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# 32. The four laws of biology

We can derive our laws of biology by bringing together: (a) our concept of biological point-energy centres working through a Gaussian surface; and (b) our line segment of molecules as in Figure 29. Our laws, which govern the intrinsic behaviour of biological populations, then arise naturally as follows …

There must always be point entities active on our line segment of molecules generating the necessary flux through our Gaussian surface, and also maintaining the components upon the line segment or both the surface and the line segment will degrade. The population would then be extinct. Therefore, there must always be at least one biological entity in existence:

- n ≥ 1.

- Law 1 (law of existence): Proviso 2

Biological populations must abide by the first law of thermodynamics, which is δQ = δW + dU. However, biological populations must also always do something more than simply dissipate heat. A population must constantly maintain its entities. Therefore, the work done can never be zero or it is extinct. So no matter how great or how small may be the changes in state, dU, the population’s entities are undertaking at any given time, they must all together do something more than simply dissipate heat, or the cycle of the generations terminates. And since the work done must always be greater than zero then we must rearrange thermodynamics’ first law equation to give:

- δW = (δQ - dU) > 0

- Law 1 (law of existence): Proviso 3

By Maxim 1 Proviso 2 of ecology, which is ∇ • M → 0, the divergence in the mass flux constantly oscillates around zero, meaning the mass flux itself oscillates around a given mean. But by Maxim 1 Proviso 1, which is ∫dm < 0, all a population’s entities will ultimately degrade. Since individual dissipation—which is the fading away of given trails of molecules—is inevitable for all, it must then be intrinsic to at least some in any given population that those selected members at some point strive to gather together all components lost, or yet to be lost, for otherwise the population will become extinct. Put simply … at least some entities must engage in the mechanical chemical work of restoring components to the line segment or tunnel of molecules.

We can express our requirement in vector terminology by saying that there will be positive divergences to which all members compulsarily contribute through being sources for dissipation as molecules and components leave all areas of the line segment. This is an inevitability imposed by the environment as expressed in ∫dm < 0. But if the population is to survive then a negative divergence or convergence must be intrinsic to at least some within the population to offset this. The convergence must draw molecules and components in to at least some areas of the line segment for at least some time periods t over T. And since biological organisms range in size from prokaryotes to dinosaurs and blue whales; and further since not all entities in any population are obliged to practice this convergence; then there is no set point at which any individual entity is obliged to cease its gathering of components before it individually dissipates. And since there is no set limit to any individual negative divergence or convergence then:

- m → ∞.

- Law 1 (law of existence): Proviso 4

And finally … by our Proviso 2 just derived of δW = (δQ - dU) > 0, which is the first law of thermodynamics, all energy reaching our Gaussian surface must have a material origin. Since all such points of origin from our line segment of molecules are subject to loss, then no matter what the scale of increases and/or decreases in mass and/or energy to which any individual entity may be subject; then since n is always at least equal to unity; and since the constraint of constant equivalence must also hold; then the average number of chemical components maintained per second over the entire population must always be greater than zero. Since this must be intrinsic to all then:

- m̅ > 0.

These four above requirements together give a first stipulation to the intrinsic nature of biological entities. They can be stated in words as:

The First Law of Biology: The Law of Existence

*n* ≥ 1; δ*W* = (δ*Q* - d*U*) > 0; *m* → ∞; *m̅* > 0

There is an entity such that it must always lift a weight; and such that it must, and by this means, at some time increase in its mass.

Since biological populations must do work in the thermodynamic sense; and since work is δW and an inexact differential; then the population must ensure that, generation after generation, its various work-performing entities remain equivalent. This can only be done by prescribing the work done and the energy to be expended by entities as they succeed to each other. This must again be done through mass, chemical components, and energy.

All biological entities—including does, bucks, crickets, bristlecone pines, and Brassica rapa—must behave in ways equivalent to “others of their kind”. They must also transmit that equivalence from one T to the succeeding T, and so therefore over all ts over every T. If not so, then the species either transforms or becomes extinct. Populations must therefore emit the same rays and pencils of rays of biological field lines or energy at the same points t over each T … and per each unit of mass and each volume element at each time point t. Only if such similarities constantly hold can any two such populations and their entities at equivalent time points be truly equivalent. We must somehow make this so … and also measurable.

Science was utterly transformed, and kick-started into the modern era, when Newton realized the absolute correspondence between the potential and the kinetic energies of bodies in orbit about a mass. If an orbiting body moves vertically in a gravitational field, then the total mechanical energy absorbed or surrendered will be equal to the sum of all the infinitesimal gravitational changes in state traversed—which is precisely the value of its kinetic energy should it instead choose to remain in orbit. By his second law of motion, F = ma, when a force acts upon a body it not only carries that body through the given distance, d, but also imparts the acceleration a. This intimate relation between force and acceleration, F and a, immediately links work and energy to gravitational fields. Gravitational fields are attractive and insistently draw towards themselves all bodies located within them.

Although dynamics was the first problem to be conquered by the modern method of scientific enquiry, it provided two different perspectives: (a) the close-up view; (b) the panoramic view. The close-up view records a body’s movements on a moment to moment and instant by instant basis. The panoramic view, by contrast, determines all the different paths by which a given object can move from its initial to its final position. This latter approach often indicates that no matter what specific path is being pursued, there are several others by which the system could have developed. Since the question then arises of why any specific path is chosen, then some way must be found to distinguish between them. The specific numerical quantity attached to any path—most usually an integral over time and expressed as joule-seconds—is then the system’s action on account of that path. Actions can then be coherently analysed. Biological populations must in this sense prescribe equivalent magnitudes of action for their equivalent successor members in the generation.

