Reciprocal Nucleopeptides as the Ancestral Darwinian Self-Replicator

Abstract Even the simplest organisms are too complex to have spontaneously arisen fully formed, yet precursors to first life must have emerged ab initio from their environment. A watershed event was the appearance of the first entity capable of evolution: the Initial Darwinian Ancestor. Here, we suggest that nucleopeptide reciprocal replicators could have carried out this important role and contend that this is the simplest way to explain extant replication systems in a mathematically consistent way. We propose short nucleic acid templates on which amino-acylated adapters assembled. Spatial localization drives peptide ligation from activated precursors to generate phosphodiester-bond-catalytic peptides. Comprising autocatalytic protein and nucleic acid sequences, this dynamical system links and unifies several previous hypotheses and provides a plausible model for the emergence of DNA and the operational code.


RNA hydrolysis rate
DNA hydrolysis rate is pH independent around neutral pH and in that case cleavage of phosphodiester linkages have a half life of 140000 years (i.e. 1.6 × 10 −13 s −1 assuming 1st order kinetics at 25 • C) and 22 years at 100 • C. RNA is much less stable, its half life being 4 years at 25 • C and 9 days at 100 • C (Wolfenden and Snider 2001). RNA is more stable at pH 4-6 compared to higher pH (Bernhardt and Tate 2012) but we assume this to be a relatively moderate effect for our suggested pH 5 and so retain these numbers i.e. 4 years. ≈ 8 × 10 −9 s −1 . We actually suggest that the system originally was XNA, possibly a mix of RNA-DNA and related molecules, meaning that stability was likely higher. But for simplicity it seems reasonable to keep this estimate of K − R ≈ 8 × 10 −9 s −1 .

Polypeptide and Polynucleotide Spontaneous Polymerisation
Many of the parameters in our model are unknown, but we can try to estimate them using simple kinetic theory. This will give some estimates or upper bound on the values we should use.
In a perfect gas, the number of collisions per unit volume between two molecules A and B is given by where ρ A and ρ B are the concentration of each reactant, r A and r B the radius of the molecules, k B the Boltzmann constant, T the temperature in Kelvin and µ AB = m A m B /(m A + m B ) their reduced mass. If m A >> m B then µ AB ≈ m B .
If the reaction is A + B → C, the equation we have to solve is where K A,B < 1 includes the activation factor. If ρ C and ρ B can be expressed in any units, ρ A must be expressed as the number of molecules per unit volume. If we express the densities in mol m −3 , we thus have is the quantity one needs to estimate.
So the collision rate of two nucleotides can be estimated as Moreover, the uncatalysed phosphodiester bond formation in solution is thought to have an activation energy E a of 21.1 kcal/mol ≈ 35k B T (Florian et al 2003). So The collision rate of 2 polypeptides can be estimated as z a,a ≈ N a (2r am ) 2 8πk B T m am /2 ≈ 3.5 × 10 8 s −1 mol −1 m 3 .
For the activation energy of polypeptide chains we take E a,R = 40 kcal/mol = 67k B T so Similarly, we can estimate the collision rate between a polynucleotide of length L and a polymerase of length to be where µ L, = L m nu m am /(Lm nu + m am ). For L = = 10 we have z ,L ≈ 1.5×10 9 s −1 mol −1 m 3 .
We then have where K L,l is an activation factor which we have conservatively estimated to be 1/1500. Moreover as all the polymerases and catalysed polypeptide chains do not vary much in length we can assume Z ,L to be independent of and L and so Z = Z ,L ≈ 10 −6 s −1 mol −1 m 3 .

RNA polymerisation rate (with concentrations)
Experiments have been carried out with the concentration of primordial polymerase (in this case an RNA molecule) present at a concentration of 2 µM , a template strand, present at 1 µM and a small primer (this is the RNA strand that will be extended, bound to the template) at 0.5 µM (Lawrence and Bartel 2003). The material to be added to the end of the primer, i.e.
activated nucleotide triphosphates, were present at large excess, i.e. 100 µM . It was found that the polynucleotide elongates by one units at a rate varying between 0.02 s −1 and 4 × 10 −5 s −1 .
So we have h R can be estimated from the collision rate of 2 nucleotides which we evaluated above to be of the order 6.6 × 10 8 s −1 mol −1 m 3 , but we must add to that a factor taking into account the correct orientation of the nucleotide and so we will take as an estimate h R = 10 6 s −1 mol −1 m 3 .
The actual value does not matter much as the limiting factor in the duplication of the polynucleotide is the rate k step .

