b (sin- (/ ()

Scylinder = 2 ( b 2 + 2 ( b (2a)

Then, Ssphere ( Sellipsoid < Scylinder .

When the volumes are set the same, then if b is given, a must be less in the cylinder.

Vsphere = 4/3 ( b 3 < Vcylinder [even when a = b].

Vellipsoid = 4/3 ( b 2 a < Vcylinder

Vsphere ( V ellipsoid ( Vcylinder [Because the ellipsoid can be enclosed in the cylinder.]

Vcylinder = ( b 2(2a) [Where 2a = height or length.]

If forms have similar dimension, the spheroid has less V (( weight or mass) than does the cylinder. If the volumes are set equal, and b is equal in both shapes, a ellipsoid > acylinder to bring the size up so that the V’s are equal. As b gets smaller during prolation, the ellipsoid obliterates most of the theorized volume, and Vellipsoid ( Vcylinder and Sellipsoid ( Scylinder (Fig. 4). Comparable changes in a for values of both ellipsoids or cylinders did not differ much. For the cylinder curve, the log radius to log S ratio is approximately -1, a direct inverse proportion. The slope of the log radius to log S for the distending ellipsoids is steeper, - 1.35 or even more, somewhat concave upwards.

[pic]

Fig. 4 Double-log plots of Sellipsoid and Scylinder .

If volumes are set equal, such as by size and metabolism constraints in nature, then a must be smaller for cylinders to bring the size down to Vcylinder = V ellipsoid . Description of cylinders [11] did not mathematically describe prolate structure.

Prolation explains the evolution of some slender mammals (even their sexual dimorphism) and the stepwise evolution of erythrocytes (from spherical cells to ellipsoids to biconcave disks, and both seem macroevolutionary. On the other hand, the evolution of birds’ eggs (from spheres) seems feasible from microevolutionary mechanisms. For the aforementioned macroevolutionary changes to have resulted by means of microevolutionary reduction, i. e., whether they can be explained by microevolution,.imposes and requires sequential changes within a chronological progression [14]. Prolation assuredly provides a foundation for subsequent prolation comprised of either a summation (= accumulation) of correlated small changes or pleiomorphic changes in shape. When controlled by the need for constant metabolism or weight (( volume), prolation can enhance fitness by altering slenderness, length, surface area, or all these together. The rate of change can be fast, slow, or uneven (Appendix 1 [15]).

Sexual dimorphism might be enhanced,

once an initial separation is made of two prolation phylogenies. The successful slender body, either male or female, may not suffer changes in proportions and specific body mass [16,17]. What led to the larger males in the first place (e.g., in ancestral mustelids) perhaps resulted from the general rule for most of carnivores that males are somewhat larger.

One relation has obvious significance. The weasel is approximately ellipsoidal, and the tunnel occupied, for much of its length, is cylindrical with a similar radius b throughout. The slender, ellipsoidal form allows the weasel to more easily twist, turn, and seize its prey, within the tight confines of a burrow with similar diameter. The deductions herein confirm that “sufficient prey” consisting of small, burrowing mammals, repays the cost of heat loss in weasels [7], and also that its metabolism is that of a thicker mammal. Log curves for length and surface area in two sexually dimorphic sexes seem parallel, and the sexes do not diverge in shape even though their lengths and areas do. The slenderization effected remarkable lengthening and expansion of body area, while V was changed little or even diminished from those of their rotund ancestors.

Briefly, I call attention to the remoteness of the interior from the surface in a spheroid. The chloroplast of an alga or hemoglobin of an erythrocyte lies deeply within those cells, and surface area is a factor in gas exchange. The depth an organ or organelle lies beneath the surface surely restricts the exchanges, even though the sphere’s minimal surface area is considered the chief hindrance. A mammalian erythrocyte, biconcave and disk-like, has two concavities approximately equal in area to two theoretical convexities, hence functionally resembles a prolate “sphere.” It is equivalent to an ellipsoid with a = c. Its volume is quite small in relation to surface. Thus, the erythrocyte in a typical mammal attains a relatively great surface area, neither by expanding its dimensions nor increasing V. The inner hemoglobin is situated near the surface. This likely explains extrusion of the mammal nucleus; the cell was no longer encumbered with the filler.

