1. Evolution as a natural process does not proceed along two axes, nor is it two processes. But the natural process of evolution reflects that in life and the universe, some attributes are easier to alter than others.
2. This produces several effects in evolution difficult to explain. These include accumulation of complexity against a loss of fitness, the punctuated pattern of the fossil record, and saturation of phylogenic possibilities as life evolves.
3. The preceding essays (see the new model of evolution) have shown how these effects can be explained without the 'two-axis' model. Only the explanations are difficult. (Such as an accumulation of small fitness gains results over time in a fitness loss. Or that genes increase fitness by forcing their hosts to become more variable, and so lose fitness.)
4. These ideas are easier to explain using a 'two-axis' model. This site is to explain the 'two-axis' approach, and invite corrections or suggestions for improving the model.

No PDF version of this file, which is large because of diagrams, but it can be provided on request. 

(Note: This essay is philosophical, more on how humans model nature than how evolution works. See other essays in new model of evolution for details of evolutionary processes leading to the 'two-axis' model.)

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6.1 Modeling Nature All models of nature contain fixed and variable traits.

6.2 Fixed and Variable Processes Fixed traits evolve from variable ones once only!

6.3 Phase Transitions Evolution of fixed traits is a phase transition.

6.4 Evolution along Two Axes We can show the effects best on a 'two-axis' model.

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Models that humans make of nature are never exactly how nature works, in all its detail and nuances.

• Mechanical models are often time irreversible, whereas in nature processes are time directional.
• Thermodynamic models are time directional but they must be enclosed inside process boundaries, whereas nature is unbounded.
• Mathematical solutions tend towards infinitely small or large quantities, whereas natural quantities are always finite even if uncountable.
• Every particle in the universe interacts with every other. But although this is known, for equations of more than several variables the math becomes unwieldy so simplifying assumptions are needed.

So, although nature is a single process, humans never use a single model of it. Rather, we must choose among available models one most apt to that segment of nature that we are trying to explain. Yet, although we have different models of nature, all models share a universal characteristic. This is that the models compose of both;

1. a property that is changing (altering in time)
2. a property against which the model is changing, that is "fixed" in time.

We can show this universal characteristic of some traits always varying against other traits that are fixed, on a diagram (Fig 1.1). Variable properties alter left-to-right along an axis that we associate with the flow of time. The "fixed", non-variable properties, we show "rotated" 90⁰ away from the plane in which time flows. The diagram is as follows: (For "trait" read property, attribute, etc.)

Variable properties are those we measure against properties we regard as fixed (we navigate against the "fixed" stars). This applies to models of evolution too;

• Species vary every generation (allele distributions alter). But species only evolve against a background of deep phylogenies, some of which have not altered in millions, or billions of years.
• The DNA code or basic structure of eukaryote cells has not altered in a billion years, or vertebrate phylogeny has existed for half a billion years. Although these properties alter imperceptibly we study evolution knowing that there will not be a sudden change in the DNA code, or an invertebrate will not give birth to a vertebrate.
• Similarly for gene flow, although we allow host fitness to expresses gene fitness, we know core sets of conserved genes (like H4 or Cytochrome C) will not be affected by host fitness. Instead of regarding genes expressing these as evolving imperceptibly in time, we consider that these genes are "fixed" properties of the host genome. These will alter, but not for a model of distribution of variable allele frequencies in a small, stable population.

Yet often overlooked (and for thousands of years was not known) is that since the beginning of the universe properties now fixed have also evolved, from a state of being variable. For example;

• When the chemical elements formed fundamental particles of matter had already stabilized, but in the very early universe, these were evolving.
• When life began on Earth the chemical composition of the elements had by then stabilized, but in the earlier era these too were forming.
• By the time modern animals evolved cell structure and the modern DNA code had stabilized, although these too were once changing and evolving.

