﻿ Three constraints: constant propagation, constant size, constant equivalence

# 25. The three constraints upon biological populations

We are now at last ready to state the constraints that act on biological organisms. There are three. They help give biological organisms the attributes that match the three dimensions of space. As do Galileo and Newton’s constraints, our three new biological constraints intersect with, and link: (a) the intrinsic to the extrinsic; and (b) the microscopic to the macroscopic. And again as do Galileo and Newton’s, they thus create a perpetual motion for biology. They will by this means help us to separate—as Newton, Galileo, and Joule did—those apparent opposites. We shall then use them to create a simplest possible case—and so a model—for biology.

The essence of our approach is to first create a gradient. The motivation is that Newton, but more particularly Lagrange, turned gravity into a potential, which also follows a gradient (Bynum et al, 1983). Temperature and pressure are other examples of physical properties that follow gradients and potentials. If two bodies each have unequal gravities or temperatures, then gradients are established … which are evidence of potentials. The gravities and the temperatures will seek to equalize and will establish a flow from high to low or hot to cold. If we symoblize our resulting gravity or temperature field by φ, then they each flow in the direction -φ where is the field’s gradient. Whatever has either raised or lowered the object or the temperature away from the mean has established that potential, and gravity or temperature then flow against that potential.

Another example is wind which always follows a pressure gradient and blows from high pressure to low. Whatever has created the point of either high or low pressure has set up a potential that eventually works against whatever created it. If we again describe the pressure field by φ, then the wind will blow in the direction -φ.

The advantage with potentials is that as soon as we have established an equilibrium state of any kind, we can then evaluate the potential in force at any point away from it. Its values along each of our selected coordinates will depend upon the degree of the resulting deformation away from our equilibrium state. We will then be able to gradually call the four theorems of the vector calculus into service and so embrace the entirety of biology and ecology.

A gradient links points or values to lines which are of one dimension. Lines then curve and form links to create areas, which are two-dimensional. Areas then form surfaces and so link to form volumes, which are three-dimensional … and all known space is three-dimensional with all other dimensions being accessible through those. Our three constraints will also be able to mediate all such transitions in dimensons for they have properties pertaining to them all. That is the power unleashed by Newton and Lagrange (Bynum et al, 1983; Bell, 1945, p. 372–3; Parker, 1998, pp. 314–316).

Figure 25: A Gaussian Surface

As in Figure 25, we declare the properties a biological population must have in order to be free from all constraints by approaching it via a Gaussian surface. We first imagine our entire Brassica rapa equilibrium age distribution population surrounded by a suitable Gaussian surface. Its total surface area encompasses all the energies absorbed and/or emitted over all the entities within the generation and over the entire generation length, T. Each individual point upon the surface is then a given time-point t. The thinner upper ray represents the energy emission of a single entity amongst the n at the time-point t1. The thicker lower ray represents a pencil of such rays from the entire population, with an energy flux of P at the time-point, t2, emitted by all the n. And … with our Gaussian surface now in place, we are ready to identify our constraints. We will eventually be able to state the perfect and perpetual biological motion that is free from all such constraints.

1. Constraint 1: Constant propagation

We first discuss the total energy, P, used by our total plant population of n entities per each second.

Our total energy flux, P, covers both:

1. the instantaneous total energy content, H, retained at any given time such as in the entities’ chemical bonds, and in gravitational, thermal etc energies; and
2. the energy flux at each moment, which is the specified throughput in, for example, the measured photosynthesis rate (or else and conversely, that the plants are consuming as they degrade the energy in seeds etc).

In honour of Alfred Russel Wallace, who published Charles Darwin’s first paper jointly with him, we now formally call this total energy flux per second, by an entire biological population, the “Wallace pressure”, P, of that population. Wallace pressure is therefore an extensive variable. It is measured in watts or joules per second. Thus the total energy flux passing through our Gaussian surface at any given t is P watts or joules per second.

Again assuming our Gaussian surface, then the first of the three constraints to which all biological organisms must be subject, and which contributes to the applicable biological constraint, is mathematically described in:

The constraint of constant propagation

0 = T0dP < P

This constraint says two important things:

1. It declares that all biological entities are organized centres of energy that must always do work in the formal scientific sense in that the Wallace pressure, P, of the population of which they are a part must always be greater than zero.
2. It implies that there exists an average value—a weighted average—for this Wallace pressure of P’ over the entire generation.

