THE CONCENTRATION THRESHOLD & EXPONENTIAL GROWTH PROBLEMS

 The Concentration Threshold and Exponential Growth Problems

(See, "Q's Response to Richard Carrier: About the Debate," for context)

(See also, "Q's Open Challenge to Richard Carrier", and "The 'Fatal Flaw' with Richard Carrier's '1 in 10^41' Argument for Abiogenesis")

Richard Carrier is Promoting Belief in Alchemy and Magical Molecules

Richard Carrier is promoting belief in alchemy and self-replicator magical molecules instead of reality. His over simplistic idealized 'toy model' paper math does not work in the real world of mass action chemistry. Two problems will suffice to illustrate: The "Concentration Threshold Problem," and the "Exponential Growth Problem."

The Concentration Threshold Problem

No chemistry can be done with a single molecule (even in the "right environment"), including a single self-replicator. Many identical copies are needed, and thus, many need to spontaneously form at the same time and place—not just a single one. This is a basic fact of chemistry. Every chemical reaction must meet a minimum critical threshold concentration. There must be enough molecules in close enough proximity (i.e., high enough concentration) in order to generate enough molecular collisions to overcome diffusion and spontaneous decay to start and sustain a chemical reaction. This is basic chemistry (for a more detailed treatment, see, e.g., Wikipedia “Collision Theory”, “Law of Mass Action”, “Reaction Rate”). How many molecules do we need?  In the real world, we need millions to billions to trillions of molecules or more just to kick-start a chemical reaction. This is a fundamental fact of chemistry (and reality!) that applies to any chemical reaction--including "self-replication".

(from Wikipedia, "Collision Theory")

The Exponential Growth Problem

But even if, for argument's sake, we were able to have an unlimited supply of reactant molecules in a high enough concentration that could somehow be maintained in order to overcome the concentration threshold problem, we would then run into the exponential growth problem, which can be illustrated, using the following statement that Richard makes in his "Biogenesis and the Laws of Evidence": 

"When replications occur in mere seconds, a hundred million years is equivalent to a billion or more years in evolutionary time. For example: over the last half billion years on average mammals could double their number every 15 years or so (losses preventing that then drive evolution over that period); if they could have done that in 15 minutes, as protobionts more than can, then they would be 4 x 24 x 365 x 15 = 525,600 times more evolved in that same period, ergo one year for a protobiont is equivalent to 35,040 years in animal-scale evolution. A hundred million years is then equivalent to 100,000,000 x 35,040 = 3,504,000,000,000 or three trillion years of equivalent evolution. That’s the math." 

There are so many things wrong with Richard's statement, but let's focus on just one: nonlinear, exponential doubling. To be sure, exponential growth by doubling does look impressive on paper, but that’s not reality. No system can sustain exponential growth indefinitely. They are limited by the very real-world constraints of finite resources and space. Even self-replicating chemical systems in labs touted as "immortal" and capable of sustaining exponential-growth "indefinitely" (provided there are unlimited resources) routinely run to completion within minutes. The advantage Richard sees in rapid replication rates simply means resources will be exhausted that much quicker. 

Here's a healthy dose of reality for Richard: The average molecular weight of an amino acid is 110 Daltons x 32 amino acids in a "Lee peptide" gives a molecular weight of 3,520 Da for a single "Lee peptide" molecule (i.e., Richard's "protobiont," falsely so-called). Using Richard's scenario above, if a "Lee peptide" were to self-replicate, doubling in number once every 15 minutes, then after a mere two days (i.e., after it is only 192 "times more evolved," by Richard's flawed reckoning), then there would be 3.14 x 10^57 "Lee peptide" molecules weighing a combined total of 2.02 x 10^31 tons—which is the weight of 827,000 planet Earths!!!  That's the math!!!

Thus, as I've repeatedly said, Richard's 'toy model' math only works on paper, but has no basis in reality. 

The Concentration, Exponential Growth, and Overcrowding Problems Explained

Noted origin of life researcher, Luisi, has repeatedly pressed these points in numerous articles and books. See, for example, chapter 3 of his well known book (Luisi, Pier Luigi (2016). The Emergence of Life. Cambridge University Press):

“3.4 Self-replication – and the concentration threshold: The going-up process in Oparin’s ladder is basically based on chemistry, and the basis of chemistry implementation is concentration. This is equivalent to saying that there is always a concentration threshold, below which the reaction does not effectively take place....[I]n enzymatic reactions, the concentration of the substrate...must be such to overwhelm dissociation forces, and so on. This may sound very trivial, but it is something that is occasionally, if not often, forgotten by theoreticians who use to work without mentioning concentration. A very common case in this field is the treatment of self-replication…." 