So, apart from fundamental constants (such as the speed of light or Plank's Constant) most properties in life and the universe we now regard as fixed were once evolving. This makes evolution of complex entities such as stars or planetary systems, but especially the evolution of life, a dual process.

1. A foreground evolution that we see unfolding before our eyes, although it can takes millions of years.
2. A background evolution in much slower cycle times, over not just millions, but billions of years since the universe began, in which the stable properties of the modern universe, and life on Earth, themselves evolved.

So, any universal model of evolution for any point in time cannot cover just evolution of the foreground properties. It must include the background processes of evolution too, in which properties once variable themselves evolve into the modern fixed properties of nature.

Every process of nature involves variable "foreground" properties altering against fixed "background" ones. This would be the case even in instances when the fixed "background" properties themselves evolved the first time. (There would be a set of even deeper 'background" properties, like a stable atomic structure.) Only strangely, whereas the model is still the same where "background" properties evolve, it is different in that "background" evolution only occurs once!

• The universe only evolved once.
• Matter only evolved once.
• On Earth the DNA code only evolved once. (There is a debate if RNA evolved once, but parts of DNA evolved twice.)

Yet, for "foreground" evolution the same cycle of change occurs many times. Matter only evolved once, but different assemblages of matter are recycled trillions of times over. Or while the DNA code only evolved once, the code sequence alters every gamete in uncountable trillions of combinations.

Another controversial difference is the teleology of the processes. Although we call the "foreground" process evolution, most changes are cyclic or oscillatory. Like the seasons, or birth and death, changes occur. But once the cycle is complete (the life cycle of a star or a species) we cannot measure the change as progress towards a goal. This often called the Red Queen Effect, because "you have to run as fast as you can just to stay in the one place". Yet, while foreground evolution does not result in scientifically measurable progress, background evolution is progressive in a historical sense.

• The life cycles of stars are non-progressive. But in the history of the universe Type I Stars form first of simpler elements. When these die they can form new elements, which can form Type II Stars, out of which complex planetary systems or life can form. So, out of a cycle (the life and death of a star) can come an event that is non-cyclic (formation of new elements).
• In biological evolution the life and death of species is cyclic. But from the life and death cycle of (only some) species and despite the Red Queen Effect, new novelties arise. These novelties give rise to whole new classes and phyla that could not in an historical sense exist before.

Again, we can show the effect on a simple diagram. At any point in time some properties change against a background set of properties, not varying at the same rate. But as the universe changes over time, certain properties do change in unique ways. This is diagrammatic, but maturing properties appear as a 'rotation' through 90⁰, from the axis of variability to axis of fixation. Fig 1.2 shows the effect.

Even so, this diagrammatic 'rotation' only involves some elements.

• Only the life and death cycle of some, not all stars (those ending in a supernova) give rise to new elements.
• Only the life and death cycle of some species, also under great evolutionary pressure we call a concatenation, evolve totally new novelties, or new classes or phyla.

Plus most cataclysmic changes in evolution on Earth were in the remote past. So, by the time intelligent beings evolve only "foreground" evolution remains for them to study as living evolution. (The fossil record supports a theory of large-scale, cataclysmic changes. But living evolution is of small-scale changes.) So, we dispute whether changes in the past were "foreground' evolution over longer times, or if different processes operated in the past than today.

But the issue is complex. Whether stars are Type I or II, the laws of physics and chemistry are the same. And whether changes in evolution are small or large, laws of natural selection or population genetics do not change either. But;

1. Natural evolutionary processes that could occur in recent times (like evolution of intelligent beings) were not possible in the remote past.
2. Events possible in the remote past (evolution of new classes and phyla) are not possible under conditions of modern saturation.

Similarly, the physical laws of the universe are invariant with time. But which processes are possible in the universe depend on its size, density, and matter-energy composition, which alter over time.

So while the laws of physics, or laws of life, do not appear to alter over time the conditions in which these laws act do alter and have a history, which is difficult to model.