There are two different ways to determine action (Gondhalekar, 2001, p. 196). One is based on kinetic energy, the other on momentum. The kinetic energy approach links the system’s development to time, while the momentum approach links it to distance. The former action is twice the value of the average kinetic energy held on its path, multiplied by the time taken. The latter is the average momentum on the path multiplied by the path’s length. Since action describes the manner in which a physical system has changed, then it almost always has one of three possible values towards which it moves: its minimum, its maximum, or a constant one. If this were not so then it would not be possible to study science.

Far and away the most commonly selected of the paths available to a system is the one that takes it to a minimum—or else the one that selects the shortest possible path. Biological populations must establish an action that enables the survival of at least some to reproductive capability; and they must also specify the energy for that action. It must in their case be derived entirely from given chemical components.

At its core, kinetic energy—which Leibniz originally named vis viva or “living force”—refers to the action attributable to a system as it changes from the perspective of the time that it spends upon the path-of-change carrying it from one energy state to another (Iltis, 1971). Similarly, momentum—proposed and championed by Newton—refers simply to the action viewed from the perspective of the distance associated with that same path. The two were originally in opposition to each other and were seen as inconsistent and incompatible. But extensive investigations ultimately revealed that they are but different views of the same change … which is what gave the dynamic approach its power (Bynum et al, 1983, pp. 122, 151).

Biological entities maintain their metabolism and physiology through the doing of useful work upon a path, and through measurable action. This work is defined, in mechanics, as the energy transferred when a force causes a body to move through a distance. It is the sum of all the increments of distance through which a given body is taken, when acted upon by a force, multiplied by the magnitude of that force of application at each instant. Work is, therefore, the product of force and distance: W = Fd, and being a manifestation of energy it is measured in joules. There must therefore be a line or boundary upon or about which it does that work.

Given the above correspondence between kinetic, potential and all other forms of energy, then all work, and therefore all energy, can be discussed in terms of the acceleration imposable upon a given body located in a given gravitational field of specified intensity. By Newton’s second law of motion, although all bodies situated in a gravitational field can fall spontaneously, it takes energy to lift them. In the language of field lines and vectors, then gravitational field lines come in from infinity and end upon masses which are sinks … and there are no sources for gravity. For a body to therefore move against gravity with no discernible source of work is equivalent to it moving to infinity. When bodies move spontaneously under gravity they therefore fall ever closer to whatever larger body is reeling them in—so giving them less far to fall. This is universal for all bodies. No body spontaneously flies gravitationally away from another.

The early thermodynamicists sought to emulate this kind of logical precision, but had great difficulty finding and discussing paths. They nevertheless rapidly developed three laws to unify and simplify their subject. Temperature was fast recognized as the measure of a given system’s “heat power” or ability to do work through imposing thermal changes in state. A thermodynamic cycle—which biological entities exploit to complete their cycles—is simply a way of converting work into energy, and vice versa, using the internal energy of that system as a ‘bank’ or ‘library’ for the storage and retrieval of energy. That internal energy is the chemical bonds of the entities, which must somehow be transmitted down the generations.

All biological entities are biological sources in the vector sense. They all dissipate. Only some amongst them can be sinks of biological activity which is to be reproductive … but neither sinks nor sources can last indefinitely. They succeed constantly to each other over the generation length, T. The biological problem is then that the number of sources exceeds the number of sinks. The sinks must therefore somehow make up all deficiencies, or the population cannot survive. This requires mass and energy.

The French physicist Sadi Carnot showed, with his comprehensive analysis of heat engines, that although all bodies must spontaneously ‘fall’ down a temperature gradient, it takes energy or heat to ‘lift’ them and make them hotter. And as a body loses in its temperature by falling down-gradient, it ever more steadily approaches the temperature of whatever colder body is sucking the energy out of it. It thereby loses not only in temperature, but also in entropy. A given quantity of thermal energy is therefore the sum of all a body’s infinitesimal and incremental changes in state as it heats or cools multiplied by the temperature at which each such change occurs. Although Carnot did not recognize entropy, as had Newton before him with respect to gravitational changes in state, Carnot gave a method for calculating the relevant thermal changes in state (Cooper, 1968; Parker, 1998; Callen, 1985; Encyclopaedia Britannica, 2002).

Since temperature was soon understood to be a form of force, and so to be a potential, it was essential to compare magnitudes. Maxwell eventually pointed out that although the basics of three laws were already in place, a further—fourth—one was required, and that should logically have preceded the first. There could be no true science of heat without a clear concept of “the same heat power”. He thus proposed a law of thermal similarity which defined a state of mutual stable equilibrium. The law defines an equivalency in potentials between the energetic and entropic states of systems.

When a thermodynamic system changes its volume, then by Boyle’s law of PV = T, the pressure changes spontaneously and conversely to match the volume, and to keep that system in thermal equilibrium with its environment. This immediately changes both the joules per unit volume, which is the energy density, and the specific volume Joule sought to measure, which is the volume occupied per each unit of mass.

It is now very important to be aware of the etymological history and intent of temperature, so we do not use it fallaciously. Assume two systems, A and B. Let them have the same volume and temperature. Let one of them now undergo some change, but such that their volmes remain the same. Temperature now describes the states of these systems such that if their volumes are indeed the same, and they have nevertheless undergone some change such that Boyle’s law is still respected, then since V = T/P then one of them has (a) a higher entropy; and also (b) a higher temperature. The one with the higher entropy is also the one with the higher temperature. The molecules in that one are moving more quickly than in the other, and they are exploring a greater number of available states within that same volume. They are undertaking more mutual collisions per unit time.