"Obviously, with one single molecule no real chemistry can be achieved. Here comes the threshold of concentration. In normal wet chemistry, in order to self-replicate, the replicator A must bind to another molecule A. The need to form the A-A complex (A2) from two A molecules is a severe constraint. First, in order to make an appreciable concentration of this complex, there must be a significant amount of A, so as to overcome the effect of diffusion; and the real difficulty arises when spontaneous decay is introduced. If the concentration of A is low enough, the population will decline no matter how large the growth rate is. This means that the concentration of A in solution must be high enough to bring about a sizable concentration of A2, so that the natural decay diffusion forces do not destroy the dimer itself." 

"The precise calculation of the critical threshold concentration of A is not easy, as we would need a good estimation of the dissociation constant Kd for the dimerization process 2A ⇄ A2. Assuming for example a Kd = 1 µM, with an initial A concentration of 10−12 M (1 pM), the conversion into the dimer would be around 0.0002% (would be 0.2% if Kd = 1 nM). In other words, there would be no significant formation of the dimer below one picomolar concentration of A. One picomolar solution of A (defined as 10−12 M) still contains (since the Avogadro number is 6.02 × 10^23) 6 × 10^11 molecules, i.e., 600 billion. And even dealing with one microliter we would have in it 6 ×10^5 (i.e., 600,000) identical copies of A. This means that we would need several hundred thousand identical copies of A in order to start replication, even restricting the volume to one microliter. In other words, in order to have an effective self-replication of a hypothetical sr-RNA, we would need a preliminary metabolism to make a useful concentration of such a molecule." 

"This is one constraint given by chemistry to the notion of self-replication. And it is not the only one….With these concentration thresholds, there are severe difficulties in the prebiotic RNA-world.” (emphasis added)

Luisi continues:

"This is one constraint given by chemistry to the notion of self-replication. replication. And it is not the only one. The other concentration problem is with regard to the quantity of basic material involved in a nonlinear growth process of replication. Suppose you were to start from a solution containing the replicator at 1µM concentration. The solution will become ca. 1M in 20 nonlinear (2, 4, 8, 16, and so on) replication steps. Imagine what it would take to do that in a small warm pond of, say, 1,000 liters; then, for a polynucleotide of 10,000 Dalton, we would have after the 20 steps, 10 tons of material, an amount really unconceivable, at least with this example. Here, we see again the threshold of concentration at another level, namely given by the fact that such an amount of material would not be realistically available in any terrestrial pond. With these concentration thresholds, there are severe difficulties in the prebiotic RNA-world,

With these concentration thresholds, there are severe difficulties in the prebiotic RNA-world, and we will come back to this point in the next chapter. Let us move to another quite different level, where concentration is, in fact, a problem. This is at the level of the first cells. In this field, terms such as overcrowding, super-concentration, and similar are often used to indicate the particular concentration of the macromolecules inside the membrane compartment. The high local concentration in the cell is an accepted notion, and is the prerequisite for their biological activity. If you squeeze the content of 1 kg of Escherichia coli in the previous large pond, you will have in that pond all DNA and RNA and enzymes – and no life. In addition, when we do in vitro protein expression using a commercial cellular extract and a given plasmid, we know that dilution by only a factor of two-three of the cellular extract inhibits the protein synthesis. Thus, it will not be enough to find a way to make ordered sequences of DNA, RNA, and proteins (still open questions), but we will have to find a way to concentrate them in a small compartment having dimensions of a few microns. As we will see in the next chapter, this is indeed one major operational problem in the “compartmentalistic” hypothesis of the origin of life."

Additional Examples from Luisi

Chessari, S., & Luisi, P. L. (2012). On evidence: the lack of evidence for prebiotic macromolecular synthesis. Origins of Life and Evolution of Biospheres, 42(5), 411-419; 

"The field of the origin of life involves many centres all around the world, with a large variety of skills and facilities. And yet, there are certain areas of great importance that are barely touched by our investigation, and one may wonder why...One of these is the prebiotic synthesis of macromolecules: proteins or nucleic acids. By this we do not mean a simple polymerization of one given monomer into a homo-polymer, or the mixture of two or more monomers into a random copolymer. We mean the synthesis of a chain with an ordered sequence, and in many identical copies, as enzymes are. And you may look into the literature, searching for the evidence of the synthesis of, say, a 30-co-oligo-peptide in a precise primary structure, and you will find nothing; the same for nucleic acids. Notice that polymerization processes of single amino acids to give, for example, poly-phe are not relevant for this discourse, as the conditions for homo-polymerization are by no means valid for co-polymerization of a mixture of two or more monomers (Luisi, 2006).