One noticeable effect of human evolution is a major change from evolution by adapting biology, to evolution by adapting culture and behavior, while leaving the biology to change imperceptibly. To reach this point, human biological evolution was very rapid, but this makes the change even more distinctive once it occurs. In the physical sciences, we would call such a transition from one state of alteration, to alteration under a different state, a "phase" transition, or phase shift. We suspect that such a phase transition occurred in human evolution, but we have no idea of;

1. what quantities are involved in the transition
2. why such a phase transition only occurred for human evolution, and not elsewhere

The new model can solve problem b) by demonstrating that phase transitions, or phase shifts (we will use the latter term) occur throughout evolution. Human evolution was just the last such transition, in a series since life began. For living matter, a phase shift is the state of a property changing in early evolution, but stabilizing in a later epoch. For example;

• The change from primal non-cellular replication to cellular replication was a phase shift, because once it occurred the first time it became a stable mode of replication, despite that it was only an evolving mode in pre-cellular life.
• Evolution of cell-tissue or the DNA code was a phase shift, because these too had to evolve the first time, but then stabilized.
• Evolution from prokaryote to eukaryote life was a phase shift, because once it evolved eukaryote life became a mature, stable form.
• Evolution of cell nucleus, photosynthesis, photoreceptors, skin, bone, vertebrae, blood, nerves, etc. were phase shifts. All these attributes had to evolve a first time, but once evolved they stabilized and were available as mature traits for subsequent evolution.

Human evolution then was not a "phase shift" in that biology was transmuted into culture, although that was the effect. It was a phase shift biologically, in that an intelligent, culturally adaptable creature had to first evolve. But once it evolved its type matured, and was able to radiate into a huge number of adaptive (mainly cultural) forms of existence.

More difficult is problem a), concerning which physical property is involved in transitions. Until we measure a property changing, the term "phase shift" is ill defined. The DNA sequence expressing the H4 protein has been stable for at least 10¹² generations. But the sequence had to evolve, so if it once exhibited a stability of less than 10¹² and then went through a transition to the new stability this was a measurable phase shift. (H4 evolved in the remote past, so we do not know when the shift occurred.) Yet many attributes have evolved in shorter evolutionary times than the subsequent stability they produced. For example;

• Life itself evolved in half a billion years, but has been a stable state on Earth 6-8 times that period.
• Some cyanobacteria have existed for three billion years, but evolved in a shorter time than that. So the DNA sequences of these microbes must have been altering faster in the past than it alters now, which would be a phase shift, if we could measure the change.
• Traits that matured in the famed Cambrian Explosion (or similar) could have evolved rapidly, in perhaps one-tenth the time (50 million years) of the subsequent stability. So sequences expressing these traits were undergoing rapid alteration until 550 million years ago, but underwent a "phase shift" to become mature, stable traits of subsequent evolution.

Even so, any alteration to rates of mutation in DNA is only measuring an effect. We could use a "selfish gene" argument that phase shifts of mutation rates from an evolving to a stable mode was a strategy of genes, to retain high sequence conservation throughout subsequent evolution. (This is the "smart gene" hypothesis, earlier.) But we could not explain all shifts in mutation rates as caused by gene selfishness.

• If the Cambrian Explosion witnessed a slow down in gene mutation rates, we must explain why so many shifts occurred in so many unrelated types at the time.
• Change from pre-biotic to biotic replication was a phase shift, but we are not sure it was a shift in rates of change in DNA. (DNA or "genes" might not have existed when life first evolved.)
• While human evolution was a phase shift, it might not be a shift in gene mutation rate. (Chimps share 98% of genes with humans, but chimp evolution was not a phase shift. Still, by being able to adapt culture without altering biology the human genome should stabilize its DNA slightly better than chimps.)

Also high sequence conservation seems more a result of selection than inherent phase shifts, because non-expressed (hence non-selected) sequences are always less conserved than expressed ones, and so they show less of a "phase shift" effect. (See also the section of phase shifts in the "smart gene" hypothesis.)