Temperature now refers to the work that thermal systems do as they seek equilibrium both with each other and with the environment. If all the states available to a given system are all at equilibrium with the same environment, then since PV = T they must all have the same temperature. No matter what their differences in pressure and in volume, they are all equivalent to each other through T. These equivalent systems can now only differ in their entropy, which measures all those differences. These two—temperature and entropy—cannot be divorced. Entropy is then the method to explain the diversity amongst otherwise equivalent systems, with temperature being an indication of their tendency to do work due to those given differences in entropy within that environment. Entropy is now of course also a way to specify given similarities between states, no matter what their temperature, pressure, or other criteria.

There exists a function (called the entropy S) of the extensive parameters of any composite system, defined for all equilibrium states and having the following property: The values assumed by the extensive parameters in the absence of an internal constraint are those that maximize the entropy over the manifold of constrained equilibrium states. (Callen, 1985, p. 27)

We can now and therefore define equivalence in terms of this entropy property. Given these various correspondences, then different thermodynamic states are equivalent when they follow the Maxwell criteria which specifies when they are in mutual stable equilibrium. By this Maxwell law then when three systems A, B and C can be brought together such that they do no work on each other and induce no changes in state, then TA = TB = TC where T is the temperature in kelvins. So if TA is a thermometer, then it can now be safely transported from B to C, and to any other, and it will measure their equivalence through their heat power, which is their ability to do work relative to each other and through temperature—which also includes their entropy. If the thermometer does not change, then their heat powers are equivalent. That transformation capability is zero in this case, and the thermometer is now a formal statement of that equivalence … but always granted the various values for mass, number, volume, velocity, and entropy displayed by the molecules in each such system as in interaction. The etymology of ‘temperature’ is therefore the denotation of an equivalence linked with entropy under given conditions of energy.

Biology now requires its own formal statement of equivalence. We know from Maxim 2, Proviso 1, the maxim of number, that h̅ = H/n—i.e. that the average individual energy over the population is also the divergence in the energy flux.

The molecules on our line segment of molecules are also subject to divergences and convergences through Maxim 1, Proviso 2, ∇ • M → 0. But if the populations are to continue, then the various divergences and convergences must be equivalent, over the generations, in terms of mass, components, and energy. Since this divergence in the mass flux means an oscillation, then any chemical components lost at any t will be replaced before the same t occurs in some succeeding cycle … and so that those components can again be lost at that same rate. The divergence in the mass flux of the chemical components that are brought in to, and pulled away from, the line segment, or tunnel, of molecules must simply tend to zero at all points and as a property of the population. The successor entities or sets of masses and components must therefore be equivalent. They must have the same action. They must evidence the same ability to do work, which is to possess equivalent numbers and masses of components; configured in equivalent ways; and to both consume and emit equivalent energies per second at every t over a complete biological cycle, T. This must be so or successive cycles cannot be completed, and no stable equilibrium age distribution is possible.

We must now specify how these sets of entities can be equivalent. And … they can only be equivalent when their capturing and their escaping tendencies follow the criteria for work and entropy established by Maxwell. These specify the equivalence in their extrinsic relations with the environment. By those criteria then since ∇ • H is the divergence in H, and so is h̅ = H/n, then any two sets of divergences at any t1 and t2 are equal when the average individual energies of those two sets of entities are the same which is:

- ∇ • H1 = ∇ • H2

- Law 2 (law of equivalence): Proviso 2

The above relation declares that the divergence of the energy, and the pencils of rays dispensing that energy, and the number of entities pertinent to a first population at t1, over its Gaussian surface, is equal to the divergence of a second over its own surface at a given t2. Thus the two populations, along with the number of their composing entities, are equivalent at those ts. By the same token, the same therefore holds at t2 and t3:

- ∇ • H2 = ∇ • H3

- Law 2 (law of equivalence): Proviso 3

But then and again … by the above Maxwell canons of energy and entropy if ∇ • H1 = ∇ • H2 while ∇ • H2 = ∇ • H3 we then immediately have:

- ∇ • H1 = ∇ • H3

… and therefore, all such distributions of entities along with their masses of chemical components held per second are equivalent at all such ts. And if such equivalencies hold for every t over T, then there is a full equivalency of those populations, for they are always and at every moment equivalently configured in equivalent numbers, and at equivalent masses and energies.

Work is again an inexact differential. It is also the statement of a path or process. And since it is a path or action it is necessary to specify the mechanical work done upon that path. By the first law of thermodynamics, this is the energy handed on through succeeding generations … which is again a path and δW, for that is the work done with the given energies, entropies, and changes in state. This H that produces that work done is also the chemical bond energies inhering in the configurations of the given chemical components, and its divergence is the sum of the capturing and escaping tendencies, but taken over the entirety of that population of entities. It is again the ∇ • H = h̅ x n = δW of Maxim 2 of ecology, the maxim of number. In other words, no matter what may be the individual work done by, or energies possessed by, any given entity, it is intrinsic to the population at large that the total work done by those entities is equivalent from one generation to the next.

We therefore now have the complete set of criteria we need to establish a biological equivalence. Our second law of biology can now be formally stated in words as:

The Second Law of Biology: The Law of Equivalence

[(δ*W*_{1} = δ*W*_{2}) ∧ (δ*W*_{2} = δ*W*_{3})] ⇒ (δ*W*_{1} = δ*W*_{3})

If a first entity can follow a path such that Law 1 is satisfied; and if a second entity can follow the same path to the same effect; then the first and second entities are equivalent.

Equivalence amongst the members of any given species is all very well, but biological entities unfortunately vary all too widely. As well as marked differences amongst individuals, worker ants and bees, for example, differ greatly from their queens and do not reproduce. This potential for diversity must be recognized and quantified or this model is utterly fruitless.