The emphasis on "many identical copies" is particularly important, and generally neglected, particularly by those adherents to the prebiotic RNA world who work on theoretical fields with no contact with real chemistry. The classic, old argument usually given by this category of researchers is the following: if you make by random process one single RNA molecule capable of self-replication, then all is solved; at least on paper. In fact, this molecule would self-reproduce and eventually give rise to a population of other self-reproducing RNA molecules which, by further self-reproduction and random evolution, will produce enzymes (ribozymes), which eventually will catalyze the synthesis of the peptide bond and of the DNA bonds.

The start of the argument is completely faulty from the point of view of basic chemistry. Even assuming that by random polymerization of mono-nucleotides you make one single molecule of a self-replicating RNA, nothing will be achieved. In solution chemistry, you do nothing with one single molecule. To have self-replication of A, you need at least to make the dimer A, A, and this must be present in a concentration which defies diffusion forces. This corresponds to a viable concentration of ca. nano- or pico-molar, which means that even in a microliter you have many thousands of identical copies of A.

Thus, good evidence of a big jump ahead in the field, would be the evidence for the prebiotic syntheses of polypeptides and/or nucleic acids with an ordered primary structure and a sizable concentration of many identical copies. 

How could we do that experimentally using prebiotic tools? As we said, there are no demonstrations in the literature. The exception, as far as we know, is given by a paper from our group, not the newest (Chessari, Thomas, and Luisi, 2006)."

Luisi, P. L. (2015). Chemistry constraints on the origin of life. Israel Journal of Chemistry, 55(8), 906-918; 

“The concentration threshold for prebiotic reactions is often not taken into account in the literature, particularly in the field of the prebiotic RNA world. In addition, this shortcoming can make  the entire prebiotic RNA world construction shaky and unreliable, including the “myth” of the perennial self-replication of an RNA macromolecule.”

“This pathway still has several unexplained points- unexplained both experimentally and conceptually. For example, we do not know how polymers of nucleic acid, DNA or RNA, could arise prebiotically. Also, we do not really know how the biogenesis of ordered sequences of DNA or RNA came about, which also means that the prebiotic origin of a self-replicating RNA is still a myth; a similar degree of uncertainty is also present (perhaps to a somewhat less extent) for proteins and enzymes the origin of the genetic code is still in a cloud, and we do not have a precise idea of the structure of the first protocells or cells, nor of the form and energetics of the first metabolic cycles.”

“6 The Concentration Threshold for the Prebiotic RNA World”

“Chemistry operates with concentration and such a trivial statement is often forgotten in the origin of life research. Particularly, if one reads some of the old papers on the RNA world, (the “original” prebiotic RNA world), one may receive the idea that one single molecule of this self-replicating RNA (sr-RNA) does everything: self-replicates and then, due to mutations, leads to ribozymes that are eventually capable of catalyzing the peptide and nucleotide bonds. The whole concept is illustrated in a simplified way in Scheme 1. 

To assume the action of a single molecule does not make any sense in normal chemistry. Suppose to have molecule A endowed, in principle, with the capability of self-replication. This means that A can make copies of itself. To do that in a normal solution, A must bind to another A molecule to make an active A₂ reactive complex (a “dimer”). This means that the concentration of A in solution must be high enough to bring about a sizable concentration of A₂, so that diffusion forces do not destroy the dimer itself. What is the critical concentration of A to allow this? The precise calculation of the critical minimal concentration of A is not easy, but a good approximation can show that there should be no significant A₂ formation below a picomolar concentration of A. One picomolar solution of A (defined as 10^-12 M) still contains (since the Avogadro number is 6.02 x 10²³) 6 x 10¹¹ molecules, that is, 600 billion. Even dealing with one microliter, we would have in it 6 x 10⁵ (i.e., 600,000) identical copies of A. This means that we would need several hundred thousand identical copies of A to start replication, even restricting the volume to one microliter. In other words, to have effective self-replication of a hypothetical sr-RNA, we would need preliminary metabolism to make such a molecule in a robust, efficient way.”