So, for our present understanding a phase shift is an event rather than a cause. (Expansion of the universe is a cause, while a phase transition from energy into matter is an event resulting from the cause.) For biological evolution a "phase shift" is an event in the history of life in which a trait once evolving matures and stabilizes.

• In the Age of Amphibians, evolution of limbs is a variable trait. But in the Age of Reptiles, limbs are a matured, homologous trait.
• In modern mammals, traits such as endothermy, four-chamber heart, and mammalian glands are matured traits. Such traits are evolving, but imperceptibly against rapid morphological and behavioral changes in modern mammals.
• During the Triassic traits now mature, stable, and homologous were in the earlier period evolving among proto-mammals, and subject to modification and selection.
• Also during the Triassic proto-mammals were evolving against a 'background' of other homologous, matured traits, but in this case ones characteristic of reptiles.

No matter how far back we find everywhere traits that evolved rapidly in a previous age stabilizing as mature traits for the next stage of evolution. Only for living tissue there is no property that can consistently quantify a difference between fixation and variability.

• DNA sequences alter mutation rates as they evolve and mature. But these rates change for many reasons, especially between the coding and non-coding sequences.
• A scale of time or generations could measure differences, but this too is misleading. Organisms like lungfish might survive fortuitously unaltered for 10⁸ generations, but their evolution is cyclic. Yet human biology has stabilized in only 10⁴ generations, but human evolution involved a phase shift.

Still, we can quantify mutation rate as an angle.

• For an "average" mutation rate of 10^-6 we express a rate 'angle' by q = tan^-1 (mutation rate x 10^-6). For a 10^-6 mutation rate q = 45⁰.
• If mutation slows by one part (to 10^-7) q jumps to 84⁰, or almost vertical.
• When the gene was evolving, if the mutation rate was one part slower than 10^-6, q falls to 5.7⁰, or near horizontal.
• For changes once per gamete (10⁰) to every thousand generations (10³) angle is virtually horizontal.
• Mutation above at 10^-7 or above will show an angle almost vertical.

We can interpret this 'angle' to mean that while a gene is within the "average" mutation rate (10^-6) it is partially variable and a partially fixed. Allele frequencies will be changing every generation along the 0⁰ horizontal axis, which is the axis of variable traits. At 45⁰ to horizontal, mutation in a gene might effect the expressed traits of a species. But once the mutation rate slows to 10^-7 or lower, it moves outside of "residence" of that species. It will mutate too slowly to appear in a small population, within a random window of selective pressure to change. Fig 5.2 (from "smart gene" hypothesis) shows the effect.

Critics have objected that this 'angle' argument is false; the tan^-1 formula is a contrivance to "force" a bipolar effect, when none is there. Yet, the effect exists because some properties are variable while background properties are "fixed".

• If we calculate gene flow over a few generations, we correctly assume that genes not altering in 10⁷ generations are "fixed" for the calculation. But that is all the tan^-1 formula asserts anyway. (The 90⁰ angle just illustrates that "this property is fixed for this calculation".)
• Yet, a property "fixed" on one scale it does not remain fixed over the life of the universe. Most properties of life and the modern universe evolve from a state of being variable. So we express a property evolving as a "rotation" through an angle from 0⁰ to 90⁰.
• All genes that now mutate at 10^-11 once did not exist. Or if they evolved by macro-mutation they still existed in the first instant at a 10⁰ mutation rate, and "rotated" through 90⁰ to reach maturity.

But genes did not physically rotate. It is just a representation. (In an alien civilization, time does not flow left to right across a page, positive rotation might not be counter-clockwise, and angles will not be measured in degrees.) But no matter how we present it, throughout the universe properties now fixed once evolved. We need to represent this effect with a model appropriate to our understanding.