And … although it is not our specific intent to create a one-to-one correspondence between biology and thermodynamics, it is nevertheless worthwhile recollecting that volume fulfills three functions in that discipline. Thermodynamics is fully general and governs all systems under energy. Volume governs the degree to which a system can expand as it releases pressure and does work through its Helmholtz energy. We can already specify a system’s size by counting; and it looks very much as if joules per unit volume is covered by the divergence in the energy flux. This leaves only the joules per unit mass which is the specific energy.

In so far as thermodynamic systems can increase in their volume, biological entities—whether seeds or foetuses—also possess a potential for expansion. At the beginning of our Brassica rapa experiment, our seeds certainly had a potential. They were located at a definite position in their distribution. They were T away from T and so at t = 0. As seeds, they were therefore suitably redolent with possibilities and potentials for diversity on account of their mass, their energy, and their position at the beginning of the cycle. That potential for diversity was coiled up in their DNA and in the reactions they could undertake. These potentials were actualized as the plants grew, fruited, and seeded. Since B. rapa’s generation length is T = 36 days, then those seed potentials resulted from their distance from the mature plants they would eventually become. This distance and potential, as well as the energy it represents, must be quantified.

It is surprisingly trivial to determine the magnitude of the potential energy a Brassica rapa seed possesses. Table 1 informs us that when in its initial condition, each B. rapa seed contains 1.017 joules of energy. When it is in its final condition, as a ripened plant, it contains 15.558 joules. Lagrange taught that a body’s potential energy is its stored capacity to do work due to its position with respect to some reference which is some state or place or condition to which it will be transported due to the energy it again stores simply on account of that position (Bynum et al, 1983, p. 335). This potential energy is a measure of the work it will do, or the energy it will evolve, due to its transition to the state or position currently possessed by its reference body. Therefore, a seed’s potential energy is simply the difference between its initial and final states, meaning that a B. rapa seed’s potential energy is immediately A = 14.541 joules, a value we can assign it simply by examining Table 1.

We now assign those joules to Brassica rapa’s DNA as a “reproductive potential”. As a seed then grows and moves from t = 0 towards T, it actualizes its potential and—as they all do—the potential will tend to zero: A → 0. This is the exact relationship between potential and kinetic energy under gravitation, the former being exchanged for the latter with the latter increasing as the former tends to zero. In similar fashion, when the seed is fully grown the potential will be exhausted, and we will have A = 0 … just as when a body has fallen through a given height, we have h = 0. The DNA will then have absorbed all possible potential energy through deployment of its capturing tendency against the escaping one imposed by the environment. This reproductive potential is the biological equivalent of the Helmholtz energy, and B. rapa will have expressed those A joules of its potential in the transformations in state that help it preserve the species by tending towards the adult plant.

This reproductive potential is clearly an important intrinsic feature of all biological entities. And since it tends to zero as they grow and develop, we therefore state a proviso to describe that transition from maximim to minimum in potential energy over the generation length, T:

- A → 0.

- Law 3 (law of diversity): Proviso 2

We now have to quantify the magnitude of the diversity this potential energy can achieve as it is exploited and tends to zero. There is a biological expansion in states, and we must reckon all those states.

We learned, from our thought experiment with a reservoir of heat energy on Black’s latent heat, that a temperature difference is a distance. It is therefore also a potential. The heat absorbed by Black’s first pail per unit of its mass, and as the ice melted, could be measured by the second pail’s rise in temperature, and also per unit mass. When the reservoir was empty its potential was exhausted; all transitions and phase changes had been completed; and the initial predicted temperature, F, had become the actual measured temperature, M, so that both A → 0 and F → M. The changes in chemical configuration the system undertook affected the final temperature distance measured. They were a part of the potential.

Both equivalence and diversity exist. And just as a thermodynamic system expands from a minimum volume to a maximum and returns, then so also does a biological cycle. We can therefore reckon that minimum and maximum as follows:

- The required set. This reflects a thermodynamic system’s minimum volume, or its maximum pressure–lowest entropy condition. We also know by Law 1, Proviso 2, that biological entities must constantly do work: δW = (δQ - dU) > 0. That work must also show itself in a measurable energy flux, and through every unit of biological mass. This must hold for all n entities in the population, and at each and every t over T. Since it is a minimum condition characteristic of this population, then this set is also its minimum and obligatory energy activity. This required set thus represents the least permissible energy configuration exhibited by whatever chemical components are available to, and are amassed by, that population.
- The allowed set. This reflects a thermodynamic system’s maximum volume, or its minimum pressure–highest entropy condition. In the transition to this state, the system does work at every point, and in different amounts. Thus in addition to the above minimum and required set of activities and configurations, over all the n entities at every t over T, there must be one or more amongst the n entities, at any given t, as are able to undertake an additional set of activities and configurations—over and above the above required set—that allow those entities to differentiate … and eventually to reproduce the population.

The transformations we now allocate to the allowed set are those that lead to population variety, to development, to sexual dimorphism, and so forth. They are transformations that require work: the disbursal of increased energies and so usage of the reproductive potential, A. These transformations will be reflected in an increase in the energy density within the population over and above any required minimum. That increased energy density will show itself in either joules per unit mass, or else joules per unit volume, or some combination of both.

The diversity that biological populations can exhibit with respect to both the required and the allowed sets must now be made quantifiable. This diversity will be of two kinds: (a) between whatever n entities exist at any t; and (b) in any given entity at different ts, over T, as it grows, develops, and undertakes any given biological cycle.