The latter of these proposals must be so because by the nature of this Gaussian surface; and by the nature of the population’s implied interaction with the environment; and as already declared in Maxim 1 of dissipation; then P can never be stationary. Since  • → 0, then the population’s mass of chemical components retained is always either increasing or decreasing, and so is always oscillating around a given and mean value. And since an equilibrium age distribution exists, then every accompanying increase in P at any given point must be matched by a converse decrease at some other point, and such that the sum of all such increments and decrements, over the entire surface, and so over that generation length, T, must be zero … or the distribution is not stable. And since the integral of dP over T must be zero, while P is always greater than zero, then a stable average value of P’ must exist. It is the weighted average flux through the whole surface, and for that generation length, T.

Our experiment measured Brassica rapa’s constraint of constant propagation at P’ = 5.013 watts or joules per second over its generation.

1. Constraint 2: Constant size

By the first law of thermodynamics, our second constraint follows immediately. The flux implied in the constraint of constant propagation must evidence itself materially. Since it is a flux, it must have a volume, which is a population count. Therefore: since the Wallace pressure P exists and always has a divergence, , which is the average individual Wallace pressure; then a distribution of all ps over the n creating the flux volume and to support that mean also exists. We can now express that average over the population as R joules per biomole where we have already defined a biomole as 1,000 biological entities, and so where R = 1,000 x . It is, therefore, an intensive variable. In honour of Darwin we title one joule per biomole a ‘darwin’.

Since the population of biological points at any time t is responsible for the energy flux generated at that time; and since the n entities are all members of an equilibrium age distribution population; then there exists:

The constraint of constant size

0 = T0dR < R

This constraint of constant size oversees “size” in two different ways. One is via the energy size or total count or volume of the points or entities in themselves; the other is via their net population size or flux density, which is therefore via the total energy flux generated by that sized a set of biological points.

The constraint of constant size tells us, through R, the quantity or the intensity of the energy attributable to each of the infinitesimally small individual points responsible for the net flux emitted by that population. But this R or average energy is also and automatically an index into the size, n, of the population responsible for any given flux, which is the number of relevant points. If P and R are known then n, the population size responsible for that flux is also known; and if R and n are known then P is known and so forth. So as P waxes and wanes over the given generation length, and over the surface, then so also do R and n wax and wane in order to maintain that flux, and that population, over that same surface and generation length. And again since all increments in R match all its decrements over that T; and further since R is always greater than zero; then a generational weighted average R’ exists as an expression of the average energy density per entity to match the earlier constraint of constant propagation, P’.

This is therefore the second of the three constraints for any given generation length, T. Its value for Brassica rapa, as measured in our experiment, is 54.012 darwins or joules per biomole.

1. Constraint 3: Constant equivalence

And then finally: by the first law of thermodynamics every transport of energy must both begin and end on some material source possessing mass. Thus for every unit of energy flux passing through our Gaussian surface that registers itself as P watts, there must exist a material substratum of specified mass at each of the time points, t, responsible for absorbing and emitting that energy, and of size R darwins. Our point sources of biological energy and flux must therefore possess a mass through which they can deploy their escaping and capturing tendencies. And since those point sources of entities are jointly doing work at P joules per second, and using R joules per biomole, then the necessary material substratum must be chemically and biochemically configured so it can evidence a work rate of W watts per kilogramme at each t, so it can satisfy the previous two constraints of propagation and size over the entire surface and generation. Each of the n point sources creating the flux, each of which has a determinable size of R darwins or joules per biomole, must therefore chemically arrange its given material components and undertake a given set of chemical transformations at each moment, so it can produce the said energy. And since that configuration and arrangement of material components must be repeated as generation after generation is repeated, the biological entities concerned—which are the points—must all at all times be equivalent at every relevant t over T. It is also an intensive variable. We thus have:

The constraint of constant equivalence

0 = T0dW < W

We once again find that there must be a value for W’, the generational average. The value for Brassica rapa is 164.720 watts per gram.

And these are the three constraints to which all biological populations must be subject. When combined, they flow invariantly … and exactly as does Galileo’s cart. They thus establish a perfect motion. We have also carefully defined them so that their integrals—the sum of all their net changes—are zero over their closed curves, and they therefore abide by the gradient theorem of line integrals.

This now means that just as Galileo and Newton can establish a perfect dynamical situation by saying “imagine a body moving in that direction with a velocity of ten metres per second”, so can we establish a perfect biological situation by saying “imagine a group of biological entities with a Wallace pressure of 5.013 watts, at 54.012 darwins, and of 164.720 watts per gram”. Just as did Galileo and Newton, we can now define the group so that any deviation from our ideal flux is proof of their biological inertia. But we must first establish what could cause departures from—and so also therefore returns to—these values.