### What Really Matters in the Search for Extraterrestrial Life?

Almost inevitably when talking about the possibility of extraterrestrial life you hear something along the lines of "there are billions of galaxies, billions of stars and billions of more planets, some of them at least must contain some sort of life". The argument–really more of an inclination–is intuitive in one sense but it misses something.

Biologists don't yet have a theory of abiogenesis (chemical evolution) but suppose we assume a purely chance-based explanation, what then is the (conditional) probability of forming the first living cell through random molecular interactions? Conditional on chance alone, that's now a very answerable question.

Even the simplest life is still very complex. The simplest extant cell, the mycoplasma genitalium a bacterium which inhabits the urinary tract in humans, contains 482 proteins and around 562,000 base pairs in its DNA. That's still a high number, as compared to what's suggested in knock out experiments which indicate a possible minimal complexity of around 250 proteins.

How complex is each protein? Some proteins are hundreds of amino acids long, let us assume then a modest average length of 150 amino acids. Each of these proteins must then meet three requirements: (a) every amino acid must form a peptide bond or the molecule will not form properly (b) every amino acid must be an L-form or the resulting polypeptide chain will not fold into a functioning protein and (c) the DNA bases must form a meaningful sequence, one that actually codes for a protein.

Peptide bonds form about as regularly as non-peptide bonds. So the probability is around half.

Likewise, the second of these conditions is about half. Amino acids (with one exception) come in two forms, L-form and D-form these are mirror images of each other known as "optical isomers" and only the left-hand form allows for a functional protein to be built.

The third of these conditions is more difficult to determine. What we need to know is the ratio of functional sequences of amino acids to non-functional sequences (i.e. those that fold into a protein to those that can't).

Amino acids have to meet up in a meaningful arrangement in order to produce a polypeptide chain, that folds into a functioning protein. There being exactly 20 different types of amino acid. The probability of a particular polypeptide chain is easy to calculate.
However, that isn’t the right probability for (c). We need to determine the probability for any functional protein, not just a specific protein. Enter Robert Sauer. Sauer performed experiments at MIT using a technique called "casette mutagenesis" to determine how tolerant proteins were to varying a given amino acid. Even taking variance into account, Sauer determined that the probability of getting a functional protein (100 amino acids long) is

Independently a Biologist named Douglas Axe formerly working at Cambridge university argued that Sauer underestimated the variance allowed by functional proteins. In a 150 amino acid long protein, estimated the probability to be

This estimation is based on his own mutagenesis experiments, published in a paper back in 2004. So now we have enough to determine the probability of a single protein forming by chance. We simply add each of the powers.

$1:10^{\left ( 45+45+74 \right )} \approx \frac{1}{10^{164}}$

That may be the probability of obtaining a single functional protein but we want to know the probability of the first organism, of 250 proteins appearing by chance. We simply multiply the probability of one, by itself 250 times.

My calculation here is not far off Fred Hoyle's estimation (though his work was based on far more guesswork). The number which is so inconceivably small, that it can in effect be ruled out, no matter how many galaxies, stars or planets in our observable universe. That may just be our answer to the Fermi paradox.

When I began reading about the origin of life it didn't take me long to discover that no one really takes "chance only" explanations seriously anymore, not to my surprise, the oft-presented textbook picture of the first cell arising in a prebiotic soup, something like presented in the Miller-Urey experiment, is entirely outdated.

Not only are there other requirements for a functioning cell beyond proteins and DNA–ATP, phosphates, lipids, sugars, vitamins, metals, some kind of proto-cell membrane–but there probably wasn't a prebiotic soup, to begin with.

Famously James Brooks and Gordon Shaw discovered that the precambrian sedimentary layers of rock didn't contain enough nitrogen to suggest the existence of an ocean rich in amino acids. Its level is less than .015 per cent and the chemicals used in the experiment are also slightly off, in a reducing atmosphere forming amino acids isn't difficult (it's an exothermic reaction). But under early earth conditions which probably contained neutral gases, carbon dioxide, nitrogen, and water vapour and also much free oxygen, these present an environment which is hostile to the production of amino acids.

That might seem bad enough but it gets even worse, many destructive chemicals are also produced in simulations of this kind of model. Biochemically irrelevant compounds, a kind of black insoluble sludge which is normally removed by the researcher. As are the presence of medium to long wavelength ultraviolet light. These obviously further diminish the odds of producing the first life but I have no way of calculating those numbers. In any case, it's doesn't matter.

Now, of course, there are other alternatives to pure chance like biochemical predestination, external self-organisation or the RNA world hypothesis, which either doesn't rely on chance at all or it plays on far fewer odds. For example in the RNA world hypothesis now the currently favoured theory, chance may bring about a much smaller molecule that can self-replicate and then a kind of pre-biotic natural selection takes over.

I'm no expert on any of this but Origin of Life research is something I'm becoming incredibly interested it. We may just have to wait and see what happens.

### Set Theory

This post is a very brief introduction to some of the basic concepts of set theory. Set theory is a branch of mathematical-logic, that has wide applications across disciplines. Its not just used in the obvious way of studying the foundations of mathematics by mathematicians but also in physics, social science, and even by philosophers as a theory of semantics for predicate logic (although you can do propositional logic without set theory).

A set is a collection of elements, or members; the notation for a set is specified by listing its components. So the set of even numbers can be represented a
$E: \left \{ 2,4,6,8 ... \right \}$$E: \left \{ x: x > 0 \wedge even\right \}$ Either of these notations is valid. Further, elements of a set can only be in that set, once. So   $E: \left \{ 2,2,2,4,4,6,8 ... \right \} = E: \left \{ 2,4,6,8 ... \right \}$ The notation used to indicate that something is an element of a set, is using the Greek symbol "epsilon". That is: $4 \epsilon S$…

### William Lane Craig and the Hartle-Hawking No Boundary Proposal

Classical standard hot Big Bang cosmology represents the universe as beginning from a singular dense point, with no prior description or explanation of classical spacetime. Quantum cosmology is different in that it replaces the initial singularity with a description in accord with some law the "quantum mechanical wave function of the universe", different approaches to quantum cosmology differ in their appeal either to describe the origin of the material content of the universe e.g., Tyron 1973, Linde 1983a, Krauss 2012 or the origin of spacetime itself e.g., Vilenkin 1982, Linde 1983b, Hartle-Hawking 1983, Vilenkin 1984.

These last few proposals by Vilenkin, Hartle-Hawking and others are solutions to the Wheeler-DeWitt equation and exist in a category of proposals called "quantum gravity cosmologies" which make cosmic applications of an approach to quantum gravity called "closed dynamic triangulation" or CDT (also known as Euclidean quantum gravity). I&#…

### Can inflation be eternal into the past?

Back in 2003 a paper appeared on the arXiv titled "Inflationary spacetimes are not past complete" that was published by Arvind Borde, Alan Guth and Alexander Vilenkin which has had considerable amounts of attention online. The theorem is rather uninteresting but simple and doesn't require a very complicated understanding of math. So I thought I'd explain the result here.

It's purpose is to demonstrate that inflationary models are geodesically incomplete into the past which they take as "synonymous to a beginning" but Vilenkin stresses that the theorem can be extended to non inflationary models so long as the condition of the theorem that the average rate of expansion is never below zero is met. These models too then are incomplete into the past. Consider the metric for an FRW universe with an exponential expansion

Where the scale factor is

Since the eternal inflation model is a "steady state cosmology" the mass density and the Hubble paramet…