Any instantaneous values for mass and energy held by the biological entities existing at any time t result from the interplay of the capturing and the escaping tendencies that the nt entities hold at that t. And if there are nt entities then by Law 2 of biology, which is the law of equivalence; as also by the equilibrium age distribution population; there is a given repetition of the moles of components held by each of the n at that t. And since all biological entities are composed of given moles of components, then the joules held per mole is also an expression of their energy density. We therefore now have three interconnected factors:

- joules per unit volume;
- joules per unit mass; and
- joules per mole of components.

Since these are all expressions of energy, then we must ensure that they are all accounted for.

The values ascribed to the nt entities existing at any t will accurately describe the population’s specific state at that time. However, since this is an equilibrium age distribution population, those same values also accurately describe the path that the population must remain on in order to arrive at the next stationary state, and to complete its cycle of generation length, T. Each stationary state therefore defines the path for each of those n sets of specified moles of components. Since energy is constantly being transacted through the capturing and escaping tendencies, then any such path demands that the numbers of entities and components on that path, along with their rates and durations, be specified.

A given population of biological entities can only continue to exist because the masses and the energies of the specified moles of components held by each entity at any given moment, t, are handed on to the next moment across the entire time span, T. Thus to state any given value for a population is also to state that population’s rate or path for all those n entities, and thus for their components. Therefore: for all possible variables X in such an equilibrium distribution population, there immediately exists a Y where if X is the stationary state or value, then Y is the path value that declares the rate in X per the given time interval, and where X = Y with their units of measure being the same. And for every complementary pair of variables X and Y, there immediately exists an x̅ and a y̅ which are the averages over the population of size n, and however those values are calculated, and whatever may be the accompanying distributions in x and y. Furthermore, each x and each y that contributes to those sums and means depends only on its own values at the earlier t, for the stationary state is also the path.

More specifically:

- If P is the flux in watts or joules per second then there also exists an H which is simply the stationary joules held by the entity, and that results in that flux; and there are also the average individual values p̅ and h̅ and their relevant distributions, which is joules per unit volume—i.e. joules per entity—over the population. More formally, P = dH/dt.
- And if M is the entities’ flow or flux of its mass of chemical components that are configured and worked in kilogrammes per second, then there exists a stationary state of U kilogrammes, which is simply the mass in kilogrammes of the number of chemical components held by the entities for the associated stationary state, and where M = U. There also exists an m̅ and a u̅, again along with their relevant distributions, and as population averages per all those components, which are then kilogrammes per entity or the mass density. More formally, M = dU/dt.
- And if Q is the number of entities maintained in moles per second, then there exists a stationary state of N moles being maintained at each t and over which Q passes as an ongoing biological charge, and where the number of moles of chemical components held at each t is q̅, which is moles of components per entity, or the components density. This will be more closely examined shortly. More formally, Q = dN/dt.

It is clearly important to record these different variables. Biological entities undertake specified chemical interactions in order both to maintain themselves and to create others of their kind. No biological entity can exist at any point in any biological cycle without being the product of some entity that earlier occupied that same temporal location in some prior cycle. The ensuing interactions are manifestations of their Gibbs energies: their abilities to acquire and then do work on, and thus transform and reconfigure, their mass of biological matter which is a given amount of substance in moles. This is in its turn a property and a demonstration of their DNA. The energy for the total work that can be done by that DNA in its biochemical aspect in its turn bequeathes a given quantity of Helmholtz energy.

When a given biological population follows the programming apparently encoded in its entities’ DNA, it is entirely permissible for the average individual mass of chemical components held over the population, m̅, to change while their configuration remains the same. There is an increase in mass, but at that same energy density. It is important to note that this value for mass can also rise without the number of moles changing simply by substituting one chemical element for another. We also note that this increase is a negative divergence or convergence of the mass flux.

And then in contradistinction to the above: it is entirely permissible for proteins to fold or unfold, at the command of DNA, in which case a given population’s mass of chemical components held remains the same, but the energy density or configuration or also their chemical type changes. Therefore, the population’s stock of energy in joules, H, depends upon (a) its numbers, n, which is the divergence in the energy flux; (b) its average individual mass of chemical components held per second, m̅, which is the divergence in the mass flux; (c) its numbers and types of components, q̅; (d) its energy density or its given state or configuration of those components. It is therefore critical, as we did in our Brassica rapa experiment, to record that energy density. We also note that this is exactly the third function of thermodynamic volume.

If a biological population changes its configuration, then it also changes its Gibbs energy—which is its potential to engage in chemical reactions—per unit of its mass. We must clearly and again quantify this for it is a change in energy density. We therefore define the “visible presence”, V, of a population as V = M/P. It is the mass flux divided by the energy flux or the divergence of the former divided by that of the latter. It is measured as kilogrammes per joule and states the kilogrammes of biological matter that a biological population and its entities must build per every joule of energy they seek to pass through a Gaussian surface in P joules per second. Visible pressence is a measure of the Gibbs energy or configuration potential per unit mass of chemical components retained. It can help to differentiate the mechanical from the nonmechanical varieties of energy because the mechanical will simply increase the mass without affecting the energy density, while the nonmechanical immediately affects the unit energy density. Visible presence helps to distinguish joules per entity from joules per unit mass from joules per mole from joules per reaction from joules per unit chemical configuration.

Visible presence means that if a population’s energy flux increases because its chemical components are changed or reconfigured, then since some potential chemical reactions are actualized, the visible presence moves oppositely and decreases. And if the DNA is reconfigured so that the net energy flux decreases, at the same value for mass of chemical components, then the visible presence increases because the potential for those reactions to be repeated increases. And similarly: if a population holds its energy values constant while its mass of components decreases, then the visible presence decreases because the components must be switching to a more intensive suite of reactions; and if it again holds energy constant, but can only do so by increasing in its mass of components, then the visible presence increases because some reactions cease to be actual and therefore become potential. We now have a measure for every permutation.