4 Conclusion

The concept of prolation, extending ellipsoidal length and surface, is an applicable rule describing and explaining many life forms and their structures, both in development and evolution.

References:

1. Kwok, L. S. The surface area of an ellipsoid revisited. J. Theoretical Biology 108, 1989, pp. 295-313.

2. Kleiber, M. The linear relation of the logarithm of body weight to logarithm of energy metabolism. Hilgardia 6, 1932, pp.315-353.

3. Zeuthen, E. Oxygen uptake as related to body size in organisms. Quart. Rev. Biology28, 1953, pp. 1-12.

4. Schmidt-Nielsen, K. Desert animals.Oxford Univ. Press, New York and Oxford. 1964.

5. Lasiewski, R. C. and W. R. Dawson. A re-examination of the relation between standard metabolic rate and body weight in birds. Condor 69, 1964, pp.13-23.

6. Long, C. A. Exponential relations of standard metabolic rates of birds and the weights of eggs. Wilson Bull., 1973, pp. 323-326.

7. Brown, J. H. and R. C. Lasiewski. Metabolism of weasels: the cost of being long and thin. Ecology 53, 1972, pp. 939-943.

8. Huxley, J. S. Constant differential growth ratios and their significance. Nature, 114, 1924, pp. 895-891.

9. Huxley, J. S. On the relation between egg-weights and body-weights in birds. Zoology. Journal of the Linnaean Society of London, 36, 1927, pp. 457-466.

10. Elias, H. and D. Schwartz Surface areas of cerebral cortex of mammals determined by stereological methods. Science, 166, 1969,111-113.

11. Kier, W. and K. K. Smith. Tongues, tentacles, and trunks: the biomechanics of muscular hydrostats. Zoology, J. Linnaean Soc.London. 83, 1985, pp.307-324.

12. Wainwright, S. A. Axis and circumference. Cambridge University Press, London. 1988.

13. Bowman, F. Introduction to elliptic functions with applications. Dover Publs, New York. 1961.

14.Bock, W. J. The synthetic explanation of macroevolutionary change—a reductionist approach. Bull.Carnegie Mus. Nat. Hist. 13, 1979, pp. 20-69.

15. Treuden, M. Personal correspondence. Dept. Mathematics, University of Wisconsin—Stevens Point. Stevens Point, Wisconsin 54481.

16. Campbell, D. R. Genetic correlation between biomass allocation to male and female functions in a natural population of Ipomopsis aggregata. Heredity. 79, 1997, pp. 606-614.

17. Moors, P. J. Sexual dimorphism in the body size of mustelids (Carnivora): the roles of food habits and breeding systems. Oikos 34, 1980, pp. 147-158.

Appendix 1.

During prolation, changes in major (a) and minor (b) semi-axis with one another with respect to time are inversely related. Since volume for a sphere Vsphere = 4/3 ( b 3 , it follows that 3V / 4 ( = a b2.

d/d t [3V / 4 ( ] = d/d t [a b 2] = (d a / d t ) b2 + a ( 2 b d b/d t ) where d/d t [3V / 4 ( ] = 0

(since V is a constant with rate of change 0). Dividing both sides by b and rewriting:

0 = b (d a / d t ) + 2 a (d b/d t) ( (1/ a ) d a / d t

= -2 (1/ b) d b/d t. (9)

The relative time rate of change of a is twice the opposite of the relative time rate of change of b. Since the length of the ellipsoid is 2 a, then the time duration is four times as much as for the change in b. If b is decreasing by 10 percent per time unit, so that d b/d t = - 0.1 b, a changes according to d a / d t = +0.2 a . This agrees with the logarithmic slopes in equation (6). The time rates of change cannot both be constant.

When b is decreasing by 1 cm per second, then a cannot be increasing at some constant rate. Instead, a would change at the rate of d a / d t = - 2 (a / b) ( - 1) = 2 a / b. If b is always decreasing at 1 centimeter per second [or perhaps 1 centimeter per 20,000 years in evolution], then a might be increasing at, say, 2 or 3 centimeters per second at one instance of time [15].

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