Only in evolution traits group into easy and hard to evolve types. We have examined this, and found that;

1. Evolution of the hard-to-alter traits leads to a loss of fitness over the lineage of the host species, whereas adaptation of easy-to-alter traits results in a fitness gain.
2. Adaptation of easy-to-alter traits is also cyclic and never saturates, whereas evolution of the hard-to-alter traits is phylogenic, and always results in saturation.
3. Evolution of the hard-to-alter traits is once only, or episodic.

This infers that "rotation" from a shallow to a steep angle would be an irreversible event. That is, the rotation would only ever be counter-clockwise. This is different from cyclic processes in evolution, like increase of decrease of length, or point mutation of a base, which are reversible. In this sense, the processes of adapting easy-to-alter and evolving hard-to-alter traits in evolution are distinct, and the 'two-axis' model is only a representation of that distinction.

If one wished to describe in nature two processes, one the inverse of another, our mathematics and language too readily provide the tools and terms to use. Language, especially Greek and Latin roots use many inverse terms.

• Properties are monophyletic or polyphyletic, homologous or analogous, divergent or convergent, irreversible or reversible, and result in macroevolution or microevolution.

Mathematics too is replete with inverse terms.

• Tan and cot, sin and cos, integration and differentiation, plus and minus, pole and zero.
• In electrical engineering circuits can be wired as either series or parallel. But the properties of these circuits, a recent invention of technology, can be almost perfectly described by the inverse terms of mathematics, many of which originated before electricity was discovered.

Such coincidences common throughout science create grave debate. We wonder if the universe is naturally bi-symmetric, and our language and mathematics reflects that, or if our mathematics and language is contrived by human prejudice and through its prism we impose a "world view" on nature.

Another problem is limited human observation. We have only seen evolution on one planet. Even then, we discovered evolution almost immediately after intelligent life (us) emerged, so the story of evolution appears to end with its own discovery.

• Simple organisms evolve until that complexity level saturates.
• Then the more complex organisms evolve in a 'layer' of complexity above the simple ones.
• This cycle repeats until the possibilities of further biological complexity become saturated.
• Then intelligent life evolves.

We cannot answer this because no human has witnessed it. Perhaps

1. We blow each other up and life starts over again.
2. Life settles to a saturated state, with further evolution only naturally among insects and microbes, with larger changes brought on by human intervention.
3. Or among trillions of planets capable of supporting life, intelligent life emerged randomly and fortuitously only on one anyway, and continues from here on a totally random path.

In these cases, all attempts (and there are many, not just this theory) to derive the "formula" of why intelligent life evolved are as vain as attempts to derive why God specially created it.

This is why if we think there is a difference, we must demonstrate that "foreground" evolution is analytically different from the "background" process objectively, independent of human perceptions, language, or use of terms. Yet in the universe, objectively, some properties are easier to alter than others. For example;

• Structures of matter such as atoms exist in stable states, and it is easier to form elements from existing structures, than form new-atomic particles.
• It is easier to combine elements into molecules, than form new elements.
• Once life evolves it is easier to mold proven materials into different shapes and textures, than evolve new materials from the start.
• Once cell types stabilize, it is easier to slightly adapt the size or function of existing cells, than evolve new cell types the first time.
• Once the DNA code stabilizes it is easier to enact evolution by varying the code, than evolve a new nuclear code again.

Only once the DNA code evolves, human perceptions again distort this bi-symmetric model of evolution. Humans have difficulty understanding why, if any DNA-coded trait is easy to evolve, DNA-coded traits will continue to group as easy or hard to evolve. We can only point out that;

1. Not all DNA combinations are equally likely. Proven sequences will tend to be maintained, so that new mutations are mostly variations to proven themes.
2. Expressed traits must be fit, and small adaptations to proven themes are usually fitter than radical departures from existing designs.

This gives fitness an additional role. We think of fitness and selection refining or improving design. But fitness and selection also retain that feature of all existence that some properties of the universe are easier to change than others. Evolution is a more efficient process overall, if it proceeds this way.