With visible presence, V, now in hand, we can now use the techniques we developed based on Black’s work with latent heat to determine the role played, in biology, by chemical configurations. These establish the population’s energy density.

We first select a reference entity, Efinal, from our population. We need one to establish—in Lagrangian style—a potential. We select Efinal where the measured energy density is at a maximum, and therefore where the potential for further transformations is at a minimum. Since all necessary transformations and reconfigurations have here been completed, the work being done per unit mass of chemical components held per second by Efinal is at its maximum. That is to say, the nonmechanical work being done is at a maximum. The Gibbs energy per unit mass must therefore be at its minimum, for it measures configuration potential … and all potentials have been used. But not only is the Gibbs energy per unit mass at its minimum, its rate of change must be zero at this point. It is about to turn positive as the work done per unit mass begins to decline from this maximum. And … if the maximum possible nonmechanical chemical work is being done at this t, then the minimum amount of mechanical chemical work per unit mass must also be being done … and its respective rate of change must similarly be zero. In this case, however, it is about to turn negative. With this information in hand we can measure p̅final, m̅final, and Vfinal, these being the average individual Wallace pressure, average individual mass, and visible presence for our specimen Efinal at that t.

Now that we have a reference point we can measure distances and quantify potentials. We next select a second entity, Einitial, from wherever energy density is instead at its minimum. The visible presence is then at its maximum. And if V is at its maximum then so also is the Gibbs energy per unit of mass of components held per second at its maximum, meaning that the quantity of transformations that Einitial can in the future undertake is at its maximum, for the Gibbs energy measures chemical potential, and this is the location of maximum potential. And if the Gibbs energy per unit mass of components held per second is at its maximum, then the nonmechanical chemical work also per unit mass of the same components held per second must be at its minimum, while the mechanical chemical work per unit mass must similarly be at its maximum. The rates of change of both are zero and about to become negative and positive respectively.

Juxtaposing our initial and final entities now allows us to analyse the population’s polar opposites in its states and configurations, and in respect of the transformations the entities can undertake at each t over T. Einitial is ready to undertake all possible transformations and changes in configuration because its visible presence is at a maximum; whereas Efinal has already completed all such configurations, for its visible presence is at a minimum. We therefore measure p̅initial, Vinitial and m̅initial.

Since this is an equilibrium distribution population, then every stationary state is also the declaration of a path. Since M = dU/dt, then U is the state, while M is the path; and similarly for P = dH/dt and Q = dN/dt. Therefore, Efinal is already in the condition that Einitial is approaching. Efinal already possesses attributes that our Einitial currently only holds in potential, and that are still encoded in its DNA. In order to become Efinal, Einitial needs only to tend from t to T. All differences between the two at each t over T are both the causes of, and are caused by, their differences in mass and energy, and which complementarily approach each other over T.

We must now find a systematic method for separating mechanical from nonmechanical chemical energy: i.e. those transformations caused (a) simply by changes in the mass flux, M; and (b) by changes in energy density and chemical configuration, V. Both cause net changes in energy, but only the nonmechanical variety cause changes in how chemical components are configured, which is again in energy density per unit mass and so in visible presence, V.

We can measure the differences we need with the “Franklin factor”, K. There are far more appropriate and elegant methods for measuring it available using unit vector normals, but we do not yet have that equipment. Our vector model is not yet properly developed. We therefore temporarily quantify the Franklin factor as the proportionate change in energy density, or configuration energy, that an organism can undertake, through nonmechanical chemical work, as it directs energy at its chemical bonds, and undertakes those transformations in state that result in a reconfiguration of its chemical bonds such that it approaches the final state. Only the numbers will change when we have fully developed this model and can use unit vector normals. But although the numbers may change, the state it describes will not. All the numbers will be clearly related to them through yet other numbers.

We can determine the Franklin factor at any t, by dividing the visible presence at that t by Vfinal and then subtracting unity: i.e. K = (Vt/Vfinal) - 1; or else by setting K = (Vt - Vfinal)/Vfinal). The Franklin factor therefore quantifies the distance between any given configuration and the final configuration ultimately adopted by the entity. These distances are changes in the Gibbs energy per unit mass, which are changes in configurations and in chemical bond structure. The Franklin factor is a measure of the relative energy density held by any given state, and therefore of its transformation potential at every t over T.

Stage |
Avg. indiv. mass, m_ |
Franklin factor, K |
Potential of energy |
Franklin energy, F |
---|---|---|---|---|

Seed |
1.171 x 10-3 grams/sec |
0.829 |
9.710 x 10-4 grams/sec |
2.142 x 10-3 grams/sec |

Leaf |
4.977 x 10-2 grams/sec |
-0.052 |
-2.567 x 10-3 grams/sec |
4.720 x 10-2 grams/sec |

Flowering |
6.503 x 10-2 grams/sec |
0.367 |
2.384 x 10-2 grams/sec |
8.888 x 10-2 grams/sec |

Fruit |
8.717 x 10-2 grams/sec |
0.074 |
6.415 x 10-4 grams/sec |
9.359 x 10-2 grams/sec |

Dry seed |
1.049 x 10-1 grams/sec |
0 |
0.000 grams/sec |
1.049 x 10-1 grams/sec |

Seed |
1.171 x 10-3 grams/sec |
0.829 |
9.710 x 10-4 grams/sec |
2.142 x 10-3 grams/sec |

