29. A molecular basis for natural selection

We now have the material we need to propose a biological system equal to Newton’s. We have our perpetual motion; we understand about axes; we have distinguished between the mechanical and the nonmechanical; and we are also completely ready for any vectors and forces we might find. And further since we also now have a set of interdependent intensive and extensive variables, we can take full advantage of the properties of homogeneous polynomials and functions, for these are all homogeneous and of degree one. We can therefore construct a suitable Euler equation in which a set of partial derivatives defines and describes the extensive variables to which they are attached; and we can also declare a Gibbs-Duhem equation to match it. We shall soon conquer natural selection and competition using our differential equations.

But if we want such an equation; and if we also want a quantum biology; then we must chase down some molecules. Our Brassica rapa experiment procured for us an equilibrium age distribution population whose properties and behaviours—particularly on the molecular scale—we must now clarify. We must firmly link the biological entities that create our equilibrium age distribution population to the molecules that compose them. Those, through their DNA, drive their behaviour.

In their paper Individual Energetics and the Equilibrium Demography of Structured Populations, Gurney et al cogently review and discuss the effects that the extrinsic demographic processes populations must contend with, because of the environment, have upon the intrinsic processes they must use to maintain themselves, and that are therefore imposed upon their members (Gurney et al, 1996) .

A word of caution is, however, necessary. As is consistent throughout the literature, the poor grasp that biologists and ecologists have on the fundamentals of energy mean that they generally fail to recognize the important distinction between its mechanical and nonmechanical forms. Thus neither Gurney et al, nor Brown et al, to whom we shall refer shortly, properly distinguish between growth and development—i.e. between the mechanical and the nonmechanical chemistries (Gurney et al, 1996; Brown et al, 2004). We shall make this important distinction for ourselves later, but we for now go meekly along.

Gurney et al discuss equilibrium age distribution populations by saying:

The equilibrium age-distribution. … The fomulation of such models has been extensively discussed by Metz and Dickmann, who show that if all individuals enter the population with the same development index (qr) and experience the same environmental conditions, then all the members of any given cohort must have equal development indexes throughout their lives. This implies that the equilibrium state of such a population can conveniently be described by the combination of an equilibrium number-at-age distribution, f(a), … and a development-at-age-function, θ(a) …. The shape of the equilibrium number-at-age distribution is related to S(a, ψ), the survival from recruitment to age a in constant environmental conditions, ψ… (Gurney et al, 1996, p. 347).

Gurney et al are helping us put some good numbers to the distribution. However, their analysis says nothing about molecules. For that we have to turn elsewhere … such as to Brown et al, who in their turn state that all of a population’s metabolic rates and energy needs will be funded through their metabolisms and the various rates thereof. They then clearly relate those rates and reactions to the chemical components concerned:

Wherever they occur, organisms transform energy to power their own activities, convert materials into uniquely organic forms, and thereby create a distinctive biological, chemical, and physical environment. Metabolism is the biological processing of energy and materials. Organisms take up energetic and material resources from the environment, convert them into other forms within their bodies, allocate them to the fitness-enhancing processes of survival, growth, and reproduction, and excrete altered forms back into the environment. Metabolism therefore determines the demands that organisms place on their environment for all resources, and simultaneously sets powerful constraints on allocation of resources to all components of fitness. The overall rate of these processes, the metabolic rate, sets the pace of life. It determines the rates of almost all biological activities.

… Metabolism is a uniquely biological process, but it obeys the physical and chemical principles that govern the transformations of energy and materials; most relevant are the laws of mass and energy balance, and thermodynamics. Much of the variation among ecosystems, including their biological structures, chemical compositions, energy and material fluxes, population processes, and species diversities, depends on the metabolic characteristics of the organisms that are present. …

In its narrow sense, stoichiometry is concerned with the proportions of elements in chemical reactions. In broader applications, such as to ecology, stoichiometry refers to the quantities, proportions, or ratios of elements in different entities, such as organisms or their environments. … All organisms have internal chemical compositions that differ from those in their environment, so they must expend metabolic energy to maintain concentration gradients across their surfaces, to acquire necessary elements, and to excrete waste products.

Fundamental stoichiometric relationships dictate the quantities of elements that are transformed in the reactions of metabolism. Biochemistry and physiology specify the quantitative relationship between the metabolic rate and the fluxes of elemental materials through an organism. The metabolic rate dictates the rates at which material resources are taken up from the environment, used for biological structure and function, and excreted as ‘‘waste’’ back into the environment. Far from being distinct ecological currencies, as some authors have implied, the currencies of energy and materials are inextricably linked by the chemical equations of metabolism. These equations specify not only the molecular ratios of elements, but also the energy yield or demand of each reaction. Ecological stoichiometry is concerned with the causes and consequences of variation in elemental composition among organisms and between organisms and their environments. Despite the overall similarity in the chemical makeup of protoplasm, organisms vary somewhat in stoichiometric ratios within individuals, among individuals of a species, and especially between different taxonomic and functional groups. …

The elemental composition of an organism is governed by the rates of turnover within an organism and the rates of flux between an organism and its environment. The concentrations of elements in ecosystems are therefore directly linked to the fluxes and turnover rates of elements in the constituent organisms (Brown et al, 2004).