But invoking fitness to explain how easy-to-enact changes are favored strikes another conceptual difficulty explaining how the hard-to-form traits evolve.

• For fitness, each successful adaptation is a slight fitness gain.
• If the change stabilizes after one or a few generations, the lineage gains fitness, because a small change to host genotype results in an enduring fitness gain without further change.
• But if the small change requires another change, then another, over thousands of generations before it stabilizes, there is evidence that over the lineage, from the first host to the last, fitness has fallen.

This last effect is very strange. There is evidence that as life becomes more complex fitness falls. It does for sex, and it appears to fall for evolution of eukaryote from prokaryote life. But it is harder to "see" how for each major increase in complexity (as for evolution of a reptile into a mammal) fitness will fall. Mathematically, this requires that a series of small positive fitness gains "add up" to an overall fitness loss. This should not be a problem, because such effects are frequent in the physical sciences, but it confounds conventional notions of evolution.

This is why we need the 'two-axis' model.

Postulation of a 'two-axis' model for evolution does not "cause" fitness to fall as a lineage of types accumulates complexity. But fitness falls in nature, and the 'two-axis' model helps explain why.  In Fig 6.3 we see two views of how small changes accumulate. In the first, we assume all changes are along the 0⁰ 'axis'. And all small changes first accumulate this way, as events in time. But as new traits, novelties or sequences mature and solidify, they "rotate" 90⁰ away from the axis of time, to indicate that they are not subject to further change. This is acknowledged anyway, as most changes are "foreground" against a background of traits that are fixed. So, the diagram shows that the individual is a composite of homoplastic traits, and deeper homologous traits, and the traits alter on different time scales. Already we see that small foreground changes will not add linearly.

• Humans evolved by a series of small changes. But actual change was only 2-4% in the human genome from a Great Ape genome.
• So, despite dramatic changes resulting in human evolution, one would barely see it on a 'two-axis' diagram. Only 2-4% of the genome changed, while 96-98% of the genome remained "fixed" along the vertical axis, not subject to further change in time.

Yet as life evolves greater complexity, more attributes "phase shift" to the vertical axis. The huge diversity of mammals is expressed from a core set of 70% of genes. And Great Ape diversity, and fast evolution into different types, is expressed from a core set of 96% of genes. So as life becomes more complex, different types can evolve, faster, from a smaller core set of genes and fixed attributes. (This data needs checking! Mammals are complex, and they can evolve quickly into a huge range of diverse types, while retaining a core set of genes and traits in common. It needs to be checked how fish, reptiles or other classes compare. Possibly once more data is known, this thesis might need updating.)

Fig 6.4 shows the problem. Complex life contains a huge genome, but most of it is unexpressed or non-coding DNA, which is tied to individuals, but is not very fit as exact copy. On the other hand the useful, core traits of complex life are expressed by a tiny fraction of DNA, most of it sharing a high homology among a variety of types. (The composite diagram would be more pronounced. For a 10^-5 "average" mutation rate genes mutating slower would be near vertical. Non-coded DNA might be altering once per generation for sexual reproduction, so it would be drawn horizontal. But different scales would show the effect differently.)

So, there are not two separate processes in evolution, and life is not evolving along two 'axes'.

• Life is evolving by one single, but its own natural processes, which humans try to explain.
• As organisms accumulate small single generation enhancements fitness rises. But as an entire lineage of organisms accumulate a series of small enhancements increasing complexity from the first to the last in the lineage, fitness falls.
• While any small change is incrementally and continuously variable, changes that have accumulated such that they result in loss of fitness over a lineage will saturate, such that an identical accumulation is not possible over another lineage.

Humans have to explain this too. They have to explain why a property that always adds as a positive (fitness) can result in a negative as a summation, and explain why a property that is infinitely variable in isolation, can saturate in summation.

The 'two-axis' model of evolution is an attempt to represent, diagrammatically, how these broader, more complex processes of evolution work. (See also notes on Complex Time)

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