“In addition, there is another concentration problem inherent to Scheme 1, and again not sufficiently considered in the literature. Suppose that indeed there would be a sr-RNA evolving according to such a scheme, that is, undergoing mutation after mutation until it evolves into a ribozyme capable of catalyzing the synthesis of DNA. How many mutations and how much time would be necessary for, say, 30 specific natural mutations (not random mutations) in a macromolecule containing, say, 70 residues? This is difficult to evaluate precisely, but certainly a few years, if not centuries. Then, there should be sustained self-replication of our sr-RNA for all this time: a replication that would take place according to the nonlinear, exponential growth of nucleic acids. Tons of nucleotides would be necessary and this would mean that after such a period of time an area equivalent to the entire Aral Lake would be full of our sr-RNA. And then? Actually, the lake would be full of all possible mutants formed concomitantly, whereby the mutant, which makes DNA, would only be one out of an extremely large number, with a dilution that would give an extremely small catalytic efficiency, if any.”

P. L. Luisi, “Why the Origin of Life is Still a Mystery”, J. Syst. Chem. 8 (2020), 1 – 8:

"self-reproducing single molecules?"

"The clearest expression of the bottom-up approach to the origin of life, is the idea to start all from a single molecular species, and this, according to most of the present literature, is RNA, an RNA species of prebiotic origin, (and I call all this "prebiotic -RNA" hypothesis). According to the original ideas, this RNA might be able to start all business, as it would be capable of replication- and mutations during replication, so as to eventually form new ribozymes capable of synthesizing the peptide and the DNA bond. From this original idea, the prebiotic RNA world has become with the years fuzzier, the later contributions looking for pristine forms of prebiotic RNA. Also, nowadays, the term RNA-world means many things, often even synthetic biology with reacting RNA species. Beautiful work again, rich of great ingenuity and originality in organic chemistry and biochemistry, but really not directly relevant for the question of the origin of life.

But coming back to the origin of life from the side of the prebiotic RNA: the main assumption of this original idea is, to start with an already fully functional molecular species, spontaneously formed prebiotically. And this is, I believe, where the all story has its main, lethal drawback. The main question about the origin of life, is to understand how ordered functional macromolecules have been formed out of the chaos of the prebiotic conditions. Not that it is unconceivable to produce under prebiotic conditions a single RNA species- but in this case you need to make spontaneously a long stereoregular sequence, capable of a very specialized functionality of enzymatic catalysis. Assuming all that from the very start, is equivalent to building a house starting from the roof. 

And a mechanism of RNA replication also implies to have ready at your disposal all four mono-nucleotides in suitable concentration- and that our prebiotic RNA is capable of catalysing the bond to all four of them. And even assuming that such a miraculous chemistry happened, what then? what about   the genetic code, namely the synthesis of ordered linear sequences of proteins and nucleic acids? and what about a cell, with its boundary and high local concentration?

To me, the insistence of most of the present research trend on RNA for the origin of life has been always a puzzle. A social problem more than a scientific one. In fact, all this should most probably be seen within the larger phenomenon in our cultural era, a century that Evelyne Fox-Keller defines as the "century of the gene".  

The interest in the prebiotic RNA is linked to the great importance attributed to self-reproduction/replication. To the point that for many of our students, now the right answer to the question "what is life?”, is: reproduction. Which is not a sensible answer, since replication (reproduction) is a consequence of life, not its origin. Or, have you have seen organisms which self-reproduce without being alive?

And as part of this cultural atmosphere, there is also the old definition of life given by the NASA and assumed by most of the RNA researchers ("life is a self-sustained chemical system capable of undergoing Darwinian evolution). This is pure tautology, aside from the fact that the notion of Darwinism applies to a population, and says nothing about a single specimen-it is not because of the Darwinian mechanism that I say that my dog is alive here and now).

Even more generally, in this century of the gene mass media but also some scientists, assume that life is equivalent to DNA. I consider the equation life= DNA as one of the most detrimental views for the research on the origin of life. We have about 50 billion tons of DNA in our planet (difficult to believe it!), -and, as far as we know, all of it has been produced by living cells.  Of course, enough DNA or nucleic acids in general must have been there to form the first viable cells- and in fact, this is another chicken and egg problem we have in the field.

The notion of the beginning of cellular life was discussed by Morowitz long ago (1992).  According to most researchers a minimal cell has to have at least 200-250 genes-a lot of DNA. And this consideration leads us again to the question of the bottom-up approach: is it conceivable that the complexity of a 250 genes cell might have arisen stepwise from an initial structure containing one or a few genes? (even though of course some mechanism of fusion of protocells can be envisaged).  

We will discuss this point in one of the next sections, dealing with the notion of minimal cell. But let us consider first a preliminary idea."