Peptide polymerisation rate
It is estimated (Wohlgemuth et al 2006) that the rate of peptide bond formation is approximately 0.001 s −1 under the following conditions: at 50 mM Mg 2+ , with the concentration of the primitive ribosome (50S subunits) at 0.6 µM , the concentration of tRNA carrying an amino acid (fMet-tRNA) at 6.6 µM and the concentration of the peptide acceptor (puromycin) at 10 mM (to ensure complete saturation of the ribosome). However, it could be as small as 10 −8 s −1 (Sievers et al 2004).
We have used mostly k + P,1 = 0.1 s −1 mol −1 m 3 , but we also considered smaller values.

Model parameter scanning
Most parameters in our model had to be estimated or inferred from known measured values.
To test the robustness of our model we have thus solved the mathematical equations for a large range of parameter values around the estimated values and we present a more detailed description of the results in this section.
The standard value of the parameters used in the paper are the following: and these are the values we have used except when specified otherwise.
Before we scan parameters, we present the values of the concentrations of the polymers of different length after the system has settled to a constant configuration. The table below contains the concentrations obtained for the parameters given above and for the initial concentration The first column corresponds to the length of the polymer. One sees that up to L = 6, the concentrations of all the polypeptide chains are identical because the polymerases are not active for such short length (l π min = 7). As the length increases the concentration of the polymerases of a given length becomes orders of magnitude larger than that of other polypeptides of the same length. In the following section we describe the impact of varying all the parameters above and show that our observation is very robust with respect to the variation of all the parameters. We show that the only parameters which have a significant impact are K − P , K + P and also λ.

Variation of L max
To perform our analysis, we have to chose the maximal length of the polymers for which we solved the equations of the model and we have chosen L max = 10. We have then solved the mathematical model for larger values of L max and found that it did not make a significant difference: the value of Q 1 and the critical values for ρ p and ρ r did not depend on L max .

Variation of l π min
We have also varied the minimum length l π min that the polymerase must have before it becomes active as a polymerase. This did not change the larger values of Q 1 but the critical concentrations ρ p and ρ r decreased with l π min , dropping from ρ p = ρ r = 1, ×10 −3 mol m −3 when l π min = 7 to ρ p = ρ r = 2, ×10 −5 mol m −3 when l π min = 4. l π min = 6 l π min = 5 l π min = 4 ρ p = ρ r Q 1 Variation of K − P The polypeptides depolymerisation rate, K − P , has a very small influence on the selection process of the polymerase. As shown in the table below, increasing K − P to 1 × 10 −8 s −1 does not change the critical concentration nor the actual selection rate Q 1 . Increasing K − P from 1 × 10 −8 s −1 to 5.1 × 10 −6 s −1 only increases the critical concentration slightly from 0.001mol m −3 to 0.0011mol m −3 . Variation of K +

R
The polymerisation of polynucleotides is essential to kick-start the polymerisation of polypeptides and so the polymerisation rate of polynucleotide K + R must be sufficiently large before polymerases can be selectively generated. We found that with K + R = 4 × 10 −8 mol −1 m 3 s −1 the critical concentration of ρ p and ρ r is 0.007 mol m −3 and for K + R = 4 × 10 −9 mol −1 m 3 s −1 it is 0.045 mol m −3 while for K + R = 3.8 × 10 −10 mol −1 m 3 s −1 is is larger than 10.  Variation of K − R and K + To confirm that it is the net polymerisation rate of polynucleotide that matters, we have varied K − R and K + R simultaneously while keeping the ratio K + R /K − R constant and we observed that the critical concentration ρ p = ρ r did not change as we varied these 2 parameters.  The peptide polymerisation rate on a polynucleotide chain also has a very small effect. When varying k + P,1 in the range 1 × 10 −8 mol −1 m 3 s −1 to 0.1 mol −1 m 3 s −1 , we found that the critical concentrations only changed by a factor of 2.