As in Table 3, the Franklin factor for a Brassica rapa seed is 0.829. This means that our n B. rapa seeds are each at this point, and at this distance from the mature plant, capable of undertaking future changes in state such that their chemical bonds will be reconfigured so they will—in the future—contain nearly 83% as much energy per unit chemical components and per second as they do now. This 83% is a measure of what they can do with the reproductive potential, A, also allocated to them in this initial state. The Franklin factor indicates the transfomations that are allowable to—and so that are a potential for—these seeds, given their current state and reproductive potential. Alternatively … if we converted all of that current potential energy to actual energy; and if gave all that actual energy to those B. rapa seeds, at this given moment; then they could be more massive as seeds … assuming they maintained the identical energy density. They could maintain 83% more in moles or amount of substance of chemical components as they each currently do. This mass supplement potential is therefore just as accurate a measure of the current biological potential accorded to them as is any comparison through their energy density. By the first law of thermodynamics these energy values are the same.

Our allowed set of activities can now be defined as the quantity of mass that the population could potentially use to do work. The potential energy available to our Brassica rapa population in its seed state can be expressed—with no loss of either scientific rigour or accuracy—as the additional mass that the population could maintain, at its present visible presence V, if each of its entities were to take on that mass, while all the time remaining at the same energy density, or chemical configuration, they currently possess as seeds. All potential changes in nonmechanical chemical energy are equivalent to specified changes in mechanical chemical energy, for again and by the first law of thermodynamics all energy is equivalent.

The Brassica rapa seed’s actual average individual mass maintained is currently 1.171 x 10-6 kgs per sec. This is its measured mass and encompasses its required set. We can now apply our Franklin factor to the seed and say that each entity’s “Franklin energy”, F, once applied, is equivalent to a total proposed and potential mass of 2.142 x 10-6 kgs per sec. That is the total mass each B. rapa entity could sustain if it were now to be granted the same amount of energy per unit mass as it will eventually have once it is an adult plant. In other words, the nonmechanical chemical energy it will in the future absorb is being expressed as a potential. It is being expressed in terms of a specified quantity of mechanical chemical energy, and so in chemical bond energy over given components. The two are ever equivalent.

We can now easily handle diversity. We can say that each Einitial or B. rapa seed has a Franklin energy or potential energy of F = 2.142 x 10-6 kgs per second simply because of its distance from the adult plant. This Franklin energy—which is also stated in the A joules difference between the initial and final states—incorporates B. rapa’s allowed set of potential chemical bonds and chemical components, and therefore its potential energy.

We now do as we did in our earlier thought experiment. We remove one joule from our reservoir and direct it at our initial entity which is a representative Brassica rapa seed. There are now three possibilities.

- The entity can increase its mechanical chemical energy. This is equivalent to Whypothetical in our Black thought experiment. If all seeds do this then the population’s mass flux, M, increases while its energy density remains invariant. The number of chemical components held per second increases as the visible presence remains unchanged, and each seed maintains the same chemical configuration while increasing in its mass. We just have bigger but identical seeds as an expression of the distance between these two states. There is no other change.
- The entity can instead diversify through utilizing nonmechanical chemical energy. This is equivalent to a phase change. The energy density now changes while the mass and number of components remains the same. All chemical components are reconfigured into a new set of bonds. There is widespread differentiation over the population, with no changes in mass. The plants in this case mature, but in bonsai-like fashion. They remain at the same mass as they originally were as seeds and go through their entire cycle as energy is directed at them, with no changes in mass.
- The entity and its population can respond with a combination of both the mechanical and the nonmechanical forms of chemical energy.

If Brassica rapa responds through either (B) or (C) above, then its Franklin factor decreases, and its Franklin energy—i.e. the additional mass it could support—necessarily and also decreases as it transforms. It also has less potential for future reconfigurations. The amount of diversity introduced, and that can be further introduced, is precisely measured by the decrease in Franklin energy, which is a decrease in potential for further changes. And since the potential for further increases in energy density declines as such changes are actualized, the projected final value for attainable mass over the population declines, and the Franklin energy, F, approaches the actual and measured mass for that population:

- F → M.

This F → M simply means that all potential configurations in chemical components available to a given sample of biological matter are gradually being exhausted.

We have now successfully replicated the Black experiment. For every joule of energy we input that elicits a change in visible presence, then the Franklin energy declines, and F—the hypothetical mass that could be attained—will ever more steadily approach M, the actual mass observed and as the population diversifies. This is a statement of potentialities and actualities. The greater the variations and diversity, the greater the initial difference between F and M over the population.

If a given biological organism has completed all configurations then its Franklin factor will be given by Kfinal = (Vfinal/Vfinal) - 1 = 0. This zero value for the Franklin factor means that the given population’s actual mass flux, M, needs in no way to be augmented to determine that population’s complete energy capability with respect to its mass of chemical components retained, for all possible potentials are now exhausted. Since no further reconfigurations can occur, the given organisms’ total energy has precisely the expected value for its mass, and F = M. And when F = M over the population, we can expect no further transformations, and all potentialities have been actualized.

By thus choosing locations in the age equilibrium distribution population where the visible presence ranges from its maximum to its minimum value, we immediately embrace all possible diversity. We have now included every possible variation across the entire population, and at every t across T. And since we now have both A → 0 and F → M, we have suitably quantified all possible variations and configurations, and so all possible biological diversity.