Since metabolism is directly related to stoichiometry, we can link the resources and the molecular components that metabolism requires directly to the equilibrium age distribution population that that metabolism is maintaining through the biological entities that create the distribution. As Gurney et al indicated in their above cited review, this requires a two-step process (Gurney et al, 1996):

  1. stating the physiological attributes of the individuals, and then
  2. linking those states to the environment and its resources.

Both of the above are ultimately molecular. We now have a doorway to molecules for all these processes are susceptible to the standard methods of chemical analysis. That is to say, both the environment’s resources, and the chemical compositions of the biological entities they lead to, can be determined in the moles and in the kilogrammes of chemical components consumed as the entities form both themselves and their distribution. We can in other words count molecules.

As Gurney et al point out, the individuals within a given population have two relevant physiological attributes: their age at any given time-point t over the generation length, T; and their state, l, at that time (Gurney et al, 1996). Gurney et al thus follow accepted scientific methods and denote the constant environment faced by a given population by ψ. The per capita rates of mortality, δ, fecundity β, and growth-plus-development, γ, over the population are then the functions δ(t, l, ψ), β(t, l, ψ) and γ(t, l, ψ) respectively. This is promising, for we should soon find gradients and everything else we need.

We now turn to the issue of the metabolism the entities individually use to maintain themselves as they corporately maintain the equilibrium age distribution population. The above three rates must be—and can only be—funded, both materially and energetically, through the various metabolic rates adopted by the entities over the generation (Brown et al, 2004). The required function to describe an entity’s metabolic state at any age t then becomes:

θ(t) = qr + ∫0tl(x, θ(x), ψ) dx,

while the shape of the accompanying equilibrium distribution of entities at each age is:

f(t) = f(0) S(t, ψ),


S(t, ψ) = e - ∫0tδ(x, θ(x), ψ) dx

We must eventually of course provide a value for f(0). It includes the replacement rate … which is an equilibrium balance not only between the natality and the mortality for the distribution, but also between the resources and the components needed to attain this. These together help determine the equilibrium age distribution of states; of metabolisms; and of resources over all ages. But that equilibrium age distribution—not just of entities but also of their components—depends upon the resources made available to the population by the environment … and in particular upon the equilibrium environment parameter, ψ*, imposed by the environment upon the population. This last is given by:

0S(x, ψ*, ψ) β(x, θ(x), ψ*) dx = 1

and which is now and in principle solvable for ψ*. We can in other words discuss biological entities in terms of their molecular components and of the resources that they consume.

The equilibrium environment parameter now states the environment’s effect, as also of its resources, upon the population. Those resources in their turn therefore determine the physiologies, the developments, and the metabolic rates over all the entities, and therefore also determine the overall mortality, fecundity, and growth-plus-development rates of δ, β, and γ.

But although we have now established the rates and the relative population sizes over all ages, we have not settled an initial size, or count, for f(0). We do not yet, therefore, have a stated absolute number density for any age. We only presently have relative abundances for both biological entities and the molecules that compose them. However … this f(0)—and so also the replacement and reproduction rates—is clearly given by:

∫∫0f(x) β(x, θ(x), ψ*) dx.

Our desired population size must therefore be independently specified. We shall do so shortly.

It is important to appreciate that the specific population size we choose for our equilibrium is of no real moment, because this being a distribution, as soon as any one count at any arbitrary t within the population has been specified, the equilibrium age distribution over all ages has also and immediately been specified. The environment parameter will also be immediately specified … and so also all the resources and all the metabolic rates needed to realize that distribution. And since this is mathematically achieved through integrations and differentiations, then the resources and the chemical components are also specified as they are each individually introduced and/or removed, and according to all variations or otherwise in the environment. As with Brassica rapa, Chorthippus brunneus, and our bristlecone pines, those resources and their variations affect the physiological states and the growth-plus-development indices of all individuals and over all ages … and they can all be measured. In other words, every molecule within our age distribution population has now been specified, as also all resources and all molecules entering and leaving at any and all ages. We are therefore free to set this f(0) at any size we wish. In the real world, the population size is of course set by the environment.

We have now stated the biological effects of all the mass fluxes and the energy fluxes passing through our Gaussian surface, and so entering into, and emerging from, this population, along with all the turnover rates and the throughputs of all the components duly incorporated into all biological entities over the entire population, and so at every time point t over the generation length T per every entity, and as it both grows and develops. This is the entirety of both its mechanical and its nonmechanical chemical work. So all we need to do now is (a) find a boundary; and then (b) measure it.