We have also found the biological analogue for thermodynamic volume. It is visible presence, V. It establishes the energy density and so the quantity of mass retained within the population per each joule and so by determining how that energy thickens and thins and in the manner of Joule. We are also now ready for the Liouville theorem and the Laplace operator, which we shall meet shortly. Biological populations at last have a potential to match any in any other science. We can therefore state:

The Third Law of Biology: The Law of Diversity

* A* → 0; *F* → *M*

The sum of all the paths that satisfy Law 2 constitutes the allowed set for the entity and its equivalents; while that which permits them to satisfy Law 1 constitutes the required set.

The third law of biology, the law of diversity, has seen an Einitial become an Efinal. But if the biological cycle is to be repeated, then any and all transformations Einitial has undertaken must be reversible. As with an oscillating spring moving between the potential and the kinetic energies, once a potential has been actualized and so tended to zero, the potential to once again undertake the said set of chemical reactions must be restored or generations cannot succeed to each other.

If the biological cycle is indeed to be repeated, then Efinal must express a set of countervailing potentials for transformations. Those must be in its DNA and must also lead it back to the same values originally possessed by Einitial. This transition between Einitial and Efinal and back again is a property of the equilibrium age distribution population. Whatever properties each of Einitial and Efinal hold they are the very properties and potentials that allow the complement to exist with its complement energy density per unit mass.

Using a Gaussian surface to approach a biological cycle means that only a small number of variables need be kept under consideration. Amongst these are the mass flux of M kilogrammes of chemical components held per second; the energy flux or Wallace pressure of P joules per second; the Helmholtz energy or reproductive potential of A joules; and the visible presence or energy density V kilogrammes per joule which incorporates the Gibbs energy per unit mass. These together establish the generation length T for the given equilibrium age distribution population, its resources, and its components.

The reproductive potential of the above third law of biology, which is the law of diversity, decreases to zero as the cycle proceeds: A → 0. The visible presence, V, also decreases to its minimum. Therefore, the visible presence must increase back to its previous maximum values so the original reproductive potential, A, is reclaimed. All relevant potentials—including the Gibbs and Helmholtz energies—are a function of the distance from t = 0 and are reclaimed exactly as the entity moves over t from tfinal to T and t = 0.

Once a given Ex or Entity X has developed and exhausted its Franklin factor, it can reproduce and create a successor Ey. By Law 2 of biology, the law of equivalence, this Ey will be equivalent to Ex and will enjoy the same mass, energy, and potentials at the same time points all across T. It will therefore behave in the same way and eventually produce an Ez which will once again be a member of the same equilibrium age distribution population and undertake the same activities. Each such successor will have the same measurable reproductive potentials and Gibbs energies, and impinge on the same Gaussian surface in the same way, with the same visible presence or Gibbs energies per unit mass and maximum and minimum values and rates of change. And if the reproductive potential, or Helmholtz energy, is indeed being reclaimed then its rate of change is positive so giving:

- dA/dt > 0

- Law 4 (law of reproduction): Proviso 2

If the reproductive potential is being reclaimed and the visible presence is increasing so that the Gibbs energy or potential for biochemical reactions per unit mass is also increasing, then a negative divergence is in force. And if a negative divergence or convergence is in force then the average individual mass of chemical components held per second by each entity over the given population must be decreasing. Negative divergence means that the mass flux is “closing in” towards a given point. Materials are condensing in and occupying smaller and smaller unit volumes and heading to a sink. And if materials are condensing in point-like fashion to a sink, then since the divergence is given by M/n, the average individual number of chemical components held per second over the population, which is m̅, is decreasing and its rate of change is negative. Therefore:

- dm̅/dt ≤ 0

- Law 4 (law of reproduction): Proviso 3

Since the mass flux is M = nm̅ while the divergence is m̅ = M/n the former is extensive, the latter intensive. They behave very differently and it is quite possibe for the mass flux to decrease while the divergence increases and conversely. Their behaviours are entirely dependent on n.

If there is a negative convergence then the number of chemical components held per entity per second is declining, and so also is the average individual energy emitted per second over the population, p̅. The energy flux must also have a negative divergence. But since, by the first law of thermodynamics, every movement of energy must have a material locus, then this negative divergence must evidence itself in some discrete and discernible progeny. There must be successor point-entities whose energies are absorbed and emitted and which also respect the Gaussian surface in both its integral and differentiable forms, and that are in their turn composed of mass and energy in repeatable reconfigurations.

Since m̅ = M/n there are three ways the average individual mass can decline:

- through a net outwards mass flux as the far more massive progenitors dissipate, leaving only the progeny; and/or
- some far smaller progeny are increasingly appearing to join the progenitors and as t tends from tfinal to T and so back to t = 0; or
- a combination of both.

Since by Maxim 1, Proviso 2, we have ∇ • M → 0, then all divergences must be restored. But the divergences of mass and energy depend upon their respective fluxes. Since all divergences depend upon n, then the number densities at each t over T must also be restored. Therefore, any and all entities lost must be replaced. And if an entity lost must be replaced, then population numbers must at some time be subject to increase. Therefore for at least some ts over T we must have:

- dn/dt ≥ 0

Our three latest provisos of dA/dt > 0, dm̅/dt ≤ 0, and dn/dt ≥ 0 can now be brought together to describe the last of our necessary and intrinsic behaviours for biological populations and their entities. As an intrinsic tendency or behaviour it is characteristic of the entire population, but need not be realized by any given member. We can now declare the fourth and last of our proposed laws of biology:

The Fourth Law of Biology: The Law of Reproduction

[(*dm̅*/*dt* ≤ 0) ∧ (*m̅* > 0)] ⇒ [(*dn/dt* ≥ 0) ∧ (*dA*/*dt* > 0)]

In the allowed set is at least one path such that mass is surrendered, and such that a further entity possessing the required set, and satisfying these four laws results.